On optimum design of frame structures

This paper considers structural optimization under a reliability constraint, where the input distribution is only partially known. Specifically, when we only know that the expected value vector and the variance-covariance matrix of the input distribution belong to a given convex set, we require that, for any realization of the input distribution, the failure probability of a structure should be no greater than a specified target value. We show that this distributionally-robus...

In this survey we consider polynomial optimization problems, asking to minimize a polynomial function over a compact semialgebraic set, defined by polynomial inequalities. This models a great variety of (in general, nonlinear nonconvex) optimization problems. Various hierarchies of (lower and upper) bounds have been introduced, having the remarkable property that they converge asymptotically to the global minimum. These bounds exploit algebraic representations of positive pol...

Optimizing over the cone of nonnegative polynomials, and its dual counterpart, optimizing over the space of moments that admit a representing measure, are fundamental problems that appear in many different applications from engineering and computational mathematics to business. In this paper, we review a number of these applications. These include, but are not limited to, problems in control (e.g., formal safety verification), finance (e.g., option pricing), statistics and ma...

Redundancy is related to the amount of functionality that the structure can sustain in the worst-case scenario of structural degradation. This paper proposes a widely-applicable concept of redundancy optimization of finite-dimensional structures. The concept is consistent with the robust structural optimization, as well as the quantitative measure of structural redundancy based on the information-gap theory. A derivative-free algorithm is proposed based on the sequential quad...

We present an efficient framework for solving constrained global non-convex polynomial optimization problems. We prove the existence of an equivalent nonlinear reformulation of such problems that possesses essentially no spurious local minima. We show through numerical experiments that polynomial scaling in dimension and degree is achievable for computing the optimal value and location of previously intractable global constrained polynomial optimization problems in high dimen...

Fast, gradient-based structural optimization has long been limited to a highly restricted subset of problems -- namely, density-based compliance minimization -- for which gradients can be analytically derived. For other objective functions, constraints, and design parameterizations, computing gradients has remained inaccessible, requiring the use of derivative-free algorithms that scale poorly with problem size. This has restricted the applicability of optimization to abstrac...

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...

Given a parametrized family of finite frames, we consider the optimization problem of finding the member of this family whose coefficient space most closely contains a given data vector. This nonlinear least squares problem arises naturally in the context of a certain type of radar system. We derive analytic expressions for the first and second partial derivatives of the objective function in question, permitting this optimization problem to be efficiently solved using Newton...

Let $\mathbf d=(d_j)_{j\in\mathbb I_m}\in\mathbb N^m$ be a finite sequence (of dimensions) and $\alpha=(\alpha_i)_{i\in\mathbb I_n}$ be a sequence of positive numbers (of weights), where $\mathbb I_k=\{1,\ldots,k\}$ for $k\in\mathbb N$. We introduce the $(\alpha\, , \,\mathbf d)$-designs i.e., $m$-tuples $\Phi=(\mathcal F_j)_{j\in\mathbb I_m}$ such that $\mathcal F_j=\{f_{ij}\}_{i\in\mathbb I_n}$ is a finite sequence in $\mathbb C^{d_j}$, $j\in\mathbb I_m$, and such that the ...

Although various structural optimization techniques have a sound mathematical basis, the practical constructability of optimal designs poses a great challenge in the manufacturing stage. Currently, there is only a limited number of unified frameworks which output ready-to-manufacture parametric Computer-Aided Designs (CAD) of the optimal designs. From a generative design perspective, it is essential to have a single platform that outputs a structurally optimized CAD model bec...