baci Webseite...FSI Problem Definition – 3-Field-Setup Kinematic condition Momentum balance see e.g. Park, Felippa, Ohayon (2001) Advanced Topics in Computational Fluid-Structure

Advanced Topics inComputational Fluid-Structure Interaction

– from Topological Changes to Bioengineering –

Institute for Computational Mechanics @ Technische Universität München

Lehrstuhl für Numerische Mechanik

Wolfgang A. WallWolfgang A. Wall& & thethe LNM LNM teamteam

B. Svetnik (2006)

EU Regional School 2010RWTH Aachen

June 6 & 7, 2010

Advanced Topics in Computational Fluid-Structure Interaction - from Topological Changes to BioengineeringWolfgang A. Wall – Institute for Computational Mechanics, TU München – http://www.lnm.mw.tum.de

Outline

• A fixed-grid approach for large deformations and topological changes

Fixed-grid FSI apprach based on XFEMA new approach for enforcing boundary conditionsEmbedded fluid meshes – hybrid FSIFluid-structure-contact interaction

• Fluid-Structure Interaction in Bioengineering


Some current FSI applications @ LNM

Respiratrory biomechanics Cellular biomechanics Cardiovascular biomechanics

Aeroelasticity/membrane wings Mesoscopic (bio-)physics TFSI in rocket nozzles

Brought to you by … bacithe parallel,

multiphysics&

multiscale researchsoftware

developed & © byInstitute for Computational Mechanics (LNM), TUM


Introduction: Deforming Vs. Fixed Fluid Grid

F Deforming fluid grid

Deforming fluid domain:

Fixed fluid grid

Deforming structure domain: Deforming structure grid


Goals

Develop a fixed-grid fluid formulation for 3D FSI & beyond

• No compromises for the structure description… Lagrangian formulation, thin-walled or bulky, material,

deformation modes, …

• Full quality at the interface• (Re-)Use of established FSI coupling schemes• Finite Element method for fluid and structure• Freedom in fluid & structural mesh sizes• Contact within FSI• Parallel, iterative solution using algebraic multigrid

techniques (AMG)

Implement in parallel in-house (LNM) multifield/multiscale research code BACI


FSI Problem Definition

Fluid-Structure-Interface Surface Coupled 2-field Problem!

Fluid

+ constitutive equations, BC

Structure

+ constitutive equations, BC


FSI Problem Definition – 3-Field-Setup

Kinematic condition

Momentum balance

see e.g. Park, Felippa, Ohayon (2001)Advanced Topics in Computational Fluid-Structure Interaction - from Topological Changes to BioengineeringWolfgang A. Wall – Institute for Computational Mechanics, TU München – http://www.lnm.mw.tum.de

From Explicit Surfaces to Embedded Interfaces

explicit fluid surface embedded discontinuity

boundary condition internal condition


Discontinuities by the XFEM

Original formulation for cracks in structures:• N. Moёs, J. Dolbow, T. Belytschko, 1999• T. Belytschko & T. Black, 19992D FSI formulations:• A. Legay, J. Chessa, T. Belytschko, 2006• A. Gerstenberger, W.A. Wall, 2008

Extended Finite Element Method (XFEM):

Enrich FE space (PUM) with appropriate functions to enhance solution:

In particular, we have jumps over the FSI interface Enrich with Heaviside function:


Enrichment strategy: thick structures

velocity field pressure field


Discontinuities by the XFEM

Extended Finite Element Method (XFEM):

Enrich FE space to extend approximation capabilities:

N. Moёs, J. Dolbow, T. Belytschko, 1999, T. Belytschko & T. Black, 1999

FE shape function

Enrichment function

Enrichment strategy for FSI:

bulky structures in contactbulky structures thin structures (multiple fluids)

Numerical integration:

Scenarios:


Example: thin structures

Standard DOFLeft Surface DOF

Right Surface DOF

Pressure Solution


Example thin structures


Example of a flow against a thin wall


Discontinuities by the XFEM - Enrichment

Implemented for linear andhigher order elements!

