Nodal Integration Technique in Meshless Method

International Journal for Numerical Methods in Engineering, 2008

Nodal integration can be applied to the Galerkin weak form to yield a particle-type method where stress and material history are located exclusively at the nodes and can be employed when using meshless or finite element shape functions. This particle feature of nodal integration is desirable for large deformation settings because it avoids the remapping or advection of the state variables required in other methods. To a lesser degree, nodal integration can be desirable because it relies on fewer stress point evaluations than most other methods. In this work, aspects regarding stability, consistency, efficiency and explicit time integration are explored within the context of nodal integration. Both small and large deformation numerical examples are provided.

Nodal integration of finite elements has been investigated recently. Compared with full integration it shows better convergence when applied to incompressible media, allows easier remeshing and highly reduces the number of material evaluation points thus improving efficiency. Furthermore, understanding it may help to create new integration schemes in meshless methods as well.

Lecture Notes in Computational Science and Engineering, 2012

A novel approach is presented to correct the error from numerical integration in Galerkin methods for meeting linear exactness. This approach is based on a Ritz projection of the integration error that allows a modified Galerkin discretization of the original weak form to be established in terms of assumed strains. The solution obtained by this method is the correction of the original Galerkin discretization obtained by the inaccurate numerical integration scheme. The proposed method is applied to elastic problems solved by the reproducing kernel particle method (RKPM) with first-order correction of numerical integration. In particular, stabilized non-conforming nodal integration (SNNI) is corrected using modified ansatz functions that fulfill the linear integration constraint and therefore conforming sub-domains are not needed for linear exactness. Illustrative numerical examples are also presented.

Computers & Structures, 2014

An efficient technique is presented for evaluation of a domain integral in which the integrand is defined by its values at a discrete set of nodes with highly varying density. The proposed technique uses quadtree and octree techniques for 2D and 3D domains, respectively, so that the background of the integration domain can be divided into a few partitions with different grades of nodal density. The integrals over all partitions are then evaluated and added together to get the value of the whole-domain integral. Some numerical examples are given to show the accuracy and efficiency of the presented method.

International Journal for Numerical Methods in Engineering, 2015

Convergent and stable domain integration that is also computationally efficient remains a challenge for Galerkin meshfree methods. High order quadrature can achieve stability and optimal convergence, but it is prohibitively expensive for practical use. On the other hand, low order quadrature consumes much less CPU but can yield non-convergent, unstable solutions. In this work, an accelerated, convergent, and stable nodal integration is developed for the reproducing kernel particle method. A stabilization scheme for nodal integration is proposed based on implicit gradients of the strains at the nodes that offers a computational cost similar to direct nodal integration. The method is also formulated in a variationally consistent manner, so that optimal convergence is achieved. A significant efficiency enhancement over a comparable stable and convergent nodal integration scheme is demonstrated in a complexity analysis and in CPU time studies. A stability analysis is also given, and several examples are provided to demonstrate the effectiveness of the proposed method for both linear and nonlinear problems.

Boundary Elements and other Mesh Reduction Methods XLI, 2018

The Meshfree Method with reduced integration (ILMF) is derived through the work theorem of structures theory. In the formulation of the ILMF, the kinematically-admissible strain field is an arbitrary rigid-body displacement; as a consequence, the domain term is canceled out and the work theorem is reduced to regular local boundary terms only. The moving least squares (MLS) approximation of the elastic field is used to construct the trial function in this local meshfree formulation. ILMF has a high performance in problems with irregular nodal arrangement leading to accurate numerical results. This paper presents the size effect of the irregularity nodal arrangement parameter (cn) on three different nodal discretization to solve the Timoshenko cantilever beam using values fixed for the local support domain (αs) and the local quadrature domain (αq). Results obtained are optimal for 2D plane stress problems when compared with the exact solution.

