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Seventh Mississippi State - UAB Conference on Differential Equations and Computational

Simulations, Electronic Journal of Differential Equations, Conf. 17 (2009), pp. 13–31.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

ftp ejde.math.txstate.edu

A COMPUTATIONAL DOMAIN DECOMPOSITION APPROACH

FOR SOLVING COUPLED FLOW-STRUCTURE-THERMAL

INTERACTION PROBLEMS

EUGENIO AULISA, SANDRO MANSERVISI, PADMANABHAN SESHAIYER

Abstract. Solving complex coupled processes involving fluid-structure-ther-

mal interactions is a challenging problem in computational sciences and engi-

neering. Currently there exist numerous public-domain and commercial codes

available in the area of Computational Fluid Dynamics (CFD), Computational

Structural Dynamics (CSD) and Computational Thermodynamics (CTD). Dif-

ferent groups specializing in modelling individual process such as CSD, CFD,

CTD often come together to solve a complex coupled application. Direct nu-

merical simulation of the non-linear equations for even the most simplified

fluid-structure-thermal interaction (FSTI) model depends on the convergence

of iterative solvers which in turn rely heavily on the properties of the coupled

system. The purpose of this paper is to introduce a flexible multilevel algo-

rithm with finite elements that can be used to study a coupled FSTI. The

method relies on decomposing the complex global domain, into several local

sub-domains, solving smaller problems over these sub-domains and then gluing

back the local solution in an efficient and accurate fashion to yield the global

solution. Our numerical results suggest that the proposed solution methodol-

ogy is robust and reliable.

1. Introduction

Engineering analysis is constantly evolving with a goal to develop novel tech-

niques to solve coupled processes that arise in multi-physics applications. The effi-

cient solution of a complex coupled system which involves FSTI is still a challenging

problem in computational mathematical sciences. The solution of the coupled sys-

tem provides predictive capability in studying complex nonlinear interactions that

arise in several applications. Some examples include a hypersonic flight, where the

structural deformation due to the aerodynamics and thermal loads leads to a sig-

nificant flow field variation or MAVs (Micro Air Vehicles) where geometric changes

possibly due to thermal effects may lead to a transient phase in which the structure

and the flow field interact in a highly non-linear fashion.

2000 Mathematics Subject Classification. 65N30, 65N15.

Key words and phrases. Fluid-structure-thermal interaction; domain decomposition;

multigrid solver.

c ?2009 Texas State University - San Marcos.

Published April 15, 2009.

Supported by grants DMS 0813825 from the National Science Foundation, and

ARP 0212-44-C399 from the Texas Higher Education Coordinating Board.

13

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14E. AULISA, S. MANSERVISI, P. SESHAIYEREJDE/CONF/17

The direct numerical simulation of this highly non-linear system, governing even

the most simplified FSTI, depends on the convergence of iterative solvers which

in turn relies on the characteristics of the coupled system. Domain decomposi-

tion techniques with non-matching grids have become increasingly popular in this

regard for obtaining fast and accurate solutions of problems involving coupled pro-

cesses. The mortar finite element method [1, 2] has been considered to be a viable

domain decomposition technique that allows coupling of different subdomains with

nonmatching grids and different discretization techniques. The method has been

shown to be stable mathematically and has been successfully applied to a variety

of engineering applications [3, 4]. The basic idea is to replace the strong continuity

condition at the interfaces between the different subdomains by a weaker one to

solve the problem in a coupled fashion. In the last few years, mortar finite ele-

ment methods have also been developed in conjunction with multigrid techniques,

[5, 6, 7, 8]. One of the great advantages of the multigrid approach is in the grid

generation process wherein the corresponding refinements are already available and

no new mesh structures are required. Also, the multigrid method relies only on

local relaxation over elements and the solution on different domains can be easily

implemented over parallel architectures.

The purpose of this paper is to introduce a flexible multigrid algorithm that

can be used to study different physical processes over different subdomains involv-

ing non-matching grids with less computational effort. In particular, we develop

the method for a model problem that involves Fluid-Structure-Thermal interaction

(FSTI). In section 2, the equations of the coupled model are discretized via the

finite element discretization. In section 3, the multigrid domain decomposition al-

gorithm to solve the discrete problem is discussed. Finally in section 4 the method

is applied to a two-dimensional FSTI application.

2. Model and governing equations

Let the computational domain Ω ⊂ ?2be an open set with boundary Γ. Let

the fluid subdomain Ωfand the solid subdomain Ωsbe two disjoint open sets with

boundary Γf and Γs, respectively and let Ω =¯Ωf∪¯Ωs.

Ωs

Ωf

Γsf

f Γ

Γ2

Γ1

Γs

Γ

Ωf

f Γ

Γ2

Γ1

Γsf

Ωs

Γs

Γ

Figure 1. Domain Ω = Ωf∪ Ωsin two different configurations.

Figure 1 presents illustrations of two sample computational domains. Γsfis the

interior boundary between Ωf and Ωs, Γe

and Γe

the only boundary which can change in time is the interior boundary Γsf. In

agreement with this assumption both subdomains Ωf and Ωsare time dependent

f= Γ ∩ Γf is the fluid exterior boundary

s= Γ ∩ Γsis the solid exterior boundary. For simplicity let us assume that

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EJDE-2009/CONF/17/A COMPUTATIONAL DOMAIN DECOMPOSITIO 15

and constrained by

¯Ωf(t) ∪¯Ωs(t) =¯Ω.