• OctTree-based detection of intersected elements

• Boundary triangulation (tri3,tri6)

• Domain tetrahedralization (tet4,tet10)

• Application of standard and enriched DOFs

U.M. Mayer, A. Gerstenberger, W.A. Wall, 2009

SolidFluid


Stabilized Fluid Formulation (nothing new here)

Time-discrete weak form:

Primary unknowns: velocity & pressure• equal-order interpolation

Stabilized, time-discrete weak form (here SUPG/PSPG/LSIC):

Time-discrete Eulerian Navier-Stokes eqs. (OST, BDF2):


Weak Form + Weak Interface Condition

Time-discret weak form (One-Step-Theta):

Challenges in 3D:• Approximation order• Interface mesh creation

2-field Fluid Formulation + Lagrange Multiplier (LM)velocity, pressure, traction:& test functions:


Interlude: Linear-Elastic Material

Strong form + material eqs.:

Weak form with classical Lagrange multiplier:

Global stiffness matrix:

Nodal unknowns: , interface nodal unknowns: , LMP approx.: ?

(+ stab. terms)


Approaches to Embedded Dirichlet Conditions

Proposed approach:

Weak form from generalized Hellinger-Reissner functional:

Nitsche’s method (adapted from Dolbow & Harari 2008):

Classical Lagrange multiplier approach:

(+ stab. terms)


Proposed Approach for Linear-Elastic Material

Uncut elements: decoupled element stresses, standard element stiffness

Weak form after integration by parts:

Intersected elements: modified element stiffness

Nodal unknowns: , element unknowns:

- stress approx.: C-1 elementwise disc.

- Q1Q-1/Q2Q-2/P1P-1 …


Convergence analysis: heat conduction equation

2d & 3d sine shaped body source leads to sine shaped solution

Heat conduction between concentric cylinders: exp., radial temp.


Interface Conditions

Stress Lagrange Multiplier

Weak form:

hybrid fluid stress:Traction Lagrange Multiplier

Weak form:

interface traction:

Interface condition: given interface velocity

- Saddle point structure - Requires interface mesh- Stable 3D Approximation?

+ No saddle point structure+ No additional interface mesh+ Stable (numerical experience)

A. Gerstenberger, W.A. Wall, 2008 A. Gerstenberger, W.A. Wall, 2010


Weak form & interface condition for incompr. NS Eqs.

3-field Hybrid/Mixed Fluid Formulation

Time-discrete weak form (One-step-Theta):

velocity, pressure, stress:& test functions:

Nodal unknowns: , element unknowns:

- stress approx.: C-1 elementwise disc.

- Q1Q1Q-1/Q2Q2Q-2/P1P1P-1 …



Intersected elements: condensed element stresses

Assembled element stiffness matrices:

Uncut elements: decoupled element stresses



• Element-wise condensation of enriched stress approximation in intersected elements

• Q1Q1/Q2Q2 stabilized (SUPG/GLS/…) fluid formulation untouched

• No extra stabilization for Lagrange multiplier (element stresses) needed

• Interface discretization can be arbitrary!

• Sparse matrix, no zero diagonal terms and only velocity + pressure unknowns parallel, iterative AMG preconditioners directly applicable

Fluid system matrix:

Define DOF sets: Standard and Enriched DOFs

Standard:

Enriched:

Summary:


Patch tests

• Linear velocity field constant viscous stress• Exact surface reaction force recovered

• Constant body force b linear hydrostatic pressure field• Exact surface reaction force recovered

Pressure approx. dictates stress approx.: Velocity-Pressure-Stress approx.: Q1Q1Q-1, Q2Q2Q-2, P2P2P-2, …


Convergence analysis: Jeffery-Hamel Flow

• Jeffery (1915), Hamel (1916), Rosenhead (1940)• Convective, radial flow between converging walls with analytic solution• All boundaries modeled via hybrid embedded Dirichlet approach• Re=85

Radial velocity:

BC:


Interface Conditions

• optimal spatial convergence rates• very good agreement with reference values

(DFG Benchmark, 1996)• tested up to Re=4000• arbitrary interface mesh good for FSI

A. Gerstenberger, W.A. Wall, An embedded Dirichlet formulation for 3D continua, CMAME, 2010


3D Inst. Computation with Prescribed Interface Motion II

Velocity field RE=50-200

(fluid: 1 layer of hex20 elements, structure: 1 or more layers of hex8 elements)

