Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows

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

November 2nd 2007

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Introduction Motivation

Motivation Multiphase flow occurs under many forms: jets, bubbles, droplets, waves, flims. Many applications, e.g.: I

water waves on rivers, lakes and oceans

I

waves interacting with vessels, structures and shores

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Introduction Motivation

Motivation Multiphase flow occurs under many forms: jets, bubbles, droplets, waves, flims. Many applications, e.g.: I

water waves on rivers, lakes and oceans

I

waves interacting with vessels, structures and shores

I

injection, casting and extrusion of polymers and liquid metals

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Introduction Motivation

Motivation Multiphase flow occurs under many forms: jets, bubbles, droplets, waves, flims. Many applications, e.g.: I

water waves on rivers, lakes and oceans

I

waves interacting with vessels, structures and shores

I

injection, casting and extrusion of polymers and liquid metals

I

bubble column chemical reactors

I

ink jet printers

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Introduction Motivation

Motivation Multiphase flow occurs under many forms: jets, bubbles, droplets, waves, flims. Many applications, e.g.: I

water waves on rivers, lakes and oceans

I

waves interacting with vessels, structures and shores

I

injection, casting and extrusion of polymers and liquid metals

I

bubble column chemical reactors

I

ink jet printers

I

ship hydrodynamics: Scientific partnership of EPFL with Team Alinghi one task: prediction of wake resistance of a given hull (up to 60% of total resistance)

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Introduction Motivation

Approaches

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Arbitrary Lagrangian-Eulerian (ALE) approach with free boundary problem: breaking waves

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Introduction Motivation

Approaches

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Arbitrary Lagrangian-Eulerian (ALE) approach with free boundary problem: breaking waves

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Volume of Fluid approach with free interface problem: interface smears out over time

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Introduction Motivation

Approaches

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Arbitrary Lagrangian-Eulerian (ALE) approach with free boundary problem: breaking waves

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Volume of Fluid approach with free interface problem: interface smears out over time

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⇒ Level Set approach with free interface Work started at CMCS-EPFL by N. Parolini and E. Burman

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Introduction Outline

Outline

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Mathematical Model

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Finite Element Methods with Interior Penalty Stabilization

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Algorithmic Aspects

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Numerical Results: Accuracy

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Applications

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Software Design Aspects

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Mathematical Model

Mathematical Model

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Mathematical Model I I

Flow Model Interface Evolution Modelling

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Finite Element Methods with Interior Penalty Stabilization

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Algorithmic Aspects

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Numerical Results: Accuracy

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Applications

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Software Design Aspects

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Mathematical Model Flow Model

Flow Model Incompressible Navier-Stokes equations with variable density and viscosity in domain Ω ⊂ IR d , d ∈ {2, 3} and time interval (0, T ): ρ∂t u + ρ(u · ∇)u − ∇ · (2µD(u)) + ∇p = f

in Ω × (0, T )

∇·u = 0

in Ω × (0, T )

u = u0

in Ω at t = 0

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Mathematical Model Flow Model

Flow Model Incompressible Navier-Stokes equations with variable density and viscosity in domain Ω ⊂ IR d , d ∈ {2, 3} and time interval (0, T ): ρ∂t u + ρ(u · ∇)u − ∇ · (2µD(u)) + ∇p = f

in Ω × (0, T )

∇·u = 0

in Ω × (0, T )

∂t ρ + u · ∇ρ = 0

in Ω × (0, T )

u = u0

in Ω at t = 0

ρ = ρ0

in Ω at t = 0

µ = µ(ρ)

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Mathematical Model Flow Model

Boundary Conditions

u = gD

on ΓD × (0, T )

2µD(u)n − pn = gN

on ΓN × (0, T )

u · n = gD · n (ωCτ u + (1 − ω)(2µD(u)n)) · τ = ωCτ gD · τ

ΓR encompasses I

Dirichlet with ω = 1

I

no-slip with ω = 1 and gD = 0

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free-slip with ω = 0 and gD = 0

on ΓR × (0, T ) on ΓR × (0, T )

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Mathematical Model Flow Model

Application to Free Surface Flow The interface Γ cuts the domain Ω into two subdomains Ω+ (t) and Ω− (t). Then:  − ρ x ∈ Ω− (t) ρ(x, t) = + ρ x ∈ Ω+ (t) and likewise for µ.

