Monolithic and partitioned approaches to determine static deformation of membrane structures due to
ponding
N.K. Narayanana,b,∗, R. W¨uchnera, J. Degrooteb,c
aChair of Structural Analysis, Technical University of Munich
bDepartment of Electromechanical, Systems and Metal Engineering, Ghent University Sint-Pietersnieuwstraat 41, 9000 Ghent, Belgium
cFlanders Make, Belgium
Abstract
This paper proposes monolithic and partitioned methods to calculate the static deformation of a membrane structure due to a given volume of pond- ing water. The partitioned methods involve coupling of a structural solver for membranes and a volume-conserving solver, modeling static incompressible fluid. Two methods of this type are proposed, either using coupling iter- ations with convergence accelerator between structural solver and volume- conserving solver or adding the linearized fluid behavior in the structural solver in addition to the external coupling iterations. The monolithic meth- ods solve the system of structural equations under hydrostatic load with the volume conservation behavior of the fluid included in the Newton-Raphson (N- R) iterations of the structural solver. One such method was already discussed in the literature and updates the free surface plane to conserve volume exactly after every N-R iteration. In the second, new monolithic method, the vol-
∗Corresponding author
Email address: [email protected](N.K. Narayanan)
ume conservation constraint is added as an additional equation and solved together with the structural equations. It was found that the partitioned method used with a quasi-Newton convergence accelerator was very robust but slower than the monolithic methods. On the other hand, the new mono- lithic method proposed in this paper was found to be both computationally efficient and robust.
Keywords: Ponding, Partitioned method, Monolithic method, Hydrostatic load, Volume-conserving solver
1. Introduction
1
Membrane structures have a unique characteristic of carrying loads by
2
undergoing significant deflection. This makes them efficient in terms of ma-
3
terial usage compared to the load capacity but also makes them vulnerable
4
to ponding. Most light weight structures are designed with sufficient gradi-
5
ent to avoid this scenario. However, there are cases where a seeding event
6
such as snowfall can create a local depression to trigger ponding. Following
7
the seeding event, based on the initial prestress, the type of cable supports
8
and the elastic property of the membrane material, rain can lead to a stable
9
or unstable water pond. The latter scenario will be fatal for the structure
10
as this will result in indefinite increase of accumulating water till the struc-
11
tural collapse or failure. Therefore, it is important to evaluate membrane
12
structures for stability under ponding. This requires a fluid-structure (FSI)
13
simulation between the membrane and the ponding water.
14
Even in the stable pond scenario, if the rain is accompanied by strong
15
winds, the wind flow around the structure may induce large oscillations. In
16
2011, during the Pukkelpop festival held in Kiewit (Belgium) [1], a strong
17
wind interacting with ponding water led to huge swaying of the large festival
18
tents, eventually resulting in the collapse of these structures. Studying such
19
cases will involve fluid-structure interaction simulation between the mem-
20
brane structure, the water and the wind flow. Imposing an initial condition
21
with a pond on a membrane requires computation of the static deformation of
22
the membrane structure under the load of a given volume of ponding water,
23
which is the main motivation of the current work.
24
Some of the other applications of this analysis include floating caps of oil
25
storage tanks [2], and optical reflector forming using ponding loads [3]. Com-
26
pared to the many other aspects of analysis on membrane structures such
27
as large deformation analysis, form finding [4], wrinkling [5] and membrane
28
wind interaction [6], the analysis involving ponding water on a membrane
29
structure is relatively rare. What makes this type of analysis in membrane
30
structures challenging is the that the shape of the ponding fluid on the struc-
31
ture is unknown. Therefore, the region of fluid loading is not known before;
32
in most cases the structure will be initially flat before the ponding analysis
33
and so cannot contain any fluid. The deformation of the structure is depen-
34
dent on structural stiffness and pressure exerted by the fluid on the structure,
35
which is in turn a function of structural deformation. Therefore, the prob-
36
lem of finding the deformed shape of a structure under the hydrostatic load
37
exerted by a given volume of ponding fluid is very non-linear.
