Download The Dirichlet-Neumann Iteration for Unsteady Thermal Fluid

The Dirichlet-Neumann Iteration for Unsteady
Thermal Fluid Structure Interaction:
Convergence Analysis for Finite Element
Discretizations
Azahar Monge∗ and Philipp Birken∗
∗
Centre for Mathematical Sciences,
Lund University,
Box 118, 22100, Lund, Sweden
June 20, 2016
Abstract
We analyze the convergence rate of the Dirichlet-Neumann iteration for
the fully discretized unsteady transmission problem. Specifically, we consider the coupling of two linear heat equations on two identical non overlapping domains with jumps in the material coefficients across these as a
model for thermal fluid structure interaction. The Laplacian is discretized
using a finite element method and the implicit Euler method is used for
the time discretization. We provide an exact formula for the spectral radius of the iteration matrix in 1D and an estimate in 2D. We then show
that these tend to the ratio of heat conductivities in the semidiscrete spatial limit, but to the ratio of the products of density and specific heat
capacity in the semidiscrete temporal one. This explains the fast convergence previously observed for cases with strong jumps in the material
coefficients. Numerical results confirm the analysis.
Keywords: Thermal Fluid Structure Interaction, Coupled Problems, Transmission Problem, Fixed Point Iteration, Dirichlet-Neumann Iteration
1
Introduction
The Dirichlet-Neumann iteration is a basic method in both domain decomposition and fluid structure interaction. In the latter case, the iteration arises in a
∗ e-mail:
[email protected]; web page: http://www.maths.lu.se/staff/azahar-monge
1
partitioned approach [8], where different codes for the sub-problems are reused
and the coupling is done by a master program which calls interface functions of
the other codes. This allows to reuse existing software for each sub-problem, in
contrast to a monolithic approach, where a new code is tailored for the coupled
equations. To satisfy the coupling conditions at the interface, the subsolvers
are iterated by providing Dirichlet- and Neumann data for the other solver in a
sequential manner.
In the domain decomposition context, the iteration has two main problems,
namely slow convergence and the need for an implementation using a red-black
colouring. The slow convergence can be improved using a relaxation procedure.
In fluid structure interaction, there are only two domains, coupled along an
interface, making the application straight forward. However, the convergence
rate is not great for the coupling between a compressible fluid and a structure
[7], which is why a lot of effort goes into convergence acceleration. On the other
hand, the Dirichlet-Neumann iteration was reported to be a very fast solver for
thermal fluid structure interaction [4].
Our prime motivation here is thermal interaction between fluids and structures, also called conjugate heat transfer. There are two domains with jumps in
the material coefficients across the connecting interface. Conjugate heat transfer plays an important role in many applications and its simulation has proved
essential [2]. Examples for thermal fluid structure interaction are cooling of
gas-turbine blades, thermal anti-icing systems of airplanes [5], supersonic reentry of vehicles from space [16, 13], gas quenching, which is an industrial heat
treatment of metal workpieces [11, 21] or the cooling of rocket nozzles [14, 15].
For the case of coupled heat equations, a one dimensional stability analysis
was presented by Giles [10]. There, an explicit time integration method was
chosen with respect to the interface unknowns. On the other hand, Henshaw and
Chand provided in [12] a method to analyze stability and convergence speed of
the Dirichlet-Neumann iteration in 2D based on applying the continuous Fourier
transform to the semi-discretized equations. Their result depends on ratios of
thermal conductivities and diffusivities of the materials. This is similar to the
situation in [6, 1] where the performance of the coupling for incompressible
fluids is affected by the added mass effect. However, in the fully discrete case
we observe that the iteration converges much faster for some choices of materials,
and that the speed of the iteration does not depend on the thermal diffusivities
in some cases. Therefore, we propose an alternative analysis here.
In principle, the convergence rate of the method is analyzed in any standard
book on domain decomposition methods, e.g. [19, 22]. There, the iteration
matrix is derived in terms of the stiffness and mass matrices of finite element
discretizations and the convergence rate is the spectral radius of that. However,
this does not provide a quantitative answer, since the spectral radius is unknown.
Computing the spectral radius is in general a non trivial task. In our context, the
material properties are discontinuous across the interface and as a consequence,
computing the spectral radius of the iteration matrix is even more difficult.
Thus, we consider a complete discretization of the coupled problem using finite elements in space and the implicit Euler method in time. Then, we compute
2
the spectral radius of the iteration matrix exactly in terms of the eigendecomposition of the resulting matrices for the one dimensional case. In the 2D case,
we provide an approximation of the convergence rate. The asymptotics of the
convergence rates when approaching the continuous case in either time or space
are also computed resulting on the ratio of the thermal conductivities in space
and the ratio of the heat capacities in time. These results are consistent with our
