Boundary element method, finite element method and thefluxspline method: a performance comparison for scalar potential problems

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1 Boundary element method, finite element method and thefluxspline method: a performance comparison for scalar potential problems A. BulcSo, C. F. Loeffler, P. C. Oliveira Departamento de Engenharia Mecdnica, Universidade Federal do Espirito Santo (UFES), av. Fernando Ferrari s/n, Goiabeiras, Vitoria, Espirito Santo, Brazil carlosloeffler@hotmail com Abstract This work presents comparisons among the numeric results obtained by the Boundary Element Method, Finite Element Method and Flux Spline Method applied to two-dimensional scalar potential problems. Cases are stationary and governed by Laplace and Poisson equations. The main purpose of this paper is to compare the accuracy of mentioned methods in computational simulations. So, the applications were selected to analyze numerical behavior when some special physical and geometrical aspects are involved, such as: discontinuities in the boundary conditions, actions concentrated on specific part on the domain, non-homogeneity in constitutive properties etc. 1 Introduction The high level of modeling of physical problems solved by the current engineering results of the use of powerful numeric methods of solution, that they use modern approaches mathematically consistent. Such methods allow to substitute the continuous domain, represented by partial differential equations, for equivalent discrete models, expressed for algebraic equations, with easy computational treatment.

2 260 Boundary Elements They are pioneers in this approach the Finite Difference Finite (DFM) and the Finite Element Method (FEM), that still today are the most important discrete numerical techniques. Recently, two other similar methods were introduced in engineering application areas: The Boundary Element Method (BEM) and the Flux Spline Method (FSM). It is verified that the specialized literature doesn't register sufficient number of works that process acting comparisons among those methods, involving precision, cost, flexibility and other aspects, so that the engineers may choose correctly the better or more suitable numerical technique for its purposes. 2 Escalar Potential Equation Consider stationary problems in two dimensions, where the physical field is scalar. Being assumed homogeneity, isotropy and physical linearity, the field can be quite expressed by the following equation: K (1) This expression is denominated by Poisson Equation, usual in the representation of stationary problems where there are presence of sources, flows or field forces inside the domain, represented by the function S. When the function S is null occurs the simplest case in the scalar field theory. The equation like that is named as Laplace Equation. 3 The Boundary Element Method Consider the Equation of Poisson (eqn 1). The formulation of BEM for such problem begins for the establishment of the Inverse integral form associated to Poisson's equation, given for: JYn,) >* -(jx'n, + JY'n^Jir (2) In former equation Q is the domain of the system, T is the boundary, ( > is the Fundamental Solution, corresponding the answer of a correlate problem defined by Poisson Equation, where it exist a singular source applied in a particular point,. This special source is represented by the Delta de Dirac function. In the point of view of the Weighted Residual Method, ( >* acts as an efficient weighted function to allow the good representation of concentrated actions and discontinuity in the behavior of the involved variables (Brebbia[1]). It was still defined other new variables: JX = K^*, JY=K^*, JX*=^ and JY* =K^. 0) 5x 3y 5x 9y

3 Boundary Elements 261 Being n% and ny the normal vectors in the directions coordinates x and y, respectively. Using the properties of the Delta of Dirac, it is possible to rewrite the equation (2) in the form: n (4) where, for subject of simplicity: J = JXn% + JYny; J = JX n^ + JY ny. The next step is to operate the right side of the equation (4), in order to also express it in the form of boundary integrals. Due to generality, the Dual Reciprocity technique is the most effective procedure. So, it is possible to express the following sentence: j-i (5) In this former equation w\ p* and W form a group of auxiliary functions, composing a similar mathematical structure to the right side of the equation (2). The function p-* and its primitive w^ are arbitrary (Partridge et al. [2]). The next step consists of the boundary discretization, with its division in elements, which <(> and normal derivative J they are approximate. It is developed, then, a system of equations starting from the location of the coincident points with the nodes of the elements (collocation method). The same procedure is made to the j points. A final system in matricial form results: H( >-GJ = S (6) 4 Flux Spline Method For the formulation of FSM it considers the Poisson Equation as follows: "SIX 9JY,»j *»j - O For a elementary volume control V^ (according to exhibition the illustration 1), the integration of governing equation produces: K«*JJ - J*i>Yj + KM - JYjAX,]= S,jAX,AY, W JX, Figure 1 - Elementary control volume.