(2.1)

In the model problem, we employ the unsteady Navier-Stokes equations describing

the flow of a fluid in a region Ωf given by:

ρf∂? u

∂t− µf∆? u + ρf(? u · ∇)? u + ∇p =?f

∇ · ? u = 0

? u =?U

in Ωf× (0,T)

in Ωf× (0,T)

on Γ1

where ρfand µfare the density and the viscosity and?f is the body force. This is

coupled with the energy equation given by,

ρcp∂T

∂t− k∆T + ρcp(? u · ∇T) = 0

T = Θ

that is solved over the whole domain Ω. In the solid region Ωsthe approximate

Euler-Bernoulli beam equation is considered. In this approximation plane cross

sections perpendicular to the axis of the beam are assumed to remain plane and

perpendicular to the axis after deformation [9] and under these hypotheses only a

mono-dimensional model is required for the normal transverse deflection field w.

We will denote by Λ the beam axis and by (ξ,η) a local reference system oriented

with the ξ-axis parallel to Λ.

in Ω × (0,T)

on Γ2

η

L

δ

Λ

Γsf

ξ

Figure 2. Domain notation for the beam domain Ωs.

As shown in Figure 2, variables δ and L are the thickness and the length of the

beam respectively, the interior boundary Γsf is in (ξ,±δ/2) for 0 ≤ ξ ≤ L and in

(L,η) for −δ/2 ≤ η ≤ δ/2.

In Γ1⊂ Γe

Neumann homogenous boundary conditions are considered on the remaining part,

Γe

temperature T, while Neumann homogenous boundary conditions are considered

on Γ \ Γ2. In ξ = 0 Dirichlet zero boundary conditions are imposed for the solid

displacement w and its derivatives. Conditions of displacement compatibility and

force equilibrium along the structure-fluid interface Γsf are satisfied.

Let?U ∈ H1/2(Γ1) be the prescribed boundary velocity over Γ1, satisfying the

compatibility condition, and Θ ∈ H1/2(Γ2) be the prescribed temperature over Γ2.

f, Dirichlet boundary conditions are imposed for the velocity field ? u;

f\ Γ1. Similarly, on Γ2⊂ Γ Dirichlet boundary conditions are imposed for the

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16E. AULISA, S. MANSERVISI, P. SESHAIYEREJDE/CONF/17

We are using Hk(Ω) to denote the space of functions with k generalized derivatives.

We set L2(Ω) = H0(Ω) and note that the derivation of these spaces can be extended

to non-integer values k by interpolation. The velocity, the pressure, the temperature

and the beam deflection (? u,p,T,w) ∈ H1(Ωf) × L2(Ωf) × H1(Ω) × H2(Λ) satisfy

the weak variational form of the unsteady fully coupled system given by the Navier-

Stokes system over Ωf,

?? u,? vf

= −?ρf? g β(T − T0),? vf?

bf

?? u −?U,? sf?Γ1= 0

?? u − ˙ w · ˆ nsf,? ssf?Γsf= 0

the energy equation over Ω

∂t,v? + a?T,v?+ c?? u;T,v?= 0

?T − Θ,s?Γ2= 0

and the Euler-Bernoulli beam equation over Ωs,

?w,vs

w(0,t) = 0,

∂ξ

In (2.2)-(2.3) the continuous bilinear forms are defined as

?

bf(? v,r) = −

Ωf

and the trilinear form as

?

where ρfand µfare the density and the viscosity of the fluid. The distributed force

in Eq. (2.2) is the Boussinesq approximation of the buoyancy force, where ? g is the

gravity acceleration, β the volumetric expansion coefficient of the fluid and T0 a

reference temperature. For T > T0the fluid expands then the density decreases and

the buoyancy force points in the direction opposite to the gravity. When T < T0

both the buoyancy force and the gravity point in the same direction. In (2.6) the

bilinear form is defined as

?

and the trilinear form

?

?ρf

∂? u

∂t,? vf? + af

?+ bf

?? vf,p?+ cf

∀? vf∈ H1(Ωf),

∀rf∈ L2(Ωf),

∀? sf∈ H−1

∀? ssf∈ H−1

?? u;? u,? vf

?

(2.2)

?? u,rf

?= 0 (2.3)

2(Γ1),

(2.4)

2(Γsf),

(2.5)

?ρcp∂T

∀v ∈ H1(Ω),

2(Γ2),

(2.6)

∀s ∈ H−1

(2.7)

?ρsδ ¨ w,vs? + as

?= ?p(? x(ξ,−δ

2),t) − p(? x(ξ,δ

∂w(0,t)

2),t),vs?∀vs,∈ H2(Λ), (2.8)

= 0,

˙ w(0,t) = 0.

