A general global-local modelling framework for the deterministic optimisation of composite structures

Abstract

This work deals with the multi-scale optimisation of composite structures by adopting a general global-local (GL) modelling strategy to assess the structure responses at different scales. The GL modelling approach is integrated into the multi-scale two-level optimisation strategy (MS2LOS) for composite structures. The resulting design strategy is, thus, called GL-MS2LOS and aims at proposing a very general formulation of the design problem, without introducing simplifying hypotheses on the laminate stack and by considering, as design variables, the full set of geometric and mechanical parameters defining the behaviour of the composite structure at each pertinent scale. By employing a GL modelling approach, most of the limitations of well-established design strategies, based on analytical or semi-empirical models, are overcome. The GL-MS2LOS makes use of the polar formalism to describe the anisotropy of the composite at the macroscopic scale (where it is modelled as an equivalent homogeneous anisotropic plate). In this work, deterministic algorithms are exploited during the solution search phase. The challenge, when dealing with such a design problem, is to develop a suitable formulation and dedicated operators, to link global and local models physical responses and their gradients. Closed-form expressions of structural responses gradients are rigorously derived by taking into account for the coupling effects when passing from global to local models. The effectiveness of the GL-MS2LOS is proven on a meaningful benchmark: the least-weight design of a cantilever wing subject to different design requirements. Constraints include maximum allowable displacements, maximum allowable strains, blending, manufacturability requirements and buckling factor.

Appendices

Appendix: A Analytic expression of laminate stiffness matrices gradient

Under the hypothesis of an orthotropic laminate, the expression of the homogenised membrane stiffness matrix in terms of the dimensionless PPs reads

$$ \begin{array}{@{}rcl@{}} \textbf{A}^{*} &=& \underbrace{ \left[\begin{array}{lll} T_{0} + 2T_{1} & -T_{0}+2T_{1} & 0\\ & T_{0} + 2T_{1} & 0\\ \mathrm{\text{sym}} & & T_{0} \end{array}\right]}_{\textbf{A}_{0}^{*}} \\ && + R_{0K}^{A^{*}}\underbrace{ \left[\begin{array}{lll} c_{4}& -c_{4} & s_{4}\\ & c_{4} & -s_{4}\\ \mathrm{\text{sym}} & & -c_{4} \end{array}\right]}_{\textbf{A}_{1}^{*}}+ R_{1}^{A^{*}}\underbrace{ \left[\begin{array}{lll} 4c_{2} & 0 & 2s_{2}\\ & -4c_{2} & 2s_{2}\\ \mathrm{\text{sym}} & & 0 \end{array}\right]}_{\textbf{A}_{2}^{*}}\\ &:=& \textbf{A}_{0}^{*} + R_{0}{\rho}_{0K}\textbf{A}_{1}^{*} + R_{1}{\rho}_{1}\textbf{A}_{2}^{*}, \end{array} $$

(A.1)

with

$$c_{2}=\cos\pi\phi_{1}, c_{4} =\cos 2\pi\phi_{1}, s_{2} = \sin\pi\phi_{1}, s_{4} = \sin 2\pi\phi_{1}. $$

Similarly, matrix H^∗ can be decomposed as

$$ \textbf{H}^{*} = \underbrace{ \left[\begin{array}{lll} T & 0\\ \text{sym} & T \end{array} \right]}_{\textbf{H}_{0}^{*}} + R {\rho}_{1}\underbrace{\left[\begin{array}{lll} c_{2}^{H^{*}} & s_{2}^{H^{*}} \\ \text{sym} & -c_{2}^{H^{*}} \end{array} \right]}_{\textbf{H}_{1}^{*}} := \textbf{H}_{0}^{*} + R {\rho}_{1} \textbf{H}_{1}^{*}, $$

(A.2)

where

$$c_{2}^{H^{*}}=\cos 2{\varPhi}^{H^{*}}, s_{2}^{H^{*}}=\sin 2{\varPhi}^{H^{*}}, {\varPhi}^{H^{*}} = {\varPhi} +{\varPhi}_{1} -\frac{\pi}{2}\phi_{1}.$$

Since quasi-homogeneity holds, B^∗ = O and D^∗ = A^∗.

Let h = t_plyn₀n_ref, with reference to (4). Therefore, the following derivatives read

$$ \begin{array}{@{}rcl@{}} \frac{\partial \textbf{A}}{\partial n_{0}} &=& \frac{h}{n_{0}}\textbf{A}^{*}, \frac{\partial \textbf{A}}{\partial {\rho}_{0K}} = h R_{0} \textbf{A}^{*}_{1}, \frac{\partial \textbf{A}}{\partial {\rho}_{1}} = h R_{1} \textbf{A}^{*}_{2},\\ \frac{\partial \textbf{A}}{\partial \phi_{1}} &=& 2\pi h R_{0}{\rho}_{0K} \left[\begin{array}{lll} -s_{4}& +s_{4} & c_{4}\\ & -s_{4} & -c_{4}\\ \mathrm{\text{sym}} & & +s_{4} \end{array}\right]\\ &&+ \pi h R_{1}{\rho}_{1} \left[\begin{array}{lll} -4s_{2}& 0& 2c_{2}\\ & +4s_{2} & 2c_{2}\\ \mathrm{\text{sym}} & & 0 \end{array}\right]. \end{array} $$

(A.3)

Similarly, for matrix D and matrix H,

$$ \begin{array}{@{}rcl@{}} \frac{\partial \textbf{D}}{\partial n_{0}} &=& \frac{h^{2}}{4}\frac{\partial \textbf{A}}{\partial n_{0}}, \frac{\partial \textbf{D}}{\partial {\rho}_{0K}} = \frac{h^{2}}{12}\frac{\partial \textbf{A}}{\partial {\rho}_{0K}}, \frac{\partial \textbf{D}}{\partial {\rho}_{1}} = \frac{h^{2}}{12}\frac{\partial \textbf{A}}{\partial {\rho}_{1}} , \frac{\partial \textbf{D}}{\partial \phi_{1}}\\ &=& \frac{h^{2}}{12}\frac{\partial \textbf{A}}{\partial \phi_{1}}, \end{array} $$

(A.4)

$$ \begin{array}{@{}rcl@{}} \frac{\partial \textbf{H}}{\partial n_{0}} &=& \frac{h}{n_{0}}\textbf{H}^{*}, \frac{\partial \textbf{H}}{\partial {\rho}_{0K}} = \textbf{O}, \frac{\partial \textbf{H}}{\partial {\rho}_{1}} = h R \textbf{H}_{1}^{*} , \frac{\partial \textbf{H}}{\partial \phi_{1}}\\ &=& -\pi h R {\rho}_{1} \left[\begin{array}{lll} -s_{2}^{H^{*}} & c_{2}^{H^{*}}\\ \text{sym} & +s_{2}^{H^{*}} \end{array}\right]. \end{array} $$

(A.5)

Finally, for orthotropic quasi-homogeneous laminates, for the generic ξ_j,

$$ \frac{\partial \textbf{K}_{e}^{lam}}{\partial \xi_{j}} = \text{diag}\left( \frac{\partial \textbf{A}_{e}}{\partial \xi_{j}}, \frac{\partial \textbf{D}_{e}}{\partial \xi_{j}}, \frac{\partial\textbf{H}_{e}}{\partial \xi_{j}}\right). $$

(A.6)

Appendix B: Analytic expression of stiffness matrix gradient

The unconstrained equilibrium system of the GFEM is of the form

$$ \hat{\textbf{K}}\hat{\textbf{u}} = \hat{\textbf{f}}, $$

(B.1)

where $\hat {\textbf {K}}\in \mathbb {M}^{n_{\text {DOF}}\times n_{\text {DOF}}}_{s}$ is the unconstrained (singular) stiffness matrix, whilst n_DOF is the number of DOFs of the GFEM before the application of the BCs.

