Isogeometric analysis for trimmed CAD surfaces using multi-sided toric surface patches

doi:10.1016/j.cagd.2020.101847

Computer Aided Geometric Design

Volume 79, May 2020, 101847

https://doi.org/10.1016/j.cagd.2020.101847 Get rights and content

Highlights

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We develop a new isogeometric method for dealing with trimmed CAD surfaces.
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The presented method can convert each trimmed element into a toric surface patch.
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The proposed method can reach the optimized convergence rate.
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The presented method integrates IGA with standard finite-element code architectures.

Abstract

We propose a new isogeometric method using Toric surface patches for trimmed CAD planar surfaces. This method converts each trimmed spline element into a Toric surface patch with conforming boundary representation and converts each non-trimmed spline element into a Bézier element. Because the Toric surface patches are a multi-sided generalization of classical Bézier surface patches, all trimmed and non-trimmed elements of a trimmed CAD surface have a unified geometric representation using Toric surface patches. Toric surface patches share the advantages of isogeometric continuum elements in that they can exactly model the geometry and can be easily implemented in standard finite-element code architectures. Several numerical examples are used to demonstrate the reliability of the proposed method.

Introduction

Isogeometric analysis (IGA) was introduced by Hughes and coworkers (Hughes et al., 2005; Cottrell et al., 2006) to bridge the gap between Computer Aided Geometric Design (CAGD) and Finite Element Analysis (FEA). The core idea of IGA is to use the same smooth and higher-order basis functions for the representation of both the geometry in CAGD and the approximation of solution fields in FEA. IGA has been successfully applied in many areas, such as structural vibrations (Kiendl et al., 2009; Goyal et al., 2013), beam and shell problems (Wang et al., 2015; Liu et al., 2016), fluid–structure interaction problems (Bazilevs et al., 2008, Bazilevs et al., 2009), structural optimization (Dede et al., 2012; Dhote et al., 2013) and fracture analysis (Borden et al., 2014; Schillinger et al., 2015) et al. IGA has been based on a variety of spline basis functions, e.g., B-splines, Non-Uniform Rational B-Splines (NURBS), T-splines (da Veiga et al., 2012; Wang et al., 2011), B-plus-plus splines (B++ Splines) (Zhu et al., 2016), hierarchical B(T)-splines (Schillinger and Rank, 2011), and PHT-splines (Li et al., 2010).

Among those spline methods, NURBS is the most popular mathematical tool in CAD/CAM. It is also a major geometric element in international product data-transfer standards such as the standard for the exchange of product model data (STEP) and the initial graphics exchange specification (IGES). However, due to the inflexibility of the tensor-product form of B-spline and NURBS, determining how to perform IGA on complex topological shapes remains a critical challenge for current IGA applications. A natural method is to use multiple tensor-product surface patches to represent the complex topological shapes. However, dividing the complex shapes into multiple tensor-product patches requires additional efforts that are usually cumbersome. An alternative method of handling complex topological shapes is to transform each complex topological shape into a watertight T-spline representation.

However, determining how to apply isogeometric analysis directly to trimmed CAD models remain a big challenge. A pioneer work of IGA for trimmed surfaces was proposed by Kim et al. in which the NURBS-enhanced FEM methodology is applied to define a suitable integration domain within the parameter space; see Kim et al., 2009, Kim et al., 2010. Another very interesting approach is the WEB-Splines, introduced by Höllig and co-workers in Hollig et al. (2001); Hollig and Reif (2003); Hollig et al. (2005); Kumar et al., 2006a, Kumar et al., 2006b; Apaydin et al. (2008), which are defined on a rectangular mesh grid. Sanches et al. (2011) also proposed an immersed B-spline method that can analyze geometrically complex domains and can locally interpolate Dirichlet boundary conditions as well. In research on imposing Dirichlet boundary conditions with Nitsche's method, Embar et al. (2010) proposed a B-spline-based embedded-domain method that can analyze complex domains. Combined with hierarchical refinement, the B-spline version of the Finite-Cell Method (FCM) presented by Rank and his coworkers Schillinger and Ruess (2015); Rank et al. (2012); Ruess et al. (2014); Zander et al. (2014) show great versatility in analyzing complex topological shapes. The approaches mentioned above can be used to analyze each component surface within a compound surface. Until now, for the trimmed elements and non-trimmed elements in a trimmed surface, there has been no unified geometric representation, which has made it difficult to integrate isogeometric analysis for trimmed CAD geometries with standard finite-element code architectures.

Toric surface patches (Warren, 1992) may provide a unified representation for both the trimmed spline elements and non-trimmed spline elements. Toric varieties were introduced in the early 1970s in algebraic geometry. Warren (Krasauskas, 2002) was probably the first to notice that the real Toric surfaces can be used in CAGD. Krasauskas (2002) created the multi-sided Bézier surface patches using basis points and predicted that further work incorporating techniques from Toric varieties may lead to practical methods for multi-sided patches. In 2002, Garcia-Puente et al. (2011) presented a kind of multi-sided surface patches, named Toric surface patches, from the theory of Toric varieties in Algebraic Geometry and Toric ideals in Combinatorics. Cottrell et al. (2009) explained the geometric meaning of control surfaces of Toric surface patches using Toric degenerations. The Toric surface patches are a multi-sided generalization of classical Bézier surface patches. Moreover, many real rational surfaces such as rational Bezier surface using in CAGD are found to be Toric surface patches.

