219 research outputs found
Initialization of lattice Boltzmann models with the help of the numerical Chapman-Enskog expansion
We extend the applicability of the numerical Chapman-Enskog expansion as a
lifting operator for lattice Boltzmann models to map density and momentum to
distribution functions. In earlier work [Vanderhoydonc et al. Multiscale Model.
Simul. 10(3): 766-791, 2012] such an expansion was constructed in the context
of lifting only the zeroth order velocity moment, namely the density. A lifting
operator is necessary to convert information from the macroscopic to the
mesoscopic scale. This operator is used for the initialization of lattice
Boltzmann models. Given only density and momentum, the goal is to initialize
the distribution functions of lattice Boltzmann models. For this
initialization, the numerical Chapman-Enskog expansion is used in this paper.Comment: arXiv admin note: text overlap with arXiv:1108.491
Numerical extraction of a macroscopic pde and a lifting operator from a lattice Boltzmann model
Lifting operators play an important role in starting a lattice Boltzmann
model from a given initial density. The density, a macroscopic variable, needs
to be mapped to the distribution functions, mesoscopic variables, of the
lattice Boltzmann model. Several methods proposed as lifting operators have
been tested and discussed in the literature. The most famous methods are an
analytically found lifting operator, like the Chapman-Enskog expansion, and a
numerical method, like the Constrained Runs algorithm, to arrive at an implicit
expression for the unknown distribution functions with the help of the density.
This paper proposes a lifting operator that alleviates several drawbacks of
these existing methods. In particular, we focus on the computational expense
and the analytical work that needs to be done. The proposed lifting operator, a
numerical Chapman-Enskog expansion, obtains the coefficients of the
Chapman-Enskog expansion numerically. Another important feature of the use of
lifting operators is found in hybrid models. There the lattice Boltzmann model
is spatially coupled with a model based on a more macroscopic description, for
example an advection-diffusion-reaction equation. In one part of the domain,
the lattice Boltzmann model is used, while in another part, the more
macroscopic model. Such a hybrid coupling results in missing data at the
interfaces between the different models. A lifting operator is then an
important tool since the lattice Boltzmann model is typically described by more
variables than a model based on a macroscopic partial differential equation.Comment: submitted to SIAM MM
Spatially Distributed Stochastic Systems: equation-free and equation-assisted preconditioned computation
Spatially distributed problems are often approximately modelled in terms of
partial differential equations (PDEs) for appropriate coarse-grained quantities
(e.g. concentrations). The derivation of accurate such PDEs starting from finer
scale, atomistic models, and using suitable averaging, is often a challenging
task; approximate PDEs are typically obtained through mathematical closure
procedures (e.g. mean-field approximations). In this paper, we show how such
approximate macroscopic PDEs can be exploited in constructing preconditioners
to accelerate stochastic simulations for spatially distributed particle-based
process models. We illustrate how such preconditioning can improve the
convergence of equation-free coarse-grained methods based on coarse
timesteppers. Our model problem is a stochastic reaction-diffusion model
capable of exhibiting Turing instabilities.Comment: 8 pages, 6 figures, submitted to Journal of Chemical Physic
Coarse Stability and Bifurcation Analysis Using Stochastic Simulators: Kinetic Monte Carlo Examples
We implement a computer-assisted approach that, under appropriate conditions,
allows the bifurcation analysis of the coarse dynamic behavior of microscopic
simulators without requiring the explicit derivation of closed macroscopic
equations for this behavior. The approach is inspired by the so-called
time-step per based numerical bifurcation theory. We illustrate the approach
through the computation of both stable and unstable coarsely invariant states
for Kinetic Monte Carlo models of three simple surface reaction schemes. We
quantify the linearized stability of these coarsely invariant states, perform
pseudo-arclength continuation, detect coarse limit point and coarse Hopf
bifurcations and construct two-parameter bifurcation diagrams.Comment: 26 pages, 5 figure
Physics-Informed Deep Neural Operator Networks
Standard neural networks can approximate general nonlinear operators,
represented either explicitly by a combination of mathematical operators, e.g.,
in an advection-diffusion-reaction partial differential equation, or simply as
a black box, e.g., a system-of-systems. The first neural operator was the Deep
Operator Network (DeepONet), proposed in 2019 based on rigorous approximation
theory. Since then, a few other less general operators have been published,
e.g., based on graph neural networks or Fourier transforms. For black box
systems, training of neural operators is data-driven only but if the governing
equations are known they can be incorporated into the loss function during
training to develop physics-informed neural operators. Neural operators can be
used as surrogates in design problems, uncertainty quantification, autonomous
systems, and almost in any application requiring real-time inference. Moreover,
independently pre-trained DeepONets can be used as components of a complex
multi-physics system by coupling them together with relatively light training.
Here, we present a review of DeepONet, the Fourier neural operator, and the
graph neural operator, as well as appropriate extensions with feature
expansions, and highlight their usefulness in diverse applications in
computational mechanics, including porous media, fluid mechanics, and solid
mechanics.Comment: 33 pages, 14 figures. arXiv admin note: text overlap with
arXiv:2204.00997 by other author
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Reactive Flows in Deformable, Complex Media
Many processes of highest actuality in the real life are described through systems of equations posed in complex domains. Of particular interest is the situation when the domain is changing in time, undergoing deformations that depend on the unknown quantities of the model. Such kind of problems are encountered as mathematical models in the subsurface, material science, or biological systems.The emerging mathematical models account for various processes at different scales, and the key issue is to integrate the domain deformation in the multi-scale context. The focus in this workshop was on novel techniques and ideas in the mathematical modelling, analysis, the numerical discretization and the upscaling of problems as described above
A time-parallel framework for coupling finite element and lattice Boltzmann methods
International audienceIn this work we propose a new numerical procedure for the simulation of time-dependent problems based on the coupling between the finite element method and the lattice Boltzmann method. The procedure is based on the Parareal paradigm and allows to couple efficiently the two numerical methods, each one working with its own grid size and time-step size. The motivations behind this approach are manifold. Among others, we have that one technique may be more efficient, or physically more appropriate or less memory consuming than the other depending on the target of the simulation and/or on the sub-region of the computational domain. Furthermore, the coupling with finite element method may circumvent some difficulties inherent to lattice Boltzmann discretization, for some domains with complex boundaries, or for some boundary conditions. The theoretical and numerical framework is presented for parabolic equations, in order to describe and validate numerically the methodology in a simple situation
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