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Topology optimization subject to additive manufacturing constraints
2021, Ebeling-Rump, Moritz, Hömberg, Dietmar, Lasarzik, Robert, Petzold, Thomas
In topology optimization the goal is to find the ideal material distribution in a domain subject to external forces. The structure is optimal if it has the highest possible stiffness. A volume constraint ensures filigree structures, which are regulated via a Ginzburg–Landau term. During 3D printing overhangs lead to instabilities. As a remedy an additive manufacturing constraint is added to the cost functional. First order optimality conditions are derived using a formal Lagrangian approach. With an Allen-Cahn interface propagation the optimization problem is solved iteratively. At a low computational cost the additive manufacturing constraint brings about support structures, which can be fine tuned according to demands and increase stability during the printing process.
Direct and inverse elastic scattering from anisotropic media
2016, Bao, Gang, Hu, Guanghui, Yin, Tao, Sun, Jiguang
Assume a time-harmonic elastic wave is incident onto a penetrable anisotropic body embedded into a homogeneous isotropic background medium. We propose an equivalent variational formulation in a truncated bounded domain and show uniqueness and existence of weak solutions by applying the Fredholm alternative and using properties of the Dirichlet-to-Neumann map in both two and three dimensions. The Fréchet derivative of the near-field solution operator with respect to the scattering interface is derived. As an application, we design a descent algorithm for recovering the interface from the near-field data of one or several incident directions and frequencies. Numerical examples in 2D are demonstrated to show the validity and accuracy of our methods.
Numerical upscaling of parametric microstructures in a possibilistic uncertainty framework with tensor trains
2022, Eigel, Martin, Gruhlke, Robert, Moser, Dieter, Grasedyck, Lars
A fuzzy arithmetic framework for the efficient possibilistic propagation of shape uncertainties based on a novel fuzzy edge detection method is introduced. The shape uncertainties stem from a blurred image that encodes the distribution of two phases in a composite material. The proposed framework employs computational homogenisation to upscale the shape uncertainty to a effective material with fuzzy material properties. For this, many samples of a linear elasticity problem have to be computed, which is significantly sped up by a highly accurate low-rank tensor surrogate. To ensure the continuity of the underlying mapping from shape parametrisation to the upscaled material behaviour, a diffeomorphism is constructed by generating an appropriate family of meshes via transformation of a reference mesh. The shape uncertainty is then propagated to measure the distance of the upscaled material to the isotropic and orthotropic material class. Finally, the fuzzy effective material is used to compute bounds for the average displacement of a non-homogenized material with uncertain star-shaped inclusion shapes.
Elastic scattering by finitely many point-like obstacles
2012, Hu, Guanghui, Sini, Mourad
This paper is concerned with the time-harmonic elastic scattering by a finite number N of point-like obstacles in Rn (n = 2, 3). We analyze the N-point interactions model in elasticity and derive the associated Green's tensor (integral kernel) in terms of the point positions and the scattering coefficients attached to them, following the approach in quantum mechanics for modeling N-particle interactions. In particular, explicit expressions are given for the scattered near and far fields corresponding to elastic plane waves or point-source incidences. As a result, we rigorously justify the Foldy method for modeling the multiple scattering by finitely many point-like obstacles for the Lame model. The arguments are based on the Fourier analysis and the Weinstein-Aronszajn inversion formula of the resolvent for the finite rank perturbations of closed operators in Hilbert spaces.
Locking free and gradient robust H(div)-conforming HDG methods for linear elasticity
2020, Fu, Guosheng, Lehrenfeld, Christoph, Linke, Alexander, Streckenbach, Timo
Robust discretization methods for (nearly-incompressible) linear elasticity are free of volume-locking and gradient-robust. While volume-locking is a well-known problem that can be dealt with in many different discretization approaches, the concept of gradient-robustness for linear elasticity is new. We discuss both aspects and propose novel Hybrid Discontinuous Galerkin (HDG) methods for linear elasticity. The starting point for these methods is a divergence-conforming discretization. As a consequence of its well-behaved Stokes limit the method is gradient-robust and free of volume-locking. To improve computational efficiency, we additionally consider discretizations with relaxed divergence-conformity and a modification which re-enables gradient-robustness, yielding a robust and quasi-optimal discretization also in the sense of HDG superconvergence.
Some inverse problems arising from elastic scattering by rigid obstacles
2012, Hu, Guanghui, Kirsch, Andreas, Sini, Mourad
In the first part, it is proved that a C2-regular rigid scatterer in R3 can be uniquely identified by the shear part (i.e. S-part) of the far-field pattern corresponding to all incident shear waves at any fixed frequency. The proof is short and it is based on a kind of decoupling of the S-part of scattered wave from its pressure part (i.e. P-part) on the boundary of the scatterer. Moreover, uniqueness using the S-part of the far-field pattern corresponding to only one incident plane shear wave holds for a ball or a convex Lipschitz polyhedron. In the second part, we adapt the factorization method to recover the shape of a rigid body from the scattered S-waves (resp. P-waves) corresponding to all incident plane shear (resp. pressure) waves. Numerical examples illustrate the accuracy of our reconstruction in R2. In particular, the factorization method also leads to some uniqueness results for all frequencies excluding possibly a discrete set.