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<big>'''A Taylor -Galerkin method for the use of finite elements in a nonlinear problem for the numerical modeling of a new Chemo-Fluid oscillator.'''</big>
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<big>'''Safety assessment of corrosion-defective natural gas pipeline under ground overload based on FEM'''</big></div>
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<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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Ting Zheng<sup>a,b</sup>, Zheng Liang<sup>a</sup>, Jie Zhang<sup>a</sup>, Liang Zhang<sup>a</sup>, Song Tang<sup>a</sup>, Xiang tao Xiao<sup>a</sup></div>
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<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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<sup>a</sup>School of Mechatronic Engineering, Southwest Petroleum University, Chengdu, Sichuan, 610500, China</div>
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<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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<sup>b</sup>School of Mechanical Engineering, Luzhou Vocational and Technical College,</div>
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<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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Luzhou, Sichuan, 646000, China</div>
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Correspondence：Zheng Liang, Email[mailto::254168368@qq.com :254168368@qq.com], School of Mechatronic Engineering, Southwest Petroleum University, Chengdu, Sichuan, 610500, China
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==Abstract==
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=ABSTRACT=
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Aiming at the safety problem of the pipeline containing corrosion defects caused by ground overload, a novel method is developed to assess the safety of buried pipelines with corrosion defects and predict the failure pressure. The effects of parameters including internal pressure, ground overload, length of the loading area, corrosion defect depth, buried depth and soil Young’s modulus, are discussed. Ground overload greatly increases the von Mises stress and strain at the corrosion defect location and decreases the internal pressure threshold. The von Mises stress and strain are an obvious nonlinear relationship with internal pressure. The high stress and strain area expand along the diagonal direction of the defect area. The local stress and strain concentration at the corrosion defect increases with the increase of ground overload, length of the loading area and corrosion defect depth, which reduces the failure pressure of the pipeline. Increasing the buried depth and soil Young’s modulus would effectively reduce local stress and strain concentration, and increase the failure pressure of the pipeline. The pipeline settlement displacement increases with the increase of internal pressure, ground overload, length of the loading area, and decreases with the increase of pipeline buried depth and soil Young’s modulus.
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'''Keywords''': Pipeline, corrosion defect, ground overload, stress, strain, failure pressure
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In this work, a finite element scheme is proposed using a method of Euler-Taylor-Galerkin described in [1], for a non-linear model which describes the behavior of a new chemo-fluidic oscillator, [2]. This model is expressed by the coupling of an ordinary differential equation describing the hydrogel dynamics, the non-linear transport equation and an auxiliary equation determining the flux volume. The numerical solution is constructed by taking a semi-discretization in time of the transport equation, employing forward-time Taylor series expansions including time derivatives of second order and third order, avoiding instabilities problems.  In this semi discrete equation, the spatial variable is approximated by the finite element formulation according to Galerkin. Some simulations are carried out taking different initial conditions for the concentration of the hydrogel. The numerical results describes the oscillatory behavior of the system as in [2], where MatLab tools are used as black box.
  
 
''Facultad de Ciencias Naturales y Matemáticas, Escuela Superior Politécnica del Litoral, Km. 30.5 Vía Perimetral, Guayaquil, Ecuador''
 
''Facultad de Ciencias Naturales y Matemáticas, Escuela Superior Politécnica del Litoral, Km. 30.5 Vía Perimetral, Guayaquil, Ecuador''
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[mailto:angela@matcom.uh.cu ''angela@matcom.uh.cu'']
 
[mailto:angela@matcom.uh.cu ''angela@matcom.uh.cu'']
  
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==1. Introduction==
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Keywords:<span style="text-align: center; font-size: 75%;"> ['''New chemo-fluidic oscillator'''</span>],  <span style="text-align: center; font-size: 75%;">['''Non-linear model'''</span>],[<span style="text-align: center; font-size: 75%;">'''Finite element'''</span>], [<span style="text-align: center; font-size: 75%;">'''hydrogel dynamics,'''</span>].
  
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Long-distance buried pipelines are an important way of oil and gas storage and transportation. Buried pipelines not only bear the gravity and internal pressure of transportation medium, but also inevitably have to withstand some ground loads, such as the roads, buildings, sediments, machineries and equipment [1]. With the continuous development of urban construction scale, unreasonable stacking overloading phenomenon often appear near the buried pipelines (i.e.stacking height, distance is unreasonable), which results in soil deformation, squeezing the buried pipeline and the superposition load at the top of the pipeline [2-3]. Ground overload may cause pipeline cross-section deformation or even rupture, which results in oil and gas leakage, hindering the cleanup pipe operation and defect detection operation, triggering and other safety accidents [4]. According to relevant reports, there were 27 pipeline damages in the transportation section of the Chengdu-Chongqing pipeline due to unreasonable stacking around pipelines in 2015 [5].
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'''Introduction'''<span style="text-align: center; font-size: 75%;">'''<span id="fnc-1">[[#fn-1|<sup>1</sup>]]</span><span id="fnc-2">[[#fn-2|<sup>2</sup>]]</span>'''</span>
  
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Many scholars have explored and studied the mechanical behavior of buried pipelines under static and dynamic pressure loads [2,6-7]. The main action form of ground overload is static positive pressure, and the load acts vertically directly above the buried pipeline [8]. Chen et al. used the classical Boussinesq method to integrate and calculate the additional load on the buried pipeline under the vertical heap load on the ground, and solved the deformation, bending moment and shear force of the buried pipeline based on the linear elastic foundation beam theory. However, this method ignores the pipe-soil interaction and cannot simulate the local deformation and buckling of the pipe [9]. Zhang et al. discussed the effect of the soil parameters of soil around buried pipelines by numerical methods and designed an underground pipeline protection device that can repair damaged pipe in time without stopping oil and gas transportation [1]. Trickey and Moore investigated the three-dimensional response of buried pipes under circular surface load using the FE method. The comparison of the new results with those of previous work by Poulos in 1974 indicated that his Mindlin solution calculations are somewhat conservative for deeply buried pipes compared with the finite-element solutions, but unconservative for a shallow buried pipe [10]. Based on the Hamilton Variation Principle (HVP) and the finite element (FE) theory, the dynamic plastic deformation of the pipeline under the dynamic compaction behavior during the ground construction was studied by Xu et al. [6]. Meesawasd et al. investigated the effects of buried depth, seismic and traffic loads for the stress distribution of buried pipelines, and demonstrated that weights, speeds and patterns of car size arrangement play their parts in causing variation of stress on the pipeline [11]. Yao et al. [12] conducted experiments and numerical simulations on raw water pipelines with ground loads. Researches showed that ground overload was not a single cause of raw water pipeline bursts, and the original state of the welds had an impact on the bursts. The failure aspects of buried pipeline caused by earthquake and degradation of metal due to corrosion were studied. In this case, corroded pipeline approaches to critical criteria in much less time [13].
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<span id='_Ref220845627'></span><span id='_Ref173317084'></span><span id='_Ref165120614'></span>Self-oscillating systems play an important role in both the natural sciences (biology or chemistry) and technology (micro electromechanical or electronic systems) because they can be coupled to other systems, one of their most important applications is the use as a system clock to trigger regular events such as circadian rhythm or electronic systems.
  
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X80 pipeline steel is currently the longest used high-steel grade pipeline material in the world. While X80 pipeline steel is being used on a large scale, accidents occasionally occur due to its high strength and low toughness characteristics. The area with volume surface defects (VSD) on pipeline is the most typical form of corrosion defects, mainly including internal corrosion defects and external corrosion defects [14]. The presence of VSD may reduce the thickness of the pipe wall and be accompanied with additional stress concentration, which usually greatly weakens the pipeline’s ability to withstand internal pressure and resist external loads, seriously threatening the safe operation of the pipeline.
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This work refers to the use of a Taylor – Galerkin method for applying finite element to the nonlinear system that controls the behavior of the new chemo oscillator – fluidic and determine its numerical modeling. The mathematical problem was raised by Páez who made a numerical approximation of the transport PDA using the well-known Line Method, however, the EDOS system provides a very rough approximation of the solution of the transport equation. <math display="inline">\left[ 1\right]</math>
  
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The pipeline safety evaluation management system based on unreasonable stacking is an important component of the pipeline integrity management system. Over the past years, the author’s team has conducted numerous pipeline integrity evaluation studies [1,8,15]. Meanwhile, developed FE-based models to assess the mechanical behavior of pipelines in geohazard areas and predict pipeline failure pressures. To date, a series of numerical models and computational codes have been developed for pipeline defect assessment [16-18]. However, to the authors’ best knowledge, none of them has considered the mechanical response and failure pressure prediction under the ground overload. This work develops a novel method to assess safety and predict the failure pressure of corrosion defect pipelines under the ground overload.
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First, this is due to the first-order discretization in space and secondly because it can propagate abrupt changes or steep fronts, which is a well-known computational problem in the numerical solution of hyperbolic PED.
  
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In this paper, a FE model is developed to analyze the stress and strain distributions around the corrosion defect on the buried pipeline under the ground overload. A failure criteria based stress are used to predict the failure pressure of the corrosion defect pipeline. The effects of parameters including internal pressure, ground overload, length of the loading area, corrosion defect depth, buried depth, and soil Young’s modulus, are discussed. These results can provide a theoretical basis and reference for the safety assessment, repair and maintenance of buried pipelines.
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In order to describe the dynamics of the chemo-fluidic oscillator a set of mathematical models was used, which I pose a challenge due to the complexity of the system, since the oscillator is affected by various nonlinearities that come from the characteristics of the hydrogel and the bidirectional coupling between the chemical and fluidic domains. The hydrogel is designed in such a way that an increase in the concentration of alcohol reduces its size and vice versa. Therefore, at low concentrations of alcohol, the hydrogel valve is closed, while high concentrations of alcohol open the valve. A bypass channel is connected to the valve inlet, to allow a continuous flow of unidirectional fluid near the hydrogel independent of whether the hydrogel valve is open or closed.
  
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==2. Numerical modeling==
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Therefore to facilitate the modeling process, the system was divided into the fluidic domain describing the behavior of volumetric flows and system pressures during operation and chemical domain which in turn divided into two parts the description of hydrogel dynamics and the modeling of the delay line , resulting in a coupled system composed of the one-dimensional transport EDP, the EDO that models the dynamic behavior of the hydrogel and the equation that determines the volume in the buffer.
  
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In this study, an X80 grade steel pipeline with corrosion defect is modeled. According to the previous analysis, the local stress state of buried corrosion defective pipelines is complicated under the action of unreasonable ground overload. Meanwhile, the focus of this paper is to conduct a safety assessment of buried corrosion defective pipelines under given conditions. Generally, the shape of corrosion defects is irregular. To quantify the shape of corrosion defects, it is necessary to reasonably simplify the model corrosion defects to apply the results to various geometric shapes [19-20]. In many industry standard specifications such as DNV and modified B31G, the maximum corrosion depth (<math display="inline">d</math>
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To perform this work following very closely what Donea did is semi-discretized the equation of nonlinear transport using Taylor's serial expansions at the time of first, second and third order to obtain a second order differential equation in the space in which we apply Galerkin's variational formulation to use the finite element method and obtain the system of linear equations that needs the value of the variable that determines the the size of the hydrogel that is obtained for each instant of time by applying the Runge -Kutta 4 method and the buffer volume that is found using numerical integration methods. <math display="inline">\left[ 2\right]</math>
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), width (<math display="inline">W</math>) and length (<math display="inline">L</math>) are used to describe the pipeline corrosion defects. If the defect depth profile is relatively smooth and does not present multiple major peaks in depth, a corrosion defect can be considered as a regular shape [20]. The shape of the corrosion defect of buried pipelines is simplified as a rectangular volume defects and rounded the corners in this paper, as shown in [[#img-1|Figure 1]], which is a common used method in literature [13,17,21].
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<div id='img-1'></div>
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Analyzing in more detail its mathematical modeling and the numerical solution of this New Chemical-Fluidic Oscillator based on intelligent hydrogels was one of the reasons for the realization of this work in addition to extending the linear method used by Donea [2] for the nonlinear transport equation and providing a solution methodology for models of new oscillators involving the nonlinear one-dimensional transport equation.
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{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
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|style="padding:10px;"| [[Image:Zheng_et_al_2021a_6100_fig1.png|426px]]
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|- style="text-align: center; font-size: 75%;"
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| colspan="1" style="padding:10px;"| '''Figure 1'''. Schematic diagram of corrosion defect pipeline
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|}
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=1.Design of a chemo-fluoscillator=
  
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Numerical analysis of the buried pipeline under the ground loads is conducted by using the FE software ABAQUS6.14. The diameter of the pipeline is 0.66m, and the wall thickness is 8mm. The length of soil along the axial is selected 30 times the pipe diameter, and the height and width are 9 times and 15 times the pipe diameter, respectively, according to the previously published article [1]. Therefore, the whole size of the soil is <math display="inline">20{\rm m}\times 10{\rm m}\times 6{\rm m}</math>, and the buried depth is 1m. Considering that the model and boundary conditions have obvious symmetry, a quarter model is used for calculation, in order to improve the calculation efficiency. The FE model of buried corrosion defect pipeline under ground overload is shown in [[#img-2|Figure 2]]. The equivalent pressure is used to describe the ground overload, and the ground overload directly acts on the soil surface above the corrosion defect pipeline, as shown in [[#img-2|Figures 2]](a) and (c). The pipeline corrosion defect is simplified as a rectangle with rounded corners, and the detailed local shape is shown in [[#img-2|Figure 2]](d).
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The new fluid chemo oscillator isbased on a negative feedback circuit containing a delay line, where negative feedback is provided by a hydrogel valve that has the ability to change its size depending on the temperature and concentration of the aqueous solution that is in contact with live with the hydrogel. In this new oscillator the temperature remains constant so the only parameter that produces a change in the size of the hydrogel is the concentration of alcohol.  
  
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The bottom boundary conditions of the model are fixed constraints, and the symmetry constraint is used on the symmetry plane (i.e. the XY plane and the ZY plane). The upper surface of the model is free, and the normal displacement of the outer end faces is restricted to prevent the soil from collapsing. The contact pair algorithm is used to simulate the interface between the outer surface of the pipe and the surrounding soil. And set the friction coefficient as 0.4 [22]. The contact algorithm is widely accepted to simulate the nonlinear behavior of pipe-soil contact, and it can truly simulate the contact force of underground pipeline and soil [2,8].
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[[Image:Draft_Alvarez_171754839-image1.png|600px]]
  
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<div id='img-2'></div>
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<span style="text-align: center; font-size: 75%;">'''Figure 1.  Photography of the manufactured chemo-fluidic oscillator circuit, filled with a highly dyed solution for better visibility of the channels. External sources of constant flow and pressure are shown schematically. Equivalent fluidic circuit. The hydrogel valve is represented by a controlled flow source.'''</span>
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{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
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|style="padding:10px;"| [[Image:Draft_Zheng_851632191-image2.png|348px]]
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|- style="text-align: center; font-size: 75%;"
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| colspan="1" style="padding:10px;"| '''Figure 2'''. Finite element model containing corrosion defect
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The oscillator is powered by three constant sources. The first is a constant flow source that supplies the system with an alcohol concentration solution. A second source provides deionized water at a constant pressure located at Node 1. Water flows through a long channel called the damping line and then mixes into Node 2 with the alcohol solution provided by , and then the mixed solution enters the channel of the long fluid that acts as a delay line. Using this channel, the solution is transported at a rate determined by the flow through the delay line and its cross-section. The end of this channel is connected to the inlet of the hydrogel valve, <math display="inline">{q}_{A}{c}_{alc}{p}_{W}{q}_{A}{q}_{2}\left( t\right)</math> whosefluid behavior is controlled by the alcohol concentration of the solution. Finally, a bypass channel connects to Node 3. to drain the liquid to a conveniently chosen constant flow rate.  <math display="inline">{q}_{B}</math>
  
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The material properties of X80 are given in [[#tab-1|Table 1]], and the stress-strain method of Ramberg-Osgood (''R-O'') power-hardening is used in the FE model to describe real stress-strain behavior of the high grades of pipe steels, as shown in Eq.(1) [23]
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The micro-fluid system will be modeled through Kirchhoff's laws, within the framework of network theory for a circuit. This approach is used by how small the dimensions of the magnitudes that govern the operation of the oscillator within the study of the micro fluids since they are in a range of micro nano and peak liter, therefore, the pressure is considered analogous to the voltage and volumetric flow rate to the electric current, as well as the lines of delay to the resistors. In this context, the oscillator can be described by the fluid network presented in Figure 2.
  
