# Caboussat Alexandre

### Professeur HES associé

### Professeur HES associé

Phone: +41 22 558 50 45

Desktop: B 2.12

Campus Battelle, Rue de la Tambourine 17, 1227 Carouge, CH

**Faculty**

Economie et services

**Main Degree Programme**

International Business Management

- Mathematics
- Statistics
- Forecasting and Decision-Making

- Méthodes Quantitatives
- Méthodologie

- Mathématiques

- Analyse Numérique

Ongoing

**Role: **Co-applicant

**Description du projet : **

Consulting mandate with industry.

**Research team within HES-SO:**
Caboussat Alexandre

**Durée du projet:**
12.05.2023

**Statut: ** Ongoing

Completed

**Role: **Main Applicant

**Financement: **
HES-SO Rectorat; FNS - Fonds national suisse

**Description du projet : **
Numerical methods for fully nonlinear equations are now gathering a lot of attention
from computational scientists. Numerical techniques range from viscosity solutions based methods, higher order finite elements, finite differences, or variational approaches.
The first application for the validation of all these methods is usually the prototypical Monge-Ampère equation. The starting point of this project is the numerical framework based on several variational approaches developed by the PI and co-authors based on least-squares or augmented Lagrangian methods.
These variational methods are designed to be applied to some elliptic fully nonlinear equations, starting with the Monge-Ampère equation, but also for some Eikonal and prescribed Jacobian equations. This project will build on this framework to develop several new algorithms, together with the corresponding theoretical convergence investigations, mainly in two directions:
1) Extension of the variational approaches. Apply the framework to a large variety of first or second order fully nonlinear elliptic equations, including Pucci's, vectorial
Eikonal equations, prescribed Jacobian equation, or the three-dimensional
Monge-Ampère equation.
Extend the methodology to the numerical approximation of the elliptic prescribed
Jacobian inequalities, and to the evolutive parabolic Monge-Ampère equation.
Study the mathematical convergence properties of the existing least-squares approach.
2) Design of adaptive finite element methods. In order to address non-smooth
problems, tackle the design of adative and mesh refinement techniques of the finite
element discretization in space of the Dirichlet problem for the Monge-Ampère equation in two dimensions of space.
In that framework, those methods are related to adaptive methods for nonlinear Stokes problems and heuristics. Extend the analysis to Eikonal equations.

**Research team within HES-SO:**
Makhlouf Shabou Basma
, Caboussat Alexandre

**Partenaires académiques: **516,International Business Management

**Durée du projet:**
01.01.2017 - 31.12.2020

**Montant global du projet: **394'554 CHF

**Statut: ** Completed

**Role: **Main Applicant

**Description du projet : **

40316.1 INNO-ICT: Geographic and socioeconomic data driven models, supporting the Swiss retail sector

**Research team within HES-SO:**
Caboussat Alexandre

**Partenaires professionnels: **Quadrane SARL

**Durée du projet:**
01.11.2019 - 30.04.2020

**Statut: ** Completed

**Role: **Main Applicant

**Description du projet : **
The project deals with the mathematical modelling and the numerical simulation of (non-cohesive) sediment transport in rivers and/or hydraulic engineering processes. The accurate modelling of sedimentation has a great impact on the behavior of free surface flows in rivers (modifications of the morphology of rivers) and in hydraulic processces (optimization of dam flush event for instance)
The design of appropriate mathematical models and numerical methods for the simulation of sedimentation processes is indeed of great importance to understand the large-time behavior of free-surface rivers flows (modification of riverbed morphology) and to guide hydraulic engineers toward a sustainable river management (optimization of dam flush event for instance). From the industrial viewpoint, the flushing of dam retention lakes is crucial for an optimal exploitation of dams, and a longest lifetime of the installation.
This project is at the crossroads between the microscopic modelling of non-cohesive sediments in rivers and the three-dimensional simulation of large-scale free-surface flows. The scientific expertise for the former is brought by the French team, who is specialized in physical modeling, theoretical fluid mechanics and mathematical physics ; on the other hand, the Swiss team is specialized in scientific computing and numerical simulation of large 3D physical and industrial processes. The collaboration between French and Swiss teams will thus be useful to couple these two approaches in a optimal and innovative way, as the two groups are very complementary. Moreover, when bringing people with different backgrounds, meeting in person is crucial ofr the initial progress of the project.
This project is in the continuity of projects sponsored by the SNF on numerical simulation of impulse waves, and development of a multiphysics numerical algorithms and solvers (see Section 6 : Collaborations existantes for more details), and undertaken in parallel to an ANR proposal led by the French team oriented on the mathematical rheology of sediments flows. This part of the collaboration will focus on the scientific computing aspects of the simulation. The deliverable will be a module of a computer software (so-called cfsFlow, developed by EPFL and transferred to Ycoor systems SA, a spin-off of EPFL, via a technology transfer). Validation will be achieved through benchmark cases in dam break flows and rivers, at the laboratory and geophysical scales. Finally, the simulation of a flush event in a dam retention lake will be considered.

