A fictitious energy approach for shape optimization

Abstract

SUMMARYThis paper deals with shape optimization of continuous structures. As in early works on shape optimization, coordinates of boundary nodes of the FE-domain are directly chosen as design variables. Convergence problems and problems with jagged shapes are eliminated by a new regularization technique: an artificial inequality constraint added to the optimization problem limits a fictitious total strain energy that measures the shape change of the design with respect to a reference design. The energy constraint defines a feasible design space whose size can be varied by one parameter, the upper energy limit. By construction, the proposed regularization is applicable to a wide range of problems; although in this paper, the application is restricted to linear elastostatic problems.

The inner structure of sensitivities in nodal based shape optimisation

“…Nodal based shape optimisation has been investigated from several viewpoints and its adequate formulation as well as the corresponding computational approach is of significant importance as demonstrated in [11,27,40]. Here, we like to take another path and add those observations which have not been explicitly considered in the cited references.…”

“…The contributions [11,40] deal with nodal based shape optimisation of free form surfaces (shells). In general, mesh distortions appear during the optimisation process.…”

“…After each shape optimisation step in normal direction, the authors apply the minimal surface regularisation (MSR) to avoid in-plane mesh distortions (in-plane regularisation). Both mentioned problems are avoided in [40] by introducing an additional constraint, which limits a fictitious energy. This energy is a measure for the design distortion of the finite element mesh.…”

“…The latter drawback of the node-based approach seems to be a matter of the past. The recent works in this area, see [11,27,40] and the references therein, report about successful application of nodal based shape optimisation to real-world problems. The jagged boundaries and convergence problems are eliminated either by using a fictitious energy approach [40] or by applying filter techniques [11,27].…”

Case study on the influence of kinematic hardening within a parameter-free and non-invasive form finding approach

“…This leads to a larger number of design parameters. Furthermore, nodalbased mesh moving strategies often suffer from distorted elements when updating the material configuration so that regularization and filter techniques have to be deployed, see Bletzinger et al [5] and Scherer et al [6].…”