In this paper, we propose an isogeometric method using Toric surface patches. The present method can convert trimmed elements in a trimmed NURBS surface to a group of Toric patches. The remaining non-trimmed elements can be changed to Toric surface patches or converted into Bézier patches using knot insertion. In doing so, all elements in a trimmed spline surface have a unified representation, which simplifies the trimmed isogeometric analysis and is helpful for integrating isogeometric analysis and traditional finite-element codes.

The layout of the paper is as follows. In Section 2, trimmed CAD geometries and the associated limitations will be presented. The basic idea of Toric surface patches is also introduced. Section 3 shows how to convert a trimmed patch into a boundary-conforming patch composed of a group of Toric surface patches and a group of Bézier elements. Note Bézier elements are also Toric surface patches. The basic idea of trimmed isogeometric analysis using Toric surface patches and the associated finite-element equations are developed and derived in Section 4. The subsequent section demonstrates the effectiveness of the proposed method in various and significant numerical examples. In the final section, the study's conclusions and an outlook are presented.

Section snippets

Trimmed NURBS surfaces

A NURBS patch of degree $(p, q)$ has the following form: $F (ξ, η) = \sum_{i = 1}^{n_{s}} \sum_{j = 1}^{n_{t}} R_{i j} (ξ, η) P_{i, j},$ where $R_{i, j} (ξ, η)$ is the basis function: $R_{i, j} (ξ, η) = \frac{N_{i, p} (ξ) M_{j, q} (η) ω_{i, j}}{Σ_{s = 1}^{n_{s}} Σ_{t = 1}^{n_{t}} N_{s, p} (ξ) M_{t, q} (η) ω_{s, t}},$ where $ω_{i, j}$ represent the weights and $P_{i, j}$ denote the control points, which form a bidirectional control net. $N_{i, p} (ξ)$ and $M_{j, q} (η)$ are the polynomial B-spline functions, which are defined on the open knot vector.

A trimmed NURBS patch Ω with boundary ∂Ω is composed of two components: a tensor-product parametric

Converting a trimmed spline surface to a group of toric patches and Bézier elements

The trimmed elements can be grouped in three categories: Type A, Type B, Type C and Type D, as shown in Fig. 5. Usually, as mentioned by Kim et al. (Kim et al., 2009), for more complicated trimmed cases with two or more trimming curves or with a highly curved trimming curve, easily resolved by using quad-tree refinements. The cases with two or more trimming curves are easy to address using Toric surface patches because Toric surface patches can interpolate an arbitrary number of Bézier boundary

Formulation

For the sake of concreteness, we shall adopt an elastostatic problem as the modeling problem using the Galerkin method. This method can also be generalized to other fields following a similar procedure.

Following the terminology in Cottrell et al. (2009), we give the strong form of the boundary-value problem:

Given $f_{i} : Ω \to R$ , $g_{i} : Γ_{D_{i}} \to R$ and $h_{i} : Γ_{N_{i}} \to R$ , find $u_{i} : \overline{Ω} \to R$ such that $σ_{i j, j} + f_{j} = 0 in Ω u_{i} = g_{i} on Γ_{D_{i}} σ_{i j} n_{j} = h_{i} on Γ_{N_{i}},$ where $Γ_{D_{i}}$ is the Dirichlet boundary and $Γ_{N_{i}}$ is the Neumann boundary.

Let the trial solution

Numerical examples

The convergence behavior of the trimmed-surface analysis is studied and compared with those of the conventional finite-element methods. Notice that how to impose Dirichlet boundary conditions will affect the numerical results. If imposing Dirichlet boundary conditions onto the control variables, the computation error cannot be ignored according to the conclusions of Wang and Xuan (2010). We followed the method proposed by Wang and Xuan (2010) to impose Dirichlet boundary conditions. Imposing

Conclusions

In this paper, isogeometric analysis for trimmed CAD surfaces using Toric surface patches is proposed. We can convert an n-sided trimmed element to a Toric surface patch. The non-trimmed spline elements can be converted into Bézier elements. Thus, a trimmed CAD surface can be converted into a group of Toric surface patches and a group of Bézier elements that exactly satisfy the Dirichlet and Neumann boundary conditions. The Toric spline patches share the advantages of isogeometric continuum

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The National Natural Science Foundation of China (No. 11872015, No. 11671068, No. 11732004) and Fundamental Research Funds for the Central Universities (No. DUT18JC47). These supports are acknowledged.

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Isogeometric analysis for trimmed CAD surfaces using multi-sided toric surface patches

Highlights

Abstract

Introduction

Section snippets

Trimmed NURBS surfaces

Converting a trimmed spline surface to a group of toric patches and Bézier elements

Formulation

Numerical examples

Conclusions

Declaration of Competing Interest

Acknowledgements

AEÜ, Int. J. Electron. Commun.

Comput. Methods Appl. Mech. Eng.

Comput. Methods Appl. Mech. Eng.

Comput. Methods Appl. Mech. Eng.

Comput. Aided Geom. Des.

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Comput. Methods Appl. Mech. Eng.

Comput. Methods Appl. Mech. Eng.

Comput. Methods Appl. Mech. Eng.

J. Comput. Appl. Math.

Comput. Methods Appl. Mech. Eng.

Comput. Methods Appl. Mech. Eng.

Comput. Methods Appl. Mech. Eng.

Comput. Methods Appl. Mech. Eng.

Comput. Methods Appl. Mech. Eng.

Comput. Aided Geom. Des.

Comput. Methods Appl. Mech. Eng.

Comput. Aided Des.