1 Modelling of thin and imperfect interfaces Public Licentiate Thesis Defence, Wednesday 13 May 2018 Respondent: Mathieu Gaborit, KTH & University of Le Mans External examiner: Associate Professor Emeline Sadoulet-Reboul, Institute FEMTO-ST, France Internal examiner: Associate Professor Karl Bolin, KTH Supervisors: Professor Olivier Dazel, University of Le Mans Professor Peter Göransson, KTH
Modelling of thin and imperfect interfaces Tools and preliminary study Opponent: Émeline Sadoulet-Reboul (FEMTO-ST, France) Examiner: Karl Bolin (KTH, Sweden) Advisors: Peter Göransson (KTH, Sweden) & Olivier Dazel (LAUM, France) M. Gaborit — Licentiate Thesis — 13 th June 2018
Context • Co-tutelle agreement between Le Mans Université and KTH • LAUM UMR CNRS 6613 ( Olivier Dazel ) • Marcus Wallenberg Lab. ( Peter Göransson ) • 6-months periods • Funding: • Le Mans Acoustique • Contributions from MWL • ÖMSE grant • Support from DENORMS COST Action 15125 • Free software: • github.com/cpplanes • github.com/OlivierDAZEL 2 • github.com/matael
Content 1. Interfaces and thin layers 2. FEM-based hybrids 3. Follow up 3
Interfaces and thin layers
Many problems but merely one solution • Noise is known to cause health-related issues 1 • Teams are working to improve our knowledge of bulk media What about the influence of the bonding zones ? 1 Babisch, W. The Noise/Stress Concept, Risk Assessment and Research Needs. Noise & Health 2002;4:1–11 World Health Organisation. Burden of Disease from Environmental Noise: Quantification of World Health Organization Europe, 2011. 106 pp. 4 • Many sound packages in use are based on poroelastic laminates Healthy Life Years Lost in Europe. Ed. by Theakston, F. OCLC: 779684347. Copenhagen, Denmark:
Far away A thin layer can be seen as a localised element Living on the edge (of the media) Interfaces are by-products of the assembly process. Interface or layer? Close by An interface zone can be thought of as a thin layer 5 Here, they are represented by mathematical relations .
Far away A thin layer can be seen as a localised element Living on the edge (of the media) Interfaces are by-products of the assembly process. Interface or layer? Close by An interface zone can be thought of as a thin layer 5 Here, they are represented by mathematical relations .
Far away A thin layer can be seen as a localised element Living on the edge (of the media) Interfaces are by-products of the assembly process. Interface or layer? Close by An interface zone can be thought of as a thin layer 5 Here, they are represented by mathematical relations .
Living on the edge (of the media) Interfaces are by-products of the assembly process. Interface or layer? Close by An interface zone can be thought of as a thin layer 5 Here, they are represented by mathematical relations . Far away A thin layer can be seen as a localised element
Working on thin layers to get to interfaces Same characteristics • They change the response 2 2 Chevillotte, F. Controlling Sound Absorption by an Upstream Resistive Layer. Applied Acoustics 2012;73:56–60. 6 • It is hard to control how they are made • Their parameters are hard to measure
Working on thin layers to get to interfaces Same characteristics 2012;73:56–60. 2 Chevillotte, F. Controlling Sound Absorption by an Upstream Resistive Layer. Applied Acoustics 6 • They change the response 2 • It is hard to control how they are made • Their parameters are hard to measure 1 . 0 Absorption Coefficient 0 . 8 0 . 6 0 . 4 0 . 2 0 . 0 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (Hz)
Working on thin layers to get to interfaces Same characteristics • They change the response 2 The advantages of films • Thickness is small enough (typically between 0.05 and 0.5 mm). 2 Chevillotte, F. Controlling Sound Absorption by an Upstream Resistive Layer. Applied Acoustics 2012;73:56–60. 6 • It is hard to control how they are made • Their parameters are hard to measure • Many sorts of films • Experimental data available for comparison
Standard tools Tools to model acoustic systems exist, let’s try to build upon them: They have their limitations: 3 Some works slightly extend the scope 7 TMM for laminates (sound packages, etc. ) FEM detailed domains and complex geometries TMM (mostly) limited to infinite plane laminates 3 FEM hates abrupt changes of mesh refinement level
