We consider an elliptic problem in a given domain Ω and a given right hand side f. The Dirichlet region is the unknown of the problem and has to be chosen in an optimal way, in order to minimize a cost functional, and in a class of admissible choices. The cost we consider is the compliance functional and the class of admissible choices consists of all one-dimensional connected sets (networks) of a given length L. Then we let L tend to infinity and look for the Γ-limit of suitably rescaled functionals, in order to identify the asymptotical distribution of the optimal networks. The asymptotically optimal shapes are discussed as well and links with average distance problems are provided.

Giuseppe Buttazzo
University of Pisa, Italy

In this talk, we are interested in the optimal shape of a pipe (inlet and outlet are fixed and the volume is prescribed). We consider an incompressible fluid, subject to the Navier-Stokes equations with classical boundary conditions on the boundary of the domain (velocity profile given on the inlet, no slip condition on the lateral boundary and outlet-pressure condition on the outlet). We are interested in the optimal shape of the pipe for the criterion "energy dissipated by the fluid"?
     In this talk, we will consider the following questions:

• Does there exist a minimizer?
• Is the cylinder the optimal shape? We prove that it is not the case. For that purpose, we explicit the first order optimality condition, thanks to adjoint state and we prove that it is impossible that the adjoint state be a solution of this over-determined system.
• More generally, how can one prove symmetry or non-symmetry in such shape optimization problems.

Antoine Henrot
University of Nancy, France

Yannick Privat
University of Nancy, France

We will discuss some questions related to the minimization of shape functionals in a family of convex subsets of IR². We will provide general first and second order optimality conditions taking into account the convexity constraint. This will be used to prove that all optimal shapes are polygons for a specific family of functionals with some adequate concavity property. We will also discuss regularity questions, in particular for the optimal shape of the second eigenvalue of the Laplace operator with measure and convexity constraints.

Michel Pierre
ENS Cachan-Bretagne and IRMAR, France

Commercial buildings are responsible for a significant fraction of the energy consumption and greenhouse gas emissions in the U.S. and worldwide. Consequently, the design, optimization and control of energy efficient buildings can have tremendous impact on energy cost and greenhouse gas emission. Buildings are complex, multi-scale, multi-physics, highly uncertain dynamic systems with wide varieties of disturbances. By itself, whole building simulation is a significant computational problem. However, when addressing additional requirements that center on design, optimization (for energy and CO₂) and control (both local and supervisory) of whole buildings, it becomes an immense challenge to develop the computational algorithms that are scalable and widely applicable to current and future building stock. In this talk we describe some fundamental engineering, design and computational challenges that can be described as shape optimization and control problems. We present model problems to illustrate the basic issues and to demonstrate how sensitivities can be used to address problems of design and control.

Jeff Borggaard
Virginia Tech, USA

John A. Burns
Virginia Tech, USA

Amit Surana
United Technologies Research Center, USA

We are concerned with shape optimization problems where the state variables are supposed to satisfy a PDE or a system thereof and the design variables are subject to bilateral pointwise constraints. In particular, we consider an all-at-once approach based on interior-point methods via the coupling of the inequality constraints by logarithmic barrier functions. The KKT conditions represent a parameter dependent nonlinear system which is solved by a multilevel path-following predictor-corrector technique with an adaptive choice of the continuation parameter along the barrier path. In particular, the predictor is a nested iteration type tangent continuation, whereas the corrector is a multilevel inexact Newton method featuring transforming null space iterations. As applications, we consider optimal shape designs of capillary barriers in microfluidic biochips, piston ducts in electrorheological shock absorbers, and microstructured biomorphic ceramics.

