Numerical simulations for the
dynamics of flexural shells
Journal Title
XX(X):1–30
©The Author(s) 0000
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DOI: 10.1177/ToBeAssigned
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Xiaoqin Shen
1
, Luisa Piersanti
2
and Paolo Piersanti
3
Abstract
In this paper we study a model describing the displacement of a linearly elastic flexural shell subjected
to given dynamic loads from the computational point of view. The model under consideration takes
the form of a set of hyperbolic variational equations posed over the space of admissible linearized
inextensional displacements, and a set of initial conditions. Since the original model is not suitable for the
implementation of a finite element method, we conduct the experiments on the corresponding penalised
model. It was recently shown that the solution to such a penalised model is a good approximation of
the solution to the original model. Numerical tests are therefore conducted on the the penalised model;
the approximation of the solution to the penalised model is obtained via Newmark’s scheme, which is
then implemented and tested for shells having the following middle surfaces: a portion of a cylinder, and
a portion of a cone. For sake of completeness, we also present the results of the numerical tests related
to a model describing the displacement of a linearly elastic elliptic membrane shell under the action of
given dynamic loads.
Keywords
Flexural shells, Elliptic membrane shells, finite element method, Newmark’s scheme, cylindrical shell,
conical shell, spherical shell
1 Introduction
Flexural shells are widely used in many applicative fields such as physics, engineering and material
science. Some remarkable applications involving the usage of such shells are: reinforced oil palm shell
and palm oil clinker concrete (PSCC) beam [1], smart composite shell panels [2], functionally graded
1
School of Sciences, Xi’an University of Technology, Xi’an, P.R. China
2
Department of Mathematics and Computer Science, University of Perugia, Perugia, Italy
3
Institute of Mathematics and Scientific Computing, Karl-Franzens-Universitat, Graz, Austria
Corresponding author:
Paolo Piersanti, Institute of Mathematics and Scientific Computing, Karl-Franzens-Universitat Graz, 36 Heinrichstrasse,
Graz, Austria
Email: paolo.piersanti@uni-graz.at
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2 Journal Title XX(X)
spherical shell panel [3], anisogrid lattice conical shells [4], and reinforced Eco-friendly coconut shell
concrete [5]. Because of its wide range of applications, the theory of flexural shells is one of the
most important branches in Mathematical Elasticity.
Unlike the static case, which was addressed by Ciarlet and his associates in the references [6],
[7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], there are very few reference about the
time-dependent case. In this direction we cite, for instance, the papers [19] and [20].
To our best knowledge, there are no references that treat the numerical simulation for well-
established models describing the dynamics of flexural shells.
In Section 2 we present some geometrical and analytical background; in Section 3 we formulate
the problem describing the displacement of a flexural shell when it is subjected to given dynamic
loads; in Section 4 we formulate the corresponding penalised problem, which is easier to treat in a
context of numerical simulations, we recall the result establishing the existence and uniqueness of
the solution of the model under consideration, and we analyse the convergence of the solution of
the penalised model to the solution of the original model; in Sections 5 and 6, we rigorously state
the algorithm that implements Newmark’s scheme for the penalised problem and we discuss the
convergence of the approximate solution it outputs to the solution of the original model; finally,
in Sections 7, 8, and 9 we perform numerical experiments in the case where the middle surface of
the linearly elastic shell under consideration is a portion of a cylinder, a portion of a cone, and a
spherical cap, respectively.
2 Geometrical preliminaries
For details about the classical notions of differential geometry recalled in this section see, e.g., [21]
or [22].
Greek indices, except ε and ν, take their values in the set {1, 2}, while Latin indices, except
when they are used for indexing sequences, take their values in the set {1, 2, 3}, and the summation
convention with respect to repeated indices is systematically used in conjunction with these two
rules. The notation E
3
designates the three-dimensional Euclidean space; the Euclidean inner
product and the vector product of u, v ∈E
3
are denoted u ⋅v and u ∧v; the Euclidean norm of
u ∈E
3
is denoted u. The notation δ
j
i
designates the Kronecker symbol.
Given an open subset Ω of R
n
, notations such as L
2
(Ω), H
m
(Ω), or H
m
0
(Ω), m ≥1, designate the
usual Lebesgue and Sobolev spaces, and the notation D(Ω)designates the space of all functions that
are infinitely differentiable over Ω and have compact support in Ω. The notation ⋅
X
designates the
norm in a normed vector space X. The dual space of a vector space X is denoted by X
∗
. Spaces of
vector-valued functions are denoted with boldface letters. Lebesgue-Bochner spaces defined over
a bounded open interval I (cf. [23]), are denoted L
p
(I; H), where H is a Banach space and
1 ≤p ≤∞. The notation ⋅
0,Ω
designates the norm of the Lebesgue space L
2
(Ω), and the notation
⋅
m,Ω
, designates the norm of the Sobolev space H
m
(Ω), m ≥1. The notation ⋅
L
p
(I;H)
designates
the norm of the Lebesgue-Bochner space L
p
(I; H). The notations ˙η and ¨η denote the first weak
derivative with respect to t ∈I and second weak derivative with respect to t ∈I of a scalar function
η defined over the interval I. The notations
˙
η and
¨
η denote the first weak derivative with respect
to t ∈I and second weak derivative with respect to t ∈I of a vector-valued function η defined over
the interval I.
