Section: Scientific Foundations
The main activities of Magique-3D in modeling are the derivation and the analysis of models that are based on mathematical physics and are suggested by geophysical problems. In particular, Magique-3D considers equations of interest for the oil industry and focus on the development and the analysis of numerical models which are well-adapted to solve quickly and accurately problems set in very large or unbounded domains as it is generally the case in geophysics.
High-Order Schemes in Space and Time. Using the full wave equation for migration implies very high computational burdens, in order to get high resolution images. Indeed, to improve the accuracy of the numerical solution, one must considerably reduce the space step, which is the distance between two points of the mesh representing the computational domain. Obviously this results in increasing the number of unknowns of the discrete problem. Besides, the time step, whose value fixes the number of required iterations for solving the evolution problem, is linked to the space step through the CFL (Courant-Friedrichs-Levy) condition. The CFL number defines an upper bound for the time step in such a way that the smaller the space step is, the higher the numbers of iterations (and of multiplications by the stiffness matrix) will be. The method that we proposed in  allows for the use of local time-step, adapted to the various sizes of the cells and we recently extended it to deal with -adaptivity  . However, this method can not yet handle dissipation terms, which prevents us for using absorbing boundary conditions or Perfectly Matched Layers (PML). To overcome this difficulty, we will first tackle the problem to used the modified equation technique  ,  ,  with dissipation terms, which is still an open problem.
We are also considering an alternative approach to obtain high-order schemes. The main idea is to apply first the time discretization thanks to the modified equation technique and after to consider the space discretization. Our approach involves -harmonic operators, which can not be discretized by classical finite elements. For the discretization of the biharmonic operator in an homogeneous acoustic medium, both C1 finite elements (such as the Hermite ones) and Discontinuous Galerkin Finite Elements (DGFE) can be used while in a discontinuous medium, or for higher-order operators, DGFE should be preferred  . This new method seems to be well-adapted to -adaptivity. Therefore, we now want to couple it to our local time-stepping method in order to deal with -adaptivity both in space and in time. We will then carry out theoretical and numerical comparisons between this technique and the classical modified equation scheme.
Once we have performant -adaptive techniques, it will be necessary to obtain error-estimators. Since we consider huge domain and complex topography, the remeshing of the domain at each time-step is impossible. One solution would be to remesh the domain for instance each 100 time steps, but this could also hamper the efficiency of the computation. Another idea is to consider only adaptivity, since in this case there is no need to remesh the domain.
Mixed hybrid finite element methods for the wave equation. The new mixed-hybrid-like method for the solution of Helmholtz problems at high frequency we have built enjoys the three following important properties: (1) unlike classical mixed and hybrid methods, the method we proposed is not subjected to an inf-sup condition. Therefore, it does not involve numerical instabilities like the ones that have been observed for the DGM method proposed by Farhat and his collaborators  ,  . We can thus consider a larger class of discretization spaces both for the primal and the dual variables. Hence we can use unstructured meshes, which is not possible with DGM method (2) the method requires one to solve Helmholtz problems which are set inside the elements of the mesh and are solved in parallel(3) the method requires to solve a system whose unknowns are Lagrange multipliers defined at the interfaces of the elements of the mesh and, unlike a DGM, the system is hermitian and positive definite. Hence we can use existing numerical methods such as the gradient conjugate method. We intend to continue to work on this subject and our objectives can be described following three tasks: (1) Follow the numerical comparison of performances of the new methods with the ones of DGM. We aim at considering high order elements such as R16-4, R32-8, ...; (2)Evaluate the performance of the method in case of unstructured meshes. This analysis is very important from a practical point of view but also because it has been observed that the DGM deteriorates significantly when using unstructured meshes; (3) Extend the method to the 3D case. This is the ultimate objective of this work since we will then be able to consider applications.
Obviously the study we propose will contain a mathematical analysis of the method we propose. The analysis will be done in the same time and we aim at establishing a priori and a posteriori estimates, the last being very important in order to adopt a solution strategy based on adaptative meshes.
