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Journal articles on the topic 'Galerkin-isogeometric method'

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Author: Grafiati

Published: 6 September 2023

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1

Yu, Shengjiao, Renzhong Feng, and Tiegang Liu. "An isogeometric discontinuous Galerkin method for Euler equations." Mathematical Methods in the Applied Sciences 40, no. 8 (December 2, 2016): 3129–39. http://dx.doi.org/10.1002/mma.4227.

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2

Michoski, C., J. Chan, L. Engvall, and J. A. Evans. "Foundations of the blended isogeometric discontinuous Galerkin (BIDG) method." Computer Methods in Applied Mechanics and Engineering 305 (June 2016): 658–81. http://dx.doi.org/10.1016/j.cma.2016.02.015.

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3

Hofer, Christoph. "Analysis of discontinuous Galerkin dual-primal isogeometric tearing and interconnecting methods." Mathematical Models and Methods in Applied Sciences 28, no. 01 (December 13, 2017): 131–58. http://dx.doi.org/10.1142/s0218202518500045.

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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.

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4

Yu, Shengjiao. "Adjoint-Based Adaptive Isogeometric Discontinuous Galerkin Method for Euler Equations." Advances in Applied Mathematics and Mechanics 10, no. 3 (June 2018): 652–72. http://dx.doi.org/10.4208/aamm.oa-2017-0046.

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5

Duvigneau, R. "Isogeometric analysis for compressible flows using a Discontinuous Galerkin method." Computer Methods in Applied Mechanics and Engineering 333 (May 2018): 443–61. http://dx.doi.org/10.1016/j.cma.2018.01.039.

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6

Wang, Kun, Shengjiao Yu, Zheng Wang, Renzhong Feng, and Tiegang Liu. "Adjoint-based airfoil optimization with adaptive isogeometric discontinuous Galerkin method." Computer Methods in Applied Mechanics and Engineering 344 (February 2019): 602–25. http://dx.doi.org/10.1016/j.cma.2018.10.033.

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7

Pezzano, Stefano, Régis Duvigneau, and Mickaël Binois. "Geometrically consistent aerodynamic optimization using an isogeometric Discontinuous Galerkin method." Computers & Mathematics with Applications 128 (December 2022): 368–81. http://dx.doi.org/10.1016/j.camwa.2022.11.004.

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8

Tran, Han Duc, and Binh Huy Nguyen. "An isogeometric SGBEM for crack problems of magneto-electro-elastic materials." Vietnam Journal of Mechanics 39, no. 2 (June 21, 2017): 135–47. http://dx.doi.org/10.15625/0866-7136/8691.

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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.

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9

Ren, Jingwen, and Hongwei Lin. "A Survey on Isogeometric Collocation Methods with Applications." Mathematics 11, no. 2 (January 16, 2023): 469. http://dx.doi.org/10.3390/math11020469.

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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.

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10

Stauffert, Maxime, and Régis Duvigneau. "Shape Sensitivity Analysis in Aerodynamics Using an Isogeometric Discontinuous Galerkin Method." SIAM Journal on Scientific Computing 43, no. 5 (January 2021): B1081—B1104. http://dx.doi.org/10.1137/20m1356269.

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11

Nguyen, B. H., X. Zhuang, P. Wriggers, T. Rabczuk, M. E. Mear, and H. D. Tran. "Isogeometric symmetric Galerkin boundary element method for three-dimensional elasticity problems." Computer Methods in Applied Mechanics and Engineering 323 (August 2017): 132–50. http://dx.doi.org/10.1016/j.cma.2017.05.011.

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12

Rahman, Sharif. "A Galerkin isogeometric method for Karhunen–Loève approximation of random fields." Computer Methods in Applied Mechanics and Engineering 338 (August 2018): 533–61. http://dx.doi.org/10.1016/j.cma.2018.04.026.

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13

HIRAI, Tetsuro, Toru TAKAHASHI, Hiroshi ISAKARI, and Toshiro MATSUMOTO. "A development of isogeometric Galerkin boundary element method for 2D periodic problem." Proceedings of Conference of Tokai Branch 2017.66 (2017): 511. http://dx.doi.org/10.1299/jsmetokai.2017.66.511.

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14

Nguyen, B. H., H. D. Tran, C. Anitescu, X. Zhuang, and T. Rabczuk. "An isogeometric symmetric Galerkin boundary element method for two-dimensional crack problems." Computer Methods in Applied Mechanics and Engineering 306 (July 2016): 252–75. http://dx.doi.org/10.1016/j.cma.2016.04.002.

