arXiv subjects

Thomas Takacs

Publications and source records attributed to Thomas Takacs.

At least 19 recordsLinked to original sources

arXiv2026

Constructing $C^1$ limit surfaces from unstructured splines via averaging and refinement

In this paper we present a construction for unstructured splines over quadrilateral meshes by iterative averaging and refinement. We represent the spline as a multi-patch B-spline, where the degrees of freedom are those B-spline coefficients on the quadrilateral patches that are not associated with interior edges and vertices of the mesh, i.e., their corresponding Greville points lie inside the patches. In every averaging step, we replace the remaining B-spline coefficients associated with interior edges and vertices by suitable averages of neighboring degrees of freedom. In the refinement step we apply regular splits to all patches by knot insertion. This process results in a subdivision scheme that, for degree $p=2$, is similar to the almost-$C^1$ spline construction from (Takacs, Toshniwal. CMAME, 2023) and behaves similar to Doo-Sabin subdivision, cf. (Doo, Sabin. CAD, 1978), and that can be defined for arbitrary degrees and regularities inside the patches. We derive two families of spline constructions, based on simple and coplanar averaging, respectively, and analyze their spectral properties when interpreted as subdivision schemes. Using this interpretation, we show that they are $C^1$ in the limit. Moreover, the coplanar averaging scheme produces splines that are $C^1$ at all vertices for every level of refinement, whereas the simple averaging is $C^1$ only in the limit. For both constructions, we have control over the subdominant eigenvalue, which has multiplicity two and can range between $\frac{1}{4}$ and $1$, with $\frac{1}{2}$ often being the desired option. The resulting basis functions form a partition of unity. Moreover, they form a non-negative partition of unity for suitably selected averaging parameters.

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arXiv2026

Robust approximation error estimates for analysis-suitable $G^1$ isogeometric multi-patch discretizations

We prove $p$-robust approximation error estimates for $H^2$-conforming isogeometric discretizations over planar multi-patch domains. Possible applications are fourth order boundary value problems, like the biharmonic equation or Kirchhoff-Love plates. Using Isogeometric Analysis, such conforming discretizations can be constructed effortlessly for the single-patch case. In order to obtain a globally $H^2$-conforming discretization in the multi-patch case, the functions must be $C^1$-smooth across the interfaces between the patches. To obtain optimal approximation properties, those $C^1$-smooth spaces must also reproduce splines of sufficiently high degree for traces and transversal derivatives at all patch interfaces. Such constructions are based on some assumptions on the geometry. We restrict ourselves to the class of analysis-suitable $G^1$ (AS-$G^1$) multi-patch domains, which is the subset of $C^0$-matching multi-patch domains that allows the definition of spline spaces that yield the necessary reproduction properties without the need to locally increase the degree. While approximation error estimates have been established for single-patch and $C^0$ isogeometric multi-patch spaces, corresponding results for the $C^1$ multi-patch setting have been missing. The resulting bounds on the approximation error depend on the geometry parameterization and on the Sobolev regularity of the target function, but are independent of the spline degree $p$.

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arXiv2024

Adaptive optimization of isogeometric multi-patch discretizations using artificial neural networks

In isogeometric analysis, isogeometric function spaces are employed for accurately representing the solution to a partial differential equation (PDE) on a parameterized domain. They are generated from a tensor-product spline space by composing the basis functions with the inverse of the parameterization. Depending on the geometry of the domain and on the data of the PDE, the solution might not have maximum Sobolev regularity, leading to a reduced convergence rate. In this case it is necessary to reduce the local mesh size close to the singularities. The classical approach is to perform adaptive h-refinement, which either leads to an unnecessarily large number of degrees of freedom or to a spline space that does not possess a tensor-product structure. Based on the concept of r-adaptivity we present a novel approach for finding a suitable isogeometric function space for a given PDE without sacrificing the tensor-product structure of the underlying spline space. In particular, we use the fact that different reparameterizations of the same computational domain lead to different isogeometric function spaces while preserving the geometry. Starting from a multi-patch domain consisting of bilinearly parameterized patches, we aim to find the biquadratic multi-patch parameterization that leads to the isogeometric function space with the smallest best approximation error of the solution. In order to estimate the location of the optimal control points, we employ a trained residual neural network that is applied to the graph surfaces of the approximated solution and its derivatives. In our experimental results, we observe that our new method results in a vast improvement of the approximation error for different PDE problems on multi-patch domains.

