A mesh refinement technique for the boundary element method based on local error analysis

Transcription

1 A mesh refinement technique for the boundary element method based on local error analysis J. J. Rodriguez <» H. Power <*> Wessex Institute of Technology, Ashurst Lodge, Ashurst, Southampton Abstract An adaptive mesh refinement technique is proposed for the boundary element method for the treatment of boundaries where Neumann conditions are prescribed. Error indicators at element level are evaluated based on a collocation scheme using an ad hoc uniform norm that compares values of the field variables at successive iterations. An h-adaptation strategy is used to subdivide elements to improve the exactness of the numerical solution. The technique is illustrated with examples for two-dimensional potential problems governed by the Laplace equation. 1 Introduction The accuracy of the boundary element method as in any numerical method depends on several factors such as the representation of the geometry, the numerical implementation, computer precision, and discretization arrangement. Refinement schemes have been devised to improve the exactness of the numerical solution as affected by the mesh discretization. Under such schemes several consecutive solutions or iterations are evaluated to transform the initial mesh into an improved one according to error assessments performed after each iteration. Three refinement strategies have gained general acceptance, namely, the h-adaptive strategy (e.g. Rencis & Mullen**, Charafi & Wrobel*), the p-adaptive strategy (e.g. Alarcon & Reverted, Yokoyama & Zhan^) and the r-adaptive strategy (e.g. Ingber & Mitra^ Kita, et al. ). Under the h-adaptive scheme, a mesh is refined by subdividing some of its elements. The p-adaptive process

2 increases the order of the interpolation functions. The r-adaptive method keeps the number and type of elements while repositioning their nodes. Some combinations of the three methods have also been advanced, in particular h-p (e.g. Rank^, Karafiat^) and h-r (e.g. Sun-Zamani^, Abe & Sakuraba*). Kita & Kamiya^ and Liapis'^ reviewed the literature on adaptive schemes applied to the boundary element method. This article presents a refinement technique based on the h-adaptive method. An ad hoc uniform norm is used to evaluate an error indicator at element level employing field variable values of two consecutive solutions of the direct collocation boundary element method. The local error indicator is based on a piecewise polynomial collocation approach, which makes it consistent with the collocation BEM used. This contrasts with Galerkinbased error indicators frequently used in most of the above references dealing with mesh refinement techniques for the collocation BEM solution. In Section 2 we introduce the formulation of the boundary element method for the 2D Laplace equation. Section 3 contains the mathematical and algorithmic aspects of the mesh refinement strategy. In Section 4 we present two sample problems. Finally, conclusions are drawn in Section 5. 2 The boundary element formulation for 2D potential problems Consider the following boundary-value problem in a two-dimensional domain Q, and its boundary F: Vw'=0 i/iq u = u ont- (1) q = q oarwhere u and q = du/dn are the potential and flux, respectively; n is the unit outward normal vector on F; u and q are the prescribed boundary conditions at boundaries F- and F-, respectively, with F = F- u F-. For a boundary value problem of the Neumann type, the boundary integral equation of the above 2D potential problem for an arbitrary source point / on the boundary is the following Fredholm equation of the second kind c-u. + j uq*dt = \ qu*dt ^2) r r where c, depends on the internal angle at the boundary point i; w*is the fundamental solution for an isotropic material, and g* is its normal derivative. They are given by 2» 1, 1 * 3w* c- = -, u = -log-, q = (3) 2n 2n r dn where 0 is the internal solid angle at point /; r is the distance from the source point i to each point on the boundary F. In general the above problem does

3 oiciiiciils Transactions on Modelling and Simulation vol 24, 1999 WIT Press, ISSN X not have a unique solution; it is necessary to prescribe at least the value of the potential at a single point. After dividing the boundary F into a mesh of N elements, equation (2) is di seretized as follows Hf ' The above integrals are evaluated at each node in succession the collocation points to obtain a system of N equations, which is expressed in matrix form as HU = GQ (5) where H and G are the influence matrices; U and Q are vectors of potentials and fluxes at the collocation points, respectively. After imposing boundary conditions, system (5) is reduced to the system of linear equations Ax = b (6) where x is a vector of the unknown values of u; b the vector of the known boundary information. The above system x can be solved using a numerical method such as the Gauss elimination procedure. As mentioned before, the accuracy of the boundary element method depends, among other factors, on the discretization approach. H-refinement schemes usually perform the following steps to improve an initial coarse mesh: the system of equations (6) is solved for the initial mesh and an error assessment is performed to subdivide some elements if a local accuracy criteria is not satisfied. A new mesh and its concomitant system of equations are built and solved. The process is repeated until no further elements are divided. 3 Error estimation and mesh construction 3.1 Mathematical foundations. Rencis and Mullen^'^ formulated a scheme derived from asymptotic errors estimates stated in Brebbia^. Two successive iterations i-1 and / are compared using an error estimator based upon the L?- norm J \u'~* u') dt, which is an error estimator for a piece wise polynomial r approximation of a Galerkin formulation of a Fredholm integral equation of the second kind^ P*^. Instead of using the above norm, we employ uniform norms that, as we shall see, are consistent with the collocation method used in this article. Our arrangement uses two uniform norms: (a) the uniform norm si potential differences Au. of meshes i-1 and / (7)

