Hauptseminar Modelling with and simulation of PDEs
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Lehrstuhl für
Numerische Mathematik
Hauptseminar
Modelling with and simulation of PDEs
A model is a physical, mathematical, or logical representation of a system
entity, phenomenon, or process. A simulation is the implementation of a model
over time. A simulation brings a model to life and shows how a particular
object or phenomenon will behave. It is useful for testing, analysis or training
where real-world systems or concepts can be represented by a model. – cit.
Lehrstuhl für Hauptseminar: Modelling with and simulation of PDEs
Numerische Mathematik
Criteria for passing the seminar
• 45–60 min. slide presentation with discussion round afterwards
• Short handout (2-4 pages) for all participants
• Block seminar
Dates of presentations: to be discussed
Final meeting
• Feedback on presentations
• Feedback on mathematical topics
Lehrstuhl für Hauptseminar: Modelling with and simulation of PDEs
Numerische Mathematik
Dates of meetings: a proposal
Dates:
• Thursday 17 June 2021
• Friday 25 June 2021
• Monday 5 July 2021
• Tuesday 13 July 2021
Time slots:
• 1PM – 4.30PM
• 3.30PM – 7PM
Lehrstuhl für Hauptseminar: Modelling with and simulation of PDEs
Numerische Mathematik
Order of topics
1. Phase separation and transport equations
2. Acoustics and absorbing boundary conditions
3. Isogeometric analysis
4. Fractional calculus
5. Discontinuous Galerkin discretizations
6. Multigrid methods
BW Barbara Wohlmuth wohlmuth@ma.tum.de 03.10.057
LM Laura Melas melas@ma.tum.de 03.10.060
UK Ustim Khristenko khristen@ma.tum.de 03.10.033
TK Tobias Köppl koepplto@ma.tum.de 03.10.035
MM Markus Muhr muhr@ma.tum.de 03.10.036
MR Mabel L. Rajendran rajendrm@ma.tum.de 03.10.058
Lehrstuhl für Hauptseminar: Modelling with and simulation of PDEs
Numerische Mathematik
Finite Volume discretization of the Cahn-Hilliard equation
Phase separation processes of two immiscible fluids at a con-
stant temperature can be modeled by means of the Cahn-
Hilliard equation, which reads as follows:
1
∂t c − ∇ · (M(c)∇µ) + ∇ · (vc) = 0
Pe
c is the concentration of one fluid, v a given velocity field,
M(c) denotes the mobility of the fluid, Pe is the Pecletnumber
and µ represents a flux accounting for the phase separation.
Four steps within a The goal of this project is to implement a numerical so-
phase separation
process of two
lution method for the Cahn-Hilliard equation using e.g.
immiscible fluids. the PDE framework DUNE or MATLAB. In particular, the
performance of the finite volume method is to be examined.
This project can be processed by two students.
Literature:
• Frank, F., Liu, C., Alpak, F. O., & Riviere, B. (2018). A finite volume/discontinuous Galerkin method
for the advective Cahn–Hilliard equation with degenerate mobility on porous domains stemming from
micro-CT imaging. Computational Geosciences, 22(2), 543-563.
• https://www.dune-project.org
Lehrstuhl für Hauptseminar: Modelling with and simulation of PDEs
Numerische Mathematik
Numerical methods for transport equations
The subject of this project is the numerical dis-
cretization of the transport equation:
∂u
+ ∇ · (vu) = S(u, t).
∂t
u stands for a concentration variable modeling a
substance transported by a velocity field v and S is
a source term modeling an external impact. Solv-
ing this PDE numerically poses some interesting
challenges, in particular if the velocities are high.
Students working on this project are supposed to
Transport of a drug in a vascular study and illustrate these challenges.
system.
Literature:
• Quarteroni, Alfio, Riccardo Sacco, and Fausto Saleri. Numerical mathematics. Vol. 37. Springer
Science & Business Media, 2010.
• Kuzmin, Dmitri. A new perspective on flux and slope limiting in discontinuous Galerkin methods for
hyperbolic conservation laws. Computer Methods in Applied Mechanics and Engineering 373 (2021):
113569.
• LeVeque, Randall J. Numerical methods for conservation laws. (1992).
