Georges Sadaka

Georges Sadaka

FreeFem++ is an open source platform to solve partial differential equations numerically, based on finite element methods. It was developed at the Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris by Frédéric Hecht in collaboration with Olivier Pironneau, Jacques Morice, Antoine Le Hyaric and Kohji Ohtsuka. The FreeFem++ platform has been developed to facilitate teaching and basic research through prototyping. FreeFem++ has an advanced automatic mesh generator, capable of a posteriori mesh adaptation; it has a general purpose elliptic solver interfaced with fast algorithms such as the multi-frontal method UMFPACK, SuperLU . Hyperbolic and parabolic problems are solved by iterative algorithms prescribed by the user with the high level language of FreeFem++. It has several triangular finite elements, including discontinuous elements. For the moment this platform is restricted to the numerical simulations of problems which admit a variational formulation. We will give in the sequel an introduction to FreeFem++ which include the basic of this software. You may find more information throw this link http://www.freefem.org/ff++.

Stéphane Mottelet, Gil Gaullier, Georges Sadaka

Comprehension of metabolic pathways is considerably enhanced by metabolic flux analysis (MFA-ILE) in isotope labeling experiments. The balance equations are given by hundreds of algebraic (stationary MFA) or ordinary differential equations (nonstationary MFA), and reducing the number of operations is therefore a crucial part of reducing the computation cost. The main bottleneck for deterministic algorithms is the computation of derivatives, particularly for nonstationary MFA. In this article we explain how the overall identification process may be speeded up by using the adjoint approach to compute the gradient of the residual sum of squares. The proposed approach shows significant improvements in terms of complexity and computation time when it is compared with the usual (direct) approach. Numerical results are obtained for the central metabolic pathways of Escherichia coli and are validated against reference software in the stationary case. The methods and algorithms described in this paper are included in the sysmetab software package distributed under an Open Source license at http://forge.scilab.org/index.php/p/sysmetab/.

Georges Sadaka

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. 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.

Georges Sadaka

The importance of the study of the propagation of a Tsunami wave came from the complex phenomenon and its natural disasters which represents a major risk for populations. To model this phenomena, we will consider a simplified Boussinesq system of Benjamin, Bona and Mahony type derived by D. Mitsotakis over a flat bottom then over a variable bottom in space and in time and apply this system, first, using a mesh generated using a photo of the Mediterranean sea, second, using a mesh generated using an imported xyz bathymetry for the sea near Java island and then we will consider a realistic example of the Tsunami wave near Java island which happened in 2006.We choose here to use FreeFem++ software which simplifies the construction of the domain, in particular, one of the advantage of FreeFem++ is that we can build a mesh using a photo and we can easily export bathymetric data in order to consider more realistic simulations where a special adapt mesh technique applied for these two methods is detailed in the sequel.