Ronald Hoppe
University of Houston, USA and University of Augsburg, Germany

We consider the multiple load topology optimization problem of minimizing the weight of a structure subject to constraints on local stresses and displacements. Our main objective is to develop a method for solving the considered class of problems to global optimality in the situation that the design variables are chosen from a discrete set of values. The considered class of problems suffers, besides the discrete design variables, from additional complications. Due to the equilibrium equations and the stress constraints the natural continuous relaxations are intrinsically non-convex and do not, in general, satisfy some constraint qualifications.
     For this class of problems we present a finitely convergent global optimization method based on the concept of non-linear branch and cut. In this method a possibly large number of continuous relaxations are solved. Using duality results from semi definite and second order cone programming the non-convex relaxations are reformulated into programs which can be efficiently solved in the branch and cut tree.
     The rate of convergence of the branch and cut method is largely determined by the quality of the relaxations. We therefore present an algorithm for generating linear cuts to strengthen the quality of the relaxations. The cuts are based on the recently developed Combinatorial Benders' cuts applied to a reformulation of the equilibrium equations as conditional linear inequalities.
     The method is developed and implemented within the PLATO-N project and is used to solve benchmark examples which are used to validate other methods and heuristics. The main aspects of the ongoing object oriented implementation of the method as well as numerical examples will be presented.

Nam Nguyen Canh
Technical University of Denmark, Lyngby

Mathias Stolpe
Technical University of Denmark, Lyngby

We consider vibrating networks of elastic strings in 3-space under boundary loadings. The problem is to optimize the topology of the network with respect to various cost functions using the concept of topological derivatives. To this end time-harmonics are introduced and topological sensitivities with respect to changes in the position and edge degree of multiple nodes are established via asymptotic analysis.
     We also present general results on self-adjoint boundary conditions and spectral properties, including inverse problems.

Günter Leugering
University of Erlangen, Germany

This paper proposes a general analytical expression for the sensitivity of the two-dimensional macroscopic elasticity tensor to topological changes of the microstructure of the underlying material. The macroscopic elasticity response is estimated by means of a homogenisation-based multi-scale constitutive theory for elasticity problems where, following closely the ideas presented in [2], the macroscopic strain and stress tensors at each point of the macroscopic continuum are defined as the volume averages of their microscopic counterparts over a Representative Volume Element (RVE) of material associated with that point. In this context and based on the axiomatic construction of multi-scale constitutive models, the proposed sensitivity leads to a symmetric fourth order tensor field over the RVE that measures how the macroscopic elasticity parameters estimated within the multi-scale framework changes when a small circular hole is introduced at the micro-scale. Its analytical formula is derived by making use of the concepts of topological asymptotic expansion and topological derivative [1, 4, 5] within the adopted multi-scale theory. These relatively new mathematical concepts allow the closed form calculation of the sensitivity, whose value depends on the solution of a set of equations over the original domain, of a given shape functional with respect to an infinitesimal domain perturbation, like the insertion of holes, inclusions or source term [3]. In the present context, the variational setting in which the underlying multi-scale theory is cast, is found to be particularly well-suited for the application of the topological derivative formalism. The final format of the proposed analytical formula is strikingly simple and can be potentially used in applications such as the synthesis and optimal design of microstructures to meet a specified macroscopic behavior. In this work, initially we describe the multi-scale constitutive framework adopted in the estimation of the macroscopic elasticity tensor. A clear variational foundation of the theory is established which is essential for the main developments to be presented later. Next, we presents the main result of the paper - the closed formula for the sensitivity of the macroscopic elasticity tensor to topological microstructural perturbations. Here, a brief description of the topological derivative concept is initially given. This is followed by its application to the problem in question which leads to the identification of the required sensitivity tensor. A simple finite element-based numerical example is also provided for the numerical verification of the analytically derived topological derivative formula. Finally, we show the potentialities of the proposed formula to the synthesis of micro-structures.

References
[1] J. Céa, S. Garreau, Ph. Guillaume, and M. Masmoudi, The shape and Topological Optimizations Connection. Computer Methods in Applied Mechanics and Engineering, 188(4), pp. 713-726, 2000.
[2] P. Germain, Q.S. Nguyen, and P. Suquet. Continuum thermodynamics. Journal of Applied Mechanics, Transactions of the ASME, 50, pp. 1010-1020, 1983.
[3] S.A. Nazarov and J. Sokolowski. Asymptotic analysis of shape functionals. Journal de Mathématiques Pures et Appliquées, 82(2), pp. 125-196, 2003.
[4] A.A. Novotny, R.A. Feijóo, C. Padra, and E. Taroco. Topological Sensitivity Analysis. Computer Methods in Applied Mechanics and Engineering, 192, pp. 803-829, 2003.
[5] J. Sokolowski and A. Zochowski. On the Topological Derivatives in Shape Optimization. SIAM Journal on Control and Optimization, 37(4), pp. 1251-1272, 1999.