38
In the literature, stability behavior under ponding has been extensively
39
discussed by Szyszkowski and Glockner [7] where they studied ponding stabil-
40
ity and deformation on spherical inflatables by solving axi-symmetric mem-
41
brane equations with the hydrostatic loads. Tuan [3] in his work focused
42
on large deformations and strains of initially flat, simply supported circular
43
membranes under gradually accumulated fluid. He used fourth-order Runge-
44
Kutta numerical integration with an iterative finite element analysis using
45
shell elements to calculate the deformation due to ponding. However, these
46
studies only involved axi-symmetric geometries. A more general approach to
47
calculate deformation due to hydrostatic follower forces on structures in the
48
finite element framework is discussed in [8] where they linearize the static
49
behaviour of incompressible fluid under gravity to obtain the symmetric load
50
stiffness matrix used in the Newton-Raphson (N-R) iterations. The symme-
51
try of the load stiffness matrices is also discussed in [9, 10] with the name
52
elasto-gravity operator. Similar work can be also found in a more recent
53
paper by Hoareau and De¨u [11, 12], where a level set approach is used for
54
numerical integration on the loaded surface to compute volume, nodal forces
55
and load stiffness matrix, where the element faces were part of a quadratic
56
hexahedral mesh. They computed the deformed shape of tanks partially filled
57
with liquid by performing volume conservation in every structural N-R itera-
58
tion with the added load stiffness matrix discussed in [8]. Since their primary
59
interest was to study deformation of tanks under hydrostatic loads, a good
60
initial geometry was available that can contain fluid, and thus relatively sim-
61
pler than the ponding analysis on large membrane tents. An example closely
62
related to ponding on membrane structures can be found in [13] where they
63
studied stability of a hydrostatic load on a flat circular membrane. They
64
used a generalized path-following scheme [14] with free surface height as a
65
controlling parameter to plot the equilibrium path of the structure. In their
66
analysis, they found several limit points on the equilibrium path when the
67
free surface height was used as a controlling parameter and suggested to use
68
the volume of the fluid instead.
69
All the studies discussed in the previous paragraph fall under the category
70
of monolithic methods to compute structural deformation under hydrostatic
71
loads. The current paper discusses two monolithic methods to calculate
72
static deformation due to a fixed volume of ponding fluid. The first method
73
which imposes conservation of volume after every structural N-R iteration,
74
similar to one discussed in [11] but a faster and robust iteration scheme is
75
used for volume conservation. Therefore, in this paper it is called monolithic
76
method with volume conservation inside structural iterations (MVCIS). This
77
is because unlike a flexible water tank which has some stiffness due to the
78
geometry, the ponding analysis involving a relatively flat and flexible mem-
79
brane structure will undergo large deformation during initial N-R iterations
80
of the structural solver, thus requiring an efficient and robust algorithm for
81
volume conservation. The main problem with this monolithic method is that
82
it enforces the volume conservation constraint exactly in non-equilibrium
83
shapes found during structural N-R iterations, which is unnecessary and in
84
some cases it led to divergence. The second method, which is a novelty,
85
solves the structural equations under hydrostatic loads with the constraint
86
that the fluid volume should be equal to the target volume. The structural
87
equations with the constraint are solved using N-R iterations by linearizing
88
the system of equations with the constraint. This way the structural equi-
89
librium equations and volume constraint are satisfied only at the end of N-R
90
iterations. The proposed method therefore is called monolithic method with
91
volume conservation as constraint (MVCC). This method was found to be
92
more robust than MVCIS, which will be shown later in Section 8.2.
93
Clearly, implementation of the above methods is only possible if the struc-
94
tural solver is accessible, but in some cases where the solver is proprietary
95
software, ponding analysis can be only performed with partitioned methods.
96
Therefore, in this paper we also a present partitioned methods to perform
97
ponding analysis. One example of this approach is presented in the work of
98
Bown et al. [15], where an in-house structural code inTENS is coupled with
99
a shallow water solver in a partitioned method to analyze ponding on ten-
100
sioned membrane structures. The partitioned methods for ponding analysis
101
presented in this paper use a volume-conserving solver instead of a transient
102
shallow water solver as used by Bown et al. The volume-conserving solver
103
models the quasi-static behavior of fluid by updating the free surface, which
104
is a plane perpendicular to gravity, to conserve a given volume of the ponding
105
fluid. In this method the structural solver and the volume-conserving solver
106
are executed sequentially inside a loop with the output of the other solver as
107
its input. The volume-conserving solver takes the displacement field of the
108
structure as input and updates the free surface plane to conserve the volume,
109
while the structural solver uses the updated hydrostatic pressure, which de-
110
pends on the new vertical height of the free surface plane to calculate a new
111
displacement field, resulting in a fixed point iteration. Convergence accel-
112
erators are used to speed up the convergence and stabilize the fixed point
113
iteration [16, 17]. The iterations are continued till the norm of the fixed point
114
residual, defined later in Section 6.1, is below certain tolerance.
115
Additionally, a second partitioned method is presented in this paper in
116
which a linearized fluid behavior is added inside the structural solver to ac-
117
celerate the fixed-point iterations. Technically, this is not a pure partitioned
118
method, since the method involves modifying a structural solver. However,
119
the method is classified as partitioned method because it still involves outer
120
fixed point iterations to solve the problem. While this method loses the ad-
121
vantage of code modularity, it has better convergence characteristics than
122
the pure partitioned method due to the inclusion of linearized behavior of
123
the fluid solver in the structural solver. However, it has one problem at the
124
first coupling iteration when the fluid volume increment is large, which will
125
be discussed in Section 8.2.
