Isogeometric analysis with strong multipatch C1-coupling
Introduction
Isogeometric analysis, first introduced by Hughes et al. (2005), has been the subject of intensive study over the past decade, in part due to its promise of more closely integrating the analysis and design phases of product development. A distinct feature of this method is that the basis functions used for the discretization of the approximation space and the geometry description can have increased smoothness, up to , where p is the polynomial degree of the basis. This is a significantly higher continuity order compared to the Lagrange polynomials commonly used in the finite element analysis, which are typically restricted to . As a result, better efficiency in terms of degrees of freedom is observed when the solution is suitably smooth, and 4th (as well as higher) order partial differential equations (PDEs) can be approximated using standard discretization methods.
Unfortunately, on complex geometries where multiple parameter spaces (patches) are joined together to describe the physical domain, there is typically a loss of continuity which occurs at the patch boundaries. This decrease of smoothness is dictated by the geometry description, where parameterizations are normally used to deal with kinks and corners in the domain. Multiple patches also need to be used to describe complex domains such as those with inclusions, since a single patch is limited by its underlying tensor product structure to describing relatively simple shapes.
Several methods have been proposed in engineering literature to deal with the decreased smoothness at the patch interfaces, such as the bending strip method or Kirchhoff–Love shells (Kiendl et al., 2010), which employs a fictitious material along the patch interface to approximately satisfy the kinematic constraints. This method was also used in Schmidt et al. (2012) to couple trimmed NURBS patches. Other approaches include mortar methods (Brivadis et al., 2015, Dornisch et al., 2015, Hesch and Betsch, 2012), where adjacent patches are related by a master-slave relation and Lagrange multipliers are used to enforce continuity in a variational sense. The most accurate and stable development to date seems to be the Nitsche's method (Guo and Ruess, 2015, Apostolatos et al., 2014, Nguyen et al., 2014, Nguyen-Thanh et al., 2017), where a mesh-dependent penalty (stabilization) term is used. These methods allow the coupling of non-conforming patches, however, they may lead to semi-definite saddle point problems in the case of mortar or Lagrange multiplier method. Nitsche's method also requires the determination of the stabilization parameters, which increases the overall cost.
A different way of ensuring that the desired continuity requirements are satisfied is to construct a basis for the approximation space, where the basis functions themselves have the required smoothness. Piecewise polynomial bases with continuity have been derived using symbolic algebra for planar bilinear patches in Kapl et al., 2015, Kapl et al., 2017a. A similar approach was used to obtain bases for planar domains in Kapl and Vitrih, 2017b, Kapl and Vitrih, 2017a. These functions span a subspace of the or spline spaces defined on each patch individually and therefore result in a reduction in the number of degrees of freedom compared to the full (with less regularity) approximation space. A possible issue with this method was noted in Collin et al. (2016), where it was shown that the over-constraining of a piecewise polynomial subspace of a given degree over certain geometries could result in a loss of approximation properties ( locking). This has led to the study of analysis-suitable (AS) parameterizations (Kapl et al., 2017b), which include bilinear and bilinear-like planar mappings.
A related problem is the construction of smooth spline spaces over unstructured meshes, such as those obtained using T-Splines or even standard finite element mesh generators. Particular attention has been devoted to the parametrization around extraordinary vertices (interior vertices on quadrilateral meshes which have valence different than 4), where the desired smoothness properties have to be imposed through additional constraints. This leads to the “capping problem”, where a ring of elements around the extraordinary vertex needs to be adjusted, for which various techniques have been proposed in Nguyen and Peters (2016), Toshniwal et al. (2017b), Karčiauskas et al. (2016). In Toshniwal et al. (2017a), smooth polar splines have been used at the extraordinary vertex, while a mathematical analysis of the dimension and basis construction on arbitrary topologies is presented in Mourrain et al. (2016). A construction using Hermite splines with optimal approximation properties is described in Wu et al. (2017), while a more general method for smooth approximations over unstructured meshes is given in Bercovier and Matskewich (2017).
