solution of large-scale porous media problems

8/6/2019 Solution of large-scale porous media problems

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Solution of large-scale porous media problems1

Manolis Papadrakakis, George M. Stavroulakis

Institute of Structural Analysis and Seismic Research, National Technical

University of Athens, 9 Iroon Polytechniou, Zografou Campus, GR-15780,Athens, Greece

1. Introduction

Soils and geomaterials, in general, have an internal pore structure that par-

tially consists of a solid phase, often called the solid matrix, and a remain-

ing part, called the void space which is filled by a single or a number of

fluid phases, like gas, water, oil, etc. The solid phase and the fluid phase(s)have different motions; due to these motions and the different material

properties of the solid and the fluid phase(s), the interaction between them

is of crucial importance, thus making the description of the mechanical be-

havior of the porous media more complicated. Moreover, this solid-fluid

interaction is particularly strong in dynamic loading and may lead to fatal

strength reduction of the porous medium, like the one that occurs during

liquefaction of loose saturated granular soils subjected to repeated cyclic

loading or during the localized failure in earth dams and wet embankments

The finite element method has been proven to be an extremely powerful

tool for the determination of both stresses and pore fluid pressure(s) distri-

bution in porous media. The first application of the finite element method

to solve Biots field equations for the consolidation problem in the planestrain condition can be found in [1]. Furthermore, a numerical analysis of

anisotropic consolidation of layered media is presented in [2], while in [3]

Biots theory is re-examined where a simplification, by ignoring the rela-

1 This work has been funded by the project Heracletus. The Project is co-

funded by the European Social Fund (75%) and National Resources (25%).


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2 Manolis Papadrakakis, George M. Stavroulakis

tive acceleration of pore fluid(s) with respect to the soil skeleton is sug-

gested. This approach was followed by many researchers and constitutes

the basic problem formulation in the present investigation.

There are various successful methods for the numerical solution of un-

coupled problems, but as far as porous media problems are concerned, the

optimum solution method is still to be found. Several methods have been

investigated [4-6] but a fully satisfactory answer has not yet been found.

The monolithic approach, where all field equations are solved simulta-

neously, is regarded as the most suitable one [7,8] for this type of coupled

problems and will be adopted in the present investigation. Solutions on pa-

rallel computing environments have also been investigated [9-11] by im-

plementing methods primarily based on frontal and multi-frontal solutiontechniques, which were found to be less attractive since the computational

overhead due to parallelization is significant [12]. In this work, a family of

innovative parallel domain decomposition algorithms based on the mono-

lithic approach will be implemented along with numerical results which

demonstrate their performance in large-scale porous media problems.

2. Porous media problem formulation

The basic equation that relates effective stresses, soil skeleton stresses and

pore pressure for a multi-phase medium (Fig. 2.1) can be written as:

'' Ta p= + m (2.1)

with

[ ]1 1 1 0 0 0=m (2.2)

where are the effective stresses, are the soil skeleton stresses and pis the pore pressure. The a coefficient takes values near unity for clay and

sand soils and can be as low as 0.5 for rocky soils.


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Solution of large-scale porous media problems 3

Fig. 2.1.A 2D porous material sample

Ifw

p anda

p represent the fluid and air pore pressure respectively and

and a represent the percentage of the pore pressure, that is due to the ex-

istence of the fluid and the air in the pores respectively, then in case of par-

tially saturated media, where the air pressure is assumed to be negligible,

the following equation holds:

( )1w w a a w w w a w wp p p p p p = + = + (2.3)

with 1=+ aw and ( )www S = , where wS is the media saturationdegree.

Darcian flow w occurs when imposing a difference in hydrostatic pres-

sure p between two material points, say A and B (Fig 2).

Fig. 2.2.Darcy flow through a medium under the action of a pressure gradient

If R is the viscous drag forces, k is the permeability matrix, having

terms defined with dimensions of [ ] [ ] [ ]masstimelength /3 (and assumingisotropic permeability, Ik k= with kbeing the permeability coefficient),then wkR = .