**Research team within HES-SO:**
Caboussat Alexandre

**Partenaires académiques: **516,International Business Management

**Durée du projet:**
01.01.2016 - 31.12.2016

**Statut: ** Completed

**Role: **Main Applicant

**Financement: **
HES-SO Rectorat; EPFL

**Description du projet : **
A numerical model for the simulation of 3D impulse waves will be developed.
It will be used to simulate experiments performed at VAW-ETHZ within a collaboration
between the two Institutes.
The starting point is a numerical model already published by the main applicant with
various co-authors, among them PhD students previously supported by the SNSF. This
model has been used for the numerical simulation of free surface flows in various
situations (bubbles, visco-elastic flows, Non-newtonian flows, etc).
The model relies on a splitting of the physical phenomena, the diffusion and transport
operators being decoupled. This project will focus on several aspects of the methods
that have to be updated to consider impulse waves. In particular, the transport step in
the numerical model will be completely redesigned. Indeed, adaptive techniques are
required for impulse waves, as we will be considering very large computational
domains that are restricted mainly by the computer memory. An octree representation
of the computational domain will be used, which will allow a very precise - adaptive -
tracking of the liquid-air free surface, with reduced memory requirements.
This updated numerical model will be used to investigate impulse waves and reproduce
experiments from VAW-ETHZ. Numerical results will also be compared to those
obtained with other models.The final output is to be able to quantify risks triggered by
the impact of falling solids in water reservoirs, such as dams or proglacial lakes, and
the overflow of lakes resulting from such impacts.

**Research team within HES-SO:**
Caboussat Alexandre

**Partenaires académiques: **516,International Business Management; Caboussat Alexandre, 516,International Business Management

**Durée du projet:**
01.10.2012 - 30.09.2015

**Montant global du projet: **52'800 CHF

**Statut: ** Completed

2024

*Computer methods in applied mechanics and engineering*,
Vol. 421, Article no 116784

**Summary:**

We present a finite elements—neural network approach for the numerical approximation of parametric partial differential equations. The algorithm generates training data from finite element simulations, and uses a data-driven (supervised) feedforward neural network for the online approximation of the solution. The objective is to ensure that the overall error of the method is below some preset tolerance, and we thus control and balance the error coming from the finite element method, and the one introduced by the neural network approximation. Two finite element methods are considered and compared; a fixed grid approach uses the same mesh for all values of the parameters, while an adaptive finite elements approach enforces the same discretization error uniformly in the parameters space. Numerical results are presented for an elliptic model problem. The fixed grid approach shows limitations in terms of error balancing and control, while the adaptive approach allows a better accuracy and more flexibility of the method. We conclude by proposing an adaptive algorithm to control the size of the training set given a network architecture and ensure that the overall error of the method is below a given tolerance.

**Summary:**

We present a numerical model for the approximation of multiphase flows with free surfaces and strong interfacial effects. The model relies on the multiphase incompressible Navier-Stokes equations, and includes surface tension effects on the interfaces between phases, and contact angles. The volume-of-fluid approach is used to track the interfaces and the free surfaces between the various phases and the ambient air. The numerical method relies on an operator splitting strategy. The space discretization relies on a two-grid approach that uses an unstructured finite element mesh for diffusion phenomena and a structured Cartesian grid for advection phenomena. An adaptive mesh refinement algorithm is incorporated to better track the interfaces and free surfaces on the finite element mesh, and approximate more accurately surface forces. The model is validated through numerical experiments, in particular for emulsion problems.

