The orientation-preservation condition, i.e., the Jacobian determinant of the deformation gradient det Vu being required to be positive, is a natural physical constraint in elasticity as well as in many other fields. ...The orientation-preservation condition, i.e., the Jacobian determinant of the deformation gradient det Vu being required to be positive, is a natural physical constraint in elasticity as well as in many other fields. It is well known that the constraint can often cause serious difficulties in both theoretical analysis and numerical computation, especially when the material is subject to large deformations. We derive a set of necessary and sufficient conditions for the quadratic iso-parametric finite element interpolation functions of cavity solutions to be orientation preserving on a class of radially symmetric large expansion accommodating triangulations. The result provides a practical quantitative guide for meshing in the neighborhood of a cavity and shows that the orientation-preservation can be achieved with a reasonable number of total degrees of freedom by the quadratic iso-parametric finite element method.展开更多
Let M be a 2-dimensional closed manifold, orientable or non-orientable. The construction of every compact locally connected subspace X of M without cut-points is analyzed. It is proved that every orientation-preservin...Let M be a 2-dimensional closed manifold, orientable or non-orientable. The construction of every compact locally connected subspace X of M without cut-points is analyzed. It is proved that every orientation-preserving (or reversing, or relatively preserving) point-wise periodic continuous self-map of X can be extended to a periodic self-homeomorphism of M (or of a 2-dimensional compact submanifold of M). In addition, every orientation-preserving (or reversing, or relatively preserving) pointwise periodic continuous self-map f of any path-connected subspace of M is proved to be a periodic self-homeomorphism, the number of the shorter-periodic points of f is shown to be finite, and generalization of Weaver’s conclusion is given.展开更多
基金supported by National Natural Science Foundation of China (Grant Nos. 11171008 and 11571022)
文摘The orientation-preservation condition, i.e., the Jacobian determinant of the deformation gradient det Vu being required to be positive, is a natural physical constraint in elasticity as well as in many other fields. It is well known that the constraint can often cause serious difficulties in both theoretical analysis and numerical computation, especially when the material is subject to large deformations. We derive a set of necessary and sufficient conditions for the quadratic iso-parametric finite element interpolation functions of cavity solutions to be orientation preserving on a class of radially symmetric large expansion accommodating triangulations. The result provides a practical quantitative guide for meshing in the neighborhood of a cavity and shows that the orientation-preservation can be achieved with a reasonable number of total degrees of freedom by the quadratic iso-parametric finite element method.
文摘Let M be a 2-dimensional closed manifold, orientable or non-orientable. The construction of every compact locally connected subspace X of M without cut-points is analyzed. It is proved that every orientation-preserving (or reversing, or relatively preserving) point-wise periodic continuous self-map of X can be extended to a periodic self-homeomorphism of M (or of a 2-dimensional compact submanifold of M). In addition, every orientation-preserving (or reversing, or relatively preserving) pointwise periodic continuous self-map f of any path-connected subspace of M is proved to be a periodic self-homeomorphism, the number of the shorter-periodic points of f is shown to be finite, and generalization of Weaver’s conclusion is given.
