International Conference on Adaptive Modeling and Simulation
ADMOS 2003
N.–E. Wiberg and P. Díez (Eds)
CIMNE, Barcelona, 2003
PARALLEL MULTILEVEL SOLUTION OF NONLINEAR
SHELL STRUCTURES
Michael Gee, Wolfgang A. Wall and Ekkehard Ramm
Institute of Structural Mechanics, University of Stuttgart
Pfaffenwaldring 7, D–70550 Stuttgart, Germany
e–mail: {ramm, gee, wall}@statik.uni–stuttgart.de
web page: http://www.uni–stuttgart.de/ibs/
Key words: three–dimensional shell, structural dynamics, multilevel preconditioning, algebraic
multigrid, iterative solvers
Abstract. The analysis of large–scale nonlinear shell problems asks for parallel simulation approaches. One crucial part of efficient and well scalable parallel FE–simulations is the solver
for the system of equations. Due to the inherent suitability for parallelization one is very much
directed towards preconditioned iterative solvers. However thin walled structures discretized by
finite elements lead to ill–conditioned system matrices and therefore performance of iterative
solvers is generally poor. This situation further deteriorates when the thickness change of the shell
is taken into account. A preconditioner for this challenging class of problems is presented combining two approaches in a parallel framework. The first is a parallel multilevel approach. A hierarchy of coarse grids is generated in a semi–algebraic sense using an aggregation concept.
Thereby the complicated and expensive explicit generation of course triangulations can be
avoided. The second approach is a mechanically motivated improvement called ’Scaled Director
Conditioning’ (SDC) and is able to remove the extra – ill conditioning that appears with three
dimensional shell formulations as compared to formulations that neglect thickness change of the
shell. It is introduced at the element level and harmonizes well with the multilevel algorithm. The
formulation of this combined preconditioning approach is given and the effects on the performance of iterative solvers is demonstrated via numerical examples.
M. Gee, W.A. Wall, E. Ramm
1 INTRODUCTION
Thin–walled shell structures have been denoted as the ‘Prima–donna’ in the hierarchy of structures’ in Ramm [27]. They have extraordinary abilities and can show a superb performance in
the right situation and when they are treated well. However in the wrong situation or with a wrong
treatment they may show an extreme sensitivity and can be very difficult to be controlled. What
is true for the structure itself sometimes also carries over to the numerical model.
The derivation of beam, plate and especially shell theories over the last centuries belongs to
the most ingenious, in particular successful achievements in mechanics. Typical representatives
of these theories are related to names like Euler, Bernoulli, Timoshenko, Kirchhoff, Love, Reissner, Mindlin and many others. Their basic ingredient is a projection of the thin–walled three–dimensional (3d) body to a one– or two–dimensional (2d) system utilizing physically motivated
simplifications, such as neglection of ’minor effects’ like thickness changes and transverse normal stresses, or, generally speaking, certain assumptions concerning the displacement field
across the thickness of the thin–walled structure.
Simplifications on the one hand lead to a considerable increase in efficiency and solvability
of the numerical model, but on the other hand, they bring along an approximation of the three–dimensional stress and strain state, a loss of generality and pitfalls such as the need for modification
of the 3d constitutive law.
One of the unsolved problems in modeling thin–walled structures is the apparently ill–conditioning of the resulting stiffness matrices as a function of the slenderness. In addition the magnitude of this effect is also dependent on the type of formulation. While formulations based on a
Kirchhoff/Love type of kinematics seem not to suffer any severe ill–conditioning it can be observed in the classical shell models with ’Reissner–Mindlin kinematics’ and – even worse – in
three–dimensional models. To give a first insight we show the condition numbers
cK max
min
(1)
of different models for a plane square thin shell with increasing lenght to thickness ratio lh in
fig. 1. The shell has in–plane dimensions of l 20 and is simply supported at two opposite
edges. The first model to be considered is a classical formulation with Reissner–Mindlin kinematics, denoted as 5–parameter formulation (5–PM), i.e. 3 translational and 2 rotational degrees
of freedom without thickness change. The second uses three translational and two relative–
translational degrees of freedom, which is also not allowing for any thickness change, and is
called 5–parameter formulation with director (5–PMd) here. The third model considered, a
7–parameter formulation (7–PM), has three translational and three relative–translational degrees
of freedom with an additional linear thickness change introduced via a hybrid formulation. In
addition, we give the results for a modeling with solid elements (brick). For further details on the
mentioned models, see section 2. For comparability all results are obtained with bilinear, respectively trilinear elements, on a pure displacement basis without any enhancement of the physical
2
M. Gee, W.A. Wall, E. Ramm
behaviour, the material properties of the applied St–Venant–Kirchhoff law are kept constant and
the Possion ratio is set to zero.
cK
10
10
10 9
a)
b)
c)
d)
5–PM (3 displ. + 2 rot.)
5–PMd (3 displ. + 2 rel.–displ.)
7–PM (3 displ. + 3 rel.–displ. + 1 strain)
brick (2 x 3 displ.)
10 8
c)
b)
10 7
d)
10 6
10 5
a)
10
4
lńh
10 3
4 10 0
1 10 1
2 10 1
2 10 2
2 10 3
1 10 4
2 10 4
Figure 1: Condition numbers for different models
While the condition of the stiffness matrix resulting from the 5–parameter formulation with
rotational degrees of freedom is insensitive to an increase of slenderness, the ill–conditioning of
all formulations based on displacements and relative displacements increases fataly.
w³0
a3
a
h³0
2
3
f + const.
Figure 2: Parametrization
Though the 5–parameter shell with rotational degrees of freedom and the one with relative–
translational degrees of freedom represent exactly the same kinematics in this example, their behaviour with respect to conditioning differs extremely. It is obvious, that only the choice of the
3
M. Gee, W.A. Wall, E. Ramm
parametrization plays the decisive role for the difference in condition. In a given certain deformation state of plates with different thicknesses, the value for the rotation of a section remains
constant with increasing slenderness, while expressing the same deformation in terms of relative
displacements of the surface to the mid–plane of the plate, the absolute value of these vanish (fig.
2). So, choosing the components of w of fig. 2 as primary unknowns leads to increasing associated
values in the stiffness matrix with the observed negative effect on the condition number.
Among possible shell formulations the three dimensional 7–parameter formulation (7–PM) is
adopted in this study, as the advantages are numerous: It includes the full three dimensional set
of strains and stresses allowing to use unmodified complete 3d constitutive laws and completely
represents (at least approximately) three dimensional structural phenomena. The modeling of
large rotations is straight–forward and it almost keeps the numerical efficiency of conventional
plate and shell formulations. It allows a good physical interpretation and easy visualization of
the main structural effects due to a pre–integration of the kinematic and static variables over the
thickness. One of the remaining challenges of such formulations is the ill–conditioning of the
resulting matrices, which is due to the parametrization mentioned above and the additional
(thickness) kinematics which are not included in conventional formulations. These ill–conditioning effects supplement the already present ill–conditioning that is independent of any formulation
and is simply a structural characteristic.
