Formulation implicite ou explicite d'un problème de dynamique

Résumé

En simulation numérique, un problème dépendant du temps peut être formulé de manière implicite ou explicite. Un problème dépendant du temps décrit une situation qui évolue ; le système est modélisé à différents instants t discrets appelés « pas de temps ». La méthode explicite consiste à déterminer la solution à t + Δt en fonction de la valeur de la fonction en t. Si la fonction à évaluer s'appelle y(t), alors le problème se formule de la manière suivante : : y(t + Δt) = F(y(t)). La méthode d'Euler est une méthode explicite. La méthode implicite consiste à déterminer la solution à t + Δt en résolvant une équation prenant en compte la valeur de la fonction en t et en t + Δt. Le problème se formule de la manière suivante : : G(y(t), y(t + Δt)) = 0. Les méthodes de Runge-Kutta sont des méthodes dites implicites-explicites (ou « imex ») car une partie est résolue par une méthode implicite et l'autre par une méthode explicite. Application en dynamique La formulation implic

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Interactive, Real Time Structural Simulation

Julien Lebet

The aim of this thesis is to develop a model that provides a fast analysis of structures subjected to large displacements. This is done by transforming the structure in a set of nite particles and by applying a method called dynamic relaxation to it. The latter simulates a dynamic behaviour of the structure by applying Newton's second law to the set of particles until the equilibrium is reached. This can be accomplished using only explicit methods and it does not require to build a global stiness matrix nor to inverse it. The position and rotations of the elements are followed during the process and their forces are always computed locally by using their linear stiness matrix. This methodology is applied to multiple type of elements and the accuracy and speed of the model is nally tested through various examples.

2020

Chargement

The quantification of uncertainties can be particularly challenging for problems requiring long-time integration as the structure of the random solution might considerably change over time. In this respect, dynamical low-rank approximation (DLRA) is very appealing. It can be seen as a reduced basis method, thus solvable at a relatively low computational cost, in which the solution is expanded as a linear combination of a few deterministic functions with random coefficients. The distinctive feature of the DLRA is that both the deterministic functions and random coefficients are computed on the fly and are free to evolve in time, thus adjusting at each time to the current structure of the random solution. This is achieved by suitably projecting the dynamics onto the tangent space of a manifold consisting of all random functions with a fixed rank. In this thesis, we aim at further analysing and applying the DLR methods to time-dependent problems.Our first work considers the DLRA of random parabolic equations and proposes a class of fully discrete numerical schemes.Similarly to the continuous DLRA, our schemes are shown to satisfy a discrete variational formulation.By exploiting this property, we establish the stability of our schemes: we show that our explicit and semi-implicit versions are conditionally stable under a ``parabolic'' type CFL condition which does not depend on the smallest singular value of the DLR solution; whereas our implicit scheme is unconditionally stable. Moreover, we show that, in certain cases, the semi-implicit scheme can be unconditionally stable if the randomness in the system is sufficiently small. The analysis is supported by numerical results showing the sharpness of the obtained stability conditions. The discrete variational formulation is further applied in our second work, which derives a-priori and a-posteriori error estimates for the discrete DLRA of a random parabolic equation obtained by the three newly-proposed schemes. Under the assumption that the right-hand side of the dynamical system lies in the tangent space up to a small remainder, we show that the solution converges with standard convergence rates w.r.t. the time, spatial, and stochastic discretization parameters, with constants independent of singular values.We follow by presenting a residual-based a-posteriori error estimation for a heat equation with a random forcing term and a random diffusion coefficient which is assumed to depend affinely on a finite number of independent random variables. The a-posteriori error estimate consists of four parts: the finite element method error, the time discretization error, the stochastic collocation error, and the rank truncation error. These estimators are then used to drive an adaptive choice of FE mesh, collocation points, time steps, and time-varying rank.The last part of the thesis examines the idea of applying the DLR method in data assimilation problems, in particular the filtering problem. We propose two new filtering algorithms. They both rely on complementing the DLRA with a Gaussian component. More precisely, the DLR portion captures the non-Gaussian features in an evolving low-dimensional subspace through interacting particles, whereas each particle carries a Gaussian distribution on the whole ambient space. We study the effectiveness of these algorithms on a filtering problem for the Lorenz-96 system.

EPFL2022

Chargement

An adaptive Rosenbrock method with applications in Electrophysiology

Giacomo Rosilho De Souza

The aim of this project is to implement the Rosenbrock method ROS3P in the C++ Finite Element library LifeV for the solution of systems of ordinary differential equations arising in electrophysiology. In the domain of electrophysiology LifeV implements cardiac cell models which can require the resolution of stiff reaction-diffusion equations. To solve this system the use of implicit methods is required. We focus in particular on one step methods. Implicit methods need the solution of a non linear system at each time step. The Rosenbrock methods derive from the linearly implicit Runge Kutta methods and just requires the solution of s linear systems per time step, where s is the number of stages. Most of the implicit Runge Kutta methods suffer from order reduction when applied to parabolic problems. For this reason we have chosen the third order Rosenbrock method ROS3P, which satisfies additional order conditions to keep the same convergence order even when applied to non linear parabolic problems. In the first part of this work we derive the Rosenbrock methods from Runge Kutta methods, explaining the time adaptivity strategy and finally giving the algorithm of our implementation in LifeV. After we present the cardiac reaction-diffusion models, deriving the mono-domain model from the bi-domain model. In the following we explain the strategy that we adopt to solve these models and how we discretize them. Finally we show some numerical experiments obtained with our approach and compare our results with the ones given by other methods.

2013

Chargement