Definition B.1

Given a matrix $\textbf {M} \in \mathbb {M}^{\mathrm {m}\times \mathrm {n}}$ and the two sets of positive natural numbers R ⊂{i∣1 ≤ i ≤m} and C ⊂{j∣1 ≤ j ≤n}, the operator $\mathfrak {R}\left (\textbf {M},R,C\right )$ returns the matrix obtained by suppressing the i th row and the j th column of M, ∀i ∈ R, and ∀j ∈ C. Similarly, $\mathfrak {R}\left (\textbf {v},R\right )$ denotes the vector obtained by suppressing the i th row of v, ∀i ∈ R.

If BCs are of the type u_j = 0 for $j\in I_{\text {BC}}\subset \{i\mid i=1,\dots ,n_{\text {DOF}}\}$, ♯I_BC = n_BC, (B.1) can be transformed in a reduced problem of the form and size considered in (13) by posing $\textbf {K}:= \mathfrak {R}\left (\hat {\textbf {\textbf {K}}},I_{\text {BC}},I_{\text {BC}}\right )$, $\textbf {u}:=\mathfrak {R}\left (\hat {\textbf {u}},I_{\text {BC}}\right )$ and $\textbf {f}:=\mathfrak {R}\left (\hat {\textbf {f}},I_{\text {BC}}\right )$.

The analytical form of ∂K/∂ξ_j can be easily determined. In fact, the expression of $\hat {\textbf {K}}$ is

$$ \hat{\textbf{K}} = \sum\limits_{e=1}^{N_{e}} \hat{\textbf{L}}_{e}^{\mathrm{T}} \int\limits_{{\varOmega}_{e}}\textbf{B}_{e}^{\mathrm{T}} \textbf{K}_{e}^{\text{lam}}\textbf{B}_{e} \mathrm{d}{\varOmega}_{e} \hat{\textbf{L}}_{e}, $$

(B.2)

where N_e is the number of elements of the GFEM, Ω_e is the integration domain for the e th element, B_e is the operator defined in (26), $\textbf {K}^{\text {lam}}_{e}$ is the element stiffness matrix defined in (2), expressed in the global frame of the GFEM, whilst $\hat {\textbf {L}}_{e}\in \mathbb {M}^{24\times n_{\text {DOF}}}$ is a linear map $\hat {\textbf {L}}_{e}: \hat {\textbf {u}}\mapsto \textbf {u}_{e}$. By deriving (B.2) with respect to the generic ξ_j, one obtains

$$ \frac{\partial \hat{\textbf{K}}}{\partial \xi_{j}} = \sum\limits_{e=1}^{N_{e}} \hat{\textbf{L}}_{e}^{\mathrm{T}} \int\limits_{{\varOmega}_{e}}\textbf{B}_{e}^{\mathrm{T}} \frac{\partial \textbf{K}_{e}^{\text{lam}}}{\partial \xi_{j}}\textbf{B}_{e} \mathrm{d}{\varOmega}_{e} \hat{\textbf{L}}_{e}. $$

(B.3)

It follows that

$$ \frac{\partial\textbf{K}}{\partial \xi_{j}} := \mathfrak{R}\left( \frac{\partial\hat{\textbf{K}}}{\partial \xi_{j}},I_{\text{BC}},I_{\text{BC}}\right). $$

(B.4)

Appendix C: Analytic expression of laminate strength matrices gradient

Matrix $\textbf {G}_{A}^{*}$ can be decomposed as

$$ \begin{array}{@{}rcl@{}} \textbf{G}_{A}^{*} &=& \underbrace{ \left[\begin{array}{lll} {\varGamma}_{0} + 2{\varGamma}_{1} & -{\varGamma}_{0}+2{\varGamma}_{1} & 0\\ & {\varGamma}_{0} + 2{\varGamma}_{1} & 0\\ \mathrm{\text{sym}} & & {\varGamma}_{0} \end{array}\right]}_{\textbf{G}_{A0}^{*}} \\ &&\\ &&+ {\varLambda}_{0}^{G_{A}^{*}} \underbrace{\left[\begin{array}{lll} c_{4}& -c_{4} & s_{4}\\ & c_{4} & -s_{4}\\ \mathrm{\text{sym}} & & -c_{4} \end{array}\right]}_{\textbf{G}_{A1}^{*}}\\ &&+ {\varLambda}_{1}^{G_{A}^{*}}\underbrace{\left[\begin{array}{lll} 4c_{2} & 0 & 2s_{2}\\ & -4c_{2} & 2s_{2}\\ \mathrm{\text{sym}} & & 0 \end{array}\right]}_{\textbf{G}_{A2}^{*}}\\&:=& \textbf{G}_{A0}^{*} + {\rho}_{0K}{\varLambda}_{0} \textbf{G}_{A1}^{*} + {\varLambda}_{1}{\rho}_{1}\textbf{G}_{A2}^{*}, \end{array} $$

(C.1)

with

$$ \begin{array}{@{}rcl@{}} c_{2}&=&\cos 2\left( \frac{\pi}{2}\phi_{1} + {\varOmega}_{1} - {\varPhi}_{1}\right),\\ c_{4} &=&\cos 4\left( \frac{\pi}{2}\phi_{1} + {\varOmega}_{0} - {\varPhi}_{0}\right), \\ s_{2} &=& \sin 2\left( \frac{\pi}{2}\phi_{1} + {\varOmega}_{1} - {\varPhi}_{1}\right),\\ s_{4} &=& \sin 4\left( \frac{\pi}{2}\phi_{1} + {\varOmega}_{0} - {\varPhi}_{0}\right). \end{array} $$

Similarly, matrix $\textbf {G}_{H}^{*}$ can be decomposed as

$$ \begin{array}{@{}rcl@{}} \textbf{G}_{H}^{*} &=& \underbrace{\left[\begin{array}{lll} {\varGamma} & 0\\ \text{sym} & {\varGamma} \end{array} \right]}_{\textbf{G}_{H0}^{*}} + {\varLambda} {\rho}_{1} \underbrace{\left[\begin{array}{lll} c_{2}^{G_{H}^{*}} & s_{2}^{G_{H}^{*}} \\ \text{sym} & -c_{2}^{G_{H}^{*}} \end{array} \right]}_{\textbf{G}_{H1}^{*}}\\ &:=& \textbf{G}_{H0}^{*} + {\varLambda} {\rho}_{1} \textbf{G}_{H1}^{*}, \end{array} $$

(C.2)

with

$$ \begin{array}{@{}rcl@{}} c_{2}^{G_{H}^{*}} &=& \cos 2\left( {\varOmega} - \frac{\pi}{2}\phi_{1} + {\varPhi}_{1}\right),\\ s_{2}^{G_{H}^{*}} &=& \sin 2\left( {\varOmega} - \frac{\pi}{2}\phi_{1} + {\varPhi}_{1}\right). \end{array} $$

Since quasi-homogeneity holds, $\textbf {G}_{B}^{*} = \textbf {O}$ and $\textbf {G}_{D}^{*} = \textbf {G}_{A}^{*}$.

