Modelling of thin and imperfect interfaces Public Licentiate Thesis - - PowerPoint PPT Presentation

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

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Modelling of thin and imperfect interfaces

Tools and preliminary study

• M. Gaborit — Licentiate Thesis — 13th June 2018

Opponent: Émeline Sadoulet-Reboul (FEMTO-ST, France) Examiner: Karl Bolin (KTH, Sweden) Advisors: Peter Göransson (KTH, Sweden) & Olivier Dazel (LAUM, France)

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/matael
• github.com/cpplanes
• github.com/OlivierDAZEL

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Content

• 1. Interfaces and thin layers
• 2. FEM-based hybrids
• 3. Follow up

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Interfaces and thin layers

Many problems but merely one solution

• Noise is known to cause health-related issues1
• Many sound packages in use are based on poroelastic laminates
• Teams are working to improve our knowledge of bulk media

What about the influence of the bonding zones ?

1Babisch, 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 Healthy Life Years Lost in Europe. Ed. by Theakston, F. OCLC: 779684347. Copenhagen, Denmark: World Health Organization Europe, 2011. 106 pp. 4

Living on the edge (of the media)

Interfaces are by-products of the assembly process. Here, they are represented by mathematical relations. Interface or layer? Close by An interface zone can be thought of as a thin layer Far away A thin layer can be seen as a localised element

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Living on the edge (of the media)

Interfaces are by-products of the assembly process. Here, they are represented by mathematical relations. Interface or layer? Close by An interface zone can be thought of as a thin layer Far away A thin layer can be seen as a localised element

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Living on the edge (of the media)

Interfaces are by-products of the assembly process. Here, they are represented by mathematical relations. Interface or layer? Close by An interface zone can be thought of as a thin layer Far away A thin layer can be seen as a localised element

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Living on the edge (of the media)

Interfaces are by-products of the assembly process. Here, they are represented by mathematical relations. Interface or layer? Close by An interface zone can be thought of as a thin layer Far away A thin layer can be seen as a localised element

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Working on thin layers to get to interfaces

Same characteristics

• It is hard to control how they are made
• Their parameters are hard to measure
• They change the response2

2Chevillotte, F. Controlling Sound Absorption by an Upstream Resistive Layer. Applied Acoustics

2012;73:56–60. 6

Working on thin layers to get to interfaces

Same characteristics

• It is hard to control how they are made
• Their parameters are hard to measure
• They change the response2

500 1000 1500 2000 2500 3000 3500 4000 Frequency (Hz) 0.0 0.2 0.4 0.6 0.8 1.0 Absorption Coefficient 2Chevillotte, F. Controlling Sound Absorption by an Upstream Resistive Layer. Applied Acoustics

2012;73:56–60. 6

Working on thin layers to get to interfaces

Same characteristics

• It is hard to control how they are made
• Their parameters are hard to measure
• They change the response2

The advantages of films

• Many sorts of films
• Thickness is small enough (typically between 0.05 and 0.5 mm).
• Experimental data available for comparison

2Chevillotte, F. Controlling Sound Absorption by an Upstream Resistive Layer. Applied Acoustics

2012;73:56–60. 6

Standard tools

Tools to model acoustic systems exist, let’s try to build upon them: TMM for laminates (sound packages, etc.) FEM detailed domains and complex geometries They have their limitations: TMM (mostly) limited to infinite plane laminates3 FEM hates abrupt changes of mesh refinement level

3Some works slightly extend the scope

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FEM and small scale elements

FEM works best on meshes without too much distortion The problem is the same for tiny embedded elements...

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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 Waves4

4Gabard, 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

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From Bloch waves to FEM

Meta-poroelastic ? Periodic inclusions, PEM matrix, thin coatings.

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Incident R0 R

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R1 Coatings Core (FEM)

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From Bloch waves to FEM

Meta-poroelastic ? Periodic inclusions, PEM matrix, thin coatings.

