Analysis-suitable volume parameterization of multi-block computational domain in isogeometric applications

doi:10.1016/j.cad.2012.10.022

Computer-Aided Design

Volume 45, Issue 2, February 2013, Pages 395-404

https://doi.org/10.1016/j.cad.2012.10.022 Get rights and content

Abstract

Parameterization of the computational domain is a key step in isogeometric analysis just as mesh generation is in finite element analysis. In this paper, we study the volume parameterization problem of the multi-block computational domain in an isogeometric version, i.e., how to generate analysis-suitable parameterization of the multi-block computational domain bounded by B-spline surfaces. Firstly, we show how to find good volume parameterization of the single-block computational domain by solving a constraint optimization problem, in which the constraint condition is the injectivity sufficient conditions of B-spline volume parameterization, and the optimization term is the minimization of quadratic energy functions related to the first and second derivatives of B-spline volume parameterization. By using this method, the resulting volume parameterization has no self-intersections, and the isoparametric structure has good uniformity and orthogonality. Then we extend this method to the multi-block case, in which the continuity condition between the neighbor B-spline volumes should be added to the constraint term. The effectiveness of the proposed method is illustrated by several examples based on the three-dimensional heat conduction problem.

Graphical abstract

Introduction

The isogeometric analysis (IGA for short) method proposed by Hughes et al. in [1] offers the possibility of bridging the gap between CAD and CAE. The approach uses the same spline representation both for the geometry and for the physical solutions, and thus avoids this costly back and forth transformations. This uniform framework provides more accurate and efficient ways to deal with complex shapes and to approximate the solutions of physical simulation problems. On the other hand, it also raises interesting geometric problems for analysis-suitable modeling tools [2], [3], [4].

It is well known that mesh generation, which generates a discrete mesh of a computational domain from a given CAD object, is a key and the most time-consuming step in finite element analysis (FEA for short). It consumes about 80% of the overall design and analysis process [5] in automotive, aerospace and ship industry. Parameterization of the computational domain in IGA, which corresponds to the mesh generation in FEA, also has some impact on the analysis result and efficiency. In particular, arbitrary refinements can be performed on the computational mesh in FEA, but in IGA if we compute with a tensor product B-splines, we can only perform refinement operations in each parametric direction by knot insertion or degree elevation. Hence, parameterization of the computational domain is also important for IGA. As it is pointed by Cottrell et al. [6], one of the most significant challenges towards isogeometric analysis is constructing trivariate spline volume parameterizations from a given CAD boundary representation.

From the viewpoint of graphics applications, volume parameterization of 3D models has been studied in [7], [8], [9]. As far as we know, there are only a few works on the parameterization of computational domains from the viewpoint of isogeometric applications. Martin et al. [10] proposed a method to fit a genus-0 triangular mesh by B-spline volume parameterization, based on discrete volumetric harmonic functions; this can be used to build computational domains for 3D IGA problems. Cohen et al. [11] proposed the concept of analysis-aware modeling, in which the parameters of CAD models should be selected to facilitate isogeometric analysis. They also demonstrated the influence of parameterization of computational domains by several examples. Escobar et al. proposed a method to construct a trivariate T-spline volume of complex genus-0 solids for isogeometric application [12]. However, the proposed method demands a surface triangulation as input data. A variational approach for constructing NURBS parameterization of swept volumes is proposed by Aigner et al. [13]. Given boundary CAD information, an approximate implicitization technique is used for parameterization of the 2D computational domain in [14]. In [15], [16], the $r$ -refinement method for generating optimal analysis-aware parameterization of the computational domain is proposed. However, it only works for specified analysis problems. A general construction method for analysis-suitable planar B-spline parameterization in the two-dimensional isogeometric problem is proposed in [17] based on harmonic mapping. In this paper, from the given boundary CAD information, the volume parameterization problem for the multi-block computational domain is studied based on the trivariate generalization of the method proposed in [16].

