Günter Leugering

Friedrich-Alexander-University Erlangen-Nürnberg | FAU · Department of Mathematics

Today’s infrastructures are mainly designed heuristically using state-of-the-art simulation software and engineering approaches. However, due to complexity, only part of the restrictions and costs that show up during the lifecycle can be taken into account. In this paper, we focus on a typical and important class of infrastructure problems, the des...

This paper is devoted to describing the asymptotic behavior of a structure consisting of thin elastic planar beams coupled at flexible joints. As the thickness of the beams tends to zero, we establish classical 1‐D beam equations for each individual structural element and transmission conditions across the joints.

The exact boundary controllability for hyperbolic systems can not be realized generally on a network with loops (see [16]). In this paper we consider the exact boundary controllability of nodal profile on a network with loops. Precisely speaking, on a network with a triangle-like loop, when nodal profiles are given at various kinds of nodes, differ...

This paper concerns a system of equations describing the vibrations of a planar network of nonlinear Timoshenko beams. The authors derive the equations and appropriate nodal conditions, determine equilibrium solutions and, using the methods of quasilinear hyperbolic systems, prove that for tree-like networks the natural initial-boundary value probl...

We consider hot forming processes, in which a metal solid body is deformed by several rolls in order to obtain a desired final shape. To minimize cutting scrap and to ensure that this shape satisfies the required tolerances as precisely as possible, we formulate an optimal control problem where we use the trajectories of the rolls as control functi...

In this paper we present a chain of mathematical models that enables the numerical simulation of the airlay process and the investigation of the resulting nonwoven material by means of virtual tensile strength tests. The models range from a highly turbulent dilute fiber suspension flow to stochastic surrogates for fiber lay-down and web formation a...

Mesocrystalline particles have been recognized as a class of multifunctional materials with potential applications in different fields. However, the internal organization of nanocomposite mesocrystals and its influence on the final properties have not yet been investigated. In this paper, a novel strategy based on electrodynamic simulations is deve...

By equivalently replacing the dynamical boundary condition by a kind of nonlocal boundary conditions, and noting a hidden regularity of solution on the boundary with a dynamical boundary condition, a constructive method with modular structure is used to get the local exact boundary controllability for 1-D quasilinear wave equations with dynamical b...

For a system that is governed by the isothermal Euler equations with friction for ideal gas, the corresponding field of characteristic curves is determined by the velocity of the flow. This velocity is determined by a second-order quasilinear hyperbolic equation. For the corresponding initial-boundary value problem with Neumann-boundary feedback, w...

A class of nonsmooth shape optimization problems for variational inequalities is considered. The variational inequalities model elliptic boundary value problems with the Signorini type unilateral boundary conditions. The shape functionals are given by the first order shape derivatives of the elastic energy. In such a way the singularities of weak s...

While the numerical discretization of one-dimensional blood flow models for vessels with viscoelastic wall properties is widely established, there is still no clear approach on how to couple one-dimensional segments that compose a network of viscoelastic vessels. In particular for Voigt-type viscoelastic models, assumptions with regard to boundary...

Mathematical modeling of biochemical pathways is an important resource in Synthetic Biology, as the predictive power of simulating synthetic pathways represents an important step in the design of synthetic metabolons. In this paper, we are concerned with the mathematical modeling, simulation, and optimization of metabolic processes in biochemical m...

We study a Dirichlet optimal control problem for a quasi-linear monotone elliptic equation, the so-called weighted p-Laplacian problem. The coefficient of the p-Laplacian, the weight u, we take as a control in BV (Ω) ∩ L∞(Ω). In this article, we use box-type constraints for the control such that there is a strictly positive lower and some upper bou...

We study a Dirichlet optimal design problem for a quasi-linear monotone p-biharmonic equation with control and state constraints. We take the coefficient of the p-biharmonic operator as a design variable in BV (Ω) . In this article, we discuss the relaxation of such problem.

We study a Dirichlet optimal design problem for a quasi-linear monotone p-biharmonic equation with control and state constraints. We take the coefficient of the p-biharmonic operator as a design variable in BV (Ω) . In this article, we discuss the relaxation of such problem.

