A Meshless Method For Generalized Linear Or Nonlinear .
Engineering Analysis with Boundary Elements 30 (2006) 515–521www.elsevier.com/locate/enganaboundA meshless method for generalized linear or nonlinearPoisson-type problemsHui Wang a, Qing-Hua Qin a,b,*baDepartment of Mechanics, Tianjin University, Tianjin 300072, People’s Republic of ChinaDepartment of Engineering, Faculty of Engineering & Information Technology, The Australian National University,Engineering Building 32, North Road, Canberra, ACT 0200, AustraliaReceived 16 June 2004; accepted 26 January 2006Available online 14 March 2006AbstractThis paper presents a meshless method, based on coupling virtual boundary collocation method (VBCM) with the radial basis functions (RBF)and the analog equation method (AEM), to analyze generalized linear or nonlinear Poisson-type problems. In this method, the AEM is used toconstruct equivalent equations to the original differential equation so that a simpler fundamental solution of the Laplacian operator, instead ofother complicated ones which are needed in conventional BEM, can be employed. While global RBF is used to approximate fictitious body forcewhich appears when the analog equation method is introduced, and VBCM are utilized to solve homogeneous solution based on the superpositionprinciple. As a result, a new meshless method is developed for solving nonlinear Poisson-type problems. Finally, some numerical experiments areimplemented to verify the efficiency of the proposed method and numerical results are in good agreement with the analytical ones. It appears thatthe proposed meshless method is very effective for nonlinear Poisson-type problems.q 2006 Elsevier Ltd. All rights reserved.Keywords: Meshless method; Nonlinear Poisson-type problems; Analog equation method; Radial basis functions; Virtual boundary collocation method1. IntroductionThe meshless numerical method has recently become analternative to the finite element method (FEM) and boundaryelement method (BEM) due to its advantages of avoidingmeshing and remeshing, effective treatment of complicatedload conditions, and avoiding mesh distortion in largedeformation problem. The meshless method is usually dividedinto two main categories: the boundary-type meshless methodand the domain-type meshless method. Since the proposedmethod in this paper belongs to the boundary-type meshlessmethod, only developments corresponding to the boundarytype meshless methods are briefly reviewed in this section.Generally, boundary element method involves onlydiscretization of boundary of the structure due to thegoverning differential equation being satisfied exactly inside* Corresponding author. Address: Department of Engineering, Faculty ofEngineering & Information Technology. The Australian National University,Engineering Building 32, North Road, Canberra, ACT 0200, Australia. Tel.:C61 2 6125 8274; fax: C61 2 6125 0506.E-mail address: qinghua.qin@anu.edu.au (Q.-H. Qin).0955-7997/ - see front matter q 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.enganabound.2006.01.009the domain leading to a relatively smaller system size withsufficient accuracy. This is an important advantage over‘domain’ type solutions such as finite element method (FEM)or finite difference method (FDM). This advantage exists onlyfor problems without body forces and having explicitfundamental solutions. When a problem involves bodyforce, a domain discretization is also required, which maycause some inconvenience in the implementation of BEM. Inorder to overcome these drawbacks, a dual reciprocity method(DRM) was introduced by Nardini and Brebbia [1] fortransferring domain integrals to the boundary ones. Later,Nowak and Brebbia [2] proposed a multiple-reciprocitymethod (MRM) which can more effectively transform domainintegrals to the boundary ones, and gave numericalverifications of this method for linear potential problemswith body forces. Cheng et al. [3] also developed a DRMbased on global interpolation function for analyzing potentialproblems. Nowak and Partridge [4] compared these twomethods and identified their advantages and drawbacks. Onthe other hand, the use of the analog equation method [5]makes it possible to treat complex problems in which there areno available explicit fundamental solutions. From thediscussion above, it is evident that the methods mentionedabove still need to divide boundary into elements and singular
