Global optimality in minimum compliance topology optimization of frames and shells by moment-sum-of-squares hierarchy

This paper studies the matrix Moment-SOS hierarchy for solving polynomial matrix optimization. Our first result is to show the tightness (i.e., the finite convergence) of this hierarchy, if the nondegeneracy condition, strict complementarity condition and second order sufficient condition hold at every minimizer, under the usual archimedeanness assumption. A useful criterion for detecting tightness is the flat truncation. Our second result is to show that every minimizer of t...

We review some features of topology optimization with a lower bound on the critical load factor, as computed by linearized buckling analysis. The change of the optimized design, the competition between stiffness and stability requirements and the activation of several buckling modes, depending on the value of such lower bound, are studied. We also discuss some specific issues which are of particular interest for this problem, as the use of non-conforming finite elements for t...

This paper studies the hierarchy of local minimums of a polynomial in the space. For this purpose, we first compute H-minimums, for which the first and second order optimality conditions are satisfied. To compute each H-minimum, we construct a sequence of semidefinite relaxations, based on optimality conditions. We prove that each constructed sequence has finite convergence, under some generic conditions. A procedure for computing all local minimums is given. When there are e...

This work presents an extended formulation of maximal stiffness design, within the framework of the topology optimization. The mathematical formulation of the optimization problem is based on the postulated principle of equal dissipation rate during inelastic deformation. This principle leads to the enforcement of stress constraints in domains where inelastic deformation would occur. During the transition from the continuous structure to the truss-like one (strut-and-tie mode...

This paper proposes a computational framework for the design optimization of stable structures under large deformations by incorporating nonlinear buckling constraints. A novel strategy for suppressing spurious buckling modes related to low-density elements is proposed. The strategy depends on constructing a pseudo-mass matrix that assigns small pseudo masses for DOFs surrounded by only low-density elements and degenerates to an identity matrix for the solid region. A novel o...

A new approach for generating stress-constrained topological designs in continua is presented. The main novelty is in the use of elasto-plastic modeling and in optimizing the design such that it will exhibit a linear-elastic response. This is achieved by imposing a single global constraint on the total sum of equivalent plastic strains, providing accurate control over all local stress violations. The single constraint essentially replaces a large number of local stress constr...

Designing lightweight yet stiff vibrating structures has been a long-standing objective in structural optimization. Here, we consider the optimization of a structural design subject to forces in the form of harmonic oscillations. We develop a unifying framework for the minimization of compliance and peak-power functions, while avoiding the assumptions of single-frequency and in-phase loads. We utilize the notion of semidefinite representable (SDr) functions and show that for ...

In this paper, the problem of load uncertainty in compliance problems is addressed where the uncertainty is described in the form of a set of finitely many loading scenarios. Computationally more efficient methods are proposed to exactly evaluate and differentiate: 1) the mean compliance, or 2) any scalar-valued function of the individual load compliances such as the weighted sum of the mean and standard deviation. The computational time complexities of all the proposed algor...

The moment-sum-of-squares (moment-SOS) hierarchy is one of the most celebrated and widely applied methods for approximating the minimum of an n-variate polynomial over a feasible region defined by polynomial (in)equalities. A key feature of the hierarchy is that, at a fixed level, it can be formulated as a semidefinite program of size polynomial in the number of variables n. Although this suggests that it may therefore be computed in polynomial time, this is not necessarily t...

This work blends the inexact Newton method with iterative combined approximations (ICA) for solving topology optimization problems under the assumption of geometric nonlinearity. The density-based problem formulation is solved using a sequential piecewise linear programming (SPLP) algorithm. Five distinct strategies have been proposed to control the frequency of the factorizations of the Jacobian matrices of the nonlinear equilibrium equations. Aiming at speeding up the overa...