(2.9)

af(? u,? v) =

Ωf

2µfD(? u) : D(? v)d? x

∀? u,? v ∈ H1(Ωf),

(2.10)

?

r∇ ·? v d? x

∀r ∈ L2(Ωf), ∀? v ∈ H1(Ωf)(2.11)

cf(? w;? u,? v) =

Ωf

ρf(? w · ∇)? u ·? v d? x

∀ ? w,? u,? v ∈ H1(Ωf),

(2.12)

a(T,v) =

Ω

k∇T · ∇v d? x

∀T,v ∈ H1(Ω)(2.13)

c(? u;T,v) =

Ω

ρcp(? u · ∇T)v d? x

∀? u ∈ H1(Ω), T,v ∈ H1(Ω),

(2.14)

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EJDE-2009/CONF/17/A COMPUTATIONAL DOMAIN DECOMPOSITIO 17

where ρ, cp, and k are the density, the heat capacity and the heat conductivity,

respectively. If the integral is over the subdomain Ωf, the fluid physical properties

ρf, cpf, and kf are used, otherwise over Ωsthe solid properties ρs, cps, and ksare

used. Furthermore in the solid region the trilinear form c(? u,T,v) is identically zero,

since the velocity ? u is zero. In (2.8) the bilinear form is defined as

?

where EI is the beam stiffness coefficient for unitary deepness. In the right hand

side of Eq. (2.8) the load due to the pressure difference between the two sides of the

beam is given. (2.8) represents the force equilibrium constraint between the two

subdomains Ωf and Ωson the common boundary Γsf. For details concerning the

function spaces, the bilinear and the trilinear forms and their properties, one may

consult [9, 10, 11]. Equations (2.4), (2.7) and (2.9) represent the exterior Dirichlet

boundary condition for the velocity, the temperature and the displacement, respec-

tively. Eq. (2.5) represents the compatibility constraints between the velocity field

? u and the time derivative of the beam deflection w on Γsf. The unitary normal ˆ nsf

points in the same direction as the local η-axis. According to Eqs. (2.1) and (2.5),

changes in the fluid and solid subdomains Ωf and Ωsshould be also considered.

as(w,v) =

Λ

EI∂2w

∂ξ2

∂2v

∂ξ2d? x

∀w,v ∈ H2(Λ),

(2.15)

2.1. Domain decomposition. Let us now introduce a non-conforming formula-

tion of the problem. Let the domain Ω be partitioned into m non-overlapping

sub-domains {Ωi}m

collection of edges of Ωiand Ωj. In the latter case, we denote this interface by Γij

which consists of individual common edges from the domains Ωiand Ωj. Let now

the fluid domain Ωf be partitioned into m non-overlapping sub-domains {Ωi

where Ωi

domain partition {Ωi}m

then Ωi

two subregions Ωi

vector, the temperature, the heat flux and the displacement (? ui,pi,? τij,Ti,qij,w) ∈

H1(Ωi

system of equations

i=1such that ∂Ωi∩ ∂Ωj(i ?= j) is either empty, a vertex, or a

f}m

i=1,

fis given by Ωi∩ Ωf. The fluid partition {Ωi

i=1, subtracting the solid region from each subdomain Ωi,

fis an empty region if Ωiis a subset of Ωs. The common boundary between

fand Ωj

f}m

i=1is obtained from the

fis denoted by Γij

f. The velocity, the pressure, the stress

f)×L2(Ωi

f)×H−1/2(Γij

f)×H1(Ωi)×H−1/2(Γij)×H2(Λ) satisfy the following

?ρf

+ ?? τij,? vi

∂? ui

∂t,? vi

f? + af

f?Γij

bf

?? ui−?U,? si

?? ui− ? uj,? sij

?? ui− ˙ w · ˆ nsf,? si

?? ui,? vi

?? ui,ri

f

?+ bf

?= 0

?? vi

f,pi?+ cf

f?

f∈ L2(Ωi

∀? si

∀? sij

sf= 0

?? ui;? ui,? vi

∀? vi

f),

f

?

f= −?ρf? g β(Ti− T0),? vi

f∈ H1(Ωi

f),

(2.16)

f

∀ri

(2.17)

f?Γi

f?Γij

sf?Γi

1= 0

f= 0

f∈ H−1/2(Γi

f∈ H−1/2(Γij

∀? si

1),

(2.18)

f),

(2.19)

sf∈ H−1/2(Γi

sf),

(2.20)

?ρcp∂Ti

∂t,vi? + a?Ti,vi?+ c?? ui;Ti,vi?+ ?? qij,? vi?Γij = 0

?Ti− Θ,si?Γi

?Ti− Tj,? sij?Γij = 0

∀vi∈ H1(Ωi), (2.21)

2= 0

∀si∈ H−1/2(Γi

∀sij∈ H−1/2(Γij),

2),

(2.22)

(2.23)

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18E. AULISA, S. MANSERVISI, P. SESHAIYEREJDE/CONF/17

?ρsδ ¨ w,vs? + as

?w,vs

?= ?p(? x(ξ,−δ

2),t) − p(? x(ξ,δ

2),t),vs?∀vs∈ H2(Λ),

(2.24)

w(0,t) = 0,

∂w(0,t)

∂ξ

= 0,

˙ w(0,t) = 0,

(2.25)

for i = 1,2,...,m where Γi

−µ∇? ui· ˆ nij

unitary external vectors normal to the subdomains Ωi

Note that in the continuous case, on the boundary Γij

and ? ujare in the same space H1/2(Γij

same remark can be applied to the stress vectors ? τijand ? τji, the temperatures Ti

and Tjand the heat fluxes qijand qji. In the rest of the paper, the variables ? u,

p and T, without the labeli, should be considered as a collection of all the local

variables ? ui, piand Ti, where

1= Γ1∩ ∂Ωi

f, Γi

2= Γ2∩ ∂Ωi, the stress vector ? τij=

f+ piˆ nij

fand the heat flux qij= −k∇Ti· ˆ nij, with nij

fand nijthe

fand Ωi, respectively.

f, the velocity vectors, ? ui

f), namely we have ? ui= ? ujpointwise. The

? u(? x,t) = ? ui(? x,t)

p(? x,t) = pi(? x,t)

T(? x,t) = Ti(? x,t)

∀ ? x ∈ Ωi

∀ ? x ∈ Ωi

∀ ? x ∈ Ωi,

f,

f,

for i = 1,2,...,m. It is straightforward to prove that the system (2.16)-(2.25)

for the local state variables (? ui,pi,? τij,Ti,qij,w), for i = 1,2,...,m, implies the

system (2.2)-(2.9) for the global state variables (? u,p,? τ,T,q,w) .

