Development of the element
Duality in the finite element analysis of shells
Lure [5] and Goldenveizer [6] have demonstrated that a perfect analogy does exist between the stress quantities and the strain quantities in the formulation of thin shell theory. Compatibility equations for strains and displacement components become equilibrium equations for stresses and stress function components when dual quantities are replaced by each other as follows:
\leftu,v,\mathrm{w},{\epsilon}_{1},{\epsilon}_{2},{\omega}_{12},{\omega}_{21},{\kappa}_{1},{\kappa}_{2},{\tau}_{12},{\tau}_{21}\right
(1)
\leftU,\phantom{\rule{0.12em}{0ex}}V,W,{M}_{2},{M}_{1},{M}_{12},{M}_{21},{N}_{2},{N}_{1},{N}_{12}\right
(2)
Where d
^{T} = u, v, w are the displacement components defined on the middle surface
{\mathbf{a}}^{T}=\left{M}_{2},{M}_{1},{M}_{12},{M}_{21},{N}_{2},{N}_{1},{N}_{21},{N}_{12}\right
are the moments and membrane forces defined per unit length of middle surface as in the classical shell theory, see for example reference [10], U, V, and W are the stress functions.
It is shown that equilibrium and compatibility are exactly satisfied with the following definition of the strain and stress:
\epsilon =\nabla d,\sigma =\nabla D
(3)
where D
^{T}=U,V,W are the stress function components defined on the middle surface
\begin{array}{l}{\xd1}^{\mathrm{T}}\mathrm{a}\phantom{\rule{0.25em}{0ex}}\mathrm{derivative}\phantom{\rule{0.25em}{0ex}}\mathrm{operator}\phantom{\rule{0.25em}{0ex}}\mathrm{of}\phantom{\rule{0.25em}{0ex}}\mathrm{matrix}\phantom{\rule{0.25em}{0ex}}\mathrm{form}\phantom{\rule{0.25em}{0ex}}\left[3\mathrm{x}8\right]\phantom{\rule{8.25em}{0ex}}\end{array}
(4)
\begin{array}{l}\mathrm{a}1\phantom{\rule{0.25em}{0ex}}=\phantom{\rule{0.25em}{0ex}}\frac{1}{2{R}_{1}}\left(\frac{\partial}{{A}_{2}\partial {\alpha}_{2}}\frac{1}{{\rho}_{1}}\right)\phantom{\rule{0.12em}{0ex}};\phantom{\rule{0.36em}{0ex}}{\mathrm{\delta}}_{1}=\frac{\partial}{{A}_{1}\partial {\alpha}_{1}}\left(\frac{1}{{R}_{1}}\right)+\frac{1}{2}\left(\frac{1}{{R}_{1}}+\frac{1}{{R}_{2}}\right)\frac{\partial}{{A}_{1}\partial {\alpha}_{1}}+\frac{1}{2{R}_{1}{R}_{2}}\left({R}_{1}\frac{\partial}{{A}_{1}\partial {\alpha}_{1}}\frac{{R}_{2}}{{\rho}_{2}}\right)\\ {\mathrm{a}}_{3}=\phantom{\rule{0.25em}{0ex}}\frac{{\partial}^{2}}{{A}_{1}\partial {\alpha}_{1}{A}_{2}\partial {\alpha}_{2}}+\frac{\partial}{{\rho}_{1}{A}_{1}\partial {\alpha}_{1}};\phantom{\rule{0.25em}{0ex}}{\mathrm{\delta}}_{2}=\frac{\partial}{{A}_{2}\partial {\alpha}_{2}}\left(\frac{1}{{R}_{2}}\right)+\frac{1}{2}\left(\frac{1}{{R}_{1}}+\frac{1}{{R}_{2}}\right)\frac{\partial}{{A}_{2}\partial {\alpha}_{2}}+\frac{1}{2{R}_{1}{R}_{2}}\left({R}_{2}\frac{\partial}{{A}_{2}\partial {\alpha}_{2}}\frac{{R}_{1}}{{\rho}_{1}}\right)\phantom{\rule{0.5em}{0ex}}\\ {\mathrm{b}}_{2}=\frac{1}{2{R}_{2}}\left(\frac{\partial}{{A}_{1}\partial {\alpha}_{1}}\frac{1}{{\rho}_{2}}\right)\phantom{\rule{0.12em}{0ex}};{\mathrm{b}}_{3}\phantom{\rule{0.25em}{0ex}}=\phantom{\rule{0.25em}{0ex}}\frac{{\partial}^{2}}{{A}_{2}\partial {\alpha}_{2}{A}_{1}\partial {\alpha}_{1}}+\frac{\partial}{{\rho}_{2}{A}_{2}\partial {\alpha}_{2}}\end{array}
α
_{1}, α
_{2}, orthogonal coordinate system
A
_{1}, A
_{2}, corresponding Lame's system
ρ
_{1}, ρ
_{2}, radius of geodesic curvature on the middle surface
Conformity and diffusivity
A displacement field satisfies conformity in a curved shell if the following continuity is insured along any edge of the shell element:
{\left({u}_{n}\right)}^{+}={\left({u}_{n}\right)}^{},{\left({u}_{n}\right)}^{+}={\left({u}_{s}\right)}^{},{\left({w}_{n}\right)}^{+}={\left({w}_{n}\right)}^{},{\left(\frac{\partial w}{\partial n}\right)}^{+}={\left(\frac{\partial w}{\partial n}\right)}^{}
(5)
Where\overrightarrow{n}and\overrightarrow{s}are the middle surface normal and tangent to the edge, respectively. A stress resultant field satisfies diffusivity if the following quantities are continuously transmitted through the boundary of the element.