Considering the soil skeleton and the fluid embraced by the usual con-

trol volume dV dxdydz= , the following equation describes the momen-tum balance relation for the soil-fluid mixture:


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( ) 0T =S u b (2.4)

with

0 0 0

0 0 0

0 0 0

T

x y z

y x z

z y x

=

S

(2.5)

( ) sww nnS += 1 is the soil-fluid mixture density, s is the soildensity, w is the fluid density, n is the porosity, u is the vector of the soil

skeleton displacements, b is the vector of body force per unit mass and the

dot refers to time differentiation. Convective terms regarding fluid flow

have been omitted as they are generally very small and can be neglected

[12].

The following equation describes the momentum balance of the fluid,

considering the fluid embraced by the usual control volume dV:

( ) 0w wp =R u b

(2.6)

while for the flow conservation of the fluid, the following equation holds:

0T wp

aQ

+ + =w m

(2.7)

By combining (2.6) and (2.7), while assuming Darcian flow of the fluid,

we obtain:

( ) 0T ww w wp

p S aQ

+ + + =k b m

(2.8)

where ( )S

w

w

w

SKna

KnSC

Q++=1 , wK is the fluid bulk modulus, SK

is the soil bulk modulus,( )

w

ww

Sdp

pdSnC = and are the soil skeleton


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strains, where, as previously, the convective terms regarding fluid flow

along with solid grain acceleration have been omitted [12].

Eqs. (2.4) and (2.8), constitute the so-called u-p formulation of the prob-

lem. This formulation exhibits some loss of accuracy for problems in

which high-frequency oscillations are important. However, these oscilla-

tions are of little importance even for earthquake analysis [12,13], thus

making this formulation very appropriate for consolidation problems and

soil-structure interaction under seismic or wave loading, as well as drain-

ing systems in landfills and reservoirs.

The discretized algebraic system occurring from Eqs. (2.4) and (2.8)

when the finite element method is applied is:

(1)

(2 )0

+ + =

T

M 0 C 0 K Q uu u f

0 0 Q S 0 H pp p f

(2.9)

where

u=B SN (2.10)

Td

= K B DB (2.11)

( )

Tp pk d

=

H N N

(2.12)

T p

wa d

= Q B mN (2.13)

( )T

u ud

= M N N (2.14)

( )1Tp pdQ

= S N N (2.15)

T pa d

= Q B mN (2.16)

( ) ( )(1)t

T Tu ud d

= + f N b N t (2.17)


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( ) ( )(2 )t

T Tp p

w wkS d d

= + f N b N q (2.18)

= +C M K (2.19)

Applying the basic Newmarks algorithm, the discretization in time of

the dynamic equation (2.9) results in the following equation at each time

step:

=eff eff K u f (2.20)

where

1oa a

= + +

eff T

M 0 C 0 K QK

0 0 Q S 0 H

(2.21)

3. Algebraic system symmetrization

The discretized equations (2.4) and (2.8) in time can be symmetrized with

the following procedure. By premultiplying the momentum balance rela-

tion for the soil-fluid mixture with a constant scalar parameter c, Eq. (2.4)

becomes:

( ) 0Tc c =S u b (3.1)

Similarly, by premultiplying the momentum balance-flow conservation

of the fluid equation with the scalar function w , Eq. (2.8) is trans-

formed to:

( ) 0T ww w w w w w wp S a pQ

=k b m

(3.2)

By implementing the space discretization procedure, described earlier,

for Eqs. (3.1) and (3.2) and taking into account the damping properties ofthe solid matrix, we obtain the following linear system:

(1)

(2)0

c c c c c + + =

T

M 0 C 0 K Q uu u f

0 0 Q S 0 H pp p f

(3.3)

where the quantities H , S , Q and (2 )f of Eq. (2.9) are replaced by:


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( )T

p p

wk d

= H N N

(3.4)

( )T

p pw dQ

= S N N

(3.5)

T p

wa d

= Q B mN (3.6)

( ) ( )(2 )t

T T

p pw w w wk S d d

= + f N b N q

(3.7)

By applying the Newmark algorithm for the solution of the above dy-

namic problem, the following effective matrix is derived:

1

0 1

1 1

o

c c c ca a

a c a c c c

a a

= + +

+ + =

eff T

eff T

M 0 C 0 K QK

0 0 Q S 0 H

M C K QK

Q S H

(3.8)

With the selection of 1ac = , effK becomes:

2

0 1 1 1 1

1 1

a a a a a

a a

+ + =

eff T

M C K QK

Q S H

(3.9)

which is obviously symmetric.

4. Domain decomposition solution methods

Domain Decomposition Methods (DDM) constitute an important category

of methods for the solution of a variety of problems in computational me-chanics. Their performance, in both serial and parallel computer environ-

ments has been demonstrated in a number of papers over the last decade.

They are basically classified as primal and dual DDM.


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Fig. 4.1.A structural domain, split in subdomains. Arrows show the traction

forces between the disconnected subdomains

The primal DDM (P-DDM) reach the solution by solving for the inter-

face displacements after elimination of the internal degrees of freedom

(d.o.f.) of the subdomains, while dual DDM (D-DDM) proceed with the

computation of the Lagrange multipliers required to enforce compatibility

between subdomains after the elimination of all the d.o.f. (internal and in-

terface) of each subdomain. In the early 90s, an important D-DDM, the Fi-

nite Element Tearing and Interconnecting (FETI) method was introduced

[14] and recently a family of P-DDM, namely the P-FETI methods wereproposed [15,16]. Since their introduction, FETI, P-FETI DDM and sever-

al variants have gained importance and today are considered as highly effi-

cient DDM. In the following sections, the basic concepts of D-DDM and

P-DDM applicable to implicit dynamic porous media problems will be de-

scribed along with some implementation considerations.

5. D-DDM with no coarse problem for implicitdynamics

The main difference of this D-DDM compared to the one-level FETI me-thod, which is the initial FETI variant proposed in [14], is the absence of

the coarse problem related to the rigid body modes of the subdomains and

consists of solving the following interface problem:

=IF d (5.1)

where


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=-1s T

IF BK B (5.2)

( )

( )

=

-1

-1

-1

1

s

Ns

K

K

K

(5.3)

=-1

s sd BK f (5.4)

=s

pf L f (5.5)

is the vector of the Lagrange multipliers, K(s)

is the stiffness matrix of

each subdomain, Lp is the so-called global to local mapping operator and

B is the so-called Lagrange mapping operator. The PCG method is used to

solve the above equation with the following preconditioners [16]:

=b b

-1 s T

I p pF B S B (5.6)

=b b

-1 s T

I p bb pF B K B (5.7)

The one-level FETI method is designed to solve problems which may

contain floating subdomains, that is subdomains with zero energy modes

which exhibit singular coefficient matrices. In that case, the local problems

of the form = T

(s) (s) (s) (s)K u f B are ill-posed and in order to guaranteetheir solvability, it is required that

( ) ( )null T

(s) (s) (s)f B K (5.8)

The above solvability condition forms a natural coarse problem which,

as stated above, is absent in this D-DDM version because it is applicable to

linear systems that have non-singular coefficient matrices and thus it does

not require the solvability condition. However, this lack of a coarse prob-

lem constitutes this D-DDM non-scalable as the error propagates slowlywhen the number of subdomains is high. In the following sections, a fami-

ly of D-DDM is discussed which use the two-level technique in order to

impose a coarse problem which ensures scalability.


