Quadrilateral isoparametric finite elements for plane elastic Cosserat bodies

4-node, 8-node and 8（4）-node quadrilateral plane isoparametric elements are used for the solution of boundary value problems in linear isotropic Cosserat elasticity. The patch test is applied to validate the finite elements. Engineering problems of stress concentration around a circular hole in plane strain condition and mechanical behaviors of heterogeneous materials with rigid inclusions and pores are computed to test the accuracy and capability of these three types of finite elements.

ISOPARAMETRIC HYPERSURFACES AND COMPLEX STRUCTURES

The main purpose of this note is to construct almost complex or complex structures on certain isoparametric hypersurfaces in unit spheres.As a consequence,complex structures on S^(1)×S^(7)×S^(6),and on S^(10)×S^(3)×S(2)with vanishing first Chern class,are built.

A second order isoparametric finite element method for elliptic interface problems

A second order isoparametric finite element method （IPFEM） is proposed for elliptic interface problems. It yields better accuracy than some existing second-order methods, when the coefficients or the flux across the immersed curved interface is discontinuous. Based on an initial Cartesian mesh, a mesh optimization strategy is presented by employing curved boundary elements at the interface, and an incomplete quadratic finite element space is constructed on the optimized mesh. It turns out that the number of curved boundary elements is far less than that of the straight one, and the total degree of freedom is almost the same as the uniform Cartesian mesh. Numerical examples with simple and complicated geometrical interfaces demonstrate the efficiency of the proposed method.

LOCKING-FREE DEGENERATED ISOPARAMETRIC SHELL ELEMENT

An 8-noded locking-free degenerated isoparametric shell element is presented. A revised interpolation for shear strain terms was constructed in natural co-ordinate system such that all necessary modes (translation, rotation and constant curvature) are preserved, which can be used to eliminate shear locking. A revised interpolation for membrane strains was produced in the local Cartesian co-ordinate system to overcome membrane locking behavior. The new 8-noded element has the proper rank, with the requisite number of zero eigenvalues each associated with a rigid mode. The element does not exhibit membrane or shear locking for large span-thickness ratio. The element does not form element mechanisms or extra spurious zero energy modes. Therefore, it can be used for both thin and thick shells.

Isoparametric Boundary Element Method Applied to the Rectangular and Delta Hydrofoils near the Free Surface

In this paper, the hydrodynamic characteristics and flow field around rectangular and delta hydrofoils, moving with a constant speed beneath the free surface are numerically studied by means of isoparametric boundary element method （IBEM）. The quantities （source and dipole strengths） and the geometry of the dements are represented by a linear distribution. Two types of three-dimensional hydrofoils （rectangular and delta） are selected with NACA4412 and symmetric Joukowski sections. Some numerical results of pressure distribution, lift, wave-making drag coefficients and velocity field around the hydrofoils are presented. Also, the wave pattern due to moving hydrofoil is predicted at different operational conditions. Comparisons are made between computational results obtained through this method and those from the experimental measurements and other numerical results which reveal good agreement.

Numerical Simulation of Mold Filling in Resin Transfer Molding Using Isoparametric Method

The mold filling of RTM was simulated based on the control volume finite element method (CV/FEM). The formulat ion using isoparametric transformation was discussed in detail and a computation al code based on isoparametric technique was developed. The simulation results w ere compared with experimental data. Different isoparametric elements, quadrilat eral and triangular, were compared in the simulation.It demonstrates that the us e of bilinear quadrilateral isoparametric elements in simulating the process can produce a higher precision and cost a less time than the use of triangular ones .

THE IMPROVED ISOPARAMETRIC TRANSFORMATION IN BEM

The midside node sensitivity of eight-node isoparametric element in 3-D BEM is investigated. The paper points out that the suggestion, based upon which the midside nodes should be located in the middle third of distance between the adjacent corners, should be followed even more strictly for the conventional isoparametric transformation (CIT) in BEM as that in FEM. A new coordinate transformation relation has been put forward to solve the singular integral problem. The computation is carried to two cases: a cubic body subjected to tensile stress and pure bending. The numerical results show that the improved isoparametric transformation (IIT) is easier and more flexible to practice.

Isoparametric Surfaces in 3-dimensional De Sitter Space and Anti-de Sitter Space

A spacelike surface M in 3-dimensional de sitter space S13 or 3-dimensional anti-de Sitter space H13 is called isoparametric, if M has constant principal curvatures. A timelike surface is called isoparametric, if its minimal polynomial of the shape operator is constant. In this paper, we determine the spacelike isoparametric surfaces and the timelike isoparametric surfaces in S13 and H13.

Isoparametric Hypersurfaces Induced by Navigation in Lorentz Finsler Geometry

Using a navigation process with the datum(F,V),in which F is a Finsler metric and the smooth tangent vector field V satisfies F(−V(x))>1 everywhere,a Lorentz Finsler metric F˜can be induced.Isoparametric functions and isoparametric hypersurfaces with or without involving a smooth measure can be defined for F˜.When the vector field V in the navigation datum is homothetic,we prove the local correspondences between isoparametric functions and isoparametric hypersurfaces before and after this navigation process.Using these correspondences,we provide some examples of isoparametric functions and isoparametric hypersurfaces on a Funk space of Lorentz Randers type.