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Incident R0 R−1 R1 Coatings Core (FEM)

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From Bloch waves to FEM

• Plane incidence, Bloch reflections
• Propagate Rl through the coatings
• Assemble FEM domain
• Add transferred Bloch load
• Balance with orthogonality
• Solve for f and Rl in q

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Incident R0 R−1 R1 Rl FEM

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From Bloch waves to FEM

• Plane incidence, Bloch reflections
• Propagate Rl through the coatings
• Assemble FEM domain
• Add transferred Bloch load
• Balance with orthogonality
• Solve for f and Rl in q

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Incident R0 R−1 R1 Rl FEM

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From Bloch waves to FEM

• Plane incidence, Bloch reflections
• Propagate Rl through the coatings
• Assemble FEM domain
• Add transferred Bloch load
• Balance with orthogonality
• Solve for f and Rl in q

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Incident R0 R−1 R1 Rl FEM

Af = b

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From Bloch waves to FEM

• Plane incidence, Bloch reflections
• Propagate Rl through the coatings
• Assemble FEM domain
• Add transferred Bloch load
• Balance with orthogonality
• Solve for f and Rl in q

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Incident R0 R−1 R1 Rl FEM

[ A C ] { f q(Rl) } = b

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From Bloch waves to FEM

• Plane incidence, Bloch reflections
• Propagate Rl through the coatings
• Assemble FEM domain
• Add transferred Bloch load
• Balance with orthogonality
• Solve for f and Rl in q

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Incident R0 R−1 R1 Rl FEM

[ A C C′ A′ ] { f q(Rl) } = { b } + { b0 b′ }

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From Bloch waves to FEM

• Plane incidence, Bloch reflections
• Propagate Rl through the coatings
• Assemble FEM domain
• Add transferred Bloch load
• Balance with orthogonality
• Solve for f and Rl in q

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Incident R0 R−1 R1 Rl FEM

[ A C C′ A′ ] { f q(Rl) } = { b } + { b0 b′ }

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Results — Comparison to pure FEM

1000 2000 3000 4000 5000 Frequency (Hz) 0.0 0.2 0.4 0.6 0.8 1.0 Absorpion Coefficient TMM Reference Coatings in FEM Proposed Method

0.2mm 20mm D

Pure FEM fails before the hybrid . The reference is pure TMM .

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FEM-based hybrids

Paper C: Protecting small elements

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A (not so) lovely example

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A (not so) lovely example

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A (not so) lovely example

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A (not so) lovely example

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A (not so) lovely example

n

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Coupling strategy

n vx vy p vn p vx vy p Sin Sout DGM with Plane Waves uses a state vector, FEM has primary/secondary fields vn p vx vy p vn Sout R p Sin Natural coupling that preserves convergence properties!

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Coupling strategy

n      vx vy p      { vn p } vx vy p Sin Sout DGM with Plane Waves uses a state vector, FEM has primary/secondary fields vn p vx vy p vn Sout R p Sin Natural coupling that preserves convergence properties!

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Coupling strategy

n      vx vy p      { vn p } vx vy p Sin Sout DGM with Plane Waves uses a state vector, FEM has primary/secondary fields [C1] { vn p } = [C2]      vx vy p      vn Sout R p Sin Natural coupling that preserves convergence properties!

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Coupling strategy

n      vx vy p      { vn p } vx vy p Sin Sout DGM with Plane Waves uses a state vector, FEM has primary/secondary fields [ 1 1 ] { vn p } = [ nx ny 1 ]      vx vy p      vn Sout R p Sin Natural coupling that preserves convergence properties!

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Coupling strategy

n vx vy p vn p vx vy p Sin Sout DGM with Plane Waves uses a state vector, FEM has primary/secondary fields [ 1 1 ] { vn p } = [ nx ny 1 ]      vx vy p      vn Sout R p Sin Natural coupling that preserves convergence properties!

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Coupling strategy

n vx vy p vn p      vx vy p      = [

• ]{

Sin Sout } DGM with Plane Waves uses a state vector, FEM has primary/secondary fields [ 1 1 ] { vn p } = [ nx ny 1 ]      vx vy p      vn Sout R p Sin Natural coupling that preserves convergence properties!

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Coupling strategy

n vx vy p vn p      vx vy p      = [

• ]{

Sin Sout } DGM with Plane Waves uses a state vector, FEM has primary/secondary fields [ 1 1 ] { vn p } = [ nx ny 1 ]      vx vy p      { vn Sout } = [R] { p Sin } Natural coupling that preserves convergence properties!

