157 research outputs found
Variational Convergence of IP-DGFEM
In this paper, we develop the theory required to perform a variational convergence analysis for discontinuous Galerkin nite element methods when applied to minimization problems. For Sobolev indices in , we prove generalizations of many techniques of classical analysis in Sobolev spaces and apply them to a typical energy minimization problem for which we prove convergence of a variational interior penalty discontinuous Galerkin nite element method (VIPDGFEM). Our main tool in this analysis is a theorem which allows the extraction of a "weakly" converging subsequence of a family of discrete solutions and which shows that any "weak limit" is a Sobolev function
A priori error for unilateral contact problems with Lagrange multiplier and IsoGeometric Analysis
In this paper, we consider unilateral contact problem without friction
between a rigid body and deformable one in the framework of isogeometric
analysis. We present the theoretical analysis of the mixed problem using an
active-set strategy and for a primal space of NURBS of degree and for
a dual space of B-Spline. A inf-sup stability is proved to ensure a good
property of the method. An optimal a priori error estimate is demonstrated
without assumption on the unknown contact set. Several numerical examples in
two- and three-dimensional and in small and large deformation demonstrate the
accuracy of the proposed method
Isogeometric Analysis on V-reps: first results
Inspired by the introduction of Volumetric Modeling via volumetric
representations (V-reps) by Massarwi and Elber in 2016, in this paper we
present a novel approach for the construction of isogeometric numerical methods
for elliptic PDEs on trimmed geometries, seen as a special class of more
general V-reps. We develop tools for approximation and local re-parametrization
of trimmed elements for three dimensional problems, and we provide a
theoretical framework that fully justify our algorithmic choices. We validate
our approach both on two and three dimensional problems, for diffusion and
linear elasticity.Comment: 36 pages, 44 figures. Reviewed versio
BPX-Preconditioning for isogeometric analysis
We consider elliptic PDEs (partial differential equations) in the framework of isogeometric analysis, i.e., we treat the physical domain by means of a B-spline or Nurbs mapping which we assume to be regular. The numerical solution of the PDE is computed by means of tensor product B-splines mapped onto the physical domain. We construct additive multilevel preconditioners and show that they are asymptotically optimal, i.e., the spectral condition number of the resulting preconditioned stiffness matrix is independent of . Together with a nested iteration scheme, this enables an iterative solution scheme of optimal linear complexity. The theoretical results are substantiated by numerical examples in two and three space dimensions
PDE-Based Parameterisation Techniques for Planar Multipatch Domains
This paper presents a PDE-based parameterisation framework for addressing the
planar surface-to-volume (StV) problem of finding a valid description of the
domain's interior given no more than a spline-based description of its boundary
contours. The framework is geared towards isogeometric analysis (IGA)
applications wherein the physical domain is comprised of more than four sides,
hence requiring more than one patch. We adopt the concept of harmonic maps and
propose several PDE-based problem formulations capable of finding a valid map
between a convex parametric multipatch domain and the piecewise-smooth physical
domain with an equal number of sides. In line with the isoparametric paradigm
of IGA, we treat the StV problem using techniques that are characteristic for
the analysis step. As such, this study proposes several IGA-based numerical
algorithms for the problem's governing equations that can be effortlessly
integrated into a well-developed IGA software suite. We augment the framework
with mechanisms that enable controlling the parametric properties of the
outcome. Parametric control is accomplished by, among other techniques, the
introduction of a curvilinear coordinate system in the convex parametric domain
that, depending on the application, builds desired features into the computed
harmonic map, such as homogeneous cell sizes or boundary layers
Moment equations for the mixed formulation of the Hodge Laplacian with stochastic loading term
We study the mixed formulation of the stochastic Hodge-Laplace problem defined on an n-dimensional domain D (nâ„1), with random forcing term. In particular, we focus on the magnetostatic problem and on the Darcy problem in the three-dimensional case. We derive and analyse the moment equations, that is, the deterministic equations solved by the mth moment (mâ„1) of the unique stochastic solution of the stochastic problem. We find stable tensor product finite element discretizations, both full and sparse, and provide optimal order-of-convergence estimates. In particular, we prove the inf-sup condition for sparse tensor product finite element space
Fast parametric analysis of trimmed multi-patch isogeometric Kirchhoff-Love shells using a local reduced basis method
This contribution presents a model order reduction framework for real-time
efficient solution of trimmed, multi-patch isogeometric Kirchhoff-Love shells.
In several scenarios, such as design and shape optimization, multiple
simulations need to be performed for a given set of physical or geometrical
parameters. This step can be computationally expensive in particular for real
world, practical applications. We are interested in geometrical parameters and
take advantage of the flexibility of splines in representing complex
geometries. In this case, the operators are geometry-dependent and generally
depend on the parameters in a non-affine way. Moreover, the solutions obtained
from trimmed domains may vary highly with respect to different values of the
parameters. Therefore, we employ a local reduced basis method based on
clustering techniques and the Discrete Empirical Interpolation Method to
construct affine approximations and efficient reduced order models. In
addition, we discuss the application of the reduction strategy to parametric
shape optimization. Finally, we demonstrate the performance of the proposed
framework to parameterized Kirchhoff-Love shells through benchmark tests on
trimmed, multi-patch meshes including a complex geometry. The proposed approach
is accurate and achieves a significant reduction of the online computational
cost in comparison to the standard reduced basis method.Comment: 43 pages, 21 figures, 3 table
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