288 research outputs found
Inertial Coupling Method for particles in an incompressible fluctuating fluid
We develop an inertial coupling method for modeling the dynamics of
point-like 'blob' particles immersed in an incompressible fluid, generalizing
previous work for compressible fluids. The coupling consistently includes
excess (positive or negative) inertia of the particles relative to the
displaced fluid, and accounts for thermal fluctuations in the fluid momentum
equation. The coupling between the fluid and the blob is based on a no-slip
constraint equating the particle velocity with the local average of the fluid
velocity, and conserves momentum and energy. We demonstrate that the
formulation obeys a fluctuation-dissipation balance, owing to the
non-dissipative nature of the no-slip coupling. We develop a spatio-temporal
discretization that preserves, as best as possible, these properties of the
continuum formulation. In the spatial discretization, the local averaging and
spreading operations are accomplished using compact kernels commonly used in
immersed boundary methods. We find that the special properties of these kernels
make the discrete blob a particle with surprisingly physically-consistent
volume, mass, and hydrodynamic properties. We develop a second-order
semi-implicit temporal integrator that maintains discrete
fluctuation-dissipation balance, and is not limited in stability by viscosity.
Furthermore, the temporal scheme requires only constant-coefficient Poisson and
Helmholtz linear solvers, enabling a very efficient and simple FFT-based
implementation on GPUs. We numerically investigate the performance of the
method on several standard test problems...Comment: Contains a number of corrections and an additional Figure 7 (and
associated discussion) relative to published versio
Supraconservative finite-volume methods for the Euler equations of subsonic compressible flow
It has been found advantageous for finite-volume discretizations of flow equations to possess additional (secondary) invariants, next to the (primary) invariants from the constituting conservation laws. The paper presents general (necessary and sufficient) requirements for a method to convectively preserve discrete kinetic energy. The key ingredient is a close discrete consistency between the convective term in the momentum equation and the terms in the other conservation equations (mass, internal energy). As examples, the Euler equations for subsonic (in)compressible flow are discretized with such supra-conservative finite-volume methods on structured as well as unstructured grids
Variational Time Integrators in Computational Solid Mechanics
This thesis develops the theory and implementation of variational integrators for computational solid mechanics problems, and to some extent, for fluid mechanics problems as well. Variational integrators for finite dimensional mechanical systems are succinctly reviewed, and used as the foundations for the extension to continuum systems. The latter is accomplished by way of a space-time formulation for Lagrangian continuum mechanics that unifies the derivation of the balance of linear momentum, energy and configurational forces, all of them as Euler-Lagrange equations of an extended Hamilton's principle. In this formulation, energy conservation and the path independence of the J- and L-integrals are conserved quantities emanating from Noether's theorem. Variational integrators for continuum mechanics are constructed by mimicking this variational structure, and a discrete Noether's theorem for rather general space-time discretizations is presented. Additionally, the algorithms are automatically (multi)symplectic, and the (multi)symplectic form is uniquely defined by the theory. For instance, in nonlinear elastodynamics the algorithms exactly preserve linear and angular momenta, whenever the continuous system does.
A class of variational algorithms is constructed, termed asynchronous variational integrators (AVI), which permit the selection of independent time steps in each element of a finite element mesh, and the local time steps need not bear an integral relation to each other. The conservation properties of both synchronous and asynchronous variational integrators are discussed in detail. In particular, AVI are found to nearly conserve energy both locally and globally, a distinguishing feature of variational integrators. The possibility of adapting the elemental time step to exactly satisfy the local energy balance equation, obtained from the extended variational principle, is analyzed. The AVI are also extended to include dissipative systems. The excellent accuracy, conservation and convergence characteristics of AVI are demonstrated via selected numerical examples, both for conservative and dissipative systems. In these tests AVI are found to result in substantial speedups, at equal accuracy, relative to explicit Newmark.
In elastostatics, the variational structure leads to the formulation of discrete path-independent integrals and a characterization of the configurational forces acting in discrete systems. A notable example is a discrete, path-independent J-integral at the tip of a crack in a finite element mesh.</p
Staggered Schemes for Fluctuating Hydrodynamics
We develop numerical schemes for solving the isothermal compressible and
incompressible equations of fluctuating hydrodynamics on a grid with staggered
momenta. We develop a second-order accurate spatial discretization of the
diffusive, advective and stochastic fluxes that satisfies a discrete
fluctuation-dissipation balance, and construct temporal discretizations that
are at least second-order accurate in time deterministically and in a weak
sense. Specifically, the methods reproduce the correct equilibrium covariances
of the fluctuating fields to third (compressible) and second (incompressible)
order in the time step, as we verify numerically. We apply our techniques to
model recent experimental measurements of giant fluctuations in diffusively
mixing fluids in a micro-gravity environment [A. Vailati et. al., Nature
Communications 2:290, 2011]. Numerical results for the static spectrum of
non-equilibrium concentration fluctuations are in excellent agreement between
the compressible and incompressible simulations, and in good agreement with
experimental results for all measured wavenumbers.Comment: Submitted. See also arXiv:0906.242
A novel structure preserving semi-implicit finite volume method for viscous and resistive magnetohydrodynamics
In this work we introduce a novel semi-implicit structure-preserving
finite-volume/finite-difference scheme for the viscous and resistive equations
of magnetohydrodynamics (MHD) based on an appropriate 3-split of the governing
PDE system, which is decomposed into a first convective subsystem, a second
subsystem involving the coupling of the velocity field with the magnetic field
and a third subsystem involving the pressure-velocity coupling. The nonlinear
convective terms are discretized explicitly, while the remaining two subsystems
accounting for the Alfven waves and the magneto-acoustic waves are treated
implicitly. The final algorithm is at least formally constrained only by a mild
CFL stability condition depending on the velocity field of the pure
hydrodynamic convection. To preserve the divergence-free constraint of the
magnetic field exactly at the discrete level, a proper set of overlapping dual
meshes is employed. The resulting linear algebraic systems are shown to be
symmetric and therefore can be solved by means of an efficient standard
matrix-free conjugate gradient algorithm. One of the peculiarities of the
presented algorithm is that the magnetic field is defined on the edges of the
main grid, while the electric field is on the faces. The final scheme can be
regarded as a novel shock-capturing, conservative and structure preserving
semi-implicit scheme for the nonlinear viscous and resistive MHD equations.
Several numerical tests are presented to show the main features of our novel
solver: linear-stability in the sense of Lyapunov is verified at a prescribed
constant equilibrium solution; a 2nd-order of convergence is numerically
estimated; shock-capturing capabilities are proven against a standard set of
stringent MHD shock-problems; accuracy and robustness are verified against a
nontrivial set of 2- and 3-dimensional MHD problems.Comment: 44 pages, 22 figures, 2 table
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