Master in Computational Science and Applications University of Rennes 1. Courses of Master 1 and Master 2 (in red : mandatory courses)

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1 Master in Computational Science and Applications University of Rennes 1 Courses of Master 1 and Master 2 (in red : mandatory courses)

2 Master 1 Computational science and modelling Numerical methods Lectures : 24h, Tutorials : 24h, Practical classes : 12h ECTS : 6 credits Semester 1 Objectives: To learn and apply numerical methods useful for solving partial differential equations and for Finite Element Methods Root-finding algorithms: dichotomy, principle of contraction, Newton-Raphson,... Approximation of functions: interpolation, piecewise polynomial interpolation, splines, ideas of FEM Numerical integration: elementary and composed quadratures, Newton-Cotes formula, Gauss quadratures, Richardson extrapolation, Romberg method, acceleration of convergence Differential equations: theory and numerical methods, application to population dynamics Matrix algorithms: factorisation, direct and iterative methods to solve linear systems and compute eigenvalues Assessment: continuous, one final paper exam in December

3 Optimization Lectures : 12h, Tutorials : 12h, Practical classes : 6h ECTS : 3 credits Semester 1 Objectives: to learn to recognize and to model optimization problems in various fields, to be able to choose good methods to solve optimization problems. Definition of optimization problems, necessary and sufficient conditions for optimability Algorithms for problems with and without constraints : gradient methods (constant step, optimal step, conjugate gradient), Lagrange multipliers, Uzawa's algorithm Introduction to linear programming: linear optimization, duality, solution of linear programming problems, simplex algorithm Introduction to graph theory: representation, algorithms for shortest path Network flow and linear programming: simplex algorithm for minimum-cost networkflow problems in networks, transport problems Assessment: continuous, one final paper exam in December

4 Operational research Lectures : 12h, Tutorials : 12h, Practical classes : 6h ECTS : 3 credits Semester 1 Objectives: to learn to recognize and to model optimization problems in various fields, to be able to choose good methods to solve optimization problems. Definition of optimization problems, necessary and sufficient conditions for optimability Algorithms for problems with and without constraints : gradient methods (constant step, optimal step, conjugate gradient), Lagrange multipliers, Uzawa's algorithm Introduction to linear programming: linear optimization, duality, solution of linear programming problems, simplex algorithm Introduction to graph theory: representation, algorithms for shortest path Network flow and linear programming: simplex algorithm for minimum-cost networkflow problems in networks, transport problems Assessment: continuous, one final paper exam in December

5 Scientific programming Lectures: 18h, Tutorials: 12h, Computer classes: 18h ECTS: 6 credits Semester 1 Objectives: To learn fundamental concepts on computers for their efficient use in numerical simulation; to be able to implement algorithms in C language (e.g. prime numbers, non-linear equations, quadratures), to apply numerical methods to solve problems from physics (planetology, quantum mechanics). To be able to test codes and valide numerical results. Participation to computer classes is essential for individual coaching and learning good programming practise. Syllabus, teaching method (flipped classes), assessment methods Introduction: Numerical simulation, HPC, Top 500, state of the art. Representation of numbers, consequences Linux: A bit of history, basic commands Emacs C language: Structure of a program, data types, condition, loops; Pointers, functions, inputs and outputs; Bit to bit operators and applications; Headers; Typedef; Structures; Matrices; Memory management, dynamical allocation Development of programs to solve mathematical problems of increasing complexity, application to integration of ODE modelling mechanical systems and physical phenomena, development of larger codes including several modules and files Introduction to software quality and optimization Individual and group projects Assessment: continuous assessment of practical exercices, MCQ on Moodle, code development on paper, final exam on paper without computer.

6 Dynamic of beam structures Lectures : 18h, Tutorials : 18h, Practical classes : 12h ECTS : 6 credits Semester 1 Objectives: To be able to model and predict static and dynamic behavior of any onedimensional structures Revision of basic concepts of Continuum Mechanics Cinematic hypotheses, constitutive law and equilibrium equations Wave and vibrations in strings: free vibrations, forced vibrations, modal analysis, orthogonality of vibration modes Deformation power and conjugate variables of a beam Waves and vibrations in straight beams: Timoshenko's model, Euler-Bernouilli's model, free vibrations, forced vibrations, modal analysis, orthogonality of vibration mode (Sturm-Liouville theorem) Waves and vibrations in curved beams. 3D elastic waves. Longitudinal waves, shear waves, Helmholtz's decomposition Viscous damping. Discretisation by virtual power principle, decomposition in a modal basis. Applications to trees. Assessment: continuous

