Levelset based fluid-structure interaction modeling with the eXtended Finite Element Method
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Transcript of Levelset based fluid-structure interaction modeling with the eXtended Finite Element Method
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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)
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2Levelset based FSI modeling with XFEM
Introduction to Fluid-Structure Interaction (FSI)
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3Levelset based FSI modeling with XFEM
Introduction to Fluid-Structure Interaction (FSI)
• Ultimate goal is Topology Optimization
• ALE-method, computationally expensive (re-meshing)
• Density-based methods, unclear interface [James, 2012]
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4Levelset based FSI modeling with XFEM
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
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5Levelset based FSI modeling with XFEM
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
![Page 6: Levelset based fluid-structure interaction modeling with the eXtended Finite Element Method](https://reader035.fdocuments.net/reader035/viewer/2022062520/56816365550346895dd43a40/html5/thumbnails/6.jpg)
6Levelset based FSI modeling with XFEM
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
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
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7Levelset based FSI modeling with XFEM
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
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8Levelset based FSI modeling with XFEM
The modelAn overview of the modeling process
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9Levelset based FSI modeling with XFEM
The modeled problem
Length tunnel: 300 μmHeight tunnel: 100 μmHeight structure: 50 μmWidth: 5 μm
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10Levelset based FSI modeling with XFEM
The modelAn overview of the process
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11Levelset based FSI modeling with XFEM
Fluid-structure interaction
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12Levelset based FSI modeling with XFEM
The modelAn overview of the process
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13Levelset based FSI modeling with XFEM
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
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14Levelset based FSI modeling with XFEM
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
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15Levelset based FSI modeling with XFEM
Levelset Method
• If the structure deforms/displaces the levelset field changes
• The levelset field depends on the structural displacements
Structural displacement u
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16Levelset based FSI modeling with XFEM
The modelAn overview of the process
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17Levelset based FSI modeling with XFEM
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
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18Levelset based FSI modeling with XFEM
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
+
=
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19Levelset based FSI modeling with XFEM
The modelAn overview of the process
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20Levelset based FSI modeling with XFEM
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
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21Levelset based FSI modeling with XFEM
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
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22Levelset based FSI modeling with XFEM
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
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23Levelset based FSI modeling with XFEM
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
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24Levelset based FSI modeling with XFEM
The modelAn overview of the process
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25Levelset based FSI modeling with XFEM
Results – Staggered schemeVelocity and displacement field – Steady state
XFEM-staggered:
COMSOL-ALE:
[-] [-]
[m/s][μm]
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26Levelset based FSI modeling with XFEM
Results – Staggered schemeVelocity and displacement field – Steady state
[-] [-]
[m/s][μm]
XFEM-staggered:
COMSOL-ALE:
≈
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27Levelset based FSI modeling with XFEM
Results – Staggered schemeVelocity and displacement field – Steady state
[-] [-]
[m/s][μm]
≈
Staggered: Residual function is ok
XFEM-staggered:
COMSOL-ALE:
≈
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28Levelset based FSI modeling with XFEM
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
![Page 29: Levelset based fluid-structure interaction modeling with the eXtended Finite Element Method](https://reader035.fdocuments.net/reader035/viewer/2022062520/56816365550346895dd43a40/html5/thumbnails/29.jpg)
29Levelset based FSI modeling with XFEM
Results – Monolithic schemeVelocity and displacement field – Exploded
XFEM:
COMSOL:
[-] [-]
[m/s][μm]
≠
Monolithic: Jacobian is not ok
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30Levelset based FSI modeling with XFEM
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)
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31Levelset based FSI modeling with XFEM
Results – Monolithic schemeJacobian check – Overview of the matrice entries
dus duf
dRf
dRs
Analytic - Desired Finite Difference - Comparison
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32Levelset based FSI modeling with XFEM
Results – Monolithic schemeJacobian check – Overview of the matrices
dus duf
dRf
dRs
Analytic - Desired Finite Difference - Comparison
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33Levelset based FSI modeling with XFEM
Results – Monolithic scheme
• 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
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34Levelset based FSI modeling with XFEM
Results – Monolithic schemeJacobian check – Schematic of 2 element problem with displacements
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35Levelset based FSI modeling with XFEM
Results – Monolithic schemeJacobian check – Schematic of 2 element problem with displacements
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36Levelset based FSI modeling with XFEM
Results – Monolithic schemeJacobian check – Schematic of 2 element problem with displacements
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37Levelset based FSI modeling with XFEM
Results – Monolithic scheme
• Displacements of structural element 1 affect zero contour in both fluid elements
Jacobian check – Schematic of 2 element problem with displacements
Presumed Actual
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38Levelset based FSI modeling with XFEM
Results – Monolithic scheme
• 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
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39Levelset based FSI modeling with XFEM
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? Monolithically, but analytic Jacobian is not numerically consistent
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
![Page 40: Levelset based fluid-structure interaction modeling with the eXtended Finite Element Method](https://reader035.fdocuments.net/reader035/viewer/2022062520/56816365550346895dd43a40/html5/thumbnails/40.jpg)
40Levelset based FSI modeling with XFEM
OutlookWhat makes this approach suitable for optimization?
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41Levelset based FSI modeling with XFEM
OutlookWhat makes this approach suitable for optimization?
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42Levelset based FSI modeling with XFEM
OutlookWhat makes this approach suitable for optimization?
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43Levelset based FSI modeling with XFEM
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? Monolithically, but Jacobian is not numerically consistent
3. What makes this approach suitable for optimization? Flexible geometry description, accurate physical behavior at interface
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
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44Levelset based FSI modeling with XFEM
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
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45Levelset based FSI modeling with XFEM
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
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46Levelset based FSI modeling with XFEM
‘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
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47Levelset based FSI modeling with XFEM
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
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48Levelset based FSI modeling with XFEM
Backup slides
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49Levelset based FSI modeling with XFEM
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
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50Levelset based FSI modeling with XFEM
Discontinuous shape functions
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51Levelset based FSI modeling with XFEM
Results – Staggered schemeThe process
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52Levelset based FSI modeling with XFEM
Results – Staggered schemeThe process
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53Levelset based FSI modeling with XFEM
Results – Staggered schemeResidual development
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54Levelset based FSI modeling with XFEM
Levelset update - Changing DOFs
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55Levelset based FSI modeling with XFEM
Results – Staggered schemePressure and displacement field – XFEM model and COMSOL
XFEM:
COMSOL:
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56Levelset based FSI modeling with XFEM
Finite Differences
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57Levelset based FSI modeling with XFEM
Non-dimensional numbers
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58Levelset based FSI modeling with XFEM
Mesh mismatch
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59Levelset based FSI modeling with XFEM
Residual equations
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60Levelset based FSI modeling with XFEM
3-field setup
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61Levelset based FSI modeling with XFEM
Projection onto fluid mesh