507 research outputs found
Discontinuous Galerkin computation of the Maxwell eigenvalues on simplicial meshes
No full textAbstractThis paper is concerned with the discontinuous Galerkin approximation of the Maxwell eigenproblem. After reviewing the theory developed in [A. Buffa, I. Perugia, Discontinuous Galerkin approximation of the Maxwell eigenproblem, Technical Report 24-PV, IMATI-CNR, Pavia, Italy, 2005 〈http://www.imati.cnr.it/∼annalisa/PS/maxwell.pdf〉], we present a set of numerical experiments which both validate the theory, and provide further insight regarding the practical performance of discontinuous Galerkin methods, particularly in the case when non-conforming meshes, characterized by the presence of hanging nodes, are employed
Analysis of a Multiscale Discontinuous Galerkin Method for Convection-Diffusion Problems
No full textWe study a multiscale discontinuous Galerkin method introduced in [T. J. R. Hughes, G. Scovazzi, P. Bochev, and A. Buffa, Comput. Meth. Appl. Mech. Engrg., 195 (2006), pp. 2761–2787] that reduces the computational complexity of the discontinuous Galerkin method, seemingly without adversely affecting the quality of results. For a stabilized variant we are able to obtain the same error estimates for the convection-diffusion equation as for the usual discontinuous Galerkin method. We assess the stability of the unstabilized case numerically and find that the inf-sup constant is positive, bounded uniformly away from zero, and very similar to that for the usual discontinuous Galerkin method
A projected super-penalty method for the C1-coupling of multi-patch isogeometric Kirchhoff plates
Get PDFThis work focuses on the development of a super-penalty strategy based on the L2-projection of suitable coupling terms to achieve C1-continuity between non-conforming multi-patch isogeometric Kirchhoff plates. In particular, the choice of penalty parameters is driven by the underlying perturbed saddle point problem from which the Lagrange multipliers are eliminated and is performed to guarantee the optimal accuracy of the method. Moreover, by construction, the method does not suffer from boundary locking, especially on very coarse meshes. We demonstrate the applicability of the proposed coupling algorithm to Kirchhoff plates by studying several benchmark examples discretized by non-conforming meshes. In all cases, we recover the optimal rates of convergence achievable by B-splines where we achieve a substantial gain in accuracy per degree-of-freedom compared to other choices of the penalty parameters.MNSThis is an Open Access article under the terms of the Creative Commons Attribution Licens
Isogeometric analysis in electromagnetics: B-splines approximation
No full textWe introduce a new discretization scheme for Maxwell equations in two space dimension. Inspired by the new paradigm of Isogeometric analysis, we
propose an algorithm based on the use of bivariate B-splines spaces suitably adapted
to electromagnetics. We construct B-splines spaces of variable interelement regularity on the parametric domain. These spaces (and their push-forwards on the physical
domain) form a De Rham diagram and we use them to solve the Maxwell source and eigen problem. Our scheme has the following features: (i) is adapted to treat complex
geometries, (ii) is spectral correct, (iii) provides regular (e.g., globally continuity) discrete solutions of Maxwell equations
Remarks on Poincare and interpolation estimates for Truncated Hierarchical B-splines
No full textThis paper should be considered as an addendum to [A. Buffa and C. Giannelli, Adaptive isogeometric methods with hierarchical splines: Error estimator and convergence, Math. Models Methods Appl. Sci. 26 (2016) 1-25] and [A. Buffa and C. Giannelli, Adaptive isogeometric methods with hierarchical splines: Optimality and convergence rates, Math. Models Methods Appl. Sci. 27 (2017) 2781-2802] where Poincare and approximation estimates are used as theoretical tools to study properties of adaptive numerical methods based on hierarchical B-splines. After noting that the support of truncated hierarchical B-splines may be disconnected (and thus no Poincare estimate can hold), we study minimal extensions of their support on suitable mesh configurations such that (i) Poincare estimates can be established on them and (ii) their overlaps stay independent of the number of levels. The Poincare estimates proposed in this note should replace the ones used in the proofs of Theorem 11 and Lemma 7 in [A. Buffa and C. Giannelli, Adaptive isogeometric methods with hierarchical splines: Error estimator and convergence, Math. Models Methods Appl. Sci. 26 (2016) 1-25] and [A. Buffa and C. Giannelli, Adaptive isogeometric methods with hierarchical splines: Optimality and convergence rates, Math. Models Methods Appl. Sci. 27 (2017) 2781-2802], respectively, in order to include the most general meshes, i.e. the cases when the support of truncated basis functions can be disconnected.MN
