77 research outputs found
Geometric integration on spheres and some interesting applications
Geometric integration theory can be employed when numerically solving ODEs or
PDEs with constraints. In this paper, we present several one-step algorithms of
various orders for ODEs on a collection of spheres. To demonstrate the
versatility of these algorithms, we present representative calculations for
reduced free rigid body motion (a conservative ODE) and a discretization of
micromagnetics (a dissipative PDE). We emphasize the role of isotropy in
geometric integration and link numerical integration schemes to modern
differential geometry through the use of partial connection forms; this
theoretical framework generalizes moving frames and connections on principal
bundles to manifolds with nonfree actions.Comment: This paper appeared in prin
Numerical integration for high order pyramidal finite elements
We examine the effect of numerical integration on the convergence of high
order pyramidal finite element methods. Rational functions are indispensable to
the construction of pyramidal interpolants so the conventional treatment of
numerical integration, which requires that the finite element approximation
space is piecewise polynomial, cannot be applied. We develop an analysis that
allows the finite element approximation space to include rational functions and
show that despite this complication, conventional rules of thumb can still be
used to select appropriate quadrature methods on pyramids. Along the way, we
present a new family of high order pyramidal finite elements for each of the
spaces of the de Rham complex.Comment: 28 page
High-order finite elements on pyramids: approximation spaces, unisolvency and exactness
We present a family of high-order finite element approximation spaces on a
pyramid, and associated unisolvent degrees of freedom. These spaces consist of
rational basis functions. We establish conforming, exactness and polynomial
approximation properties.Comment: 37 pages, 3 figures. This work was originally in one paper, then
split into two; it has now been recombined into one paper, with substantial
changes from both of its previous form
Fast integral equation methods for the Laplace-Beltrami equation on the sphere
Integral equation methods for solving the Laplace-Beltrami equation on the
unit sphere in the presence of multiple "islands" are presented. The surface of
the sphere is first mapped to a multiply-connected region in the complex plane
via a stereographic projection. After discretizing the integral equation, the
resulting dense linear system is solved iteratively using the fast multipole
method for the 2D Coulomb potential in order to calculate the matrix-vector
products. This numerical scheme requires only O(N) operations, where is the
number of nodes in the discretization of the boundary. The performance of the
method is demonstrated on several examples
Revisiting the Jones eigenproblem in fluid-structure interaction
The Jones eigenvalue problem first described by D.S. Jones in 1983 concerns
unusual modes in bounded elastic bodies: time-harmonic displacements whose
tractions and normal components are both identically zero on the boundary. This
problem is usually associated with a lack of unique solvability for certain
models of fluid-structure interaction. The boundary conditions in this problem
appear, at first glance, to rule out {\it any} non-trivial modes unless the
domain possesses significant geometric symmetries. Indeed, Jones modes were
shown to not be possible in most domains (see article by T. Harg\'e
1990). However, we should in this paper that while the existence of Jones modes
sensitively depends on the domain geometry, such modes {\it do} exist in a
broad class of domains. This paper presents the first detailed theoretical and
computational investigation of this eigenvalue problem in Lipschitz domains. We
also analytically demonstrate Jones modes on some simple geometries
The cellular dynamics of bone remodeling: a mathematical model
The mechanical properties of vertebrate bone are largely determined by a
process which involves the complex interplay of three different cell types.
This process is called {\it bone remodeling}, and occurs asynchronously at
multiple sites in the mature skeleton. The cells involved are bone resorbing
osteoclasts, bone matrix producing osteoblasts and mechanosensing osteocytes.
These cells communicate with each other by means of autocrine and paracrine
signaling factors and operate in complex entities, the so-called bone
multicellular units (BMU). To investigate the BMU dynamics in silico, we
develop a novel mathematical model resulting in a system of nonlinear partial
differential equations with time delays. The model describes the osteoblast and
osteoclast populations together with the dynamics of the key messenger molecule
RANKL and its decoy receptor OPG. Scaling theory is used to address parameter
sensitivity and predict the emergence of pathological remodeling regimes. The
model is studied numerically in one and two space dimensions using finite
difference schemes in space and explicit delay equation solvers in time. The
computational results are in agreement with in vivo observations and provide
new insights into the role of the RANKL/OPG pathway in the spatial regulation
of bone remodeling
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