344 Boundary Element Technology - WIT Press · 2014-05-13 · Similar to FEM, mesh refinement...

Error estimation and adaptive mesh

refinement in boundary element methods

N. Kamiya, E. Kita, M. Koide

Department of Mechano-Informatics and Systems,

Nagoya University, Nagoya, 464-01 Japan

ABSTRACT

This paper reviews recent studies on error estimation and adaptive mesh refinement schemesin boundary element methods (BEM). The adaptive methods discussed here are constructed oferror estimation, adaptive tactics and mesh refinement. The error estimation is the process toevaluate BEM solution error and its schemes will be classified into residual type, interpolationerror type, boundary integral equation error type and so on. Mesh refinement is the processto refine initially prescribed mesh uniformly or selectively, which will be divided into h-, p-,r- and their combinations. Adaptive tactics are the way to search the element to be refinedand to select relevant refinement scheme.

INTRODUCTION

In design and manufacturing processes of machines and structures, computer aided design(CAD) and computer aided engineering (CAE) are employed widely. They mainly employfinite element methods (FEM) for various required analyses but more recently boundary ele-ment methods (BEM) also have been recognized as a powerful alternate. When the objectunder consideration is governed by linear and homogeneous differential equation, the BEMneeds only its boundary discretization and therefore, data input is easier and more efficient.Since computational accuracy depends strongly on the mesh discretization employed, adap-tive meshes or elements are necessary for sufficiently accurate solution without consultingexperienced user.

In the earlier and primary studies, boundary element adaptive methods followed those de-veloped for FEM. FEM and BEM have their individual theoretical backgrounds and therefore,adequate adaptive method must be developed for BEM. For example, FEM is based on thevariational principle or the Galerkin method, while BEM is usually formulated by collocationof the integral equation. Consequently, error estimation scheme for FEM can not be appliedto BEM directly. Besides, for the elliptic differential equation, discretization error in FEMhas the local property and computational accuracy is improved fairly by refining bigger errorelements. On the other hand, discretization error in BEM has global property due to useof the fundamental solution of influence function type, which makes difficult to constructadaptive boundary elements. These difficulties were revealed by the recent studies but theremay be some more difficulties to be overcome. The aim of this paper is to review the recentstudies on the boundary element adaptive elements and to reveal the insufficiency of theexisting methods in order to develop more effective ones.

Transactions on Modelling and Simulation vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X

344 Boundary Element Technology

The adaptive methods are constructed of error estimation, adaptive tactics and meshrefinement, having roles of:

Error estimation: Discretization error of BEM solution is estimated and then, if thedesired accuracy can be obtained, the process is completed and otherwise, it continuesto the next step.Adaptive tactics: The element is selected to be refined and then, the mesh refine-ment scheme to be employed is determined.Mesh refinement: The data of the refined mesh is generated.

The methods will be discussed from these points of view.

BOUNDARY ELEMENT ANALYSIS

In this section, in order to facilitate the following discussion, boundary element analysis willbe explained briefly for the two-dimensional potential problem[5, 6].

Boundary integral equation

Let us consider the region H with its boundary F. With respect to the potential u andits normal derivative (flux) q, the governing equation and the boundary condition of thetwo-dimensional potential problem can be expressed as:

V^u = 0 in H , u = u on F« , q = —- = q on F« (1)on

where F* and F? are the potential-specified and the flux-specified boundaries, respectively,and 7i the unit outer normal vector on the boundary and (~) the specified boundary value.

We take the fundamental solution u* which satisfies

V*u* 4-5 = 0 (5 :Dirac's delta function) (2)

to obtain the following boundary integral equation:

iq*}dT = 0 (3)

where c(p) the parameter dependent on where the source point p is placed; c(p) = 1 whenp is inside H, c(p) = 0 when outside H, c(p) = 1/2 when on the smooth boundary.

