Towards learning Lattice Boltzmann collision operators

In this work we explore the possibility of learning from data collision operators for the Lattice Boltzmann Method using a deep learning approach. We compare a hierarchy of designs of the neural network (NN) collision operator and evaluate the performance of the resulting LBM method in reproducing time dynamics of several canonical flows. In the current study, as a first attempt to address the learning problem, the data was generated by a single relaxation time BGK operator. We demonstrate that vanilla NN architecture has very limited accuracy. On the other hand, by embedding physical properties, such as conservation laws and symmetries, it is possible to dramatically increase the accuracy by several orders of magnitude and correctly reproduce the short and long time dynamics of standard fluid flows

The arbitrary order mixed mimetic finite difference method for the diffusion equation

We propose an arbitrary-order accurate mimetic finite difference (MFD) method for the approximation of diffusion problems in mixed form on unstructured polygonal and polyhedral meshes. As usual in the mimetic numerical technology, the method satisfies local consistency and stability conditions, which determines the accuracy and the well-posedness of the resulting approximation. The method also requires the definition of a high-order discrete divergence operator that is the discrete analog of the divergence operator and is acting on the degrees of freedom. The new family of mimetic methods is proved theoretically to be convergent and optimal error estimates for flux and scalar variable are derived from the convergence analysis. A numerical experiment confirms the high-order accuracy of the method in solving diffusion problems with variable diffusion tensor. It is worth mentioning that the approximation of the scalar variable presents a superconvergence effect

The non-conforming virtual element method for the Stokes equations

We present the non-conforming Virtual Element Method (VEM) for the numerical approximation of velocity and pressure in the steady Stokes problem. The pressure is approximated using discontinuous piecewise polynomials, while each component of the velocity is approximated using the nonconforming virtual element space. On each mesh element the local virtual space contains the space of polynomials of up to a given degree, plus suitable non-polynomial functions. The virtual element functions are implicitly defined as the solution of local Poisson problems with polynomial Neumann boundary conditions. As typical in VEM approaches, the explicit evaluation of the non-polynomial functions is not required. This approach makes it possible to construct nonconforming (virtual) spaces for any polynomial degree regardless of the parity, for two-and three-dimensional problems, and for meshes with very general polygonal and polyhedral elements. We show that the non-conforming VEM is inf-sup stable and establish optimal a priori error estimates for the velocity and pressure approximations. Numerical examples confirm the convergence analysis and the effectiveness of the method in providing high-order accurate approximations

Toward learning Lattice Boltzmann collision operators

In this work, we explore the possibility of learning from data collision operators for the Lattice Boltzmann Method using a deep learning approach. We compare a hierarchy of designs of the neural network (NN) collision operator and evaluate the performance of the resulting LBM method in reproducing time dynamics of several canonical flows. In the current study, as a first attempt to address the learning problem, the data were generated by a single relaxation time BGK operator. We demonstrate that vanilla NN architecture has very limited accuracy. On the other hand, by embedding physical properties, such as conservation laws and symmetries, it is possible to dramatically increase the accuracy by several orders of magnitude and correctly reproduce the short and long time dynamics of standard fluid flows

A Numerical Study of One-Step Models of Polymerization: Frontal Versus Bulk Mode

In free-radical polymerization, a monomer-initiator mixture is converted into a polymer. Depending on initial and boundary conditions, free-radical polymerization can occur either in a bulk mode (BP) or in a frontal mode (FP) via a propagating self-sustaining reaction front. The main goal of this paper is to study the role that bulk polymerization plays in frontal polymerization processes for various one-step kinetics models. We use numerical simulations to study the influence of reaction kinetics on one-dimensional frontal polymerization. We show that the long-time behavior of systems modeled with discontinuous distributed kinetics (e.g. step-function kinetics) significantly departs from the long-time behavior of systems modeled with Arrhenius kinetics. The difference is due to slow BP in the initial mixture of reagents, which influences both the speed and the long-time stability of the reaction front. Further, we show that for distributed kinetics a “true” FP is only possible for a steadily propagating, traveling-wave reaction front. When a front propagates in a pulsating mode, we demonstrate the existence of pockets of unreacted monomer behind the front. These pockets evolve via a bulk polymerization mechanism. A mathematical model of one-step free-radical frontal polymerization is identical to the model of gasless combustion, so bulk reactions play a role in the latter context, as well. However, fronts propagate much faster in combustion than in polymerization, and slow bulk reactions in regions ahead of the burning front can generally be neglected