Academic literature on the topic 'Galerkin-isogeometric method'

In this paper, we present the analysis of the discontinuous Galerkin dual-primal isogeometric tearing and interconnecting method (dG-IETI-DP) for a multipatch discretization in two-space dimensions where we only consider vertex primal variables. As model problem, we use the Poisson equation with globally constant diffusion coefficient. The dG-IETI-DP method is a combination of the dual-primal isogeometric tearing and interconnecting method (IETI-DP) with the discontinuous Galerkin (dG) method. We use the dG method only on the interfaces to couple different patches. This enables us to handle non-matching grids on patch interfaces as well as segmentation crimes (gaps and overlaps) between the patches. The purpose of this paper is to derive quasi-optimal bounds for the condition number of the preconditioned system with respect to the maximal ratio [Formula: see text] of subdomain diameter and mesh size. Moreover, we show that the condition number is independent of the number of patches, but depends on the mesh sizes of neighboring patches [Formula: see text] and the parameter [Formula: see text] in the dG penalty term.

The isogeometric symmetric Galerkin boundary element method is applied for the analysis of crack problems in two-dimensional magneto-electro-elastic domains. In this method, the field variables of the governing integral equations as well as the geometry of the problems are approximated using non-uniform rational B-splines (NURBS) basis functions. The key advantage of this method is that the isogeometric analysis and boundary element method deal only with the boundary of the domain. To verify the accuracy of the proposed method, numerical examples for crack problems in infinite and finite domains are examined. It is observed that the computed generalized stress intensity factors obtained by the proposed method agree well with the exact solutions and other references.

Isogeometric analysis (IGA) is an effective numerical method for connecting computer-aided design and engineering, which has been widely applied in various aspects of computational mechanics. IGA involves Galerkin and collocation formulations. Exploiting the same high-order non-uniform rational B-spline (NURBS) bases that span the physical domain and the solution space leads to increased accuracy and fast computation. Although IGA Galerkin provides optimal convergence, IGA collocation performs better in terms of the ratio of accuracy to computational time. Without numerical integration, by working directly with the strong form of the partial differential equation over the physical domain defined by NURBS geometry, the derivatives of the NURBS-expressed numerical solution at some chosen collocation points can be calculated. In this study, we survey the methodological framework and the research prospects of IGA. The collocation schemes in the IGA collocation method that affect the convergence performance are addressed in this paper. Recent studies and application developments are reviewed as well.

