We consider structural optimization (SO) under uncertainty formulated as a mathematical game between two players -- a "designer" and "nature". The first player wants to design a structure that performs optimally, whereas the second player tries to find the worst possible conditions to impose on the structure. Several solution concepts exist for such games, including Stackelberg and Nash equilibria and Pareto optima. Pareto optimality is shown not to be a useful solution concept. Stackelberg and Nash games are, however, both of potential interest, but these concepts are hardly ever discussed in the literature on SO under uncertainty. Based on concrete examples of topology optimization of trusses and finite element-discretized continua under worst-case load uncertainty, we therefore analyze and compare the two solution concepts. In all examples, Stackelberg equilibria exist and can be found numerically, but for some cases we demonstrate nonexistence of Nash equilibria. This motivates a view of the Stackelberg solution concept as the correct one. However, we also demonstrate that existing Nash equilibria can be found using a simple so-called decomposition algorithm, which could be of interest for other instances of SO under uncertainty, where it is difficult to find a numerically efficient Stackelberg formulation.

We develop a topology optimization method including high-cycle fatigue as a constraint. The fatigue model is based on a continuous-time approach, which uses the concept of a moving endurance surface as a function of the stress history and back stress evolution. The development of damage only occurs when the stress state lies outside the endurance surface. Furthermore, an aggregation function, which approximates the maximum fatigue damage, is implemented. As the optimization workflow is sensitivity-based, the fatigue sensitivities are determined using an adjoint sensitivity analysis. The capabilities of the presented approach are tested on numerical models where the problem is to maximize the stiffness subject to high-cycle fatigue constraints.

In scoliosis, kypholordos and wedge properties of the vertebrae should be involved in determining how stress is distributed in the vertebral column. The impact is logically expected to be maximal at the apex.

The paper concerns robustness with respect to uncertain loading in topology optimization problems with essentially arbitrary objective functions and constraints. Using a game theoretic framework we formulate problems, or games, defining Nash equilibria. In each game a set of topology design variables aim to find an optimal topology, while a set of load variables aim to find the worst possible load. Several numerical examples with uncertain loading are solved in 2D and 3D. The games are formulated using global stress, mass and compliance as objective functions or constraints.

When subjected to periodic loading, elastic systems containing contact interfaces might exhibit frictional slip which ceases after some loading cycles. In such cases, it is said that the system shakes down. For elastic discrete systems presenting complete contacts, it has been proved that Melan’s theorem, originally proposed for elastic-plastic problems, offers a sufficient condition for the system to shake down, provided that the contact is of an uncoupled type. In the present paper, the application of Melan’s theorem is speculated for systems involving plasticity and friction. A finite element example of an elastic-plastic solid containing a frictional crack is discussed.

This paper presents a general mathematical structure for design optimization problems, where state problem functionals are used as design objectives.It extends to design optimization the general model of physical theories pioneered by Tonti (1972, 1976) and Oden and Reddy (1974, 1983). It turns out that the classical structural optimization problem of compliance minimization is a member of the treated general class of problems. Other particular examples, discussed in the paper, are related to Darcy-Stokes flow and pipe flow models. A main novel feature of the paper is the unification of seemingly different design problems, but the general mathematical structure also explains some previously not fully understood phenomena. For instance, the self-penalization property of Stokes flow design optimization receives an explanation in terms of minimization of a concave function over a convex set.

A goal function approach is used to derive an extension of Murray’s law that includes effects of nonlinear mechanics of the artery wall. The artery is modeled as a thin-walled tube composed of different species of nonlinear elastic materials that deform together. These materials grow and remodel in a process that is governed by a target state defined by a homeostatic radius and a homeostatic material composition. Following Murray’s original idea, this target state is defined by a principle of minimum work. We take this work to include that of pumping and maintaining blood, as well as maintaining the materials of the artery wall. The minimization is performed under a constraint imposed by mechanical equilibrium. We derive a condition for the existence of a cost-optimal homeostatic state. We also conduct parametric studies using this novel theoretical frame to investigate how the cost-optimal radius and composition of the artery wall depend on flow rate, blood pressure, and elastin content.

Structures designed by topology optimization (TO) are frequently sensitive to loads different from the ones accounted for in the optimization. In extreme cases this means that loads differing ever so slightly from the ones it was designed to carry may cause a structure to collapse. It is therefore clear that handling uncertainty regarding the actual loadings is important. To address this issue in a systematic  manner is one of the main goals in the field of robust TO. In this work we present a deterministic robust formulation of TO for maximum stiffness design which accounts for uncertain variations around a set of nominal loads. The idea is to find a design which minimizes the maximum compliance obtained as the loads vary in infinite, so-called uncertainty sets. This naturally gives rise to a semi-infinite optimization problem, which we here reformulate into a non-linear, semi-definite program. With appropriate numerical algorithms this optimization problem can be solved at a cost similar to that of solving a standard multiple load-case TO problem with the number of loads equal to the number of spatial dimensions plus one, times the number of nominal loads. In contrast to most previously suggested methods, which can only be applied to small-scale problems, the presented method is – as illustrated by a numerical example – well-suited for large-scale TO problems.

Phenomena such as biological growth and damage evolution can be thought of as time evolving processes, the directions of which are governed by descendent of certain goal functions. Mathematically this means using a dynamical systems approach to optimization. We extend such an approach by introducing a field quantity, representing nutrients or other non-mechanical stimuli, that modulate growth and damage evolution. The derivation of a generic model is systematic, starting from a Lyaponov-type descent condition and utilizing a Coleman-Noll strategy. A numerical algorithm for finding stationary points of the resulting dynamical system is suggested and applied to two model problems where the influence of different levels of nutrient sensitivity are observed. The paper demonstrates the use of a new modeling technique and shows its application in deriving a generic problem of growth and damage evolution. (C) 2014 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim

The paper concerns worst-case compliance optimization by finding the structural topology with minimum compliance for the loading due to the worst possible acceleration of the structure and attached non-structural masses. A main novelty of the paper is that it is shown how this min-max problem can be formulated as a non-linear semi-definite programming (SDP) problem involving a small-size constraint matrix and how this problem is solved numerically. Our SDP formulation is an extension of an eigenvalue problem seen previously in the literature; however, multiple eigenvalues naturally arise which makes the eigenvalue problem non-smooth, whereas the SDP problem presented in this paper provides a computationally tractable problem. Optimized designs, where the uncertain loading is due to acceleration of applied masses and the weight of the structure itself, are shown in two and three dimensions and we show that these designs satisfy optimality conditions that are also presented.