Game theory approach to robust topology optimization with uncertain loading

Self 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.

Section: Numerical Solution Of Nash Gamesmentioning

confidence: 99%

Game formulations for structural optimization under uncertainty

Thore

Grundström

Klarbring

2019

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“…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%

“…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%

“…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. 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 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%

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 of compliant mechanisms with uncertainties in output stiffness

Cardoso

Silva

Beck

2019

Summary This work addresses the topology optimization approach to design robust compliant mechanisms with respect to uncertainties in the output stiffness, when compared to the traditional deterministic approach. To this end, two formulations are proposed: probabilistic and nonprobabilistic. The probabilistic formulation minimizes a joint objective function of expected output displacement plus a measure of its standard deviations, for given statistical distribution of the output stiffness. The nonprobabilistic formulation is written as minimization of a joint function of the median of output displacements, plus the width of the intervals that contains the extreme values of the output displacements, for a given interval of output stiffness. The Monte Carlo simulation method is used to evaluate expected values and standard deviations of output displacements in the probabilistic formulation and to assess results obtained with the deterministic approach. It is shown that both formulations lead to designs where output displacements are less sensitive to variations of output stiffness when compared to the traditional deterministic approach. Furthermore, as an additional benefit, it is observed that large variations of output stiffness can hinder the appearance of one‐node connected hinges, usually found in the deterministic design of compliant mechanisms.