High-order direct parametrisation of invariant manifolds for model order reduction of finite element structures: application to generic forcing terms and parametrically excited systems

Abstract

The direct parametrisation method for invariant manifolds is used for model order reduction of forced-damped mechanical structures subjected to geometric nonlinearities. Nonlinear mappings are introduced, allowing one to pass from the degrees of freedom of the finite-element model to the normal coordinates. Arbitrary orders of expansions are considered for the unknown mappings and the reduced dynamics, which are then solved sequentially through the homological equations for both autonomous and time-dependent terms. It is emphasised that the two problems share a similar structure, which can be used for an efficient implementation of the non-autonomous added terms. Special emphasis is also put on the new resonance conditions arising due to the presence of the external forcing frequencies, which allow predicting phenomena such as parametric excitation and isolas formation. The method is then applied to structures of academic and industrial interest. First, the large amplitude vibrations of a forced-damped cantilever beam are studied. This example highlights that high-order non-autonomous terms are compulsory to correctly estimate the maximum vibration amplitude experienced by the structure. The birth of isolated solutions is also illustrated on this example. The cantilever is then used to show how quadratic coupling creates conditions for the excitation of the parametric instability, and that this feature is correctly embedded in the reduction process. A shallow arch excited with multi-modal forcing is then studied to detail different forcing effects. Finally, the approach is validated on a structure of industrial relevance, i.e. a comb-driven micro-electro-mechanical resonator. The accuracy and computational performance reported suggest that the proposed methodology can accurately predict the nonlinear dynamic response of a large class of nonlinear vibratory systems.

Appendices

Weak formulation of the invariance equations

As remarked in Sect. 3, an alternative representation of the nonlinear change of coordinates is obtained by defining the nonlinear mappings between the normal coordinates and the continuous fields (here displacements and velocities), rather than operating from the nodal values from the FE discretisation. This alternative formulation will thus provide a more general setting, where one could for example use another numerical method for the space discretisation of the problem.

For the purpose of deriving this generalisation, let us first restart from the finite element approximation, which expressed the involved fields as a linear combination of a finite set of basis functions:

$$\begin{aligned} \varvec{u}_h = \sum _{i=1}^{\textrm{N}} \varvec{l}_{i}\textrm{U}_i, \qquad \varvec{w}_h = \sum _{i=1}^{\textrm{N}} \varvec{l}_{i}\textrm{W}_i, \end{aligned}$$

(128)

with $\varvec{l}_{i}$ nodal shape functions. For the present treatment, the same shape functions are used to interpolate both displacement and test fields. This choice is adopted only to simplify the presentation. We can then introduce a coordinate change between normal coordinates and continuous fields of the type:

$$\begin{aligned} \varvec{u}_h = \varvec{\varPsi }_{h}(\textbf{z}), \end{aligned}$$

(129)

with:

$$\begin{aligned} \varvec{\varPsi }_{h}(\textbf{z}) = \sum _{p=0}^{o} [\varvec{\varPsi }_{h,\mathcal {I}}(\textbf{z})]_{p}. \end{aligned}$$

(130)

We can therefore expand each tensor order as:

$$\begin{aligned}{}[\varvec{\varPsi }_{h}(\textbf{z})]_{p} = \sum _{\mathcal {I}} \varvec{\varPsi }^{(p)}_{h,\mathcal {I}} \pi _{\mathcal {I}}^{(p)}, \end{aligned}$$

(131)

within this setting, each mapping field $\varvec{\varPsi }^{(p)}_{h,\mathcal {I}}$ is a continuous field since it is defined pointwise over the entire domain $\textrm{B}$. We can then express each mapping field using a finite element approximation identical to that used for the displacement field itself:

$$\begin{aligned} \varvec{\varPsi }^{(p)}_{h,\mathcal {I}} = \sum _{i=1}^{\textrm{N}} \Psi ^{(p)}_{h,\mathcal {I},i} \varvec{l}_i, \end{aligned}$$

(132)

where $\Psi ^{(p)}_{h,\mathcal {I},i}$ corresponds to the nodal values of the mappings. We remark that these nodal values are mathematically equivalent to the entries of the mapping vectors used on the discretised problem presented in the work. This representation is exploited for the computation of the left-hand sides of the homological equations, without the need to compute the complete nonlinearity tensors, which is faster and more memory efficient. To better appreciate it, let us report the weak form associated with the first-order autonomous problem:

$$\begin{aligned}&\int _{\textrm{B}} \rho \, \dot{\varvec{v}}_{h} \cdot \varvec{w}_{h} \, \textrm{dB} + \int _{\textrm{B}} \varvec{\varepsilon }(\varvec{u}_{h}):\mathcal {A}:\varvec{\varepsilon }(\varvec{w}_{h}) \,\textrm{dB} \nonumber \\&+ \int _{\textrm{B}} \frac{1}{2} \varvec{\gamma }(\varvec{u}_{h},\varvec{u}_{h}):\mathcal {A}:\varvec{\varepsilon }(\varvec{w}_{h}) + \varvec{\varepsilon }(\varvec{u}_{h}):\mathcal {A}:\varvec{\gamma }(\varvec{u}_{h},\varvec{w}_{h}) \,\textrm{dB} \nonumber \\&{+} \int _{\textrm{B}} \frac{1}{2} \varvec{\gamma }(\varvec{u}_{h},\varvec{u}_{h}):\mathcal {A}:\varvec{\gamma }(\varvec{u}_{h},\varvec{w}_{h}) \,\textrm{dB} = 0, \quad \forall \, \varvec{w}_{h} {\in } \mathcal {C}_{h}(\textbf{0}), \end{aligned}$$

(133a)

$$\begin{aligned}&\int _{\textrm{B}} \rho \, \dot{\varvec{u}}_h \cdot \varvec{w}_{h} \,\textrm{dB} = \int _{\textrm{B}} \rho \, \varvec{v}_h \cdot \varvec{w}_{h} \textrm{dB} \quad \forall \, \varvec{w}_{h} \in \mathcal {C}_{h}(\textbf{0}), \end{aligned}$$

(133b)

where the finite-dimensional velocity field $\dot{\varvec{u}}_h=\varvec{v}_h$ has been introduced. Time derivatives are taken exactly as for the discrete problem:

$$\begin{aligned} \dot{\varvec{u}}_h = \nabla _{\textbf{z}}\varvec{\varPsi }_{h}(\textbf{z})\varvec{f}(\textbf{z}), \quad \dot{\varvec{v}}_h = \nabla _{\textbf{z}}\varvec{\varUpsilon }_{h}(\textbf{z})\varvec{f}(\textbf{z}), \end{aligned}$$

(134)

And upon substitution of mapping fields and their time derivatives into Eq. (133), we retrieve again the invariance equations, which are now formulated at the level of the weak form of the partial differential equation:

$$\begin{aligned}&\int _{\textrm{B}} \rho \, \nabla _{\textbf{z}}\varvec{\varUpsilon }_{h}(\textbf{z})\varvec{f}(\textbf{z}) \cdot \varvec{w}_{h} \, \textrm{dB} + \int _{\textrm{B}} \varvec{\varepsilon }(\varvec{\varPsi }_{h}(\textbf{z})):\mathcal {A}:\varvec{\varepsilon }(\varvec{w}_{h}) \,\textrm{dB} \nonumber \\&+~\int _{\textrm{B}} \!\frac{1}{2} \varvec{\gamma }(\varvec{\varPsi }_{h}(\textbf{z}),\varvec{\varPsi }_{h}(\textbf{z}))\!:\!\mathcal {A}\!:\!\varvec{\varepsilon }(\varvec{w}_{h}) \!\! +\!\! \varvec{\varepsilon }(\varvec{\varPsi }_{h}(\textbf{z}))\!:\!\mathcal {A}\!:\!\varvec{\gamma }(\varvec{\varPsi }_{h}(\textbf{z}),\varvec{w}_{h}) \textrm{dB} \nonumber \\&+ \int _{\textrm{B}} \frac{1}{2} \varvec{\gamma }(\varvec{\varPsi }_{h}(\textbf{z}),\varvec{\varPsi }_{h}(\textbf{z})):\mathcal {A}:\varvec{\gamma }(\varvec{\varPsi }_{h}(\textbf{z}),\varvec{w}_{h}) \,\textrm{dB} = 0, \quad \forall \, \varvec{w}_{h} \in \mathcal {C}_{h}(\textbf{0}), \end{aligned}$$

(135a)

$$\begin{aligned}&\int _{\textrm{B}} \rho \, \nabla _{\textbf{z}}\varvec{\varPsi }_{h}(\textbf{z})\varvec{f}(\textbf{z}) \cdot \varvec{w}_{h} \,\textrm{dB} = \int _{\textrm{B}} \rho \, \varvec{\varUpsilon }_{h}(\textbf{z}) \cdot \varvec{w}_{h} \textrm{dB}, \quad \forall \, \varvec{w}_{h} \in \mathcal {C}_{h}(\textbf{0}), \end{aligned}$$

(135b)

These two invariance equations are completely equivalent to Eq. (29), if the finite element approximation is identical and if the same integration rules are used. The interesting feature of Eq. (135) is that it represents, for each monomial of the asymptotic expansion, a linear elliptic partial differential equation, already discretised in a finite-dimensional space. As a result, all matrices and left-hand sides can be computed from the integration of Eq. (135) without the need to export the full nonlinearity tensor of the system. Also, Eq. (135) can be used in a more general setting and one could enlarge the scope and select another numerical method than the finite element procedure to solve out the elliptic PDEs. Furthermore, no constraint on the finite element scheme is required, hence any Galerkin method can be applied.

