This paper deals with the development of h-version adaptive mesh refinement and recovery strategy using variable-node elements and its application to various engineering field problems with 2D quadrilateral and 3D hexahedral models.
The finite element method (FEM) is widely used for solving partial differential equations over complex domains. The quality and accuracy of the numerical results of FEM depend much upon the discretization of the domain and the type of elements. A relatively finer mesh is necessary in the areas of higher gradients of the variables and a rather coarser mesh where the gradient distribution is relatively uniform. It is generally true that the initial mesh is not optimal for these well-graded meshes. The adaptive scheme generates a new mesh based on the solution obtained form an earlier mesh. The h-refinement is one of the most commonly used adaptive strategies that reduce the element sizes to create a finer mesh where the initial finite element model is not adequate for the prescribed error tolerance.
The quadrilateral and hexahedral variable-node elements are best utilized in the adaptive h-refinement strategy by bisecting the element to be refined into four or eight subdivided elements. The variable-node elements that have variable mid-side nodes on edges or faces are effectively used in overcoming some problems in connecting the different layer patterns the transition zone between the refined and coarse meshes. The use of elements with variable-nodes eliminates the necessity of imposing constraints on irregular (hanging) nodes in order to enforce the inter-element compatibility. For an economical analysis of transient problems, in which the locations where the mesh refinements are needed change time to time not only the mesh refinement but also the mesh recovery is needed. In the region in which the error is greater than the permissible refinement error, the mesh is locally refined by subdivision. Reversely, in some parts of the domain having the error smaller than the permissible recovery error, the mesh is locally recovered (coarsened) by combination.
In this paper, a systematic arrangement of the quadrilateral and hexahedral variable-node element formulation and adaptive h-refinement strategy is presented. Especially, a modified recovery technique of gradients adequate for quadrilateral and hexahedral variable-node elements and proper selection of error norms for each engineering field problems are proposed. One-dimensional array of storage for hierarchical structures meaning refinement history (e.g. quadtrees and octrees) is proposed to guarantee the recovery process. Element-based storage structures that describe neighbor relation and inherent relation is also proposed to improve the effectiveness of the refinement/recovery scheme.
The h-adaptive refinement/recovery strategy that is presented in this paper is applied to the axisymmetric structural analysis, 3D heat conduction analysis and 3D flow analysis using variable-node elements newly developed for each analyses. In order to examine the performance of this adaptive algorithm, some practical numerical examples are carried out with proper methods for recovery of gradients, error norm and refinement error and the results show the good performance of quadrilateral and hexahedral variable-node elements and the effectiveness and validity of h-adaptive refinement/recovery scheme.
