On the Performance of Certain Iterative Solvers for Coupled Systems Arising in Discretization of Non-Newtonian Flow Equations

On the Performance of Certain

Iterative Solvers for Coupled Systems

Arising in Discretization of

Bericht 62 (2004)

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Prof. Dr. Dieter Prätzel-Wolters

Institutsleiter

O. Nevanlinna and R. Rannacher (assoc. eds.) Jyv¨askyl¨a, 24–28 July 2004

ON THE PERFORMANCE OF CERTAIN ITERATIVE

SOLVERS FOR COUPLED SYSTEMS ARISING IN

DISCRETIZATION OF NON-NEWTONIAN FLOW

EQUATIONS

Oleg Iliev

, Joachim Linn

, Mathias Moog

†

, Dariusz Niedziela

and Vadimas

Starikovicius

_{Fraunhofer Institute for Industrial Mathematics (ITWM),}

Gottlieb-Daimler-Str., Geb.49, D-67663 Kaiserslautern, Germany

e-mails: [email protected], [email protected], [email protected], [email protected],

web page: http://www.itwm.fhg.de

†

_{MAGMA Giessereitechnologie GmbH}

Kackertstr. 11 D-52072 Aachen, Germany

e-mail: [email protected]

Key words: Performance of iterative solvers, Preconditioners, Non-Newtonian flow.

Abstract.

Iterative solution of large scale systems arising after discretization and

lin-earization of the unsteady non-Newtonian Navier–Stokes equations is studied. cross WLF

model is used to account for the non-Newtonian behavior of the fluid. Finite volume

method is used to discretize the governing system of PDEs. Viscosity is treated explicitely

(e.g., it is taken from the previous time step), while other terms are treated implicitly.

Different preconditioners (block–diagonal, block–triangular, relaxed incomplete LU

factor-ization, etc.) are used in conjunction with advanced iterative methods, namely, BiCGStab,

CGS, GMRES. The action of the preconditioner in fact requires inverting different blocks.

For this purpose, in addition to preconditioned BiCGStab, CGS, GMRES, we use also

algebraic multigrid method (AMG). The performance of the iterative solvers is studied

with respect to the number of unknowns, characteristic velocity in the basic flow, time

step, deviation from Newtonian behavior, etc. Results from numerical experiments are

presented and discussed.

1

INTRODUCTION

Efficient numerical simulation of non-Newtonian flows is a non-trivial task. While the

numerical methods for Newtonian flows are, in general, well studied, this is not the case for

the non-Newtonian fluids. Here we discuss numerical solution of a class of flow problems

of so called generalized Newtonian fluids. In this case the viscosity µ depends on the

temperature and on the rate of the strain tensor (in certain cases also on the pressure):

µ = µ(T,

| ˙γ| , p). As a result, the momentum equations are strongly coupled through their

viscous terms (recall, that for Newtonian fluids the coupling of momentum equations is

only through the convective terms and the pressure). In the Newtonian case, projection

methods for decoupling momentum and continuity equations are often used (for details

and further references on fractional time step projection methods of Chorin type, or on

SIMPLE-type algorithms, see, e.g., [12, 18, 10, 9]). Decoupling methods might be not

efficient in the non-Newtonian case when the momentum equations are strongly coupled.

An alternative of the segregated (decoupled) solvers are the so called coupled solvers (see,

e.g., [18, 7, 8, 16, 2]). In this case momentum and (transformed) continuity equation

are solved together. In the current paper we study the performance of certain iterative

solvers for solving such coupled systems, arising in discretization of non-Newtonian flow

equations.

The remainder of the paper is organized as follows. Next section presents the governing

equations. The discretization is discussed in the third section. Iterative methods and

preconditioners for solving coupled system of equations are explained in the fourth section.

Fifth section is devoted to numerical experiments and their analysis. Finally, conclusions

are drawn in the last section.

2

GOVERNING EQUATIONS

Consider unsteady non-isothermal Navier-Stokes equations describing weakly

com-pressible flow of a liquid with variable viscosity (e.g., fluids described by cross WLF

model).

Momentum equations are written as :

∂(ρu

_i

)

∂t

+ div(ρu

i

u) =

−

∂p

∂x

_i

−

2

3

∂

∂x

_i

(µdiv(u)) + div(2µ ˙γ

i

)

(1)

for the velocity vector components, u

_i

, i = 1, 2, 3. Here ρ is the density, µ is the viscosity,

and

˙γ =

1

2

(

∇u + (∇u)

T

_).

₍₂₎

Continuity equation is written as:

∂ρ

Energy equation, written with respect to enthalpy h, is:

∂(ρ h)

∂t

+ (

u,

∇T ) = div(κ ∇T ) + µ Φ

V

+ L

∂f

_s

∂t

.

(4)

The last two terms account for the dissipative heating and for latent heat of the phase

change, respectively.

Equations of state for density and for viscosity have to be added to the above system.

The density depends on the pressure and on the temperature:

ρ = ρ(p, T ).

(5)

The viscosity depends on the temperature and on the rate of the strain tensor.

Ac-counting for a pressure dependence is also desirable in certain cases.

µ = µ(T,

| ˙γ| , p).

(6)

Particular rheology model and pvT model can be selected from the SIGMASOFT c



material database (see www.sigmasoft.de).

The unknowns, to be determined from the above system of equations, are temperature,

velocity vector, density, pressure and viscosity:

T, 

u, ρ, p, µ.

The above system describes, in particular, flow and solidification of liquid polymer.

