Multi-patch Domains Research Articles

On the use of elliptic PDEs for the parameterisation of planar multipatch domains

AbstractThis paper presents a parameterisation framework based on (inverted) elliptic PDEs for addressing the planar parameterisation problem of finding a valid description of the domain’s interior given no more than a spline-based description of its boundary contours. The framework is geared towards isogeometric analysis (IGA) applications wherein the physical domain is comprised of more than four sides, hence requiring more than one patch. We adopt the concept of harmonic maps and propose several PDE-based problem formulations capable of finding a valid map between a convex parametric multipatch domain and the piecewise-smooth physical domain with an equal number of sides. In line with the isoparametric paradigm of IGA, we treat the parameterisation problem using techniques that are characteristic for the analysis step. As such, this study proposes several IGA-based numerical algorithms for the problem’s governing equations that can be effortlessly integrated into a well-developed IGA software suite. We augment the framework with mechanisms that enable controlling the parametric properties of the outcome. Parametric control is accomplished by, among other techniques, the introduction of a curvilinear coordinate system in the convex parametric domain, for which more general elliptic PDEs are adopted. Depending on the application, parametric control allows for building desired features into the computed map, such as homogeneous cell sizes or boundary layers.

A Broken FEEC Framework for Electromagnetic Problems on Mapped Multipatch Domains

We present a framework for the structure-preserving approximation of partial differential equations on mapped multipatch domains, extending the classical theory of finite element exterior calculus (FEEC) to discrete de Rham sequences which are broken, i.e., fully discontinuous across the patch interfaces. Following the Conforming/Nonconforming Galerkin (CONGA) schemes developed in Campos Pinto and Sonnendrücker (Math Comput 85:2651–2685, 2016) and Campos Pinto and Güçlü (Broken-FEEC discretizations and Hodge Laplace problems. arXiv:2109.02553, 2022), our approach is based on: (i) the identification of a conforming discrete de Rham sequence with stable commuting projection operators, (ii) the relaxation of the continuity constraints between patches, and (iii) the construction of conforming projections mapping back to the conforming subspaces, allowing to define discrete differentials on the broken sequence. This framework combines the advantages of conforming FEEC discretizations (e.g. commuting projections, discrete duality and Hodge–Helmholtz decompositions) with the data locality and implementation simplicity of interior penalty methods for discontinuous Galerkin discretizations. We apply it to several initial- and boundary-value problems, as well as eigenvalue problems arising in electromagnetics. In each case our formulations are shown to be well posed thanks to an appropriate stabilization of the jumps across the interfaces, and the solutions are extremely robust with respect to the stabilization parameter. Finally we describe a construction using tensor-product splines on mapped cartesian patches, and we detail the associated matrix operators. Our numerical experiments confirm the accuracy and stability of this discrete framework, and they allow us to verify that expected structure-preserving properties such as divergence or harmonic constraints are respected to floating-point accuracy.

Adaptive isogeometric methods with C1 (truncated) hierarchical splines on planar multi-patch domains

Isogeometric analysis is a powerful paradigm which exploits the high smoothness of splines for the numerical solution of high order partial differential equations. However, the tensor-product structure of standard multivariate B-spline models is not well suited for the representation of complex geometries, and to maintain high continuity on general domains special constructions on multi-patch geometries must be used. In this paper, we focus on adaptive isogeometric methods with hierarchical splines, and extend the construction of [Formula: see text] isogeometric spline spaces on multi-patch planar domains to the hierarchical setting. We replace the hypothesis of local linear independence for the basis of each level by a weaker assumption, which still ensures the linear independence of hierarchical splines. We also develop a refinement algorithm that guarantees that the assumption is fulfilled by [Formula: see text] splines on certain suitably graded hierarchical multi-patch mesh configurations, and prove that it has linear complexity. The performance of the adaptive method is tested by solving the Poisson and the biharmonic problems.

Stable discretizations and IETI-DP solvers for the Stokes system in multi-patch IgA

We are interested in a fast solver for the Stokes equations, discretized with multi-patch Isogeometric Analysis. In the last years, several inf-sup stable discretizations for the Stokes problem have been proposed, often the analysis was restricted to single-patch domains. We focus on one of the simplest approaches, the isogeometric Taylor–Hood element. We show how stability results for single-patch domains can be carried over to multi-patch domains. While this is possible, the stability strongly depends on the shape of the geometry. We construct a Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP) solver that does not suffer from that effect. We give a convergence analysis and provide numerical tests.