2.2. ALE formulation and time discretization. In order to account for the

changing nature of the fluid and solid subdomains, we wish to define a dynamic

mesh for the space discretization. However, to avoid extreme distortion, we choose

to move the mesh independently of the fluid velocity in the interior of Ωf. Such

a scheme, called arbitrary Lagrangian-Eulerian (ALE) formulation, is commonly

applied when studying fluid-structure interaction [12, 13, 14, 15]. Inside the solid

region each point is moving according to the time derivative of the displacement w.

The grid velocity ? ugcan be any velocity satisfying the following constraints

? ug= 0

? ug= ? u

on Γ,

on Γsf,

(2.26)

(2.27)

(2.28)

? ug(? x(ξ,η),t) = ˙ w(ξ,t) ˆ nsf

in Ωs.

If the grid velocity is known as a function of time, the trajectory inside the domain

Ω of a generic point of coordinate ? x(t) can be traced by solving the integral equation

? x(t1) = ? x(t0) +

?t1

t0

? ug(? x(t?),t?)dt?.

(2.29)

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EJDE-2009/CONF/17/A COMPUTATIONAL DOMAIN DECOMPOSITIO19

The Lagrangian derivative of the velocity field ? u(? x(t),t), evaluated along the point

trajectory, is given by

D? u(? x(t),t)

Dt

=∂? u

∂t+∂? u

=∂? u

∂x1

dx1

dt

ug1+∂? u

+∂? u

∂x2

dx2

dt

∂t+∂? u

=∂? u

∂t+ (? ug· ∇)? u.

∂x1

∂x2

ug2

(2.30)

Then the Eulerian derivative can be written as the difference between the La-

grangian derivative and the corresponding grid velocity advection term

∂? u

∂t=D? u(? x(t),t)

Dt

− (? ug· ∇)? u.

(2.31)

In the same way the Eulerian derivative of the temperature T can be expressed

as the difference between the Lagrangian derivative and the corresponding grid

velocity advection term

∂T

∂t=DT(? x(t),t)

Dt

− (? ug· ∇)T .

(2.32)

Using (2.31)-(2.32) in system (2.16)-(2.25), the Navier-Stokes and energy conserva-

tion equations become

?ρf

= −?ρf? g β(Ti− T0),? vi

D? ui(? x(t),t)

Dt

,? vi

f? + af

?? ui,? vi

f

?+ bf

?? vi

f,pi?+ cf

?? ui− ? ui

g;? ui,? vi

f

?+ ?? τij,? vi

f?Γij

f

f?

(2.33)

?ρcpDTi(? x(t),t)

Dt

,vi? + a?Ti,vi?+ c?? ui− ? ui

g;Ti,vi?+?? qij,? vi

f?Γij = 0.

Given the initial velocity field ? u0the Lagrangian derivatives can be discretized in

time using a simple first order integration method.

The structural equation is discretized in time by a using Newmark integration

scheme [9]. In this method the displacement and its time derivative are approxi-

mated according to

wt+1= wt+ ∆t ˙ wt+1

2∆t2¨ wt+γ,

(2.34)

˙ wt+1= ˙ wt+ ∆t ¨ wt+α,

(2.35)

where

¨ wt+θ= (1 − θ) ¨ wt+ θ ¨ wt+1,

(2.36)

and α and γ are parameters that determine the stability and accuracy of the

method. We chose α = γ = 0.5, which is known as the constant-average accelera-

tion method. This technique is stable for each time step ∆t and conserves energy

for free vibration problem. The use of (2.34)-(2.36) in (2.24) gives the following

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20E. AULISA, S. MANSERVISI, P. SESHAIYER EJDE/CONF/17

approximation

?ρsδ (a3wt+1),vs? + as

= ?ρsδ(a3wt+ a4 ˙ wt+ a5¨ wt),vs?,

a3=

?wt+1,vs

2

γ ∆t2,

?− ?(pt+1(? x(ξ,−δ

2

γ ∆t,

2),t) − pt+1(? x(ξ,δ

2),t),vs?

a4=

a5=1

γ− 1,

(2.37)

where all the terms in the right hand side, wt, ˙ wtand ¨ wt, are evaluated at the pre-

vious time step. Note that the calculation of the right hand side requires knowledge

of the initial conditions w0, ˙ w0and ¨ w0. In practice, one does not have ¨ w0. As an

approximation, it can be calculated from (2.24)

?ρsδ ¨ w0,vs? = ?p0(? x(ξ,−δ

for given initial displacement w0and initial pressure p0. At the end of each time

step, the new velocity ˙ wt+1and acceleration ¨ wt+1are computed using

¨ wt+1= a3(wt+1− ws) − a4 ˙ wt− a5¨ wt,

˙ wt+1= ˙ wt+ a2¨ wt+ a1¨ wt1,

a1= α∆t, a2= (1 − α)∆t .

3. System discretization

2),t) − p0(? x(ξ,δ

2),t),vs? − as

?w0,vs

?,

(2.38)

3.1. Multilevel domain decomposition. In this section a multilevel domain

decomposition methodology will be described for the whole domain Ω and the

corresponding variable T. The same consideration can be directly extended to the

fluid domain Ωf and the corresponding variables ? u and p.