The objective of this contribution is to show, that this 7–PM–drawback can be easily overcome
by the use of a scaled parametrization, the Scaled Director Conditioning approach (SDC). The
condition of the resulting stiffness matrices comes to the same range as such of formulations using
rotational degrees of freedom. Further we outline the construction of a parallel algebraic multigrid preconditioner which is then advantagously combined with the SDC–approach. We will
show the effect on the performance of iterative solution strategies applying this combination of
techniques by studying several examples.
2 SHELL FORMULATION
In the following a brief sketch of our three–dimensional shell formulation will be given just
in order to have the prerequisites to introduce our Scaled Director Conditioning approach. Detailed presentations and discussions on this formulation can be found in [3], [4], [5], [8].
One of the characteristic features of a shell formulation, compared to a completely three–dimensional treatment of the shell continuum, is the fact that the geometric description of the initial and
deformed shell body is projected on a two–dimensional reference space, usually the mid–surface
of the shell (fig. 3). In our case the displacement field in the third dimension is approximated
assuming a linear variation of the displacements across the thickness
x + r ) q3 a3 .
4
(2)
M. Gee, W.A. Wall, E. Ramm
Here, x is the initial position vector of an arbitrary point in the shell space, r is the position
vector of the corresponding point in the mid surface and a 3 is the so–called director. Following
an isoparametric concept, the position vector in the current (deformed) configuration is defined
accordingly
x + r ) q3 a3 .
q3 + z
(3)
reference configuration
current configuration
a3
q2 + h
a2
a1
q3 + z
u
a3 w
v
x
a3
q1 + c
r
a1
x
a2
q2 + h
r
e3
e2
q1 + c
e1
Figure 3: Kinematics of 7 parameter model
In the following a bar denotes values defined in the current configuration. It should be mentioned at this point that due to the extra strain parameter introduced in the sequel, the resulting
elements can — strictly speaking — not be denoted as isoparametric, as this strain parameter does
not show up in the description of the shell geometry. The displacement u of an arbitrary point
P is obtained as
u + v ) q3 w ;
v+ r*r ;
w + a3 * a3 ,
(4)
where w denotes the relative displacement field between the mid and the upper surface. This
approximate kinematic modelling of the three–dimensional shell structure can be interpreted as
‘semi–discretization’ in thickness direction. In addition an independent strain field is introduced,
utilizing the Enhanced Assumed Strain (EAS) Method (Simo and Rifai [29]). The extra strains
~
E supplement the ‘compatible’ strains E u obtained from the kinematic equation to the total strain
field E
~
E + Eu ) E ;
Eu + 1 ǒFT @ F * g Ǔ ,
2
5
(5)
M. Gee, W.A. Wall, E. Ramm
where F denotes the deformation gradient. The, with respect to the displacements, ‘compatible’
part of the strain tensor E u + E uij a i Ě a i can be expressed in terms of the displacements and covariant base vectors g i
E uij + 1 ǒu, i @ g j ) u, j @ g i ) u, i @ u, jǓ
2
(6)
with
g a + ēxa + x, a ;
ēq
a i + r, i ;
a
g 3 + a 3 :+ h 1
Ť
2 a1
a2
,
a 2Ť
(7)
where h is the thickness of the shell.
The exclusive use of the resulting strain components E uij leads to the 6 parameter model, men~
tioned previously. The additional degree of freedom b , leading to a 7 parameter model, is now
~
supplemented by the additional strain field E introduced in eq. (5), affecting only the transversal
normal strain E 33 :
~
E 33 + E u33 ) E 33 + a 3 @ w ) 1 w @ w ) q 3 b
2
~
(8)
With this extension, both the constant and the linear part of the strain distribution in thickness
direction are now completely three–dimensional which is the decisive feature of the 7–parameter–model.
E 11
a u11
b u11
E 12
a u12
b u12
E 13
a u13
b u13
E 22
2.1
+
a u22
) q3
b u22
E 23
a u23
b u23
E 33
a u33
b
(9)
~
Conditioning of thin–walled structures
To develop an approach that is able to effectively improve the condition number of the global
stiffness matrix while keeping all the desired merits of our three–dimensional formulation, it is
necessary to gain additional understanding of the physical model. In order to get this insight we
have a closer look at the spectral analysis of the plane shell example from section 1 and fig. 1.
In fig. 4 the whole spectra of the eigenvalues for the 5–parameter formulation with rotational
degrees of freedom (5–PM), for a modelling with solid elements (bricks) and for the 7–parameter
formulation (7–PM) is shown for an extreme lenght to thickness ratio lńh + 10000. The Poisson
ratio is chosen to n + 0.3 and the Young’s modulus to E + 1 @ 10 3. When looking at fig. 4 one
notices a characteristic jump appearing in the spectra of the three dimensional shell and the mod-
6
M. Gee, W.A. Wall, E. Ramm
elling with bricks which does not appear with the 5–parameter formulation. Further it is obvious,
that the relative displacements used in the 7–parameter formulation do not introduce any additional harm to the condition, compared to the modeling with bricks. Also, the extra strain
introduced to allow for a linear strain field through the thickness does not have any negative effect.
10 8 i
c)
20.0
10 6
b)
10 4
20.0
10 2
0.002
10 0
10
a)
condition numbers:
a) 5–PM c k + 2.282E)04
b) brick c k + 9.096E)09
c) 7–PM c k + 2.216E)10
*2
10 *4
0
100
200
300
400
500
Figure 4: Eigenvalue spectra for different models
7
600
i
700
M. Gee, W.A. Wall, E. Ramm
10 8 i
10 6
10 4
10 2
7–PM
10
0
10 2
i
10 4
0
100
200
300
400
500
600
700
Figure 5: Spectrum and eigenmodes for 7 parameter formulation
In fig. 5 selected eigenmodes of the three dimensional shell are visualized and associated to
their eigenvalue. For a better visualization of the modes, the shell thickness is scaled graphically.
The lowest eigenvalues obviously correspond to the classical bending modes. The last eigenvalue
before the jump is a pure in–plane mode and all higher eigenvalues after the jump are pure shear
and thickness–change modes closely related to the extensible director of the model. This confirms the expectation that the poor condition of three–dimensional shell models as compared to
classical models is caused by the high ratio of transverse to in–plane normal stiffnesses. So in
order to improve the condition this ratio has to be somehow lowered. Apparently this high ratio
stems from the fact that in–plane and out–of–plane length scales differ so much. By adjusting the
8
M. Gee, W.A. Wall, E. Ramm
different length scales and making them independent of the slenderness by the use of a scaled
director, it is possible to achieve the properties of classical formulations with respect to condition
numbers, and to maintain all the advantages of the use of a relative displacement field.