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6. D-DDM-S and D-DDM-P: D-DDM with an artificialcoarse problem based on optional admissibleconstraints

These D-DDM are based on a FETI variant which imposes an artificial

coarse problem and was proposed in [17]. These D-DDM are constructed

by applying the two-level technique on the previous D-DDM with no

coarse problem. By applying the two-level technique [19] to Eq. (5.1),

the following interface problem is needed to be solved:

=IPF Pd (6.1)

where

( )1

= T TI IP I C C F C C F (6.2)

As in the D-DDM with no coarse problem, the preconditioners used for the

implementation of this method with the PCG method are the same with the

D-DDM and are given by Eqs. (5.6), (5.7).

The introduction of matrix C is equivalent to imposing a set of optional

admissible constraints [17]. These constraints adhere to the following equ-

ation:

10

ss NT

s

=

= =f(s) (s)

C B u

(6.3)

wheref

(s)u are the exact solutions of the local problems

= T(s) (s) (s) (s)

K u f B and are called optional because they are not re-quired for the solution of these local problems. In order to impose these

constraints, a starting vector is chosen equal to [17]

( )1

0

= T TI C C F C C d (6.4)

Due to the nature of these constraints, any properly chosen matrix C

with linearly independent columns will exhibit superior convergence prop-

erties compared to the D-DDM with no coarse problem [17].For the case of D-DDM-S, it is proposed that matrix C should be equal

to:

= IC G (6.5)

where


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( ) ( ) ( ) ( ) = 1 1 Ns Ns

IG B R B R (6.6)

( ) ( )( )null=s sR K

(6.7)

and( )s

K

is the coefficient matrix of a subdomain for the corresponding

static structural problem with all its displacement boundary conditions re-

moved. Calculating the null space of( )sK

can be done using geometric-

algebraic algorithms, as proposed in [18], which are very cost-effective

and robust.

For the case of D-DDM-P, it is proposed that matrix C should be equalto:

=I

C E (6.8)

where

( ) ( ) ( ) ( ) = 1 1 Ns Ns

I H HE B R B R (6.9)

( ) ( )( )null=s sHR H

(6.10)

and

( )s

H

is the permeability coefficient matrix of subdomains with alltheir pore boundary conditions removed. The null space of

( )sH is equiva-

lent to the null space of a structural problem with one d.o.f. per node and is

equal to a one-column vector with equal values at each position (ie. unity).

7. P-DDM-S and P-DDM-P: P-DDM with an artificialcoarse problem based on optional admissibleconstraints

These P-DDM belong to the PFETI family of methods, originally intro-

duced in [15,16]. They consist of the PSM, preconditioned with the firstestimate for the interface unknowns of the domain obtained from the first

iteration of the D-DDM family with an artificial coarse problem based on

optional admissible constraints. Thus, this P-DDM family is in fact a PSM

algorithm with the following preconditioner:


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( )( )1

1 =

-1 -1 -1

b b

T s s T T T s

p b I b pS L S S B C C F C C B S L

(7.1)

As in the case of the D-DDM family with an artificial coarse problem

based on optional admissible constraints, any properly chosen matrix C

with linearly independent columns will exhibit superior convergence prop-

erties compared to the PSM because of the implicit introduction of a

coarse problem.

For the case of P-DDM-S, the preconditioner used is

( )( )1

1 =

-1 -1 -1

b b

T s s T T T s

p b I b pS L S S B G G F G G B S L

(7.2)

while for the P-DDM-P is

( )( )1

1 =

-1 -1 -1

b b

T s s T T T s

p b I I I I I b pS L S S B E E F E E B S L

(7.3)

8. DDM computational issues

Both D-DDM-S and D-DDM-P methods, along with their corresponding

primal counterparts P-DDM-S and P-DDM-P, are variants of the DDM

family with an artificial coarse problem based on optional admissible con-straints, varying on the selection of matrix C. This selection defines a

number of properties of the method, such as the size of the coarse problem,

the projector evaluation time, storage requirements, number of iterations

and computational time.

