A mesh regularization scheme to update internal control points for isogeometric shape design optimization

https://doi.org/10.1016/j.cma.2014.11.045Get rights and content

Highlights

  • Efficient mesh regularization scheme that avoids mesh entanglement.

  • Bijective mapping due to the convexity of Dirichlet energy functional.

  • High quality of domain parameterization in isogeometric shape design optimization.

  • Uniformity achieved by minimizing the variance of Jacobian.

  • Orthogonality using a dimensionless measure instead of stretching energy functional.

Abstract

This paper presents a variational method to update internal control points in isogeometric shape optimization. The important properties of domain parameterization such as bijective mapping between parametric and physical domains, uniform mesh, and orthogonal mesh are enforced simultaneously. The bijective mapping is achieved by minimizing a Dirichlet energy functional. To prevent the divergent behavior of the minimizing process due to the severely distorted initial mesh, a constraint is introduced to enforce the positive Jacobian of mapping from parametric to physical domains. In spite of adding the constraint that might increase computational costs, the proposed method is more efficient due to the convexity of Dirichlet energy functional, compared with the other unconstrained methods. Also, it turns out that the proposed method is more effective to achieve the bijective mapping, especially near a concave boundary. The uniform parameterization of the domain is achieved by minimizing the Dirichlet energy functional and the orthogonality of mesh is performed by minimizing a dimensionless functional. The required design sensitivity of the employed functional and constraint is derived with respect to the position of internal control points. The developed scheme of mesh regularization is effective to maintain the high quality of domain parameterization during the shape optimization process.

Introduction

To resolve the discrepancy between the finite element model and the computer aided design (CAD) model, Hughes et al.  [1] developed the isogeometric analysis (IGA) method, which is an analysis framework employing the same basis function as used in the CAD system. Instead of the mesh quality in the conventional finite element analysis (FEA), an analogous concept of model quality exists within the IGA (Cohen et al.  [2]). The model quality indicates the parameterization quality of the physical domain, which has been shown to play a significant role in the accuracy of the isogeometric analysis as mesh quality does in the conventional finite element analysis. Lipton et al.  [3] investigated the effect of degenerating control net on the accuracy of IGA. Also, Xu et al.  [4] showed that the quality of domain parameterization significantly affects the solution accuracy, based on the IGA of heat conduction problems. One of the major causes of the invalid parameterization of domain, which indicates negative Jacobian of mapping from parametric to physical domain exists in the domain, is the large variation of the physical domain during the simulation process. Especially in the shape design optimization procedure, it is still an inherent challenge to update domain parameterization after boundary variation, in order to avoid mesh distortion and maintain high quality of domain parameterization for reliable response and design sensitivity analysis.

In the shape design optimization based on the conventional finite element analysis, re-meshing is unavoidable if the boundary variation is significantly large and complicated. However, re-meshing should not be frequently used, since the addition of finite elements may lead to the sudden variation of the objective function or violation of the constraints, which prevents smooth convergence to an optimal shape. To minimize the use of remeshing, Belegundu and Rajan  [5] choose the magnitude of a set of fictitious loads applied on the structure as design variables, and the nodal displacements obtained by solving the linear elasticity problem under these fictitious loads are added to the initial mesh to update the design. Yao and Choi  [6] consider the design variation on the boundary as prescribed displacement, and find the nodal displacements of internal nodes by solving the linear elasticity problems to update the internal mesh. However, Hsu and Chang  [7] show that using homogeneous material properties in the aforementioned methods utilizing the solution of linear elasticity problem may result in severe mesh distortion due to over-stiffening or under-stiffening of some elements. To overcome this problem, they proposed a method to perform two consecutive linear elastic finite element analyses. The first analysis is performed with the homogeneous elastic property, and the second analysis is performed based on the non-uniform elastic property determined, using a fully stressed design method, from the result of the first analysis. However, in boundaries with large variations, it is fundamentally unavoidable to incrementally increase the boundary variation; and for each increment, two linear elastic analyses are required to be performed, which may significantly increase the computational cost of the shape design optimization.

In the isogeometric analysis framework, there have been several researches to construct high quality domain parameterization, based on the boundary geometry obtained from CAD system. Xu et al.  [4] suggest a mesh regularization scheme using constrained optimization to distribute the internal control points, considering the uniformity, orthogonality and injectivity of the domain parameterization. The uniformity and orthogonality are realized through minimizing the bending and stretching energy functionals, respectively. Also, the injectivity is realized through imposing the positivity of the Jacobian control scalars. Since it requires large computational costs to evaluate the Jacobian control scalars, they utilize a method that approximately tests the sign of the Jacobian control scalars based on conic-hull hodograph. The conic-hull hodograph based method is efficient, but is not usable if the cone of the given boundary is non-transverse due to its complicated geometry. For detailed explanations of the methods to prevent self-intersection of the CAD object using the Jacobian control scalar and conic-hull hodograph, interested readers may refer to Gain and Dodgson  [8]. Also, this mesh regularization scheme using constrained optimization is applied to three dimensional solid model in Xu et al.  [9], [10]. Wang and Qian  [11] maximize the minimal Jacobian control scalar to obtain valid domain parameterization of a trivariate B-spline solid through constrained optimization method. They also utilize the constraint aggregation strategy to reduce the number of constraints. Instead of using constraints on the Jacobian control scalar to impose the positivity of the Jacobian in the whole domain, Xu et al.  [12] suggest the unconstrained optimization method, based on the harmonic mapping theory. They derive a functional from the Laplace’s equation, under the assumption that Jacobian is non-vanishing in the domain. However, a stationary solution of minimizing this functional is not guaranteed to satisfy the Laplace’s equation, and gives a significantly deteriorated domain parameterization, especially near the concave boundary. This drawback is investigated in this paper, through comparison with the results, obtained from minimizing the Dirichlet energy functional. Furthermore, they obtain the sensitivity of the suggested functionals with respect to the position of the internal control point by using the finite difference method, which requires much larger computational costs than evaluating the analytic sensitivity. Also, due to the dimensional heterogeneity among the functionals combined to the objective function as weighted sum, selecting suitable weights remains a great challenge.