3 THE SCALED DIRECTOR CONDITIONING (SDC)
Lenght scales between in–plane and out–of–plane dimensions can be numerically adjusted by
a scaling of the parametrization. Such an approach is pursued in the following and as shown in
fig. 6 the length scales are adjusted here via the use of a scaled director. Therefore the name
‘Scaled Director Conditioning’ (SDC) is adopted for this idea.
3
w
l
2
a 3
a3
u
a 3 + a 3
a 3
w
a3
w + w
a3
a3
a1
2
u
a2
v
b
l
a 3 [ l [ b
2
2
x
x
r
e3
e1
a 3 + a 3
a3 w
1
r
3
e2
a3
a1 a2
1
2
Figure 6: Kinematics of 7–parameter shell model with scaled director
In order not to change the physical behaviour of the model this director scaling is employed
only in the sense of a preconditioning. For this, the scaling has to be introduced into the formulation in a consistent way. Hence, the conditioning of the left hand side is changed before entering any solver and afterwards the scaling is removed from the solution. From the above it gets
obvious, that has to be chosen in a way that the scaled element thickness corresponds approximately to the distance between adjacent nodes.
From the physical interpretation of this principal idea it becomes clear, that this approach can
also be applied to other kinds of element formulations, such as solids, shell–like solids and multilayer models, e.g. as given in Braun et al. [6]. In the following the major steps of incorporating
the Scaled Director Conditioning into the three–dimensional shell model are described. A detailed description and discussion of the unmodified base formulation can be found in Bischoff
[3].
9
M. Gee, W.A. Wall, E. Ramm
3.1
Modified kinematics and basis vectors
The scaling variable is introduced via the base vectors of the shell middle surface
a
a 3 + h 1
2 Ťa 1
a a + r, a
a2
a 2Ť
(10)
in the reference configuration and affects only the vector in the thickness direction of the shell.
The base vectors of the current (deformed) configuration are obtained similarly with respect to
the in–plane directions and via the scaled difference displacements w of the shell body according
to the director a 3 :
a 3 + a 3 ) w a a + r, a
(11)
The base vectors g a of the shell body can then be expressed in terms of the unmodified base vectors of the middle surface and the scaled director.
g a + a a ) q a 3,a
3
3.2
g 3 + a 3
(12)
Algorithmic strains
The purely algorithmic Green–Lagrange strains are computed from the modified base vectors
and metric tensors of the shell body without further modifications.
~
) E ij
E ij + E u,
ij
(13)
+ 1 ǒg i @ g j * g i @ g j Ǔ
E u,
ij
2
(14)
E ij + E u,
ij
(15)
;
(i, j) 0 (3, 3)
~
) E 33 + a 3 @ w ) 1 w @ w ) q 3 b E 33 + E u,
33
2
~
(16)
It should be noted here, that these algorithmic strains are to be used in the B–operator for the
construction of the tangential stiffness and the algorithmic internal forces. The true physical
strains can easily be computed from the known unmodified metric.
10
M. Gee, W.A. Wall, E. Ramm
3.3
Modified integration, tangential stiffness and internal forces
The modifications which are necessary in the integration of the constitutive tensor over the
shell thickness are the removal of the scaling factor from the so–called shell–shifter m^
1
ŕ ( · ) dV + ŕ ŕ( · ) m
^
dq 3
(g 1 g 2) @ g 3
1
m+
,
Ťa 1 a 2Ť
^
dA ;
A *1
V
(17)
and from the linear and quadratic terms within the preintegration of the constitutive tensor
1
D ijkl,
+
K
ŕǒq Ǔ
3
ǒ Ǔ m dq3 ,
K
C ijkl, h
2
K
^
K Ů {0, 1, 2} .
(18)
*1
To allow for nonlinear constitutive equations, the constitutive tensor C ijkl and the Piola–Kirchhoff stresses S ij are formed using the true Green–Lagrange strains and unmodifed base vectors.
Both are transformed into scaled algorithmic values by applying
j, k, l, mnpq
C ijkl, + a i,
m an ap aq C
(19)
j, mn ,
S ij, + a i,
m an S
(20)
and
transforming the unscaled to the scaled base vectors
with a i,
j
g j + a i,
gi ;
j
+
a i,
j
1
0
0
0
1
0
0
0
.
(21)
The integration of the material tensor through the shell thickness (18) results in the consistent
relationship between algorithmic strain and stress resultants n, a representing constant and m, b
linear varying contributions with respect to thickness direction.
n ij,
m ij,
+
D ijkl,
D ijkl,
0
1
D ijkl,
D ijkl,
1
2
11
@
a ij
b ij
.
(22)
M. Gee, W.A. Wall, E. Ramm
A more detailed description of the scaling factors to the coefficients of the material tensor is
1 D 1111 1 D 1112
c0
c0
1 D 1212
c0
1 D (@)
c1
1 D (@)
c1
1 D (@)
c2
1 (@)
c0
1 (@)
c0
1 (@)
c1
1 (@)
c0
1 (@)
c1
1 (@)
c1
1 (@)
c2
1 (@)
c1
1 (@)
c2
1 (@)
c2
1 (@)
c2
1 (@)
c3
1 (@)
c2
1 (@)
c3
1 (@)
c4
D ijkl, +
1 (@)
c1
1 (@)
c1
1 (@)
c2
1 (@)
c1
1 (@)
c2
1 (@)
c3
1 (@)
c2
1 (@)
c2
1 (@)
c1
1 (@)
c1
1 (@)
c2
1 (@)
c1
1 (@)
c2
1 (@)
c3
1 (@)
c2
sym.
1 (@)
c2
1 (@)
c2
1 (@)
c3
1 (@)
c2
1 (@)
c3
1 (@)
c4
1 (@)
c3
1 (@)
c3
1 (@)
c4
1 (@)
c1
1 (@)
c1
1 (@)
c2
1 (@)
c1
1 (@)
c2
1 (@)
c3
1 (@)
c2
1 (@)
c2
1 (@)
c3
1 (@)
c2
1 (@)
c2
1 (@)
c3
1 (@)
c2
1 (@)
c3
1 (@)
c4
1 (@)
c3
1 (@)
c3
1 (@)
c4
1 (@)
c3
1 (@)
c3
1 (@)
c4
1 (@)
c3
1 (@)
c4
1 (@)
c5
1 (@)
c4
1 (@)
c4
1 (@)
c5
a 11
a 12
a 13
a 22
a 23
a 33
b 11
(23)
b 12
b 13
1 (@) 1 (@) 1 (@) b
22
c2
c3
c4
1 (@) 1 (@) b
23
c4
c5
~
1 (@) b
33
c6
where the appropriate entries D ijklare replaced by (@) for clarity. One can clearly see the distribution of factors *n , n Ů {0AAA6} appearing in the material tensor, which is preintegrated in
thickness direction.
B
+
v1
v2
v3
w 1
w 2
w 3
1(@)
1(@)
(.)
1(@)
1(@)
1(@)
(.)
