Meshless local radial point interpolation to three-dimensional wave equation with Neumann's boundary conditions

In this article, the meshless local radial point interpolation (MLRPI) method is applied to simulate three-dimensional wave equation subject to given appropriate initial and Neumann's boundary conditions. The main drawback of methods in fully 3-D problems is the large computational costs. In the MLRPI method, all integrations are carried out locally over small quadrature domains of regular shapes such as a cube or a sphere. The point interpolation method with the help of radial basis functions is proposed to form shape functions in the frame of MLRPI. The local weak formulation using Heaviside step function converts the set of governing equations into local integral equations on local subdomains where Neumann's boundary condition is imposed naturally. A two-step time discretization technique with the help of the Crank-Nicolson technique is employed to approximate the time derivatives. Convergence studies in the numerical example show that the MLRPI method possesses reliable rates of convergence.     

基于无网格径向点插值方法的简谐激励下的连续体结构拓扑优化

针对用有限元法进行连续体结构拓扑优化时需不断重构网格来处理网格畸变和网格移动,且存在数值计算不稳定等问题,基于无网格径向点插值方法(Radial Point Interpolation Method,RPIM)对简谐激励下的连续体结构进行拓扑优化.选取节点的相对密度作为设计变量,以结构动柔度最小化为目标函数,基于带惩罚的各向同性固体微结构(Solid Isotropic Microstructure with Penalization,SIMP)模型建立简谐激励下的优化模型;采用伴随法求解得到目标函数的敏度分析公式;利用优化准则法求解优化模型.经典的二维连续体结构拓扑优化算例证明该方法的可行性和有效性.     

An element-free smoothed radial point interpolation method (EFS-RPIM) for 2D and 3D solid mechanics problems

This paper presents a novel element-free smoothed radial point interpolation method (EFS-RPIM) for solving 2D and 3D solid mechanics problems. The idea of the present technique is that field nodes and smoothing cells (SCs) used for smoothing operations are created independently and without using a background grid, which saves tedious mesh generation efforts and makes the pre-process more flexible. In the formulation, we use the generalized smoothed Galerkin (GS-Galerkin) weak-form that requires only discrete values of shape functions that can be created using the RPIM. By varying the amount of nodes and SCs as well as their ratio, the accuracy can be improved and upper bound or lower bound solutions can be obtained by design. The SCs can be of regular or irregular polygons. In this work we tested triangular, quadrangle, $n$-sided polygon and tetrahedron as examples. Stability condition is examined and some criteria are found to avoid the presence of spurious zero-energy modes. This paper is the first time to create GS-Galerkin weak-form models without using a background mesh that tied with nodes, and hence the EFS-RPIM is a true meshfree approach. The proposed EFS-RPIM is so far the only technique that can offer both upper and lower bound solutions. Numerical results show that the EFS-RPIM gives accurate results and desirable convergence rate when comparing with the standard finite element method (FEM) and the cell-based smoothed FEM (CS-FEM).     

Application of the dual reciprocity boundary integral equation technique to solve the nonlinear Klein-Gordon equation

This paper aims to obtain approximate solutions of the Nonlinear Klein-Gordon (NLKG) equation by employing the Boundary Integral Equation (BIE) method and the Dual Reciprocity Boundary Element Method (DRBEM). This method is improved by using a predictor-corrector scheme to the nonlinearity which appears in the problem. We employ the time stepping scheme to approximate the time derivative, and the Linear Radial Basis Functions (LRBFs), are used in the Dual Reciprocity (DR) technique. To confirm the accuracy of the new approach, the numerical results of a Double-Soliton and a problem with inhomogeneous terms are compared with analytical solutions and for the examples possessing single and periodic waves, two conserved quantities associated to the (NLKG) equation, the energy and the momentum are investigated.