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Mathematical Model Flow Model

Application to Free Surface Flow The interface Γ cuts the domain Ω into two subdomains Ω+ (t) and Ω− (t). Then:  − ρ x ∈ Ω− (t) ρ(x, t) = + ρ x ∈ Ω+ (t) and likewise for µ. Interface conditions at interface Γ: [[u]]Γ = 0,

[[2µD(u)nΓ − pnΓ ]]Γ = σκnΓ

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Mathematical Model Interface Evolution Modelling

Interface Evolution Modelling The Level Set Method:  − ρ φ<0 ρ(φ) = , ρ+ φ > 0 ∂t φ + u · ∇φ = 0

 µ(φ) =

µ− µ+

φ<0 φ>0

in Ω × (0, T )

φ = φ0

in Ω at t = 0

φ = φin

on Σin

where Σin = {(x, t) ∈ ∂Ω × (0, T ) : u(x, t) · n < 0}

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Mathematical Model Interface Evolution Modelling

Level Set Properties

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Interface related quantities ∇φ |∇φ| κ = −∇ · nΓ

nΓ =

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Mathematical Model Interface Evolution Modelling

Level Set Properties

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Interface related quantities ∇φ |∇φ| κ = −∇ · nΓ

nΓ =

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Initial condition: Not unique, |∇φ| ≈ 1 ⇒ signed distance function

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Reinitialization: |∇φ| = 1 is not preserved under advection with u

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Finite Element Methods with Interior Penalty Stabilization

Finite Element Methods with Interior Penalty Stabilization I I

Mathematical Model Finite Element Methods with Interior Penalty Stabilization I I I I

Preliminaries Interior Penalty Formulation of Flow Equations An Unfitted Scheme for the Stokes Problem Interior Penalty Formulation of Advection Equation

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Algorithmic Aspects

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Numerical Results: Accuracy

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Applications

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Software Design Aspects

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Finite Element Methods with Interior Penalty Stabilization Preliminaries

Time Discretization and Linearization ρ∂t u + ρ(u · ∇)u − ∇ · (2µD(u)) + ∇p = f

in Ω × (0, T )

∇·u = 0

in Ω × (0, T )

Discretizing in time with BDF2 and linearizing the convective term yields the Oseen problem: αu + (β · ∇)u − ∇ · (2µD(u)) + ∇p = f

in Ω

∇·u = 0

in Ω

3ρ(t n+1 ) 2∆t β = ρ(t n+1 )ˆ un+1

µ = µ(t n+1 )

p = p(t n+1 )

f = f(t n+1 ) +

α=

u = u(t n+1 ) ρ(t n+1 ) (4u(t n ) − u(t n−1 )) 2∆t

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Finite Element Methods with Interior Penalty Stabilization Preliminaries

Variational Formulation Find u ∈ V = {w ∈ [H 1 (Ω)]d : w|ΓD = 0 and (w · n)|ΓR = 0}, p ∈ L2 (Ω) and φ ∈ L2 (Ω) such that B[(u, p), (v, q)] = f (v),

∀(v, q) ∈ V × Q,

where B[(u, p), (v, q)] =(αu, v) + ((β · ∇)u, v) + (2µD(u), D(v)) + hω ∗ u, viΓR − (p, ∇ · v) + (q, ∇ · u), f (v) =(f, v) + hgN , viΓN − hσκnΓ , viΓ + hω ∗ gD , viΓR with ω ∗ = ωCτ /(1 − ω).

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Finite Element Methods with Interior Penalty Stabilization Interior Penalty Formulation of Flow Equations

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 [[∇p]], [[∇q]] , jp (p, q) := γp max{hf |β| , µ} ΓI added to the incompressibility constraint equation.

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Finite Element Methods with Interior Penalty Stabilization Interior Penalty Formulation of Flow Equations

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 [[∇p]], [[∇q]] , jp (p, q) := γp 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)

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Finite Element Methods with Interior Penalty Stabilization Interior Penalty Formulation of Flow Equations

More stabilization terms Stabilization for dominant convection at high Reynolds numbers:   h2 jβ (u, v) := γβ f [[β · ∇u]], [[β · ∇v]] , |β| ΓI

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Finite Element Methods with Interior Penalty Stabilization Interior Penalty Formulation of Flow Equations

More stabilization terms Stabilization for dominant convection at high Reynolds numbers:   h2 jβ (u, v) := γβ f [[β · ∇u]], [[β · ∇v]] , |β| ΓI Additional control of the incompressibility condition: jdiv (u, v) := hγdiv hf2 |β| [[∇ · u]], [[∇ · v]]iΓI , added to the momentum equation.