numerical experiments. In particular, the speed of the iteration when approaching the continuous case in space does not depend on the thermal diffusivities.
These results describe the convergence rates of the thermal interaction between
two different materials using the Dirichlet-Neumann iteration and they can be
used to design an efficient simulation beforehand.
An outline of the paper now follows. In Section 2, we define the problem to
be solved in terms of the partial differential equations, boundary conditions and
interface conditions. We also give a description of the discretization. In Section
3, we explain the Dirichlet-Neumann model. The analysis of the semidiscrete
case is mentioned in section 4. Our analysis for the discrete case of the model
problem using Dirichlet-Neumann interface conditions is presented in Section
5. In Section 6, we present numerical results that show the theoretical stability
analysis.
2
Model Problem
The unsteady transmission problem is as follows, where we consider a domain
Ω ⊂ Rd which is cut into two subdomains Ω1 ∪ Ω2 = Ω with transmission
conditions at the interface Γ = Ω1 ∩ Ω2 :
αm
∂um (x, t)
− ∇ · (λm ∇um (x, t)) = fm (x, t), t ∈ [t0 , tf ], x ∈ Ωm ⊂ Rd , m = 1, 2,
∂t
um (x, t) = 0, t ∈ [t0 , tf ], x ∈ ∂Ωm \Γ,
u1 (x, t) = u2 (x, t), x ∈ Γ,
λ2
∂u2 (x, t)
∂u1 (x, t)
= −λ1
, x ∈ Γ,
∂n2
∂n1
um (x, 0) = u0m (x), x ∈ Ωm .
(1)
where nm is the outward normal to Ωm for m = 1, 2, and we consider d = 1, 2
(see figure 1).
The constants λ1 and λ2 describe the thermal conductivities of the materials
on Ω1 and Ω2 respectively. D1 and D2 represent the thermal diffusivities of the
materials and they are defined by
Dm =
λm
, with αm = ρm Cm
αm
3
(2)
Figure 1: Illustration of the domains for the model problem (1). On the left,
domains when d = 1. On the right, domains when d = 2.
where ρm represents the density and Cm the heat capacity of the material placed
in Ωm , m = 1, 2. From now on we assume that f1 = f2 = 0 in order to simplify
the analysis.
In the one-dimensional case (when d = 1), we discretize this problem using
a finite element method (FEM) with a constant mesh width of ∆x = 1/(N + 1)
with N being the number of interior space discretization points in the intervals
Ωm , m = 1, 2. If instead we consider (1) with d = 2, we will use a constant
mesh width with respect to both spatial components (∆y := ∆x) resulting in N 2
interior space discretization points in both Ω1 and Ω2 . A FEM will be applied
for the space discretization over triangular elements. We use the implicit Euler
method for the time discretization for both d = 1, 2.
2.1
Finite Element Discretization
In this section we describe the FEM formulation of (1). Let Vm := H01 (Ωm ). A
semi-discretization in space of the first two equations in (1) can be defined via a
Galerkin approximation of the spaces Vm by finite dimensional subspaces Vm,h
for m = 1, 2. Then, the semi-discrete approximate problem reads as follows:
Given u0m,h ∈ Vm,h being a suitable approximation of the initial data u0m , find
um,h ∈ Vm,h such that for each t ∈ [t0 , tf ],
Z
αm
Ωm
d
um,h vm,h dx − λm
dt
Z
∆um,h vm,h dx = 0 ∀vm,h ∈ Vm,h , x ∈ Ωm , m = 1, 2,
Ωm
u1,h = u2,h on Γ
(3)
We suppose that the interface Γ does not cut any element (see figure 2).
This implies that a global triangulation of Ω induces the two triangulations of
Ω1 and Ω2 that are compatible on Γ (they share the same nodes on Γ).
It is now useful to consider a global finite dimensionalP
subspace Vh in H01 (Ω).
Let {φj } be a nodal basis of Vh and consequently uh (t) = j uj (t)φj . Therefore,
applying integration by parts to the first equation in (3) in order to remove the
Laplacian operator, we can write the resulting discrete systems in the following
compact form:
4
Figure 2: Splitting of Ω and finite element triangulation.
(1)
(1)
(1)
(1)
(4)
(2)
(2)
(2)
(2)
(5)
M1 u̇I + MIΓ u̇Γ + A1 uI + AIΓ uΓ = 0,
M2 u̇I + MIΓ u̇Γ + A2 uI + AIΓ uΓ = 0.
(1)
(2)
Here, the unknown coefficient functions uI and uI correspond to the
interior nodes on Ω1 and Ω2 respectively and uΓ corresponds to the nodes at
the interface Γ. Am and Mm are the stiffness and the mass matrices for the
interior nodes on Ωm , m = 1, 2 and they are given by
Z
(Am )ij = λm
∇φi ∇φj dx, x ∈ Ωm ,
(6)
φi φj dx, x ∈ Ωm .
(7)
Ωm
Z
(Mm )ij = αm
Ωm
The required data from the interface is inserted in the equations by the
(m)
(m)
matrices AIΓ and MIΓ , m = 1, 2 given by (6) and (7) as well, but with i
running over the interior nodes of Ωm and j over the nodes at Γ.
Finally, if φj is a nodal basis function for a node on Γ we observe that
the normal derivatives in the fourth equation of (1) can be written as linear
functionals using Green’s formula [22, pp. 3]. Thus,
Z
Z
∂um
λm
φj dS = λm
(∆um φj + ∇um ∇φj )dx
Γ ∂nm
Ω
Z
Z m
d
= αm
um φj + λm
∇um ∇φj dx, m = 1, 2.
dt
Ωm
Ωm
(8)
Letting j run over the nodes on Γ we obtain the following compact expression
equivalent to the fourth equation in (1):
5
(2)
(2) (2)
(2)
(2) (2)
(1)
(1) (1)
(1)
(1) (1)
MΓΓ u̇Γ + MΓI u̇I + AΓΓ uΓ + AΓI uI = −MΓΓ u̇Γ − MΓI u̇I − AΓΓ uΓ − AΓI uI .
(9)
(m)
(m)
Here, AΓΓ and MΓΓ are the stiffness and the mass matrix with respect to
the nodes located at Γ and they are given by (6) and (7) with i, j running over
the nodes at Γ. Finally, the required data from domains Ω1 and Ω2 is inserted
(m)
(m)
in the equation (9) by the matrices AΓI and MΓI , m = 1, 2 given by (6) and
(7) with i running over the nodes at Γ and j over the interior nodes of Ωm .
Thus, equation (9) completes the system (4)-(5). We now reformulate the
coupled equations (4), (5) and (9) into an ODE for the vector of unknowns
(1)
(2)
u = (uI , uI , uΓ )T
M̃u̇ + Ãu = 0,
(10)
where

