Plates and shells are very common in nature and thus they inspired engineers that used both from the very beginning of structural mechanics. Shells offer a diversity of possible shapes and geometries, some of them with simple curvature and most of them with double curvature. Many times they are assembled in complex structural systems, in many applications they contain many stiffeners as in the case of aircraft fuselages.

In general the design of such structural elements requires the calculation of stresses, strains and displacements for the design loads. Strains and stresses are related by the so-called constitutive law. The simplest one consists of the linear elasticity. Despite its simplicity many structures are designed for working precisely within the elastic domain. Other designs require considering more complex behaviors (e.g. non-linear elasticity due to material or geometrical non linearities, elastoplastic behaviors usually encountered in material forming – forging, bending,... –, or complex multiphysics behaviors as the ones encountered in composites manufacturing processes implying change of phases, crystallization, polymerization,... coupled with rich thermomechanical mechanisms).

Design problems always involve the solution of a set of partial differential equations in the degenerate domain of the plate or the shell with appropriate initial and boundary conditions. These domains are degenerated because one of its characteristic dimensions (the thickness in the present case) is much lower that the other characteristic dimensions. We will understand the consequences of such degeneracy later. When analytical solutions are neither available nor possible because the geometrical or behavior complexities, the solution must be calculated by invoking any of the available numerical techniques (finite elements, finite differences, finite volumes, methods of particles,...).

In the numerical framework the solution will be only obtained in a discrete number of points, usually called nodes, distributed in the domain. From the solution at those points, it can be interpolated at any other point in the domain. In general regular nodal distributions are preferred because they offer the best accuracy. In the case of degenerated plate or shell domains one could expect that if the solution evolves significantly in the thickness direction, a large enough number of nodes must be distributed along the thickness direction to ensure the accurate representation of the field evolution in that direction. In that case, a regular nodal distribution in the whole domain will imply the use of an extremely large number of nodes with the consequent impact on the numerical solution efficiently.

When simple behaviors and domains were considered semi-analytical models can be considered [1]. For addressing more complex scenarios plate and shell theories were developed allowing, through the introduction of some hypotheses, reducing the 3D complexity to the 2D related to the problem now formulated by considering the in-plane coordinates. The use of these theories have been extended gradually for addressing larger and more complex geometries (anisotropic laminates,...) and behaviors.

There are thousand of papers concerning the proposal and application of plate and shell models (the interested reader can refer to the recent reviews [2, 3] and the references therein). Some models are based on the introduction of kinematic hypotheses in the thickness (e.g. [4] among many others). Transverse shear can be also taken into account [5]. Recent zig-zag representations [6, 7], layer-wise models [8–10] and solid-shell approaches [11, 12], allow addressing accurately more complex scenarios, by increasing the computational complexity slightly. Stiffeners require an appropriate coupling of beam and shell models in order to perform calculations at a moderate computational cost [13].

However, as soon as richer physics are involved in the models and the considered geometries differ of those ensuring the validity of the different reduction hypotheses, efficient simulations are compromised. For example in composites manufacturing processes of large parts many reactions and thermal processes inducing significant evolutions on the thermomechanical fields in the thickness occur. These inhomogeneities are at the origin of residual stresses and the associated distortion of the formed parts [14].

In these circumstances as just indicated the reduction from the 3D model to a 2D simplified one is not obvious, and 3D simulations appear many times as the only valid route for addressing such models, that despite the fact of being defined in degenerated geometries (plate or shell) they seem requiring a fully 3D solution. However in order to integrate such calculations (fully 3D and implying an impressive number of degrees of freedom) in usual design procedures, a new efficient (fast and accurate) solution procedure is needed.

The Saint Venant’s principle was extensively used in the Ladeveze’s works for defining elegant and efficient 3D simplified models [15]. This technique was then generalized to dynamics [16]. This technique allowed significant reduction of computational complexity.

Later, a new discretization technique based on the use of separated representations was proposed for addressing space-time nonlinear models [17] and then it was generalized for defining general separated representations of solutions involving conformational coordinates [18], space and time and even parameters considered as extra-coordinates. The interested reader can refer to the recent reviews [19–22] and the references therein.