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Finite Element Methods with Interior Penalty Stabilization Interior Penalty Formulation of Flow Equations

More stabilization terms Stabilization for dominant convection at high Reynolds numbers:   h2 jβ (u, v) := γβ f [[β · ∇u]], [[β · ∇v]] , |β| ΓI Additional control of the incompressibility condition: jdiv (u, v) := hγdiv hf2 |β| [[∇ · u]], [[∇ · v]]iΓI , added to the momentum equation. γβ and γdiv are dimensionless constants γβ = γdiv = 0 allowed for low Re

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Finite Element Methods with Interior Penalty Stabilization Interior Penalty Formulation of Flow Equations

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 ,

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Finite Element Methods with Interior Penalty Stabilization Interior Penalty Formulation of Flow Equations

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, with P = I − n ⊗ n, Bh [(u, p), (v, q)] = (αu, v) + ((β · ∇)u, v) + (2µD(u), D(v)) − hω P(2µD(u)n), PviΓR − hω Pu, P(2µD(v)n)iΓR + hCn u · n, v · niΓR + hωCτ Pu, PviΓR − (p, ∇ · v) + (q, ∇ · u) + jβ (u, v) + jdiv (u, v) + jp (p, q)

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Finite Element Methods with Interior Penalty Stabilization Interior Penalty Formulation of Flow Equations

Right Hand Side and Weak Boundary Conditions Right Hand Side fh (v) = (f, v) + hgN , viΓN − hσκnΓ , viΓ − hωPgD , P(2µD(v)n)iΓR + hCn gD · n, v · niΓR + hωCτ PgD , PviΓR

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Finite Element Methods with Interior Penalty Stabilization Interior Penalty Formulation of Flow Equations

Right Hand Side and Weak Boundary Conditions Right Hand Side fh (v) = (f, v) + hgN , viΓN − hσκnΓ , viΓ − hωPgD , P(2µD(v)n)iΓR + hCn gD · n, v · niΓR + hωCτ PgD , PviΓR Weak Boundary Conditions Cn = γn max{|β| , µ/h} Cτ = γτ µ/h + max{−β · n, 0} All γi are dimensionless constants.

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Finite Element Methods with Interior Penalty Stabilization An Unfitted Scheme for the Stokes Problem

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.

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Finite Element Methods with Interior Penalty Stabilization An Unfitted Scheme for the Stokes Problem

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

in Ωi , i = 1, 2 in Ω

u = 0 on ∂Ω \ Γ Interface conditions at interface Γ: [[u]]Γ = 0,

[[2µD(u)nΓ − pnΓ ]]Γ = σκnΓ

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Finite Element Methods with Interior Penalty Stabilization An Unfitted Scheme for the Stokes Problem

Finite Element Spaces: P1 -P0 Ω

Ω1

Γ

s

Ω2

T1 ∪ T2 T1 T2 Vh,1 Vh,2 Qh,1 Qh,2

XX XX XX XX X   XX XX XX    X XX  X XX X X XX

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Finite Element Methods with Interior Penalty Stabilization An Unfitted Scheme for the Stokes Problem

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

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Finite Element Methods with Interior Penalty Stabilization An Unfitted Scheme for the Stokes Problem

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,

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

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Finite Element Methods with Interior Penalty Stabilization An Unfitted Scheme for the Stokes Problem

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

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Finite Element Methods with Interior Penalty Stabilization An Unfitted Scheme for the Stokes Problem

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

Γ

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Finite Element Methods with Interior Penalty Stabilization An Unfitted Scheme for the Stokes Problem

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

Ωi

Z ph ∇ · vh dx +

{ph }[[vh · n]] ds, Γ

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Finite Element Methods with Interior Penalty Stabilization An Unfitted Scheme for the Stokes Problem

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.