{\left({N}_{n}\right)}^{+}={\left({N}_{n}\right)}^{},{\left({\overline{N}}_{s}\right)}^{+}={\left({N}_{s}+{M}_{n}s/{R}_{s}\right)}^{+}={\left({\overline{N}}_{s}\right)}^{},{\left({M}_{n}\right)}^{+}={\left({M}_{n}\right)}^{}
(6)
{\left({K}_{n}\right)}^{+}={Q}_{n}+\left(?{M}_{n}s\right)/?s={\left({K}_{n}\right)}^{}
And the local jumps of the twisting moment at vertex k is as follows:
{Z}_{k}={\left({M}_{n}s\right)}_{k}^{+}{\left({M}_{n}s\right)}_{k}^{}
Nguyen Dang Hung [4] has presented a boundary duality theorem which states that: ‘If displacements conformity is satisfied, stress resultants diffusivity is also satisfied when the same fields are used for displacement and stress functions’. In other words, let us choose a shape function for the displacement field d, making use of some appropriate connectors (nodal displacements) on the boundary, such that (5) is satisfied; if the same shape function is chosen for the stress function field D, that is similar assumptions on the field and corresponding connectors (nodal stress functions), then equilibrium conditions on the boundary (6) are automatically satisfied.
In the case of planar shell\left(\frac{1}{{R}_{1}}=\frac{1}{{R}_{2}}=0\right), there is no coupling between membrane stress components and bending moments and if Cartesian coordinates are used \left(\frac{1}{{\rho}_{1}}=\frac{1}{{\rho}_{2}}=0\right), the derivative operator ∇ is reduced to a simpler for
{\nabla}^{T}=\left[\begin{array}{cccccc}\hfill \frac{\partial}{\partial x}\hfill & \hfill 0\hfill & \hfill \frac{\partial}{2\partial y}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{\partial}{\partial y}\hfill & \hfill \frac{\partial}{2\partial x}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{{\partial}^{2}}{\partial {x}^{2}}\hfill & \hfill \frac{{\partial}^{2}}{\partial {y}^{2}}\hfill & \hfill 2\frac{{\partial}^{2}}{\partial x\partial y}\hfill \end{array}\right]
(7)
The dual quantities become
{d}^{T}=\leftu,v,\mathrm{w}\right;\phantom{\rule{0.24em}{0ex}}{\epsilon}^{T}=\left{\epsilon}_{x},{\epsilon}_{y},\frac{{\gamma}_{\mathit{xy}}}{2},{\kappa}_{x},{\kappa}_{y},{\kappa}_{\mathit{xy}}\right
(8)
{D}^{T}=\leftU,V,F\right;\phantom{\rule{0.5em}{0ex}}{\sigma}^{T}=\left{M}_{y},{M}_{x},{M}_{\mathit{xy}},{N}_{y},{N}_{x},{N}_{\mathit{xy}}\right
(9)
Where we recognize U and V as Southwell's stress functions (for bending effects) and F as Airy's stress function (for the membrane part). The boundary duality theorem identifies the problem of finding stress functions U and V and of expressing their continuity across the interface with the problem encountered in the derivation of conforming displacement fields for membrane stretching [11]:
\begin{array}{l}\left(\mathrm{u},\mathrm{v}\right)\phantom{\rule{0.25em}{0ex}}\mathrm{continuous}\phantom{\rule{0.25em}{0ex}}\mathrm{entails}\phantom{\rule{0.25em}{0ex}}\left(\mathrm{U},\phantom{\rule{0.25em}{0ex}}\mathrm{V}\right)\phantom{\rule{0.25em}{0ex}}\mathrm{continuous}\phantom{\rule{0.25em}{0ex}}\mathrm{i}.\mathrm{e}.\phantom{\rule{0.25em}{0ex}}\mathrm{force}\phantom{\rule{0.25em}{0ex}}\mathrm{components}\phantom{\rule{0.25em}{0ex}}{\mathrm{M}}_{\mathrm{n}},\phantom{\rule{0.25em}{0ex}}{\mathrm{K}}_{\mathrm{n}}\mathrm{continuous}\\ \mathrm{and}\phantom{\rule{0.25em}{0ex}}{\mathrm{Z}}_{\mathrm{k}}=\phantom{\rule{0.25em}{0ex}}0\phantom{\rule{0.25em}{0ex}}\mathrm{at}\phantom{\rule{0.25em}{0ex}}\mathrm{the}\phantom{\rule{0.25em}{0ex}}\mathrm{vertex}\phantom{\rule{0.25em}{0ex}}\mathrm{k}.\end{array}
(10)
Conversely, according to the same theorem, the problem of finding a continuous stress field (Nx
_{,}
Ny, Nxy) and expressing the continuity of the components (Nn, Nns) across the interface of a membrane element is the same as the problems of finding a conforming transversal displacement in plate bending.