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{| class="formulaSCP" style="width: 100%; text-align: center;"
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[[Image:Draft_Alvarez_171754839-image2.png|600px]]
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{| style="text-align: center; margin:auto;"
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| <math>\varepsilon =\frac{\sigma }{E}+K{\left(\frac{\sigma }{{\sigma }_u}\right)}^n</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (1)
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|}
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where <math display="inline"> \varepsilon</math> is a strain, <math display="inline"> \rho  </math> is stress, <math display="inline">E  </math> is Young’s modulus, <math display="inline">\sigma_y  </math> is yield strength, <math display="inline"> \sigma_u </math> is ultimate tensile strength, and <math display="inline">K </math> and <math display="inline">n</math> are ''R-O ''model parameters, which depend on the materials. It is 0.079 and 12 for X80 steel, respectively.
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<span style="text-align: center; font-size: 75%;">'''Figure 2. Equivalent fluidic circuit. The hydrogel valve is represented by a controlled flow source.'''</span>
  
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<div class="center" style="font-size: 75%;">'''Table 1'''. Mechanical properties of X80 pipeline steels</div>
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=2.Mathematical model of chemical oscillator - fluidic.=
  
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<div id='tab-1'></div>
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The mathematical problem to be studied was raised by Páez et al. en , and consists in finding functions : such that: <math display="inline">\left[ 1\right] {l}_{v}\left[ 0,T\right] \times \left[ 0,1\right] \rightarrow {R}_{+}</math>
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{| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;"
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{| class="formulaSCP" style="width: 100%; text-align: center;"  
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|-style="text-align:center"
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|-
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! style="vertical-align: top;" |Steel !! Yield strength <br> (MPa) !! Ultimate tensile  <br> strength (MPa) !! Young’s modulus <br>(GPa) !! style="vertical-align: top;" |Poisson’s ratio
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| <math>\begin{matrix}\frac{d{l}_{v}}{dt}\left( t\right) =\gamma \left[ C\left( t,1\right) \right] \left\{ {l}_{eq}\left[ C\left( t,1\right) -{l}_{v}\left( t\right) \right] \right. \\{l}_{v}\left( 0\right) ={l}_{v}^{0}\end{matrix}\Longrightarrow \begin{matrix}{l}_{v}^{0}\, ,\, {C}^{0}\left( x\right) ={C}_{ini}\left( x\right) \, \, \forall x\in \left( 0\, ,\, \left. 1\right] \right. \quad \quad \quad \, \, \left( 1\right) \\{l}_{v}^{n+1}=F\left( {l}_{v}^{n}\, ,{C}^{n}\left( 1\right) ;\Delta t\right) ,n=0,1,2,...,N\end{matrix}</math>
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|-style="text-align:center"
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|  style="text-align: center;"|X80
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|  style="text-align: center;"|534.1
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|  style="text-align: center;"|718.2
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|  style="text-align: center;"|200
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|  style="text-align: center;"|0.3
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|}
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{|class="formulaSCP" style="width: 100%; text-align: center;"  
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Pipeline backfill soil is a heterogeneous soil with high compressibility, viscoelasticity and low shear resistance [3]. Under ground overload, the buried pipeline is mainly subjected to the top soil pressure of the backfill generated by the self-weight of the backfill, which increases with the depth of the pipeline buried depth. The soil adopts the Drucker Prager constitutive model, and the shear strength of the soil is reflected by selecting the friction angle and cohesive force. The change of the yield surface is adjusted by setting the cohesive force [24]. It is suitable for pressure-sensitive soil, because it can well reflect the unequal tension and compression characteristics of rock and soil, and is widely used. The parameters of the soil around the pipe are given in [[#tab-2|Table 2]] [25].
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<div class="center" style="font-size: 75%;">'''Table 2'''. Parameters of model for soil</div>
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<div id='tab-2'></div>
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{| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;"
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|-style="text-align:center"
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! Young’s modulus<br> (MPa) !! style="vertical-align: top;" |Poisson’s ratio !! Density <br>(kg/m<sup>3</sup>) !!  Friction Angle <br> (°) !! Flow Stress <br> ratio !! style="vertical-align: top;" |Cohesion <br>(KPa)
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| style="text-align: center;"|20
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| <math>\begin{matrix}\frac{\partial C}{\partial t}\left( t,x\right) +v\left( {l}_{v}\left( t\right) \right) \frac{\partial C}{\partial x}\left( t,x\right) =0\, \, ,\, \forall \left( t,x\right) \in \left( 0,\left. T\right] \times \left( 0,\left. 1\right] \right. \right. \, ;\quad \\C\left( 0,x\right) ={C}_{ini}\left( x\right) \, ,\quad \forall x\in \left[ 0,{L}_{d}\right] ;\\C\left( t,0\right) ={C}_{a}\left( t\right) \, ,\quad \quad \forall t\geq \left[ 0,T\right] ;\end{matrix}\quad (2)\quad</math>
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|  style="text-align: center;"|0.35
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|  style="text-align: center;"|1840
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|  style="text-align: center;"|30
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|  style="text-align: center;"|1
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|  style="text-align: center;"|29.3
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Failure models of the buried pipeline are varied. Typical failure modes of the buried pipeline include strength failure, buckling failure and excessive deformation [10]. Under ground overload, large local stress and deformation will occur in the corrosion defect pipe. The main failure modes for corrosion defect pipeline are local strength failure and excessive deformation. Generally, the failure criterion of the pipeline is based on the elastic failure and plastic failure adopted the stress based failure criterion for defect pipeline failure pressure prediction under cyclic loads [7].
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The data in this problem are <math display="inline">\Upsilon</math> the functions , ,, , , and where is the unknown value to be determined from the function at the end ; which must also be determined <math display="inline">{l}_{eq}\, v\, {C}_{ini}\, {C}_{a}{l}_{v}^{0}C\left( t,1\right) C\left( t,x\right) x=</math><math>1</math> simultaneously, in this problem, the function : defined by: <math display="inline">{V}_{buff}\left( t\right) \left[ 0,T\right] \rightarrow {R}^{+}</math>
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In this paper, failure criteria based stress is used to describe the failure of corroded pipes. The pipeline failure is considered to occur when the max von Mises stress exceeds the allowable value at the location of the pipeline corrosion defect. The failure criterion is:
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{| class="formulaSCP" style="width: 100%; text-align: center;"  
 
{| class="formulaSCP" style="width: 100%; text-align: center;"  
 
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| <math>{V}_{\_vuff}\left( t\right) :=\int_{0}^{t}{q}_{2}\left[ {l}_{v}\left( s\right) \right] ds-</math><math>{q}_{A}t\, ;\forall t\in \left( 0,\left. T\right] ,\right. \quad \quad (3)\quad \,</math>  
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{| style="text-align: center; margin:auto;"
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| <math>{\sigma }_{mises}=\frac{1}{\sqrt{2}}\sqrt{{\left({\sigma }_1-{\sigma }_2\right)}^2+{\left({\sigma }_2-{\sigma }_3\right)}^2+{\left({\sigma }_3-{\sigma }_1\right)}^2}>\left[{\sigma }_u\right]</math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (2)
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|}
 
|}
  
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where <math display="inline"> {\sigma }_{mises} </math> is von Mises stress, <math display="inline"> {\sigma }_{1,2,3} </math> refer to three principal stress on the pipeline, and where <math display="inline"> {\sigma }_{u} </math> is the ultimate tensile strength.
 
  
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The element type of the pipe and the surrounding soil adopts eight-node brick element with the reduced integral unit to improve the convergence efficiency for geometrically nonlinear modeling. In order to obtain a reasonable grid that ensure that the simulation results are sufficiently stable and reliable, multiple sets of grid independence tests are conducted. Meanwhile, local grid refinement is used in the corrosion defect area of the pipeline and the ground overload area, and the sparse grid is used in the area far from the important area. The maximum von Mises stress for different element number are shown in [[#tab-3|Table 3]], and the loads and corrosion defect size are shown in [[#tab-4|Table 4]]. When the number of elements exceeds 112785, the maximum Mises stress area stabilizes. According to the above analysis, the number of elements finally selected 112785, and the smallest grid size sets as 1mm<math display="inline">\times  </math>1mm<math display="inline">\times </math>0.8mm. In particular, in order to ensure the accuracy of the calculation, a four-layer grid is selected for the thickness of the corrosion defect wall, and highlighted in dark red in [[#img-2|Figure 2]](d).
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Where it is a known function and is a constant also known. <math display="inline">{q}_{2}{q}_{A}</math>
  
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<div class="center" style="font-size: 75%;">'''Table 3'''. Maximum von Mises stress for different mesh number </div>
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==2.1 Semi - discretization in problem time (2) ==
  
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<div id='tab-3'></div>
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The transport equation is considered
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{| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;"
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{| class="formulaSCP" style="width: 100%; text-align: center;"  
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|-style="text-align:center"
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! style="vertical-align: top;" | Number of elements !! Maximum von Mises<br> stress (MPa)
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|-
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| style="text-align: center;"|30305
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| <math>\frac{\partial C}{\partial t}\left( t,x\right) =-v\left( {l}_{v}\left( t\right) \right) \frac{\partial C}{\partial x}\left( t,x\right) \, \, ,\, \forall \left( t,x\right) \in \left( 0,\left. T\right] \times \left( 0,\left. 1\right] \right. \right. \, ;</math>
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|  style="text-align: center;"|552.1
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|-
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|  style="text-align: center;"|54105
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|  style="text-align: center;"|556.2
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−
|-
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|  style="text-align: center;"|86522
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|  style="text-align: center;"|559.6
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−
|-
+
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|  style="text-align: center;"|112785
+
−
|  style="text-align: center;"|578.1
+
−
|-
+
−
|  style="text-align: center;"|128569
+
−
|  style="text-align: center;"|578.4
+
−
|-
+
−
|  style="text-align: center;"|153674
+
−
|  style="text-align: center;"|578.6
+
 
|}
 
|}
  
  
−
The model uses single-factor analysis to discuss the safety assessment and failure pressure prediction of buried pipelines with corrosion defects under ground overload conditions. The basic parameters of the model are shown in [[#tab-4|Table 4]].
+
If it is denoted by the value of the function evaluated on the node, then a schema in finite differences very simple to approximate the temporal derivative in would be the one obtained by the serial expansion of Taylor, in the first order, around the point : <math display="inline">{C}^{n}\left( x\right) C\left( t,x\right) t=</math><math>{t}^{n}\left( 4\right) {t}^{n+1}={t}^{n}+\Delta t</math>
−
 
+
{| class="formulaSCP" style="width: 100%; text-align: center;"  
−
<div class="center" style="font-size: 75%;">'''Table 4'''. Base condition parameters </div>
+
−
 
+
−
<div id='tab-4'></div>
+
−
{| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;"
+
−
|-style="text-align:center"
+
−
!  colspan='2'  style="text-align: center;"|Variable !! Value
+
−
|-
+
−
|  rowspan='2' style="text-align: left;"|Load (MPa)
+
−
|  style="text-align: center;"|Ground overload (OLD)
+
−
|  style="text-align: center;"|1
+
−
|-
+
−
|  style="text-align: center;"|Internal pressure (IP)
+
−
|  style="text-align: center;"|7
+
−
|-
+
−
|  style="text-align: left;"|Loading area (m)
+
−
|  style="text-align: center;"|width<math display="inline">\times </math>length
+
−
|  style="text-align: center;"|0.8<math display="inline">\times </math>1.8
+
−
|-
+
−
|  style="text-align: left;"|Buried depth (m)
+
−
|  style="text-align: center;"|H
+
−
|  style="text-align: center;"|1
+
 
|-
 
|-
−
| style="text-align: left;"|Corrosion defect (mm)
+
| <math>C\left( {t}^{n}+\Delta t,x\right) =C\left( {t}^{n}+\Delta t\right) +\Delta t\left[ \frac{\partial C}{\partial t}\right] \left( t,x\right) \left| t=\right. </math><math>\left. {t}^{n}\right. +O\left( \Delta t\right)</math>  
−
| style="text-align: center;"|d<math display="inline">\times </math>L<math display="inline">\times </math>W
+
−
|  style="text-align: center;"|5.6<math display="inline">\times </math>40<math display="inline">\times </math>32
+
 
|}
 
|}
  
−
==3 Result and discussion==
 
  
−
===3.1 Model verification results===
+
From here, despising him you have to: <math display="inline">O\left( \Delta t\right)</math>
−
 
+
{| class="formulaSCP" style="width: 100%; text-align: center;"  
−
Being compared with theoretical calculations or experimental test data is an effective and reliable comparison verification method. In this paper, the model validation is conducted by using the experimental data obtained on an X60 steel pipeline containing a single corrosion defect, which was published by Freire [21]. The outer diameter of the pipe is 324 mm and nine groups of corrosion defect dimension pipes are used for comparative verification, as shown in [[#tab-5|Table 5]].
+
−
 
+
−
<div class="center" style="font-size: 75%;">'''Table 5'''. Parameters of corrosion specimens</div>
+
−
 
+
−
<div id='tab-5'></div>
+
−
{| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;"
+
−
|-style="text-align:center"
+
−
! Case !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9
+
−
|-
+
−
|  style="text-align: center;"|''t ''(mm)
+
−
|  style="text-align: center;"|9.79
+
−
|  style="text-align: center;"|9.74
+
−
|  style="text-align: center;"|9.80
+
−
|  style="text-align: center;"|9.66
+
−
|  style="text-align: center;"|9.71
+
−
|  style="text-align: center;"|9.71
+
−
|  style="text-align: center;"|9.91
+
−
|  style="text-align: center;"|9.94
+
−
|  style="text-align: center;"|9.79
+
−
|-
+
−
|  style="text-align: center;"|''l ''(mm)
+
−
|  style="text-align: center;"|500
+
−
|  style="text-align: center;"|528
+
−
|  style="text-align: center;"|256
+
−
|  style="text-align: center;"|306
+
−
|  style="text-align: center;vertical-align: top;"|350
+
−
|  style="text-align: center;vertical-align: top;"|395
+
−
|  style="text-align: center;vertical-align: top;"|433
+
−
|  style="text-align: center;vertical-align: top;"|467
+
−
|  style="text-align: center;vertical-align: top;"|484
+
−
|-
+
−
|  style="text-align: center;"|''t* ''(mm)
+
−
|  style="text-align: center;"|2.80
+
−
|  style="text-align: center;"|2.60
+
−
|  style="text-align: center;"|2.72
+
−
|  style="text-align: center;"|2.9
+
−
|  style="text-align: center;vertical-align: top;"|2.78
+
−
|  style="text-align: center;vertical-align: top;"|2.80
+
−
|  style="text-align: center;vertical-align: top;"|2.60
+
−
|  style="text-align: center;vertical-align: top;"|2.72
+
−
|  style="text-align: center;vertical-align: top;"|2.80
+
 
|-
 
|-
−
| style="text-align: center;"|''w'' (mm)
+
| <math>\left[ \frac{\partial C}{\partial t}\right] \left( t,x\right) =\frac{{C}^{n+1}\left( x\right) -{C}^{n}\left( x\right) }{\Delta t};</math>
−
|  style="text-align: center;"|95.3
+
−
|  style="text-align: center;"|95.3
+
−
|  style="text-align: center;"|95.3
+
−
|  style="text-align: center;"|95.3
+
−
|  style="text-align: center;"|95.3
+
−
|  style="text-align: center;"|95.3
+
−
|  style="text-align: center;"|95.3
+
−
|  style="text-align: center;"|95.3
+
−
|  style="text-align: center;"|95.3
+
 
|}
 
|}
  
−
<div class="center" style="font-size: 75%;">Note：''t'' represents the diameter of pipe and ''t* ''represents the thickness wall of pipe</div>
 
  
 +
which is the well-known forward-time (Euler) scheme.
  