Let h = t_plyn₀n_ref, with reference to (4). Therefore, the following derivatives read

$$ \begin{array}{@{}rcl@{}} \frac{\partial \textbf{G}_{A}}{\partial n_{0}} &=& \frac{h}{n_{0}}\textbf{G}_{A}^{*}, \frac{\partial \textbf{G}_{A}}{\partial {\rho}_{0K}} = h {\varLambda}_{0} \textbf{G}^{*}_{A1}, \frac{\partial \textbf{G}_{A}}{\partial {\rho}_{1}} = h {\varLambda}_{1} \textbf{G}^{*}_{A2},\\ \frac{\partial \textbf{G}_{A}}{\partial \phi_{1}} &=& 2\pi h {\varLambda}_{0}{\rho}_{0K} \left[\begin{array}{lll} -s_{4}& +s_{4} & c_{4}\\ & -s_{4} & -c_{4}\\ \mathrm{\text{sym}} & & +s_{4} \end{array}\right]\\ &&+ \pi h {\varLambda}_{1}{\rho}_{1} \left[\begin{array}{lll} -4s_{2}& 0& 2c_{2}\\ & +4s_{2} & 2c_{2}\\ \mathrm{\text{sym}} & & 0 \end{array}\right]. \end{array} $$

(C.3)

Similarly, for matrix G_D and matrix G_H,

$$ \begin{array}{@{}rcl@{}} \frac{\partial \textbf{G}_{D}}{\partial n_{0}} &=& \frac{h^{2}}{4}\frac{\partial \textbf{G}_{A}}{\partial n_{0}}, \frac{\partial \textbf{G}_{D}}{\partial {\rho}_{0K}} = \frac{h^{2}}{12}\frac{\partial \textbf{G}_{A}}{\partial {\rho}_{0K}}, \frac{\partial \textbf{G}_{D}}{\partial {\rho}_{1}}\\ &=& \frac{h^{2}}{12}\frac{\partial \textbf{G}_{A}}{\partial {\rho}_{1}} , \frac{\partial \textbf{G}_{D}}{\partial \phi_{1}} = \frac{h^{2}}{12}\frac{\partial \textbf{G}_{A}}{\partial \phi_{1}}, \end{array} $$

(C.4)

$$ \begin{array}{@{}rcl@{}} \frac{\partial \textbf{G}_{H}}{\partial n_{0}} = \frac{h}{n_{0}}\textbf{G}_{H}^{*}, \frac{\partial \textbf{G}_{H}}{\partial {\rho}_{0K}} &=& \textbf{O}, \frac{\partial \textbf{G}_{H}}{\partial {\rho}_{1}} = h {\varLambda} \textbf{G}_{H1}^{*},\\ \frac{\partial \textbf{H}}{\partial \phi_{1}} &=& -\pi h {\varLambda} {\rho}_{1} \left[\begin{array}{lll} -s_{2}^{G_{H}^{*}} & c_{2}^{G_{H}^{*}}\\ \text{sym} & s_{2}^{G_{H}^{*}} \end{array}\right]. \end{array} $$

(C.5)

Finally, for the generic variable ξ_j, for an orthotropic quasi-homogeneous laminate,

$$ \frac{\partial \textbf{G}}{\partial \xi_{j}} = \text{diag}\left( \frac{\partial \textbf{G}_{A}}{\partial \xi_{j}}, \frac{\partial \textbf{G}_{D}}{\partial \xi_{j}}, \frac{\partial \textbf{G}_{H}}{\partial \xi_{j}} \right). $$

(C.6)

Appendix D: Analytic expression of geometric stiffness matrix gradient and buckling factor gradient

The generalised eigenvalue problem for the LFEM can be stated as follows:

$$ \left( \textbf{K}^{\flat} + \lambda \textbf{K}_{\sigma}^{\flat} \right)\pmb{\psi}_{\flat} = \textbf{0}, $$

(D.1)

or, passing to work,

$$ \pmb{\psi}_{\flat}^{T}\left( \textbf{K}^{\flat} + \lambda \textbf{K}_{\sigma}^{\flat} \right)\pmb{\psi}_{\flat} = 0, $$

(D.2)

where $\textbf {K}^{\flat } \in \mathbb {M}^{n_{\text {IN}}^{\flat }\times n_{\text {IN}}^{\flat }}_{s++}$ is the (reduced) stiffness matrix of the LFEM, $\textbf {K}_{\sigma }^{\flat }\in \mathbb {M}^{n_{\text {IN}}^{\flat }\times n_{\text {IN}}^{\flat }}_{s}$ is the geometric stiffness matrix of the LFEM, and λ and ψ_♭ are the eigenvalue and eigenvector, respectively, non-trivial solution of problem (D.1). Note that $\textbf {K}_{\sigma }^{\flat }$ is not, in general, positive-definite.

As usually done in classical buckling eigenvalue analyses, $\textbf {K}_{\sigma }^{\flat }$ is calculated from the stress field solution of the (static) equilibrium boundary problem of the LFEM subject to the same BCs of the original eigenvalue buckling problem. However, in the framework of the considered GL modelling approach, the equilibrium boundary problem of the LFEM is of the Dirichlet’s type: non-zero displacements are imposed at some DOFs, which can be collected in the set $I_{\text {BC}}^{\flat }$ whose cardinality is $n_{\text {BC}}^{\flat }$. On the other hand, the unknown DOFs are collected in the set $I_{\text {IN}}^{\flat }$ whose cardinality is $n_{\text {IN}}^{\flat }$. Moreover, BCs depend on the displacement field solution of the GFEM equilibrium problem (13). Therefore, the key point is to express properly the equilibrium displacement boundary problem (and the related derivatives) for the LFEM.

Definition D.1

Given a matrix $\textbf {M} \in \mathbb {M}^{\mathrm {m}\times \mathrm {n}}$ and the two sets of positive natural numbers R ⊂{i∣1 ≤ i ≤m} and C ⊂{j∣1 ≤ j ≤n}, the operator $\mathfrak {Z}(\textbf {M},R,C)$ returns a matrix obtained by annihilating the i th row and the j th column of M, ∀i ∈ R, and ∀j ∈ C. Similarly, $\mathfrak {Z}\left (\textbf {v},R\right )$ denotes the vector obtained by annihilating the i th component of vector $\textbf {v}\in \mathbb {M}^{n\times 1}$, ∀i ∈ R. Operator $\mathfrak {Z}(\cdot )$ preserves the dimensions of its argument.

