14,699 research outputs found
Stabilized four-node tetrahedron with nonlocal pressure for modeling hyperelastic materials
Non-linear hyperelastic response of reinforced elastomers is modeled using a novel three-dimensional
mixed finite element method with a nonlocal pressure field. The element is unconditionally convergent
and free of spurious pressure modes. Nonlocal pressure is obtained by an implicit gradient technique and
obeys the Helmholtz equation. Physical motivation for this nonlocality is shown. An implicit finite element
scheme with consistent linearization is presented. Finally, several hyperelastic examples are solved to
demonstrate the computational algorithm including the inf–sup and verifications test
The design of conservative finite element discretisations for the vectorial modified KdV equation
We design a consistent Galerkin scheme for the approximation of the vectorial
modified Korteweg-de Vries equation. We demonstrate that the scheme conserves
energy up to machine precision. In this sense the method is consistent with the
energy balance of the continuous system. This energy balance ensures there is
no numerical dissipation allowing for extremely accurate long time simulations
free from numerical artifacts. Various numerical experiments are shown
demonstrating the asymptotic convergence of the method with respect to the
discretisation parameters. Some simulations are also presented that correctly
capture the unusual interactions between solitons in the vectorial setting
Momentum maps for mixed states in quantum and classical mechanics
This paper presents the momentum map structures which emerge in the dynamics
of mixed states. Both quantum and classical mechanics are shown to possess
analogous momentum map pairs. In the quantum setting, the right leg of the pair
identifies the Berry curvature, while its left leg is shown to lead to more
general realizations of the density operator which have recently appeared in
quantum molecular dynamics. Finally, the paper shows how alternative
representations of both the density matrix and the classical density are
equivariant momentum maps generating new Clebsch representations for both
quantum and classical dynamics. Uhlmann's density matrix and Koopman-von
Neumann wavefunctions are shown to be special cases of this construction.Comment: 20 pages; no figures. To appear in J. Geom. Mec
A Hybrid Godunov Method for Radiation Hydrodynamics
From a mathematical perspective, radiation hydrodynamics can be thought of as
a system of hyperbolic balance laws with dual multiscale behavior (multiscale
behavior associated with the hyperbolic wave speeds as well as multiscale
behavior associated with source term relaxation). With this outlook in mind,
this paper presents a hybrid Godunov method for one-dimensional radiation
hydrodynamics that is uniformly well behaved from the photon free streaming
(hyperbolic) limit through the weak equilibrium diffusion (parabolic) limit and
to the strong equilibrium diffusion (hyperbolic) limit. Moreover, one finds
that the technique preserves certain asymptotic limits. The method incorporates
a backward Euler upwinding scheme for the radiation energy density and flux as
well as a modified Godunov scheme for the material density, momentum density,
and energy density. The backward Euler upwinding scheme is first-order accurate
and uses an implicit HLLE flux function to temporally advance the radiation
components according to the material flow scale. The modified Godunov scheme is
second-order accurate and directly couples stiff source term effects to the
hyperbolic structure of the system of balance laws. This Godunov technique is
composed of a predictor step that is based on Duhamel's principle and a
corrector step that is based on Picard iteration. The Godunov scheme is
explicit on the material flow scale but is unsplit and fully couples matter and
radiation without invoking a diffusion-type approximation for radiation
hydrodynamics. This technique derives from earlier work by Miniati & Colella
2007. Numerical tests demonstrate that the method is stable, robust, and
accurate across various parameter regimes.Comment: accepted for publication in Journal of Computational Physics; 61
pages, 15 figures, 11 table
Implementation of a low-mach number modification for high-order finite-volume schemes for arbitrary hybrid unstructured meshes
An implementation of a novel low-mach number treatment for high-order finite-volume schemes using arbitrary hybrid unstructured meshes is presented in this paper. Low-Mach order modifications for Godunov type finite-volume schemes have been implemented successfully for structured and unstructured meshes, however the methods break down for hybrid mesh topologies containing multiple element types. The modification is applied to the UCNS3D finite-volume framework for compressible flow configurations, which have been shown as very capable of handling any type of grid topology. The numerical methods under consideration are the Monotonic Upstream-Centered Scheme for Conservation Laws (MUSCL) and the Weighted Essentially Non-Oscillatory (WENO) schemes for two-dimensional mixed-element type unstructured meshes. In the present study the HLLC Approximate Riemann Solver is used with an explicit TVD Runge-Kutta 3rd-order method due to its excellent scalability. These schemes (up to 5th-order) are applied to well established two-dimensional and three-dimensional test cases. The challenges that occur when applying these methods to low-mach flow configurations is thoroughly analysed and possible improvements and further test cases are suggested
Entropy Stable Finite Volume Approximations for Ideal Magnetohydrodynamics
This article serves as a summary outlining the mathematical entropy analysis
of the ideal magnetohydrodynamic (MHD) equations. We select the ideal MHD
equations as they are particularly useful for mathematically modeling a wide
variety of magnetized fluids. In order to be self-contained we first motivate
the physical properties of a magnetic fluid and how it should behave under the
laws of thermodynamics. Next, we introduce a mathematical model built from
hyperbolic partial differential equations (PDEs) that translate physical laws
into mathematical equations. After an overview of the continuous analysis, we
thoroughly describe the derivation of a numerical approximation of the ideal
MHD system that remains consistent to the continuous thermodynamic principles.
The derivation of the method and the theorems contained within serve as the
bulk of the review article. We demonstrate that the derived numerical
approximation retains the correct entropic properties of the continuous model
and show its applicability to a variety of standard numerical test cases for
MHD schemes. We close with our conclusions and a brief discussion on future
work in the area of entropy consistent numerical methods and the modeling of
plasmas
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