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15

Nguyen, Lieu B., Chien H. Thai, Ngon T. Dang, and H. Nguyen-Xuan. "Transient analysis of laminated composite plates using NURBS-based isogeometric analysis." Vietnam Journal of Mechanics 36, no. 4 (November 27, 2014): 267–81. http://dx.doi.org/10.15625/0866-7136/36/4/4129.

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We further study isogeometric approach for response analysis of laminated composite plates using the higher-order shear deformation theory. The present theory is derived from the classical plate theory (CPT) and the shear stress free surface conditions are naturally satisfied. Therefore, shear correction factors are not required. Galerkin weak form of response analysis model for laminated composite plates is used to obtain the discrete system of equations. It can be solved by isogeometric approach based on the non-uniform rational B-splines (NURBS) basic functions. Some numerical examples of the laminated composite plates under various dynamic loads, fiber orientations and lay-up numbers are provided. The accuracy and reliability of the proposed method is verified by comparing with analytical solutions, numerical solutions and results from Ansys software.

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16

Liu, Zhaowei, Andrew McBride, Prashant Saxena, and Paul Steinmann. "Assessment of an isogeometric approach with Catmull–Clark subdivision surfaces using the Laplace–Beltrami problems." Computational Mechanics 66, no. 4 (July 15, 2020): 851–76. http://dx.doi.org/10.1007/s00466-020-01877-3.

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Abstract An isogeometric approach for solving the Laplace–Beltrami equation on a two-dimensional manifold embedded in three-dimensional space using a Galerkin method based on Catmull–Clark subdivision surfaces is presented and assessed. The scalar-valued Laplace–Beltrami equation requires only $$C^0$$ C 0 continuity and is adopted to elucidate key features and properties of the isogeometric method using Catmull–Clark subdivision surfaces. Catmull–Clark subdivision bases are used to discretise both the geometry and the physical field. A fitting method generates control meshes to approximate any given geometry with Catmull–Clark subdivision surfaces. The performance of the Catmull–Clark subdivision method is compared to the conventional finite element method. Subdivision surfaces without extraordinary vertices show the optimal convergence rate. However, extraordinary vertices introduce error, which decreases the convergence rate. A comparative study shows the effect of the number and valences of the extraordinary vertices on accuracy and convergence. An adaptive quadrature scheme is shown to reduce the error.

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17

Ouadefli, Lahcen El, Omar El Moutea, Abdeslam El Akkad, Ahmed Elkhalfi, Sorin Vlase, and Maria Luminița Scutaru. "Mixed Isogeometric Analysis of the Brinkman Equation." Mathematics 11, no. 12 (June 17, 2023): 2750. http://dx.doi.org/10.3390/math11122750.

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This study focuses on numerical solution to the Brinkman equation with mixed Dirichlet–Neumann boundary conditions utilizing isogeometric analysis (IGA) based on non-uniform rational B-splines (NURBS) within the Galerkin method framework. The authors suggest using different choices of compatible NURBS spaces, which may be considered a generalization of traditional finite element spaces for velocity and pressure approximation. In order to investigate the numerical properties of the suggested elements, two numerical experiments based on a square and a quarter of an annulus are discussed. The preliminary results for the Stokes problem are presented in References.

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18

Zhang, Futao, Yan Xu, Falai Chen, and Ruihan Guo. "Interior Penalty Discontinuous Galerkin Based Isogeometric Analysis for Allen-Cahn Equations on Surfaces." Communications in Computational Physics 18, no. 5 (November 2015): 1380–416. http://dx.doi.org/10.4208/cicp.010914.180315a.

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AbstractWe propose a method that combines Isogeometric Analysis (IGA) with the interior penalty discontinuous Galerkin (IPDG) method for solving the Allen-Cahn equation, arising from phase transition in materials science, on three-dimensional (3D) surfaces consisting of multiple patches. DG ideology is adopted at patch level, i.e., we employ the standard IGA within each patch, and employ the IPDG method across the patch interfaces. IGA is very suitable for solving Partial Differential Equations (PDEs) on (3D) surfaces and the IPDG method is used to glue the multiple patches together to get the right solution. Our method takes advantage of both IGA and the IPDG method, which allows us to design a superior semi-discrete (in time) IPDG scheme. First and most importantly, the time-consuming mesh generation process in traditional Finite Element Analysis (FEA) is no longer necessary and refinements, includingh-refinement andp-refinement which both maintain the original geometry, can be easily performed at any level. Moreover, the flexibility of the IPDG method makes our method very easy to handle cases with non-conforming patches and different degrees across the patch interfaces. Additionally, the geometrical error is eliminated (for all conic sections) or significantly reduced at the beginning due to the geometric flexibility of IGA basis functions, especially the use of multiple patches. Finally, this method can be easily formulated and implemented. We present our semi-discrete IPDG scheme after generally describe the problem, and then briefly introduce the time marching method employed in this paper. Theoretical analysis is carried out to show that our method satisfies a discrete energy law, and achieves the optimal convergence rate with respect to theL2norm. Furthermore, we propose an elliptic projection operator on (3D) surfaces and prove an approximation error estimate which are vital for us to obtain the error estimate in theL2norm. Numerical tests are given to validate the theory and gauge the good performance of our method.