Let us introduce a finite element discretization in each subdomain Ωithrough

the mesh parameter h which tends to zero. Let {Ωi

discretized domain Ωh. Now, by starting at the multigrid coarse level l = 0, we

subdivide each Ωi

meshes Ti,0

levels can be built to reach the finite element meshes Ti,n

level l = n. At the coarse level, as at the generic multigrid level l, the triangulation

over two adjacent subdomains, Ωi

constraints along the common interfaces Γij

their construction one may consult [16, 17, 18].

By using this methodology we have constructed a sequence of meshes for each

multigrid level in a standard finite element fashion with compatibility enforced

across all the element interfaces built over midpoint refinements. In every subdo-

main Ωi

a global solution over Ωh, consisting mesh solutions at different levels over different

subdomains. Let Ωi,l

the multigrid level l. It should be noted that the multigrid levels at which the so-

lution is computed over adjacent subdomains, Ωi,l

each other (l ?= k), with no compatibility enforced across the common interface Γij

In Figure 3 an example of non-conforming mesh partitioning is given for a square

domain Ωh(Figure 3.a). The square is divided in four subdomains (Figure 3.b) and

3 conforming level meshes are considered (l = 0,1,2). The coarse mesh (Figure 3.c)

h}m

i=1be the partition of the

hand consequently Ωhinto triangles or rectangles by families of

h. Based on a simple element midpoint refinement different multigrid

h

at the top finest multigrid

hand Ωj

h, obeys the finite element compatibility

h. For details on multigrid levels and

hthe energy equations can be solved over a different level mesh, generating

hbe the subdomain i where the solution will be computed at

hand Ωj,k

h, may be different from

h.

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EJDE-2009/CONF/17/A COMPUTATIONAL DOMAIN DECOMPOSITIO 21

Ωh

Ω1,0

hh

2,0

Ω

3,2

h

ΩΩ4,1

h

h

1

ΩΩh

2

3

Ωh

Ωh

4

Τ1,0

h

2,0

Τh

h

3,0

Τ

h

Τ4,0

h

Τ

h

Τ

Τhh

Τ

1,22,2

3,2 4,2

h

Τ

Τhh

Τ

h

Τ

1,1 2,1

4,1 3,1

(a)(b)

(d)

(e)

(c)

(f)

Figure 3. Domain (a), domain partitioning (b),conforming coarse

level mesh (c), midpoint refinement generation of two finer con-

forming level meshes (d),(e), non-conforming mesh (f).

is given, while the other two (Figures 3.d-e) are generated by successive midpoint

refinements. In Figure 3.f one possible non-conforming mesh configuration is il-

lustrated. Note that in Figure 3.f only across the common boundary Γ1,2

element compatibility is obtained since the two subdomain meshes Ω1,0

are chosen at the same level (l = 0). On the other three common boundaries, Γ1,3

Γ2,,4

h

and Γ3,4

ered between two adjacent subdomains. It should be noticed that on the common

boundaries the nodes of the coarsest grid are always included in the nodes of the

finest one. This is the key point for discretization and resolution of (2.16)-(2.25) and

it is always true for different level meshes generated by using midpoint refinements.

Finite element approximation spaces can be generated regularly, as function of

the characteristic length h over each multigrid level l resulting in different approxi-

mation spaces over the solution meshes Ωi,l

labels i,l the solution over the corresponding subdomains, i.e., for the temperature

Ti,l, and with no labels the extended solution over Ω, i.e. Th for the extended

temperature. Note that the temperature Ti,lis computed over each Ωi,l

corresponding level l, but the extended solution Th on the top level n is defined

over all Ωhin a standard and regular way. There may be parts of the domain where

the solution is not computed at the top level but a projection operator In

coarse level l to the top level n can always be used to interpolate the solution over

Ωh. The extended temperature is therefore defined by

h

and Ω2,0

finite

hh

h,

h, compatibility is not enforced since different level meshes are consid-

h. In the rest of the paper we denote with

hat the

lfrom the

Th(? x,t) = In

lTi,l(? x,t),

Page 10

22E. AULISA, S. MANSERVISI, P. SESHAIYER EJDE/CONF/17

for all ? x ∈ Ωi

the extended velocity and pressure by using the same operator In

? uh(? x,t) = In

ph(? x,t) = In

for all ? x ∈ Ωi

subdomains of Ωhf. These extended functions take the same values over the coarse

and the top mesh at those nodes included in both meshes.

h, i = 1,2,...,m. We can easily generalize the notations and evaluate

las

l? ui,l(? x,t),

lpi,l(? x,t),

(3.1)

(3.2)

hf, i = 1,2,..,m, where {Ωi

hf}m

i=1is the partition over the discrete

3.2. Non-conforming finite element discretization. Let Xl

Sl

Xl

families of finite element spaces to satisfy appropriate stability and approximation

properties [19, 20] that will allow us building a regular conforming approxima-

tion. We indicate with Pl

Pl

tain by choosing the family of Hermitian elements, Zh(Λ) ⊂ H2(Λ), for which the

interpolation functions are continuous with non-zero derivatives up to order two.