126
The outline of the paper is as following. In Section 2, the mathematical
127
formulation of the ponding problem is presented, which involves non-linear
128
equilibrium equations of the membrane structure and equilibrium equations
129
of the fluid. This is followed by constitutive equations for the isotropic
130
plane-stress linear elastic and hyper-elastic material model. Subsequently,
131
the two solver components used in the analysis are presented in Section 3:
132
the structural solver with membrane elements and the volume-conserving
133
solver, which models the quasi-static behavior of the fluid. The linearization
134
of the fluid loading used in N-R iterations of three of the discussed methods
135
is explained in Section 4. Section 5 and Section 6 discuss the monolithic and
136
partitioned methods for ponding analysis, respectively. The procedure for
137
integration on a discretized surface required by the various methods is pre-
138
sented in Section 7. Finally, in Section 8 the proposed methods are analyzed
139
and compared using numerical examples.
140
2. Mathematical formulation
141
Consider a membrane structure, denoted by ∂Ωs, containing a certain
142
volume Vt of incompressible fluid of specific weight γf. The fluid region is
143
denoted by Ωf, which is enclosed by free surface of the fluid (∂Ωf) and wetted
144
surface of the membrane (∂Ωf s). This system has two components: fluid
145
and membrane structure. To find the static deformation due to ponding, the
146
equilibrium equations of both fluid and membrane have to be solved along
147
with the constraint that the volume of fluid is equal to Vt.
148
Figure 1: Ponding on a membrane structure.
2.1. Fluid equations
149
Under static conditions, the free surface of the fluid is always flat and
150
perpendicular to gravity. For the sake of brevity, we assume that gravity is
151
along negative z-direction and thereforeez is the unit normal at any point on
152
the free surface. Additionally, the pressure pat any point on the free surface
153
is zero (relative to atmosphere). This boundary condition with the fluid
154
equilibrium equation at static conditions, given in Eq.(1) and the constraint
155
that the volume of fluid region Ωf should be equal to Vt, forms the system
156
of equations for the fluid at rest,
157
∇p=−γfez ∀x∈Ωf (1)
p= 0 ∀x∈∂Ωf (2)
Z
Ωf
dV =Vt. (3)
However, the system of equations Eqs. (1), (2) and (3) requires volume
158
discretization of Ωf. This is avoided by expressing it in terms of surface
159
quantities. To that end, integrating Eq.(1) with Eq.(2) as boundary condition
160
results in the familiar hydrostatic loading on the wetted surface,p=−γf(z−
161
zf) withz =x·ez, ∀x∈∂Ωf s and zf as the z-coordinate of the free surface.
162
Furthermore, with the absence of shear stress under hydrostatic condition the
163
traction at the wetted surface can be written in terms of pressure and the
164
unit normal ¯nas t=pn. For the membrane surface which is not in contact¯
165
with the fluid (∂Ωs\∂Ωf s), the pressure relative to the atmosphere is zero.
166
Consequently, in the absence of any other external load the traction is equal
167
to the zero vector in ∂Ωs\∂Ωf s. The volume conservation constraint Eq. (3)
168
can also be expressed in terms of a surface integral of infinitesimal vertical
169
volume elements dV = (z−zf)ez·n¯ dS, which results in the following set
170
of equations for the fluid in terms of surface quantities,
171
t =−γf(z−zf) ¯n ∀x∈∂Ωf s, (4) t =0 ∀x∈∂Ωs\∂Ωf s, Z
∂Ωf s
(z−zf)ez·n¯ dS =Vt, (5)
2.2. Structural equations
172
The membrane structure shown in Fig. 1 is in static equilibrium with the
173
ponding fluid. Therefore, by applying the principle of virtual work for the
174
structure in its current configuration we can write,
175
Z
∂Ωs
tσ :δdS
| {z }
δWint
− Z
∂Ωs
t·δudS
| {z }
δWext
= 0, ∀δu∈ Cu (6)
whereσ is the Cauchy stress tensor and= 12 ∇xδu+∇Txδu
is the virtual
176
Eulerian strain tensor, with ∇x•= ∂x∂•, δu as virtual displacement field and
177
Cu is the kinematically admissible space of smooth enough functions. The
178
thickness of the membrane is denoted byt, which need not be constant. The
179
equation has two terms: the internal virtual work (δWint), and the exter-
180
nal virtual work (δWext). In the total Lagrangian formulation, the internal
181
virtual work is written in the reference configuration as,
182
δWint= Z
∂Ω0s
t S :δE dS0, (7)
whereSis the second Piola-Kirchhoff stress tensor andδE= 12 δFTF +FTδF
183
is the virtual Green-Lagrange strain tensor, with δF = ∇Xδu and F =
184
∇Xu. The operator ∇X• = ∂X∂• is defined as the gradient of a field with
185
respect to the material coordinate X. In a typical displacement based ap-
186
proach [18], the internal and external work are expressed in terms of the
187
displacement field as unknown. To that end, the stress tensor at any point
188
is expressed as a function of the strain tensor which is in turn written as a
189
function of the displacement field using the strain definition above.
