Volume 20, Issue 3-4

- The trace spaces H 1/2 and H 1/2 00 play a key role in the FETI and mortar families of domain decomposition methods. However, a direct numerical evaluation of these norms is usually avoided. On the other hand, and for stability issues, the subspace of functions for which their jumps across the interfaces of neighbouring subdomains belong to these trace spaces yields a more suitable framework than the standard broken Sobolev space. Finally, the nullity of these jumps is usually imposed via Lagrange multipliers and using the pairing of the trace spaces with their duals. A direct computation of these pairings can be performed using the Riesz-canonical isometry. In this work we consider all these ingredients and introduce a domain decomposition method that falls into the FETI-DP mortar family. The application is to the incompressible Stokes problem and we see that continuous bounds are replicated at the discrete level. As a consequence, no stabilization is required. Some numerical tests are finally presented.

- This paper describes an implementation in freefem++ of a mimetic scheme due to Yu.Kuznetsov. This is done by a minor modification of the program which implements the totally discontinuous Raviart-Thomas element RT0 because both elements have similar degrees of freedom. The method is tested in combination with characteristic-Galerkin unwinding on the rotating hill problem.

- The Arbitrary Lagrangian-Eulerian framework allows to compute free surface flows with the Finite Element functions defined on a fittedmesh which follows the globalmotion of the fluid domain. We describe here how freefem++ can be used to implement this method, and we provide two and three dimensional illustrations in the context of water waves.

- Heat treatment is the controlled heating and cooling of metals used to alter their physical and mechanical properties. This is a very enabling process that can facilitate machining, formability and restore ductility, after a cold working operation, improving the product’s performances by increasing its strength or other desirable characteristics. This paper summarizes a mathematical model for the induction-conduction hardening of a specific steel workpiece: the steering rack of an automobile. The whole formulation takes into account both electromagnetic effects that lead to heating the workpiece and the corresponding thermomechanical effects that cause the hardening, Joule’s heating being the main responsible of the coupling between them. Some numerical simulations are presented; these numerical results have been computed using the freefem++ software.

-The purpose of this paper is to explain the phenomenon of symmetry breaking for optimal functions in functional inequalities by the numerical computations of some well chosen solutions of the corresponding Euler-Lagrange equations. For many of those inequalities it was believed that the only source of symmetry breaking would be the instability of the symmetric optimizer in the class of all admissible functions. But recently, it was shown by an indirect argument that for some Caffarelli-Kohn-Nirenberg inequalities this conjecture was not true. In order to understand this new symmetry breaking mechanism we have computed the branch of minimal solutions for a simple problem. A reparametrization of this branch allows us to build a scenario for the new phenomenon of symmetry breaking. The computations have been performed using freefem++.

-This is a short presentation of the freefem++ software. In Section 1, we recall most of the characteristics of the software, In Section 2, we recall how to to build the weak form of a partial differential equation (PDE) from the strong form. In the 3 last sections, we present different examples and tools to illustrated the power of the software. First we deal with mesh adaptation for problems in two and three dimension, second, we solve numerically a problem with phase change and natural convection, and the finally to show the possibilities for HPC we solve a Laplace equation by a Schwarz domain decomposition problem on parallel computer.

Motivated by experiments on the acoustics of submerged granular media, we have developed a new fictitious domain method that combines the acoustic wave equation with the equations of motion of a rigid body. The coupling is realized through the natural boundary conditions of the wave equation on the surface of the body and enforced with boundary Lagrange multipliers. The method is validated using analytic solutions for a movable rigid sphere in a sound wave. Results show that the method is promising because numerical results and analytic solutions are close for long wavelengths, although some deviations occur when the wavelength is smaller to the sphere radius. This method permits us to use a discrete element method for particle motion and interaction, and finite elements for wave propagation in the fluid.

- In this document, we present a parallel implementation in freefem++ of scalable two-level domain decomposition methods. Numerical studies with highly heterogeneous problems are then performed on large clusters in order to assert the performance of our code.

-We consider here different family of Boussinesq systems in two space dimensions. These systems approximate the three-dimensional Euler equations and consist of three coupled nonlinear dispersive wave equations that describe propagation of long surface waves of small amplitude in ideal fluids over a horizontal bottom and which was studied in [7,9,10].We present here a freefem++ code aimed at solving numerically these systems where a discretization using P1 finite element for these systems was taken in space and a second order Runge-Kutta scheme in time.We give the detail of our code where we use a mesh adaptation technique. An optimization of the used algorithm is done and a comparison of the solution for different Boussinesq family is done too. The results we obtained agree with those of the literature.

- Vortex rings are generated in numerous practical applications (biological propulsion, internal combustion engines) by the sudden injection of a fluid. Measurements of such flows often provide partial information, generally limited to the exterior of the vortex ring. We propose in this paper a new algorithm for the reconstruction of the stream function inside a bounded domain, where a vortex ring is supposed to form.We derive a quasi-Newton Broyden- Fletcher-Goldfarb-Shanno (BFGS) algorithm for this problem, using a cost function that describes the matching between the reconstructed field and given values of normal derivatives on the boundary of the domain. The algorithm computes the best parameters of the vortex model that is used to describe the vorticity distribution inside the vortex ring. Several vortex models are considered. The method is implemented in freefem++ using a P1 finite-element spatial discretization. Validation tests using analytical models (Hill’s spherical vortex) and Navier-Stokes direct numerical simulation data show good agreement between reconstructed and reference fields.