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This work deals with shape optimization of an elastic body in sliding contact (Signorini) with a rigid foundation. The mechanical problem is written under its augmented Lagrangian formulation, then solved using a classical iterative approach. For practical reasons we are interested in applying the optimization process with respect to an intermediate solution produced by the iterative method. Due to the projection operator involved at each iteration, the iterate solution is not classically shape differentiable. However, using an approach based on directional derivatives, we are able to prove that it is conically differentiable with respect to the shape, and express sufficient conditions for shape differentiability. Finally, from the analysis of the sequence of conical shape derivatives of the iterative process, conditions are established for the convergence to the conical derivative of the original contact problem.

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This paper presents a new algorithm for the numerical solution of the Navier-Stokes equations coupled with the convection-diffusion equation. After establishing convergence of the semidiscrete formulation at each time step, we introduce a new iterative scheme based on a projection method called the coupled prediction scheme. We show that even though the predicted temperature is advected by a velocity prediction which is not necessarily divergence free, the theoretical time accuracy of the global scheme is conserved. From a numerical point of view, this new approach gives a faster and more efficient algorithm compared to the usual fixed-point approaches. Introduction.Heat transfer is an important factor in many fluid dynamics applications. Whenever there is a temperature difference between the fluid and the confining area, heat will be transferred and the flow will be affected in nontrivial ways. Natural convection is such an example in which the driving forces are density variations and gravity (see Jiji [28], for instance). Natural convection flows are observed in different situations such as geophysics, weather, ocean movement and are also exploited in numerous applications such as double-glazed windows, cooling in electronic devices, building insulation, etc.The model is generally described using the Boussinesq approximation. In this approximation, the density of the fluid is assumed to be constant and the gravitational source force (the buoyancy term in the momentum equation) depends on the temperature (Martynenko and Khramtsov [34]).Typically, in the Boussinesq approximation, the coupling between the fluid and the temperature appears through two terms: a source term depending linearly on the temperature, and a convective term based on the velocity of the fluid (see system (2.1)). In this paper we propose a reinforcement of this coupling by adding an explicit dependency to the temperature for the viscosity and the diffusion coefficients. Moreover, since the assumptions on the source term for the momentum equation are not essential (Remark 2.1), we will consider a more general source term. Owing to this departure from the usual Boussinesq equations, the proposed model can be viewed more generally as a thermally coupled Navier-Stokes problem.Thermally coupled incompressible flow problems present two major difficulties requiring special attention: solving the incompressible Navier-Stokes equations on very fine three-dimensional meshes in a reasonable computational time is a difficult task; the strong coupling between the Navier-Stokes and convection-diffusion equations often leads to very complex time dependent dynamics requiring efficient solvers.

Summary We introduce a new flexible mesh adaptation approach to efficiently compute a quantity of interest by the finite element method. Efficiently, we mean that the method provides an evaluation of that quantity up to a predetermined accuracy at a lower computational cost than other classical methods. The central pillar of the method is our scalar error estimator based on sensitivities of the quantity of interest to the residuals. These sensitivities result from the computation of a continuous adjoint problem. The mesh adaptation strategy can drive anisotropic mesh adaptation from a general scalar error contribution of each element. The full potential of our error estimator is then reached. The proposed method is validated by evaluating the lift, the drag, and the hydraulic losses on a 2D benchmark case: the flow around a cylinder at a Reynolds number of 20.

In neural structures with complex geometries, numerical resolution of the Poisson-Nernst-Planck (PNP) equations is necessary to accurately model electrodiffusion. This formalism allows one to describe ionic concentrations and the electric field (even away from the membrane) with arbitrary spatial and temporal resolution which is impossible to achieve with models relying on cable theory. However, solving the PNP equations on complex geometries involves handling intricate numerical difficulties related either to the spatial discretization, temporal discretization or the resolution of the linearized systems, often requiring large computational resources which have limited the use of this approach. In the present paper, we investigate the best ways to use the finite elements method (FEM) to solve the PNP equations on domains with discontinuous properties (such as occur at the membrane-cytoplasm interface). 1) Using a simple 2D geometry to allow comparison with analytical solution, we show that mesh adaptation is a very (if not the most) efficient way to obtain accurate solutions while limiting the computational efforts, 2) We use mesh adaptation in a 3D model of a node of Ranvier to reveal details of the solution which are nearly impossible to resolve with other modelling techniques. For instance, we exhibit a non linear distribution of the electric potential within the membrane due to the non uniform width of the myelin and investigate its impact on the spatial profile of the electric field in the Debye layer.

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The operator splitting approach applied to the Navier-Stokes equations, gave rise to various numerical methods for the simulations of the dynamics of fluids. The separate work of Chorin and Temam on this subject gave birth to the so-called projection methods. The basic projection schemes, either the incremental or non-incremental variant (see [1]) induces an artificial Neumann boundary condition on the pressure. By getting rid of this boundary condition on the pressure, the so-call rotational incremental pressure-correction scheme as proposed by Timmermans et al. [2] for Newtonian fluids with constant viscosity gives a consistent equation for the pressure. In this work we propose a family of projection methods for generalized Newtonian fluids based on an extension of the rotational projection scheme. Called shear rate projections, these methods produces consistent pressure when applied to generalized Newtonian fluids. Accuracy of the methods will be illustrated using a manufactured solution. Numerical experiments for the flow past a cylinder, with a Carreau rheological model, will also be presented.