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19

Nguyen, B. H., S. S. Nanthakumar, Y. Q. He, H. D. Tran, K. Hackl, and X. Zhuang. "Forward and inverse problems in piezoelectricity using isogeometric symmetric Galerkin boundary element method and level set method." Engineering Analysis with Boundary Elements 113 (April 2020): 118–32. http://dx.doi.org/10.1016/j.enganabound.2019.12.020.

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20

Friedrich, Léo A. J. "Time- and Frequency-Domain Steady-State Solutions of Nonlinear Motional Eddy Currents Problems." J 4, no. 1 (January 8, 2021): 22–48. http://dx.doi.org/10.3390/j4010002.

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This paper presents a comparison of different time- and frequency-domain solvers for the steady-state simulation of the eddy current phenomena, due to the motion of a permanent magnet array, occurring in the soft-magnetic stator core of electrical machines that exhibits nonlinear material characteristics. Three different dynamic solvers are implemented in the framework of the isogeometric analysis, namely the traditional time-stepping backward-Euler technique, the space-time Galerkin approach, and the harmonic balance method, which operates in the frequency domain. Two-dimensional electrical machine benchmarks, consisting of both slotless and slotted stator core, are considered to establish the accuracy, convergence, and computational efficiency of the presented solvers.

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21

Friedrich, Léo A. J. "Time- and Frequency-Domain Steady-State Solutions of Nonlinear Motional Eddy Currents Problems." J 4, no. 1 (January 8, 2021): 22–48. http://dx.doi.org/10.3390/j4010002.

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This paper presents a comparison of different time- and frequency-domain solvers for the steady-state simulation of the eddy current phenomena, due to the motion of a permanent magnet array, occurring in the soft-magnetic stator core of electrical machines that exhibits nonlinear material characteristics. Three different dynamic solvers are implemented in the framework of the isogeometric analysis, namely the traditional time-stepping backward-Euler technique, the space-time Galerkin approach, and the harmonic balance method, which operates in the frequency domain. Two-dimensional electrical machine benchmarks, consisting of both slotless and slotted stator core, are considered to establish the accuracy, convergence, and computational efficiency of the presented solvers.

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22

EVANS, JOHN A., and THOMAS J. R. HUGHES. "ISOGEOMETRIC DIVERGENCE-CONFORMING B-SPLINES FOR THE STEADY NAVIER–STOKES EQUATIONS." Mathematical Models and Methods in Applied Sciences 23, no. 08 (April 22, 2013): 1421–78. http://dx.doi.org/10.1142/s0218202513500139.

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We develop divergence-conforming B-spline discretizations for the numerical solution of the steady Navier–Stokes equations. These discretizations are motivated by the recent theory of isogeometric discrete differential forms and may be interpreted as smooth generalizations of Raviart–Thomas elements. They are (at least) patchwise C0 and can be directly utilized in the Galerkin solution of steady Navier–Stokes flow for single-patch configurations. When applied to incompressible flows, these discretizations produce pointwise divergence-free velocity fields and hence exactly satisfy mass conservation. Consequently, discrete variational formulations employing the new discretization scheme are automatically momentum-conservative and energy-stable. In the presence of no-slip boundary conditions and multi-patch geometries, the discontinuous Galerkin framework is invoked to enforce tangential continuity without upsetting the conservation or stability properties of the method across patch boundaries. Furthermore, as no-slip boundary conditions are enforced weakly, the method automatically defaults to a compatible discretization of Euler flow in the limit of vanishing viscosity. The proposed discretizations are extended to general mapped geometries using divergence-preserving transformations. For sufficiently regular single-patch solutions subject to a smallness condition, we prove a priori error estimates which are optimal for the discrete velocity field and suboptimal, by one order, for the discrete pressure field. We present a comprehensive suite of numerical experiments which indicate optimal convergence rates for both the discrete velocity and pressure fields for general configurations, suggesting that our a priori estimates may be conservative. These numerical experiments also suggest our discretization methodology is robust with respect to Reynolds number and more accurate than classical numerical methods for the steady Navier–Stokes equations.

