Interior Penalty Finite Element Approximation of Navier ...

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Introduction Numerics Results

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Application to Free Surface Flows Christoph Winkelmann Public Defense CMCS - EPFL

December 5th 2007

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Understanding the Title Physics Mathematics

Understanding the Title

I

Interior Penalty Finite Element Approximation

I

of Navier-Stokes Equations

I

and Application to Free Surface Flows

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Understanding the Title Physics Mathematics

Understanding the Title

I

Interior Penalty Finite Element Approximation

I

of Navier-Stokes Equations

I

and Application to Free Surface Flows physical phenomena - in everyday life

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Understanding the Title Physics Mathematics

Understanding the Title

I

Interior Penalty Finite Element Approximation

I

of Navier-Stokes Equations mathematical model - on paper

I

and Application to Free Surface Flows physical phenomena - in everyday life

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Understanding the Title Physics Mathematics

Understanding the Title

I

Interior Penalty Finite Element Approximation numerical method - in computer

I

of Navier-Stokes Equations mathematical model - on paper

I

and Application to Free Surface Flows physical phenomena - in everyday life

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Understanding the Title Physics Mathematics

What are Free Surface Flows?

Definition: Motion of two or more fluids (liquids/gases) that do not mix and are hence separated by a free surface

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Understanding the Title Physics Mathematics

What are Free Surface Flows?

Definition: Motion of two or more fluids (liquids/gases) that do not mix and are hence separated by a free surface Examples: I

water and air

I

oil in water

I

more exotic combinations

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Understanding the Title Physics Mathematics

What are Free Surface Flows?

Definition: Motion of two or more fluids (liquids/gases) that do not mix and are hence separated by a free surface Examples: I

water and air

I

oil in water

I

more exotic combinations

Why do we study free surface flows?

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Understanding the Title Physics Mathematics

Applications: Jets 2 fluids: water and air free surface: surface of the jet

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Understanding the Title Physics Mathematics

Applications: Jets 2 fluids: water and air free surface: surface of the jet application: e.g. metal casting

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Understanding the Title Physics Mathematics

Applications: Jets 2 fluids: water and air free surface: surface of the jet application: e.g. metal casting

ink jet printers?

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Understanding the Title Physics Mathematics

Applications: Droplets

free surface: droplet surface

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Understanding the Title Physics Mathematics

Applications: Droplets

free surface: droplet surface application: e.g. ink jet printers

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Understanding the Title Physics Mathematics

Applications: Droplets

free surface: droplet surface application: e.g. ink jet printers gas in liquid instead of liquid in gas: bubbles application: e.g. bubble column chemical reactors

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Understanding the Title Physics Mathematics

Applications: Waves

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Understanding the Title Physics Mathematics

Applications: Waves application: namics

e.g. ship hydrody-

origin of this research project: scientific partnership of EPFL with team Alinghi

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Understanding the Title Physics Mathematics

The Navier-Stokes Equations Aim: Describe the motion, i.e. the evolution of the velocity u and the pressure p in function of time and space.

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Understanding the Title Physics Mathematics

The Navier-Stokes Equations Aim: Describe the motion, i.e. the evolution of the velocity u and the pressure p in function of time and space. I

Momentum conservation - Newton’s law F = m a Forces involved: friction, pressure, gravitation, surface tension ρ∂t u + ρ(u · ∇)u − ∇ · (2µD(u)) + ∇p = ρg + δΓ σκnΓ describes evolution of u and p

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Understanding the Title Physics Mathematics

The Navier-Stokes Equations Aim: Describe the motion, i.e. the evolution of the velocity u and the pressure p in function of time and space. I

Momentum conservation - Newton’s law F = m a Forces involved: friction, pressure, gravitation, surface tension ρ∂t u + ρ(u · ∇)u − ∇ · (2µD(u)) + ∇p = ρg + δΓ σκnΓ describes evolution of u and p