2023

**Summary:**

Optimal transportation of raw material from suppliers to customers is an issue in supply chain that we address here with a continuous model. A least-squares method is designed to solve the prescribed Jacobian problem that arises in optimal transportation in two dimensions of space. An iterative algorithm allows to decouple the variational aspects of the problem from the nonlinearities and from the weak treatment of the boundary conditions. Numerical experiments illustrate the transport of material in several configurations.

**Summary:**

We address the numerical solution of the Dirichlet problem for the two-dimensional elliptic Monge–Ampère equation using a least-squares/relaxation approach. The relaxation algorithm allows the decoupling of the differential operators from the nonlinearities of the equation, within a splitting approach. The approximation relies on mixed low order finite element methods with regularization techniques. In order to account for data singularities in non-smooth cases, we introduce an adaptive mesh refinement technique. The error indicator is based an independent formulation of the Monge–Ampère equation under divergence form, which allows to explicit a residual term. We show that the error is bounded from above by an a posteriori error indicator plus an extra term that remains to be estimated. This indicator is then used within the existing least-squares framework. The results of numerical experiments support the convergence of our relaxation method to a convex classical solution, if such a solution exists. Otherwise they support convergence to a generalized solution in a least-squares sense. Adaptive mesh refinement proves to be efficient, robust, and accurate to tackle test cases with singularities.

**Summary:**

We present a multi-physics model for the approximation of the coupled system formed by the heat equation and the Navier-Stokes equations with solidification and free surfaces. The computational domain is the union of two overlapping regions: a larger domain to account for thermal effects, and a smaller region to account for the fluid flow. Temperature-dependent surface effects are accounted for via surface tension and Marangoni forces. The volume-of-fluid approach is used to track the free surfaces between the metal (liquid or solidified) and the ambient air. The numerical method incorporates all the physical phenomena within an operator splitting strategy. The discretization relies on a two-grid approach that uses an unstructured finite element mesh for diffusion phenomena and a structured Cartesian grid for advection phenomena. The model is validated through numerical experiments, the main application being laser melting and polishing.

**Summary:**

In origami theory, the problem of rigid maps consists in finding a paper folding from the two-dimensional space onto the three-dimensional space. This problem is an example of a first-order fully nonlinear equation. In this article, we present a general variational framework to solve the problem of rigid maps with Dirichlet boundary conditions. The numerical framework relies on the introduction of a regularized objective function and the penalization of the constraints. A splitting algorithm is advocated for the corresponding flow problem. The iterations sequence consists of local nonlinear problems and a global linear variational problem at each step. Numerical experiments validate the efficiency of the method for piecewise smooth exact solutions.

2022

**Summary:**

We consider a least-squares/relaxation finite element method for the numerical solution of the prescribed Jacobian equation. We look for its solution via a least-squares approach. We introduce a relaxation algorithm that decouples this least-squares problem into a sequence of local nonlinear problems and variational linear problems. We develop dedicated solvers for the algebraic problems based on Newton’s method and we solve the differential problems using mixed low-order finite elements. Various numerical experiments demonstrate the accuracy, efficiency and the robustness of the proposed method, compared for instance to augmented Lagrangian approaches.

**Summary:**

We present a numerical model for the simulation of 3D sediment transport in a Newtonian flow with free surfaces. The Navier–Stokes equations are coupled with the transport, deposition, and resuspension of the particle concentrations, and a volume-of-fluid approach to track the free surface between water and air. The numerical method relies on operator splitting to decouple advection and diffusion phenomena, and a two-grid method. An appropriate combination of characteristics, finite volumes, and finite elements methods is advocated. The numerical model is validated through comparisons with numerical experiments, sediment flushing, shear flow erosion, and the formation of dunes.

2021

*Journal of computational and applied mathematics*,
2022, vol. 407, article no. 113997, pp. 1-21

**Summary:**

Orthogonal maps are two-dimensional mappings that are solutions of the so-called origami problem obtained when folding a paper. These mappings are piecewise linear, and the discontinuities of their gradient form a singular set composed of straight lines representing the folding edges. The proposed algorithm relies on the minimization of a variational principle discussed in Caboussat et al. (2019). A splitting algorithm for the corresponding flow problem derived from the first-order optimality conditions alternates between local nonlinear problems and linear elliptic variational problems at each time step. Anisotropic adaptive techniques allow to obtain refined triangulations near the folding edges while keeping the number of vertices as low as possible. Numerical experiments validate the accuracy and efficiency of the adaptive method in various situations. Appropriate convergence properties are exhibited, and solutions with sharp edges are recovered.