\left({N}_{n},{N}_{\mathit{ns}},\right)\phantom{\rule{0.24em}{0ex}}\left(F,\frac{\partial F}{\partial n}\right)\phantom{\rule{0.24em}{0ex}}\left(w,\phantom{\rule{0.12em}{0ex}}\frac{\partial w}{\partial n}\right)\phantom{\rule{0.12em}{0ex}}\mathrm{Continuous}
(11)
Figure 1a presents a triangular element with quadratic displacement field along the edges; 12 nodal displacements are necessary to satisfy the conformity.
Figure 1b shows a bent plate finite element with 12 nodal stress functions as connectors; this triangle ensures continuity of the normal moment Mn (linear along an edge), the equivalent shear force Kn (constant along an edge) and the local jumps of twisting moment Zk at each corner. If forces are taken as nodal values rather than stress functions, the corresponding 12 connectors are those exhibited in Figure 1c.
This system of nodal forces is adopted in reference [12] for the formulation of an equilibrium model for plate bending with linear assumptions for the moment field.
Figure 2a represents a conforming element for plate bending with quadratic assumptions on the vertical deflection w along the edges.
Figure 2b shows the dual membrane element with nodal values of the Airy' stress function; it is equivalent to the element shown on Figure 2c where the resultants.
R
_{
x
} = ∫ (l. N
_{
x
} + m. N
_{
xy
})dx; R
_{
y
} = ∫ (m. N
_{
y
} + l. N
_{
xy
})dy have been chosen as nodal values.
(l, m are the components of the unit normal to the edge).
Hybrid finite elements and métis finite elements
Let us now consider the following mixed hybrid functional:
Where IT is the modified complementary energy functional for the membrane effect:
{I}_{T}={\displaystyle \sum _{N}}\left({\displaystyle \underset{\mathit{AN}}{\int}}\frac{1}{2}\phantom{\rule{0.12em}{0ex}}{T}_{\mathit{ijkl}}.{N}_{\mathit{ij}}{N}_{\mathit{kl}}\right)\mathit{dA}{\displaystyle \underset{{\Gamma}_{N}}{\int}}{n}_{j}.{N}_{\mathit{ij}}\phantom{\rule{0.12em}{0ex}}{\tilde{u}}_{i}\phantom{\rule{0.12em}{0ex}}\mathit{ds}+{\displaystyle \underset{{\Gamma}_{\mathit{\sigma N}}}{\int}}\mathit{nj}.{\overline{N}}_{\mathit{ij}}{\tilde{u}}_{i}\phantom{\rule{0.12em}{0ex}}\mathit{ds})
(13)
and IB is the modified potential energy functional for the bending effect:
\begin{array}{l}{I}_{B}={\displaystyle \sum _{N}({\displaystyle {\int}_{{A}_{N}}}\left(\frac{1}{2}B{i}_{\mathit{jkl}}.{K}_{\mathit{ij}}{K}_{\mathit{kl}}\mathit{pw}\right)\mathit{dA}}+{\displaystyle \int \Gamma uN\left[{\stackrel{\phantom{\rule{0.96em}{0ex}}~}{M}}_{n}\frac{\partial \overline{w}}{\partial n}\stackrel{\phantom{\rule{0.36em}{0ex}}~}{{K}_{n}}\overline{w}+\frac{\partial}{\partial s}\left({\stackrel{~}{M}}_{\mathit{ns}}\overline{w}\right)\right]}\mathit{ds}\\ \phantom{\rule{0.48em}{0ex}}{\displaystyle {\int}_{{\Gamma}_{N}}}\left[{\stackrel{~}{M}}_{n}\frac{\partial w}{\mathit{an}}{\stackrel{~}{K}}_{n}w+\frac{\partial}{\partial s}\left(\stackrel{~}{{M}_{\mathit{ns}}}w\right)\right]\left(\right)close=")">\mathit{ds}\end{array}\n
(14)
In these expressions,