Simulation of polymer moulding and solidification is of special interest for us, and the

main aim of our investigations is to accelerate the flow solver in SIGMASOFT c

. In

fact, a relatively small time step has to be used in simulating polymer solidification due

to the existing free boundaries during the filling of the mold and during solidification.

The most CPU consuming part of the calculations is the simulation of the flow at each

time step. Motivated by this, below we consider the isothermal case which represents the

main difficulties to be overcome. Denote by C

_u

and D

_u

the operators corresponding to

the convective and to the viscous terms in the momentum equations. Obviously, C

_u

=

C

_u

(

u), D

_u

= D

_u

(µ(T,

| ˙γ| , p)) . Further, denote by G and B the operators corresponding

to the gradient and divergence. In these notations we can write Navier-Stokes equations

as follows:

∂(ρ

u)

∂t

+ C

u



u

− D

u



u + G p = 0,

(7)

B 

u = 0.

(8)

Note, that the weak compressibility does not play essential role in the liquid polymer

flows we consider, therefore we will not pay special attention to it. Although we solve

the weakly compressible equations, here we will discuss the incompressible case, what is

enough for our purposes. For a more detailed study on the methods for weakly

compress-ible flows (in the case when this is important) we refer, e.g., to [6, 5, 9] and references

therein.

3

Discretization

Finite volume method on a staggered grid is applied for discretizing in space the above

system of equations (see, e.g., [13] or [10]). Particular form of this space discretization

is not discussed here, instead we concentrate on discretiaztion in time and later on, on

iterative methods for solving the discretized system.

3.1

Fractional time step projection methods

Let us shortly consider fractional time step methods which suggest solving the

momen-tum equations in two steps. Projection methods (e.g., Chorin type methods, see, e.g.,

[12] for a detailed discussion) treat pressure explicitely at the first step, what allows to

decouple momentum and continuity equations (the last is used for obtaining an equation

with respect to the pressure or to the pressure correction). A variant of such a projection

method looks as follows:

(ρ

u)

n+12

_{+ τ}

C

_u



u

n+12

− D

_u

_

_u

n+12



=(ρ 

u)

n

− τ G p

n

,

(ρ

u)

n+1

− (ρu)

n+12

₌

−τ



_{G p}

n+1

− G p

n



_,

B (ρ

u)

n+1

=0.

Here we use the superscript

n

to denote the values at the old time level, and

n+1

to

denote the values at the new time level. Notation τ stands for the time step, τ = t

n+1

−t

n

.

The discrete operators are denoted by the same notations as the continuous ones. Viscosity

in the diffusive terms and velocity in the convective terms are treated explicitely.

The sum of the first and the second equations above gives an discretization of

momen-tum equations. Applying divergence operator (in our case denoted by B) to the second

equation, and using the third (continuity) equation, we obtain

τ BGδp =

−B(ρu)

n+1

+ B(ρ

u)

n+12

_{= B(ρ}

_u)

n+12

_,

where δp = p

n+1

− p

n

. The result of the above Chorin type discretization is a

Poisson-type equation for the pressure correction, which is decoupled from the momentum

equa-tions. There exist extensive mathematical literature, concerning first and second order

fractional time step discretization, incremental and non-incremental form of equations,

stability, splitting of the boundary conditions, etc. Some recent results, as well as further

references, can be found, e.g., in [4, 14, 1]. Note the Newtonian case, the discretized in

this way momentum equations are not coupled, and can be solved consecutively. This is

not the case for the non-Newtonian fluids. An attempt to discretize explicitly off-diagonal

viscous terms will lead to severe restriction on the time step.

3.2

Fractional time step coupled method

In the coupled methods the pressure is treated implicitly in the momentum equations.

Consider such an approach.

(ρ

u)

n+12

_{=(ρ }

_u)

n

− τC

_u

_

_u

n

− τ G p

n

_,

(ρ

u)

n+1

− (ρu)

n+12

_{= τ}



_D

_u

_

_u

n+1

− G p

n+1

_{+ G p}

n



_,

B (ρ

u)

n+1

=0.

Again a Poisson type equation for the pressure correction can be obtained from the

second and the third equations, but in this case the system remains coupled, i.e. the

second and the transformed third equation have to be solved simultaneously. In fact, in

this case we decouple only convective and viscous transport, but keep coupled viscous and

pressure forces.

An interesting fractional time step discretization is suggested in [19], but it is not

discussed here.

3.3

Implicit discretization

If an implicit discretization is used instead of the fractional time step approach, we

obtain

(ρ

u)

n+1

+ τ

C

_u



u

n+1

− D

_u



u

n+1



=(ρ 

u)

n

− τ G p

n+1

,

B (ρ

u)

n+1

=0.

4

Iterative methods for the discretized system

In this section we consider the iterative methods for solving the coupled system of

discretized equations. Such a system arises after the implicit discretization, or after the

fractional time step coupled discretization.

With an obvious change of notations we

rewrite the large scale system of linear algebraic equations to be solved at each time step

t = t

n+1

as follows:



A τ G

B

0

 



u

p



=



S

_u

0



.

(9)

We will use also notations

Lv = b

with v = (

u, p)

t

.

4.1

Iterative projection methods

Iterative projection methods like SIMPLE are discussed in this subsection.

Good

systematization and detailed references for these methods can be found, e.g., in the book

of Turek [18]. One way to derive such methods in the pure incompressible case is as

following. Solving the first equation gives us



u = A

−1

(S

_u

− τGp) .