A domain in R
n
is a bounded and connected open subset Ω of R
n
, whose boundary ∂Ω is
Lipschitz-continuous, the set Ω being locally on a single side of ∂Ω.
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Shen, Piersanti and Piersanti 3
Let ω be a domain in R
2
, let y =(y
α
) denote a generic point in ω, and let ∂
α
∶=∂∂y
α
and
∂
αβ
∶=∂
2
∂y
α
∂y
β
. A mapping θ ∈C
1
(ω; E
3
)is an immersion if the two vectors
a
α
(y)∶=∂
α
θ(y)
are linearly independent at each point y ∈ω. Then the image θ(ω)of the set ω under the mapping
θ is a surface in E
3
, equipped with y
1
, y
2
as its curvilinear coordinates. Given any point y ∈ω, the
vectors a
α
(y)span the tangent plane to the surface θ(ω)at the point θ(y), the unit vector
a
3
(y)∶=
a
1
(y)∧a
2
(y)
a
1
(y)∧a
2
(y)
is normal to θ(ω)at θ(y), the three vectors a
i
(y)form the covariant basis at θ(y), and the three
vectors a
j
(y)defined by the relations
a
j
(y)⋅a
i
(y)=δ
j
i
form the contravariant basis at θ(y); note that the vectors a
β
(y)also span the tangent plane to
θ(ω)at θ(y)and that a
3
(y)=a
3
(y).
The first fundamental form of the surface θ(ω)is defined by means of its covariant components
a
αβ
∶=a
α
⋅a
β
=a
βα
∈C
0
(ω),
or by means of its contravariant components
a
αβ
∶=a
α
⋅a
β
=a
βα
∈C
0
(ω).
Note that the symmetric matrix field (a
αβ
) is the inverse of the matrix field (a
αβ
), that
a
β
=a
αβ
a
α
and a
α
=a
αβ
a
β
, and that the area element along θ(ω)is given at each point θ(y), y ∈ω,
by
a(y)dy, where
a ∶=det(a
αβ
)∈C
0
(ω).
Given an immersion θ ∈C
2
(ω; E
3
), the second fundamental form of the surface θ(ω)is defined
by means of its covariant components
b
αβ
∶=∂
α
a
β
⋅a
3
=−a
β
⋅∂
α
a
3
=b
βα
∈C
0
(ω),
or by means of its mixed components
b
β
α
∶=a
βσ
b
ασ
∈C
0
(ω),
and the Christoffel symbols associated with the immersion θ are defined by
Γ
σ
αβ
∶=∂
α
a
β
⋅a
σ
=Γ
σ
βα
∈C
0
(ω).
The Gaussian curvature at each point θ(y), y ∈ω, of the surface θ(ω)is defined by
κ(y)∶=
det(b
αβ
(y))
det(a
αβ
(y))
=det
b
β
α
(y)
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4 Journal Title XX(X)
(the denominator in the above relation does not vanish since θ is assumed to be an immersion).
Note that the Gaussian curvature κ(y) at the point θ(y) is also equal to the product of the two
principal curvatures at this point.
A surface θ(ω)defined by means of an immersion θ ∈C
2
(ω; E
3
)is said to be elliptic if its Gaussian
curvature is everywhere >0 in ω, or equivalently, if there exists a constant κ
0
such that
0 <κ
0
≤κ(y) for all y ∈ω.
Given an immersion θ ∈C
2
(ω; E
3
)and a vector field η =(η
i
)∈C
1
(ω; R
3
), the vector field
˜
η ∶=η
i
a
i
can be viewed as a displacement field of the surface θ(ω), thus defined by means of its covariant
components η
i
over the vectors a
i
of the contravariant bases along the surface. If the norms
η
i
C
1
(ω)
are small enough, the mapping (θ +η
i
a
i
)∈C
1
(ω; E
3
) is also an immersion, so that the
set (θ +η
i
a
i
)(ω)is also a surface in E
3
, equipped with the same curvilinear coordinates as those
of the surface θ(ω), called the deformed surface corresponding to the displacement field
˜
η =η
i
a
i
.
One can then define the first fundamental form of the deformed surface by means of its covariant
components
a
αβ
(η)∶=(a
α
+∂
α
˜
η)⋅(a
β
+∂
β
˜
η),
and the second fundamental form of the deformed surface by means of its covariant components
b
αβ
(η)∶=∂
α
(a
β
+∂
β
˜
η)⋅
(a
1
+∂
1
˜
η)∧(a
2
+∂
2
˜
η)
(a
1
+∂
1
˜
η)∧(a
2
+∂
2
˜
η)
The linear part with respect to
˜
η in the difference
1
2
(a
αβ
(η)−a
αβ
)is called the linearized change
of metric tensor associated with the displacement field η
i
a
i
, the covariant components of which are
then given by
γ
αβ
(η)=
1
2
(a
α
⋅∂
β
˜
η +∂
α
˜
η ⋅a
β
)
=
1
2
(∂
β
η
α
+∂
α
η
β
)−Γ
σ
αβ
η
σ
−b
αβ
η
3
=γ
βα
(η).