T{i}_{\mathit{jkl}}=\left[T\right]=\frac{1}{\mathit{Eh}}\left[\begin{array}{l}1\phantom{\rule{0.96em}{0ex}}v\phantom{\rule{0.72em}{0ex}}0\\ v\phantom{\rule{0.96em}{0ex}}1\phantom{\rule{0.84em}{0ex}}0\\ 0\phantom{\rule{1.32em}{0ex}}0\phantom{\rule{0.48em}{0ex}}2\left(1+v\right)\end{array}\right]\mathrm{is}\phantom{\rule{0.25em}{0ex}}\mathrm{the}\phantom{\rule{0.25em}{0ex}}\mathrm{elastic}\phantom{\rule{0.25em}{0ex}}\mathrm{compliance}\phantom{\rule{0.25em}{0ex}}\mathrm{of}\phantom{\rule{0.25em}{0ex}}\mathrm{the}\phantom{\rule{0.25em}{0ex}}\mathrm{stretching}\phantom{\rule{0.25em}{0ex}}\mathrm{effect}
{B}_{\mathit{ijkl}}=\left[B\right]=\frac{E{h}^{3}}{12\left(1{\nu}^{2}\right)}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill \nu \hfill & \hfill 0\hfill \\ \hfill \nu \hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 2\left(1\nu \right)\hfill \end{array}\right]\phantom{\rule{0.25em}{0ex}}\mathrm{is}\phantom{\rule{0.25em}{0ex}}\mathrm{the}\phantom{\rule{0.25em}{0ex}}\mathrm{elastic}\phantom{\rule{0.25em}{0ex}}\mathrm{compliance}\phantom{\rule{0.25em}{0ex}}\mathrm{of}\phantom{\rule{0.25em}{0ex}}\mathrm{the}\phantom{\rule{0.25em}{0ex}}\mathrm{bending}\phantom{\rule{0.25em}{0ex}}\mathrm{effect}
AN, the domain of element
N, its boundary
\Gamma \sigma N, portion of where tensions nj\overline{N}\mathit{ij}={\overline{T}}_{i} are prescribed
p, normal pressure
h, thickness of the shell
One may notice that in (13) there are two unknown fields: the stress field Nij must be defined and in equilibrium over the domain AN.
The displacement field {\tilde{u}}_{i} is defined on the boundary Γ
_{
N
} in such a way that conformity (i.e., the two first relations of 5) is insured. This functional was first proposed by Pian [13],[14] for a hybrid stress finite element formulation. On the other hand, functional (14) possesses two other unknown fields: the deflection w must be defined continuously in AN
_{;} the stress field \left[\begin{array}{ccc}\hfill {\tilde{M}}_{n}\hfill & \hfill {\tilde{K}}_{n}\hfill & \hfill \tilde{Z}\hfill \end{array}\right] must be defined on the boundary Γ
_{
N
} in such a way that diffusivity (i.e., three last relations of 6) is satisfied.
This functional is adopted by Jones [15] for a hybrid displacement finite element formulation.
Nguyen Dang Hung [7],[8] has shown that if the boundary conforming field {\tilde{u}}_{i} of (13) defined on the boundary Γ
_{
N
} can be extended over AN in other words if the two unknown fields in (13) are well defined everywhere in the closed domain {\overline{A}}_{N}, the hybrid stress element belongs to a special class called ‘mongrel displacement element’ which leads to important advantage in energy convergence. In the same way, if the two unknown fields of functional (14) are well defined in the closed domain {\overline{A}}_{N} (i.e., equilibrium boundary field \left[\begin{array}{ccc}\hfill {\tilde{M}}_{n}\hfill & \hfill {\tilde{K}}_{n}\hfill & \hfill \tilde{Z}\hfill \end{array}\right] can be extended everywhere in {\overline{A}}_{N}), the corresponding hybrid displacement element for plate bending becomes a ‘mongrel stress element’ with the same properties concerning the convergence.
