Appendix
1.1 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 (i, i), 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.