M1

M̃ =  0
(1)
MΓI
2.2
0
M2
(2)
MΓI


(1)
MIΓ
A1


(2)
 , Ã =  0
MIΓ
(1)
(2)
(1)
MΓΓ + MΓΓ
AΓI
0
A2
(2)
AΓI

(1)
AIΓ

(2)
.
AIΓ
(1)
(2)
AΓΓ + AΓΓ
Time Discretization
Applying the implicit Euler method with time step ∆t to the system (10), we
(1),n+1
(2),n+1
get for the vector of unknowns un+1 = (uI
, uI
, un+1
)T
Γ
Aun+1 = M̃un ,
(11)
where

M1 + ∆tA1

A = M̃ + ∆tà = 
0
(1)
(1)
MΓI + ∆tAΓI
(1)
0
M2 + ∆tA2
(2)
(2)
MΓI + ∆tAΓI
(2)
(1)

(1)
(1)
MIΓ + ∆tAIΓ
(2)
(2) 
MIΓ + ∆tAIΓ  ,
MΓΓ + ∆tAΓΓ
(2)
with MΓΓ = MΓΓ + MΓΓ and AΓΓ = AΓΓ + AΓΓ .
3
Fixed Point Iteration
We now employ a standard Dirichlet-Neumann iteration to solve the discrete
system (11). This corresponds to alternately solving the discretized equations
6
of the transmission problem (1) on Ω1 with Dirichlet data on Γ and the discretization of (1) on Ω2 with Neumann data on Γ.
Therefore, from (11) one gets for the k-th iteration the two equation systems
(1),n+1,k+1
(M1 + ∆tA1 )uI
(1)
(1)
(1),n
= −(MIΓ + ∆tAIΓ )un+1,k
+ M1 uI
Γ
(1)
+ MIΓ unΓ ,
(12)
Âûk+1 = M̂un − bk ,
(13)
to be solved in succession. Here,
M2 + ∆tA2
(2)
(2)
MΓI + ∆tAΓI
 =
(2)
(2)
MIΓ + ∆tAIΓ
(2)
(2)
MΓΓ + ∆tAΓΓ
!
, M̂ =
0
(1)
MΓI
(2)
M2
(2)
MΓI
MIΓ
MΓΓ
!
,
and
bk =
0
(1)
(1)
(1),n+1,k+1
(MΓI + ∆tAΓI )uI
(1)
(1)
+ (MΓΓ + ∆tAΓΓ )un+1,k
Γ
(2),n+1,k+1
uI
un+1,k+1
Γ
, ûk+1 =
= unΓ . The iteration is terminated
with some initial condition, here un+1,0
Γ
k+1
according to the standard criterion kuΓ − ukΓ k ≤ τ where τ is a user defined
tolerance [3].
One way to analyze this method is to write it as a splitting method for
(11) and try to estimate the spectral radius of that iteration. However, the
results obtained in this way are much too inaccurate. For that reason, we now
rewrite (12)-(13) as an iteration for un+1
to restrict the size of the space to the
Γ
(1),n+1,k+1
dimension of uΓ . To this end, we isolate the term uI
from (12) and
(2),n+1,k+1
uI
from the first equation in (13):
(1),n+1,k+1
= (M1 + ∆tA1 )−1 (−(MIΓ + ∆tAIΓ )un+1,k
+ M1 uI
Γ
(2),n+1,k+1
= (M2 + ∆tA2 )−1 (−(MIΓ + ∆tAIΓ )un+1,k+1
+ M2 uI
Γ
uI
uI
(1)
(1)
(2)
(2)
(1),n
(1)
+ MIΓ unΓ ),
(14)
(2),n
(2)
+ MIΓ unΓ ).
(15)
Inserting (14) and (15) into the second equation of (13) one obtains the
iteration un+1,k+1
= Σun+1,k
+ ψ n , with iteration matrix
Γ
Γ
Σ = −S(2)
7
−1 (1)
S
,
(16)
!
,
where
(m)
(m)
(m)
(m)
(m)
(m)
S(m) = (MΓΓ + ∆tAΓΓ ) − (MΓI + ∆tAΓI )(Mm + ∆tAm )−1 (MIΓ + ∆tAIΓ ),
(17)
for m = 1, 2 and ψ n contains terms that depend only on the solutions at the
previous time step. Notice that Σ is a discrete version of the Steklov-Poincaré
operator.
Thus, the Dirichlet-Neumann iteration is a linear iteration and the rate of
convergence is described by the spectral radius of the iteration matrix Σ.
4
Semi Discrete Case
Before we present in the next section an analysis for the fully discrete equations just presented, we want to describe previous results which analyze the
behaviour of the Dirichlet-Neumann iteration for the transmission problem in
the semi discrete case. Here, one applies the implicit Euler method for the time
discretization on both equations in (1) but keeps the space continuous. Then,
Henshaw and Chand applied in [12] the Fourier transform in space in order to
transform the second order derivatives into algebraic expressions. This converts
the partial differential equations into a system of purely algebraic equations.
Once we have a coupled system of algebraic equations, we can insert one into
the other and obtain the convergence rate, called amplification factor β in [12].
They then derive the approximation
λ1
β≈
λ2
5
r
D2
.
D1
(18)
Analysis
In this section, we study the iteration matrix Σ for two specific linear FEM
discretizations in 1D and 2D. We will give an exact formula which computes the
convergence rates in 1D and we will propose an approximation for the 2D case.
The behaviour of the rates when approaching both the continuous case in time
and space are also given.
5.1
Specific Analysis in 1D
In this subsection we analyze the iteration matrix for the 1D case. Specifically,
we use Ω1 = [0, 1], Ω2 = [1, 2] and the standard piecewise-linear polynomials
φk (x) :=





x−xk−1
xk −xk−1 ,
xk+1 −x
xk+1 −xk ,
0,
8

if xk−1 < x ≤ xk 

if xk < x ≤ xk+1


otherwise
(19)
as test functions.
T
∈ RN where the only
If we consider ej = 0 · · · 0 1 0 · · · 0
nonzero entry is located at the j-th position, the discretization matrices are
given by