Discretization, error and residual

By discretizing the boundary F by M boundary elements F^, Eqn.(3) leads to

M' = 0 (4)

and then, u and q on F^ are approximated by the linear combination of their nodal valuesmultiplied by the interpolation functions. Denoting the approximate solutions of u and q asu and q, respectively, the potential error e% and the flux error e% are expressed as:

e* =u-u , eq = q -q (5)


Boundary Element Technology 345

When u and q are substituted into Eqn.(4) instead of u and q, Eqn.(4) is not, in general,satisfied and the residual R arises;

M(6)

For the collocation BEM, which is the most popular formulation, the residual R of Eqn.(6)is taken as 0 on the selected boundary points (collocation points);

R(pi) = 0 (z = 1,- ••, N) (7)

where N is total number of the collocation points. These are solved as a system of equationsby introducing the prescribed boundary condition.

On the other hand, for the Galerkin formulation, the product of the residual R and (global)interpolation function <f> is integrated on the boundary, which is then taken as 0;

These are also solved as a system of equations. It is worth while to notice that boundaryintegral appears twice in the above equation and this is less efficient than the collocationformulation.

Besides, for the Galerkin BEM, it is proved that the solution error can be bounded by theresidual of the boundary integral equation[23, 22, 44];

l (9)

where c% and 0% are the constants independent of e(p,) and R(pt) and ||-|| denotes the norm.

ERROR ESTIMATION

In this section, the error estimation schemes will be classified; residual type, interpolationerror type, boundary integral equation error type and so on.

Residual type

As mentioned above, it is proved for Galerkin BEM that, the solution error is bounded by theresidual of boundary integral equation. Therefore, the residual can be employed as a measureof the solution error. The following error norms are employed; LI norm[60, 62, 59, 72],H*(L?) norm[13, 16, 15, 14, 43, 30, 42, 40, 41], tf */* norm[23, 22] and H* norm[44, 45, 46],which are dependent on the space of the solution.They have individual advantages as well as disadvantages. Hsiao et al.[23, 22] compared

the results by H® and H*f* norms and pointed out the followings; firstly, # (£2) norm isrelatively easy to calculate and may give appropriate error estimation when the boundary andthe data are smooth enough. When they are non-smooth, however, H* norm underestimatesthe actual solution error and in this case, H^^ norm may give better error estimation.

For the collocation BEM, this scheme has a great difficulty; as shown by Eqn.(7), theresidual is always zero on the initial collocation points and therefore, special treatments arenecessary. Parreira[15, 14, 43, 41] and Rank[44, 45, 46] take the other collocation points forthe residual estimation points, which are located on the center of the element. Besides, Sunand Zamani[61, 60, 62, 59, 72] and Abe[2, 1, 3] employ the relationship between the residualand the error, which is derived from the boundary integral equation.



Interpolation error type

Exact solution is predicted by approximating the current BEM solution by the higher orderinterpolation function than the initial and then, the solution error is defined as the differencebetween these two solutions[12, 35, 37, 36, 47, 48, 49, 51, 50, 63, 64]. This scheme is simpleand easy to implement

Boundary integral equation error type

The relationship between the error and the residual is derived from boundary integral equationand then, it is discretized to determine the error[8, 9, 10, 11, 34, 25, 28, 26, 29, 27, 32, 33,31, 38, 56, 57]. Since, however, for the collocation BEM, the residual disappears on theinitial collocation points and the resulting error is zero, the other collocation points must betaken for the residual estimation.

Other schemes

In addition to the above mentioned, the following schemes are presented.

Mullen and Rencis: In this method[39, 51], the problem is solved twice; first by N andsecond by 27V boundary elements. The latter solution is expected to be more accurate thanthe former and their difference defines the solution error. The estimated error is to be relatedto the element length for the mesh refinement.Guiggiani et al.: When employing the conforming quadratic element, the solution isdependent on where the node between both end ones (middle node) is located. In themethod of Guiggiani et al.[18, 17, 19, 20], BEM analysis is carried out twice when the middlenode is located on the center of the element and when it is slightly apart from the center;the solution error is defined by their difference.Yuuki et al.: In this method[69, 70, 68, 71, 67], the error is defined as the differencebetween the solution obtained by the ordinary BEM (Direct Singular Method, DSM) andthat by the non-singular BEM (Direct Regular Method, DRM), in which source points arelocated outside the region.Charafi et al.: In this method[12], after BEM analysis for the initial mesh, an arbitraryelement is divided into equal two elements and the relevant reanalysis is carried out skilfully.The error is defined by the difference between both solutions. The reanalysis is carried outover each element and the bigger error elements are refined.

