International Conference on Spectral and High Order Methods
12th - 16th July 2021 | Vienna, Austria
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, 07:38:17am CET
MS 3a: shape and topology optimization
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
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.
12:00pm - 12:30pm
A reduced data-driven shape optimisation framework with application in naval design
1SISSA, Italy; 2SISSA, Italy
We deal with the implementation of a data-driven shape optimisation pipeline in a naval architecture application. We adopt reduced order models in order to improve the efficiency of the overall optimisation. We applied the above mentioned pipeline to a realistic cruise ship geometry in order to reduce the total drag. We start defining the design space, generated by deforming an initial shape in a parametric way using free form deformation. The evaluation of the performance of each new hull is determined by simulating the flux via finite volume discretisation of a two-phase (water and air) fluid. Since the fluid dynamics model may result very expensive—especially dealing with complex industrial geometries—we propose also a dynamic mode decomposition enhancement to reduce the computational cost of a single numerical simulation. The real-time computation is finally achieved by means of proper orthogonal decomposition with Gaussian process regression technique.
12:30pm - 1:00pm
Stochastic Approximation for Optimization in Shape Spaces
Models incorporating uncertain inputs, such as random forces or material properties, are of increasing interest in shape optimization, but the study of efficient algorithms to solve such problems is still in its infancy. In this work, we present a novel approach based on stochastic approximation to solve stochastic shape optimization problems. Our approach is the extension of the classical stochastic gradient method, dating back to Robbins and Monro (1951), to infinite-dimensional shape manifolds. The basic paradigm of stochastic approximation is the use of partial function information combined with decreasing step sizes to asymptotically dampen error. Thanks to its many applications, particularly in machine learning, there have been a number of recent advancements in the understanding of the method's convergence. We use this as a basis to prove convergence of the method on Riemannian manifolds. Furthermore, we make the connection to shape spaces. The method is demonstrated on a model shape optimization problem. Uncertainty arises in the form of a random partial differential equation, where underlying probability distributions of the random coefficients and inputs are assumed to be known. We verify conditions for convergence for the model problem and demonstrate the method numerically. Challenges and open questions will also be addressed.
1:00pm - 1:30pm
Pre-Shape Calculus: A Unified Framework for Simultaneous Shape and Mesh Quality Optimization
Trier University, Germany
Deformations of the computational mesh arising from shape optimization rou-
tines usually lead to decrease of mesh quality or even destruction of the mesh.
We propose a theoretical framework using pre-shapes to generalize classical
shape optimization and -calculus. We define pre-shape derivatives and derive
according structure and calculus theorems. In particular, tangential directions
are featured in pre-shape derivatives, in contrast to classical shape derivatives
featuring only normal directions. Techniques from classical shape optimiza-
tion and -calculus are shown to carry over to this framework. An optimization
problem class for mesh quality is introduced, which is numerically solvable
by use of pre-shape derivatives. This class of problems has the property to
leave the classical shape optimization solution invariant. Building on this,
we present pre-shape gradient system modifications, which permit simulta-
neous shape optimization while improving mesh quality. The computational
burden of our approach is limited, since additional solution of possibly larger
(non-)linear systems for regularized shape gradients is not necessary. Also,
our approach does not depend on the choice of metrics representing shape
gradients. We implement and compare pre-shape gradient regularization ap-
proaches for a hard to solve 2D problem.
1:30pm - 2:00pm
Fireshape: a shape optimization toolbox for Firedrake
1University of Leicester, United Kingdom; 2Courant Institute of Mathematical Sciences, New York, USA
We present Fireshape, an open-source and automated shape optimization toolbox for the finite element software Firedrake. Fireshape is based on isoparametric finite elements and allows users with minimal shape optimization knowledge to tackle with ease challenging shape optimization problems constrained to partial differential equations.
Fireshape is available at https://github.com/fireshape/fireshape . Fireshape’s documen- tation comprises several tutorials and is available at https://fireshape.readthedocs.io/en/ latest/ .
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