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Finite Element Methods with Interior Penalty Stabilization An Unfitted Scheme for the Stokes Problem

Approximation Properties kv k2±1/2,h,Γ :=

X

∓1 hK kv k20,ΓK

K ∈Gh

|||(v, q)|||2 := kvk20,Ω +k∇vk20,Ω +k{2µD(v)n}k2−1/2,h,Γ +k[[v]]k21/2,h,Γ X + kqk20,Ω + k[[q n − 2µD(v)n]]k2−1/2,h,∂K ∪ΓK K ∈T

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Finite Element Methods with Interior Penalty Stabilization An Unfitted Scheme for the Stokes Problem

Approximation Properties kv k2±1/2,h,Γ :=

X

∓1 hK kv k20,ΓK

K ∈Gh

|||(v, q)|||2 := kvk20,Ω +k∇vk20,Ω +k{2µD(v)n}k2−1/2,h,Γ +k[[v]]k21/2,h,Γ X + kqk20,Ω + k[[q n − 2µD(v)n]]k2−1/2,h,∂K ∪ΓK K ∈T

Theorem 2.5.1 There are interpolation operators (Ih∗ , Ch∗ ) s. t. |||(v − Ih∗ v, q − Ch∗ q)||| ≤ Ch(kvk2,Ω∗ + kqk1,Ω∗ ), ∀(v, q) ∈ [H01 (Ω) ∩ H 2 (Ω∗ )]2 × L20 (Ω) ∩ H 1 (Ω∗ ).

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Finite Element Methods with Interior Penalty Stabilization An Unfitted Scheme for the Stokes Problem

The inf-sup Condition Lemma 2.5.3 For any S : Ω → IR d×d s.t. S|Ki =const: k{S n}k2−1/2,h,Γ ≤ CI kSk20,Ω∗ Weighted average necessary here to allow interface to be arbitrarily close to element boundary.

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Finite Element Methods with Interior Penalty Stabilization An Unfitted Scheme for the Stokes Problem

The inf-sup Condition Lemma 2.5.3 For any S : Ω → IR d×d s.t. S|Ki =const: k{S n}k2−1/2,h,Γ ≤ CI kSk20,Ω∗ Weighted average necessary here to allow interface to be arbitrarily close to element boundary. Theorem 2.5.2 Let (uh , ph ) ∈ Vh × Qh , then cs |||(uh , ph )||| ≤

sup (vh ,qh )∈Vh ×Qh

cs is at least linear in µmax /µmin .

Bh [(uh , ph ), (vh , qh )] . |||(vh , qh )|||

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Finite Element Methods with Interior Penalty Stabilization An Unfitted Scheme for the Stokes Problem

A Priori Error Estimate

Property 2.5.1 Assume that the weak solution (u, p) resides in [H 2 (Ω∗ )]d × [H 1 (Ω∗ ) ∩ L20 (Ω)]; then the finite element solution satisfies the error estimate |||(u − uh , p − ph )||| ≤ ch(kuk2,Ω∗ + kpk1,Ω∗ ).

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Finite Element Methods with Interior Penalty Stabilization Interior Penalty Formulation of Advection Equation

Interior Penalty Formulation of Advection Equation ∂t φ + u · ∇φ = 0 Discretizing in time with BDF2 yields: αφ + β · ∇φ = f

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Finite Element Methods with Interior Penalty Stabilization Interior Penalty Formulation of Advection Equation

Interior Penalty Formulation of Advection Equation ∂t φ + u · ∇φ = 0 Discretizing in time with BDF2 yields: αφ + β · ∇φ = f Space discretization with continuous finite elements and interior penalty stabilization: Find φ ∈ Vhk such that for all ψ ∈ Vhk (αφ, ψ) + (β · ∇φ) − hβ · n φ, ψi∂Ωin + hγφ hf2 |β · n| [[∇φ]]f , [[∇ψ]]f iΓI = (f , ψ) − hβ · n φin , ψi∂Ωin

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Algorithmic Aspects

Algorithmic Aspects I

Mathematical Model

I

Finite Element Methods with Interior Penalty Stabilization Algorithmic Aspects

I

I I I I I

Iterative Schemes Linear Algebraic Solvers Treatment of Stabilization Terms Incremental Matrix Update Level Set Reinitialization

I

Numerical Results: Accuracy

I

Applications

I

Software Design Aspects

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Algorithmic Aspects Iterative Schemes

Iterative Schemes

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solve Navier-Stokes and level set advection equations alternating until convergence

I

Navier-Stokes linearized around previous iteration

I

use BDF2-extrapolation as initial guess for convection velocity

I

simple splitting, fixpoint iterations or Aitken relaxation

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Algorithmic Aspects Linear Algebraic Solver

Linear Algebraic Solver I

GMRES + ILU

I

adaptive reuse of the ILU preconditioner: strategy based on the concept of amortization of the costs for recomputing the preconditioner

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Algorithmic Aspects Linear Algebraic Solver

Linear Algebraic Solver I

GMRES + ILU

I

adaptive reuse of the ILU preconditioner: strategy based on the concept of amortization of the costs for recomputing the preconditioner

I

recompute preconditioner for system N + 1 if   N X 1 cN > cPC + ci N i=1

ci : number of iterations for system i, cPC cost of preconditioner recomputation in iteration equivalents.

