7 research outputs found
A family of quadrilateral finite elements
We present a novel family of quadrilateral finite elements, which
define global spaces over a general quadrilateral mesh with vertices of
arbitrary valency. The elements extend the construction by (Brenner and Sung,
J. Sci. Comput., 2005), which is based on polynomial elements of tensor-product
degree , to all degrees . Thus, we call the family of
finite elements Brenner-Sung quadrilaterals. The proposed quadrilateral
can be seen as a special case of the Argyris isogeometric element of (Kapl,
Sangalli and Takacs, CAGD, 2019). The quadrilateral elements possess similar
degrees of freedom as the classical Argyris triangles. Just as for the Argyris
triangle, we additionally impose continuity at the vertices. In this
paper we focus on the lower degree cases, that may be desirable for their lower
computational cost and better conditioning of the basis: We consider indeed the
polynomial quadrilateral of (bi-)degree~, and the polynomial degrees
and by employing a splitting into or polynomial
pieces, respectively.
The proposed elements reproduce polynomials of total degree . We show that
the space provides optimal approximation order. Due to the interpolation
properties, the error bounds are local on each element. In addition, we
describe the construction of a simple, local basis and give for
explicit formulas for the B\'{e}zier or B-spline coefficients of the basis
functions. Numerical experiments by solving the biharmonic equation demonstrate
the potential of the proposed quadrilateral finite element for the
numerical analysis of fourth order problems, also indicating that (for )
the proposed element performs comparable or in general even better than the
Argyris triangle with respect to the number of degrees of freedom
Triangular G1 interpolation by 4-splitting domain triangles
International audienceA piecewise quintic G1 spline surface interpolating the vertices of a triangular surface mesh of arbitrary topological type is presented. The surface has an explicit triangular Bezier representation, is affine invariant and has local support. The twist compatibility problem which arises when joining an even number of polynomial patches G1 continuously around a common vertex is solved by constructing C2-consistent boundary curves. Piecewise C1 boundary curves and a regular 4-split of the domain triangle make shape parameters available for controlling locally the boundary curves. A small number of free inner control points can be chosen for some additional local shape effects
Triangular Bézier Surfaces with Approximate Continuity
When interpolating a data mesh using triangular Bézier patches, the requirement of C¹ or G¹ continuity imposes strict constraints on the control points of adjacent patches. However, fulfillment of these continuity constraints cannot guarantee that the resulting surfaces have good shape. This thesis presents an approach to constructing surfaces with approximate C¹/G¹ continuity, where a small amount of discontinuity is allowed between surface normals of adjacent patches.
For all the schemes presented in this thesis, although the resulting surface has C¹/G¹ continuity at the data vertices, I only require approximate C¹/G¹ continuity along data triangle boundaries so as to lower the patch degree.
For functional data, a cubic interpolating scheme with approximate C¹ continuity is presented. In this scheme, one cubic patch will be constructed for each data triangle and upper bounds are provided for the normal discontinuity across patch boundaries.
For a triangular mesh of arbitrary topology, two interpolating parametric schemes are devised. For each data triangle, the first scheme performs a domain split and constructs three cubic micro-patches; the second scheme constructs one quintic patch for each data triangle. To reduce the normal discontinuity, neighboring patches across data triangle boundaries are adjusted to have identical normals at the middle point of the common boundary. The upper bounds for the normal discontinuity between two parametric patches are also derived for the resulting approximate G¹ surface.
In most cases, the resulting surfaces with approximate continuity have the same level of visual smoothness and in some cases better shape quality