1(@)
1(@)
1(@)
(.)
1(@)
0
0
0
0
1(@)
0
1(@)
0
1(@)
0
0
0
a 11
a 12
a 13
a 22
(@)
(@)
(@)
1(@)
1(@)
1(@)
a 23
0
(@)
(@)
0
(@)
(@)
0
(@)
(@)
0
(@)
0
0
(@)
0
(@)
1(@)
1(@)
(@)
1(@)
(@)
(@)
1(@)
1(@)
(@)
1(@)
(@)
a 33
0
(@)
0
(@)
1(@)
1(@)
(@)
1(@)
(@)
0
0
0
0
0
0
b 33
12
b 11
b 12
b 13
b 22
b 23
~
(24)
M. Gee, W.A. Wall, E. Ramm
The tangential B–Operator (24) for the displacement–compatible part of the strain field holds
~
factors m , m Ů {0, 1}, the incompatible strains E 33 are scaled by 1. From equations (26)
^
and (31) we can derive, that the tangential stiffness matrix K eT is scaled by mixed polynoms in
*n , n Ů {0AAA4}.
3.4
Incremental solution
After variation and linearization of the modified Hu–Washizu functional, which is the basis
of the 7–parameter formulation ([3] and references therein), we result in the scaled incremental
system of equations
K T
L T
L
~
@
Dd f
+
D D
R
*
,
(25)
dA ,
(26)
~
0
R
with
K T +
ŕE
u,
@
h,d
S u,
dA )
h,d
Ae
+
ŕE
u,
h,dd
Ae
ŕB
T
ŕE
~
h,a @
~
S h,d dA +
~
ŕE
u,
@
h,d
S u,
dA +
h,a
ŕM
T
@ D @ B dA ,
(27)
ŕB
T
@ D @ M dA ,
(28)
@ D @ M dA ,
(29)
Ae
ŕE
~
h,a @
~
S h,a dA +
Ae
R +
u,
h
Ae
Ae
D +
T
d
Ae
Ae
L +
ŕB , @ S
@ D @ B dA )
Ae
L T +
@ S u,
dA
h
ŕE
ŕM
T
Ae
u,
@
h,d
R +
~
S u,
dA ,
h
Ae
ŕE
~
Ae
13
h,a @
~
S h dA .
(30)
M. Gee, W.A. Wall, E. Ramm
By eliminating the incremental strains D from (25) at the element level, the scaled director
conditioned incremental system of equations results in
ǒKT * L T @ D~ *1
@ L Ǔ @ Dd + f * R ) L T @ D @ R ,
~
K
@ Dd + f
^
*1
(31)
~
* R
^
where the solution Dd consists of the (unmodified) displacements Dv of the shell mid–surface
and the scaled relative displacements Dw + D( w) of the scaled director as shown in fig. 6.
The scaling factor is removed from the solution with
Dd +
Dv
Dv
+
(32)
Dw Dw
only for the purpose of postprocessing and visualization of the results. The nonlinear solution
procedure itself merely operates on scaled variables.
3.5
Spectral analysis
10 8 i
b)
20.0
10 6
10 4
20.0
0.002
10 2
a)
c)
10 0
a) 5–PM
10 *2
c k + 2.282E)04
b) 7–PM c k + 2.216E)10
c) 7–PM c k + 2.357E)04 i
10 *4
0
100
200
300
400
500
Figure 7: Eigenvalue spectra with SDC – conditioning
14
600
700
M. Gee, W.A. Wall, E. Ramm
Once again we recall the thin plane shell example used in fig. 4, this time showing the spectra
of eigenvalues for the 5–parameter– , the 7–parameter– and the 7–parameter formulation with
scaled director in fig. 7. From this investigation it gets very obvious that the ’spectral gap’ between the 5 – and the 7 parameter formulation could be closed by our SDC approach. In Fig. 8
we compare condition numbers c K for increasing slenderness using the example given in Fig. 1.
cK
10 10
10 9
a) 5–PM
b) 7–PM
c) 7–PM
b)
10 8
10 7
10 6
10 5
a)
10
4
10
3
c)
lh
4 10 0
1 10 1
2 10 1
2 10 2
2 10 3
1 10 4
2 10 4
Figure 8: Evolution of condition numbers with SDC – conditioning
The eigenvalues of the 7–parameter formulation with scaled director correspond to those of
the formulation using rotational degrees of freedom (5–parameter model), although in the latter
the additional thickness change modes are not included at the high end of the spectrum. The argument, that shell formulations based on relative displacements are inherently worse conditioned
than classical shells, as e.g. stated in Simo et al. [28], can therefore be invalidated. Also the low
convergence rates and numerical instabilities reported in [28] could not be validated by the authors, as simulations with very large in–plane to thickness ratios perform well within common
computer precisions (8 byte floating point values).
Three remarks shall be added here that are important to the assessment of the given condition
numbers:
D All given examples refer to pure bi–(tri–)linear displacement based finite elements without any
enhancement as mentioned before. Due to the severe locking effects appearing in this case, the
bending modes are artificially much too stiff. This means that the eigenvalues of the low bending
modes are too high and for this reason the condition numbers of all models are much better then
they would be, when applying higher order shape functions and/or hybrid element formulations.
15
M. Gee, W.A. Wall, E. Ramm
Having this in mind, the achieved c K [ 10 4 for all shell formulations must still be called a severe
ill–conditioning.
D The SDC approach can easily be introduced to any hybrid formulation. As it was shown in section 3 the introduction to the Enhanced Assumed Strain (EAS) Method (Simo and Rifai [29]) is
straight forward, as it is the appliance of SDC in combination with Assmued Natural Strain methods (ANS) [2].
D The SDC approach works with negligible cost with respect to computing time and no cost at
all with respect to memory requirements, as it affords only a few additional scalar multiplications
in the element integration procedures and directly affects the tangential stiffness matrices. Technically, it can therefore be combined with other common preconditioning techniques without any
conflict.
However, it is necessary to thoroughly choose a proper combination of SDC and other preconditioners to achieve best performance in the context of iterative solution strategies. In the following we present an approach for a multilevel preconditioner to complement the SDC approach,
as it enhances solution behaviour in the long wave portion of the error and solution, respectively.
4 PARALLEL MULTILEVEL PRECONDITIONER
The process of subdividing a large linear system of equations into smaller problems, whose
exact or approximate solution can be used to build a preconditioner to an iterative solver, is called
a domain decomposition preconditioner. It would exceed the extents and mistake the purpose of
this contribution to give an extensive overview of the various techniques in this area of methods.
Hence, we restrict ourselves to the Additiv–Schwartz method [10] that forms the basic algorithm
underlying the semi–algebraic multilevel preconditioner adopted in the following. For detailed
discussion on domain decomposition techniques we like to refer to (Smith et al. [30]), (Quarteroni and Valli [26]) and references therein.