I

Volume conservation - incompressibility ∇·u=0

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Understanding the Title Physics Mathematics

The Level Set Approach Aim: Describe the motion of the interface Γ Mass conservation 6ρ

uΓ

x -

∂t ρ + u · ∇ρ = 0 µ = µ(ρ)

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Understanding the Title Physics Mathematics

The Level Set Approach Aim: Describe the motion of the interface Γ Mass conservation

Signed distance function

6ρ

6φ

u-

uΓ

x -

Γ

x -

∂t ρ + u · ∇ρ = 0

∂t φ + u · ∇φ = 0

µ = µ(ρ)

ρ = ρ(φ) µ = µ(φ)

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Interior Penalty Formulation of Flow Equations An Unfitted Scheme for the Stokes Problem Algorithmic and Software Contributions

Idea of Discretization Problem: A computer of limited size cannot compute in limited time the solution (u, p, φ) at an infinite number of points and times.

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Interior Penalty Formulation of Flow Equations An Unfitted Scheme for the Stokes Problem Algorithmic and Software Contributions

Idea of Discretization Problem: A computer of limited size cannot compute in limited time the solution (u, p, φ) at an infinite number of points and times. Solution: Approximation of the solution at a finite number of discrete points and times:

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Interior Penalty Formulation of Flow Equations An Unfitted Scheme for the Stokes Problem Algorithmic and Software Contributions

Idea of Discretization Problem: A computer of limited size cannot compute in limited time the solution (u, p, φ) at an infinite number of points and times. Solution: Approximation of the solution at a finite number of discrete points and times:

Finite Elements: geometric elements (triangles) of finite size

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Interior Penalty Formulation of Flow Equations An Unfitted Scheme for the Stokes Problem Algorithmic and Software Contributions

Transformation

ρ∂t u + ρ(u · ∇)u − ∇ · (2µD(u)) + ∇p = ρg + δΓ σκnΓ

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Interior Penalty Formulation of Flow Equations An Unfitted Scheme for the Stokes Problem Algorithmic and Software Contributions

Transformation

ρ∂t u + ρ(u · ∇)u − ∇ · (2µD(u)) + ∇p = ρg + δΓ σκnΓ discretization in time and linearization

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Interior Penalty Formulation of Flow Equations An Unfitted Scheme for the Stokes Problem Algorithmic and Software Contributions

Transformation

ρ∂t u + ρ(u · ∇)u − ∇ · (2µD(u)) + ∇p = ρg + δΓ σκnΓ discretization in time and linearization αu + (β · ∇)u − ∇ · (2µD(u)) + ∇p = f with α = ρn+1 /∆t, u = un+1 , β = ρn+1 un , p = p n+1 , and f = ρn+1 un /∆t + ρn+1 g + δΓ σκnΓ

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Interior Penalty Formulation of Flow Equations An Unfitted Scheme for the Stokes Problem Algorithmic and Software Contributions

Interior Penalty Formulation of Flow Equations

Idea: penalize jumps of derivatives of continuous FE functions Stabilization of otherwise unstable FE spaces, to get rid of spurious pressure modes: hf3 jp (p, q) := γp [[∇p]], [[∇q]] , max{hf |β| , µ} ΓI added to the incompressibility constraint equation.

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Interior Penalty Formulation of Flow Equations An Unfitted Scheme for the Stokes Problem Algorithmic and Software Contributions

Interior Penalty Formulation of Flow Equations

Idea: penalize jumps of derivatives of continuous FE functions Stabilization of otherwise unstable FE spaces, to get rid of spurious pressure modes: hf3 jp (p, q) := γp [[∇p]], [[∇q]] , max{hf |β| , µ} ΓI added to the incompressibility constraint equation. γp is a dimensionless constant γp = 0 allowed for inf-sup stable FE spaces (Taylor-Hood)

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Interior Penalty Formulation of Flow Equations An Unfitted Scheme for the Stokes Problem Algorithmic and Software Contributions

