MSc Thesis presentation – Thijs Bosma – December 4th 2013 Levelset based fluid-structure interaction modeling with the eXtended Finite Element Method MSc 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 work Goals 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 work Goals 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 work 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 Does the approximated solution describe the physics of the system? How can the problem be solved efficiently? What makes this approach suitable for optimization?

Content The model The solvers Results staggered Results monolithic 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 model An overview of the modeling process

The modeled problem Length tunnel: 300 μm Height tunnel: 100 μm Height structure: 50 μm Width: 5 μm

The model An overview of the process

Fluid-structure interaction

The model An overview of the process

Levelset Method LSF zero contour 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 6000 ft. contour φ(x)<0 φ(x)=0 φ(x)>0 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 φ(x)<0 φ(x)=0 φ(x)>0

Levelset Method If the structure deforms/displaces the levelset field changes The levelset field depends on the structural displacements Structural displacement u

The model An overview of the process

eXtended Finite Element Method Discontinuity turns off part of the element 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

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 model An overview of the process

The Solver The Newton-Raphson method for non-linear problems R(un) is residual error function u0 is initial solution How to get to solution from initial solution?

The Solver The Newton-Raphson method for non-linear problems 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 Solver Staggered Monolithic The monolithic and the staggered approach Staggered Monolithic Fluid and structure are solved separately Complex FSI coupling terms in Jacobian are ignored Residual error complete Fluid and structure solved simultaneously Complete Jacobian is used Residual error complete

The Solver Staggered Monolithic Staggered: check the Residual function The monolithic and the staggered approach Staggered Monolithic Inefficient Unsuitable for optimization Guarantees a steady state solution Efficient Suitable for optimization Difficult to find steady state solution Staggered: check the Residual function Monolithic: check the Jacobian

The model An overview of the process

Results – Staggered scheme Velocity and displacement field – Steady state XFEM-staggered: [-] [-] COMSOL-ALE: [m/s] [μm]

Results – Staggered scheme Velocity and displacement field – Steady state XFEM-staggered: [-] [-] ≈ COMSOL-ALE: [m/s] [μm]

Results – Staggered scheme Velocity and displacement field – Steady state XFEM-staggered: [-] [-] ≈ ≈ COMSOL-ALE: Staggered: Residual function is ok [m/s] [μm]

Goal 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 Does the approximated solution describe the physics of the system? Yes, based on qualitative check! How can we efficiently solve the system? What makes this approach suitable for optimization?

Results – Monolithic scheme Velocity and displacement field – Exploded XFEM: ≠ [-] [-] COMSOL: Monolithic: Jacobian is not ok [m/s] [μm]

Results – Monolithic scheme Jacobian check – Test Case Finite differences (FD) FD is expensive, but reliable Four element problem, all elements intersected 3 problems discovered – 1 discussed After discretization Jacobian is a matrix

Results – Monolithic scheme Jacobian check – Overview of the matrice entries dus duf dRs dRf Analytic - Desired Finite Difference - Comparison

Results – Monolithic scheme Jacobian check – Overview of the matrices dus duf dRs dRf Analytic - Desired Finite Difference - Comparison

Results – Monolithic scheme Jacobian check – Schematic of 2 element problem 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

Results – Monolithic scheme Jacobian check – Schematic of 2 element problem with displacements

Results – Monolithic scheme Jacobian check – Schematic of 2 element problem with displacements

Results – Monolithic scheme Jacobian check – Schematic of 2 element problem with displacements

Results – Monolithic scheme Jacobian check – Schematic of 2 element problem with displacements Displacements of structural element 1 affect zero contour in both fluid elements Presumed Actual

Results – Monolithic scheme Jacobian check – Schematic of 2 element problem with displacements 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

Goal 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 Does the approximated solution describe the physics of the system? Yes, based on qualitative check! How can we efficiently solve the system? Monolithically, but analytic Jacobian is not numerically consistent What makes this approach suitable for optimization?

Outlook What makes this approach suitable for optimization?

Outlook What makes this approach suitable for optimization? Move the extra beam to the top wall

Outlook What makes this approach suitable for optimization?

Goal 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 Does the approximated solution describe the physics of the system? Yes, based on qualitative check! How can we efficiently solve the system? Monolithically, but Jacobian is not numerically consistent 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

* Loosely translated by Thijs Bosma ‘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

How to describe the behavior of the system? The modeled problem The physical configuration 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?

Discontinuous shape functions

Results – Staggered scheme The process

Results – Staggered scheme The process

Results – Staggered scheme Residual development

Levelset update - Changing DOFs

Results – Staggered scheme Pressure and displacement field – XFEM model and COMSOL XFEM: BACKUP SLIDE PRESSURE COMSOL:

Finite Differences

Non-dimensional numbers BACKUP

Mesh mismatch

Residual equations BACKUP

3-field setup

Projection onto fluid mesh