4 (b) and the uniform norm of potentials u'- of mesh / = maxw', It has been shown in Golberg & Chen^' ^^ that the error between the exact solution u for equation (2) and the approximation u given by the collocation method is bounded by \\u - w L < khf (9) where «- w = max w - u\; /z = max /z^, y = 1 -N, with /%/ the size of element y; is a constant independent of /z; /=/?+1, where p is the degree of the interpolation functions (/?=! for linear elements). From (9) it is possible to deduce an error bound for the approximate solutions u^ and u' of two successive iterations After discretization, and generalising heuristically, the error bound for each element j is given by or d D Equation (12) applies to the boundaries where the flux q is prescribed, i.e. where the potential u is calculated using the boundary element method. From equation (9) we found that at the i* iteration the following inequality holds (12) where h'- has been obtained by dividing the element h'^ in m equal parts, i.e. h',=h^/m. By the reverse triangle inequality, we have and considering that at the I"* iteration the norm of its solution, i.e. II L'J ' lloo, is a fraction of the norm «^, then we can write the above inequality as with «I given as a tolerance. From equation (13) and (15) we have w-w'. _>f «', _ (15)

5 Boundary Elements Transactions on Modelling and Simulation vol 24, 1999 WIT Press, ISSN X max u'; < k m (16) To have an estimation of the number m of subdivisions, we will regard the inequalities (12) and (16) as equalities, i.e. maxw. -Uj (17) and maxw, ~ (18) m from which the following expression for the number of subdivisions can be obtained. It will be considered to be valid for the i+1 iteration: (*;)'+(*;-')' max M. u, =r.l ' / ^maxw./ The smaller number of subdivisions is obtained when the element of the previous iteration was divided by half. i.e. m=2. In our case, for linear interpolation, i.e. 1=2, we have maxw'. -u-' (20) / ma> V ^./ We use this expression to keep the number of subdivisions as small as possible. 3.2 Algorithmic approach. Primordial meshes. Following Rencis and Mullen*^ our technique begins with the construction of a primordial uniform mesh containing a reasonable small number of elements. The mesh should be sufficient to represent the complexity of the boundary. During the first iteration, the boundary element method is applied to resolve the N unknown variables at the collocation points. Therewith, an enhanced primordial mesh is created with twice the number of elements by dividing each initial one into two subelements of equal length. If linear elements are used, only N new nodes are incorporated. At this stage, knowing the values of the field variables u and q at all collocation points of the primordial mesh from boundary conditions or the first BEM solution, the potentials u» for the new N nodes are calculated using the discretized boundary element equation (19) 1 M r 7=1 r (21)

6 296 Boundary Elements Transactions on Modelling and Simulation vol 24, 1999 WIT Press, ISSN X No attempt is made to assign fluxes g, to the new nodes. Under this scheme only the potentials will be used to evaluate the error indicators at the elements where Neumann boundary conditions are prescribed. Now the enhanced primordial mesh contains 2N elements with known variables at its nodes except for the aforementioned fluxes. Second mesh. Subsequently, the initial boundary conditions are again prescribed to all the nodes of the enhanced primordial mesh. The original N old nodes are the same, with their known or unknown values. The new N nodes acquire values from the given flux conditions. Therefore, each new node also contains an unknown value of the potential. This second mesh is again solved using the boundary element method to obtain the unknown potentials u at the 2N collocation nodes. Error assessment. Elements are now considered by pairs. The scheme is only applied to the elements where Neumann boundary conditions are given. Every two consecutive elements are examined evaluating the following expression obtained from the previous analysis. It takes into account the potentials at their nodes in the enhanced primordial mesh and in the second mesh mov,,-second enhanced primer. mdx w, u; r penh.primor.l max \u ' V ' where j is the boundary of two consecutive elements; m*'"* is the number of new element divisions for the pair of elements considered; «*""""* is the potential in the boundary j of the second mesh; u y**"**?"""* is the potential in the boundary j of the enhanced primordial mesh; and is a given tolerance. The above maxima are computed from the values of the potentials at the elements collocation points. If with this choice no divisions are obtained, the maxima are again calculated by looking at the value of the potentials at k inter-element points. We have chosen k equal to 1 plus the number of Gauss points used in the numerical integration of the elements. Equation (21) is used to obtain the value of the potential at these points. The internal points are selected to be different from the Gauss points to avoid the superconvergence problem rendered when the error is measured over Prior to constructing the third mesh, an enhanced second mesh is built dividing each element k of every pairy which satisfies the above subdivision (V.^1 criteria, into d^ = Nint - (22) + 1 subelements, where Nint is the 'nearest integer' function. The process applied to the primordial mesh is here

