Meshfree methods

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

Enhanced Meshfree Methods for Numerical Solution of Local and Nonlocal Theories of Solid Mechanics

Abstract: Author(s): Pasetto, Marco | Advisor(s): Chen, Jiun-Shyan | Abstract: Achieving good accuracy while keeping a low computational cost in numerical simulationsof problems involving large deformations, material fragmentation and crack propagations stillremains a challenge in computational mechanics. For these classes of problems, meshfreediscretizations of local and nonlocal approaches, have been shown to be effective as they avoidsome of the common issues associated with mesh-based techniques, such as the need for re-meshing due to excessive mesh distortion. Nonetheless, other issues remain.In the framework of local mechanics, the semi-Lagrangian reproducing kernel particlemethod (RKPM) has been proved to be particularly effectively for material damage and frag-mentation, as by reconstructing the field approximations in the current configuration it doesnot require the deformation gradient to be positive definite. This, however, results in a highcomputational cost.Furthermore, for crack propagation problems, the use of classical local mechanics presentsmany challenges, such as the need of accurately representing the singular stress field at crack tips. The peridynamic nonlocal theory circumvents these issues by reformulating solid mechanics in terms of integral equations. In engineering applications, a simple node-based discretization of peridynamics is typically employed. This approach is limited to first order convergence and often lacks the symmetry of interaction of the continuous form. The latter can be recovered through the use of the peridynamic weak form, which however involves costly double integration.First, we first propose, in the context of local mechanics, a blending-based spatial couplingscheme to transition from the computationally cheaper Lagrangian RKPM to the semi-Lagrangian RKPM. Next, we introduce an RK approximation to the field variables in strong form peridynam-ics to increase the order of convergence of peridynamic numerical solutions. Then, we develop an efficient n-th order symmetrical variationally consistent nodal integration scheme for RK enhanced weak form peridynamics.Lastly, we propose a Waveform Relaxation Newmark algorithm for time integration ofthe semi-discrete systems arising from meshfree discretizations of local and nonlocal dynamicsproblems. This scheme retains the unconditional stability of the implicit Newmark scheme withthe advantage of the lower computational cost of explicit time integration schemes.Numerical examples demonstrate the effectiveness of the proposed approaches.

Independent cover meshless method using a polynomial approximation

Abstract: In this paper, a novel independent cover meshless method (ICMM) is presented for the analyses of two dimensional linear elastic solids and the simulation of crack propagation. In the ICMM, the Delaunay criterion is employed to dynamically construct the non-overlapping independent nodal cover for each discrete node of analysis domain, and the virtual crack closure technique is used to calculate the stress intensity factors. The ICMM has the definite nodal cover (or nodal influence domain) and employs the general polynomial function to interpolate the nodal cover, which results in a simple formulation and numerical implementation, and facilitates the accurate numerical integration of the equilibrium equation and the enforcement of the boundary conditions. It well solves the problem of the general meshless methods interpolated by rational functions, and has a wide application prospective in practical engineering. Several representative numerical examples demonstrate the convergence, accuracy and robustness of the present method.

Pressure‐stabilized maximum‐entropy methods for incompressible Stokes

Abstract: We present a parameter-free stable maximum-entropy method for incompressible Stokes flow. Derived from a least-biased optimization inspired by information theory, the meshfree maximum-entropy method appears as an interesting alternative to classical approximation schemes like the finite element method. Especially compared to other meshfree methods, e.g. the moving least-squares method, it allows for a straightforward imposition of boundary conditions. However, no Eulerian approach has yet been presented for real incompressible flow, encountering the convective and pressure instabilities. In this paper, we exclusively address the pressure instabilities caused by the mixed velocity-pressure formulation of incompressible Stokes flow. In a preparatory discussion, existing stable and stabilized methods are investigated and evaluated. This is used to develop different approaches towards a stable maximum-entropy formulation. We show results for two analytical tests, including a presentation of the convergence behaviour. As a typical benchmark problem, results are also shown for the leaky lid-driven cavity. Together with the information-flux method of [1] for convection-dominated problems, we see this as the last step towards a maximum-entropy method capable of simulating full incompressible flow problems.

Modeling groundwater flow by the element-free Galerkin (EFG) method

Abstract: The element-free Galerkin (EFG) method is one of meshless methods, which is a very powerful, efficient and accurate method of modeling problems of fluid or solid mechanics with complex boundary shapes and large changes in boundary conditions. This paper reports the theory and the first-known application of the EFG method to groundwater flow modeling. The EFG method constructs shape functions based on moving least square (MLS) approximations, which do not require any element but only a set of nodes. Thus, the EFG method eliminates time-consuming mesh generation procedure with irregular shaped boundaries. The coupled EFG-FEM technique was used to treat Dirichlet boundary conditions. A computer code EFGGW was developed for the problems of steady-state and transient groundwater flow in homogeneous or heterogeneous aquifers. Solutions by the EFG method were similar in accuracy to that by the FEM. The main advantages of the method are the convenience of node generation and the enforced implementation of boundary conditions.

Kernel-Based Discretization for Solving Matrix-Valued PDEs

Abstract: In this paper, we discuss the numerical solution of certain matrix-valued partial differential equations. Such PDEs arise, for example, when constructing a Riemannian contraction metric for a dynamical system given by an autonomous ODE. We develop and analyse a new meshfree discretisation scheme using kernel-based approximation spaces. However, since these pproximation spaces have now to be matrix-valued, the kernels we need to use are fourth order tensors. We will review and extend recent results on even more general reproducing kernel Hilbert spaces. We will then apply this general theory to solve a matrix-valued PDE and derive error estimates for the approximate solution. The paper ends with applications to typical examples from dynamical systems