−
Verification and analysis by comparison is an effective means to verify the reliability of the developed failure pressure prediction finite element model. The existing methods for failure pressure prediction of corrosion-defective pipelines include ASME B31Gand DNV. Most of these codes are based on semi-empirical solutions of fracture mechanics and have been recognized by the industry. In this work, the failure pressure obtained through the developed model is also compared with the predicted by two assessment methods: DNV and B31G model. [[#img-3|Figure 3]] shows the comparison of the failure pressure prediction results among the established FE model, the published experimental test data, DNV and B31G modeling results. It can be clearly seen that the prediction result of the B31G method is obviously conservative, and the prediction result of the DNV method is closer to the experimental value. The relative error between experimental test data and the FE model results is within 5%. In addition, the relative error of the FE model results is less than 4%, compared with DNV model results. Therefore, the above verification results show that the established failure pressure prediction FE model can obtain reliable results.
+
If the EDP is now evaluated in you have for each EDO: <math display="inline">\left( 4\right) t=</math><math>{t}^{n}n=0,1,...,N\,</math>
−
 
+
{| class="formulaSCP" style="width: 100%; text-align: center;"  
−
<div id='img-3'></div>
+
−
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: 45%;"
+
 
|-
 
|-
−
|style="padding:10px;"| [[Image:Zheng_et_al_2021a_4727_fig3.png|454px]]
+
| <math>\left[ \frac{\partial C}{\partial t}\right] \left( t,x\right) \left| t={t}^{n}\right. =</math><math>-{\left[ v\left[ {l}_{v}\left( t\right) \right] \right] }_{t={t}^{n}}{\left. \left[ \frac{\partial C}{\partial x}\right] \left( t,x\right) \right| }_{t={t}^{n}}\, ,\, \forall x\in \left( 0\, ,\left. 1\right] \right. \quad \left( 5\right)</math>
−
|- style="text-align: center; font-size: 75%;"
+
−
| colspan="1" style="padding:10px;"| '''Figure 3'''. Validation of model prediction failure pressure results
+
 
|}
 
|}
  
−
===3.2 Internal pressure effect===
 
  
−
The maximum von Mises stress and strain around the corrosion defect on X80 steel pipeline under different internal pressure (IP) for the presence and absence of ground overload (OLD) are shown in [[#img-4|Figures 4]](a) and (b). Both the maximum von Mises stress and strain of pipeline corrosion defects have been significantly increased, because of the presence of ground overload. When there is only internal pressure and the internal pressure is below 5MPa, the corrosion defects pipeline is in the elastic stage. When the internal pressure is greater than 5MPa, the maximum von Mises stress of the pipeline exceeds the yield stress and plastic deformation occurs. However, when the pressure is 1 MPa, the maximum von Mises stress of the pipeline exceeds the yield stress and plastic deformation occurs for the presence of ground overload. Meanwhile, both the maximum von Mises stress and strain of pipeline corrosion defects show obvious nonlinear behavior with the increase of internal pressure for two load conditions (i.e. IP+OLD). When the internal pressure is 1-3MPa, the external load plays a leading role in the pipeline deformation, and the overall ability of the pipeline to resist deformation is weak. Therefore, the plastic deformation around the corrosion-defective pipeline decreases with the increase of internal pressure. When the internal pressure is greater than 3MPa, the pipeline plastic deformation around the corrosion-defective pipeline increases with the increase of internal pressure. Therefore, the presence of ground overload decreases the internal pressure threshold, exceeding that local plastic deformation around corrosion defects on the pipeline occurs.
+
Now, using the schema type forward – time la expression is transformed into:  <math display="inline">\left( 5\right)</math>
−
 
+
{| class="formulaSCP" style="width: 100%; text-align: center;"  
−
Along the axial direction of the pipeline, the pipeline is divided into three areas in the case of ground overload, namely the overload zone, the transition zone and the non-overload zone, respectively. The pipeline settlement displacement distribution along the axial direction under different internal pressures is shown in [[#img-4|Figure 4]](c). The maximum settlement displacement decreases with the increase of the internal pressure, which shows that the internal pressure improves the stiffness of the pipeline and increases the ability to resisting deformation of the pipeline. The maximum settlement displacement is located below the ground overload area, and there is an obvious transition zone. The settlement displacement gradually decreases with the increase of the axial length of the pipeline, and the minimum settlement is in the non-overloaded zone.
+
−
 
+
−
[[#img-4|Figure 4]](d) shows that the von Mises stress in the overload area is greater than the other two areas, and the maximum stress value appears in the center of the corrosion defect in the overload zone. As the previous analysis, the stress distribution along the axial direction of the pipeline also exhibits nonlinear behavior with different internal pressure. In the transition area, the stress transition becomes less obvious with the increase of internal pressure except the internal pressure is 1MPa. In the non-overload zone, the stress increases with the increase of internal pressure.
+
−
 
+
−
<div id='img-4'></div>
+
−
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: 45%;"
+
 
|-
 
|-
−
|style="padding-top:10px;"| [[Image:Draft_Zheng_851632191-picture- 1.svg|center|600px]]
+
| <math>\frac{{C}^{n+1}\left( x\right) -{C}^{n}\left( x\right) }{\Delta t}=-v\left( {l}_{v}^{n}\right) \frac{\partial {C}^{n}}{\partial x}\left( x\right) \, ,\, 0\leq n\leq N\, ,\, \forall x\in \left( 0\, ,\left. 1\right] \right. \quad (6)</math>
−
|- style="text-align: center; font-size: 75%;"
+
−
| colspan="1" style="padding:10px;"| '''Figure 4'''. Mechanical behavior of X80 corrosion-defective pipeline under different internal pressure
+
 
|}
 
|}
  
  
−
[[#img-5|Figure 5]] shows the von Mises stress and strain distributions around the corrosion defect on the X80 steel pipeline under different internal pressure and ground overload. It should be noted that the unit of stress is Pa, and PEEQ is dimensionless. It can be seen that the stress concentration occurs in the corrosion defect area, and the maximum stress develops from the root of the corrosion area to the middle with the increase of the internal pressure. The stress distribution develops from the root to the middle along the length of the corrosion defect with the internal pressure increases. When the internal pressure is greater than 6MPa, low stress areas appear on both sides of the corrosion defect width direction, and the range of the low stress areas increases with the increase of internal pressure. According to the strain distribution, when the internal pressure is less than 3MPa, the maximum strain decreases with the increase of internal pressure. When the internal pressure is greater than 3MPa, the maximum strain increases with the increase of internal pressure, and the maximum value appears in the middle and diagonal area of corrosion defect.
+
Here is the value in which it is determined using Ruge -Kutta 4 applied to the initial problem <math display="inline">{l}_{v}^{n}{l}_{v}(t)t=</math><math>{t}^{n}\left( 1\right)</math>
  
−
Meanwhile, the failure pressure of X80 steel pipes with corrosion defects for the presence and absence of ground overload are compared. The failure pressure of the pipeline with the presence of ground overload is 16.35 MPa, and it can reach 17.1 MPa in the absence of ground overload. This indicates that ground overload can decrease the failure pressure of pipelines with corrosion defects and affect the safety production pressure of pipelines. Under the simultaneous action of corrosion defects and ground overload, the internal pressure threshold is reduced.
+
Where, in general, is the unknown value at the border for time. For this schema, the known constant is the initial iteration. In the context of the finite differences method, the expression produces an unstable numerical scheme by approx approximation of the spatial derivative term using a centered scheme, i.e.  <math display="inline">{C}^{n}\left( 1\right) C\left( t\right) =</math><math>1x=1t={t}^{n}{l}_{v}^{0}\left( 6\right)</math>
−
 
+
{| class="formulaSCP" style="width: 100%; text-align: center;"  
−
<div id='img-5'></div>
+
−
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: 45%;"
+
 
|-
 
|-
−
|style="padding:10px;"| [[Image:Draft_Zheng_851632191-picture- 2.svg|center|600px]]
+
| <math>\frac{\partial {C}^{n}}{\partial x}\left( x\right) =\frac{{C}_{m+1\, }^{n}-{C}_{m}^{n}}{\Delta x}</math>
−
|- style="text-align: center; font-size: 75%;"
+
−
| colspan="1" style="padding:10px;"| '''Figure 5'''. Distributions of maximum von Mises stress and strain around X80 corrosion- defective pipelines under different internal pressure
+
 
|}
 
|}
  
−
===3.3 Ground overload effect===
 
  
−
[[#img-6|Figure 6]](a) shows that both the maximum von Mises stress and strain around the corrosion defects of pipeline increase with the increase of ground overload. The von Mises stress and plastic deformation increase approximately linear upward trend with the increase of the ground overload when the ground overload exceeds 0.8 MPa. This indicates that ground overload has a direct effect on pipeline safety.
+
Lto instability arises because the partial derivative relative to the spatial coordinate is evaluated at a time level earlier than the time level where the term temporal derivative is evaluated. Therefore, a stable schema can be obtained if the two derived <math display="inline">nn</math> terms s <math display="inline">\frac{\partial C}{\partial t}</math> and <math display="inline">\frac{\partial C}{\partial x}</math> s evaluate at the same time level (at least a second order in ). In this order of ideas, Donea states that the easiest way to make the evaluation of both terms of the expression at the same time level, is by expressing the approximation in difference for the term of temporal derivative at the time level. One way to achieve this is through a forward-looking Taylor serial expansion over time, including second- and third-order derivatives. That is, from the expansion <math display="inline">n\Delta t\left[ 1\right] \left( 6\right) nn</math>
−
 
+
{| class="formulaSCP" style="width: 100%; text-align: center;"  
−
As shown in [[#img-6|Figure 6]](b), when the ground overload is greater than a certain level, a certain amount of warping deformation occurs at the center position of the corrosion defect, because of the large deformation of the soil squeezing pipe, as shown in the enlarged image [[#img-6|Figure 6]](b).
+
−
 
+
−
In the overload zone, the stress increases with the ground overload increasing, and the maximum value appears at the center of the corrosion zone ([[#img-6|Figure 6]](c)). In the transition zone, the stress distribution is an asymmetric S shape. The stress near the overload zone decreases with the increase of ground load. However, the stress near the non-overload zone increases with the increases of ground overload. In the non-overload zone, the stress distribution increases with the increases of ground overload, but tends to the same stable value at the far end of the pipeline.
+
−
 
+
−
[[#img-6|Figure 6]](d) shows the von Mises stress distribution of symmetrical section of the corrosion defect on X80 steel pipeline under different ground overload. It can be seen that there is a local high stress area at the top of the pipeline section (0°). The size and stress value of this area increases with the increase of ground overload. At the same time, the stress near the cross section 90°(270°) of the pipeline is the smallest, and the stress at the bottom of the pipeline smaller than the top. Therefore, ground overloads very harmful for the safe operation of pipelines, especially for pipelines with corrosion defects.
+
−
 
+
−
<div id='img-6'></div>
+
−
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: 40%;"
+
 
|-
 
|-
−
|style="padding-top:10px;"| [[Image:Draft_Zheng_851632191-picture- 4.svg|center|800px]]
+
| <math>C\left( {t}^{n}+\Delta t,\, x\right) =C\left( {t}^{n},x\right) +\Delta t{\left. \left[ \frac{\partial C}{\partial x}\right] \left( t,x\right) \right| }_{t={t}^{n}}+</math><math>\frac{\Delta {t}^{2}}{2}{\left. \left[ \frac{{\partial }^{2}C}{\partial {x}^{3}}\right] \left( t,x\right) \right| }_{t={t}^{n}}+</math><math>\frac{\Delta {t}^{3}}{6}{\left. \left[ \frac{{\partial }^{3}C}{\partial {x}^{2}}\right] \left( t,x\right) \right| }_{t={t}^{n}}+</math><math>O\left( \Delta {t}^{3}\right)</math>
−
|- style="text-align: center; font-size: 75%;"
+
−
| colspan="1" style="padding:10px;"| '''Figure 6'''. Mechanical behavior of X80 corrosion-defective pipelines under different ground overloads
+
 
|}
 
|}
  
  
−
In [[#img-7|Figure 7]], it can be seen that the stress concentration appears in the corrosion defect area, and the stress concentration obviously increases with the increase of the ground overload. When the ground overload is 2MPa, the maximum stress concentration reaches 688.6MPa, which is close to the ultimate tensile strength, as highlighted in dark red in [[#img-7|Figure 7]](a). When the load is less than 0.8MPa, the stress distribution develops from both sides of the corrosion defect to the middle along the width direction of the corrosion defect. The stress distribution develops from the center of the corrosion defect to the diagonal direction along the length of the corrosion defect when the ground overload is greater than 0.8MPa. The plastic deformation distribution also has the same trend. When the ground overload reaches 2MPa, the maximum plastic deformation appears at the center of the corrosion defect, as highlighted in dark red in [[#img-7|Figure 7]](b).
+
The term is cleared and (considering again the notation that was introduced for the forward-time <math display="inline">{\left. \left[ \frac{\partial C}{\partial x}\right] \left( t,x\right) \right| }_{t={t}^{n}}</math>scheme) d bypreciatingthe <math display="inline">O\left( \Delta {t}^{3}\right)</math>  term and using discretization over time, for the transport equation can be replaced by the following: <math display="inline">\left( 5\right) \left( 4\right)</math>
−
 
+
{| class="formulaSCP" style="width: 100%; text-align: center;"  
−
<div id='img-7'></div>
+
−
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: 45%;"
+
 
|-
 
|-
−
|style="padding-top:10px;"| [[Image:Draft_Zheng_851632191-picture- 5.svg|center|600px]]
+
| <math>\frac{{C}^{n+1}\left( x\right) -{C}^{n}\left( x\right) }{\Delta t}-\frac{\Delta {t}^{2}}{6}\, {\left[ \frac{{\partial }^{3}C}{\partial {t}^{3}}\right] }^{n}\left( x\right) =</math><math>-v\left( {l}_{v}^{n}\right) \frac{\partial {C}^{n}}{\partial x}\left( x\right) +</math><math>\frac{\Delta t}{2}\left[ \frac{{\partial }^{2}{C}^{n}}{\partial {t}^{2}}\right] \left( x\right) \, ,\, \forall x\, \left( 0\, ,\left. 1\right] \right. \quad \quad \quad \quad \, \left( 9\right)</math>
−
|- style="text-align: center; font-size: 75%;"
+
−
| colspan="1" style="padding:10px;"| '''Figure 7'''. Distributions of von Mises stress and strain at corrosion defects of X80 buried pipelines under different ground overloads
+
 
|}
 
|}
  
−
The magnitude of the ground load will also affect the failure pressure of the corrosion-defective pipeline. According to failure criterion Eq.(2), the predicted value of the failure pressure of the corrosion-defective pipeline under different loads is shown in [[#tab-6|Table 6]]. The results show that the predicted failure pressure decreases with the increase of the ground overload. When the ground overload is greater than 2MPa, the failure pressure decreases significantly.
 
