The DRM-MD integral equation method for the numerical solution of convection-diffusion

Transcription

1 The DRM-MD integral equation method for the numerical solution of convection-diffusion equation V. Popov & H. Power Wessex Institute of Technology, Ashurst Lodge, Ashurst, Abstract This work presents a multi-domain decomposition integral equation method for the numerical solution of domain dominant problems, for which it is known that the standard boundary element method (BEM) is at a disadvantage in comparison with classical domain schemes, such as finite difference (FDM) and finite element (FEM) methods. As in the recently developed Green element method (GEM), in the present approach the original domain is divided into several sub-domains. In each of them the corresponding Green's integral representational formula is applied, and on the interfaces of the adjacent subregions the full matching conditions are imposed. In contrast with the GEM, where in each sub-region the domain integrals are computed by the use of cell integration, here those integrals are transformed into surface integrals at the contour of each sub-region via the dual reciprocity method (DRM), using some of the most efficient interpolation functions in the literature on mathematical interpolation. As in the FEM and GEM, the obtained global matrix system possesses a banded structure. However in contrast with these two methods (GEM and non-hermitian FEM), here one is able to solve the system for the complete internal nodal variables, ie the field variables and their derivatives, without any additional interpolation. Finally some examples showing the accuracy, the efficiency, and the flexibility of the method for the solution of the convection-diffusion equation are presented. 1 Introduction The boundary element method (BEM) is now a well established numerical technique in engineering. The basis of the method is to transform the original partial differential equation (PDE), or system of PDEs that define a given physical problem, into an equivalent integral equation (or system) by

2 68 Boundary Element Research In Europe means of the corresponding Green's theorem and its fundamental solution. In this way some or all of the field variables and their derivatives are only necessary to be defined at the boundary. Further increase in the number of applications of the BEM has been hampered by the need to operate with relatively complex fundamental solutions or by the difficulties encountered when these solutions cannot be expressed in a closed form, and also when the technique is applied to nonlinear and time-dependent problems. In the BEM for this kind of problems, it is common to use an integral representation formula based upon a partial differential equation for which a closed form expression of the fundamental solution is known, and express the remaining terms of the original equation as domain integrals. It is known that in these cases the BEM is in disadvantage in comparison with the classical domain schemes, such as the finite difference (FDM) and thefiniteelement (FEM) methods. In the early BEM analysis the evaluation of domain integrals was done using cell integration, a technique, which while effective and general, made the method lose its boundary only nature, which is one of its attractive features, introducing additional internal discretization. Although good results can be obtained using the cell integration technique, this approach for certain applications is of several orders of magnitude more time consuming than domain methods. This computational cost mainly depends on the fact that the solution at each surface or internal point must involve the evaluation of the complete surface integrals, so the matrix of the system of equations is fully populated. Several methods have been developed to take domain integrals to the boundary in order to eliminate the need for internal cells (boundary only BEM formulations). One of the most popular to date is the dual reciprocity method (DRM) introduced by Nardini and Brebbia [1]. The DRM uses the concept of particular solutions, but instead of solving for the particular solution and the homogeneous solution separately, it applies the divergence theorem to the domain integral terms and converts the domain integrals into equivalent boundary integrals. A major problem encountered with the DRM is that the resulting algebraic system consists of a series of matrix multiplications of fully populated matrices. When only few internal points are required in the DRM, the resulting computing time is in general smaller than the one required by the cell integration scheme, but still being costly in comparison with domain approaches. Besides, in complex problems these two approaches have been limited only to small values of the non-linear terms, or small Peclet numbers in the case of the convection-diffusion equation. When dealing with the BEM for large problems, it is common practise to use the method of domain decomposition, in which the original domain is divided into subdomains, and on each of them the full integral representation formulae are applied. At the interfaces of the adjacent subdomains, the corresponding full matching conditions are imposed. While the BEM matrices, which arise in the single domain formulation, are fully populated, the subdomain formulation leads to blocks banded matrix systems with

