numerical solution of fluid mechanics problems with pure ... · numerical solution of fluid...

Recent Advances in PDEs: Analysis, Numerics and Control

Sevilla, January 25-27, 2017

Numerical solution of fluid

mechanics problems with pure

Lagrange-finite element methods

Marta Benıtez(1), Alfredo Bermudez(2) and Pedro Fontan(3)

(1) Department of Mathematics. Universidade da Coruna and ITMATI. Spain

(2) Department of Applied Mathematics. Universidade de Santiago deCompostela and ITMATI. Spain

(3) Instituto Tecnologico de Matematica Industrial (ITMATI). Spain

Contents

1 Introduction

2 The framework of continuum mechanics3 Navier-Stokes equations

Eulerian formulationLagrangian formulation

4 Pure Lagrange-Galerkin schemes

Central difference schemesNewmark schemes

5 Computer implementation (FEniCS)6 Numerical experiments

Test 1. Academic problem: order of convergenceTest 2. Free boundary flow. Sloshing modes.Test 3. Free boundary flow. Dam-break problemTest 4. Flow past a cylinder

Introduction

Convection-diffusion equations

Continuum thermo-mechanics: energy and, in the case of a mixture,mass of species conservation equations (scalar)

Continuum thermo-mechanics: motion or momentum equation(vector)

Applications: fluid mechanics, solid mechanics, fluid-structureinteraction, chemistry, combustion

Applications: financial mathematics (pricing derivatives, Black-Scholesequation) (scalar)

A vector nonlinear convection-diffusion problem: the motion equationof continuum mechanics

Find two fields v : T =⋃

t∈(0,T )(Ωt × t) −→ V and T : T −→ Lin such that

ρ(x, t)

(∂v

∂t(x, t) + gradv(x, t)v(x, t)

)− div T (x, t) = b(x, t) x ∈ Ωt,

v(x, t) = vd(x, t) x ∈ ΓDt ,

T (x, t)n(x, t) = h(x, t) x ∈ ΓNt ,

v(x, 0) = v0(x) x ∈ Ω0.

ρ: density, v: spatial velocity, T : Cauchy stress tensor,V: model vector space, Lin: vector space of (second order) tensors.

Ω

p

A

z = Z(p, t)

Ωt = X(Ω, t)

x = X(p, t) = Y (z, t)

X(·, t)

Y (·, t)

Z(·, t)

Numerical methods for vector convection-diffusion problems

⋆ Eulerian methods (E):Projection methodsA. Chorin, R. Temam, 1968

Petrov-Galerkin, SUPG, least square,...I. Christie, D.F. Griffiths, A.R. Mitchell, O.C.

Zienkiewicz, 1976.

A.N. Brooks, T.J.R. Hughes, 1982,...

Splitting methodsR. Glowinski (book 1984, review paper 2004)

⋆ ALE methods:J. Donea, A. Huerta, J.-Ph. Ponthot and A.

Rodrıguez-Ferran (review paper, 2004)

⋆ Semi-Lagrangian methods (SL)

When combined with finite element methods they are usually called,

Characteristics-finite element methodsLagrange-Galerkin methods

J. P. Benque, B. Ibler, A. Keramisi, G. Labadie, 1980.

J. Douglas Jr. and T.F. Russel , 1981.

O. Pironneau, 1982, 1992.

R. Bermejo, 1991.

K. Boukir, Y. Maday, B. Metivet and E.

Razafindrakoto, 1997.

H. Rui and M. Tabata, 2002.

A. Bermudez, M. Rodrıguez and C. Vazquez, 2006.

K. Chrysafinos and N.J. Walkington, 2008.

R. Bermejo and L. Saavedra, 2012.

L. Yang, G. Wei, H. Siriguleng and F. Zhichao, 2013.

⋆ Lagrangian methods (L)

Free Lagrange method (FL)M.J. Fritts MJ, W.P. Crowley WP, H.E. Trease, 1970,1985.

Smooth Particle Hydrodynamics Method (SPH)R.A. Gingold, J.J. Monaghan, 1997.S. Koshizuka S, Y. Oka, 1996.