All these properties are directly related to the size of matrix C. With( )s

dofn being the number of d.o.f. per subdomain, the size of matrix C, for

a 3D continuum problem in the case of D-DDM-S and P-DDM-S, is( ) 6sdofn per subdomain, where as in the case of D-DDM-P and P-DDM-

P, the corresponding size per subdomain is only( ) 1sdofn . This means

that the size of the porous coarse problem with 200 subdomains grows bythree orders of magnitude for the D-DDM-S and P-DDM-S variants, as

opposed to only two orders of magnitude when using D-DDM-P and P-

DDM-P. This difference of one order of magnitude has a significant im-

pact on the efficiency of the D-DDM-P and P-DDM-P methods compared

to the D-DDM-S and P-DDM-S methods. These computational issues are

clearly demonstrated in the following section


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9. Numerical example

In order to assess the efficiency of the previously discussed methods, their

performance is compared in a cubic soil consolidation problem subjected

to a surface step load. The domain is discretized with 8 node hexahedral

finite elements with 3 d.o.f. per node for the soil skeleton and 1 d.o.f. for

the pore pressure and is solved using the monolithic u-p formulation pre-

sented in section 2. The boundary conditions of this test case along with

the loading are shown in the 2D cut of Fig. 9.1.

15m

15m

p=0

ux=uy=uz=0

ux=uy=0

Fig. 9.1. Test case: A 2D cut showing boundary and loading conditions

The resulting linear system to be solved at each time increment has

115,320 d.o.f. and, in order to investigate the scalability of the various me-

thods, a parametric analysis was carried out with respect to the number of

subdomains. The solution was obtained for the first time increment of the

Newmark algorithm and with number of subdomains ranging from 45 to

300. Two characteristic subdivisions are shown in Fig. 9.2. The time step

for this test case was 10-1

seconds.


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Fig. 9.2. Partitioning of the domain in 45 and 125 subdomains

Figs. 9.3 and 9.4 depict the required number of iterations and the com-

puting time to reach a solution tolerance of 10-4

of the DDM considered for

the first time step of the Newmark time integration algorithm and for dif-

ferent number of subdomains. The initialization time required for the com-

putation of the projection matrix P (Eq. (6.2)) and for the subdomain facto-

rization of the coefficient matrix Keff(Eq. (3.9)) is shown in Fig. 9.5.

Fig. 9.3. Number of iterations for each time step for different number of subdo-

mains


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Fig. 9.4. Computation time in seconds per iteration for different number of sub-

domains

Fig. 9.5.Initialization time in seconds for different number of subdomains

The overall performance of the methods for the optimum number ofsubdomains is shown in Table 1. N is the optimum number of subdomains,

T1 stands for the required time in seconds for the first time step, T 2 corres-

ponds to the initialization time, T3 is the total time and T4 denotes the time

required per DDM iteration. Finally, Fig. 9.6 demonstrates the best overall

performance of the methods for one time step of the Newmark time inte-


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gration algorithm with t=10-2 and convergence tolerance of the DDM

equal to 10-4

.

Table 9.1. Performance of the methods for the optimum number of subdomains

Method N T1 [sec] T2 [sec] T3 [sec] T [sec]

D-DDM-S 125 334 3,920 4,254 22

D-DDM-P 216 315 2,193 2,507 15

P-DDM-S 216 215 3,699 3,914 17

P-DDM-P 216 342 2,210 2,552 18

Noptimum number of subdomains, T1 required time for the first time step, T2 in-

itialization time, T3 total time, T4 time required per DDM iteration

Fig. 9.6Overall performance of the methods

10. References

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[3] O. C. Zienkiewicz, C. T. Chang, P. Bettess, Drained, undrained, consolidatingand dynamic behavior assumptions in soils, Geotechnique, 30, No. 4, 385-395

(1980)


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[4] K. C. Park, C. A. Felippa, Partitioned analysis of coupled systems, Computa-

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solution of large-scale porous media problems

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