  
−
<div class="center" style="font-size: 75%;">'''Table 6'''. Failure pressure of X80 corrosion-defective pipelines under different ground overloads</div>
+
For each. The second and third derived terms that appear in this expression can be determined first by successive differentiation of the equation and then by evaluating over time. The calculation for the second derivative is illustrated below: On the one hand <math display="inline">n=</math><math>0,1,...,N\left( 4\right) t={t}^{n}</math>
−
 
+
{| class="formulaSCP" style="width: 100%; text-align: center;"  
−
<div id='tab-6'></div>
+
−
{| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;"
+
−
|-style="text-align:center"
+
−
! OLD (MPa) !! Failure pressure (MPa)
+
 
|-
 
|-
−
| style="text-align: center;"|0.4
+
| <math>\frac{{\partial }^{2}C}{\partial {t}^{2}}={v}^{2}\frac{{\partial }^{2}C}{\partial {t}^{2}}-</math><math>\frac{dv}{dt}\frac{\partial C}{\partial x}\quad \quad \quad \quad \quad \quad \quad \quad \quad \left( 10\right)</math>
−
| style="text-align: center;"|16.91
+
|}
 +
{|class="formulaSCP" style="width: 100%; text-align: center;"  
 
|-
 
|-
−
| style="text-align: center;"|0.8
+
| <math>\frac{{\partial }^{3}C}{\partial {t}^{3}}={v}^{2}\frac{{\partial }^{2}}{\partial {t}^{2}}\cdot \left( \frac{\partial C}{\partial t}\right) +</math><math>3v\frac{dv}{dt}\left( \frac{{\partial }^{2}C}{\partial {x}^{2}}\right) -\frac{\partial C}{\partial x}\cdot \left( \frac{{d}^{2}v}{d{t}^{2}}\right) \quad \quad \quad \quad \, \, \left( 11\right)</math>
−
|  style="text-align: center;"|16.64
+
−
|-
+
−
|  style="text-align: center;"|1.2
+
−
|  style="text-align: center;"|16.22
+
−
|-
+
−
|  style="text-align: center;"|1.6
+
−
|  style="text-align: center;"|16.13
+
−
|-
+
−
|  style="text-align: center;"|2.0
+
−
|  style="text-align: center;"|15.63
+
 
|}
 
|}
  
−
===3.4 Loading area effect===
 
  
−
Both the maximum von Mises stress and strain around the corrosion defect of pipeline increase with the increase of the length of the loading area ([[#img-8|Figure 8]](a)). Moreover, there is also a significant non-linear relation between the above two. When the length of the loading area for less than 2.2 meters, the maximum stress and plastic deformation show an approximately linear upward trend. While it shows a downward trend with the length of loading area for greater than 2.2 meters. This indicates that when the ground overload area exceeds a certain range, the impact on pipeline corrosion defects is reduced. Therefore, the pipeline should be laid as far as possible to avoid areas with frequent human activities and frequent geological loads.
+
Now, combining and, evaluated in , with the equation you have  <math display="inline">\left( 10\right) \left( 11\right) t=</math><math>{t}^{n}\left( 9\right)</math>
−
 
+
{| class="formulaSCP" style="width: 100%; text-align: center;"  
−
In [[#img-8|Figure 8]](b), the pipeline settlement displacement distribution along the axial direction increases with the length of the loading area increasing. When the length of the loading area reaches 3m, the maximum settlement displacement of the pipeline reaches 0.083m. As shown in [[#img-8|Figure 8]](c), when the length of the loading area reaches 2.2 m, the maximum stress appears at the center of the corrosion defect is 583.1MPa. Near the ground overload zone in the transition zone, the stress increases with the length of the loading area increasing. However, near the non-ground overload zone, the stress decreases with the length of the load increasing.
+
−
 
+
−
In [[#img-8|Figure 8]](d), the stress distribution approximately is apple-shaped, and the stress at the top of the pipeline is greater than the bottom. Meanwhile, high and low pressure zones alternately appear in the width direction on both sides of the pipeline corrosion defects. The minimum stress appears at both sides of the pipe. In general, there is a nonlinear relationship between the overall stress distribution law and the length of the loading area, which is very similar to [[#img-8|Figure 8]](a).
+
−
 
+
−
<div id='img-8'></div>
+
−
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: 45%;"
+
 
|-
 
|-
−
|style="padding-top:10px;"| [[Image:Draft_Zheng_851632191-picture- 6.svg|center|600px]]
+
| <math>\frac{{C}^{n+1}\left( x\right) -{C}^{n}\left( x\right) }{\Delta t}-\frac{\Delta {t}^{2}}{6}\, {\left[ v\left( {l}_{v}^{n}\right) \frac{{\partial }^{2}}{\partial {x}^{2}}\left( \frac{\partial {C}^{n}}{\partial t}\right) \left( x\right) +3v\left( {l}_{v}^{n}\right) \frac{dv\left( {l}_{v}^{n}\right) }{dt}\cdot \frac{{\partial }^{2}{C}^{n}}{\partial {x}^{2}}\left( x\right) -\frac{{d}^{2}v\left( {l}_{v}^{n}\right) }{d{t}^{2}}\cdot \frac{{\partial C}^{n}}{\partial x}\left( x\right) \right] }^{1}=</math><math>-v\left( {l}_{v}^{n}\right) \frac{\partial {C}^{n}}{\partial x}\left( x\right) +</math><math>\frac{\Delta t}{2}\left[ v\left( {l}_{v}^{n}\right) \frac{{\partial }^{2}{C}^{n}}{\partial {t}^{2}}\left( x\right) -\right. </math><math>\left. \frac{dv\left( {l}_{v}^{n}\right) }{dt}\cdot \frac{\partial {C}^{n}}{\partial x}\left( x\right) \right] \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \, \left( 12\right) \quad \quad \quad \quad</math>
−
|- style="text-align: center; font-size: 75%;"
+
−
| colspan="1" style="padding:10px;"| '''Figure 8'''. Mechanical behavior of X80 corrosion-defective pipeline under different length of the loading area
+
 
|}
 
|}
−
 
+
{|class="formulaSCP" style="width: 100%; text-align: center;"  
−
 
+
−
In [[#img-9|Figure 9]], when the length of the loading area is 1m, the stress concentration mainly is located at the center and both sides of the corrosion defect area. In other states, the stress concentration is mainly concentrated in the corrosion defect area. According to the strain distribution, the plastic deformation is mainly distributed at the bottom of the corrosion defect area, and the high plastic deformation is mainly distributed diagonally along the bottom of the corrosion defect. When the length of loading area less than 2.2m, the plastic deformation increases with the increase of the length of the loading area. However, the plastic deformation shows a downward trend with the length of the length of loading area, when the length of loading area greater than 2.2m.
+
−
 
+
−
<div id='img-9'></div>
+
−
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: 50%;"
+
 
|-
 
|-
−
|style="padding-top:10px;"| [[Image:Draft_Zheng_851632191-picture- 7.svg|center|600px]]
+
| <math>Para\, todo\, x\in \left( 0\, ,\left. 1\right] \right. \, y\, cada\, n=</math><math>0,1,...,N\quad \quad \quad \quad \quad</math>
−
|- style="text-align: center; font-size: 75%;"
+
−
| colspan="1" style="padding:10px;"| '''Figure 9'''. Distributions of maximum von Mises stress and strain around X80 corrosion- defective pipelines under different length of the loading area
+
 
|}
 
|}
  
  
−
[[#tab-7|Table 7]] shows the relationship between the predicted failure pressure and the length of the loading area. When the length of the loading area is less than 1.8m, the predicted failure pressure value remains unchanged. When the length of the loading area is greater than 2.2m, the failure pressure decreases with the length of the loading area increasing. Therefore, within a certain range, the length of the ground overload area has little effect on the failure pressure of the pipeline.
+
The expression (12) is similar to that proposed by Leveque when generating stabilized numerical methods by dictating a diffuse term to the nonlinear transport equation. However, it should be noted that the term appears as part of the approximation by difference for the partial derivative of C with respect to time, evaluated at level n. On the other hand, following closely what Donea suggested the term of the third-order partial derivative that appears in Taylor's serial expansion is expressed in a mixed space-time form. This mixed form of the derivative will allow the use of type finite elements with a simple modification of the usual and consistent mass matrix quite similarly as is done in the context of weighted residues of petrov-Galerkin. <math display="inline">\epsilon \frac{{\partial }^{2}{C}^{n}}{\partial {x}^{2}}\left( 12\right) \frac{{\partial }^{2}{C}^{n}}{\partial {x}^{2}}\left[ 1\right] {C}^{0}\left[ 5\right]</math>
  
−
<div class="center" style="font-size: 75%;">'''Table 7'''. Failure pressure of X80 corrosion-defective pipelines under different length of ground overload</div>
+
By developing the specified products and grouping the terms we have that the problem raised in (11) is semi discretized over time and for each the next problem arises, <math display="inline">n</math>
  
−
<div id='tab-7'></div>
+
Given <math display="inline">\left[ 0,1\right] \ni x\, \rightarrow {C}^{0}\left( x\right) =</math><math>{C}_{ini}\left( x\right)</math> '',''  find , such that: <math display="inline">{C}^{n}{\left( x\right) }_{0\leq n\leq N},\forall x\in \left[ 0,1\right]</math>
−
{| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;"
+
{| class="formulaSCP" style="width: 100%; text-align: center;"  
−
|-style="text-align:center"
+
−
! Length (m) !! Failure pressure (MPa)
+
 
|-
 
|-
−
| style="text-align: center;"|1.0
+
| <math>\frac{{\partial }^{2}{C}^{n+1}\left( x\right) }{\partial {x}^{2}}-{\alpha }_{1}{C}^{n+1}\left( x\right) =</math><math>{\alpha }_{2}\frac{{\partial }^{2}{C}^{n}\left( x\right) }{\partial {x}^{2}}+{\alpha }_{3}\frac{\partial {C}^{n}\left( x\right) }{\partial x}-</math><math>{\alpha }_{1}{C}^{n}\left( x\right) \quad \quad \, \, (13)</math>
−
| style="text-align: center;"|16.35
+
|}
 +
{|class="formulaSCP" style="width: 100%; text-align: center;"  
 
|-
 
|-
−
| style="text-align: center;"|1.4
+
| <math>{\propto }_{1}=\frac{6}{\Delta {t}^{2}{v}^{2}}\, \, ;\, \, {\alpha }_{2}=-2-\frac{3\Delta t}{v}\frac{dv}{dt}\, ;\, \, {\propto }_{3}=</math><math>-\frac{\Delta t}{{v}^{2}}\frac{{d}^{2}v}{d{t}^{2}}-\frac{3}{{v}^{2}}\frac{dv}{dt}-</math><math>\frac{6}{\Delta tv}\quad \quad \quad</math>
−
| style="text-align: center;"|16.35
+
|}
 +
{|class="formulaSCP" style="width: 100%; text-align: center;"  
 
|-
 
|-
−
| style="text-align: center;vertical-align: top;"|1.8
+
| <math>\forall x\in \left( 0,\left. 1\right] \quad \quad \quad \quad \, \, {C}^{n}\left( 0\right) =\right. </math><math>\left. {C}_{a}^{n}={C}_{a}\left( {t}^{n}\right) \right. \quad \quad \quad \, \,</math>
−
|  style="text-align: center;"|16.35
+
−
|-
+
−
|  style="text-align: center;"|2.2
+
−
|  style="text-align: center;"|16.49
+
−
|-
+
−
|  style="text-align: center;"|2.6
+
−
|  style="text-align: center;"|16.32
+
−
|-
+
−
|  style="text-align: center;"|3
+
−
|  style="text-align: center;"|16.13
+
 
|}
 
|}
  
−
===3.5 Defect depth effect===
 
  
−
Both the maximum von Mises stress and strain around the corrosion defect of pipeline increases with the increase of corrosion defect depths, as shown in [[#img-10|Figure 10]](a). However, the maximum von Mises stress and strain of the corrosion defect pipeline shows an slowly increase with the increase of corrosion defect depths, and then rapidly increase when the corrosion depth is greater than 60%. Therefore, in the case of avoiding ground overload as much as possible, pipeline defects should be checked regularly to prevent the depth of corrosion defects further increasing, so as to ensure the safe operation of the pipeline.
+
To be able to solve the differential equation of Second Order posed in (15) we need 2 conditions, but the problem only provides us with a condition so it was necessary to impose asecond condition to solve the problem, which we induc from the mathematical model for the problem of transport proper to the chemo-fluid oscillator:
−
 
+
{| class="formulaSCP" style="width: 100%; text-align: center;"  
−
In [[#img-10|Figure 10]](b), the pipeline settlement displacement along the axial direction does not vary significantly with the depth of corrosion defects increasing, which indicates that the depth of corrosion defects has little effect on the settlement displacement of the pipeline, under certain load conditions. Among them, the corrosion defect area has obvious warpage deformation, when the corrosion depth is 80%. As shown in [[#img-10|Figure 10]](c), the maximum stress increases with the increase of the corrosion depth in the overload zone, and maximum stress occurs at the center of the corrosion defect area.
+
−
 
+
−
The stress distribution is approximately apple-shaped under different corrosion defect depths, as shown in [[#img-10|Figure 10]](d). The stress value of the high stress area on the top of the pipeline increases with the increase of the corrosion defect depth. However, the stress value of the low stress area on both sides of the pipeline top decreases with the increase of the corrosion defect depth. The stress at the top of the pipeline is greater than the bottom, and the stress at 105° (255°) corresponding to both sides of the pipeline, is the smallest.
+
−
 
+
−
<div id='img-10'></div>
+
−
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: 45%;"
+
 
|-
 
|-
−
|style="padding-top:10px;"|  [[Image:Draft_Zheng_851632191-picture- 8.svg|center|600px]]
+
| <math>\left\{ \begin{matrix}\frac{d{l}_{v}}{dt}\left( t\right) =\gamma \left[ C\left( t,1\right) \right] \left\{ {l}_{eq}\left[ C\left( t,1\right) -{l}_{v}\left( t\right) \right] \right. \\{l}_{v}\left( 0\right) ={l}_{v}^{0}\end{matrix}\Longrightarrow \begin{matrix}{l}_{v}^{0}\, ,\, {C}^{0}\left( x\right) ={C}_{ini}\left( x\right) \, \, \forall x\in \left( 0\, ,\, \left. 1\right] \right. \quad \quad \quad \, \, \left( 14\right) \\{l}_{v}^{n+1}=F\left( {l}_{v}^{n}\, ,{C}^{n}\left( 1\right) ;\Delta t\right) ,n=0,1,2,...,N\end{matrix}\right.</math>
−
|- style="text-align: center; font-size: 75%;"
+
−
| colspan="1" style="padding:10px;"| '''Figure 10'''. Mechanical behavior of X80 corrosion-defective pipeline under different defect depths
+
 
|}
 
|}
  
  
−
In [[#img-11|Figure 11]], the stress concentration appears in the corrosion defect area, and the stress concentration increases with the increase of the defect depth. When the corrosion defect depth is 20%, the stress concentration is not obvious. When the depth of the corrosion defect is greater than 40%, the stress concentration will obviously extend from the center of the corrosion area along the diagonal direction of the corrosion defect and occurs in the entire corrosion area, as highlighted in dark red in [[#img-11|Figure 11]](a). At the same time, it can be seen that there is a low stress area along the width and the axial direction around the corrosion area, which also proves the stress distribution law of [[#img-11|Figures 11]](c) and (d). The plastic deformation of the corrosion defect area on the pipeline also has the same trend.
+
Deduction of the condition in <math display="inline">x=</math><math>1</math>
  
−
<div id='img-11'></div>
+
From the EDP:
−
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: 45%;"
+
{| class="formulaSCP" style="width: 100%; text-align: center;"  
 
|-
 
|-
−
|style="padding-top:10px;"|  [[Image:Draft_Zheng_851632191-picture- 9.svg|center|600px]]
+
| <math>v\left[ {l}_{v}\left( t\right) \right] \frac{\partial C}{\partial x}\left( t,x\right) =</math><math>-\frac{\partial C}{\partial t}\left( t,x\right) \quad \quad \quad \quad \, \, \left( 15\right)</math>
−
|- style="text-align: center; font-size: 75%;"
+
−
| colspan="1" style="padding:10px;"| '''Figure 11'''. Distributions of maximum von Mises stress and strain around X80 corrosion- defective pipelines under different defect depths
+
 
|}
 
|}
  
  
−
[[#tab-8|Table 8]] shows the relationship between corrosion depths and predicted failure pressure under ground overload. The results show that the failure pressure of the pipeline decreases with the corrosion depth increasing. Therefore, pipeline defect detection should be carried out in time to avoid a further increase in pipeline corrosion depth, which results in pipeline failure pressure decreasing.
+
semi-discretization of <math display="inline">\frac{\partial C}{\partial t}\, \Longrightarrow \frac{{C}^{n+1}\left( x\right) -{C}^{n}\left( x\right) }{\Delta t}</math>
−
 
+
{| class="formulaSCP" style="width: 100%; text-align: center;"  
−
<div class="center" style="font-size: 75%;">'''Table 8'''. Failure pressure of X80 corrosion-defective pipelines under different defect depths</div>
+
|-
−
 