Particle finite element method (PFEM)S.R. Idhelson E. Onate, F. del Pin, 2004.

Pure Lagrangian methods (PL)

The equations are rewritten in the reference configuration by using themotion to change from Eulerian to Lagrangian variables.Then they are solved by using discretization methods.

M. Benıtez, A. Bermudez. Numerical Analysis of a second-orderpure Lagrange-Galerkin method for convection-diffusion problems. PartI: time discretization. SIAM J. Numer. Anal., 50:858-882, 2012.

M. Benıtez, A. Bermudez. Numerical Analysis of a second-orderpure Lagrange-Galerkin method for convection-diffusion problems. PartII: fully discretized scheme and numerical results. SIAM J. Numer.Anal., 50:2824-2844, 2012.

M. Benıtez, A. Bermudez. Pure Lagrangian and semi-Lagrangianfinite element methods for the numerical solution ofconvection-diffusion problems. Int. Numer. Anal. Mod., 11:271-287,2014.

Pure Lagrangian methods (PL) (cont)

M. Benıtez, A. Bermudez. Pure Lagrangian and semi-Lagrangianfinite element methods for the numerical solution of Navier-Stokesequations. Appl. Numer. Math., 95:62-81, 2015.

M. Benıtez, A. Bermudez. Second order pure Lagrange-Galerkinmethods for fluid-structure interaction problems. SIAM J. Sci. Comput.37:744–777, 2015.

The formalism of continuum mechanics is used to introduce both PLand SL methods for fluid and fluid-structure problems

The framework of continuum mechanics

Motion. Reference mapping.

m

p

Ω

ΓN

ΓD

X(·, t)

P (·, t)

n

x = X(p, t)

ΓNt = X(ΓN , t)

ΓDt = X(ΓD , t)

Ωt = X(Ω, t)

Notations concerning motions

X(p, 0) = p =⇒ Ω = X(Ω, 0)material configuration = initial spatial configuration

T := (x, t) : x ∈ Ωt, t ∈ [0, T ] −→ trajectory of the motion

P (x, t) −→ reference mapping: P (·, t) = (X(·, t))−1.

The framework of continuum mechanics

Motion. Reference mapping.

m

p

Ω

ΓN

ΓD

X(·, t)

P (·, t)

n

x = X(p, t)

ΓNt = X(ΓN , t)

ΓDt = X(ΓD , t)

Ωt = X(Ω, t)

Notations concerning motions

v(x, t) := X(P (x, t), t) ∀(x, t) ∈ T −→ velocity

F (p, t) := ∇pX(p, t) −→ deformation gradient

u(p, t) = X(p, t)− p −→ displacement

Motion referred to the material configuration

Let Ψ be a spatial (or Eulerian) field

X(·, t)

p

Ω

x

Ωt

Ψ(·, t)R

V

Lin

Ψm(., t) := Ψ(X(., t), t)

Notation

m: material(or Lagrangian)

The Navier-Stokes equations

Navier-Stokes equations in Eulerian coordinates (x ∈ Ωt)

ρ(x, t)(∂v

∂t(x, t) + gradv(x, t)v(x, t)) + gradπ(x, t)

− divη(x, t)( grad v(x, t) + gradvt(x, t))

= b(x, t),

divv(x, t) = g(x, t),

for x ∈ Ωt and t ∈ (0, T ),

v(x, 0) = v0(x) for x ∈ Ω,

v(x, t) = vd(x, t) on ΓDt ,(

−π(x, t)I + η(x, t)( grad v(x, t) + gradvt(x, t)))n(x, t) = h on ΓN

t .

Constitutive law. Newtonian fluid:

T = −πI + 2ηD,

D =1

2( gradv + gradvt) (strain rate)

Change of variable to material coordinates: x = X(p, t), p ∈ Ω

vm(p, t) = v(X(p, t), t) =∂X

∂t(p, t) =

∂u

∂t(p, t) ⇒

(v)m(p, t) = (vm)·(p, t) =∂2X

∂t2(p, t) =

∂2u

∂t2(p, t).

