Zhongxuan Luo

Recurrence surfaces on arbitrary quadrilateral mesh

In this paper, L- and W-surface are constructed via recurrence scheme. The derivation formulae, basis function properties, envelope theorems of L-surface and an equivalent representation of L-surface into Bezier surface are presented. Especially, a C2 bicubic spline surface on a cross-cut grid partition consisting of quadrilaterals is given, which can be used to replace the biquintic surface in most of the applications. The new method has been applied in the field of contour design of human body and aircraft.

Constructing cubature formulae of degree 5 with few points

This paper is devoted to construct a family of fifth degree cubature formulae for n-cube with symmetric measure and n-dimensional spherically symmetrical region. The formula forn-cube contains at most n^2+5n+3 points and for n-dimensional spherically symmetrical region contains only n^2+3n+3 points. Moreover, the numbers can be reduced to n^2+3n+1 and n^2+n+1 if n=7 respectively, the latter of which is minimal.

Convergence analysis of a family of 14-node brick elements

In this paper, we will give convergence analysis for a family of 14-node elements which was proposed by Smith and Kidger (1992). The 14 DOFs are taken as the values at the eight vertices and the six face-centroids. For second-order elliptic problems, we will show that among all the Smith-Kidger 14-node elements, Type 1, Type 2 and Type 5 elements provide optimal-order convergent solutions while Type 6 element gives one-order lower convergent solutions. Motivated by our proof, we also find that the order of convergence of the Type 6 14-node nonconforming element improves to be optimal if we change the DOFs into the values at the eight vertices and the integration values on the six faces. We also show that Type 1, Type 2 and Type 5 keep the optimal-order convergence if the integral DOFs on the six faces are adopted.

New nonconforming finite elements on arbitrary convex quadrilateral meshes

In this paper, we construct new nonconforming finite elements on the meshes consisting of arbitrary convex quadrilaterals, especially for the quadratic and cubic cases. For each case, we first define a quadrilateral element that adopts edge moments as the degrees of freedom (DoFs), and then enforce a linear constraint on this element. We have, for the quadratic case, eight degrees of freedom per element and, for the cubic case, eleven DoFs per element, respectively. The dimensions and the bases of different types for the global finite element spaces are provided. We consider the approximations of two-dimensional second order elliptic problems for both of these elements. Error estimates with optimal convergence order in both broken H 1 norm and L 2 norm are given. Moreover, we consider the discretization of the Stokes equations adopting our quadratic element to approximate each component of the velocity, along with piecewise discontinuous P 1 element for the pressure. This mixed scheme is stable and optimal error estimates both for the velocity and the pressure are also achieved. Numerical examples verify our theoretical analysis.

On the singularity of multivariate Hermite interpolation

In this paper, we study the singularity of multivariate Hermite interpolation of type total degree. We present two methods to judge the singularity of the interpolation schemes considered and by methods to be developed, we show that all Hermite interpolation of type total degree on m=d+k points in R^d is singular if d>=2k. And then we solve the Hermite interpolation problem on [emailxa0protected]?d+3 nodes completely. Precisely, all Hermite interpolations of type total degree on [emailxa0protected]?d+1 points with d>=2 are singular; only three cases for m=d+2 and one case for m=d+3 can produce regular Hermite interpolation schemes, respectively. Besides, we also present a method to compute the interpolation space for Hermite interpolation of type total degree.

A novel local/global approach to spherical parameterization

This paper proposes a novel local/global spherical parameterization for the genus-zero triangular mesh, which naturally extends the planar approach to the spherical case. In our method, we derive two fitting matrices (conformal and isometric) in 3D space. By optimizing the so-called spring energy, the spherical results are achieved by solving a nonlinear system with spherical constraints. Intuitively, it represents the stitching together of the 1-ring patches to form a unit sphere. Moreover, the derivation of the 3D fitting matrices can also be applied to planar triangles directly, so that we can obtain a class of novel planar approaches (conformal, isometric, authalic) to the problem of flattening triangular meshes. In order to enhance robustness of the proposed spherical method, a stretch operator is introduced for dealing with high-curvature models. Numerical results demonstrate that our method is simple, efficient and convergent, and it outperforms several state-of-the-art methods in terms of trading-off the distortions of angle, area and stretch. Furthermore, it achieves better visualization in texture mapping.

ARAP++: an extension of the local/global approach tomesh parameterization 1 Project supported by the National Natural Science Foundation of China (Nos. 61432003, 61572105, 11171052, and 61328206)

Mesh parameterization is one of the fundamental operations in computer graphics (CG) and computeraided design (CAD). In this paper, we propose a novel local/global parameterization approach, ARAP++, for singleand multi-boundary triangular meshes. It is an extension of the as-rigid-as-possible (ARAP) approach, which stitches together 1-ring patches instead of individual triangles. To optimize the spring energy, we introduce a linear iterative scheme which employs convex combination weights and a fitting Jacobian matrix corresponding to a prescribed family of transformations. Our algorithm is simple, efficient, and robust. The geometric properties (angle and area) of the original model can also be preserved by appropriately prescribing the singular values of the fitting matrix. To reduce the area and stretch distortions for high-curvature models, a stretch operator is introduced. Numerical results demonstrate that ARAP++ outperforms several state-of-the-art methods in terms of controlling the distortions of angle, area, and stretch. Furthermore, it achieves a better visualization performance for several applications, such as texture mapping and surface remeshing.