+
| <math>\Rightarrow v\left[ {l}_{v}\left( {t}^{n+1}\right) \right] \frac{\partial C}{\partial x}\left( {t}^{n+1},x\right) =</math><math>-\left[ \frac{{C}^{n+1}\left( x\right) -{C}^{n}\left( x\right) }{\Delta t}\quad \right]</math>
−
<div id='tab-8'></div>
+
|}
−
{| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;"
+
{|class="formulaSCP" style="width: 100%; text-align: center;"  
−
|-style="text-align:center"
+
|-
−
!Defect depth !! Failure pressure (MPa)
+
| <math>\Rightarrow v\left[ {l}_{v}^{n+1}\right] \frac{\partial {C}^{n+1}}{\partial x}\left( x\right) =</math><math>-\frac{{C}^{n+1}\left( x\right) }{\Delta t}+\frac{{C}^{n}\left( x\right) }{\Delta t}</math>
−
|-style="text-align:center"
+
|}
−
| 20%
+
{|class="formulaSCP" style="width: 100%; text-align: center;"  
−
| 18.03
+
|-
−
|-style="text-align:center"
+
| <math>\Rightarrow v\left[ {l}_{v}^{n+1}\right] \frac{\partial {C}^{n+1}}{\partial x}\left( x\right) +</math><math>\frac{{C}^{n+1}\left( x\right) }{\Delta t}=\frac{{C}^{n}\left( x\right) }{\Delta t}\quad \quad \left( 16\right)</math>
−
| 40%
+
−
| 17.28
+
−
|-style="text-align:center"
+
−
| 60%
+
−
| 16.35
+
−
|-style="text-align:center"
+
−
| 80%
+
−
| 14.17
+
 
|}
 
|}
  
−
===3.6 Buried depth effect===
 
  
−
As shown in [[#img-12|Figure 12]](a), both the maximum von Mises stress and strain of corrosion defects on pipeline decrease with the increase of buried depth. When the buried depth is less than 1m, the maximum von stress decreases significantly with the increase of the buried depth, while the depth exceeds 1m, this downward trend decreases. This indicates that the protection capability of the pipeline does not increase significantly with the buried depth continues increased, when the buried depth of the pipeline reaches a certain value. The plastic deformation of the pipeline has the same changing trend with the buried depth of the pipeline increasing. Therefore, the appropriate depth of backfill should be selected, according to reasonable topography and landforms to decrease the probability of pipeline failure.
+
That by evaluating her in, we have a Condition of Robin. <math display="inline">x=</math><math>1</math>
  
−
In [[#img-12|Figure 12]](b), when the buried depth is 0.4m, the pipeline settlement reaches the largest, and a certain degree of warping deformation occurs at the center of the corrosion defect location of the pipeline. In the overload zone and non-overload zone, the stress decreases with the increase of the buried depth, as shown in [[#img-12|Figure 12]](c). In the transition zone, the magnitude of the reduction of the stress value is very obvious, and the stress value is basically the same. Therefore, the shallower the buried depth, the easier to damage.
+
==2.2 Taylor Method - Galerkin==
  
−
There is a high stress area that appears around the corrosion defect area, as shown in [[#img-12|Figure 12]](d). Low-stress areas and high-stress areas alternately appear along the width direction on both sides of the corrosion defect, the stress value of the high-stress area on both sides of the corrosion defect increases with the buried depth increasing. The minimum stress appears at about 90° (270°) of the pipe section. The stress at the bottom of the pipe is less than that at the top, so the top of the pipe is most easily damaged.
+
Considering the internal product on the range : <math display="inline">{L}_{2}\left( 0,1\right)</math>
−
 
+
{| class="formulaSCP" style="width: 100%; text-align: center;"  
−
<div id='img-12'></div>
+
−
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: 45%;"
+
 
|-
 
|-
−
|style="padding-top:10px;"| [[Image:Draft_Zheng_851632191-picture-10.svg|center|600px]]
+
| <math>{\left\langle u,v\right\rangle }_{{L}_{2}}:=\int_{0}^{1}u\left( x\right) \cdot v\left( x\right) dx\quad \quad \quad \quad \quad \left( 17\right)</math>
−
|- style="text-align: center; font-size: 75%;"
+
−
| colspan="1" style="padding:10px;"| '''Figure 12'''. Mechanical behavior of X80 corrosion-defective pipeline under different buried depths
+
 
|}
 
|}
−
 
+
{|class="formulaSCP" style="width: 100%; text-align: center;"  
−
 
+
−
It can be seen that the stress concentration appears in the corrosion defect area, as highlighted in dark red in[[#img-13|Figure 13]](a). And the stress level decreases with the increase of the buried depths. The stress concentration area gradually decreases along the width and diagonal directions of the corrosion area. From the strain distribution, the plastic deformation distributes at the bottom of the corrosion defect, and the maximum plastic deformation occurs at the center of the bottom of the corrosion defect. The plastic deformation decreases with the buried depth increasing. The shallower the buried pipeline, the easier it to fail under ground overload. Therefore, buried pipelines should be buried more deeply in areas where ground overload may occur.
+
−
 
+
−
<div id='img-13'></div>
+
−
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: 45%;;"
+
 
|-
 
|-
−
|style="padding-top:10px;"|  [[Image:Draft_Zheng_851632191-picture-11.svg|center|600px]]
+
| <math>\left\langle \frac{{\partial }^{2}{c}_{\left( x\right) }^{n+1}}{{\partial x}^{2}}-\right. </math><math>\left. {\alpha }_{1}{c}_{\left( x\right) }^{n+1}-{\alpha }_{2}\frac{{\partial }^{2}{c}_{\left( x\right) }^{n}}{{\partial x}^{2}}+\right. </math><math>\left. {\propto }_{3}\frac{{\partial c}_{\left( x\right) }^{n}}{\partial x}-{\alpha }_{1}{c}_{\left( x\right) }^{n}\, ,\, v\left( x\right) \right\rangle =</math><math>0\, \, \left( 18\right) \, \,</math>
−
|- style="text-align: center; font-size: 75%;"
+
−
| colspan="1" style="padding:10px;"| '''Figure 13'''. Distributions of maximum von Mises stress and strain around X80 corrosion- defective pipelines under different buried depths
+
 
|}
 
|}
  
  
−
[[#tab-9|Table 9]] shows the relationship between the buried depth of the pipeline and the predicted failure pressure under ground overload. The failure pressure decreases, as the buried depth of the pipeline increases. Properly increasing the buried depth of the pipeline is beneficial to increase the failure pressure value of the pipeline.
+
Applying the definition of the internal product in the spaces, <math display="inline">{L}_{2}</math> with border conditions of Dirilecht and Robin, applying the integration formula in parts and replacing the functions in a way that we have for the problem: <math display="inline">\left( 18\right) \sum _{j=1}^{M}{c}_{j}^{n+1}\frac{{d\varphi }_{j}\left( 1\right) }{dx}{\varphi }_{i}\left( 1\right) -</math><math>\sum _{j=1}^{M}{c}_{j}^{n+1}\int_{0}^{1}\frac{d{\varphi }_{j}\left( x\right) }{dx}\frac{d{\varphi }_{i}\left( x\right) }{dx}dx\, +</math><math>{\alpha }_{1}\int_{0}^{1}\left( \sum _{j=1}^{M}{c}_{j}^{n+1}{\varphi }_{j}\left( x\right) \right) \, {\varphi }_{i}\left( x\right) dx\quad =</math><math>-{\alpha }_{2}\sum _{j=1}^{M}{c}_{j}^{n}{\frac{d{\varphi }_{j}\left( 1\right) }{dx}}_{j}{\varphi }_{i}\left( 1\right) \, +</math><math>{\alpha }_{2}\sum _{j=1}^{M}{c}_{j}^{n}\int_{0}^{1}\frac{d{\varphi }_{j}\left( x\right) }{dx}\frac{d{\varphi }_{i}\left( x\right) }{dx}dx\quad +</math><math>{\alpha }_{3}\sum _{j=1}^{M}{c}_{j}^{n}\frac{d{\varphi }_{j}\left( x\right) }{dx}{\varphi }_{i}\left( x\right) dx\, \, -</math><math>{\alpha }_{1}\sum _{j=1}^{M}{c}_{j}^{n}\int_{0}^{1}{\varphi }_{i}\left( x\right) {\varphi }_{j}\left( x\right) dx\quad \quad \quad \, \, \forall {\varphi }_{i},{\varphi }_{j}\in {V}_{h}\quad \quad \quad \left( 19\right)</math>
  
−
<div class="center" style="font-size: 75%;">'''Table 9'''. Failure pressure of X80 corrosion-defective pipelines under different buried depths</div>
+
==2.3 Finite Element Method ==
  
−
<div id='tab-9'></div>
+
We will consider a discretization of finite elements as explained above, but adapted to the working <math display="inline">\left[ 0,1\right]</math>  interval. In effect the partition of this interval corresponds to the longitudinal discretization of the delay line channel into length elements for Being , the partition represents a mesh of points that we will denote by . <math display="inline">\left( M-\right. </math><math>\left. 1\right) \, {I}_{j}=\left[ {x}_{j},{x}_{j+1}\right] {h}_{j}={x}_{j+1}-{x}_{j}j=</math><math>1,2,3,...,M.{h}_{i}=h{x}_{j}{\tau }_{h}</math>
−
{| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;"
+
 
−
|-style="text-align:center"
+
If we write <math display="inline">\left( 19\right)</math> using the elementary formulation we would have:
−
! Buried depth (m) !! Failure pressure (MPa)
+
{| class="formulaSCP" style="width: 100%; text-align: center;"  
−
|-style="text-align:center"
+
|-
−
|  0.4
+
| <math>\left[ \begin{matrix}\frac{d{\varphi }_{i}^{,}\left( 1\right) }{dx}\cdot {\varphi }_{i}\left( 1\right) &\frac{d{\varphi }_{i}^{,}\left( 1\right) }{dx}\cdot {\varphi }_{i+1}\left( 1\right) \\\frac{d{\varphi }_{i+1}^{,}\left( 1\right) }{dx}\cdot {\varphi }_{i}\left( 1\right) &\frac{d{\varphi }_{i+1}^{,}\left( 1\right) }{dx}\cdot {\varphi }_{i+1}\left( 1\right) \end{matrix}\right] \left[ \begin{matrix}{C}_{i}^{n+1}\\{c}_{i+1}^{n+1}\end{matrix}\right] -</math><math>\left[ \begin{matrix}\frac{d{\varphi }_{i}^{,}\left( 0\right) }{dx}\cdot {\varphi }_{i}\left( 1\right) &\frac{d{\varphi }_{i}^{,}\left( 0\right) }{dx}\cdot {\varphi }_{i+1}\left( 1\right) \\\frac{d{\varphi }_{i+1}^{,}\left( 0\right) }{dx}\cdot {\varphi }_{i}\left( 1\right) &\frac{d{\varphi }_{i+1}^{,}\left( 0\right) }{dx}\cdot {\varphi }_{i+1}\left( 1\right) \end{matrix}\right] \left[ \begin{matrix}{C}_{i}^{n+1}\\{c}_{i+1}^{n+1}\end{matrix}\right] +</math><math>\frac{1}{{h}_{i}}\left[ \begin{matrix}1&-1\\-1&1\end{matrix}\right] \left[ \begin{matrix}{C}_{i}^{n+1}\\{c}_{i+1}^{n+1}\end{matrix}\right] +</math><math>\frac{{\alpha }_{1}}{6}{h}_{i}\left[ \begin{matrix}2&1\\1&2\end{matrix}\right] \left[ \begin{matrix}{C}_{i}^{n+1}\\{c}_{i+1}^{n+1}\end{matrix}\right] +</math><math>\frac{-{\propto }_{311}}{2}\left[ \begin{matrix}-1&1\\-1&1\end{matrix}\right] \left[ \begin{matrix}{C}_{i}^{n+1}\\{c}_{i+1}^{n+1}\end{matrix}\right] \left[ {c}_{M}^{n+1}\right] =</math><math>{\propto }_{2}\left[ \begin{matrix}\frac{d{\varphi }_{i}^{,}\left( 1\right) }{dx}\cdot {\varphi }_{i}\left( 1\right) &\frac{d{\varphi }_{i}^{,}\left( 1\right) }{dx}\cdot {\varphi }_{i+1}\left( 1\right) \\\frac{d{\varphi }_{i+1}^{,}\left( 1\right) }{dx}\cdot {\varphi }_{i}\left( 1\right) &\frac{d{\varphi }_{i+1}^{,}\left( 1\right) }{dx}\cdot {\varphi }_{i+1}\left( 1\right) \end{matrix}\right] \left[ \begin{matrix}{C}_{i}^{n}\\{c}_{i+1}^{n}\end{matrix}\right] -</math><math>{\propto }_{2}\left[ \begin{matrix}\frac{d{\varphi }_{i}^{,}\left( 0\right) }{dx}\cdot {\varphi }_{i}\left( 0\right) &\frac{d{\varphi }_{i}^{,}\left( 0\right) }{dx}\cdot {\varphi }_{i+1}\left( 0\right) \\\frac{d{\varphi }_{i+1}^{,}\left( 0\right) }{dx}\cdot {\varphi }_{i}\left( 0\right) &\frac{d{\varphi }_{i+1}^{,}\left( 1\right) }{dx}\cdot {\varphi }_{i+1}\left( 0\right) \end{matrix}\right] \left[ \begin{matrix}{C}_{i}^{n}\\{c}_{i+1}^{n}\end{matrix}\right] +</math><math>\frac{-{\propto }_{2}}{{h}_{i}}\left[ \begin{matrix}1&-1\\-1&1\end{matrix}\right] \left[ \begin{matrix}{C}_{i}^{n}\\{c}_{i+1}^{n}\end{matrix}\right] +</math><math>\frac{{\propto }_{322}}{2}\left[ \begin{matrix}-1&1\\-1&1\end{matrix}\right] \left[ \begin{matrix}{C}_{i}^{n}\\{c}_{i+1}^{n}\end{matrix}\right] +</math><math>\frac{{\propto }_{1}{h}_{i}}{6}\left[ \begin{matrix}2&1\\1&2\end{matrix}\right] \left[ \begin{matrix}{C}_{i}^{n}\\{c}_{i+1}^{n}\end{matrix}\right] \quad \quad \quad \quad \quad \quad \left( 20\right)</math>
−
|  16.06
+
−
|-style="text-align:center"
+
−
|  0.6
+
−
|  16.20
+
−
|-style="text-align:center"
+
−
|  0.8
+
−
|  16.32
+
−
|-style="text-align:center"
+
−
| 1.0
+
−
|  16.35
+
−
|-style="text-align:center"
+
−
| 1.2
+
−
|  16.7
+
 
|}
 
|}
  
−
===3.7 Soil Young’s modulus effect===
+
=3.Numerical experimentation.=
  
−
In [[#img-14|Figure 14]](a), both the maximum von Mises stress and strain of pipeline corrosion defects decrease with the increase of soil Young’s modulus. Moreover, the maximum Mises stress and plastic deformation shows two different downward trends with the increase of soil Young’s modulus. When the soil Young’s modulus is greater than 20, the stress and plastic deformation change rate becomes lower, which indicates that the ability to resisting deformation of the soil will not change significantly with the soil Young’s modulus increase, when the soil Young’s modulus is over certain value.
+
After raising the mathematical part for the model and obtaining the system of equations, a program was designed in Matlab for the coupled system where several tests were performed with their respective numerical adjustments based on the theoretical definitions explained in chapters 2 and 3.
  
