A general framework for robust topology optimization under load-uncertainty including stress constraints

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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.

“…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%

Game formulations for structural optimization under uncertainty

Thore

Grundström

Klarbring

2019

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“…A different approach is considered herein, when compared to Holmberg et al 12 and Thore et al, 17 which is not based on the worst-case model. In this work, stress constraints are defined as a weighted sum between expectation and standard deviation of von Mises equivalent stresses, where the number of standard deviations is chosen a priori by the designer to satisfy a specified level of robustness to the optimal topology.…”

Section: Introductionmentioning

confidence: 99%

“…In Holmberg et al, 12 a worst-case RTO approach is developed through a game theoretic framework, where applied load with known maximum intensity and unknown direction is modeled as a non-probabilistic uncertainty; it is demonstrated that von Mises stresses do not exceed stress limit, imposed by the designer, considering any loading condition inside predefined ellipsoid in design space. In Thore et al, 17 a very efficient approach is presented for solving worst-case RTO problems considering that objective function and applied constraints are quadratic functions of an uncertain load; this allows the solution of problems with Euclidean norm for the stress constraints considering an ellipsoidal uncertainty model, like the presented problem of worst-case compliance minimization under worst-case stress constraints.…”

Section: Introductionmentioning

confidence: 99%

“…A careful literature review resulted in 3 recent works providing contributions in this subject. 8,12,17 In Luo et al, 8 an RBTO approach is formulated to solve continuum problems where the objective is the minimum volume subjected to stress constraints under uncertainties in applied loads and material properties; it is demonstrated that the singularity phenomenon is even more challenging in the reliability-based formulation, and a target reliability index relaxation is proposed for helping alleviate this difficulty. In Holmberg et al, 12 a worst-case RTO approach is developed through a game theoretic framework, where applied load with known maximum intensity and unknown direction is modeled as a non-probabilistic uncertainty; it is demonstrated that von Mises stresses do not exceed stress limit, imposed by the designer, considering any loading condition inside predefined ellipsoid in design space.…”

Section: Introductionmentioning

confidence: 99%

“…A review of the literature reveals many papers dealing with continuum topology optimization formulations to solve stochastic problems and find optimal structures in the presence of uncertainties. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] Two main techniques are often used to address design optimization problems under uncertainties: reliability-based design optimization and robust design optimization. 18 In the reliability-based design optimization approach, uncertainties affect only the design constraints, which are written in such a way as to warrant a minimal level of safety for the optimal design.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Topology optimization of continuum structures with stress constraints and uncertainties in loading

Silva

Beck

Cardoso

2017

SummaryTopology optimization using stress constraints and considering uncertainties is a serious challenge, since a reliability problem has to be solved for each stress constraint, for each element in the mesh. In this paper, an alternative way of solving this problem is used, where uncertainty quantification is performed through the first-order perturbation approach, with proper validation by Monte Carlo simulation. Uncertainties are considered in the loading magnitude and direction. The minimum volume problem subjected to local stress constraints is formulated as a robust problem, where the stress constraints are written as a weighted average between their expected value and standard deviation. The augmented Lagrangian method is used for handling the large set of local stress constraints, whereas a gradient-based algorithm is used for handling the bounding constraints. It is shown that even in the presence of small uncertainties in loading direction, different topologies are obtained when compared to a deterministic approach. The effect of correlation between uncertainties in loading magnitude and direction on optimal topologies is also studied, where the main observed result is loss of symmetry in optimal topologies.

Robust topology optimization under multiple independent uncertainties of loading positions

Wang

Gao

2020

The topology optimization problem of a continuum structure is further investigated under the independent position uncertainties of multiple external loads, which are now described with an interval vector of uncertain-but-bounded variables. In this study, the structural compliance is formulated with the quadratic Taylor series expansion of multiple loading positions. As a result, the objective gradient information to the topological variables can be evaluated efficiently upon an explicit quadratic expression as the loads deviate from their ideal application points. Based on the minimum (largest absolute) value of design sensitivities, which corresponds to the most sensitive compliance to the load position variations, a two-level optimization algorithm within the non-probabilistic approach is developed upon a gradient-based optimization method. The proposed framework is then performed to achieve the robust optimal configurations of four benchmark examples, and the final designs are compared comprehensively with the traditional topology optimizations under the loading point fixation. It will be observed that the present methodology can provide a remarkably different structural layout with the auxiliary components in the design domain to counteract the load position uncertainties. The numerical results also show that the present robust topology optimization can effectively prevent the structural performance from a noticeable deterioration than the deterministic optimization in the presence of load position disturbances.