MESH REFINEMENT

Similar to FEM, mesh refinement schemes for BEM are classified into the h-, p-, r- and theircombination schemes.

h-refinement scheme

In the h-refinement scheme, total number of elements is increased but the order of interpo-lation function remains invariant. The algorithm is simple and easy to implement on BEMsolver. Since, however, the global matrix formulation is necessary at each mesh refinement,the computational cost becomes high. For overcoming this difficulty, recently, h-hierarchicalrefinement scheme is presented[15, 14, 43, 19, 20], which employs h-hierarchical interpo-lation function and the solution accuracy is improved although the mesh refinement is notachieved in the sense of the ordinary h-refinement, i.e., the total number of the elements isnot increased. Most of the coefficient matrices at the initial analysis can be used after meshrefinement in this case, which contributes the improvement of the computational efficiency.



Kita and Kamiya[35, 37, 36] presented an alternate h-refmement in which not only thebigger error elements are divided but also the elements with relatively smaller error are com-bined. Combination of the sufficiently accurate elements prevents increase of total numberof elements.

p-refinement scheme

In p-refinement scheme, the initially assumed mesh is not changed during entire iterativeprocess but the order of the interpolation function is increased uniformly or selectively. Theway of increasing the order is classified into the "ordinary type" using the ordinary inter-polation function[63, 64] and the "hierarchical type" using the p-hierarchical interpolationfunction[4, 8, 9, 10, 38, 11, 42, 40, 52]: the most attention focuses on the latter. In thehierarchical type, the smaller of the coefficient matrices is calculated at each mesh refinementthan that in the ordinary type.

The most popular hierarchical interpolation functions are the Legendre polynomials[9, 11,38, 42, 40] and the Peano's family[4, 8, 10, 52], which, as Parreira[40] indicated, havethe following features; the Legendre polynomials leads to the matrices with lower conditionnumbers but yields larger numerical integration errors due to their oscillatory character. Onthe other hand, in the Peano's family, interelement continuity on a continuous model is easierto be implemented and more accurate numerical integration can be achieved. Therefore, heconcluded that the Peano's family seems more convenient to use.

r-refinement scheme

In r-refinement scheme, both the total number of elements and the order of the interpolationfunction are kept invariant but the mesh points are relocated within fixed topology in orderto minimize the error of the global mesh.

Combination scheme

The above-mentioned schemes have individual advantages as well as disadvantages and thecombination schemes are devised to reduce their disadvantages. The existing combinationschemes are hp- and hr-refinements.

The hp-refinement scheme is presented by Rank[44, 45, 46] and Stephan et al.[21, 58].Rank employed the h-refinement on the elements adjacent to a singular point and the p-refinement on the other.

The hr-refinement scheme is presented by Sun and Zamani[61, 60, 62, 59, 72] and Yuukiet al.[67]. In their schemes, firstly, the initially assumed mesh is refined by the h-refinementscheme and then, more in detail by the r-refinement scheme.

ADAPTIVE TACTICS

In this step, the elements to be refined and the relevant adaptive scheme are determinedaccording to the result of the previous error estimation. The algorithm is strongly dependenton the error estimation scheme, the mesh refinement scheme and the policy of the programmeramong others. In the following, the principal studies will be shown.

Adaptive tactics for h-refinement

Reference value method: One refines the elements with bigger error than the prescribedreference value, which may be the average error or the product of the maximum error and aparameter %(0 < 77 < 1). In the former, the solution error at each element is minimized on



the average but we often encounter too fine final mesh. In the contrary, such a difficulty maynot happen in the latter while it is very difficult to specify the value of 77 adequately.

Error convergency method: In the Galerkin BEM, it was proved that the solution errorconverges to zero in proportion to the square of the length of the elements[65]. Therefore,the following relation holds for the solutions u* and ii*+* at the k and (k 4- l)st. iterations,

/V+i - uJr

where /i* is the standard length of the element at the &th iteration and K is a parameter. Fromthis relation, Rends et al.[39, 51, 50] derived a way to calculate the number of subelements(refined elements) for obtaining the desired accuracy. In their descending study[47, 48], theyalso presented that the number of subelements is determined in proportion to the magnitudeof the error, while its theoretical background is not evident.