A direct consequence was the separated representations involving the space coordinates. Thus in plate domains an in-plane-out-of-plane decomposition was proposed for solving flow problems in laminates [20], then for solving thermal problems in extruded geometries [23], elasticity problems [24] and coupled multiphycisc problems [25]. In those cases the 3D solution was obtained from the solution of a sequence of 2D problems (the ones involving the in-plane coordinates) and 1D problems (the ones involving the coordinate related to the plate thickness).

It is important emphasizing the fact that these approaches are radically different to standard plate and shell approaches. We proposed a 3D solver able to compute the different unknown fields without the necessity of introducing any hypothesis. The most outstanding advantage is that 3D solutions can be obtained with a computational cost characteristic of standard 2D solutions. Moreover, as noticed in [24] no locking effects were found, possibly because the fully 3D solution accomplished.

In this work we will generalize the just referred approach considered in the case of plate domains for calculating the fully 3D solution of the elastic problem in shell domains. The 3D solution will be calculated again from the solution of a sequence of 2D and 1D problems thanks to the in-plane-out-of-plane separated representation.

It is important to note that in this paper we are not addressing a new shell modeling, and by this reason we do not need neither establishing a precise state of the art on shell theories nor comparing our approach with the solutions obtained by using shell models. As we are proposing a new procedure for calculating 3D solutions (keeping a computational complexity characteristic of 2D solution procedures) we will compare our solutions with the ones obtained by considering the fully 3D elastic solution in the shell geometries computed with standard 3D solvers (e.g. finite elements).

Before generalizing the technique proposed in [24] for treating elastic problems defined in shell domains we are summarizing it.

### In-plane-out-of-plane separated representation of elastic problems defined in plate domains

We proposed in [24] and original in-plane-out-of-plane decomposition of the 3D elastic solution in a plate geometry. The elastic problem was defined in a plate domain \Xi =\Omega \times \mathcal{I} with (*x*_{1},*x*_{2})∈*Ω*, \Omega \subset {\mathcal{R}}^{2} and {x}_{3}\in \mathcal{I}, \mathcal{I}=[0,H]\subset \mathcal{R}, being *H* the plate thickness. The separated representation of the displacement field **u**=(*u*_{1},*u*_{2},*u*_{3}) reads:

\mathbf{u}({x}_{1},{x}_{2},{x}_{3})=\left(\begin{array}{c}{u}_{1}({x}_{1},{x}_{2},{x}_{3})\\ {u}_{2}({x}_{1},{x}_{2},{x}_{3})\\ {u}_{3}({x}_{1},{x}_{2},{x}_{3})\end{array}\right)\approx \sum _{i=1}^{N}\left(\begin{array}{c}{P}_{1}^{i}({x}_{1},{x}_{2})\xb7{T}_{1}^{i}\left({x}_{3}\right)\\ {P}_{2}^{i}({x}_{1},{x}_{2})\xb7{T}_{2}^{i}\left({x}_{3}\right)\\ {P}_{3}^{i}({x}_{1},{x}_{2})\xb7{T}_{3}^{i}\left({x}_{3}\right)\end{array}\right)

(1)

where {P}_{k}^{i}, *k*=1,2,3, are functions of the in-plane coordinates (*x*_{1},*x*_{2}) whereas {T}_{k}^{i}, *k*=1,2,3, are functions involving the thickness coordinate *x*_{3}. In [24] we compared the first modes of such separated representations with the kinematic hypotheses usually considered in plate theories. Similar behavior was noticed in the case of elastic solutions in shell domains with respect to classical shell theories.

Expression (1) can be written in a more compact form by using the Hadamard product:

\mathbf{u}({x}_{1},{x}_{2},{x}_{3})\approx \sum _{i=1}^{N}{\mathbf{P}}^{i}({x}_{1},{x}_{2})\circ {\mathbf{T}}^{i}\left({x}_{3}\right)

(2)

where vectors **P**^{i} and **T**^{i} contains functions {P}_{k}^{i} and {T}_{k}^{i} respectively.

