23 citations · 37 across the 15 of their papers we have counts for
15 papers
Shape optimisation for adaptive -refinement: the one-dimensional case with residual based error estimators
Philip J. Herbert
We consider -refinement for the finite element discretisation of a Poisson problem. The goal of -refinement is to reposition the nodes of a computational mesh in order to bet…
Finite element approximation of the enthalpy formulation for Stefan problems on evolving surfaces
Philip J. Herbert, Thomas Sales, Chandrasekhar Venkataraman
We propose, and analyse, a spatially discrete evolving surface finite element method for the approximation of the enthalpy formulation of the two-phase Stefan problem posed on an e…
Small deformations of a near cylindrical tube for the Canham-Helfrich Energy with applications to biological membranes
Charles M. Elliott, Carsten Gräser, Philip J. Herbert
In this article we develop a quadratic energy which approximates the Canham-Helfrich energy for a tube-like surface with clamped boundary and area constraint. The energy is suited…
Combining diffuse and sharp interface methods in shape optimisation
Philip J. Herbert, Michael Hinze, Christian Kahle
We develop a concept for the numerical treatment of shape optimization problems based on the combination of phase field and sharp interface methods. On the one hand, phase field me…
Global convergence of -steepest descent for PDE constrained shape optimisation with semilinear elliptic equations in function space
Klaus Deckelnick, Philip J. Herbert, Michael Hinze
We prove global convergence in function space for the steepest descent method in shape optimisation with semilinear elliptic partial differential equations. Steepest descent is rea…
Finite element approximation to the non-stationary quasi-geostrophic equation
Dohyun Kim, Amiya K. Pani, Eun-Jae Park
In this paper, C1-conforming element methods are analyzed for the stream function formulation of a single layer non-stationary quasi-geostrophic equation in the ocean circulation m…
Shape optimization in with geometric constraints: a study in distributed-memory systems
Philip J. Herbert, Jose A. Pinzon Escobar, Martin Siebenborn
In this paper we present a shape optimization scheme which utilizes the alternating direction method of multipliers (ADMM) to approximate a direction of steepest descent in $W^{1,\…
The Stochastic Steepest Descent Method for Robust Optimization in Banach Spaces
Neil K. Chada, Philip J. Herbert
Stochastic gradient methods have been a popular and powerful choice of optimization methods, aimed at minimizing functions. Their advantage lies in the fact that that one approxima…
Shape Optimisation with : A connection between the steepest descent and Optimal Transport
Philip J. Herbert
In this work, we discuss the task of finding a direction of optimal descent for problems in Shape Optimisation and its relation to the dual problem in Optimal Transport. This link…
PDE constrained shape optimisation with first-order and Newton-type methods in the topology
Klaus Deckelnick, Philip J. Herbert, Michael Hinze
We present a general shape optimisation framework based on the method of mappings in the topology. We propose steepest descent and Newton-like minimisation algorithm…
An evolving space framework for Oseen equations on a moving domain
Ana Djurdjevac, Carsten Gräser, Philip J. Herbert
This article considers non-stationary incompressible linear fluid equations in a moving domain. We demonstrate the existence and uniqueness of an appropriate weak formulation of th…
A novel approach to shape optimisation with Lipschitz domains
Klaus Deckelnick, Philip J. Herbert, Michael Hinze
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 w…
A formula for membrane mediated point particle interactions on near spherical biomembranes
Charles M. Elliott, Philip J. Herbert
We consider a model of a biomembrane with attached proteins. The membrane is represented by a near spherical continuous surface and attached proteins are described as discrete rigi…
Small deformations of spherical biomembranes
Charles M. Elliott, Luke Hatcher, Philip J. Herbert
In this contribution to the proceedings of the 11th Mathematical Society of Japan (MSJ) Seasonal Institute (July 2018) we give an overview of some recent work on a mathematical mod…
Second order splitting of a class of fourth order PDEs with point constraints
Charles M. Elliott, Philip J. Herbert
We formulate a well-posedness and approximation theory for a class of generalised saddle point problems with a specific form of constraints. In this way we develop an approach to a…