516H. Wang, Q.-H. Qin / Engineering Analysis with Boundary Elements 30 (2006) 515–521integral is also involved. In order to remove this disadvantage,many boundary collocation methods were proposed during thepast decades. Sun et al. [6] proposed a virtual boundarycollocation and element methods to avoid the singularity offundamental solutions [7] and applied this method to solvemany engineering problems successfully. However, they didnot include a proper process for dealing with body sources.The appearance of radial basis functions (RBFs) provides thepossibility of developing a real meshless method. With theconcept of RBFs, Chen and Tanaka [8] presented a boundaryknot method (BKM) based on the nonsingular generalsolution. Later, Chen and Hon [9] gave numerical investigations on convergence of BKM. More recently, Wang et al.[10] presented a meshless method for treating linear heatconduction problem.This paper presents a boundary collocation method bycombining virtual boundary collocation method [10] withRBF approximation and the analog equation method (AEM)[5] for analyzing generalized nonlinear Poisson-type problems. Firstly, AEM is used to convert the original governingdifferential equation into an equivalent standard Poissonequation and then RBF approximation and virtual boundarycollocation method (VBCM), are, respectively, employed toconstruct particular and homogeneous solution parts of theequivalent equation. Then, the superposition of the particularsolution and the homogeneous solution can yield the fullsolution of the original system. Satisfying all equations of theoriginal problem at collocation points can generate a linear ornonlinear system of equations, from which all unknowncoefficients can be solved. Finally, to verify the performanceof the method presented in the paper, three numericalexamples including standard Poisson, Helmholtz and nonlinear Poisson problems are considered.The construction of the paper is as follows. In Section 2, wegive a general definition of generalized Poisson-type equation.Then, a detailed numerical implementation is described andsome important factors of the proposed algorithm are discussedin Section 3. Section 4 provides some numerical results todemonstrate the effectiveness of the proposed meshless methodin analyzing generalized nonlinear Poisson-type equations.Finally, some conclusions and discussions are presented inSection 5.on G1(2a)† Neumann boundary condition for the boundary normalgradient:q Z q on G2(2b)† mixed boundary condition:b1 u C b2 q Z b3on G3The boundary value problem (BVP) under consideration is ageneralized nonlinear Poisson problem. The physical field u isgoverned by the following partial differential equation(1)satisfying following boundary conditions:† Dirichlet boundary condition related to unknown potentialfield:(2c)dwhere X2R is the position vector, d is the dimension ofdomain U which has a piecewise smooth boundary G. P2represents the Laplacian operator, and q stands for theboundary normal flux defined by qZKðvu vnÞ; n i arecomponents of the unit outward normal vector n to theboundary G(GZG1gG2gG3); u and q are specified values onthe boundary, respectively, b i (iZ1,2,3) are knowncoefficients.It is noted that Eq. (1) applies to a wide range of engineeringproblems such as heat conduction, mass transfer, wavepropagation, and so on, as different problems can becharacterized by different right-hand term f of Eq. (1). Forexample, Eq. (1) was reduced to a standard Poisson equationand Helmholtz equation when the function f is a function ofposition vector X or a linear function of physical field u,respectively. When the function f is a nonlinear function ofphysical field u and its derivatives, it may represent a soapbubble problem or other complex problems. Therefore, Eq. (1)can represent most engineering problems by properly choosingfunction f.3. The analog equation method and virtual boundarycollocation method with RBFs approximation3.1. The analog equation methodThe boundary value problem defined by Eqs. (1) and (2a)–(2c) can be converted into a standard Poisson-type equationusing the analog equation method [5]. For this purpose,suppose u(X) is the sought solution to the BVP, which is acontinuously differentiable function with up to two orders in U.If the Laplacian operator is applied to this function, namelyV2 u Z bðXÞ2. Problem definitionV2 u Z f ðX;u;u;x ;u;y ;u;xx ;u;xy ;u;yy Þ in Uu Z u (3)then, Eq. (3) indicates that the solution of Eq. (1) can beestablished by solving the linear Eq. (3) under the sameboundary conditions (2a)–(2c), if the fictitious sourcedistribution b(X) is known. The solution procedure is detailedbelow.Firstly, the solution of Eq. (3) can be written as a sum of thehomogeneous solution uh and the particular solution up, that is:u Z uh C up(4)Accordingly, uh and up, respectively, satisfy2V up ðXÞ Z bðXÞ(5)
H. Wang, Q.-H. Qin / Engineering Analysis with Boundary Elements 30 (2006) 515–521and:8 2V uh Z 0 u Z uKu hp qh Z qKq p :b1 uh C b2 qh Z b3 Kb1 up Kb2 qpsolutions u j can be written asin Uon G1on G2u j Z(6)on G3The next step of the proposed approach is to evaluate theparticular solution by RBF approximation. To this end, theright-hand term of Eq. (5) can be approximated byMXaj fj ðXÞ(7)rj5rj2C425(12)for a 2D problem, andu j Z3.2. RBF approximation for the particular solution upbðXÞ Z517rj2 rj5C6 30(13)for a 3D problem. Since the inhomogeneous term b(X) is anunknown ‘body force’ depending on the unknown functionu(X), the coefficients aj cannot be determined directly throughsolving Eq. (7). However, this problem can be tackled in thefollowing way.3.3. VBCM for the homogeneous solutionjZ1where MZICB, I and B are the number of interpolation pointsinside the domain and on its boundary, respectively. aj arecoefficients to be determined and fj are a set of approximatingfunctions.Similarly, the particular solution up is also approximated inthe formup ðXÞ ZMXaj u j ðXÞ(8)jZ1where u j are a corresponding set of particular solutions.Correspondingly, boundary flux can be expressed as:MXvup ðXÞvu j ðXÞZKqp ðXÞ ZKajvnvnjZ1To obtain a weak solution of Laplace problem (6), Nfictitious source points Yi (iZ1,2,.,N) on the virtual boundaryand the same number collocation points on the physical one areselected, respectively (see Fig. 1). The virtual boundary isoutside the domain under consideration. Moreover, assume thatthere is a virtual source 4i (1%i%N) at each fictitious sourcepoint.According to the superposition principle, the potential uhand the boundary normal gradient at an arbitrary field point Xin the domain or on the boundary can be expressed by a linearcombination of fundamental solutions in terms of fictitioussource placed on the virtual boundary, respectively [7–11], thatis(9)NXu ðX;Yi Þ4i(14)Because the particular solution up satisfies Eq. (5), the key tothis approximation is the assumption of a corresponding set ofapproximating particular solutions fj, which, for the case ofLaplacian operator, satisfy:NXvuvu ðX;Yi Þqh ðXÞ ZK h ZK4ivnvniZ1(15)V2 u j ðXÞ Z fj ðXÞwhere u* is the fundamental solution of the Laplacian operatoruh ðXÞ Z(10)The effectiveness and accuracy of the interpolationdepends on the choice of the approximating functions fj.Global interpolation functions, such as Langrange polynomials, Fourier sine and cosine series, RBFs of polynomialtype and thin plate spline (TPS) may be used [11–17] forthis purpose. In this paper, the functions fj in Eq. (7) areselected to be locally polynomial RBFs in terms of powerseries of a distance function rj. Because even powers of rjare not RBF’s [18], and artificially created singularities canprobably be encountered in some cases [11], the local RBFscan be taken asfj ðXÞZ 1 C rj3iZ1u ðX;YÞ Z(16)for a 2D problem, andu ðX;YÞ Z14pr(17)for a 3D problem.oyooo oo o oo ooo(11)where r(X,Xj)Zrj(X)ZjXKXjj denotes the distance from thesource point Xj to the field point X. The functions in Eq.(11) have shown to be most convenient to implement intostandard computer program.Using Eqs. (10) and (11), the approximating particular11ln2p rðX;YÞVirtual boundaryoFictitious source pointsPhysicalboundary CollocationpointsxFig. 1. Illustration of a computational domain and point discretization on thephysical and virtual boundary.