−
The amount of settlement displacement decreases with the increase of soil Young’s modulus, as shown in [[#img-14|Figure 14]](b). When the soil Young’s modulus is 10MPa, the change rate of settlement displacement increases significantly, and the maximum settlement displacement reaches 0.1m. The stress distribution along the axial direction of the pipeline is shown in [[#img-14|Figure 14]](c), and the maximum stress occurs in the middle of the corrosion defect location. When the soil Young’s modulus is 10MPa, the stress value in the load zone is the largest. Meanwhile, in the non-loaded zone, the stress distribution is higher at a longer distance, because the soil is softer.
+
<span id='_Hlk40781132'></span>Each experiment details the change in the initial <math display="inline">C\left( 0,x\right) =</math><math>{c}_{ini}\left( x\right)</math> concentration which is a unique experimental value for the operation of the hydrogel that is not listed as data in Páez and has been imposed on it according to the physical model. <math display="inline">\left[ 1\right]</math>
  
−
As shown in [[#img-14|Figure 14]](d), a high stress area appears around the corrosion defect area. When the soil Young’s modulus is 10MPa，an obvious high-stress area appears on both sides along the width direction of the corrosion defect. The overall stress distribution is similar to that described in the previous section. Low-stress areas and high-stress areas alternately appear along the width direction on both sides of the corrosion defect, and the stress value of the high-stress area on both sides of the corrosion defect increases with soil Young’s modulus increasing. Therefore, the appropriate backfill soil should be selected according to the reasonable topography and geomorphology.
+
'''Experiment 1 '''
  
−
<div id='img-14'></div>
+
Values are taken in the space of <math display="inline">M=</math><math>20</math> (nodes); ;   ; ; Courant 0.1 <math display="inline">dx=</math><math>\frac{1}{M-1}Courant=\frac{dt}{dx}dt=courant\cdot dx</math>
−
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: 45%;"
+
−
|-
+
−
|style="padding-top:10px;"| [[Image:Draft_Zheng_851632191-picture-12.svg|center|600px]]
+
−
|- style="text-align: center; font-size: 75%;"
+
−
| colspan="1" style="padding:10px;"| '''Figure 14'''. Mechanical behavior of X80 corrosion-defective pipeline under different soil Young’s modulus
+
−
|}
+
  
 +
A constant value was taken as a condition which in this case is the experimental value with which the numerical part worked in the initial study. <math display="inline">C\left( 0,x\right) =</math><math>{c}_{ini}\left( x\right) {c}_{ini}\left( x\right) =2.2635</math>
  
−
The von Mises stress and plastic deformation distribution both sides along with the width and length directions of the corrosion defect substrate area decreases with the increase of soil Young’s modulus, as shown in [[#img-15|Figure 15]]. When the soil Young’s modulus is 10MPa, the maximum stress and plastic deformation reaches 632.9MPa and 0.01789, respectively. The maximum value of plastic deformation appears at the center of the corrosion defects, as highlighted in dark red in [[#img-15|Figure 15]](b). The higher the soil Young’s modulus, the harder the soil, and the stronger the ability to resisting deformation of soils under ground overload. Therefore, choosing a high soil Young’s modulus for pipeline backfilling will help to reduce the possibility of pipeline damage.
+
The values of those derived from the concentration found in are replaced by forward differences (Euler) for .<math display="inline">{\alpha }_{3}t\geq 0</math>
  
−
<div id='img-15'></div>
+
[[Image:Draft_Alvarez_171754839-image3.png|600px]]
−
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: 45%;"
+
−
|-
+
−
|style="padding-top:10px;"|  [[Image:Draft_Zheng_851632191-picture-13.svg|center|600px]]
+
−
|- style="text-align: center; font-size: 75%;"
+
−
| colspan="1" style="padding:10px;"| '''Figure 15'''. Distributions of maximum von Mises stress and strain around X80 corrosion- defective pipelines under different soil Young’s modulus
+
−
|}
+
  
 +
<span style="text-align: center; font-size: 75%;">'''Figure 3. Response of the chemo-fluidicoscillator  modeled by the system.'''</span>
  
−
The type of stratum soil also affects the failure pressure of the pipeline. [[#tab-10|Table 10]] shows the predicted failure pressure values of corrosion defect pipelines under different soil Young’s modulus. The results show that the failure pressure increases with the increase of the soil Young’s modulus. When the soil Young’s modulus is 10MPa, the failure pressure of the pipeline is only 15.71MPa, which indicates that it is easier to damage. Therefore, the soil type should avoid a soft soil layer when laying the pipeline.
+
Periodic behavior can be observed in the Hidrogel which is what allows the device to function as an oscillator with negative feedback resulting from the increase and decrease of the alcohol mixture in the hydrogel chamber but at the beginning of the flow of mixture of alcohol and water there is instability in the wave fronts that are then regularized as the process of opening and closing the valve progresses that because the hydrogel suffers deformation greater than the length of the camera
  
−
<div class="center" style="font-size: 75%;">'''Table 10'''. Failure pressure of X80 corrosion-defective pipelines under different soil Young’s modulus</div>
+
[[Image:Draft_Alvarez_171754839-image4.png|600px]]
  
−
<div id='tab-10'></div>
+
<span style="text-align: center; font-size: 75%;">'''Figurineat 4. Response of the chemo-fluidic oscillator modeled by state variables.'''</span>
−
{| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;"  
+
−
|-style="text-align:center"
+
−
! Young’s modulus (MPa) !! Failure pressure (MPa)
+
−
|-style="text-align:center"
+
−
|  10
+
−
|  15.71
+
−
|-style="text-align:center"
+
−
| 20
+
−
|  16.35
+
−
|-style="text-align:center"
+
−
|  30
+
−
|  16.49
+
−
|-style="text-align:center"
+
−
| 40
+
−
|  16.45
+
−
|-style="text-align:center"
+
−
|  50
+
−
|  16.47
+
−
|}
+
  
−
==4. Conclusions==
+
==Experiment 2==
  
−
A FE model is established to analyze the local stress and strain distributions around the corrosion defect on the pressure pipeline under ground overload. The effects of internal pressure, ground overload, length of the loading area, defect depth, buried depth and soil Young’s modulus on mechanical behavior and failure pressure of buried corrosion defect pipelines are studied. The following conclusions are obtained:
+
In this trial, a quadratic profile was taken as an initial condition since there is a mixture of water and alcohol in the canal resulting from deionized water flows and alcohol but not reaching the minimum concentration level so that the hydrogel reacts and begins to compress due to the increase in alcohol, periodic behavior occurs , but with less disturbance at the beginning of the process allowing a more stable and smooth concentration flow on the wave fronts.
  
−
# The presence of ground overload results in the local von Mises stress and strain around the corrosion defect location increases greatly nonlinearly, cause the internal pressure threshold reduce. Stress and strain increase nonlinearly with the increase of internal pressure.
+
[[Image:Draft_Alvarez_171754839-image5.png|600px]]
−
# The von Mises stress, stress area, strain, strain area and settlement displacement of buried corrosion defect pipelines increase, and the failure pressure of pipeline decreases, with the increase of ground overload, length of the loading area and corrosion defect depth. However, the von Mises stress, stress area, strain, strain area and settlement displacement of buried corrosion defect pipelines decrease, and he failure pressure of the pipeline increases, with the increase of buried depth and soil Young’s modulus. High stress and high strain are located in the corrosion defect area and develop from the corrosion center along the diagonal direction of the corrosion defect.
+
  
 +
<span style="text-align: center; font-size: 75%;">'''Figurineat 5. Response of the chemo-fluidic  oscillator modeled by state variables.'''</span>
  
−
Pipeline safety assessment under ground overload is an indispensable part of the pipeline integrity management program. The research results of this paper can be used to determine the local stress and strain of corrosion defects of different sizes under ground overload. Quantitative analysis of the stress, strain and failure pressure of buried pipelines with corrosion defects under ground overload will provide important suggestions on safe operating conditions and improve pipeline integrity management for the industry.
+
[[Image:Draft_Alvarez_171754839-image6.png|600px]]
  
−
==Acknowledgement==
+
<span style="text-align: center; font-size: 75%;">'''Figure 3. Response of the chemo-fluidicoscillator  modeled by the system.'''</span>
  
−
This research work was supported by National Key Research and Development Program of China (2016YFC0802102-03); Luzhou City Science and Technology Planning Project (2018-GYF-6); Sichuan Science and Technology Project: (20MZGC0090); Open Fund of National Joint Engineering Research Center for abrasion control and molding of metal materials (HKDNM201806); SWPU Youth Research and Innovation Team (2018CXTD12).
+
<span id='_Hlk41429379'></span>
  
−
==References==
+
=4. CONCLUSIONS AND RECOMMENDATIONS.=
−
<div class="auto" style="text-align: left;width: auto; margin-left: auto; margin-right: auto;font-size: 85%;">
+
−
[1] Zhang J., Liang Z., Zhao G. Mechanical behaviour analysis of a buried steel pipeline under ground overload. Engineering failure analysis, 63:131-145, 2016.
+
  
−
[2] Alzabeebee S., Chapman D.N., Faramarzi A. A comparative study of the response of buried pipes under static and moving loads. Transportation Geotechnics, 15:39-46, 2018.
+
The first observation that can be made is that the method applied to the system composed of <math display="inline">(15),\left( 16\right) \, y\, \left( 17\right) \,</math> equations reproduces the dynamics of the original numerical model of Paez, with small differences in amplitude and period, but that it is able to produce stable periodic signals for parameter configuration without needing any external forging, which means that the oscillating behavior is self-excited. <math display="inline">\left[ 1\right]</math>
  
−
[3] Naeini S.A., Mahmoudi E., Shojaedin M.M., Misaghian M. Mechanical response of buried High-Density Polyethylene pipelines under normal fault motions. KSCE Journal of Civil Engineering, 20(6):2253-2261, 2016.
+
The second observation is that from a numerical approximation of the linear transport equation (constant velocity) based on the Euler-Taylor-Galerkin method for discretization over time and the Finite Elements Method for discretization in the space posed by Donea, by applying the same method with some variations in the initial conditions and border equation in the equation of the transport of the nonlinear system that is also coupled to a nonlinear ordinary differential equation that governs the behavior of the hydrogel and another equation that controls the volume of the buffert resulting in a complex system of solving, satisfactory results were obtained in relation to its oscillation and its periodic and dimensioned movement. It is very important to note that this numerical model makes it possible to observe that if we change the initial value there will be a variation at the beginning of the hydrogel's operation that would be in the stationary regimen but that after <math display="inline">{C}_{ini}</math> this initial regimen the hydrogel shows an oscillating and periodic behavior typical of Paez's initial analysis.
  
−
[4] Saadeldin R., Hu Y., Henni A. Numerical analysis of buried pipes under field geo-environmental conditions. International Journal of Geo-Engineering, 6(6), 2015.
+
The third observation is that this numerical analysis applied to the chemo-fluidic oscillator analytically contributes to the understanding that the valve composed of the hydrogel is extremely sensitive to the variation of the parameters, i.e. it has the ability to drastically change its volume under small variations of special thermodynamic parameters.
  
−
[5] Qiao Y.Y. Study on the influence of heap loading on the force of buried gas pipelines and safety evaluation methods. MSc Thesis, Chengdu University of Technology, 2017.
+
In this research another option could be given to mathematically model the domains of the oscillator, in the fluidic domain given by the flow network, the transport of concentration through the delay line and in the chemical domain given by the behavior of the hydrogel and the concentration of alcohol that produce a smooth dynamic system in parts.
  
−
[6] Xu T., Mao J., Liu Y., Yao A. Dynamic response and limit analysis of buried gas pipeline under ground consolidation load. Journal of Vibroengineering, 19(7):4989-5008, 2017.
+
It is recommended to use for future studies a mathematical refinement method such as the Galerkin-Discontinuous method to be able to model in more detail the behavior of the Chemo-fluidic Oscillator.
  
−
[7] Fang H., Yang K., Li B., Tan P., Wang F., Du X. Experimental and numerical study on mechanical analysis of buried corroded concrete pipes under static traffic loads. Applied Sciences, 9(23):5002, 2019.
+
=5. References.=
  
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[8] Liang Z., Yang Q., Zhang J., Zhu B. Mechanical analysis of buried polyethylene pipelines underground overload. Journal of Failure Analysis and Prevention, 19(1):193-203, 2019.
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<math display="inline">\left[ 1\right]</math> <span style="text-align: center; font-size: 75%;"> Jhosep Paez, A new self-extracting chemo-fluidic oscillator based on stimuli sensitive hydrogels: Mathematical model and dynamic behavior.</span>
  
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[9] Cai Z.Y, Zhang C.Y, Huang M.S. Analysis of the influence of ground surcharge on buried pipelines with variable cross sections. Rock and Soil Mechanics 32(S1):408-412, 2011.
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<math display="inline">\left[ 2\right]</math> <span style="text-align: center; font-size: 75%;"> Jean Donea, A Taylor – Galerkin Method for Convective Transport problems.</span>
  
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[10] Trickey S.A., Moore I.D. Three-dimensional response of buried pipes under circular surface loading. Journal of geotechnical and geoenvironment engineering, 133(2):219-223, 2007.
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<math display="inline">\left[ 3\right]</math> <span style="text-align: center; font-size: 75%;"> P.D. Lax and B. Wendroff , "On the stability of difference Schemes", Comm. Pure Appl . Math .13,217(1960).</span>
  
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[11] Meesawasd N., Boonyasiriwat C., Kongnuan S., Chamchod F. Finite element modeling for stress analysis of a buried pipeline under soil and traffic loads. IEEE, 2016, pp. 385-390.
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<math display="inline">\left[ 4\right]</math> <span style="text-align: center; font-size: 75%;"> P. D Lax and B.Wendroff , "Difference Schemes for Hyperbolic equations with high order of accuracy". comm, Pure, Appl. Math. 13381 (1964).</span>
  
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<math display="inline">\left[ 5\right]</math> <span style="text-align: center; font-size: 75%;"> C.E. Leith, in Methods in Computational. Physics, Vol 4, P.1, Academic Press, New York, 1965.</span>
  
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[13] Kolbadi S.M.S., Hassani N., Mahdi S., Kolbadi S., Shiravand M.R. Effect of ground motions on nonlinear seismic behavior of corroded buried gas pipeline. American Journal of Civil Engineering, 3(2):9-13, 2015.
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<math display="inline">\left[ 6\right] \,</math> <span style="text-align: center; font-size: 75%;">J.K. Dukowicz and J.D. Ramshaw, "Tensor Viscosity Method for Convection in numerical fluid dinamics" J.compu . Phys , 32.71-79 (1979).</span>
  
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+
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</div>
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Latest revision as of 10:53, 28 July 2021


ABSTRACT

In this work, a finite element scheme is proposed using a method of Euler-Taylor-Galerkin described in [1], for a non-linear model which describes the behavior of a new chemo-fluidic oscillator, [2]. This model is expressed by the coupling of an ordinary differential equation describing the hydrogel dynamics, the non-linear transport equation and an auxiliary equation determining the flux volume. The numerical solution is constructed by taking a semi-discretization in time of the transport equation, employing forward-time Taylor series expansions including time derivatives of second order and third order, avoiding instabilities problems. In this semi discrete equation, the spatial variable is approximated by the finite element formulation according to Galerkin. Some simulations are carried out taking different initial conditions for the concentration of the hydrogel. The numerical results describes the oscillatory behavior of the system as in [2], where MatLab tools are used as black box.

Facultad de Ciencias Naturales y Matemáticas, Escuela Superior Politécnica del Litoral, Km. 30.5 Vía Perimetral, Guayaquil, Ecuador

ialvarez@espol.edu.ec

Facultad de Ciencias Naturales y Matemáticas, Escuela Superior Politécnica del Litoral, Km. 30.5 Vía Perimetral, Guayaquil, Ecuador

davetera@espol.edu.ec

Facultad de Ciencias Naturales y Matemáticas, Escuela Superior Politécnica del Litoral, Km. 30.5 Vía Perimetral, Guayaquil, Ecuador

wangulo@espol,edu.ec

Facultad de Ciencias Naturales y Matemáticas, Escuela Superior Politécnica del Litoral, Km. 30.5 Vía Perimetral, Guayaquil, Ecuador

dany@espol.edu.ec

Facultad de Ciencias de la Computación y Matemáticas, Universidad de la Habana, calle 25 y Vedado , La Habana , Cuba

angela@matcom.uh.cu

Keywords: [New chemo-fluidic oscillator], [Non-linear model],[Finite element], [hydrogel dynamics,].

Introduction12

Self-oscillating systems play an important role in both the natural sciences (biology or chemistry) and technology (micro electromechanical or electronic systems) because they can be coupled to other systems, one of their most important applications is the use as a system clock to trigger regular events such as circadian rhythm or electronic systems.

This work refers to the use of a Taylor – Galerkin method for applying finite element to the nonlinear system that controls the behavior of the new chemo oscillator – fluidic and determine its numerical modeling. The mathematical problem was raised by Páez who made a numerical approximation of the transport PDA using the well-known Line Method, however, the EDOS system provides a very rough approximation of the solution of the transport equation. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left[ 1\right]}


First, this is due to the first-order discretization in space and secondly because it can propagate abrupt changes or steep fronts, which is a well-known computational problem in the numerical solution of hyperbolic PED.

In order to describe the dynamics of the chemo-fluidic oscillator a set of mathematical models was used, which I pose a challenge due to the complexity of the system, since the oscillator is affected by various nonlinearities that come from the characteristics of the hydrogel and the bidirectional coupling between the chemical and fluidic domains. The hydrogel is designed in such a way that an increase in the concentration of alcohol reduces its size and vice versa. Therefore, at low concentrations of alcohol, the hydrogel valve is closed, while high concentrations of alcohol open the valve. A bypass channel is connected to the valve inlet, to allow a continuous flow of unidirectional fluid near the hydrogel independent of whether the hydrogel valve is open or closed.

Therefore to facilitate the modeling process, the system was divided into the fluidic domain describing the behavior of volumetric flows and system pressures during operation and chemical domain which in turn divided into two parts the description of hydrogel dynamics and the modeling of the delay line , resulting in a coupled system composed of the one-dimensional transport EDP, the EDO that models the dynamic behavior of the hydrogel and the equation that determines the volume in the buffer.

