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applied
sciences
Article
Low-Rank Approximation of Frequency Response Analysis of
Perforated Cylinders under Uncertainty
Harri Hakula 1,* and Mikael Laaksonen 2
Citation: Hakula, H.; Laaksonen, M.
Low-Rank Approximation of
Frequency Response Analysis of
Perforated Cylinders under
Uncertainty. Appl. Sci. 2022,12, 3559.
https://doi.org/10.3390/
app12073559
Academic Editors: Dimitrios Aggelis
and Giuseppe Lacidogna
Received: 11 February 2022
Accepted: 29 March 2022
Published: 31 March 2022
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1Department of Mathematics and Systems Analysis, Aalto University, Otakaari 1, FI-00076 Espoo, Finland
2Wärtsilä Finland, Hiililaiturinkuja 2, FI-00180 Helsinki, Finland; Mikael.Laaksonen@wartsila.com
*Correspondence: Harri.Hakula@aalto.fi
Abstract:
Frequency response analysis under uncertainty is computationally expensive. Low-rank
approximation techniques can significantly reduce the solution times. Thin perforated cylinders, as
with all shells, have specific features affecting the approximation error. There exists a rich thickness-
dependent boundary layer structure, leading to local features becoming dominant as the thickness
tends to zero. Related to boundary layers, there is also a connection between eigenmodes and
the perforation patterns. The Krylov subspace approach for proportionally damped systems with
uncertain Young’s modulus is compared with the full system, and via numerical experiments, it
is shown that the relative accuracy of the low-rank approximation of perforated shells measured
in energy depends on the dimensionless thickness. In the context of frequency response analysis,
it then becomes possible that, at some critical thicknesses, the most energetic response within the
observed frequency range is not identified correctly. The reference structure used in the experiments
is a trommel screen with a non-regular perforation pattern with two different perforation zones.
The low-rank approximation scheme is shown to be feasible in computational asymptotic analysis of
trommel designs when the proportional damping model is used.
Keywords: frequency response; eigenproblem; perforated shells; p-version; uncertainty quantification
1. Introduction
Cylindrical shell structures are common in engineering. Within computational me-
chanics, they belong to the class of thin solids, the class which arguably remains computa-
tionally one of the most challenging ones. Here, the focus is on perforated cylinders—more
specifically, trommel screens and their frequency response. Perforated cylinders are used,
for instance, in mass dryers and as screening devices, and are naturally subjected to large
loads, which are often of impact type or otherwise concentrated [
1
,
2
]. Ideally, one would
like to be able to analyse perforated structures as homogenised, where the effect of the
perforation pattern is transferred to material parameters, the so-called effective material
parameters [3–5].
The structural models used in shell analysis are dimensionally reduced, where the
thickness is treated as a (dimensionless) parameter. Somewhat surprisingly, there is an
intricate interplay between the perforation patterns and the thickness, which sets limits to
homogenisation both under static loading and in dynamical eigenmode setting; see [
6
,
7
].
In this study, no attempts at homogenisation are made, and all simulations are carried out
in exact geometry.
Frequency response analysis of thin solids is a multiparametric problem. In the
structures considered here, uncertainty is included in the material properties; hence, the
parameters can be divided into the stochastic parameters defining the material properties
(Young’s modulus), and the deterministic parameters, i.e., the dimensionless thickness and
the frequency considered. From the point of view of the analysis, no changes are necessary
even if the perforation patterns were randomly perturbed as well. Many of these issues
Appl. Sci. 2022,12, 3559. https://doi.org/10.3390/app12073559 https://www.mdpi.com/journal/applsci
Appl. Sci. 2022,12, 3559 2 of 17
have been addressed in a previous study [
8
]. Modern stochastic analysis of multiparametric
problems leads to a large number of simulations. Low-rank approximation can be a highly
effective way of reducing the computational complexity. The Krylov subspace approach
pioneered by Freund is the one considered here [
9
]. Naturally, it becomes necessary to
establish limits within the parameter space when such approximations are applicable in
the problem class of interest.
Frequency response analysis is of interest in many engineering fields, and hence the body
of literature is not only vast but also scattered over a large number of journals. On structural
health monitoring alone, two review articles cover nearly 300 publications [
10
,
11
]. In this paper,
the underlying mathematical models are taken to be linear, and the materials isotropic
in order to simplify the discussion. As examples of more complicated configurations in
related problems, Wang et al. consider vibration composite beams under random loads [
12
]
and Vu et al. corrugated cylinders and nonlinear buckling [
13
]. A very recent paper by
Fu et al. [
14
] provides references to papers on complex vibration problems and related
uncertainty quantification. A well-known paper by Worden et al. [
15
] highlights specific
problematic issues in the uncertainty quantification of nonlinear systems. In the context
of collocation methods such as those used here, the fundamental question is whether the
interpolation property of the system is preserved. It is reasonable to expect that, within
linear elasticity, this is never violated.
The two foundational papers in introducing the Krylov subspace method in this
context are by Eid et al. [
16
] and Han [
17
]. Three more widely cited papers building on
the connection with the proportionally damped systems are Johnson and Wojtkiewicz [
18
],
Wu et al. [19]
, and Sepahvand and Marburg [
20
], where also material uncertainties are
included in a similar way as here. Wu et al. have continued with a series of
papers [21,22]
.
Moreover, Delissen et al. [
23
] have studied this approach in the context of topology optimi-
sation, where also the computational complexity can increase significantly.