Since no external nodal forces are applied, the equilibrium equation of the LFEM reads

$$ \hat{\textbf{K}}^{\flat}\hat{\textbf{u}}_{0}^{\flat} = \hat{\textbf{f}}_{0}^{\flat}, $$

(D.3)

where $\hat {\textbf {K}}^{\flat } \in \mathbb {M}^{n_{\text {DOF}}^{\flat }\times n_{\text {DOF}}^{\flat }}_{s}$ is the (singular) stiffness matrix of the LFEM, $\hat {\textbf {u}}_{0}^{\flat } \in \mathbb {M}^{n_{\text {DOF}}^{\flat }\times 1}$ is the vector collecting the set of DOFs of the LFEM static analysis (both imposed displacements and unknown ones), and $\hat {\textbf {f}}_{0}^{\flat } \in \mathbb {M}^{n_{\text {DOF}}^{\flat }\times 1}$ is the vector of the unknown nodal forces (occurring at nodes where BCs on generalised displacements are applied). In the above expressions, $n_{\text {DOF}}^{\flat } = n_{\text {BC}}^{\flat } + n_{\text {IN}}^{\flat }$.

As discussed in Wu et al. (2007) and Reddy (2005), problem (D.3) can be solved after a proper rearranging. In particular, if $\hat {u}_{s}$ (for some s) is assigned, one must set $\hat {K}_{ss} = 1$, $\hat {K}_{is} = \hat {K}_{si} = 0$ for i≠s and subtract to the right-hand side the s th column of the (unmodified) stiffness matrix, multiplied by $\hat {u}_{s}$. After this operation, the new system can be reduced, as usually, and the unknown nodal displacements can be determined.

Remark D.1

Let $A\subset \{i\mid i=1,\dots ,n\}$ and $B\subset \{i\mid i=1,\dots ,n\}$ be two sets such that A ∩ B = ∅ and ♯(A + B) = n. Therefore, $\textbf {u}=\mathfrak {Z}(\textbf {u},A) \oplus \mathfrak {Z}(\textbf {u},B), \ \forall \textbf {u} \in \mathbb {M}^{n\times 1}$.

By applying Remark D.1 to $\hat {\textbf {u}}_{0}^{\flat }$ and $\hat {\textbf {f}}_{0}^{\flat }$, one obtain

$$ \begin{array}{@{}rcl@{}} \hat{\textbf{u}}_{0}^{\flat} &=& \mathfrak{Z}(\hat{\textbf{u}}_{0}^{\flat},I_{\text{BC}}^{\flat}) \oplus \mathfrak{Z}(\hat{\textbf{u}}_{0}^{\flat}, I_{\text{IN}}^{\flat}) := \hat{\textbf{u}}^{\flat} + \hat{\textbf{u}}^{\flat}_{\text{BC}},\\ \hat{\textbf{f}}_{0}^{\flat} &=& \mathfrak{Z}(\hat{\textbf{f}}_{0}^{\flat},I_{\text{BC}}^{\flat}) \oplus \mathfrak{Z}(\hat{\textbf{f}}_{0}^{\flat}, I_{\text{IN}}^{\flat}) := \hat{\textbf{0}} + \hat{\textbf{f}}^{\flat}_{\text{BC}}, \end{array} $$

(D.4)

where only vector $\hat {\textbf {u}}^{\flat }_{\text {BC}}$ is known. Therefore, problem (D.3) becomes

$$ \hat{\textbf{K}}^{\flat}\hat{\textbf{u}}^{\flat} + \hat{\textbf{K}}^{\flat}\hat{\textbf{u}}^{\flat}_{\text{BC}} = \hat{\textbf{0}} + \hat{\textbf{f}}^{\flat}_{\text{BC}}. $$

(D.5)

To solve for the unknown part of $\hat {\textbf {u}}^{\flat }_{0}$, the operator $\mathfrak {R}$ of Definition B.1 must be applied to (D.5): the resulting reduced system reads

$$ \textbf{K}^{\flat}\textbf{u}^{\flat} + \textbf{K}_{\text{BC}}^{\flat}\textbf{u}_{\text{BC}}^{\flat} = \textbf{0}, $$

(D.6)

where $\mathbf {K}^{\flat } := \mathfrak {R}(\hat {\textbf {K}}^{\flat }, I_{\text {BC}}^{\flat }, I_{\text {BC}}^{\flat })$, $\textbf {u}^{\flat } := \mathfrak {R}(\hat {\textbf {u}}_{0}^{\flat }, I_{\text {BC}}^{\flat })$, $\textbf {K}_{\text {BC}}^{\flat } := \mathfrak {R}(\hat {\textbf {K}}, I_{\text {BC}}^{\flat }, I_{\text {IN}}^{\flat })$, $\textbf {u}_{\text {BC}}^{\flat } := \mathfrak {R}(\hat {\textbf {u}}_{0}^{\flat }, I_{\text {IN}}^{\flat })$, and $\mathfrak {R}(\hat {\textbf {f}}^{\flat }_{\text {BC}}, I_{\text {BC}}^{\flat }) = \textbf {0}$. Inasmuch as $\textbf {u}_{\text {BC}}^{\flat }$ depends on the GFEM solution, it is convenient to introduce the linear map

$$ \textbf{P}:\textbf{u}\mapsto \textbf{u}_{\text{BC}}^{\flat}, \textbf{P}\textbf{u} = \textbf{u}_{\text{BC}}^{\flat}, \textbf{P} \in \mathbb{M}^{n_{\text{BC}}^{\flat} \times \left( n_{\text{DOF}}-n_{\text{BC}}\right)}, $$

(D.7)

whose aim is to determine the BCs to be imposed to the LFEM (in terms of nodal displacements), starting from the solution of the static analysis carried out on the GFEM. In particular, the number of nodes belonging to the boundary of the LFEM, where BCs are applied, is different (usually larger) than the number of nodes located on the same boundary where the known displacement field is extracted from the GFEM results. Furthermore, meshes may be completely dissimilar, as shown in Fig. 12 (boundary of LFEM in red, GFEM mesh in black).

Fig. 12

Differences between GFEM and LFEM meshes

The construction of P can be done according to the steps listed in Algorithm 1, whose structure refers to the notation provided in Fig. 12.

Taking into account for the above aspects, (D.6) reads

$$ \textbf{K}^{\flat}\textbf{u}^{\flat} + \textbf{K}_{\text{BC}}^{\flat} \textbf{P} \textbf{u} = \textbf{0}. $$

(D.10)

Consider, now, the augmented version of (D.2):

$$ \begin{array}{@{}rcl@{}} &&\pmb{\psi}_{\flat}^{\mathrm{T}}\left( \textbf{K}^{\flat} + \lambda \textbf{K}_{\sigma}^{\flat} \right)\pmb{\psi}_{\flat} + \pmb{\mu}^{\mathrm{T}} \left( \textbf{K}^{\flat}\textbf{u}^{\flat} + \textbf{K}_{\text{BC}}^{\flat} \textbf{P} \textbf{u} \right)\\ &&+ \textbf{w}^{\mathrm{T}}\left( \textbf{K}\textbf{u}-\textbf{f}\right)=0, \end{array} $$