For the Additiv–Schwartz preconditioner we introduce some definitions, that are necessary
to describe the decomposition procedure. Let W(0) be the discretized domain being decomposed
into n (0) overlapping subdomains W(0)
where indizes W(@) in brackets indicate the grid in the muli
be a restriction operator to be
tilevel hierachy, which will be used later in this section. Let R (0)
i
(0)
applied to the vector d returning the part of the vector defined in the interior of a subdomain
W(0)
, that is
i
d (0)
Wi
+
R (0)
i
d (0)
+( I O)
d (0)
W
i
d (0)
WńWi
.
(33)
and its counterpart, the prolongator P (0)
+ R (0)T
operate excluThe restriction operator R (0)
i
i
i
sively inside one level of a grid hierachy in contrast to the interlevel transfer operators R (j)1)
and
(j)
P (j)1)
which
will
be
introduced
later.
The
iterative
procedure
of
the
1–Level
Additiv–Schwartz
(j)
method in iteration step k can then be written as
16
M. Gee, W.A. Wall, E. Ramm
(0)
n
ȡdk)1 ² dk ) ȍ
ǒf * K dkǓȣ
B *1
ȧ
ȧ
i
Ȣ
Ȥ
i+1
(0)
(34)
with the correction operator
ǒ
B (0)*1
+ P i ǒR i K P iǓ
i
*1
Ǔ
(0)
(35)
Ri
of subdomain i. The Additiv–Schwartz method is adopted here due to its inherent high parallelity.
All corrections derived by the B *1
can be applied independently at the same time for all subdoi
mains Wi . As it is a stationary Richardson Method and can be seen as a generalization of the block
Jacobi iteration method (while the Multiplicative–Schwartz iteration represents a generalization
of the block Gauss–Seidel method), it suffers from poor convergence rates. In most applications,
the Additiv–Schwartz procedure would therefore not be used as a solver itself but as a preconditioner to some Krylov subspace method. In this case, it does not make sense to calculate the B *1
i
too accurate, so an approximate subdomain solution – a subdomain preconditioner – can be defined as
ǒ
Ǔ
M (0)*1
+ B (0)*1
+ P i K *1
W Ri
i
i
~
~
i
(0)
.
(36)
The overall preconditioner on W(0) then resolves to
ȍ M(0)*1
.
i
n (0)
M (0)*1
+
(37)
i+1
Various types of standard preconditioners, e.g. incomplete factorizations or basic iterative
~
methods can be applied to form subdomain preconditioners K *1
W .
i
To motivate the construction of a multilevel approach let us assume an error
e k + K *1ǒ f * K d k) ,
(38)
as being (nearly) constant over a certain region of W that completly includes a subdomain Wi .
If K is derived by a finite element discretization of a standard second order elliptic PDE, then the
row–sums of K away from boundary conditions is zero as is the product K e k. The local correction obtained by a preconditioner or an exact solve on Wi is therefore also (nearly) zero.
17
M. Gee, W.A. Wall, E. Ramm
ȡ00
ȧ
c i +ȧ0
ȧ
ȧ0
Ȣ0
0 0 0 0ȣ
0 0 0 0
~
ȧ
ȧ k
0 K *1
Wi 0 0 K e [ 0
0
0
0
0
0
0
ȧ
0ȧ
0Ȥ
(39)
Such a (nearly) constant part of the error is called smooth and is ’invisible’ to the subdomain
solve or preconditioner. To overcome this circumstance, a hierarchy of coarser discretizations
W(j) , j Ů [1,. .., N * 1] is introduced, N denoting the total number of grids, which overlap the
entire discrete domain W(0).
In iteration k of the iterative solver: r (0)k + ǒf * K W(0) d kǓ
ȍ P(0)i K*1
R (0)
r (0)k
W
i
N (0)
c (0)
+
~
Presmoothing
(0)
i+1
ǒr(0)k * KW(0) c(0)Ǔ
r (1) + R (1)
(0)
ȍ P(1)i K*1
R (1)
r (1)
W
i
N (1)
c (1)
+
Restriction of Residuum to coarser grid
~
Presmoothing
(1)
i+1
r (2)
ǒr(1)k * KW(1) c(1)Ǔ
+ R (2)
(1)
Restriction of Residuum to coarser grid
.
.
c (N*1) + K *1
r (N*1)
W(N*1)
Exact solve on coarsest grid
c (N*2) + c (N*2) ) P (N*1)
c (N*1) Prolongation of correction to finer grid
(N*2)
.
.
ǒr(1) * KW
ȍ P(1)i K*1
R (1)
W
i
N (1)
c (1)
c (0)
+
c (1) )
+
i+1
(0)
c ) P (1)
(0)
+
c (0) )
(1)
c (1)
(1)
c (1)Ǔ
Postsmoothing
Prolongation of correction to finer grid
ǒr(0)k * KW
ȍ P(0)i K*1
R (0)
W
i
N (0)
c (0)
~
~
(0)
(0)
c (0)Ǔ
i+1
Figure 9: Multilevel Schwarz preconditioner
18
Postsmoothing
M. Gee, W.A. Wall, E. Ramm
By these coarse grids, the local exchange of information between distant points is complemented using less variables as on the fine grid. In the case of the underlying Additiv–Schwartz procedure a coarse grid correction can be added by
c (j*1)
+
P (j)
(j*1)
ǒ R(j)(j*1) K(j*1) P(j)(j*1) Ǔ
*1
R (j)
r (j*1) ,
(j*1)
(40)
where R (j)
and P (j)
+ R (j)T
are the – still undefined – restriction and prolongation opera(j*1)
(j*1)
(j*1)
tor between the fine and the coarse grid as mentioned previously. As on the finest grid, on each
higher level only an approximate solution is computed.
K (j)*1
Ǔ
+ ǒ R (j)
K (j*1) P (j)
(j*1)
(j*1)
*1
+ ȍ M(j)*1
i
n (j)
[
M (j)*1
(41)
i+1
The procedure in fig. 9 algorithmically represents a classical V–cycle multigrid preconditioner that can also be seen as a combination of the Additiv–Schwarz method used within levels,
embedded in a Multiplicative–Schwartz decomposition between levels.
4.1
Aggregation multigrid
The interlevel restriction and prolongation operators R (j)
and P (j)
+ R (j)T
are derived
(j*1)
(j*1)
(j*1)
using an aggregation concept [32], [33], [7].
