A Method of Grid Optimization for Finite Element Methods*

The finite element method has become an important tool of engineering analysis and design. Much of the popularity of the method is due to the freedom that it allows in the construction of the discretized model. It has long been realized, however, that this freedom must be carefully exercised since the quality of the finite element solution greatly depends on how the discretization is performed. Several methods have been proposed to improve the discretized model in an iterative manner. One such method is presented in this paper. A natural way of improving the quality of finite element solutions is to increase the number of degrees of freedom. The process is normally performed after an initial solution is already available. Several schemes have been devised to introduce the new degrees of freedom in a selective manner in order to produce the greatest possible improvement of the previous solution. This calls for the definition of criteria to identify the regions of the domain where the finite element approximation is poorer. The new degrees of freedom are added in these regions by either increasing the order of polynomial approximation inside elements, the so-called p-method, or by subdivision of elements, or h-method. The process is continued until a specified accuracy is achieved. The quality of the finite element solution may be improved also by optimizing the disposition of the nodes. Analysts often rely on their experience to construct grids that make an efficient use of the available degrees of freedom. It is also possible to improve the quality of existing meshes iteratively using predefined guidelines for the redistribution of the nodes. This paper considers the development of such guidelines.