(10)

Substituting in the second equation, we obtain

τ BA

−1

Gp = BA

−1

S

_u

.

(11)

Because A

−1

is a full matrix, direct solving of the above equation is not possible.

Instead, an iterative procedure (preconditioned Richardson method) can be written as

p

i+1

= p

i

− M

−1

τ BA

−1

Gp

i

− BA

−1

S

_u



(12)

This means that at each iteration we solve

M δp =

−

τ BA

−1

Gp

i

− BA

−1

S

_u



(13)

The preconditioner should be spectrally close to τ BA

−1

G, and at the same time easily

invertable. The usual choice is

M = τ BD

−1

G,

where D is a diagonal matrix, so that M is identical to discretization of a second order

elliptic operator. For example, in SIMPLE the choice is

D = diagA.

(14)

(i) Constant viscosity:

A =

⎛

⎜

⎝

A

11

0

0

0

A

22

0

0

0

A

33

⎞

⎟

⎠

.

(ii) Variable viscosity:

A =

⎛

⎜

⎝

A

11

A

12

A

13

A

21

A

22

A

23

A

31

A

32

A

33

⎞

⎟

⎠

.

The above representation shows that in the case of strong off-diagonal blocks the

ap-proximation of A with an a diagonal matrix D should not be good. It should be noted

that even in the Newtonian case (block-diagonal matrix A) the decoupling approach might

not be successful - an example are flow computations on stretched grids (see, e.g., [18]).

4.2

Iterative methods for the coupled system

An alternative for the decoupling approach are coupled solvers. There are different

coupled solvers. Few of them work with non transformed system (e.g., Vanka approach

[18]), others first transform the system and after that solve. In the CFD literature the

coupled solvers are often applied as smoothers within nonlinear multigrid flow solvers (see,

e.g. [18, 8, 7]). The coupled solvers require much more memory, compared to segregated

solvers. Some software developers (e.g. Fluent) provide both types of solvers. In general,

coupled solvers have more restricted area of applicability. It should be mentioned that

some authors consider coupled solvers having advantages for steady state solutions, and

decoupled techniques being preferable for unsteady problems (see, e.g., [18, 15]). However,

very few particular cases are completely analyzed and further studies have to be performed

in order to confirm or to reject the above statement. Also, there are observations that

the coupled solvers are more robust for some flow problems, and less robust for others,

but here also additional work is needed in order to study and classify the advantages of

the two approaches.

In this paper we discuss coupled solvers based on proper preconditioning the coupled

system (9). Detailed discussion and further references concerning coupled solvers can be

found, for example, in [16, 2]. The specific of our approach is described below. A right

block triangular preconditioner is used in [16] (the statement in that paper is that left,

right, or double sided preconditioning are almost identical). The preconditioner they use

is in the form

M =



A R

0

S



.

so that

LM

−1

=



I

−(R + τG)S

−1

BA

−1

−BA

−1

RS

−1



.

We do not discuss here the specific choice of operators R and S, the interested reader will

find them in [16].

Several preconditioners for (9) are carefully analyzed in the recent paper [2]. Among

them are block Gauss-Seidel preconditioner, a preconditioner based on congruence

trans-formation, a two-sided block incomplete preconditioner, etc.

For example, the block

Guass-Seidel preconditioner looks as

M =



D

1

0

B

D

2



.

so that

M

−1

L =



D

₁−1

A

−D

−1₂

B(D

−1₁

− I)

τ D

−1₁

G

−τD

₂−1

BD

−1₁

G



.

For the particular choice of operators D

1

and D

2

, and for discussion on some other

pre-conditioners, we refer to [2].

In general (see, e.g., [3]) applying a preconditioned iterative method for solving a

system of equations is equivalent to applying a non-preconditioned method for solving

the transformed system. The above approaches [16, 2] are also such examples. Instead

of using this approach, we use a two stage approach for solving (9). At the first stage we

transform the system using a matrix like above mentioned preconditioners. In the second

stage, instead of using unpreconditioned iterative method, we use a preconditioned one.

Of course, this two-stage procedure can be written and analyzed as an one-stage one. We

postpone such an analysis for another paper, here we concentrate on the algorithmical

part and on the numerical study of the performance of the preconditioners. So, at the

first stage we use the matrix

M

−1

=



D

0

−BD I



.

so that

M

−1

L =



DA

τ DG

B

− BDA −τBDG



.

The aim of this transformation is to obtain ”good” blocks at the main diagonal of the

transformed system. It is clear, that the choice D = A

−1

will lead to a block triangular

system, but the operator BDG will have a full matrix in this case. So, like in SIMPLE,

we select

D = (diag

{A})

−1

Thus we obtain an transformed system

˜

Figure 1: Geometry for evaluating the linear solvers

where ˜

L is not a block triangular matrix, however the blocks on the main diagonal are

easily invertable (they are similar to a discretization of Poisson equation). This system

can be also written down as



˜

A τ ˜

G

˜

B

Λ

˜

 



u

p



=



˜

b

1

b

2



.

(16)

At the second stage we apply block triangular preconditioners to the transformed system.