We discuss a regularization of state-constrained optimal control problem via a Henig relaxation of ordering cones. Considering a state-constrained optimal control problem, the pointwise state constraint is replaced by an inequality condition involving a so-called Henig dilating cone. It is shown that this class of cones provides a reasonable solid...

Unit operations and product design are the two most important pillars of chemical engineering. Product design is the formation, formulation, handling, manufacturing, and characterization of complex multiphase products with specific properties and is thus at the core of mesoscale science and engineering. The applications define the required product...

We consider a system of scalar nonlocal conservation laws on networks that model a highly re-entrant multi-commodity manufacturing system as encountered in semiconductor production. Every single commodity is mod-eled by a nonlocal conservation law, and the corresponding PDEs are coupled via a collective load, the work in progress. We illustrate the...

We consider shape optimization for objects illuminated by light. More precisely, we focus on time-harmonic solutions of the Maxwell system in curl-curl-form scattered by an arbitrary shaped rigid object. Given a class of cost functionals, including the scattered energy and the extinction cross section, we develop an adjoint-based shape optimization...

In this review article the theoretical foundations for shape-topological sensitivity analysis of elastic energy functional in bodies with nonlinear cracks and inclusions are presented. The results obtained can be used to determine the location and the shape of inclusions which influence in a desirable way the energy release at the crack tip. In con...

In this paper, we study an optimal control problem for the mixed boundary value problem for an elastic body with quasistatic evolution of an internal damage variable. We suppose that the evolution of microscopic cracks and cavities responsible for the damage is described by a nonlinear parabolic equation. A density of surface traction p acting on a...

We consider an optimal control problem for nonlinear degenerate elliptic problems involving an anisotropic p-Laplacian and Dirichlet boundary conditions. We take the matrix-valued coefficients A(x) of such system as a control in \(L^{p/2}(\varOmega ;\mathbb {R}^{\frac{N(N+1)}{2}})\). One of the important features of the admissible controls is the f...

Pipeline networks for gas transportation often contain circles. For such networks it is more difficult to determine the stationary states than for networks without circles. We present a method that allows to compute the stationary states for subsonic pipe flow governed by the isothermal Euler equations for certain pipeline networks that contain cir...

We study a wave equation in one space dimension with a general diffusion coefficient which degenerates on part of the boundary. Degeneracy is measured by a real parameter $\mu_a>0$. We establish observability inequalities for weakly (when $\mu_a \in [0,1[$) as well as strongly (when $\mu_a \in [1,2[$) degenerate equations. We also prove a negative...

We prove the Eshelby theorem for an ellipsoidal piezoelectric inclusion in an infinite piezoelectric material. Explicit formulas for the link and polarization matrices are derived. Passing to the limits with respect to parameters in the corresponding equations, the result is extended to cases when either the inclusion or the surrounding material is...

In this article the authors continue the discussion in [9] about inverse problems for second order elliptic and hyperbolic equations on metric trees from boundary measurements. In the present paper we prove the identifiability of varying densities of a planar tree-like network of strings along with the complete information on the graph, i.e. the le...

This work presents the application of a Fully Implicit Method for Ostwald Ripening (FIMOR) for simulating the ripening of ZnO quantum dots (QDs). Its stable numerics allow FIMOR to employ the full exponential term of the Gibbs–Thomson equation which significantly outperforms the common Taylor-approximation at typical QD sizes below 10 nm. The imple...

We consider the shape-topological control of a singularly perturbed variational inequality. As the reference geometry-dependent state problem the paper addresses a heterogeneous medium with a micro-object (defect) and a macro-object (crack) modeled in 2d and illustrated analytically in 1d. The corresponding nonlinear optimization problem subject to...

The class of nonsmooth shape optimization problems for variational inequalities is con-sidered. The variational inequalities model elliptic boundary value problems with the Signorini type unilateral boundary conditions. The shape functionals are given by the first order shape derivatives of the elastic energy. In such a way the singularities of wea...

In this article we study an optimal control problem for a nonlinear monotone Dirichlet problem where the controls are taken as matrix-valued coefficients in $L^\infty(\Omega;\mathbb{R}^{N\times N} )$. For the exemplary case of a tracking cost functional, we derive first order optimality conditions. This is the first part out of two articles. This f...

