240 research outputs found
A Space-time Smooth Artificial Viscosity Method For Nonlinear Conservation Laws
We introduce a new methodology for adding localized, space-time smooth,
artificial viscosity to nonlinear systems of conservation laws which propagate
shock waves, rarefactions, and contact discontinuities, which we call the
-method. We shall focus our attention on the compressible Euler equations in
one space dimension. The novel feature of our approach involves the coupling of
a linear scalar reaction-diffusion equation to our system of conservation laws,
whose solution is the coefficient to an additional (and artificial)
term added to the flux, which determines the location, localization, and
strength of the artificial viscosity. Near shock discontinuities, is
large and localized, and transitions smoothly in space-time to zero away from
discontinuities. Our approach is a provably convergent, spacetime-regularized
variant of the original idea of Richtmeyer and Von Neumann, and is provided at
the level of the PDE, thus allowing a host of numerical discretization schemes
to be employed. We demonstrate the effectiveness of the -method with three
different numerical implementations and apply these to a collection of
classical problems: the Sod shock-tube, the Osher-Shu shock-tube, the
Woodward-Colella blast wave and the Leblanc shock-tube. First, we use a
classical continuous finite-element implementation using second-order
discretization in both space and time, FEM-C. Second, we use a simplified WENO
scheme within our -method framework, WENO-C. Third, we use WENO with the
Lax-Friedrichs flux together with the -equation, and call this WENO-LF-C.
All three schemes yield higher-order discretization strategies, which provide
sharp shock resolution with minimal overshoot and noise, and compare well with
higher-order WENO schemes that employ approximate Riemann solvers,
outperforming them for the difficult Leblanc shock tube experiment.Comment: 34 pages, 27 figure
Plasmonic nanoparticle monomers and dimers: From nano-antennas to chiral metamaterials
We review the basic physics behind light interaction with plasmonic
nanoparticles. The theoretical foundations of light scattering on one metallic
particle (a plasmonic monomer) and two interacting particles (a plasmonic
dimer) are systematically investigated. Expressions for effective particle
susceptibility (polarizability) are derived, and applications of these results
to plasmonic nanoantennas are outlined. In the long-wavelength limit, the
effective macroscopic parameters of an array of plasmonic dimers are
calculated. These parameters are attributable to an effective medium
corresponding to a dilute arrangement of nanoparticles, i.e., a metamaterial
where plasmonic monomers or dimers have the function of "meta-atoms". It is
shown that planar dimers consisting of rod-like particles generally possess
elliptical dichroism and function as atoms for planar chiral metamaterials. The
fabricational simplicity of the proposed rod-dimer geometry can be used in the
design of more cost-effective chiral metamaterials in the optical domain.Comment: submitted to Appl. Phys.
Fourier-Galerkin method for 2D solitons of Boussinesq equation
Abstract We develop a Fourier-Galerkin spectral technique for computing the stationary solutions of 2D generalized wave equations. To this end a special complete orthonormal system of functions in L 2 (−∞, ∞) is used for which product formula is available. The exponential rate of convergence is shown. As a featuring example we consider the Proper Boussinesq Equation (PBE) in 2D and obtain the shapes of the stationary propagating localized waves. The technique is thoroughly validated and compared to other numerical results when possible
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 stochastic continuum damage model for dynamic fracture analysis of quasi-brittle materials using asynchronous Spacetime Discontinuous Galerkin (aSDG) method
The microstructural design has an essential effect on the fracture response of brittle materials. We present a stochastic bulk damage formulation to model dynamic brittle fracture. This model is compared with a similar interfacial model for homogeneous and heterogeneous materials. The damage models are rate-dependent, and the corresponding damage evolution includes delay effects. The evolution equation specifies the rate at which damage tends to its quasi-static limit. The relaxation time of the model introduces an intrinsic length scale for dynamic fracture and addresses the mesh sensitivity problem of earlier damage models with much less computational efforts. The ordinary differential form of the damage equation makes this remedy quite simple and enables capturing the loading rate sensitivity of strain-stress response. A stochastic field is defined for material cohesion and fracture strength to involve microstructure effects in the proposed formulations. The statistical fields are constructed through the Karhunen-Loeve (KL) method.An advanced asynchronous Spacetime Discontinuous Galerkin (aSDG) method is used to discretize the final system of coupled equations. Local and asynchronous solution process, linear complexity of the solution versus the number of elements, local recovery of balance properties, and high spatial and temporal orders of accuracy are some of the main advantages of the aSDG method.Several numerical examples are presented to demonstrate mesh insensitivity of the method and the effect of boundary conditions on dynamic fracture patterns. It is shown that inhomogeneity greatly differentiates fracture patterns from those of a homogeneous rock, including the location of zones with maximum damage. Moreover, as the correlation length of the random field decreases, fracture patterns resemble angled-cracks observed in compressive rock fracture. The final results show that a stochastic bulk damage model produces more realistic results in comparison with a homogenizes model
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