An approximate [formula omitted] multi-patch space for isogeometric analysis with a comparison to Nitsche’s method

We present an approximately C1-smooth multi-patch spline construction which can be used in isogeometric analysis (IGA). The construction extends the one presented in [1] for two-patch domains. A key property of IGA is that it is simple to achieve high order smoothness within a single patch. However, to represent more complex geometries one often uses a multi-patch construction. In this case, the global continuity for the basis functions is in general only C0. Therefore, to obtain C1-smooth isogeometric functions, a special construction for the basis is needed. Such spaces are of interest when solving numerically fourth-order problems, such as the biharmonic equation or Kirchhoff–Love plate/shell formulations, using an isogeometric Galerkin method.Isogeometric spaces that are globally C1 over multi-patch domains can be constructed as in [2–6]. The constructions require geometry parametrizations that satisfy certain constraints along the interfaces, so-called analysis-suitable G1 parametrizations. To allow C1 spaces over more general multi-patch parametrizations, one needs to increase the polynomial degree and/or to relax the C1 conditions. Thus, we define function spaces that are not exactly C1 but only approximately. We adopt the construction for two-patch domains, as developed in [1], and extend it to more general multi-patch domains.We employ the construction for a biharmonic model problem and compare the results with Nitsche’s method. We compare both methods over complex multi-patch domains with non-trivial interfaces. The numerical tests indicate that the proposed construction converges optimally under h-refinement, comparable to the solution using Nitsche’s method. In contrast to weakly imposing coupling conditions, the approximate C1 construction is explicit and no additional terms need to be introduced to stabilize the method/penalize the jump of the derivative at the interface. Thus, the new proposed method can be used more easily as no parameters need to be estimated.

Practical unstructured splines: Algorithms, multi-patch spline spaces, and some applications to numerical analysis

In this work, we show how some recent advances on simplex spline spaces can be used to construct a polynomial-reproducing space of unstructured splines on multi-patch domains of arbitrary shape and topology. The traces of these functions on the subdomain boundaries reproduce the usual traces of standard polynomial bases used in discontinuous Galerkin (DG) approximations, allowing to borrow many theoretical and practical tools from these methods. Concurrently, we recast some theoretical results on the construction and evaluation of spaces of simplex splines into an explicit, algorithmic form. Together, these efforts allow to formulate a practical, efficient and fully unstructured multi-patch discontinuous Galerkin - isogeometric analysis (DG-IGA) scheme that bridges the gap between some current multi-patch isogeometric analysis (IGA) approaches and the more traditional mesh-based interior penalty discontinuous Galerkin (IPDG) method. We briefly discuss the advantages of this unified framework for time-explicit hyperbolic problems, and we present some interesting numerical examples using the acoustic wave equation.

Tree–cotree decomposition of isogeometric mortared spaces in H(curl) on multi-patch domains

When applying isogeometric analysis to engineering problems, one often deals with multi-patch spline spaces that have incompatible discretisations, e.g. in the case of moving objects. In such cases mortaring has been shown to be advantageous. This contribution discusses the appropriate B-spline spaces needed for the solution of Maxwell’s equations in the functions space H(curl) and the corresponding mortar spaces. The main contribution of this paper is to show that in formulations requiring gauging, as in the vector potential formulation of magnetostatic equations, one can remove the discrete kernel subspace from the mortared spaces by the graph-theoretical concept of a tree–cotree decomposition. The tree–cotree decomposition is done based on the control mesh, it works for non-contractible domains, and it can be straightforwardly applied independently of the degree of the B-spline bases. Finally, the simulation workflow is demonstrated using a realistic model of a rotating permanent magnet synchronous machine.

Stochastic isogeometric analysis on arbitrary multipatch domains by spline dimensional decomposition

This paper presents a new stochastic method by integrating spline dimensional decomposition (SDD) of a high-dimensional random function and isogeometric analysis (IGA) on arbitrary multipatch geometries to solve stochastic boundary-value problems from linear elasticity. The method, referred to as SDD-mIGA, involves (1) analysis-suitable T-splines with significant approximating power for geometrical modeling, random field discretization, and stress analysis; (2) Bézier extraction operator for isogeometric mesh refinement; and (3) a novel Fourier-like expansion of a high-dimensional output function in terms of measure-consistent orthonormalized splines. The proposed method can handle arbitrary multipatch domains in IGA and uses standard least-squares regression to efficiently estimate the SDD expansion coefficients for uncertainty quantification applications. Analytical formulae have been derived to calculate the second-moment properties of an SDD-mIGA approximation for a general output random variable of interest. Numerical results, including those obtained for a 54-dimensional, industrial-scale problem, demonstrate that a low-order SDD-mIGA is capable of efficiently delivering accurate probabilistic solutions when compared with the benchmark results from crude Monte Carlo simulation.