The main theoretical contribution of this work is that, through carefully designed
experiments, it is shown that the relative accuracy of the low-rank approximation of
perforated shells depends on the dimensionless thickness. This is a consequence of the
boundary layers—that is, local features becoming dominant as the thickness tends to zero.
This is precisely the same phenomenon which affects homogenisation. Determining the
limits of low-rank approximation requires extensive and possibly expensive simulations.
Here, it is shown how computational asymptotic analysis becomes feasible if the damping
models and the related low-rank approximation scheme are chosen in the computationally
most advantageous way.
The reference problem is the trommel screen. Trommel screens are devices used in the
grading of raw materials and solid waste classification. Consider a perforated cylinder as
in Figure 1, but tilted so that the smaller holes are higher, and the whole cylinder rotating.
As the raw materials are fed into the trommel from the top, the perforation structure
separates objects of different sizes. What makes this device an interesting example is
that the perforation patterns are not regular either in size or distribution. For shells of
revolution, the eigenmodes have integer-valued wave numbers in the angular direction if
the material properties are isotropic, and since this condition is not satisfied, the frequency
responses of trommel screens can be more varied. The quantity of interest is the total energy
of the response, since there are no guarantees for the convergence of local features such as
pointwise displacement. In Figure 1, some of the effects of the deterministic parameters
are illustrated.
The rest of the paper is as follows: First, the shell eigenproblem and the related fre-
quency response problem are reviewed also in the stochastic setting; next, some recent
results on the free vibration of perforated results are discussed, including the lack of pre-
dictive power in the general perforation setting. In Section 4, the low-rank approximation
approach is outlined, including the powerful preconditioner necessary for the iterative
construction of the reduced basis; finally, an extensive set of numerical experiments is
discussed, before the conclusions in Section 6.
Appl. Sci. 2022,12, 3559 3 of 17
(a) (b) (c) (d)
Figure 1.
Trommel screen. Shell of revolution generated by a profile function
φ(x) =
1. Transverse
deflection fields under angular excitation of wave number
=
20. Effect of the deterministic param-
eters
t
(thickness) and
f
(frequency) is illustrated, (
a
)
t=
1
/
100,
f=
5, (
b
)
t=
1
/
100,
f=
40,
(
c
)
t=
1
/
1000,
f=
5, (
d
)
t=
1
/
1000,
f=
40. Notice how, from (
a
) to (
b
), the maximal intensity
moves from the area of the large perforations to the small ones, but from (
c
) to (
d
), the same effect
does not take place. In (
d
), the solution is locally dominant and the energy is concentrated on the
boundary layers.
2. Preliminaries
This section introduces both the shell eigenproblems and the collocation scheme, which
is used to compute the frequency responses in the cases where the material parameters,
e.g., Young’s modulus, are random. The discussion below follows closely that in [8].
2.1. Shell Eigenproblems
The free vibration problem for a general shell under the assumption of a time harmonic
displacement field leads to the following abstract eigenvalue problem: Find
u∈R3
and
ω2∈Rsuch that (S u =ω2M u
+boundary conditions. (1)
Above,
u={u
,
v
,
w}
represents the shell displacement field, while
ω2
represents the
square of the eigenfrequency. In the abstract setting,
S
and
M
are differential operators
representing deformation energy and inertia, respectively. In the discrete setting, they refer
to corresponding stiffness and mass matrices.
Reduction in Thickness
In the case of a shell of revolution with constant thickness
t
, the free vibration problem
for a general shell (1) can be stated as the following eigenvalue problem: Find
u(t)
and
ω2(t)∈Rsuch that
(tAMu(t) + tASu(t) + t3ABu(t) = ω2(t)M(t)u(t)
+boundary conditions. (2)
Again,
u(t)
represents the shell displacement field, while
ω2(t)
represents the square
of the eigenfrequency. The differential operators
AM
,
AS
and
AB
account for membrane,
shear, and bending potential energies, respectively, and are independent of
t
. Finally,
M(t)
is the inertia operator, which in this case can be split into the sum
M(t) = t Ml+t3Mr
,
with Ml(displacements) and Mr(rotations) independent of t.
Appl. Sci. 2022,12, 3559 4 of 17
The finite element method is based on the variational formulation of problem
(2)
.
Accordingly, when the space
V
of admissible displacements is introduced, the problem
becomes: Find (u(t),ω2(t)) ∈V×Rsuch that
t am(u(t),v) + t as(u(t),v) + t3ab(u(t),v) =
ω2(t)m(t;u(t),v)∀v∈V,(3)
where
am(·
,
·)
,
as(·
,
·)
,
ab(·
,
·)
and
m(t
;
·
,
·)
are the bilinear forms associated with the oper-
ators
AM
,
AS
,
AB
and
M(t)
, respectively. Obviously, the space
V
and the three bilinear
forms depend on the chosen shell model (see, for instance, [24]).
The energy norm
||| · |||
is defined in a natural way in terms of the deformation energy
E(u):=|||u||| :=pA(u,u)
. Similarly, for (squared) bending, membrane, and shear
energies, e.g.,
B(u):=t2AB(u
,
u)
. The shell model used throughout the paper, the Reissner–
Naghdi model, is outlined in Appendix A.
2.2. Frequency Response Analysis under Uncertainty
The stochastic version of the shell eigenproblem is obtained by introducing uncertain-
ties in the material properties or geometry of the shell. The key properties of the problem
and a framework for frequency response analysis under uncertainty are outlined in this
section. For computing the expected value as well as higher moments of the quantity of
interest, a stochastic collocation algorithm is proposed.
