Load-path optimisation of funicular networks

Abstract

This paper describes the use of load-path optimisation for discrete, doubly curved, compression-only structures, represented by thrust networks. The load-path of a thrust network is defined as the sum of the internal forces in the edges multiplied by their lengths. The presented approach allows for the finding of the funicular solution for a network layout defined in plan, that has the lowest volume for the given boundary conditions. The compression-only thrust networks are constructed with Thrust Network Analysis by assigning force densities to the network’s independent edges. By defining a load-path function and deriving its associated gradient and Hessian functions, optimisation routines were used to find the optimum independent force densities that minimised the load-path function subject to compression-only constraints. A selection of example cases showed a dependence of the optimum load-path and force distribution on the network topology. Appropriate selection of the network pattern encouraged the flow of compression forces by avoiding long network edges with high force densities. A general, non-orthogonal network example showed that structures of high network indeterminacy can be investigated both directly for weight minimisation, and for the understanding of efficient thrust network patterns within the structure.

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Acknowledgements

The authors would like to acknowledge William Baker for presenting the load-path concept to the authors and for solving the optimal h / l ratio for the continuous parabola.

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Corresponding author

Correspondence to A. Liew.

Appendix

Appendix

Gradient and Hessian derivation of the load-path function

To calculate the gradient and Hessian of the external load-path function in Eq. (31), the following common matrix calculus properties are used, which follow [23] Differential Calculus notation,

$$\begin{aligned} \frac{\mathrm {d}{\mathbf {A}}\mathbf {X}{\mathbf {B}}}{\mathrm {d}\mathbf {X}}= & {} {\mathbf {B}}^\mathrm {T}\otimes {\mathbf {A}} \end{aligned}$$
(35a)
$$\begin{aligned} \frac{\mathrm {d}{\mathbf {A}}^{-1}}{\mathrm {d}{\mathbf {A}}}= & {} -({\mathbf {A}}^{-T}\otimes {\mathbf {A}}^{-1}), \end{aligned}$$
(35b)

Note that this derivative is essentially a fourth-order tensor, whose generic component represents the derivative of the inverse of a matrix with respect to its matrix. Let matrix functions \(f:\mathbb {R}^{n\times k}\rightarrow \mathbf {R}^{m\times p}\text { and }g:\mathbb {R}^{n\times k}\rightarrow \mathbb {R}^{p\times q}\) then

$$\begin{aligned} \begin{aligned} \frac{\mathrm {d}f(\mathbf {X})\cdot g(\mathbf {X})}{\mathrm {d}\mathbf {X}}=(g(\mathbf {X})^{\mathrm {T}}\otimes \mathbf {I}_{\mathrm {m}})f'(\mathbf {X})+(\mathbf {I}_{\mathrm {q}}\otimes f(\mathbf {X}))g'(\mathbf {X}), \end{aligned} \end{aligned}$$
(35c)

where \(\mathbf {I}_{\mathrm {m}}\) and \(\mathbf {I}_{\mathrm {q}}\) are the \(m\times m, q\times q\) identity matrices, and \(\otimes\) is the Kronecker matrix operator, defined as the complete multiplication between two matrices i.e. if \({\mathbf {A}}\) \((m\times n)\) and \({\mathbf {B}}\) \((p\times q)\) matrices, then \({\mathbf {A}}\otimes {\mathbf {B}}\) is an \((mp\times nq)\) matrix. Using the chain rule on the load-path function, Equation (31) gives

$$\begin{aligned} \frac{\mathrm {d}f(\mathbf {q}_{\mathrm {id}})}{\mathrm {d}\mathbf {q}_{\mathrm {id}}}=\frac{\mathrm {d}f(\mathbf {q}_{\mathrm {id}})}{\mathrm {d}\mathbf {Q}}\cdot \frac{\mathrm {d}\mathbf {Q}}{\mathrm {d}\mathbf {q}_{\mathrm {id}}} \end{aligned}$$
(36)

