The first part of the project concerns the efficiency of numerical methods. We shall increase the method order in space to provide the same accuracy with very much fewer unknowns. Nevertheless, methods suffer a major drawback when dealing with curved boundaries: domain approximations reduce any numerical method to second-order. The DG method uses isoparametric elements, leading to local transformations and additional computational effort. Moreover, curved meshes are mandatory, leading to the development of a complex machinery for meshing. A new technology provides the optimal order was experimented for 2D linear convection diffusion problems. We shall extend the technology for complex 3D geometries with a wide class of boundary conditions.

Very high-order implicit time schemes for non-stationary equations are not unconditionally stable and the time step has to be adjusted with respect to the smallest cell size requiring very small-time step. Formulation based on time-space discretization allows designing an unconditional scheme, relaxing the condition on the time step. In [A2] the authors developed a 2nd-order implicit version for the Maxwell equation, and we seek for an extension using the time- space coupling discretization.

Personal Computers turn to be more powerful by adding three ingredients: the vectorial units, many-core processors, and a hierarchical cache system. There exist techniques to efficiently use the powerful capacities, and efficient numerical schemes have to be designed within such context. The Alternative Direction Implicit (ADI) method splits a 2D or 3D problem into a series of independent 1D problems coupled with an iterative process and was revisited in [P5]. The studies mentioned above develop a new paradigm but without taking advantage of the recent up-to-date hardware capacities. We shall design and implement a new ADI formulation taking advantage of the new hardware architecture. Many 1D problem are solved with threads available on the computer at the same time providing very much faster code.

In the context of fluid-structure interfaces or moving boundaries, the Eulerian formulation is not adequate: the interfaces cut the cells, leading to discrepancies in interface location adding some diffusion. The SPH method is well-adapted for compressible or weakly compressible fluids with moving interface. The convolution kernel has to respect the unit partition to provide a consistent scheme. Unfortunately, consistency is hard to achieve, leading to the so-called "E0-error" and several techniques have been tested to reduce the lack of consistency. On one hand, we propose a new SPH formulation using the finite volume approach but using a new convolution compatible with the space deformation introduced by the particle displacement. The consistency will offer a better approximation since the scheme respects the physics of the problem.

The second part of the project concerns the applications. Thermoplastic polymer manufacturing processes require a large amount of energy for heating/melting, forming and cooling along the production line. Numerical tools are designed to optimize the process based on numerical simulation with 2nd-order finite volume schemes. They provide temperature predictions and guide experimental assessment procedure. A major drawback in the currently codes is the use of highly refined meshes, leading to large computational time. The development of more efficient tools with accuracy improvement and optimal implementation will provide real-time applications that we shall insert in the control loop of the cooling process.

Coastal structures located in tsunami-prone regions shall be designed to withstand the tsunami effects without compromising the safety or wellbeing of the population. The trade-offs between the structural performance and the constructive costs required a reliable characterization of the hydraulic phenomena. Numerical software based on finite volume methods are commonly accepted to tackle the deep ocean shallow flow, while complementary methods, such as the Smoothed Particle Hydrodynamics, are capable to reproduce the more complex tsunami-structure interaction since both dynamic and mechanic aspects of fluids are solved in the same model. Yet, SPH is a non-consistent method and suffers from strong numerical dissipation. The development of a consistent SPH scheme will contribute to establish an efficient tool to numerically assess the impact of tsunami waves against coastal structures

The retina is the visible part of the central nervous system available to be directly imaged using non-invasive optical devices. A common method to assess the progression of diabetic macular edema (DME) or neurodegenerative diseases is to monitor their retinal thickness with optical coherence tomography (OCT). Using OCT, patients with DME can be identified, when contrasted with healthy controls, as they exhibit an increased retinal thickness. However, OCT is still unable to directly assess changes at the cellular level, so it is crucial to study in detail the behaviour of the electromagnetic wave as it travels through the sample. Simulating the full complexity of the retina requires a more rigorous approach that can be achieved by solving Maxwell’s equations. We developed a methodology based in a 3D multilayer Monte Carlo computational model of the human retina. The optical properties of each layer were obtained by solving the Maxwell’s equations for 3D domains representative of small regions of those layers, using a Discontinuous Galerkin Finite Element Method (DG-FEM). We now aim to derive an advanced computational model of OCT imaging in the retina, based on time-space explicit DG-FEM methods, to study normal ageing and neurodegenerative diseases without resorting to morphological alterations.