2.2.1. The Stochastic Eigenproblem
Let
D⊂R2
denote the computational domain (see Appendix A). Assume further that
the Young’s modulus is a random field expressed in a series of the form
E(x,y,ξ) = E0(x,y) +
M
∑
m=1
Em(x,y)ξm,(x,y)∈D, (4)
where
M∈N
and
ξ= (ξ1
,
ξ2
,
. . .
,
ξM)
is a vector of mutually independent random
variables. For simplicity, it is assumed here that each
ξm
is uniformly distributed in some
interval of
R
and, therefore, after suitable rescaling, the random vector
ξ
takes values
in the domain
Γ:= [−
1, 1
]M
. A parametrisation of the form
(4)
can be seen as derived
from a Karhunen–Loéve expansion of the underlying random field
E(x
,
y
,
ξ)
; see, e.g., [
25
],
in which case the functions
Em(x
,
y)
are the eigenfunctions of some covariance function
C(x
,
y)
. These eigenfunctions are known explicitly only in some simple geometric settings,
but can also be resolved numerically as eigenvectors of the sampled covariance matrix;
see, e.g., Chapter 8 in Ref. [
26
] and [
25
,
27
]. The series has to be truncated for numerical
computations, of course. In the sequel, Mterms are used in applications.
The standard assumption is that
E(x
,
y
,
ξ)
is strictly uniformly positive and uniformly
bounded, i.e., there exist Emin,Emax >0 such that
Emin ≤ess inf
(x,y)∈DE(x,y,ξ)≤ess sup
(x,y)∈D
E(x,y,ξ)≤Emax ∀ξ∈Γ. (5)
Given an integrable function
v=v(ξ)
, denote by
EΓ[v] =
2
−MRΓv(ξ)dξ
its expecta-
tion taken over the parameter space Γ.
The uncertainty is introduced to the problem via material properties—more specif-
ically, by representing the Young’s modulus using the stochastic model
(4)
above. As a
consequence, the eigenpairs of the free vibration problem depend on
ξ∈Γ
. Obtaining the
stochastic version of the eigenvalue problem becomes simple: one only has to replace the
differential operators in
(2)
with their stochastic counterparts and assume that the resulting
equation holds for every realisation of
ξ∈Γ
. In variational form, the problem is stated as:
Find functions u(t,·):Γ→Vand ω2(t,·):Γ→Rsuch that for all ξ∈Γ
Appl. Sci. 2022,12, 3559 5 of 17
tam(ξ;u(t,ξ),v) + tas(ξ;u(t,ξ),v) + t3ab(ξ;u(t,ξ),v)
=ω2(t,ξ)m(t;u(t,ξ),v)∀v∈V. (6)
Here,
am(ξ
;
·
,
·)
,
as(ξ
;
·
,
·)
and
ab(ξ
;
·
,
·)
are stochastic equivalents of the deterministic
bilinear forms in (3).
The stochastic stiffness matrix
S(ξ)
is obtained after integration and assembly. Us-
ing (4), one has
S(ξ) = S(0)+
M
∑
m=1
S(m)ξm,ξ∈Γ, (7)
where each matrix
S(m)
corresponds to a term in the series. Notice that there is no depen-
dence on the parameter vector ξ∈Γin the mass matrix M.
2.2.2. Frequency Response Analysis
In frequency response analysis, the idea is to study the effects of excitation or applied
force in the frequency domain. It is clear that the uncertainty in the eigenproblem is strongly
connected to the frequency response.
The starting point is the equation of motion for the system [28]:
M¨v(ξ) + C˙v(ξ) + S(ξ)v(ξ) = f, (8)
where, given
ξ∈Γ
,
M
is the mass matrix,
C
the viscous damping matrix,
S(ξ)
the
stochastic stiffness matrix,
f
the (deterministic) force vector, and
v(ξ)
the displacement
vector. The matrices
M
and
S(ξ)
are defined as above. The viscous damping matrix
C
is a
linear combination of
M
and
S(ξ)
. This choice is often referred to as proportional Rayleigh
damping, and is central to the low-rank approximation scheme considered below.
In the case of harmonic excitation, a steady-state solution is sought. The angular
frequency is
ω=
2
πf
, where
f
is the ordinary frequency. The force and the corresponding
response have harmonic function representations as
f=ˆ
f(ω)eiωt,
v(ξ) = ˆv(ξ,ω)eiωt.(9)
Taking the first and second derivatives of Equation
(8)
and substituting using
(9)
leads to
−ω2Mˆv(ξ,ω)eiωt+iωCˆv(ξ,ω)eiωt+S(ξ)ˆv(ξ,ω)eiωt=ˆ
f(ω)eiωt, (10)
finally reducing to a linear system of equations:
(−ω2M+iωC+S(ξ)) ˆv(ξ,ω) = ˆ
f(ω). (11)
In principle, any quantity of interest can be derived from the solution
ˆv(ξ
,
ω)
. Often,
in engineering practice, maximal deflections at specific locations are used. For shells,
the presence of boundary layers makes it difficult to guarantee satisfactory convergence. A
mathematically reasonable choice is to take the mechanical energy defined by the natural
energy norm as the quantity of interest, even though, in practice, it may be difficult
to measure.