2019

**Summary:**

Orthogonal maps are the solutions of the mathematical model of paper-folding, also called the origami problem. They consist of a system of first-order fully nonlinear equations involving the gradient of the solution. The Dirichlet problem for orthogonal maps is considered here. A variational approach is advocated for the numerical approximation of the maps. The introduction of a suitable objective function allows us to enforce the uniqueness of the solution. A strategy based on a splitting algorithm for the corresponding flow problem is presented and leads to decoupling the time-dependent problem into a sequence of local nonlinear problems and a global linear variational problem at each time step. Numerical experiments validate the accuracy and the efficiency of the method for various domains and meshes.

**Summary:**

The numerical solution of the Dirichlet problem for an elliptic Pucci’s equation in two dimensions of space is addressed by using a least-squares approach. The algorithm relies on an iterative relaxation method that decouples a variational linear elliptic PDE problem from the local nonlinearities. The approximation method relies on mixed low order finite element methods. The least-squares framework allows to revisit and extend the approach and the results presented in [Caffarelli, Glowinski, 2008] to more general cases. Numerical results show the convergence of the iterative sequence to the exact solution, when such a solution exists. The robustness of the approach is highlighted, when dealing with various types of meshes, domains with curved boundaries, nonconvex domains, or non-smooth solutions.

2018

**Summary:**

In this article, we address the numerical solution of the Dirichlet problem for the three-dimensional elliptic Monge–Ampère equation using a least-squares/relaxation approach. The relaxation algorithm allows the decoupling of the differential operators from the nonlinearities. Dedicated numerical solvers are derived for the efficient solution of the local optimization problems with cubicly nonlinear equality constraints. The approximation relies on mixed low order finite element methods with regularization techniques. The results of numerical experiments show the convergence of our relaxation method to a convex classical solution if such a solution exists; otherwise they show convergence to a generalized solution in a least-squares sense. These results show also the robustness of our methodology and its ability at handling curved boundaries and non-convex domains.

**Summary:**

We present a numerical model for the simulation of 3D poly-dispersed sediment transport in a Newtonian flow with free surfaces. The physical model is based on a mixture model for multiphase flows. The Navier–Stokes equations are coupled with the transport and deposition of the particle concentrations, and a volume-of-fluid approach to track the free surface between water and air. The numerical algorithm relies on operator-splitting to decouple advection and diffusion phenomena. Two grids are used, based on unstructured finite elements for diffusion and an appropriate combination of the characteristics method with Godunov’s method for advection on a structured grid. The numerical model is validated through numerical experiments. Simulation results are compared with experimental results in various situations for mono-disperse and bi-disperse sediments, and the calibration of the model is performed using, in particular, erosion experiments.

**Summary:**

We consider the Dirichlet problem for a partial differential equation involving the Jacobian determinant in two dimensions of space. The problem consists in finding a vector-valued function such that the determinant of its gradient is given pointwise in a bounded domain, together with essential boundary conditions. This problem was initially considered in Dacorogna and Moser [Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (1990), pp. 1--26], and several theoretical generalizations have been derived since. In this work, we design a numerical algorithm for the approximation of the solution of such a problem for various kinds of boundary data. The proposed method relies on an augmented Lagrangian algorithm with biharmonic regularization, and low order mixed finite element approximations. An iterative method allows us to decouple the nonlinearity and the differential operators. Numerical experiments show the capabilities of the method for benchmarks and then for more demanding test problems.

