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An efficient automatic mesh generation algorithm for planar isogeometric analysis using high-order rational Bézier triangles

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Abstract

Isogeometric analysis (IGA) is a numerical method that is receiving increasing attention in the last decade. The main goal of IGA is to closely couple geometric modeling with numerical analysis. To that end, in IGA both components use the same geometric representation, e.g., Bézier and NURBS curves and surfaces. However, in many cases, the geometric representation in a CAD system cannot be directly employed in numerical simulation, as only the boundary of the geometry is parametrized. In these cases, an interior parametrization must be constructed before performing isogeometric analysis. This paper presents an algorithm for generation of unstructured geometrically exact meshes composed of rational Bézier triangles of arbitrary degree, and applies it to plane models described by NURBS curves in a Boundary-Representation (B-Rep) scheme. The proposed algorithm respects input discretization and is capable of generating high quality coarse meshes even when high curvature segments are considered. An efficient high-order smoothing step is employed to avoid tangled elements and to improve element quality. The proposed algorithm attains superior performance when compared to a well-known algorithm in the literature, and performs well in the case of complex geometries. High quality meshes were obtained in all examples analyzed. Furthermore, our implementation is efficient, written in C++, and is available as an open-source software

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Acknowledgements

The authors acknowledge the financial support provided by CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior) in PDSE (Programa de Doutorado Sanduíche no Exterior) process 88881.190187/2018-01, and by CNPq (Conselho Nacional de DesenvolvimentoCientífico e Tecnológico).

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Authors and Affiliations

  1. Computer Graphics, Virtual Reality and Animation Group (CRAb), Universidade Federal do Ceará, Fortaleza, CE, Brazil

    Elias Saraiva Barroso, Joaquim Bento Cavalcante-Neto & Creto Augusto Vidal

  2. Aerospace Engineering Sciences, University of Colorado at Boulder, Boulder, CO, USA

    John Andrew Evans

  3. Laboratório de Mecânica Computacional e Visualização, Universidade Federal do Ceará, Fortaleza, CE, Brazil

    Evandro Parente Jr.

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Appendix

Appendix

1.1 A—Bivariate Bernstein polynomials and derivatives

The dynamic programming concept is used to define an efficient algorithm to evaluate all basis functions appearing in Eq. (12), avoiding waste of resources. The array indexing scheme used to store basis functions is shown in Fig. 3b.

During the evaluation of \(B^{p}\) basis, the terms \(B^{p-1}_{i-1,j,k}\), \(B^{p-1}_{i,j-1,k}\) and \(B^{p-1}_{i,j,k-1}\) vanish, respectively, when \(i = 0\), \(j = 0\) and \(k = 0\). These terms are located in the \(i^{th}\) column, where \(B^{p-1}_{i-1,j,k}\) is in the same position of the current element, and in the \((i+1){th}\) column, where \(B^{p-1}_{i, j,k-1}\) and \(B^{p-1}_{i,j-1,k}\) are to the right of the current element. Hence, for each intermediary degree, processing the evaluation of the basis functions in reverse order (i from 0 to p) does not require auxiliary memory. These considerations guide the evaluation of basis functions in the algorithm presented in Fig. 28. The inputs are a triangle of degree p and barycentric coordinates r and s and the output are the basis functions stored in the \(\mathbf {b}\) array.

Fig. 28
figure 28

Bézier triangle basis function code

Full size image

The algorithm for the evaluation of the first derivatives is presented in Fig. 29, where the derivatives are stored in dr and ds arrays. The same considerations used to construct the former algorithm are used here. Note that the same memory used to store partial derivatives in the s direction (array ds) can be used to store and access basis functions (array \(\mathbf {b}\)).

Fig. 29
figure 29

Bivariate Bernstein polynomials first derivative code

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1.2 B—Gmsh parameters

See Fig. 29, Table 8.

Table 8 Gmsh (version 4.4.1) parameters used in Example 5.2
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Barroso, E.S., Evans, J.A., Cavalcante-Neto, J.B. et al. An efficient automatic mesh generation algorithm for planar isogeometric analysis using high-order rational Bézier triangles. Engineering with Computers 38, 4387–4408 (2022). https://doi.org/10.1007/s00366-022-01613-w

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