Dans ce travail, nous introduisons différentes formulations faibles basées sur des méthodes de Galerkin continues en temps pour plusieurs types de problèmes, pilotés par des équations aux dérivées partielles dans l’espace et le temps. Notre approche repose sur une discrétisation simultanée et arbitraire de l’espace et du temps. L’analyse isogéométrique (IGA) est utilisée comme outil de discrétisation à la place de la méthode classique des éléments finis (FEM) afin de bénéficier des propriétés de continuité des fonctions B-splines et NURBS. Un état de l’art détaillé est présenté pour introduire le concept de ces deux méthodes et pour montrer les travaux déjà réalisés dans la littérature concernant les méthodes espace-temps d’une part, et l’IGA d’une autre part. Les méthodes seront combinées et appliquées à différents types de problèmes mécaniques. Ces problèmes sont principalement des problèmes d’ingénierie tels que l’élastodynamique, la thermomécanique et les problèmes viscoélastiques. On compare différents types de formulations variationelles et différentes configurations de discrétisation. On montre que dans le cas de problèmes ayant des solutions discontinues en temps comme les problèmes d’impact, l’utilisation conjointe d’une formulation avec des fonctions test dérivées en temps et des termes de stabilisation de type moindres carrés permettent de contrôler les oscillations numériques souvent observées pour ce type de problèmes. De plus, nous introduisons une nouvelle technique de stabilisation qui peut être utilisée facilement pour des problèmes non linéaires. Celle-ci est basée sur la condition de consistence de l’accélération, nous l’appelons donc Galerkin avec consistence sur l’accélération. Les problèmes étudiés prennent donc à la fois des formes linéaires et non linéaires. Nous résolvons des problèmes en petites et en grandes déformations : que ce soit pour l’élastodynamique, la thermomécanique ou pour les problèmes de type viscoélastique. Des matériaux compressibles et incompressibles sont considérés. La convergence de la méthode est étudiée numériquement et comparée aux méthodes existantes. Nous vérifions autant que possible les propriétés de conservation de la formulation et les comparons aux propriétés de conservation des méthodes classiques telles que la FEM équipée d’un schéma HHT en temps. Les résultats numériques montrent que les méthodes espace-temps sont plus conservatives en énergie que les méthodes classiques pour les problèmes d’élastodynamique. Différents tests de convergence sont menés et des taux de convergence optimaux sont obtenus à chaque fois, montrant l’efficacité de la méthode. Nous montrons en outre que des schémas hétérogènes et asynchrones peuvent être construits d’une manière très simple, ouvrant à de nombreuses possibilités avec les méthodes espace-temps. Enfin, les performances observées sur différents problèmes et la polyvalence de l’approche suggèrent que les méthodes IGA espace-temps ont un fort potentiel dans le domaine de la simulation numérique en ingénierie
In this work, we introduce different weak formulations based on time continuous Galerkin methods for several types of problems, governed by partial differential equations in space and time. Our approach is based on a simultaneous and arbitrary discretization of the space and time. The Isogeometric Analysis (IGA) is employed instead of the classical Finite Element Method (FEM) in order to take advantage of the continuity properties of B-splines and NURBS functions. A detailed state of the art is narrated first to introduce the concept of both of these methods and to show the work already done in literature regarding the space-time methods on a first basis, and the IGA on a second basis. Then, the methods are applied to different types of mechanical problems. These problems are mainly engineering problems such as elastodynamics, thermomechanics, and history dependant behaviors (viscoelasticity). We compare different types of variational formulations and different discretizations. We show that in the case of problems having discontinuous solutions such as impact problems, the use of both a formulation with derived in time test functions and additional least square terms makes it possible to avoid the spurious numerical oscillations often observed for these type of problems. Furthermore, we introduce a new stabilization technique that can be used easily for non-linear problems. It is based on the consistency condition of the acceleration, so we call it Galerkin with Acceleration Consistency (GAC). The problems investigated take both linear and non-linear forms. We solve elastodynamics, thermomechanics and viscoelatic type problems at small and finite strains. Both compressible and incompressible materials are considered. The convergence of the method is numerically studied and compared with existing methods. We verify, where applicable, the conservation properties of the formulation and compare them to the conservation properties of the classical methods such as the FEM equipped with an HHT scheme for the time discretization. The numerical results show that space-time methods are more energy conserving than classical methods for the elastodynamic problems. Different convergence tests are leaded and optimal convergence rates are obtained, showing the efficiency of the method. We show furthermore that heterogeneous and asynchroneous schemes can be built in a very simple manner, opening up many possibilities while dealing with space-time methods. Finally, the performances observed on different problems and the versatility of the approach suggest that ST IGA methods have a strong potential for advanced simulations in engineering

Thesis deals with solving the problems of continuum mechanics by method of Isogeometric analysis. This relatively young method combines the advantages of precise NURBS geometry and robustness of the classical finite element method. The method is described on procedure of solving a plane Poissons boundary value problem. Solver is implemented in MatLab and algorithms are attached to the text.

L’Analyse isogéométrique (AIG) est une méthode innovante de résolution numérique des équations différentielles, proposée à l’origine par Thomas Hughes, Austin Cottrell et Yuri Bazilevs en 2005. Cette technique de discrétisation est une généralisation de l’analyse par éléments finis classiques (AEF), conçue pour intégrer la conception assistée par ordinateur (CAO), afin de combler l’écart entre la description géométrique et l’analyse des problèmes d’ingénierie. Ceci est réalisé en utilisant des B-splines ou des B-splines rationnelles non uniformes (NURBS), pour la description des géométries ainsi que pour la représentation de champs de solutions inconnus.L’objet de cette thèse est d’étudier la méthode isogéométrique dans le contexte des problèmes hyperboliques en utilisant les fonctions B-splines comme fonctions de base. Nous proposons également une méthode combinant l’AIG avec la méthode de Galerkin discontinue (GD) pour résoudre les problèmes hyperboliques. Plus précisément, la méthodologie de GD est adoptée à travers les interfaces de patches, tandis que l’AIG traditionnelle est utilisée dans chaque patch. Notre méthode tire parti de la méthode de l’AIG et la méthode de GD.Les résultats numériques sont présentés jusqu’à l’ordre polynomial p= 4 à la fois pour une méthode deGalerkin continue et discontinue. Ces résultats numériques sont comparés pour un ensemble de problèmes de complexité croissante en 1D et 2D
Isogeometric Analysis (IGA) is a modern strategy for numerical solution of partial differential equations, originally proposed by Thomas Hughes, Austin Cottrell and Yuri Bazilevs in 2005. This discretization technique is a generalization of classical finite element analysis (FEA), designed to integrate Computer Aided Design (CAD) and FEA, to close the gap between the geometrical description and the analysis of engineering problems. This is achieved by using B-splines or non-uniform rational B-splines (NURBS), for the description of geometries as well as for the representation of unknown solution fields.The purpose of this thesis is to study isogeometric methods in the context of hyperbolic problems usingB-splines as basis functions. We also propose a method that combines IGA with the discontinuous Galerkin(DG)method for solving hyperbolic problems. More precisely, DG methodology is adopted across the patchinterfaces, while the traditional IGA is employed within each patch. The proposed method takes advantageof both IGA and the DG method.Numerical results are presented up to polynomial order p= 4 both for a continuous and discontinuousGalerkin method. These numerical results are compared for a range of problems of increasing complexity,in 1D and 2D