Because neither the number of terms in the separated representation of the displacement field nor the dependence on *x*_{3} of functions {T}_{k}^{i} are assumed *a priori*, the approximation is flexible enough for representing the fully 3D solution, being obviously more general than theories assuming particular *a priori* evolutions in the thickness direction *x*_{3}.

Let’s consider a linear elasticity problem on a plate domain \Xi =\Omega \times \mathcal{I}. The weak formulation reads:

{\int}_{\Xi}\epsilon {\left({\mathbf{u}}^{\ast}\right)}^{T}\xb7\mathbf{K}\xb7\epsilon \left(\mathbf{u}\right)d\mathbf{x}={\int}_{\Xi}{\mathbf{u}}^{\ast}\xb7{\mathbf{f}}_{d}d\mathbf{x}+{\int}_{{\Gamma}_{N}}{\mathbf{u}}^{\ast}\xb7{\mathbf{F}}_{d}d\mathbf{x},\phantom{\rule{1em}{0ex}}\forall {\mathbf{u}}^{\ast}

(3)

where **K** is the generalized 6×6 Hooke tensor, **f**_{
d
} represents the volumetric body forces while **F**_{
d
} represents the traction applied on the boundary *Γ*_{
N
}. The separation of variables introduced in Eq. (1) yields the following expression for the derivatives of the displacement components *u*_{
i
}, *i*=1,2,3:

\frac{\partial {u}_{i}}{\partial {x}_{j}}\approx \sum _{k=1}^{k=N}\frac{\partial {P}_{i}^{k}}{\partial {x}_{j}}\xb7{T}_{i}^{k}

(4)

for *j*=1,2; and

\frac{\partial {u}_{i}}{\partial {x}_{3}}\approx \sum _{k=1}^{k=N}{P}_{i}^{k}\xb7\frac{\partial {T}_{i}^{k}}{\partial {x}_{3}}

(5)

from which we can obtain the separated vector form of the strain tensor *ε*:

\phantom{\rule{1em}{0ex}}\epsilon \left(\mathbf{u}\right({x}_{1},{x}_{2},{x}_{3}\left)\right)\approx \sum _{k=1}^{N}\left(\begin{array}{c}\frac{\partial {P}_{1}^{k}}{\partial {x}_{1}}\xb7{T}_{1}^{k}\\ \frac{\partial {P}_{2}^{k}}{\partial {x}_{2}}\xb7{T}_{2}^{k}\\ {P}_{3}^{k}\xb7\frac{\partial {T}_{3}^{k}}{\partial {x}_{3}}\\ \frac{\partial {P}_{1}^{k}}{\partial {x}_{2}}\xb7{T}_{1}^{k}+\frac{\partial {P}_{2}^{k}}{\partial {x}_{1}}\xb7{T}_{2}^{k}\\ \frac{\partial {P}_{3}^{k}}{\partial {x}_{1}}\xb7{T}_{3}^{k}+{P}_{1}^{k}\xb7\frac{\partial {T}_{1}^{k}}{\partial {x}_{3}}\\ \frac{\partial {P}_{3}^{k}}{\partial {x}_{2}}\xb7{T}_{3}^{k}+{P}_{2}^{k}\xb7\frac{\partial {T}_{2}^{k}}{\partial {x}_{3}}\end{array}\right).

(6)

Depending on the number of non-zero elements in the **K** matrix, the development of *ε*(**u**^{∗})^{T}·**K**· *ε*(**u**) involves different number of terms, 21 in the case of an isotropic material and 41 in the case of general anisotropic behaviors.