E.A. de Souza Neto
Swansea University, UK

R.A. Feijóo
LNCC Brazil

S.M. Giusti
LNCC Brazil

Andre Novotny
LNCC Brazil

Using the Tube formulation [1] we derive the shape variational formulation [2] for some Euler flow (associated with the Shape metric developed as a generalisation of the Courant metric [3]). In order to derive a shape differentiable criteria we introduce a new perimeter concept based on the s-sobolev norm (s < 1/2) of the characteristic functions of bounded perimeter sets, see [3], [4].

References
[1] J.P. Zolésio. Variational Formulation for Incompressible Euler equation by Weak Shape evolution. Intern. Series of Num. Math., 133, pp. 309-323, Birkhauser, 1999.
[2] J.P. Zolésio. Control of Moving Domains, Shape Stabilization and Variational Tube Formulation, Intern. Series of Num. Math., 153, pp. 329-382, Birkhauser, 2007.
[3] M.C. Delfour and J.P. Zolésio. Shapes and Geomeries, volume 4 of Advances in Design and Control, SIAM, Philadelphia, 2001.
[4] M.C. Delfour and J.P. Zolésio. Curvatures and skeletons in shape optimization. Z. Angew. Math. Mech., 76(3), pp. 198-203, 1996.


Jean Paul Zolésio
CNRS & INRIA, Sophia-Antipolis, France

In structural topology optimization, level set method has emerged as a powerful method. In the level set-based approach, the surface of a solid structure is represented as an iso-surface of zero level of an implicit scalar function. Through the use of the classical shape derivative analysis, normal velocity field can be obtained as a decent gradient direction to generate the propagation of the surface boundary of the solid for optimization.
     In this presentation, we review recent development in two major aspects of the approach: numerical computation and applications. The classical level set method operates on discretization of the implicit function, often through a distance transform, using special finite difference methods developed to solve the Hamilton-Jacobi partial differential equation to propagate the surface. Alternatively, tensor basis or radial basis functions (RBFs) are used to describe the implicit scalar function. These analytical representations result in an implicit's parameterization, yielding programming algorithms to directly solve the topology optimization problem.
     The level set representation has topological flexibility and inherent capabilities of geometric, physical and material modeling, incorporating dimension, shape, topology, material properties, and even micro-structures for design and optimization of the heterogeneous solids. The method has found a wide range of applications in the design of multi-material structures, materials design, and compliant mechanism synthesis. These applications will be reviewed, and other applications in tissue modeling, flexonics, and MEMS will be discussed.

Michael Y. Wang
The Chinese University of Hong Kong

In the talk we shall show how the need for the application of the Steklov-Poincaré operator and its asymptotic expansions arises naturally in topology optimization for contact problems. We shall describe approaches used in obtaining explicit analytical forms of these expansions. Finally the use of the Steklov-Poincaré operator for formulating new type of boundary conditions in time-dependent problems will be discussed.

Jan Sokolowski
University of Nancy, France

Antoni Zochowski
Polish Academy of Sciences, Warsaw, Poland

In this talk we discuss two relaxation approaches in topology optimization and their limitations, namely convex relaxation in the space of functions of bounded variation and nonconvex phase-field approaches. We discuss their advantages and limitations in various respects and obtain new theoretical insight. Moreover we present some application studies in imaging and stress-constrained design.

Martin Burger
University of Muenster, Germany

Geometric flows, in which hypersurfaces move such that an energy, involving surface and bending terms, decreases appear in many situations in the natural sciences and in geometry. Classic examples are mean curvature, surface diffusion and Willmore flows. Computational methods to approximate such flows are based on one of three approaches (i) parametric methods, (ii) phase field methods or (iii) level set methods. The first tracks the hypersurface, whilst the other two implicitly capture the hypersurface. A key problem with the first approach, apart from the fact that it does not naturally deal with changes of topology, is that in many cases the mesh has to be redistributed after every few time steps to avoid coalescence of mesh points.
     In this talk we present a variational formulation of the parametric approach, which leads to an unconditionally stable, fully discrete approximation. In addition, the scheme has very good properties with respect to the distribution of mesh points, and if applicable volume conservation. We illustrate this for (anisotropic) mean curvature and (anisotropic) surface diffusion of closed curves in IR². We extend these flows to curve networks in IR². Here the triple junction conditions, that have to hold where three curves meet at a point, are naturally approximated in the discretization of our variational formulation. Our approach naturally generalises to the corresponding flows on curves in IR^d, d ≤ 3, as well as to curves on two-dimensional manifolds in IR³. Finally, we extend these approximations to flows on closed hypersurfaces and to surface clusters in IR³.

