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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)
2Levelset based FSI modeling with XFEM
Introduction to Fluid-Structure Interaction (FSI)
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]
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
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
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
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
8Levelset based FSI modeling with XFEM
The modelAn overview of the modeling process
9Levelset based FSI modeling with XFEM
The modeled problem
Length tunnel: 300 μmHeight tunnel: 100 μmHeight structure: 50 μmWidth: 5 μm
10Levelset based FSI modeling with XFEM
The modelAn overview of the process
11Levelset based FSI modeling with XFEM
Fluid-structure interaction
12Levelset based FSI modeling with XFEM
The modelAn overview of the process
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
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)>0
φ(x)=0
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
16Levelset based FSI modeling with XFEM
The modelAn overview of the process
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
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
+
=
19Levelset based FSI modeling with XFEM
The modelAn overview of the process
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
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
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
ud
Rd
ud
Rdud
Rd
ud
Rd
J
f
f
s
s
stag
ud
Rdud
Rd
J0
0
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
24Levelset based FSI modeling with XFEM
The modelAn overview of the process
25Levelset based FSI modeling with XFEM
Results – Staggered schemeVelocity and displacement field – Steady state
XFEM-staggered:
COMSOL-ALE:
[-] [-]
[m/s]
[μm]
26Levelset based FSI modeling with XFEM
Results – Staggered schemeVelocity and displacement field – Steady state
[-] [-]
[m/s]
[μm]
XFEM-staggered:
COMSOL-ALE:
≈
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:
≈
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
29Levelset based FSI modeling with XFEM
Results – Monolithic schemeVelocity and displacement field – Exploded
XFEM:
COMSOL:
[-] [-]
[m/s]
[μm]
≠
Monolithic: Jacobian is not ok
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)
31Levelset based FSI modeling with XFEM
Results – Monolithic schemeJacobian check – Overview of the matrice entries
dus duf
dRf
dRs
Analytic - Desired Finite Difference - Comparison
32Levelset based FSI modeling with XFEM
Results – Monolithic schemeJacobian check – Overview of the matrices
dus duf
dRf
dRs
Analytic - Desired Finite Difference - Comparison
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
34Levelset based FSI modeling with XFEM
Results – Monolithic schemeJacobian check – Schematic of 2 element problem with displacements
35Levelset based FSI modeling with XFEM
Results – Monolithic schemeJacobian check – Schematic of 2 element problem with displacements
36Levelset based FSI modeling with XFEM
Results – Monolithic schemeJacobian check – Schematic of 2 element problem with displacements
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
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
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
40Levelset based FSI modeling with XFEM
OutlookWhat makes this approach suitable for optimization?
41Levelset based FSI modeling with XFEM
OutlookWhat makes this approach suitable for optimization?
42Levelset based FSI modeling with XFEM
OutlookWhat makes this approach suitable for optimization?
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
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
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
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
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
48Levelset based FSI modeling with XFEM
Backup slides
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
50Levelset based FSI modeling with XFEM
Discontinuous shape functions
51Levelset based FSI modeling with XFEM
Results – Staggered schemeThe process
52Levelset based FSI modeling with XFEM
Results – Staggered schemeThe process
53Levelset based FSI modeling with XFEM
Results – Staggered schemeResidual development
54Levelset based FSI modeling with XFEM
Levelset update - Changing DOFs
55Levelset based FSI modeling with XFEM
Results – Staggered schemePressure and displacement field – XFEM model and COMSOL
XFEM:
COMSOL:
56Levelset based FSI modeling with XFEM
Finite Differences
57Levelset based FSI modeling with XFEM
Non-dimensional numbers
58Levelset based FSI modeling with XFEM
Mesh mismatch
59Levelset based FSI modeling with XFEM
Residual equations
60Levelset based FSI modeling with XFEM
3-field setup
61Levelset based FSI modeling with XFEM
Projection onto fluid mesh