Another common approach for constructing smooth surfaces is through the use of subdivision surfaces, for which applications, in particular to thin-shell analysis have seen increased interest, see Cirak et al., 2000, Wei et al., 2016. However, while subdivision surfaces provide a convenient way to construct smooth surfaces, they are not yet widely used for engineering applications. Alternatively, T-Splines have been used more extensively for modeling surface and plane geometries (Bazilevs et al., 2010, Dörfel et al., 2010), using both unstructured and hierarchical meshes. Both the T-Spline and subdivision surface meshes require particular attention near the extraordinary vertices where the smoothness or approximation properties may be reduced when used in the analysis. Moreover, several approaches have been developed for discretizing the interior of the domain from a boundary triangulation or CAD surface geometry using T-Splines (Escobar et al., 2011, Zhang et al., 2012), or Bézier tetrahedra (Engvall and Evans, 2017, Xia and Qian, 2017).
In this work, we follow a constructive approach for the approximation space, while leaving the geometric parametrization unchanged. We consider general geometries which are not limited to planar or bilinear mappings. A suitable basis is given in terms of Bézier–Bernstein polynomials, whose coefficients are numerically computed based on the given geometry. The problem of locking is overcome by localized degree elevation along the patch boundaries which restores the optimal convergence of the approximation in terms of degrees of freedom. To ensure that the basis functions have local support, a minimal determining set (MDS) (Bercovier and Matskewich, 2017) is computed. The method is also applied to non-planar surfaces, with the underlying assumption that the surfaces have (normal vector) continuity along the patch interfaces. This allows for the study of manifold-based 4th order PDEs on smooth surfaces as in Majeed and Cirak (2017), Dedè and Quarteroni (2015), Nguyen (2016).
The remainder of the paper is organized as follows: in Section 2 we describe the main ideas of applying the continuity constraints to the basis construction for two arbitrary planar patches. The extension of the method to surfaces is given in Section 3 and to multiple patches in Section 4. The procedure of degree elevation along the patch boundaries to overcome locking is described in Section 5. A method based on the MDS to ensure that the basis functions have local support is discussed in Section 6. Numerical results for several 2nd and 4th order PDEs are given in Section 7, followed by concluding remarks in Section 8.
Section snippets
Smooth isogeometric functions in 2D
We first describe the construction of continuous isogeometric functions for two coplanar patches. Suppose that the two patch subdomains are denoted by and , and we assume that each is parametrized by a geometry mapping . In standard isogeometric analysis, the mapping G is defined as a linear combination of NURBS basis functions , where are the control points corresponding to the given basis functions. The space
coupling for surfaces in space
The coupling method, discussed in the previous section can be extended to surfaces. Suppose the geometric mapping in the space is given by where are the control points correspond to patch (ℓ), and we assume that the surface is at least at the patch interface. For an arbitrary parametric point on patch (ℓ), its corresponding position on the surface is: and the Jacobian is
Multi-patch coupling
Multi-patch coupling requires more computational effort than the two patch coupling because it involves more than one common boundary. The main ideas are nevertheless similar. For multi-patch coupling, we need to construct Type 2 basis functions which are continuous across the common boundaries. Similarly to the two patch case, the values of the relevant coefficients of the Type 2 basis functions can be obtained by solving a homogeneous linear system .
We first select and index the
Degree elevation at patch interface
As discussed in Collin et al. (2016), for certain geometries the over-constraining of the solution space would lead to a suboptimal order of approximation. Consequently, the error in the approximation would not decrease even when the mesh is refined. Thus the convergence rate is restricted, and this circumstance is known as locking. It was observed that this phenomenon could be mitigated by either adopting lower order continuity or by increasing the polynomial degree globally. These
Support localization using MDS
We note that a standard orthogonal nullspace that considers all the coefficients along the common interface would result in basis functions with support over all the boundary elements associated with the common interface. For complex geometries or fine discretizations, this will lead to ill-conditioning and non-banded stiffness matrices with a large number of non-zero entries. Therefore it is necessary to localize the Type 2 basis functions by computing a sparse nullspace. Unfortunately, it is
Numerical examples
In this section, we will show with several examples the efficiency of the method described in the previous sections. In particular, we demonstrate the accuracy of coupling with problems involving fourth order partial differential equations such as Kirchhoff–Love shell models and Cahn–Hilliard phase field applications.
