An approach of finite element grid optimization is proposed as an application of a shape design sensitivity analysis. Change of the mesh is described by design velocity fields that can be obtained by a piecewise linear interpolation from the nodal positions. Sensitivity of functionals with respect to the change of the nodal points has been formulated explicitly for a thick plate and a thick shell. For design sensitivity analysis, the variational approach according to the material derivative and domain methods is utilized. For a given topology of finite element mesh, to get optimum mesh, it is suggeseted that strain energy be maximized for static problems and eigenvalues minimized for eingenvalue problems with respect to the nodal positions.
Several examples are presented to show typical applications for structural optimization. Specialized sensitivity formulas are derived for a few types of structural elements. The first motivating problem is a grid optimization in the Timoshenko-beam and the Mindlin plate for eigenvalue analysis. From the result it is shown that the optimal grid is dependent on mode shapes as expected; for both beam and plate the meshes near a nodal point or a nodal line become denser than near an anti-node. Numerical examples for the Timoshenko beams and the Mindlin plates are obtained. The proposed approach is shown to be a feasible method that can be used for shape or configuration designs where large distortion of meshes often destroys optimization process.
As practical examples, eigenvalue problems for a square plate with line supports or stiffeners are treated. The fundamental or the second natural frequency, of which some are repeated, is to be maximized with respect to the location of supports or stiffeners. The grid optimization is adopted to bring in convergency during the optimization process. Two kinds of optimization for shape and grid are performed at the same time within the algorithm. The results show that the proposed method of optimization is effective to get the optimal structures. In the other example, the optimal rib position of an automobile bonnet model is found and demonstrated. The effect of rib thickness to the change of mode shape is also studied.
Through the numerical study, the sensitivities are found rather sensitive to the accuracy of finite element analysis. The choice of finite element is also found critical. For the analysis of the thick plate, an assumed strain method is shown to be good. For the thick plate theory dealing with a thin plate, the effect of locking should be removed for the sensitivity analysis. A meshless method of analysis can be a good alternative for shape optimization and a future topic of research. For practical applications of the present approach, further study is necessary for the criteria of optimum grid and combination of remeshing with the overall procedure of shape optimization utilizing commercial packages.