3 Boundary Element Research In Europe 69 one block for each subdomain and overlaps between blocks when subdomains have a common interface. When using continuous elements of high orders (more than constant), the application of the matching conditions at common interfaces, i.e. the matrix assembly, leads to an over-determined system of algebraic equations (as more subdomains are defined the bigger the over-determination). Several schemes are known that reduce the over-determined system to a closed system. The simplest possible scheme is obtained by expressing the derivatives of the field variables at the common nodes between more than two subdomains in terms of the variables themselves by using the interpolation functions or by afinitedeference approximation. A way of avoiding of this problem is by using discontinuous elements in such common nodes, obtaining in this way a closed matrix system at the expense of having a larger number of unknown variables. In the limit of a very large number of subdomains, the resulting internal mesh pattern looks like a finite element grip. The implementation of the subdomain BEM formulation in this limiting case, i.e. a very large number of subdomains, including cells integration at each subdomain has been called by Taigbenu and collaborators the Green element method (GEM) (see Taigbenu [2], and Taigbenu et al [3]), in which a finite deference approximation is used to reduce the over-determined system of equations. In these conditions, the resultant coefficient matrix is as sparse as that encountered in FEM and therefore its solution is as efficient as in that domain approach, and the results are as accurate as those of the BEM. Recently Popov and Power [4] found that the DRM approach can be substantially improved by using domain decomposition. Their idea of using domain decomposition to improve the accuracy of the DRM approach, was inspired by the work of Kansa and Carlson [5] on the radial basis function data approximations, where they observed that the best approximation is obtained when the original domain is split into matching subdomains. This work will proceed with the idea of Popov and Power of improving the DRM approach by using domain decomposition. However, here a large number of subdomains will be used, i.e. multi-domain decomposition, in such a way that the proposed numerical scheme will be equivalent to the GEM, but without the need to resort to cell integration at each element. Besides the above difference between the present approach and the GEM, here the resulting over-determined system of algebraic equations for the complete set of nodal variables, i.e. the field variables and their derivatives, will be solved directly without the need of additional interpolations. Due to the main ideas involved in the proposed approach, i.e. the dual reciprocity method and the multi-domain decomposition technique, from now on it will be referred to as the dual reciprocity multi-domain method (DRM-MD).

4 70 Boundary Element Research In Europe 2 Basic Concepts of the DRM-MD approach Let us consider the linear convection-diffusion equation, where the transport of a substance with a concentration c (kg m~^) in a compressible or incompressible flow field with variable velocity V (m s~*), with production term P (kg m~^s~*), coefficient of diffusion D (m* s~*) and reaction constant A; (s~*), satisfies the equation: +P-kc = 0 (1) % ^ In addition, in the case of incompressible flow, the continuity equation f*-0 (2) OXi also has to be satisfied. Equation (1) can be rewritten in terms of the Laplacian operator, with the convective, production and reaction terms as non-homogeneous terms of the Laplacian operator (source term), i.e.: where the source term b will be defined as: (3) b = ( VJ 4- c - - P 4- /cc ), (4) From the Green's integral representation formula, it is found that the concentration at a point x of the i^ subdomain bounded by the contour I\ that enclose the domain fji, is given by: X(x)c(x)+ f q'(x,y)c(y)dty- I c"(x,y)q(y)dty = I c*(x,y)b(y)dtty t/l«i r-\ "It / ft i/a*i / (-) (5) for i = 1,2,..., M, where M is the total number of subdomains. In the above equations it is written x instead of x and y instead of y, and for convenience this notation will be used from here on. Here, c*(#,%/) is the fundamental solution of the Laplace equation, q(y) = dc(y)/dn and q*(x,y) = dc*(x,y)/dn. Notice that in equation (5) all the integrals are over the contour of the subdomain i except for the one corresponding to the term b(y), which represents the sum of the non-homogeneous terms in the equation (3). At each interface, the flux leaving one subdomain has to be equal to the flux entering the other. Therefore, it is necessary that the following flux matching conditions hold at the m^ interface of the subdomains i and 24-1: (6)