Multivariate tensor notation

Most quantities adopted in the present work are coefficients of multivariate polynomials. In this Section we recall the tensorial representation introduced in [35] to derive such quantities.

A generic vector of multivariate polynomials $\textbf{a}(\textbf{z})$ can be expressed as a summation of terms of increasing order from zero, up to the maximum order of the expansion o:

$$\begin{aligned} \textbf{a}(\textbf{z}) = \sum _{p=0}^{o} [\textbf{a}(\textbf{z})]_{p}. \end{aligned}$$

(136)

for each expansion order, polynomial terms are explicitly defined as:

$$\begin{aligned}{}[\textbf{a}(\textbf{z})]_{p} = \sum _{i_1=1}^{n} \sum _{i_2=1}^{n} \ldots \sum _{i_p=1}^{n} \textbf{a}^{(p)}_{\{i_1 i_2\ldots i_p\}} {z}_{i_1} {z}_{i_2} \ldots {z}_{i_p},\nonumber \\ \end{aligned}$$

(137)

with n the number of variables of the polynomial expansion. Alternatively, the following expansion can be used:

$$\begin{aligned}{}[\textbf{a}(\textbf{z})]_{p} = \sum _{\mathcal {I}} \textbf{a}^{(p)}_{\mathcal {I}} \pi _{\mathcal {I}}^{(p)}, \end{aligned}$$

(138)

where indexes and variables products are collected with the following notation:

$$\begin{aligned} \mathcal {I}= \{ i_1 i_2 \ldots i_{p} \}, \qquad \pi _{\mathcal {I}}^{(p)} = {z}_{i_1} {z}_{i_2} \ldots {z}_{i_p}. \end{aligned}$$

(139)

For a given polynomial with order p, there exist a finite number of indexes combinations $\mathcal {I}$ corresponding to the expansion monomials. Such combinations are collected in sets ${\mathcal {H}}^{(p)}$.

As a natural consequence of the above notation, vectors belonging to an asymptotic expansion are in general expressed as:

$$\begin{aligned} \textbf{a}^{(p)}_{\mathcal {I}} = \textbf{a}^{(p)}_{\{i_1 i_2\ldots i_p\}}, \end{aligned}$$

(140)

or alternatively, instead of writing vectors using the compact bold-typed notation, it is possible to express vector components using the following notation:

$$\begin{aligned} \textrm{a}_{s\mathcal {I}}^{(p)} = \textrm{a}_{s\{i_1 i_2\ldots i_p\}}^{(p)}, \end{aligned}$$

(141)

where the first subscript s stands for the s-th vector entry. If multiple vectors are present for a given monomial and order, as for instance when multiple forcing eigenvalues are present, a further index may be needed. Expansion are in general expressed as:

$$\begin{aligned} \textbf{a}^{(p)}_{j,\mathcal {I}} = \textbf{a}^{(p)}_{j,\{i_1 i_2\ldots i_p\}}, \end{aligned}$$

(142a)

$$\begin{aligned} \textrm{a}_{j,s\mathcal {I}}^{(p)} = \textrm{a}_{j,s\{i_1 i_2\ldots i_p\}}^{(p)}, \end{aligned}$$

(142b)

where index j refers to the j-th forcing eigenvalue.

Nonlinearity tensors computation

An important remark on dynamical systems stemming from partial differential equations is that their discretised algebraic formulation is equivalent to their finite-dimensional weak formulation up to numerical integration errors. This has already been underlined in Appendix A, where a finite-dimensional weak form for the invariance equation is derived. The natural result is that nonlinearities can be computed through integration over the domain. Let us consider the expressions reported in Eqs. (38d)–(38e). The most efficient technique to compute such term is simply to derive them through numerical integration of their associated weak formulation. Regarding the quadratic operator, its assemblage is performed as:

