Topic
Meshfree methods
About: Meshfree methods is a research topic. Over the lifetime, 2216 publications have been published within this topic receiving 69596 citations.
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TL;DR: In this paper, a weaker weak formulation based on the edges of triangular cells obtained by a Delaunay triangulation is considered, framing the so-called edge-based smoothed domains.
15 citations
TL;DR: In this article, the Trefftz method with cylindrical basis functions was used to solve the Laplace equation with non-harmonic boundary conditions in complicated irregular domains in 3D.
Abstract: The method of fundamental solutions (MFS) and the Trefftz method are two powerful boundary meshless methods for solving boundary value problems governed by homogeneous partial differential equations. High accuracy can be obtained when we employ these two methods to solve equations with harmonic boundary conditions. However, dealing with equations with non-harmonic boundary conditions in irregular domains remains a challenge. Despite the long history of these two methods, each one has its disadvantages in numerical implementation. Recent advances in the Trefftz method using the multiple scale technique has made significant improvement in reducing the condition number. As a result, the Trefftz method has become more effective for solving challenging problems. Meanwhile, there has also been progress in selecting the source points in the MFS using the Leave-One-Out Cross Validation (LOOCV) method. In this paper, we propose a simple and yet effective approach to further improve the selection of source points of the MFS in 3D. Equipped with these new techniques, we compare these two methods for solving the Laplace equation with non-harmonic boundary conditions in complicated irregular domains in 3D. In this paper, we only consider the Trefftz method with cylindrical basis functions.
15 citations
TL;DR: This paper extends for the first time Schaback’s linear discretization theory to nonlinear operator equations, relying heavily on the methods in Bohmer's 2010 book, with results covering error bounds and convergence rates.
15 citations
TL;DR: It is proved that the convergence of meshfree collocation methods for the terminal value problems of fully nonlinear parabolic partial differential equations in the framework of viscosity solutions can be proved.
Abstract: We prove the convergence of meshfree collocation methods for the terminal value problems of fully nonlinear parabolic partial differential equations in the framework of viscosity solutions, provided that the basis function approximations of the terminal condition and the nonlinearities are successful at each time step. A numerical experiment with a radial basis function demonstrates the convergence property.
15 citations
Journal Article•
TL;DR: This work discusses rank instability resulting from the nodal integration of Galerkin type meshfree method as well as the ill-conditioning type instability in the radial basis collocation method, and demonstrates how meshfree methods can be applied to problems difficult to be modeled by the conventional finite element methods.
Abstract: Meshfree methods have been developed based on Galerkin type weak formulation and strong formulation with collocation. Galerkin type formulation in conjunction with the compactly supported approximation functions and polynomial reproducibility yields algebraic convergence, while strong form collocation method with nonlocal approximation such as radial basis functions offers exponential convergence. In this work, we discuss rank instability resulting from the nodal integration of Galerkin type meshfree method as well as the ill-conditioning type instability in the radial basis collocation method. We present the recent advances in resolving these difficulties in meshfree methods, and demonstrate how meshfree methods can be applied to problems difficult to be modeled by the conventional finite element methods due to their intrinsic regularity constraints.
15 citations