The Localized Method of Fundamental Solution for Two Dimensional Signorini Problems

In this work,the localized method of fundamental solution(LMFS)is extended to Signorini problem.Unlike the traditional fundamental solution(MFS),the LMFS approximates the field quantity at each node by using the field quantities at the adjacent nodes.The idea of the LMFS is similar to the localized domain type method.The fictitious boundary nodes are proposed to impose the boundary condition and governing equations at each node to formulate a sparse matrix.The inequality boundary condition of Signorini problem is solved indirectly by introducing nonlinear complementarity problem function(NCP-function).Numerical examples are carried out to validate the reliability and effectiveness of the LMFS in solving Signorini problems.

Electroelastic Analysis of Two-Dimensional Piezoelectric Structures by the Localized Method of Fundamental Solutions

Accurate and efficient analysis of the coupled electroelastic behavior of piezoelectric structures is a challenging task in the community of computational mechanics.During the past few decades,the method of fundamental solutions(MFS)has emerged as a popular and well-established meshless boundary collocation method for the numerical solution of many engineering applications.The classical MFS formulation,however,leads to dense and non-symmetric coefficient matrices which will be computationally expensive for large-scale engineering simulations.In this paper,a localized version of the MFS(LMFS)is devised for electroelastic analysis of twodimensional(2D)piezoelectric structures.In the LMFS,the entire computational domain is divided into a set of overlapping small sub-domains where the MFS-based approximation and the moving least square(MLS)technique are employed.Different to the classical MFS,the LMFS will produce banded and sparse coefficient matrices which makes the method very attractive for large-scale simulations.Preliminary numerical experiments illustrate that the present LMFM is very promising for coupled electroelastic analysis of piezoelectric materials.

Domain-Decomposition Localized Method of Fundamental Solutions for Large-Scale Heat Conduction in Anisotropic Layered Materials

The localized method of fundamental solutions(LMFS)is a relatively new meshless boundary collocation method.In the LMFS,the global MFS approxima-tion which is expensive to evaluate is replaced by local MFS formulation defined in a set of overlapping subdomains.The LMFS algorithm therefore converts differential equations into sparse rather than dense matrices which are much cheaper to calcu-late.This paper makes thefirst attempt to apply the LMFS,in conjunction with a domain-decomposition technique,for the numerical solution of steady-state heat con-duction problems in two-dimensional(2D)anisotropic layered materials.Here,the layered material is decomposed into several subdomains along the layer-layer inter-faces,and in each of the subdomains,the solution is approximated by using the LMFS expansion.On the subdomain interface,compatibility of temperatures and heatfluxes are imposed.Preliminary numerical experiments illustrate that the proposed domain-decomposition LMFS algorithm is accurate,stable and computationally efficient for the numerical solution of large-scale multi-layered materials.

Localized Method of Fundamental Solutions for Acoustic Analysis Inside a Car Cavity with Sound-Absorbing Material

This paper documents the first attempt to apply a localized method of fundamental solutions(LMFS)to the acoustic analysis of car cavity containing soundabsorbing materials.The LMFS is a recently developed meshless approach with the merits of being mathematically simple,numerically accurate,and requiring less computer time and storage.Compared with the traditional method of fundamental solutions(MFS)with a full interpolation matrix,the LMFS can obtain a sparse banded linear algebraic system,and can circumvent the perplexing issue of fictitious boundary encountered in the MFS for complex solution domains.In the LMFS,only circular or spherical fictitious boundary is involved.Based on these advantages,the method can be regarded as a competitive alternative to the standard method,especially for high-dimensional and large-scale problems.Three benchmark numerical examples are provided to verify the effectiveness and performance of the present method for the solution of car cavity acoustic problems with impedance conditions.

Localized space-tim e method of fundamental solutions for three-dimensional transient diffusion problem

A localized space-time method of fundamental solutions(LSTMFS)is extended for solving three-dimensional transient diffusion problems in this paper.The interval segmentation in temporal direction is developed for the accurate simulation of long-time-dependent diffusion problems.In the LSTMFS,the whole space-time domain with nodes arranged i divided into a series of overlapping subdomains with a simple geometry.In each subdomain,the conventional method of fundamental solutions is utilized and the coefficients associated with the considered domain can be explicitly determined.By calculating a combined sparse matrix system,the value at any node inside the space-time domain can be obtained.Numerical experi-ments demonstrate that high accuracy and efficiency can be simultaneously achieved via the LSTMFS,even for the problems defined on a long-time and quite complex computational domain.

Localized MFS for three‐dimensional acoustic inverse problems on complicated domains

This paper proposes a semi‐analytical and local meshless collocation method,the loca-lized method of fundamental solutions(LMFS),to address three‐dimensional(3D)acoustic inverse problems in complex domains.The proposed approach is a recently developed numerical scheme with the potential of being mathematically simple,nu-merically accurate,and requiring less computational time and storage.In LMFS,an overdetermined sparse linear system is constructed by using the known data at the nodes on the accessible boundary and by making the remaining nodes satisfy the governing equation.In the numerical procedure,the pseudoinverse of a matrix is solved via the truncated singular value decomposition,and thus the regularization techniques are not needed in solving the resulting linear system with a well‐conditioned matrix.Numerical experiments,involving complicated geometry and the high noise level,confirm the ef-fectiveness and performance of the LMFS for solving 3D acoustic inverse problems.

Local MFS Matrix Decomposition Algorithms for Elliptic BVPs in Annuli

We apply the local method of fundamental solutions(LMFS)to boundary value problems(BVPs)for the Laplace and homogeneous biharmonic equations in annuli.By appropriately choosing the collocation points,the LMFS discretization yields sparse block circulant system matrices.As a result,matrix decomposition algorithms(MDAs)and fast Fourier transforms(FFTs)can be used for the solution of the systems resulting in considerable savings in both computational time and storage requirements.The accuracy of the method and its ability to solve large scale problems are demonstrated by applying it to several numerical experiments.