2.2.3. Stochastic Collocation
There are many options for the selection of the collocation operator. Although a sim-
pler formulation would also suffice for the numerical experiments below, it is advantageous
to aim for efficient computation even in high-dimensional parameter spaces. With this in
mind, an anisotropic Smolyak-type collocation operator defined with respect to a finite
multi-index set (see, e.g., [29–32]) is taken as the model.
Appl. Sci. 2022,12, 3559 6 of 17
For the sake of accuracy, it is standard practice to choose the collocation points to be the
abscissae of orthogonal polynomials; see [
33
]. The collocation method is formulated using
Legendre polynomials, which are the optimal choice when the input random variables are
uniform. In the case of, for example, Gaussian random variables, one should use Hermite
polynomials instead, but otherwise the collocation method remains the same.
Let
Lp
be the univariate Legendre polynomial of degree
p
. Denote by
{χ(p)
k}p
k=0
the zeros of
Lp+1
and by
{w(p)
k}p
k=0
the associated Gauss–Legendre quadrature weights.
The one-dimensional Lagrange interpolation operators I(m)
pare defined via
I(m)
pv(ξm) =
p
∑
k=0
v
χ(p)
k`(p)
k(ξm), (12)
where {`(p)
k}p
k=0are the related Lagrange basis polynomials of degree p.
Now, let
A ⊂ NM
0
be a finite set of multi-indices. For
α
,
β∈ A
write
α≤β
if
αm≤βm
for all
m=
1,
. . .
,
M
. Further, let us assume that
A
is monotone in the following sense:
∃α∈ A such that β≤α⇒β∈ A. The sparse collocation operator is defined as
IA:=∑
α∈A
M
O
m=1I(m)
αm−I(m)
αm−1, (13)
where I(m)
−1:=0. The collocation points are of the form
χ(α)
γ=χ(α1)
γ1, . . . , χ(αM)
γM∈Γ(14)
for some
γ∈NM
0
such that
γ≤α∈ A
. Similarly, the tensorised quadrature weights
w(α)
γ=w(α1)
γ1···w(αM)
γM
are defined. Statistics, such as the expected value and variance,
for the collocated solution may now be computed by applying the quadrature rule
EΓ
"M
O
m=1I(m)
αmv#=∑
γ≤α
vχ(α)
γw(α)
γ(15)
on the terms in (13).
The accuracy of the collocated approximation is ultimately determined by the smooth-
ness of the solution as well as the choice of the multi-index set
A ⊂ NM
0
. For a detailed
analysis, refer to [30,32]. In this paper, tensor product grids defined by
A={α∈NM
0|αm≤p,m=1, . . . , M},p∈N0, (16)
are used.
3. Free Vibration of Perforated Shells
Shells of revolution have symmetry-induced clusters of eigenmodes. With the excep-
tion of torsion modes, in free vibration, every eigenmode is a member of a cluster of size
at least two. This is a simple consequence of the eigenmodes having integer-valued wave
numbers in the angular direction. In a non-perforated case, this wave number increases
with the rate ∼1/ 4
√tas t→0.
For regular perforation patterns, the asymptotics for the lowest eigenmode are given
in terms of a conjecture.
Conjecture 1
(Asymptotics of Perforated Parabolic Shells of Revolution, [
7
])
.
Given a
g×g
regular perforation pattern, there exists a critical dimensionless thickness
tc
at which the angular
wave number
kc≈g/
2. For thicknesses
t<tc
, the asymptotics of the lowest eigenmode do not
conform to those of the non-perforated case.
Appl. Sci. 2022,12, 3559 7 of 17
Geometrically, this means that there exists a thickness at which, within the subspace
containing the lowest eigenmode, at least one pair of eigenmodes is in perfect alignment
with the perforation grid so that one mode acts (has maximal amplitudes) on the holes and
the other in the areas in between.
In many structures, the perforation patterns may be locally regular—for instance,
in shells made out of panels with local perforation patterns. Once the angular symmetry is
removed, the eigenmodes are no longer pure, but linear combinations of periodic functions
in the angular direction. Removal of axial symmetry affects the local axial amplitudes only.
Here, the trommel screens with precisely these properties are considered. In Figure 1, an
example of the axial effect (Figure 1a,b) and the saturation with high wave numbers in
response (Figure 1d) are illustrated. Currently, even a case of two adjacent but different
regular perforation grids is open in the sense that the asymptotics cannot be predicted.
4. Low-Rank Approximation
In this section, the choice of the Rayleigh damping model and its effect on the low-
rank approximation are motivated. Let us recall the equation of motion (8) for the system
integrated at a given collocation point
ξ
. Set
C=ζ(αS(ξ) + βM)
, where the parameters
α>
0,
β>
0,
α+β=
1, and
ζ>
0 are chosen on the basis of experimental results and
previous experience, and can vary for the same structural model depending on the external
conditions. These facts pose an additional requirement for simulation—in other words,
the free variation of
α
and
β
without having to repeat the reduction procedure. This choice
of the damping is often referred to as proportional damping and it follows immediately
that it preserves the eigenspace of the original undamped problem. Another option would
be to use simple diagonal damping.
Given an eigenspace preserving damping model, the task is to find a suitable subspace
V
of (small) rank
r
such that the solution of the reduced system where the subscript
r
denotes reduced matrix, e.g., Mr=VTMV,
Mr¨v(ξ) + Cr˙v(ξ) + Sr(ξ)v(ξ) = f, (17)
is sufficiently close to the solution of the full problem.