Am =
λm
∆x2
2

 −1



0
−1
0
..
2
..
.
.
..
.
−1


4
1




 , Mm = αm  1

6 

−1 
2
0
4
..
.
0
..
.
..
.
1



,

1 
4
2αm
λm
(m)
, AΓΓ =
, m = 1, 2.
6
∆x2
λ1
λ2
α1
α2
(1)
(2)
(1)
(2)
AIΓ = −
eN , AIΓ = −
e1 , MIΓ =
eN , MIΓ =
e1 ,
2
2
∆x
∆x
6
6
α2 T
λ1 T
λ2 T
α1 T
(2)
(1)
(2)
(1)
e , MΓI =
e , AΓI = −
e , AΓI = −
e .
MΓI =
6 N
6 1
∆x2 N
∆x2 1
(m)
MΓΓ =
(m)
(m)
(m)
where ∆x = 1/(N + 1) and Am , Mm ∈ RN ×N , AIΓ , MIΓ ∈ RN ×1 and AΓI ,
(m)
MΓI ∈ R1×N for m = 1, 2.
Note that the iteration matrix Σ is just a real number in this case and thus
its spectral radius is its modulus. The goal now is to compute S(1) and S(2) .
Inserting the corresponding matrices specified in (17) we have
(1)
S
=
S(2) =
2
α1
λ1
− ∆t
−
eTN (M1 + ∆tA1 )−1 eN
6
∆x2
2
α1
λ1
α1
λ1
1
+ ∆t
−
∆t
=
−
αN
N,
3
∆x2
6
∆x2
(20)
2
α2
λ2
− ∆t
eT1 (M2 + ∆tA2 )−1 e1
6
∆x2
2
α2
λ2
α2
λ2
2
=
+ ∆t
−
− ∆t
α11
,
3
∆x2
6
∆x2
(21)
α1
λ1
+ ∆t
3
∆x2
α2
λ2
+ ∆t
3
∆x2
−
m
where αij
represent the entries of the matrices (Mm + ∆tAm )−1 for i, j =
1, ..., N , m = 1, 2. Observe that the matrices (Mm + ∆tAm ), m = 1, 2 are tridiagonal Toeplitz matrices but their inverses are full matrices. The computation
of the exact inverses is based on a recursive formula which runs over the entries
1
2
[9] and consequently, it is non trivial to compute αN
N and α11 this way.
Due to these difficulties, we propose to rewrite the matrices (Mm +∆tAm )−1 ,
m = 1, 2 in terms of their eigendecomposition:
9
(Mm + ∆tAm )−1
−1
αm ∆x2 − 6λm ∆t 2αm ∆x2 + 6λm ∆t αm ∆x2 − 6λm ∆t
,
,
= tridiag
6∆x2
3∆x2
6∆x2
= VΛ−1
m V, for m = 1, 2.
(22)
where the matrix V has the eigenvectors of Mm + ∆tAm as a columns and
the matrix Λm is a diagonal matrix having the eigenvalues of Mm + ∆tAm as
entries. These are known and given e.g. in [17, pp. 514-516]
2
k=1 sin
λm
j =
1
3∆x2
1
vij = P
N
kπ
N +1
sin
ijπ
N +1
for i, j = 1, ..., N,
2αm ∆x2 + 6λm ∆t + (αm ∆x2 − 6λm ∆t) cos
jπ
N +1
(23)
for j = 1, ..., N m = 1, 2.
2
−1
1
and (M2 +
The entries αN
N and α11 of the matrices (M1 + ∆tA1 )
−1
∆tA2 ) , respectively, are now computed through their eigendecomposition resulting in
iπN
sin2 N
+1
s1
= PN
,
PN
2
2
iπ
sin
(iπ∆x)
sin
i=1
i=1
N +1
PN
1
αN
N =
1
i=1 λ1i
(24)
sin2 Niπ
+1
s2
= PN
,
PN
2
2
iπ
sin
(iπ∆x)
sin
i=1
i=1
N +1
(25)
3∆x2 sin2 (iπ∆x)
2αm ∆x2 + 6λm ∆t + (αm ∆x2 − 6λm ∆t) cos(iπ∆x)
(26)
PN
2
α11
=
1
i=1 λ2i
with
sm =
N
X
i=1
for m = 1, 2. Now, inserting (24) and (25) into (20) and (21) we get for S(1)
and S(2) ,
(1)
S
=
α1 ∆x2 + 3λ1 ∆t
3∆x2
−
(α1 ∆x2 − 6λ1 ∆t)2
s1
,
PN
2
36∆x4
i=1 sin (iπ∆x)
10
(27)
S(2) =
α2 ∆x2 + 3λ2 ∆t
3∆x2
−
(α2 ∆x2 − 6λ2 ∆t)2
s2
.
PN
2
36∆x4
sin
(iπ∆x)
i=1
(28)
With this we obtain an explicit formula for the spectral radius of the iteration
matrix Σ as a function of ∆x and ∆t:
ρ(Σ) = |Σ| = |S(2)
−1 (1)
α2 ∆x2 + 3λ2 ∆t
(α2 ∆x2 − 6λ2 ∆t)2
s1
−
PN
2
4
2
3∆x
36∆x
i=1 sin (iπ∆x)
!
α1 ∆x2 + 3λ1 ∆t
(α1 ∆x2 − 6λ1 ∆t)2
s2
·
.
−
PN
2
3∆x2
36∆x4
i=1 sin (iπ∆x)
=
S |
!−1
(29)
PN
To simplify this, the finite sum i=1 sin2 (iπ∆x) can be computed. We first
rewrite the sum of squared sinus terms into a sum of cosinus terms using the
identity sin2 (x/2) = (1 − cos(x))/2. Then, the resulting sum can be converted
into a geometric sum using Euler’s formula:
N
X
N
1 − ∆x 1 X
cos(2jπ∆x)
−
2∆x
2 j=1
j=1


