Partial differential equation

Related publications (100)

The goal of this report is to study the method introduced by Adomian known as the Adomian Decomposition Method (ADM), which is used to find an approximate solution to nonlinear partial differential equations (PDEs) as a series expansion involving the recursive solution of linear PDEs. We first describe the method, giving two specific examples with different nonlinearities and show exactly how the method works for these problems. Some analytical convergence results are then given, along with numerical solutions to the examples demonstrating these convergence results. A discussion of parameters inside of these nonlinearities follows, both for polynomial nonlinearities and for the more complicated hyperbolic sine nonlinearity problem. Finally, we compare the ADM with the Picard method, pointing out some important differences and demonstrating them by solving the given examples with both methods and comparing the results.

2013

2012

Isogeometric Collocation Method for Partial Differential Equations on Surfaces

Lucas Daniel Amoudruz

This project aims to study the concept of collocation method for isogeometric analysis with NURBS. We first introduce the isogeometric concept and its main advantages compared to finite elements methods. We then present the isogeometric collocation method and compare it to the isogeometric Galerkin method in terms of computational cost and accuracy. Elliptic problems and parabolic problems (linear and non-linear) are considered. The convergence rates of the collocation method are numerically verified and the comparisons with the Galerkin method tend to show a better efficiency of the collocation method for even degrees of NURBS basis.

2015

Sur quelques équations aux dérivées partielles en lien avec le lemme de Poincaré

David Valentin Strütt

In this thesis, we study two distinct problems. The first problem consists of studying the linear system of partial differential equations which consists of taking a k-form, and applying the exterior derivative 'd' to it and add the wedge product with a 1-form 'a'. The study of this differential operator is linked to the study of the multiplication by a two form, that is the system of linear equations where we take a k-form and apply the exterior wedge product by 'da', the exterior derivative of 'a'. We establish links between the partial differential equation and the linear system. The second problem is a generalization of the symmetric gradient and the curl equation. The equation of a symmetric gradient consists of taking a vector field, apply the gradient and then add the transpose of the gradient, whereas in the curl equation we subtract the transpose of the gradient. Both can be seen as an equation of the form A * grad u + (grad u)t * A, where A is a symmetric matrix for the case of the symmetric gradient and skew symmetric for the curl equation. We generalize to the case where A verifies no symmetry assumption and more significantly add a Dirichlet condition on the boundary.

EPFL2018

ON A GENERALIZATION OF THE POINCARE LEMMA TO EQUATIONS OF THE TYPE dw plus a Lambda w = f

David Valentin Strütt

We study the system of linear partial differential equations given by dw + a Lambda w = f, on open subsets of R-n, together with the algebraic equation da Lambda u = beta, where a is a given 1-form, f is a given (k + 1)-form, beta is a given k + 2-form, w and u are unknown k-forms. We show that if rank[da] >= 2(k+1) those equations have at most one solution, if rank[da] equivalent to 2m >= 2(k + 2) they are equivalent with beta = df + a Lambda f and if rank[da] equivalent to 2m >= 2(n - k) the first equation always admits a solution. Moreover, the differential equation is closely linked to the Poincare lemma. Nevertheless, as soon as a is nonexact, the addition of the term a Lambda w drastically changes the problem.

2018

Incremental Model Identification of Gas-Liquid Reaction Systems with Unsteady-State Diffusion

