Alain Cartalade

Adjoint state method for fractional diffusion: Parameter identification

Fractional partial differential equations provide models for sub-diffusion, among which the fractal Mobile-Immobile Model (fMIM) is often used to represent solute transport in complex media. The fMIM involves four parameters, among which we have the order of an integro-differential operator that accounts for the possibility for solutes to be sequestered during very long times. To guess fMIM parameters from experiments, an accurate method consists in optimizing an objective function that measures how much model solutions deviate from data. We show that solving an adjoint problem helps accurate computing of the gradient of such an objective function, with respect to the parameters. We illustrate the method by applying it on experimental data issued from tracing tests in porous media.

Identifying Space-Dependent Coefficients and the Order of Fractionality in Fractional Advection–Diffusion Equation

Tracer tests in natural porous media sometimes show abnormalities that suggest considering a fractional variant of the advection–diffusion equation supplemented by a time derivative of non-integer order. We are describing an inverse method for this equation: It finds the order of the fractional derivative and the coefficients that achieve minimum discrepancy between solution and tracer data. Using an adjoint equation divides the computational effort by an amount proportional to the number of freedom degrees, which becomes large when some coefficients depend on space. Method accuracy is checked on synthetical data, and applicability to actual tracer test is demonstrated.

Lattice Boltzmann simulations of 3D crystal growth

A lattice-Boltzmann (LB) scheme, based on the Bhatnagar-Gross-Krook (BGK) collision rules is developed for a phase-field model of alloy solidification in order to simulate the growth of dendrites. The solidification of a binary alloy is considered, taking into account diffusive transport of heat and solute, as well as the anisotropy of the solid-liquid interfacial free energy. The anisotropic terms in the phase-field evolution equation, the phenomenological anti-trapping current (introduced in the solute evolution equation to avoid spurious solute trapping), and the variation of the solute diffusion coefficient between phases, make it necessary to modify the equilibrium distribution functions of the LB scheme with respect to the one used in the standard method for the solution of advection-diffusion equations. The effects of grid anisotropy are removed by using the lattices D3Q15 and D3Q19 instead of D3Q7. The method is validated by direct comparison of the simulation results with a numerical code that uses the finite-difference method. Simulations are also carried out for two different anisotropy functions in order to demonstrate the capability of the method to generate various crystal shapes.

Nonlinear random-walk approach to concentration-dependent contaminant transport in porous media.

We propose a nonlinear random-walk model to describe the dynamics of dense contaminant plumes in porous media. A coupling between concentration and velocity fields is found so that transport displays non-Fickian features. The qualitative behavior of the pollutant spatial profiles and moments is explored with the help of Monte Carlo simulation, within a continuous-time random-walk approach. Model outcomes are then compared with experimental measurements of variable-density contaminant transport in homogeneous and saturated vertical columns.

On anisotropy function in crystal growth simulations using Lattice Boltzmann equation

In this paper, we present the ability of the Lattice Boltzmann (LB) equation, usually applied to simulate fluid flows, to simulate various shapes of crystals. Crystal growth is modeled with a phase-field model for a pure substance, numerically solved with a LB method in 2D and 3D. This study focuses on the anisotropy function that is responsible for the anisotropic surface tension between the solid phase and the liquid phase. The anisotropy function involves the unit normal vectors of the interface, defined by gradients of phase-field. Those gradients have to be consistent with the underlying lattice of the LB method in order to avoid unwanted effects of numerical anisotropy. Isotropy of the solution is obtained when the directional derivatives method, specific for each lattice, is applied for computing the gradient terms. With the central finite differences method, the phase-field does not match with its rotation and the solution is not any more isotropic. Next, the method is applied to simulate simultaneous growth of several crystals, each of them being defined by its own anisotropy function. Finally, various shapes of 3D crystals are simulated with standard and nonstandard anisotropy functions which favor growth in {100}-, {110}- and {111}-directions.