5

NUMERICAL EXPERIMENTS

Numerical experiments were performed in order to evaluate the performance of the

linear solvers applied in the simulation of liquid polymer flow. We choose a simple test

geometry shown in Figure 1 consisting of a u-shaped pipe with a contraction and an

expansion in the middle. It is well known that the plastic melts do not obey Newtonian

behavior, especially near the solidification temperature. A shear thinning behavior is

commonly observed for plastic melts. This leads to a necessity of using specific material

models for the viscosity including temperature, shear rate and pressure. Several rheology

models are available in the material database of SIGMASOFT c

, for instance the

Cross-WLF model [11]. It belongs to so called generalized Newtonian models, and it is used in

our simulations. During the process the viscosity changes by several orders of magnitude.

We solve an unsteady problem, starting simulations from the liquid being in rest,

and calculating till the steady state is reached. At each time step the heat equation is

decoupled from the system due to an explicit treatment of the dissipative terms and of

the velocity in the convective term of the heat equation. The coupled momentum and

continuity equations are transformed (as it was described at the end of the preceding

section) and after that solved by a preconditioned iterative method. Several methods and

preconditioners are applied. More precisely, the used iterative methods are

• CGS: conjugate gradient squared method;

• BiCGStab: bi-conjugate gradient method, stabilized;

• GMRES(m): generalized minimal residuum method

Note, GMRES is the preferred method in [16, 2]. As it will be shown below, in general

GMRES is not suitable for us. The simulations show that good convergence can be

achieved only using long sequences, what is impossible in 3D simulations due to the

memory limitations. Note, that [16, 2] use ’long enough’ sequences, but they solve more

academic examples.

Concerning the choice of the preconditioners. We have used three preconditioners for

the transformed system (16). These are block diagonal, upper triangular (denoted by T

−

in Tables below), and lower triangular (denoted by T L

−). That is,



M

11

0

0

M

22



,

T =



M

11

M

12

0

M

22



,

T L =



M

11

0

M

21

M

22



Following variants for the blocks in the preconditioners were used:

• M

12

= τ ˜

G;

M

21

= ˜

B;

• M

11

= ILU

_A˜

;

M

11

= BRILU

_A˜

for various α (calculated for diagonal blocks);

M

11

= ˜

A

−1

• M

22

= ILU

_Λ˜

;

M

22

= RILU

_Λ˜

for various α;

M

22

= ˜

Λ

−1

In the case when the diagonal blocks have to be inverted, either CGS, Jacobi, GMRES,

each preconditioned by RILU(α) or Block RILU(α), were used. Additionally, Algebraic

multigrid, AMG, [17] was also used for inverting ˜

Λ.

We investigated the performance of the linear solvers with respect to grid size, flow

type, accuracy, Re number and the size of the time step. More precisely, the following

tests are performed:

Grid:

coarse, 4584 fluid cells

finer, 38040 fluid cells

flow type:

Newtonian ( µ = const.)

Non-Newtonian (variable µ)

accuracy:

lower, = 10

−3

higher, = 10

−8

velocity:

slow, ∆p = 10

faster, ∆p = 30

time step:

smaller

larger

Figure 2: Viscosity

Here ∆p stands for the pressure difference between the inlet and outlet. The Figure 2

gives an impression about the non-Newtonian case. Variations in the viscosity are plotted

there (recall, that viscosity is a constant for Newtonian fluids). Variations in the viscosity

here are moderate. In the simulations of solidification of real 3-D plastic parts (results are

not shown here, the behavior of the linear solvers is similar) the variation in the viscosity

was several orders of magnitude.

Below we show several tables with results from simulations. The block diagonal

pre-conditioner in all cases performed worse compared to the triangular ones, therefore it is

not included in the tables.

First, we show four tables presenting results obtained without the usage of AMG. At

a last table we show what can be achieved using AMG.

As it can be seen in Table 2, GMRES with short restart sequences does not provide good

results. The best results here are achieved by BiCGStab preconditioned with an upper

triangular matrix, where the block M

11

is a block RILU(0.8) factorization for the main

sub-blocks of ˜

A, and ˜

Λ is approximately inverted by two Jacobi iterations preconditioned

with RILU(0.4) factorization. Performing more or less Jacobi iterations gives worser

results.

GMRES was not used on the coarse grid (Table 3) because of the bad results it showed

on the fine grid. The best iterative method and the best preconditioner on the coarse grid

are the same as on the finer grid.

Preconditioner

CGS

BiCGStab

GMRES(10)

Iter.

CPU

Iter

CPU

Iter.

CPU

T-BILU-ILU

228

31

232

31

> 500

T-BRILU0.8-RILU0.4

188

26

149

20

> 500

T-BRILU0.8-Jac2RILU0.4

138

23

115

18

> 500

T-CGS6BRILU0.8-RILU0.4

72

282

70

284

122

398

TL-BILU-ILU

224

37

213

34

> 500

TL-BRILU0.8-RILU0.4

180

30

142

23

> 500

TL-BRILU0.8-Jac2RILU0.4

138

27

115

22

> 500

TL-CGS6BRILU0.8-RILU0.4

60

247

46

195

140

308

Table 2: Number of iteration and CPU time for Non-Newtonian fast flow on fine grid,

= 10

−8

.

Preconditioner

CGS

BiCGStab

GMRES(10)

Iter.

CPU

Iter

CPU

Iter.

CPU

T-BILU-ILU

79

1.1

67

0.9

T-BRILU0.8-RILU0.4

62

0.9

58

0.8

T-BRILU0.8-Jac2RILU0.4

54

0.9

47

0.7

T-CGS6BRILU0.8-RILU0.4

47

7.6

46

7.4

TL-BILU-ILU

79

1.3

68

1.1

TL-BRILU0.8-RILU0.4

61

1.0

56

0.9

TL-BRILU0.8-Jac2RILU0.4

52

1.0

44

0.8

TL-CGS6BRILU0.8-RILU0.4

43

6.9

33

5.6

Table 3: Number of iteration and CPU time for Non-Newtonian fast flow on coarse grid,

= 10

−8

.