4.1. Krylov Subspace Construction
A remarkably efficient choice for the subspace is the Krylov subspace [
9
]. Using stan-
dard notation from numerical linear algebra, a Krylov basis of dimension nis denoted as
colspan(V) = Kn(−S−1(ξ)M,S−1(ξ)f). (18)
The reduced system in Equation (17) matches the first
n
moments of the original
full-order system [
16
]. Here, the usefulness of proportional damping becomes apparent; it
is not needed in its explicit form, and therefore the construction is valid for any choice of
α
and β.
4.2. Deflation Preconditioning
The Krylov subspace has to be constructed for every collocation point
ξ
. In order
to simplify the notation, the dependence on
ξ
explicit is not made explicit. It is neces-
sary to solve systems in terms of bending
SB
and membrane
SM
(membrane and shear
summed together)
(SM+t2SB)v=b, (19)
for every thickness
t
. In order to avoid factorisation of
S
at every
ξ
, it would be natural to
consider iterative methods. Unfortunately, shell problems are singularly perturbed and the
parameter-independent preconditioning of such systems is an open problem. However,
if one considers a sequence of problems, it is possible to transform the problem as follows:
Let SM=LLT(Cholesky decomposition); then,
Appl. Sci. 2022,12, 3559 8 of 17
L(I+t2L−1SBL−T)LTv=b, (20)
where the subspace defined by
L−1SBL−T
is invariant over all thicknesses
t
. The inner
systems
(I+t2L−1SBL−T)ˆ
v=ˆ
b
, have their spectra bounded by 1 from below, making
it sufficient to collect the largest eigenvectors into subspace
W
, say. Once the subspace
W
has been constructed, the deflated conjugate gradient method can be applied [
34
,
35
].
Here, collecting Lanczos vectors is not sufficient since the construction of the Krylov
subspace leads to multiple right-hand sides. This can be taken still further by constructing
the subspace only once, corresponding to the point
ξ= (
0,
. . .
, 0
)
, and using this as an
approximative preconditioner for every other collocation point, leading to only a small
number of extra iteration steps.
Remark 1.
Since the system can be singularly perturbed, i.e.,
SM
is not necessarily invertible, it is
possible to use the stabilised version
((SM+eSB) + (t2−e)SB)v=b, (21)
where e∈[0, t2]. The choice of optimal eis problem-dependent.
Remark 2.
This kind of preconditioner requires multiple sparse triangular solves. In high-level
systems such as Matlab and Mathematica, the built-in solver routines typically have superior
performance, and the benefits of the approach proposed here are difficult to realise [36].
5. Numerical Experiments: Trommel Screen
By design, the perforation patterns of trommel screens are not uniform, which makes
the problem more interesting since the existing asymptotic results on eigenmodes are not
directly applicable here since they rely on the material uniformity assumption. In the refer-
ence case (see Figure 2), the representative parabolic profile function is
φ(x) = 1.
In all cases,
x∈[−π
,
π]
so that the 2D computational domain is
D= [−π
,
π]×[
0, 2
π]\{perforations}
.
The hole coverage is very high, 55%, with one half of the shell with a regular 20
×
10 grid,
and another with two 5
×
5 triangular pattern panels. Even though the mesh appears rather
coarse, the
p
-version of the finite element method is used with uniform
p=
4, which is
sufficient to prevent numerical locking effects from dominating the solution.
The quantity of interest is the relative accuracy of the low-rank approximation over a
range of thicknesses. As a byproduct of this, one can obtain asymptotic rates for quantities
of interest such as the total energy or the frequency of the dominant response as the
thickness tends to zero. The size of the Krylov subspace is determined dynamically via
orthogonalisation as vectors are added. Using notation of (18),
K4
was found to be adequate
over the whole set of experiments.
Material constants adopted for all simulations are:
E=
2.069
×
10
11 MPa
,
ν=
1
/
3,
and
ρ=
7868
kg m−3
, unless otherwise specified. For damping, the selected weights are
simply
α=β=
1
/
2, and
ζ=
1
/
2000, which is as in earlier work [
8
]. The deterministic
parameter ranges of
t∈[
1
/
100, 1
/
1000
]
(the so-called practical range), and frequency
f∈[
5, 240
]Hz
(chosen experimentally) with angular frequency
ω=
2
πf
, are covered. The
loads or excitation functions act only on the transverse direction
w
. The first three are Fourier
modes in the angular direction and constant in the axial direction, i.e.,
ˆ
f(x
,
y) = Ccos(Ky)N
,
where
C=
1000 is a scaling parameter, and
K∈ {
5, 10, 20
}
is the angular wave number.
The fourth loading is a concentrated load
ˆ
f(x
,
y) = C2exp(−
100
(x2+ (y−π)2)) N
. The
concentrated load is not a point load, since, formally, the convergence of the finite element
method is not established for point loads. The random Young’s modulus depends only
on the axial direction and has the form
E(x
,
y
,
ξ) = E0(x) + ∑M
m=1Em(x)ξm
,
(x
,
y)∈D
,
ξ∈[−
1, 1
]M
, where
E0(x) = E
(constant), and
Em(x) = √λmEsin(mx)
with decaying
sequence of coefficients λm=1/(m+1)4.
The angular wave numbers are chosen as factors of the regular grid size
=
20 in the
angular direction. In particular, the results on free vibration indicate (Conjecture 1) that
Appl. Sci. 2022,12, 3559 9 of 17
the performance profiles for
K=
10 should mark a transition to dominant local (boundary
layer) features as the thickness t→0 [7].
