Meshfree methods

About: Meshfree methods is a research topic. Over the lifetime, 2216 publications have been published within this topic receiving 69596 citations.

A global RBF-QR collocation technique for solving two-dimensional elliptic problems involving arbitrary interface

Abstract: In recent years, several numerical methods have been developed for differential equations with discontinuous coefficients, which are known as interface problems. Our aim is to introduce an efficient global meshless method for solving elliptic problems with arbitrary interface. Meshless methods based on infinitely smooth radial basis functions (RBFs) can achieve spectral convergence rate for solving partial differential equations. RBF-QR is a newly developed method which allows numerically stable computations with RBFs for all values of the free shape parameter $$\varepsilon$$ . In this paper, an appropriate RBF-QR collocation technique is applied for two-dimensional elliptic interface problems, including Poisson, Helmholtz and elasticity problems. The Helmholtz interface problem is studied, by taking into account the fact that the solutions of such problems might be oscillatory. The elasticity interface problem is discussed as an application of elliptic interface problem in vector form. Several problems involving complex interface geometries and discontinuous variable coefficients are presented to confirm the efficiency and stability of RBF-QR method for this class of problems in the case of arbitrary small values of $$\varepsilon$$ .

Thermomechanical compliant actuator design using meshless topology optimization

Abstract: A new topology optimization method for designing thermomechanical compliant actuators using meshless methods is presented. The meshless method is employed to discretize the design domain and the bulk density field, which can avoid the undesirable mesh distortion when the standard finite element method (FEM) is applied. The method for imposing boundary conditions in the meshless method is discussed in details. An interpolation scheme is applied to indicate the dependence of material mechanical and thermal properties on pseudo-densities which are distributed to the related integration points. Furthermore, the design sensitivity analysis is performed by incorporating the adjoint approach into the meshless method. The optimization problem is then solved by the method of moving asymptotes (MMA). The availability of the proposed method is demonstrated by using a widely studied compliant structure.

Parallel meshfree computation for parabolic equations on graphics hardware

Abstract: The objective of this study is to develop a parallel algorithm of the meshfree method for solving parabolic equations. We assume that parallel computation is performed using graphics hardware, and the algorithm's design depends on the architecture of the device. The meshfree method employed in this study is the modified version of the radial point interpolation method, and the matrix assembly process in the algorithm is divided into smaller subprocesses that can be computed independently in parallel.