Further, we show results for a non-Newtonian flow with higher variation in the viscosity

(Table 4). This is a nonisothermal flow.

Figure 3 shows the behavior of some linear solvers during all the time steps. It can be

seen, that the convergence is more or less uniform during the time stepping.

In certain cases there is no need to solve very accurately the linear system arising

in discretization of the unsteady PDEs. In Table 5 we show results obtained when low

accuracy is required. As it can be seen, in this case the lower triangular preconditioner

with an exact inverting of the viscous block shows the best results. This mean that the

convergence of the iterative methods is different when using the two different triangular

preconditioners. The lower triangular preconditioner ensures fast reduction of the residual

at the beginning, but later on the convergence is slower compared to that obtained with

the upper triangular preconditioner.

It should be noted that for many industrial problems approximate solution of the linear

system of equations is reasonable. In our case the results obtained with = 1e

− 3 and

Preconditioner

CGS

BiCGStab

GMRES

Iter.

CPU

Iter

CPU

Iter.

CPU

T-BILU-ILU

185

25.2

140

18.4

1063

90.4

T-BRILU0.8-RILU0.4

129

17.6

113

14.8

889

75.6

T-BRILU0.8-Jac2RILU0.4

124

20.4

116

18.4

2628

263.9

T-CGS6BRILU0.8-RILU0.4

90

262.5

83

239.7

133

220.2

TL-BILU-ILU

172

28.3

150

23.8

393

88.6

TL-BRILU0.8-RILU0.4

135

22.1

114

18.0

209

46.2

TL-BRILU0.8-Jac2RILU0.4

125

24.0

132

24.6

186

43.5

TL-CGS6BRILU0.8-RILU0.4

52

162.2

42

125.4

73

125.6

Table 4: Number of iteration and CPU time for nonisothermal Non-Newtonian fast flow on coarse grid,

= 10

−8

_.

= 1e

− 8 in many cases show less than 10 pro cent difference.

Preconditioner

CGS

BiCGStab

GMRES

Iter.

CPU

Iter

CPU

Iter.

CPU

T-BILU-ILU

45

6.2

16

2.0

60

5.1

T-BRILU0.8-RILU0.4

17

2.3

16

2.0

36

3.1

T-BRILU0.8-Jac2RILU0.4

28

4.6

15

2.4

121

12.3

T-CGS6BRILU0.8-RILU0.4

18

51.8

12

32.1

29

44.6

TL-BILU-ILU

43

7.1

27

4.2

56

9.3

TL-BRILU0.8-RILU0.4

16

2.7

14

2.2

41

5.9

TL-BRILU0.8-Jac2RILU0.4

49

9.4

29

5.4

43

6.9

TL-CGS6BRILU0.8-RILU0.4

1

2.7

1

1.4

1

4.0

Table 5: Number of iteration and CPU time for nonisothermal Non-Newtonian fast flow on coarse grid,

= 10

−3

_.

Finally, we show some results obtained using algebraic multigrid for inverting ˜

Λ. It

can be seen that the results with AMG are superior to other results.

The results obtained with test geometry have been confirmed with several industrial

plastic parts. The speedup of the combination of RILU and AMG preconditioner lead to

a speedup up to factor 4 for complicated geometries with more than 100.000 fluid cells.

6

SUMMARY

The performance of different preconditioners and iterative methods was studied for

Venturi geometry, for fast and slow, Newtonian and Non-Newtonian flow, on coarse and

on finer grid, for = 10

−3

and = 10

−8

, using different time steps.

0 100 200 300 400 500 600 0 500 1000 1500 2000 2500 time level

Number of iterations in SolveViscousBlock

T_BRILU0.8_Jac2RILU0.4 BiCGstab

T_BRILU0.8_Jac2RILU0.4 CGS

T_BILU_ILU CGS

Figure 3: Number of iterations at each time step

The results for Newtonian and Non-Newtonian flows are very close in our calculations.

A reason for this might be that we always treat the viscous block (momentum equations)

in a coupled way. A more detailed comparison with full segregated solution is in progress

now and will be a subject of another paper. Another reason might be the relatively small

time step used in our calculations.

We have simulated relatively slow flows, and we did not observe significant differences

between our slower and faster regimes.

The choice of the velocities in our case was

motivated by some practical applications (packing phase of injection molding), for other

applications the behavior of the solvers might be different.

BiCGStab has shown the most robust behavior, it was fastest in our tests. GMRES

needs long sequences (e.g., 100) for a good convergence, and this is inappropriate for 3-D

industrial applications.

Lower triangular preconditioner with inverting the viscous block was the best choice

for lower accuracy ( = 10

−3

, = 10

−4

).

Upper triangular preconditioner, with inverting the Laplacian by Algebraic Multigrid

solver was the best choice for high accuracy ( = 10

−8

).

The presented here results concern the performance of the linear solvers for a particular

finite volume discretization and for a particular class of flows. Further studies are planned

to better understand the performance of the solvers for other geometries, other

non-Newtonian flows and other discretizations.