Isogeometric shape design optimization has several benefits over the conventional one based on finite element analysis (Cho and Ha  [13]; Yoon et al.  [14]): First, it provides more accurate sensitivity of complex geometries including higher order effects such as normal vector and curvature information. Second, it vastly simplifies the design modification of complex geometries without communicating with the CAD description of geometry during the optimization process. Since the geometric properties are embedded into the NURBS basis functions and control points, design modifications are easily obtainable through adjusting control points which represent the geometric model. For more flexible shape representation, Qian  [15] and Nagy et al.  [16] employ weights as well as positions of control points as design variables. In a two dimensional planar NURBS surfaces or in three dimensional solid models, the movement of the internal control points does not change the shape, but does change the domain parameterization. Thus, it is significant to update the internal control points after boundary variation to maintain high quality domain parameterization during the shape design optimization. Several methods have been proposed to maintain the injectivity of the domain parameterization during the isogeometric shape design optimization, through enforcing positivity of Jacobian control scalars when negative Jacobian appears in the domain. Qian and Sigmund  [17] maximize the minimum value of the Jacobian control scalar until it becomes positive, based on min–max optimization formulation. Manh et al.  [18] present quasi-conformal mapping and spring-based mesh model, which are simple and computationally inexpensive but are not suitable for constructing the high quality of domain parameterization, to update the internal control points. Also, if negative Jacobian appears in the domain during the shape design optimization process, they minimize the winslow functional to realize conformal mapping under the constraint for the Jacobian control scalars to be positive. Nörtoft and Gravesen  [19] improve the quality of the boundary parameterization through regularizing the optimization problem by adding a measure of the boundary parameterization quality. They investigate how this regularization measure influences the optimization process and optimal design. The weighted sum of the measures which are the pressure drop of flow through pipe and the other one for improving the quality of boundary parameterization is utilized as the objective function of shape design optimization. However, its greatest challenge lies in the choice of the regularization weight to control the individual contribution of each measure to the objective function.

In Section  2, we describe the construction of NURBS basis functions, and explain the components that affect the domain parameterization of two dimensional planar NURBS surfaces. In Section  3, we explain the mesh regularization scheme, which is covered by two parts. The first part is to achieve the bijective mapping between the parametric and physical domain through minimizing the Dirichlet energy functional, and the second part is to improve the mesh quality in terms of uniformity and orthogonality. Also, sensitivity formulas of the suggested functionals and constraint with respect to the position of the internal control point are analytically derived. In Section  4, we employ the mesh regularization scheme to update internal control points during the shape design optimization process of heat conduction problems. The efficiency and applicability of the mesh regularization scheme are demonstrated. Finally, we draw conclusions, which present the importance of the suggested mesh regularization scheme to avoid mesh distortion and maintain high quality domain parameterization during the shape design optimization process.

Section snippets

B-spline basis function

In the IGA, the solution space is expressed in terms of the same basis functions used to describe the geometry. The IGA has several advantages over the conventional FEA such as geometric exactness and ease of refinements due to the use of NURBS basis functions. Consider a set of knots ξ in one dimensional space, ξ={ξ1,ξ2,,ξn+p+1}, where p and n are the order of basis function and the number of control points, respectively. The B-spline basis functions are recursively constructed as Ni0(ξ)={1

Mesh regularization in isogeometric analysis

There are three basic requirements for general mesh regularization:

  • (1)

    Bijective mapping between parametric and physical domains,

  • (2)

    Uniformity of domain parameterization, and

  • (3)

    Orthogonality of mesh.

Isogeometric shape design optimization

In this section, the proposed mesh regularization scheme is applied to update the internal control points for the domain variation process after the boundary variations. The domain variation process is divided into two steps: initialization and mesh regularization. During the initialization step, the internal control points are distributed using the discrete Coons method  [4]. The discrete Coons method determines the positions of the internal control points, based on those of the boundary

Conclusions

In this paper, to update the internal control points during the isogeometric shape design optimization, we develop a mesh regularization scheme that avoids the well-known mesh entanglement and keeps the high quality of domain parameterization. The features of proposed domain parameterization are in two folds. First, we perform an effective bijective mapping between the parametric and the physical domains using the convexity of the Dirichlet energy functional. Second, the quality of domain

Acknowledgments

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2010-0018282). The authors would also like to thank Ms. Inyoung Cho at Korea University for editing assistance.

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