67 research outputs found
Analysis and computations for a model of quasi-static deformation of a thinning sheet arising in superplastic forming
We consider a mathematical model for the quasi-static deformation of a thinning sheet. The model couples a first-order equation for the thickness of the sheet to a prescribed curvature equation for the displacement of the sheet. We prove a local in time existence and uniqueness theorem for this system when the sheet can be written as a graph. A contact problem is formulated for a sheet constrained to be above a mould. Finally we present some computational results
Primal-dual active set methods for Allen-Cahn variational inequalities with non-local constraints
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Modelling and simulation of cell migration in heterogeneous media
Cell motility is a complex process comprising, among others, cytoskeletal remodelling and growth, active force generation, chemical transport and reaction of multiple substances inside and outside the cell as well as on the cell membrane, anisotropic material behavior of the cell and of the extracellular matrix (ECM), and proteolytic cleavage of the ECM. We are working on an evolving finite element based computational framework that is able to incorporate models and methods for all the key features of cell motility. The approach exploits a mesoscopic point of view and aims at a comprehensive 3D model allowing for the simulation of cell motility at reasonable computational costs. In the presentation we will first sketch the overall framework, addressing all the respective models and methods, and will then focus on some essential subunits
Allen-Cahn and Cahn-Hilliard variational inequalities solved with Optimization Techniques
Parabolic variational inequalities of Allen-Cahn and Cahn-
Hilliard type are solved using methods involving constrained optimization. Time discrete variants are formulated with the help of Lagrange multipliers for local and non-local equality and inequality constraints. Fully discrete problems resulting from finite element discretizations in space are solved with the help of a primal-dual active set approach. We
show several numerical computations also involving systems of parabolic variational inequalities
Relating phase field and sharp interface approaches to structural topology optimization
A phase field approach for structural topology optimization which allows for topology
changes and multiple materials is analyzed. First order optimality conditions are
rigorously derived and it is shown via formally matched asymptotic
expansions that these conditions converge to classical first order conditions obtained in
the context of shape calculus. We also discuss how to deal with triple junctions where
e.g. two materials and the void meet. Finally, we present several
numerical results for mean compliance problems and a cost involving the least square error
to a target displacement
Optimal control of the propagation of a graph in inhomogeneous media
We study an optimal control problem for viscosity solutions of a HamiltonâJacobi equation describing the propagation of a one-dimensional graph with the control being the speed function. The existence of an optimal control is proved together with an approximate controllability result in the -norm. We prove convergence of a discrete optimal control problem based on a monotone finite difference scheme and describe some numerical results
Phase-field approaches to structural topology optimization
The mean compliance minimization in structural topology optimization is solved with the help of a phase field approach. Two steepest descent approaches based on L2- and H-1 gradient flow dynamics are discussed.
The resulting flows are given by Allen-Cahn and Cahn-Hilliard type dynamics coupled to a linear elasticity system. We finally compare numerical results obtained from the two different approaches
Existence of solution to a system of PDEs modeling the crystal growth inside lithium batteries
The life-cycle of electric batteries depends on a complex system of
interacting electrochemical and growth phenomena that produce dendritic
structures during the discharge cycle. We study herein a system of 3 partial
differential equations combining an Allen--Cahn phase-field model (simulating
the dendrite-electrolyte interface) with the Poisson--Nernst--Planck systems
simulating the electrodynamics and leading to the formation of such dendritic
structures. We prove novel existence, uniqueness and stability results for this
system and use it to produce simulations based on a finite element code.Comment: 27 pages, 22 figures, free software and open source code availabl
Cahn--Hilliard inpainting with the double obstacle potential
The inpainting of damaged images has a wide range of applications, and many different mathematical methods have been proposed to solve this problem. Inpainting with the help of Cahn{Hilliard models has been particularly successful, and it turns out that Cahn{Hilliard inpainting with the double obstacle potential can lead to better results compared to inpainting with a smooth double well potential. However, a mathematical analysis of this approach is missing so far. In this paper we give first analytical results for a Cahn--Hilliard double obstacle inpainting model regarding existence of global solutions to the time-dependent problem and stationary solutions to the time-independent problem without constraints on the parameters involved. With the help of numerical results we show the effectiveness of the approach for binary and grayscale images
Preconditioning nonlocal multi-phase flow
We propose an efficient solver for saddle point problems arising from finite element approximations of nonlocal multi-phase Allen-Cahn variational inequalities. The solver is seen to behave mesh independently and to have only a very mild dependence on the number of phase field variables. In addition we prove convergence, in three GMRES iterations, of the approximation of the two phase problem, regardless of mesh size or interfacial width. Numerical results are presented that illustrate the competitiveness of this approach
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