MathMods :: Joint MSc

Sem3 TWU Applied PDEs

Sem3 TWU Applied PDEs

Applications  @  TUW  30 ECTS credits

Advanced modelling and numerics for applied PDEs

TBD

 

Below you can find information about the subjects for this semester.

  • Numerical simulation and scientific computing [6 credits]

    Numerical simulation and scientific computing

    • ECTS credits 6
    • Semester 3
    • University Vienna University of Technology
    • Objectives

       

      This course will provide the students with the essential basic skills to tackle the myriad challenges in computational science and engineering. As such, the course is highly interdisciplinary as it covers several different topics in applied mathematics, scientific computing and computer science. (Due to the interdisciplinary nature, each/some topical block(s) could be handled by several departments; this would also help in balancing the load introduced by the examination which this course will certainly require. However, given the topics, probably 1-2 Institutes should be able to handle this, but it could be extended to involve more Institutes if desired.)

    • Topics

       

      This course will cover basics in computer architectures, performance optimization, scientific programming with C/C++/Python, partial differential equations, finite difference method, solvers, research software engineering, shared- and distributed-memory parallelization with OpenMP and MPI on multi-core and cluster systems, ray tracing and surface tracking methods.


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  • High performance computing & Introduction to Python programming [6 credits]

    High performance computing & Introduction to Python programming

    • ECTS credits 6
    • Semester 3
    • University Vienna University of Technology
    • Objectives

       

      Basic knowledge of current HPC architectures and communication networks, problems and challenges. Mastering advanced features of MPI and/or other interfaces for HPC. Knowledge of problems, algorithms, solutions and tools for HPC.

    • Topics

       

      Overview of current HPC architectures and communication networks, problems, algorithms and solutions (with project/exercises); advanced MPI programming (with project/exercise), tools, performance models, libraries (with project/exercise). Python overview. Packages explained. Python basics (Variables, Control Structures, Functions, ...). Important numerical packages (Numpy, Scipy, Pandas, Matplotlib). IPython and the IPython/Jupyter. Notebook. File I/O (Text files, NetCDF, HDF5, GeoTIFF, Shapefiles). Iterators, Generators, Object oriented programming.


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  • Modelling with PDEs [6 credits]

    Modelling with PDEs

    • ECTS credits 6
    • Semester 3
    • University Vienna University of Technology
    • Objectives

       

      Introduction to several applied differential equation models: discussion of modelling, analytic and numerical aspects.

    • Topics

       

      Fluid dynamical models (Euler, Navier-Stokes, vortex models), traffic flow models, theory of elasticity, hyperbolic conservation laws, nonlinear waves (KdV, solitons, invers scattering theory), image processing models (nonlinear diffusion filter, shock filter), models for pattern formation (reaction-diffusion equations, Turing instability), evolution of thin films.


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  • Numerical methods in fluid dynamics [5 credits]

    Numerical methods in fluid dynamics

    • ECTS credits 5
    • Semester 3
    • University Vienna University of Technology
    • Objectives

       

      Introduction to advanced methods of computational fluid dynamics

    • Topics

       

      Methods for a treatment of convection-diffusion equations, projection methods for incompressible and compressible Navier-Stokes equations, treatment of complex geometries, turbulence modelling.


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  • Finite element methods for multi-physics [5 credits]

    Finite element methods for multi-physics

    • ECTS credits 5
    • Semester 3
    • University Vienna University of Technology
    • Objectives

       

      In this lecture the current state of numerical simulation using the Finite-Element (FE) method for coupled field problems, which appear within the design process of modern mechatronic systems (e.g., sensors and actuators), is presented. In combination to the theory of magnetic, mechanical and acoustic fields with all their couplings, we will discuss practical applications occuring during the design and optimization of modern mechatronic systems (numerical simulation of electromagnetic actuators, of piezoelectric positioning drives, vibrational induced sound generated by machines and automobiles, etc.)