190
The external virtual work is generally written in terms of the quantities
191
in the current configuration which depends on the traction field t from the
192
ponding fluid resulting in the final expression of the principle of virtual work
193
that needs to be satisfied at equilibrium,
194
Z
∂Ω0s
t S :δE dS0− Z
∂Ωs
t·δudS = 0. (8) 2.3. Constitutive models
195
The relation between stress and strain tensor is described using the con-
196
stitutive model or material law. In the numerical example presented in the
197
paper, two types of hyper-elastic materials are used: the Saint-Venant Kirch-
198
hoff material law, given in Eq. (9) and the incompressible Mooney-Rivlin
199
material law, given in Eq. (10). The former material law is applicable for
200
large displacements and small strains cases, while the latter is applicable for
201
large displacements and finite strains [19].
202
SSV = 2µE+λtr(E)I, (9)
203
SM R =βC−1+ 2∂ΨM R(C)
∂C , (10)
where the subscripts SV and M R stand for Saint-Venant Kirchhoff and
204
Mooney Rivlin, respectively. As clear from Eq.(9), the relation between the
205
2nd Piola Kirchhoff and the Green-Lagrange strain tensor E= 12 FTF −I
206
is linear. The two constants appearing in Eq. (9) are called Lam´e constants
207
which are related to the material properties, Young’s modulus E and Pois-
208
son’s ratio ν as
209
λ= νE
(1 +ν)(1−2ν), µ = E
2(1 +ν). (11)
The relation between stress and strain tensor for the incompressible Mooney-
210
Rivlin material on the other hand is non-linear. The expression of the 2nd
211
Piola-Kirchhoff stress tensor is given in terms of the Cauchy-Green strain
212
tensor (C =FTF), strain energy function ΨM R and Lagrange multiplier β.
213
The most commonly used expression of the strain energy function is written
214
in terms of the first invariant (I1) and second invariant (I2) of the Cauchy-
215
Green strain tensor,
216
ΨM R =c1(I1−3) +c2(I2−3), (12) whereI1 =tr(C) and I2 = 12 I12−tr(CTC)
with material constantsc1 and
217
c2 [20].
218
For the plane stress case the 2nd Piola-Kirchhoff stress tensor can be fur-
219
ther simplified because the components along the thickness direction vanish,
220
leading to a form:
221
S =
S11 S12 0 S21 S22 0
0 0 0
, (13)
where direction 3 is normal to the membrane surface (thickness direction).
222
Moreover, by using the small thickness assumption of the membrane as com-
223
pared to the other spatial dimensions, the off-diagonal components along the
224
thickness direction can be neglected. As a result, the Cauchy-Green tensor
225
and Green-Lagrange strain tensor have the following simplified forms:
226
C=
C11 C12 0 C21 C22 0 0 0 C33
,E=
E11 E12 0 E21 E22 0 0 0 E33
. (14)
Using the condition S33 = 0, we get the linear stress-strain relation for
227
Saint-Venant Kirchhoff material expressed in Voigt notation,
228
S11 S22 S12
= E
1−ν2
1 ν 0
ν 1 0
0 0 1−ν2
E11 E22 2E12
. (15)
Similarly, for the case of the Mooney-Rivlin material law the value of
229
β is determined by using the plane stress condition S33 = 0 and using the
230
Cauchy-Green tensor of the form given in Eq. (14). When the obtained value
231
of β is substituted in Eq. (10), we get the following relation between stress
232
and strain:
233
S11 S22 S12
=2c1
1 1 0
− 2c1 (C11C22−C122 )2
C22 C11
−C12
+ 2c2
C11C22−C122
1 1 0
+ 2c2
1− C11C22−C122 2
(C11+C22)
C22 C11
−C12
. (16)
3. Solver components
234
The deformed shape of the membrane is found when the virtual work
235
equation in Eq. (8) is satisfied together with the fluid equations given in
236
Eq. (4) and (5). The virtual work equation is solved using a structural solver
237
with the load boundary condition from Eq. (4), while the volume conservation
238
constraint is implemented using a volume-conserving solver. Three of the four
239
methods discussed in this paper find the solution by coupling these solvers to
240
determine the deformed shape. The only exception is the second monolithic
241
method (MVCC), where the structural equations are modified to include the
242
volume conservation constraint without using the volume conservation solver,
243
which will be discussed in Section 5.2.