5 Boundary Element Research In Europe 71 Transactions on Modelling and Simulation vol 19, 1998 WIT Press, ISSN X Besides the above conditions, the concentration at each interface needs to be continuous, i.e.: Ira Ira ^ ' In the present approach the DRM approximation will be used to transform the domain integrals, at each subdomain, into surface integrals at their contours. After applying the DRM approach, the matrix form of the integral representation formula at each subdomain can be written as: H<c - G;q = (HiC - GiQ) F~*b (8) where the thin plate spline interpolant (see Golberg and Chen [6]) has been used in the domain interpolation, and its corresponding particular solution in the DRM approach. It is important to remember that in the above equation q stands for the derivative of the concentration at the collocation point in the direction normal to the element, and not for the total flux. Therefore at a point on a corner of the subdomain contour, we will have two derivatives, one in each direction of the normal vector at each side of the corner. The derivatives of the concentration are approximated, as it is usually done in the DRM approach, as: With the above DRM approximation for the derivative of the concentration, we can express the equation (4) in terms of the concentration. In this way the equation (8) represents a matrix system for the normal derivative of the concentration at the surface points, and for the concentration at the surface and internal points in each subdomain (or element). Each of the local matrix systems, i.e. for each i = 1, 2,..., M, has to be assembled with its neighbouring systems according to the concentration and flux matching conditions given before, resulting in this way in an over-determined banded global matrix system. Instead of using a finite difference approximation to reduce the resulting over-determined system of equations, as it is done in the GEM, here we will solve the complete over-determined system. In that regard, several algorithms are known in the mathematical literature which can solve numerically this kind of system, being the conjugate gradient method and the least-square approach the most popular. Here the over-determined system will be solved in the least-square sense, due to the simplicity of its formulation and its previous success in application in several numerical solutions of boundary integral equations. In standard BEM solutions, this scheme is very expensive due to the number of operations needed in the multiplications of fully populated matrices. However, in this case it can be done very efficiently by considering the banded structure of the matrices, and the simetricity in respect to the

6 72 Boundary Element Research In Europe Figure 1: Division of a domain into subdomain. main diagonal of the obtained matrix. The resultant matrix is not only symmetric but is also positive definite. In contrast with the FEM, and in less sense with the GEM, in the present approach there is no major restriction in the arrangement and forms of subdomains, or elements, and even it is possible to have a subdomain defined by several contour elements (more than four) matched with smaller subdomains that are only defined by four or three contours elements (see Figure i). Finally, it is important to observe that all the matrices, i.e., H, G, C,Q,F, F~*, and df/ctej, are geometrically defined in terms of the distances between the source points and thefieldpoints or thefieldpoints and the DRM collocation points. Therefore, those matrices are identical for all the subdomains that have the same size and shape, for which it is important to define the same local numeration of the nodes, in order to evaluate those matrices at each of those subdomains by evaluating them in one of the subdomains. Figure 2 shows the computer mesh (subdomains) which authors used in a problem of design of trenches in a multi-layered landfill. The final matrix of the least-square solution for this case is shown in Figure 3.

7 Boundary Element Research In Europe 73 Transactions on Modelling and Simulation vol 19, 1998 WIT Press, ISSN X [Mill [ Ml 1 1 III I I MM 1 I \y/ / / 0 ( ) ) Figure 2: Computer mesh (subdomains) used in a problem of design of trenches in a multi-layered landfill rrs 1 1 r i % " V* " VN - < > tt;\. ' "'.'"' "..- - ' \ "'. "' '^ -^ " 'S S : Vx\ n : --:x Figure 3: The final matrix of the least-square solution, using continuous elements.