John Barrett
Imperial College London, UK

Harald Garcke
University of Regensburg, Germany

Robert Nürnberg
Imperial College London, UK

In this talk I will present several approaches to the computation of partial differential equations. In particular I will describe the surface finite element method, a method for PDEs defined on implicit surfaces and a diffuse interface method.

Charlie Elliott
University of Warwick, UK

This talk deals with optimization of the free boundary in external Bernoulli problems. The goal is to find the shape of the free boundary of a-priori given properties. As a control variable the shape of the inner component (inclusion) will be used. We prove the existence of a solution to this problem for a class of inclusions. The main attention will be paid to numerical realization of the free boundary (state) problems. This will be done by the so-called pseudo-solid approach. Finally, results of several model examples will be shown.

Jaroslav Haslinger
Charles University, Prague, Czech Republic

As known, the structure of the continuous spectrum in a periodic waveguide permits for gaps, i.e. open intervals, which have the endpoints on the continuous spectrum but can include points of the discrete spectrum only. Since a gap forbids the propagation of waves with frequencies in the corresponding range, the creation of such gaps is of importance in engineering of wave filters and dampers.
     In the talk it will be shown that a small periodic perturbation of a straight cylindrical waveguide, either singular, or regular, produces a gap in the continuous spectrum of a boundary value problem for a scalar differential equation.
     With a different idea and based on the asymptotic analysis in junctions of domains, such gaps are found out in a periodic elastic waveguide. The approach rests upon a special, weighted and anisotropic, inequality of Korn's type.

Serguei A. Nazarov
University of St. Petersburg, Russia

The shape derivative of the cost functional in a Bernoulli-type problem is characterized. The technique to calculate the derivative of the cost does not use the shape derivative of the state variable and is achieved under mild regularity conditions on the boundary of the domain.

Jaroslav Haslinger
Charles University, Prague, Czech Republic

Kazufumi Ito
North Carolina State University, USA

Tomas Kozubek
VSB-Technical University Ostrava, Czech Republic

Karl Kunisch
University of Graz, Austria

Gunther Peichl
University of Graz, Austria

Automatic (or algorithmic) differentiation [1] is a technique developed to differentiate computer codes exactly (up to numerical floating-point precision) and with minimal user intervention. The technique exploits the fact that every computer program executes a sequence of elementary arithmetic operations. A set of independent variables is defined in the beginning of the computation. In the case of shape optimization [2] the independent variables can be for example the geometrical design (CAD) parameters or the mesh nodal co-ordinates. The chain rule of differentiation is then successively applied to every elementary operation throughout the computation. In so called forward mode of automatic differentiation the derivative information is propagated forward in the execution chain.
     We have implemented a simple lightweight dynamic sparse forward derivative propagation technique. Dynamic means that we automatically at run time capture the relationships between the dependent and the independent variables (index domains of the dependent variables). By sparse we mean that the derivative information of each intermediate variable is saved in a sparse representation. This allows the computation of non-zero partial derivatives only, without the need of separate sparsity pattern detection and graph coloring [3] phases. Details of our implementation are given in [4] present a similar technique in [5], using an additional library called SparsLinC for the sparse storage of the derivatives.
     Our implementation is based on the operator overloading technique of C++ programming language. Thanks to the operation overloading the code exploiting automatic differentiation is mostly identical to the one that uses regular real variables, since the compiler takes care of calling the appropriate functions implementing the derivative computation. Thus the work of converting an original simulator into one that computes also the geometrical sensitivities of the solution comprises mostly of replacing the variables with their AD counterparts where needed.
     This automatic index domain capturing makes the technique easy on the developer of the code, since the developer does not have to manually keep track of the dependencies of the variables, which could require a special structure from the code. This is especially important when the technique is applied to an existing solver.
     Applicability of the implementation is demonstrated by shape optimization examples.
     An existing antenna simulation software, based on EFIE integral equation formulation of time-harmonic Maxwell equations and discretized using Galerkin's method, has been differentiated with respect to the geometry using the presented technique [4]. Virtually no changes to the program structure had to be made. Computational performance of the original simulation code and an automatically differentiated version were compared, and it was found out that assembly time of the system matrix was only a little over two times slower when the regular double and complex<double> types were replaced by their AD counterparts. The computation time naturally increases in the number of derivatives that have to be computed, but the increase is linear only if all the independent variables affect all the system matrix elements.
     We have applied so called pseudo-solid approach to solve a shape optimization problem governed by free boundary problems of Bernoulli type [6]. This approach treats the free boundary problem as a coupled non-linear problem, which is solved using Newton iteration with an exact Jacobian. The location of the free boundary is then optimized by adjusting the shape of another boundary. To do this, the non-linear state problem is differentiated with respect to the geometry. Implementation of the solver for this optimization problem is greatly facilitated by the use of the presented automatic differentiation technique.