7 Differential equations and transport phenomena Lectures : 24h, Tutorials : 24h ECTS : 6 credits Semester 1 Objectives: the goal of this lecture is to discuss the resolution of ordinary differential equations from a statistical physics point of view, namely through the use of transport equations, seen as partial differential equations governing the collective behaviour of many particles' systems. It is proved indeed that one can describe such systems either by ordinary differential equations posed in large dimensions, or by transport equations posed in low dimensional spaces. We discuss these aspects from a modelling point of view, but also from a functioanl analytic viewpoint. We also discuss numerical aspects. Differential equations Cauchy-Lipschitz theorem, flux of a vector field, numerical methods Introduction to partial differential equations Transport equations (method of characteristics, invariants) Finite difference methods (consistancy, order, stability, Lax equivalence theorem, Von Neumann's analysis) Statistical physics Assessment: continuous

8 Finite elements methods for computational science Lectures : 24h, Practical classes : 24h ECTS : 6 credits Semester 2 Objectives: To be able to write the weak formulation of partial differential equation problems and solve it using Finite Elements Methods, to develop a FEM code, to modify an existing FEM computer package, to choose adapted type of finite elements, to analyse errors and convergence Introduction to the variational formulation of partial differential equation problems, existence and uniqueness of solutions, convergence of the FEM Analysis and construction of a FEM code for the discretisation of a 2D boundary problem How to choose the variational formulation and the finite elements. P1, P2, Q1, Q2 finite elements How to use some functionalities of the MELINA research library Meshes and numbering Elementary calculations using reference element (e.g. mass and rigidity matrices) Assembling sparse matrices Essential boundary conditions Solution of linear systems with sparse matrices Construction of a main program in C language and a directive file to control functionalities of the MELINA library Graphical plot of results Assessment: continuous, group project

9 Numerical solution of partial differential equations Lectures : 24h, Tutorials : 24h, Practical classes : 12h ECTS : 6 credits Semester 2 Objectives: Modelling of classical physical phenomena using partial differential equations, definition of the mathematical equations and numerical resolution. Introduction to fundamental models: stationary (Laplace equation), propagation (transport, waves), diffusion (heat) 2D Laplace equation:variational formulation for different boundary problems, application of Lax-Milgram theorem, FEM 1D transport equation: method of characteristics, finite difference methods Assessment: continuous, project

10 Multiphysics modelling Lectures : 18h, Practical classes : 18h ECTS : 3 credits Semester 2 Objectives: to teach competences needed in high technology to solve efficiently coupled problems Partial differential equations: elliptic, parabolic, hyperbolic Conservation equations. Interpretation and modelling Diffusioon and propagation mechanisms. Coupled mechanical systems Application to physical systems: heat diffusion, diffusion-convection, wave equations. Diffusion-convection-propagation coupling Resolution of equations with the FEM Comsol Multiphysics Solution of integrated problems with different couplings: diffusion of particles in a fluid, thermomechanical problems, wave diffraction, diphasic flows (gas-fluid and particle-fluid). Magneto-elastic coupling (piezo-electric) Assessment: continuous, project

11 Fluids and solids in interaction Lectures : 24h, Tutorials : 24h, Practical classes : 18h ECTS : 6 credits Semester 2 Objectives: To be able to model systems with several degrees of freedom with various types of active and passive connections, using Kane-Levinson formalism. I. Vibration of Plates and Shells. 1. Plates in plane motion. Plates out-of-plane motion. 2. Axisymmetric shells. Circular cylindrical shells. II. Fluid-structure interaction. 1. Linear approximation of the fluid equations. Sound waves. 2. Linear sloshing dynamics. Vibroacoustic coupling. 3. Inertial coupling in continuous systems. Modal projection. Added mass. III. Numerical method with Comsol Multiphysics software. 1. Finite element Method in modal analysis and dynamics. 2. Modal projection in dynamics. IV. Various applications. 1. Coupling dynamics in Microfluidics. Biomimetics. 2. Viscous damping in liquid containers. Free and forced oscillations. 3. Acoustic radiation of structures (cantilever beams, plates and shells). Assessment: continuous, project.

12 Project Duration : 2 months ECTS: 3 credits Semester 2 Objectives: The project is a first experience in a professional environment, with application of theoretical concepts learnt during the Master 1 on a larger scale problem. It can take place in a research laboratory at the University of Rennes 1 (mainly Institute of Mathematics, Institute of Physics) or elsewhere (IFREMER, INRA, CHU de Nantes) or in a private company (e.g. Armen Instruments, EDF), under the joint supervision of a tutor and an academic of the University of Rennes 1. : Bibliographical study of new topics; modeling; transcription in mathematical equations; finding solutions by analytical or numerical methods; learning or improving practice of a numerical tools or programming language; validation and critical analysis of results; developing perspectives. The minimum duration is 2 full months, from the end of April to the end of June, after the final assessments in April. The project can be extended over July and August, until the start of the Master 2. Assessment: report (20 pages + 10 pages appendix), 15 minute presentation + 10 minutes discussion with the jury.