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23

Durga Rao, S. S., M. Dinachandra, and Sethuraman Raju. "Free vibration analysis of FGM plates with internal defects using extended isogeometric hybrid collocation–Galerkin method." International Journal for Computational Methods in Engineering Science and Mechanics 19, no. 6 (November 2, 2018): 405–11. http://dx.doi.org/10.1080/15502287.2018.1534154.

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24

EVANS, JOHN A., and THOMAS J. R. HUGHES. "ISOGEOMETRIC DIVERGENCE-CONFORMING B-SPLINES FOR THE DARCY–STOKES–BRINKMAN EQUATIONS." Mathematical Models and Methods in Applied Sciences 23, no. 04 (January 31, 2013): 671–741. http://dx.doi.org/10.1142/s0218202512500583.

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We develop divergence-conforming B-spline discretizations for the numerical solution of the Darcy–Stokes–Brinkman equations. These discretizations are motivated by the recent theory of isogeometric discrete differential forms and may be interpreted as smooth generalizations of Raviart–Thomas elements. The new discretizations are (at least) patchwise C0 and can be directly utilized in the Galerkin solution of Darcy–Stokes–Brinkman flow for single-patch configurations. When applied to incompressible flows, these discretizations produce pointwise divergence-free velocity fields and hence exactly satisfy mass conservation. In the presence of no-slip boundary conditions and multi-patch geometries, the discontinuous Galerkin framework is invoked to enforce tangential continuity without upsetting the conservation or stability properties of the method across patch boundaries. Furthermore, as no-slip boundary conditions are enforced weakly, the method automatically defaults to a compatible discretization of Darcy flow in the limit of vanishing viscosity. The proposed discretizations are extended to general mapped geometries using divergence-preserving transformations. For sufficiently regular single-patch solutions, we prove a priori error estimates which are optimal for the discrete velocity field and suboptimal, by one order, for the discrete pressure field. Our estimates are in addition robust with respect to the parameters of the Darcy–Stokes–Brinkman problem. We present a comprehensive suite of numerical experiments which indicate optimal convergence rates for both the discrete velocity and pressure fields for general configurations, suggesting that our a priori estimates may be conservative. The focus of this paper is strictly on incompressible flows, but our theoretical results naturally extend to flows characterized by mass sources and sinks.

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25

Wang, Haitao, Lei Kou, and Hongkang Zhu. "Hydro-Mechanical Coupling of Cement-Based Slurry Grouting in Saturated Geomaterials." Mathematics 11, no. 13 (June 27, 2023): 2877. http://dx.doi.org/10.3390/math11132877.

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A mathematical model is proposed to simulate the fully hydro-mechanical coupling of two-phase cement-based slurry migration in saturated deformable geomaterials from microscopic to macroscopic scale. The mass conservation equations and the momentum balance equations for cement-based slurry and geomaterials are derived based on the thermodynamically constrained averaging theory (TCAT). The Galerkin discretization of the governing equations of hydro-mechanical coupling are developed by the isogeometric analysis (IGA) integrated with the Bézier extraction operator, and the numerical calculation is implemented with the generalized backward Euler method. The presented modeling is verified by comparison of the numerical calculation with the experimental tests, and the pore fluid pressure of the stratum and the slurry concentration of cement-based slurry migration in saturated deformable geomaterials are discussed. The modeling presented provides an effective alternative method to simulate cement-based slurry migration and explore isothermal multiphase coupled problems.

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26

Grm, Aleksander. "Analysis of Generic IGBEM for Lifting Hydrofoils." Journal of Maritime & Transportation Science 3, no. 3 (June 2020): 205–15. http://dx.doi.org/10.18048/2020.00.15.

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Today the most crucial aspect in the preliminary vessel design stage is to make it as green/blue as possible. One of the exciting goals is the minimisation of vessel resistance. The use of hydrofoils to reduce the vessel draught and consequently, reduction in the vessel resistance is today one of the hottest design topics, especially for catamaran passenger vessels. In the present work, we discuss the issues related to the implementation of Isogeometric Analysis (IGA) Boundary Element Method (BEM) for the calculation of the hydrodynamic properties of lifting hydrofoils. The use of IGBEM allows numerical calculation of foil hydrodynamic properties without the traditional step of mesh generation using the CAD geometry directly. The analysis relies on the NURBS basis function with the generic Galerkin approach allowing identical solutions procedures for 2D or 3D problems. Method accuracy and computational times for a different number of Degrees of Freedom (DOF) in 2D are investigated.

