Analysis of the Time Increment for the Diffusion Equation with Time-Varying

Mechanics, Materials Science & Engineering, December 2016

ISSN 2412-5954

Analysis of the Time Increment for the Diffusion Equation with Time-Varying Heat Source from the Boundary Element Method12 Roberto Pettres1, a 1

Federal University of Parana, Program of Pos-graduate in Numerical Methods in Engineering. Curitiba, Brazil.

a

pettres@ufpr.br DOI 10.2412/mmse.8.968.954

Keywords: Boundary Element Method, diffusion equation, time increment, transient analyses.

ABSTRACT. In this paper a Boundary Element Formulation for the one-dimensional transient heat flow problem is presented. The formulation employs a time-independent fundamental solution; consequently, a domain integral appears in the integral equations, which contains the potential time derivative and the time-dependent heat source term of the governing equation. Linear elements are used for the domain discretization. The time marching scheme is implemented with finite difference approximations. The performance of the formulation was assessed comparing the numerical results with an analytical solution. Convergence of the numerical results is evaluated with varying size time-increment during analysis.

1. Introduction. The first records dealing with the origin of the Boundary Element Method (BEM) date from the year 1823, in a publication by the Norwegian mathematician Niels Henrik Abel on the tautochronous problem ('equal time') [1]. In this work, Abel portrayed to the method as a technique based on integral equations to solve problems based on partial differential equations. This method received attention from several researchers and it took another eight decades of studies for the method to receive the first classical theory of integral equations developed by Fredholm in 1903 [2].

Still in the twentieth century, several authors used the technique of integral equations and made important contributions to the evolution of the method, being called the Boundary Element Method from the works of Brebbia [3], which presented a formulation based on integral equations and in tecniques of weighted residues.

Nowadays, the BEM has been used to solve a growing number of problems in solids mechanics, electromagnetism, heat diffusion [4], among others, and in certain formulations, it ends up counting on the coupling of other numerical methods, such as the Finite Differences Method (FDM) [5]. In this work, coupled to the BEM, the FDM is used to solve the heat diffusion equation with a heat generation term variable in time and a study is performed on the convergence behavior of formulation when using variables time increment values, counting on a fundamental solution independent of time.

At the end of the work the results are presented.

2. Mathematical and Geometric Model. The mathematical model chosen for this study is Diffusion Equation with a source term, given by (eq. (1)) [6]:

(1)

The Authors. Published by Magnolithe GmbH. This is an open access article under the CC BY-NC-ND license http://creativecommons.org/licenses/by-nc-nd/4.0/

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The geometric model is a unidimensional bar of unit length with a variable heat source in time, under the boundary conditions given in (2) and (3) and initial in (4).

Essentials

(2)

Naturals

(3)

Initials

(4)

2.1 Problem formulation from BEM. Being an approximate solution to the problem, which does not meet the boundary conditions, two types of residues or errors are generated: i) in

(domain):

(5)

ii) in

(contour):

(6)

iii) in

(contour):

(7)

The basic sentence of weighted residues is written as:

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(8)

The weighting functions

,

and

, can be chosen conveniently, aiming to simplify the problem.

Integrating the integral containing the Laplacian twice by parts (8), obtains:

(9)

Replacing (9) in (8):

(10)

Making

and at some time

and

in (10), obtains the resulting equation called

inverse formulation of weighted residues:

(11)

Using and applying the properties of the Dirac Delta function [6] to match the differential (12), can obtain the effect at the x field point of a concentrated source applied at the source point. Then, substituting (12) into (11), obtain the equation (13):

(12)

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(13)

In the BEM, the weighting function is the solution of the equivalent singular problem, that is, the Green function [7] for the differential operator. Thus, called the fundamental solution, can be interpreted as the effect at the field point x of a concentrated source applied at the source point. For the one-dimensional case, the fundamental solution [8] is given by:

(14)

Replacing (14) in (13), obtain:

(15)

Making and in (15) and defining and defining the essential contour conditions according to (16), obtain the constitutive equation of the BEM (17) for the proposed problem.

(16)

(17)

In the first integrating of equation (17) a temporal derivative is present. As the fundamental solution used in this work is independent of time, it is necessary to use some technique or numerical model for the process of march in time.

2.2 Numerical model of march in time. Several approaches have been proposed for the application of the BEM in parabolic problems, where it is used as solution of the equivalent singular problem, a solution independent of time. In this type of formulation, it is necessary to use advance in time methods, because of the integral that contains the differential term in time. Among the commonly used methods, coupled to the BEM is the Finite Differences Method (FDM).

The coupling of the FDM and the BEM was first proposed by Brebbia [3] for the diffusion equation, implemented and investigated by Curran, Cross and Lewis [9], who found that this method produces only accurate results if the approximation used for time derivative presents precision. Curran, Cross and Lewis investigated the use of a higher-order approximation for time derivative, concluding that the use of this approach improved the accuracy of the method, but led to a deterioration in the convergence behavior of the model.

