Efficient Adaptive Finite Element Mesh Generation for Dynamics

Abstract

The finite element method has become the most widely used method of structural analysis and recently, the method has often been applied to complex dynamic and nonlinear structural analyses problems. Even for these complex problems, where the responses are hard to predict, finite element analyses yield reliable results if appropriate element types and meshes are used. However, the dynamic and nonlinear behaviors of a structure often include large deformations in various portions of the structure and if the same mesh is used throughout the analysis, some elements may deform to shapes beyond the reliable limits; thus dynamically adapting finite element meshes are needed in order for the finite element analyses to be accurate. In addition, to satisfy the users requirement of quick real run time of finite element programs, the algorithms must be computationally efficient. This paper presents an adaptive finite element mesh generation scheme for dynamic analyses of structures that may adapt at each time step. Representative strain values are used for error estimates and combinations of the h-method(node movement) and the r-method(element division) are used for mesh refinements. A coefficient that depends on the shape of an element is used to limit overly distorted elements. A simple frame example shows the accuracy and computational efficiency of the scheme. The aim of the study is to outline the adaptive scheme and to demonstrate the potential use in general finite element analyses of dynamic and nonlinear structural problems commonly encountered.

An Adaptive Mesh Generation Scheme for the Finite Element Method

“…(8), the representative error of element i is weighted by the element's area with respect to the total area and thus the relative order of errors for all elements are computed to identify elements to be adaptively changed. Previous studies have shown that this is computationally very efficient for achieving this (Yoon, 2012;Yoon, 2013).…”

“…(6) is only used to define the error and various attempts are generally devoted to estimating errors that resemble Eq.(6). The error defined below are based on the representative strains of element based on the standard deviations of the strains at the Gauss points in the element computed during the previous finite element analysis (Jeong and Yoon, 2003;Yoon and Park, 2010). The rationale for this is that as solution approaches the exact solution, the strains within an element must approach a constant value and thus the standard deviations must approach zero at the Gauss points.…”

“…The dynamic analysis considered is in time domain, based on direct integration. The results are based on the basic procedure outlined in reference Yoon (2013) but the data presented are from diverse refinements on the software and tuning of parameters more clearly described in this paper. A case study using four node quadrilateral isoparametric elements for plane stain is considered.…”

“…The mesh generating scheme combines the r-method (moving an existing node) and the h-method (dividing an element into smaller elements) using a dispersion parameter (de Las Casa, 1988;Yoon, 2005). The scheme includes a check for limiting distortion for each element using shape factor that is easily computed for a particular shape (Yoon and Park, 2010). The dynamic analysis considered is in time domain, based on direct integration.…”

Adaptive Finite Element Mesh Generation for Dynamic Planar Problems

“…Thus, the finite element mesh for these types of analyses must be dynamically adaptive, and considering the rapid process of analysis in real time, the dynamically adaptive finite element mesh generating schemes must be computationally efficient. When the method is applied to dynamics analyzed in the time domain, the meshes may need to be modified at each time step as the finite element results of an analysis largely depend on the mesh and the element types used (Heesom and Mahdjoubi, 2001;Jang and Lee, 2011;Yoon and Park, 2010).…”