On Quasi-Interpolation Operators in Spline Spaces
No full textWe propose the construction of a class of L2 stable quasi-interpolationoperators onto the space of splines on tensor-product meshes, in any space dimension. The estimate we propose is robust with respect to knot repetition and to knot ?vicinity? (up to p+1 knots), so it applies to the most general scenario in which the B-spline functions are known to be well defined.Fil: Buffa, Annalisa. Consiglio Nazionale delle Ricerche. Istituto di Matematica Applicata e Tecnologie Informatiche; ItaliaFil: Garau, Eduardo Mario. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; ArgentinaFil: Giannelli, Carlotta. Università degli Studi di Firenze; ItaliaFil: Sangalli, Giancarlo. Consiglio Nazionale delle Ricerche. Istituto di Matematica Applicata e Tecnologie Informatiche; Itali
An interior penalty coupling strategy for isogeometric non-conformal Kirchhoff-Love shell patches
Get PDFThis work focuses on the coupling of trimmed shell patches using Isogeometric Analysis, based on higher continuity splines that seamlessly meet the requirement of Kirchhoff-Love-based discretizations. Weak enforcement of coupling conditions is achieved through the symmetric interior penalty method, where the fluxes are computed using their correct variationally consistent expression that was only recently proposed and is unprecedentedly adopted herein in the context of coupling conditions. The constitutive relationship accounts for generically laminated materials, although the proposed tests are conducted under the assumption of uniform thickness and lamination sequence. Numerical experiments assess the method for an isotropic and a laminated plate, as well as an isotropic hyperbolic paraboloid shell from the new shell obstacle course. The boundary conditions and domain force are chosen to reproduce manufactured analytical solutions, which are taken as reference to compute rigorous convergence curves in the L 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} , H 1 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} , and H 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} norms, that closely approach optimal ones predicted by theory. Additionally, we conduct a final test on a complex structure comprising five intersecting laminated cylindrical shells, whose geometry is directly imported from a STEP file. The results exhibit excellent agreement with those obtained through commercial software, showcasing the method's potential for real-world industrial applications.MN
The Immersed Boundary Conformal Method for Kirchhoff–Love and Reissner–Mindlin shells
Get PDFThis work utilizes the Immersed Boundary Conformal Method (IBCM) to analyze linear elastic Kirchhoff-Love and Reissner-Mindlin shell structures within an immersed domain framework. Immersed boundary methods involve embedding complex geometries within a background grid, which allows for great flexibility in modeling intricate shapes and features despite simplicity of the approach. The IBCM method introduces additional layers conformal to boundaries, allowing for the strong imposition of Dirichlet boundary conditions and facilitating local refinement. In this study, the construction of boundary layers is combined with high degree spline-based approximation spaces to further increase efficiency. The Nitsche method, employing non-symmetric average operators, is used to couple the boundary layers with inner patch, while stabilizing the formulation with minimal penalty parameters. High-order quadrature rules are applied for integration over cut elements and patch interfaces. Numerical experiments demonstrate the efficiency and accuracy of the proposed formulation, highlighting its potential for complex shell structures modeled through Kirchhoff-Love and Reissner- Mindlin theories. These tests include the generation of conformal interfaces, the coupling Kirchhoff-Love and Reissner-Mindlin theories, and the simulation of a cylindrical shell with through-the-thickness crack
IsoGeometric Analysis: A New Paradigm in the Numerical Approximation of PDEs Cetraro, Italy 2012 Preface
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