-3-2-10123
0
1
2
3
4
5
6
X
Y
(a) Computational domain, D.
-3-2-10123
0
1
2
3
4
5
6
X
Y
(b) Triangulated D.
Figure 2.
Trommel design: Shell of revolution with
φ(x) =
1, i.e., uniform radius = 1. Hole coverage
is 55%. The boundaries at x=±πare clamped, holes are free, and y=0 and y=2πare periodic.
5.1. Effect of Boundary Layers
For parabolic shells, there exist axial layers with a characteristic length scale
∼√t
and
angular layers
∼4
√t
. In Figures 3and 4, transverse deflection profiles for one Fourier load
(
K=
20) and the concentrated load are shown, respectively. In the Fourier load, the large
perforations dictate the profile, except in the case of
t=
1
/
1000 and
f=
40. However,
for the concentrated load, the angular wave structure and its dependence on the thickness
is clearly visible. In both Figures 3d and 4d for
t=
1
/
1000 and
f=
40, the local features
dominate completely.
It also worth mentioning that even though different types of solution features domi-
nate in different parameter configurations, the energies may not necessarily differ signifi-
cantly. In both of the examples, the relative change is greatest within the thick (
t=
1
/
100)
case, a fact which is practically impossible to deduce from the transverse displacement
field only.
(a)t=1/100, f=5 (b)t=1/100, f=40
Figure 3. Cont.
Appl. Sci. 2022,12, 3559 10 of 17
(c)t=1/1000, f=5 (d)t=1/1000, f=40
Figure 3.
Fourier loading with
K=
20. Relative transverse displacement field with temperature
colours. Observed energies: (a) 0.000167373, (b) 0.000849719, (c) 0.01163, (d) 0.0108526.
(a)t=1/100, f=5 (b)t=1/100, f=40
(c)t=1/1000, f=5 (d)t=1/1000, f=40
Figure 4.
Concentrated load. Relative transverse displacement field with temperature colours.
Observed energies: (a) 2.66805, (b) 8.45301 (c) 145.964, (d) 98.8892.
Appl. Sci. 2022,12, 3559 11 of 17
5.2. Low-Rank Approximation Concerns
As demonstrated above, for perforated shells, the interactions between loading, per-
foration patterns, and deterministic parameters, such as thickness and frequency in this
context, are complex. In particular, in those situations where the local features become
dominant, it is intuitively clear that the low-rank approximation may lead to unexpected
errors. On the other hand, in the global feature range of problems, there is no reason not to
employ low-rank approximations. Given the mesh of Figure 2with uniform
p=
6, one
ends up with a system of 481,140 degrees of freedom after the boundary conditions are
deployed. Remarkably, the Krylov basis has a rank
=
4 after orthogonalisation for every
instance tested here.
In Figure 5, the frequency responses of the cases
K=
20 and the concentrated load
at
t=
1
/
1000 are shown. For the Fourier case, the relative errors in the total energy are
fairly large, but the frequencies correct, with the exception of the secondary peak at the
low frequencies. In the concentrated load case, the low-rank approximation fails and the
dominant frequency is not correct.
0 50 100 150 200
0.00
0.02
0.04
0.06
0.08
0.10
Frequency
TotalEnergy
(a)K=20, t=1/1000.
20 40 60 80 100
0.00
0.02
0.04
0.06
0.08
0.10
Frequency
TotalEnergy
(b) Detail with std.
0 50 100 150 200
0
200
400
600
800
1000
Frequency
TotalEnergy
(c) Concentrated, t=1/1000.
10 20 30 40 50
0
200
400
600
800
1000
1200
Frequency
TotalEnergy
(d) Detail with std.
Figure 5.
Frequency response. Comparison of the full (solid line) and low-rank approximation
(dashed line) over a range of frequencies. In the detail plot, one standard deviation (std) from the
collocation solution (
M=
6) is added. Green indicates higher energies for the full and red for the low
rank with one standard deviation.
Let us further focus on the case of Figure 5b. In Figure 6, the convergence of the
low-rank approximation as the number of basis vectors grows has been added in two
cases. The basis has a rank
=
4, so the sequence rank
=
2, 3,4 is complete. In the case of
K=
5,
t=
1
/
100, already rank
=
3 is sufficient. However, in Figure 6b, strong variation
within the sequence is observed. In particular, in the case of rank
=
3, the observed
energy is very small
=
0.001 exactly at the frequency
f=
70, where the full version has its
maximum =0.072
. In this case, the
L2
-norms of the different vector field components of
the solution (see Appendix A.2) reveal that, in the low-rank approximation, the boundary
layers in the axial direction have amplitudes that are an order of magnitude smaller than
those of the full solution. These data are tabulated in Figure 6c.
Appl. Sci. 2022,12, 3559 12 of 17
20 40 60 80 100
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Frequency
TotalEnergy
(a)K=5, t=1/100.
20 40 60 80 100
0.00
0.05
0.10
0.15
0.20
Frequency
TotalEnergy
(b)K=20, t=1/1000.
Solution u v w θ ψ
Full 2.4 ×10−73.0 ×10−75.4 ×10−62.6 ×10−57.9 ×10−5
Rank =3 2.4 ×10−86.7 ×10−88.3 ×10−83.1 ×10−61.2 ×10−5
(c)L2-norms of the vector field components in K=20, t=1/1000 at f=70.