Lehrstuhl für Hauptseminar: Modelling with and simulation of PDEs
Numerische Mathematik
Acoustics and absorbing boundary conditions
Thinking of the simulation of e.g. an earthquake
at a specific location, one does not wish (or can)
simulate the wave propagation around the whole
Fig: Wave reflection vs. Transparent
globe. Instead one will truncate the simulation
boundary domain Ω at some artificial (!) boundary. Now,
what happens if the seismic wave hits that bound-
ary? Classical boundary conditions like Dirich-
let or Neumann result in reflections that are -
for artificial boundaries - unphysical. A trans-
parent boundary condition is needed. Similar to
d’Alembert’s formula for waves, such conditions
can be derived by decomposing a wave into in-
bound and outbound parts.
Fig: Artificial truncation of Ω
Literatur:
• Kaltenbacher, Manfred. Numerical simulation of mechatronic sensors and actuators. Vol. 2. Berlin:
Springer, 2007.
• Shevchenko, I., and B. Wohlmuth. ”Self-adapting absorbing boundary conditions for the wave
equation.” Wave motion 49.4 (2012): 461-473.
Lehrstuhl für Hauptseminar: Modelling with and simulation of PDEs
Numerische Mathematik
Isogeometric analysis
Isogeometric analysis aims to bring together com-
puter aided design (CAD) and finite element anal-
ysis (FEA). Coming from CAD, objects in engi-
neering are often represented by spline-surfaces
or volumes. IGA makes use of that by choosing
no classical polynomials but the same splines as
Fig: B-spline FEM-basisfunctions
basisfunctions not just for the geometry represen-
tation but also the FEA. This yields advantages
like exact geometry representation (of e.g. a cir-
cle) or easily accessable high order finite elements.
Disadvantages like denser matrices or restrictions
to tensorial grids have to be taken into account.
Fig: 3D spline geometry with
displacement
Literatur:
• Cottrell, J. Austin, Thomas JR Hughes, and Yuri Bazilevs. Isogeometric analysis: toward integration of
CAD and FEA. John Wiley & Sons, 2009.
• Vázquez, Rafael. ”A new design for the implementation of isogeometric analysis in Octave and Matlab:
GeoPDEs 3.0.” Computers & Mathematics with Applications 72.3 (2016): 523-554.
Lehrstuhl für Hauptseminar: Modelling with and simulation of PDEs
Numerische Mathematik
Isogeometric analysis & Acoustics
Some possible topics related to isogeometric analysis ...
• Foundations of B-splines, NURBS and isogeometric finite elements
→ Relates to (projective) geometry, FEM-meshing theory and practice
→ Geometry representation and scalar PDE solutions with GeoPDEs
• Linear elasticity - continuum mechanics and structural analysis
→ Relates to vector analysis, tensors and mathematical modelling
→ Derivation of vectorial deformation PDEs, Simulation using GeoPDEs
... can be mixed very well with acoustic topics (possibly in teams)
• The acoustic wave equation - derivation and numerical methods
→ Relates to mathematical modeling, FD/FEM and time integration
→ Derivation as pressure equation, integration of time dependent PDE
• Absorbing boundary conditions - derivation and implementation
→ Relates to differential calculus and/or more advanced implementations
→ Focus can be put on theory and/or implementation
Lehrstuhl für Hauptseminar: Modelling with and simulation of PDEs
Numerische Mathematik
Fractional calculus
Many natural complex phenomena like long range
interactions, memory or hereditary effects, when
integer order differential models face their limita-
tions, are often modelled with non-integer (frac-
tional) differential equations. Non-locality of such
problems leads to extra computational and mem-
ory costs. So, special techniques have to be
considered. Lack of basic properties like chain
and product rule poses challenges in the analysis.
Two applications are proposed: time-fractional
diffusion (TFD) or fractional Laplacian problems.
Task 1: Show the existence and uniqueness of
Fig: Evolution of time fractional heat
equation the solution for TFD. Task 2: Implement a finite
differences numerical scheme for TFD. Task 3:
Numerical solution of the fractional Laplacian.
Literatur:
• Diethelm, Kai. The Analysis of Fractional Differential Equations: An Application-Oriented Exposition
using Differential Operators of Caputo Type. Springer, 2010.