In IGA, the parameterization of a computational domain is determined by control points, knot vectors and the degrees of B-spline objects. For three-dimensional IGA problems, the knot vectors and the degree of the computational domain are determined by the given boundary surfaces. That is, given boundary surfaces, the quality of parameterization of the computational domain is determined by the positions of inner control points. Hence, finding a good placement of the inner control points inside the computational domain is a key issue. A basic requirement of the resulting volume parameterization for IGA is that it doesn’t have self-intersections, so that it is an injective map from the parameterization domain to the computational domain. In order to get more accurate simulation results, the isoparametric structure in the computational domain should be as uniform as possible and have orthogonal isoparametric surfaces [18]. In this paper, we study the volume parameterization problem of the multi-block computational domain in the isogeometric version, i.e., how to generate analysis-suitable parameterization of the multi-block computational domain bounded by B-spline surfaces. Our main contributions are:

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A constraint optimization framework is proposed to generate a multi-block volume parameterization without self-intersections, and the resulting isoparametric structure has good uniformity and orthogonality;
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Some classical results in the field of differential geometry related to parametric surfaces are generalized to the case of trivariate parametric volumes, such as the orthogonal conditions of isoparametric structure and the $C^{1}$ conditions between B-spline volumes.
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We test the volume parameterization results on the heat conduction problem to show the effectiveness of the proposed method.

The remainder of the paper is organized as follows. Section 2 presents some preliminaries on B-spline volume parameterization. Section 3 describes the constraint optimization method for single-block volume parameterization of the 3D computational domain. Section 4 presents a volume parameterization framework for the multi-block computational domain based on the proposed methods in Section 3. Section 5 tests the volume parameterization results on the heat conduction problem to show the effectiveness of the proposed method. Finally, we conclude this paper in Section 6.

Section snippets

Preliminaries

For a parameterization $σ$ from $P : [a, b] \times [c, d] \times [e, f]$ to $Ω \subset R^{3}$ , we define the boundary surfaces as the image of ${a} \times [c, d] \times [e, f]$ , ${b} \times [c, d] \times [e, f]$ , ${c} \times [a, b] \times [e, f]$ , ${d} \times [a, b] \times [e, f]$ , ${e} \times [a, b] \times [c, d]$ , ${f} \times [a, b] \times [c, d]$ by $σ$ . We say that $σ$ defines a regular boundary if these surfaces do not intersect pairwise, except at their boundary curves and if they have no self-intersection points.

We consider the following trivariate B-spline parameterization $σ : (ξ, η, ζ) \in P ≔ [a, b] \times [c, d] \times [e, f] \mapsto σ (ξ, η, ζ) ≔ \sum_{\binom{\binom{0 \leq i \leq l}{0 \leq j \leq m}}{0 \leq k \leq n}} c_{i, j,}$

Constraint optimization method for volume parameterization of computational domains

In this section, we aim at finding injective volume parameterization of the computational domain with a uniform and orthogonal isoparametric net.

Volume parameterization of multi-block computational domain

In this section, we propose a volume parameterization framework for the multi-block computational domain based on the proposed methods in Section 3.

Examples and comparison

Starting from a trivariate B-spline volume as the computational domain, a general framework of an isogeometric solver for the 3D heat conduction problem (13) has been implemented as a plugin in the AXEL¹ platform, yielding a B-spline volume as the solution field. The proposed multi-block constraint optimization methods are implemented as a part of the isogeometric toolbox of the project EXCITING.²

In this paper, we test the different

Conclusion

Analysis-suitable volume parameterization of the computational domain plays an important role in isogeometric analysis as mesh generation in finite element analysis. In this paper, the volume parameterization problem of the multi-block computational domain in isogeometric applications is studied. This problem is solved in a constraint optimization framework, in which the constraint condition is the injectivity sufficient conditions of B-spline volume parameterization, and the optimization term

Acknowledgments

The authors wish to thank all anonymous referees for their valuable comments and suggestions. The authors are supported by the 7th Framework Program of the European Union, project SCP8-218536 “EXCITING”. The first author is partially supported by the National Nature Science Foundation of China (Nos. 61004117, and 61272390), the Zhejiang Provincial Natural Science Foundation of China (No. Y1090718), the Defense Industrial Technology Development Program (A3920110002), the Scientific Starting

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Analysis-suitable volume parameterization of multi-block computational domain in isogeometric applications

Abstract

Graphical abstract

Introduction

Section snippets

Preliminaries

Constraint optimization method for volume parameterization of computational domains

Volume parameterization of multi-block computational domain

Examples and comparison

Conclusion

Acknowledgments

Computer Methods in Applied Mechanics and Engineering

Computer Methods in Applied Mechanics and Engineering

Computer-Aided Design

Computer Methods in Applied Mechanics and Engineering

Computer Aided Geometric Design

Computer Methods in Applied Mechanics and Engineering

Computer Methods in Applied Mechanics and Engineering

Computer Aided Geometric Design

Computer Aided Geometric Design

Computer Aided Geometric Design

Local refinement of analysis-suitable T-splines

Computer Methods in Applied Mechanics and Engineering