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27

Liu, Wei, Zhengdong Huang, and Yunhua Liu. "An isogeometric analysis approach for solving the Reynolds equation in textured piston ring – cylinder liner contacts." Engineering Computations 37, no. 9 (April 20, 2020): 3045–78. http://dx.doi.org/10.1108/ec-03-2019-0076.

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Purpose The purpose of this study is to propose an isogeometric analysis (IGA) approach for solving the Reynolds equation in textured piston ring cylinder liner (PRCL) contacts. Design/methodology/approach The texture region is accurately and conveniently expressed by non-uniform rational B-splines (NURBS) besides hydrodynamic pressure and the oil film density ratio is represented in this mathematical form. A quadratic programming method combined with a Lagrange multiplier method is developed to address the cavitation issue. Findings The comparison with the results solved by an analytical method has verified the effectiveness of the proposed approach. In the study of the PRCL contact with two-dimensional circular dimple textures, the solution of the IGA approach shows high smoothness and accuracy, and it well satisfies the complementarity condition in the case of cavitation presence. Originality/value This paper proposes an IGA approach for solving the Reynolds equation in textured PRCL contacts. Its novelty is reflected in three aspects. First, NURBS functions are simultaneously used to express the solution domain, texture shape, hydrodynamic pressure and oil density ratio. Second, the streamline upwind/Petrov–Galerkin method is adopted to create a weak form for the Reynolds equation that takes the oil density ratio as a first-order unknown variable. Third, a quadratic programming approach is developed to impose the complementarity conditions between the hydrodynamic pressure and the oil density ratio.

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28

Aimi, Alessandra, and Ariel Surya Boiardi. "IGA-Energetic BEM: An Effective Tool for the Numerical Solution of Wave Propagation Problems in Space-Time Domain." Mathematics 10, no. 3 (January 22, 2022): 334. http://dx.doi.org/10.3390/math10030334.

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The Energetic Boundary Element Method (BEM) is a recent discretization technique for the numerical solution of wave propagation problems, inside bounded domains or outside bounded obstacles. The differential model problem is converted into a Boundary Integral Equation (BIE) in the time domain, which is then written into an energy-dependent weak form successively discretized by a Galerkin-type approach. Taking into account the space-time model problem of 2D soft-scattering of acoustic waves by obstacles described by open arcs by B-spline (or NURBS) parametrizations, the aim of this paper is to introduce the powerful Isogeometric Analysis (IGA) approach into Energetic BEM for what concerns discretization in space variables. The same computational benefits already observed for IGA-BEM in the case of elliptic (i.e., static) problems, is emphasized here because it is gained at every step of the time-marching procedure. Numerical issues for an efficient integration of weakly singular kernels, related to the fundamental solution of the wave operator and dependent on the propagation wavefront, will be described. Effective numerical results will be given and discussed, showing, from a numerical point of view, convergence and accuracy of the proposed method, as well as the superiority of IGA-Energetic BEM compared to the standard version of the method, which employs classical Lagrangian basis functions.

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29

Mika, Michal L., Thomas J. R. Hughes, Dominik Schillinger, Peter Wriggers, and René R. Hiemstra. "A matrix-free isogeometric Galerkin method for Karhunen–Loève approximation of random fields using tensor product splines, tensor contraction and interpolation based quadrature." Computer Methods in Applied Mechanics and Engineering 379 (June 2021): 113730. http://dx.doi.org/10.1016/j.cma.2021.113730.

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30

Gantner, Gregor, Dirk Praetorius, and Stefan Schimanko. "Adaptive isogeometric boundary element methods with local smoothness control." Mathematical Models and Methods in Applied Sciences 30, no. 02 (February 2020): 261–307. http://dx.doi.org/10.1142/s0218202520500074.

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In the frame of isogeometric analysis, we consider a Galerkin boundary element discretization of the hyper-singular integral equation associated with the 2D Laplacian. We propose and analyze an adaptive algorithm which locally refines the boundary partition and, moreover, steers the smoothness of the NURBS ansatz functions across elements. In particular and unlike prior work, the algorithm can increase and decrease the local smoothness properties and hence exploits the full potential of isogeometric analysis. We prove that the new adaptive strategy leads to linear convergence with optimal algebraic rates. Numerical experiments confirm the theoretical results. A short appendix comments on analogous results for the weakly-singular integral equation.

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31

Raval, Krunal, Carla Manni, and Hendrik Speleers. "Tchebycheffian B-splines in isogeometric Galerkin methods." Computer Methods in Applied Mechanics and Engineering 403 (January 2023): 115648. http://dx.doi.org/10.1016/j.cma.2022.115648.