2016

*Splitting methods in communication, imaging, science, and engineering*
(pp. 677-729). 2016,
Cham : Springer

**Summary:**

We explore the benefits of operator splitting algorithms in the context of computational fluid dynamics. In particular, we exploit their capacity in handling free surface flows and a large variety of physical phenomena in a flexible way. A mathematical and computational framework is presented for the numerical simulation of free surface flows, where the operator splitting strategy allows to separate inertial effects from the other effects. The characteristics method on a fine structured grid is put forward to accurately approximate the inertial effects while continuous piecewise polynomial finite element subordinated to a coarser subdivision made of simplices is advocated for the other effects. In addition, the splitting strategy also allows to be modular and change the rheological model for the fluid in a straightforward manner.We will emphasize this flexibility by treating Newtonian flows, viscoelastic flows and multi-phase immiscible incompressible Newtonian flows based on multiple densities. The numerical framework is thoroughly presented; the test case of the filling of a cylindrical tube, with potential die swell in an extrusion process is taken as the main illustration of the advantages of operator splitting.

2015

*Europa Star Première : le journal de l'écosystème horloger suisse*,
2015, vol. 17, no 5 (30 septembre), pp. 20-21

**Summary:**

Au-delà de la seule mécanique horlogère, les modélisations mathématiques permettent de reproduire et maîtriser toujours davantage des phénomènes tels la dilatation, les frottements, la polarisation ou encore l’acoustique des montres.

*Chinese Annals of Mathematics, Series B*,
2015, vol. 36, no. 5, pp. 689-702

**Summary:**

The steady Eikonal equation is a prototypical first-order fully nonlinear equation. A numerical method based on elliptic solvers is presented here to solve two different kinds of steady Eikonal equations and compute solutions, which are maximal and minimal in the variational sense. The approach in this paper relies on a variational argument involving penalty, a biharmonic regularization, and an operator-splitting-based time-discretization scheme for the solution of an associated initial-value problem. This approach allows the decoupling of the nonlinearities and differential operators. Numerical experiments are performed to validate this approach and investigate its convergence properties from a numerical viewpoint.

**Summary:**

In this article, we discuss a numerical method for the computation of the minimal and maximal solutions of a steady scalar Eikonal equation. This method relies on a penalty treatment of the nonlinearity, a biharmonic regularization of the resulting variational problem, and the time discretization by operator-splitting of an initial value problem associated with the Euler-Lagrange equations of the regularized variational problem. A low-order finite element discretization is advocated since it is well-suited to the low regularity of the solutions. Numerical experiments show that the method sketched above can capture efficiently the extremal solutions of various two-dimensional test problems and that it has also the ability of handling easily domains with curved boundaries.

2014

*International journal for numerical methods in fluids. 2014. Vol. 76, Issue 12, pp. 1004–1024*,

*In : Fitzgibbon, W. (ed.) et al. Modeling, simulation and optimization for science and technology. Berlin : Springer, 2014, pp. 23-40. Computational methods in applied sciences, vol. 34.*. 2014

2013

*ESAIM : control, optimisation and calculus of variations. 2013. Vol.?29, no.?3, p.?780-810*,

2012

2023

*Proceedings of the XIth International Conference on Adaptive Modeling and Simulation (ADMOS 2023)*

**Summary:**

We consider a hybrid approach for the approximation of the solution to parametric partial differential equations based on finite elements and deep neural net-works. Finite element simulations with adaptive mesh refinement are used to generate input data for the training of a neural network. A deep feedforward neural network is then used to approximate the solution of the partial differential equation. We aim at balancing the numerical errors introduced by the finite element method and the neural network approximation respectively. Numerical results are presented for the transport equation.

2022

*Proceedings of the 8th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2022)*

**Summary:**

A numerical model for the simulation of multiphase flows with free surfaces is presented. The model allows to incorporate in a unified manner several phases ranging from incompressible Newtonian flows, Oldroyd-B viscoelastic flows and neo-Hookean elastic solids deformations. We advocate a Eulerian modeling of the multiphase flows, relying on the volume fraction of liquid, describing multiple phases with those different rheologies. One advantage of the Eulerian approach is to allow for large deformations of elastic solids, and changes of topologies. The numerical framework relies on an operator splitting strategy and a two-grid method. The numerical model is validated with a numerical experiment based on the collision between two elastic bodies with free surfaces.

