57 research outputs found
Discrete hyperbolic curvature flow in the plane
Hyperbolic curvature flow is a geometric evolution equation that in the plane
can be viewed as the natural hyperbolic analogue of curve shortening flow. It
was proposed by Gurtin and Podio-Guidugli (1991) to model certain wave
phenomena in solid-liquid interfaces. We introduce a semidiscrete finite
difference method for the approximation of hyperbolic curvature flow and prove
error bounds for natural discrete norms. We also present numerical simulations,
including the onset of singularities starting from smooth strictly convex
initial data.Comment: 23 pages, 10 figure
An unconditionally stable finite element scheme for anisotropic curve shortening flow
summary:Based on a recent novel formulation of parametric anisotropic curve shortening flow, we analyse a fully discrete numerical method of this geometric evolution equation. The method uses piecewise linear finite elements in space and a backward Euler approximation in time. We establish existence and uniqueness of a discrete solution, as well as an unconditional stability property. Some numerical computations confirm the theoretical results and demonstrate the practicality of our method
Optimal control of the propagation of a graph in inhomogeneous media
We study an optimal control problem for viscosity solutions of a HamiltonāJacobi 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
A novel approach to shape optimisation with Lipschitz domains
This article introduces a novel method for the implementation of shape
optimisation with Lipschitz domains. We propose to use the shape derivative to
determine deformation fields which represent steepest descent directions of the
shape functional in the topology. The idea of our approach is
demonstrated for shape optimisation of -dimensional star-shaped domains,
which we represent as functions defined on the unit -sphere. In this
setting we provide the specific form of the shape derivative and prove the
existence of solutions to the underlying shape optimisation problem. Moreover,
we show the existence of a direction of steepest descent in the
topology. We also note that shape optimisation in this context is closely
related to the Laplacian, and to optimal transport, where we highlight
the latter in the numerics section. We present several numerical experiments in
two dimensions illustrating that our approach seems to be superior over a
widely used Hilbert space method in the considered examples, in particular in
developing optimised shapes with corners
Shape optimisation in the topology with the ADMM algorithm
We present a general shape optimisation framework based on the method of
mappings in the topology. We propose steepest descent and
Newton-like minimisation algorithms for the numerical solution of the
respective shape optimisation problems. Our work is built upon previous work of
the authors in (Deckelnick, Herbert, and Hinze, ESAIM: COCV 28 (2022)), where a
framework for star-shaped domains is proposed. To illustrate our
approach we present a selection of PDE constrained shape optimisation problems
and compare our findings to results from so far classical Hilbert space methods
and recent -approximations
- ā¦