Boundary conditions. The construction of efficient absorbing conditions is very important for solving wave equations, which are generally set in unbounded or very large domains. The efficiency of the conditions depends on the type of waves which are absorbed. Classical conditions absorb propagating waves but recently new conditions have been derived for both propagating and evanescent waves in the case of flat boundaries. MAGIQUE-3D would like to develop new absorbing boundary conditions whose derivation is based on the full factorization of the wave equation using pseudodifferential calculus. By this way, we can take the complete propagation phenomenon into account which means that the boundary condition takes propagating, grazing and evanescent waves into account, and then the absorption is optimized. Moreover our approach can be applied to arbitrarily-shaped regular surfaces.
We intend to work on the development of interface conditions that can be used to model rough interfaces. One approach, already applied in electromagnetism  , consists in using homogenization methods which describes the rough surface by an equivalent transmission condition. We propose to apply it to the case of elastodynamic equations written as a first-order system. In particular, it would be very interesting to investigate if the rigorous techniques that have been used in  ,  can be applied to the theory of elasticity. This type of investigations could be a way for MAGIQUE-3D to consider medical applications where rough interfaces are often involved. Indeed, we would like to work on the modelling and the numerical simulation of ultrasonic propagation and its interaction with partially contacting interfaces, for instance bone/titanium in the context of an application to dentures, in collaboration with G. Haiat (University of Paris 7).
In the context of wave propagation problems, we are investigating physical problems which involves multiple scales. Due to the presence of boundary layers (and/or thin layers, rough interfaces, geometric singularities), the direct numerical simulation (DNS) of these phenomenas involves a large numbers of degrees of freedom and high performance computing is required. The aim of this work is to develop credible alternatives to the DNS approach.
Performing a multi-scale asymptotic analysis, we derive approximate models whose solution can be computed for a low computational cost. We study these approximate models mathematically (well-posedness, uniform error estimates) and numerically (we compare the solution of these approximate models to the solution of the initial model computed with high performance computating).
We are mostly interested in the following problems.
Eddy current modeling in the context of electrothermic applications for the design of electromagnetic devices in collaboration with laboratories Ampre, Laplace, INRIA Team MC2, IRMAR, and F.R.S.-FNRS;
ultrasonic wave propagation through bone-titanium media in medicine in collaboration with INRIA Team MC2, and MSME;
asymptotic modeling of multi perforate plates in turbo reactors in collaboration with Cerfacs, INSA-Toulouse, Onera and Snecma in the framework of the ANR APAM.
Nonlinear problems in fluid dynamics. In order to model heat transfers, fluid-solid interactions, in particular landslides and tsunamis induced by earthquakes, tremors induced by fluid motions in volcanoes, sharp solid-to-fluid transitions in some planets, it is of crucial importance to develop efficient parallel solvers on multicore/multi-processor supercomputing platforms. High order finite volumes introducing compact schemes or spectral-like integrations as well as high order finite elements and their related high order boundary conditions are needed to take into account, at the same time, discontinuities in geological structures, sharp variations and shocks in fluid velocities and properties (density, pressure and temperature), and the coupling between both codes. Discrete Galerkin techniques, spectral finite volumes or finite-volume techniques should be taken into account in compact schemes in order to reduce drastically the memory storage involved and compute larger models. Viscous compressible and incompressible codes need to be solved using non-conforming meshes between solid and fluid, and large linear systems need to be solved on very huge multi-CPU/multi-GPU supercomputers. Moving meshes close to the interface between solids and fluids should be taken into account by dynamic or adaptive remeshing. Furthermore we developped PML for the full compressible Navier-Stokes system of equations  using finite-differences discretization in curvilinear coordinates and we are planning to extend PML conditions to both compressible and incompressible viscous flows in the context of high order finite volumes or Discontinuous galerkin methods.
Another direction that we would like to consider would be the use of solitons in nonlinear problems. Indeed, a soliton is an interesting tool for modeling and explaining some nonlinear phenomena. For example tsumanis are sometimes explained by the emergence of solitons created by earth tremor. Strain solitons can be also used to explain the propagation of breaking in solids  . Therefore it would be interesting to investigate more this issue.