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2.2.1 Numerical model of advance in time using FDM. The FDM has the purpose of obtaining the rate of variation from one instant to the next, being an approximation to the value of the derivative at the point when becomes extremely small. Thus, the derivative at time present in equation (17) is approximated by the quotient of the variation of the potentials by the corresponding time interval, according to equation (18).

(18) Replacing (18) in (17), obtain:

(19)

Using the approximation from the FDM the original equation becomes an equation with solution obtained iteratively, for a number n of iterations over time.

The term source F (x, t) also evolves in time, presenting a contribution portion of the problem domain that causes influence in the contour. For an internal solution where the evaluated point belongs to the domain, it is possible to determine the solution from equation (19) counting with domain cells due to integrals in . Thus, from the integral equation (19) arrive at a system of linear algebraic equations by the discretization of the domain in cells. The contour integrals are transformed into sums of integrals on each cell, passing to a solution in terms of the nodal points.

2.3 Domain discretization. Divided into cells the domain (Figure 1), is possibile obtain a representation of this domain in an exact or approximate form, depending on the coincidence or not of the nodes and the approximate function chosen for each cell.

Fig. 1. Domain discretization.

Each cell associates one or more points called "functional nodes" or "nodal points" and the values of the associated variables are called "nodal values". Throughout each cell the problem variables are approximated by polynomial (constant, linear, quadratic, ...) functions that are defined as a function of the number of nodal points chosen (1, 2, 3, ...).

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In this work was opted for linear approximation functions, assuming that the variation from one node to the next, presents a linear behavior and the functions are defined according to equations (20) and (21) and illustrated by Figure 2.

(20)

(21)

Fig. 2. Linear approximation functions.

By approaching the geometry of in linear cells, one can discretize the domain exactly by matching to coincide the i + 1 node of the cell with node i of cell +1.

2.4 Linear equations sistem. Discretizing the equation (19) and transforming in a summation of functions, have:

(22)

Grouping similar terms and using matrix notation, one can write equation (22) as follows:

(23)

Where H and G are matrices with contour coefficients, M is a vector containing the contributions of F(x, t) and so of derivative in the next time step (m+1) and D is the vector containing the derivative MMSE Journal. Open Access www.mmse.xyz

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in the current step (m). The vectors M and D represent the influence due to the domain integrals in (22).

From the initial conditions of the problem, it is known, can use equation (23) to obtain the solution of the problem advanced in time by making as a pseudo-initial condition for the next time step. Taken as a time-forward constant, the matrices H and G and the vectors M and D are assembled, storing the subsequent calculations in the iterative process.

2.4.1 Numerical solution with advance in time. To obtain the solution of the problem, consider the contribution of all the cells in the assembly of the system of equations formed according to equation (23). The boundary conditions can be applied to form a solution system this way: (24)

where A is the coefficient matrix containing terms relative to the matrices H, G and vector M; is the vector of unknown nodal values at the moment

;

y is a vector constructed from known values of the previous time step containing the contributions of vector D. For a problem with time-dependent boundary conditions, the solution needs to be reformulated and updated at each time step. This update can be performed using as initial pseudo-conditions, the conditions obtained after the moment an internal solution is constructed, repeating the process at each iteration of . This time-advance procedure only involves integrating the domain at a given time, so ideally a domain integral only needs to be calculated once.

For a problem with time-independent boundary conditions, as addressed by this work, at each step of time, it is only necessary to upgrade and resolve the system to . However, in the present model, only the essential contour conditions, potential condition, remain constant, since the natural conditions, flow condition, are time dependent, being recalculated at each iteration by updating the model.

From (24) obtain the vector of unknowns x is:

(25)

After the determination of the vector of unknowns, can obtain the variables in points belonging to the domain of the problem.

3. Computational implementation. The formulation adopted was implemented in commercial software Matlab R2011. In the simulations, the following initial condition was used: (26)

The coefficient of thermal conductivity was defined as defined by equation (27).

, 100 time steps and the term source was

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The size of the time increment was initially defined by the stability criterion, which is relative to the domain cell size and thermal conductivity coefficient k of the material, which, according to [10], is defined as follows:

(28)

where

is the critical time increment.

4. Results. The initial results obtained were compared to the analytical solution of the problem, which according to [6] is given by:

(29)

In order to obtain the level of correlation between the numerical and analytical results, a statistical inference study was performed and the correlation coefficient R2 (Pearson's square) was calculated between the two solutions.

Fig. 3. Comparison between the analytical solution and the BEM for potential (a) and for flow (b).

Results showed that for the proposed problem, the relation presented in (28) produces accurate results for the flow values, R2 = 1, but satisfactory for the potential values, R2 = 0.97806. Analysing both results, it is observed that the model's response is accurate when points belonging to the contour are analysed (where the flows are obtained), because it deals only with contour values, as the name of the method suggests, already for the calculated potential At one point in the domain, the model has a small error. This type of error is related to the size of the time increment , the type of approximation MMSE Journal. Open Access www.mmse.xyz

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function used for the cells and their dimensions (length), requiring a specific time interval for the process of diffusion of heat throughout the cell extension (Figure 4).

Fig. 4. Insufficient time increase (1) for heat diffusion and sufficient (2).

