We study distributed and boundary integral expressions of Eulerian and Fréchet shape derivatives for several classes of nonsmooth domains such as open sets, Lipschitz domains, polygons and ...curvilinear polygons, semiconvex and convex domains. For general shape functionals, we establish relations between distributed Eulerian and Fréchet shape derivatives in tensor form for Lipschitz domains, and infer two types of boundary expressions for Lipschitz and C1-domains. We then focus on the particular case of the Dirichlet energy, for which we compute first and second order distributed shape derivatives in tensor form. Depending on the type of nonsmooth domain, different boundary expressions can be derived from the distributed expressions. This requires a careful study of the regularity of the solution to the Dirichlet Laplacian in nonsmooth domains. These results are applied to obtain a matricial expression of the second order shape derivative for polygons.
Nous étudions les expressions intégrales distribuées et les expressions intégrales de bord des dérivées de forme eulériennes et de Fréchet pour plusieurs classes de domaines non réguliers tels que les domaines ouverts, Lipschitz, convexes et semi-convexes, ainsi que les polygones et les polygones curvilinéaires. Dans le cas de fonctionnelles de forme générales, nous montrons certaines relations entre les expressions tensorielles des dérivées de forme distribuées eulériennes et de Fréchet pour les domaines Lipschitz, et nous en déduisons deux types d'expressions intégrales de bord dans le cas des domaines Lipschitz et C1. Par la suite, nous nous concentrons sur le cas particulier de l'énergie de Dirichlet, pour laquelle nous calculons les expressions tensorielles des dérivées de forme distribuées de premier et de second ordre. En fonction du type de domaine non régulier, différentes expressions intégrales de bord peuvent être obtenues à partir des expressions distribuées, ce qui requiert une étude minutieuse de la régularité de la solution du laplacien avec conditions aux limites de Dirichlet dans les domaines non réguliers. Nous appliquons ensuite ces résultats pour obtenir une expression matricielle de la dérivée seconde de forme dans le cas particulier des polygones.
The eigenmodes of resonating structures, e.g., electromagnetic cavities, are sensitive to deformations of their shape. In order to compute the sensitivities of the eigenpair with respect to a scalar ...parameter, we state the Laplacian and Maxwellian eigenvalue problems and discretize the models using isogeometric analysis. Since we require the derivatives of the system matrices, we differentiate the system matrices for each setting considering the appropriate function spaces for geometry and solution. This approach allows for a straightforward computation of arbitrary higher order sensitivities in a closed-form. In our work, we demonstrate the application in a setting of small geometric deformations, e.g., for the investigation of manufacturing uncertainties of electromagnetic cavities, as well as in an eigenvalue tracking along a shape morphing.
We present a fully-coupled monolithic formulation of the fluid-structure interaction of an incompressible fluid on a moving domain with a nonlinear hyperelastic solid. The arbitrary ...Lagrangian–Eulerian description is utilized for the fluid subdomain and the Lagrangian description is utilized for the solid subdomain. Particular attention is paid to the derivation of various forms of the conservation equations; the conservation properties of the semi-discrete and fully discretized systems; a unified presentation of the generalized-
α
time integration method for fluid-structure interaction; and the derivation of the tangent matrix, including the calculation of shape derivatives. A NURBS-based isogeometric analysis methodology is used for the spatial discretization and three numerical examples are presented which demonstrate the good behavior of the methodology.
In this paper, we introduce a new method for applying the implicit function theorem to find nontrivial solutions to overdetermined problems with a fixed boundary (given) and a free boundary (to be ...determined). The novelty of this method lies in the kind of perturbations considered. Indeed, we work with perturbations that exhibit different levels of regularity on each boundary. This allows us to construct solutions (whose given boundary and free boundary exhibit different regularities) that would have been out of reach via more simple perturbation techniques. Another benefit of this method lies in the improvement of the regularity gap that we get between the free boundary and the boundary of the given domain (this can be interpreted as a “smoothing effect”). Moreover, we show how to employ this method to construct solutions to both the Bernoulli free boundary problem and the two-phase Serrin’s overdetermined problem near radially symmetric configurations. Finally, some geometric properties of the solutions, such as symmetry and convexity, are also discussed.
•Consistent augmented Lagrangian scheme to handle the volume constraint of the compliance optimization problem.•Hilbertian extension of the velocity field adopted within a radial basis parameterized ...level-set method.•The mesh defined by the centers of the radial basis functions may be decoupled from the finite element mesh.•Numerical implementation developed in Python, using resources from the FEniCS project.
This work addresses the structural compliance minimization problem through a level-set-based strategy that rests upon radial basis functions with compact support combined with Hilbertian velocity extensions. A consistent augmented Lagrangian scheme is adopted to handle the volume constraint. The linear elasticity model and the variational problem associated with the computation of the velocity field are tackled by the finite element method using resources from the FEniCS project. The parameterization mesh constituted by the centers of the radial basis functions may be decoupled from the finite element mesh. A numerical investigation is conducted employing a Python implementation and five benchmark structures. Aimed at isolating the distinct aspects that compose the proposed strategy, the experiments provide grounds to analyze and put into perspective the inherent decisions and the related elements.
We provide theoretical analyses and numerical comparisons of boundary-based and volumetric shape derivative expressions of linear objective functionals encountered in topology optimization of linear ...elastic structures. The two expressions yield identical results if the domain is smooth and the governing equation is solved exactly; however, the finite element approximation of the expressions for less regular domains yield different results. We first review the two expressions to show that the volumetric shape derivative places weaker regularity requirements, which, unlike the requirements for boundary-based shape derivatives, are satisfied in most finite element approximations. We then analyze the error in the degree-k polynomial finite element approximations of the two expressions; we show that, for sufficiently regular problems, the boundary-based and volumetric shape derivatives provide kth and 2k-th order accurate approximations, respectively, of the true shape derivative. We finally assess, through numerical examples, the practical implications of using the volumetric vs boundary-based shape derivatives in topology optimization problems; we demonstrate that methods based on the volumetric shape derivative yield more robust solutions to topology optimization problems.
•Shape derivatives can be formulated as volumetric or boundary based integrals.•Volumetric shape derivatives require less regularity and converge faster for FEM.•Volumetric shape derivatives provide practical advantages for level set optimization.
We discuss automating the calculation of weak shape derivatives in the Unified Form Language (ACM TOMS 40(2):9:1–9:37
2014
) by introducing an appropriate additional step in the pullback from ...physical to reference space that computes Gâteaux derivatives with respect to the coordinate field. We illustrate the ease of use with several examples.
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