Let Jibe the set of the j-indices of all the neighboring regions Ωjsurrounding

the subdomain Ωiand Ji

Ωj

Sl

stress vector, the temperature, the heat flux and the beam displacement over the

corresponding subdomains. The variable state (? ui,l,pi,l,? τij,l,Ti,l,qij,l,w) satisfies

the Navier -Stokes discrete system

h(Ωf) ⊂ H1(Ωf),

h(Ωf) ⊂ L2(Ωf), Rl

h|Γ⊂ H1/2(Γ) be the approximation spaces. At each level mesh l we chose the

h(Γf) = Xl

h|Γf⊂ H1/2(Γf), Xl

h(Ω) ⊂ H1(Ω) and Rl

h(Γ) =

h(Γf) ⊂ H−1/2(Γf) the dual space of Rl

h(Γ). The beam space discretization is ob-

h(Γf) and with

h(Γ) ⊂ H−1/2(Γ) the dual space of Rl

fbe the set of the j-indices of all the neighboring regions

fsurrounding the subdomain Ωi

h(Ωi

f. Let (? ui,l,pi,l,? τij,l,Ti,l,qij,l,w) be in Xl

h(Ωi) × Pl

h(Ωi

f)×

f) × Pl

h(Γij

f) × Xl

h(Γij) × H2(Λ) be the velocity, the pressure, the

?ρf

+ cf

D? ui,l(? xh,t)

Dt

?? ui,l− ? ui,l

= −?ρf? g β(Ti,l− T0),? vi,l

bf

?? ui,l−?U,? si,l

?? ui,l− ? uj,k,? sij,l

?? ui,l− ˙ wh· ˆ nsf,? si,l

,? vi,l

f? + af

?? ui,l,? vi,l

f

?+ bf

f?Γij

f∈ Xl

f∈ L2(Ωi

∀? si,l

∀? sij,l

∀? si,l

?? vi,l

f,pi,l?

g;? ui,l,? vi,l

f

?+ ?? τij,l,? vi,l

f?

?= 0

fh

h(Ωi

∀? vi,l

f),

(3.3)

?? ui,l,ri,l

f

∀ri,l

f),

(3.4)

f?Γi

f?Γij

1h= 0

f∈ Pl

h(Γi

1),

(3.5)

fh= 0

f

∈ Pl

sf∈ Pl

h(Γij

f),

(3.6)

sf?Γi

sfh= 0

h(Γi

sf),

(3.7)

over Ωi

ffor all j ∈ Ji

?ρcpDTi,l(? xh,t)

+ c?? ui,l− ? ui,l

fand i = 1,2,...,m, the energy equation

Dt

,vi,l? + a?Ti,l,vi,l?

g;Ti,l,vi,l?+ ?? qij,l,? vi,l?Γij

?Ti,l− Θ,si,l?Γi

?Ti,l− Tj,k,? sij,l?Γij

h= 0

∀si,l∈ Pl

∀sij,l∈ Pl

∀vi,l∈ Xl

h(Γi

h(Ωi),

(3.8)

2h= 0

h= 0

2),

(3.9)

h(Γij),

(3.10)

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EJDE-2009/CONF/17/A COMPUTATIONAL DOMAIN DECOMPOSITIO 23

over Ωifor all j ∈ Ji

equation

fand i = 1,2,...,m, and the discrete Euler-Bernoulli beam

?ρsδ ¨ wh,vsh? + as

= ?ph(? xh(ξ,−δ

?wh,vsh

2),t) − ph(? xh(ξ,δ

∂wh(0,t)

?

2),t),vsh?

vsh∈ Z(Λ),

(3.11)

wh(0,t) = 0,

∂ξ

= 0,

˙ wh(0,t) = 0.

(3.12)

over Λ. On the shared boundaries Γij

the two different spaces Pl

the weak form by

fhthe stress vectors, ? τij,land ? τji,k, belong to

h, and their equivalence should be considered in

hand Pk

?? τij,l,? vi,l?Γij

fh= −?? τji,k,? vi,l?Γij

fh.

(3.13)

Similarly qij,land qji,kbelong to the two different spaces Pl

hand Pk

h, and

?qij,l,vi,l?Γij

h= −?qji,k,vi,l?Γij

h.

(3.14)

Since different level meshes are generated by using midpoint refinement the bound-

ary vector spaces are nested, with Rl

the traces of the test functions ? vi,land vi,lon the coarsest boundary can be decom-

posed as a linear combination of traces of test functions ? vj,k

boundary. Under these hypotheses, Eqs. (3.13) and (3.14) become

?

?qij,l,vi,l?Γij

a

which means that in (3.3) and (3.8) the Lagrange multipliers ?? τij,l,? vi,l?Γij

?qij,l,vi,l?Γij

on the boundary and then restricted over the coarsest one. Eqs. (3.15) and (3.16),

define the restriction operator Rl

In (3.6) and (3.10) the equalities between ? ui,land ? uj,kon the boundary Γij

between Ti,land Tj,kon the boundary Γij

since the velocities and the temperatures belong to the two corresponding different

vector spaces Rl

nested with Rl

equations if the velocities or the temperatures belong to the coarsest spaces Rl

Rl

h⊆ Rk

hand Rl

h⊆ Rk

h, assuming l ≤ k. Then

a

and vj,k

a

on the finest

?? τij,l,? vi,l?Γij

f= −?? τji,k,

a

?

wa? vj,k

a?Γji

fh:= −?? τji,k,Rl

k? vj,k?Γji

fh,

(3.15)

h= − < qji,k,wavj,k

a?Γji

h:= −?qji,k,Rl

kvj,k?Γji

h,

(3.16)

f

and

hcan be always discretized and computed on the finest grid available

kfrom the finest to the coarsest vector space.

fh, and

h, must be considered in their weak form

h, Rk

h⊆ Rk

hand Rl

hand Rl

h, Rk

h⊆ Rk

hrespectively. Assuming l ≤ k, the spaces are

h. The weak equations may turn into pointwise

hor

h, respectively.