13 Master 2 Computational science and modelling Object programming in C++ Lectures: 24h, Computer classes: 24h ECTS : 6 credits Semester 3 Objectives: Knowledge of the main features of object-oriented programming and their usage in C++ langage. Creation of programs and handling of existing C++ packages in an autonomous way. Fundamental concepts of object programming C++ : objects and classes, encapsulation, templates, hierarchy and inheritance, polymorphism, input/output Advanced concepts of object programming in C++ : exceptions, generic programming Standard library STL Modification of an existing C++ package Assessment: one exam on paper and on computer, individual project

14 Practice of finite element computer packages Lectures: 18h, Computer classes: 30h ECTS : 6 credits Semester 3 Objectives: To introduce the mathematical key concepts for an efficient use of Finite Elements software packages in various problems issued from modelling; to train students in the practise of such Finite Elements software packages used in research and companies. The course focuses on the weak (or variational) formulations of partial differential equation problems. A special emphasis is placed on the way boundary and interface conditions are taken into account. Notice that weak formulation of boundary value problems is a pre-requisite to the use of the Finite Element Method. Other practical aspects of time dependent or non linear problems are also covered. We also consider other aspects of the Finite Element discretisation such as the choice of the Finite Elements subspaces, the properties of the linear system resulting from the discretisation, and the choice of the linear system solver. All these topics are studied through classical problems issued from electromagnetism, optics, structural mechanics, fluid mechanics, thermal stress, etc. Numerical modelling of physical devices using Ansys Setting up a calculation, numerical methods, geometry, choice of finite elements, meshing, calculation, analysis of outputs Modelling of a physical problem, exploiting symmetry, adapted finite elements and meshes Critical analysis of results and validity Applications to problems in mechanics, thermics, electricity, fluids, and couplings between theses fields Introduction to Ansys Workbench Assessment: continuous, project, one paper exam; project with Ansys

15 Mathematical tools for computer-aided geometric design Lectures: 24h, Computer classes: 24h ECTS : 6 credits Semester 3 Objectives: Knowledge of fundamental principles of CAGD environments One variable case: Bézier curves B-splines, spline curves and associated algorithms, NURBS Applications: interpolation, approximation Two variables case: Polynomials on rectangles (tensor product) or triangles ; representation in the Bernstein basis Interpolation, finite elements, interpolation error Splines defined as tensor products of one-variable splines Polynomial splines defined by triangular pieces: construction of surfaces globally of class C1 by assembling local Hermite finite elements Implementation of certain methods using Fortran or C language, graphical tools, and Matlab for pre- and post-processing Assessment: paper exam, project

16 Parallel computing and on GPU Lectures: 12h, Computer classes: 16h ECTS : 3 credits Semester 3 Objectives: Introduction to programming scientific codes on parallel machines and on graphical processors Basic principles Open-MP MPI Open CL / CUDA Applications in individual projects Assessment: continuous, projects, paper exam

17 Mathematical modelling of propagation phenomena Lectures: 24h, Tutorial: 24h ECTS : 6 credits Semester 3 Objectives: To model physical propagation phenomena using partial differential equations, to study characteristics of corresponding mathematical problems, to determine adequate numerical approaches to solve problems Models of wave propagation in electromagnetism, optics, elastodynamics, acoustic, surface waves Propagation in closed and open wave guides, diffraction of electromagnetic waves by an object, formation of hydraulic jumps Modelling, mathematical study, numerical simulation, applications Existence, unicity of solutions, spectral problems, radiation condition, Riemann problem, Numerical solution using finite differences, finite volumes, finite elements and integral equation methods Several subjects will be studied in more details

18 Inverse problems Lectures: 16h, Computer classes: 16h ECTS : 3 credits Semester 3 Objectives: Introduction to concepts and methods in inverse problems with application to real cases Optimisation and inversion Matrix methods to solve linear and quasi-linear problems Regression as an inverse problem Underdetermined and over-determined problems (Generalised least squares, Moore- Penrose Inverse) Tomography as an inversion problem Uncertainty on model parameters Monte-Carlo Markov chain as an inversion tool for non-linear problems Assessment: coursework, project