A NEURAL SOLVER FOR VARIATIONAL PROBLEMS ON CAD GEOMETRIES WITH APPLICATION TO ELECTRIC MACHINE SIMULATION

This work presents a deep learning-based framework for the solution of partial differential equations on complex computational domains described with computer-aided design tools. To account for the underlying distribution of the training data caused by spline-based projections from the reference to the physical domain, a variational neural solver equipped with an importance sampling scheme is developed, such that the loss function based on the discretized energy functional obtained after the weak formulation is modified according to the sample distribution. To tackle multi-patch domains possibly leading to solution discontinuities, the variational neural solver is additionally combined with a domain decomposition approach based on the Discontinuous Galerkin formulation. The proposed neural solver is verified on a toy problem and then applied to a real-world engineering test case, namely that of electric machine simulation. The numerical results show clearly that the neural solver produces physics-conforming solutions of significantly improved accuracy.

Planar multi-patch domain parameterization for isogeometric analysis based on evolution of fat skeleton

In this paper, we propose a new algorithm for computing planar multi-patch domain parameterizations which are suitable for consequent numerical computations based on isogeometric analysis. Firstly, a so-called fat skeleton of the given domain, represented by a multi-patch NURBS parameterization, is constructed, which determines the topology of a multi-patch NURBS parameterization suitable for the given domain. Then, we formulate an evolution process which successively transforms the initial shape represented by the fat skeleton to the target shape represented by the given domain. Moreover, a multi-patch optimization is applied in every time iteration to make the multi-patch NURBS parameterization as uniform and orthogonal as possible. The method is constructed such that it is fully automatic and it solves the biggest problems of the domain parameterization problem, i.e., it automatically determines a suitable topology of a multi-patch NURBS parameterization for the given domain and it also automatically solves the boundary correspondence problem. The functionality of the proposed method is demonstrated on several examples.

Construction of approximate [formula omitted] bases for isogeometric analysis on two-patch domains

In this paper, we develop and study approximately smooth basis constructions for isogeometric analysis over two-patch domains. One key element of isogeometric analysis is that it allows high order smoothness within one patch. However, for representing complex geometries, a multi-patch construction is needed. In this case, a C0-smooth basis is easy to obtain, whereas C1-smooth isogeometric functions require a special construction. Such spaces are of interest when solving numerically fourth-order PDE problems, such as the biharmonic equation and the Kirchhoff–Love plate or shell formulation, using an isogeometric Galerkin method.With the construction of so-called analysis-suitable G1 (in short, AS-G1) parametrizations, as introduced in Collin et al. (2016), it is possible to construct C1 isogeometric spaces which possess optimal approximation properties, cf. (Kapl et al., 2019). These geometries need to satisfy certain constraints along the interfaces and additionally require that the regularity r and degree p of the underlying spline space satisfy 1≤r≤p−2. The problem is that most complex geometries are not AS-G1 geometries. Therefore, we define basis functions for isogeometric spaces by enforcing approximate C1 conditions following the basis construction from Kapl et al. (2017). For this reason, the defined function spaces are not exactly C1 but only approximately.We study the convergence behavior and define function spaces that converge optimally under h-refinement, by locally introducing functions of higher polynomial degree and lower regularity. The convergence rate is optimal in several numerical tests performed on domains with non-trivial interfaces. While an extension to more general multi-patch domains is possible, we restrict ourselves to the two-patch case and focus on the construction over a single interface.

Cs-smooth isogeometric spline spaces over planar bilinear multi-patch parameterizations

The design of globally Cs-smooth (s ≥ 1) isogeometric spline spaces over multi-patch geometries with possibly extraordinary vertices, i.e. vertices with valencies different from four, is a current and challenging topic of research in the framework of isogeometric analysis. In this work, we extend the recent methods Kapl et al. Comput. Aided Geom. Des. 52–53:75–89, 2017, Kapl et al. Comput. Aided Geom. Des. 69:55–75, 2019 and Kapl and Vitrih J. Comput. Appl. Math. 335:289–311, 2018, Kapl and Vitrih J. Comput. Appl. Math. 358:385–404, 2019 and Kapl and Vitrih Comput. Methods Appl. Mech. Engrg. 360:112684, 2020 for the construction of C1-smooth and C2-smooth isogeometric spline spaces over particular planar multi-patch geometries to the case of Cs-smooth isogeometric multi-patch spline spaces of degree p, inner regularity r and of a smoothness s ≥ 1, with p ≥ 2s + 1 and s ≤ r ≤ p − s − 1. More precisely, we study for s ≥ 1 the space of Cs-smooth isogeometric spline functions defined on planar, bilinearly parameterized multi-patch domains, and generate a particular Cs-smooth subspace of the entire Cs-smooth isogeometric multi-patch spline space. We further present the construction of a basis for this Cs-smooth subspace, which consists of simple and locally supported functions. Moreover, we use the Cs-smooth spline functions to perform L2 approximation on bilinearly parameterized multi-patch domains, where the obtained numerical results indicate an optimal approximation power of the constructed Cs-smooth subspace.

