Constructing IGA-suitable planar parameterization from complex CAD boundary by domain partition and global/local optimization

In this paper, we propose a general framework for constructing IGA-suitable planar B-spline parameterizations from given complex CAD boundaries consisting of a set of B-spline curves. Instead of forming the computational domain by a simple boundary, planar domains with high genus and more complex boundary curves are considered. Firstly, some pre-processing operations including B\'ezier extraction and subdivision are performed on each boundary curve in order to generate a high-quality planar parameterization; then a robust planar domain partition framework is proposed to construct high-quality patch-meshing results with few singularities from the discrete boundary formed by connecting the end points of the resulting boundary segments. After the topology information generation of quadrilateral decomposition, the optimal placement of interior B\'ezier curves corresponding to the interior edges of the quadrangulation is constructed by a global optimization method to achieve a patch-partition with high quality. Finally, after the imposition of C1=G1-continuity constraints on the interface of neighboring B\'ezier patches with respect to each quad in the quadrangulation, the high-quality B\'ezier patch parameterization is obtained by a C1-constrained local optimization method to achieve uniform and orthogonal iso-parametric structures while keeping the continuity conditions between patches. The efficiency and robustness of the proposed method are demonstrated by several examples which are compared to results obtained by the skeleton-based parameterization approach

A family of C1C^1 quadrilateral finite elements

We present a novel family of $C^1$ quadrilateral finite elements, which define global $C^1$ 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 $p\geq 6$, to all degrees $p \geq 3$. Thus, we call the family of $C^1$ finite elements Brenner-Sung quadrilaterals. The proposed $C^1$ 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 $C^2$ 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~$5$, and the polynomial degrees $p=3$ and $p=4$ by employing a splitting into $3\times3$ or $2\times2$ polynomial pieces, respectively. The proposed elements reproduce polynomials of total degree $p$. 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 $p\in\{3,4,5\}$ 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 $C^1$ quadrilateral finite element for the numerical analysis of fourth order problems, also indicating that (for $p=5$) the proposed element performs comparable or in general even better than the Argyris triangle with respect to the number of degrees of freedom

Automation of isogeometric formulation and efficiency consideration

This paper deals with automation of the isogeometric finite element formulation. Isogeometric finite element is implemented in AceGen environment, which enables symbolic formulation of the element code and the expressions are automatically opti- mized. The automated code is tested for objectivity regarding numerical efficiency in a numeric test with the Cooke membrane. This test shows that automatic code generation optimizes the isogeometric quadrilateral element with linear Bezier splines to the degree of only twelve percent overhead against standard displacement quadrilateral element of four nodes. Additionaly, the automated isogeometric element code is tested on a set of standard benchmark test cases to further test the accurancy and efficiency of the pre- sented isogeometric implementation. The isogeometric displacement brick element with quadratic Bezier splines is in all tests compared to a collection of standard displacement element formulations and a selection of EAS elements. The presented results show su- perior behaviour of the isogeometric displacement brick element with quadratic Bezier splines for coarse meshes and best convergence rate with mesh refinement in most test cases. Despite all optimization of the element code the biggest disadvantage of the isogeo- metric model remains the time cost of the isogeometric analysis. Thus, when considering the ratio between solution error and solution time, the use of stable EAS elements, likeTSCG12, remains preferable

Development of a multiblock procedure for automated generation of two-dimensional quadrilateral meshes of gear drives

This article describes a new multiblock procedure for automated generation of two-dimensional quadrilateral meshes of gear drives. The typical steps of the multiblock schemes have been investigated in depth to obtain a fast and simple way to mesh planar sections of gear teeth, allowing local mesh refinement and minimizing the appearance of distorted elements in the mesh. The proposed procedure is completed with two different mesh quality enhancement techniques. One of them is applied before the mesh is generated, and reduces the distortion of the mesh without increasing the computational time of the meshing process. The other one is applied once the mesh is generated, and reduces the distortion of the elements by means of a mesh smoothing method. The performance of the proposed procedure has been illustrated with several numerical examples, which demonstrate its ability to mesh different gear geometries under several meshing boundary conditions