244
3.1. Structural solver
245
In this section we will briefly discuss how the structural solver solves the
246
virtual work equation Eq. (8), given a traction field tfrom the ponding fluid.
247
Using the stress-strain relation given in Eqs. (15) and (16) and the strain
248
definition discussed before, we can express the internal virtual work in terms
249
of only the displacement field as unknown. The resultant expression will be
250
non-linear for large displacements, regardless of the use of linear or non-linear
251
material law, as C and E are non-linear functions of the displacement field.
252
Moreover, if the traction or the resultant external forces on the structure
253
depends on the deformed state, the external virtual work will be also a non-
254
linear function of u. In fact, the hydrostatic forces on the structure due to
255
ponding is one such example of so-called follower forces.
256
The finite element discretization of the internal and external virtual work
257
gives a non-linear residual equation, Eq. (17) as a function of the nodal dis-
258
placement vector ˆu, where the displacement and virtual displacement field
259
are approximated using the shape function matrix N as u≈uh =Nuˆ and
260
δu ≈ δuh = Nδu, respectively. The superscriptˆ •h represents the approx-
261
imation of a given field with finite element discretization and the accent ˆ•
262
represents the associated nodal vector for the approximation. The resultant
263
residual equation from the discretized virtual work expression can be written
264
as,
265
fˆext( ˆu)−fˆint( ˆu) = 0, (17) where ˆfint is the internal nodal forces and ˆfext is the external nodal forces.
266
The structural solver used in the current work is implemented in an open-
267
source FEM code called KRATOS [21], which uses the N-R algorithm to
268
solve the vector equation given in Eq. (17), where at every iteration we solve
269
a linear system
270
Ktan∆ ˆu= ˆr (18) to obtain the update in the nodal displacements ∆ ˆu, where ˆr = ˆfext−fˆint
271
is the out of balance force vector or residual vector, and Ktan is the tangent
272
stiffness matrix, where
273
Ktan =Kmem−Kl. (19) The matrix Kmem is the familiar global tangent stiffness matrix from
274
membrane elements, interested readers can refer to [22] for more details. The
275
second matrix in Eq. (19) is called the load stiffness matrix, which depends
276
on the type of follower load [8, 23]. In the current paper, the follower tangent
277
stiffness matrixKldepends on the type of approach, whether it is partitioned
278
or monolithic.
279
3.2. Volume-conserving solver
280
The ponding fluid on membrane structures (generally water) is always
281
incompressible and therefore the volume of the ponding fluid is always con-
282
served. Moreover, under static conditions the free surface of the fluid is flat
283
and perpendicular to gravity i.e. normal to the z-direction in Fig. 1. These
284
properties of the ponding fluid under static conditions are used to update the
285
free surface position using an algorithm called the volume-conserving solver.
286
The volume-conserving solver consists of two components: a volume calcu-
287
lation algorithm and an iterative algorithm to conserve a given volume by
288
moving the flat and horizontal free surface vertically. The volume of the fluid
289
enclosed by ∂Ωf ∪∂Ωf s in Fig. 1 can be calculated using the relation,
290
Vf = Z
∂Ωf s
(z−zf)ez·n¯ dS (20) as already discussed in Section 2.1, and its derivative with respect to zf,
291
∂Vf
∂zf = Z
∂Ωf s
−ez·n¯ dS =Af, (21) where Af is the area of the free surface. In the current work, the leap-
292
frogging Newton’s method is used for conserving a given volume. This
293
method is discussed in detail in [24]. It consists of a Newton step followed
294
by a pseudo secant step, as shown in Fig. 2. The main advantage of this
295
method is that it has cubic convergence at a simple root with computa-
296
tional efficiency comparable to that of Newton’s method. Newton’s method
297
and leap-frogging Newton method were tested for volume conservation with
298
some axi-symmetric geometries and it was found that the leap-frogging New-
299
ton was much more robust and had faster convergence rate than Newton’s
300
method. Hence, it was chosen over the other. The equations used for iter-
301
ation to conserve volume are given in Eqs. (22) and (23), with the function
302
f(zfm) being the volume residual (Vfm −Vt), and f0(zfm) its derivative with
303
respect tozf, where the superscript•m denotes the iteration number and the
304
accent ˇ• is used to specify quantities at the intermediate position. It should
305
be noted that the denominator in Eq. (23) can cause rounding-off problems,
306
as it could become very small quickly. To avoid this problem, using the
307
machine precision npre we add another stopping criteria for the iterations,
308
|f(zfm)−f(ˇzfm)|=|Vfm−Vˇfm|<10npre−1.
309
Figure 2: Leap froging Newton’s algorithm adopted from [24], withzf† as the root of the functionf(zf).