Prescribed displacement d(t)

slip

no-slip

Traction free


3D Inst. Computation with Prescribed Interface Motion II

Velocity field RE=50-200


(fluid: 1 layer of hex20 elements, structure: 1 layer of hex8 elements)


3D Inst. Computation with Prescribed Interface Motion

(fluid: 1 layer of hex8 elements, structure: 2 or more layers of hex8 elements)

Velocity field RE=50 +/- 10

Cylinder moves to the right: lower relative flow speed no vortex shedding

Cylinder moves to the left: higher relative flow speed vortex shedding


Moving Interfaces

(fluid & structure: 1 layer of hex8 elements

Velocity field, RE=50 +/- 10

Velocity field, RE=100 +/- 10


3D Inst. Computation with Prescribed Interface Motion

U_max = 2.2U_mean = 1Channel height h = 1Kinematic viscosity = 0.001

Re = 200


High Reynolds number flow

U_max = 7.2U_mean = 2Channel height h = 0.41, Object height h_o = 0.2Kinematic viscosity = 0.0001

Re = 2*0.2/0.0001 = 4000




Re = 2*0.2/0.0001 = 4000




Re = 2*0.2/0.0001 = 4000


FSI Problem Definition – 3-Field-Setup

Kinematic condition

Momentum balance


Structure discretization

Strong Form (body forces omitted):

Weak Form without FSI coupling equations:

Time discretization Beta-Newmark:

- Material geometric and material non-linearities possible

- Lagrange formulation for large deformation/large strains

Spatial discretization:


Structure-Interface Coupling

Weak Form:

Interface Condition:

Mortar: Interface is slave side

Square matrixRectangular matrixFinal discrete system:


FSI Problem Definition – 3-field-setup

Kinematic condition

Traction condition


FSI System: Partitioned Solution

Partitioned, Iterative Dirichlet-Neumann Coupling (Aitken relax.) w. Matching Nodes


FSI System: Monolithic Solution

Monolithic system without extra interface mesh


Channel Flow with Bending Structure

Velocity field

(RE = 20, fluid: hex20 elements, structure: hex8 elements, hyper-elastic, EAS21 formulation)


3D Instationary FSI Computation

(fluid: 1 layer of hex20 elements, structure: 1 layer of hex8 elements)

Velocity field RE=50-200 velocity field


3D Instationary FSI Computation

Velocity field RE=100 (using channel width)

(fluid: 1 layer of hex8 elements, structure: 2 layers of hex8 elements, hyper-elastic, full EAS formulation)


Preparing for real applications

(fluid: 1 layer of hex20 elements, structure: 2 layers of hex8 elements)

Symmetric shear flow + elastic structure


FSI Example 2: 3D Instationary FSI Computation

Quasi-stationary symmetric shear flow + elastic ring (1st principal stress)

Oscillating shear flow + elastic ring


Preparing for real applications

asymmetric shear flow + elastic ring

(fluid: 1 layer of hex8 elements, structure: 2 layers of hex8 elements, hyper-elastic, full EAS formulation)


… and take good care of your structure!


Extensions 2: Hybrid ALE-XFEM approach

Starting point: Presented FSI approach

Basic idea: Add intermediate (moving) fluid mesh

“Classic” Moving-Mesh-Fluid-Structure CouplingXFEM Fluid-Fluid Coupling


Example: Flow Around The Cylinder

Surface fitted, anisotropic mesh can efficientlyresolve boundary layers

Mesh construction simplified


Extensions: Automatic Adaptivity vs. Hybrid Approach

Re = 20

(2D)


2D Benchmark Computations

Velocity Field

Pressure Field

0



Velocity Field



Pressure Field


“Spider Silk” project

Formation and behavior of hydrogels in protein solution in a microfluidic channel.