To perform this work following very closely what Donea did is semi-discretized the equation of nonlinear transport using Taylor's serial expansions at the time of first, second and third order to obtain a second order differential equation in the space in which we apply Galerkin's variational formulation to use the finite element method and obtain the system of linear equations that needs the value of the variable that determines the the size of the hydrogel that is obtained for each instant of time by applying the Runge -Kutta 4 method and the buffer volume that is found using numerical integration methods. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left[ 2\right]}


Analyzing in more detail its mathematical modeling and the numerical solution of this New Chemical-Fluidic Oscillator based on intelligent hydrogels was one of the reasons for the realization of this work in addition to extending the linear method used by Donea [2] for the nonlinear transport equation and providing a solution methodology for models of new oscillators involving the nonlinear one-dimensional transport equation.

1.Design of a chemo-fluoscillator

The new fluid chemo oscillator isbased on a negative feedback circuit containing a delay line, where negative feedback is provided by a hydrogel valve that has the ability to change its size depending on the temperature and concentration of the aqueous solution that is in contact with live with the hydrogel. In this new oscillator the temperature remains constant so the only parameter that produces a change in the size of the hydrogel is the concentration of alcohol.

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Figure 1. Photography of the manufactured chemo-fluidic oscillator circuit, filled with a highly dyed solution for better visibility of the channels. External sources of constant flow and pressure are shown schematically. Equivalent fluidic circuit. The hydrogel valve is represented by a controlled flow source.

The oscillator is powered by three constant sources. The first is a constant flow source that supplies the system with an alcohol concentration solution. A second source provides deionized water at a constant pressure located at Node 1. Water flows through a long channel called the damping line and then mixes into Node 2 with the alcohol solution provided by , and then the mixed solution enters the channel of the long fluid that acts as a delay line. Using this channel, the solution is transported at a rate determined by the flow through the delay line and its cross-section. The end of this channel is connected to the inlet of the hydrogel valve, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {q}_{A}{c}_{alc}{p}_{W}{q}_{A}{q}_{2}\left( t\right)} 

whosefluid behavior is controlled by the alcohol concentration of the solution. Finally, a bypass channel connects to Node 3. to drain the liquid to a conveniently chosen constant flow rate.  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {q}_{B}}


The micro-fluid system will be modeled through Kirchhoff's laws, within the framework of network theory for a circuit. This approach is used by how small the dimensions of the magnitudes that govern the operation of the oscillator within the study of the micro fluids since they are in a range of micro nano and peak liter, therefore, the pressure is considered analogous to the voltage and volumetric flow rate to the electric current, as well as the lines of delay to the resistors. In this context, the oscillator can be described by the fluid network presented in Figure 2.

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Figure 2. Equivalent fluidic circuit. The hydrogel valve is represented by a controlled flow source.

2.Mathematical model of chemical oscillator - fluidic.

The mathematical problem to be studied was raised by Páez et al. en , and consists in finding functions : such that: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left[ 1\right] {l}_{v}\left[ 0,T\right] \times \left[ 0,1\right] \rightarrow {R}_{+}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{matrix}\frac{d{l}_{v}}{dt}\left( t\right) =\gamma \left[ C\left( t,1\right) \right] \left\{ {l}_{eq}\left[ C\left( t,1\right) -{l}_{v}\left( t\right) \right] \right. \\{l}_{v}\left( 0\right) ={l}_{v}^{0}\end{matrix}\Longrightarrow \begin{matrix}{l}_{v}^{0}\, ,\, {C}^{0}\left( x\right) ={C}_{ini}\left( x\right) \, \, \forall x\in \left( 0\, ,\, \left. 1\right] \right. \quad \quad \quad \, \, \left( 1\right) \\{l}_{v}^{n+1}=F\left( {l}_{v}^{n}\, ,{C}^{n}\left( 1\right) ;\Delta t\right) ,n=0,1,2,...,N\end{matrix}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \begin{matrix}\frac{\partial C}{\partial t}\left( t,x\right) +v\left( {l}_{v}\left( t\right) \right) \frac{\partial C}{\partial x}\left( t,x\right) =0\, \, ,\, \forall \left( t,x\right) \in \left( 0,\left. T\right] \times \left( 0,\left. 1\right] \right. \right. \, ;\quad \\C\left( 0,x\right) ={C}_{ini}\left( x\right) \, ,\quad \forall x\in \left[ 0,{L}_{d}\right] ;\\C\left( t,0\right) ={C}_{a}\left( t\right) \, ,\quad \quad \forall t\geq \left[ 0,T\right] ;\end{matrix}\quad (2)\quad


The data in this problem are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \Upsilon} 

the functions , ,, , , and where is the unknown value to be determined from the function at the end ; which must also be determined Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {l}_{eq}\, v\, {C}_{ini}\, {C}_{a}{l}_{v}^{0}C\left( t,1\right) C\left( t,x\right) x=}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 1 

simultaneously, in this problem, the function : defined by: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {V}_{buff}\left( t\right) \left[ 0,T\right] \rightarrow {R}^{+}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {V}_{\_vuff}\left( t\right) :=\int_{0}^{t}{q}_{2}\left[ {l}_{v}\left( s\right) \right] ds-

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {q}_{A}t\, ;\forall t\in \left( 0,\left. T\right] ,\right. \quad \quad (3)\quad \,


Where it is a known function and is a constant also known. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {q}_{2}{q}_{A}}


2.1 Semi - discretization in problem time (2)

The transport equation is considered

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{\partial C}{\partial t}\left( t,x\right) =-v\left( {l}_{v}\left( t\right) \right) \frac{\partial C}{\partial x}\left( t,x\right) \, \, ,\, \forall \left( t,x\right) \in \left( 0,\left. T\right] \times \left( 0,\left. 1\right] \right. \right. \, ;


If it is denoted by the value of the function evaluated on the node, then a schema in finite differences very simple to approximate the temporal derivative in would be the one obtained by the serial expansion of Taylor, in the first order, around the point : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {C}^{n}\left( x\right) C\left( t,x\right) t=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {t}^{n}\left( 4\right) {t}^{n+1}={t}^{n}+\Delta t

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): C\left( {t}^{n}+\Delta t,x\right) =C\left( {t}^{n}+\Delta t\right) +\Delta t\left[ \frac{\partial C}{\partial t}\right] \left( t,x\right) \left| t=\right.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. {t}^{n}\right. +O\left( \Delta t\right)


From here, despising him you have to: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle O\left( \Delta t\right)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left[ \frac{\partial C}{\partial t}\right] \left( t,x\right) =\frac{{C}^{n+1}\left( x\right) -{C}^{n}\left( x\right) }{\Delta t};


which is the well-known forward-time (Euler) scheme.

If the EDP is now evaluated in you have for each EDO: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left( 4\right) t=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {t}^{n}n=0,1,...,N\,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left[ \frac{\partial C}{\partial t}\right] \left( t,x\right) \left| t={t}^{n}\right. =

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -{\left[ v\left[ {l}_{v}\left( t\right) \right] \right] }_{t={t}^{n}}{\left. \left[ \frac{\partial C}{\partial x}\right] \left( t,x\right) \right| }_{t={t}^{n}}\, ,\, \forall x\in \left( 0\, ,\left. 1\right] \right. \quad \left( 5\right)


Now, using the schema type forward – time la expression is transformed into: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left( 5\right)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{C}^{n+1}\left( x\right) -{C}^{n}\left( x\right) }{\Delta t}=-v\left( {l}_{v}^{n}\right) \frac{\partial {C}^{n}}{\partial x}\left( x\right) \, ,\, 0\leq n\leq N\, ,\, \forall x\in \left( 0\, ,\left. 1\right] \right. \quad (6)


Here is the value in which it is determined using Ruge -Kutta 4 applied to the initial problem Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {l}_{v}^{n}{l}_{v}(t)t=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {t}^{n}\left( 1\right)


Where, in general, is the unknown value at the border for time. For this schema, the known constant is the initial iteration. In the context of the finite differences method, the expression produces an unstable numerical scheme by approx approximation of the spatial derivative term using a centered scheme, i.e. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {C}^{n}\left( 1\right) C\left( t\right) =} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 1x=1t={t}^{n}{l}_{v}^{0}\left( 6\right)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{\partial {C}^{n}}{\partial x}\left( x\right) =\frac{{C}_{m+1\, }^{n}-{C}_{m}^{n}}{\Delta x}


Lto instability arises because the partial derivative relative to the spatial coordinate is evaluated at a time level earlier than the time level where the term temporal derivative is evaluated. Therefore, a stable schema can be obtained if the two derived Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle nn} 

terms s Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{\partial C}{\partial t}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{\partial C}{\partial x}}
s evaluate at the same time level (at least a second order in ). In this order of ideas, Donea states that the easiest way to make the evaluation of both terms of the expression at the same time level, is by expressing the approximation in difference for the term of temporal derivative at the time level. One way to achieve this is through a forward-looking Taylor serial expansion over time, including second- and third-order derivatives. That is, from the expansion Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n\Delta t\left[ 1\right] \left( 6\right) nn}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): C\left( {t}^{n}+\Delta t,\, x\right) =C\left( {t}^{n},x\right) +\Delta t{\left. \left[ \frac{\partial C}{\partial x}\right] \left( t,x\right) \right| }_{t={t}^{n}}+

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{\Delta {t}^{2}}{2}{\left. \left[ \frac{{\partial }^{2}C}{\partial {x}^{3}}\right] \left( t,x\right) \right| }_{t={t}^{n}}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{\Delta {t}^{3}}{6}{\left. \left[ \frac{{\partial }^{3}C}{\partial {x}^{2}}\right] \left( t,x\right) \right| }_{t={t}^{n}}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): O\left( \Delta {t}^{3}\right)


The term is cleared and (considering again the notation that was introduced for the forward-time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\left. \left[ \frac{\partial C}{\partial x}\right] \left( t,x\right) \right| }_{t={t}^{n}}} scheme) d bypreciatingthe Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle O\left( \Delta {t}^{3}\right)} 

 term and using discretization over time, for the transport equation can be replaced by the following: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left( 5\right) \left( 4\right)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{C}^{n+1}\left( x\right) -{C}^{n}\left( x\right) }{\Delta t}-\frac{\Delta {t}^{2}}{6}\, {\left[ \frac{{\partial }^{3}C}{\partial {t}^{3}}\right] }^{n}\left( x\right) =

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -v\left( {l}_{v}^{n}\right) \frac{\partial {C}^{n}}{\partial x}\left( x\right) + Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{\Delta t}{2}\left[ \frac{{\partial }^{2}{C}^{n}}{\partial {t}^{2}}\right] \left( x\right) \, ,\, \forall x\, \left( 0\, ,\left. 1\right] \right. \quad \quad \quad \quad \, \left( 9\right)


For each. The second and third derived terms that appear in this expression can be determined first by successive differentiation of the equation and then by evaluating over time. The calculation for the second derivative is illustrated below: On the one hand Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0,1,...,N\left( 4\right) t={t}^{n}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{\partial }^{2}C}{\partial {t}^{2}}={v}^{2}\frac{{\partial }^{2}C}{\partial {t}^{2}}-

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{dv}{dt}\frac{\partial C}{\partial x}\quad \quad \quad \quad \quad \quad \quad \quad \quad \left( 10\right)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{\partial }^{3}C}{\partial {t}^{3}}={v}^{2}\frac{{\partial }^{2}}{\partial {t}^{2}}\cdot \left( \frac{\partial C}{\partial t}\right) +

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 3v\frac{dv}{dt}\left( \frac{{\partial }^{2}C}{\partial {x}^{2}}\right) -\frac{\partial C}{\partial x}\cdot \left( \frac{{d}^{2}v}{d{t}^{2}}\right) \quad \quad \quad \quad \, \, \left( 11\right)


Now, combining and, evaluated in , with the equation you have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left( 10\right) \left( 11\right) t=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {t}^{n}\left( 9\right)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{C}^{n+1}\left( x\right) -{C}^{n}\left( x\right) }{\Delta t}-\frac{\Delta {t}^{2}}{6}\, {\left[ v\left( {l}_{v}^{n}\right) \frac{{\partial }^{2}}{\partial {x}^{2}}\left( \frac{\partial {C}^{n}}{\partial t}\right) \left( x\right) +3v\left( {l}_{v}^{n}\right) \frac{dv\left( {l}_{v}^{n}\right) }{dt}\cdot \frac{{\partial }^{2}{C}^{n}}{\partial {x}^{2}}\left( x\right) -\frac{{d}^{2}v\left( {l}_{v}^{n}\right) }{d{t}^{2}}\cdot \frac{{\partial C}^{n}}{\partial x}\left( x\right) \right] }^{1}=

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -v\left( {l}_{v}^{n}\right) \frac{\partial {C}^{n}}{\partial x}\left( x\right) + Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{\Delta t}{2}\left[ v\left( {l}_{v}^{n}\right) \frac{{\partial }^{2}{C}^{n}}{\partial {t}^{2}}\left( x\right) -\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. \frac{dv\left( {l}_{v}^{n}\right) }{dt}\cdot \frac{\partial {C}^{n}}{\partial x}\left( x\right) \right] \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \, \left( 12\right) \quad \quad \quad \quad

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): Para\, todo\, x\in \left( 0\, ,\left. 1\right] \right. \, y\, cada\, n=

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0,1,...,N\quad \quad \quad \quad \quad


The expression (12) is similar to that proposed by Leveque when generating stabilized numerical methods by dictating a diffuse term to the nonlinear transport equation. However, it should be noted that the term appears as part of the approximation by difference for the partial derivative of C with respect to time, evaluated at level n. On the other hand, following closely what Donea suggested the term of the third-order partial derivative that appears in Taylor's serial expansion is expressed in a mixed space-time form. This mixed form of the derivative will allow the use of type finite elements with a simple modification of the usual and consistent mass matrix quite similarly as is done in the context of weighted residues of petrov-Galerkin. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \epsilon \frac{{\partial }^{2}{C}^{n}}{\partial {x}^{2}}\left( 12\right) \frac{{\partial }^{2}{C}^{n}}{\partial {x}^{2}}\left[ 1\right] {C}^{0}\left[ 5\right]}


By developing the specified products and grouping the terms we have that the problem raised in (11) is semi discretized over time and for each the next problem arises, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle n}


Given Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left[ 0,1\right] \ni x\, \rightarrow {C}^{0}\left( x\right) =} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {C}_{ini}\left( x\right) 

,  find , such that: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {C}^{n}{\left( x\right) }_{0\leq n\leq N},\forall x\in \left[ 0,1\right]}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{\partial }^{2}{C}^{n+1}\left( x\right) }{\partial {x}^{2}}-{\alpha }_{1}{C}^{n+1}\left( x\right) =

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\alpha }_{2}\frac{{\partial }^{2}{C}^{n}\left( x\right) }{\partial {x}^{2}}+{\alpha }_{3}\frac{\partial {C}^{n}\left( x\right) }{\partial x}- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\alpha }_{1}{C}^{n}\left( x\right) \quad \quad \, \, (13)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\propto }_{1}=\frac{6}{\Delta {t}^{2}{v}^{2}}\, \, ;\, \, {\alpha }_{2}=-2-\frac{3\Delta t}{v}\frac{dv}{dt}\, ;\, \, {\propto }_{3}=

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -\frac{\Delta t}{{v}^{2}}\frac{{d}^{2}v}{d{t}^{2}}-\frac{3}{{v}^{2}}\frac{dv}{dt}- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{6}{\Delta tv}\quad \quad \quad

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \forall x\in \left( 0,\left. 1\right] \quad \quad \quad \quad \, \, {C}^{n}\left( 0\right) =\right.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. {C}_{a}^{n}={C}_{a}\left( {t}^{n}\right) \right. \quad \quad \quad \, \,


To be able to solve the differential equation of Second Order posed in (15) we need 2 conditions, but the problem only provides us with a condition so it was necessary to impose asecond condition to solve the problem, which we induc from the mathematical model for the problem of transport proper to the chemo-fluid oscillator:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left\{ \begin{matrix}\frac{d{l}_{v}}{dt}\left( t\right) =\gamma \left[ C\left( t,1\right) \right] \left\{ {l}_{eq}\left[ C\left( t,1\right) -{l}_{v}\left( t\right) \right] \right. \\{l}_{v}\left( 0\right) ={l}_{v}^{0}\end{matrix}\Longrightarrow \begin{matrix}{l}_{v}^{0}\, ,\, {C}^{0}\left( x\right) ={C}_{ini}\left( x\right) \, \, \forall x\in \left( 0\, ,\, \left. 1\right] \right. \quad \quad \quad \, \, \left( 14\right) \\{l}_{v}^{n+1}=F\left( {l}_{v}^{n}\, ,{C}^{n}\left( 1\right) ;\Delta t\right) ,n=0,1,2,...,N\end{matrix}\right.