In this paper, such is the case for the mixed hybrid planar shell element described in this paper as well for membrane as for bending effects.
Selfdual metis planar shell element
Let us make the following assumptions concerning the four unknown fields of the hybrid mixed functional (12):
w={\displaystyle \sum _{m=0}^{M}}{\displaystyle \sum _{n=0}^{n}{\beta}_{\mathit{mn}}}{x}^{mn}{y}^{n};F={\displaystyle \sum _{m=0}^{M}}{\displaystyle \sum _{n=0}^{n}\beta {\text{'}}_{\mathit{mn}}}{x}^{mn}{y}^{n}
(15)
\begin{array}{l}\stackrel{~}{u}={\alpha}_{1}+{\alpha}_{2}x+{\alpha}_{3}y+{\alpha}_{4}{x}^{2}+{\alpha}_{5}\mathit{xy}+{\alpha}_{6}{y}^{2}\\ \stackrel{~}{v}={\alpha}_{7}+{\alpha}_{8}x+{\alpha}_{9}y+{\alpha}_{10}{x}^{2}+{\alpha}_{11}\mathit{xy}+{\alpha}_{12}{y}^{2}\\ \stackrel{~}{V}=\alpha {\text{'}}_{7}+\alpha {\text{'}}_{8}x+\alpha {\text{'}}_{9}y+\alpha {\text{'}}_{10}{x}^{2}+\alpha {\text{'}}_{11}\mathit{xy}+\alpha {\text{'}}_{12}{y}^{2}\\ \stackrel{~}{U}=\alpha {\text{'}}_{1}+\alpha {\text{'}}_{2}x+\alpha {\text{'}}_{3}y+\alpha {\text{'}}_{4}{x}^{2}+\alpha {\text{'}}_{5}\mathit{xy}+\alpha {\text{'}}_{6}{y}^{2}\end{array}
(16)
Where β
_{
mn
}, β' _{
mn
}, α
_{
i
}, α' _{
i
} are the interpolation parameters,
M is the maximum degree of the polynomial (15).
Let us adopt for the membrane element the natural system of nodal displacements shown on Figure 1a and for the bending effect the natural system of nodal stress functions shown on Figure 1b. It appears that we will have a mongrelmixed planar shell element. Assumptions (16) indicate that the boundary fields \left(\begin{array}{cc}\hfill \tilde{u}\hfill & \hfill \tilde{v}\hfill \end{array}\right) and \left(\begin{array}{cc}\hfill \tilde{U}\hfill & \hfill \tilde{V}\hfill \end{array}\right) are defined everywhere in {\overline{A}}_{N} and conformity and diffusivity are both satisfied with the system of 24 nodal values shown on Figure 3.
Dual quantity < w > is a sort of mean vertical deflection of the shell.
This element, denominated ‘HYTCOQ’ constitutes a selfdual metis planar shell element because the strain field and the stress field for the membrane and the bending effects are respectively dual quantities of each other in the sense discussed in the section Duality in the finite element analysis of shells. The details of the stiffness matrix formation for the membrane and bending effects are respectively given in references [16],[17].
Here, we merely observe that, for the bending effect, we have formulated the element in such a way that nodal generalized displacements replace nodal forces as unknowns; in this way, no special modifications are required to run this element on existing codes written for displacement elements.
On the other hand, we notice that the system of nodal displacements of this element (Figure 3b) is well suited for easy connection of adjacent elements. In particular, the normal slope is locally defined on the edge of the element; this avoids the drawback frequently encountered with flat shell elements when the slopes are defined at the corners.
One may summarize here the nature of HYTCOQ:

(a)
The membrane part possesses displacement metis stretching element with quadratic displacement field defined on the vertexes. The equilibrium stress field (which is derived from a polynomial Airy's function) is defined only inside the element. It appears that when the degree of the Airy's function such that at least M = 4, stress field is quadratic the normal rule for kinematic stability is respected. In these conditions, the displacement hybrid or metis formulation does not imply spurious modes, this element leads to good behavior in convergence and precision according to the numerical tests realized in LTAS. Recently (2013), a new examination is performed and it appears that this element leads to very good performance in terms of convergence, precision, and numerical stability [18].

(b)
The bending part possesses stress metis element with linear moment field defined on the vertexes. The vertical defection (which is derived from a polynomial function) is defined only inside the element. This bending stress metis element was examined intensively, and the very good results are described in the paper [17].
As both stretching and bending effects are represented by very good elements and there exists no interaction effect due to the flat geometry, we must expect to a good performance of the present selfdual planar shell element.