3.3. Solution strategy. Although the equation system (3.3)-(3.12) is fully cou-

pled, its solution is achieved with an iterative strategy, where the three systems of

equations are solved separately and in succession, always using the latest informa-

tion, until convergence is reached. An iterative multigrid solver is used for both the

Navier-Stokes and the energy equation systems since the number of unknowns could

be quite large. For the solution of the beam equation a direct LU decomposition is

used.

At each iteration, the linearized Navier-Stokes system is assembled, using the

latest updated value of the temperature T and the latest updated value of the

grid velocity ? ugin the nonlinear term cf

?? u −? ug;? u,? vf

?. In the nonlinear term, the

Page 12

24 E. AULISA, S. MANSERVISI, P. SESHAIYEREJDE/CONF/17

first of the two velocity ? u is considered explicitly. On the boundary Γsf Dirichlet

boundary conditions are imposed according to the latest updated value of the beam

displacement time derivative ˙ w. A V-cycle multigrid algorithm is used to obtain a

new update solution for the pressure p and the velocity ? u. Then the energy equation

system is assembled, using the previously evaluated velocity and grid velocity in the

advection term c?? u−? ug;T,v?. A multigrid V-cycle is solved and updated values of

load field is computed using the previous evaluated pressure p. Since the number of

the subdomain unknowns is limited an direct LU decomposition solver can be used

for computing the new displacement w and its time derivatives. The grid velocity

is then computed according to Eqs. (2.26)-(2.28) and the grid nodes are advected

along the corresponding characteristic lines. The whole procedure is repeated until

convergence is finally reached.

The Navier-Stokes system (3.3)-(3.7) is solved using a fully coupled iterative

multigrid solver [21] with a Vanka-type smoother. Multigrid solvers for coupled

velocity/pressure system compute simultaneously the solution for both the pressure

and the velocity field, and they are known to be one of the best class of solvers for

laminar Navier-Stokes equations (see for examples [17, 18]). An iterative coupled

solution for the linearized discretized Navier-Stokes system requires the solution of

a large number of sparse saddle point problems. In order to optimally solve the

equation system (3.3)-(3.7), involving the unknown stress vector ? τij, we use the

block Gauss-Seidel method, where each block consists of a small number of degrees

of freedom (for details see [18, 21, 22, 23, 24]). The characteristic feature of this

type of smoother is that in each smoothing step a large number of small linear

systems of equations has to be solved. Each block of equations corresponds to

all the degrees of freedom which are connected to few elements. For example, for

conforming finite elements, the block may consist of all the elements, containing

some pressure vertices. Thus, a smoothing step consists of a loop over all the

blocks, solving only the equations involving the unknowns inside the elements that

are around the considered pressure vertices. The velocity and pressure variables

are updated many times in one smoothing step.

The Vanka smoother employed in our multigrid solver involves the solution of

a small number of degrees of freedom given by the conforming Taylor-Hood finite

element discretization used. For this kind of element the pressure is computed only

at the vertices while the velocity field is computed also at the midpoints. Over the

internal part of the generic subregion Ωi

our Vanka-block consists of an element and all its neighboring elements. We solve

for all the degrees of freedom inside the block, with boundary condition taken on

the external boundaries.

For example, in Figure 4, our block consists of four vertex points and 12 mid-

points to be solved, for a total of 36 unknowns. We have also used different blocks

with different performances but we have found this particular block to be very ro-

bust and reliable even at high Reynolds numbers. Examples of computations with

this kind of solver can be found in [5, 6, 17, 18].

The fact that the solution is searched locally allows us to solve for ? τijonly near

the common boundary Γij

inside the subdomains by using Vanka smoother. When solving block elements near

the common boundary, distinction should be made if the block element is inside

the temperature T are found. Finally the beam equation system is built, where the

fh, where there are no boundary elements,

fh, where different level meshes are coupled, and to solve

Page 13

EJDE-2009/CONF/17/A COMPUTATIONAL DOMAIN DECOMPOSITIO25

a

b

Figure 4. Unknowns (black circles) and boundary conditions

(white circles) are shown for both the velocity field ? uh (a) and

the pressure ph(b) for our particular Vanka-block smoother.

the coarse or the fine subregion. Let us consider the two adjacent subdomains Ωi,l

and Ωj,k

where the nodes of the coarsest mesh are included in the finest one, the traces of

the two velocities ui,land uj,kare equal and belong to the coarsest vector spaces

Rl

use the standard solution technique imposing the trace of the coarsest velocity ui,l

as Dirichlet boundary condition for the block equation system. We have also seen

that each Lagrange multiplier in the coarsest system can be represented has a linear

combination of Lagrange multipliers of the finest system, thus when solving over

boundary blocks on the coarsest subregion Ωi,l

(3.3) is replaced with (3.15), where the stress tensor vector ? τji,kis evaluated over

the finest subdomain, by using the pressure, pj,k, and velocity field, ? vj,k. In this

way a constant and reciprocal exchange of information between the two subdomains

is ensured.