ˇ
zfm =zmf − f(zmf )
f0(zfm) (22)
zfm+1 =zmf − f(zfm)2
f0(zfm) f(zfm)−f(ˇzfm) (23) Using Eqs. (20)-(23) we can write an algorithm for the volume-conserving
310
solver. As already discussed, the leap-frogging Newton algorithm is robust
311
and has good convergence characteristics. However, for certain cases it would
312
update the free surface position below the wetted surface. As a result, the
313
algorithm would fail to update the free surface in the next step since the
314
calculated volume and free surface area would be zero. One example of such
315
a case is shown in Fig. 3, where the intermediate position of the free surface
316
plane after the nth iteration goes below the membrane surface i.e. ˇzfm < zf∗
317
or ∆ˇzfm < zf∗−zf with Vf(zf∗) = 0. As evident from Fig. 3a the limit value
318
of the slope is given by fmin0 <(Vfm−0)/(zmf −zf∗). Therefore, the standard
319
leap-frogging Newton algorithm is modified to limit the slope to avoid such
320
cases and the final proposed algorithm can be found in Algorithm 1.
321
Algorithm 1 Modified Leap-frogging Newton’s method for volume conser- vation.
1: m= 0
2: while
Vfm−Vt
Vt
> εand |Vfm−Vˇfm|<10npre−1 and m < mmax do
3: Calculate f(zfm) =Vfm−Vt
4: Calculate f0(zfm) =Amf
5: if ∆ˇzfm+1 <(zf∗−zmf ) then
6: Af =Vfm/(zmf −z∗f)
7: end if
8: Calculate ˇzmf using Eq. (22). Move the plane to ˇzmf .
9: Calculate f(ˇzfm) = ˇVfm−Vt.
10: Calculate zm+1f using Eq. (23). Move the plane to zfm+1.
11: m =m+ 1
12: end while
(a) (b)
Figure 3: An example where the standard leap-frogging Newton volume conservation algorithm will fail, when implemented without any condition on the slope of the volume residual: a) volume of the ponding fluidVf vs position of the free surfacezf showing the minimum slope to avoid failure of the algorithm , b) corresponding membrane structure and the free surface update that will move the free surface below the membrane surface.
4. Linearization of the fluid loading
322
The ponding fluid interacts with the structure by applying traction on
323
the wetted region, which manifests as an external nodal force vector ˆfext on
324
the structure. It will be explained in this section that the external nodal
325
force vector is a non-linear function of the displacement field. Therefore, by
326
including the linearized behavior of the fluid loading in the N-R iteration
327
convergence speed can be greatly improved. In terms of implementation
328
this means including a load stiffness matrix, mentioned in Section 3.1 in
329
the N-R iterations of the structural solver. In this section, we present the
330
full linearization of the fluid loading and discuss the load stiffness matrices
331
associated with the different contributions to the load behavior of the fluid.
332
The derived load stiffness matrices will be used fully or partially depending
333
on the method.
334
The expression of the nodal force vector can be obtained by considering
335
the discretized virtual external work δWexth associated with δWext in Eq. (6)
336
and using the traction t from Eq. (4),
337
δWexth = Z
∂Ωf s
δuh· −γf(zh −zf) ¯nh dS+ Z
∂Ωs\∂Ωf s
δuh·0dS
=−γf Z
η
Z
ξ
δuh·(zh−zf)nh dξdη
=δuˆT
−γf Z
η
Z
ξ
(zh−zf)NTnh dξdη
| {z }
fˆext
=δuˆTfˆext. (24)
In the equations above it can be seen that the domain of integration is
338
changed to the parametric space ξ−ηof the discretized wetted surface with
339
base vectorsgξh =xh,ξ andgηh =xh,η, where•,r = ∂•∂r for any parameterr. This
340
transformation of integration domain uses the definitions of a normal vector
341
at any point on the discretized wetted surfacenh =gξh×gηh =
gξh×ghη n¯h
342
and an infinitesimal surface area dS =
gξh×gηh
dξdη. It can observed that
343
the discretized virtual external work and consequently nodal external force
344
vector is a non-linear function of nodal displacement vector ˆu as
345
nh =gξh×ghη =xh,ξ( ˆu)×xh,η( ˆu), z =xh( ˆu)·ez,
and zf = zf( ˆu) from the volume conservation constraint. Linearizing the
346
discretized virtual work we get,
347
δWexth + ∆δWext[∆uh] =− Z
η
Z
ξ
δuh·γf(zh−zf)nh dξdη
−γf Z
η
Z
ξ
(zh−zf)δuh· ∆nh[∆uh]dξdη
| {z }
∆δWext∆n[∆uh]
−γf Z
η
Z
ξ
δuh·∆zh[∆uh]nh dξdη
| {z }
∆δWext∆z[∆uh]
+γf Z
η
Z
ξ
δuh·∆zf[∆uh]nh dξη
| {z }
∆δWext∆zf[∆uh]
. (25)
The linear part of change in external work due to ∆uh can be split into
348
three components: ∆δWext∆n[∆uh], ∆δWext∆z[∆uh] and ∆δWext∆zf[∆uh]. The
349
second term in rhs of Eq. (25) ∆δWext∆n[∆uh] accounts for the change in
350
normal vector due to the wetted surface movement with constant hydrostatic
351
pressure. The effect of change in hydrostatic pressure from the movement
352
of the wetted surface alone is represented by ∆δWext∆z[∆uh] . Finally, the
353
contribution from the movement of the free surface to conserve volume is
354
captured by ∆δWext∆zf[∆uh]. The three components of the change in external
355
virtual work can be written in the form ofδuˆTKl•∆ ˆusuch that the linear part
356
of change in the nodal force vector due to the nodal displacement increment
357
∆ ˆu can be written as ∆ ˆf•[∆ ˆu] = Kl•∆ ˆu, where superscript • represents
358
the three contributions that we discussed before. Thus, we have three load
359
stiffness matrices: Kl∆n,Kl∆z and Kl∆zf.
