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).

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Session Overview
Session
MS 3c: shape and topology optimization
Time:
Wednesday, 14/July/2021:
2:00pm - 4:00pm

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
2:00pm - 2:30pm

Higher order topological derivatives and their approximation

Kevin Sturm1, Peter Gangl2, Phillip Baumann1

1TU Wien, Austria; 2TU Graz, Austria

In this talk we discuss the higher topological expansion of a shape function subject to the Laplace equation. The design variable enters in the right hand side of the Laplace equation. We present a closed form

formula of the topological derivative for a tracking type cost function. For the derivation

we use the compound layer method, which introduces so-called corrector equations,

which we propose to approximate with the boundary element method. Using this higher

order expansion we propose an acceleration of a level-set algorithm, where the design

variable is represented by a level-set function.



2:30pm - 3:00pm

High-Fidelity Gradient-Free Optimization of Low-Pressure Turbine Cascades

Anthony Aubry, Hamid Karbasian, Brian Vermeire

Concordia University, Canada

This paper explores aerodynamic shape optimization of Low Pressure Turbine (LPT) blades using Implicit Large Eddy Simulation (ILES) coupled with Mesh Adaptive Direct Search (MADS) optimization algorithm. Aerodynamic shape optimization in the aerospace industry relies heavily on the adjoint method combined with steady-state Computational Fluid Dynamics (CFD) solvers, namely the Reynolds-Averaged Navier-Stokes (RANS) approach. While this methodology has proven quite useful and efficient for common optimization problem, its effectiveness when faced with separated turbulent flow is less known. As LPT designs move increasingly towards high lift configurations to reduce engine weight, accurately predicting transition and turbulent separation for each optimization cycles becomes a necessity. The T106D blade and test case with an exit Mach number of 0.4 and Reynolds number 80,000 is used as the initial conditions before optimization, because of the presence of turbulent transition and separation on the suction side. First, the ILES framework employing the Flux Reconstruction (FR) approach is presented along with the benefits of its implementation on modern many-core hardware, specifically Graphical Processing Units (GPUs). The formulation of the MADS algorithm is then presented and compared against gradient-based methods and other gradient-free methods. A Bezier curve is used to modify the original blade camber line, with the use of 4 control points, permitting a wide range of shapes to be obtained. Then, the results of the baseline case selected are presented with a comparison with experimental results to validate the methodology selected. The results of optimization cycles using 2 different objective functions are then presented and compared with the original blade. Results demonstrate that a total pressure loss coefficient reduction of about 16% was achieved in the first optimization, while a tangential force increase of more than 25% was achieved with the second optimization. Finally, recommendations are made as to how this methodology could be successfully applied in industrial applications and what future research should focus on.



3:00pm - 3:30pm

A spectral discretization scheme for quasi-periodic boundary integral equations and shape optimization of transmission gratings.

Rubén David Aylwin1, Jose Pinto1, Gerardo Silva-Oelker2, Carlos Jerez-Hanckes1

1Universidad Adolfo Ibáñez, Santiago, Chile; 2Universidad Tecnológica Metropolitana, Santiago, Chile

We present a spectral Galerkin scheme for the treatment of boundary integral equations associated with integral representations of solutions to the Helmholtz transmission problem in multi-layered, periodic domains, which is subsequently employed in the shape optimization of transmission gratings in view of their diffraction efficiency for applications in chirped pulse amplification (CPA), beam splitting, among others.

Representing the unknowns of the arising boundary integral equations through a quasi-periodic Fourier series expansion, and choosing our discrete spaces as truncated quasi-periodic Fourier series, we are able to prove the well-posedness of the continuous and discrete problems, as well as super-algebraic convergence for our discrete subspaces. Our results are then confirmed through several numerical examples.

Then, we turn to the shape optimization of transmission gratings for the maximization of their diffraction efficiency in applications that demand for high diffraction efficiencies. Representing our geometries through finite trigonometric series allows for the restatement of the optimization problem in a parametric setting, where first order derivatives of the diffraction efficiencies on our parameters can be computed through appropriate shape derivatives of the scattered and transmitted fields. Computation of these derivatives through the adjoint method further enhances the efficiency of the method. The applicability of the proposed strategy is then displayed by considering the design of several multi-layered gratings for applications in CPA, exemplifying the advantages of designing optical devices in this manner, mainly (i) a precise computation of the diffraction efficiencies and associated quantities of interest (as well as of the respective gradients) at low computational costs and (ii) the emergence of complex structures without the necessity of relying on designer experience or zero-order methods (which become unmanageable for a large number of optimization parameters).



 
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