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 s...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.展开更多
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 the...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.展开更多
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 ge...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.展开更多
基金This work was supported by JSPS KAKENHI Grant Nos.19K14590,21K18301,Japan.
文摘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.
基金The work of B.Li was supported by the US National Science Foundation(NSF)through grants DMS-0451466 and DMS-0811259the US Department of Energy through grant DE-FG02-05ER25707+2 种基金the Center for Theoretical Biological Physics through the NSF grants PHY-0216576 and PHY-0822283J.Lowengrub gratefully acknowledges support from the US National Science Foundation Divisions of Mathematical Sciences(DMS)and Materials Research(DMR)The work of A.Voigt and A.Ratz was supported by the 6th Framework program of EU STRP 016447 and German Science Foundation within the Collaborative Research Program SFB 609.
文摘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.
基金All authors acknowledge support from the Leverhulme Trust Research Project Grant(RPG-2014-149)Thework of CV,VS and AMwas partially supported by the Engineering and Physical Sciences Research Council,UK grant(EP/J016780/1)This work(AM)has also received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 642866.The work of CV is partially supported by an EPSRC Impact Accelerator Account award.The authors(FWY,CV,VS,AM)thank the Isaac Newton Institute for Mathematical Sciences for its hospitality during the programme(Coupling Geometric PDEs with Physics for Cell Morphology,Motility and Pattern Formation,EPSRC EP/K032208/1).AM was partially supported by Fellowships from the Simons Foundation.AM is a Royal Society Wolfson Research Merit Award Holder generously funded by the Royal Society and the Wolfson Foundation.
文摘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.