2,314 research outputs found
Importance and effectiveness of representing the shapes of Cosserat rods and framed curves as paths in the special Euclidean algebra
We discuss how the shape of a special Cosserat rod can be represented as a
path in the special Euclidean algebra. By shape we mean all those geometric
features that are invariant under isometries of the three-dimensional ambient
space. The representation of the shape as a path in the special Euclidean
algebra is intrinsic to the description of the mechanical properties of a rod,
since it is given directly in terms of the strain fields that stimulate the
elastic response of special Cosserat rods. Moreover, such a representation
leads naturally to discretization schemes that avoid the need for the expensive
reconstruction of the strains from the discretized placement and for
interpolation procedures which introduce some arbitrariness in popular
numerical schemes. Given the shape of a rod and the positioning of one of its
cross sections, the full placement in the ambient space can be uniquely
reconstructed and described by means of a base curve endowed with a material
frame. By viewing a geometric curve as a rod with degenerate point-like cross
sections, we highlight the essential difference between rods and framed curves,
and clarify why the family of relatively parallel adapted frames is not
suitable for describing the mechanics of rods but is the appropriate tool for
dealing with the geometry of curves.Comment: Revised version; 25 pages; 7 figure
A RBF partition of unity collocation method based on finite difference for initial-boundary value problems
Meshfree radial basis function (RBF) methods are popular tools used to
numerically solve partial differential equations (PDEs). They take advantage of
being flexible with respect to geometry, easy to implement in higher
dimensions, and can also provide high order convergence. Since one of the main
disadvantages of global RBF-based methods is generally the computational cost
associated with the solution of large linear systems, in this paper we focus on
a localizing RBF partition of unity method (RBF-PUM) based on a finite
difference (FD) scheme. Specifically, we propose a new RBF-PUM-FD collocation
method, which can successfully be applied to solve time-dependent PDEs. This
approach allows to significantly decrease ill-conditioning of traditional
RBF-based methods. Moreover, the RBF-PUM-FD scheme results in a sparse matrix
system, reducing the computational effort but maintaining at the same time a
high level of accuracy. Numerical experiments show performances of our
collocation scheme on two benchmark problems, involving unsteady
convection-diffusion and pseudo-parabolic equations
The nuclear dimension of C*-algebras
We introduce the nuclear dimension of a C*-algebra; this is a noncommutative
version of topological covering dimension based on a modification of the
earlier concept of decomposition rank. Our notion behaves well with respect to
inductive limits, tensor products, hereditary subalgebras (hence ideals),
quotients, and even extensions. It can be computed for many examples; in
particular, it is finite for all UCT Kirchberg algebras. In fact, all classes
of nuclear C*-algebras which have so far been successfully classified consist
of examples with finite nuclear dimension, and it turns out that finite nuclear
dimension implies many properties relevant for the classification program.
Surprisingly, the concept is also linked to coarse geometry, since for a
discrete metric space of bounded geometry the nuclear dimension of the
associated uniform Roe algebra is dominated by the asymptotic dimension of the
underlying space.Comment: 33 page
Stable Generalized Finite Element Method (SGFEM)
The Generalized Finite Element Method (GFEM) is a Partition of Unity Method
(PUM), where the trial space of standard Finite Element Method (FEM) is
augmented with non-polynomial shape functions with compact support. These shape
functions, which are also known as the enrichments, mimic the local behavior of
the unknown solution of the underlying variational problem. GFEM has been
successfully used to solve a variety of problems with complicated features and
microstructure. However, the stiffness matrix of GFEM is badly conditioned
(much worse compared to the standard FEM) and there could be a severe loss of
accuracy in the computed solution of the associated linear system. In this
paper, we address this issue and propose a modification of the GFEM, referred
to as the Stable GFEM (SGFEM). We show that the conditioning of the stiffness
matrix of SGFEM is not worse than that of the standard FEM. Moreover, SGFEM is
very robust with respect to the parameters of the enrichments. We show these
features of SGFEM on several examples.Comment: 51 pages, 4 figure
A Finite Element Method With Singularity Reconstruction for Fractional Boundary Value Problems
We consider a two-point boundary value problem involving a Riemann-Liouville
fractional derivative of order \al\in (1,2) in the leading term on the unit
interval . Generally the standard Galerkin finite element method can
only give a low-order convergence even if the source term is very smooth due to
the presence of the singularity term x^{\al-1} in the solution
representation. In order to enhance the convergence, we develop a simple
singularity reconstruction strategy by splitting the solution into a singular
part and a regular part, where the former captures explicitly the singularity.
We derive a new variational formulation for the regular part, and establish
that the Galerkin approximation of the regular part can achieve a better
convergence order in the , H^{\al/2}(0,1) and -norms
than the standard Galerkin approach, with a convergence rate for the recovered
singularity strength identical with the error estimate. The
reconstruction approach is very flexible in handling explicit singularity, and
it is further extended to the case of a Neumann type boundary condition on the
left end point, which involves a strong singularity x^{\al-2}. Extensive
numerical results confirm the theoretical study and efficiency of the proposed
approach.Comment: 23 pp. ESAIM: Math. Model. Numer. Anal., to appea
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