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Fully and Semi-Automated Shape Differentiation in NGSolve
TLDR
This approach combines the mathematical Lagrangian approach for differentiating PDE-constrained shape functions with the automated differentiation capabilities of NGSolve to allow for either a more custom-like or black-box–like behaviour of the software.Abstract:
In this paper we present a framework for automated shape differentiation in the finite element software NGSolve. Our approach combines the mathematical Lagrangian approach for differentiating PDE constrained shape functions with the automated differentiation capabilities of NGSolve. The user can decide which degree of automatisation is required and thus allows for either a more custom-like or black-box-like behaviour of the software.
We discuss the automatic generation of first and second order shape derivatives for unconstrained model problems as well as for more realistic problems that are constrained by different types of partial differential equations. We consider linear as well as nonlinear problems and also problems which are posed on surfaces. In numerical experiments we verify the accuracy of the computed derivatives via a Taylor test. Finally we present first and second order shape optimisation algorithms and illustrate them for several numerical optimisation examples ranging from nonlinear elasticity to Maxwell's equations.read more
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Boundary integral formulations of eigenvalue problems for elliptic differential operators with singular interactions and their numerical approximation by boundary element methods
Markus Holzmann,Gerhard Unger +1 more
TL;DR: The self-adjointness of these operators is shown, and equivalent formulations for the eigenvalue problems involving boundary integral operators are derived for the numerical computations of the discrete eigenvalues and the corresponding eigenfunctions by boundary element methods.
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cashocs: A Computational, Adjoint-Based Shape Optimization and Optimal Control Software
TL;DR: Cashocs as discussed by the authors is a new software package written in Python, which automatically solves PDE constrained optimization problems in the context of optimal control and shape optimization by discretizing the continuous adjoint approach.
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Multi-objective free-form shape optimization of a synchronous reluctance machine.
TL;DR: The presented approach is based on the mathematical concept of shape derivatives and allows to obtain new motor designs without the need to introduce a geometric parametrization and an extension of the free-form shape optimization algorithm to the case of multiple objective functions and an approximate Pareto front is shown.
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An unfitted finite element method for two-phase Stokes problems with slip between phases
TL;DR: In this paper, an isoparametric finite element approach of the CutFEM or Nitsche-XFEM family is presented for simulation of two-phase Stokes problems with slip between phases.
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A finite element method for two-phase flow with material viscous interface
TL;DR: In this paper, a model of two-phase flow with an immersed material viscous interface and a finite element method for numerical solution of the resulting system of PDEs is studied.
References
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