The linear part with respect to
˜
η in the difference (b
αβ
(η)−b
αβ
)is called the linearized change
of curvature tensor associated with the displacement field η
i
a
i
, the covariant components of which
are then given by
ρ
αβ
(η)=(∂
αβ
˜
η −Γ
σ
αβ
∂
σ
˜
η)⋅a
3
=∂
αβ
η
3
−Γ
σ
αβ
∂
σ
η
3
−b
σ
α
b
σβ
η
3
+b
σ
α
(∂
β
η
σ
−Γ
τ
βσ
η
τ
)+b
τ
β
(∂
α
η
τ
−Γ
σ
ατ
η
σ
)
+(∂
α
b
τ
β
+Γ
τ
ασ
b
σ
β
−Γ
σ
αβ
b
τ
σ
)η
τ
=ρ
βα
(η).
Let us now recall the definition of the time-dependent version of the linearised change of metric
tensor γ
αβ
. Consider the operator
˜γ
αβ
∶L
2
(0, T ; H
1
(ω)×H
1
(ω)×L
2
(ω))→L
2
(0, T ; L
2
(ω)),
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Shen, Piersanti and Piersanti 5
defined by
˜γ
αβ
(η)(t)∶=γ
αβ
(η(t))for all η ∈L
2
(0, T ; H
1
(ω)×H
1
(ω)×L
2
(ω)),
for almost all (a.a. in what follows) t ∈(0, T ). This operator is well-defined, linear, and continuous
(cf., [24]).
Let us also recall the definition of the time-dependent version of the linearised change of curvature
tensor ρ
αβ
. Consider the operator
˜ρ
αβ
∶L
2
(0, T ; H
1
(ω)×H
1
(ω)×H
2
(ω))→L
2
(0, T ; L
2
(ω)),
defined by
˜ρ
αβ
(η)(t)∶=ρ
αβ
(η(t))for all η ∈L
2
(0, T ; H
1
(ω)×H
1
(ω)×H
2
(ω)),
for a.a. t ∈(0, T ). This operator is clearly well-defined, linear, and continuous (cf., [24]).
3 A natural model for time-dependent flexural shells
Let ω be a domain in R
2
with boundary γ, and let γ
0
be a non-empty relatively open subset of γ.
Let I be an interval of the form (0, T ), with T <∞.
For each ε >0, we define the sets
Ω
ε
∶=ω ×]−ε, ε[ and Γ
ε
±
∶=ω ×{±ε},
we let x
ε
=(x
ε
i
)designate a generic point in the set Ω
ε
, and let ∂
ε
i
∶=∂∂x
ε
i
. Hence we have x
ε
α
=y
α
and ∂
ε
α
=∂
α
. Define, also, the set
Γ
ε
0
∶=γ
0
×[−ε, ε],
which is thus a subset of the lateral face of the undeformed reference configuration.
In all that follows, we are given an injective immersion θ ∈C
3
(ω; E
3
)and ε >0, and we consider
a shell with middle surface θ(ω) and with constant thickness 2ε. This means that the reference
configuration of the shell is the set Θ(Ω
ε
), where the mapping Θ ∶Ω
ε
→E
3
is defined by
Θ(x
ε
)∶=θ(y)+x
ε
3
a
3
(y)at each point x
ε
=(y, x
ε
3
)∈Ω
ε
.
Note that the injectivity assumption is made here for physical reasons, but that is otherwise not
needed in the proofs. One can then show (cf. Theorem 3.1-1 of [21] or Theorem 4.1-1 of [22]) that,
if the thickness ε >0 is small enough, such a mapping Θ ∈C
2
(Ω
ε
; E
3
)is a C
2
-diffeomorphism from
Ω
ε
onto Θ(Ω
ε
), hence is in particular an injective immersion, in the sense that the three vectors
g
ε
i
(x
ε
)∶=∂
ε
i
Θ(x
ε
)
are linearly independent at each point x
ε
∈Ω
ε
; these vectors then constitute the covariant basis at
the point Θ(x
ε
), while the three vectors g
j,ε
(x
ε
), defined by the relations
g
j,ε
(x
ε
)⋅g
ε
i
(x
ε
)=δ
j
i
,
constitute the contravariant basis at the same point.
It will be implicitly assumed in what follows that the immersion θ ∈C
3
(ω; E
3
) is injective and
that ε >0 is small enough so that Θ ∶Ω
ε
→E
3
is a C
2
-diffeomorphism onto its image.
We henceforth assume that the shell is made of a homogeneous and isotropic linearly elastic
material and that its reference configuration Θ(Ω
ε
)is a natural state, i.e., is stress free. As a result
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