In this paper, a subdivision scheme which generalizes a surface scheme in previous papers to volume meshes is designed.The scheme exhibits significant control over shrink-age/size of volumetric models.It also has the ...In this paper, a subdivision scheme which generalizes a surface scheme in previous papers to volume meshes is designed.The scheme exhibits significant control over shrink-age/size of volumetric models.It also has the ability to conveniently incorporate boundaries and creases into a smooth limit shape of models.The method presented here is much simpler and easier as compared to MacCracken and Joy's.This method makes no restrictions on the local topology of meshes.Particularly,it can be applied without any change to meshes of non-manifold topology.展开更多
We consider the design of an effective and reliable adaptive finite element method(AFEM)for the nonlinear Poisson-Boltzmann equation(PBE).We first examine the two-term regularization technique for the continuous probl...We consider the design of an effective and reliable adaptive finite element method(AFEM)for the nonlinear Poisson-Boltzmann equation(PBE).We first examine the two-term regularization technique for the continuous problem recently proposed by Chen,Holst and Xu based on the removal of the singular electrostatic potential inside biomolecules;this technique made possible the development of the first complete solution and approximation theory for the Poisson-Boltzmann equation,the first provably convergent discretization and also allowed for the development of a provably convergent AFEM.However,in practical implementation,this two-term regularization exhibits numerical instability.Therefore,we examine a variation of this regularization technique which can be shown to be less susceptible to such instability.We establish a priori estimates and other basic results for the continuous regularized problem,as well as for Galerkin finite element approximations.We show that the new approach produces regularized continuous and discrete problemswith the samemathematical advantages of the original regularization.We then design an AFEM scheme for the new regularized problem and show that the resulting AFEM scheme is accurate and reliable,by proving a contraction result for the error.This result,which is one of the first results of this type for nonlinear elliptic problems,is based on using continuous and discrete a priori L¥estimates.To provide a high-quality geometric model as input to the AFEM algorithm,we also describe a class of feature-preserving adaptive mesh generation algorithms designed specifically for constructing meshes of biomolecular structures,based on the intrinsic local structure tensor of the molecular surface.All of the algorithms described in the article are implemented in the Finite Element Toolkit(FETK),developed and maintained at UCSD.The stability advantages of the new regularization scheme are demonstrated with FETK through comparisons with the original regularization approach for a model problem.The convergence and accuracy of the overall AFEMalgorithmis also illustrated by numerical approximation of electrostatic solvation energy for an insulin protein.展开更多
文摘In this paper, a subdivision scheme which generalizes a surface scheme in previous papers to volume meshes is designed.The scheme exhibits significant control over shrink-age/size of volumetric models.It also has the ability to conveniently incorporate boundaries and creases into a smooth limit shape of models.The method presented here is much simpler and easier as compared to MacCracken and Joy's.This method makes no restrictions on the local topology of meshes.Particularly,it can be applied without any change to meshes of non-manifold topology.
基金supported in part by NSF Awards 0715146,0821816,0915220 and 0822283(CTBP)NIHAward P41RR08605-16(NBCR),DOD/DTRA Award HDTRA-09-1-0036+1 种基金CTBP,NBCR,NSF and NIHsupported in part by NIH,NSF,HHMI,CTBP and NBCR.The third,fourth and fifth authors were supported in part by NSF Award 0715146,CTBP,NBCR and HHMI.
文摘We consider the design of an effective and reliable adaptive finite element method(AFEM)for the nonlinear Poisson-Boltzmann equation(PBE).We first examine the two-term regularization technique for the continuous problem recently proposed by Chen,Holst and Xu based on the removal of the singular electrostatic potential inside biomolecules;this technique made possible the development of the first complete solution and approximation theory for the Poisson-Boltzmann equation,the first provably convergent discretization and also allowed for the development of a provably convergent AFEM.However,in practical implementation,this two-term regularization exhibits numerical instability.Therefore,we examine a variation of this regularization technique which can be shown to be less susceptible to such instability.We establish a priori estimates and other basic results for the continuous regularized problem,as well as for Galerkin finite element approximations.We show that the new approach produces regularized continuous and discrete problemswith the samemathematical advantages of the original regularization.We then design an AFEM scheme for the new regularized problem and show that the resulting AFEM scheme is accurate and reliable,by proving a contraction result for the error.This result,which is one of the first results of this type for nonlinear elliptic problems,is based on using continuous and discrete a priori L¥estimates.To provide a high-quality geometric model as input to the AFEM algorithm,we also describe a class of feature-preserving adaptive mesh generation algorithms designed specifically for constructing meshes of biomolecular structures,based on the intrinsic local structure tensor of the molecular surface.All of the algorithms described in the article are implemented in the Finite Element Toolkit(FETK),developed and maintained at UCSD.The stability advantages of the new regularization scheme are demonstrated with FETK through comparisons with the original regularization approach for a model problem.The convergence and accuracy of the overall AFEMalgorithmis also illustrated by numerical approximation of electrostatic solvation energy for an insulin protein.
