Alternating optimization of design and stress for stress-constrained topology optimization

Handling stress constraints is an important topic in topology optimization. In this paper, we introduce an interpretation of stresses as optimization variables, leading to an augmented Lagrangian formulation. This formulation takes two sets of optimization variables, i.e., an auxiliary stress variable per element, in addition to a density variable as in conventional density-based approaches. The auxiliary stress is related to the actual stress (i.e., computed by its definition) by an equality constraint. When the equality constraint is strictly satisfied, an upper bound imposed on the auxiliary stress design variable equivalently applies to the actual stress. The equality constraint is incorporated into the objective function as linear and quadratic terms using an augmented Lagrangian form. We further show that this formulation is separable regarding its two sets of variables. This gives rise to an efficient augmented Lagrangian solver known as the alternating direction method of multipliers (ADMM). In each iteration, the density variables, auxiliary stress variables, and Lagrange multipliers are alternatingly updated. The introduction of auxiliary stress variables enlarges the search space. We demonstrate the effectiveness and efficiency of the proposed formulation and solution strategy using simple truss examples and a dozen of continuum structure optimization settings.

The PPADMM Method for Solving Quadratic Programming Problems

In this paper, a preconditioned and proximal alternating direction method of multipliers (PPADMM) is established for iteratively solving the equality-constraint quadratic programming problems. Based on strictly matrix analysis, we prove that this method is asymptotically convergent. We also show the connection between this method with some existing methods, so it combines the advantages of the methods. Finally, the numerical examples show that the algorithm proposed is efficient, stable, and flexible for solving the quadratic programming problems with equality constraint.

Improving the Optimality Verification and the Parallel Processing of the General Knapsack Linear Integer Problem

The chapter presents a new approach to improve the verification process of optimality for the general knapsack linear integer problem. The general knapsack linear integer problem is very difficult to solve. A solution for the general knapsack linear integer problem can be accurately estimated, but it can still be very difficult to verify optimality using the brach and bound related methods. In this chapter, a new objective function is generated that is also used as a more binding equality constraint. This generated equality constraint can be shown to significantly reduce the search region for the branch and bound-related algorithms. The verification process for optimality proposed in this chapter is easier than most of the available branch and bound-related approaches. In addition, the proposed approach is massively parallelizable allowing the use of the much needed independent parallel processing.

Dot Product Equality Constrained Attitude Determination from Two Vector Observations: Theory and Astronautical Applications

In this paper, the attitude determination problem from two vector observations is revisited, incorporating the redundant equality constraint obtained by the dot product of vector observations. Analytical solutions to this constrained attitude determination problem are derived. It is found out that the studied two-vector attitude determination problem by Davenport q-method under the dot product constraint has deterministic maximum eigenvalue, which leads to its advantage in error/perturbation analysis and covariance determination. The proposed dot product constrained two-vector attitude solution is applied then to solve several engineering problems. Detailed simulations on spacecrafts attitude determination indicate the efficiency of the proposed theory.