Spectral finite elements for a mixed formulation in computational acoustics taking flow effects into account

Transcription

1 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

2 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

3 Acoustics in Flowing Media Euler s equations Idea of decomposition Mean quantities Alternating quantities (disturbances)

4 Acoustics in Flowing Media Pierce equation (just for simple flows) PML in time domain (Imbo Sim, Poster on Monday)

5 Acoustics in Flowing Media Acoustic perturbation equations 1 Subset of linearized Euler equations Support just acoustic modes no entropy and vorticity modes Fully considers convection refraction 1 R. Ewert and W. Schröder. Acoustic perturbation equations based on flow decomposition via source filtering. Journal of Computational Physics, 2003

6 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

7 Acoustics (no flow) Mixed formulation for conservation equations Discrete spaces 1 Lagrange polynomial space of order N Mapping: Piola transform 1 G. Cohen & S. Fauqueux, Mixed Finite Elements with Mass-Lumping for the Transient Wave Equation Journal of Computational Acoustics, 2000

8 Acoustics (no flow) Properties of Piola transform Preserves the normal component! Term with gradient Term with divergence

9 Acoustics (no flow) Spectral finite elements

10 Acoustics (no flow) Consequences of the Choice of Spaces Elements Semidiscrete Galerkin formulation

11 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

12 Acoustics (no flow) Comparison Pressure Field

13 Acoustics (no flow) Comparison for time domain computations Conservation equations

14 Acoustics Perturbation Equation Formulation Spaces Piola transform

15 Acoustics Perturbation Equation Semidiscrete Galerkin formulation Example Initial condition Flow

16 Acoustics in Flowing Media Results (cartesian grid)

17 Acoustics in Flowing Media Results: Long time simulation (Cartesian grid)

18 Acoustics Perturbation Equation Formulation Spaces Piola transform

19 Acoustics in Flowing Media Stabilization Central flux term Reverse integration by parts on

20 Acoustics in Flowing Media Stabilization Averaging leads to Add penalty (dissipative) term

21 Acoustics in Flowing Media Results: Long time simulation (Cartesian grid, penalty term)

22 Acoustics in Flowing Media Results: Long time simulation (Cartesian grid, penalty + flux term)

23 Acoustics in Flowing Media Results: Long time simulation (deformed grid)

24 Acoustics in Flowing Media Results: Long time simulation (deformed grid, penalty term)

25 Acoustics in Flowing Media Results: Long time simulation (deformed grid, penalty + flux term)

26 Acoustics in Flowing Media Results: Long time simulation (deformed grid) Spurious waves

27 Acoustics in Flowing Media Shear flow

28 CAA (Computational Aeroacoustics) Air foil URANS CFD computations (Fluent, Michele Degenaro, AIT, Vienna) Mach number about 0.3

29 CAA (Computational Aeroacoustics) Acoustic sources Lighthill analogy Test function RHS of Lighthill s equation Acoustic Perturbation equation (APE) Lamb vector RHS of APE

30 CAA (Computational Aeroacoustics) Arbitrary flow Without flow With flow

31 Multi-Model Approach General idea Non-matching grid interface (Mortar framework) PML layer Acoustic perturbation equation Pierce equation

32 Multi-Model Approach Interface conditions Continuity of pressure Continuity of normal component of particle velocity Lagrange multiplier

33 Multi-Model Approach Formulation Acoustic perturbation equation Pierce equation Continuity of pressure in a weak sense

34 The End Thank you for your attention!