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Algorithmic Aspects Treatment of Stabilization Terms

Reducing Assembly Cost: Idea

j(p, q) =

X K ∈Th

2 hK

Z γ[[∇p]] · [[∇q]]ds ∂K

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Algorithmic Aspects Treatment of Stabilization Terms

Reducing Assembly Cost: Idea

j(p, q) =

X K ∈Th

2 hK

Z γ[[∇p]] · [[∇q]]ds ∂K

for the right scaling:

γ=

hK max(hK |β| , µ)

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Algorithmic Aspects Treatment of Stabilization Terms

Reducing Assembly Cost: Idea

j(p, q) =

X K ∈Th

2 hK

Z γ[[∇p]] · [[∇q]]ds ∂K

for the right scaling:

γ=

hK max(hK |β| , µ)

⇒ the related matrix J could be rebuilt all the time As j is “only” a stabilization term, this is not necessary. Instead, we rebuild it only when rebuilding the preconditioner.

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Algorithmic Aspects Treatment of Stabilization Terms

Reducing Assembly Cost: Result −4

x 10

rebuild stab. always rebuild stab. with preconditioner

CPU time / M

6

5

4

3 2

10

3

4

10

10 M

CPU time per degree of freedom M vs. M

5

10

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Algorithmic Aspects Treatment of Stabilization Terms

Reducing the Stencil Idea: Splitting [[∇p]]f · [[∇q]]f = (∇p + · nf )(∇q + · nf ) + (∇p − · nf )(∇q − · nf ) {z } | → jsG (p,q)

−(∇p · nf )(∇q · nf ) − (∇p − · nf )(∇q + · nf ) | {z } +



→ −jX (p,q)

Overrelaxation:  j(p, q) = θjsG (p, q) − (θ − 1)jsG (p, q) + jX (p, q) | {z } | {z } → matrix → rhs vector More nonlinear iterations (factor 1.5. . . 2), less nonzeros in matrix (factor ≈ d)

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Algorithmic Aspects Incremental Matrix Update in Two Fluid Flow Problems

Incremental Matrix Update in Two Fluid Flow Problems

I

ρ and µ vary in time, even from one nonlinear iteration to the next one

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Algorithmic Aspects Incremental Matrix Update in Two Fluid Flow Problems

Incremental Matrix Update in Two Fluid Flow Problems

I

ρ and µ vary in time, even from one nonlinear iteration to the next one

I

diffusion matrix is not constant any more (µ)

I

mass matrix is not constant any more (ρ)

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Algorithmic Aspects Incremental Matrix Update in Two Fluid Flow Problems

Incremental Matrix Update in Two Fluid Flow Problems

I

ρ and µ vary in time, even from one nonlinear iteration to the next one

I

diffusion matrix is not constant any more (µ)

I

mass matrix is not constant any more (ρ)

I

reassembling them as the convective term is expensive

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Algorithmic Aspects Incremental Matrix Update in Two Fluid Flow Problems

Incremental Matrix Update in Two Fluid Flow Problems

I

ρ and µ vary in time, even from one nonlinear iteration to the next one

I

diffusion matrix is not constant any more (µ)

I

mass matrix is not constant any more (ρ)

I

reassembling them as the convective term is expensive

Reassembling is unnecessary, because only those matrix entries associated to elements close to the interface may change.

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Algorithmic Aspects Incremental Matrix Update in Two Fluid Flow Problems

Solution Notation: φprev at previous assembly, φcurr at current assembly Take the mass term as an example. Note that we can write ρprev ,curr := ρ(φprev ,curr ) = ρ− + (ρ+ − ρ− )H(φprev ,curr ) where H is the Heaviside function.