Lecture Notes in Computational Science and Engineering, 2017

There are many application papers that solve elliptic boundary value problems by meshless methods, and they use various forms of generalized stiffness matrices that approximate derivatives of functions from values at scattered nodes x 1 ,. . ., x M ∈ Ω ⊂ R d. If u * is the true solution in some Sobolev space S allowing enough smoothness for the problem in question, and if the calculated approximate values at the nodes are denoted byũ 1 ,. .. ,ũ M , the canonical form of error bounds is max 1≤ j≤M |u * (x j) −ũ j | ≤ ε u * S where ε depends crucially on the problem and the discretization, but not on the solution. This contribution shows how to calculate such ε numerically and explicitly, for any sort of discretization of strong problems via nodal values, may the discretization use Moving Least Squares, unsymmetric or symmetric RBF collocation, or localized RBF or polynomial stencils. This allows users to compare different discretizations with respect to error bounds of the above form, without knowing exact solutions, and admitting all possible ways to set up generalized stiffness matrices. The error analysis is proven to be sharp under mild additional assumptions. As a byproduct, it allows to construct worst cases that push discretizations to their limits. All of this is illustrated by numerical examples. 1 INTRODUCTION 2 with N ≥ M, whatever the underlying PDE problem is, and the N × M matrix A with entries a k j can be called a generalized stiffness matrix. Users solve the system somehow and then get valuesũ 1 ,. . .,ũ M that satisfy M ∑ j=1 a k jũ j ≈ f k , 1 ≤ k ≤ N, but they should know how far these values are from the values u * (x j) of the true solution of the PDE problem that is supposed to exist. The main goal of this paper is to provide tools that allow users to assess the quality of their discretization, no matter how the problem was discretized or how the system was actually solved. The computer should tell the user whether the discretization is useful or not. It will turn out that this is possible, and at tolerable computational cost that is proportional to the complexity for setting up the system, not for solving it.

Engineering Analysis with Boundary Elements, 2017

This paper introduces a weak meshless procedure combined with a multi-resolution numerical integration and its comparison with a strong local meshless formulation for approximating displacement and strain modeled in the form of Elliptic Boundary Value Problems (EBVPs) in one-and two-dimensional spaces. Assets and losses of both strong and weak meshless approaches are considered in detail. The meshless weak formulation considered in the current paper is the well-known Element Free Galerkin (EFG) method whereas the Local Radial Basis Functions Collocation Method (LRBFCM) is taken as a strong formulation. First aspect of the current work is implementation of the new numerical integration techniques introduced in Siraj-ul-Islam et al. ( ) and Aziz et al. (2011) [1,2] in the EFG method and its comparison with numerical integration based on standard Gaussian quadrature, adaptive integration and stabilized nodal integration techniques used in the context of EFG and other allied weak meshless formulations. Second aspect of the current work is analysis of comparative performance of the localized versions of strong and weak meshless formulations. Standard numerical tests are conducted to validate performance of both the approaches.

International Journal for Numerical Methods in Engineering, 2000

Domain integration by Gauss quadrature in the Galerkin mesh-free methods adds considerable complexity to solution procedures. Direct nodal integration, on the other hand, leads to a numerical instability due to under integration and vanishing derivatives of shape functions at the nodes. A strain smoothing stabilization for nodal integration is proposed to eliminate spatial instability in nodal integration. For convergence, an integration constraint (IC) is introduced as a necessary condition for a linear exactness in the mesh-free Galerkin approximation. The gradient matrix of strain smoothing is shown to satisfy IC using a divergence theorem. No numerical control parameter is involved in the proposed strain smoothing stabilization. The numerical results show that the accuracy and convergent rates in the mesh-free method with a direct nodal integration are improved considerably by the proposed stabilized conforming nodal integration method. It is also demonstrated that the Gauss integration method fails to meet IC in mesh-free discretization. For this reason the proposed method provides even better accuracy than Gauss integration for Galerkin mesh-free method as presented in several numerical examples.

2021

In this paper, we firstly introduce a nodal-integration-based finite element method. The method allows the use of first-order tetrahedral elements without suffering from the volumetric locking problem. The most important advantage of tetrahedral meshes is that they can be automatically generated for complex geometries using existing reliable meshing tools. The method is then applied to 3 types of applications. The first application is a large displacement, large strains elastic-plastic simulation on a notched specimen. The second application is an elastic-plastic bending problem. And the last example concerns the numerical simulation of the thermo-mechanical problem. In all the cases, the solution given by the nodal-integration-based FEM is compared to more classical FEM results.