Sven-Joachim Kimmerle

An Electrohydrodynamic Equilibrium Shape Problem for Polymer Electrolyte Membranes in Fuel Cells

We present a novel, thermodynamically consistent, model for the charged-fluid flow and the deformation of the morphology of polymer electrolyte membranes (PEM) in hydrogen fuel cells. The solid membrane is assumed to obey linear elasticity, while the pore is completely filled with protonated water, considered as a Stokes flow. The model comprises a system of partial differential equations and boundary conditions including a free boundary between liquid and solid. Our problem generalizes the well-known Nernst-Planck-Poisson-Stokes system by including mechanics. We solve the coupled non-linear equations numerically and examine the equilibrium pore shape. This computationally challenging problem is important in order to better understand material properties of PEM and, hence, the design of hydrogen fuel cells.

Optimal control of an elastic crane-trolley-load system - a case study for optimal control of coupled ODE-PDE systems

ABSTRACT We present a mathematical model of a crane-trolley-load model, where the crane beam is subject to the partial differential equation (PDE) of static linear elasticity and the motion of the load is described by the dynamics of a pendulum that is fixed to a trolley moving along the crane beam. The resulting problem serves as a case study for optimal control of fully coupled partial and ordinary differential equations (ODEs). This particular type of coupled systems arises from many applications involving mechanical multi-body systems. We motivate the coupled ODE-PDE model, show its analytical well-posedness locally in time and examine the corresponding optimal control problem numerically by means of a projected gradient method with Broyden-Fletcher-Goldfarb-Shanno (BFGS) update.

Optimal Control of Mean Field Models for Phase Transitions

Abstract Various models prescribe precipitation due to phase transitions. On a macroscopic level the well-known Lifshitz-Slyozov-Wagner (LSW) models and its discrete analogons, so-called mean field models, prescribe the size evolution of precipitates for two-phase systems. For industrial tasks it is desirable to control the resulting distribution of droplet volume. While there are optimal control results for phase-field models and for nonlinear hyperbolic conservation laws, it seems that control problems for LSW equations and mean field models, including measure-valued solutions or switching conditions, have not been considered so far. We formulate the model for this important new control problem and present first numerical results.

Macroscopic diffusion models for precipitation in crystalline gallium arsenide

Based on a thermodynamically consistent model for precipitation in gallium arsenide crystals including surface tension and bulk stresses by Dreyer and Duderstadt [DD08], we propose two different mathematical models to describe the size evolution of liquid droplets in a crystalline solid. The first model treats the diffusion-controlled regime of interface motion, while the second model is concerned with the interface-controlled regime of interface motion. Our models take care of conservation of mass and substance. These models generalise the well-known MullinsSekerka model [MS63] for Ostwald ripening. We concentrate on arsenic-rich liquid spherical droplets in a gallium arsenide crystal. Droplets can shrink or grow with time but the centres of droplets remain fixed. The liquid is assumed to be homogeneous in space. Due to different scales for typical distances between droplets and typical radii of liquid droplets we can derive formally so-called mean field models. For a model in the diffusion-controlled regime we prove this limit by homogenisation techniques under plausible assumptions. These mean field models generalise the Lifshitz-Slyozov-Wagner model, see [LS61], [Wag61], which can be derived from the Mullins-Sekerka model rigorously, see [Nie99], [NO01], and is wellunderstood. Mean field models capture the main properties of our system and are well adapted for numerics and further analysis. We determine possible equilibria and discuss their stability. Numerical evidence suggests in which case which one of the two regimes might be appropriate to the experimental situation.

A study of structure-exploiting SQP algorithms for an optimal control problem with coupled hyperbolic and ordinary differential equation constraints

In this article, structure-exploiting optimisation algorithms of the sequential quadratic programming (SQP) type are considered for optimal control problems with control and state constraints. Our approach is demonstrated for a 1D mathematical model of a vehicle transporting a fluid container. The model involves a fully coupled system of ordinary differential equations (ODE) and nonlinear hyperbolic first-order partial differential equations (PDE), although the ideas for exploiting the particular structure may be applied to more general optimal control problems as well. The time-optimal control problem is solved numerically by a full discretisation approach. The corresponding nonlinear optimisation problem is solved by an SQP method that uses exact first and second derivative information. The quadratic subproblems are solved using an active-set strategy. In addition, two approaches are examined that exploit the specific structure of the problem: (A) a direct method for the KKT system, and (B) an iterative method based on combining the limited-memory BFGS method with the preconditioned conjugate gradient method. Method (A) is faster for our model problem, but can be limited by the problem size. Method (B) opens the door for a potential extension of the truck-container model to three space dimensions.