Research and projects
Research and projects
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Research Interests
Research Interests
Finite element methods
Continuum Mechanics and Computational Fluid Dynamics
Numerical linear algebra / randomized linear algebra
Mixed precision algorithms
High Performance of Computing
Mathematics for ML and DL & ML and DL for Mathematics
In this project, we consider finite element method for viscoleasticity model problems. We present advanced a priori stability and error analyses for generalised Maxwell solid models and fractional order viscoelasticity models.
Elastic Solid
Elastic Solid
Newtonian Fluid
Newtonian Fluid
Viscoelastic solid
Viscoelastic solid
Physics-informed neural networks for solving integro-differential equations with applications to viscoelasticity
Physics-informed neural networks for solving integro-differential equations with applications to viscoelasticity
We develop a Physics-Informed Neural Network (PINN) framework for solving integro-differential equations (IDEs) in the space-time domain without relying on traditional numerical methods such as finite difference or finite element methods. By fully approximating differential operators using automatic differentiation, our approach enables a mesh-free solution strategy. We reformulate IDEs as PDEs with ODE constraints by introducing internal variables, significantly improving computational efficiency. The proposed framework is applied to the generalised Maxwell model for viscoelastic materials and is further extended to inverse problems for identifying mechanical response parameters.
Schematic of a PINN solving the space-time Volterra integral equation
Schematic of a PINN solving the space-time Volterra integral equation
Fast linear solvers for non-symmetric systems in incompressible CFD-simulations
Fast linear solvers for non-symmetric systems in incompressible CFD-simulations
We investigate efficient linear solvers for linear systems that arise in incompressible fluid flow simulations using the Compatible Discrete Operator (CDO) scheme. The study focuses on algebraic multigrid (AMG) and Krylov subspace methods to improve solver performance. Specifically, it evaluates different AMG cycle strategies, including K-cycle, to enhance robustness and efficiency. The proposed solver framework is tested on benchmark problems, demonstrating improved convergence rates and scalability for large-scale CFD simulations. The findings provide insights into the optimal choice of solvers for high-fidelity fluid simulations.
Numerical methods for saddle point systems:
Golub-Kahan bidiagonalization
Augmented Lagrangian Uzawa method
Algebraic transformation
Our proposed linear solvers are implemented and available at code_satrune (https://www.code-saturne.org).
2D Burggraf flow with Re= 100: pressure field (left) and velocity field (right).
2D Burggraf flow with Re= 100: pressure field (left) and velocity field (right).
Next generation of CFD
Next generation of CFD
SoNICS Project
The SoNICS (So Naturally Innovative CFD Solver) project is a collaborative initiative between ONERA and Safran, aimed at enhancing CFD simulations for aerospace applications. The project focuses on developing efficient numerical solvers and high-performance computing (HPC) strategies to improve the accuracy and computational efficiency of aerodynamic simulations. Key research areas include advanced linear solvers, preconditioning techniques, and acceleration methods tailored for large-scale industrial CFD codes.
NextSim Project
The NextSim project, funded by the EU-HPC Joint Undertaking (JU) under the EU Horizon 2020 programme, aims to revolutionize CFD tools for HPC. With contributions from Spain, France, and Germany, the project focuses on enhancing CODA, a next-generation CFD solver developed by ONERA, DLR, and Airbus. By improving numerical efficiency, data management, and HPC implementation, NextSim addresses critical aerospace challenges, enabling faster, more accurate, and cost-effective aerodynamic simulations.
Laminar flow past a cylinder
Laminar flow past a cylinder
Advanced linear algebra with randomized and mixed precision algorithms
Advanced linear algebra with randomized and mixed precision algorithms
The development of advanced linear algebra techniques using randomized algorithms and mixed precision arithmetic has the potential to significantly improve the efficiency and scalability of solving large-scale linear systems. This work introduces an advanced GMRES framework that incorporates randomized algorithms and mixed precision arithmetic to enhance both efficiency and robustness. The framework begins with a randomized Gram-Schmidt process for orthogonalizing Krylov basis vectors, improving numerical stability while reducing computational costs. Despite not being strictly l2 orthonormal, the basis' random projection onto a lower-dimensional space ensures stable convergence, even in extreme-scale problems. The method is extended to flexible GMRES variants with deflation, which integrate harmonic Ritz pairs and singular value decomposition (SVD) based restarting strategies. To optimize performance further, mixed precision arithmetic is applied, where lower precision is used for less critical computations, while higher precision is preserved for key operations such as matrix-vector multiplications. While deflated restarting in low precision can cause stagnation, augmentation techniques are employed to mitigate this and preserve accuracy.
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