Interior Penalty Finite Element Scheme k ≥ 1: polynomial degree, consider Whk = [Vhk ]d × Vhk , equal order interpolation continuous FE spaces. Find (u, p) ∈ Whk , such that Bh [(u, p), (v, q)] = fh (v) ∀(v, q) ∈ Whk ,

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Interior Penalty Formulation of Flow Equations An Unfitted Scheme for the Stokes Problem Algorithmic and Software Contributions

Interior Penalty Finite Element Scheme k ≥ 1: polynomial degree, consider Whk = [Vhk ]d × Vhk , equal order interpolation continuous FE spaces. Find (u, p) ∈ Whk , such that Bh [(u, p), (v, q)] = fh (v) ∀(v, q) ∈ Whk , where Bh [(u, p), (v, q)] = (αu, v) + ((β · ∇)u, v) + (2µD(u), D(v)) − (p, ∇ · v) + (q, ∇ · u) + jp (p, q) fh (v) = (f, v) + weak boundary conditions

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Interior Penalty Formulation of Flow Equations An Unfitted Scheme for the Stokes Problem Algorithmic and Software Contributions

An Unfitted Scheme for the Stokes Problem If mesh is not fitted to interface, deterioration of convergence can be expected. Goal: Suggest and analyze a scheme that does not have this problem.

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Interior Penalty Formulation of Flow Equations An Unfitted Scheme for the Stokes Problem Algorithmic and Software Contributions

Introduction Numerics Results

An Unfitted Scheme for the Stokes Problem If mesh is not fitted to interface, deterioration of convergence can be expected. Goal: Suggest and analyze a scheme that does not have this problem. Simplified model problem: Stationary Stokes problem in IR 2 : −∇ · (2µi D(u)) + ∇p = f ∇·u=0 u=0

in Ωi , i = 1, 2 in Ω on ∂Ω \ Γ

Interface conditions at interface Γ: [[u]]Γ = 0,

[[2µD(u)nΓ − pnΓ ]]Γ = σκnΓ

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Interior Penalty Formulation of Flow Equations An Unfitted Scheme for the Stokes Problem Algorithmic and Software Contributions

Finite Element Spaces: P1 -P0 Ω1

Ω

Γ

s

Ω2

T1 ∪ T2 T1 T2 Vh,1

XX XX XX X X X X X X X XX

Vh,2

X XX X XX X X XX

Qh,1 Qh,2

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Interior Penalty Formulation of Flow Equations An Unfitted Scheme for the Stokes Problem Algorithmic and Software Contributions

Bilinear Form Bh [(uh , ph ), (vh , qh )] =ah (uh , vh ) + bh (ph , vh ) − bh (qh , uh ) + J(uh , ph , vh , qh ),

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Interior Penalty Formulation of Flow Equations An Unfitted Scheme for the Stokes Problem Algorithmic and Software Contributions

Bilinear Form Bh [(uh , ph ), (vh , qh )] =ah (uh , vh ) + bh (ph , vh ) − bh (qh , uh )

ah (uh , vh ) =

2 Z X

+ J(uh , ph , vh , qh ), Z 2µD(uh ) : D(vh )dx − {2µD(uh )n}·[[vh ]] ds

Ωi

Γ

Zi=1 Z − {2µD(vh )n} · [[uh ]] ds + γ1 h−1 [[uh ]] · [[vh ]] ds, Γ

Christoph Winkelmann Public Defense

Γ

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Interior Penalty Formulation of Flow Equations An Unfitted Scheme for the Stokes Problem Algorithmic and Software Contributions