Isogeometric analysis with [formula omitted] hierarchical functions on planar two-patch geometries

Adaptive isogeometric methods for the solution of partial differential equations rely on the construction of locally refinable spline spaces. A simple and efficient way to obtain these spaces is to apply the multi-level construction of hierarchical splines, that can be used on single-patch domains or in multi-patch domains with C0 continuity across the patch interfaces. Due to the benefits of higher continuity in isogeometric methods, recent works investigated the construction of spline spaces with global C1 continuity on two or more patches. In this paper, we show how these approaches can be combined with the hierarchical construction to obtain global C1 continuous hierarchical splines on two-patch domains. A selection of numerical examples is presented to highlight the features and effectivity of the construction.

Multi-patch NURBS formulation for anisotropic variable angle tow composite plates

Conventional methods result in sharp variation of stiffness in analysis of variable angle tow (VAT) composite plates, which can be enhanced using high-order continuity of non-uniform rational B-spline (NURBS). To this end, a Bézier extraction-based isogeometric (IGA) model based on the higher-order shear deformation theory (HSDT) is developed for the parametric instability study of anisotropic VAT composite plates. Bézier extraction-based IGA approach expresses exactly the orientation function of VAT composites for multi-patch geometries, which provides accurate solution for general-shape plates. The accuracy and convergence of the developed framework is compared with recently formulated methods. Primary results shows that the developed concept solves VAT problems with much lower number of equations than variable kinematic models. Further, response characteristics of analyzed anisotropic VAT plates are presented and discussed considering various fiber orientation functions, boundary conditions and multi-patch geometry. Based on results, Bézier control mesh is more accurate than the NURBS-based IGA solution due to considering the continuity of strain field along coupled boundaries. Anisotropic VAT plates have severe deformation couplings and the current concept accurately captures this feature even with courser meshes. The interaction between boundary conditions and fiber orientations along edges affects mainly the dynamic instability regions (DIR) of modeled VAT plates, dislike to straight fiber ones. The dynamic instability opening (DIO) of VAT plates relies on the fiber orientation functions and boundaries. In anisotropic VAT plates, maximum frequency case and maximum buckling force case would not necessarily be the same. Modes shapes are variable of external force intensity and therefore the Rayleigh-Ritz based methods are not appropriate for dynamic stability analysis of VAT plates. It is essential to change the NURBS control mesh to the Bézier type to offer the required accuracy for the multi-patch domains.

Discontinuous Galerkin isogeometric analysis for segmentations generating overlapping regions

In the Isogeometric Analysis (IGA) framework, the computational domain has very often a multipatch representation. The multipatch domain can be obtained by a volume segmentation of a boundary represented domain, e.g. provided by a Computer Aided Design model. Typically, small gaps and overlapping regions can appear at the patch interfaces of such multipatch representations. In the current work, we consider multipatch representations having only small overlapping regions between the patches. We develop a Discontinuous Galerkin (DG)-IGA method that can be immediately applied to these representations. Our method appropriately connects the fluxes of the one face of the overlapping region with the flux of the opposite face. We provide a theoretical justification of our approach by splitting the whole error into two components: the first is related to the incorrect representation of the patches (consistency error) and the second to the approximation properties of the IGA space. We show bounds for both components of the error. We verify the theoretical error estimates in a series of numerical examples.

Isogeometric collocation on planar multi-patch domains

We present an isogeometric framework based on collocation to construct a C2-smooth approximation of the solution of the Poisson’s equation over planar bilinearly parameterized multi-patch domains. The construction of the used globally C2-smooth discretization space for the partial differential equation is simple and works uniformly for all possible multi-patch configurations. The basis of the C2-smooth space can be described as the span of three different types of locally supported functions corresponding to the single patches, edges and vertices of the multi-patch domain. For the selection of the collocation points, which is important for the stability and convergence of the collocation problem, two different choices are numerically investigated. The first approach employs the tensor-product Greville abscissae as collocation points, and shows for the multi-patch case the same convergence behavior as for the one-patch case [1], which is suboptimal in particular for odd spline degree. The second approach generalizes the concept of superconvergent points from the one-patch case (cf. [2–4]) to the multi-patch case. Again, these points possess better convergence properties than Greville abscissae in case of odd spline degree.