N
X
1 − ∆x 1
=
− Re 
e2ijπ∆x 
2∆x
2
j=1
!
2iπ∆x
2iN π∆x
e
1−e
1
1 − ∆x 1
− Re
=
=
2iπ∆x
2∆x
2
1−e
2∆x
sin2 (jπ∆x) =
(30)
Inserting (30) into (29) we get after some manipulations
|Σ| =
6∆x(α1 ∆x2 + 3λ1 ∆t) − (α1 ∆x2 − 6λ1 ∆t)2 s1
.
6∆x(α2 ∆x2 + 3λ2 ∆t) − (α2 ∆x2 − 6λ2 ∆t)2 s2
(31)
We could not find a way of simplifying the finite sum (26) because ∆x
depends on N (i.e., ∆x = 1/(N + 1)). However, (31) is a computable formula
that gives exactly the convergence rates of the Dirichlet-Neumann iteration for
given ∆x, ∆t, αm and λm , m = 1, 2.
We are now interested in the asymptotics of (31). In particular, we want
to know the behaviour of (31) when ∆t or ∆x tend to 0, so that we can relate this to the results of the semidiscrete analysis in section 4. However, the
denominator of (26) becomes zero when ∆x tends to 0. To solve this problem,
we reformulate (31) in terms of c := ∆t/∆x2 . Figure 2 illustrates the relation
between computing c → 0 or ∆t → 0 and c → ∞ or ∆x → 0.
11
1
c=∞
c=1
c=0
0.9
0.8
0.7
∆x
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
∆t
0.6
0.7
0.8
0.9
1
Figure 3: Relation between computing c → 0 or ∆t → 0 and c → ∞ or ∆x → 0.
From figure 3 one can observe that for a fixed ∆x, if we choose a ∆t that
enforces the condition ∆t << ∆x2 (i.e, c → 0), we approach the green line.
Similarly, for a fixed ∆t, if we choose a ∆x that enforces the condition ∆t >>
∆x2 (i.e, c → ∞), we approach the blue line.
Thus, we multiply (31) by 1/∆x2 both in the numerator and denominator
and get
6∆x(α1 + 3λ1 c) − (α1 − 6λ1 c)2 s01
6∆x(α2 + 3λ2 c) − (α2 − 6λ2 c)2 s02
|Σ| =
(32)
where
s0m =
N
X
i=1
3 sin2 (iπ∆x)
2αm + 6λm c + (αm − 6λm c) cos(iπ∆x)
(33)
for m = 1, 2.
Finally, computing the limits of (32) when c → 0 and c → ∞ we get
lim |Σ| =
c→0
3 sin2 (iπ∆x)
i=1 α1 (2+cos(iπ∆x))
PN
sin2 (iπ∆x)
α22 i=1 α23(2+cos(iπ∆x))
6∆xα1 − α12
6∆xα2 −
PN
PN
α1 6∆x − i=1
=
PN
α2 6∆x − i=1
3 sin2 (iπ∆x)
2+cos(iπ∆x)
3 sin2 (iπ∆x)
=
2+cos(iπ∆x)
(34)
lim |Σ| = lim
c→∞
c→∞
3 sin2 (iπ∆x)
i=1 6λ1 c(1−cos(iπ∆x))
PN
sin2 (iπ∆x)
(6λ2 c)2 i=1 6λ23c(1−cos(iπ∆x))
6∆x3λ1 c − (6λ1 c)2
6∆x3λ2 c −
PN
λ1 6∆x − 6 i=1
=
PN
λ2 6∆x − 6 i=1
12
PN
3 sin2 (iπ∆x)
1−cos(iπ∆x)
λ
= 1 =: δ.
3 sin2 (iπ∆x)
λ2
1−cos(iπ∆x)
(35)
α1
=: γ,
α2
From the results obtained in (34) and (35) we can observe that strong jumps
in the physical properties of the materials placed in Ω1 and Ω2 will imply fast
convergence. This is the case when modelling thermal fluid structure interaction,
where often a fluid with low thermal conductivity and density is coupled with
a structure having higher thermal conductivity and density.
On the other hand, in the domain decomposition context, the coupling iteration will be slow because the coefficients αm and λm , m = 1, 2 are typically
continuous across subdomains.
Finally, we can also observe that the result obtained in (35) does not agree
with the semidiscrete analysis described in (18). However, the semidiscrete
analysis was performed in 2D and the discrete analysis in 1D.
5.2
Specific Analysis in 2D
In this section we extend the analysis presented in the previous section to the
2D case. The subdomains are here Ω1 = [0, 1] × [0, 1] and Ω2 = [1, 2] × [0, 1]. An
equidistant grid is chosen i.e, ∆x = ∆y = 1/(N +1). For the FEM discretization,
we use triangular elements distributed as sketched in figure 4 and the following
pyramidal test functions
φk (x, y) =



















x+y
∆x − 1,
y
∆x ,
∆x−x
∆x ,
1 − x+y
∆x ,
∆x−y
∆x ,
x
∆x ,
if x = (x, y) ∈ Region 1,
if
x ∈ Region 2,
if
x ∈ Region 3,
if
x ∈ Region 4,
if
x ∈ Region 5,
if
x ∈ Region 6,
otherwise.
0,
5
6
4
6
1
2
3
6
6
5
1
4
2
5
4
1
3
5
1
3
26 5
(36)
4
2
3
4
3
1
2
Figure 4: Sketch of the regions for the pyramidal test functions defined in (39).
The discretization matrices are given in this case by
13

Am
λm
=
∆x2
B
−I

 −I



0
B
..
.
0
..


4




 where B =  −1


..

. −I 
0
−I B
.
−1
4
..
.
0
..
.
..
.
−1





−1 
4
and I ∈ RN ×N is an identity matrix. Note that each block of the matrices
2
2
Am ∈ RN ×N has size N × N .

N
N2
0




 N1 N . . .


Mm = αm 


..
..

.
. N2 
0
N1 N

−1/12
0
0

.
..
 1/4
−1/12
N1 = 

..
..

.
.
0
0
1/4 −1/12
5/6
−1/12

0


..

 −1/12
.
5/6
,
where N = 


..
..

.
.
−1/12 
0
−1/12
5/6



−1/12
1/4
0



.



0
−1/12 . .
.
 , and N2 = 



.
.
..
..


1/4 
0
0 −1/12
2
2
Each block of the matrices Mm ∈ RN ×N has size N × N as well. As
T
2
before, we consider here Ej = 0 · · · 0 I 0 · · · 0
∈ RN ×N where
the only nonzero block is the j-th block of size N × N . Thus,
(1)
AIΓ = −
λ1
λ2
(2)
(1)
(2)
EN , AIΓ = −
E1 , MIΓ = α1 EN N1 , MIΓ = α2 E1 N2 ,
2
∆x
∆x2

(m)
MΓΓ
5/12

 −1/24
= αm 


0
(m)
−1/24
5/12
..
.
0
..