• Baleanu, Dumitru and Diethelm, Kai and Scalas, Enrico and Trujillo, Juan J. Fractional calculus:
models and numerical methods. Vol. 3. World Scientific, 2012.Lehrstuhl für Hauptseminar: Modelling with and simulation of PDEs
Numerische Mathematik
Discontinuous Galerkin discretizations
Discontinuous Galerkin (DG) discretizations are
based on piecewise polynomial but discontinu-
ous approximations and allow for non-conforming
grids and locally varying polynomial approxima-
tion orders. Moreover, they feature high order ac-
curacy. The built-in flexibility of these schemes is
useful in complex two and three dimensional prob-
Fig: Non-conforming grid lems that feature multi-scale phenomena. On the
other hand, one of the main drawback of the DG
techniques is that the approach is applied elemen-
twise, and therefore the proliferation of degrees of
freedom cannot be kept under control. However
since the elements are discontinuous, DG meth-
ods are highly parallelizable, which is demanding
Fig: DG solution
in complex numerical simulations.
Literatur:
• Rivière. Discontinuous Galerkin methods for solving elliptic and parabolic equations: Theory and
implementation, 2008Lehrstuhl für Hauptseminar: Modelling with and simulation of PDEs
Numerische Mathematik
Algebraic Multigrid
Multigrid methods are among the most efficient
techniques to solve the linear systems of equa-
tions stemming from the discretization of differ-
ential problems. Algebraic Multigrid (AMG) is
a class of multigrid methods that is based on a
hierarchic and purely matrix-based approach. Be-
Fig: Matrix hierarchy
sides the smoothers and the coarse-level correc-
tion strategies at the basis of the V- and W-cycle,
we have to construct suitable coarsening strate-
gies and define how to construct the associated
transfer operators. There are two main techniques
in the literature: classical and smoothed aggrega-
tion algebraic multigrid.
Fig: V-cycle
Literatur:
• Briggs, Henson and Mc Cormick. A multigrid tutorial, 2000
• Vaněk, Mandel and Brezina. Algebraic multigrid by smoothed aggregation for second and fourth order
elliptic problems. Computing , 56(3):179–196. 1996Lehrstuhl für Hauptseminar: Modelling with and simulation of PDEs
Numerische Mathematik
Algebraic Multigrid & Discontinuous Galerkin discretizations
Some possible topics related to discontinuous Galerkin discretizations ...
• Foundations of DG discretizations
→ Non-conforming grids, DG FE discretizations, PDE solutions
→ Stability and convergence analysis
• Linear elasticity and elastodynamics - continuum mechanics
→ Mathematical modelling for static and dynamic problems in continuum
mechanics
Some possible topics related to algebraic multigrid ...
• Foundations of AMG methods
→ Coarsening strategies and transfer operators: classical (C-colouring)
scheme and smoothed aggregation
• AMG as efficient solver for the discretization of PDE problems
→ AMG for conforming/discontinous Galerkin discretizations as stand-alone
solver or preconditioner
→ AMG for elliptic problems (e.g. Laplace, linear elasticity, ...)Lehrstuhl für Hauptseminar: Modelling with and simulation of PDEs
Numerische Mathematik
HyTeG: A matrix-free multigrid solver for HPC
HyTeG is a matrix-free multigrid framework special-
ized in solving finite-element discretizations on tetra-
hedral grids. It uses a hybrid-grid structure, which
combines the efficiency of uniform grids with the flex-
iblity of unstructured grids to take todays hardware to
its limits. Possible projects within this framework are
Fig: A hybrid-grid. • the implementation of a diagonal preconditioner
for a linear Cahn-Hilliard scheme,
• writing a preconditioner for the
Schur-complement of a nonlinear Cahn-Hilliard
scheme or
• implementing the GMRES method inside of
HyTeG.
Fig: Uniform grids allow stencils. Strong background in C++ is recommended!
Literatur:
• Kohl, Nils, et al. ”The HyTeG finite-element software framework for scalable multigrid solvers.”
• Brenner, Susanne C., et al. ”A robust solver for a mixed finite element method for the Cahn–Hilliard
equation.”You can also read
