Stéphane Clain

Stéphane Clain, Emmanuel Franck, Victor Michel-Dansac

We present an adaptation of the so-called structural method \cite{CMM23} for Hamiltonian systems, and redesign the method for this specific context, which involves two coupled differential systems. Structural schemes decompose the problem into two sets of equations: the physical equations, which describe the local dynamics of the system, and the structural equations, which only involve the discretization on a very compact stencil. They have desirable properties, such as unconditional stability or high-order accuracy. We first give a general description of the scheme for the scalar case (which corresponds to e.g. spring-mass interactions or pendulum motion), before extending the technique to the vector case (treating e.g. the $n$-body system). The scheme is also written in the case of a non-separable system (e.g. a charged particle in an electromagnetic field). We give numerical evidence of the method's efficiency, its capacity to preserve invariant quantities such as the total energy, and draw comparisons with the traditional symplectic methods.

Armando Coco, Alessandro Coclite, Stéphane Clain et al.

Unfitted boundary methods are widely used to numerically solve partial differential equations (PDEs) on irregular domains, avoiding the computational burden of generating boundary-conforming grids. In the finite-difference framework, structured Cartesian grids offer advantages such as ease of implementation and efficient parallelization, while geometry is represented implicitly, for instance, through level-set functions. In this setting, ghost point methods are commonly employed to enforce boundary conditions by introducing additional relations between interior and ghost nodes. However, constructing these relations becomes challenging for high-order accurate discretizations, which often rely on wide stencils that can reduce computational efficiency and degrade performance in large-scale parallel simulations. In this work, we investigate alternative ghost-point discretizations based on compact stencils. We introduce a formulation based on a boundary operator that locally approximates the boundary condition near each ghost node, replacing it with linear relations involving both interior and ghost points. The operator is constructed via least-squares reconstruction, allowing flexible stencil configurations while preserving the desired order of accuracy. Several strategies for selecting and adapting compact stencils are proposed, guided by conditioning criteria and iterative refinement procedures to improve global stability. Numerical experiments on various geometries and convection-diffusion regimes demonstrate the effectiveness of the proposed approach, showing that it maintains high accuracy even in the presence of boundary layers and improves stencil compactness and conditioning of the resulting linear systems.

S. Clain, M. T. Malheiro, G. J. Machado et al.

We design and analyse a new numerical method to solve ODE system based on the structural method. We compute approximations of solutions together with its derivatives up to order $K$ by solving an entire block corresponding to $R$ time steps. We build the physical relations that connect the function and derivative approximations at each time step by using the ODE and its derivatives, and develop the structural equations that establish linear relations between the function and its derivative over the whole block of $R$ times steps. The non-linear system is solved and provide very accurate approximations with nice spectral resolution properties.

Diogo Lopes, António Ramires Fernandes, Stéphane Clain

There are several numerical models that describe real phenomena being used to solve complex problems. For example, an accurate numerical breast model can provide assistance to surgeons with visual information of the breast as a result of a surgery simulation. The process of finding the model parameters requires numeric inputs, either based in medical imaging techniques, or other measures. Inputs can be processed by iterative methods (inverse elasticity solvers). Such solvers are highly robust and provide solutions within the required degree of accuracy. However, their computational complexity is costly. On the other hand, machine learning based approaches provide outputs in real-time. Although high accuracy rates can be achieved, these methods are not exempt from producing solutions outside the required degree of accuracy. In the context of real life situations, a non accurate solution might present complications to the patient. We present an hybrid parameter estimation method to take advantage of the positive features of each of the aforementioned approaches. Our method preserves both the real-time performance of deep-learning methods, and the reliability of inverse elasticity solvers. The underlying reasoning behind our proposal is the fact that deep-learning methods, such as neural networks, can provide accurate results in the majority of cases and they just need a fail-safe system to ensure its reliability. Hence, we propose using a Multilayer Neural Networks (MNN) to get an estimation which is in turn validated by a iterative solver. In case the MNN provides an estimation not within the required accuracy range, the solver refines the estimation until the required accuracy is achieved. Based on our results we can conclude that the presented hybrid method is able to complement the computational performance of MNNs with the robustness of iterative solver approaches.

Diogo Lopes, Stéphane Clain, António Ramires Fernandes

Bio-mechanical breast simulations are based on a gravity free geometry as a reference domain and a nonlinear mechanical model parameterised by physical coefficients. As opposed to complex models proposed in the literature based on medical imagery, we propose a simple but yet realistic model that uses a basic set of measurements easy to realise in the context of routinely operations. Both the mechanical system and the geometry are controlled with parameters we shall identify in an optimisation procedure. We give a detailed presentation of the model together with the optimisation method and the associated discretisation. Sensitivity analysis is then carried out to evaluate the robustness of the method.