2
−1/2




 , A(m) = λm  −1/2
ΓΓ

2 
..
∆x

.
−1/24 
−1/24 5/12
0
.
(m)
where MΓΓ and AΓΓ ∈ RN ×N for m = 1, 2.
(1)
AΓI = −
λ1 T
λ2 T
(2)
(1)
(2)
E , AΓI = −
E , MΓI = α1 ETN N1 , MΓI = α2 ET1 N2 ,
∆x2 N
∆x2 1
for m = 1, 2.
In the two-dimensional case, the iteration matrix Σ is a matrix of size N ×N .
As in the 1D case, one computes S(1) and S(2) by inserting the corresponding
matrices specified above in (17) obtaining
14
2
..
.
0
..
..
.
.
−1/2



,

−1/2 
2
(1)
(1)
(1)
(1)
(1)
(1)
(2)
(2)
(2)
(2)
(2)
(2)
S(1) = (MΓΓ + ∆tAΓΓ ) − (MΓI + ∆tAΓI )(M1 + ∆tA1 )−1 (MIΓ + ∆tAIΓ )
1
1 5
1
λ1 ∆t
1
tridiag − , 2, −
= α1 tridiag − , , −
+
24 12 24
∆x2
2
2
λ
∆t
λ1 ∆t
1
−1
T
I EN (M1 + ∆tA1 ) EN α1 N1 −
I
− α1 N1 −
∆x2
∆x2
(37)
S(2) = (MΓΓ + ∆tAΓΓ ) − (MΓI + ∆tAΓI )(M2 + ∆tA2 )−1 (MIΓ + ∆tAIΓ )
1 5
1
λ2 ∆t
1
1
+
tridiag − , 2, −
= α2 tridiag − , , −
24 12 24
∆x2
2
2
λ2 ∆t
λ2 ∆t
−1
T
− α2 N2 −
I E1 (M2 + ∆tA2 ) E1 α2 N2 −
I
∆x2
∆x2
(38)
In this case the iteration matrix Σ is not easy to compute for several reasons.
First of all, the matrices M1 +∆tA1 and M2 +∆tA2 are sparse block tridiagonal
matrices, and consequently, their inverses are not straight forward to compute.
A block-by-block algorithm for inverting a block tridiagonal matrix is explained
in [20]. However, the algorithm is based on the iterative application of the Schur
complement [23], and it results in a sequence of block matrices and inverses of
block matrices that we did not find possible to compute exactly. Moreover, the
diagonal blocks of M1 +∆tA1 and M2 +∆tA2 are tridiagonal but their inverses
are full matrices [9].
Due to these difficulties, we propose here to approximate Σ. One can observe
that M1 +∆tA1 and M2 +∆tA2 are strictly diagonally dominant matrices, and
therefore, we propose to approximate them by their block diagonal. The same
reasoning is used to approximate the diagonal block matrices of M1 + ∆tA1
and M2 + ∆tA2 by their diagonal. Thus,
S(1) ≈
(2)
S
≈
2 −1
2λ1 ∆t
λ1 ∆t
4λ1 ∆t
5α1
α1
5α1
+
+
+
I
−
I
12
∆x2
12
∆x2
6
∆x2
2(5α1 ∆x2 + 24λ1 ∆t)2 − (α1 ∆x2 + 12λ1 ∆t)2
=
I
24∆x2 (5α1 ∆x2 + 24λ1 ∆t)
(39)
2 −1
5α2
2λ2 ∆t
α2
λ2 ∆t
5α2
4λ2 ∆t
+
I−
+
+
I
12
∆x2
12
∆x2
6
∆x2
2(5α2 ∆x2 + 24λ2 ∆t)2 − (α2 ∆x2 + 12λ2 ∆t)2
=
I
24∆x2 (5α2 ∆x2 + 24λ2 ∆t)
(40)
Thus, we obtain an estimate of the spectral radius of the iteration matrix Σ:
15
ρ(Σ) ≈
(5α2 ∆x2 + 24λ2 ∆t)(2(5α1 ∆x2 + 24λ1 ∆t)2 − (α1 ∆x2 + 12λ1 ∆t)2 )
=: σ
(5α1 ∆x2 + 24λ1 ∆t)(2(5α2 ∆x2 + 24λ2 ∆t)2 − (α2 ∆x2 + 12λ2 ∆t)2 )
(41)
Furthermore, computing the limits of (41) when ∆t → 0 and ∆x → 0 we get
lim ρ(Σ) ≈
5α2 ∆x2 (2(5α1 ∆x2 )2 − (α1 ∆x2 )2 )
α1
=
=: γ,
5α1 ∆x2 (2(5α2 ∆x2 )2 − (α2 ∆x2 )2 )
α2
(42)
lim ρ(Σ) ≈
24λ2 ∆t(2 · 242 λ21 ∆t2 − 122 λ21 ∆t2 )
λ1
=: δ.
=
24λ1 ∆t(2 · 242 λ22 ∆t2 − 122 λ22 ∆t2 )
λ2
(43)
∆t→0
∆x→0
Thus, the asymptotic behavior of the convergence rates for the 2D case is
consistent with the 1D case.
However, the asymptotics for the discrete case when one approaches the
continuous case in space does not depend on the thermal diffusivities D1 and
D2 as the semidiscrete analysis observes in (18). We will come back to this
point in the following section.
6
Numerical Results
In this section we present a set of numerical experiments designed to show how
(31) computes the convergence rates in the 1D case and the validity of (41) as
an estimator for the rates in the 2D case of the coupled problem formulated
above. We also show that the theoretical asymptotics deduced both in (34) and
(35) and in (42) and (43) match with the numerical experiments.
We first compare |Σ| for the 2D case with β in (18) and the observed convergent rates for different thermal diffusivities and conductivities D1 , D2 , λ1 and
λ2 . Input data and results for the different cases are shown in table 1.
From table 1, one can observe that the semidiscrete estimator β and ρ(Σ)
match for the specific choice of ∆t = 0.01 and ∆x = 1/320. However, the limit
of the discrete estimator (41) when ∆x → 0, i.e, δ, does not agree with β. Case
A in table 1 was introduced to visualize this result. One observes that β does
not estimate the convergence rates for the choice ∆t = 100 and ∆x = 1/10.
This result lead us to the conclusion that the semidiscrete estimator β does
not represent the asymptotics of the discrete estimator (43) approaching the
continuous case in space.
Figures 5 and 6 show the cases 1 and 2 specified in table 1 for 1D and 2D,
respectively. The circles correspond to ρ(Σ), the crosses to the experimental
convergence rates and the dashed line to the corresponding asymptotic (γ when
∆t → 0 and δ when ∆x → 0). Note that both the 1D and 2D cases have a
similar behavior. However, the convergence rates are computed exactly in the
1D case and estimated in the 2D case. We observe from the 2D plots that the
16
Table 1: The first four columns contain the input parameters for the different
test cases. β and ρ(Σ) are the semidiscrete and the discrete estimator respectively (in 2D). Moreover, β and ρ(Σ) have been computed for ∆t = 0.01 and