Acknowledgments. This work is partially supported by HYKE: a Research Training

Network (RTN) on ’HYperbolic and Kinetic Equations : Asymptotics, Numerics,

Analy-sis’, financed by the European Union in the 5th Framework Programme ”Improving the

˜

A

Lambda

˜

Iterative

Iterations

Time

Preconditioner

Preconditioner

Method

[s]

ILU

ILU

CGS

182

25.6

ILU

ILU

BiCGstab

194

26.3

RILU 0.8

RILU 0.4

CGS

124

17.5

RILU 0.8

RILU 0.4

BiCGstab

103

14.0

RILU 0.8

Jac 2 RILU 0.4

CGS

105

17.7

RILU 0.8

Jac 2 RILU 0.4

BiCGstab

84

13.7

CGS 6 RILU 0.8

RILU 0.4

CGS

82

172.5

CGS 6 RILU 0.8

RILU 0.4

BiCGstab

68

144.8

RILU 0.8

AMG V

CGS

44

≈ 6.9

RILU 0.8

AMG V

BiCGstab

41

≈ 5.0

Table 6: Number of iteration and CPU time for nonisothermal Non-Newtonian flow on fine grid,

= 10

−8

.

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Fraunhofer ITWM

The PDF-fi les of the following reports

are available under:

www.itwm.fraunhofer.de/rd/presse/

berichte

1. D. Hietel, K. Steiner, J. Struckmeier

A Finite - Volume Particle Method for

Compressible Flows

We derive a new class of particle methods for ser va tion laws, which are based on numerical fl ux functions to model the in ter ac tions between moving particles. The der i va tion is similar to that of classical Finite-Volume meth ods; except that the fi xed grid structure in the Fi nite-Volume method is sub sti tut ed by so-called mass pack ets of par ti cles. We give some numerical results on a shock wave solution for Burgers equation as well as the well-known one-dimensional shock tube problem.

(19 pages, 1998)

2. M. Feldmann, S. Seibold

Damage Diagnosis of Rotors: Application

of Hilbert Transform and

Multi-Hypothe-sis Testing

In this paper, a combined approach to damage diag-nosis of rotors is proposed. The intention is to employ signal-based as well as model-based procedures for an im proved detection of size and location of the damage. In a fi rst step, Hilbert transform signal processing niques allow for a computation of the signal envelope and the in stan ta neous frequency, so that various types of non-linearities due to a damage may be identifi ed and clas si fi ed based on measured response data. In a second step, a multi-hypothesis bank of Kalman Filters is employed for the detection of the size and location of the damage based on the information of the type of damage pro vid ed by the results of the Hilbert trans-form.

Keywords: Hilbert transform, damage diagnosis, Kal-man fi ltering, non-linear dynamics

(23 pages, 1998)

3. Y. Ben-Haim, S. Seibold

Robust Reliability of Diagnostic

Multi-Hypothesis Algorithms: Application to

Rotating Machinery

Damage diagnosis based on a bank of Kalman fi l-ters, each one conditioned on a specifi c hypothesized system condition, is a well recognized and powerful diagnostic tool. This multi-hypothesis approach can be applied to a wide range of damage conditions. In this paper, we will focus on the diagnosis of cracks in rotating machinery. The question we address is: how to optimize the multi-hypothesis algorithm with respect to the uncertainty of the spatial form and location of cracks and their re sult ing dynamic effects. First, we formulate a measure of the re li abil i ty of the diagnos-tic algorithm, and then we dis cuss modifi cations of the diagnostic algorithm for the max i mi za tion of the reliability. The reliability of a di ag nos tic al go rithm is measured by the amount of un cer tain ty con sis tent with no-failure of the diagnosis. Un cer tain ty is quan ti ta tive ly represented with convex models.

Keywords: Robust reliability, convex models, Kalman fi l ter ing, multi-hypothesis diagnosis, rotating machinery,

4. F.-Th. Lentes, N. Siedow

Three-dimensional Radiative Heat Transfer

in Glass Cooling Processes

For the numerical simulation of 3D radiative heat fer in glasses and glass melts, practically applicable math e mat i cal methods are needed to handle such prob lems optimal using workstation class computers. Since the ex act solution would require super-computer ca pa bil i ties we concentrate on approximate solu-tions with a high degree of accuracy. The following approaches are stud ied: 3D diffusion approximations and 3D ray-tracing meth ods.

(23 pages, 1998)

5. A. Klar, R. Wegener

A hierarchy of models for multilane

vehicular traffi c

Part I: Modeling

In the present paper multilane models for vehicular traffi c are considered. A mi cro scop ic multilane model based on reaction thresholds is developed. Based on this mod el an Enskog like kinetic model is developed. In particular, care is taken to incorporate the correla-tions between the ve hi cles. From the kinetic model a fl uid dynamic model is de rived. The macroscopic coef-fi cients are de duced from the underlying kinetic model. Numerical simulations are presented for all three levels of description in [10]. More over, a comparison of the results is given there.

(23 pages, 1998)

Part II: Numerical and stochastic

investigations

In this paper the work presented in [6] is continued. The present paper contains detailed numerical inves-tigations of the models developed there. A numerical method to treat the kinetic equations obtained in [6] are presented and results of the simulations are shown. Moreover, the stochastic correlation model used in [6] is described and investigated in more detail. (17 pages, 1998)

6. A. Klar, N. Siedow

Boundary Layers and Domain De com po

si tsion for Radsiatsive Heat Transfer and Dsif fu

sion Equa tions: Applications to Glass Man u

-fac tur ing Processes

In this paper domain decomposition methods for ra di a tive transfer problems including conductive heat transfer are treated. The paper focuses on semi-trans-parent ma te ri als, like glass, and the associated condi-tions at the interface between the materials. Using asymptotic anal y sis we derive conditions for the cou-pling of the radiative transfer equations and a diffusion approximation. Several test cases are treated and a problem appearing in glass manufacturing processes is computed. The results clearly show the advantages of a domain decomposition ap proach. Accuracy equivalent to the solution of the global radiative transfer solu-tion is achieved, whereas com pu ta solu-tion time is strongly reduced.