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Algorithmic Aspects Incremental Matrix Update in Two Fluid Flow Problems

Solution Notation: φprev at previous assembly, φcurr at current assembly Take the mass term as an example. Note that we can write ρprev ,curr := ρ(φprev ,curr ) = ρ− + (ρ+ − ρ− )H(φprev ,curr ) where H is the Heaviside function. Then we have ρcurr

= ρ− + (ρ+ − ρ− )H(φcurr ) = ρprev + (ρ+ − ρ− )(H(φcurr ) − H(φprev )) | {z } ρδ

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Algorithmic Aspects Incremental Matrix Update in Two Fluid Flow Problems

Solution Notation: φprev at previous assembly, φcurr at current assembly Take the mass term as an example. Note that we can write ρprev ,curr := ρ(φprev ,curr ) = ρ− + (ρ+ − ρ− )H(φprev ,curr ) where H is the Heaviside function. Then we have ρcurr

= ρ− + (ρ+ − ρ− )H(φcurr ) = ρprev + (ρ+ − ρ− )(H(φcurr ) − H(φprev )) | {z } ρδ

But ρδ has a more compact support: only where φ changed sign. (ρcurr u, v) = (ρprev u, v) + (ρδ u, v)

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Algorithmic Aspects Incremental Matrix Update in Two Fluid Flow Problems

Solution Notation: φprev at previous assembly, φcurr at current assembly Take the mass term as an example. Note that we can write ρprev ,curr := ρ(φprev ,curr ) = ρ− + (ρ+ − ρ− )H(φprev ,curr ) where H is the Heaviside function. Then we have ρcurr

= ρ− + (ρ+ − ρ− )H(φcurr ) = ρprev + (ρ+ − ρ− )(H(φcurr ) − H(φprev )) | {z } ρδ

But ρδ has a more compact support: only where φ changed sign. (ρcurr u, v) = (ρprev u, v) + (ρδ u, v) This applies also to the diffusion term.

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Algorithmic Aspects Incremental Matrix Update in Two Fluid Flow Problems

Results −4

3.8

x 10

standard matrix reassembly incremental matrix update

3.6 3.4

CPU time / M

3.2 3 2.8 2.6 2.4 2.2 2 1.8 2 10

3

4

10

10 M

CPU time per degree of freedom M vs. M

5

10

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Algorithmic Aspects Level Set Reinitialization

Level Set Reinitialization Choice of φ I

physics is determined only by sign of φ

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Algorithmic Aspects Level Set Reinitialization

Level Set Reinitialization Choice of φ I

physics is determined only by sign of φ

I

advection solving difficulty is determined by ∇φ

I

⇒ choose signed distance function: |∇φ| = 1

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Algorithmic Aspects Level Set Reinitialization

Level Set Reinitialization Choice of φ I

physics is determined only by sign of φ

I

advection solving difficulty is determined by ∇φ

I

⇒ choose signed distance function: |∇φ| = 1

I

advected distance function is not a distance function any more

I

⇒ reinitialize φ to a distance function from time to time

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Algorithmic Aspects Level Set Reinitialization

Level Set Reinitialization Choice of φ I

physics is determined only by sign of φ

I

advection solving difficulty is determined by ∇φ

I

⇒ choose signed distance function: |∇φ| = 1

I

advected distance function is not a distance function any more

I

⇒ reinitialize φ to a distance function from time to time

Criteria for Algorithm I

Efficiency: O(N) or O(N log (N)) operations for N nodes

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Algorithmic Aspects Level Set Reinitialization

Level Set Reinitialization Choice of φ I

physics is determined only by sign of φ

I

advection solving difficulty is determined by ∇φ

I

⇒ choose signed distance function: |∇φ| = 1

I

advected distance function is not a distance function any more

I

⇒ reinitialize φ to a distance function from time to time

Criteria for Algorithm I

Efficiency: O(N) or O(N log (N)) operations for N nodes

I

Accuracy: Mass conservation error from reinitialization should be same order as error from discretization of interface (order h2 for P1 functions)

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Algorithmic Aspects Level Set Reinitialization

Chosen Solution 11111111111111111111 00000000000000000000 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111 00000000000000000000 11111111111111111111

Interface Local Projection (Parolini 2004) + Fast Marching (Sethian 1996) ⇒ local accuracy and global efficiency Contribution: Reformulation of distance computation in fast marching method on unstructured meshes in any dimension, providing an efficient implementation and geometric insight.