Benoit Till Cretegny

Identification of kinetic models and estimation of reaction and mass-transfer parameters can be performed using the extent-based identification method, whereby each chemical/physical process is treated individually. This method is used here to analyze gas-liquid systems under unsteady-state mass transfer. Such a situation is common in the case of diffusion-controlled reactions and can be modeled by the film theory. In both the gas and liquid bulks, mass-balance relations describe the species dynamics as ordinary differential equations (ODE) and serve as boundary conditions for the film. On the other hand, the dynamic accumulation in the film is described by Fick's second lay. The resulting partial differential equation (PDE) system is solved by discretization and rearrangement into ODEs. Kinetic models are assessed and the corresponding parameters are estimated using extents of reaction. The estimation of diffusion coefficients follows a tow-steps procedure. First, the extents of mass transfer are computed from measurements in the two bulks. Diffusion coefficients are then estimated individually by fitting each extent of mass transfer to the extent obtained by solving the corresponding PDE. Comparison of the estimated diffusion coefficients with their literature values serves to validate the models identified in the two bulks. The estimation of both kinetic parameters and diffusion coefficients is investigated for gas-liquid reaction systems with unsteady-state diffusion. The approach is illustrated with simulated examples.

2013

Méthodes numériques liées à la distribution d'alumine dans une cuve d'électrolyse d'aluminium

Thomas Frédéric Hilke

Metallic aluminium plays a key role in our modern economy. Primary metallic aluminium is produced by the transformation of aluminium oxide using the Hall-Héroult industrial process. This process, which requires enormous quantities of energy, consists in performing the electrolysis of an aluminium oxide solute in large pots and with hundreds of thousands of amperes of electrical current. The topic of this thesis is the study of some selected aspects of the modelisation of the electrolysis process from the point of view of numerical simulation. This thesis is divided in two parts. The first part is focused on the numerical modelisation of the alumina powder dissolution and transport in the electrolytic bath as a function of the bath temperature. We provide a mathematical model for the transport and dissolution of the alumina powder, followed by its time and space discretisation by means of a finite element method. Finally, we study the behavior of this numerical model in the case of an industrial electrolysis pot. The second part is devoted to the development of a numerical scheme for the approximation of the fluid flows in an electrolysis pot. The scheme relies on a Fourier basis decomposition of the unknowns. The amplitude of each Fourier component satisfies a partial differential equation which is explicitely derived. The solution of this equation is approximated by means of a finite element method. Finally, the approximate fluid flow obtained with this new method is compared with the solution provided by the reference model in an industrial

EPFL2019

We are interested in the well posedness of quasilinear partial differential equations of order two. Motivated by the study of the Einstein equation in relativity theory, there are a number of works dedicated to the local well-posedness issue for the quasilinear wave equation. We will focus on local well-posedness for the wave equation ; more precisely we are looking at the smallest Sobolev index such that the local well-posedness holds true for initial data in this space. In 2005, D. Tataru and Hart. F. Smith provided the current best upper bound for the smallest index in low dimension. In 1998, Hans Lindblad constructed a counter example for s=3 in dimension three, thus revealing the sharpness of Tataru and Smith's criteria in this dimension. Here, our goal is to obtain sharp counterexamples to local well-posedness for quasilinear wave equations of geometric character. First, we check how the construction by Lindblad translates to dimension two. Next, we shall try to see if a similar breakdown result applies to the vanishing mean curvature problem in Minkowski space. Finally, as a more long term goal, we may try to find explicit singular solutions of this problem, starting with smooth data, by following the constructions of Krieger-Schlag-Tataru.