To find \(\frac{\mathrm {d}\mathbf {Q}}{\mathrm {d}\mathbf {q}_{\mathrm {id}}}\), notice that the diagonal matrix \(\mathbf {Q}\) can mathematically be written as a function of vector \(\mathbf {q}\) with

$$\begin{aligned} \mathbf {Q}=\sum _{i=1}^m\mathbf {E}_{\mathrm {i}}\mathbf {q}{\mathbf {e}}_{\mathrm {i}}^{\mathrm {T}}, \end{aligned}$$
(37)

where \(\mathbf {E}_{\mathrm {i}}\) is an \(m\times m\) matrix with all its entries zero except for identity in (ii), and \({\mathbf {e}}_{\mathrm {i}}\) is an \(m\times 1\) vector with identity on the \(i{\mathrm {th}}\) element and zero everywhere else. This derivative then becomes

$$\begin{aligned} \frac{\mathrm {d}\mathbf {Q}}{\mathrm {d}\mathbf {q}_{\mathrm {id}}}=&\frac{\mathrm {d}\sum _{i=1}^m\mathbf {E}_{\mathrm {i}}\mathbf {q}{\mathbf {e}}_{\mathrm {i}}^{\mathrm {T}}}{\mathrm {d}\mathbf {q}_{\mathrm {id}}}= \frac{\mathrm {d}\sum _{i=1}^m\mathbf {E}_{\mathrm {i}}{\mathbf {K}}\mathbf {q}_{\mathrm {id}}{\mathbf {e}}_{\mathrm {i}}^{\mathrm {T}}}{\mathrm {d}\mathbf {q}_{\mathrm {id}}}= \sum _1^m{\mathbf {e}}_{\mathrm {i}}\otimes \mathbf {E}_{\mathrm {i}}{\mathbf {K}} \end{aligned}$$
(38)
$$\begin{aligned} =&\sum _1^m\mathbf {I}_{m^2\times m}(m(i-1)+i,i){\mathbf {K}}={{\varvec{\Omega }}}{\mathbf {K}}, \end{aligned}$$
(39)

where the matrix \({{\varvec{\Omega }}}\) is constructed by adding unity in the aforementioned slots \(\forall i\), \(1\le i\le m\) of the \(m^2\times m\) matrix \(\mathbf {I}\).

Now the derivative \(\frac{\mathrm {d}f(\mathbf {q}_{\mathrm {id}})}{\mathrm {d}\mathbf {Q}}\) of Eq. (36) is

$$\begin{aligned} \frac{\mathrm {d}f(\mathbf {q}_{\mathrm {id}})}{\mathrm {d}\mathbf {Q}}=&-({\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}\otimes {\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}})({\mathbf {D}}_{\mathrm {i}}^{-1}\otimes {\mathbf {D}}_{\mathrm {i}}^{-1})({\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}\otimes {\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}})+ \mathbf {x}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {C}}_{\mathrm {b}}^{\mathrm {T}}\otimes \mathbf {x}^{\mathrm {T}}{\mathbf {C}}^{\mathrm {T}} \nonumber \\&+\mathbf {y}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {C}}_{\mathrm {b}}^{\mathrm {T}}\otimes \mathbf {y}^{\mathrm {T}}{\mathbf {C}}^{\mathrm {T}} -\frac{\mathrm {d}\,{\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {D}}_{\mathrm {b}}\mathbf {z}_{\mathrm {b}}}{\mathrm {d}\,\mathbf {Q}} \nonumber \\ =&-{\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}\otimes {\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}+ \mathbf {x}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {C}}_{\mathrm {b}}^{\mathrm {T}}\otimes \mathbf {x}^{\mathrm {T}}{\mathbf {C}}^{\mathrm {T}} \nonumber \\&+ \mathbf {y}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {C}}_{\mathrm {b}}^{\mathrm {T}}\otimes \mathbf {y}^{\mathrm {T}}{\mathbf {C}}^{\mathrm {T}} -\frac{\mathrm {d}{\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {D}}_{\mathrm {b}}\mathbf {z}_{\mathrm {b}}}{\mathrm {d}\mathbf {Q}} \end{aligned}$$
(40)

Note that the term \({\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {D}}_{\mathrm {b}}\) is a function of \(\mathbf {q}_{\mathrm {id}}\) if \({\mathbf {C}}_{\mathrm {i}}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}\) and is singular, i.e its spectral decomposition has a diagonal matrix with at least one zero diagonal element. The invariance of this term, in the case that \({\mathbf {C}}_{\mathrm {i}}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}\) is invertible, holds by

$$\begin{aligned} {\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}({\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}\mathbf {Q}{\mathbf {C}}_{\mathrm {i}})^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}\mathbf {Q}{\mathbf {C}}_{\mathbf {b}}\mathbf {z}_{\mathrm {b}} =\;&{\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}({\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}\mathbf {Q}{\mathbf {C}}_{\mathrm {i}})^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}\mathbf {Q}{\mathbf {C}}_{\mathrm {i}}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}({\mathbf {C}}_{\mathrm {i}}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}})^{-1}{\mathbf {C}}_{\mathrm {b}}\mathbf {z}_{\mathrm {b}} \nonumber \\ =\;&{\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}({\mathbf {C}}_{\mathrm {i}}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}})^{-1}{\mathbf {C}}_{\mathrm {b}}\mathbf {z}_{\mathrm {b}}. \end{aligned}$$
(41)