In an existing finite element code, it is difficult and numerically expensive to construct coarse
discretizations in the general case without any restrictions concerning the use of unstructured
grids or the modeling of complex geometries. Hence, we form an artificial hierarchy of coarse
problems from the fine grid information, the fine grid system matrix and the physical characteristics of the modelled problem. This is done by first decomposing the set of nodes to mutually disjoint subsets called aggregates (fig. 10). These aggregates can and should be derived from the
sparsity pattern of the underlying stiffness matrix, as the aggregation procedure has to be repeated
on higher levels where no explicit discretization data is available. There exist several algorithms
for parallel aggregation [31], which mainly differ in the treatment of aggregates near borders of
subdomains. Aggregates may overlap the border between subdomains, which can improve convergence rates in the solution process but which in construction is more expensive and leads to
more communication overhead in the multilevel procedure. The prolongation operator
constructed from an aggregate should achieve low energy of the coarse grid basis functions and
therefore is chosen to include the rigid body modes (kernel of the piece of system matrix
associated with the aggregate) of an aggregate exactly, disregarding the contribution of elements
between the aggregates. The discrete representation of the rigid body modes is dependent on the
type of finite elements.
19
M. Gee, W.A. Wall, E. Ramm
Processor 1
coarse grid node and
degrees of freedom
Processor 2
fine grid shell node
and director
Figure 10: Mutually disjoint aggregates
For the SDC–conditioned 7–parameter shell element this results in the intergrid prolongator
P (1)
(0)ij
(x, a 3 , x (1))
+
x 3 * x (1)
3
* x 2 ) x (1)
2
0
x 1 * x (1)
1
1
0
0
0
1
(1)
0 * x3 ) x3
0
0
1
0
0
0
0
a 3;3
* a 3;2
0
0
0
* a 3;3
0
a 3;1
0
0
0
a 3;2
* a 3;1
0
0
x 2 * x (1)
* x 1 ) x (1)
2
1
0
(42)
where x and a 3 are shell mid surface coordinates and scaled director of the fine grid node i and
x (1) are the coordinates of the coarse grid node j chosen to be the center of the aggregate. The
rows of P (1)
are related to the degrees of freedom of the shell node consisting of the three dis(0)ij
placements v followed by three relative displacements w of the director, while the columns are
associated with three displacements v and three rotational degrees of freedom f of the coarse grid
node.
20
M. Gee, W.A. Wall, E. Ramm
Aggregation is repeated on higher levels and transfer operators P (l)
ô l Ů [2, N * 1] be(l*1)
tween coarse grids result from
P (l)
(l*1)ij
(x (l*1), x (l))
+
x (l*1)
* x (l)
* x (l*1)
) x (l)
3
3
2
2
1
0
0
0
1
(l*1) ) x (l)
0 * x3
3
0
0
1
0
0
0
0
0
0
x (l*1)
* x (l)
1
1
x (l*1)
* x (l)
2
2
* x (l*1)
) x (l)
1
1
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
, (43)
where rows and columns are related to degrees of freedom v and f of the finer and coarser grid,
respectively. In [33] it is shown, that such prolongators do not provide sufficient smoothness of
coarse basis functions as they neglect contribution of elements between aggregates. It is proposed
to apply a simple damped Jacobi smoother to the columns of P (l)
ô l Ů [1, N * 1] to obtain
(l*1)
the final prolongator
+ ǒI * wD *1 K (l*1)Ǔ P (l)
ô l Ů [1, N * 1] ,
P (l)
(l*1)
(l*1)
^
(44)
where D + diag(K (l*1)) and w is the damping parameter. By this convergence and stability can
be very much improved but are sensitive to a proper choice of w, which is not a straight forward
selection [33]. Furthermore, the smoothing of the prolongator produces an unneglectable amount
of fill–in to the sparsity pattern which then carries over to the coarse grid system matrices and
therefore leads to an intense increase in computational and communication cost per iteration. The
efficiency of the implementation of the sparse matrix operations then plays the decisive role
whether this increase in convergence pays off in terms of a decrease in overall solving time.
Prolongators between coarse grids P (l)
ô l Ů [2, N * 1] can also be obtained by decom(l*1)
(l*1) as it is an orthonormal basis of it, and the upper
posing P (l*1)
+
Q
R
,
where
Q
replaces
P
(l*2)ij
(l*2)ij
triangular matrix R is used to construct the next higher level prolongator P (l)
to which it carries
(l*1)
over the range of functions (here, the rigid body modes) that can be represented exactly [34].
It is not the intention of this contribution to give a complete overview of methods, so we like to
refer to [23] and [12] and references therein as further interesting approaches and overview of
methods in the context of smoothed aggregation multigrid.
21
M. Gee, W.A. Wall, E. Ramm
5 EXAMPLES
5.1
Geometrically nonlinear honeycomb
A honeycomb as described in fig. 11 is chosen to demonstrate the skills and limitations of the
SDC method. The material properties are E + 2.06 @ 10 12 , n + 0.3 and ò + 8 @ 10 3 . It is
deformed by a prescribed Dirichlet boundary condition of the top nodes moving downwards by
a total value of 0.6. The 7–parameter formulation with and without SDC preconditioning was
applied. The solution is obtained using a Chung–Hulbert implicit time integration [9] with 500
time steps and an average of about 4 Newton iterations per step. The resulting system of equations
has a size of 56,952 degrees of freedom and a parallel conjugate gradient solver [17] on 4 processors was used. The solution was calculated to an accuracy of 10 *9 in the error norm. Nine sets
of combinations of the SDC method, a diagonal preconditioning and an Additiv–Schwarz incomplete factorization with fill–in level of 0 and 1 were applied in a sequence of analyses. The total
number of iterations and the solving and preconditioning time taken by the solver were summed
over the entire simulation (fig. 12). To correctly interprete the results, it has to be taken into account, that the honeycomb has very stiff bending properties. The low range of the eigenvalues
is therefore not too small and the ill–conditioning of the system is primarily due to the slenderness
of the single finite elements.
0.3
0.002
2.0
Figure 11: Geometry and final state of honeycomb
22
M. Gee, W.A. Wall, E. Ramm
We can see the large effect of the SDC pre–preconditioning on the iteration numbers. As there
is no cost of the SDC method in the solution process, the total solution time is reduced proportionally to the number of iterations.
solving time [CPU–h]
6 10 5
20.0
5 10 5
15.0
4 10 5
iterations
3 10 5
10.0
2 10 5
5.0
1 10 5
(a) (b) (c) (d) (e) (f) (g) (h) (i)
(a) SDC
(b) diag. pre–precond.
(c) SDC / diag. pre–precond.
(d) Additive–Schwartz–ILU(0)
(e) SDC / AS–ILU(0)
(a) (b) (c) (d) (e) (f) (g) (h) (i)
(f) AS–ILU(0) / diag. pre–precond.
(g) SDC / AS–ILU(0) / diag. pre–precond.
(h) AS–ILU(1) / diag. pre–precond.
(i) SDC / AS–ILU(1) / diag. pre–precond.
Figure 12: Summed number of iterations and solving time for honeycomb
The second insight is that additionally applied diagonal preconditioning can further reduce
solving time and leads to the fastest solution (fig. 12(a) and 12(c)). Further, a combination of the
SDC method and the Additiv–Schwartz–ILU(0) and the Additiv–Schwartz–ILU(1) also improves performance (fig. 12(f) and 12(g)), though the extra cost of the incomplete factorization
does not pay–off compared to the combination in fig. 12(c). Applying such costly preconditioners
as ILU(n) in the absence of low bending modes is obviously ’over–preconditioning’. If there are
low bending modes, which would be the standard case for thin–walled structures, the use of these
more sophisticated preconditioners can be necessary, as we will see in our next example.