EPFL2021

This thesis is devoted to the derivation of error estimates for partial differential equations with random input data, with a focus on a posteriori error estimates which are the basis for adaptive strategies. Such procedures aim at obtaining an approximation of the solution with a given precision while minimizing the computational costs. If several sources of error come into play, it is then necessary to balance them to avoid unnecessary work. We are first interested in problems that contain small uncertainties approximated by finite elements. The use of perturbation techniques is appropriate in this setting since only few terms in the power series expansion of the exact random solution with respect to a parameter characterizing the amount of randomness in the problem are required to obtain an accurate approximation. The goal is then to perform an error analysis for the finite element approximation of the expansion up to a certain order. First, an elliptic model problem with random diffusion coefficient with affine dependence on a vector of independent random variables is studied. We give both a priori and a posteriori error estimates for the first term in the expansion for various norms of the error. The results are then extended to higher order approximations and to other sources of uncertainty, such as boundary conditions or forcing term. Next, the analysis of nonlinear problems in random domains is proposed, considering the one-dimensional viscous Burgers' equation and the more involved incompressible steady-state Navier-Stokes equations. The domain mapping method is used to transform the equations in random domains into equations in a fixed reference domain with random coefficients. We give conditions on the mapping and the input data under which we can prove the well-posedness of the problems and give a posteriori error estimates for the finite element approximation of the first term in the expansion. Finally, we consider the heat equation with random Robin boundary conditions. For this parabolic problem, the time discretization brings an additional source of error that is accounted for in the error analysis. The second part of this work consists in the analysis of a random elliptic diffusion problem that is approximated in the physical space by the finite element method and in the stochastic space by the stochastic collocation method on a sparse grid. Considering a random diffusion coefficient with affine dependence on a vector of independent random variables, we derive a residual-based a posteriori error estimate that controls the two sources of error. The stochastic error estimator is then used to drive an adaptive sparse grid algorithm which aims at alleviating the so-called curse of dimensionality inherent to tensor grids. Several numerical examples are given to illustrate the performance of the adaptive procedure.

EPFL2016

Developing structure-oriented models for enzymatic hydrolysis of lignocellulosic biomass

Jessica Charlène Rohrbach

With the unsustainable use of fossil fuels increasing strains on human institutions and ecosystems, the development of a renewable energy alternative is of paramount importance. Second generation biorefineries, based on the production of fuels and chemicals from lignocellulosic biomass, appear as attractive alternative to their non-renewable counterparts. Within the multiplicity of existing valorisation routes, enzymatic hydrolysis of lignocellulosic biomass into its constituent sugars has generated considerable interest, notably in the context of biofuels production. However, major hurdles stemming from the intricate structure of the lignocellulosic feedstocks still impede large-scale deployment of this process. The multiplicity of highly intertwined spatiotemporal factors impacting the enzymatic hydrolysis of lignocellulosic biomass represents a challenge to rationally design efficient hydrolysis process. Here, we propose a theory-based modelling framework relying on pore-diffusion and surface reaction to explore the effects of recognised bottlenecks on the enzymatic hydrolysis of lignocellulosic substrates. The model is based on a set of partial differential equations describing the evolution of the substrate morphology to investigate the interplay between experimental conditions and the physical characteristics of biomass particles as the reaction proceeds. The overall quantity of cellulase present in the hydrolysis mixture is carefully considered to investigate its interplay with the available accessible cellulose surface. Also, non-uniformity in terms of cellulose accessibility and cellulose digestibility are introduced in the model to weight their influence on observed hydrolysis rates. Finally, deactivation mechanisms are considered through unproductive adsorption of cellulases on both cellulose and lignin fractions, with the existence of such phenomena alleged to be critical in the efficiency of the hydrolysis process. Based on predictions of our model, we were able to confirm the critical role of cellulose accessibility, as defined by the combination of particle size, porosity and accessible cellulose surface, in dictating early reaction rates for a range of pretreated beech wood substrates. While high biomass loading should be favoured to improve enzyme penetration in the substrates, high enzyme loadings going beyond the initial number of cellulose adsorption appeared beneficial in notably two cases: (i) to promote internal diffusion in large particles and (ii) counteract undesired enzyme adsorption on lignin. For the latter, a relatively low increase in enzyme loading was sufficient to offset the resulting slowdown. We also showed that the existence of structural heterogeneities, and in particular non-uniform pore volume distribution within the lignocellulosic samples, contribute to the rate slowdown observed at later stage of the hydrolysis, while not explaining it in its entirety. Unproductive adsorption to cellulose, coupled to decrease in the cellulase efficiency at the cellulose surface, appeared as major contributor to the rate slowdown. Overall, we show how the use of a theory-based model can help decouple and evaluate the effects of key factors in the enzymatic hydrolysis of lignocellulosic biomass. As such, our model can help pave the way towards efficient integrated rational design strategies for enzyme process engineering for biomass conversion.

EPFL2021