Simulation of:• Non-Newtonian fluid

(shear thinning, Carreau-Yasuda model)

• FSI• Intermolecular forces• Growing structures• Contact of submerged

structures …

Araneus diadematus


Macromolecular Interaction on a Mesoscopic Level

Intrasolid interaction: Behaviour of atoms and molecules within the

structure is described by material laws W( )

Intersolid interaction: Behaviour of atoms and molecules between

the structures as well as the atomic and ionic influence

of the medium on the structure are described by

various macromolecular interaction potentials

Ionic influence of electrical double layer s in the liquid medium on thestructures

Molecular dynamics :- Intrasolid interaction :- Intersolid interaction :


Numerical Examples Suspension of Microspheres

Suspension of spider silk nano/microspheres in shear flow including

macromolecular attraction and repulsion:

Stability of suspension necessary for the production of

drug delivery systems, coating of thin films


Macromolecular Interaction Potentials Example

Half sphere is pushed towards a block:

Long-range attraction and short-range repulsion

modelled by a Lennard-Jones potential :

Resulting force :


Contact and Interaction Example

Half sphere is pushed towards a block:

Long-range (Van-der-Waals) attraction is described by a

Lennard-Jones potential

Macroscopic contact is performed instead of very short-ranged repulsion


Fluid-structure-contact interaction (FSCI)


Schematic form of monolithic FSCI system

dual mortar contact formulation condensation of Lagrange multipliers only displacement DOFs

contact does not restrict the application of FSI coupling algorithms state-of-the-art monolithic or partitioned solution schemes

contact modificationsto structure block

condensation of thefluid stress unknownsat element level

U. M. Mayer et al., 3D fluid-structure-contact interaction based on a combined XFEM FSIand dual mortar contact approach, Computational Mechanics, 46 (2010), pp. 53-67


Ring and rigid obstacle

one-body contact (rigid obstacle) elastic ring (E=1000, =0.4, =5) Newtonian fluid (=0.01, =1) parabolic velocity profile at inflow laminar flow (Re≈70)


Beam and rigid obstacle

one-body contact (rigid obstacle) elastic beam (E=500, =0.4, =5) Newtonian fluid (=0.01, =1) parabolic velocity profile at inflow laminar flow (Re≈70)




XFEM - FSI Interface Handling and Enrichment

Enrichment of several interfaces per fluid elements

for intermolecular interaction and contact of mesoscopic structures


FSI Example 3: Fixed-Grid FSI + Contact Extension

Mayer, Popp, Gerstenberger, Wall, Comp.Mech., 2010

4x SlowMotionRealtime




Two-phase flows

Ursula Rasthofer, Florian Henke, Volker Gravemeier

incompressible Navier-Stokes equations

level set equation


Other XFEM applications: turbulent combustion

Turbulent combustion, Emmy-Noether Research GroupV. Gravemayer, F. Henke, F. v.d. Bos, W.A. Wall

Flamefronts can be modeled as sharp or fuzzy moving interfaces


Summary / conclusion for XFSI

Developed and implemented parallel, 3D FSI based on fixed Eulerian FE fluid grids

• No limitation on complexity of structure (shape, material, deformation,…)

• Sharply defined interface w. embedded Dirichlet conditions• Local condensation of Lagrange multipliers• iterative, parallel solution w. AMG precond. for fluid and structure

• Influence of “fictitious” fluid domain eliminated• No incompressibility constraint on structure• No artificial viscosity

• Fluid solved on fixed Eulerian grid• No mesh distortion + update algorithm• Any fluid element type possible

(hex, tet, wedge,…)

• Simple extension to hybrid (fixed/ALE) meshes

• Use established FSI coupling schemes

• Extended to mesoscopic biophysics (FSI including contact, Brownian dynamics)

• Developed approach is being applied to a number of other multifield problems

… and don’t forget to take good care of your structure!


Outline

• A fixed-grid approach for large deformations and topological changes

Fixed-grid FSI apprach based on XFEMA new approach for enforcing boundary conditionsEmbedded fluid meshes – hybrid FSIFluid-structure-contact interaction

• Fluid-Structure Interaction in Bioengineering


Biological suspenions

Microscopic modeling of blood and damage of RBC“Blood is a very special fluid”(Mephistopheles in J.W. Goethe’s Faust)

[Li et al., 2007]


Some modeling & computational complexities

Goal: prediction of individual rupture risk of AAA

• Patient-specific geometries• Realistic material properties• Various nonlinearities• Unknown zero pressure configuration • Large size parallel FSI with

Pulsatile, high flow rate, locally turbulentLarge stiffness jump artery wall / ILTWomersley inflow profilesImpedance outflow BCs (Windkessel)Moving BCs in FSIMixed meshes in fluid/structure

