In this paper, we study numerical approximations of a recently proposed phase field model for the vesicle membrane deformation governed by the variation of the elastic bending energy. To overcome the challenges of hig...In this paper, we study numerical approximations of a recently proposed phase field model for the vesicle membrane deformation governed by the variation of the elastic bending energy. To overcome the challenges of high order nonlinear differential systems and the nonlinear constraints associated with the problem, we present the phase field bending elasticity model in a nested saddle point formulation. A mixed finite element method is then employed to compute the equilibrium configuration of a vesicle membrane with prescribed volume and surface area. Coupling the approximation results for a related linearized problem and the general theory of Brezzi-Rappaz-Raviart, optimal order error estimates for the finite element approximations of the phase field model are obtained. Numerical results are provided to substantiate the derived estimates.展开更多
One-dimensional(1D)nanomaterials easily bend due to perturbations from their surroundings or their own behaviors.This phenomenon not only impacts the performances of various devices but has also been employed to devel...One-dimensional(1D)nanomaterials easily bend due to perturbations from their surroundings or their own behaviors.This phenomenon not only impacts the performances of various devices but has also been employed to develop a variety of new functional devices,in which the bending energies of the nanomaterials determine the device performances.However,measuring the energies of such nanomaterials is extremely difficult.Herein,pseudo-break imaging of 1D nanomaterials has been proposed and realized on individual carbon nanotubes(CNTs),in which a CNT appears to break and has a fracture but is actually intact.This imaging approach provides the values of the bending energies of the CNTs with an accuracy of 1–50 eV.Furthermore,this imaging approach can manipulate the bending shapes and energies of CNTs.This work presents a protocol for bending analysis and manipulation,which are vital to fundamental and applied studies of 1D nanomaterials.展开更多
文摘In this paper, we study numerical approximations of a recently proposed phase field model for the vesicle membrane deformation governed by the variation of the elastic bending energy. To overcome the challenges of high order nonlinear differential systems and the nonlinear constraints associated with the problem, we present the phase field bending elasticity model in a nested saddle point formulation. A mixed finite element method is then employed to compute the equilibrium configuration of a vesicle membrane with prescribed volume and surface area. Coupling the approximation results for a related linearized problem and the general theory of Brezzi-Rappaz-Raviart, optimal order error estimates for the finite element approximations of the phase field model are obtained. Numerical results are provided to substantiate the derived estimates.
基金the National Natural Science Foundation of China(Nos.51971157 and 12102307)Shenzhen Science and Technology Program(Nos.JCYJ20210324115412035 and ZDSYS20210813095534001)+1 种基金Tianjin Science Fund for Distinguished Young Scholars(No.19JCJQJC61800)the Natural Science Foundation of Hubei Province,China(No.2021CFB138).
文摘One-dimensional(1D)nanomaterials easily bend due to perturbations from their surroundings or their own behaviors.This phenomenon not only impacts the performances of various devices but has also been employed to develop a variety of new functional devices,in which the bending energies of the nanomaterials determine the device performances.However,measuring the energies of such nanomaterials is extremely difficult.Herein,pseudo-break imaging of 1D nanomaterials has been proposed and realized on individual carbon nanotubes(CNTs),in which a CNT appears to break and has a fracture but is actually intact.This imaging approach provides the values of the bending energies of the CNTs with an accuracy of 1–50 eV.Furthermore,this imaging approach can manipulate the bending shapes and energies of CNTs.This work presents a protocol for bending analysis and manipulation,which are vital to fundamental and applied studies of 1D nanomaterials.
