240 research outputs found

    A Space-time Smooth Artificial Viscosity Method For Nonlinear Conservation Laws

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    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 CC-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 C(x,t)C(x,t) 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, C(x,t)C(x,t) 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 CC-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 CC-method framework, WENO-C. Third, we use WENO with the Lax-Friedrichs flux together with the CC-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

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    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

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    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

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    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

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    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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