MathMods :: Joint MSc

 

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This is our 2020 curriculum. For the new structure, valid as of the 2021 intake, click here

Sem3 TWU Applied PDEs for 2020 intake

Sem3 TWU Applied PDEs for 2020 intake

Applications  @  TUW  30 ECTS credits

Advanced modelling and numerics for applied PDEs

The specialization track “Advanced modelling and numerics for Applied PDEs” is specifically devoted to a selected methodological area which is of great impact in engineering: partial differential equations (PDEs). The reference group at TUW provides a more advanced knowledge concerning numerical methods compared to Semester 2. This is done in the two compulsory units “Numerical simulations and scientific computing” and “High performance computing & Introduction to Python programming”. At the same time, the compulsory unit “Modelling with PDEs” provides advanced background on the derivation of continuum PDE models in several domains of physics and engineering. The students will then have a wide set of optional choices, either in the direction of computational fluid mechanics (“Numerical Methods in Fluid Dynamics”, “Finite Element Methods for Multi- Physics”, “Computing Turbulent Flows with CFD codes”) or towards a deeper theoretical background (“Functional Analysis 2”) or probabilistic methods (“Stochastic Analysis”). The MSc thesis at TUW can be done in collaboration with the group held by Prof. Jan Maas at IST Austria (associated partner).

 

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
    • University Vienna University of Technology
    • Semester 3
    • Lecturer 1 J. Schöberl
    • 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
    • University Vienna University of Technology
    • Semester 3
    • Lecturer 1 J. L. Träff
    • Lecturer 2 W. Karel
    • 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
    • University Vienna University of Technology
    • Semester 3
    • Lecturer 1 A. Arnold
    • Lecturer 2 M. Melenk
    • 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
    • University Vienna University of Technology
    • Semester 3
    • Lecturer 1 H. C. Kuhlmann
    • 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
    • University Vienna University of Technology
    • Semester 3
    • Lecturer 1 M. Kaltenbacher
    • 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
    • University Vienna University of Technology
    • Semester 3
    • Lecturer 1 H. Steinrück
    • 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
    • University Vienna University of Technology
    • Semester 3
    • Lecturer 1 Schmock
    • 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
    • University Vienna University of Technology
    • Semester 3
    • Lecturer 1 H. Woracek
    • Lecturer 2 M. Kaltenbäck
    • Lecturer 3 M. Blümlinger
    • 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
    • University Vienna University of Technology
    • Semester 3

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Home Structure for 2020 intake Sem3 TWU Applied PDEs for 2020 intake

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