QUASI-CONFORMING FINITE ELEMENT APPROXIMATION FOR A FOURTH ORDER VARIATIONAL INEQUALITY WITH DISPLACEMENT OBSTACLE

In this paper, unconventional quasi-conforming finite element approximation for a fourth order variational inequality with displacement obstacle is considered, the optimal order of error estimate O(h) is obtained which is as same as that of the conventional finite elements.

A TRIANGULAR QUASI-CONFORMING FINITE ELEMENT FOR TRANSIENT DYNAMIC ANALYSIS

A type of 3 node triangular element is constructed by the Quasi-conforming method, which may be used to solve the equation of a type of inverse problem of wave propagation after Laplace transformation △u-A^2u=0. The strains in the element are approximated by an expohential function and the string-net function between neigh- bouring elements is apporoximated by one dimensional general solution of the equation. Furthermore the strain field satisfies the equation, and therefore in the derivation of the element formulation, no shape function is needed. In this sense, it is a kind of hybrid element. Compared with the ordinary linear triangular element, the new one features higher precision with coarse meshes. Some numerical tests are presented.

18-DOF Triangular Quasi-Conforming Element for Couple Stress Theory

The basic idea of quasi-conforming method is that the strain-dis-placement equations are weakened as well as the equilibrium equations.In this paper,an 18-DOF triangular element for couple stress theory is proposed within the framework of quasi-conforming technique.The formulation starts from truncated Taylor expansion of strains and appropriate interpolation functions are chosen to calculate strain integration.This element satisfies C0 continuity with second order accuracy and weak C1 continuity simultaneously.Numerical examples demonstrate that the proposed model can pass the C0??1 patch test and has high accuracy.The element does not exhibit extra zero energy modes and can capture the scale effects of microstructure.

THE DISPLACEMENT FUNCTION OF QUASI-CONFORMING ELEMENT AND ITS NODE ERROR

Based on the strain formulation of the quasi-conforming finite element, displacement functions are constructed which have definite physical meaning, and a conclusion can be obtained that the coefficients of the constant and the linear strain are uniquely determined, and the quasi-conforming finite element method is convergent to constant strain. There are different methods for constructing the rigid displacement items, and different methods correspond to different order node errors, and this is different from ordinary displacement method finite element.

NONLINEAR QUASI-CONFORMING FINITE ELEMENT METHOD

The nonlinear quasi-conforming FEM is presented based on the basic concept of the quasi- -conforming finite element. First, the incremental principle of stationary potential energy is discussed, Then, the formulation process of the nonlinear quasi-conforming FEM is given. Lastly, two computational examples of shells are given.

LINEAR AND GEOMETRICALLY NONLINEAR ANALYSIS WITH 4-NODE PLANE QUASI-CONFORMING ELEMENT WITH INTERNAL PARAMETERS

A linear 4-node quadrilateral quasi-conforming plane element with internal parameters is proposed. The element preserves advantages of the quasi-conforming technique, including an explicit stiffness matrix, which can be applied to nonlinear problems. The weak patch test guarantees the convergence of the element. Then the linear element is extended to the geometri- cally nonlinear analysis in the framework of Total Lagrangian （TL） formulation. The numerical tests indicate that the present element is accurate and insensitive to mesh distortion.

Assumed stress quasi-conforming triangular element for couple stress theory

In this paper, a 3-node triangular element for couple stress theory is proposed based on the assumed stress quasi-conforming method. The formulation starts from polynomial approx- imation of stresses. Then the stress-function matrix is treated as the weighted function to weaken the strain-displacement equations. Finally, the string-net functions are introduced to calculate strain integration and the stress smooth technique is adopted to improve the stress accuracy. Numerical results show that the proposed new model can pass the Co- 1 patch test with excellent precision, does not exhibit extra zero energy modes and can cap- ture the scale effects of microstructure.

Feature matching using quasi-conformal maps

We present a fully automatic method for finding geometrically consistent correspondences while discarding outliers from the candidate point matches in two images. Given a set of candidate matches provided by scale-invariant feature transform(SIFT) descriptors, which may contain many outliers, our goal is to select a subset of these matches retaining much more geometric information constructed by a mapping searched in the space of all diffeomorphisms. This problem can be formulated as a constrained optimization involving both the Beltrami coefficient(BC) term and quasi-conformal map, and solved by an efficient iterative algorithm based on the variable splitting method. In each iteration, we solve two subproblems, namely a linear system and linearly constrained convex quadratic programming. Our algorithm is simple and robust to outliers. We show that our algorithm enables producing more correct correspondences experimentally compared with state-of-the-art approaches.

A note on extremal quasi-conformal mappings

An extremal quasi-conformal mapping f of a domain D is said to be of non-landslide type if the set Ef(δ):= {z∈D:|μf(z)|≤||μ|| ∞ -δ} has no interior points for any δ > 0. In this paper,we construct a quasi-conformal mapping f of the unit disc D such that its Teichmu¨ller equivalence class [f] contains infinitely many extremal mappings of non-landslide type. The relation between extremal mappings of non-landslide type and locally extremal mappings is also discussed.

Quasi-Conformally Flat Submanifolds with Parallel Mean Curvature Vector

This paper studied the conditions such that an n-dimensional complete space-like submanifold M with a parallel mean curvature vector field in an indefinite space form nppSc+()(p2,n3) is a totally umbilical submanifold. By means of Laplacian estimation and the choice of a diagonal frame field, the following theorem is proved: if M is quasi-conformally flat and 2,HcRic(M)d?M1(1)轾--犏臌nt2()-cH, then M is a totally umbilical submanifold.

Integrated photonic systems based on transformation optics enabled gradient index devices

Integrated photonics is expected to play an increasingly important role in optical communications,imaging,computing and sensing with the promise for significant reduction in the cost and weight of these systems.Future advancement of this technology is critically dependent on an ability to develop compact and reliable optical components and facilitate their integration on a common substrate.Here we reveal,with the utility of the emerging transformation optics technique,that functional components composed of planar gradient index materials can be designed and readily integrated into photonic circuits.The unprecedented design flexibility of transformation optics allows for the creation of a number of novel devices,such as a light source collimator,waveguide adapters and a waveguide crossing,which have broad applications in integrated photonic chips and are compatible with current fabrication technology.Using the finite-difference time-domain method,we perform full-wave numerical simulations to demonstrate their superior optical performance and efficient integration with other components in an on-chip photonic system.These components only require spatially-varying dielectric materials with no magnetic properties,facilitating low-loss,broadband operation in an integrated photonic environment.

On the structure of Fatou domains

Let U be a multiply-connected fixed attracting Fatou domain of a rational map f.We prove that there exist a rational map g and a completely invariant Fatou domain V of g such that(f,U) and(g,V) are holomorphically conjugate,and each non-trivial Julia component of g is a quasi-circle which bounds an eventually superattracting Fatou domain of g containing at most one postcritical point of g.Moreover,g is unique up to a holomorphic conjugation.