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MOOC# Numerical Analysis for Engineers

Description

Ce cours contient les 7 premiers chapitres d'un cours donné aux étudiants bachelor de l'EPFL. Il est basé sur le livre "Introduction à l'analyse numérique", J. Rappaz M. Picasso, Ed. PPUR. Il aborde des outils de base et la question de la résolution numérique d'équations différentielles.

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Related concepts (99)

Related courses (119)

Related publications (226)

Lectures in this MOOC (64)

Finite element method

The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. The FEM is a general numerical method for solving partial differential equations in two or three space variables (i.e., some boundary value problems).

Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts.

Free surface

In physics, a free surface is the surface of a fluid that is subject to zero parallel shear stress, such as the interface between two homogeneous fluids. An example of two such homogeneous fluids would be a body of water (liquid) and the air in the Earth's atmosphere (gas mixture). Unlike liquids, gases cannot form a free surface on their own. Fluidized/liquified solids, including slurries, granular materials, and powders may form a free surface. A liquid in a gravitational field will form a free surface if unconfined from above.

The student will learn state-of-the-art algorithms for solving differential equations. The analysis and implementation of these algorithms will be discussed in some detail.

Ce cours présente une introduction aux méthodes d'approximation utilisées pour la simulation numérique en mécanique des fluides.
Les concepts fondamentaux sont présentés dans le cadre de la méthode d

Continuum conservation laws (e.g. mass, momentum and energy) will be introduced. Mathematical tools, including basic algebra and calculus of vectors and Cartesian tensors will be taught. Stress and de

Interpolation: Problem StatementMOOC: Numerical Analysis for Engineers

Introduces the problem of interpolation and demonstrates challenges with adding more points.

The Wrong Method: Polynomial InterpolationMOOC: Numerical Analysis for Engineers

Explores the inefficiencies of the incorrect method for polynomial interpolation.

Lagrange Interpolation: Case 2MOOC: Numerical Analysis for Engineers

Explains Lagrange interpolation with M=2, covering base polynomials and linear independence.

Interpolation de Lagrange: General CaseMOOC: Numerical Analysis for Engineers

Explains the Lagrange interpolation method for arbitrary n and constructing polynomials through given points.

Interpolation of a Continuous FunctionMOOC: Numerical Analysis for Engineers

Explores the interpolation of a continuous function by a polynomial.

Removing geometrical details from a complex domain is a classical operation in computer aided design for simulation and manufacturing. This procedure simplifies the meshing process, and it enables faster simulations with less memory requirements. However, depending on the partial differential equation that one wants to solve in the geometrical model of interest, removing some important geometrical features may greatly impact the solution accuracy. For instance, in solid mechanics simulations, such features can be holes or fillets near stress concentration regions. Unfortunately, the effect of geometrical simplification on the accuracy of the problem solution is often neglected, because its analysis is a time-consuming task that is often performed manually, based on the expertise of engineers. It is therefore important to have a better understanding of the effect of geometrical model simplification, also called defeaturing, to improve our control on the simulation accuracy along the design and analysis phase.In this thesis, we formalize the process of defeaturing, and we analyze its impact on the accuracy of solutions of some partial differential problems. To achieve this goal, we first precisely define the error between the problem solution defined in the exact geometry, and the one defined in the simplified geometry. Then, we introduce an a posteriori estimator of the energy norm of this error. This allows us to reliably and efficiently control the error coming from the addition or the removal of geometrical features. We subsequently consider a finite element approximation of the defeatured problem, and the induced numerical error is integrated to the proposed defeaturing error estimator. In particular, we address the special case of isogeometric analysis based on (truncated) hierarchical B-splines, in possibly trimmed and multipatch geometries. In this framework, we derive a reliable a posteriori estimator of the overall error, i.e., of the error between the exact solution defined in the exact geometry, and the numerical solution defined in the defeatured geometry.We then propose a two-fold adaptive strategy for analysis-aware defeaturing, which starts by considering a coarse mesh on a fully-defeatured computational domain. On the one hand, the algorithm performs classical finite element mesh refinements in a (partially) defeatured geometry. On the other hand, the strategy also allows for geometrical refinement. That is, at each iteration, the algorithm is able to choose which missing geometrical features should be added to the simplified geometrical model, in order to obtain a more accurate solution.Throughout the thesis, we validate the presented theory, the properties of the aforementioned estimators and the proposed adaptive strategies, thanks to an extensive set of numerical experiments.

Aluminium is a metal sought in the industry because of its various physical properties. It is produced by an electrolysis reduction process in large cells. In these cells, a large electric current goes through the electrolytic bath and the liquid aluminium. This electric current generates electromagnetic forces that set the bath and the aluminium into motion. Moreover, large quantities of carbon dioxide gas are produced through chemical reactions in the electrolytic bath: the presence of these gases alleviates the density of the liquid bath and changes the dynamics of the flow. Accurate knowledge of this fluid flow is essential to improve the efficiency of the whole process.The purpose of this thesis is to study and approximate the interaction of carbon dioxide with the fluid flow in the aluminium electrolysis process.In the first chapter of this work, a mixture-averaged model is developed for mixtures of gas and liquid. The model is based on the conservation of mass and momentum equations of the two phases, liquid and gas. By combining these equations, a system is established that takes into account the velocity of the liquid-gas mixture, the pressure, the gas velocity and the local gas concentration as unknowns.In the second chapter, a simplified problem is studied theoretically. It is shown that under the assumption that the gas concentration is small, the problem is well-posed. Moreover, we prove a priori error estimates of a finite element approximation of this problem.In the third chapter, we compare this liquid-gas model with a water column reactor experiment. Finally, the last chapter shows that the fluid flow is changed in aluminium electrolysis cells when we take into account the density of the bath reduced by carbon dioxide. These changes are quantified as being of the order of 30% and explain partially the differences between previous models and observations from Rio Tinto Aluminium engineers.

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.