106 research outputs found
Fast and spectrally accurate summation of 2-periodic Stokes potentials
We derive a Ewald decomposition for the Stokeslet in planar periodicity and a
novel PME-type O(N log N) method for the fast evaluation of the resulting sums.
The decomposition is the natural 2P counterpart to the classical 3P
decomposition by Hasimoto, and is given in an explicit form not found in the
literature. Truncation error estimates are provided to aid in selecting
parameters. The fast, PME-type, method appears to be the first fast method for
computing Stokeslet Ewald sums in planar periodicity, and has three attractive
properties: it is spectrally accurate; it uses the minimal amount of memory
that a gridded Ewald method can use; and provides clarity regarding numerical
errors and how to choose parameters. Analytical and numerical results are give
to support this. We explore the practicalities of the proposed method, and
survey the computational issues involved in applying it to 2-periodic boundary
integral Stokes problems
Fast integral equation methods for the modified Helmholtz equation
We present a collection of integral equation methods for the solution to the
two-dimensional, modified Helmholtz equation, u(\x) - \alpha^2 \Delta u(\x) =
0, in bounded or unbounded multiply-connected domains. We consider both
Dirichlet and Neumann problems. We derive well-conditioned Fredholm integral
equations of the second kind, which are discretized using high-order, hybrid
Gauss-trapezoid rules. Our fast multipole-based iterative solution procedure
requires only O(N) or operations, where N is the number of nodes
in the discretization of the boundary. We demonstrate the performance of the
methods on several numerical examples.Comment: Published in Computers & Mathematics with Application
Numerical methods for computing Casimir interactions
We review several different approaches for computing Casimir forces and
related fluctuation-induced interactions between bodies of arbitrary shapes and
materials. The relationships between this problem and well known computational
techniques from classical electromagnetism are emphasized. We also review the
basic principles of standard computational methods, categorizing them according
to three criteria---choice of problem, basis, and solution technique---that can
be used to classify proposals for the Casimir problem as well. In this way,
mature classical methods can be exploited to model Casimir physics, with a few
important modifications.Comment: 46 pages, 142 references, 5 figures. To appear in upcoming Lecture
Notes in Physics book on Casimir Physic
A finite element approach to self-consistent field theory calculations of multiblock polymers
Self-consistent field theory (SCFT) has proven to be a powerful tool for
modeling equilibrium microstructures of soft materials, particularly for
multiblock polymers. A very successful approach to numerically solving the SCFT
set of equations is based on using a spectral approach. While widely
successful, this approach has limitations especially in the context of current
technologically relevant applications. These limitations include non-trivial
approaches for modeling complex geometries, difficulties in extending to
non-periodic domains, as well as non-trivial extensions for spatial adaptivity.
As a viable alternative to spectral schemes, we develop a finite element
formulation of the SCFT paradigm for calculating equilibrium polymer
morphologies. We discuss the formulation and address implementation challenges
that ensure accuracy and efficiency. We explore higher order chain contour
steppers that are efficiently implemented with Richardson Extrapolation. This
approach is highly scalable and suitable for systems with arbitrary shapes. We
show spatial and temporal convergence and illustrate scaling on up to 2048
cores. Finally, we illustrate confinement effects for selected complex
geometries. This has implications for materials design for nanoscale
applications where dimensions are such that equilibrium morphologies
dramatically differ from the bulk phases
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