∆x = 1/320 in cases 1-4 and for ∆t = 100 and ∆x = 1/10 for case A. Finally,
γ and δ are the resulting limits when approaching the continuous case in time
and space of the discrete estimator.
Case
1
2
3
4
A
D1
1
0.5
1
0.5
0.5
D2
0.5
1
0.5
1
1
λ1
0.3
0.3
1
1
0.3
λ2
1
1
0.3
0.3
1
β
0.21
0.42
2.36
4.71
0.42
−0.7
δ
0.3
0.3
3.33
3.33
0.3
|Σ|
Conv. Rate
γ
−0.2
log
log
γ
0.15
0.6
1.67
6.67
0.6
−0.15
|Σ|
Conv. Rate
γ
−0.75
−0.8
−0.85
−0.9
−6
ρ(Σ)
0.21
0.42
2.36
4.71
0.301
−0.25
−0.3
−5.5
−5
−4.5
−4
−6
log(∆t)
−5.5
−5
−4.5
−4
log(∆t)
(a) Case 1.
(b) Case 2.
Figure 5: Cases 1 and 2 from table 1 in the 1D case. The circles correspond to
|Σ|, the crosses to the experimental convergence rates and the dashed line to γ.
The curves are restricted to the discrete values ∆t = 1e−4/50, 2·1e−4/50, ..., 50·
1e − 4/50 and ∆x = 1/20. One observes how γ describes the behaviour of the
convergence rates when we enforce the condition ∆t/∆x2 << 1.
approximation predicts the convergence rates quite well because the difference
with respect to the experimental rates is really small.
Now we want to test if the convergence rates when approaching the continuous case in space have dependency on the thermal diffusivities D1 and D2 as
predicted in [12] or they do not as computed in our analysis. For that, figure 7
shows the cases 1 and 2 from table 1 for the 1D (the 2D case has a very similar
behavior). Here, the thermal conductivities λ1 and λ2 are the same in both
plots but the thermal diffusivities are switched (meaning that D1 in case 1 corresponds to D2 in case 2 and D2 in case 1 corresponds to D1 in case 2). We can
observe that the asymptotics of the convergence rates do not vary in both plots.
This result is consistent with [18] where a similar behaviour was observed for
17
−0.7
σ
Conv. Rate
γ
−0.2
log
−0.75
log
−0.15
σ
Conv. Rate
γ
−0.8
−0.85
−0.25
−0.3
−0.9
−6
−5.5
−5
−4.5
−4
−6
−5.5
−5
log(∆t)
−4.5
−4
log(∆t)
(a) Case 1.
(b) Case 2.
Figure 6: Cases 1 and 2 from table 1 in the 2D case. The circles correspond
to σ, the crosses to the experimental convergence rates and the dashed line to
γ. The curves are restricted to the discrete values ∆t = 1e − 4/50, 2 · 1e −
4/50, ..., 50 · 1e − 4/50 and ∆x = 1/20. As in the 1D case, here one observes
how γ describes the behaviour of the convergence rates when ∆t → 0.
the 2D version of the coupled unsteady transmission problem discretized with
finite differences. Finally, observe that the convergence rate does not vary a lot
when we decrease ∆x. For a farly large choice of ∆x (for instance ∆x = 1/10),
the convergence rates are already quite close to δ.
−0.5
−0.5
−0.6
−0.6
−0.7
−0.7
log
−0.4
log
−0.4
−0.8
−0.8
−0.9
−1
−1.1
−1.1
−0.9
|Σ|
Conv. Rate
δ
−1
−0.9
−0.8
−0.7
|Σ|
Conv. Rate
δ
−1
−1.1
−1.1
−0.6
log(∆x)
−1
−0.9
−0.8
−0.7
−0.6
log(∆x)
(a) Case 1.
(b) Case 2.
Figure 7: Cases 1 and 2 from table 1 in the 1D case. The circles correspond
to |Σ|, the crosses to the experimental convergence rates and the dashed line to
γ. The curves are restricted to the discrete values ∆x = 1/3, 1/4, ..., 1/10 and
∆t = 100. One observes how δ describes the behaviour of the convergence rates
when we enforce the condition ∆t/∆x2 >> 1.
Notice that the method does not converge for some cases. Nevertheless, |Σ|
describes the convergence rate and γ and δ their asymptotics. To show this,
figures 8 and 9 show the cases 3 and 4 of table 1 for the 1D and 2D cases
18
respectively. We have plotted the convergence rates with respect to ∆t. Here
one keeps the same values for D1 and D2 and varies λ1 and λ2 .
|Σ|
Conv. Rate
γ
0.85
log
log
0.3
0.25
0.2
0.15
−6
|Σ|
Conv. Rate
γ
0.8
0.75
−5.5
−5
−4.5
0.7
−6
−4
−5.5
log(∆t)
−5
−4.5
−4
log(∆t)
(a) Case 3.
(b) Case 4.
Figure 8: Cases 3 and 4 from table 1 in the 1D case. The curves are restricted to
the discrete values ∆t = 1e − 4/50, 2 · 1e − 4/50, ..., 50 · 1e − 4/50 and ∆x = 1/20.
γ describes the behaviour of the convergence rates when ∆t → 0.
σ
Conv. Rate
γ
0.85
log
log
0.3
0.25
0.2
0.15
−6
σ
Conv. Rate
γ
0.8
0.75
−5.5
−5
−4.5
0.7
−6
−4
log(∆t)
−5.5
−5
−4.5
−4
log(∆t)
(a) Case 3.
(b) Case 4.
Figure 9: Cases 3 and 4 from table 1 in the 2D case. The curves are restricted to
the discrete values ∆t = 1e − 4/50, 2 · 1e − 4/50, ..., 50 · 1e − 4/50 and ∆x = 1/20.
γ describes the behaviour of the convergence rates when ∆t → 0.
Similary, δ predicts the asymptotics of cases 3 and 4 when we plot the
convergence rates with respect to ∆x as one can check in figure 9 for the 1D
case (2D case behaves very similar).
Before ending this section, we want to present three real data examples. We
consider here the thermal interaction between air at 273K with steel at 900K,
water at 283K with steel at 900K and air at 273K with water at 283K. Physical
properties of the materials and resulting asymptotics for these three cases are
shown in table 2 and 3 respectively.
Figures 11, 12 and 13 show the convergence rates in 1D for the interactions
19
0.6
0.6
|Σ|
Conv. Rate
δ
0.58
|Σ|
Conv. Rate
δ
