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!