Moreover,

− div xT (X(p, t), t) =− Div(Tm(p, t)F−t(p, t) detF (p, t)

) 1

detF (p, t),

where Div := div p.

Change of variable to material coordinates: x = X(p, t), p ∈ Ω

vm(p, t) = v(X(p, t), t) =∂X

∂t(p, t) =

∂u

∂t(p, t) ⇒

(v)m(p, t) = (vm)·(p, t) =∂2X

∂t2(p, t) =

∂2u

∂t2(p, t).

Moreover,

− div xT (X(p, t), t) =− Div(Tm(p, t)F−t(p, t) detF (p, t)

) 1

detF (p, t),

where Div := div p.

→ the first Piola-Kirchhoff stress tensor

Navier-Stokes equations in Lagrangian coordinates (p ∈ Ω)

Find two functions u : Ω× [0, T ] −→ V and πm : Ω× [0, T ] −→ R such that

ρ0u− Div((

− πmI + ηm(∇uF−1 + F−t(∇u)t))detFF−t

)

= bm detF in Ω× (0, T ),

detFF−t : ∇u = gm detF in Ω× (0, T ),

u(p, 0) = v0(p) in Ω,

u(p, t) = vd,m(p, t) on ΓDτ × [0, T ],

(−πmI + ηm(∇uF−1 + F−t

(∇u)t

))F−t

m = |F−t(p, t)m(p)|hm

on ΓN × [0, T ],

where ρ0(p) = ρ(X(p, t), t) det F (p, t) is the (given) reference density.

Recall that F (p, t) := ∇X(p, t) = I +∇u(p, t).

Weak formulation

Find two functions u(·, t) ∈ H1(Ω) and πm(·, t) ∈ L2(Ω) such that

∫

Ωρ0u · z+

∫

Ω

(−πmI + ηm

(∇uF−1 + F−t (∇u)t

))F−t

: ∇zdetF =

∫

Ωbm · zdetF

+

∫

ΓN

|F−tm|detFhm · z dAp ∀z ∈ H

1ΓD(Ω),

∫

ΩdetFF−t : ∇u q =

∫

Ωgm detF q ∀q ∈ L2(Ω).

Pure Lagrange-Galerkin schemes

Approximation of the second time derivative (three-point central formula)

u(p, t) ≈u(p, t+∆t)− 2u(p, t) + u(p, t−∆t)

∆t2+O

(∆t2

)

Approximation of the first time derivative (two-point central formula)

∇u(p, t) ≈∇u(p, t+∆t)−∇u(p, t−∆t)

2∆t+O

(∆t2

)

1. Central Pure Lagrangian-Galerkin: (CPLG) −→ t = tn+1/2 (Linear scheme)

∫

Ω

ρ0

(un+3/2∆t − 2u

n+1/2∆t + u

n−1/2∆t

∆t2

)· z−

∫

Ω

πn+1/2m,∆t detF

n+1/2∆t (F

n+1/2∆t )−t

: ∇z+ η

∫

Ω

∇u

n+3/2∆t −∇u

n−1/2∆t

2∆t(F

n+1/2∆t )−1

+(Fn+1/2∆t )−t (∇u

n+3/2∆t )t − (∇u

n−1/2∆t )t

2∆t

detF

n+1/2∆t (F

n+1/2∆t )−t : ∇z

=

∫

Ω

bn+1/2 X

n+1/2∆t · zdetF

n+1/2∆t

+

∫

ΓN

|(Fn+1/2∆t )−t

m| detFn+1/2∆t h

n+1/2 Xn+1/2∆t · z ∀z ∈ H

1ΓD (Ω).

where Xn+1/2∆t (p) := p+ u

n+1/2∆t (p), F

n+1/2∆t (p) := I +∇u

n+1/2∆t (p) and

0 ≤ n ≤ N − 1.

Time discretization (divergence condition)

∫

ΩdetF

n+1/2∆t (F

n+1/2∆t )−t :

(∇u

n+3/2∆t −∇u

n−1/2∆t

2∆t

)q = 0

∀q ∈ L2(Ω).