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arXiv2023

IGA Using Offset-based Overlapping Domain Parameterizations

Isogeometric analysis (IGA) is a numerical method that connects computer-aided design (CAD) with finite element analysis (FEA). In CAD the computational domain is usually represented by B-spline or NURBS patches. Given a NURBS parameterization of the domain, an isogeometric discretization is defined on the domain using the same NURBS basis as for the domain parameterization. Ideally, such an isogeometric discretization allows an exact representation of the underlying CAD model. CAD models usually represent only the boundary of the object, thus, for planar domains, it is given as a collection of curves. Finding a suitable parameterization of the interior is one of the major issues in IGA, similar to the mesh generation process in FEA. The objective of this parameterization problem is to obtain a set of patches, which exactly represent the boundary of the domain and which are parameterized regularly and without self-intersections. This can be achieved by segmenting the domain into patches which are matching along interfaces, or by covering the domain with overlapping patches. In this paper we follow the second approach. To construct from a given boundary a planar parameterization suitable for IGA, we propose an offset-based domain parameterization algorithm. Given a boundary curve, we obtain an inner curve by generalized offsetting. Those two curves define a ring-shaped patch, which has a hole that can be covered by a multi-cell domain. Consequently, the domain is represented as a union of two overlapping subdomains which are both regularly parameterized. On such a configuration, one can employ the overlapping multi-patch method introduced in (Kargaran, J\"uttler, Kleiss, Mantzaflaris, Takacs; CMAME, 2019), to solve PDEs on the given domain. The performance of the proposed method is reported in several numerical examples, considering different shapes of the domain.

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arXiv2023

Construction of Smooth Isogeometric Function Spaces on Singularly Parameterized Domains

We aim at constructing a smooth basis for isogeometric function spaces on domains of reduced geometric regularity. In this context an isogeometric function is the composition of a piecewise rational function with the inverse of a piecewise rational geometry parameterization. We consider two types of singular parameterizations, domains where a part of the boundary is mapped onto one point and domains where parameter lines are mapped collinearly at the boundary. We locally map a singular tensor-product patch of arbitrary degree onto a triangular patch, thus splitting the parameterization into a singular bilinear mapping and a regular mapping on a triangular domain. This construction yields an isogeometric function space of prescribed smoothness. Generalizations to higher dimensions are also possible and are briefly discussed in the final section.

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arXiv2023

Imposing nonlocal boundary conditions in Galerkin-type methods based on non-interpolatory functions

The imposition of inhomogeneous Dirichlet (essential) boundary conditions is a fundamental challenge in the application of Galerkin-type methods based on non-interpolatory functions, i.e., functions which do not possess the Kronecker delta property. Such functions typically are used in various meshfree methods, as well as methods based on the isogeometric paradigm. The present paper analyses a model problem consisting of the Poisson equation subject to non-standard boundary conditions. Namely, instead of classical boundary conditions, the model problem involves Dirichlet- and Neumann-type nonlocal boundary conditions. Variational formulations with strongly and weakly imposed inhomogeneous Dirichlet-type nonlocal conditions are derived and compared within an extensive numerical study in the isogeometric framework based on non-uniform rational B-splines (NURBS). The attention in the numerical study is paid mainly to the influence of the nonlocal boundary conditions on the properties of the considered discretisation methods.

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arXiv2023

Approximation properties over self-similar meshes of curved finite elements and applications to subdivision based isogeometric analysis

In this study we consider domains that are composed of an infinite sequence of self-similar rings and corresponding finite element spaces over those domains. The rings are parameterized using piecewise polynomial or tensor-product B-spline mappings of degree $q$ over quadrilateral meshes. We then consider finite element discretizations which, over each ring, are mapped, piecewise polynomial functions of degree $p$. Such domains that are composed of self-similar rings may be created through a subdivision scheme or from a scaled boundary parameterization. We study approximation properties over such recursively parameterized domains. The main finding is that, for generic isoparametric discretizations (i.e., where $p=q$), the approximation properties always depend only on the degree of polynomials that can be reproduced exactly in the physical domain and not on the degree $p$ of the mapped elements. Especially, in general, $L^\infty$-errors converge at most with the rate $h^2$, where $h$ is the mesh size, independent of the degree $p=q$. This has implications for subdivision based isogeometric analysis, which we will discuss in this paper.

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arXiv2023

Isogeometric $C^1$ mortar method

We present an isogeometric mortar method for the discretization of the biharmonic equation posed on multi-patch domains. We assume only $C^0$-conformity at interfaces and employs a mortar approach to weakly enforce $C^1$-continuity across patch interfaces. Discrete inf-sup stability is ensured by selecting a Lagrange multiplier space consisting of splines of degree reduced by two compared to the primal space, with increased smoothness or merged elements near vertices. We prove optimal a priori error estimates and confirm the theoretical findings with a series of numerical experiments.