• Prospective simulation before surgery:– 48h time slot


„the early freak show“

Sizes not to scale!Advanced Topics in Computational Fluid-Structure Interaction - from Topological Changes to BioengineeringWolfgang A. Wall – Institute for Computational Mechanics, TU München – http://www.lnm.mw.tum.de

Fluid-Structure Interaction

• Large scale & parallel• Womersley inflow profile• Windkessel impedance BCs• Prestressed structure• Newtonian incomp. fluid

(valid in large vessels only)

• Efficient monolithic FSI solverbased on algebraic multigrid:

Algebraic multigrid coarse approximation:


Methodology

Patient specificgeometry

3D reconstructedgeometry

• Three-dimensionalgeometry is reconstructedfrom a patients CT scan.

Modeling

• Reduced-dimensional FSI model built form reporteddata in literature.

Reduced-D extractedgeometry

3D

1D

0D

(Formaggia et al. 2006, Blanco et al. 2010, Vignon et al. 2006)


BC – reduced dimensional models

Three-dimensional-problem volumetric flow rate is applied on one-D side.

One-dimensional-problem pressure is applied as a traction on the 3D side.

Coupling of full and reduced dimensional models

Advantage: Applying traction on the three-dimensional side preserves theWomersley velocity profile on corresponding surface.



+3 element windkessel BC

Reflective BC =

1D extractedgeometry

Boundaryconditions

Cardiac output




Results

Left Common Iliac

1.3760

0.7890

0.0789

Final

1.500

0.700

0.070

Initial

Right Common Iliac

Initial FinalDesign variables

1.37801.500C [Pa-1.mm3]

0.78700.700R2 [Pa.s.mm-3]

0.07870.070R1 [Pa.s.mm-3]

Left Common Iliac Right Common Iliac Systolic Pressure Diastolic Pressure


Virtual Lung Model

www.thaimed.us

Alveoli

Lower airways

Alveolar epithelial cells

Lung Parenchyma

Tan et al.


Virtual Lung Model

Global lung models

Recruitment and derecruitment

Noisy ventilation

Evaluation of different ventilation protocols

Alveoli

Lower airways

Alveolar epithelial cells

Lung Parenchyma

Tan et al.

In vivo local alveolar stresses and strains

Biological effects


Reason for study

Acute Lung Injury (ALI) and AcuteRespiratory Distress Syndrome (ARDS)

• Causes: sepsis, aspiration, trauma…

• Treatment: mechanical ventilation

• Mortality: ~40%

Ventilator-Induced Lung Injury (VILI)

• Exacerbation of lung injury due to impropermethods of ventilation

• Mechanical problem

• Not so easy to fix – ventilation/perfusion

Low compliance


Volume-Coupled FSI – Motivation

3200

350

7

31

Volume[ml]

----0.28300 x 106Alveoli

0.0459000.414.19 x 106Alveolar duct

34191.092050T. Bronchus

43502.6181Trachea

Reynolds No.

Area[cm2]

DiameterNumber


Segmentation and meshing

Geometry segmented from 0.7mm resolution CT scans

Outlet extrusion and meshing


Governing equations

Solver

• Solved in our in house multi-physics research code BACI

• MPI parallelization

• SUPG/PSPG stabilized finite element method

• One step theta time integration scheme

• Resulting linear system solved using GMRES and multilevel preconditioning

Numerical method


Physiological boundary conditions

Geometric space filling or mathematical defined trees

Acinar condition for current tree

From 1D fluid mechanics for oscillatory flow in an elastic tube,the impedance at the upstream end can be written as a function of the downstream end (Olufsen 2000)


• Implemented in our research code BACI

• Dirichlet to Neumann approach

• Any downstream tree can be applied

• Efficient – no inner iterations required on the boundary

Coupling

Impedance of whole peripheral tree

Inverse Fourier transform

Convolution with flowrate history

Couple to 3D outflow boundary


Pressure drop – Whole conducting airway

• Varies a lot in literature – Different methods and definitions

• Present results show nonlinear behavior

• Geometry is “King” – simplified models do not give correct results


Pure fluid results

Pressure is elevated due to peripheral airways

Hypothetical disease scenarios

Hysteresis of the pressure flow loop


FSI lung model

• Simulated in our in-house FSI research code

• Monolithic approach, previously deemed better than segregated approaches for biomedical applications (kϋttler et al 2009)

Navier-Stokes

Nonlinear elastodynamics

Interface coupling


The airway wall

• The mechanical properties of the airway wall still remain to be fully elucidated (Kamm, 1999)

• Here we use E=60kPa, which covers the highly variable range in literature

• Fibre directions?