Besides, in the method of Yokoyama et al.[53, 54, 55, 66], it was assumed that the solu-tion error converged in proportion to (h*Y , where the parameter /3 was determined by thenumerical experiment.

Adaptive tactics for hierarchical p-refinement

Alarcon et al.: In this scheme[4, 8, 9, 10, 11, 38, 52], the initial analysis is carried out byconforming linear elements and next, the accuracy is improved by adding the p-hierarchicalfunctions to the initial interpolation. Denoting the initial solutions as u and q, we write theimproved solutions u and q as

where A^+i and Nn+i are the hierarchical interpolation functions and 0, .4-1 and bn+i areunknown coefficients, which are calculated by applying the algorithm employed in FEM[73]with some modification.

Parreira: In the above-mentioned scheme, the solution error is reduced by mesh refinementbut, in this scheme[42, 40, 41], the residual of the integral equation is reduced; the residual ofEqn.(6) is expressed by the product of the hierarchical function and the unknown coefficient.

The unknown coefficient a -n is determined by the collocation method or the Galerkinmethod.

Adaptive tactics for r-refinement

The algorithm for this scheme is defined as an optimization problem. The object functionexpressing the total error of the employed mesh is defined by the maximum error norm[24]or by the residual of the integral equation[l, 2, 3, 59, 60, 61, 62, 72] and is minimized byappropriate mesh redistribution. Carey and Kennon[7], Sun and Zamani[61, 60, 62, 59, 72],Abe[l, 3] and Yuuki et a I. [6 7] employ the grading function in order to co-relate the error andthe mesh distributions.

Other tactics

In the above-mentioned various schemes, the bigger error elements are refined, which is be-cause the error is thought to have the local property. This assumption, in FEM, is encouragedby the local property of certain kinds of governing differential equations and the interpolation



function but, in BEM, it is not guaranteed because the local error has some influence onthe global solution due to the use of the fundamental solution of influence function type.Kamiya and his co-workers[34, 25, 28, 26, 29, 27, 32, 33, 31] paid attention to this pointand proposed "extended error indicator", which was defined by integrating the product ofthe solution error and the corresponding fundamental solution at each element and therefore,the global property of the BEM solution error was reflected to the mesh refinement. Theyemployed it to the h-refinement scheme, in which the elements to be refined are determinedby the magnitude of the extended error indicator not by the local error level.

CONCLUSION

This paper discussed the error estimation and the relevant adaptive mesh refinement schemesof BEM analysis. The adaptive process is composed of the error estimation, the adaptivetactics and the mesh refinement. Therefore, the discussion was performed from these threepoints of view.

In section 3, the error estimation schemes were classified into the residual type, the inter-polation error type, the boundary integral equation error type and so on. The residual typeestimates the solution error by the residual of the integral equation while, in the others, thesolution error is estimated directly. In the residual type and the boundary integral equationerror type, the residual estimation process is necessary. In the collocation BEM, however,since the residual is always zero on the initial collocation points, one must employ the specialtreatments shown in the section.

In section 4, we discussed the mesh refinement schemes, which were classified into h-,p-, r- and their combination (hp- and hr-) schemes. It seems difficult to conclude which isthe best because they have individual advantages as well as disadvantages. However, thehp-refinement scheme seems to be the most promising among them because of its highererror convergency. In the h-refinement, the bigger error elements are usually divided intoequal length elements but, in some problems, unequal length elements seems more adequate,which requires more study on the hr- refinement. In principle, the hpr-refinement schemewould be the best.

In section 5, we discussed the adaptive tactics, in which the element to be refined andthe relevant refinement scheme are determined. They are strongly dependent on the errorestimation scheme, the mesh refinement scheme and the programmer's policy and favorite.

The error estimation and the adaptive mesh refinement schemes for BEM have been studiedfor about past two decades. Recently, attention is becoming to aim to developing the originalscheme more adequate to BEM. Since, however, the adaptive method is composed of threemain parts, as mentioned above, it will be necessary to develop better-balanced schemes.

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