(D.11)

where μ≠0 and w≠0 are the arbitrarily-defined adjoint vectors. By deriving (D.11) with respect to the generic design variable ξ_j, one obtains:

$$ \begin{array}{@{}rcl@{}} \frac{\partial \lambda}{\partial \xi_{j}} &=& \frac{\lambda}{\pmb{\psi}_{\flat}^{\mathrm{T}} \textbf{K}^{\flat} \pmb{\psi}_{\flat}} \left[ \pmb{\psi}_{\flat}^{\mathrm{T}} \left( \frac{\partial \textbf{K}^{\flat}}{\partial \xi_{j}} + \lambda \frac{\partial \textbf{K}_{\sigma}^{\flat}}{\partial \xi_{j}} \right)\pmb{\psi}_{\flat} \right.\\ &&\qquad\qquad\quad + \pmb{\mu}^{\mathrm{T}}\left( \frac{\partial \textbf{K}^{\flat}}{\partial \xi_{j}}\textbf{u}^{\flat} + \textbf{K}^{\flat}\frac{\partial \textbf{u}^{\flat}}{\partial \xi_{j}}+ {\cdots} \right. \\ &&\qquad\qquad\quad\left. {\cdots} + \frac{\partial \textbf{K}^{\flat}_{\text{BC}}}{\partial \xi_{j}}\textbf{P}\textbf{u} + \textbf{K}^{\flat}_{\text{BC}}\textbf{P}\frac{\partial \textbf{u}}{\partial \xi_{j}} \right)\\ &&\qquad\qquad\quad\left.+ \textbf{w}^{\mathrm{T}}\left( \frac{\partial \textbf{K}}{\partial \xi_{j}}\textbf{u} + \textbf{K}\frac{\partial \textbf{u}}{\partial \xi_{j}} \right) \right]. \end{array} $$

(D.12)

As discussed in Setoodeh et al. (2009), the geometric stiffness matrix of the generic shell element can be expressed as:

$$ {\textbf{K}_{\sigma}^{\flat}}_{e} = \sum\limits_{i=1}^{8} r_{0ei}^{\flat} \overline{\textbf{K}}_{i}, $$

(D.13)

where $r_{0ei}^{\flat }$ are the components of vector r of (2), resulting from the static analysis of (D.10) carried out on the LFEM, whilst $\overline {\textbf {K}}_{i}\in \mathbb {M}^{24\times 24}_{s}$ are matrices depending only on the geometry of the element. The algorithm for retrieving the expression of each matrix $\mathbf {\overline {K}}_{i}$ for a shell element with four nodes and six DOFs per node (like the SHELL181 ANSYS^®; shell element), whose kinematics is described in the framework of the FSDT, is presented. Of course, this algorithm must be executed off-line, i.e. before the optimisation process, once the element type has been selected.

The expressions of $\mathbf {\overline {K}}_{i}$ for a square SHELL181 element of side L are provided here below. Each matrix $\mathbf {\overline {K}}_{i}$ is a symmetric and sparse partitioned matrix, composed of symmetric blocks. Only non-null terms are provided in the following:

$$ \begin{array}{@{}rcl@{}} \mathbf{\overline{K}}_{1} &=& \frac{1}{8} \left[\begin{array}{cc} \left [ \begin{array}{cc} \hat{\mathbf{A}}_{1}& \hat{\mathbf{B}}_{1} \\ \hat{\mathbf{B}}_{1}^{\mathrm{T}} & \hat{\mathbf{C}}_{1} \end{array} \right] & \left [ \begin{array}{cc} -\hat{\mathbf{A}}_{1} & -\hat{\mathbf{B}}_{1} \\ -\hat{\mathbf{B}}_{1}^{\mathrm{T}} & -\hat{\mathbf{C}}_{1} \end{array} \right] \\ \left [ \begin{array}{cc} -\hat{\mathbf{A}}_{1} & -\hat{\mathbf{B}}_{1} \\ -\hat{\mathbf{B}}_{1}^{\mathrm{T}} & -\hat{\mathbf{C}}_{1} \end{array} \right] & \left [ \begin{array}{cc} \hat{\mathbf{A}}_{1} & \hat{\mathbf{B}}_{1} \\ \hat{\mathbf{B}}_{1}^{\mathrm{T}} & \hat{\mathbf{C}}_{1} \end{array} \right] \end{array} \right],\\ &&\text{with}, \ \hat{\mathbf{A}}_{1}=\hat{\mathbf{A}}_{1}^{\mathrm{T}}, \ \hat{\mathbf{C}}_{1}=\hat{\mathbf{C}}_{1}^{\mathrm{T}}, \ \text{ and } \\ \hat{\mathrm{A}}_{1}^{(1,1)}&=&-3, \ \hat{\mathrm{A}}_{1}^{(1,2)}=-1, \ \hat{\mathrm{A}}_{1}^{(2,2)}=1, \ \hat{\mathrm{A}}_{1}^{(3,3)}=2,\\ \hat{\mathrm{B}}_{1}^{(1,1)}&=&1, \ \hat{\mathrm{B}}_{1}^{(1,2)}=1, \ \hat{\mathrm{B}}_{1}^{(2,1)}=-1,\ \hat{\mathrm{B}}_{1}^{(2,2)}\\&=&-1, \ \hat{\mathrm{B}}_{1}^{(3,3)}=-2,\\ \hat{\mathrm{C}}_{1}^{(1,1)}&=&-3, \ \hat{\mathrm{C}}_{1}^{(1,2)}=1, \ \hat{\mathrm{C}}_{1}^{(2,2)}=1, \ \hat{\mathrm{C}}_{1}^{(3,3)}=2, \end{array} $$

(D.14)

$$ \begin{array}{@{}rcl@{}} \mathbf{\overline{K}}_{2} &=& \frac{1}{8} \left[ \begin{array}{cc} \left [ \begin{array}{cc} \hat{\mathbf{A}}_{2}& \hat{\mathbf{B}}_{2} \\ \hat{\mathbf{B}}_{2}^{\mathrm{T}} & \hat{\mathbf{C}}_{2} \end{array} \right] & \left [ \begin{array}{cc} -\hat{\mathbf{A}}_{2} & -\hat{\mathbf{B}}_{2} \\ -\hat{\mathbf{B}}_{2}^{\mathrm{T}} & -\hat{\mathbf{C}}_{2} \end{array} \right] \\ \left[ \begin{array}{cc} -\hat{\mathbf{A}}_{2} & -\hat{\mathbf{B}}_{2} \\ -\hat{\mathbf{B}}_{2}^{\mathrm{T}} & -\hat{\mathbf{C}}_{2} \end{array} \right] & \left [\begin{array}{cc} \hat{\mathbf{A}}_{2} & \hat{\mathbf{B}}_{2} \\ \hat{\mathbf{B}}_{2}^{\mathrm{T}} & \hat{\mathbf{C}}_{2} \end{array} \right] \end{array} \right],\\ &&\text{ with}, \ \hat{\mathbf{A}}_{2}=\hat{\mathbf{A}}_{2}^{\mathrm{T}}, \ \hat{\mathbf{C}}_{2}=\hat{\mathbf{C}}_{2}^{\mathrm{T}}, \ \text{ and} \\ \hat{\mathrm{A}}_{2}^{(1,1)}&=&1, \ \hat{\mathrm{A}}_{2}^{(1,2)}=-1, \ \hat{\mathrm{A}}_{2}^{(2,2)}=-3, \ \hat{\mathrm{A}}_{2}^{(3,3)}=2,\\ \hat{\mathrm{B}}_{2}^{(1,1)}&=&1, \ \hat{\mathrm{B}}_{2}^{(1,2)}=1, \ \hat{\mathrm{B}}_{2}^{(2,1)}=-1,\ \hat{\mathrm{B}}_{2}^{(2,2)}\\ &=&-1, \ \hat{\mathrm{B}}_{2}^{(3,3)}=2,\\ \hat{\mathrm{C}}_{2}^{(1,1)}&=&1, \ \hat{\mathrm{C}}_{2}^{(1,2)}=1, \ \hat{\mathrm{C}}_{2}^{(2,2)}=-3, \ \hat{\mathrm{C}}_{2}^{(3,3)}=2, \end{array} $$