FEM and small scale elements FEM works best on meshes without too much distortion The problem is the same for tiny embedded elements ... 8
FEM-based hybrids
FEM-based hybrids Both papers B and C propose to mix different methods based on the existing scale separation Paper B Thin layers removed from the FEM domain Paper C FEM usage limited to dense, detailed areas In both cases, other methods are used for the remaining regions: • Transfer Matrix Method, • Discontinuous Galerkin Method with Plane Waves 4 4 Gabard, G. Discontinuous Galerkin Methods with Plane Waves for Time-Harmonic Problems. Journal of Computational Physics 2007;225:1961–1984. 9
FEM-based hybrids Paper B: Condensing thin layers 10
From Bloch waves to FEM Meta-poroelastic ? Periodic inclusions, PEM matrix, thin coatings. ? Incident R 0 R 1 R 1 Coatings Core (FEM) 11
From Bloch waves to FEM Meta-poroelastic ? Periodic inclusions, PEM matrix, thin coatings. ? Incident R 0 R 1 Coatings Core (FEM) 11 R − 1
• Propagate R l through the coatings • Solve for f and R l in q From Bloch waves to FEM • Plane incidence, Bloch reflections • Assemble FEM domain • Add transferred Bloch load • Balance with orthogonality ? Incident R 0 R 1 R l FEM 12 R − 1
• Solve for f and R l in q From Bloch waves to FEM • Plane incidence, Bloch reflections • Assemble FEM domain • Add transferred Bloch load • Balance with orthogonality ? Incident R 0 R 1 R l FEM 12 • Propagate R l through the coatings R − 1
• Solve for f and R l in q From Bloch waves to FEM • Plane incidence, Bloch reflections • Assemble FEM domain • Add transferred Bloch load • Balance with orthogonality ? Incident R 0 R 1 R l FEM 12 • Propagate R l through the coatings R − 1 Af = b
• Solve for f and R l in q From Bloch waves to FEM R 0 f A C FEM R l • Plane incidence, Bloch reflections R 1 Incident ? • Balance with orthogonality • Add transferred Bloch load • Assemble FEM domain 12 • Propagate R l through the coatings R − 1 ] { } [ = b q ( R l )
• Solve for f and R l in q From Bloch waves to FEM R l b 0 0 b f • Plane incidence, Bloch reflections C A FEM 12 R 1 R 0 Incident ? • Balance with orthogonality • Add transferred Bloch load • Assemble FEM domain • Propagate R l through the coatings R − 1 [ ] { } { } { } = + C ′ A ′ q ( R l ) b ′
From Bloch waves to FEM R l b 0 0 b f • Plane incidence, Bloch reflections C A FEM 12 R 1 • Add transferred Bloch load R 0 Incident ? • Balance with orthogonality • Assemble FEM domain • Propagate R l through the coatings • Solve for f and R l in q R − 1 [ ] { } { } { } = + C ′ A ′ q ( R l ) b ′
Results — Comparison to pure FEM 0.2mm The reference is pure TMM . . fails before the hybrid Pure FEM 20mm 13 TMM Reference Coatings in FEM Proposed Method 1 . 0 D 0 . 8 Absorpion Coefficient 0 . 6 0 . 4 0 . 2 0 . 0 0 1000 2000 3000 4000 5000 Frequency (Hz)
FEM-based hybrids Paper C: Protecting small elements 14
A (not so) lovely example 15
A (not so) lovely example 16
A (not so) lovely example 16
A (not so) lovely example 16
A (not so) lovely example n 16
Coupling strategy v n Natural coupling that preserves convergence properties! S in p R S out v n p v y v x p vector, FEM has primary/secondary fields n DGM with Plane Waves uses a state S out S in p v y v x p v n p v y v x 17
Coupling strategy v x S in S out DGM with Plane Waves uses a state vector, FEM has primary/secondary fields v n p v y v y p v n S out R p S in Natural coupling that preserves convergence properties! p v x n p v x v y 17 v n p { }
Coupling strategy v x S in S out DGM with Plane Waves uses a state vector, FEM has primary/secondary fields n v n p v y v y p v n S out R p S in Natural coupling that preserves convergence properties! p 17 v x p v n v y p v x { } [ C 1 ] = [ C 2 ] { }
Coupling strategy 0 0 0 1 n v n p n x n y 0 0 vector, FEM has primary/secondary fields 1 v x v y p v n S out R p S in Natural coupling that preserves convergence properties! 1 17 DGM with Plane Waves uses a state p v y v x p v n v x p v y S out S in ] [ ] { } [ = { }
Coupling strategy v x n n y 0 0 0 1 v y v n p v n S out R p S in Natural coupling that preserves convergence properties! p n x 17 S out p S in 1 v x p v n DGM with Plane Waves uses a state vector, FEM has primary/secondary fields 1 0 p v y v x 0 v y ] [ ] { } [ =
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