Boundary condition for the displacement

un+3/2∆t (p) = u

n−1/2(p) + 2∆tvn+1/2d

(X

n+1/2∆t (p)

)on ΓD,

and 0 ≤ n ≤ N − 1.

Initial conditions for the displacement

We take Ω−∆t/2 as reference configuration instead of Ω0. Then,

u−1/2∆t (p) := 0,

u1/2∆t (p) := ∆tv0

(p+ v

0(p)∆t

2

).

In order to keep the global second order of the scheme it is importantto build an approximation of u1/2 of order 3 as the previous one.Further details can be seen in

→ M. Benıtez, A. Bermudez. Pure Lagrangian and semi-Lagrangian

finite element methods for the numerical solution of Navier-Stokes

equations. Appl. Numer. Math., 95:62-81, 2015.

⋆ Post-processing. Second-order central formulas

Approximation of the material description of the velocity

vn+1m (p) = u

n+1(p) ≃ vn+1m,∆t(p) :=

un+3/2∆t (p)− u

n+1/2∆t (p)

∆t.

Motion approximation at times tn+1N−1n=0

Xn+1(p) ≃ Xn+1∆t (p) :=

un+3/2∆t (p) + u

n+1/2∆t (p)

2+ p.



vn+1m (p) = u

n+1(p) ≃ vn+1m,∆t(p) :=

un+3/2∆t (p)− u

n+1/2∆t (p)

∆t.


Xn+1(p) ≃ Xn+1∆t (p) :=

un+3/2∆t (p) + u

n+1/2∆t (p)

2+ p.

Approximation of the spatial description of the velocity

vn+1(Xn+1(p)) = v

n+1m (p) ≃ v

n+1m,∆t(p).



vn+1m (p) = u

n+1(p) ≃ vn+1m,∆t(p) :=

un+3/2∆t (p)− u

n+1/2∆t (p)

∆t.


Xn+1(p) ≃ Xn+1∆t (p) :=

un+3/2∆t (p) + u

n+1/2∆t (p)

2+ p.


vn+1(Xn+1(p)) = v

n+1m (p) ≃ v

n+1m,∆t(p).

Approximate pressure in spatial coordinates

πn+1/2(Xn+1/2(p)) = πn+1/2m (p) ≃ π

n+1/2m,∆t (p).

Space discretization

Numerical experiments in 2D (3D in progress).

P1+bubble/P1 finite elements for displacement/pressure in the fluid

Restarting the CPLG scheme

Let us assume that we decide to restart the scheme at time tr−1/2 being

1 ≤ r ≤ N − 1. Firstly we also compute ur+1/2∆t,h and then v

r∆t,h.

The new reference domain: Ωtr−1/2(a priori unknown) −→ is

determined by using the approximate motion Xr−1/2∆t,h .

Initial conditions have to be carefully chosen in order to not to loseaccuracy:

u(y, tr−1/2) = 0,

u(y, tr+1/2) = ∆tvr∆t,h

(y + v

r∆t,h(y)

∆t

2

).

Further details in → M. Benıtez, A. Bermudez. Pure Lagrangian and

semi-Lagrangian finite element methods for the numerical solution of

Navier-Stokes equations. Appl. Numer. Math., 95:62-81, 2015.

Newmark family of integration schemes → (β, γ)

un+1 = u

n+∆tun+∆t2(βun+1 +

(1

2− β

)un

)+O(∆t3)

(1

6− β

)+O(∆t4)

un+1 = u

n +∆t(γun+1 + (1− γ)un

)+O(∆t2)

(1

2− γ

)+O(∆t3)

⋆ These relationships are obtained by using Taylor expansions

Newmark family of integration algorithms → (β, γ)

un+1 = u

n+∆tun+∆t2(βun+1 +

(1

2− β

)un

)+O(∆t3)

(1

6− β

)+O(∆t4)

un+1 = u

n +∆t(γun+1 + (1− γ)un

)+O(∆t2)

(1

2− γ

)+O(∆t3)

Stability of Newmark schemes:

⋆ If γ ≥ 1/2 and 2β − γ ≥ 0: unconditional stability

⋆ If γ ≥ 1/2 and 2β − γ < 0: conditional stability

⋆ If γ < 1/2: instability

Optimal accuracy: β = 1/6, γ = 1/2

Optimal accuracy compatible with unconditional stability: β = 1/4,γ = 1/2

2. Linearized Newmark Pure Lagrange-Galerkin schemes(LNPLG) −→ t = tn+1 (linear schemes)∫

Ω

ρ0

(un+1∆t − u

n∆t −∆t(1− γ)un

∆t

∆tγ

)· z−

∫

Ω

πn+1m,∆t det F

n+1∆t (Fn+1

∆t )−t : ∇z

+

∫

Ω

ηn+1 Xn+1∆t ∇u

n+1∆t (Fn+1

∆t )−1 det Fn+1∆t (Fn+1

∆t )−t : ∇z

+

∫

Ω

ηn+1 Xn+1∆t (Fn+1

∆t )−t(∇un+1∆t )t det Fn+1

∆t (Fn+1∆t )−t : ∇z

=

∫

Ω

bn+1 Xn+1

∆t · zdet Fn+1∆t

+

∫

ΓN

|(Fn+1∆t )−t

m| det Fn+1∆t h

n+1/2 Xn+1∆t · z ∀z ∈ H

1ΓD(Ω).

whereXn+1

∆t (p) := p+ un+1∆t (p), Fn+1

∆t (p) := I +∇un+1∆t (p),

un+1∆t (p) = u

n∆t(p) + ∆tun

∆t(p) +∆t2

2un∆t(p), (Newmark with β = 0)

Linearized Newmark Pure Lagrangian schemes(LNPLG) −→ t = tn+1

Divergence condition.

∫

Ω

det Fn+1∆t (Fn+1

∆t )−t : ∇un+1∆t q =

∫

Ω

gn+1 Xn+1∆t det Fn+1

∆t q ∀q ∈ L2(Ω).

Boundary condition for the material velocity

un+1∆t (p) = v

n+1d

(Xn+1

∆t (p))

on ΓD.

Update acceleration and displacement at time tn+1 by using un+1∆t

un+1∆t =

un+1∆t − u

n∆t −∆t(1− γ)un

∆t

∆tγ,

un+1∆t = u

n∆t +∆tun

∆t +∆t2(βun+1

∆t +

(1

2− β

)un∆t

).

Initial conditions

Displacement and velocity:

u0∆t(p) = 0, u

0∆t(p) = v

0(p).

Initial acceleration is computed from equations at initial time:

∫

Ωρ0u0

∆t · z+

∫

Ω

(−π0

∆tI + η0(gradv0 +

(grad v0

)t))

: grad z =

∫

Ωb0 · z ∀z ∈ H

10(Ω),

∫

Ω

(Div u0

∆t − grad v0 · ( grad v0)t)q

=

∫

Ω

((∂g

∂t

)0

+ grad g0 · v0

)q ∀q ∈ L2(Ω).

Notice that F 0 = I and ((F−t)·)0 = −(gradv0

)t.

For spatial discretization we also use piecewise P1-bubble/P1 finite elements

Motion approximation

Xn+1(p) = p+ un+1(p) ≃ Xn+1

∆t,h(p) := p+ un+1∆t,h(p).


Xn+1(p) = p+ un+1(p) ≃ Xn+1

∆t,h(p) := p+ un+1∆t,h(p).


vn+1(Xn+1(p)) = u

n+1(p) ≃ vn+1(Xn+1

∆t,h(p)) = un+1∆t,h(p).


Xn+1(p) = p+ un+1(p) ≃ Xn+1

∆t,h(p) := p+ un+1∆t,h(p).


vn+1(Xn+1(p)) = u

n+1(p) ≃ vn+1(Xn+1

∆t,h(p)) = un+1∆t,h(p).

Approximation of the spatial description of the acceleration

an+1(Xn+1(p)) = u

n+1(p) ≃ an+1(Xn+1

∆t,h(p)) = un+1∆t,h(p).


