Null space gradient flows for constrained optimization with applications to shape optimization

Feppon, Florian; Allaire, Grégoire; Dapogny, Charles

doi:10.1051/cocv/2020015

Cited by 30 publications

(40 citation statements)

References 58 publications

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“…The constrained shape optimization problem (2.5) is solved thanks to the null space algorithm developed in [44]. The latter calculates a deformation θ of a given shape Γ, such that the value of the objective function J(Γ) is decreased, while ensuring that the constraints g i and h j are "better satisfied".…”

Section: Null Space Gradient Flow Optimization Algorithmmentioning

confidence: 99%

“…where γ is a regularization length scale typically set to a few times the mesh element size. In a similar spirit, the transpose T of (3.7) and (3.8) is different from the usual matrix transpose T in (3.3) and (3.4), see section 2.1 in [44] for the mathematical definition. 2.…”

Section: Null Space Gradient Flow Optimization Algorithmmentioning

confidence: 99%

“…Another salient aspect of our methodology is the use of the null space optimization algorithm developed in one of our other previous works [44]. This algorithm is a generalization of gradient flow strategies for constrained optimization problems which is well-suited for the shape optimization applications of this paper.…”

Section: Introductionmentioning

confidence: 99%

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Topology optimization of thermal fluid–structure systems using body-fitted meshes and parallel computing

Feppon

Allaire

Dapogny³

et al. 2020

Journal of Computational Physics

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An efficient framework is described for the shape and topology optimization of realistic threedimensional, weakly-coupled fluid-thermal-mechanical systems. At the theoretical level, the proposed methodology relies on the boundary variation of Hadamard for describing the sensitivity of functions with respect to the domain. From the numerical point of view, three key ingredients are used: (i) a level set based mesh evolution method allowing to describe large deformations of the shape while maintaining an adapted, highquality mesh of the latter at every stage of the optimization process; (ii) an efficient constrained optimization algorithm which is very well adapted to the infinite-dimensional shape optimization context; (iii) efficient preconditioning techniques for the solution of large finite element systems in a reasonable computational time. The performance of our strategy is illustrated with two examples of coupled physics: respectively fluid-structure interaction and convective heat transfer. Before that, we perform three other test cases, involving a single physics (structural, thermal and aerodynamic design), for comparison purposes and for assessing our various tools: in particular, they prove the ability of the mesh evolution technique to capture very thin bodies or shells in 3D.

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Section: Null Space Gradient Flow Optimization Algorithmmentioning

confidence: 99%

Section: Null Space Gradient Flow Optimization Algorithmmentioning

confidence: 99%

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Topology optimization of thermal fluid–structure systems using body-fitted meshes and parallel computing

Feppon

Allaire

Dapogny³

et al. 2020

Journal of Computational Physics

Self Cite

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“…All the involved finite element computations are performed within the FreeFem++ environment [39] -whether they are related to the resolution of the physical system (4.1) or to that of our variational problem (4.9). The resolution of the constrained optimization program (4.2) is carried out by the discretization of a specific gradient flow in the spirit of [56], see the details of our algorithm in our recent work [37]. When it comes to representing shapes and their evolution in the course of the iterative resolution of (4.2), the level set based mesh evolution method of [4,36] is used, as a convenient combination of the "classical" level set method for shape and topology optimization [2,61] and the mmg open-source mesh modification algorithm [21].…”

Section: Shape Optimization Of Linearly Elastic Structuresmentioning

confidence: 99%

A variational formulation for computing shape derivatives of geometric constraints along rays

2020

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In the formulation of shape optimization problems, multiple geometric constraint functionals involve the signed distance function to the optimized shape Ω. The numerical evaluation of their shape derivatives requires to integrate some quantities along the normal rays to Ω, a challenging operation to implement, which is usually achieved thanks to the method of characteristics. The goal of the present paper is to propose an alternative, variational approach for this purpose. Our method amounts, in full generality, to compute integral quantities along the characteristic curves of a given velocity field without requiring the explicit knowledge of these curves on the spatial discretization; it rather relies on a variational problem which can be solved conveniently by the finite element method. The well-posedness of this problem is established thanks to a detailed analysis of weighted graph spaces of the advection operator β ⋅ ∇ associated to a C1 velocity field β. One novelty of our approach is the ability to handle velocity fields with possibly unbounded divergence: we do not assume div(β) ∈ L∞. Our working assumptions are fulfilled in the context of shape optimization of C2 domains Ω, where the velocity field β = ∇dΩ is an extension of the unit outward normal vector to the optimized shape. The efficiency of our variational method with respect to the direct integration of numerical quantities along rays is evaluated on several numerical examples. Classical albeit important implementation issues such as the calculation of a shape’s curvature and the detection of its skeleton are discussed. Finally, we demonstrate the convenience and potential of our method when it comes to enforcing maximum and minimum thickness constraints in structural shape optimization.

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“…The algorithm can be seen as a null-space gradient method, see e.g. [33,4]. It has been described in [6] without proof of convergence; its local convergence has been proven in [9].…”

Section: 3mentioning

confidence: 99%

A gradient-type algorithm for constrained optimization with application to microstructure optimization

Barbarosie¹,

Toader²,

Lopes³

2020

Discrete &Amp; Continuous Dynamical Systems - B

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We propose a method to optimize periodic microstructures for obtaining homogenized materials with negative Poisson ratio, using shape and/or topology variations in the model hole. The proposed approach employs worst case design in order to minimize the Poisson ratio of the (possibly anisotropic) homogenized elastic tensor in several prescribed directions. We use a minimization algorithm for inequality constraints based on an active set strategy and on a new algorithm for solving minimization problems with equality constraints, belonging to the class of null-space gradient methods. It uses first order derivatives of both the objective function and the constraints. The step is computed as a sum between a steepest descent step (minimizing the objective functional) and a correction step related to the Newton method (aiming to solve the equality constraints). The linear combination between these two steps involves coefficients similar to Lagrange multipliers which are computed in a natural way based on the Newton method. The algorithm uses no projection and thus the iterates are not feasible; the constraints are only satisfied in the limit (after convergence). A local convergence result is proven for a general nonlinear setting, where both the objective functional and the constraints are not necessarily convex functions.

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Null space gradient flows for constrained optimization with applications to shape optimization

Cited by 30 publications

References 58 publications

Topology optimization of thermal fluid–structure systems using body-fitted meshes and parallel computing

Topology optimization of thermal fluid–structure systems using body-fitted meshes and parallel computing

A variational formulation for computing shape derivatives of geometric constraints along rays

A gradient-type algorithm for constrained optimization with application to microstructure optimization

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