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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 ma ...

We consider finite element error approximations of the steady incompressible Navier-Stokes equations defined on a randomly perturbed domain, the perturbation being small. Introducing a random mapping, these equations are transformed into PDEs on a fixed reference domain with random coefficients. Under suitable assumptions on the random mapping and the input data, in particular the so-called small data assumption, we prove the well-posedness of the problem. We assume then that the mapping depends affinely on

A multiphase-field model for the description of coalescence in a binary alloy is solved numerically using adaptive finite elements with high aspect ratio. The unknown of the multiphase-field model are the three phase fields (solid phase 1, solid phase 2, and liquid phase), a Lagrange multiplier and the concentration field. An Euler implicit scheme is used for time discretization, together with continuous, piecewise linear finite elements. At each time step, a linear system corresponding to the three phases

A numerical method based on an adaptive octree space discretization for the simulation of displacement of free surfaces is proposed and applied to 3D free surface flow problems. A VOF approach is combined with a mass-conserving semi-Lagrangian time-steppin ...

A posteriori error estimates for the heat equation in two space dimensions are presented. A classical discretization is used, Euler backward in time, and continuous, piecewise linear triangular finite elements in space. The error is bounded above and below by an explicit error estimator based on the residual. Numerical results are presented for uniform triangulations and constant time steps. The quality of our error estimator is discussed. An adaptive algorithm is then proposed. Successive Delaunay triangul

An adaptive finite element algorithm to compute transonic viscous flows around a wing is presented. The adaptive criteria is based on an anisotropic error estimator in the 115 semi-norm, justified for an advection-diffusion problem with stabilized finite e ...

A model which takes into account the main phenomena occurring during the laser-cladding process is proposed. For a given laser power, beam radius, powder jet geometry, and clad height, this model evaluates two other processing parameters, namely, the laser-beam velocity and the powder feed rate. It considers the interactions between the powder particles, the laser beam, and the molten pool. The laser power reaching the surface of the workpiece is estimated and, assuming this power is used to remelt the subs

A shape optimization algorithm is presented that estimates the ice thickness distribution within a three-dimensional, shallow glacier, given a transient surface geometry and a mass-balance distribution. The approach is based on the minimization of the surf ...

We propose a simple stopping criterion for the conjugate gradient (CG) algorithm in the framework of anisotropic, adaptive finite elements for elliptic problems. The goal of the adaptive algorithm is to find a triangulation such that the estimated relative ...

Parameter identification is important in many engineering processes, and benefits from quick evaluations with reduced models. We present here a framework to solve inverse problems for parametric partial differential equations. The reduced model is built on a neural network method for the numerical approximation of a given parametric partial differential equation. Training data are generated thanks to adaptive finite element simulations. A supervised feedforward neural network is then used for the online app

In this paper we present briefly two possible approaches for modelling the laser cladding process. The first approach is 2D and consists in finding the shape of the molten pool given the fraction of the laser power which is available at the surface of the workpiece and given the amount of powder sticking to the molten pool. The powder reaching the workpiece is assumed to melt instantaneously at the surface of the molten pool, it is then mixed into the molten pool according to the computed velocity field. Be

An a posteriori upper bound is derived for the nonstationary convection-diffusion problem using the Crank-Nicolson scheme and continuous, piecewise linear stabilized finite elements with large aspect ratio. Following Lozinski et al. (2009) [13], a quadrati ...

In this paper, we present a mathematical modeling of some magnetohydrodynamic effects arising in an aluminum production cell as well as its numerical approximation by a finite element method. We put the emphasis on the magnetic effects which live in the whole three dimensional space and which are solved numerically with a domain decomposition method.

In this article, a finite element error analysis is performed on a class of linear and nonlinear elliptic problems with small uncertain input. Using a perturbation approach, the exact (random) solution is expanded up to a certain order with respect to a pa ...

Spaces of continuous piecewise linear finite elements are considered to solve a Poisson problem, and several numerical methods are investigated to recover second derivatives. Numerical results on 2D and 3D isotropic and anisotropic meshes indicate that the ...

A mixture model to take into account the flow of small carbon dioxide bubbles dissolved in a liquid is presented. The model describes the evolution of the velocity fields (mixture and gas), the pressure and the volume fraction of gas. The system of equatio ...

An adaptive algorithm is proposed for solving the Stokes problem with continuous, piecewise linear stabilized finite elements and meshes with high aspect ratio. Anisotropic, a posteriori error estimates in the H-1 x L-2 norm are derived, the constant being ...

The time-dependent Stokes problem is solved using continuous, piecewise linear finite elements and a classical stabilization procedure. Four order-one methods are proposed for the time discretization. The first one is nothing but the Euler backward scheme and requires a large linear system involving the velocity and pressure unknowns to be solved. The other three schemes allow velocity and pressure computations to be decoupled, namely the pressure-matrix method, a method based on an inexact LU factorization

In this paper, order one finite elements together with Galerkin least-squares (GLS) methods are used for solving a three-held Stokes problem arising from the numerical study of viscoelastic hows. Stability and convergence results are established, even when the solvent viscosity is small compared to the viscosity due to the polymer chains. An iterative algorithm decoupling velocity-pressure and stress calculations is proposed. The link with the modified elastic viscous split stress (EVSS) method studied in (

A numerical method for the solution to the density-dependent incompressible Navier-Stokes equations modeling the flow of N immiscible incompressible liquid phases with a free surface is proposed. It allows to model the flow of an arbitrary number of liquid ...

Marco Picasso

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Numerical Approximation Of Orthogonal Maps

MATHICSE Technical Report : A posteriori error estimation for the steady Navier-Stokes equations in random domains

Adaptive finite elements with high aspect ratio for the computation of coalescence using a phase-field model

An octree-based adaptive semi-Lagrangian VOF approach for simulating the displacement of free surfaces

Adaptive finite elements for a linear parabolic problem

An anisotropic adaptive finite element algorithm for transonic viscous flows around a wing

A Simple but Realistic Model for Laser Cladding

Estimating the ice thickness of shallow glaciers from surface topography and mass-balance data with a shape optimization algorithm

A stopping criterion for the conjugate gradient algorithm in the framework of anisotropic adaptive finite elements

A framework to solve inverse problems for parametric PDEs using adaptive finite elements and neural networks

Laser-Powder-Material Interactions in the Laser Cladding Process

An adaptive algorithm for the Crank-Nicolson scheme applied to a time-dependent convection-diffusion problem

Some Mathematical and Numerical Aspects in Aluminum Production

A posteriori error estimations for elliptic partial differential equations with small uncertainties

A Numerical Study Of Some Hessian Recovery Techniques On Isotropic And Anisotropic Meshes

A mixture model for a dilute dispersion of gas in a liquid flow: Mathematical analysis and application to aluminium electrolysis

An adaptive algorithm for the Stokes problem using continuous, piecewise linear stabilized finite elements and meshes with high aspect ratio

Stability of time-splitting schemes for the Stokes problem with stabilized finite elements

GLS and EVSS methods for a three-field Stokes problem arising from viscoelastic flows

Numerical simulation of 3D free surface flows, with multiple incompressible immiscible phases. Applications to impulse waves