Hyperelastic strain energy function

Neohookean formulation according to Holzapfel (2000)


FSI – Normal breathing

• Enlarged low velocity regions compared with rigid wall simulations

• Cross-sectional area changes

Undeformedconfiguration

Deformed configuration


Pressure and stress

Pressure increase leads to elevated stresses in the wall

Pressure Stress


Principal Stresses

• Wall is under both compression and tension

• Significant variation over the thickness

• Interesting implications for mechanotransduction

• Bending in the wall


Stress field within the wall

• Slice from the 3rd generation

• Field is predominantly aligned in the circumferential direction

• Localised region of high compressive stress


Hypothetical disease

Pressure 3rd principal stress

• Flow into downstream regions impeded, representative of downstream alveolar derecruitment


Embedded FSI

• Same geometry as previously presented, however now we have the parenchymaltissue surrounding the airways.

Parenchymal tissue properties

• E~6kPa

• Experimentally based

• compressible


Influence of parenchymal tissue

• Fivefold reduction in stress

• Distribution through the wall and throughout the geometry is different


Some interesting features

• Stress distribution in the vicinity of bifurcations changes due to the influence of the surrounding tissue

• Different rates of vessel inflation


Multi-Scale Approach (FE )

boundary conditions

homogenized parameters

representative volume elements (RVE) of the micro-structure associated with macro-scaleGauss points (at hot spots)

nested solution of nonlinear BVP on both scales (simultaneous simulation)

macro-scale deformation state defines boundary conditions for micro-scale4

volume averaging on micro-scale provides homogenized parameters formacro-scale simulation

3D nonlinear dynamic framework (Wiechert and Wall 2010, CMAME)

-

-

-

micro-scalemacro-scale

2


• Algebraic variational multiscale – multigridmethod (Gravemeier, 2009)

• VMLES turbulence approach

• Meshes 1.6 million DOFs and 3.6 million DOFs

• Laryngeal jet

Turbulence modelling in the larynx and bronchial tree


Laryneal jet

This has an influence in the trachea


Pressure drop

Effect of turbulence & upper airways on pressure

Only in the trachea is pressure and flow influenced

Pressure in the bronchial tree is “unaffected” by the upper airways


Mean Velocity- laminar vs. AVM3

There are some differences, but from a physiological perspective these are negligible.

Centreline through glottis and trachea Centreline in the right lobe


Instantaneous fluctuations

Further investigation required

• Turbulent statistics

• Is turbulence important for ventilation?


Segmented lung lobes

Flow and pressure at any point in the tree via 0D model of entire tree

Acinar volume based continuous splitting of sub-lobe based on tree growing

Artificial tree and Acini Generation

Meshing of lung lobes

• Bridges the scale between segmented airway tree and the acini of the lung –volume coupling

• Each airway has an associated 3D volume on its end representing an acinus

Trees and acinar volumes


Coupling of Airway and Parenchyma Models

coupling of airway wall deformation and airflow (FSI)

coupling of fluidflow throughdeforming outletand volume changeof surroundingparenchyma

-

-






Numerical Examples


Numerical Examples


Numerical Examples


• Recruitment - aeration

• Noisy ventilation modes – Max, min, random

• Flow distribution - PET

What are we going to do with it?

Collapsed region (different compliance)

Healthy tissue


Nanoparticles

• What are nanoparticles?

• They have both adverse and beneficialeffects for the lungs.

Inflammation

“Nano” drugs

• Are “potent”

Have a high mass to area ratio


DEF

Cycle Time


Deposition efficiency


- FIN -

Denis Diderots "Encylopédie", 1751

Questions?

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Documents

baci Webseite...FSI Problem Definition – 3-Field-Setup Kinematic condition Momentum balance see e.g. Park, Felippa, Ohayon (2001) Advanced Topics in Computational Fluid-Structure