Mesh Optimization: Some Results in 3d Elasticity

Though the mathematical theory of classical linear elasticity is well established, there still lack some ingredients toward the numerical solution of real technological problems. In this paper we address one of these critical ingredients, namely the automatic construction of three-dimensional meshes in arbitrary geometries. Several methods exist for this purpose, but further improvements are still required to achieve the needed robustness and generality. We present and discuss the idea of mesh optimization, namely the manipulation of the mesh geometry and topology so as to maximize some suitable quality measure. The e®ects of mesh optimization in the ̄nite element solution of a linear thermoelastic problem are evaluated. Finally, we report on a recent method that couples mesh optimization with a posteriori error estimation ideas, so that mesh re ̄nement in regions of high stress gradients is achieved through optimization using a suitable space-varying metric. Numerical results for this last techniques are restricted to two dimensions, as a 3D implementation is under way.