8 74 Boundary Element Research In Europe 3 Numerical examples To compare the performance of the proposed DRM-MD formulation with other BEM formulations previously reported in the literature, let us consider the following one-dimensional convection-diffusion problem with a variable velocityfieldwith a governing equation: satisfying the boundary conditions c(0) = Co and c(l) = C\. The convective velocity field is % = ln^_t_&(%_l); T/,=0 (11) GO z corresponding to the flow of a hypothetical compressible fluid with a density variation inversely proportional to the velocity field, i.e., p Therefore, equation (10) can be rewritten as The analytical solution of the above boundary value problem for D = I m^s~* is given by: f K / /nr 7. \ ^ This problem has been considered as benchmark problem in the DRM literature (Partridge [7]). For its numerical analysis this one-dimensional problem will be considered here as a two-dimensional in a rectangular domain with dimensions 1 x 0.7 m, with 0 < x < I and 0.35 <y< 0.35 subject to the following boundary conditions: (12) and c(0,%) = Co = 300; c(l,%) = Ci = 10 (13) dn '%="0 35 ' dn The numerical results obtained with the DRM-MD approach based upon the thin plate splines interpolation function, were compared with the results obtained by Qiu et al. [8] using two different approaches: conventional DRM formulation with the use of the classical radial basis approximating function, 1 -f /?, and a standard boundary element formulation with cell integration (BEM-CI). Both formulations were based on the Laplace fundamental solution. Two different values of the parameter k were considered, k = 5, and k 40, for which the velocity field takes the limiting values Vx = ± 2.5 and V* = ± 20, respectively. For k = 5, all three formulations produced good results. For k = 40, the DRM formulation

9 Boundary Element Research In Europe 75 Transactions on Modelling and Simulation vol 19, 1998 WIT Press, ISSN X analytical solution A DRM n BEM-CI O DRM-MD X Figure 4: Concentration profiles compared with the analytical solution (k = 40). used 20 quadratic elements with 54 internal points, the BEM-CI employed 38 linear elements with 28 linear rectangular cells and the DRM-MD used 80 subdomains, each made of four linear continuous elements with one internal point. The results for the internal concentration are shown in Figure 4. As can be observed, due to the abrupt change in the value of the concentration, it is not possible to distinguish the differences in the results obtained with each approach, However, if the same results are given in semi-logarithmic plot, Figure 5, it can be observed that the DRM-MD follows the analytical solution almost exactly, while the other two methods do not follow the shape of the concentration profile, having their maximum deviation in the region of the smaller values of the concentration, and besides, the DRM shows oscillations. In Figure 5, the results obtained with the DRM-MD formulation based on the classical radial basis approximating function, 1 -f R> are also presented. The results obtained with this scheme also follow the analytical solution, but, as expected, with less accuracy than those obtained with the use of the thin plate spline function. While the results obtained with the DRM approach cannot be improved by increasing the number of internal points due to the problem of convergence, the results obtained with the BEM-CI can be improved by increasing the number of internal cells, at the expense of the computing time. The results reported in Figure 5 for the cell integration technique and the DRM-MD required approximately the same CPU-time, 25 seconds on an IBM RS/6000 station for the BEM-CI and 32 seconds for the DRM-MD approach, when both approaches were solved using standard Gauss elimination with pivoting, having, in the case of the DRM-MD approach 57

10 76 Boundary Element Research In Europe Transactions on Modelling and Simulation vol 19, 1998 WIT Press, ISSN X O analytical solution A DRM a BEM-CI o DRM-MD* O DRM-MD A A X Figure 5: Log-plot of obtained concentration profiles compared with the analytical solution (& = 40), (* formulation based on the classical radial basis approximation function). internal nodes, each shared between four subdomains, with three unknowns for each node, i.e. the concentration and its two main directional derivatives. The size of the matrix of the DRM-MD approach was 415 by 415. It is important to point out that in this case it has been only taken advantage of its banded matrix structure when dealing with the reduction of the over-determined system of equations, and not on the solution of the matrix system. When the same system was solved using the Cholesky decomposition, where care has been taken not to do any operations on the matrix elements outside the bandwidth of the matrix (solver scheme that will be called the modified Cholesky decomposition), the required time for solution of the problem was 10 seconds. The authors of this work used this method of solving the system just to provide comparison on the efficiency of the proposed algorithm when the final banded structure of the matrix is taken into account or not, which does not mean that this is the most efficient way to solve this type of matrix. As the matrix is well conditioned, the use of an iterative solver would represent the best option, particularly for large problems. Therefore, it can be concluded that by using a solver that considers the structure of the matrix of the system, the computational time required by the DRM-MD approach will be considerably shorter than the one required by the BEM-CI approach. The case for A; = 40 was also solved with the DRM-MD approach using four discontinuous elements, instead of continuous elements, at the contour of each of the 80 subdomains. The system of equations obtained with discontinuous elements is not over-determined as is the case for continuous