References
[1] A. Griewank. Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, SIAM, 2000.
[2] J. Haslinger and R.A.E. Mäkinen. Introduction to Shape Optimization: Theory, Approximation, and Computation, SIAM, 2003.
[3] A.H. Gebremedhin, F. Manne, and A. Pothen. What Color is Your Jacobian? Graph Coloring for Computing Derivatives, Siam Review, Vol. 47, pp. 629-705, 2005.
[4] J.I. Toivanen, R.A.E Mäkinen, S. Järvenpää, P. Ylä-Oijala, and J. Rahola. Electromagnetic Sensitivity Analysis and Shape Optimization Using Method of Moments and Automatic Differentiation, Submitted to IEEE Transactions on Antennas & Propagation.
[5] C.H. Bischof, P.M. Khademi, A. Buaricha, and C. Alan. Efficient Computation of Gradients and Jacobians by Dynamic Exploitation of Sparsity in Automatic Differentiation. Optimization Methods and Software, Vol. 7, pp. 1-39, 1996.
[6] J.I. Toivanen, R.A.E. Mäkinen, and J. Haslinger. Shape Optimization of Systems Governed by Bernoulli Free Boundary Problems, to appear in Computer Methods in Applied Mechanics and Engineering.


Raino Mäkinen
University of Jyväskylä, Finland

Jukka I. Toivanen
University of Jyväskylä, Finland

In aerodynamic shape optimization and elsewhere it is often claimed that the adjoint approach makes the computation of the gradient independent of the number of unknowns. However, the need to compute so called "mesh sensitivities" in practical applications has so far proven to be a major obstacle to an industry size large scale aerodynamic shape optimization.
     As a remedy, we present shape calculus techniques which result in a shape derivative that exists only on the surface of the aircraft and can be computed without any sensitivities at all. By staying in the analytical framework all the time, we arrive at a gradient formulation which is independent of any parameterization like free-form, b-splines, or CAD. Thus, a large deformation of the shape is possible while the gradient is still very cheap to compute.
     The resulting loss of regularity is treated by considering the shape Hessian, which is derived for a Stokes flow. We also show shape Hessian approximations for the Navier-Stokes and Euler equations using operator symbols.

Nicolas Gauger
Humboldt University, Berlin, Germany

Caslav Ilic
German Aerospace Center (DLR), Braunschweig, Germany

Stephan Schmidt
University of Trier, Germany

Volker Schulz
University of Trier, Germany

On the one hand, solutions of free boundary problems can be characterized as stationary shapes of the energy of the system.
     Consequently, an energy variational approach and shape optimization techniques are useful tools for free surface computations. On the other hand, tracking type formulations might be an alternative as well. Here, tracking the Neumann condition while the Dirichlet boundary condition is left for the definition of the state equation or vice versa might be possible choices. In the talk, the well-posedness is investigated for the energy variational approach and the Neumann type tracking formulation.
     Despite of the fact, that nature of the two-norm discrepancy differs for both formulations, the same sufficient criterion for coercivity is derived. Efficient numerical methods are available via wavelet based BEM both for 2D and 3D problems.