Figure 6.
Low-rank approximation. Convergence of the low-rank approximation as the number of
the Krylov basis vectors grows. Solid line: full, dotted: rank
=
2, , dot-dashed: rank
=
3, dashed:
rank =4. The θ-components explain the difference in the observed energies at f=70.
An interesting question is whether stochastic modelling of the material parameters
could recover or contain the right solution in the case shown in Figure 5d. In order to test
this, the collocation with
M=
6 and second-order polynomials (729 collocation points
in tensor quadrature) was used with pointwise low-rank approximation. It appears that
the approximation error is orders of magnitude larger than anything that could result
from simple perturbations of the material parameters. This is not necessarily a general
phenomenon; in Figure 7, an example where this compensation actually happens is shown.
10 20 30 40 50
0
50
100
150
200
250
300
350
Frequency
TotalEnergy
(a)t=0.0033.
10 20 30 40 50
0
20
40
60
80
100
Frequency
TotalEnergy (Std)
(b) Standard deviation.
Figure 7.
Concentrated load. Low-rank approximation results in a spurious peak at
f=
25. Inter-
estingly, at
M=
6 for the dominant peak at
f=
15, one standard deviation compensates for the
difference between the full and low approximation.
It has to be emphasised that the message here is that, indeed, there are extreme
configurations where the low-rank approximations may fail. It is reasonable to expect that,
in usual design situations, these cases can be identified a priori.
5.3. Asymptotics
As stated above, there are two different interesting asymptotics related to the most
energetic response that follow directly from this study. These are the observed frequency
and the total energy as functions of the thickness. Two features of the experimental setup
are central. First, the energy contributions of the regular grid part and the triangular
pattern panels are parameter-dependent; second, for a given loading, there exists a crit-
ical thickness
tc
at which the solution is aligned with the regular grid so that maximal
amplitudes are exactly in between the perforation rows leading to the energetic response.
Appl. Sci. 2022,12, 3559 13 of 17
For thicknesses
tc
, local bending occurs at the scale of the smallest perforations and thus
total energy growth saturates.
In Figures 8and 10, both sets of asymptotics are shown. Let us first concentrate
on the observed frequencies. Both
K=
5 and the concentrated load have one dominant
parameter-dependent response over the whole range, with respective rates of
∼
1 and
∼
1/2.
However, for
K=
10 and
K=
20, there are two competing ones, with the transition to
the higher-frequency one occurring at some critical thickness
tc
. For
K=
5, this transition
would also eventually occur, but with a critical thickness below the range considered here.
Moreover, for
K=
10, this transition is sharp; for
K=
20, it is more gradual; in fact,
in Figure 8c, around
t=
3
/
1000, the low-rank approximation picks higher frequencies
already before the transition. This is explained with graphs in Figure 9.
0.001 0.002 0.005 0.010
5
10
20
50
Thickness
Frequency
(a)K=5.
0.001 0.002 0.005 0.010
5
10
50
100
Thickness
Frequency
(b)K=10.
0.001 0.002 0.005 0.010
5
10
50
100
500
1000
Thickness
Frequency
(c)K=20.
0.001 0.002 0.005 0.010
10
20
30
Thickness
Frequency
(d) Concentrated.
Figure 8.
Most energetic response. Observed frequency as a function of the thickness. Red colour
indicates low-rank approximation if there is a discrepancy. All trend lines for Fourier loads have a
rate ∼1; for the concentrated load, the rate is ∼1/2.
0 50 100 150 200
0.000
0.001
0.002
0.003
0.004
Frequency
TotalEnergy
(a)t=0.0028.
0 50 100 150 200
0.000
0.001
0.002
0.003
0.004
0.005
Frequency
TotalEnergy
(b)t=0.0027.
Figure 9.
Fourier
K=
20: Comparison of the full (solid line) and low-rank approximation (dashed
line) over a range of frequencies. Low-rank approximation picks a higher frequency.
The energy asymptotics of Figure 10 tell a similar story. For
K=
10, the sharpness of
the transition leads to an almost monotone increase in the observed total energy. In the case
of
K=
5, the large holes dominate first, leading to a higher initial rate of energy increase.
A similar change in rate is also seen for
K=
20, but here, the reason is saturation in the
higher frequencies with local features dominating.
Appl. Sci. 2022,12, 3559 14 of 17
0.001 0.002 0.005 0.010
1
10
100
1000
Thickness
TotalEnergy
(a)K=5.
0.001 0.002 0.005 0.010
0.005
0.010
0.050
0.100
0.500
1
5
Thickness
TotalEnergy
(b)K=10.
0.001 0.002 0.005 0.010
0.001
0.010
0.100
1
Thickness
TotalEnergy
(c)K=20.
0.001 0.002 0.005 0.010
50
100
500
1000
5000
Thickness
TotalEnergy
(d) Concentrated.
Figure 10.
Most energetic response. Observed total energy as a function of the thickness. Comparison
of the full (solid line) and low-rank approximation (dashed line) over a range of thicknesses. Trend line
rates (negative) for Fourier loads: K=5: ∼4, K=10: ∼3, K=20: ∼3, for the concentrated load: ∼2.
5.4. Other Considerations
For the iterative scheme discussed above, it is the largest thickness which is the prob-
lematic one, since automatically also the condition number is the largest one. For the
computationally most efficient case with only one factorisation over the whole set, the iter-
ation counts for the Krylov basis for
t=
1
/
100 are
(
33, 29, 25, 23
)
, i.e., constant in relative
terms. This is also the worst-case result. In the current implementation, this approach is
50-times slower (!) than the built-in solver in Mathematica 13 on MacOS 12.2.