360
Additionally, it is well known that a constant pressure and hydrostatic
361
pressure loading on large displacement cases are conservative [25]. Therefore,
362
the associated load stiffness matrices are symmetric. The proof of symmetry
363
for the constant pressure can be found in [23] and for hydrostatic pressure
364
with constant fluid volume can be found in [8, 9]. The symmetric part of
365
the load stiffness matrices is obtained by performing integration by parts and
366
some algebraic manipulations. The interested readers are encouraged to refer
367
to [8] for detailed derivation. In the derivation, they list five conditions at the
368
boundary of the wetted surface Γ to have symmetric load stiffness matrices.
369
If atleast one of the conditions is satisfied it would lead to symmetric load
370
stiffness matrices. Among these, either (i)p= 0 or (ii)δu= 0 on Γ is always
371
satisfied in ponding scenario, see Fig. 4. With the derivation of the symmetric
372
load stiffness matrices already given in some of the previous work [8, 9]
373
and more recently [11], we directly state the linear part of the change in
374
discretized external virtual work containing only symmetric terms:
375
(a) (b)
Figure 4: Two scenarios of ponding on a membrane structure where the boundary condi- tions at the wetted surface Γ lead to symmetric load stiffness matrices: a) the membrane structure is partially filled (p= 0 at Γ) and b) the membrane structure is fully filled and the free surface moves above the fixed boundary (δu=0at Γ).
∆δWexth [∆uh] =−1 2γf
Z
η
Z
ξ
(zh−zf) δuh,ξ ×gηh−δuh,η×gξh
·∆uh dξdη +1
2γf Z
η
Z
ξ
(zh−zf)δuh· ghη ×∆uh,ξ −gξh×∆uh,η dξdη
−1 2γf
Z
η
Z
ξ
δuh· nh⊗ez+ez⊗nh
·∆uh dξdη
− γf Af
Z
η
Z
ξ
δuh·nh dξdη Z
η
Z
ξ
nh·∆uh dξdη, (26) where the terms in the first three lines are the sum of the contributions from
376
change in normal and change in hydrostatic pressure from the movement
377
of the wetted surface alone i.e ∆δWext∆n + ∆δWext∆z. The last term is the
378
contribution from the free surface movement to conserve volume (∆δWext∆zf),
379
which is obtained by substituting the expression of the linear part of the free
380
surface movement ∆zf[∆uh] given in Eq. (27) in the expression of ∆δWext∆zf
381
(a) (b)
Figure 5: Volume conservation using the linear part of the change in volume from mem- brane deformation: a) the linear part of change in volume ∆Vf due to deformation of the wetted surface, indicated in red color, b) free surface update ∆zf by considering a cylinder of volume ∆Vf with base area equal to the free surface area and height equal to ∆zf.
in Eq. (25). The linear part of the free surface movement can be obtained
382
by an observation that only the normal component of the wetted surface
383
displacement contributes to volume change and dividing the obtained volume
384
change (∆Vf) by the free surface area gives the linear part of the free surface
385
movement, as illustrated in Fig. 5.
386
∆zf[∆uh] =−∆Vf Af =−
R
η
R
ξ ∆uh ·nh dξdη
Af (27)
To obtain the load stiffness matrices the external virtual work expression
387
in Eq. (26) can be written in terms of the associated nodal vectors and shape
388
function matrix, which leads to the following expression:
389
∆δWext[∆uh] =−δuˆT1 2γf
Z
η
Z
ξ
(zh−zf) N,ξTΩhηN −N,ηTΩhξN
dξdη ∆ ˆu +δuˆT1
2γf Z
η
Z
ξ
(zh −zf) NTΩhηN,ξ −NTΩhξN,η
dξdη ∆ ˆu
−δuˆT1 2γf
Z
η
Z
ξ
NT nh⊗ez+ez⊗nh
N dξdη ∆ ˆu
−δuˆT γf Af
Z
η
Z
ξ
NTnh dξdη Z
η
Z
ξ
NTnh dξdη T
∆ ˆu.