19 Modelling in Earth science Lectures: 12h, Computer classes: 12h ECTS : 3 credits Semester 3 Objectives: Introduction to numerical modelling and simulation in geophysics for nonspecialists, no prerequisite in geophysics is required. Why do we need modelling in geosciences? Geodynamics and Plate tectonics Fundamental equations used in Geosciences (thermal solver, Stokes solver) Simple study cases and analytical solutions Using numerical methods to understand geological processes Project: resolution of a geological problem using a FD 2D numerical code (e.g.: evolution of a subduction plate in the Earth mantle). Assessment: project

20 Machine learning for biology Lectures : 18h ECTS : 3 crédits Semester : 3 Objectives: Use, validate, compare Machine Learning tools Use Big Data modules in R and/or Python Linear model for regression Visualization of data in many dimensions Linear methods for supervised classification, aggregation of models, random forests, boosting, kernel methods, VSM,... Graphical methods Neuronal networks, deep learning Assessment : group project

21 Professional integration Lectures : 16h, Tutorials : 10, Practical classes: 8h ECTS : 3 crédits Semester : 3 Objective : competences and techniques to enhance recruitment success in the field of computational science 1) Human ressources Competence portfolio, definition of a professional project CV, application letters, personal file Professional network, modern tools of communication Monitoring job prospection Simulation of a job interview 2) Scientific English Why and how to communicate in science Writing up an abstract Preparing a scientific talk Presentation of a personal scientific research project in English, simulation of a seminar

22 Case study Lectures : 12h, Computer classes : 12h ECTS : 3 crédits Semester : 4 Objectives: To solve a real scale problem from start to finish : scientific background, motivation, mathematical analysis of the model, choice of numerical methods, code development, numerical simulation, presentation of results, validation, interpretation. Every year, a different subject will be proposed by a partner from a scientific institution or a private company : rocket science (CNES PERSEUS project), impact of tsunamis, medical imaging, optronics,...)

23 Thermomechanics and applications Lectures: 24h, Tutorials: 12h ECTS : 3 credits Semester 3 Objectives: To promote understanding of nonlinear behavior of materials under mechanical and thermal loadings To engage interest in coupling effects between mechanics and biology of living tissues Finite transformations of material elements: curves, surfaces and volumes Conservation laws: mass, momentum, angular momentum, energy Constitutive laws and general principles: entropy inequality, generalised standard materials Hyperelastic media: application to isotropic and anisotropic materials Coupled thermoelasticity: classical models, influence on wave propagation Biomechanics Assessment: continuous, project

24 Estimation of parameters and optimization Lectures: 14h, Tutorials: 14h, Computer classes: 8h ECTS : 3 credits Semester 4 Objectives: Introduction to estimation of parameters and optimisation of dynamical systems Basic tools and convex optimisation methods for control and identification problems Theory of optimal control and identification of linear and non-linear dynamical systems, Pontryagin maximum principle, Hamilton-Jacobi theory Numerical methods for the resolution of control problems, application in physics, chemistry and biology Assessment: paper exam, group project

25 Monte-Carlo molecular simulation Lectures: 16h, Computer classes: 8h ECTS : 3 credits Semester 4 Objectives: Molecular simulations to predict macroscopic properties in fundamental and applied science and for the understanding of microscopic processes in matter. No pre-requisite of physics is necessary. Fundamental concepts of molecular and mesoscopic simulation: molecular dynamics and Monte-Carlo method Statistical physics, micro-macro transition, multi-scale simulations, calculation of macroscopic properties: free enthalpy, surface tension, diffusion coefficient, viscocity, adsorption properties Exploring microscopic processes: radial distribution, average force potential, density profile, etc Direct application to nanotechnology: confinement of fluids in nanopores; phase transitions, greenhouse gas capture and removal Assessment: project

26 Project and internship Duration: 4 to 6 months, from mid-march to end of September ECTS : 18 credits Semester 4 Objectives: Preparation and first long-term experience in a professional environment The student defines an internship in modelling and computational science in a private company or large research centre. Many subjects are provided by the Master CSA network of industrial partners. The students are allowed to choose an internship in an autonomous way as long as the objectives and content of the project correspond to those of the Master's degree. An university tutor is assigned to the project. The student prepares a 2 month project with the tutors, from the company/research centre and from the university, to complete his education in order to be fully operational on the first day of the internship. The preparation might include: bibliography, computing languages, computer softwares. It must also include a first case study: modelling, development of mathematical and/or numerical solution of a relevant subject. The student writes a report (20 pages + annexes) and gives an oral presentation (15 minutes) before the start of the internship. If the report or presentation is not satisfactory, the student must restart the process. During the internship, the student must send monthly reports to the university tutor. A visit on site or by video-conference is organized in the middle of the internship. Assessment: The student writes a final report (50 pages + annexes) and gives an oral presentation (30 minutes).