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32

Wang, Liang, Chunguang Xiong, Xinpeng Yuan, and Huibin Wu. "Discontinuous Galerkin Isogeometric Analysis of Convection Problem on Surface." Mathematics 9, no. 5 (February 28, 2021): 497. http://dx.doi.org/10.3390/math9050497.

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The objective of this work is to study finite element methods for approximating the solution of convection equations on surfaces embedded in R3. We propose the discontinuous Galerkin (DG) isogeometric analysis (IgA) formulation to solve convection problems on implicitly defined surfaces. Three numerical experiments shows that the numerical scheme converges with the optimal convergence order.

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33

Mantzaflaris, Angelos, Bert Jüttler, Boris N. Khoromskij, and Ulrich Langer. "Low rank tensor methods in Galerkin-based isogeometric analysis." Computer Methods in Applied Mechanics and Engineering 316 (April 2017): 1062–85. http://dx.doi.org/10.1016/j.cma.2016.11.013.

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34

Donatelli, Marco, Carlo Garoni, Carla Manni, Stefano Serra-Capizzano, and Hendrik Speleers. "Symbol-Based Multigrid Methods for Galerkin B-Spline Isogeometric Analysis." SIAM Journal on Numerical Analysis 55, no. 1 (January 2017): 31–62. http://dx.doi.org/10.1137/140988590.

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35

Garoni, Carlo, Carla Manni, Stefano Serra-Capizzano, Debora Sesana, and Hendrik Speleers. "Spectral analysis and spectral symbol of matrices in isogeometric Galerkin methods." Mathematics of Computation 86, no. 305 (August 3, 2016): 1343–73. http://dx.doi.org/10.1090/mcom/3143.

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36

Zhang, Futao, Yan Xu, and Falai Chen. "Discontinuous Galerkin Methods for Isogeometric Analysis for Elliptic Equations on Surfaces." Communications in Mathematics and Statistics 2, no. 3-4 (December 2014): 431–61. http://dx.doi.org/10.1007/s40304-015-0049-y.

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37

Loli, Gabriele, Monica Montardini, Giancarlo Sangalli, and Mattia Tani. "An efficient solver for space–time isogeometric Galerkin methods for parabolic problems." Computers & Mathematics with Applications 80, no. 11 (December 2020): 2586–603. http://dx.doi.org/10.1016/j.camwa.2020.09.014.

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38

Schneckenleitner, Rainer, and Stefan Takacs. "IETI-DP methods for discontinuous Galerkin multi-patch Isogeometric Analysis with T-junctions." Computer Methods in Applied Mechanics and Engineering 393 (April 2022): 114694. http://dx.doi.org/10.1016/j.cma.2022.114694.

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39

Hofer, Christoph. "Parallelization of continuous and discontinuous Galerkin dual–primal isogeometric tearing and interconnecting methods." Computers & Mathematics with Applications 74, no. 7 (October 2017): 1607–25. http://dx.doi.org/10.1016/j.camwa.2017.06.051.

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Schillinger, Dominik, John A. Evans, Alessandro Reali, Michael A. Scott, and Thomas J. R. Hughes. "Isogeometric collocation: Cost comparison with Galerkin methods and extension to adaptive hierarchical NURBS discretizations." Computer Methods in Applied Mechanics and Engineering 267 (December 2013): 170–232. http://dx.doi.org/10.1016/j.cma.2013.07.017.

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Schillinger, Dominik, John A. Evans, Alessandro Reali, Michael A. Scott, and Thomas J. R. Hughes. "Isogeometric Collocation: Cost Comparison with Galerkin Methods and Extension to Adaptive Hierarchical NURBS Discretizations." PAMM 13, no. 1 (November 29, 2013): 107–8. http://dx.doi.org/10.1002/pamm.201310049.

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Niiranen, J., V. Balobanov, J. Kiendl, and SB Hosseini. "Variational formulations, model comparisons and numerical methods for Euler–Bernoulli micro- and nano-beam models." Mathematics and Mechanics of Solids 24, no. 1 (November 20, 2017): 312–35. http://dx.doi.org/10.1177/1081286517739669.