Figure 4 illustrates two cases of the heat diffusion process. In case (1), it is observed that only part of the cell was influenced by the heat diffusion. This is because the established model uses a discrete time interval, suddenly stopping the heat flow, causing the diffusion process to be insufficient, since it does not count on the total dissipation of such energy on the whole cell, adding error in the integration stage of the Cells. In case (2), it is observed that the relationship between the length of the domain cell and the size of the time increment was adequate and sufficient for the process of diffusion of heat in the cell.

In this way, chooses to determine the increment of time that would provide the highest level of correlation between the numerical and analytical response (R2 = 1). For this, the size of the increment of time in the interval was varied , adding 0.01 to each iteration. The results obtained in this analysis are illustrated by Figure 5:

Fig. 5. Time increment analysis.

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According to [10], the higher the values used to , the greater will be the local truncation errors associated to temporal discretization, a fact that is observed in the previous figure for the highest values of . However, for this one-dimensional analysis, the increase in theoretical time ( ), including smaller values to the same, presented results lower than the one obtained numerically ( ), based on the values obtained for R2 = 0.97806 and R2 = 0.98456 respectively.

This result expresses the convergence behaviour of the proposed mathematical model, indicating that there is a maximum limit value of correlation between the numerical and analytical solution, associated to the increment of time that represents the lowest error level of the formulation.

Values for the thermal conductivity coefficient of material k between 0.2 and 2 were tested, with some cases shown in Figure 6. The convergence of the model was obtained using k = 1.6, with the result R2 = 0.99635 when using the increase of numerical time in relation to the theoretical that presented for this estimator the value 0.99634 as illustrated by Figure 7 (b).

Fig. 6. Time increment analyses to k = 0.2 (a), k = 0.5 (b) and k = 1.0 (c).

Fig. 7. Time increment analyses to k = 1.5 (a), k = 1.6 (b) and k = 1.7 (c).

Figure 7 (c) shows that, for values of k > 1.6, the theoretical increment presents better results by reference to the coefficient R2.

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For the proposed formulation, the correlation level between the variables is higher when using a relatively higher time increment than the theoretical one, being limited by the set illustrated by Figure 8 (a), in analyzes where the coefficient of thermal conductivity of the material belongs to the range 0.2 k 1.6.

Fig. 8. Analysis of the time increment: (a) set of values for set.

and (b) ampliation of the image of the

Figure 8 (b) illustrates the limit for k (1.6), indicating that for the k > 1.6 values, the time increment that presents the best results is the theoretical time increment, the numeric of the set being limited by the line in red.

Summary. The results obtained for the proposed problem indicated that theoretically proposed values for time increment provide solutions with a reasonable correlation level when analysing a point belonging to the domain. It was also verified that, for the proposed formulation, the level of correlation between the variables can be higher when using a relatively higher time increment than theoretical in analyses where the coefficient of thermal conductivity of the material belongs to the range 0.2 k 1.6, Making the mathematical model more efficient and presenting a lower level of error.

The highest level of correlation obtained was 0.99635 with the use of time increment equal to 0.078616 of the numerical model, being higher than the value 0.99634 obtained from the theoretical time increment, 0.078125, when using k = 1.6. For analyses in which k > 1.6, it was verified that the use of theoretical time increment presents better results and it is suggested its use in applications where k is defined in such a way.

Also, from these results, this work demonstrates the effectiveness of the BEM for the proposed problem and the potential of the use of the fundamental solution independent of time for the transient case.

References [1] Simmons, G. F. (1987). Calculus with Analytical Geometry Vol. 2. McGraw Hill.

[2] Jacobs, D. (1979). The State of the Art in Numerical Analysis, Academic Press, New York, USA. [3] Brebbia, C. A. (1978). The boundary element method for engineers. Pentech Press, London.

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[4] Pettres, R.; Lacerda, L. A.; Carrer, J.A.M. (2015) A boundary element formulation for the heat equation with dissipative and heat generation terms. Engineering Analysis with Boundary Elements, vol. 51, Feb., pp 191-198.

[5] Kreyszig, E. (2006). Advanced Engineering Mathematics 9th Edition. Wiley, Ohio.

[6] Greenberg, M. D. (1998). Advanced Engineering Mathematics (2nd Edition). Prentice-Hall, New Jersey.

-Hall, New Jersey.

[8] Vladimirov, V. S. (1979). Generalized Functions in Mathematical Physics. Nauka Publishers, Moscow.

[9] Curran, D. A. S., Cross, M. and Lewis, B. A. (1980). Solution of parabolic differential equations by the boundary element method using discretisation in time - Applied Mathematical Modelling, vol.

4, pp 398 400.

[10] Wrobel, L. C. (1981). Potential and Viscous Flow Problems Using the Boundary Element Method, U.K. Ph.D. Thesis, University of Southampton.

Cite the paper Roberto Pettres (2016). Analysis of the Time Increment for the Diffusion Equation with Time-Varying Heat Source from the Boundary Element Method. Mechanics, Materials Science & Engineering, Vol 7. doi:10.2412/mmse.8.968.954

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