The Krylov basis has to be computed separately for every thickness, and using the
fastest option took on average two seconds on a modern workstation. Naturally, solving
a resulting 4
×
4 system is orders of magnitude faster to solve once the reduction has
been performed.
The collocation solutions were performed on Aalto University cluster Triton using the
low-rank approximation.
6. Conclusions
Low-rank approximation techniques are highly effective in reducing the computa-
tional complexity of frequency response analysis. In parameter-dependent problems such
as shell problems, where boundary layer-induced local features may actually dominate
the response, it is possible that the approximation error is so large that the observed
frequency response is misleading. This curious phenomenon should not be alarming
since it is likely to occur only in cases where either the shell is very thin or the loading is
challenging—for instance, highly concentrated.
The addition of uncertainties to models increases the need for efficient solution tech-
niques, including efficient low-rank approximation. As always, effective implementations
depend on the underlying platforms and the current state of the art in this context is still
very much evolving.
Appl. Sci. 2022,12, 3559 15 of 17
Author Contributions:
Conceptualisation, H.H. and M.L.; methodology, M.L.; software, H.H.;
validation, H.H. and M.L.; formal analysis, M.L.; investigation, H.H. and M.L.; writing—original draft
preparation, H.H.; writing—review and editing, H.H.; visualisation, H.H.; supervision, H.H.; project
administration, H.H. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.
Acknowledgments:
We acknowledge the computational resources provided by the Aalto Science-
IT project.
Conflicts of Interest: The authors declare no conflicts of interest.
Appendix A. Parabolic Shell of Revolution
The cylinder is the simplest parabolic shell. In this appendix, the shell model for
general shells of revolution and the strains for the special case are outlined.
Appendix A.1. Shell Geometry
Thin shells of revolution can formally be characterised as domains in R3of type
Ω={x+zn(x)|x∈ω,−d/2 <z<d/2}, (A1)
where
d
is the (constant) thickness of the shell,
ω
is a (mid)surface of revolution, and
n(x)
is the unit normal to
ω
. For realistic geometries, one has to consider the principal curvature
coordinates, where only four parameters, the radii of principal curvature
R1
,
R2
, and the
so-called Lamé parameters,
A1
,
A2
, which relate coordinate changes to arc lengths, are
needed to specify the curvature and the metric on
ω
. There are other options, however. The
model above can be simplified by assuming that
ω
can be unfolded as a rectangular domain
expressed in the coordinates
x1
and
x2
. Let us denote this computational domain with
D
(Figure 2). In the sequel, the thickness
d
is replaced with the dimensionless thickness
t=d/L, where L∼diam(D).
Let us consider a cylindrical shell generated by a function
f1(x1) =
1,
x1∈[−x0
,
x0]
,
x0>
0. In this case, the product of the Lamé parameters (metric),
A1(x1)A2(x1) =
1,
and the reciprocal curvature radii are 1/R1(x1) = 0 and 1/R2(x1) = 1, since
A1(x1) = q1+ [ f0
1(x1)]2,A2(x1) = f1(x1), (A2)
and
R1(x1) = −A1(x1)3
f00
1(x1),R2(x1) = A1(x1)A2(x1). (A3)
Appendix A.2. Reissner–Naghdi Shell Model
The two-dimensional shell model applied is the Reissner–Naghdi [
37
], where the
transverse deflections are approximated with low-order polynomials. The resulting vector
field has five components
u= (u
,
v
,
w
,
θ
,
ψ)
, where the first three are the displacements
and the latter two are the rotations in the axial and angular directions, respectively. Here,
the convention that the computational domain
D
is given by the surface parametrisation
and the axial/angular coordinates, which are denoted by xand y, has been adopted.
Deformation energy A(u,u)is divided into bending, membrane, and shear energies,
denoted by subscripts B,M, and S, respectively.
A(u,u) = t2AB(u,u) + AM(u,u) + AS(u,u). (A4)
Bending, membrane, and shear energies are given as
Appl. Sci. 2022,12, 3559 16 of 17
t2AB(u,u) = t2ZDhν(κ11(u) + κ22 (u))2+ (1−ν)
2
∑
i,j=1
κij (u)2iA1A2dx dy, (A5)
AM(u,u) = 12 ZDhν(β11(u) + β22(u))2+ (1−ν)
2
∑
i,j=1
βij (u)2iA1A2dx dy, (A6)
AS(u,u) = 6(1−ν)ZDh(ρ1(u)2+ρ2(u))2iA1A2dx dy, (A7)
where
ν
is the Poisson ratio (constant). The scaling
E/(
12
(
1
−ν2))
, where
E
is the Young’s
modulus, has been omitted.
Using the identities above, the bending, membrane, and shear strains [
37
],
κij
,
βij
, and
ρi, respectively (with the curvature tensor values already inserted), can be written as
κ11 =∂θ
∂x,κ22 =∂ψ
∂y,κ12 =1
2∂ψ
∂x+∂θ
∂y−∂v
∂x,
β11 =∂u
∂x,β22 =∂v
∂y+w,β12 =1
2∂v
∂x+∂u
∂y,
ρ1=∂w
∂x−θ,ρ2=∂w
∂y−v−ψ.
(A8)
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