Bilinear Form Bh [(uh , ph ), (vh , qh )] =ah (uh , vh ) + bh (ph , vh ) − bh (qh , uh )

ah (uh , vh ) =

+ J(uh , ph , vh , qh ), Z 2µD(uh ) : D(vh )dx − {2µD(uh )n}·[[vh ]] ds

2 Z X Ωi

Γ

Zi=1 Z − {2µD(vh )n} · [[uh ]] ds + γ1 h−1 [[uh ]] · [[vh ]] ds, Γ

bh (ph , vh ) = −

Γ 2 Z X i=1

Z ph ∇ · vh dx +

Ωi

Christoph Winkelmann Public Defense

{ph }[[vh · n]] ds, Γ

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Interior Penalty Formulation of Flow Equations An Unfitted Scheme for the Stokes Problem Algorithmic and Software Contributions

Bilinear Form Bh [(uh , ph ), (vh , qh )] =ah (uh , vh ) + bh (ph , vh ) − bh (qh , uh )

ah (uh , vh ) =

+ J(uh , ph , vh , qh ), Z 2µD(uh ) : D(vh )dx − {2µD(uh )n}·[[vh ]] ds

2 Z X Ωi

Γ

Zi=1 Z − {2µD(vh )n} · [[uh ]] ds + γ1 h−1 [[uh ]] · [[vh ]] ds, Γ

bh (ph , vh ) = −

Γ 2 Z X i=1

J(uh , ph , vh , qh ) =

Z ph ∇ · vh dx +

Ωi

XZ K ∈T

{ph }[[vh · n]] ds, Γ

γ2 hK [[ph n − 2µD(uh )n]]·

∂K ∪ΓK

[[qh n − 2µD(vh )n]]ds. Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Interior Penalty Formulation of Flow Equations An Unfitted Scheme for the Stokes Problem Algorithmic and Software Contributions

Formulation Find (ui , pi ) ∈ Vh,i × Qh,i , i = 1, 2 such that uh = (u1 , u2 ), ph = (p1 , p2 ) satisfy Z Bh [(uh , ph ), (vh , qh )] = (f, vh ) + σκhvh · ni ds Γ XZ + γ2 hK σκn·[[qh n−2µD(vh )n]] ds ∀(vh , qh ) ∈ Vh ×Qh K ∈T

ΓK

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Interior Penalty Formulation of Flow Equations An Unfitted Scheme for the Stokes Problem Algorithmic and Software Contributions

Formulation Find (ui , pi ) ∈ Vh,i × Qh,i , i = 1, 2 such that uh = (u1 , u2 ), ph = (p1 , p2 ) satisfy Z Bh [(uh , ph ), (vh , qh )] = (f, vh ) + σκhvh · ni ds Γ XZ + γ2 hK σκn·[[qh n−2µD(vh )n]] ds ∀(vh , qh ) ∈ Vh ×Qh K ∈T

ΓK

Weighted averages as key ingredients: κi = ⇒

|Ki | , |K |

{a} = κ1 a|Ω1 + κ2 a|Ω2 , κ1 + κ2 = 1,

Christoph Winkelmann Public Defense

hai = κ2 a|Ω1 + κ1 a|Ω2 , [[ab]] = {a}[[b]] + [[a]]hbi.

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Interior Penalty Formulation of Flow Equations An Unfitted Scheme for the Stokes Problem Algorithmic and Software Contributions

A Priori Error Estimate

Assume that the weak solution (u, p) resides in [H 2 (Ω∗ ) ∩ H01 (Ω)]d × [H 1 (Ω∗ ) ∩ L20 (Ω)], Ω∗ = Ω1 ∪ Ω2 ; then the finite element solution satisfies the error estimate ku − uh k1,Ω + kp − ph k0,Ω ≤ ch(kuk2,Ω∗ + kpk1,Ω∗ ).