518H. Wang, Q.-H. Qin / Engineering Analysis with Boundary Elements 30 (2006) 515–521λ a 2 b2λaλbgaCenterbequations is nearly zero [21]. The similarity ratio l is generallyselected to be in the range of 1.8–4.0 for internal problems and0.6–0.8 for external problems in practical computation [6,10].With regard to the condition number of solution matrix, itmay be influenced by the location and shape of virtualboundary as well as the number of fictitious source points. Ingeneral, coefficient matrix formed by using the virtualboundary collocation method has a large condition number.In this case, the standard Gauss elimination or LU decomposition solvers may be invalid and the singular valuedecomposition (SVD) is recommended to keep results stablefor ill-conditioning systems [20].3.4. The construction of solution systemBased on the procedure described above, the solution uZu(X) we are seeking to Eq. (3) can be obtained asFig. 2. Rectangular and circular virtual boundaries of a simple rectangulardomain.uðXÞ ZNXiZ1It should be mentioned that effectiveness of VBCM dependsstrongly on following three aspects:"NX4i u ðX;Yi Þ CMXaj u j ðXÞ(18)jZ1Mvu j ðXÞvu ðX;Yi Þ XC4iajvnvnjZ1#– shape of virtual boundary;– distance between the virtual and physical boundary;– investigation of conditioning number of solution matrix.q ZKFirst, any shape of the virtual boundaries can be,theoretically, used in the calculation. However, due to thelimitation of computer’s inherent precision, the shape of thevirtual boundary may influence the numerical accuracy ofthe output results. It is proved that circular virtual boundary[19] and similar virtual boundary [10] are suitable for VBCM.Based on these two schemes, for example, the shapes of avirtual boundary can be chosen as either rectangle or circle fora rectangular domain (see Fig. 2). The effectiveness of the twoschemes is assessed in Section 4.The distance between fictitious source points and physicalboundary has also an important effect on the accuracy of theVBCM. In order to determine the location of virtual boundary,a parameter l representing the ratio between characteristiclength of the virtual and physical boundary is introduced [6,10]which is defined by:which are also the solutions of Eq. (1). Differentiating Eq. (18)yieldsFrom the point of view of computation, accuracy of thenumerical results will become worse if the distance between ofthe virtual boundary and physical boundary is too close (i.e. thesimilarity ratio l is close to one), as that may cause problemsdue to singularity of the fundamental solutions. Conversely,round-off error in C/Fortran floating point arithmetic may be aserious problem when the source points are far from the realboundary. In that case, the coefficient matrix of the system of(20)NMXvu j ðXÞv2 uvu ðX;Yi Þ XZ4iCajvxm vxnvxm vxnvxm vxniZ1jZ1(21)where m,nZ1,2 for 2D plane problems and m,nZ1,2,3 for 3Dcases.Finally, in order to determine unknowns aj and 4i, Eq. (18)should satisfy the governing differential Eq. (1) at Minterpolation points. Besides, Eqs. (18) and (19) should satisfycorresponding boundary conditions (2a)–(2c) at N collocationq 0you 1characteristic length of virtual boundarycharacteristic length of physical boundary(19)NMXvu j ðXÞvuvu ðX;Yi Þ XZ4iCajvxmvxvxmmiZ1jZ1u 7/6lZiZ10.2q 0x1.0Fig. 3. Geometry of the rectangular domain and specified boundary conditions.