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Coupling strategy

n vx vy p vn p      vx vy p      = [

• ]{

Sin Sout } DGM with Plane Waves uses a state vector, FEM has primary/secondary fields [ 1 1 ] { vn p } = [ nx ny 1 ]      vx vy p      { vn Sout } = [R] { p Sin } Natural coupling that preserves convergence properties!

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Follow up

Stochastics around the corner Modelling of thin and imperfect interfaces

The next two years will be on the stochastic side and more in depth

• n modelling coupling conditions

There is a need for simpler models with few parameters and equations

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Stochastics around the corner Modelling of thin and imperfect interfaces

The next two years will be on the stochastic side and more in depth

• n modelling coupling conditions

There is a need for simpler models with few parameters and equations

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Stochastics around the corner Modelling of thin and imperfect interfaces

The next two years will be on the stochastic side and more in depth

• n modelling coupling conditions

There is a need for simpler models with few parameters and equations

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Stochastics around the corner Modelling of thin and imperfect interfaces

The next two years will be on the stochastic side and more in depth

• n modelling coupling conditions

There is a need for simpler models with few parameters and equations

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Follow up

Paper A: Simplified model for thin films

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Transfer matrix and jumps

Fields on each side of a layer are related by a transfer matrix. This matrix can be written as a matrix exponential: kd 1 and a 1st

• rder expansion makes it a set of jumps:

s 0 I d s d

+ hypotheses on fields decoupling

Sparser matrix! x z d ≪ 1

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Transfer matrix and jumps

Fields on each side of a layer are related by a transfer matrix. This matrix can be written as a matrix exponential: kd ≪ 1 and a 1st

• rder expansion makes it a set of jumps:

s(0) ≈ (I − dα) s(d)

+ hypotheses on fields decoupling

Sparser matrix! x z d ≪ 1

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Transfer matrix and jumps

Fields on each side of a layer are related by a transfer matrix. This matrix can be written as a matrix exponential: kd ≪ 1 and a 1st

• rder expansion makes it a set of jumps:

s(0) ≈ (I − dα) s(d)

+ hypotheses on fields decoupling → Sparser matrix!

x z d ≪ 1

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Transfer matrix and jumps

Fields on each side of a layer are related by a transfer matrix. This matrix can be written as a matrix exponential: kd ≪ 1 and a 1st

• rder expansion makes it a set of jumps:

s(0) ≈ (I − dα) s(d)

+ hypotheses on fields decoupling → Sparser matrix!

20

Transfer matrix and jumps

Fields on each side of a layer are related by a transfer matrix. This matrix can be written as a matrix exponential: kd ≪ 1 and a 1st

• rder expansion makes it a set of jumps:

s(0) ≈ (I − dα) s(d)

+ hypotheses on fields decoupling → Sparser matrix!

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Results

Rigid backing High angle (thin assumption) PEM + Rigid backing High angle (thin assumption) + Resonances (mechanical effects) In the following graphs : the lighter, the better.

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Results

Rigid backing High angle (thin assumption) PEM + Rigid backing High angle (thin assumption) + Resonances (mechanical effects) In the following graphs : the lighter, the better.

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Results

500 1000 1500 2000 2500 3000 3500 4000 Frequency (Hz) 20 40 60 80 Incidence Angle θ 0.697 1.394 2.091 2.788 ×10−2 500 1000 1500 2000 2500 3000 3500 4000 Frequency (Hz) 20 40 60 80 Incidence Angle θ 1.355 2.711 4.066 5.421 ×10−3

Rigid PEM + Rigid

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Follow up

What to expect next

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Towards the PhD

Multiple collaborations started since the beginning will continue More to come on:

• local effects at interfaces
• homogenisation of boundary conditions
• uncertainties propagation in laminates

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Towards the PhD

Multiple collaborations started since the beginning will continue More to come on:

• local effects at interfaces
• homogenisation of boundary conditions
• uncertainties propagation in laminates

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Thanks! — Tack! — Merci !

gaborit@kth.se

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