Xn+1(p) = p+ un+1(p) ≃ Xn+1

∆t,h(p) := p+ un+1∆t,h(p).


vn+1(Xn+1(p)) = u

n+1(p) ≃ vn+1(Xn+1

∆t,h(p)) = un+1∆t,h(p).

Approximation of the spatial description of the acceleration

an+1(Xn+1(p)) = u

n+1(p) ≃ an+1(Xn+1

∆t,h(p)) = un+1∆t,h(p).

Approximate pressure in spatial coordinates

πn+1(Xn+1(p)) = πn+1m (p) ≃ πn+1(Xn+1

∆t,h(p)) = πn+1m,∆t,h(p).

Let us assume that we decide to restart the scheme at time tr being1 ≤ r ≤ N − 1. Then we also compute u

r∆t,h.

The new reference domain: Ωtr (a priori unknown) is approximated byusing Xr

∆t,h.

Xl∆t,hXr

∆t,h

Xltr,∆t,h

p

Ω

y

Ωtr

x

Ωtl

New initial conditions:

ur∆t,h(y) = 0,

ur∆t,h(y) = v

r∆t,h(y) in Ωtr ,

ur∆t,h(y) = a

r∆t,h(y) in Ωtr .

Remark

We get a Linearized Newmark Semi-Lagrange-Galerkin scheme (LNSLG)by restarting the LNPLG scheme at each time step

Remark

For β = 0 and γ = 1/2 the linearized Newmark pureLagrange-Galerkin (LNPLG) scheme (which is conditionally stable)can be rewritten only in terms of displacements.

It coincides with the CPLG scheme. However, the reference domainand the initial conditions are slightly different.

Computer implementation

FEniCS project (https://fenicsproject.org)

FEniCS project is a set of computational libraries focusing on the solutionof partial differential equations by finite element methods. Main features:

Open source.

Domain specific language UFL (Unified Form Language), built on topof the Python language, to represent variational formulations in anotation close to mathematical notation.

FFC (FEniCS Form Compiler) compiler for finite element variationalforms. Generates efficient low level C++ code from high level UFLdescription.

Automatic functional differentiation.

M. S. Alnaes, J. Blechta, J. Hake, A. Johansson, B. Kehlet,

A. Logg, C. Richardson, J. Ring, M. E. Rognes and G. N.

Wells, The FEniCS Project Version 1.5, Arch. of Num. Soft., vol. 3,2015.

Test 1: an academic example

Ω = (0, 1)× (0, 1), T = 2

ρ = 1, η = 10−3. Functions b, g, v0 and vd are taken such that theexact solution is

π(x, y) = 10(2x − 1)(2y − 1)u(x, y, t) = 10tetx2(x− 1)2y(y − 1)(2y − 1)v(x, y, t) = −10tetx(x− 1)(2x − 1)y2(y − 1)2.

For this test we analyze the rates of convergence of four linear methods:three pure-Lagrangian and one semi-Lagrangian

Central Pure Lagrange-Galerkin scheme (CPLG)

Second order for displacement and velocity

First order for pressure

Linearized Newmark Pure Lagrange-Galerkin scheme for β = 1/6 andγ = 1/2 (LNPLG2) and LNSLG

Second order for displacement, velocity and acceleration

Second order for pressure

Linearized Newmark Pure Lagrange-Galerkin scheme for β = 1/6 andγ = 1 (LNPLG1)

First order for displacement, velocity and acceleration

First order for pressure

Velocity. Time discretization error in l∞(L2h) norm (Nx = Ny = 125)

101

102

103

104

105

10−10

10−8

10−6

10−4

10−2

100

N: number of time step

Err

or

l∞(Lh2(Ω

tn

)) error curve (velocity)

CPLGLNPLG

2

LNSLG2

y=C/N2

LNPLG1

y=C/N

Pressure. Time discretization error in l∞(L2h) norm (Nx = Ny = 125)

101

102

103

104

105

10−10

10−8

10−6

10−4

10−2

100

N: number of time step

Err

or

l∞(Lh2(Ω

tn

)) error curve (pressure)