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Interior Penalty Formulation of Flow Equations An Unfitted Scheme for the Stokes Problem Algorithmic and Software Contributions

Algorithmic and Software Contributions

I

Preconditioning strategy with reusable preconditioner for sequences of linear systems with evolving matrices in general

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Interior Penalty Formulation of Flow Equations An Unfitted Scheme for the Stokes Problem Algorithmic and Software Contributions

Algorithmic and Software Contributions

I

Preconditioning strategy with reusable preconditioner for sequences of linear systems with evolving matrices in general

I

Investigation of strategies for reducing time and memory cost of interior penalty methods

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Interior Penalty Formulation of Flow Equations An Unfitted Scheme for the Stokes Problem Algorithmic and Software Contributions

Algorithmic and Software Contributions

I

Preconditioning strategy with reusable preconditioner for sequences of linear systems with evolving matrices in general

I

Investigation of strategies for reducing time and memory cost of interior penalty methods

I

Intuitive and efficient reformulation of distance computation in fast marching schemes for reinitialization on unstructured meshes in any dimension

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Interior Penalty Formulation of Flow Equations An Unfitted Scheme for the Stokes Problem Algorithmic and Software Contributions

Algorithmic and Software Contributions

I

Preconditioning strategy with reusable preconditioner for sequences of linear systems with evolving matrices in general

I

Investigation of strategies for reducing time and memory cost of interior penalty methods

I

Intuitive and efficient reformulation of distance computation in fast marching schemes for reinitialization on unstructured meshes in any dimension

I

Suggestion and implementation of an appropriate choice of mathematical abstractions for a state-of-the art numerical code

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Laminar Flow around a Cylinder Rising Bubble with Surface Tension The End

Laminar Flow around a Cylinder Benchmark by Sch¨afer/Turek, in 2D/(3D)

Benchmark quantities: drag and lift coefficient, pressure drop discretization lower bound upper bound P1 -P1 γp = 0.1 P2 -P2 γp = 0.01 P2 -P1

N

207030 52569 207786 52302 156047 39330

h

0.005 0.01 0.01 0.02 0.01 0.02

Christoph Winkelmann Public Defense

cD 5.5700 5.5900 5.5686 5.5603 5.5780 5.5766 5.5775 5.5742

cL 0.0104 0.0110 0.0094 0.0075 0.0105 0.0108 0.0105 0.0107

∆p 0.1172 0.1176 0.11777 0.11781 0.11753 0.11765 0.11751 0.11745

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Laminar Flow around a Cylinder Rising Bubble with Surface Tension The End

Rising Bubble with Surface Tension Benchmark paper by Hysing et. al.

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Laminar Flow around a Cylinder Rising Bubble with Surface Tension The End

Rising Bubble with Surface Tension Benchmark paper by Hysing et. al. Two test cases: I

ellipsoidal bubble (high surface tension)

high surface tension Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Laminar Flow around a Cylinder Rising Bubble with Surface Tension The End

Rising Bubble with Surface Tension Benchmark paper by Hysing et. al. Two test cases: I

ellipsoidal bubble (high surface tension)

I

skirted ellipsoidal-cap bubble (low surface tension)

high surface tension Christoph Winkelmann Public Defense

low surface tension IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Laminar Flow around a Cylinder Rising Bubble with Surface Tension The End

Rising Bubbles

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Laminar Flow around a Cylinder Rising Bubble with Surface Tension The End

Results

Benchmark quantities (in function of time): I

Center of mass of the bubble

I

Circularity of the bubble (∈ (0, 1])

I

Mean velocity of the bubble

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Laminar Flow around a Cylinder Rising Bubble with Surface Tension The End

Results

Benchmark quantities (in function of time): I

Center of mass of the bubble

I

Circularity of the bubble (∈ (0, 1])

I

Mean velocity of the bubble

Reliable reproduction of benchmark values and competitive convergence orders using Taylor-Hood finite elements

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Laminar Flow around a Cylinder Rising Bubble with Surface Tension The End

Two Rising Bubbles

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Laminar Flow around a Cylinder Rising Bubble with Surface Tension The End

. . . and this is . . .

Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows

Introduction Numerics Results

Laminar Flow around a Cylinder Rising Bubble with Surface Tension The End

Sy no Frooge? *

* Any questions? / Y a-t-il des questions? Christoph Winkelmann Public Defense

IP-FEM for NS eq. and Application to Free Surface Flows