(28) where Ωhξ and Ωhη are the skew matrices associated with the cross product
390
of the base vectors gξh and ghη, respectively. Finally, the symmetric load
391
stiffness matrix associated with each part can be extracted by comparing
392
with the expression δuˆTKl•∆ ˆu,
393
Kl∆n =−1 2γf
Z
η
Z
ξ
(zh−zf) N,ξTΩhηN −N,ηTΩhξN dξdη
+1 2γf
Z
η
Z
ξ
(zh−zf) NTΩhηN,ξ−NTΩhξN,η dξdη
Kl∆z =−1 2γf
Z
η
Z
ξ
NT nh⊗ez+ez⊗nh
N dξdη (29) Kl∆zf =− γf
Af Z
η
Z
ξ
NTnh dξdη Z
η
Z
ξ
NTnh dξdη T
∆ ˆu. (30) 5. Monolithic methods for ponding analysis
394
5.1. Monolithic method with volume conservation inside structural iterations
395
(MVCIS)
396
Having discussed the linearized equations for the fluid loading in the pre-
397
vious section, we can now start using it in the different methods for ponding
398
analysis. The first method which is classified as monolithic method involves
399
volume conservation inside the structural N-R iterations. In this method we
400
solve the structural equations by incorporating the full linearized equations
401
of the fluid loading inside the N-R iterations of the structural solver. In terms
402
of implementation this means we use the load stiffness matrices discussed in
403
Section 4 along with the nested iterations of the volume-conserving solver to
404
update the free surface after every N-R iteration. The complete algorithm is
405
written in Algorithm. 2
Algorithm 2 Monolithic method with volume conservation inside structural iterations (MVCIS)
1: n = 0
2: Find zf0 using Algorithm 1 with Vt as input argument
3: while
fˆext−fˆint
> ε and n < nmax do
4: Update: Kmem, Kl∆n, Kl∆z, Kl∆zf, ˆfext and ˆfint using ˆun and znf
5: Solve:
Kmem−Kl∆n−Kl∆z−Kl∆zf
∆ ˆun+1 = ˆfext−fˆint
6: Update displacement: ˆun+1 = ˆun+ ∆ ˆun+1
7: Update structure: ˆxn+1 = ˆX+ ˆun+1
8: Update free surface using Algorithm 1 with Vt as input argument
9: n =n+ 1
10: end while
406
5.2. Monolithic method with volume conservation as a constraint (MVCC)
407
The problem of determining the static deformation of a structure un-
408
der the load of fixed volume of fluid in a monolithic approach with volume
409
conservation as constraint g( ˆu, zf) can be stated as follows:
410
fˆint( ˆu)−fˆext( ˆu, zf) = 0, (31)
g( ˆu, zf) =Vf( ˆu, zf)−Vt= 0. (32) Here, we introduce an additional independent variablezf, the z-coordinate
411
of the free surface, which allows the volume conservation constraint to be in-
412
corporated in the system of equations. The system of equations given in Eqs.
413
(31) and (32) can be solved using N-R algorithm, where the linearized form
414
at iteration n+ 1 can be written as:
415
∂fˆint( ˆun)
∂uˆ − ∂fˆext uˆn, znf
∂uˆ
!
∆ ˆun+1−∂fˆext uˆn, zfn
∂zf ∆zfn+1 = ˆfext uˆn, znf
−fˆint( ˆun) (33)
∂g uˆn, zfn
∂uˆ ∆ ˆun+1+∂g uˆn, znf
∂zf ∆zfn+1 =Vt−V uˆn, zfn
. (34)
In Eq. (33), ∂f∂ˆˆintu is the familiar global membrane tangent stiffness matrix
416
Kmem [22]. The second term, ∂f∂ˆˆextu in the equation is the sum Kl∆n+Kl∆z,
417
discussed in Section 4. The derivative ∂∂zfˆext
f can be obtained by differentiating
418
fˆext given in Eq. (24) with respect to free surface height. We have not
419
discussed this before as zf was not an independent variable. The obtained
420
expression is given in Eq. (35). To obtain the terms in Eq. (34), we only need
421
to calculate the derivative of the fluid volume with respect to the variables,
422
sinceVtis constant. The first term in the left-hand side of Eq. (34) represents
423
the change in fluid volume with respect to the nodal displacement vector.
424
This can be obtained by substituting ∆uh =N∆ ˆuin the expression of ∆Vf
425