11 Boundary Element Research In Europe 77 elements, but the resulting matrix system is bigger. The system was solved in two different ways. Thefirstway was by solving the original system using Gauss elimination with pivoting, which required 209 seconds, and the second way was in the least square sense in order to be able to use the modified Cholesky decomposition solver, which led to a solution in 14 seconds. This big save in time can be easily understood if one reconsiders the structure of the matrix, which was of size 792 by 792, presented in Figure 6. The results for the concentration and its derivative inside the domain compared with the analytical solution, using continuous and discontinuous elements, are presented in the Tables 1 and 2 respectively. It can be seen that the use of discontinuous elements led to a better result in the first part of the domain where concentration and derivatives have higher values, while the use of continuous elements gave better result in the second part of the domain. The accuracy in both cases is comparable and when an appropriate solver is used, then the difference in the solution time for problems of moderate size is not significant Figure 6: The bandwidth of the matrix of the system when discontinuous elements are used.

12 78 Boundary Element Research In Europe Table 1: Concentration values at internal points obtained using the DRM- MD, continuous and discontinuous elements, compared with the analytical solution for & = 40. X analytical contin. elem discont. elem Table 2: Results for the derivative of the concentration obtained using the DRM-MD with thin plate spline function, for continuous and discontinuous elements, compared with the analytical solution for k = 40. X analytical , contin. elem discont. elem

13 4 Conclusions Boundary Element Research In Europe 79 By using the Green's integral representation formula with the idea of multidomain decomposition, the DRM domain integral transformation to the boundary, the least-square approach for solving the resulting over-determined system and the use of some of the most efficient interpolation functions available in the mathematical literature, a significant improvement in the accuracy and speed of integral equation approach, to solve numerically domain dominant problems, was achieved. The main advantages of the proposed scheme are: the domain integrals at the subdomains are transformed into surface integrals at their contours, and the resulting approximated surface integrals are evaluated analytically, there are no major restrictions on the arrangement and the forms of the subdomains, the problem of near-singular integration is minimized, the matrix coefficients of geometrically similar subdomains are calculated only once, thefinalmatrix system is banded, symmetric and positive definite, a hermitian solution is obtained with the same computational effort of previous non-hermitian one. The computational aspect of the present numerical algorithm can be further improved by using a more efficient solver that considers the blockdiagonal structure of the matrix system, and the use of an adaptive scheme for the definition of the subdomains. References [1] D. Nardini and C. A. Brebbia, * A new approach to free vibration analysis using boundary elements', Applied Mathematical Modelling 7, (1983). [2] A. E. Taigbenu, 'The Green Element Method', Int. J. for Numerical Methods in Eng. 38, (1995). [3] A. E. Taigbenu and O. O. Onyejekwe, 'Green element simulations of the transient nonlinear unsaturated flow equation', Applied Mathematical Modelling, 19, (1995). [4] V. Popov and H. Power, 'A domain decomposition in the dual reciprocity approach', Boundary Element Communications 7/1, 1-5 (1996).

14 80 Boundary Element Research In Europe [5] E. J, Kansa and R. E. Carlson, 'Radial basis functions: A class of gripfree, scattered data approximations', Computational Fluid Dynamics Journal 3/4, (1995). [6] M. A. Golberg and C. S. Chen, 'A bibliography on radial basis functions approximation', Boundary Elements Communications 7, (1996). [7]P. W. Partridge, C. A. Brebbia and L. C. Wrobel, The Dual Reciprocity Boundary Element Method, Computational Mechanics Publications, Southampton & Elsevier Applied Science, New York, [8] Z. H. Qiu, L. C. Wrobel and H. Power, 'An evaluation of boundary element schemes for convection-diffusion problems', Boundary Elements XV) edt. C. A. Brebbia and J. J. Remcis, Computational Mechanics Publications (1993).