(D.15)

$$ \begin{array}{@{}rcl@{}} \mathbf{\overline{K}}_{3} &=& \frac{1}{2} \left[ \begin{array}{cc} \left [ \begin{array}{cc} \hat{\mathbf{A}}_{3}& \mathbf{O} \\ \text{sym} & \hat{\mathbf{C}}_{3} \end{array} \right] & \left [ \begin{array}{cc} -\hat{\mathbf{A}}_{3} & \mathbf{O} \\ \text{sym} & -\hat{\mathbf{C}}_{3} \end{array} \right] \\ \left[ \begin{array}{cc} -\hat{\mathbf{A}}_{3} & \mathbf{O} \\ \text{sym} & -\hat{\mathbf{C}}_{3} \end{array} \right] & \left [ \begin{array}{cc} \hat{\mathbf{A}}_{3} & \mathbf{O} \\ \text{sym} & \hat{\mathbf{C}}_{3} \end{array} \right] \end{array} \right], \\ &&\text{with}, \ \hat{\mathbf{A}}_{3}=\hat{\mathbf{A}}_{3}^{\mathrm{T}}, \ \hat{\mathbf{C}}_{3}=\hat{\mathbf{C}}_{3}^{\mathrm{T}}, \ \text{and}\\ \hat{\mathrm{A}}_{3}^{(1,2)}&=&-1, \ \hat{\mathrm{A}}_{3}^{(3,3)}=1,\\ \hat{\mathrm{C}}_{3}^{(1,2)}&=&-1, \ \hat{\mathrm{C}}_{3}^{(3,3)}=-1, \end{array} $$

(D.16)

$$ \mathbf{\overline{K}}_{4} = \mathbf{\overline{K}}_{5} = \mathbf{\overline{K}}_{6} = \mathbf{O}, $$

(D.17)

$$ \begin{array}{@{}rcl@{}} \mathbf{\overline{K}}_{7} &=& \frac{1}{72} \left[ \begin{array}{cc} \left [ \begin{array}{cc} -24\hat{\mathbf{K}}_{7}-12\hat{\mathbf{A}}_{7}+4\hat{\mathbf{C}}_{7}& 24\hat{\mathbf{K}}_{7}+12\hat{\mathbf{B}}_{7}+2\hat{\mathbf{C}}_{7} \\ 24\hat{\mathbf{K}}_{7}-12\hat{\mathbf{B}}_{7}+2\hat{\mathbf{C}}_{7} & -24\hat{\mathbf{K}}_{7}+12\hat{\mathbf{A}}_{7}+4\hat{\mathbf{C}}_{7} \end{array} \right] &\left [ \begin{array}{cc} 12\hat{\mathbf{K}}_{7}+6\hat{\mathbf{B}}_{7}+\hat{\mathbf{C}}_{7} & -12\hat{\mathbf{K}}_{7}-6\hat{\mathbf{A}}_{7}+2\hat{\mathbf{C}}_{7} \\ -12\hat{\mathbf{K}}_{7}+6\hat{\mathbf{A}}_{7}+2\hat{\mathbf{C}}_{7} & 12\hat{\mathbf{K}}_{7}-6\hat{\mathbf{B}}_{7}+\hat{\mathbf{C}}_{7} \end{array} \right] \\ \left[ \begin{array}{cc} 12\hat{\mathbf{K}}_{7}-6\hat{\mathbf{B}}_{7}+\hat{\mathbf{C}}_{7} & -12\hat{\mathbf{K}}_{7}+6\hat{\mathbf{A}}_{7}+2\hat{\mathbf{C}}_{7} \\ -12\hat{\mathbf{K}}_{7}-6\hat{\mathbf{A}}_{7}+2\hat{\mathbf{C}}_{7} & 12\hat{\mathbf{K}}_{7}+6\hat{\mathbf{B}}_{7}+\hat{\mathbf{C}}_{7} \end{array} \right] & \left [ \begin{array}{cc} -24\hat{\mathbf{K}}_{7}+12\hat{\mathbf{A}}_{7}+4\hat{\mathbf{C}}_{7}& 24\hat{\mathbf{K}}_{7}-12\hat{\mathbf{B}}_{7}+2\hat{\mathbf{C}}_{7} \\ 24\hat{\mathbf{K}}_{7}+12\hat{\mathbf{B}}_{7}+2\hat{\mathbf{C}}_{7} & -24\hat{\mathbf{K}}_{7}-12\hat{\mathbf{A}}_{7}+4\hat{\mathbf{C}}_{7} \end{array} \right] \end{array} \right],\\ &&\text{with}, \ \hat{\mathbf{K}}_{7}=\hat{\mathbf{K}}_{7}^{\mathrm{T}}, \ \hat{\mathbf{A}}_{7}=\hat{\mathbf{A}}_{7}^{\mathrm{T}}, \ \hat{\mathbf{B}}_{7}=-\hat{\mathbf{B}}_{7}^{\mathrm{T}}, \ \hat{\mathbf{C}}_{7}=\hat{\mathbf{C}}_{7}^{\mathrm{T}}, \ \text{and}\\ \hat{\mathrm{K}}_{7}^{(1,3)}&=&1, \ \hat{\mathrm{A}}_{7}^{(1,5)}=L, \ \hat{\mathrm{B}}_{7}^{(1,5)}=-\hat{\mathrm{B}}_{7}^{(5,1)}=-L, \ \hat{\mathrm{C}}_{7}^{(4,6)}=L^{2}, \end{array} $$