This work is concerned with the use of isogeometric analysis based on Non- Uniform Rational B-Splines (NURBS) to develop efficient and robust numerical techniques to deal with the problems of incompressibility in the fields of solid and fluid mechanics. Towards this, two types of formulations, mixed Galerkin and least-squares, are studied. During the first phase of this work, mixed Galerkin formulations, in the context of isogeometric analysis, are presented. Two-field and three-field mixed variational formulations - in both small and large strains - are presented to obtain accurate numerical solutions for the problems modelled with nearly incompressible and elasto-plastic materials. The equivalence of the two mixed formulations, for the considered material models, is derived; and the computational advantages of using two-field formulations are illustrated. Performance of these formulations is assessed by studying several benchmark examples. The ability of the mixed methods, to accurately compute limit loads for problems involving elastoplastic material models; and to deal with volumetric locking, shear locking and severe mesh distortions in finite strains, is illustrated. Later, finite element formulations are developed by combining least-squares and isogeometric analysis in order to extract the best of both. Least-squares finite element methods (LSFEMs) based on the use of governing differential equations directly - without the need to reduce them to equivalent lower-order systems - are developed for compressible and nearly incompressible elasticity in both the small and finite strain regimes; and incompressible Navier-Stokes. The merits of using Gauss-Newton scheme instead of Newton-Raphson method to solve the underlying nonlinear equations are presented. The performance of the proposed LSFEMs is demonstrated with several benchmark examples from the literature. Advantages of using higher-order NURBS in obtaining optimal convergence rates for non-norm-equivalent LSFEMs; and the robustness of LSFEMs, for Navier-Stokes, in obtaining accurate numerical solutions without the need to incorporate any artificial stabilisation techniques, are demonstrated.

Until recently, interior penalty methods have been applied to elliptic operators using an approach based on the mass matrix of finite elements that possess a constant Jacobian. In the case of isogeometric analysis, such an approach would not always guarantee coercivity of the bilinear form and thus numerical stability of the solution. In this paper, optimal interior penalty parameters [1] are described and applied to the symmetric interior penalty scheme of the discontinuous Galerkin isogeometric spatial discretisation of the neutron diffusion equation. The numerical accuracy of the proposed method is compared against a standard continuous Bubnov-Galerkin isogeometric spatial discretisation of the neutron diffusion equation. Numerical consistency and order of error-convergence is verified by means of the method of manufactured solutions. Numerical results are also presented for a two-dimensional pin-cell test case, based upon the OECD/NEA C5G7 quarter core MOX fuel assembly benchmark for nuclear reactor physics parameters of interest.

The aim of this work is to derive a formulation for linear two-dimensional elasticity using just one degree of freedom. With the Airy stress function, a measure without further physical meaning is chosen to this single degree of freedom. The corresponding Airy equation requires higher order basis functions for the discretization of the formulation [1]. Isogeometric structural analysis (IGA) is based on shape functions of the system in Computer-Aided design (CAD) software [2]. These shape functions can fulfill the requirement of high continuity and therefore the formulation is obtained through IGA methods. Non-Uniform Rational B-splines (NURBS) are used to discretize the domain and to solve the occurring differential equations within the Galerkin method [3]. The received one-degree of freedom formulation allows to compute stresses as direct solution of the underlying system of equations. Numerical examples demonstrate the accuracy for a quadratic plate under standard, but also under complex loading. For constant or linear loading functions only one element is sufficient to receive the exact solution – a general advantage of using higher order basis functions. The correct convergence behaviour of the proposed formulation is proved by the -error norm for a complex load situation. Here, only a few refinement steps yield a good approximation with a very small error of the stresses.