Assuming that \({\mathbf {C}}_{\mathrm {i}}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}\) is not invertible, using Eq. (35c), the derivative of \({\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {D}}_{\mathrm {b}}\mathbf {z}_{\mathrm {b}}\) is then

$$\begin{aligned}&\frac{\mathrm {d}[{\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}({\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}\mathbf {Q}{\mathbf {C}}_{\mathbf {i}})^{-1}]\cdot ({\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}\mathbf {Q}{\mathbf {C}}_{\mathrm {b}}\mathbf {z}_{\mathrm {b}})}{\mathrm {d}\mathbf {Q}} \nonumber \\&\quad = -(\mathbf {z}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {b}}^{\mathrm {T}}\otimes \mathbf {I}_1)({\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}\otimes {\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}) + (\mathbf {I}_1\otimes {\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1})(\mathbf {z}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {C}}_{\mathrm {b}}^{\mathrm {T}}\otimes {\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}) \nonumber \\&\quad = -(\mathbf {z}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}})\otimes ({\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}) \nonumber + (\mathbf {z}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {C}}_{\mathrm {b}}^{\mathrm {T}})\otimes ({\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}) \nonumber \\&\quad = [\mathbf {z}_{\mathrm {b}}^{\mathrm {T}}({\mathbf {C}}_{\mathrm {b}}^{\mathrm {T}}-{\mathbf {D}}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}})]\otimes ({\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}). \end{aligned}$$
(42)

Thus,

$$\begin{aligned} \frac{\mathrm {d}f(\mathbf {Q})}{\mathrm {d}\mathbf {Q}}=&-{\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}\otimes {\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}} + \mathbf {x}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {C}}_{\mathrm {b}}^{\mathrm {T}}\otimes \mathbf {x}^{\mathrm {T}}{\mathbf {C}}^{\mathrm {T}}+\mathbf {y}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {C}}_{\mathrm {b}}^{\mathrm {T}}\otimes \mathbf {y}^{\mathrm {T}}{\mathbf {C}}^{\mathrm {T}} \nonumber \\&- [\mathbf {z}_{\mathrm {b}}^{\mathrm {T}}({\mathbf {C}}_{\mathrm {b}}^{\mathrm {T}}-{\mathbf {D}}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}})]\otimes ({\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}). \end{aligned}$$
(43)

Inserting Eqs. (43) and (39) into (36) gives the \(1\times k\) gradient of the load-path

$$\begin{aligned} {{\varvec{\nabla }}}f(\mathbf {q}_{\mathrm {id}}) =&\frac{\mathrm {d}f(\mathbf {Q})}{\mathrm {d}\mathbf {Q}}\cdot \frac{\mathrm {d}\mathbf {Q}}{\mathrm {d}\mathbf {q}_{\mathrm {id}}} \nonumber \\ =&[-({\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}\otimes {\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}) + \mathbf {x}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {C}}_{\mathrm {b}}^{\mathrm {T}}\otimes \mathbf {x}^{\mathrm {T}}{\mathbf {C}}^{\mathrm {T}}+\mathbf {y}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {C}}_{\mathrm {b}}^{\mathrm {T}}\otimes \mathbf {y}^{\mathrm {T}}{\mathbf {C}}^{\mathrm {T}} \nonumber \\&- [\mathbf {z}_{\mathrm {b}}^{\mathrm {T}}({\mathbf {C}}_{\mathrm {b}}^{\mathrm {T}}-{\mathbf {D}}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}})]\otimes ({\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}})]\cdot \Omega {\mathbf {K}} \nonumber \\ =&{\sum }_{i=1}^m\left[ \mathbf {x}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {C}}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {e}}_{\mathrm {i}}\mathbf {x}^{\mathrm {T}}{\mathbf {C}}^{\mathrm {T}}\mathbf {E}_{\mathrm {i}}{\mathbf {K}} +\; \mathbf {y}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {C}}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {e}}_{\mathrm {i}}\mathbf {y}^{\mathrm {T}}{\mathbf {C}}^{\mathrm {T}}\mathbf {E}_{\mathrm {i}}{\mathbf {K}} - {\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}{\mathbf {e}}_{\mathrm {i}}{\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}\mathbf {E}_{\mathrm {i}}{\mathbf {K}} \right. \nonumber \\&\left. +\; \mathbf {z}_{\mathrm {b}}^{\mathrm {T}}({\mathbf {D}}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}-{\mathbf {C}}_{\mathrm {b}}^{\mathrm {T}}){\mathbf {e}}_{\mathrm {i}} {\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}\mathbf {E}_{\mathrm {i}}{\mathbf {K}}\right] . \end{aligned}$$
(44)