4 262 Boundary Elements In the discretization procedure used is assumed that inside each control volume the flows J can vary linearly along each direction. It is interesting to use elementary adimensional coordinates, in the form: J =A-ri + B, where * = &_ZJo_ and B = JQ (9) ATI This technique of discretization is denominated of Flux-Spline and already in that point, due to the fact of the angular coefficient A to absorb source terms sources and multidimensional influences, it reveals the largest effectiveness when compared to the traditional procedures that, assuming constant flow, it takes to the well-known central difference scheme. It is necessary to take account the hypothesis of lineal flow, equation (9), and to process integration for the control volume in subject. With the condition that the dependent variable (() is stored in the center of the control volume r, it is possible to obtain the quadratic shape potential function in the considered direction: * *MiHl <"» The unidimensional procedure is generalized being considered the flow expressed of similar way in other directions like it was shown for the direction T. The joining among adjacent control volumes is made being imposed the continuity of the function (J>, and of the flows in the interface of the same ones. This way, being applied the continuity condition of ( ) in X and Y, in a volume control i j is obtained the following discretizated equation for (() : j+, - AX, + DJYy AX, + DJX,,,. AY^ + DJXy AY. + ^ - JX,,,j). AY. Values of coefficients that appear in last equation can be obtained in Oliveira [3 ]. The flows in each direction are calculated being assumed initially A that J are null. The system of algebraic equations for (() is, then, solved in iterative form through the solution of tri-diagonal matrices in each direction coordinate X and Y (Line-by-Line), being used Thomas' algorithm (TDMA). With that result it returns to the calculation of the flows to the convergence. 5 The Finite Element Method There are several formulations of the Method of the Finite Elements, but the most common and important way begins with the establishment of Weak integral form associated to the Poisson governing equation, given for:

5 Boundary Elements 263 _K f * * + * * dq + KfodT = ("SW (12) J dx ox dy dy J J QL ' -U r a Under the point of view of several important approaches, <j> it can take interpretations as weighted function, admissible function or virtual potential, according to the case (Zienkiewicz, [4]). The FEM establishes the division of the domain O in discrete subdomains (the finite elements) whose points that define them are denominated nodal points. Usually it is assumed the basic variable $ and the auxiliary function f are approximated by the same polynomial interpolation function, whose distribution inside the element is given with reference to the values in the nodal points <Dj and O J, in the following way: Derivatives of<dj and Oj are calculated through interpolation functions in agreement with the coordinate directions. Making the convenient substitutions the integral expression at finite element level, and being considered that the nodal values are not integration variables, the previous expression, equation (12), can be arranged, placing the auxiliary parameters <X> in evidence in such a way that: -O^k Jaj,a%,,dQ + k = 0 (14) It is convenient to define the equation (14) in an appropriate matricial form, where thefirstintegral is expressed for K\ as being an elementary matrix of properties; the second integral is expressed for f \ being a vector of nodal forces related to boundary interaction of the finite element; and the expressed third integral p% represents a vector of nodal forces relative to the external domain actions. The global arrangement of the system is made with base in the compatibility among the nodal potential values, resulting in a matricial system, given for: K<D = f-p (*5) 6 Examples 6.1 First example The first example is governed by equation of Laplace. It consists of a seepage problem, which its characteristics are given in the illustration (2):

6 264 Boundary Elements Figure 2 - Seepage problem: geometry and boundary conditions. In this problem in subject is particularly difficult to get a good computational response due to the discontinuity in the value of the flow in the superior part of the porous medium. Next results are shown for the flow along the superior boundary obtained by BEM and FSM simulations. BEM used elements with constant interpolation. The results for both methods are shown in the figure 3(a) and 3(b): B.E ^t* 0.0 F V o.o 4*,-*, r-^-f 0.0 -< 3.5 i &B.E.... ": 2.5 * _ 10 * * ^ * BE Figure 3a - Boundary Elements Method results j " _ (1 FV ~ 32DJJ/I _ * - # Figure 3b - Flux Spline Method results. ;:; ; ' ; / ;» The agreement among the values obtained by both methods is quite good, but undoubtedly a small advantage for BEM can be admitted, not just for the best reception of the singularity, but also for the use of less refined meshes and more simpler interpolation shape.