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Algorithmic Aspects Level Set Reinitialization

Test Case Notation: φbef before reinitialization, φaft after reinitialization “distorted” circle as test case: φbef = f (x 2 + y 2 ) − f (r 2 ), f (ξ) = ξ 2x+1/2

φ0 φ1 The true distance function is obtained with f (ξ) = ξ 1/2 .

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Algorithmic Aspects Level Set Reinitialization

Accuracy

I

Ωbef = {x : φbef (x) < 0}

I

Ωaft = {x : φaft (x) < 0}

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Algorithmic Aspects Level Set Reinitialization

Accuracy

I

Ωbef = {x : φbef (x) < 0}

I

Ωaft = {x : φaft (x) < 0}

I

relative mass error: em = (|Ωaft | − |Ωbef |)/|Ωbef |

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Algorithmic Aspects Level Set Reinitialization

Accuracy

I

Ωbef = {x : φbef (x) < 0}

I

Ωaft = {x : φaft (x) < 0}

I

relative mass error: em = (|Ωaft | − |Ωbef |)/|Ωbef |

I

Ωδ = {x : φbef (x)φaft (x) < 0}

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Algorithmic Aspects Level Set Reinitialization

Accuracy

I

Ωbef = {x : φbef (x) < 0}

I

Ωaft = {x : φaft (x) < 0}

I

relative mass error: em = (|Ωaft | − |Ωbef |)/|Ωbef |

I

Ωδ = {x : φbef (x)φaft (x) < 0}

I

sign change error: es = |Ωδ |

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Algorithmic Aspects Level Set Reinitialization

Accuracy −1

10

−2

10

mass error sign change error 2 Ch

−3

error

10

−4

10

−5

10

−6

10

−2

−1

10

10 h

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Algorithmic Aspects Level Set Reinitialization

Accuracy −1

10

−2

10

mass error sign change error 2 Ch

−3

error

10

−4

10

−5

10

−6

10

−2

−1

10

10 h

⇒ optimally accurate reinitialization

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Algorithmic Aspects Level Set Reinitialization

Efficiency 1

10

CPU time C N log(N) CN 0

CPU time

10

−1

10

−2

10

2

10

3

4

10

10 N

5

10

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Algorithmic Aspects Level Set Reinitialization

Efficiency 1

10

CPU time C N log(N) CN 0

CPU time

10

−1

10

−2

10

2

10

3

4

10

10 N

⇒ optimally efficient reinitialization

5

10

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Numerical Results: Accuracy

Numerical Results: Accuracy

I

Mathematical Model

I

Finite Element Methods with Interior Penalty Stabilization

I

Algorithmic Aspects Numerical Results: Accuracy

I

I I I I

(2D stationary Navier-Stokes) (2D time dependent Navier-Stokes) 3D time dependent Navier-Stokes (Level Set Advection)

I

Applications

I

Software Design Aspects

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Numerical Results: Accuracy Navier-Stokes with Constant Density and Viscosity

Three Dimensional Time Dependent Problem Test with exact solution by Ethier/Steinman: pe = − a2 /2 e −2νd

2t

e 2ax1 + e 2ax2 + e 2ax3 + 2 sin(ax1 + dx2 ) cos(ax3 + dx1 ) + 2 sin(ax2 + dx3 ) cos(ax1 + dx2 ) + 2 sin(ax3 + dx1 ) cos(ax2 + dx3 )

ue = − a e

−νd 2 t



e ax1 sin(ax2 + dx3 ) + e ax3 cos(ax1 + dx2 ) ! e ax2 sin(ax3 + dx1 ) + e ax1 cos(ax2 + dx3 ) e ax3 sin(ax1 + dx2 ) + e ax2 cos(ax3 + dx1 )

a = π/4, d = π/2, Ω = [−1, 1]3 , T = 0.1, ν = µ/ρ, ρ = 1 u = ue

on ∂Ω × [0, T ] ∪ Ω × {0}

P1 -P1 , ∆t = 0.025, γβ = 0.02, γdiv = γp = 0.2

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Numerical Results: Accuracy Navier-Stokes with Constant Density and Viscosity

Numerical Results: Velocity Field

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Numerical Results: Accuracy Navier-Stokes with Constant Density and Viscosity