CPLGLNPLG

2

LNSLG2

y=C/N2

LNPLG1

y=C/N

Velocity. Spatial discretization error in l∞(L2h) norm (Nx = Ny = 125)

101

102

10−5

10−4

10−3

10−2

10−1

100

Nx=N

y: number of degrees of freedom in each spatial direction (h=1/N

x)

Err

or

l∞(L2h(Ω

tn

)) error curve (velocity)

CPLGLNPLG

2

LNSLG2

y=C/Nx2

y=C/Nx

Pressure. Spatial discretization error in l∞(L2h) norm (Nx = Ny = 125)

101

102

10−5

10−4

10−3

10−2

10−1

100

Nx=N

y: number of degrees of freedom in each spatial direction (h=1/N

x)

Err

or

l∞(L2h(Ω

tn

)) error curve (pressure)

CPLGLNPLG

2

LNSLG2

y=C/Nx2

y=C/Nx

Total computing time per 1 s simulation

Test 2. Free boundary flow: sloshing modes

Water in a rectangular container

Body force:

b1(x1, x2, t) = ρAg sin(2πNt)

b2(x1, x2, t) = −ρg

where the frequency N has been taken close to resonant frequencies:N = 0,89 Hz, 1,67 Hz. and A = 0,01 m

A. Huerta, K. Liu, Viscous flow with large free surface motion,Comp.Meth. Appl. Mech. and Eng. 69 (1988) 277-324.

H. Zhou, J. F. Li and T. S. Wang , Simulation of liquid sloshing incurved-wall containers with arbitrary LagrangianEulerian method, Int. J.Numer. Meth. Fluids 57 (2008) 437452.

Numerical results

Pure Lagrangian methods without remeshing

Numerical results. N = 0,89 Hz, ρ = 1000, η = 0.001. CPLG.

Figura: Modulus of the velocity and streamlines


0 1 2 3 4 5 6 7 8 9−0.25

0

0.25

0.5

t: time

Dim

ensi

onle

ss w

ave

heig

ht

Time history of the wave height at the wall for µ=1.d−3

left wallright wall

Figura: Time histories of the heights at the walls


Figura: Modulus of the velocity and streamlines


0 2 4 6 8 10−0.06

−0.04

−0.02

0

0.02

0.04

0.06

t: time

Dim

ensi

onle

ss w

ave

heig

ht

Time history of the wave height at the wall

left wallright wall


Numerical results. N = 0,89 Hz, ρ = 1000, η = 1. CPLG.

0 5 10 15 20 25 30 35 40

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t: time

Dim

ensi

onle

ss w

ave

heig

ht

Time history of the wave height at the wall

left wallright wall

Figura: Time histories of the heights at the walls. Longer time interval

Numerical results with Newmark Pure Lagrange-Galerkin (NPLG).

Optimal accuracy: β = 1/6, γ = 1/2 (to be denoted NPLG3)

Optimal accuracy compatible with unconditional stability: β = 1/4,γ = 1/2 (to be denoted NPLG2))

Numerical results with NPLG. N = 0,89 Hz, ρ = 1000, η = 0.001.

0 1 2 3 4 5 6 7 8−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

t: time

Dim

ensi

onle

ss w

ave

heig

ht

Time history of the wave height at the wall for µ=1.d−3, ∆ t=0.02

left wall CPLGright wall CPLGleft wall NPLG

2

right wall NPLG2

left wall NPLG3

right wall NPLG3


Numerical results with NPLG. N = 0,89 Hz, ρ = 1000, η = 0.001.