In order to increase the convergence rate, the considered Vanka-type smoother

has been coupled with a standard V-cycle multigrid algorithm. The multigrid does

not change the nature of the solver, but allows the information to travel faster

among different parts of the domain. A rough global solution is evaluated on the

coarsest mesh l = 0 and projected on the finer grid l = 1, where Vanka-loops are

performed improving its details. The updated solution is then projected on the

mesh level l = 2 and improved. The procedure is repeated until the finest mesh

is reached. Solving the equation system in fine meshes improves solution details,

but at the same time reduces the communication speed over the domain. However,

this does not affect the global convergence rate since a considerable information

exchange among different parts of the domain has been already done when solving

in coarser mesh levels. The analysis of the convergence of the Vanka type multigrid

solvers and the associated invertibility of the discrete system for the Navier-Stokes

can be found in [25, 5]. All these considerations can be directly extended to the

energy equation solver, where the same element block is considered. For an interior

block, as in Figure 4-a, the only unknowns are the values of the temperature at

fh

fhwith l ≤ k. We have already seen, that in our particular discretization,

h. Thus, when solving boundary blocks over the finest subregion Ωj,k

fh, we can

fh, the Lagrange multiplier term in

Page 14

26E. AULISA, S. MANSERVISI, P. SESHAIYEREJDE/CONF/17

the vertices and in the midpoints, for a total of 12 variables, while for a boundary

block the heat flux qij,land qji,kshould be also solved over the common boundary

Γij

h.

4. Numerical Experiments

In this section we test the FSTI non-conforming multilevel formulation and its

solution strategy.

Ωs

Ωf

Γsf

_ _ _

Ω = Ω + Ω

f

fs

Γ = Γ + Γ − Γ

s sf

x

y

Figure 5. Computational domain.

As shown in Figure 5 let the rectangular region Ω = [4m] × [2m] be the com-

putational domain with boundary Γ and Ωf and Ωs be the fluid and the solid

subdomain respectively. The solid region Ωs consists of a beam, clamped at the

point (1m,0), with length equal to 0.5m and thickness equal to 0.04m. The fluid

and the solid boundaries, Γfand Γsare the contours of the two shaded regions and

their intersection is labeled by Γsf.

On the left side of the domain inflow boundary conditions are imposed for the

velocity field ? u = (u1,u2) with parabolic profile u1= 0.1y (2 − y)m/s and u2= 0.

On the right side of the domain outflow boundary conditions are imposed while on

the remaining part of the boundary non-slip conditions are set. The temperature

is set equal to zero in the inlet region and to 100oC on the solid boundary where

the beam is clamped. Adiabatic conditions are imposed on the rest of the domain.

The initial conditions for both the temperature and the velocity field are zero.

The fluid and the solid properties are chosen in order to produce a large defor-

mation of the beam. This choice implies strong interactions among all the parts

of the system and test the reliability of the solver in challenging situations. In

the Navier-Stokes system, the fluid density ρf, the viscosity µf, the volumetric

expansion coefficient β and the reference temperature T0are equal to 100kg/m3,

0.01Kg/ms, 0.01K−1and 0oC, respectively. In the temperature equation the solid

density ρsis 200kg/m3, while the heat capacity cpand the heat conductivity K, are

100J/Kg K and 10W/mK in the fluid region, and 10J/Kg K and 400W/mK in

the solid region respectively. The stiffness for unitary length of the beam is equal

to 1kgm2/s2.

In all the simulations the same time step ∆T = 0.01s is used, for a total of 500

time steps (5 seconds). Only the four level meshes, l0,l1,l2and l3, are considered

and in Figure 6, the two different level meshes, l0and l3, are shown.

Page 15

EJDE-2009/CONF/17/ A COMPUTATIONAL DOMAIN DECOMPOSITIO 27

Figure 6. The coarse mesh level l0(left) and finest level mesh l3(right).

The coarse mesh level l0has 207 elements, while the mesh level l3obtained after

three consecutive midpoint refinements has 13248 elements. The one-dimensional

mesh on the beam axis follows the same midpoint refinement algorithm used for the

two-dimensional computational domain Ω. On the coarse level l0 three elements

are available, while on the fine grid l3after 3 refinements, the number of elements

becomes 24. Since the number of unknowns is quite small, (24 + 1) × 2 = 50, the

solution of the beam equation is always evaluated on the finest mesh using a direct

LU decomposition.

The results obtained with our coupled model (case B) are first compared with

the results obtained for the same geometry with a rigid beam, EI = ∞, and with

zero buoyancy force, β = 0 (case A). All the computations are done at the time

t = 5s and over the finest level l3.

Figure 7. Beam bending and grid distortion (left); velocity field

map (right).

In Figure 7 on the left, the beam bending and the corresponding grid deformation

are displayed, showing the strong influence of the pressure load on the beam shape.

Figure 7 on the right shows the velocity field map and clearly indicates that the

stationary solution is not reached since new vortices are constantly created and

advected towards the outflow region.

In Figure 8 the velocity field (top), the pressure (bottom-left) and the tempera-

ture (bottom right) profiles, evaluated over the section y = 0.5 for 0 ≤ x ≤ 4, are

shown for the two cases EI = ∞, β = 0 (case A) and EI = 1, β = 0.01 (case B).

The combined effect of the beam deflection and the buoyancy force modify consid-

erably all the profiles, pointing out how sensitive is the interaction among all the

parts of the system.

The number of unknowns (velocity field, pressure, temperature and displace-

ment), involved in the computation at the mesh level l3is quite large, approxima-

tively 94000. However Figure 7 on the right shows that the only part of the system,