Boundary knot method for Laplace and biharmonic problems

Summary The boundary knot method (BKM) [1] is a meshless boundary-type radial basis function (RBF) collocation scheme, where the nonsingular general solution is used instead of fundamental solution to evaluate the homogeneous solution, while the dual reciprocity method (DRM) is employed to approximation of particular solution. Despite the fact that there are not nonsingular RBF general solutions available for Laplace and biharmonic problems, this study shows that the method can be successfully applied to these problems. The high-order general and fundamental solutions of Burger and Winkler equations are also first presented here. Introduction: As a boundary-type RBF scheme, the method of fundamental solution (MFS), also known as regular boundary elements, attains refresh attentions in recent years [2]. Because of the use of singular fundamental solution, the MFS requires a controversial fictitious boundary outside physical domain, which effectively blocks its practical use for complex geometry problems. Chen and Tanaka [1] recently developed a boundary knot method (BKM), where the perplexing artificial boundary is eliminated via the nonsingular general solution. Just like the MFS and dual reciprocity BEM (DR-BEM) [3], the BKM also applies the DRM to approximate the particular solution. The method is symmetric, spectral convergence, integration-free, meshfree and easy to learn and implement, and successfully applied to Helmholtz, convectiondiffusion, and Winkler plate problems. Unfortunately, the nonsingular RBF general solutions of Laplace and biharmonic operators are a constant rather than the RBF. Based on some physical investigations, this paper presented a few simple strategies to apply the BKM to these problems without losing its merits. BKM for Laplacian: For a complete description of the BKM see ref. 2. Here we begin with a Laplace problem ( ) x f u = ∇ 2 , Ω ∈ x , (1) ( ) ( ) x R x u = , u S x ⊂ , ( ) ( ) x N n x u = ∂ ∂ , T S x ⊂ , (2a,b) where x means multi-dimensional independent variable, and n is the unit outward normal. The governing equation (1) can be restated as ( ) u x f u u δ δ + = + ∇ 2 2 or ( ) u x f u u δ δ − = − ∇ 2 2 , (3a,b) where δ is an artificial parameter. Eqs. (3a,b) are respectively Helmholtz and diffusion-reaction equations. Their zero and high order general solutions [3] are ( ) ( ) ( ) r J r Q r u m n m n m m γ γ + − + + − = 1 2 1 2 # , and ( ) ( ) ( ) r I r Q r u m n m n m m τ τ + − + + − = 1 2 1 2 # , n≥2, (4) where n is the dimension of the problem; Qm=Qm-1/(2*m*γ), Q0=1; m denotes the order of general solution; J and I represent the Bessel and modified Bessel function of the first kind. The solution of the problem can be split as the homogeneous and particular solutions p h u u u + = , (5) The latter satisfies the governing equation but not boundary conditions. To evaluate the particular solution, the inhomogeneous term is approximated by ( ) ( ) j L N