Optimal Sizing and Shape Optimization in Structural Mechanics
We consider an industrial application consisting of the mass minimization of a frame in an injection moulding machine. This frame has to compensate the forces acting on the mould inside the machine and has to fulfill certain critical constraints. The deformation of that frame with constant thickness is described by the plain stress state equations for linear elasticity. If the thickness varies then we use a generalized plain stress state with constant thickness in the coarse grid elements. These direct problems are solved by an adaptive multigrid solver. The mass minimization problem leads to a constrained minimization problem for a non-linear functional which will be solved by some standard optimization algorithm which requires the gradients with respect to design parameters. For the shape optimization problem, we assume that the machine components consist of simple geometrical primitives determined by a few design parameters. Therefore, we calculate the gradient in the shape optimization by means of numerical differentiation which requires the solution of approximately 4 direct problems per design parameter. The adaptive solver guarantees the detection of critical regions automatically, and ensures a good approximation to the exact solution of the direct problem. This rather slow approach can be significantly accelerated by using the adjoint method to express the gradient. It will be combined with a direct implementation of several terms that appear after applying the chain rule to the gradient.
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