Risk Averse Shape Optimization

Conti, Sergio; Held, Harald; Pach, Martin; Rumpf, Martin; Schultz, Rüdiger

doi:10.1137/090754315

Cited by 31 publications

(20 citation statements)

References 45 publications

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“…Shape sensitivity analysis as introduced by Sokolowski and Zolésio [31] [2,6] so that the maximum number of holes is prescribed by the initialization. In order to allow the level-set method to create holes in 2D, the topological derivative is sometimes used to identify and remove rather inactive interior material parts [7,20,24].…”

Section: Related Workmentioning

confidence: 99%

A phase-field model for compliance shape optimization in nonlinear elasticity

2010

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Abstract. Shape optimization of mechanical devices is investigated in the context of large, geometrically strongly nonlinear deformations and nonlinear hyperelastic constitutive laws. A weighted sum of the structure compliance, its weight, and its surface area are minimized. The resulting nonlinear elastic optimization problem differs significantly from classical shape optimization in linearized elasticity. Indeed, there exist different definitions for the compliance: the change in potential energy of the surface load, the stored elastic deformation energy, and the dissipation associated with the deformation. Furthermore, elastically optimal deformations are no longer unique so that one has to choose the minimizing elastic deformation for which the cost functional should be minimized, and this complicates the mathematical analysis. Additionally, along with the non-uniqueness, buckling instabilities can appear, and the compliance functional may jump as the global equilibrium deformation switches between different bluckling modes. This is associated with a possible non-existence of optimal shapes in a worst-case scenario. In this paper the sharp-interface description of shapes is relaxed via an Allen-Cahn or Modica-Mortola type phase-field model, and soft material instead of void is considered outside the actual elastic object. An existence result for optimal shapes in the phase field as well as in the sharp-interface model is established, and the model behavior for decreasing phase-field interface width is investigated in terms of Γ-convergence. Computational results are based on a nested optimization with a trust-region method as the inner minimization for the equilibrium deformation and a quasi-Newton method as the outer minimization of the actual objective functional. Furthermore, a multi-scale relaxation approach with respect to the spatial resolution and the phase-field parameter is applied. Various computational studies underline the theoretical observations. Mathematics Subject Classification. 49Q10, 74P05, 49J20.

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Section: Related Workmentioning

confidence: 99%

A phase-field model for compliance shape optimization in nonlinear elasticity

2010

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“…Alternative objectives to the compliance such as the L 2 -norm of the internal stresses or the expected excess compliance for a probability distribution of loads are much less understood but also possible [11,12].…”

Section: (A) Introduction To Compliance Minimizationmentioning

confidence: 99%

Optimal fine-scale structures in compliance minimization for a uniaxial load

Kohn

Wirth

2014

Proc. R. Soc. A.

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We consider the optimization of the topology and geometry of an elastic structure O ⊂ R 2 subjected to a fixed boundary load, i.e. we aim to minimize a weighted sum of material volume Vol(O), structure perimeter Per(O) and structure compliance Comp(O) (which is the work done by the load). As a first simple and instructive case, this paper treats the situation of an imposed uniform uniaxial tension load in two dimensions. If the weight ε of the perimeter is small, optimal geometries exhibit very fine-scale structure which cannot be resolved by numerical optimization. Instead, we prove how the minimum energy scales in ε, which involves the construction of a family of near-optimal geometries and thus provides qualitative insights. The construction is based on a classical branching procedure with some features unique to compliance minimization. The proof of the energy scaling also requires an ansatzindependent lower bound, which we derive once via a classical convex duality argument (which is restricted to two dimensions and the uniaxial load) and once via a Fourier-based refinement of the Hashin-Shtrikman bounds for the effective elastic moduli of composite materials. We also highlight the close relation to and the differences from shape optimization with a scalar PDE-constraint and a link to the pattern formation observed in intermediate states of type-I superconductors.

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“…where u(y; z) ∈ V for all y ∈ Γ solves (2.2). Such optimization problems are considered in, for example, [12,56] and in the context of shape optimization in [23]. As we have mentioned, optimization problems of the form (2.3) arise when one must decide on the control action prior to observing the outcome.…”

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confidence: 99%

A Trust-Region Algorithm with Adaptive Stochastic Collocation for PDE Optimization under Uncertainty

Kouri¹,

Heinkenschloss²,

Ridzal³

et al. 2013

SIAM J. Sci. Comput.

106

116

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Abstract. The numerical solution of optimization problems governed by partial differential equations (PDEs) with random coefficients is computationally challenging because of the large number of deterministic PDE solves required at each optimization iteration. This paper introduces an efficient algorithm for solving such problems based on a combination of adaptive sparse-grid collocation for the discretization of the PDE in the stochastic space and a trust-region framework for optimization and fidelity management of the stochastic discretization. The overall algorithm adapts the collocation points based on the progress of the optimization algorithm and the impact of the random variables on the solution of the optimization problem. It frequently uses few collocation points initially and increases the number of collocation points only as necessary, thereby keeping the number of deterministic PDE solves low while guaranteeing convergence. Currently an error indicator is used to estimate gradient errors due to adaptive stochastic collocation. The algorithm is applied to three examples, and the numerical results demonstrate a significant reduction in the total number of PDE solves required to obtain an optimal solution when compared with a Newton conjugate gradient algorithm applied to a fixed high-fidelity discretization of the optimization problem.

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Risk Averse Shape Optimization

Cited by 31 publications

References 45 publications

A phase-field model for compliance shape optimization in nonlinear elasticity

A phase-field model for compliance shape optimization in nonlinear elasticity

Optimal fine-scale structures in compliance minimization for a uniaxial load

A Trust-Region Algorithm with Adaptive Stochastic Collocation for PDE Optimization under Uncertainty

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