In the same fashion, to determine the Hessian of the function, the following chain rule is used

$$\begin{aligned} {{\varvec{\nabla }}}^2f(\mathbf {q}_{\mathrm {id}})=\frac{\mathrm {d}{{\varvec{\nabla }}}f}{\mathrm {d}\mathbf {Q}}\cdot \frac{\mathrm {d}\mathbf {Q}}{\mathrm {d}\mathbf {q}_{\mathrm {id}}}. \end{aligned}$$
(45)

Note that one can temporarily ignore the sums of the gradient, since \(\mathrm {d}\sum =\sum \mathrm {d}\), and consider them in the final chain-rule calculation. The first two terms of Equation (44) vanish in the Hessian, leaving only the derivative

$$\begin{aligned} \frac{\mathrm {d}{\varvec{\nabla }}f}{\mathrm {d}\mathbf {Q}}=&- \dfrac{\mathrm {d}\,{\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}{\mathbf {e}}_{\mathrm {i}}{\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}\mathbf {E}_{\mathrm {i}}{\mathbf {K}}}{\mathrm {d}\mathbf {Q}} \nonumber \\&+ \dfrac{\mathrm {d}\,\mathbf {z}_{\mathrm {b}}^{\mathrm {T}}({\mathbf {D}}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}} - {\mathbf {C}}_{\mathrm {b}}^{\mathrm {T}}){\mathbf {e}}_{\mathrm {i}}{\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}\mathbf {E}_{\mathrm {i}}{\mathbf {K}}}{\mathrm {d}\mathbf {Q}}. \end{aligned}$$
(46)

The first term of Eq. (46) by the multiplication rule is equivalent to

$$\begin{aligned}&\dfrac{\mathrm {d}({\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}{\mathbf {e}}_{\mathrm {i}})({\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}\mathbf {E}_{\mathrm {i}}{\mathbf {K}})}{\mathrm {d}\mathbf {Q}}\end{aligned}$$
(47)
$$\begin{aligned}&\quad =-({\mathbf {K}}^{\mathrm {T}}\mathbf {E}_{\mathrm {i}}{\mathbf {C}}_{\mathrm {i}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {p}}_{\mathrm {z}}\otimes \mathbf {I}_1)[{\mathbf {e}}_{\mathrm {i}}^{\mathrm {T}}{\mathbf {C}}_{\mathrm {i}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}\otimes {\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}]\end{aligned}$$
(48)
$$\begin{aligned}&\qquad -(\mathbf {I}_{\mathrm {k}\times \mathrm {k}}\otimes {\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}{\mathbf {e}}_{\mathrm {i}})[{\mathbf {K}}^{\mathrm {T}}\mathbf {E}_{\mathrm {i}}{\mathbf {C}}_{\mathrm {i}}{\mathbf {D}}_{\mathrm {i}}^{-1} {\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}\otimes {\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}]\end{aligned}$$
(49)
$$\begin{aligned}&\quad =-\left[ {\mathbf {K}}^{\mathrm {T}}\mathbf {E}_{\mathrm {i}}{\mathbf {C}}_{\mathrm {i}}{\mathbf {D}}_{\mathrm {i}}^{-1}({\mathbf {p}}_{\mathrm {z}}{\mathbf {e}}_{\mathrm {i}}^{\mathrm {T}}{\mathbf {C}}_{\mathrm {i}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}+\;{\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}{\mathbf {e}}_{\mathrm {i}}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}})\otimes {\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}\right] . \end{aligned}$$
(50)

The second derivative of Eq. (46) is slightly more complicated and is thus split in two parts

$$\begin{aligned}&\frac{\mathrm {d}\,\mathbf {z}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}{\mathbf {e}}_{\mathrm {i}}{\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1} {\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}\mathbf {E}_{\mathrm {i}}{\mathbf {K}}}{\mathrm {d}\mathbf {Q}} \end{aligned}$$
(51)
$$\begin{aligned}&\frac{\mathrm {d}\,\mathbf {z}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {C}}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {e}}_{\mathrm {i}}{\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}\mathbf {E}_{\mathrm {i}}{\mathbf {K}}}{\mathrm {d}\mathbf {Q}}. \end{aligned}$$
(52)