5.2
Bending of a tube
The tube in fig. 13 is loaded by an equillibrium group of area loads which is increased linearily.
Again, a Chung–Hulbert time integration with a full Newton–Raphson equillibrium iteration and
a parallel preconditioned conjugate gradient solver were used. The tube is discretized with 3000
bilinear 3d shell elements which results in a system of equations with 18,174 unknowns.
23
M. Gee, W.A. Wall, E. Ramm
0.0005
1.0
0.1
E + 2.06 @ 10 12
n + 0.3
ò + 8.0 @ 10 3
Figure 13: Geometry and final state of tube
In fig. 14 the iteration numbers and solving time over all solver calls are given for this example
for the case of a standard one–level Additiv–Schwarz preconditioner and the presented algebraic
multilevel preconditioner using 3 coarse grids (i.e. 4 levels). An incomplete factorization ILU(1)
was chosen as subdomain preconditioner in all cases. Results are given for combinations with
and without our SDC method. It can be seen how iteration numbers increase when buckling of
the system appears. Partially this is due to the increasing condition number, but also due to the
disability of the coarse grid shape functions to represent the contribution from the geometric tangential stiffness of the system.
Combined with the multilevel preconditioner, the effect of the SDC method is less distinct
compared to the one–level case. Hence, coarse grid parametrization was chosen to have rotational
degrees of freedom instead of a director and therefore, as shown in section 1, are relatively insensitive according to slenderness. Hence, coarse grids naturally show properties which were
introduced to the director formulation through the SDC method. Therefore applying the SDC
preconditioner does only improve condition on the finest grid. (If coarse grids were chosen to
be of the same parametrization as the fine grid, the effect of the SDC method would be comparable to the one–level case.)
The SDC method is not only affecting the solution of the system of equations but also the Newton’s iteration of dynamic equillibrium. Therefore, fig. 14 shows smaller total number of solver
calls for the case of the SDC method applied.
24
M. Gee, W.A. Wall, E. Ramm
800
iterations
50
600
[s]
40
30
400
no SDC
20
200
0
0
800
no SDC
SDC
10
SDC
solver calls
solver calls
0
0
100
200
100
200
4–level 4–subdomains multilevel Additiv–Schwarz with ILU(1) on subdomains
iterations
no SDC
50
600
[s]
no SDC
40
30
400
SDC
20
SDC
200
10
0
0
solver calls
solver calls
0
0
100
200
100
200
1–level 4–subdomains Additiv–Schwarz with ILU(1) on subdomains
Figure 14: Iterations and solving time for tube example
5.3
Tube intersection
An intersection of two steel tubes is clamped on two flanges and loaded by an increasing area load
on the remaining flanges until failure by buckling appears [1] (fig. 15). The discretization of the
problem results in a system of linear equations with 131.388 unknowns and is solved applying
the algebraic multilevel preconditioned conjugate gradient method. An incomplete factorization
ILU(2) is used as subdomain preconditioner on all levels including the coarsest grid, except for
the 5–level preconditioner, were system matrices on the coarsest grid were small enough to apply
an exact solve. In fig. 15 iteration numbers and residuum over iterations are given for a selected
25
M. Gee, W.A. Wall, E. Ramm
time step were buckling appears. Best convergence rate is achieved by the 5–level multigrid preconditioner with the direct solve on coarsest grid. It should be mentioned, that the increase in convergence rates by the introduction of additional course grid corrections does not carry over one
to one to a decrease in solving time because of the additional computational cost per iteration.
In this example fastest solution in terms of wall–clock time was achieved applying the 4–level
preconditioner in parallel on 8 processors (the 8 domains are indicated as colors of the deformed
configuration in fig. 15).
A combination with SDC preconditioning was not used here because the elements’ length to
thickness ratio already is very good due to the fine discretization.
30
200
30
3
E + 2.1 @ 10 6
n + 0.3
ò + 7.8 @ 10 *6
clamped
6
400
clamped
level unknowns
0 131.388
1
34.452
2
10.428
3
4.026
4
2.334
10 3
no. iterations
ø r ø2
1228
10 0
1089
695
10 *3
565
400
10
*6
5 lev.
4 lev.
400
3 lev.
2 lev.
800
iterations
1 lev.
1200
1
2
3
4
5
number of levels
Figure 15: Tube intersection
6 Conclusions
In this article we have reflected the ill–conditioning characteristics inherent to thin–walled
structural shell problems. It was shown that shell models based on relative displacements beside
26
M. Gee, W.A. Wall, E. Ramm
their numerous advantages tend to be even worse conditioned then those based on rotational degrees of freedom. We have shown that this parametrization–dependent disadvantage of such 3D
models can be eliminated by the Scaled Director Conditioning approach (SDC). This means that
the ’spectral gap’ between the different models can easily be closed. This SDC approach is
introduced in a three dimensional, geometrically and materially nonlinear shell formulation and
is applied along with a hybrid finite element discretization.
In addition the SDC approach is combined with an semi–algebraic multilevel preconditioner
based on an aggregation concept. As all coarse grids are constructed from the geometry and the
system matrix of the fine mesh, no explicit coarse grid discretizations have to be supplied and
the preconditioner can easily be integrated into an existing parallel finite element implementation
and be applied to even very complex shell problems. Examples show, that its skills to cover large
geometrically nonlinear effects are limited raising demand for further research in this direction.
The combination of these two methods results in a stable iterative solution process capable of
achieving good performance for a wide range of large–scale problems in nonlinear shell dynamics and statics. Hence, the presented approach seems to be an attractive option when iterative solvers should be used for this class of problems.
7 REFERENCES
[1] G. Alefeld, I. Lenhardt, Th. Rottner, K. Schweizerhof, Verbesserung und Beschleunigung
von Lösungsverfahren für Versagensanalysen in der Strukturmechanik, Theorie und Parallelisierung: Schlußbericht des BMBF Vorhabens 01IR413A3, Institute of Mechanics, Institute of Applied Mathematics, University of Karlsruhe, (1997).
[2] K.–J. Bathe, E.N. Dvorkin, A Four–Node Plate Bending Element Based on Mindlin/Reissner Theory, Int. J. Num. Meth. Eng., 21, (1985), 367–383.
[3] M. Bischoff, Theorie und Numerik einer dreidimensionalen Schalenformulierung. Ph.D.–
Dissertation, Report No. 30, Institute of Structural Mechanics, University of Stuttgart,
(1999).
[4] M. Bischoff, E. Ramm, Solid–Like Shell or Shell–Like Solid Formulation? A Personal View.
