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 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 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 Acoustic Conservation Equations Wave Equation Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
Acoustics (no flow) Mixed formulation for conservation equations Lagrange polynomial space of order N Discrete spaces 1 Mapping: Piola transform 1 G. Cohen & S. Fauqueux, Mixed Finite Elements with Mass-Lumping for the Transient Wave Equation Journal of Computational Acoustics, 2000 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 Defomed mesh Setup 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 Test function RHS of Lighthill’s equation Lamb vector Acoustic Perturbation equation (APE) RHS of APE 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 PML layer Non-matching grid interface (Mortar framework) Acoustic perturbation equation Pierce equation 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! Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment, RICAM, 2011
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