Meshfree methods

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

A greedy sparse meshless method for solving heat conduction problems

Abstract: In this paper, we introduce a greedy approximation algorithm for solving the transient heat conduction problem. This algorithm can overcome on some of challenges of the full meshless kernel-based methods such as ill-conditioning and computational cost associated with the dense linear systems that arise. In addition, the greedy algorithm allows to control the consistency error by explicit calculation. First, the space derivatives of the heat conduction equation are discretized to a finite number of test functional equations, and a greedy sparse discretization is applied for approximating the linear functionals. Each functional is stably approximated by some few trial points with an acceptable accuracy. Then a time-stepping method is employed for the time derivative. Stability of the scheme is also discussed. Finally, numerical results are presented in three test cases. These experiments show that greedy approximation approach is accurate and fast, and yields the better conditioning in contrast with the fully meshless methods.

A novel quadratic Edge-based Smoothed Conforming Point Interpolation Method (ES-CPIM) for elasticity problems

Abstract: his paper formulates an edge-based smoothed conforming point interpolation method (ES-CPIM) for solid mechanics using the triangular background cells. In the ES-CPIM, a technique for obtaining conforming PIM shape functions (CPIM) is used to create a continuous and piecewise quadratic displacement field over the whole problem domain. The smoothed strain field is then obtained through smoothing operation over each smoothing domain associated with edges of the triangular background cells. The generalized smoothed Galerkin weak form is then used to create the discretized system equations. Numerical studies have demonstrated that the ES-CPIM possesses the following good properties: (1) ES-CPIM creates conforming quadratic PIM shape functions, and can always pass the standard patch test; (2) ES-CPIM produces a quadratic displacement field without introducing any additional degrees of freedom; (3) The results of ES-CPIM are generally of very high accuracy.

An adaptive variational multiscale element free Galerkin method for convection–diffusion equations

Abstract: For very strong convection-dominated problems, stabilized meshless methods such as variational multiscale element-free Galerkin (VMEFG) method may still produce over- and under-shootings near the boundary or interior layers. In this paper, an adaptive VMEFG method is presented to solve convection–diffusion equations with convection-dominated. The adaptive algorithm based on background integration cell locates high gradient region with Zienkiewicz–Zhu indicator and refine the nodes in the region to improve the computational accuracy of VMEFG method. Meanwhile, this adaptive algorithm can also be used in element-free Galerkin (EFG) method. To compare and verify the validity of the proposed adaptive VMEFG method in convection-dominated problem, seven case studies are calculated by the adaptive VMEFG and EFG methods. The numerical experiments show that the proposed adaptive algorithm can not only refine the singularity regions well, but also is simple, effective and efficient for convection-dominated problem.

High-Order Non-Oscillatory Schemes Using Meshfree Interpolating Moving Least Squares Reconstruction for Hyperbolic Conservation Laws

Abstract: Meshfree methods have attracted much attention for the development and their applications in the recent years. The methods are commonly formulated using the Moving Least Squares (MLS) methods. The interpolation version of the methods is determined by introducing the singular weight functions for constructing the shape functions and called as Interpolating Moving Least Squares (IMLS) methods. Since the shape functions of the IMLS interpolants satisfy the Kronecker delta, the IMLS methods have the property of nodal interpolation. For more information of the IMLS method the explicit formulae of the derivatives of the IMLS interpolants are derived. The methods are applied to a linear scalar conservation law with the Euler and Lax-Wendroff time discretizations. The higher order schemes are presented employing a Taylor series expansion. The field variables and their successive derivatives are reconstructed using the IMLS methods. An analysis of the L_2-norm of this method is given. The Weighted Essentially Non-Oscillatory (WENO) schemes are adopted in the new schemes to prevent spurious oscillation. Our new methods based on staggered grids are discretized on space and the central Runge-Kutta schemes are used for time integration. Numerical results show that our new methods achieve the expected accuracy from an analysis of L_2-norm. Representative simulations show that the proposed methods are applicable to hyperbolic conservation laws.

Numerical solution of the space-time fractional diffusion equation: Alternatives to finite differences

Abstract: One of the ongoing issues with fractional-order diffusion models is the design of efficient numerical schemes for the space and time discretizations. Until now, most models have relied on a low-order finite difference (FD) method to discretize both the fractional-order space and time derivatives. Some numerical schemes using low-order finite elements (FE) have also been proposed. Both the FD and FE methods have long been used to solve integer-order partial differential equations. These low-order schemes generally require many grid points to obtain an accurate solution but that is offset by their local or piecewise nature. They lead to systems of linear equation defined by large sparse matrices that can be handled easily. However, fractional-order derivatives are non-local differential operators and the resulting matrices are full as the global behavior of the solution has to be taken into account. The computational efficiency of FD and FE schemes is thus severely impaired when going from an integer-order to a fractional-order equation. In this talk, we consider high-order, global numerical methods like the pseudo-spectral (PS) and radial basis functions (RBF) methods to discretize the space and time fractional derivatives. These methods appear to be a better choice as they naturally take the global behavior of the solution into account and use a limited number of degrees of freedom. Our PS scheme relies on an expansion of the model solution in terms of Chebyshev polynomials in both space and time. Our RBF scheme relies on the QR algorithm to analytically remove the ill-conditioning that appears when the number of nodes increases or when basis functions are made increasingly flat. We will present a flexible approach that allows the combination of a Chebyshev PS expansion in time with either FD, FE, PS or RBF discretizations in space. The proposed method can also accommodate an exponential tempering of the space and/or time fractional derivatives. A number of examples of numerical solutions of the space-time fractional diffusion equation are presented with various combinations of the time and space derivatives. The proposed numerical scheme is shown to be both efficient and flexible.