2021

*Proceedings of the 10th International Conference on Adaptive Modeling and Simulation (ADMOS 2021)*

**Summary:**

Parabolic fully nonlinear equations may be found in various applications,for instance in optimal portfolio management strategy. We focus here on a canonical parabolic Monge-Ampère equationin two space dimensions. A numerical method has beeninvestigated in[1]. The goal is toextend themethodology by coupling atime steppingsemi-implicit methodthat relies on a conservative formulation of the Monge-Ampère equationwith mesh adaptation.The parabolic Monge-Ampère equation can be expressed as astronglynonlinear, heat-type, parabolic equation, where the nonlinear diffusion function is expressed asa function ofthe cofactor matrix of the Hessianmatrixof the solution. We linearize this diffusion operator and advocatea semi-implicit time-stepping algorithm.In particular, we use the time-evolutive equation to reach a stationarysolutioncorresponding to a solutionof the elliptic Monge-Ampère equation. A loworder, piecewise linear,finite element method is used for spacediscretization, together with a mixed approach for the approximation of the second derivatives. The error is bounded above by an error indicator plus an extra term that can be disregarded in special cases. A mesh adaptivity strategy based on these estimates is then implemented within thetime-stepping algorithmfor the nonlinear equation. Numerical experiments exhibit appropriate convergence orders and arobust behavior. Adaptive mesh refinement proves to be efficient and accurate to tackle test cases with singularities. Inparticular, we consider equations with exact solutionswith singularitieson the boundary of the domain, or with right-hand sides involving Dirac functions.

*Proceedings of the 14th World Congress on Computational Mechanics (WCCM) and ECCOMAS Congress 2020*

**Summary:**

A mathematical model coupling the heat and fluid flow with solidification and free surfaces is presented. The numerical method relies on an operator splitting strategy, and a two-grid method. The free surfaces are tracked with a volume-of-fluid approach. A special emphasis is laid on the modeling of surface tension forces on the free surface. A comparison between approaches is highlighted, and a mesh convergence analysis is presented. Finally, the model is validated with the simulation of a static laser melting process.

2019

*Proceedings of the II International Conference on Simulation for Additive Manufacturing Sim-AM 2019*

**Summary:**

We present a multi-physics model for the approximation of the coupled system formed by the temperature-dependent Navier-Stokes equations with free surfaces. The main application is the industrial process of shallow laser surface melting (SLSM), for laser polishing of metal surfaces. We consider incompressible flow equations with solidification, and we model the laser source through physically-consistent boundary conditions. We incorporate Marangoni effects in the surface tension model to drive internal motion in the liquid metal. The numerical method relies on an operator splitting strategy and a two-grid approach. A proof of concept of the numerical model is achieved through a static laser melting process.

*Proceedings of the European numerical mathematics and advanced applications conference 2019*

**Summary:**

Parabolic fully nonlinear equations may be found in various applications, for instance in optimal portfolio management strategy. A numerical method for the approximation of a canonical parabolic Monge-Ampère equation is investigated in this work. A second order semi-implicit time-stepping method is presented, coupled to safeguarded Newton iterations A low order finite element method is used for space discretization. Numerical experiments exhibit appropriate convergence orders and a robust behavior.

2015

*Numerical mathematics and advanced applications - ENUMATH 2013 Proceedings of ENUMATH 2013 : the 10th European Conference on Numerical Mathematics and Advanced Applications, Lausanne, August 2013*

**Summary:**

We address the numerical solution of the Dirichlet problem for a partial differential equation involving the Jacobian determinant in two dimensions of space. The problem consists in finding a vector-valued function such that the determinant of its gradient is given point-wise in a bounded domain, together with essential boundary conditions. The proposed numerical algorithm relies on an augmented Lagrangian algorithm with biharmonic regularization, and low order mixed finite element approximations. An iterative method allows to decouple the local nonlinearities and the global variational problem that involves a biharmonic operator. Numerical experiments validate the proposed method.

2014

*In: Proceedings of the 11th World Congress on Computational Mechanics (WCCM XI), 5th European Conference on Computational Mechanics (ECCM V) and 6th European Conference on Computational Fluid Dynamics (ECFD VI), July 20-25, 2014, Barcelona, Spain. [S.l.] : IACM ; ECCOMAS, 2014. Pp.?5381-5391*

*In : Proceedings of the 20th International Conference on Computing in Economics. [S.l.] : The Society for Computational Economics, 2014. 13 p.*

Achievements

2022

2022 ;
*Formation continue*

**Collaborateurs: **
Caboussat Alexandre

Obtention du diplôme de formation continue Executive MBA