434 research outputs found
IGA-based Multi-Index Stochastic Collocation for random PDEs on arbitrary domains
This paper proposes an extension of the Multi-Index Stochastic Collocation
(MISC) method for forward uncertainty quantification (UQ) problems in
computational domains of shape other than a square or cube, by exploiting
isogeometric analysis (IGA) techniques. Introducing IGA solvers to the MISC
algorithm is very natural since they are tensor-based PDE solvers, which are
precisely what is required by the MISC machinery. Moreover, the
combination-technique formulation of MISC allows the straight-forward reuse of
existing implementations of IGA solvers. We present numerical results to
showcase the effectiveness of the proposed approach.Comment: version 3, version after revisio
The INTERNODES method for the treatment of non-conforming multipatch geometries in Isogeometric Analysis
In this paper we apply the INTERNODES method to solve second order elliptic
problems discretized by Isogeometric Analysis methods on non-conforming
multiple patches in 2D and 3D geometries. INTERNODES is an interpolation-based
method that, on each interface of the configuration, exploits two independent
interpolation operators to enforce the continuity of the traces and of the
normal derivatives. INTERNODES supports non-conformity on NURBS spaces as well
as on geometries. We specify how to set up the interpolation matrices on
non-conforming interfaces, how to enforce the continuity of the normal
derivatives and we give special attention to implementation aspects. The
numerical results show that INTERNODES exhibits optimal convergence rate with
respect to the mesh size of the NURBS spaces an that it is robust with respect
to jumping coefficients.Comment: Accepted for publication in Computer Methods in Applied Mechanics and
Engineerin
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
Matrix-free weighted quadrature for a computationally efficient isogeometric -method
The -method is the isogeometric method based on splines (or NURBS, etc.)
with maximum regularity. When implemented following the paradigms of classical
finite element methods, the computational resources required by the method
are prohibitive even for moderate degree. In order to address this issue, we
propose a matrix-free strategy combined with weighted quadrature, which is an
ad-hoc strategy to compute the integrals of the Galerkin system. Matrix-free
weighted quadrature (MF-WQ) speeds up matrix operations, and, perhaps even more
important, greatly reduces memory consumption. Our strategy also requires an
efficient preconditioner for the linear system iterative solver. In this work
we deal with an elliptic model problem, and adopt a preconditioner based on the
Fast Diagonalization method, an old idea to solve Sylvester-like equations. Our
numerical tests show that the isogeometric solver based on MF-WQ is faster than
standard approaches (where the main cost is the matrix formation by standard
Gaussian quadrature) even for low degree. But the main achievement is that,
with MF-WQ, the -method gets orders of magnitude faster by increasing the
degree, given a target accuracy. Therefore, we are able to show the
superiority, in terms of computational efficiency, of the high-degree
-method with respect to low-degree isogeometric discretizations. What we
present here is applicable to more complex and realistic differential problems,
but its effectiveness will depend on the preconditioner stage, which is as
always problem-dependent. This situation is typical of modern high-order
methods: the overall performance is mainly related to the quality of the
preconditioner
Multipatch Approximation of the de Rham Sequence and its Traces in Isogeometric Analysis
We define a conforming B-spline discretisation of the de Rham complex on
multipatch geometries. We introduce and analyse the properties of interpolation
operators onto these spaces which commute w.r.t. the surface differential
operators. Using these results as a basis, we derive new convergence results of
optimal order w.r.t. the respective energy spaces and provide approximation
properties of the spline discretisations of trace spaces for application in the
theory of isogeometric boundary element methods. Our analysis allows for a
straightforward generalisation to finite element methods
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Mini-Workshop: Mathematical Foundations of Isogeometric Analysis
Isogeometric Analysis (IgA) is a new paradigm which is designed to merge two so far disjoint disciplines, namely, numerical simulations for partial differential equations (PDEs) and applied geometry. Initiated by the pioneering 2005 paper of one of us organizers (Hughes), this new concept bridges the gap between classical finite element methods and computer aided design concepts.
Traditional approaches are based on modeling complex geometries by computer aided design tools which then need to be converted to a computational mesh to allow for simulations of PDEs. This process has for decades presented a severe bottleneck in performing efficient simulations. For example, for complex fluid dynamics applications, the modeling of the surface and the mesh generation may take several weeks while the PDE simulations require only a few hours.
On the other hand, simulation methods which exactly represent geometric shapes in terms of the basis functions employed for the numerical simulations bridge the gap and allow from the beginning to eliminate geometry errors. This is accomplished by leaving traditional finite element approaches behind and employing instead more general basis functions such as B-Splines and Non-Uniform Rational B-Splines (NURBS) for the PDE simulations as well. The combined concept of Isogeometric Analysis (IgA) allows for improved convergence and smoothness properties of the PDE solutions and dramatically faster overall simulations.
In the last few years, this new paradigm has revolutionized the engineering communities and triggered an enormous amount of simulations and publications mainly in this field. However, there are several profound theoretical issues which have not been well understood and which are currently investigated by researchers in Numerical Analysis, Approximation Theory and Applied Geometry
SoftIGA: soft isogeometric analysis
We extend the softFEM idea to isogeometric analysis (IGA) to reduce the
stiffness (consequently, the condition numbers) of the IGA discretized problem.
We refer to the resulting approximation technique as softIGA. We obtain the
resulting discretization by first removing the IGA spectral outliers to reduce
the system's stiffness. We then add high-order derivative-jump penalization
terms (with negative penalty parameters) to the standard IGA bilinear forms.
The penalty parameter seeks to minimize spectral/dispersion errors while
maintaining the coercivity of the bilinear form. We establish dispersion errors
for both outlier-free IGA (OF-IGA) and softIGA elements. We also derive
analytical eigenpairs for the resulting matrix eigenvalue problems and show
that the stiffness and condition numbers of the IGA systems significantly
improve (reduce). We prove a superconvergent result of order for
eigenvalues where characterizes the mesh size and specifies the order
of the B-spline basis functions. To illustrate the main idea and derive the
analytical results, we focus on uniform meshes in 1D and tensor-product meshes
in multiple dimensions. For the eigenfunctions, softIGA delivers the same
optimal convergence rates as the standard IGA approximation. Various numerical
examples demonstrate the advantages of softIGA over IGA
A p-multigrid method enhanced with an ILUT smoother and its comparison to h-multigrid methods within Isogeometric Analysis
Over the years, Isogeometric Analysis has shown to be a successful
alternative to the Finite Element Method (FEM). However, solving the resulting
linear systems of equations efficiently remains a challenging task. In this
paper, we consider a p-multigrid method, in which coarsening is applied in the
approximation order p instead of the mesh width h. Since the use of classical
smoothers (e.g. Gauss-Seidel) results in a p-multigrid method with
deteriorating performance for higher values of p, the use of an ILUT smoother
is investigated. Numerical results and a spectral analysis indicate that the
resulting p-multigrid method exhibits convergence rates independent of h and p.
In particular, we compare both coarsening strategies (e.g. coarsening in h or
p) adopting both smoothers for a variety of two and threedimensional
benchmarks
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