(D.18)

$$ \begin{array}{@{}rcl@{}} \mathbf{\overline{K}}_{8} &=& \frac{1}{72} \left[ \begin{array}{cc} -12\hat{\mathbf{K}}_{8}+6\hat{\mathbf{A}}_{8}+2\hat{\mathbf{C}}_{8} \left [ \begin{array}{cc} 2\mathbf{I}& \mathbf{I} \\ \mathbf{I} & 2\mathbf{I} \end{array} \right] & 12\hat{\mathbf{K}}_{8}+6\hat{\mathbf{B}}_{8}+\hat{\mathbf{C}}_{8} \left [ \begin{array}{cc} \mathbf{I}& 2\mathbf{I} \\ 2\mathbf{I} & \mathbf{I} \end{array} \right] \\ 12\hat{\mathbf{K}}_{8}-6\hat{\mathbf{B}}_{8}+\hat{\mathbf{C}}_{8} \left [ \begin{array}{cc} \mathbf{I}& 2\mathbf{I} \\ 2\mathbf{I} & \mathbf{I} \end{array} \right] & -12\hat{\mathbf{K}}_{8}-6\hat{\mathbf{A}}_{8}+2\hat{\mathbf{C}}_{8} \left [ \begin{array}{cc} 2\mathbf{I}& \mathbf{I} \\ \mathbf{I} & 2\mathbf{I} \end{array} \right] \end{array} \right]\\ &&\text{with}, \ \hat{\mathbf{K}}_{8}=\hat{\mathbf{K}}_{8}^{\mathrm{T}}, \ \hat{\mathbf{A}}_{8}=\hat{\mathbf{A}}_{8}^{\mathrm{T}}, \ \hat{\mathbf{B}}_{8}=-\hat{\mathbf{B}}_{8}^{\mathrm{T}}, \ \hat{\mathbf{C}}_{8}=\hat{\mathbf{C}}_{8}^{\mathrm{T}}, \ \text{and}\\ \hat{\mathrm{K}}_{8}^{(2,3)}&=&1, \ \hat{\mathrm{A}}_{8}^{(2,4)}=L, \ \hat{\mathrm{B}}_{8}^{(2,4)}=-\hat{\mathrm{B}}_{8}^{(4,2)}=L, \ \hat{\mathrm{C}}_{8}^{(5,6)}=L^{2}, \end{array} $$

(D.19)

The expressions of matrices $\mathbf {\overline {K}}_{i}$ reported above are supposed independent from the aspect ratio of the element. Of course, this assumption is justified if and only if the mesh of the FE model is structured and regular as much as possible (i.e. composed by pseudo-square elements). Accordingly, the singular form of the geometric stiffness matrix reads

$$ \begin{array}{@{}rcl@{}} \hat{\textbf{K}}_{\sigma}^{\flat} &=& \sum\limits_{e=1}^{N_{e}^{\flat}} \hat{\textbf{L}}_{e}^{\flat\mathrm{T}} \sum\limits_{i=1}^{8} r_{0ei}^{\flat} \overline{\textbf{K}}_{i} \hat{\textbf{L}}_{e}^{\flat}\\ &=& \sum\limits_{e=1}^{N_{e}^{\flat}} \hat{\textbf{L}}_{e}^{\flat\mathrm{T}} \sum\limits_{i=1}^{8} \left( \textbf{K}_{e}^{\text{lam}}\textbf{B}_{e} \textbf{u}_{e0}^{\flat}\right)_{i} \overline{\textbf{K}}_{i} \hat{\textbf{L}}_{e}^{\flat}\\ &=& \sum\limits_{e=1}^{N_{e}^{\flat}} \hat{\textbf{L}}_{e}^{\flat\mathrm{T}} \sum\limits_{i=1}^{8} \left( \textbf{K}_{e}^{\text{lam}}\textbf{B}_{e} \hat{\textbf{L}}_{e}^{\flat} \hat{\textbf{u}}_{0}^{\flat} \right)_{i} \overline{\textbf{K}}_{i} \hat{\textbf{L}}_{e}^{\flat}, \end{array} $$

(D.20)

and the non-singular counterpart can be obtained as

$$ \textbf{K}_{\sigma}^{\flat} := \mathfrak{R}\left( \hat{\textbf{K}}_{\sigma}^{\flat}, I_{\text{BC}}^{\flat}, I_{\text{BC}}^{\flat} \right). $$

(D.21)

Consider the following quantity:

$$ \begin{array}{@{}rcl@{}} &&\hat{\pmb{\psi}}^{\mathrm{T}}_{\flat} \frac{\partial \hat{\textbf{K}}_{\sigma}^{\flat}}{\partial \xi_{j}} \hat{\pmb{\psi}}_{\flat}\\ &=&\hat{\pmb{\psi}}^{\mathrm{T}}_{\flat} \left( \sum\limits_{e=1}^{N_{e}^{\flat}} \hat{\textbf{L}}_{e}^{\mathrm{T}} \left( \sum\limits_{i=1}^{8} \frac{\partial}{\partial \xi_{j}}\left( \textbf{K}_{e}^{\text{lam}}\textbf{B}_{e} \textbf{u}_{e0}^{\flat}\right)_{i} \overline{\textbf{K}}_{i}\right) \hat{\textbf{L}}_{e}^{\flat}\right)\hat{\pmb{\psi}}_{\flat}\\ &=& \sum\limits_{e=1}^{N_{e}^{\flat}} \hat{\pmb{\psi}}^{\mathrm{T}}_{\flat}\hat{\textbf{L}}_{e}^{\mathrm{T}} \left( \sum\limits_{i=1}^{8} \frac{\partial}{\partial \xi_{j}}\left( \textbf{K}_{e}^{\text{lam}}\textbf{B}_{e} \textbf{u}_{e0}^{\flat}\right)_{i} \overline{\textbf{K}}_{i}\right) \hat{\textbf{L}}_{e}^{\flat}\hat{\pmb{\psi}}_{\flat}\\ &=&\sum\limits_{e=1}^{N_{e}^{\flat}} \pmb{\psi}_{e\flat}^{\mathrm{T}}\left( \sum\limits_{i=1}^{8} \frac{\partial}{\partial \xi_{j}}\left( \textbf{K}_{e}^{\text{lam}}\textbf{B}_{e} \textbf{u}_{e0}^{\flat}\right)_{i} \overline{\textbf{K}}_{i}\right) \pmb{\psi}_{e\flat}\\ &=&\sum\limits_{e=1}^{N_{e}^{\flat}} \sum\limits_{i=1}^{8} \frac{\partial}{\partial \xi_{j}}\left( \textbf{K}_{e}^{\text{lam}}\textbf{B}_{e} \textbf{u}_{e0}^{\flat}\right)_{i} \pmb{\psi}_{e\flat}^{\mathrm{T}}\overline{\textbf{K}}_{i} \pmb{\psi}_{e\flat}\\ &=&\sum\limits_{e=1}^{N_{e}^{\flat}} \textbf{s}_{e\flat}^{\mathrm{T}} \frac{\partial}{\partial \xi_{j}}\left( \textbf{K}_{e}^{\text{lam}}\textbf{B}_{e} \textbf{u}_{e0}^{\flat}\right) \end{array} $$

$$ \begin{array}{@{}rcl@{}} &=& \underbrace{ \sum\limits_{e=1}^{N_{e}^{\flat}} \textbf{s}_{e\flat}^{\mathrm{T}} \frac{\partial \textbf{K}_{e}^{\text{lam}}}{\partial \xi_{j}}\textbf{B}_{e} \hat{\textbf{L}}_{e}^{\flat}}_{\hat{\textbf{a}}^{\mathrm{T}}} \hat{\textbf{u}}_{0}^{\flat} + \underbrace{\sum\limits_{e=1}^{N_{e}^{\flat}} \textbf{s}_{e\flat}^{\mathrm{T}} \textbf{K}_{e}^{\text{lam}}\textbf{B}_{e} \hat{\textbf{L}}_{e}^{\flat}}_{\hat{\textbf{b}}^{\mathrm{T}}} \frac{\partial \hat{\textbf{u}}_{0}^{\flat}}{\partial \xi_{j}}\\ &:=& \hat{\textbf{a}}^{\mathrm{T}} \hat{\textbf{u}}_{0}^{\flat} + \hat{\textbf{b}}^{\mathrm{T}} \frac{\partial \hat{\textbf{u}}_{0}^{\flat}}{\partial \xi_{j}}, \end{array} $$

(D.22)

with $\textbf {s}_{e\flat } :=\{\pmb {\psi }_{e\flat }^{\mathrm {T}}\overline {\textbf {K}}_{i} \pmb {\psi }_{e\flat }\mid i=1,\dots ,8\}$.