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arXiv2023

$C^1$-smooth isogeometric spline functions of general degree over planar mixed meshes: The case of two quadratic mesh elements

Splines over triangulations and splines over quadrangulations (tensor product splines) are two common ways to extend bivariate polynomials to splines. However, combination of both approaches leads to splines defined over mixed triangle and quadrilateral meshes using the isogeometric approach. Mixed meshes are especially useful for representing complicated geometries obtained e.g. from trimming. As (bi-)linearly parameterized mesh elements are not flexible enough to cover smooth domains, we focus in this work on the case of planar mixed meshes parameterized by (bi-)quadratic geometry mappings. In particular we study in detail the space of $C^1$-smooth isogeometric spline functions of general polynomial degree over two such mixed mesh elements. We present the theoretical framework to analyze the smoothness conditions over the common interface for all possible configurations of mesh elements. This comprises the investigation of the dimension as well as the construction of a basis of the corresponding $C^1$-smooth isogeometric spline space over the domain described by two elements. Several examples of interest are presented in detail.

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arXiv2022

Locally refined quad meshing for linear elasticity problems based on convolutional neural networks

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. 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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arXiv2022

An approximate $C^1$ multi-patch space for isogeometric analysis with a comparison to Nitsche's method

We present an approximately $C^1$-smooth multi-patch spline construction which can be used in isogeometric analysis (IGA). A key property of IGA is that it is simple to achieve high order smoothness within a single patch. 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 $C^0$. Therefore, to obtain $C^1$-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 $C^1$ over multi-patch domains can be constructed as in (Collin, Sangalli, Takacs; CAGD, 2016) and (Kapl, Sangalli, Takacs; CAGD, 2019). The constructions require so-called analysis-suitable $G^1$ parametrizations. To allow $C^1$ spaces over more general multi-patch parametrizations, we need to increase the polynomial degree and relax the $C^1$ conditions. We adopt the approximate $C^1$ construction for two-patch domains, as developed in (Weinm\"uller, Takacs; CMAME, 2021), 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 $C^1$ construction is explicit and no additional terms need to be introduced to stabilize the method. Thus, the new proposed method can be used more easily as no parameters need to be estimated.

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arXiv2022

Almost-$C^1$ splines: Biquadratic splines on unstructured quadrilateral meshes and their application to fourth order problems

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-$C^1$ are biquadratic splines on fully unstructured quadrilateral meshes (without restrictions on placements or number of extraordinary vertices). They are $C^1$ smooth almost everywhere, that is, at all vertices and across most edges, and in addition almost (i.e. approximately) $C^1$ smooth across all other edges. Thus, the splines form $H^2$-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-$C^1$ splines are described in an explicit B\'ezier-extraction-based framework that can be easily implemented. Numerical tests suggest that the basis is well-conditioned and exhibits optimal approximation behavior.

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arXiv2021

Adaptive Refinement for Unstructured T-Splines with Linear Complexity

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.

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arXiv2021

Construction of approximate $C^1$ 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 $C^0$-smooth basis is easy to obtain, whereas $C^1$-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 $G^1$ (in short, AS-$G^1$) parametrizations, as introduced in (Collin, Sangalli, Takacs; CAGD, 2016), it is possible to construct $C^1$ isogeometric spaces which possess optimal approximation properties. 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 $1 \leq r \leq p-2$. The problem is that most complex geometries are not AS-$G^1$ geometries. Therefore, we define basis functions for isogeometric spaces by enforcing approximate $C^1$ conditions following the basis construction from (Kapl, Sangalli, Takacs; CAGD, 2017). For this reason, the defined function spaces are not exactly $C^1$ 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.

