Post on 16-Oct-2020
Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
Spectral finite elements for a mixed
formulation in computational acoustics
taking flow effects into account
Manfred Kaltenbacher
in cooperation with
A. Hüppe, I. Sim (University of Klagenfurt), G. Cohen and S. Imperial
(INRIA, Paris) and B. Wohlmuth (TU Munich)
Alps-Adriatic University of Klagenfurt, Austria
Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
Overview
Physical modeling Pierce equation
Acoustic perturbation equation
FE formulation (no flow) Acoustic conservation equations
Mixed formulation
Spectral elements
Comparison to wave equation with pFEM
FE formulation (with flow) Acoustic perturbation equations
Occurring instabilities
Stabilization (flux term and dissipative term)
Application to aeroacoustics
Multi-Model approach
Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
Acoustics in Flowing Media
Euler’s equations
Idea of decomposition
Mean quantities
Alternating quantities (disturbances)
Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
Acoustics in Flowing Media
Pierce equation (just for simple flows)
PML in time domain (Imbo Sim, Poster on Monday)
Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
Acoustics in Flowing Media
Acoustic perturbation equations1
Subset of linearized Euler equations
Support just
acoustic modes
no entropy and vorticity modes
Fully considers
convection
refraction
1R. Ewert and W. Schröder. Acoustic perturbation equations based on flow decomposition via source filtering.
Journal of Computational Physics, 2003
Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
Acoustics (no flow)
Conservation equations
Linear acoustic wave equation
Investigated Methods
- h-FEM → mesh refinement
- p-FEM & s-FEM → increase order of approximation
Wave Equation Acoustic Conservation Equations
Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
Acoustics (no flow)
Mixed formulation for conservation equations
Discrete spaces1
1G. Cohen & S. Fauqueux, Mixed Finite Elements with Mass-Lumping for the Transient Wave Equation
Journal of Computational Acoustics, 2000
Piola transform
Lagrange polynomial space of order N
Mapping:
Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
Acoustics (no flow)
Properties of Piola transform
Preserves the normal component!
Term with gradient
Term with divergence
Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
Acoustics (no flow)
Spectral finite elements
Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
Acoustics (no flow)
Consequences of the Choice of Spaces
Elements
Semidiscrete Galerkin formulation
Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
Acoustics (no flow)
Example
Excitation with a sine pulse of main wavelength λ
Reference solution obtained with h= λ/120 and Δt= 1/(f 200)
Computational mesh with mean element size of λ/5 and Δt<=λ /(2*c)
Time Stepping:
h-FEM & p-FEM: Implicit Newmark scheme
s-FEM: Explicit leapfrog time stepping
Setup Defomed mesh
Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
Acoustics (no flow)
Comparison
Pressure Field
Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
Acoustics (no flow)
Comparison for time domain computations
Conservation equations
Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
Acoustics Perturbation Equation
Formulation
Spaces
Piola transform
Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
Acoustics Perturbation Equation
Semidiscrete Galerkin formulation
Example
Initial condition
Flow
Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
Acoustics in Flowing Media
Results (cartesian grid)
Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
Acoustics in Flowing Media
Results: Long time simulation (Cartesian grid)
Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
Acoustics Perturbation Equation
Formulation
Spaces
Piola transform
Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
Acoustics in Flowing Media
Stabilization
Central flux term
Reverse integration by parts on
Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
Acoustics in Flowing Media
Stabilization
Averaging leads to
Add penalty (dissipative) term
Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
Acoustics in Flowing Media
Results: Long time simulation (Cartesian grid, penalty term)
Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
Acoustics in Flowing Media
Results: Long time simulation (Cartesian grid, penalty + flux term)
Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
Acoustics in Flowing Media
Results: Long time simulation (deformed grid)
Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
Acoustics in Flowing Media
Results: Long time simulation (deformed grid, penalty term)
Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
Acoustics in Flowing Media
Results: Long time simulation (deformed grid, penalty + flux term)
Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
Acoustics in Flowing Media
Results: Long time simulation (deformed grid)
Spurious waves
Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
Acoustics in Flowing Media
Shear flow
Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
CAA (Computational Aeroacoustics)
Air foil
URANS CFD computations (Fluent, Michele Degenaro, AIT, Vienna)
Mach number about 0.3
Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
CAA (Computational Aeroacoustics)
Acoustic sources
Lighthill analogy
Acoustic Perturbation equation (APE)
RHS of Lighthill’s equation
Test function
RHS of APE
Lamb vector
Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
CAA (Computational Aeroacoustics)
Arbitrary flow Without flow
With flow
Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
Multi-Model Approach
General idea
Acoustic perturbation equation Pierce equation
PML layer Non-matching grid interface
(Mortar framework)
Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
Multi-Model Approach
Interface conditions
Continuity of pressure
Continuity of normal component of particle velocity
Lagrange multiplier
Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
Multi-Model Approach
Formulation
Acoustic perturbation equation
Pierce equation
Continuity of pressure in a weak sense
Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
The End
Thank you for
your attention!