Karsten Eppler
TU Dresden, Germany

Helmut Harbrecht
University of Bonn, Germany

Topology extends its applicability from the classical optimal structural design, to the design of compliant mechanisms. Compliant mechanism is a single-piece flexible structure that is flexible enough to deliver motion according to design and stiff enough to bear loads. A multi-purpose compliant mechanism is a multi-functional flexible structure that delivers different motions according to different input load cases. From both sources, namely the balance between flexibility to deliver motion and stiffness as well as the several functions, the problem is a multi-objective optimization one. Several multi-objective topology optimization problems are formulated. Due to local minima that arise using iterative local search methods, a hybrid algorithm utilizing evolutionary strategies such as Differential Evolution and Particle Swarms, is proposed. Representative numerical experiments are presented. The results of this work can be used for the design of multi-functional micro-mechanisms for elastic mechanical problems or multiphysics.

Nikos Kaminakis
Technical University of Crete, Greece

G.E. Stavroulakis
Carolo Wilhelmina Technical University, Braunschweig, Germany

Extrusion is a manufacturing process used by the plastics industry to create continuous objects of a fixed cross-sectional profile. Plastic melt is pushed through a die, which forms the polymer into the desired profile shape. In industrial applications, extrusion is a large-scale process, financially attractive only for mass products. This is due to the time-consuming running-in experiments that must precede each production cycle. Up to 15 iterations may be needed before the ultimate shape of the extrusion die, satisfying all quality criteria of the product, is determined [1]. This causes significant costs, considering the demands on machines and raw material. As part of the Cluster of Excellence "Integrative Production Technologies for High-Wage Countries" at the RWTH Aachen University, there is a joint effort by the Chair for Computational Analysis of Technical Systems (CATS) and the Institute of Plastics Processing (IKV) to shorten the manual running-in process by the means of numerical shape optimization, making this process significantly less costly and mostly automatic. From a numerical point of view, the extrusion process is very much suited, since it can be described by steady Stokes equations without major loss of accuracy. The drawback, however, is the need for accurate modeling of the plastics behaviour, which generally calls for shear-thinning or even viscoelastic models, as well as the need for 3D computations, leading to large computational grids.
     The optimization is performed with the flow solver XNS, developed at CATS. XNS uses the finite element method with Galerkin/Least-Squares stabilization, can utilize various parallel machines (Apple Xserve, Cray XD1, IBM Blue Gene, etc.), and is able to exploit the common communication interfaces for distributed-memory systems (SHMEM and MPI). XNS has been coupled with an optimization framework, creating the basis for the integration with a variety of optimizers [2].
     The presentation will start with the experiences we made with the direct simulation of the plastic melts, mainly focussing on the validation of the new implementation of the Carreau model, which is often used to model the shear-thinning behaviour of plastic melts, in XNS. In addition, a first 2D optimization test case will be shown, representing a flow through a slit. The emphasis will be on the parametrization of the geometry, realized in this case through non-uniform rational B-splines (NURBS).

References
[1] W. Michaeli, T. Schmitz, T. Baranowski, and B. Fink. Automatic optimisation of extrusion dies. The Polymer Processing Society 23rd Annual Meeting, Bahia do Salvador, May 27-31, 2007.
[2] M. Nicolai and M. Behr. Portable optimization framework for serial and parallel machines. Second European Conference on Computational Optimization, Montpellier, April 3, 2007.


M. Behr
RWTH Aachen University, Germany

Stefanie Elgeti
RWTH Aachen University, Germany

B. Fink
RWTH Aachen University, Germany

W. Michaeli
RWTH Aachen University, Germany

M. Nicolai
RWTH Aachen University, Germany

M. Probst
RWTH Aachen University, Germany

C. Windeck
RWTH Aachen University, Germany

A general Mumford-Shah type approach for the simultaneous reconstruction of functional and geometrical information from indirect data is presented. A level-set based reconstruction algorithm is discussed and applied to problem of finding and segmenting density and activity data in X-ray tomography and SPECT models.

