Conference Agenda

Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).

Please note that all times are shown in the time zone of the conference. The current conference time is: 1st Dec 2022, 06:50:25am CET

 
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Session Overview
Session
MS 3b: shape and topology optimization
Time:
Wednesday, 14/July/2021:
11:50am - 1:50pm

Session Chair: Alberto Paganini
Session Chair: Kevin Sturm
Virtual location: Zoom 4


Session Abstract

Shape optimization arises naturally in many industrial applications. Most common examples

are improving the design of product components like airplane fuselages and boat hulls, as well

as solving shape identification inverse problems, for instance in the framework of electrical

impedance tomography. In these applications, the underlying shape optimization problem is

constrained to partial differential equations (PDEs).

The key aspect of shape optimization is that the control variable is the shape of a domain.

There are several competing approaches to shape optimization, and these differ mostly by

the control variable discretization they employ, that is, by how they represent shapes.

A commonality of standard shape optimization methods is that, beside technical differ-

ences, they boil down to representing shapes via polytopes. This simplifies the coupling

between shape representations and solvers for PDE-constraints, but it inherently limits the

order of accuracy of the resulting algorithms. To overcome this limitation, a new shape

optimization trend is the use of higher-order discretizations of shapes. New approaches rely

on isogeometric analysis, isoparametric finite elements, and radial basis functions, among

others.

This minisymposium comprises sessions dedicated to these new and more accurate

shape optimization techniques. Its contributions will range from the convergence analysis of

higher-order shape discretization to new optimization algorithms and software developments,

with some contributions dedicated to industrial application.


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Presentations
11:50am - 12:20pm

Numerical approximation and optimization of the Canham-Helfrich elastic bending energy

Michael Neunteufel, Joachim Schöberl, Kevin Sturm

TU Wien, Austria

The Canham-Helfrich elastic bending energy on closed hypersurfaces involves the curvature of the manifold. The Hellan-Herrmann-Johnson (HHJ) method introduces a moment tensor for computing fourth order elliptic problems as a mixed method, which has been recently generalized to nonlinear shells.

In this talk we discuss the computation of the Weingarten tensor on discrete surfaces using the HHJ method. As the mapping of these elements requires the Piola transformation the shape derivative produces additional terms, which need to be taken into account. The derivative of the Weingarten tensor and the therein involved jump terms are discussed.

The method is implemented in the finite element library Netgen/NGSolve (www.ngsolve.org). Finally, we present numerical results.



12:20pm - 12:50pm

Adjoint based methods to compute higher order topological derivatives with an application to elasticity

Phillip Baumann, Kevin Sturm

TU Wien, Austria

The goal of this talk is to give a comprehensive and short review on how to compute the first and second order topological derivative and potentially higher order topological derivatives for PDE constrained shape functionals.

We employ the adjoint and averaged ajoint variable within the Lagrangian framework and compare three different adjoint based methods to compute higher order topological derivatives.

To illustrate the methodology proposed in this paper, we then apply the methods to a linear elasticity model.



12:50pm - 1:20pm

Computing multiple solutions of topology optimisation problems

Ioannis Papadopoulos1, Patrick Farrell1, Thomas Surowiec2

1University of Oxford, United Kingdom; 2Philipps-Universität Marburg, Germany

Topology optimisation finds the optimal material distribution of a continuum in a domain, subject to PDE and volume constraints. The models often result in a PDE, volume and inequality constrained, nonconvex, infinite-dimensional optimisation problem. These problems can exhibit many local minima. In practice, heuristics are used to obtain the global minimum, but these can fail even in the simplest of cases. In this talk, we will introduce the deflated barrier method, an algorithm that solves such problems and can systematically discover many of these local minima. We will present examples which include finding 42 solutions of the topology optimisation of a fluid satisfying the Navier-Stokes equations and more recent work involving the three-dimensional topology optimisation of a fluid in Stokes flow discretised with the Brezzi-Douglas-Marini finite element. We also discuss preconditioners for solving the linear systems arising in three-dimensional problems that are robust to the order of the finite element discretisation.



1:20pm - 1:50pm

Nonlinear Conjugate Gradient Methods for Shape Optimization

Sebastian Blauth

Fraunhofer ITWM, Germany

Shape optimization based on shape calculus has received a lot of attention in recent years, particularly regarding the development, analysis, and modification of efficient optimization algorithms. In this talk we continue these advancements and present novel nonlinear conjugate gradient (NCG) methods for the solution of shape optimization problems constrained by partial differential equations. The methods’ efficiency and low memory requirements make them particularly attractive for large-scale shape optimization problems. Moreover, as the NCG methods are based on Steklov-Poincaré-type metrics, they are well-suited for numerical discretization. We compare the performance of the NCG methods to the already established gradient descent and limited memory BFGS methods on several numerical benchmark problems. The corresponding results confirm that the NCG methods are efficient and attractive additions to already established gradient-based shape optimization algorithms.



 
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