17,472 research outputs found
Direct and Inverse Computational Methods for Electromagnetic Scattering in Biological Diagnostics
Scattering theory has had a major roll in twentieth century mathematical
physics. Mathematical modeling and algorithms of direct,- and inverse
electromagnetic scattering formulation due to biological tissues are
investigated. The algorithms are used for a model based illustration technique
within the microwave range. A number of methods is given to solve the inverse
electromagnetic scattering problem in which the nonlinear and ill-posed nature
of the problem are acknowledged.Comment: 61 pages, 5 figure
Immersed Boundary Smooth Extension: A high-order method for solving PDE on arbitrary smooth domains using Fourier spectral methods
The Immersed Boundary method is a simple, efficient, and robust numerical
scheme for solving PDE in general domains, yet it only achieves first-order
spatial accuracy near embedded boundaries. In this paper, we introduce a new
high-order numerical method which we call the Immersed Boundary Smooth
Extension (IBSE) method. The IBSE method achieves high-order accuracy by
smoothly extending the unknown solution of the PDE from a given smooth domain
to a larger computational domain, enabling the use of simple Cartesian-grid
discretizations (e.g. Fourier spectral methods). The method preserves much of
the flexibility and robustness of the original IB method. In particular, it
requires minimal geometric information to describe the boundary and relies only
on convolution with regularized delta-functions to communicate information
between the computational grid and the boundary. We present a fast algorithm
for solving elliptic equations, which forms the basis for simple, high-order
implicit-time methods for parabolic PDE and implicit-explicit methods for
related nonlinear PDE. We apply the IBSE method to solve the Poisson, heat,
Burgers', and Fitzhugh-Nagumo equations, and demonstrate fourth-order pointwise
convergence for Dirichlet problems and third-order pointwise convergence for
Neumann problems
High-order implicit palindromic discontinuous Galerkin method for kinetic-relaxation approximation
We construct a high order discontinuous Galerkin method for solving general
hyperbolic systems of conservation laws. The method is CFL-less, matrix-free,
has the complexity of an explicit scheme and can be of arbitrary order in space
and time. The construction is based on: (a) the representation of the system of
conservation laws by a kinetic vectorial representation with a stiff relaxation
term; (b) a matrix-free, CFL-less implicit discontinuous Galerkin transport
solver; and (c) a stiffly accurate composition method for time integration. The
method is validated on several one-dimensional test cases. It is then applied
on two-dimensional and three-dimensional test cases: flow past a cylinder,
magnetohydrodynamics and multifluid sedimentation
Fast and Accurate Computation of Time-Domain Acoustic Scattering Problems with Exact Nonreflecting Boundary Conditions
This paper is concerned with fast and accurate computation of exterior wave
equations truncated via exact circular or spherical nonreflecting boundary
conditions (NRBCs, which are known to be nonlocal in both time and space). We
first derive analytic expressions for the underlying convolution kernels, which
allow for a rapid and accurate evaluation of the convolution with
operations over successive time steps. To handle the onlocality in space,
we introduce the notion of boundary perturbation, which enables us to handle
general bounded scatters by solving a sequence of wave equations in a regular
domain. We propose an efficient spectral-Galerkin solver with Newmark's time
integration for the truncated wave equation in the regular domain. We also
provide ample numerical results to show high-order accuracy of NRBCs and
efficiency of the proposed scheme.Comment: 22 pages with 9 figure
A simple and efficient BEM implementation of quasistatic linear visco-elasticity
A simple, yet efficient procedure to solve quasistatic problems of special
linear visco-elastic solids at small strains with equal rheological response in
all tensorial components, utilizing boundary element method (BEM), is
introduced. This procedure is based on the implicit discretisation in time (the
so-called Rothe method) combined with a simple "algebraic" transformation of
variables, leading to a numerically stable procedure (proved explicitly by
discrete energy estimates), which can be easily implemented in a BEM code to
solve initial-boundary value visco-elastic problems by using the Kelvin
elastostatic fundamental solution only. It is worth mentioning that no inverse
Laplace transform is required here. The formulation is straightforward for both
2D and 3D problems involving unilateral frictionless contact. Although the
focus is to the simplest Kelvin-Voigt rheology, a generalization to Maxwell,
Boltzmann, Jeffreys, and Burgers rheologies is proposed, discussed, and
implemented in the BEM code too. A few 2D and 3D initial-boundary value
problems, one of them with unilateral frictionless contact, are solved
numerically
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