Minmax topology optimization

Brittain, Kevin; Silva, Mariana; Tortorelli, Daniel A.

doi:10.1007/s00158-011-0715-y

Cited by 23 publications

(6 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The load vector is given by

bold-italicf false(bold-italicr false) = {bold-italicQ}^{sans-serifT} bold-italicr,

where r satisfies ‖ r ‖ ≤ 1, in which ‖·‖ is the Euclidean norm, and Q is a given matrix. This is an example of the so‐called ellipsoidal model used by many researchers to model load uncertainty . The compliance is now given by

c false(bold-italicx, bold-italicr false) = bold-italicf {false(bold-italicr false)}^{sans-serifT} {bold-italicK}^{- 1} false(bold-italicx false) bold-italicf false(bold-italicr false) = {bold-italicr}^{sans-serifT} bold-italicQ bold-italicK {false(bold-italicx false)}^{- 1} {bold-italicQ}^{sans-serifT} bold-italicr .

Since c is convex as a function of r , the maximizers are found among the extreme points of the set

false{bold-italicr \in {double-struckR}^{d} false| false‖ bold-italicr false‖ \leq 1 false}

.…”

Section: Topology Optimization Under Load‐uncertaintymentioning

confidence: 99%

See 1 more Smart Citation

Game formulations for structural optimization under uncertainty

Thore

Grundström

Klarbring

2019

Numerical Meth Engineering

View full text Add to dashboard Cite

Summary 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.

show abstract

“…The load vector is given by

bold-italicf false(bold-italicr false) = {bold-italicQ}^{sans-serifT} bold-italicr,

c false(bold-italicx, bold-italicr false) = bold-italicf {false(bold-italicr false)}^{sans-serifT} {bold-italicK}^{- 1} false(bold-italicx false) bold-italicf false(bold-italicr false) = {bold-italicr}^{sans-serifT} bold-italicQ bold-italicK {false(bold-italicx false)}^{- 1} {bold-italicQ}^{sans-serifT} bold-italicr .

Since c is convex as a function of r , the maximizers are found among the extreme points of the set

false{bold-italicr \in {double-struckR}^{d} false| false‖ bold-italicr false‖ \leq 1 false}

.…”

Section: Topology Optimization Under Load‐uncertaintymentioning

confidence: 99%

“…This is an example of the so-called ellipsoidal model used by many researchers to model load uncertainty. 3,4,6,[8][9][10][31][32][33][34][35][36][37][38][39][40][41] The compliance is now given by…”

Section: Topology Optimization Under Load-uncertaintymentioning

confidence: 99%

Game formulations for structural optimization under uncertainty

Thore

Grundström

Klarbring

2019

Numerical Meth Engineering

View full text Add to dashboard Cite

show abstract

“…These problems may, assuming small deformations and using certain load parametrizations, be cast as generalized eigenvalue problems (Brittain et al 2012;Cherkaev and Cherkaev 2008;Takezawa et al 2011) or as semidefinite programming problems ; Thore et al 2015). In this paper however we propose a much more general game theoretic framework for TO under loaduncertainty including a wide range of different objective functions and constraints.…”

Section: Introductionmentioning

confidence: 99%

“…Appendix A). Choosing the compliance as the objective function and a suitable loading parametrization one can retrieve the generalized eigenvalue problems (Brittain et al 2012;Cherkaev and Cherkaev 2008;Takezawa et al 2011;Kobelev 1993) or semi-definite programming problems Thore et al 2015) mentioned above. When applicable these formulations may be more efficient, but compared to the proposed game theory framework they are very limited in that they only apply to zero-sum games with certain choices of functions and parametrizations.…”

Section: Introductionmentioning

confidence: 99%

Game theory approach to robust topology optimization with uncertain loading

Holmberg

Thore

Klarbring

2016

Struct Multidisc Optim

View full text Add to dashboard Cite

The paper concerns robustness with respect to uncertain loading in topology optimization problems. Using a game theoretic framework we formulate problems, or games, defining generalized 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.

show abstract

“…Recently multiscale design of material and structure has seen a strong progress with the use of optimization methods, namely topology optimization based methodologies. The works in [2][3][4][5][6][7][8][9][10][11][12][13][14][15] describe some of these recent developments. To formulate and solve the multiscale optimal design problem, we follow a hierarchical approach, as described in [16].…”

Section: Introductionmentioning

confidence: 99%