Continuum structural topological optimizations with stress constraints based on an active constraint technique

Abstract

SUMMARYStress-related problems have not been given the same attention as the minimum compliance topological optimization problem in the literature. Continuum structural topological optimization with stress constraints is of wide engineering application prospect, in which there still are many problems to solve, such as the stress concentration, an equivalent approximate optimization model and etc. A new and effective topological optimization method of continuum structures with the stress constraints and the objective function being the structural volume has been presented in this paper. To solve the stress concentration issue, an approximate stress gradient evaluation for any element is introduced, and a total aggregation normalized stress gradient constraint is constructed for the optimized structure under the r th load case. To obtain stable convergent series solutions and enhance the control on the stress level, two p-norm global stress constraint functions with different indexes are adopted, and some weighting p-norm global stress constraint functions are introduced for any load case. And an equivalent topological optimization model with reduced stress constraints is constructed,being incorporated with the rational approximation for material properties, an active constraint technique, a trust region scheme, and an effective local stress approach like the qp approach to resolve the stress singularity phenomenon. Hence, a set of stress quadratic explicit approximations are constructed, based on stress sensitivities and the method of moving asymptotes. A set of algorithm for the one level optimization problem with artificial variables and many possible non-active design variables is proposed by adopting an inequality constrained nonlinear programming method with simple trust regions, based on the primal-dual theory, in which the non-smooth expressions of the design variable solutions are reformulated as smoothing functions of the Lagrange multipliers by using a novel smoothing function. Finally, a twolevel optimization design scheme with active constraint technique, i.e. varied constraint limits, is proposed to deal with the aggregation constraints that always are of loose constraint (non active constraint) features in the conventional structural optimization method. A novel structural topological optimization method with stress constraints and its algorithm are formed, and examples are provided to demonstrate that the proposed method is feasible and very effective.

Topological Optimization of Bi-Directional Progressive Structures with Dynamic Stress Constraints under Aperiodic Load

“…This algorithm solves the topology optimization to minimize the volume of the structure under stress constraints. Rong [12] established a topology optimization method for a continuum structure with stress gradient constraints, solved the problem of stress concentration, and proposed a topology optimization algorithm through P-norm and Lagrange multipliers. Fan Zhao [13] and others proposed the following.…”

Stress-constrained topology optimization using approximate reanalysis with on-the-fly reduced order modeling

“…To reduce the computational complexity, local stress constraints are usually transformed into a global stress constraint or some form of cluster-based stress constraints by using a condensation function that approximates the value of the maximum local function. Currently, the two most common types of condensation functions are the Kieisselmeier-Steinhauser function(K-S) [4,[12][13][14][15]17] and the P-norm function [9,[16][17][18], and they can merge a mass of local stresses into a single constraint formula.…”

Stress-Constrained Topology Optimization Using Approximate Reanalysis with on-the-Fly Reduced Order Modeling

“…To reduce the computational complexity, local stress constraints are usually transformed into a global stress constraint or some form of cluster-based stress constraints by using a condensation function that approximates the value of the maximum local function. Currently, the two most common types of condensation functions are the Kreisselmeier-Steinhauser function (K-S) [4,[12][13][14][15][16] and the P-norm function [9,[16][17][18], and they can merge a mass of local stresses into a single constraint formula. 3.…”

“…Highly nonlinear nature of the stress constraint. Any change in density value of the adjacent region will significantly affect the stress level within some key regions [13,17] such as depressions and corners. This requires the optimization procedure to be able to effectively reduce or eliminate the stress concentration phenomenon, and that the solution algorithm must maintain numerical consistency with the optimization procedure in order to ensure stable convergence of the overall procedure [19,20].…”