Topology Optimization with Stress Constraints

Abstract

This thesis contains contributions to the development of topology optimization techniques capable of handling stress constraints. The research that led to these contributions was motivated by the need for topology optimization techniques more suitable for industrial applications. Currently, topology optimization is mainly used in the initial design phase, and local failure criteria such as stress constraints are considered in additional post-processing steps. Consequently, there is often a large gap between the topology optimized design and the final design for manufacturing. Taking into account stress constraints directly into the topology optimization process would reduce this gap. Several difficulties arise in topology optimization with local stress constraints which complicate solving the optimization problem directly. \chap{litreview} discusses these difficulties, and reviews solutions that have been applied. Two fundamental difficulties are: (i) the presence of singular optima, which are true optima inaccessible to standard nonlinear programming techniques, and (ii) the fact that the stress is a local state variable, which typically leads to a large number of constraints. Currently, the conventional strategy to circumvent these difficulties is to apply (i) constraint relaxation, which perturbs the feasible domain to make singular optima accessible, followed by (ii) constraint aggregation to transform the typically large number of relaxed constraints into a single or few global constraints thereby reducing the order of the problem. Although there is no consensus on the exact choice of aggregation and relaxation functions and their numerical implementation, in general, this approach introduces two additional parameters to the problem: an aggregation and a relaxation parameter. Following this approach, one solves an alternative optimization problem with the aim of finding a solution to the original stress-constrained topology optimization. The feasible domain of this alternative optimization problem is related to the original feasible domain via these parameters. In Chapter 2, we investigated the parameter dependence of this alternative optimization problem on an elementary two-bar truss problem. It was found that the location of the global optimum of this alternative optimization problem with respect to the true optimum depends in a non-trivial way on these problem parameters (in their range of application); i.e., for a given parameter set, it is difficult to predict the influence of changing one of the parameter values, and if this change will result in a feasible domain in which the global optimum is closer to the true optimum. This complicates determining optimal parameter values \emph{a priori} which, in addition, are problem-dependent. In Chapter 3, we investigated the effect of design parameterization, and relaxation techniques in stress-constrained topology optimization. An elementary numerical example was considered, representing a situation as might occur in density-based topology optimization. As previously observed in truss optimization, we found that a global optimum of the relaxed optimization problem may not converge to the true optimum as the relaxation parameter is decreased to zero. In this thesis, we present two novel approaches: a unified aggregation and relaxation approach in Chapter 4, and the damage approach in Chapter 5. In the unified aggregation and relaxation approach, we applied constraint aggregation such that it simultaneously perturbs the feasible domain, and makes singular optima accessible. Consequently, conventional relaxation techniques become unnecessary when applying constraint aggregation following this approach. The main advantage is that the problem only depends on a single parameter, which reduces the parameter dependency of the problem. The damage approach is presented as a viable alternative for conventional methodologies. Following the damage approach stress constraint violation is penalized by degrading material where the stress exceeds the allowable stress. Material degradation affects the overall performance of the structure, and therefore, the optimizer promotes a design without stress constraint violation. Similar to conventional constraint aggregation techniques a large number of local constraints can be controlled by imposing a single or a few global constraints. Both novel approaches are validated on elementary truss examples and tested on numerical examples in density-based topology optimization. In contrast to the conventional strategy of relaxation followed by aggregation, there exists a clear relationship between the perturbed feasible domain and the original unperturbed feasible domain in terms of a single problem parameter.