Efficient numerical simulation of time-harmonic wave equations - - PowerPoint PPT Presentation

Efficient numerical simulation of time-harmonic wave equations

• Prof. Tuomo Rossi
• Dr. Dirk Pauly

Ph.Lic. Sami K¨ ahk¨

• nen

Ph.Lic. Sanna M¨

• nk¨
• l¨

a M.Sc. Tuomas Airaksinen M.Sc. Anssi Pennanen M.Sc. Jukka R¨ abin¨ a Department of Mathematical Information Technology University of Jyv¨ askyl¨ a, Jyv¨ askyl¨ a, Finland

Collaboration

The research group has active collaboration with, e.g.,

• Dr. Erkki Heikkola

(Numerola Oy)

• Dr. Jari Toivanen

(University of Stanford)

• Prof. Roland Glowinski

(University of Houston)

• Prof. Norbert Weck,
• Prof. Karl Josef Witsch,
• Prof. Axel Klawonn,
• Dr. Oliver Rheinbach

(University of Duisburg-Essen)

On-going research project with Department of Physics, University of Jyv¨ askyl¨ a: efficient simulation methods for modeling the fluid flow in porous materials.

• K. Mattila, J. Hyv¨

aluoma, T. Rossi, M. Aspn¨ as, J. Westerholm, An efficient swap algorithm for the lattice Boltzmann method, Computer Physics Communications (2007).

• K. Mattila, J. Hyv¨

aluoma, J. Timonen, T. Rossi, Comparison of implementations

• f the lattice-Boltzmann method, Computers & Mathematics with Applications

(2008).

Contents

Physical phenomena Mathematical model (partial differential equations) Discretization methods Efficient solvers Simulation results

Applications on various fields of science and engineering

• Fluid dynamics: computational acous-

tics, echo sounding,

• Elastodynamics: deformation of elastic

structures, seismic waves,

• Electromagnetics:

antenna applica- tions,

• Medical and biological systems: mod-

eling the structure of human tissue (medical imaging), cochlea in the in- ner ear.

Example of coupled time-harmonic wave equations: fluid-structure interaction

Structure domain Ωs

Γ Ωs Ωf

Fluid domain Ωf

−ω2ρsus − ∇ · σ(us) = gs in Ωs, −ω2 c2 p − ∇2p = gf in Ωf, ∂p ∂nf + ω2ρfus · ns = 0

• n Γ,

σ(us)ns − pnf = 0

• n Γ.

Numerical simulation

• Simulation tools are used to avoid dangerous, expensive and time-consuming ex-

periments.

• Computer simulation is an efficient tool in testing and optimizing of model param-

eters.

• The design process can be dramatically improved and the development cycle short-

ened with computer aided modeling.

• Solving problems arising from real life applications demands a considerable amount
• f time and memory

– need to use efficient methods, – utilization of modern parallel computing platforms (Playstation 3 CELL-processors, Nvidia CUDA graphics processors).

Discretization methods

• Traditionally: low-order Finite Element Method (FEM),

– Especially for heterogeneous materials, sufficient accuracy requires dense dis- cretization = ⇒ large-scale systems to be solved. – Solutions with high frequency suffer from numerical dispersion. – For time-harmonic wave equations it is challenging to construct efficient iterative solution techniques.

• Novel approaches:

– element-based methods with higher-order polynomial basis, e.g., Spectral Element Method (SEM), – Discrete Exterior Calculus (DEC).

Efficient solution techniques

• Fictitious domain,

– The computational domain is embedded into a larger and simpler domain. – a tetrahedral finite element mesh, which is locally adapted to the bound- ary of the obstacle D [B¨

• rgers -90].

– linear finite elements with mass lump- ing.

Wave scattering by a submarine, 200 wavelenghts per diameter, 20 nodes per wavelenght.

• Domain decomposition,

– The computational domain is divided into subdomains. – Instead of solving the whole PDE-system, the problems in the subdomains are solved separately. – The sequence of the subproblems converge to the solution of the original prob- lem.

• Multigrid preconditioners

– Shifted-Laplacian operator (damped Helmholtz operator), BSL = −∇ ·

1 ρ(x)∇ − (β1 + β2i) k(x)2 ρ(x) , as a preconditioner. Vibrations in a crankshaft

• Exact controllability

– Searching for the periodic solution of the time-dependent wave equations. – Initial conditions e0 and e1 are the control variables, and we minimize

J(e0, e1,Y(e0, e1)) = 1 2

• YN − e0

T K

• YN − e0
• + 1

2

∂YN

∂t − e1

T

M

∂YN

∂t − e1

• .

– Y contains the nodal values and solves the system of wave equations in [0, T].

Recent/selected journal publications

• T. Airaksinen, E. Heikkola, A. Pennanen, J. Toivanen, An algebraic multigrid based shifted-Laplacian preconditioner for

the Helmholtz equation, Journal of Computational Physics (2007).

• T. Airaksinen, E. Heikkola, J. Toivanen, Active noise control in a stochastic domain based on a finite element model,

Journal of Sound and Vibration (submitted).

• T. Airaksinen, A. Pennanen, J. Toivanen, A damping preconditioner for time-harmonic wave equations in fluid and

elastic material, Journal of Computational Physics (2009).

• T. Airaksinen, S. M¨
• nk¨
• l¨

a, Comparison between shifted-Laplacian preconditioning and controllability method for com- putational acoustics, Journal of Computational and Applied Mathematics (to appear).

• R. Glowinski, T. Rossi, A mixed formulation and exact controllability approach for the computation of the periodic

solutions of the scalar wave equation. (I) Controllability problem formulation and related iterative solution’, C. R.

• Math. Acad. Sci. Paris, (2006).
• E. Heikkola, S. M¨
• nk¨
• l¨

a, A. Pennanen, T. Rossi, Controllability method for acoustic scattering with spectral elements, Journal of Computational and Applied Mathematics (2007).

• E. Heikkola, S. M¨
• nk¨
• l¨

a, A. Pennanen, T. Rossi, Controllability method for the Helmholtz equation with higher order discretizations, Journal of Computational Physics (2007).

• S. M¨
• nk¨
• l¨

a, Time-harmonic solution for acousto-elastic interaction with controllability and spectral elements, Journal

• f Computational and Applied Mathematics, (to appear).
• D. Pauly, Generalized Electro-Magneto Statics in Nonsmooth Exterior Domains, Analysis, (2007).
• D. Pauly, Complete Low Frequency Asymptotics for Time-Harmonic Generalized Maxwell Equations in Nonsmooth

Exterior Domains, Asymptotic Analysis, (2008).