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As a first step, variational formulations and governing equations with boundary conditions are derived for a pair of Euler–Bernoulli beam bending models following a simplified version of Mindlin’s strain gradient elasticity theory of form II. For both models, this leads to sixth-order boundary value problems with new types of boundary conditions that are given additional attributes singly and doubly, referring to a physically relevant distinguishing feature between free and prescribed curvature, respectively. Second, the variational formulations are analyzed with rigorous mathematical tools: the existence and uniqueness of weak solutions are established by proving continuity and ellipticity of the associated symmetric bilinear forms. This guarantees optimal convergence for conforming Galerkin discretization methods. Third, the variational analysis is extended to cover two other generalized beam models: another modification of the strain gradient elasticity theory and a modified version of the couple stress theory. A model comparison reveals essential differences and similarities in the physicality of these four closely related beam models: they demonstrate essentially two different kinds of parameter-dependent stiffening behavior, where one of these kinds (possessed by three models out of four) provides results in a very good agreement with the size effects of experimental tests. Finally, numerical results for isogeometric Galerkin discretizations with B-splines confirm the theoretical stability and convergence results. Influences of the gradient and thickness parameters connected to size effects, boundary layers and dispersion relations are studied thoroughly with a series of benchmark problems for statics and free vibrations. The size-dependency of the effective Young’s modulus is demonstrated for an auxetic cellular metamaterial ruled by bending-dominated deformation of cell struts.

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Xia, Yang, and Pan Guo. "A time discontinuous Galerkin isogeometric analysis method for non-Fourier thermal wave propagation problem." Engineering Computations ahead-of-print, ahead-of-print (August 31, 2019). http://dx.doi.org/10.1108/ec-08-2018-0377.

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Purpose Numerical instability such as spurious oscillation is an important problem in the simulation of heat wave propagation. The purpose of this study is to propose a time discontinuous Galerkin isogeometric analysis method to reduce numerical instability of heat wave propagation in the medium subjected to heat sources, particularly heat impulse. Design/methodology/approach The essential vectors of temperature and the temporal gradients are assumed to be discontinuous and interpolated individually in the discretized time domain. The isogeometric analysis method is applied to use its property of smooth description of the geometry and to eliminate the mesh-dependency. An artificial damping scheme with proportional stiffness matrix is brought into the final discretized form to reduce the numerical spurious oscillations. Findings The numerical spurious oscillations in the simulation of heat wave propagation are effectively eliminated. The smooth description of geometry with spline functions solves the mesh-dependency problem and improves the numerical precision. Originality/value The time discontinuous Galerkin method is applied within the isogeometric analysis framework. The proposed method is effective in the simulation of the wave propagation problems subjecting to impulse load with numerical stability and accuracy.

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Elasmi, Mehdi, Christoph Erath, and Stefan Kurz. "Non-symmetric isogeometric FEM-BEM couplings." Advances in Computational Mathematics 47, no. 5 (August 19, 2021). http://dx.doi.org/10.1007/s10444-021-09886-3.

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AbstractWe present a coupling of the Finite Element and the Boundary Element Method in an isogeometric framework to approximate either two-dimensional Laplace interface problems or boundary value problems consisting of two disjoint domains. We consider the Finite Element Method in the bounded domains to simulate possibly non-linear materials. The Boundary Element Method is applied in unbounded or thin domains where the material behavior is linear. The isogeometric framework allows to combine different design and analysis tools: first, we consider the same type of NURBS parameterizations for an exact geometry representation and second, we use the numerical analysis for the Galerkin approximation. Moreover, it facilitates to perform h- and p-refinements. For the sake of analysis, we consider the framework of strongly monotone and Lipschitz continuous operators to ensure well-posedness of the coupled system. Furthermore, we provide a priori error estimates. We additionally show an improved convergence behavior for the errors in functionals of the solution that may double the rate under certain assumptions. Numerical examples conclude the work which illustrate the theoretical results.

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Benedusi, Pietro, Paola Ferrari, Carlo Garoni, Rolf Krause, and Stefano Serra-Capizzano. "Fast Parallel Solver for the Space-time IgA-DG Discretization of the Diffusion Equation." Journal of Scientific Computing 89, no. 1 (September 1, 2021). http://dx.doi.org/10.1007/s10915-021-01567-z.

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AbstractWe consider the space-time discretization of the diffusion equation, using an isogeometric analysis (IgA) approximation in space and a discontinuous Galerkin (DG) approximation in time. Drawing inspiration from a former spectral analysis, we propose for the resulting space-time linear system a multigrid preconditioned GMRES method, which combines a preconditioned GMRES with a standard multigrid acting only in space. The performance of the proposed solver is illustrated through numerical experiments, which show its competitiveness in terms of iteration count, run-time and parallel scaling.

