1,734 research outputs found
General tooth boundary conditions for equation free modelling
We are developing a framework for multiscale computation which enables models
at a ``microscopic'' level of description, for example Lattice Boltzmann, Monte
Carlo or Molecular Dynamics simulators, to perform modelling tasks at
``macroscopic'' length scales of interest. The plan is to use the microscopic
rules restricted to small "patches" of the domain, the "teeth'', using
interpolation to bridge the "gaps". Here we explore general boundary conditions
coupling the widely separated ``teeth'' of the microscopic simulation that
achieve high order accuracy over the macroscale. We present the simplest case
when the microscopic simulator is the quintessential example of a partial
differential equation. We argue that classic high-order interpolation of the
macroscopic field provides the correct forcing in whatever boundary condition
is required by the microsimulator. Such interpolation leads to Tooth Boundary
Conditions which achieve arbitrarily high-order consistency. The high-order
consistency is demonstrated on a class of linear partial differential equations
in two ways: firstly through the eigenvalues of the scheme for selected
numerical problems; and secondly using the dynamical systems approach of
holistic discretisation on a general class of linear \textsc{pde}s. Analytic
modelling shows that, for a wide class of microscopic systems, the subgrid
fields and the effective macroscopic model are largely independent of the tooth
size and the particular tooth boundary conditions. When applied to patches of
microscopic simulations these tooth boundary conditions promise efficient
macroscale simulation. We expect the same approach will also accurately couple
patch simulations in higher spatial dimensions.Comment: 22 page
Evaluating the stability of numerical schemes for fluid solvers in game technology
A variety of numerical techniques have been explored to solve the shallow water equations in real-time water simulations for computer graphics applications. However, determining the stability of a numerical algorithm is a complex and involved task when a coupled set of nonlinear partial differential equations need to be solved. This paper proposes a novel and simple technique to compare the relative empirical stability of finite difference (or any grid-based scheme) algorithms by solving the inviscid Burgersâ equation to analyse their respective breaking times. To exemplify the method to evaluate numerical stability, a range of finite difference schemes is considered. The technique is effective at evaluating the relative stability of the considered schemes and demonstrates that the conservative schemes have superior stability
A FFT-based numerical implementation of mesoscale field dislocation mechanics: Application to two-phase laminates
International audienceIn this paper, we present an enhanced crystal plasticity elasto-viscoplastic fast Fourier transform (EVPFFT) formulation coupled with a phenomenological Mesoscale Field Dislocation Mechanics (MFDM) theory here named MFDM-EVPFFT formulation. In contrast with classic CP-EVPFFT, the model is able to tackle plastic flow and hardening due to polar dislocation density distributions or geometrically necessary dislocations (GNDs) in addition to statistically stored dislocations (SSDs). The model also considers GND mobility through a GND density evolution law numerically solved with a recently developed filtered spectral approach, which is here coupled with stress equilibrium. The discrete Fourier transform method combined with finite differences is applied to solve both lattice incompatibility and Lippmann-Schwinger equations in an augmented Lagrangian numerical scheme. Numerical results are presented for two-phase laminate composites with plastic channels and elastic second phase. It is shown that both GND densities and slip constraint at phase boundaries influence the overall and local hardening behavior. In contrast with the CP-EVPFFT formulation, a channel size effect is predicted on the shear flow stress with the present MFDM-EVPFFT formulation. The size effect originates from the progressive formation of continuous screw GND pileups from phase boundaries to the channel center. The effect of GND mean free path on local and global responses is also examined for the two-phase composite
A high-order compact finite difference scheme and precise integration method based on modified Hopf-Cole transformation for numerical simulation of n-dimensional Burgers' system
This paper introduces a modification of n-dimensional Hopf-Cole
transformation to the n-dimensional Burgers' system. We obtain the
n-dimensional heat conduction equation through the modification of the
Hopf-Cole transformation. Then the high-order exponential time differencing
precise integration method (PIM) based on fourth-order Taylor approximation in
combination with a spatially global sixth-order compact finite difference (CFD)
scheme is presented to solve the equation with high accuracy. Moreover,
coupling with the Strang splitting method, the scheme is extended to
multi-dimensional (two,three-dimensional) Burgers' system, which also possesses
high computational efficiency and accuracy. Several numerical examples verify
the performance and capability of the proposed scheme. Numerical results show
that the proposed method appreciably improves the computational accuracy
compared with the existing numerical method. In addition, the two-dimensional
and three-dimensional examples demonstrate excellent adaptability, and the
numerical simulation results also have very high accuracy in medium Reynolds
numbers
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