7 Boundary Elements Second example In this second example the equation of Laplace is still approached, confronting the performance of the BEM front to FEM, in a problem of heat conduction. In the figure 4 are shown the characteristics of problem. In this example both FEM and BEM methods used isoparametric quadratic elements in their meshes. It is stood out that for FEM the program was used Ansys 5.4, with the following elements: plane35 (triangular element of 6 nodes) and plane?? (square element of 8 nodes). Figure 4 - Physical representation and different discretization used. For comparison among the methods are shown graphs, in logarithmic scale, contends the averages of the percentual error in the numeric response with reference of analytic solution for all discrete points. In vertical axis are plotted the value of global percentual error and in the horizontal axis are plotted the number of elements =- = ,,-rr, 0.00 \ BEM FEM Figure5- (a) Potential, nodal points on the boundary; (b) Potential, nodal points inside the domain; (c) Potential Derivative (radial direction), nodal points inside the domain. It is wide the advantage the quality of the results of the BEM in that analysis, particularly in respect to the calculation of values inside the domain. Really, that excellent behavior of the BEM is general in simulation of problems governed by the Equation of Laplace.

8 266 Boundary Elements 6.3 Third example In the third example, the numeric difficulty involved it consists of the discontinuity of the potential prescribed in the horizontal edge of the semicircle shown in the illustration.6. In the point where there is discontinuity of the value of the potential, for prescription of the boundary conditions in both methods the mean value of the potentials was adopted prescribed in this face. A Y Figure 6 - Physical representation of the problem. In the figure 7 are depicted the results. Shape quadratic functions are used in BEM and FEM discretization. Once again it is noticed a wide advantage in the results obtained with to BEM front to FEM in that class of problems. 100^, _ =-- = - A.~=^ = 0.10 ^ ^ - lt_ -=~^ = \ BEM FEM Figure 7 -(a) Potential, nodal points on the boundary (b) Potential Derivative (normal direction), nodal points on the boundary; (c) Potential, nodal points inside the domain. 6.4 Fourth example It consists of the Poisson's problem. A square membrane, totally fastened in its boards, is loaded in a central section of its surface, as shown in figure 8. In that case they are compared the performance of the BEM in comparison to FSM. The results are shown in the figure (9), where the percentual error is calculated in comparison the analytic solution. M j\ a = 9b.l. b = 3 b.l. & P = 1 N/b.l. T(pre-tragao) 11 N/b.l. Figure 8- Square membrane.

9 Boundary Elements 267 The discretizations used were, respectively, 14, 42 and 126 constant boundary elements and 9, 81 and 729 finite volumes. The scale logarithmic exhibits the convergence of the methods with the refinement. The best performance of FSM is just verified when the mesh is not very refined. Such behavior modifies with the increase of the number of entities, where it is noticed a better performance of the BEM. It should be considered that in this example were used constant boundary elements, while the interpolant function of FSM is quadratic. More ahead an analysis is presented in which both methods are used with elements of same order Number of Elements Figure 9 - Error in displacement of central point of the membrane. 6.5 Fifth Example In this problem governed by the equation of Poisson, is applied a body action equal to 10 units (force/u.l.^) constant along the whole physical domain, and the material property is unitary. Figure 10 shows the aspect of the problem. Figure 10 - Different discretizations used in circular membrane problem. In following illustration, are presented the graphs contends the average of the percentual error obtained for this analysis.