In this paper we study an optimal control problem for a nonlinear monotone Dirichlet problem where the controls are taken as the matrix-valued coefficients in $L^\infty(\Omega;\mathbb{R}^{N\times N} )$. Given a suitable cost function, the objective is to provide a substantiation of the first order optimality conditions using the concept of converge...

In optimal control loops delays can occur, for example through transmission via digital communication channels. Such delays influence the state that is generated by the implemented control. We study the effect of a delay in the implementation of L 2-norm minimal Neumann boundary controls for the wave equation. The optimal controls are computed as s...

A review of results on first order shape-topological differentiability of energy functionals for a class of variational inequalities of elliptic type is presented.The velocity method in shape sensitivity analysis for solutions of elliptic unilateral problems is established in the monograph (Sokołowski and Zolésio, Introduction to Shape Optimization...

We propose a novel approach to adaptivity in FEM based on local sensitivities for topological mesh changes. To this end, we consider refinement as a continuous operation on the edge graph of the finite element discretization, for instance by splitting nodes along edges and expanding edges to elements. Thereby, we introduce the concept of a topologi...

We summarize recent theoretical results as well as numerical results on the feedback stabilization of first order quasilinear hyperbolic systems (on networks). For the stabilization linear feedback controls are applied at the nodes of the network. This yields the existence and uniqueness of a C 1-solution of the hyperbolic system with small C 1-nor...

In this part a variety of important applications in the field of PDE constrained optimization and optimal control is presented. The results range from physical and chemical over electronic to biomedical and -technological applications. Furthermore, an introduction to a new collection of prototypical problems in PDE constrained optimization is given...

We consider optimal control problems governed by nonlinear hyperbolic conservation laws at junctions and analyze in particular the Fréchet-differentiability of the reduced objective functional. This is done by showing that the control-to-state mapping of the considered problems satisfies a generalized notion of differentiability. We consider both,...

We consider the processes of particle nucleation, growth, precipitation and ripening via modeling by nonlinear 1-D hyperbolic partial integro differential equations. The goal of this contribution is to provide a concise predictive forward modeling of the processes including appropriate goal functions and to establish a mathematical theory for the o...

A study was conducted to propose a model of an elastic body with Timoshenko thin inclusions with possible delamination. The purpose of this study was to formulate correct differential and variational statements of the corresponding problem and to analyze the convergence of solutions if the inclusion rigidity parameters tended to zero and infinity....

In this contribution the optimal boundary control problem for a first order nonlinear, nonlocal hyperbolic PDE is studied. Motivated by various applications ranging from re-entrant manufacturing systems to particle synthesis processes, we establish the regularity of solutions for $W^{1,p}$-data. Based on a general $L^2$ tracking type cost functiona...

We derive an asymptotic one-dimensional model of a thin piezoelectric rod by means of a dimension reduction procedure. The rod is made from a heterogeneous material with a possibly varying cross-section and distorted ends. Asymptotically exact error estimates are derived. Representation formulas for effective moduli are established and, for a concr...

An elastic body weakened by small cracks is considered in the framework of unilateral variational problems in linearized elasticity. The frictionless contact conditions are prescribed on the crack lips in two spatial dimensions, or on the crack faces in three spatial dimensions. The weak solutions of the equilibrium boundary value problem for the e...

We present an asymptotic approach to find optimal rotations of orthotropic material inclusions inside an isotropic linear elastic matrix. We compute approximate optimal solutions with respect to compliance and a stress based cost functional. We validate the local and global quality of the candidate solutions by means of finite element based paramet...

We study an optimal control problem for the mixed boundary value problem for an elastic body with quasistatic evolution of an internal damage variable. We use the damage field $\zeta=\zeta(t,x)$ as an internal variable which measures the fractional decrease in the stress-strain response. When $\zeta=1$ the material is damage-free, when $\zeta=0$ th...

Problems of optimization and optimal control subject to constraints governed by partial differential equations (PDEs) arise in a huge variety of industrial, technological, economic, medical and environmental applications. The questions that appear in this field range from shape and topology optimization problems, such as optimal design of the wing...