Deduction of the condition in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 1


From the EDP:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): v\left[ {l}_{v}\left( t\right) \right] \frac{\partial C}{\partial x}\left( t,x\right) =

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -\frac{\partial C}{\partial t}\left( t,x\right) \quad \quad \quad \quad \, \, \left( 15\right)


semi-discretization of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \frac{\partial C}{\partial t}\, \Longrightarrow \frac{{C}^{n+1}\left( x\right) -{C}^{n}\left( x\right) }{\Delta t}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \Rightarrow v\left[ {l}_{v}\left( {t}^{n+1}\right) \right] \frac{\partial C}{\partial x}\left( {t}^{n+1},x\right) =

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -\left[ \frac{{C}^{n+1}\left( x\right) -{C}^{n}\left( x\right) }{\Delta t}\quad \right]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \Rightarrow v\left[ {l}_{v}^{n+1}\right] \frac{\partial {C}^{n+1}}{\partial x}\left( x\right) =

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -\frac{{C}^{n+1}\left( x\right) }{\Delta t}+\frac{{C}^{n}\left( x\right) }{\Delta t}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \Rightarrow v\left[ {l}_{v}^{n+1}\right] \frac{\partial {C}^{n+1}}{\partial x}\left( x\right) +

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{C}^{n+1}\left( x\right) }{\Delta t}=\frac{{C}^{n}\left( x\right) }{\Delta t}\quad \quad \left( 16\right)


That by evaluating her in, we have a Condition of Robin. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 1


2.2 Taylor Method - Galerkin

Considering the internal product on the range : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {L}_{2}\left( 0,1\right)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\left\langle u,v\right\rangle }_{{L}_{2}}:=\int_{0}^{1}u\left( x\right) \cdot v\left( x\right) dx\quad \quad \quad \quad \quad \left( 17\right)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left\langle \frac{{\partial }^{2}{c}_{\left( x\right) }^{n+1}}{{\partial x}^{2}}-\right.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. {\alpha }_{1}{c}_{\left( x\right) }^{n+1}-{\alpha }_{2}\frac{{\partial }^{2}{c}_{\left( x\right) }^{n}}{{\partial x}^{2}}+\right. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. {\propto }_{3}\frac{{\partial c}_{\left( x\right) }^{n}}{\partial x}-{\alpha }_{1}{c}_{\left( x\right) }^{n}\, ,\, v\left( x\right) \right\rangle = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 0\, \, \left( 18\right) \, \,


Applying the definition of the internal product in the spaces, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {L}_{2}} 

with border conditions of Dirilecht and Robin, applying the integration formula in parts and replacing the functions in a way that we have for the problem: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left( 18\right) \sum _{j=1}^{M}{c}_{j}^{n+1}\frac{{d\varphi }_{j}\left( 1\right) }{dx}{\varphi }_{i}\left( 1\right) -}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \sum _{j=1}^{M}{c}_{j}^{n+1}\int_{0}^{1}\frac{d{\varphi }_{j}\left( x\right) }{dx}\frac{d{\varphi }_{i}\left( x\right) }{dx}dx\, + Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\alpha }_{1}\int_{0}^{1}\left( \sum _{j=1}^{M}{c}_{j}^{n+1}{\varphi }_{j}\left( x\right) \right) \, {\varphi }_{i}\left( x\right) dx\quad = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): -{\alpha }_{2}\sum _{j=1}^{M}{c}_{j}^{n}{\frac{d{\varphi }_{j}\left( 1\right) }{dx}}_{j}{\varphi }_{i}\left( 1\right) \, + Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\alpha }_{2}\sum _{j=1}^{M}{c}_{j}^{n}\int_{0}^{1}\frac{d{\varphi }_{j}\left( x\right) }{dx}\frac{d{\varphi }_{i}\left( x\right) }{dx}dx\quad + Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\alpha }_{3}\sum _{j=1}^{M}{c}_{j}^{n}\frac{d{\varphi }_{j}\left( x\right) }{dx}{\varphi }_{i}\left( x\right) dx\, \, - Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\alpha }_{1}\sum _{j=1}^{M}{c}_{j}^{n}\int_{0}^{1}{\varphi }_{i}\left( x\right) {\varphi }_{j}\left( x\right) dx\quad \quad \quad \, \, \forall {\varphi }_{i},{\varphi }_{j}\in {V}_{h}\quad \quad \quad \left( 19\right)


2.3 Finite Element Method

We will consider a discretization of finite elements as explained above, but adapted to the working Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left[ 0,1\right]} 

 interval. In effect the partition of this interval corresponds to the longitudinal discretization of the delay line channel into length elements for Being , the partition represents a mesh of points that we will denote by . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left( M-\right. }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left. 1\right) \, {I}_{j}=\left[ {x}_{j},{x}_{j+1}\right] {h}_{j}={x}_{j+1}-{x}_{j}j= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 1,2,3,...,M.{h}_{i}=h{x}_{j}{\tau }_{h}


If we write Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left( 19\right)} 

using the elementary formulation we would have:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left[ \begin{matrix}\frac{d{\varphi }_{i}^{,}\left( 1\right) }{dx}\cdot {\varphi }_{i}\left( 1\right) &\frac{d{\varphi }_{i}^{,}\left( 1\right) }{dx}\cdot {\varphi }_{i+1}\left( 1\right) \\\frac{d{\varphi }_{i+1}^{,}\left( 1\right) }{dx}\cdot {\varphi }_{i}\left( 1\right) &\frac{d{\varphi }_{i+1}^{,}\left( 1\right) }{dx}\cdot {\varphi }_{i+1}\left( 1\right) \end{matrix}\right] \left[ \begin{matrix}{C}_{i}^{n+1}\\{c}_{i+1}^{n+1}\end{matrix}\right] -

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \left[ \begin{matrix}\frac{d{\varphi }_{i}^{,}\left( 0\right) }{dx}\cdot {\varphi }_{i}\left( 1\right) &\frac{d{\varphi }_{i}^{,}\left( 0\right) }{dx}\cdot {\varphi }_{i+1}\left( 1\right) \\\frac{d{\varphi }_{i+1}^{,}\left( 0\right) }{dx}\cdot {\varphi }_{i}\left( 1\right) &\frac{d{\varphi }_{i+1}^{,}\left( 0\right) }{dx}\cdot {\varphi }_{i+1}\left( 1\right) \end{matrix}\right] \left[ \begin{matrix}{C}_{i}^{n+1}\\{c}_{i+1}^{n+1}\end{matrix}\right] + Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{1}{{h}_{i}}\left[ \begin{matrix}1&-1\\-1&1\end{matrix}\right] \left[ \begin{matrix}{C}_{i}^{n+1}\\{c}_{i+1}^{n+1}\end{matrix}\right] + Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{\alpha }_{1}}{6}{h}_{i}\left[ \begin{matrix}2&1\\1&2\end{matrix}\right] \left[ \begin{matrix}{C}_{i}^{n+1}\\{c}_{i+1}^{n+1}\end{matrix}\right] + Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{-{\propto }_{311}}{2}\left[ \begin{matrix}-1&1\\-1&1\end{matrix}\right] \left[ \begin{matrix}{C}_{i}^{n+1}\\{c}_{i+1}^{n+1}\end{matrix}\right] \left[ {c}_{M}^{n+1}\right] = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\propto }_{2}\left[ \begin{matrix}\frac{d{\varphi }_{i}^{,}\left( 1\right) }{dx}\cdot {\varphi }_{i}\left( 1\right) &\frac{d{\varphi }_{i}^{,}\left( 1\right) }{dx}\cdot {\varphi }_{i+1}\left( 1\right) \\\frac{d{\varphi }_{i+1}^{,}\left( 1\right) }{dx}\cdot {\varphi }_{i}\left( 1\right) &\frac{d{\varphi }_{i+1}^{,}\left( 1\right) }{dx}\cdot {\varphi }_{i+1}\left( 1\right) \end{matrix}\right] \left[ \begin{matrix}{C}_{i}^{n}\\{c}_{i+1}^{n}\end{matrix}\right] - Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\propto }_{2}\left[ \begin{matrix}\frac{d{\varphi }_{i}^{,}\left( 0\right) }{dx}\cdot {\varphi }_{i}\left( 0\right) &\frac{d{\varphi }_{i}^{,}\left( 0\right) }{dx}\cdot {\varphi }_{i+1}\left( 0\right) \\\frac{d{\varphi }_{i+1}^{,}\left( 0\right) }{dx}\cdot {\varphi }_{i}\left( 0\right) &\frac{d{\varphi }_{i+1}^{,}\left( 1\right) }{dx}\cdot {\varphi }_{i+1}\left( 0\right) \end{matrix}\right] \left[ \begin{matrix}{C}_{i}^{n}\\{c}_{i+1}^{n}\end{matrix}\right] + Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{-{\propto }_{2}}{{h}_{i}}\left[ \begin{matrix}1&-1\\-1&1\end{matrix}\right] \left[ \begin{matrix}{C}_{i}^{n}\\{c}_{i+1}^{n}\end{matrix}\right] + Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{\propto }_{322}}{2}\left[ \begin{matrix}-1&1\\-1&1\end{matrix}\right] \left[ \begin{matrix}{C}_{i}^{n}\\{c}_{i+1}^{n}\end{matrix}\right] + Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{{\propto }_{1}{h}_{i}}{6}\left[ \begin{matrix}2&1\\1&2\end{matrix}\right] \left[ \begin{matrix}{C}_{i}^{n}\\{c}_{i+1}^{n}\end{matrix}\right] \quad \quad \quad \quad \quad \quad \left( 20\right)

3.Numerical experimentation.

After raising the mathematical part for the model and obtaining the system of equations, a program was designed in Matlab for the coupled system where several tests were performed with their respective numerical adjustments based on the theoretical definitions explained in chapters 2 and 3.

Each experiment details the change in the initial Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C\left( 0,x\right) =} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {c}_{ini}\left( x\right) 

concentration which is a unique experimental value for the operation of the hydrogel that is not listed as data in Páez and has been imposed on it according to the physical model. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left[ 1\right]}


Experiment 1

Values are taken in the space of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle M=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): 20 

(nodes);  ;   ;  ; Courant 0.1 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle dx=}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): \frac{1}{M-1}Courant=\frac{dt}{dx}dt=courant\cdot dx


A constant value was taken as a condition which in this case is the experimental value with which the numerical part worked in the initial study. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C\left( 0,x\right) =} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {c}_{ini}\left( x\right) {c}_{ini}\left( x\right) =2.2635


The values of those derived from the concentration found in are replaced by forward differences (Euler) for .Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\alpha }_{3}t\geq 0}


600px

Figure 3. Response of the chemo-fluidicoscillator modeled by the system.

Periodic behavior can be observed in the Hidrogel which is what allows the device to function as an oscillator with negative feedback resulting from the increase and decrease of the alcohol mixture in the hydrogel chamber but at the beginning of the flow of mixture of alcohol and water there is instability in the wave fronts that are then regularized as the process of opening and closing the valve progresses that because the hydrogel suffers deformation greater than the length of the camera

600px

Figurineat 4. Response of the chemo-fluidic oscillator modeled by state variables.

Experiment 2

In this trial, a quadratic profile was taken as an initial condition since there is a mixture of water and alcohol in the canal resulting from deionized water flows and alcohol but not reaching the minimum concentration level so that the hydrogel reacts and begins to compress due to the increase in alcohol, periodic behavior occurs , but with less disturbance at the beginning of the process allowing a more stable and smooth concentration flow on the wave fronts.

600px

Figurineat 5. Response of the chemo-fluidic oscillator modeled by state variables.

600px

Figure 3. Response of the chemo-fluidicoscillator modeled by the system.

4. CONCLUSIONS AND RECOMMENDATIONS.

The first observation that can be made is that the method applied to the system composed of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle (15),\left( 16\right) \, y\, \left( 17\right) \,} 

equations reproduces the dynamics of the original numerical model of Paez, with small differences in amplitude and period, but that it is able to produce stable periodic signals for parameter configuration without needing any external forging, which means that the oscillating behavior is self-excited. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left[ 1\right]}


The second observation is that from a numerical approximation of the linear transport equation (constant velocity) based on the Euler-Taylor-Galerkin method for discretization over time and the Finite Elements Method for discretization in the space posed by Donea, by applying the same method with some variations in the initial conditions and border equation in the equation of the transport of the nonlinear system that is also coupled to a nonlinear ordinary differential equation that governs the behavior of the hydrogel and another equation that controls the volume of the buffert resulting in a complex system of solving, satisfactory results were obtained in relation to its oscillation and its periodic and dimensioned movement. It is very important to note that this numerical model makes it possible to observe that if we change the initial value there will be a variation at the beginning of the hydrogel's operation that would be in the stationary regimen but that after Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {C}_{ini}} 

this initial regimen the hydrogel shows an oscillating and periodic behavior typical of Paez's initial analysis.

The third observation is that this numerical analysis applied to the chemo-fluidic oscillator analytically contributes to the understanding that the valve composed of the hydrogel is extremely sensitive to the variation of the parameters, i.e. it has the ability to drastically change its volume under small variations of special thermodynamic parameters.

In this research another option could be given to mathematically model the domains of the oscillator, in the fluidic domain given by the flow network, the transport of concentration through the delay line and in the chemical domain given by the behavior of the hydrogel and the concentration of alcohol that produce a smooth dynamic system in parts.

It is recommended to use for future studies a mathematical refinement method such as the Galerkin-Discontinuous method to be able to model in more detail the behavior of the Chemo-fluidic Oscillator.

5. References.

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 Jhosep Paez, A new self-extracting chemo-fluidic oscillator based on stimuli sensitive hydrogels: Mathematical model and dynamic behavior.

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 Jean Donea, A Taylor – Galerkin Method for Convective Transport problems.

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 P.D. Lax and B. Wendroff , "On the stability of difference Schemes", Comm. Pure Appl . Math .13,217(1960).

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 P. D Lax and B.Wendroff , "Difference Schemes for Hyperbolic equations with high order of accuracy". comm, Pure, Appl. Math. 13381 (1964).

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 C.E. Leith, in Methods in Computational. Physics, Vol 4, P.1, Academic Press, New York, 1965.

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J.K. Dukowicz and J.D. Ramshaw, "Tensor Viscosity Method for Convection in numerical fluid dinamics" J.compu . Phys , 32.71-79 (1979).

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 "Explicit and implicit FEM -FCT algorithms with flux linearization". Joural of computational Physics 2009, Vol228.

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 Forward-Back word Scheme .and its use in a Limited Area Model. Feder Mesinger.

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 Courant Number. Introduction to the Ivan Aldear Fluid Simulation 2012.

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 "Explicit algorithm for The Advetive Equation in the FTCS" Numerical Model of the Atmosphere 2017. Marcelo Barreiro.

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 A. Lenk, R.G. Ballas, R. Werthschutzky , G.Piefer , Electromechanical Systems in Microtechnology and Mecatronics electrical , Mechanical and Acoustic ,Their interactions and aplications ,Springer -Verlay , Berlin 2010 .

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 R.K. Gary, A.Dixit , P. Yadov ,Basic Electronics ,Laxmi Publications , New Delhi 2009.

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 A. Richter, J.Wenzel , K Kristchmer , Mechanically adjustable chemostats based on Stimulli-Responsive polymers , Sensor Actuat . B Chem.125 (2) (2007) 569 -573 .

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 K.W.Morton and A.K.Parrott " Genaralized Galerkin Methods for Hiperbolic equations", J.Comprest.Phys .36,249-270 (1980).

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J.K. Dukowics and J.D. Ramshow , " Tensor Viscosity Method For Convection in Numerical fluid dynamics ", J. Comput . Phys. 32, 71-79 (1979).

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 R.J. Leveque, Finite Difference Methods for Ordinary and partial.

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 I. Christies, D.F.Griffiths , A.R. Mitchell and O.C. Zienkicwicz , Finite element methods for second order differential equations with significant First derivatives ; Int. J. number. Methods ing.10, 1389-1396.
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