H. Wang, Q.-H. Qin / Engineering Analysis with Boundary Elements 30 (2006) 515–521519Table 1Effect of the shape and location of the virtual boundary to numerical results and condition numberLocation, Parameter 4585.09.5!1091.14564.8!1071.1458Data in first row denote condition number and second row potential value.points on the physical boundary:problem, and a nonlinear Poisson problem are considered andtheir results are compared with the analytical results.8NMXX 2 4VuðX;YÞCaj V2 u j ðXÞ Z f ð4i ;aj Þ ii iZ1jZ1 NM XX 4uðX;YÞCaj u j ðXÞZ u ii iZ14.1. Example 1: 2D standard Poisson problemsjZ1NMXX 4i q ðX;Yi Þ Caj q j ðXÞ Z q iZ1jZ1 NM XX 4½buðX;YÞCbqðX;YÞ Caj ½b1 u j ðXÞ C b2 q j ðXÞ Z b3 i1i2i:iZ1jZ1(22)As a result, a system of equations in terms of the unknowncoefficients aj and fi can be written asFðxÞ Z 0(23)where the sought vector x contains the unknown coefficients ajand 4i.Once these unknown coefficients are determined, thepotential field u and its normal derivative q at any point Xinside the domain or on its boundary can be determined byusing Eqs. (18) and (19).As mentioned before, Eq. (23) may be either linear ornonlinear depending on the right-hand term f. For the linear ornonlinear system considered in this analysis, both the singularvalue decomposition (SVD) method and Newton–Raphsoniteration method appeared in [22] are employed in ourcalculation.Consider a classic Poisson problem whose domain is arectangle of size 1!0.2 (Fig. 3). The right-hand function isf(x,y)ZKx. The upper and lower boundaries are assumed to beinsulated, whereas Dirichlet boundary conditions are appliedon the remaining boundaries as shown in Fig. 3.The exact solution of this problem is:7 1u Z K x36 6In this example, the effect of the shape and location of thevirtual boundary is considered. To this end, 12 collocationpoints are selected on the physical boundary, which is the sameas the number of fictitious source points on the virtualboundary. Total 27 internal interpolation points are used,which is uniformly distributed in the domain. The numericalresults at point (0.5,0.1) and condition number of coefficientmatrix are shown in Table 1. The corresponding exact solutionis 1.1458.It can be seen from Table 1 that the use of circular virtualboundary can benefit to weaken the effect of the singularity offundamental solutions on the solution matrix and can reachgood accuracy when parameter l ranges from 2.5 to 5.0.0.74. Numerical experimentsIn order to demonstrate the efficiency and accuracy ofproposed method, three benchmark numerical examples of astandard Poisson equation, an inhomogeneous Helmholtzy1q 0Relative error %0.6N 12N 20N 360.50.40.30.20.1u 1ou 1 sin1q 01xFig. 4. Geometry of square domain and boundary conditions.000.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9x1Fig. 5. Relative error on potential between numerical results and exact solutionsalong the line yZ0.5 when the number of fictitious source points varies andMZ16.
520H. Wang, Q.-H. Qin / Engineering Analysis with Boundary Elements 30 (2006) 515–521ooooyo o oo ox4.3. Example 3: nonlinear Poisson problemsM 9 ooo proposed meshless method. Results of relative errors areshown in Fig. 5 for different number of fictitious source points,from which we can see that the relative error decreases as thedensity of fictitious source points increases.o Finally, consider a generalized nonlinear Poisson problemin the 1!1 rectangular domain whose governing equation isdefined by 2vuZ 2y C x4V2 uðx;yÞ CvyM 25with following boundary conditions:N 12Fig. 6. Demonstration of fict
This paper presents a boundary collocation method by combining virtual boundary collocation method [10] with RBF approximation and the analog equation method (AEM) [5] for analyzing generalized nonlinear Poisson-type pro-blems. Firstly, AEM is used to convert the original governing
the boundary element method (BEM) [1,2]. Generally, meshless (meshfree) methods can be divided to some types as follows: Those meshfree methods which are constructed as weak form like for example [3–12]: meshless local Petrov-Galerkin (MLPG), th
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The Generalized Finite Element Method (GFEM) presented in this paper combines and extends the best features of the finite element method with the help of meshless formulations based on the Partition of Unity Method. Although an input finite element mesh is used by the pro- . Probl
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