On the Chern conjecture for isoparametric hypersurfaces

For a closed hypersurface Mn?Sn+1(1)with constant mean curvature and constant non-negative scalar curvature,we show that if tr(Ak)are constants for k=3,...,n-1 and the shape operator A,then M is isoparametric.The result generalizes the theorem of de Almeida and Brito(1990)for n=3 to any dimension n,strongly supporting the Chern conjecture.

Topology and Curvature of Isoparametric Families in Spheres

An isoparametric family in the unit sphere consists of parallel isoparametric hypersurfaces and their two focal submanifolds.The present paper has two parts.The first part investigates topology of the isoparametric families,namely the homotopy,homeomorphism,or diffeomorphism types,parallelizability,as well as the Lusternik-Schnirelmann category.This part extends substantially the results of Wang(J Differ Geom 27:55-66,1988).The second part is concerned with their curvatures;more precisely,we determine when they have non-negative sectional curvatures or positive Ricci curvatures with the induced metric.

CLASSIFICATIONS OF DUPIN HYPERSURFACES IN LIE SPHERE GEOMETRY

This is a survey of local and global classification results concerning Dupin hypersurfaces in S^(n)(or R^(n))that have been obtained in the context of Lie sphere geometry.The emphasis is on results that relate Dupin hypersurfaces to isoparametric hypersurfaces in spheres.Along with these classification results,many important concepts from Lie sphere geometry,such as curvature spheres,Lie curvatures,and Legendre lifts of submanifolds of S^(n)(or R^(n)),are described in detail.The paper also contains several important constructions of Dupin hypersurfaces with certain special properties.

Mobius Isoparametric Hypersurfaces in S^(n+1) with Two Distinct Principal Curvatures

A hypersurface x: M→S^(n+1) without umbilic point is called a Mbius isoparametric hypersurface if its Mbius form Φ=-ρ^(-2)∑_i(ei(H)+∑_j(h_(ij)-Hδ_(ij))e_j(logρ))θ_i vanishes and its Mbius shape operator S=ρ^(-1)(S-Hid) has constant eigenvalues. Here {e_i} is a local orthonormal basis for I=dx·dx with dual basis {θ_i}, II=∑_(ij)h_(ij)θ_iθ_J is the second fundamental form, H=1/n∑_i h_(ij), ρ~2=n/(n-1)(‖II‖~2-nH^2) and S is the shape operator of x. It is clear that any conformal image of a (Euclidean) isoparametric hypersurface in S^(n+1) is a Mbius isoparametric hypersurface, but the converse is not true. In this paper we classify all Mbius isoparametric hypersurfaces in S^(n+1) with two distinct principal curvatures up to Mbius transformations. By using a theorem of Thorbergsson [1] we also show that the number of distinct principal curvatures of a compact Mbius isoparametric hypersurface embedded in S^(n+1) can take only the values 2, 3, 4, 6.

Conformal isoparametric hypersurfaces with two distinct conformal principal curvatures in conformal space

The conformal geometry of regular hypersurfaces in the conformal space is studied.We classify all the conformal isoparametric hypersurfaces with two distinct conformal principal curvatures in the conformal space up to conformal equivalence.

ISOPARAMETRIC HYPERSURFACES WITH FOUR DISTINCT PRINCIPAL CURVATURES

I. PREPARATION AND LEMMASWe start recalling theMain Theorem of Abresch. Given an isoparametric hypersurface in Sn+1 with g= 4, then the pair （m, m+ ）-w. r. g, we may assume that m ≤m+, satisfies one of the three conditions below:

Isoparametric hypersurfaces in Funk spaces

Funk metrics are a kind of important Finsler metrics with constant negative flag curvature. In this paper, it is proved that any isoparametric hypersurface in Funk spaces has at most two distinct principal curvatures. Moreover, a complete classification of isoparametric families in a Funk space is given.

Blaschke Isoparametric Hypersurfaces in the Conformal Space Q_1^(n+1),Ⅰ

Let x ：M - Qn＋l be a regular hypersurface in the conformal space Qn＋I. We classify all the space-like Blaschke isoparametric hypersurfaces with two distinct Blaschke eigenvalues in the conformal space up to the conformal equivalence.

Classifications of Isoparametric Hypersurfaces in Randers Space Forms

In this paper,we give the complete classifications of isoparametric hypersurfaces in Randers space forms.By studying the principal curvatures of anisotropic submanifolds in a Randers space(N,F)with the navigation data(h,W),we find that a Randers space form(N,F,dμBH)and the corresponding Riemannian space(N,h)have the same isoparametric hypersurfaces,but in general,their isoparametric functions are different.We give a necessary and sufficient condition for an isoparametric function of(N,h)to be isoparametric on(N,F,dμBH),from which we get some examples of isoparametric functions.

On Mbius Form and Mbius Isoparametric Hypersurfaces

An umbilic-free hypersurface in the unit sphere is called MSbius isoparametric if it satisfies two conditions, namely, it has vanishing MSbius form and has constant MSbius principal curvatures. In this paper, under the condition of having constant MSbius principal curvatures, we show that the hypersurface is of vanishing MSbius form if and only if its MSbius form is parallel with respect to the Levi-Civita connection of its MSbius metric. Moreover, typical examples are constructed to show that the condition of having constant MSbius principal curvatures and that of having vanishing MSbius form are independent of each other.

Willmore Submanifolds in the Unit Sphere via Isoparametric Functions

In this paper,we show that both focal submanifolds of each isoparametric hypersurface in the sphere with six distinct principal curvatures are Willmore,hence all focal submanifolds of isoparametric hypersurfaces in the sphere are Willmore.