Numerical Results: Relative L2 Error vs. h 0

10

ep,0 µ=1 e

u,0

µ=1 −5

ep,0 µ=10

−1

10

e

u,0

µ=10−5

error

C h2 −2

10

−3

10

−4

10

−2

10

−1

10 h

0

10

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Applications

Applications

I

Mathematical Model

I

Finite Element Methods with Interior Penalty Stabilization

I

Algorithmic Aspects

I

Numerical Results: Accuracy Applications

I

I I I

I

Laminar Flow around a Cylinder Rising Bubble with Surface Tension Two Rising Bubbles Undergoing Topology Change

Software Design Aspects

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Applications Laminar Flow around a Cylinder

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

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

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Applications Rising Bubble with Surface Tension

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

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Applications Rising Bubble with Surface Tension

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)

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Applications Rising Bubble with Surface Tension

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)

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

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Applications Rising Bubble with Surface Tension

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)

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

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Applications Rising Bubble with Surface Tension

Results Bubble shapes at t = 3

high surface tension

low surface tension

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Applications Rising Bubble with Surface Tension

Results Bubble shapes at t = 3

high surface tension ⇒ movies

low surface tension

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Software Design Aspects

Software Design Aspects

I

Mathematical Model

I

Finite Element Methods with Interior Penalty Stabilization

I

Algorithmic Aspects

I

Numerical Results: Accuracy

I

Applications Software Design Aspects

I

I I

The Life Project Contributions to Life

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Software Design Aspects The Life Project

The Life Project LIFE: LIbrary of Finite Elements a C++ template library developed by C. Prud’homme

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Software Design Aspects The Life Project

The Life Project LIFE: LIbrary of Finite Elements a C++ template library developed by C. Prud’homme I

generic w.r.t. polynomial degree, dimension and numerical type

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Software Design Aspects The Life Project

The Life Project LIFE: LIbrary of Finite Elements a C++ template library developed by C. Prud’homme I

generic w.r.t. polynomial degree, dimension and numerical type

I

expression template language for variational formulation:

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Software Design Aspects The Life Project

The Life Project LIFE: LIbrary of Finite Elements a C++ template library developed by C. Prud’homme I

generic w.r.t. polynomial degree, dimension and numerical type

I

expression template language for variational formulation:

Assembly of the x-part of the matrix for b(v , p) = −(∇ · v, p): form( spaceU, spaceP, matrixDerX ) = integrate( elements(mesh), integrationMethod, -dx(vx)*idt(p) );

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Software Design Aspects Contributions to Life

Contributions to Life

I

Linear Algebra Backend: decouple construction from solution of linear systems

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Software Design Aspects Contributions to Life

Contributions to Life

I

Linear Algebra Backend: decouple construction from solution of linear systems

I

Linear Functionals and Operators: decouple functionals and operators from their linear algebraic representations as matrices and vectors

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Software Design Aspects Contributions to Life

Contributions to Life

I

Linear Algebra Backend: decouple construction from solution of linear systems

I

Linear Functionals and Operators: decouple functionals and operators from their linear algebraic representations as matrices and vectors

I

Oseen Problem as Versatile Framework: decouple generic problem (Oseen) and its construction from application context (stationary/time dependent, one/two fluid, iterative schemes for time discretization and nonlinearity, . . . )

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Conclusions

Conclusions Contributions: I

Interior penalty approximation of multiphase flows

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Conclusions

Conclusions Contributions: I

Interior penalty approximation of multiphase flows

I

Original analysis of an unfitted finite element scheme for the two fluid Stokes problem

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Conclusions

Conclusions Contributions: I

Interior penalty approximation of multiphase flows

I

Original analysis of an unfitted finite element scheme for the two fluid Stokes problem

I

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

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Conclusions

Conclusions Contributions: I

Interior penalty approximation of multiphase flows

I

Original analysis of an unfitted finite element scheme for the two fluid Stokes problem

I

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

I

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

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Conclusions

Conclusions Contributions: I

Interior penalty approximation of multiphase flows

I

Original analysis of an unfitted finite element scheme for the two fluid Stokes problem

I

Preconditioning strategy with reusable preconditioner for sequences of 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

Interior Penalty Finite Element Approximation of Navier-Stokes Equations and Applications to Free Surface Flows Conclusions

Conclusions Contributions: I

Interior penalty approximation of multiphase flows

I

Original analysis of an unfitted finite element scheme for the two fluid Stokes problem

I

Preconditioning strategy with reusable preconditioner for sequences of 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

Interior Penalty Finite Element Approximation of Navier ...

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