7 Boundary Elements Transactions on Modelling and Simulation vol 24, 1999 WIT Press, ISSN X repeated: the potentials w, for the new discretized boundary element equation (21). 2^k-\^* nodes are calculated using the Third and further meshes. The third mesh i is assembled prescribing the initial boundary conditions to all nodes of the enhanced second mesh i-1. The mesh / is once more solved using the boundary element method to obtain the unknown potentials u at its collocation nodes. Again, elements are now considered by pairs. For every two elements where Neumann boundary conditions are given, the following equation is evaluated to determine the number of divisions for the next mesh /+7. Equation (22) now takes the form max w'y -LI (23) where j is the boundary of two consecutive elements; i-1 is the previous enhanced mesh; i is the current mesh; /+/ is next mesh; m'^ is the number of new element divisions for the pair of elements considered; u'- is the potential in boundary j of mesh /; is the given tolerance. It should be noted that with the choice of our criteria in the evaluation of equation (23), the value of ra'*' is independent of the size of the elements. The construction of meshes continues until either no further subdivisions are needed i.e. when m'^ is zero for all the j pairs considered in one iteration or when the global termination criteria <c' (24) is attained over the whole boundary F, where e'«1 is another tolerance parameter. A serious drawback of the present approach is that for each new mesh, the corresponding algebraic system of equations coming from equation (4) has to be solved. This calculation is repeated every time for larger and larger systems. This difficulty is usually avoided by the use of hierarchical interpolation functions. The hierarchical nature of the approximation is reflected in the structure of the new system of equations, leading to an efficient solution algorithm^. In this work instead of using hierarchical interpolation functions to elude the above problem, we consider that the desired values of the solution are the obtained values of the potential at the elements with no further subdivisions. We use them to rearrange the algebraic system (6), resulting in a smaller overdetermined system. We solve it in the least-square sense due to the simplicity of its formulation and the

8 298 Boundary Elements success obtained when applying it to several applications of numerical solutions of the boundary integral equation^. 4 Numerical Examples 4.1 Torsion problem. Figure 1 shows a circular section under torsion. The example is broadly explained in Paris & Canas^. The analytical solution to the problem is u = a\ rh sin0 I ') where u is the warping function. In our example a=2 and b=l. y Figure 1: Circular section under torsion. An initial mesh was constructed with 10 equal length elements along line ABC and 10 along line CD A as shown in Figure 2. Figure 2. Initial mesh for the torsion problem The refinement process converges after the 4* iteration. New are evenly distributed along the two semicircles as seen in Figure 3. elements

9 Boundary Elements 299 Transactions on Modelling and Simulation vol 24, 1999 WIT Press, ISSN X Figure 3. Elements after 4 ' iteration Figure 4 shows the diference between the analitical and BEM solutions at the 1^ and the 4* iteration between nodes 8 and 15. It can be seen that at the 4* iteration the analitical and numerical solutions clearly overlap. 5 T Distance along boundary (1st. Iteration) Distance along boundary (4th. Iteration) Figure 4. Warping function between initial nodes 8 and 15

10 300 Boundary Elements 4.2 Heat Transfer in an L-shaped domain. Figure 5 depicts four stages resulting from the application of our h-adaptive scheme to an L-shape domain with the prescribed conditions. Only the nodes of the initial mesh and the new nodes obtained at each iteration are shown. u=10 q=o q=0 Initial mesh u=0 3"* iteration 4* iteration iteration Figure 5. Mesh divisions for the L-shaped problem Most new elements are placed around the E corner, particularly at the final 4* and 5^ iterations. This problem has previously been studied by Kita et al., and a similar one by Paris & Canas^, obtaining an analogous concentration of elements around the E corner. 5 Conclusions An h-adaptive refinement scheme has been reported based on local error analysis. The technique applies to the boundaries where Neumann conditions are prescribed. The errors are assessed with an ad-hoc uniform norm in accordance with the collocation BEM used. Convergence overestimation is also addressed by the method; when the accuracy of the first error indicator is reached, a second indicator is evaluated bypassing the Gauss points of the numerical integration. Additionally, when an element is not divided, its nodal variables are not included in the next solution, resulting in an abridged system of equations. The method has proved to be efficient and reliable when applied to 2D Laplace problems.

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