In this paper, we consider a finite volume approach for modelling multiphase flow coupled to geochemistry in porous media. Reactive multiphase flows are modelled by a highly nonlinear system of degenerate partial differential equations coupled with algebraic and ordinary differential equations. We propose a fully implicit scheme using a direct substitution approach (DSA) implemented in the framework of the parallel open-source platform DuMuX. We focus on the particular case where porosity changes due to mineral dissolution/precipitation are taken into account. This alteration of the porosity can have significant effects on the permeability and the tortuosity. The accuracy and effectiveness of the implementation of permeability/porosity and tortuosity/porosity relationships related to mineral dissolution/precipitation for single-phase and two-phase flows are demonstrated through numerical simulations.

We focus on the numerical solver of unilateral cracks by the Schwarz Method with Total Overlap. The aim is to isolate the treatment at the vicinity of the cracks from other regions of the computational domain. This avoids any direct interaction between specific approximations one may use around the singularities born at the tips of the cracks and more standard methods employed away from the cracks. We apply an iterative sub-structuring technique to capture the small structures by insulating the cracks into patches and making a zoom around each of them. The macro-problem is in turn set on the whole domain. As for the classical Schwarz method, the communication between the micro (local) and macro (global) levels is achieved iteratively through some suitable boundary conditions. The micro problem is fed by Dirichlet data along the (outer) boundary of the patches. The specificity of our approach is that the macro problem inherits transmission conditions. Although they are expressed across the cracks, the final algebraic system to invert is blind to the discontinuities of the solution. In fact, the stiffness matrix turns out to be the one related to a safe domain, as if cracks were closed or the unilateral singularities were switched off. Only the right hand side is affected by what happens at the vicinity of the cracks. This enables users to run one of many efficient algorithms found in the literature to solve the linear macro-problem. In the other hand side, in spite of the still bad conditioning and the non-linearity of the unilateral micro problems, they are reduced in size and may be inverted properly by convex optimization algorithms. A successful convergence analysis of this variant of the Schwarz Method is performed after adapting to the unilateral non-linearity the variational tools developed by P. L. Lions.

We propose a multiscale approach for a nonstandard higher-order PDE based on the $p$(·)-Kirchhoff energy. We first use the topological gradient approach for a semi-linear case in order to detect important objects of the image. We consider a fully nonlinear $p$(·)-Kirchhoff equation with variable-exponent functions that are chosen adaptively based on the map provided by the topological gradient in order to preserve important features of the image. Then, we consider the split Bregman method for the numerical implementation of the proposed model. We compare our model with other classical variational approaches such as the TVL and bi-harmonic restoration models. Finally, we present some numerical results to illustrate the effectiveness of our approach.

In this paper, we use the topological and shape gradient framework, to optimize a current carrying multicables. The geometry of the multicables is modeled as a coated inclusions with different conductivities and the problem we are interested in is the location of the inclusions to get a suitable thermal environnent. We solve numerically the optimization problem using topological and shape gradient strategy. Finally, we present some numerical experiments.

We present a monolithic algorithm for solving fluid-structure interaction. The Updated Lagrangian framework is used for the incompressible neo-hookean structure and Arbitrary Lagrangian Eulerian coordinate is employed for the Navier-Stokes equations. The algorithm uses a global mesh for the fluid-structure domain which is compatible with the fluid-structure interface. At each time step, a non-linear system is solved in a domain corresponding to the precedent time step. It is a semi-implicit algorithm in the sense that the velocity, the pressure are computed implicitly, but the domain is updated explicitly. Using one velocity field defined over the fluid-structure mesh, and globally continuous finite elements, the continuity of the velocity at the interface is automatically verified. The equation of the continuity of the stress at the interface does not appear in this formulation due to action and reaction principle. The stability in time is proved. A second algorithm is introduced where at each time step, only a linear system is solved in order to find the velocity and the pressure. Numerical experiments are presented.

The question of a mathematical representation and theoretical overcoming by optimised therapeutic strategies of drug-induced drug resistance in cancer cell populations is tackled here from the point of view of adaptive dynamics and optimal population growth control, using integro-differential equations. Combined impacts of external continuous-time functions, standing for drug actions, on targets in a plastic (i.e., able to quickly change its phenotype in deadly environmental conditions) cell population model, represent a therapeutical control to be optimised. A justification for the introduction of the adaptive dynamics setting, retaining such plasticity for cancer cell populations, is firstly presented in light of the evolution of multicellular species and disruptions in multicellularity coherence that are characteristics of cancer and of its progression. Finally, open general questions on cancer and evolution in the Darwinian sense are listed, that may open innovative tracks in modelling and treating cancer by circumventing drug resistance. This study sums up results that were presented at the international NUMACH workshop, Mulhouse, France, in July 2018.