Conclusions
coupling of basis functions on the multi-patch domains is presented in this study. Globally smooth basis functions can be constructed provided the patches satisfy a continuity condition. The basis functions are obtained by imposing continuity constraints on their graph surfaces. We propose to overcome locking by performing partial degree elevation. As a result, the elements associated with the common boundary will possess basis functions of different degree, in particular, the
Acknowledgements
The authors would like the acknowledge the financial support of the German Academic Exchange Program (DAAD).
References (46)
- et al.
Stable generalized finite element method (SGFEM)
Comput. Methods Appl. Mech. Eng.
(2012) - et al.
Isogeometric analysis using T-splines
Computational Geometry and Analysis
Comput. Methods Appl. Mech. Eng.
(2010) - et al.
Isogeometric mortar methods
Isogeometric Analysis Special Issue
Comput. Methods Appl. Mech. Eng.
(2015) - et al.
Analysis-suitable multi-patch parametrizations for isogeometric spaces
SI: New Developments Geometry
Comput. Aided Geom. Des.
(2016) - et al.
Isogeometric analysis for second order partial differential equations on surfaces
Isogeometric Analysis Special Issue
Comput. Methods Appl. Mech. Eng.
(2015) - et al.
Adaptive isogeometric analysis by local h-refinement with T-splines
Computational Geometry and Analysis
Comput. Methods Appl. Mech. Eng.
(2010) - et al.
Isogeometric unstructured tetrahedral and mixed-element Bernstein–Bézier discretizations
Comput. Methods Appl. Mech. Eng.
(2017) - et al.
A new approach to solid modeling with trivariate T-splines based on mesh optimization
Comput. Methods Appl. Mech. Eng.
(2011) - et al.
Isogeometric analysis of the Cahn–Hilliard phase-field model
Comput. Methods Appl. Mech. Eng.
(2008) - et al.
Nitsche's method for a coupling of isogeometric thin shells and blended shell structures
Isogeometric Analysis Special Issue
Comput. Methods Appl. Mech. Eng.
(2015)
Isogeometric analysis and domain decomposition methods
Comput. Methods Appl. Mech. Eng.
Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement
Comput. Methods Appl. Mech. Eng.
Isogeometric analysis with geometrically continuous functions on planar multi-patch geometries
Special Issue on Isogeometric Analysis: Progress and Challenges
Comput. Methods Appl. Mech. Eng.
Dimension and basis construction for analysis-suitable two-patch parameterizations
Geometric Modeling and Processing 2017
Comput. Aided Geom. Des.
Space of -smooth geometrically continuous isogeometric functions on planar multi-patch geometries: dimension and numerical experiments
Comput. Math. Appl.
Space of -smooth geometrically continuous isogeometric functions on two-patch geometries
Comput. Math. Appl.
Isogeometric analysis with geometrically continuous functions on two-patch geometries
High-Order Finite Element and Isogeometric Methods
Comput. Math. Appl.
Generalizing bicubic splines for modeling and IGA with irregular layout
SPM 2015
Comput. Aided Des.
Stable generalized finite element method and associated iterative schemes; application to interface problems
Comput. Methods Appl. Mech. Eng.
The bending strip method for isogeometric analysis of Kirchhoff–Love shell structures comprised of multiple patches
Comput. Methods Appl. Mech. Eng.
IETI-isogeometric tearing and interconnecting
Comput. Methods Appl. Mech. Eng.
Isogeometric analysis using manifold-based smooth basis functions
Special Issue on Isogeometric Analysis: Progress and Challenges
Comput. Methods Appl. Mech. Eng.
Dimension and bases for geometrically continuous splines on surfaces of arbitrary topology
Comput. Aided Geom. Des.
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