Levelset based fluid-structure interaction modeling with the eXtended Finite Element Method

1Levelset based FSI modeling with XFEM

Levelset based fluid-structure interaction modeling with the eXtended Finite Element MethodMSc Thesis presentation – Thijs Bosma – December 4th 2013

Supervisors:Matthijs Langelaar(DUT)Fred van Keulen(DUT)Kurt Maute(CU)


Introduction to Fluid-Structure Interaction (FSI)


Introduction to Fluid-Structure Interaction (FSI)

• Ultimate goal is Topology Optimization

• ALE-method, computationally expensive (re-meshing)

• Density-based methods, unclear interface [James, 2012]


Introduction to my workGoals of the research

Model: Levelset based based geometry description for fluid-structure interaction (FSI) problems with

eXtended Finite Element Method (XFEM) approximation

Goal: Develop an efficient solver scheme that finds the steady state solution of the FSI problem

(simultaneously for fluid and structure), such that it can be used in an optimization framework


Introduction to my workGoals of the research

Model: Levelset based based geometry description for fluid-structure interaction (FSI) problems with

eXtended Finite Element Method (XFEM) approximation




Introduction to my work

1. Does the approximated solution describe the physics of the system?

2. How can the problem be solved efficiently?3. What makes this approach suitable for optimization?

Goals of the research

Model: Levelset based geometry description for fluid-structure interaction (FSI) problems with eXtended

Finite Element Method (XFEM) approximation




Content

• The model• The solvers

• Monolithic and staggered solver• Results staggered

• 1) Does the approximated solution describe the physics of the system?

• Results monolithic• 2) How can the problem be solved efficiently?

• Outlook• 3) What makes this approach suitable for optimization?

• Conclusions/Recommendations


The modelAn overview of the modeling process


The modeled problem

Length tunnel: 300 μmHeight tunnel: 100 μmHeight structure: 50 μmWidth: 5 μm


The modelAn overview of the process


Fluid-structure interaction




Levelset Method

• Zero contour of signed distance function φ(x) describes the interface

• Shortest distance from a point in the domain to the interface determines levelset field (LSF)

LSF zero contour


Levelset Method

• Divides the domain in 3 parts:• Fluid (φ(x)<0)• Zero contour (φ(x)=0)• Structure

(φ(x)>0)• Concept similar to

elevation map of Boulder, CO, USA

6000 ft. contour

φ(x)<0

φ(x)>0φ(x)=0


Levelset Method

• If the structure deforms/displaces the levelset field changes

• The levelset field depends on the structural displacements

Structural displacement u




eXtended Finite Element Method

• Approximation/discretization technique, based on FEM

• Only find solution at discrete points in domain (nodes)

• Assume solution and allow discontinuous solution between nodes

• Discontinuity is transition from fluid to structure

Discontinuity turns off part of the element


eXtended Finite Element Method

• LSF zero contour determines location of discontinuity

• Two meshes• Approximation introduces

Residual error • Residual is function of

solution and LSF• If error is zero,

approximated solution is found

+

=




The Solver

• R(un) is residual error function

• u0 is initial solution• How to get to solution

from initial solution?

The Newton-Raphson method for non-linear problems


The Solver

• Iteratively using the ‘slope’ is an efficient and accurate way

• Slope can be found analytically, but is difficult

• J is the slope of function R, called Jacobian

• Principle holds for N dimensions

The Newton-Raphson method for non-linear problems


The Solver

Staggered• Fluid and structure are solved

separately• Complex FSI coupling terms

in Jacobian are ignored• Residual error complete

Monolithic

The monolithic and the staggered approach

• Fluid and structure solved simultaneously

• Complete Jacobian is used• Residual error complete

f

f

s

f

f

s

s

s

mono

udRd

udRd

udRd

udRd

J

f

f

s

s

stag

udRd

udRd

J0

0


The Solver

Staggered• Inefficient• Unsuitable for optimization• Guarantees a steady state

solution

Monolithic

The monolithic and the staggered approach

• Efficient• Suitable for optimization• Difficult to find steady state

solution

Staggered: check the Residual functionMonolithic: check the Jacobian




Results – Staggered schemeVelocity and displacement field – Steady state

XFEM-staggered:

COMSOL-ALE:

[-] [-]

[m/s][μm]



[-] [-]

[m/s][μm]

XFEM-staggered:

COMSOL-ALE:

≈



[-] [-]

[m/s][μm]

≈

Staggered: Residual function is ok

XFEM-staggered:

COMSOL-ALE:

≈


Goal

1. Does the approximated solution describe the physics of the system? Yes, based on qualitative check!

2. How can we efficiently solve the system?3. What makes this approach suitable for optimization?

Goal: Develop an efficient solver scheme that finds the steady state solution of the FSI

problem (simultaneously for fluid and structure), such that it can be used in an

optimization framework


Results – Monolithic schemeVelocity and displacement field – Exploded

XFEM:

COMSOL:

[-] [-]

[m/s][μm]

≠

Monolithic: Jacobian is not ok


Results – Monolithic scheme

• FD is expensive, but reliable

• Four element problem, all elements intersected

• 3 problems discovered – 1 discussed

• After discretization Jacobian is a matrix

Jacobian check – Test Case Finite differences (FD)


Results – Monolithic schemeJacobian check – Overview of the matrice entries

dus duf

dRf

dRs

Analytic - Desired Finite Difference - Comparison


Results – Monolithic schemeJacobian check – Overview of the matrices

dus duf

dRf

dRs

Analytic - Desired Finite Difference - Comparison



• Location zero contour structure depends on displacements

• Zero contour fluid depends on orthogonal distance to zero contour

• Zero contours determine

what part is deleted from solution

Jacobian check – Schematic of 2 element problem


Results – Monolithic schemeJacobian check – Schematic of 2 element problem with displacements







• Displacements of structural element 1 affect zero contour in both fluid elements

Jacobian check – Schematic of 2 element problem with displacements

Presumed Actual



• Displacements of element 1 affect zero contour in both elements

• Secondary coupling introduced between intersected elements through LSM

• Secondary coupling not incorporated in

analytic Jacobian

Jacobian check – Schematic of 2 element problem with displacements


Goal


2. How can we efficiently solve the system? Monolithically, but analytic Jacobian is not numerically consistent

3. What makes this approach suitable for optimization?





OutlookWhat makes this approach suitable for optimization?






Goal


2. How can we efficiently solve the system? Monolithically, but Jacobian is not numerically consistent

3. What makes this approach suitable for optimization? Flexible geometry description, accurate physical behavior at interface





Conclusions

• The staggered setup has qualitatively shown that the steady state solution is comparable with the solution from ALE-based method

• The FSI problem can not be solved with a monolithic setup yet• Jacobian is not numerically consistent

• Flexible geometry description with physically relevant results


Recommendations

• More elaborate and quantitative validation of the results should performed

• The analytic Jacobian needs to be improved• Secondary coupling• Two other issues

• Topology Optimization


‘The primary product of science is failure, but failure teaches us where not to go in the

future’

– Vincent Icke, physics professor University of Leiden in DWDD 27/11/2013*

Thanks for the attention!

* Loosely translated by Thijs Bosma


References

• James, K.A. and Martins, J.R. (2012). An isoparametric approach to level set topology optimization using a body fitted finite element mesh. Computers & Structures, 90-91:97-106


Backup slides


The modeled problem

• Abstract blood vessel with valve

• 2D horizontal tunnel with structure fixed at bottom

• Fluid flows from left to right

• Steady state• Fluid applies force on

structure• Structure changes flow

path

Dimensions in μm

How to describe the behavior of the system?

The physical configuration


Discontinuous shape functions


Results – Staggered schemeThe process


Results – Staggered schemeThe process


Results – Staggered schemeResidual development


Levelset update - Changing DOFs


Results – Staggered schemePressure and displacement field – XFEM model and COMSOL

XFEM:

COMSOL:


Finite Differences


Non-dimensional numbers


Mesh mismatch


Residual equations


3-field setup


Projection onto fluid mesh

Documents

Levelset based fluid-structure interaction modeling with the eXtended Finite Element Method