We propose a model for a two-dimensional elastic body with a thin elastic inclusion modeled by a beam equation. Moreover, we assume that a delamination of the inclusion may take place resulting in a crack. Nonlinear boundary conditions are imposed at the crack faces to prevent mutual penetration between the faces. Both variational and differential...

We propose a novel approach to adaptive refinement in FEM based on local sensitivities for node insertion. To this end, we consider refinement as a continuous graph operation, for instance by splitting nodes along edges. Thereby, we introduce the concept of the topological mesh derivative for a given objective function. For its calculation, we rely...

Commonly applied models to study vocal fold vibrations in combination with air flow distributions are self-sustained physical models of the larynx consisting of artificial silicone vocal folds. Choosing appropriate mechanical parameters and layer geometries for these vocal fold models while considering simplifications due to manufacturing restricti...

We introduce the Griffith shape functional as the distributed shape derivative of the elastic energy evaluated in a domain with a crack, with respect to the crack length. We are interested in the dependence of this functional on domain perturbations far from the crack. As a result, the directional shape and topological derivatives of the nonsmooth...

We investigate the propagation of cracks in 2-d elastic domain-s, which are subjected to quasi-static loading scenarios. As we take cohesive effects along the crack path into account and impose a non-penetration con-dition, inequalities appear in the constitutive equations describing the elastic behavior of a domain with crack. In contrast to exist...

The paper concerns the analysis of equilibrium problems for 2D elastic bodies with thin inclusions modeled in the framework of Timoshenko beams. The first focus is on the well-posedness of the model problem in a variational setting. Then delaminations of the inclusions are considered, forming a crack between the elastic body and the inclusion. Nonl...

In this paper we study an optimal control problem associated to a linear degenerate elliptic equation with mixed boundary conditions. We adopt the square symmetric matrix of coefficients A(x)=[aij(x)]i,j=1,...,N as a control in L1(S). The characteristic feature of the class of admissible controls is the fact that eigenvalues of the coefficient matr...

We consider the subcritical gas flow through star-shaped pipe networks. The gas flow is modeled by the isothermal Euler equations with friction. We stabilize the isothermal Euler equations locally around a given stationary state on a finite time interval. For the stabilization we apply boundary feedback controls with time-varying delays. The delays...

Our goal is to design brittle composite materials yielding maximal energy dissipation for a given static load case. We focus on the effect of variation of fiber shapes on resulting crack paths and thus on the fracture energy. To this end, we formulate a shape optimization problem, in which the cost function is the fracture energy and the state prob...

This paper concerns the generalization and regularization of a nonlinear scalarization method by Pascoletti and Serafini, [14], to abstract partially ordered Banach spaces. In particular, we put emphasis on the possibility that the ordering cone might have an empty interior. All this is motivated by applying the method to a vector optimization prob...

The paper is concerned with the control of the shape of rigid and elastic inclusions and crack paths in elastic bodies. We provide the corresponding problem formulations and analyze the shape sensitivity of such inclusions and cracks with respect to different perturbations. Inequality type boundary conditions are imposed at the crack faces to provi...

We present an overview on recent results concerning hyperbolic systems on networks. We present a summary of theoretical results on existence, uniqueness and stability. The established theory extends previously known results on the Cauchy problem for nonlinear, 2×2 hyperbolic balance laws. The proofs are based on Wave-Front Tracking and therefore we...

The production of nanoscaled particulate products with exactly pre-defined characteristics is of enormous economic relevance. Although there are different particle formation routes they may all be described by one class of equations. Therefore, simulating such processes comprises the solution of nonlinear, hyperbolic integro-partial differential eq...

We consider the feedback stabilization of quasilinear hyperbolic systems on star-shaped networks. We present boundary feedback controls with varying delays. The delays are given by C1-functions with bounded derivatives. We obtain the existence of unique C1-solutions on a given finite time interval. In order to measure the system evolution, we intro...