(24 pages, 1998)

7. I. Choquet

Heterogeneous catalysis modelling and

numerical simulation in rarifi ed gas fl ows

Part I: Coverage locally at equilibrium

A new approach is proposed to model and simulate nu mer i cal ly heterogeneous catalysis in rarefi ed gas fl ows. It is developed to satisfy all together the follow-ing points:

1) describe the gas phase at the microscopic scale, as required in rarefi ed fl ows,

2) describe the wall at the macroscopic scale, to avoid prohibitive computational costs and consider not only crystalline but also amorphous surfaces,

3) reproduce on average macroscopic laws correlated with experimental results and

4) derive analytic models in a systematic and exact way. The problem is stated in the general framework of a non static fl ow in the vicinity of a catalytic and non porous surface (without aging). It is shown that the exact and systematic resolution method based on the Laplace trans form, introduced previously by the author to model col li sions in the gas phase, can be extended to the present problem. The proposed approach is applied to the mod el ling of the Eley Rideal and Langmuir Hinshel wood re com bi na tions, assuming that the coverage is locally at equilibrium. The models are developed con sid er ing one atomic species and extended to the general case of sev er al atomic species. Numerical calculations show that the models derived in this way reproduce with accuracy be hav iors observed experimentally.

(24 pages, 1998)

8. J. Ohser, B. Steinbach, C. Lang

Effi cient Texture Analysis of Binary Images

A new method of determining some characteristics of binary images is proposed based on a special linear fi l ter ing. This technique enables the estimation of the area fraction, the specifi c line length, and the specifi c integral of curvature. Furthermore, the specifi c length of the total projection is obtained, which gives detailed information about the texture of the image. The in fl u ence of lateral and directional resolution depend-ing on the size of the applied fi lter mask is discussed in detail. The technique includes a method of increasing di rec tion al resolution for texture analysis while keeping lateral resolution as high as possible.

(17 pages, 1998)

9. J. Orlik

Homogenization for viscoelasticity of the

integral type with aging and shrinkage

A multi phase composite with periodic distributed in clu sions with a smooth boundary is considered in this con tri bu tion. The composite component materials are sup posed to be linear viscoelastic and aging (of the non convolution integral type, for which the Laplace trans form with respect to time is not effectively ap pli -ca ble) and are subjected to isotropic shrinkage. The free shrinkage deformation can be considered as a fi cti-tious temperature deformation in the behavior law. The pro ce dure presented in this paper proposes a way to de ter mine average (effective homogenized) viscoelastic and shrinkage (temperature) composite properties and the homogenized stress fi eld from known properties of the components. This is done by the extension of the as ymp tot ic homogenization technique known for

[2], [9]) have considered homogenization for vis coelas -tic i ty of the differential form and only up to the fi rst de riv a tive order. The integral modeled viscoelasticity is more general then the differential one and includes almost all known differential models. The homogeni-zation pro ce dure is based on the construction of an asymptotic so lu tion with respect to a period of the composite struc ture. This reduces the original problem to some auxiliary bound ary value problems of elastic-ity and viscoelasticelastic-ity on the unit periodic cell, of the same type as the original non-homogeneous problem. The existence and unique ness results for such problems were obtained for kernels satisfying some constrain conditions. This is done by the extension of the Volterra integral operator theory to the Volterra operators with respect to the time, whose 1 ker nels are space linear operators for any fi xed time vari ables. Some ideas of such approach were proposed in [11] and [12], where the Volterra operators with kernels depending addi-tionally on parameter were considered. This manuscript delivers results of the same nature for the case of the space operator kernels.

(20 pages, 1998)

10. J. Mohring

Helmholtz Resonators with Large Aperture

The lowest resonant frequency of a cavity resona-tor is usually approximated by the clas si cal Helmholtz formula. However, if the opening is rather large and the front wall is narrow this formula is no longer valid. Here we present a correction which is of third or der in the ratio of the di am e ters of aperture and cavity. In addition to the high accuracy it allows to estimate the damping due to ra di a tion. The result is found by apply-ing the method of matched asymptotic expansions. The correction contains form factors de scrib ing the shapes of opening and cavity. They are computed for a num-ber of standard ge om e tries. Results are compared with nu mer i cal computations.

(21 pages, 1998)

11. H. W. Hamacher, A. Schöbel

On Center Cycles in Grid Graphs

Finding “good” cycles in graphs is a problem of great in ter est in graph theory as well as in locational analy-sis. We show that the center and median problems are NP hard in general graphs. This result holds both for the vari able cardinality case (i.e. all cycles of the graph are con sid ered) and the fi xed cardinality case (i.e. only cycles with a given cardinality p are feasible). Hence it is of in ter est to investigate special cases where the problem is solvable in polynomial time. In grid graphs, the variable cardinality case is, for in stance, trivially solvable if the shape of the cycle can be chosen freely. If the shape is fi xed to be a rectangle one can ana-lyze rectangles in grid graphs with, in sequence, fi xed di men sion, fi xed car di nal i ty, and vari able cardinality. In all cases a complete char ac ter iza tion of the optimal cycles and closed form ex pres sions of the optimal ob jec tive values are given, yielding polynomial time algorithms for all cas es of center rect an gle prob lems. Finally, it is shown that center cycles can be chosen as rectangles for small car di nal i ties such that the center cy cle problem in grid graphs is in these cases plete ly solved.

