MESH OPTIMIZATION: SOME RESULTS IN 3D ELASTICITY
Gustavo C. Busgalia, Enzo A. Dari, and Pablo D. Zavattieri
Gustavo C. Busgalia, Enzo A. Dari, and Pablo D. Zavattieri
Esami-proc
Abstract.
Though the mathematical theory of classical linear elastic, ity 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 effects of mesh optimization in the finite 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 refinement 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.
The mathematical theory of linear elasticity being well established, it is the geometry of the solution domain that still makes it dificult to obtain an accurate approximation of the exact solution in real-life problems. Thin, walled structural components resist straightforward finite element treatment because of mesh degeneration and locking, and plate/shell theory is certainly the adequate mathematical and numerical tool to avoid this dificulty (the reader is referred to other lectures of this conference for state-of-the-art developments in plate/shell theory). There remain, of course, many problems in linear elasticity with domains that are truly three-dimensional. In these cases the bottleneck in the analysis is the construction of a three-dimensional mesh fitting in the domain under consideration. Much research e ort has been devoted, during the last years, to the development of effective algorithms for the generation of grids in general 3D domains. For this process to be automatic, the current choice is that of unstructured meshes of tetrahedra. Significant progress has been achieved concerning the robustness and exibility of unstructured algorithms (frontal methods, Delaunay-based methods, octree-based methods, and variants of them). As a consequence, it has been possible to mesh complex domains, allowing for massive application of finite element or finite volume solvers to industrial problems.
Among the several dificulties that still remain to be solved, this presentation concerns that associated with the geometrical quality of the resulting meshes. Badly distorted elements and over- or under-refined regions are not unusual in current 3D meshes. A complete, automatic control on the mesh quality is a primary goal in the present state of the field.
The application of optimization techniques during the generation procedure has proved useful for this purpose. This can be done during both the element-creation step [1, 2, 3, 4] and/or the a posteriori mesh improving stage [5,6,7,8,9,10,11,12,13]. In Section 2 we include the description of a complete quality-based mesh improving method, addressing the optimization of nodal positions and of the connectivity structure of the mesh. In particular, the node-repositioning technique consists of a non-differentiable optimization algorithm over the space of nodal positions. The connectivity changes are based on local cluster reconnection, a technique that can be seen as the extension of diagonal-swapping but involves significant complexity in 3D[3,14,15]. In Section we address the question of how much does the quality of the mesh affect 3D solid mechanic calculations. For this purpose, we first solve a simple academic problem with analytical solution, which is an oversimplified version of a more interesting technological problem. The knowledge of the exact solution allows us to evaluate the numerical errors and look at the effect of optimizing the mesh. In Section 4 we report some sample calculations concerning the stress analysis of a nuclear fuel pellet. A linear thermoelastic model is used, and the geometries we treat are pellets with different crack sizes. No exact solution is available, but a consistent 20-30% reduction in the number of conjugate-gradient iterations evidences a better conditioning of the linear system upon optimization.
Finally, we discuss the coupling of our mesh optimization procedure with adaptivity. A mesh quality based upon a solution-adapted variable metric is proposed that automatically leads to suitable refinement and stretching. The adaptive method that results [16] is a robust, optimization-based, variant of the ideas of Peraire, Morgan and Peiro [1,2,17], further developed recently by Dompierre et al [18]. Though the ideas are easily extendable to three-dimensional problems, technical issues arise when dealing with boundaries in 3D that have delayed the implementation, so that examples are presented in two space dimensions.
1. Introduction
The mathematical theory of linear elasticity being well established, it is the geometry of the solution domain that still makes it dificult to obtain an accurate approximation of the exact solution in real-life problems. Thin, walled structural components resist straightforward finite element treatment because of mesh degeneration and locking, and plate/shell theory is certainly the adequate mathematical and numerical tool to avoid this dificulty (the reader is referred to other lectures of this conference for state-of-the-art developments in plate/shell theory). There remain, of course, many problems in linear elasticity with domains that are truly three-dimensional. In these cases the bottleneck in the analysis is the construction of a three-dimensional mesh fitting in the domain under consideration. Much research e ort has been devoted, during the last years, to the development of effective algorithms for the generation of grids in general 3D domains. For this process to be automatic, the current choice is that of unstructured meshes of tetrahedra. Significant progress has been achieved concerning the robustness and exibility of unstructured algorithms (frontal methods, Delaunay-based methods, octree-based methods, and variants of them). As a consequence, it has been possible to mesh complex domains, allowing for massive application of finite element or finite volume solvers to industrial problems.
Among the several dificulties that still remain to be solved, this presentation concerns that associated with the geometrical quality of the resulting meshes. Badly distorted elements and over- or under-refined regions are not unusual in current 3D meshes. A complete, automatic control on the mesh quality is a primary goal in the present state of the field.
The application of optimization techniques during the generation procedure has proved useful for this purpose. This can be done during both the element-creation step [1, 2, 3, 4] and/or the a posteriori mesh improving stage [5,6,7,8,9,10,11,12,13]. In Section 2 we include the description of a complete quality-based mesh improving method, addressing the optimization of nodal positions and of the connectivity structure of the mesh. In particular, the node-repositioning technique consists of a non-differentiable optimization algorithm over the space of nodal positions. The connectivity changes are based on local cluster reconnection, a technique that can be seen as the extension of diagonal-swapping but involves significant complexity in 3D[3,14,15]. In Section we address the question of how much does the quality of the mesh affect 3D solid mechanic calculations. For this purpose, we first solve a simple academic problem with analytical solution, which is an oversimplified version of a more interesting technological problem. The knowledge of the exact solution allows us to evaluate the numerical errors and look at the effect of optimizing the mesh. In Section 4 we report some sample calculations concerning the stress analysis of a nuclear fuel pellet. A linear thermoelastic model is used, and the geometries we treat are pellets with different crack sizes. No exact solution is available, but a consistent 20-30% reduction in the number of conjugate-gradient iterations evidences a better conditioning of the linear system upon optimization.
Finally, we discuss the coupling of our mesh optimization procedure with adaptivity. A mesh quality based upon a solution-adapted variable metric is proposed that automatically leads to suitable refinement and stretching. The adaptive method that results [16] is a robust, optimization-based, variant of the ideas of Peraire, Morgan and Peiro [1,2,17], further developed recently by Dompierre et al [18]. Though the ideas are easily extendable to three-dimensional problems, technical issues arise when dealing with boundaries in 3D that have delayed the implementation, so that examples are presented in two space dimensions.