The authors consider the problem of boundary feedback stabilization of the 1D Euler gas dynamics locally around stationary states and prove the exponential stability with respect to the H 2-norm. To this end, an explicit Lyapunov function as a weighted and squared H 2-norm of a small perturbation of the stationary solution is constructed. The autho...

In this contribution the identification of new reaction conditions for the production of nearly monodisperse silicon nanoparticles via the pyrolysis of monosilane in a hot wall reactor is considered. For this purpose a full finite volume model has been combined with a state-of-the-art trust-region optimisation algorithm for process control. Verifie...

We consider a thin elastic plate with piezo patches mounted on top of it. Electrodes are located on the upper and, depending on the devices, at the lower surface of the patches. This piezo actuator is coupled to an elastic body. We develop an asymptotic procedure to derive a two‐dimensional approximation of the entire structure. As a result, we obt...

We study a Dirichlet optimal control problem associated with a linear elliptic equation the coefficients of which we take as controls in $L^1(\Omega)$. In particular, when the coefficient matrix is taken to satisfy the decomposition $B(x)=\rho(x)A(x)$ with a scalar function $\rho$, we allow the $\rho$ to degenerate. Such problems are related to var...

Part I Optimization of Water Supply Networks.- Modelling and Numerical Simulation of Pipe Flow Problems in Water Supply Systems.- Simulation and Continuous Optimization.- Mixed Integer Optimizationof Water Supply Networks.- Nonlinear and Mixed Integer Linear Programming.- Part II Optimal Control of Sewer Networks.- Optimal Control of Sewer Networks...

This paper concerns a system of nonlinear wave equations describing the vibrations of a 3-dimensional network of elastic strings. The authors derive the equations and appropriate nodal conditions, determine equilibrium solutions, and, by using the methods of quasilinear hyperbolic systems, prove that for tree networks the natural initial, boundary...

We present topology optimization of piezoelectric loudspeakers using the SIMP method and topology gradient based methods along with analytical and numerical results.

We present the analysis for finding optimal locations and rotations of anisotropic material inclusions in a matrix material by using the polarization matrix. We compare different types of cost funcþionale, in particular local ones, and show their respective differences. We use the Eshelby theorem and the representation of stresses based on the link...

This special volume focuses on optimization and control of processes governed by partial differential equations. The contributors are mostly participants of the DFG-priority program 1253: Optimization with PDE-constraints which is active since 2006. The book is organized in sections which cover almost the entire spectrum of modern research in this...

We compare the quality and generation performance of the optimal control sequence produced by the software frameworks BlueM.MPC and Lamatto.

A hydrodynamic process model based on shallow water equations is discretized on 1D-networks with the method of finite volumes. Based on the finite volumes we replace algebraic coupling conditions by a consistent finite volume junction model. We use discrete adjoint computation for one step Runge-Kutta schemes to generate fast and robust gradients f...

The aim of the workshop was to study geometrical objects and their sensitivities in applications based on partial differential equations or differential variational inequalities. Focus topics comprised analytical investigations, numerical developments, issues in applications as well as new and future directions. Particular emphasis was put on: (i)...

We consider second order problems on metric graphs under given boundary and nodal conditions. We consider the problem of changing the topology of the underlying graph in that we replace a multiple node by an imported subgraph, or, in reverse, concentrate a subgraph to a single node or delete or add edges, respectively. We wish to do so in some opti...

A shape optimization problem in three spatial dimensions for an elasto-dynamic piezoelectric body coupled to an acoustic chamber is introduced. Well-posedness of the problem is established and first order necessary optimality conditions are derived in the framework of the boundary variation technique. In particular, the existence of the shape gradi...

The ability of velocity methods to describe changes in topology by creating defects such as holes is investigated. For shape optimization, energy-type objective functions are considered, which depend on the geometry by means of the state variables. The ...

The optimization and manufacturing of an auxetic structure is presented. An inverse homogenization method is used to obtain the optimized geometry shown in the figure. The resulting structure is then produced using selective electron beam melting. The numerically predicted properties are experimentally verified.

The visual appearance of the artificial world is largely governed by films or composites containing particles with at least one dimension smaller than a micron. Over the past century and a half, the optical properties of such materials have been scrutinized and a broad range of colorant products, based mostly on empirical microstructural improvemen...