Using Eqs. (35a) and (35b), the derivative (52) becomes

$$\begin{aligned} -\mathbf {z}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {C}}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {e}}_{\mathrm {i}}({\mathbf {K}}^{\mathrm {T}}\mathbf {E}_{\mathrm {i}}{\mathbf {C}}_{\mathrm {i}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}} \otimes {\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}). \end{aligned}$$
(53)

Using the multiplication rule on Eq. (51) gives

$$\begin{aligned}&\frac{\mathrm {d}(\mathbf {z}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}{\mathbf {e}}_{\mathrm {i}})({\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1} {\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}\mathbf {E}_{\mathrm {i}}{\mathbf {K}})}{\mathrm {d}\mathbf {Q}}\end{aligned}$$
(54)
$$\begin{aligned}&\quad = ({\mathbf {K}}^{\mathrm {T}}\mathbf {E}_{\mathrm {i}}{\mathbf {C}}_{\mathrm {i}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {p}}_{\mathrm {z}}\otimes \mathbf {I}_1)[({\mathbf {e}}_{\mathrm {i}}^{\mathrm {T}}{\mathbf {C}}_{\mathrm {i}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}})\otimes \mathbf {z}_{\mathrm {b}}^{\mathrm {T}}({\mathbf {C}}_{\mathrm {b}}^{\mathrm {T}}-{\mathbf {D}}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}})]\end{aligned}$$
(55)
$$\begin{aligned}&\qquad - (\mathbf {I}_{\mathrm {k}\times \mathrm {k}}\otimes \mathbf {z}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}{\mathbf {e}}_{\mathrm {i}})[{\mathbf {K}}^{\mathrm {T}}\mathbf {E}_{\mathrm {i}}{\mathbf {C}}_{\mathrm {i}} {\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}\otimes {\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}]\end{aligned}$$
(56)
$$\begin{aligned}&\quad = {\mathbf {K}}^{\mathrm {T}}\mathbf {E}_{\mathrm {i}}{\mathbf {C}}_{\mathrm {i}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {p}}_{\mathrm {z}}{\mathbf {e}}_{\mathrm {i}}^{\mathrm {T}}{\mathbf {C}}_{\mathrm {i}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}\otimes \mathbf {z}_{\mathrm {b}}^{\mathrm {T}} ({\mathbf {C}}_{\mathrm {b}}^{\mathrm {T}}-{\mathbf {D}}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}})\end{aligned}$$
(57)
$$\begin{aligned}&\qquad -\mathbf {z}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}{\mathbf {e}}_{\mathrm {i}}({\mathbf {K}}^{\mathrm {T}}\mathbf {E}_{\mathrm {i}}{\mathbf {C}}_{\mathrm {i}}{\mathbf {D}}_{\mathrm {i}}^{-1} {\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}\otimes {\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}) \end{aligned}$$
(58)

Putting (50), (58) and (53) together gives

$$\begin{aligned} \frac{\mathrm {d}{\varvec{\nabla }}f}{\mathrm {d}\mathbf {Q}} =&{\sum }_{i=1}^m \left[ {\mathbf {K}}^{\mathrm {T}}\mathbf {E}_{\mathrm {i}}{\mathbf {C}}_{\mathrm {i}}{\mathbf {D}}_{\mathrm {i}}^{-1}\left[ \mathbf {z}_{\mathrm {b}}^{\mathrm {T}}({\mathbf {C}}_{\mathrm {b}}^{\mathrm {T}}-{\mathbf {D}}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}) {\mathbf {e}}_{\mathrm {i}}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}+\;{\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}{\mathbf {e}}_{\mathrm {i}}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}} + {\mathbf {p}}_\mathrm {z}{\mathbf {e}}_{\mathrm {i}}^{\mathrm {T}} {\mathbf {C}}_{\mathrm {i}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}\right] \right. \nonumber \\&\left. \otimes\; ({\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}) + {\mathbf {K}}^{\mathrm {T}}\mathbf {E}_{\mathrm {i}}{\mathbf {C}}_{\mathrm {i}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {p}}_{\mathrm {z}}{\mathbf {e}}_{\mathrm {i}}^{\mathrm {T}}{\mathbf {C}}_{\mathrm {i}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}} \otimes\; \mathbf {z}_{\mathrm {b}}^{\mathrm {T}}({\mathbf {C}}_{\mathrm {b}}^{\mathrm {T}}- {\mathbf {D}}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}) \right] \end{aligned}$$
(59)

