A Structure-Preserving Numerical Method for the Fourth-Order Geometric Evolution Equations for Planar Curves

For fourth-order geometric evolution equations for planar curves with the dissipation of the bending energy,including the Willmore and the Helfrich flows,we consider a numerical approach.In this study,we construct a structure-preserving method based on a discrete variational derivative method.Furthermore,to prevent the vertex concentration that may lead to numerical instability,we discretely introduce Deckelnick’s tangential velocity.Here,a modification term is introduced in the process of adding tangential velocity.This modified term enables the method to reproduce the equations’properties while preventing vertex concentration.Numerical experiments demonstrate that the proposed approach captures the equations’properties with high accuracy and avoids the concentration of vertices.

Geometric Evolution Laws for Thin Crystalline Films: Modeling and Numerics

Geometrical evolution laws are widely used in continuum modeling of surface and interface motion in materials science.In this article,we first give a brief review of various kinds of geometrical evolution laws and their variational derivations,with an emphasis on strong anisotropy.We then survey some of the finite element based numerical methods for simulating the motion of interfaces focusing on the field of thin film growth.We discuss the finite element method applied to front-tracking,phase-field and level-set methods.We describe various applications of these geometrical evolution laws to materials science problems,and in particular,the growth and morphologies of thin crystalline films.

A Robust and Efficient Adaptive Multigrid Solver for the Optimal Control of Phase Field Formulations of Geometric Evolution Laws

We propose and investigate a novel solution strategy to efficiently and accurately compute approximate solutions to semilinear optimal control problems,focusing on the optimal control of phase field formulations of geometric evolution laws.The optimal control of geometric evolution laws arises in a number of applications in fields including material science,image processing,tumour growth and cell motility.Despite this,many open problems remain in the analysis and approximation of such problems.In the current work we focus on a phase field formulation of the optimal control problem,hence exploiting the well developed mathematical theory for the optimal control of semilinear parabolic partial differential equations.Approximation of the resulting optimal control problemis computationally challenging,requiring massive amounts of computational time and memory storage.The main focus of this work is to propose,derive,implement and test an efficient solution method for such problems.The solver for the discretised partial differential equations is based upon a geometric multigrid method incorporating advanced techniques to deal with the nonlinearities in the problem and utilising adaptive mesh refinement.An in-house twogrid solution strategy for the forward and adjoint problems,that significantly reduces memory requirements and CPU time,is proposed and investigated computationally.Furthermore,parallelisation as well as an adaptive-step gradient update for the control are employed to further improve efficiency.Along with a detailed description of our proposed solution method together with its implementation we present a number of computational results that demonstrate and evaluate our algorithms with respect to accuracy and efficiency.A highlight of the present work is simulation results on the optimal control of phase field formulations of geometric evolution laws in 3-D which would be computationally infeasible without the solution strategies proposed in the present work.

A curve flow evolved by a fourth order parabolic equation

We study a fourth order curve flow, which is the gradient flow of a functional describing the shapes of human red blood cells. We prove that for any smooth closed initial curve in ?2, the flow has a smooth solution for all time and the solution subconverges to a critical point of the functional.

On the Kahler-Ricci Flow on Projective Manifolds of General Type

This note concerns the global existence and convergence of the solution for Kahler-Ricci flow equation when the canonical class, Kx, is numerically effective and big. We clarify some known results regarding this flow on projective manifolds of general type and also show some new observations and refined results.

Gradient Flow for the Helfrich Functional

The author studies the L2 gradient flow of the Helfrich functional, which is a functional describing the shapes of human red blood cells. For any λi ≥ 0 and co, the author obtains a lower bound on the lifespan of the smooth solution, which depends only on the concentration of curvature for the initial surface.