0.58
log
0.56
log
0.56
0.54
0.54
0.52
0.52
0.5
−1.1
−1
−0.9
−0.8
−0.7
0.5
−1.1
−0.6
−1
−0.9
log(∆x)
−0.8
−0.7
−0.6
log(∆x)
(a) Case 3.
(b) Case 4.
Figure 10: Cases 3 and 4 from table 1 in the 1D case. The curves are restricted
to the discrete values ∆x = 1/3, 1/4, ..., 1/10 and ∆t = 100. δ describes the
behaviour of the convergence rates when ∆x → 0.
Table 2: Physical properties of the materials. λ is the thermal conductivity, ρ
the density, C the specific heat capacity and α = ρC.
Material
Air
Water
Steel
λ (W/mK)
0.0243
0.58
48.9
ρ (kg/m3 )
1.293
999.7
7836
C (J/kgK)
1005
4192.1
443
α (J/K m3 )
1299.5
4.1908e6
3471348
Table 3: The limits of the convergence rates when ∆t → 0 (γ) and ∆x → 0 (δ).
Case
Air-Steel
Water-Steel
Air-Water
γ
3.7434e-4
1.2073
3.1008e-4
δ
4.9693e-4
0.0119
0.0419
between air-steel, water-steel and air-water respectively. On the left we always
plot the rates with respect to the variation of ∆t and for a fixed ∆x. On the
right we plot the behaviour of the rates for a fixed ∆t and varying ∆x.
From figure 11 and 13 we can observe that the convergence rates are really
fast (factor of ∼ 1e − 4) when there exist strong jumps in the coefficient of
the materials. For instance, when performing the thermal coupling between air
and steel the Dirichlet-Neumann iteration only needs two iterations to achieve
a tolerance of 1e − 10.
Finally, the thermal coupling between water and steel shows us the importance of choosing carefully the spatial resolution (∆x) and timestep (∆t) for
20
−3
−3.38
|Σ|
Conv. Rate
γ
−3.39
|Σ|
Conv. Rate
δ
−3.2
−3.4
log
log
−3.4
−3.41
−3.6
−3.42
−3.8
−3.43
−1
−0.5
0
0.5
−4
1
−1.6
−1.4
−1.2
−1
−0.8
−0.6
log(∆x)
log(∆t)
(a) The curves are restricted to the discrete values ∆t = 10/40, 2 · 10/40, ..., 40 ·
10/40 and ∆x = 1/20.
(b) The curves are restricted to the discrete values ∆x = 1/3, 1/4, ..., 1/30 and
∆t = 1e8.
Figure 11: Air-Steel thermal interaction with respect ∆t on the left and ∆x on
the right.
−1.5
0.1
|Σ|
Conv. Rate
δ
0.08
0.06
log
log
0.04
−2
0.02
0
|Σ|
Conv. Rate
γ
−0.02
−0.04
−1
−0.5
0
0.5
−2.5
1
−1.6
−1.4
−1.2
−1
−0.8
−0.6
log(∆x)
log(∆t)
(a) The curves are restricted to the discrete values ∆t = 10/40, 2 · 10/40, ..., 40 ·
10/40 and ∆x = 1/20.
(b) The curves are restricted to the discrete values ∆x = 1/3, 1/4, ..., 1/30 and
∆t = 1e8.
Figure 12: Water-Steel thermal interaction with respect ∆t on the left and ∆x
on the right.
solving our problem. If we choose a pair ∆t, ∆x such that ∆t/∆x2 ∼ 0, the
rates will behave as γ and the numerical method will be divergent as is shown
in figure 12a. However, if we choose a pair ∆t, ∆x such that ∆t/∆x2 ∼ ∞, the
rates will behave as δ and the numerical method will converge fast as is shown
in figure 12b.
7
Summary and Conclusions
We have described the Dirichlet-Neumann iteration for the coupling of two heat
equations on two identical domains. In particular, the coupled PDE were dis21
−1
−3.3
−3.35
|Σ|
Conv. Rate
γ
|Σ|
Conv. Rate
δ
−1.2
−1.4
log
log
−3.4
−3.45
−1.6
−3.5
−1.8
−3.55
−1
−0.5
0
0.5
−2
1
−1.6
−1.4
−1.2
−1
−0.8
−0.6
log(∆x)
log(∆t)
(a) The curves are restricted to the discrete values ∆t = 10/40, 2 · 10/40, ..., 40 ·
10/40 and ∆x = 1/20.
(b) The curves are restricted to the discrete values ∆x = 1/3, 1/4, ..., 1/30 and
∆t = 1e8.
Figure 13: Air-Water thermal interaction with respect ∆t on the left and ∆x
on the right.
cretized into a system of algebraic equations. Afterwards, a fixed point iteration
was performed and the iteration matrix was found in both 1D and 2D cases. An
exact formula describing the convergence rates is derived for the 1D case and
an approximation is proposed for the 2D case. Finally, the limits of the convergence rates when approaching the continuous case either in space (δ := λ1 /λ2 )
or time (γ := α1 /α2 ) are computed. In the numerical results, we have presented
four different test cases and three real data cases which show how the computed
asymptotics predict the behaviour of the convergence rates.
From the first four test cases we conclude that ∆x does not strongly affect the
convergence rates. However, they are affected by ∆t. Moreover, our results show
that when approaching the continuous case in space (∆x → 0) the convergence
rates do not depend on the thermal diffusivities D1 and D2 as predicted in [12]
for the semidiscrete case. We found the analysis in [12] to be correct and are not
sure where the discrepancy comes from, this is subject of further investigation.
From the last three real data cases we conclude that having strong jumps in
the coefficients of the coupled PDEs will imply fast convergence. We have also
shown how the performance of the coupled method can be affected by carefully
choosing the pair ∆x, ∆t and the relation between them.
Finally, in the domain decomposition context, the coupling will be slow
because the coefficients αm and λm , m = 1, 2 are kept continuous over all the
subdomains, and therefore, γ, δ ∼ 1.
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