Finally, using Eq. (45) one gets the \(k\times k\) Hessian matrix of the load-path

$$\begin{aligned} {\varvec{\nabla }}^2f(\mathbf {q}_{\mathrm {id}}) =&{\sum }_{i=1}^m\left[ {\mathbf {K}}^{\mathrm {T}}\mathbf {E}_{\mathrm {i}}{\mathbf {C}}_{\mathrm {i}}{\mathbf {D}}_{\mathrm {i}}^{-1}\left[ \mathbf {z}_{\mathrm {b}}^{\mathrm {T}}({\mathbf {C}}_{\mathrm {b}}^{\mathrm {T}}-{\mathbf {D}}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}) {\mathbf {e}}_{\mathrm {i}}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}+{\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1} {\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}{\mathbf {e}}_{\mathrm {i}}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}+{\mathbf {p}}_{\mathrm {z}}{\mathbf {e}}_{\mathrm {i}}^{\mathrm {T}} {\mathbf {C}}_{\mathrm {i}} {\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}\right] \right. \nonumber \\&\left. \otimes\; ({\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}) + {\mathbf {K}}^{\mathrm {T}}\mathbf {E}_{\mathrm {i}}{\mathbf {C}}_{\mathrm {i}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {p}}_{\mathrm {z}}{\mathbf {e}}_{\mathrm {i}}^{\mathrm {T}}{\mathbf {C}}_{\mathrm {i}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}} \otimes \mathbf {z}_{\mathrm {b}}^{\mathrm {T}}({\mathbf {C}}_{\mathrm {b}}^{\mathrm {T}}- {\mathbf {D}}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}) \right] \cdot {\varvec{\Omega }} {\mathbf {K}} \nonumber \\ =&{\sum }_{j=1}^m{\sum }_{i=1}^m\left[ {\mathbf {K}}^{\mathrm {T}}\mathbf {E}_{\mathrm {i}}{\mathbf {C}}_{\mathrm {i}}{\mathbf {D}}_{\mathrm {i}}^{-1}\left[ \mathbf {z}_{\mathrm {b}}^{\mathrm {T}} ({\mathbf {C}}_{\mathrm {b}}^{\mathrm {T}}-{\mathbf {D}}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}) {\mathbf {e}}_{\mathrm {i}}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}+{\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}} {\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}{\mathbf {e}}_{\mathrm {i}}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}+{\mathbf {p}}_{\mathrm {z}}{\mathbf {e}}_{\mathrm {i}}^{\mathrm {T}}{\mathbf {C}}_{\mathrm {i}}{\mathbf {D}}_{\mathrm {i}}^{-1} {\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}\right] {\mathbf {e}}_{\mathrm {j}} \right. \nonumber \\&\left. \otimes ({\mathbf {p}}_{\mathrm {z}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}\mathbf {E}_{\mathrm {j}}{\mathbf {K}}) +({\mathbf {K}}^{\mathrm {T}}\mathbf {E}_{\mathrm {i}}{\mathbf {C}}_{\mathrm {i}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {p}}_{\mathrm {z}}{\mathbf {e}}_{\mathrm {i}}^{\mathrm {T}}{\mathbf {C}}_{\mathrm {i}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}{\mathbf {e}}_{\mathrm {j}}) \otimes \mathbf {z}_{\mathrm {b}}^{\mathrm {T}}({\mathbf {C}}_{\mathrm {b}}^{\mathrm {T}}-{\mathbf {D}}_{\mathrm {b}}^{\mathrm {T}}{\mathbf {D}}_{\mathrm {i}}^{-1}{\mathbf {C}}_{\mathrm {i}}^{\mathrm {T}}\mathbf {E}_{\mathrm {i}}{\mathbf {K}}) \right] . \end{aligned}$$
(60)

All authors confirm/declare that they have no conflict of interests with respect to the submitted research project.

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Liew, A., Pagonakis, D., Van Mele, T. et al. Load-path optimisation of funicular networks. Meccanica 53, 279–294 (2018). https://doi.org/10.1007/s11012-017-0714-1

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Keywords

  • Load-path optimisation
  • Minimum volume problem
  • Thrust Network Analysis
  • Structural optimisation
  • Thrust network topology