In W. Wunderlich (editor), Solids, Structures and Coupled Problems in Engineering, Proc.
of the ECCM ’99, TU Munich, (1999).
[5] M. Bischoff, E. Ramm, On the Physical Significance of Higher Order Kinematics and Static
Variables in a Three–Dimensional Shell Formulation. Int. J. Solids and Structures, 37,
(2000), 6933–6960.
[6] M. Braun, M. Bischoff, E. Ramm, Nonlinear Shell Formulations for Complete Three–Dimensional Constitutive Laws Including Composites and Laminates, Computational Mechanics, 15, (1994), 1–18.
27
M. Gee, W.A. Wall, E. Ramm
[7] M. Brezina, P. Vanek, One black–box iterative solver, UCD/CCM Report 106, Center for
Computational Mathematics, University of Colorado at Denver, 1997.
[8] N. Büchter, E. Ramm, 3d–Extension of Nonlinear Shell Equations Based on the Enhanced
Assumed Strain Concept, In: C. Hirsch (editor), Comp. Mech. Appl. Sciences, Elsevier
(1992), 55–62.
[9] J. Chung, G.M. Hulbert, A time integration algorithm for structural dynamics with improved
numerical dissipation: The generalized–a method, Journal of Applied Mechanics, 60,
(1993), 371–375.
[10] M. Dryja, O.B. Widlund, An additive variant of the Schwartz alternating method for the case
of many subregions, Ultracomputer Note 131, Department of Computer Science, Courant
Institute, 1987.
[11] R.M. Ferencz, T.J.R. Hughes, Iterative Finite Element Solutions in Nonlinear Solid Mechanics, in: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, Vol. VI, Elsevier Science B.V., (1998).
[12] J. Fish, V. Belsky, Generalized Aggregation Multilevel Solver, Int. J. Num. Meth. Eng., 40,
(1997), 4341–4361.
[13] M. Gee, W.A. Wall, E. Ramm, Parallel iterative solvers in nonlinear shell analysis, ZAMM
– Zeitschrift für Angewandte Mathematik und Mechanik, Proc. of the GAMM Conference,
Göttingen, Germany, (2000).
[14] W. Hackbusch, Iterative Lösung großer schwachbesetzter Gleichungssysteme, B.G. Teubner, Stuttgart, (1993).
[15] A. Haufe, H. Menrath, E. Ramm, A Comparison of Plasticity Models for Thin–Walled Reinforced Concrete Structures, Proceedings of the COMPLAS VI, Eds.: E. Onate, D. R. J.
Owen, E. Hinton, Barcelona/Spain, 2000.
[16] T.J.R. Hughes, I. Levit, J. Winget, An element–by–element solution algorithm for problems
of structural and solid mechanics, Comp. Meth. Appl. Mech. Engrg., 36, (1983), 241–254.
[17] S.A. Hutchinson, L.V. Prevost, J.N. Shadid, C. Tong, R.S. Tuminaro, Aztec User’s Guide
Version 2.0, Massively Parallel Computing Research Laboratory, Sandia National Laboratories, Albuquerque, NM 87185, (1998).
[18] Center for Applied Scientific Computing, Lawrence Livermore National Laboratory,
HYPRE – high performance preconditioners, http://www.llnl.gov/CASC/hypre
[19] G. Kirchhoff, Über das Gleichgewicht und die Bewegung einer elastischen Scheibe, Journal
für reine angewandte Mathematik 40, (1850), 51–58.
28
M. Gee, W.A. Wall, E. Ramm
[20] D. Kuhl, E. Ramm, Generalized Energy–Momentum Method for non–linear adaptive shell
dynamics, Com. Meth. in Appl. Mech. Engrg., 178, (1999), 343–366.
[21] A.E.H. Love, On the Small Vibrations and Deformations of Thin Elastic Shells, Philosophical Transactions of the Royal Society 179, (1888), 491.
[22] P. Le Tallec, Domain decomposition methods in computational mechanics, Computational
Mechanics Advances, 1, (1994), pp. 121–220.
[23] J. Mandel, M. Brezina, P. Vanek, Energy Optimization of Algebraic Multigrid Bases, Computing 62 , (1999), 205–228.
[24] R.D. Mindlin, Influence of Rotatory Inertia and Shear on Flexural Motions of Isotropic
Elastic Plates, Journal of Applied Mechanics 18, (1951), 31–38.
[25] A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics, Springer–Verlag, New York,
USA, (2000).
[26] A. Quarteroni, A. Valli, Domain Decomposition Methods for Partial Differential Equations, Oxford Univ. Press, Oxford, GB, (1999).
[27] E. Ramm, Primadonna–quotation in: Form und Tragverhalten in ”Heinz Isler Schalen” (E.
Ramm & E. Schnuck) Karl Krämer Verlag, Stuttgart (1986) 29–34.
[28] J.C. Simo, S. Rifai, D.D. Fox, On a Stress Resultant Geometrically Exact Shell Model. Part
IV: Variable Thickness Shells with Through–the–Thickness Stretching. Comp. Meth. Appl.
Mech. Engrg., 81, (1990), 91–126.
[29] J.C. Simo, S. Rifai, A Class of Mixed Assumed Strain Methods and the Method of Incompatible Modes. Int. J. Num. Meth. Engrg., 29, (1990), 1595–1638.
[30] B. Smith, P. Bjorstad, W. Gropp, Domain decomposition. Cambridge University Press,
(1996).
[31] R.S. Tuminaro and C. Tong, Parallel Smoothed Aggregation Multigrid: Aggregation Strategies on Massively Parallel Machines, Proceedings of SuperComputing 2000, J. Donnelley
(ed.), 2000.
[32] P. Vanek, J. Mandel and M. Brezina, Algebraic Multigrid on unstructured meshes, UCD/
CCM Report 34, Center for Computational Mathematics, University of Colorado at Denver,
1994.
[33] P. Vanek, J. Mandel and M. Brezina, Algebraic Mulitgrid based on Smoothed Aggregation
for Second and Fourth Order Elliptic Problems, Computing, v. 56, pp. 179–196, 1996.
29
M. Gee, W.A. Wall, E. Ramm
[34] P. Vanek, J. Mandel and M. Brezina, Convergence analysis of algebraic multigrid based on
smoothed aggregation, UCD/CCM Report 126, Center for Computational Mathematics,
University of Colorado at Denver, 1998.
[35] W.A. Wall, D.P. Mok, E. Ramm, Simulation of nonlinear shells interacting with instationary
flows, Proc. IASS–IACM 2000 – Fourth International Colloquium on Computation for
Shells & Spatial Structures, IASS–IACM, Chania – Crete, Greece, (2000).
[36] W.A. Wall, M. Gee, E. Ramm, The challenge of a three–dimensional shell formulation – the
conditioning problem –, Proc. IASS–IACM 2000 – Fourth International Colloquium on
Computation for Shells & Spatial Structures, IASS–IACM, Chania – Crete, Greece, (2000).
30