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Montardini, Monica, Giancarlo Sangalli, Rainer Schneckenleitner, Stefan Takacs, and Mattia Tani. "A IETI-DP method for discontinuous Galerkin discretizations in Isogeometric Analysis with inexact local solvers." Mathematical Models and Methods in Applied Sciences, June 9, 2023. http://dx.doi.org/10.1142/s0218202523500495.

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Deng, Feng, Wenshan Yu, Xu Liang, and Shengping Shen. "The existence and uniqueness theorem for linear flexoelectricity and application to the Galerkin approximation." Mathematics and Mechanics of Solids, March 29, 2023, 108128652311643. http://dx.doi.org/10.1177/10812865231164330.

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Flexoelectricity represents the electric polarization response to strain gradient and is governed by a boundary value problem (BVP) of fourth-order partial differential equations (PDEs). Due to the complicities of the BVP, we are frequently prevented from searching for analytical formulas of solutions and have to make a detour to numerical approximations. The well-posedness of the BVP has not been rigorously confirmed yet, although it serves as the cornerstone of numerical methods for flexoelectricity. We address this issue by establishing a theorem indicating a unique solution to the electric Gibbs variational principle (EGVP) that is a weak form of the BVP. The theorem leads us to bear out the solvability and error-estimation formula for some Galerkin methods, e.g., the finite element method (FEM) and isogeometric analysis (IGA). The reliability of the numerical theory is further verified by numerical experiments of benchmark problem, where the convergence rate given by FEM results agrees well with that predicted by error-estimation formulas. This study confirms the well-posedness of the flexoelectric theory and further establishes computational flexoelectricity on rigorous mathematical foundations instead of numerical tests.

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Obohat, M. Amin, Ehsan Tahvilian, M. Erden Yildizdag, and Ahmet Ergin. "Three-dimensional multi-patch isogeometric analysis of composite laminates with a discontinuous Galerkin approach." Proceedings of the Institution of Mechanical Engineers, Part M: Journal of Engineering for the Maritime Environment, June 18, 2020, 147509022092573. http://dx.doi.org/10.1177/1475090220925734.

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In this study, a three-dimensional discontinuous Galerkin isogeometric analysis framework is presented for the analysis of composite laminates. Non-uniform rational B-splines are employed as basis functions for both geometric and computational implementations. From a practical point of view, modeling with multiple non-uniform rational B-spline patches is required in many different applications due to the complexity of computational domains. Then, a special numerical technique is necessary to couple different non-uniform rational B-spline patches to carry out the isogeometric analysis. In this study, therefore, one of the discontinuous Galerkin methods, namely, symmetric interior penalty Galerkin formulation is utilized to deal with multi-patch isogeometric analysis applications. In order to show the applicability of the proposed framework, composite laminates under sinusoidally distributed load with different stacking sequences are studied in the numerical examples. The predicted results are compared with those obtained by the three-dimensional elasticity solutions and various numerical models available in the literature.

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Gantner, Gregor, Dirk Praetorius, and Stefan Schimanko. "Stable Implementation of Adaptive IGABEM in 2D in MATLAB." Computational Methods in Applied Mathematics, June 9, 2022. http://dx.doi.org/10.1515/cmam-2022-0050.

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Abstract We report on the Matlab program package IGABEM2D which provides an easily accessible implementation of adaptive Galerkin boundary element methods in the frame of isogeometric analysis and which is available on the web for free download. Numerical experiments with IGABEM2D underline the particular importance of adaptive mesh refinement for high accuracy in isogeometric analysis.

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Schröder, Jörg, Thomas Wick, Stefanie Reese, Peter Wriggers, Ralf Müller, Stefan Kollmannsberger, Markus Kästner, et al. "A Selection of Benchmark Problems in Solid Mechanics and Applied Mathematics." Archives of Computational Methods in Engineering, September 18, 2020. http://dx.doi.org/10.1007/s11831-020-09477-3.

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Abstract In this contribution we provide benchmark problems in the field of computational solid mechanics. In detail, we address classical fields as elasticity, incompressibility, material interfaces, thin structures and plasticity at finite deformations. For this we describe explicit setups of the benchmarks and introduce the numerical schemes. For the computations the various participating groups use different (mixed) Galerkin finite element and isogeometric analysis formulations. Some programming codes are available open-source. The output is measured in terms of carefully designed quantities of interest that allow for a comparison of other models, discretizations, and implementations. Furthermore, computational robustness is shown in terms of mesh refinement studies. This paper presents benchmarks, which were developed within the Priority Programme of the German Research Foundation ‘SPP 1748 Reliable Simulation Techniques in Solid Mechanics—Development of Non-Standard Discretisation Methods, Mechanical and Mathematical Analysis’.

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