(15 pages, 1998)

a multiple objective optimisation ap proach

For some decades radiation therapy has been proved successful in cancer treatment. It is the major task of clin i cal radiation treatment planning to realize on the one hand a high level dose of radiation in the cancer tissue in order to obtain maximum tumor control. On the other hand it is obvious that it is absolutely neces-sary to keep in the tissue outside the tumor, particularly in organs at risk, the unavoidable radiation as low as possible.

No doubt, these two objectives of treatment planning - high level dose in the tumor, low radiation outside the

tumor - have a basically contradictory nature. Therefore, it is no surprise that inverse mathematical models with dose dis tri bu tion bounds tend to be infeasible in most cases. Thus, there is need for approximations pro mis ing between overdosing the organs at risk and un der dos ing the target volume.

Differing from the currently used time consuming it er a tive approach, which measures de vi a tion from an ideal (non-achievable) treatment plan us ing re cur sive ly trial-and-error weights for the organs of in ter est, we go a new way trying to avoid a priori weight choic es and con sid er the treatment planning problem as a mul-tiple ob jec tive linear programming problem: with each organ of interest, target tissue as well as organs at risk, we as so ci ate an objective function measuring the maxi-mal de vi a tion from the prescribed doses.

We build up a data base of relatively few effi cient so lu tions rep re sent ing and ap prox i mat ing the variety of Pare to solutions of the mul ti ple objective linear programming problem. This data base can be easily scanned by phy si cians look ing for an ad e quate treat-ment plan with the aid of an appropriate on line tool. (14 pages, 1999)

13. C. Lang, J. Ohser, R. Hilfer

On the Analysis of Spatial Binary Images

This paper deals with the characterization of mi cro -scop i cal ly heterogeneous, but macro-scopically homo-geneous spatial structures. A new method is presented which is strictly based on integral-geometric formulae such as Crofton’s intersection formulae and Hadwiger’s recursive defi nition of the Euler number. The corre-sponding al go rithms have clear advantages over other techniques. As an example of application we consider the analysis of spatial digital images produced by means of Computer Assisted Tomography. (20 pages, 1999)

14. M. Junk

On the Construction of Discrete Equilibrium

Distributions for Kinetic Schemes

A general approach to the construction of discrete equi lib ri um distributions is presented. Such distribution func tions can be used to set up Kinetic Schemes as well as Lattice Boltzmann methods. The general prin-ciples are also applied to the construction of Chapman Enskog dis tri bu tions which are used in Kinetic Schemes for com press ible Navier-Stokes equations.

(24 pages, 1999)

Stokes equations

The relation between the Lattice Boltzmann Method, which has recently become popular, and the Kinetic Schemes, which are routinely used in Computational Flu id Dynamics, is explored. A new discrete veloc-ity model for the numerical solution of Navier-Stokes equa tions for incompressible fl uid fl ow is presented by com bin ing both the approaches. The new scheme can be interpreted as a pseudo-compressibility method and, for a particular choice of parameters, this interpretation carries over to the Lattice Boltzmann Method. (20 pages, 1999)

16. H. Neunzert

Mathematics as a Key to Key Technologies

The main part of this paper will consist of examples, how mathematics really helps to solve industrial lems; these examples are taken from our Institute for Industrial Mathematics, from research in the Tech no-math e mat ics group at my university, but also from ECMI groups and a company called TecMath, which orig i nat ed 10 years ago from my university group and has already a very suc cess ful history.

(39 pages (4 PDF-Files), 1999)

17. J. Ohser, K. Sandau

Considerations about the Estimation of the

Size Distribution in Wicksell’s Corpuscle

Prob lem

Wicksell’s corpuscle problem deals with the estima-tion of the size distribuestima-tion of a populaestima-tion of particles, all hav ing the same shape, using a lower dimensional sampling probe. This problem was originary formulated for particle systems occurring in life sciences but its solution is of actual and increasing interest in materials science. From a mathematical point of view, Wicksell’s problem is an in verse problem where the interest-ing size distribution is the unknown part of a Volterra equation. The problem is often regarded ill-posed, because the structure of the integrand implies unstable numerical solutions. The ac cu ra cy of the numerical solutions is considered here using the condition num-ber, which allows to compare different numerical meth-ods with different (equidistant) class sizes and which indicates, as one result, that a fi nite section thickness of the probe reduces the numerical problems. Fur-thermore, the rel a tive error of estimation is computed which can be split into two parts. One part consists of the relative dis cret i za tion error that increases for in creas ing class size, and the second part is related to the rel a tive statistical error which increases with decreasing class size. For both parts, upper bounds can be given and the sum of them indicates an optimal class width depending on some specifi c constants. (18 pages, 1999)

18. E. Carrizosa, H. W. Hamacher, R. Klein,

S. Nickel

Solving nonconvex planar location

prob-lems by fi nite dominating sets

It is well-known that some of the classical location prob lems with polyhedral gauges can be solved in poly no mi al time by fi nding a fi nite dominating set, i. e. a fi nite set of candidates guaranteed to contain at least