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arXiv2020

A family of $C^1$ quadrilateral finite elements

We present a novel family of $C^1$ quadrilateral finite elements, which define global $C^1$ spaces over a general quadrilateral mesh with vertices of arbitrary valency. The elements extend the construction by (Brenner and Sung, J. Sci. Comput., 2005), which is based on polynomial elements of tensor-product degree $p\geq 6$, to all degrees $p \geq 3$. Thus, we call the family of $C^1$ finite elements Brenner-Sung quadrilaterals. The proposed $C^1$ quadrilateral can be seen as a special case of the Argyris isogeometric element of (Kapl, Sangalli and Takacs, CAGD, 2019). The quadrilateral elements possess similar degrees of freedom as the classical Argyris triangles. Just as for the Argyris triangle, we additionally impose $C^2$ continuity at the vertices. In this paper we focus on the lower degree cases, that may be desirable for their lower computational cost and better conditioning of the basis: We consider indeed the polynomial quadrilateral of (bi-)degree~$5$, and the polynomial degrees $p=3$ and $p=4$ by employing a splitting into $3\times3$ or $2\times2$ polynomial pieces, respectively. The proposed elements reproduce polynomials of total degree $p$. We show that the space provides optimal approximation order. Due to the interpolation properties, the error bounds are local on each element. In addition, we describe the construction of a simple, local basis and give for $p\in\{3,4,5\}$ explicit formulas for the B\'{e}zier or B-spline coefficients of the basis functions. Numerical experiments by solving the biharmonic equation demonstrate the potential of the proposed $C^1$ quadrilateral finite element for the numerical analysis of fourth order problems, also indicating that (for $p=5$) the proposed element performs comparable or in general even better than the Argyris triangle with respect to the number of degrees of freedom.

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arXiv2020

A super-smooth $C^1$ spline space over planar mixed triangle and quadrilateral meshes

In this paper we introduce a $C^1$ spline space over mixed meshes composed of triangles and quadrilaterals, suitable for FEM-based or isogeometric analysis. In this context, a mesh is considered to be a partition of a planar polygonal domain into triangles and/or quadrilaterals. The proposed space combines the Argyris triangle, cf. (Argyris, Fried, Scharpf; 1968), with the $C^1$ quadrilateral element introduced in (Brenner, Sung; 2005) and (Kapl, Sangalli, Takacs; 2019) for polynomial degrees $p\geq 5$. The space is assumed to be $C^2$ at all vertices and $C^1$ across edges, and the splines are uniquely determined by $C^2$-data at the vertices, values and normal derivatives at chosen points on the edges, and values at some additional points in the interior of the elements. The motivation for combining the Argyris triangle element with a recent $C^1$ quadrilateral construction, inspired by isogeometric analysis, is two-fold: on one hand, the ability to connect triangle and quadrilateral finite elements in a $C^1$ fashion is non-trivial and of theoretical interest. We provide not only approximation error bounds but also numerical tests verifying the results. On the other hand, the construction facilitates the meshing process by allowing more flexibility while remaining $C^1$ everywhere. This is for instance relevant when trimming of tensor-product B-splines is performed. In the presented construction we assume to have (bi)linear element mappings and piecewise polynomial function spaces of arbitrary degree $p\geq 5$. The basis is simple to implement and the obtained results are optimal with respect to the mesh size for $L^\infty$, $L^2$ as well as Sobolev norms $H^1$ and $H^2$.

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arXiv2019

Manifold-based B-splines on unstructured meshes

We introduce new manifold-based splines that are able to exactly reproduce B-splines on unstructured surface meshes. Such splines can be used in isogeometric analysis (IGA) to represent smooth surfaces of arbitrary topology. Since prevalent computer-aided design (CAD) models are composed of tensor-product B-spline patches, any IGA suitable construction should be able to reproduce B-splines. To achieve this goal, we focus on univariate manifold-based constructions that can reproduce B-splines. The manifold-based splines are constructed by smoothly blending together polynomial interpolants defined on overlapping charts. The proposed constructions automatically reproduce B-splines in regular parts of the mesh, with no extraordinary vertices, and polynomial basis functions in the remaining parts of the mesh. We study and compare analytically and numerically the finite element convergence of several univariate constructions. The obtained results directly carry over to the tensor-product case.

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arXiv2018

Isogeometric analysis with $C^1$ functions on unstructured quadrilateral meshes

In the context of isogeometric analysis, globally $C^1$ isogeometric spaces over unstructured quadrilateral meshes allow the direct solution of fourth order partial differential equations on complex geometries via their Galerkin discretization. The design of such smooth spaces has been intensively studied in the last five years, in particular for the case of planar domains, and is still task of current research. In this paper, we first give a short survey of the developed methods and especially focus on the approach [26]. There, the construction of a specific $C^1$ isogeometric spline space for the class of so-called analysis-suitable $G^1$ multi-patch parametrizations is presented. This particular class of parameterizations comprises exactly those multi-patch geometries, which ensure the design of $C^1$ spaces with optimal approximation properties, and allows the representation of complex planar multi-patch domains. We present known results in a coherent framework, and also extend the construction to parametrizations that are not analysis-suitable $G^1$ by allowing higher-degree splines in the neighborhood of the extraordinary vertices and edges. Finally, we present numerical tests that illustrate the behavior of the proposed method on representative examples.

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