Wolfgang Ring
University of Graz, Austria

Recently, optimization has become an integral part of the aerodynamic design process chain. However, because of uncertainties with respect to the flight conditions and geometry uncertainties, a design optimized by a traditional design optimization method seeking only optimality may not achieve its expected performance. Robust optimization deals with optimal designs, which are robust with respect to small (or even large) perturbations of the optimization setpoint conditions. That means, the optimal designs computed should still be good designs, even if the input parameters for the optimization problem formulation are changed by a non-negligible amount. Thus even more experimental or numerical effort can be saved.
     In this talk, we aim at an improvement of existing simulation and optimization technology, so that numerical uncertainties are identified, quantized and included in the overall optimization procedure, thus making robust design in this sense possible. Beside the scalar valued uncertainties in the flight conditions we consider the shape itself as an uncertainty source and apply a Karhunen-Loève expansion to approximate the infinite-dimensional probability space. To overcome the curse of dimensionality an adaptively refined sparse grid is used in order to compute statistics of the solution. These investigations are part of the current German research program MUNA.

Claudia Schillings
University of Trier, Germany

Volker Schulz
University of Trier, Germany

We propose a method for computing transport and diffusion on a moving surface. The idea is based on a diffuse interface model in which a bulk diffusion-advection equation is solved in a thin layer that contains the surface. The conserved quantity in the bulk domain is the concentration weighted by a density which vanishes on the boundary of the thin layer. The discrete equations are then formulated on a moving narrow band. We conclude with some numerical experiments.

Vanessa Styles
University of Sussex, UK

In the framework of optimization approach to brittle fracture, we study shape and topology optimization problems for cracks with contact and kink. The problems deal with finding optimal parameters for geometrical variables which represent an unknown crack in a domain. Thus, for a predefined crack path, the one-parametric shape optimization problem with respect to the length-parameter along the path describes appearance and quasi-static propagation of a crack, for instance, during the delamination process. The respective two-parametric optimization problem refers to unknown shape parameters of the crack length and of the angle of its kink at the fixed bifurcation point.
     For proper modeling of solids with cracks we get the variational formulation of crack problems accounting conditions of contact between the crack faces, which results in unilaterally constrained state problems. The other necessary ingredient includes kinematic description of cracks as codimensional-one open manifolds. We manage the crack evolution with the help of homeomorphic maps and implicit surface functions solving a linear transport equation with given a-priori velocity field. To calculate a solution, we reformulate the state-constrained optimization problem on the set of extremal points using differentiability properties of the cost function. This approach requires the shape and topology sensitivity analysis of crack problems with respect to perturbations of geometric variables of the crack.
     Numerical tests demonstrate that the suggested optimization approach refines the classic Griffith law of fracture in the cases where the latter one was not applicable.

References
[1] M. Hintermüller, V.A. Kovtunenko, and K. Kunisch. An optimization approach for the delamination of a composite material with non-penetration, in: Free and Moving Boundaries: Analysis, Simulation and Control, R. Glowinski and J.-P. Zolésio (Eds.), Lecture Notes Pure Appl. Math. 252, pp. 331-348, Chapman & Hall/CRC, Boca Raton, FL, 2007.
[2] A.M. Khludnev and V.A. Kovtunenko. Analysis of Cracks in Solids, WIT-Press, Southampton, Boston, 2000.
[3] A.M. Khludnev, V.A. Kovtunenko, and A. Tani. Evolution of a crack with kink and non-penetration, Research Report 07/001, Keio University, 2007.
[4] V.A. Kovtunenko. Primal-dual methods of shape sensitivity analysis for curvilinear cracks with non-penetration, IMA J. Appl. Math., 71, pp. 635-657, 2006.
[5] V.A. Kovtunenko. Interface cracks in composite orthotropic materials and their delamination via global shape optimization, Optim. Eng., 7, pp. 173-199, 2006.
[6] V.A. Kovtunenko, K. Kunisch, and W. Ring. Perturbation and motion of cracks based on level sets and velocities, SFB F003 Bericht 308, Graz, 2004.
[7] V.A. Kovtunenko and I.V. Sukhorukov. Optimization formulation of the evolutionary problem of crack propagation under quasibrittle fracture, Appl. Mech. Tech. Phys., 47(5), pp. 704-713, 2006.


Victor A. Kovtunenko
University of Graz, Austria and Russian Academy of Sciences, Novosibirsk, Russia

The control problem with multidimensional integral functional under wave type constraints for control is considered. Next a type of deformation with control of the domain is described and then we define suitable shape functional. Having defined trajectory and control of deformation dual dynamic programming tools are applied to derive optimality condition for the shape functional with respect to that deformation.

Andrzej Nowakowski
University of Lodz, Poland