0 5 10 15 20 25 30−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

t: time

Dim

ensi

onle

ss w

ave

heig

ht

Time history of the wave height at the wall for µ=1.d−3, NPLG2

left wall ∆ t=0.02right wall ∆ t=0.02left wall ∆ t=0.24right wall ∆ t=0.24left wall ∆ t=0.3right wall ∆ t=0.3


Test 3. Free boundary flow: dam break problem

Data:

Cavity: [0, 3.5] × [0, 7].L = 3,5 cm

ρ = 1 g/cm3,η = 0.01 g/(cm · s),g = 980 cm/s2

Initial velocity: 0 cm/s

at t = 0

v2 = 0

v1 = 0

Problem definition

References⋆ J.C. Martin, W.J. Moyce. An experimental study of the collapse of

liquid columns on a rigid horizontal plane. Philos Trans R Soc London, SerA. 244, 312-324 (1952)

⋆ C.W. Hirt, B.D. Nichols. Volume of fluid (VOF) method for thedynamics of free boundaries. J. Comput. Phys. 39,201-225 (1981)

⋆ B. Ramaswamy, M. Kawahara. Lagrangian finite element analysisapplied to viscous free surface fluid flow. Int J Numer Methods Fluids.7,953-984 (1987)

⋆ P. Hansbo. The characteristic streamline diffusion method for thetime-dependent in compressible NavierStokes equations. Comput Meth ApplMech Eng 99,171-186 (1992)

⋆ E. Walhorn, A. Kolke, B. Hubner, D. Dinkler. Fluid-structurecoupling within a monolithic model involving free surface flows. Computersand Structures. 83, 2100-2111 (2005)

⋆ Wolfgang A. Wall, Steffen Genkinger, Ekkehard Ramm. A

strong coupling partitioned approach for fluidstructure interaction with free

surfaces. Computers and Fluids. 36, 169-183 (2007)

Numerical results

Pure Lagrangian methods without remeshing

Dam break problem (2860 vertices, ∆t = 0.001)

0 0.5 1 1.5 2 2.5 3 3.51

1.5

2

2.5

3

3.5

4

4.5

time t (2g/L)1/2

x(t)

/L

History of the displacement

CPLGWolfgang A. Wall et al.E. Walhorn et al.P. HansboRamaswamy & KawaharaHirt & Nichols (experimental)Martin & Moyce (experimental)

Pressure

Modulus of the velocity

Dam break problem (2860 vertices, ∆t = 0.001)

Figure: Deformed mesh after 0.2 s.

Test 4. External flows: flow past a cylinder

In order to work with the same particles all the time we redefine theproblem with periodic boundary condition at the input/exitboundaries of a large enough computational domain.

The input velocity is imposed on the upper and lower boundaries

Flow past a cylinder

Density= 1. Dynamic viscosity: 0.01, 0.001

Flow past a cylinder of diameter 0.1

Computational domain: [0, 20] × [0, 0.5]

Input velocity: 1

Flow past a cylinder (η = 0.01, n. of vertices: 38910, ∆t = 0.001)

Figure: Modulus of the velocity, streamlines


0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

v1(0.2,⋅): horizontal velocity for x=0.2

y

(LG) methodFluent

0.86 0.88 0.9 0.92 0.94 0.96 0.98 1−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

v1(1,⋅): horizontal velocity for x=1

y

(LG) methodFluent

Figure: Comparison with ANSYS-FLUENT


0.85 0.9 0.95 1−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

v1(−0.2,⋅): horizontal velocity for x=−0.2

y

(LG) methodFluent

0.9 0.92 0.94 0.96 0.98 1−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

v1(−1,⋅): horizontal velocity for x=−1

y

(LG) methodFluent

Figure: Comparison with ANSYS-FLUENT


Figure: Modulus of the velocity, streamlines at t = tn, t = tn + 2,respectively


Figure: Modulus of the velocity, streamlines at t = tn + 4, t = tn + 6,respectively

Recent Advances in PDEs: Analysis, Numerics and Control

Sevilla, January 25-27, 2017

Numerical solution of fluid

mechanics problems with pure

Lagrange-finite element methods

Marta Benıtez(1), Alfredo Bermudez(2) and Pedro Fontan(3)

(1) Department of Mathematics. Universidade da Coruna and ITMATI. Spain

(2) Department of Applied Mathematics. Universidade de Santiago deCompostela and ITMATI. Spain

(3) Instituto Tecnologico de Matematica Industrial (ITMATI). Spain

numerical solution of fluid mechanics problems with pure ... · numerical solution of fluid...

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