Recent experiments revealed many new phenomena of the macroscopic domain patterns in the stress-induced phase transformation of a superelastic polycrystalline NiTi tube during tensile loading. The new phenomena includ...Recent experiments revealed many new phenomena of the macroscopic domain patterns in the stress-induced phase transformation of a superelastic polycrystalline NiTi tube during tensile loading. The new phenomena include deformation instability with the formation of a helical domain, domain topology transition from helix to cylinder, domain-front branching and loading-path dependence of domain patterns. In this paper, we model the polycrystal as an elastic continuum with nonconvex strain energy and adopt the non-local strain gradient energy to account for the energy of the diffusive domain front. We simulate the equilibrium domain patterns and their evolution in the tubes under tensile loading by a non-local Finite Element Method （FEM）. It is revealed that the observed loading-path dependence and topology transition of do- main patterns are due to the thermodynamic metastability of the tube system. The computation also shows that the tube-wall thickness has a significant effect on the domain patterns： with fixed material properties and interfacial energy density, a large tube-wall thickness leads to a long and slim helical domain and a severe branching of the cylindrical-domain front.展开更多
This paper considers the finite element approximation to parabolic optimal control problems with measure data in a nonconvex polygonal domain.Such problems usually possess low regularity in the state variable due to t...This paper considers the finite element approximation to parabolic optimal control problems with measure data in a nonconvex polygonal domain.Such problems usually possess low regularity in the state variable due to the presence of measure data and the nonconvex nature of the domain.The low regularity of the solution allows the finite element approximations to converge at lower orders.We prove the existence,uniqueness and regularity results for the solution to the control problem satisfying the first order optimality condition.For our error analysis we have used piecewise linear elements for the approximation of the state and co-state variables,whereas piecewise constant functions are employed to approximate the control variable.The temporal discretization is based on the implicit Euler scheme.We derive both a priori and a posteriori error bounds for the state,control and co-state variables.Numerical experiments are performed to validate the theoretical rates of convergence.展开更多
文摘Recent experiments revealed many new phenomena of the macroscopic domain patterns in the stress-induced phase transformation of a superelastic polycrystalline NiTi tube during tensile loading. The new phenomena include deformation instability with the formation of a helical domain, domain topology transition from helix to cylinder, domain-front branching and loading-path dependence of domain patterns. In this paper, we model the polycrystal as an elastic continuum with nonconvex strain energy and adopt the non-local strain gradient energy to account for the energy of the diffusive domain front. We simulate the equilibrium domain patterns and their evolution in the tubes under tensile loading by a non-local Finite Element Method （FEM）. It is revealed that the observed loading-path dependence and topology transition of do- main patterns are due to the thermodynamic metastability of the tube system. The computation also shows that the tube-wall thickness has a significant effect on the domain patterns： with fixed material properties and interfacial energy density, a large tube-wall thickness leads to a long and slim helical domain and a severe branching of the cylindrical-domain front.
文摘This paper considers the finite element approximation to parabolic optimal control problems with measure data in a nonconvex polygonal domain.Such problems usually possess low regularity in the state variable due to the presence of measure data and the nonconvex nature of the domain.The low regularity of the solution allows the finite element approximations to converge at lower orders.We prove the existence,uniqueness and regularity results for the solution to the control problem satisfying the first order optimality condition.For our error analysis we have used piecewise linear elements for the approximation of the state and co-state variables,whereas piecewise constant functions are employed to approximate the control variable.The temporal discretization is based on the implicit Euler scheme.We derive both a priori and a posteriori error bounds for the state,control and co-state variables.Numerical experiments are performed to validate the theoretical rates of convergence.