The separated representation constructor proceeds by computing a term of the sum at each iteration. Assuming that the first *n*-1 modes (terms of the finite sum) of the solution were already computed, **u**^{n-1}(*x*_{1},*x*_{2},*x*_{3}) with *n*≥1, the solution enrichment reads:

{\mathbf{u}}^{n}({x}_{1},{x}_{2},{x}_{3})={\mathbf{u}}^{n-1}({x}_{1},{x}_{2},{x}_{3})+{\mathbf{P}}^{n}({x}_{1},{x}_{2})\circ {\mathbf{T}}^{n}\left({x}_{3}\right)

(7)

where both vectors **P**^{n} and **T**^{n} containing functions {P}_{i}^{n} and {T}_{i}^{n} (*i*=1,2,3) depending on (*x*_{1},*x*_{2}) and *x*_{3} respectively, are unknown at the present iteration. The test function **u**^{∗} reads **u**^{∗}=**P**^{∗}∘**T**^{n}+**P**^{n}∘**T**^{∗}.

The introduction of Eq. (7) into (3) results in a non-linear problem. We proceed by considering the simplest linearization strategy, an alternated directions fixed point algorithm, that proceeds by calculating **P**^{n,k} from **T**^{n,k-1} and then by updating **T**^{n,k} from the just calculated **P**^{n,k} where *k* refers to the step of the non-linear solver. The iteration procedure continues until convergence, that is, until reaching the fixed point ∥**P**^{n,k}∘**T**^{n,k}-**P**^{n,k-1}∘**T**^{n,k-1}∥<*ε*, that results in the searched functions **P**^{n,k}→**P**^{n} and **T**^{n,k}→**T**^{n}. Then, the enrichment step continues by looking for the next mode **P**^{n+1}∘**T**^{n+1}. The enrichment stops when the model residual becomes small enough.

When **T**^{n} is assumed known, we consider the test function **u**^{⋆} given by **P**^{⋆}∘**T**^{n}. By introducing the trial and test functions into the weak form and then integrating in\phantom{\rule{0.1em}{0ex}}\mathcal{I} because all the functions depending on the thickness coordinate are known, we obtain a 2D weak formulation defined in *Ω* whose discretization (by using a standard discretization strategy, e.g. finite elements) allows computing **P**^{n}.

Analogously, when **P**^{n} is assumed known, the test function **u**^{⋆} is given by **P**^{n}∘**T**^{⋆}. By introducing the trial and test functions into the weak form and then integrating in *Ω* because all the functions depending on the in-plane coordinates (*x*_{1},*x*_{2}) are at present known, we obtain a 1D weak formulation defined in\phantom{\rule{0.1em}{0ex}}\mathcal{I} whose discretization (using any technique for solving standard ODE equations) allows computing **T**^{n}.

The problems related to the solution of functions **P**^{n} and **T**^{n} are defined in Appendix A.

As discussed in [24] this separated representation allows computing 3D solutions while keeping a computational complexity characteristic of 2D solution procedures. If we consider a hexahedral domain discretized using a regular structured grid with *N*_{1}, *N*_{2} and *N*_{3} nodes in the *x*_{1}, *x*_{2} and *x*_{3} directions respectively, usual mesh-based discretization strategies imply a challenging issue because the number of nodes involved in the model scales with *N*_{1}·*N*_{2}·*N*_{3}, however, by using the separated representation and assuming that the solution involves *N* modes, one must solve about *N* 2D problems related to the functions involving the in-plane coordinates (*x*_{1},*x*_{2}) and the same number of 1D problems related to the functions involving the thickness coordinate *x*_{3}. The computing time related to the solution of the one-dimensional problems can be neglected with respect to the one required for solving the two-dimensional ones. Thus, the resulting complexity scales as *N*·*N*_{1}·*N*_{2}. By comparing both complexities we can notice that as soon as *N*_{3}≫*N* the use of separated representations leads to impressive computing time savings, making possible the solution of models never until now solved, and even using light computing platforms.

In [24] we considered the simplest approximations of functions involving the in-plane coordinates {P}_{k}^{i}({x}_{1},{x}_{2}) by considering bilinear quadrilateral finite elements and piecewise linear 1D elements for approximating functions involving the thickness coordinate *x*_{3}, {T}_{k}^{i}\left({x}_{3}\right). Richer approximations were analyzed in [26].

In the present work we are generalizing the just described separated representation for solving 3D models defined in shell domains.