Remark D.2

Consider the scalar product v^Tu of two vectors $\textbf {u},\textbf {v}\in \mathbb {M}^{n\times 1}$. If u, A and B satisfies conditions of Remark D.1, then: $\textbf {v}^{\mathrm {T}}\textbf {u} = \textbf {v}^{\mathrm {T}}\mathfrak {Z}(\textbf {u},A) \oplus \textbf {v}^{\mathrm {T}}\mathfrak {Z}(\textbf {u},B) = \mathfrak {R}(\textbf {v},A)^{\mathrm {T}}\mathfrak {R}(\textbf {u},A) \oplus \mathfrak {R}(\textbf {v},B)^{\mathrm {T}}\mathfrak {R}(\textbf {u},B)$.

By applying Remarks D.1 and D.2 to both $\hat {\pmb {\psi }^{\flat }}$ and $\hat {\textbf {u}}^{\flat }_{0}$ of (D.22), considering that $\mathfrak {R}(\hat {\pmb {\psi }}^{\flat }, I_{\text {IN}}^{\flat }) = \textbf {0}$, one obtains:

$$ \begin{array}{@{}rcl@{}} \pmb{\psi}^{\mathrm{T}}_{\flat} \frac{\partial \textbf{K}_{\sigma}^{\flat}}{\partial \xi_{j}} \pmb{\psi}_{\flat} &=& \mathfrak{R}(\hat{\textbf{a}},I_{\text{IN}}^{\flat})^{\mathrm{T}}\mathfrak{R}(\hat{\textbf{u}}_{0}^{\flat},I_{\text{IN}}^{\flat}) + \mathfrak{R}(\hat{\textbf{a}},I_{\text{BC}}^{\flat})^{\mathrm{T}} \mathfrak{R}(\hat{\textbf{u}}_{0}^{\flat},I_{\text{BC}}^{\flat})+ \mathfrak{R}(\hat{\textbf{b}},I_{\text{IN}}^{\flat})^{\mathrm{T}}\mathfrak{R}\left( \frac{\partial \hat{\textbf{u}}_{0}^{\flat}}{\partial \xi_{j}},I_{\text{IN}}^{\flat}\right)+ \cdots\\ &&{\cdots} + \mathfrak{R}(\hat{\textbf{b}},I_{\text{BC}}^{\flat})^{\mathrm{T}}\mathfrak{R}\left( \frac{\partial \hat{\textbf{u}}_{0}^{\flat}}{\partial \xi_{j}},I_{\text{BC}}^{\flat}\right) \\ &=&\mathfrak{R}(\hat{\textbf{a}},I_{\text{IN}}^{\flat})^{\mathrm{T}}\textbf{u}^{\flat}_{\text{BC}} + \mathfrak{R}(\hat{\textbf{a}},I_{\text{BC}}^{\flat})^{\mathrm{T}}\textbf{u}^{\flat}+ \mathfrak{R}(\hat{\textbf{b}},I_{\text{IN}}^{\flat})^{\mathrm{T}}\frac{\partial \textbf{u}^{\flat}_{\text{BC}}}{\partial \xi_{j}}+ \mathfrak{R}(\hat{\textbf{b}},I_{\text{BC}}^{\flat})^{\mathrm{T}}\frac{\partial \textbf{u}^{\flat}}{\partial \xi_{j}} \\ &=& \mathfrak{R}(\hat{\textbf{a}},I_{\text{IN}}^{\flat})^{\mathrm{T}} \textbf{P} \textbf{u} + \mathfrak{R}(\hat{\textbf{a}},I_{\text{BC}}^{\flat})^{\mathrm{T}}\textbf{u}^{\flat}+ \mathfrak{R}(\hat{\textbf{b}},I_{\text{IN}}^{\flat})^{\mathrm{T}} \textbf{P} \frac{\partial \textbf{u}}{\partial \xi_{j}}+ \mathfrak{R}(\hat{\textbf{b}},I_{\text{BC}}^{\flat})^{\mathrm{T}}\frac{\partial \textbf{u}^{\flat}}{\partial \xi_{j}}, \end{array} $$

(D.23)

By injecting (D.23) into (D.11), and by choosing μ and w such that the terms multiplying ∂u^♭/∂ξ_j and ∂u/∂ξ_j vanish, one finally obtains:

$$ \left\lbrace \begin{array}{l} \frac{\partial \lambda}{\partial \xi_{j}} = \frac{\lambda}{\pmb{\psi}_{\flat}^{\mathrm{T}} \textbf{K}^{\flat} \pmb{\psi}_{\flat}} \left[ \pmb{\psi}_{\flat}^{\mathrm{T}} \frac{\partial \textbf{K}^{\flat}}{\partial \xi_{j}} \pmb{\psi}_{\flat} + \lambda \left( \mathfrak{R}(\hat{\textbf{a}},I_{\text{IN}}^{\flat})^{\mathrm{T}} \textbf{P} \textbf{u} + \mathfrak{R}(\hat{\textbf{a}},I_{\text{BC}}^{\flat})^{\mathrm{T}}\textbf{u}^{\flat} \right)+ \pmb{\mu}^{\mathrm{T}}\left( \frac{\partial \textbf{K}^{\flat}}{\partial \xi_{j}}\textbf{u}^{\flat} +\frac{\partial \textbf{K}^{\flat}_{\text{BC}}}{\partial \xi_{j}}\textbf{P}\textbf{u}\right) + \textbf{w}^{\mathrm{T}}\frac{\partial \textbf{K}}{\partial \xi_{j}}\textbf{u}\right], \\ j=1,\dots,n_{\text{vars}}, \\ \textbf{K}^{\flat} \pmb{\mu} = -\lambda \mathfrak{R}(\hat{\textbf{b}},I_{\text{BC}}^{\flat}),\\ \textbf{K} \textbf{w} = -\textbf{P}^{\mathrm{T}} \left[ \lambda \mathfrak{R}(\hat{\textbf{b}},I_{\text{IN}}^{\flat}) + \textbf{K}_{\text{BC}}^{\flat T} \pmb{\mu} \right]. \end{array} \right. $$

(D.24)

Equation (D.24) represents the gradient of the buckling factor of the LFEM subject to non-null imposed BCs, which are related to the displacement field solution of static analysis performed on the GFEM.

The last term of the first formula in (D.24) is the coupling effect between GFEM and LFEM and is non-zero ∀j = 1,⋯ ,n_vars. Conversely, the other terms are non-zero if and only if the design variable ξ_j is defined in the LFEM domain.