Strong multipatch [formula omitted]-coupling for isogeometric analysis on 2D and 3D domains

The solution spaces of isogoemetric analysis (IGA) constructed from p degree basis functions allow up to Cp−1 continuity within one patch. However, for a multi-patch domain, the continuity is only C0 at the boundaries between the patches. In this study, we present the construction of basis functions of degree p≥2 which are C1 continuous across the common boundaries shared by the patches. The new basis functions are computed as a linear combination of the C0 basis functions on the multi-patch domains. An advantage of the proposed method is that for the new basis functions, the continuity within a patch is preserved, without additional treatment of the functions in the interior of the patch.We apply continuity constraints to the new basis functions to enforce C1 continuity, where the constraints are developed according to the concept of “matched Gk-constructions always yield Ck-continuous isogeometric elements” discussed in Groisser and Peters, (2015). However, for certain geometries, the over-constrained solution space will lead to C1 locking (Collin and Sangalli, 2016). We discuss and show the usage of partial degree elevation to overcome this problem. We demonstrate the potential of the C1 basis functions for IGA applications through several examples involving biharmonic equations.

Overlapping multi-patch structures in isogeometric analysis

In isogeometric analysis (IGA) the domain of interest is usually represented by B-spline or NURBS patches, as they are present in standard CAD models. Complex domains can often be represented as a union of simple overlapping subdomains, parameterized by (tensor-product) spline patches. Numerical simulation on such overlapping multi-patch domains is a serious challenge in IGA. To obtain non-overlapping subdomains one would usually reparameterize the domain or trim some of the patches. Alternatively, one may use methods that can handle overlapping subdomains.In this paper, we propose a non-iterative, robust and efficient method defined directly on overlapping multi-patch domains. Consequently, the problem is divided into several sub-problems, which are coupled in an appropriate way. The resulting system can be solved directly in a single step. We compare the proposed method with iterative Schwarz domain decomposition approaches and observe that our method reduces the computational cost significantly, especially when handling subdomains with small overlaps.Summing up, our method significantly simplifies the domain parameterization problem, since we can represent any domain of interest as a union of overlapping patches without the need to introduce trimming curves/surfaces. The performance of the proposed method is demonstrated by several numerical experiments for the Poisson problem and linear elasticity in two and three dimensions.

Isogeometric analysis on non-matching segmentation: discontinuous Galerkin techniques and efficient solvers

One challenging task in Isogeometric Analysis (IGA) is the segmentation of domains, which are represented via their boundaries, e.g. arising from a computer aided design model, into suitable blocks and the construction of their spline parametrizations. This yields a multipatch description of the domain. However, errors can occur during the computation of the associated control points when constructing the spline parametrizations of a patch. This leads to the derivation of non-matching interface parametrizations, i.e. gap and overlapping regions can appear between the patch interfaces. In this paper, we develop discontinuous Galerkin techniques in the context of IGA for solving elliptic diffusion problems on multipatch domains which can include gap and overlapping regions. This work is the continuation of two previous works, where we presented a complete discretization analysis for the case of having only gap regions. Here we focus on a unified presentation of the method for general gap and overlapping regions, and the development of efficient domain decomposition solvers based on tearing and interconnecting technology. In addition, we investigate the influence of the size of the gap/overlapping regions on the performance of the proposed solvers. Comparisons are shown and implementation issues are discussed on actual numerical examples.

Solving the triharmonic equation over multi-patch planar domains using isogeometric analysis

We present a framework for solving the triharmonic equation over bilinearly parameterized planar multi-patch domains by means of isogeometric analysis. Our approach is based on the construction of a globally C 2 -smooth isogeometric spline space which is used as discretization space. The generated C 2 -smooth space consists of three different types of isogeometric functions called patch, edge and vertex functions. All functions are entirely local with a small support, and numerical examples indicate that they are well-conditioned. The construction of the functions is simple and works uniformly for all multi-patch configurations. While the patch and edge functions are given by a closed form representation, the vertex functions are obtained by computing the null space of a small system of linear equations. Several examples demonstrate the potential of our approach for solving the triharmonic equation.