We study the isothermal Euler equations with friction and consider non-stationary solutions locally around a stationary subcritical state on a finite time interval. The considered control system is a quasilinear hyperbolic system with a source term. For the corresponding initial-boundary value problem we prove the existence of a continuously differ...

We investigate the evolution and propagation of cracks in 2-d elastic domains, which are subjected to quasi-static loading scenarios. In addition to the classical variational formulation, where the standard potential energy is minimized over the cracked domain under physical conditions characterizing the behavior of the material close to the crack...

Today, the prevention and treatment of voice disorders is an ever-increasing health concern. Since many occupations rely on verbal communication, vocal health is necessary just to maintain one's livelihood. Commonly applied models to study vocal fold vibrations and air flow distributions are self sustained physical models of the larynx composed of...

In brittle composite materials, failure mechanisms like debonding of the matrix-fiber interface or fiber breakage can result in crack deflection and hence in the improvement of the damage tolerance. More generally it is known that high values of fracture energy dissipation lead to toughening of the material. Our aim is to investigate the influence...

We investigate the occurrence of self-penalization in topology optimization problems for piezoceramic-mechanical composites. Our main goal is to give physical interpretations for this phenomenon, i.e., to study the question why for various problems intermediate material values are not optimal in the absence of explicit penalization of the pseudo de...

In this paper we consider the control of cracks in elastic bodies with rigid inclusion. We first describe the problem statement, provide an equivalent formulation as a variational inequality and prove existence and uniqueness of solutions. Furthermore, we consider this problem as a limiting problem when the elasticity parameters of the inclusion te...

We analyze the subcritical gas flow through fan-shaped networks of pipes, that is, through tree-shaped networks with exactly one node where more than two pipes meet. The gas flow in pipe networks is modeled by the isothermal Euler equations, a hyperbolic PDE system of balance laws. For this system we analyze stationary states and classical nonstati...

Optimal control problems (OCPs) for the Navier–Stokes equations have been the subject of extensive study in recent years. A systematic mathematical and numerical analysis of OCPs of different types (e.g., having Dirichlet, Neumann, and distributed controls) for the steady-state Navier–Stokes system was given by Abergel and Temam (1990), Fursikov, G...

The book focuses on all of these aspects from two perspectives. First, a rigorous and mostly self-contained mathematical theory of PDEs on reticulated domains together with well-posedness for the governing optimal control problems is described. The concept of optimal control problems for PDEs in varying such domains, and hence in varying Banach spa...

In this chapter, we are mainly interested in optimal L ∞(Ω)-control of the coefficients of an elliptic Dirichlet problem. We prove an existence result for this problem using the direct method of Calculus of Variations, and then we provide a sensitivity analysis of this problem on a reticulated structured with respect to the domain perturbation and...

In this chapter, we study the asymptotic behavior of the following class of the parabolic optimal control problems (OCPs) Ie(ue,ye)=ò0TòW+(ye- q0)2 d x d t + ò0TòGeu2e d x¢ d t ® inf,I_{\varepsilon}(u_{\varepsilon},y_{\varepsilon})=\int_0^T\int_{\Omega^{+}}(y_{\varepsilon}- q_0)^2 \,\mathrm{d} x\,\mathrm{d} t \, +\,\int_0^T\int_{\Gamma_{\varepsilo...

In this chapter, we continue the discussion of the optimal control problems (OCPs) described in Chap. 3. Our main focus here is a proper regularization and approximation of these problems. In order to clarify the significance of these notions, we begin with the following abstract extremal problem.

In this chapter, we study a class of optimal control problems (OCPs) for a linear elliptic equation in a domain Ω ε ⊂ℝ n (thick multistructure), whose boundary ∂Ω ε contains a very highly oscillating part with respect to ε, as ε→0. We consider this problem assuming that there are two types of the controls active via Neumann and Dirichlet boundary c...

The main interest in this book is in mathematical models of optimal control problems (OCPs) that depend on some small parameter ε. In many mathematical problems, which come from natural or engineering sciences, industrial applications, or abstract mathematical questions all by themselves, some parameters appear (small or large, of geometric or cons...