58 research outputs found
The Argyris isogeometric space on unstructured multi-patch planar domains
Multi-patch spline parametrizations are used in geometric design and
isogeometric analysis to represent complex domains. We deal with a particular
class of planar multi-patch spline parametrizations called
analysis-suitable (AS-) multi-patch parametrizations (Collin,
Sangalli, Takacs; CAGD, 2016). This class of parametrizations has to satisfy
specific geometric continuity constraints, and is of importance since it allows
to construct, on the multi-patch domain, isogeometric spaces with optimal
approximation properties. It was demonstrated in (Kapl, Sangalli, Takacs; CAD,
2018) that AS- multi-patch parametrizations are suitable for modeling
complex planar multi-patch domains.
In this work, we construct a basis, and an associated dual basis, for a
specific isogeometric spline space over a given AS-
multi-patch parametrization. We call the space the Argyris
isogeometric space, since it is across interfaces and at all
vertices and generalizes the idea of Argyris finite elements to tensor-product
splines. The considered space is a subspace of the entire
isogeometric space , which maintains the reproduction
properties of traces and normal derivatives along the interfaces. Moreover, it
reproduces all derivatives up to second order at the vertices. In contrast to
, the dimension of does not depend on the domain
parametrization, and admits a basis and dual basis which possess
a simple explicit representation and local support.
We conclude the paper with some numerical experiments, which exhibit the
optimal approximation order of the Argyris isogeometric space and
demonstrate the applicability of our approach for isogeometric analysis
A family of quadrilateral finite elements
We present a novel family of quadrilateral finite elements, which
define global spaces over a general quadrilateral mesh with vertices of
arbitrary valency. The elements extend the construction by (Brenner and Sung,
J. Sci. Comput., 2005), which is based on polynomial elements of tensor-product
degree , to all degrees . Thus, we call the family of
finite elements Brenner-Sung quadrilaterals. The proposed quadrilateral
can be seen as a special case of the Argyris isogeometric element of (Kapl,
Sangalli and Takacs, CAGD, 2019). The quadrilateral elements possess similar
degrees of freedom as the classical Argyris triangles. Just as for the Argyris
triangle, we additionally impose continuity at the vertices. In this
paper we focus on the lower degree cases, that may be desirable for their lower
computational cost and better conditioning of the basis: We consider indeed the
polynomial quadrilateral of (bi-)degree~, and the polynomial degrees
and by employing a splitting into or polynomial
pieces, respectively.
The proposed elements reproduce polynomials of total degree . We show that
the space provides optimal approximation order. Due to the interpolation
properties, the error bounds are local on each element. In addition, we
describe the construction of a simple, local basis and give for
explicit formulas for the B\'{e}zier or B-spline coefficients of the basis
functions. Numerical experiments by solving the biharmonic equation demonstrate
the potential of the proposed quadrilateral finite element for the
numerical analysis of fourth order problems, also indicating that (for )
the proposed element performs comparable or in general even better than the
Argyris triangle with respect to the number of degrees of freedom
Scaled boundary isogeometric analysis with C1 coupling for Kirchhoff plate theory
Although isogeometric analysis exploits smooth B-spline and NURBS basis
functions for the definition of discrete function spaces as well as for the
geometry representation, the global smoothness in so-called multipatch
parametrizations is an issue. Especially, if strong C1 regularity is required,
the introduction of function spaces with good convergence properties is not
straightforward. However, in 2D there is the special class of analysis-suitable
G1 (AS-G1) parametrizations that are suitable for patch coupling. In this
contribution we show that the concept of scaled boundary isogeometric analysis
fits to the AS-G1 idea and the former is appropriate to define C1-smooth basis
functions. The proposed method is applied to Kirchhoff plates and its
capability is demonstrated utilizing several numerical examples. Its
applicability to non-trivial and trimmed shapes is demonstrated
Construction of analysis-suitable planar multi-patch parameterizations
Isogeometric analysis allows to define shape functions of global
continuity (or of higher continuity) over multi-patch geometries. The
construction of such -smooth isogeometric functions is a non-trivial
task and requires particular multi-patch parameterizations, so-called
analysis-suitable (in short, AS-) parameterizations, to ensure
that the resulting isogeometric spaces possess optimal approximation
properties, cf. [7]. In this work, we show through examples that it is possible
to construct AS- multi-patch parameterizations of planar domains, given
their boundary. More precisely, given a generic multi-patch geometry, we
generate an AS- multi-patch parameterization possessing the same
boundary, the same vertices and the same first derivatives at the vertices, and
which is as close as possible to this initial geometry. Our algorithm is based
on a quadratic optimization problem with linear side constraints. Numerical
tests also confirm that isogeometric spaces over AS- multi-patch
parameterized domains converge optimally under mesh refinement, while for
generic parameterizations the convergence order is severely reduced
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
Recommended from our members
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
Isogeometric continuity constraints for multi-patch shells governed by fourth-order deformation and phase field models
This work presents numerical techniques to enforce continuity constraints on
multi-patch surfaces for three distinct problem classes. The first involves
structural analysis of thin shells that are described by general Kirchhoff-Love
kinematics. Their governing equation is a vector-valued, fourth-order,
nonlinear, partial differential equation (PDE) that requires at least
-continuity within a displacement-based finite element formulation. The
second class are surface phase separations modeled by a phase field. Their
governing equation is the Cahn-Hilliard equation - a scalar, fourth-order,
nonlinear PDE - that can be coupled to the thin shell PDE. The third class are
brittle fracture processes modeled by a phase field approach. In this work,
these are described by a scalar, fourth-order, nonlinear PDE that is similar to
the Cahn-Hilliard equation and is also coupled to the thin shell PDE. Using a
direct finite element discretization, the two phase field equations also
require at least a -continuous formulation. Isogeometric surface
discretizations - often composed of multiple patches - thus require constraints
that enforce the -continuity of displacement and phase field. For this,
two numerical strategies are presented: For this, two numerical strategies are
presented: A Lagrange multiplier formulation and a penalty method. The
curvilinear shell model including the geometrical constraints is taken from
Duong et al. (2017) and it is extended to model the coupled phase field
problems on thin shells of Zimmermann et al. (2019) and Paul et al. (2020) on
multi-patches. Their accuracy and convergence are illustrated by several
numerical examples considering deforming shells, phase separations on evolving
surfaces, and dynamic brittle fracture of thin shells.Comment: In this version, typos were fixed, Chapter 6.4 is added, Table 1 is
updated, and clarifying explanations and remarks are added at several place
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