10 268 Boundary Elements i :.. -r ^ - \ - ~ - _ n:^-^ ,,,-n BEM FEM Figure 11 - (a) Potential Derivative of the Potential (normal direction), nodal points localized on the boundary; (b) Potential, nodal points inside the domain; (c) Potential Derivative (radial direction), nodal points inside the domain. Results shown in figure 11 also show the better performance of BEM front to FEM in this problem, in despite of the difference among behavior of two methods is not very considerable like in Laplace's simulations. Indeed, here it is used Dual Reciprocity procedure, which mathematically introduces approximation in domain action representation. However, considering the good results, they mean that the error in Dual Reciprocity model have small significance and the Boundary Element approach is not affected negatively. 6.6 Sixth example The physical domain of this analysis is a square shown in figure 12. It subjects the null essential boundary conditions along its boards. There is an application of a domain action constant in all its extension. p = 10 For^a/u.l.^ X u = < 10 u.c. Figure 12 - Representation of the problem and different discretization. Figure 13 presents the graphs contends the results. In this case were used linear boundary elements, while quadratic finite elements are used in FEM analysis. Once more the BEM models show better results that FEM ones. Here the good performance of BEM probably results of its capacity to represent behavior of functions in corners.

11 Boundary Elements ^ ^ : V _ ^ n T, BEM FEM Figure 13 -(a) Potential Derivative (normal direction), nodal points on the boundary; (b) Potential, nodal points inside the domain; (c) Potential Derivative (module), nodal points inside the domain. 6.7 Seventh example This problem represents a physical situation where heat generation rods of square cross section (material k2) with heat generation are embedded centrally in a slab (material kl), the surfaces of which are maintained with a constant temperature [5]. In the present simulation the value of conductivity ratio kl/k2 is 100. It must be noted that problems of this kind have numerical difficulties by finite difference approach. The comparison presented here will be done among BEM and FSM. For the first method, it is adopted 40 quadratic boundary elements and 20 poles in source region. For Flux-Spline was used finite volumes. The high level of refinement of FSM allows to take its response like a reference solution. The physical problem is shown in figure 14. u=0 u=0 q=0 u=0 u=0 q=0 Figure 14 - Representation of the seventh example. An excellent agreement it is observed in figure 15 among the temperature solution in both numerical methods. In spite of the high difference among the physical properties of the sub-areas, the numerical behavior was similar. It must be noticed that Dual Reciprocity performance was very good, because this problem is not a typical case for boundary technique approach. It should be noticed that for the FSM, increasing the relation among the physical properties of the sub-regions, an increase of the number of necessary iterations is made to reach the preset stop criteria. Because of this, in such cases, the distribution of the profiles of temperatures and flows become more intricate, requesting like this a larger number of iterations.

12 270 Boundary Elements Figure 15 - Representation of temperature inside the physical domain. Lines mean isothermal paths. Left side: FSM solution;right side:bem solution. Conclusions The problems presented in this work showed the excellence of the BEM results and its superiority in comparison to the other methods in that problem category. Such problems don't possess any private difficulty that turn them inadequate to FEM and FSM. In fact, what happens it is a larger consistency of the discrete model generated by the BEM in these cases. It should be noticed that even in the cases where domain actions exist, the use of the BEM, with the aproximation of the Dual Reciprocity procedure, it produced superior results. References 1. Brebbia, C.A.; Telles, J.C.F. & Wrobel, L.C. Boundary Element Techniques, Theory and Applications in Engineering, Springer-Verlag, New York, Partridge, P.W., Brebbia, C.A. & Wrobel, L.C. The Dual Reciprocity, Boundary Element Method, Computacional Mechanics Publications, London, Oliveira, P. C. Esquema Flux Spline Aplicado em Cavidades Abertas com Convercsao Natural, UNICAMP, Tese de Doutorado, S&o Paulo., Zienkiewicz, O.C. e Taylor, R.L.; The Finite Element Method, Volume I, Basic Formulation and Linear Problems, McGraw-Hill Book Company, London, 1988, 5. Kelkar, K. M., Iterative Method for the Numerical Prediction of Heat Transfer in Problems Involving Large Differences in Thermal Conductivities, Numerical Heat Transfer, Part B, vol 17, pp , 1990.