    • Topics

       

      The lecture starts with a detailed discussion of the Finite-Element (FE) method including the main aspects for computer implementation. Thereby, a python program will be available, which contains the main routines for computation of mechanical problems. In a next step, the current state of numerical simulation using the Finite-Element (FE) method for coupled field problems, which appear within the design process of modern sensors and actuators, is presented. In combination to the theory of magnetic, mechanical and acoustic fields with all their couplings, we will discuss practical applications occurring during the design and optimization of modern mechatronic systems (numerical simulation of electromagnetic actuators, of piezoelectric positioning drives, vibrational induced sound generated by machines and automobiles, etc.).


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  • Calculating Turbulent Flows with CFD-Codes [3 credits]

    Calculating Turbulent Flows with CFD-Codes

    • ECTS credits 3
    • Semester 3
    • University Vienna University of Technology
    • Objectives

       

      The aim of this course is an introduction to the application of CFD programs for the calculation of simple turbulent flows. This course will provide a basis for the further processing of projects using CFD methods and will be jointly supervised by representatives of various working groups using CFD methods. Therefore, an introduction to various CFD programs (FLUENT, CCM +, OpenFOAM) is also given.

    • Topics

       

      Application and comparison commercial CFD program (FLUENT, CCM +) with an open source software program (OpenFOAM) in the calculation of (turbulent) flows. The effect of different turbulence models on the flow or heat transfer will be investigated and the results will be verified by data from the literature. Geometry creation, grid generation, adjustment and selection of flow models and parameters, turbulence models and wall treatment, presentation and interpretation of results.


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  • Stochastic analysis in financial and actuarial mathematics [7 credits]

    Stochastic analysis in financial and actuarial mathematics

    • ECTS credits 7
    • Semester 3
    • University Vienna University of Technology
    • Objectives

       

      Introduction to stochastic analysis as needed for continuous-time financial and actuarial mathematics.

    • Topics

       

      Definition and properties of multi-dimensional normal distribution, definition and elementary properties of Brownian motion, existence and Hölder continuity of Brownian motion using the Kolmogorov-Chentsov continuity criterion, filtrations, stopping times, progressive measurability, path properties, martingales, uniform integrability, Vitali's convergence theorem, sub- and supermartingales, maximum inequality, Doob's inequality for p-integrable submartingales, Doob's optional sampling theorem with applications, local martingales and examples, integration of predictable step processes, p-variation of functions, quadratic variation and covariation process of continuous local martingales, Kunita-Watanabe inequality, stochastic integration for continuous local martingales and generalization for continuous semimartingales.


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  • Functional analysis 2 [6 credits]

    Functional analysis 2

    • ECTS credits 6
    • Semester 3
    • University Vienna University of Technology
    • Objectives

       

      Solution of mathematical problems by functional analytic methods.

    • Topics

       

      Banach Algebras, Spectral theory of linear operators, spectral resolution for normal and unbounded selfadjoint operators, semi-groups of operators, Hille-Yoshida theorem, distributions, selected topics.


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  • German (level A2.1) [2 credits]

    German (level A2.1)

    • ECTS credits 2
    • Semester 3
    • University Vienna University of Technology

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Home Program structure Semester 3 TUW Advanced mod. and numerics for applied PDEs

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University of L'Aquila, Italy (UAQ)

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Autonomous University of Barcelona, Catalonia - Spain (UAB)

Departament de Matemàtiques, Edifici Cc - Campus UAB 08193 Bellaterra – Catalonia

Hamburg University of Technology, Germany (TUHH)

Institute of Mathematics
Schwarzenberg-Campus 3, Building E-10
D-21073 Hamburg - Germany

University of Nice - Sophia Antipolis, France (UNS)

Laboratoire J.A.Dieudonné
Parc Valrose, France-06108 NICE Cedex 2

Vienna Univ. of Technology, Austria (TUW)

Technische Universität Wien
Institute of Analysis & Scientific Computing
Wiedner Hauptstr. 8, 1040 Vienna - Austria

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