Isogeometric Analysis generalizes classical finite element analysis and intends to integrate it with the field of Computer-Aided Design. A central problem in achieving this objective is the ...reconstruction of analysis-suitable models from Computer-Aided Design models, which is in general a non-trivial and time-consuming task. In this article, we present a novel spline construction, that enables model reconstruction as well as simulation of high-order PDEs on the reconstructed models. The proposed almost-C1splines are biquadratic splines on fully unstructured quadrilateral meshes (without restrictions on placements or number of extraordinary vertices). They are C1 smooth at all regular and extraordinary vertices. Moreover, they are C1 smooth across all edges between regular vertices and C0 smooth across all edges that are adjacent to an extraordinary vertex. The splines thus form H2-nonconforming analysis-suitable discretization spaces. This is the lowest-degree unstructured spline construction that can be used to solve fourth-order problems. The associated spline basis is non-singular and has several B-spline-like properties (e.g., partition of unity, non-negativity, local support), the almost-C1 splines are described in an explicit Bézier-extraction-based framework that can be easily implemented. Numerical tests suggest that the basis is well-conditioned and exhibits optimal approximation behaviour.
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.
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.
•Constructing approximate C1 smooth bases for isogeometric two-patch domains.•Proof of convergence rates of the jump of the normal derivative across the interface.•Optimal convergence of the isogeometric discretization under h-refinement verifying the results on several numerical examples.
We present a framework for the construction of a globally C1-smooth isogeometric spline space over a particular class of G1-smooth multi-patch surfaces called analysis-suitable G1 (in short AS-G1) ...multi-patch surfaces. The class of AS-G1 multi-patch surfaces consists of those G1-smooth multi-patch surfaces which allow the construction of C1-smooth isogeometric spline spaces with optimal polynomial reproduction properties, see 1. Our method extends the work 2, which is limited to the case of planar AS-G1 multi-patch parameterizations, to the case of AS-G1 multi-patch surfaces. The C1-smooth isogeometric spline space is generated as the span of locally supported and explicitly given basis functions of three different types that correspond to the patches, interfaces and vertices of the considered AS-G1 multi-patch surface. We further present simple and practical methods for the design of AS-G1 multi-patch surfaces and demonstrate the potential of the C1-smooth spline space for solving fourth order partial differential equations over AS-G1 multi-patch surfaces on the basis of the biharmonic equation. The obtained numerical results indicate convergence rates of optimal order in the L2-norm and in the H1- and H2-seminorms.
We present an adaptive refinement algorithm for T-splines on unstructured 2D meshes. While for structured 2D meshes, one can refine elements alternatingly in horizontal and vertical direction, such ...an approach cannot be generalized directly to unstructured meshes, where no two unique global mesh directions can be assigned. To resolve this issue, we introduce the concept of direction indices, i.e., integers associated to each edge, which are inspired by theory on higher-dimensional structured T-splines. Together with refinement levels of edges, these indices essentially drive the refinement scheme. We combine these ideas with an edge subdivision routine that allows for I-nodes, yielding a very flexible refinement scheme that nicely distributes the T-nodes, preserving global linear independence, analysis-suitability (local linear independence) except in the vicinity of extraordinary nodes, sparsity of the system matrix, and shape regularity of the mesh elements. Further, we show that the refinement procedure has linear complexity in the sense of guaranteed upper bounds on a) the distance between marked and additionally refined elements, and on b) the ratio of the numbers of generated and marked mesh elements.
•Refinement algorithm for T-splines on unstructured meshes.•Procedure is based on theory for higher-dimensional structured meshes.•Characterization of basis functions as traces of higher-dimensional structured splines.•Proof of global linear independence of the basis functions and linear complexity of the refinement.
In this paper we propose a method to generate suitably refined finite element meshes using neural networks. As a model problem we consider a linear elasticity problem on a planar domain (possibly ...with holes) having a polygonal boundary. We impose boundary conditions by fixing the position of a part of the boundary and applying a force on another part of the boundary. The resulting displacement and distribution of stresses depend on the geometry of the domain and on the boundary conditions. When applying a standard Galerkin discretization using quadrilateral finite elements, one usually has to perform adaptive refinement to properly resolve maxima of the stress distribution. Such an adaptive scheme requires a local error estimator and a corresponding local refinement strategy. The overall costs of such a strategy are high. We propose to reduce the costs of obtaining a suitable discretization by training a neural network whose evaluation replaces this adaptive refinement procedure. We set up a single network for a large class of possible domains and boundary conditions and not on a single domain of interest. The computational domain and boundary conditions are interpreted as images, which are suitable inputs for convolution neural networks. In our approach we use the U-net architecture and we devise training strategies by dividing the possible inputs into different categories based on their overall geometric complexity. Thus, we compare different training strategies based on varying geometric complexity. One of the advantages of the proposed approach is the interpretation of input and output as images, which do not depend on the underlying discretization scheme. Another is the generalizability and geometric flexibility. The network can be applied to previously unseen geometries, even with different topology and level of detail. Thus, training can easily be extended to other classes of geometries.
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