$$\begin{aligned}&\textrm{G}_{\mathcal {I}}^{(p)} =\sum _{k=1}^{p-1} \,\varvec{G}\left( \varvec{\varPsi }^{(k)}_{\{i_1\ldots i_{k}\}}, \varvec{\varPsi }^{(p-k)}_{\{i_{k+1}\ldots i_{p}\}} \right) \nonumber \\&\,=\sum _{k=1}^{p-1} \frac{1}{2} \int _{\textrm{B}} \,\varvec{\gamma }\left( \varvec{\varPsi }^{(k)}_{h,\{i_1\ldots i_{k}\}},\varvec{\varPsi }^{(p-k)}_{h,\{i_{k+1}\ldots i_{p}\}}\right) :\mathcal {A}:\varvec{\varepsilon }(\varvec{w}_{h}) \nonumber \\&\, + \varvec{\varepsilon }\left( \varvec{\varPsi }^{(k)}_{h,\{i_1\ldots i_{k}\}}\right) :\mathcal {A}:\varvec{\gamma }\left( \varvec{\varPsi }^{(p-k)}_{h,\{i_{k+1}\ldots i_{p}\}},\varvec{w}_{h}\right) \nonumber \\&\, + \varvec{\varepsilon }\left( \varvec{\varPsi }^{(p-k)}_{h,\{i_{k+1}\ldots i_{p}\}}\right) :\mathcal {A}:\varvec{\gamma }\left( \varvec{\varPsi }^{(k)}_{h,\{i_1\ldots i_{k}\}},\varvec{w}_{h}\right) \,\textrm{dB}, \end{aligned}$$

(143)

where the operator has been recasted in symmetric form. Similar treatment is performed for the $\varvec{H}(\textbf{U},\textbf{U},\textbf{U})$ operator:

$$\begin{aligned} \textbf{H}_{\mathcal {I}}^{(p)} = \sum _{k=1}^{p-2}\sum _{l=1}^{p-k-1}&\,\varvec{H}\left( \varvec{\varPsi }^{(k)}_{\{i_1\ldots i_{k}\}},\varvec{\varPsi }^{(l)}_{\{i_{k+1}\ldots i_{k+l}\}},\right. \nonumber \\&\quad \left. \varvec{\varPsi }^{(p-k-l)}_{\{i_{k+l+1}\ldots i_{p}\}}\right) = \nonumber \\ \sum _{k=1}^{p-2}\sum _{l=1}^{p-k-1} \frac{1}{6} \int _{\textrm{B}}&\, \varvec{\gamma }\left( \varvec{\varPsi }^{(k)}_{h,\{i_1\ldots i_{k}\}},\varvec{\varPsi }^{(l)}_{h,\{i_{k+1}\ldots i_{k+l}\}} \right) :\nonumber \\&\quad \mathcal {A}:\varvec{\gamma }\left( \varvec{\varPsi }^{(p-k-l)}_{h,\{i_{k+l+1}\ldots i_{p}\}},\varvec{w}_{h} \right) \nonumber \\&\, + \varvec{\gamma }\left( \varvec{\varPsi }^{(p-k-l)}_{h,\{i_{k+l+1}\ldots i_{p}\}}, \varvec{\varPsi }^{(k)}_{h,\{i_1\ldots i_{k}\}} \right) :\nonumber \\&\quad \mathcal {A}:\varvec{\gamma }\left( \varvec{\varPsi }^{(l)}_{h,\{i_{k+1}\ldots i_{k+l}\}},\varvec{w}_{h} \right) \nonumber \\&\, + \varvec{\gamma }\left( \varvec{\varPsi }^{(l)}_{h,\{i_{k+1}\ldots i_{k+l}\}},\varvec{\varPsi }^{(p-k-l)}_{h,\{i_{k+l+1}\ldots i_{p}\}} \right) :\nonumber \\&\quad \mathcal {A}:\varvec{\gamma }\left( \varvec{\varPsi }^{(k)}_{h,\{i_1\ldots i_{k}\}},\varvec{w}_{h} \right) \,\textrm{dB}, \end{aligned}$$

(144)

Non-autonomous gradients

In Eq. (32), terms involving a gradient with respect to the normal coordinate require a dedicated treatment. In particular, two new terms depending on such gradient are involved. The first involves the gradient of the non-autonomous mappings:

(145a)

(145b)

where the shortcut notation has been introduced to refer to a term with order strictly larger than one and smaller than p. Expanding the product in the right-hand side, one can rewrite:

(146a)

(146b)

The second term at the right-hand side of Eq. (146) can be explicitly rewritten as:

(147a)

(147b)

where all the terms have been grouped and expanded onto the monomials $\pi _{\mathcal {I}}^{(p)}$, thus making appear new coefficients $\bar{\varvec{\mu }}_{\mathcal {I}}^{(p)}$ and $\bar{\varvec{\nu }}_{\mathcal {I}}^{(p)}$, the full expressions of which can be written as:

$$\begin{aligned} \bar{\varvec{\mu }}_{\mathcal {I}}^{(p)} =&\, \sum _{s=1}^{2\textrm{n}} \sum _{k=2}^{p} \sum _{l=0}^{p-k} \hat{\varvec{\Psi }}^{(p-k+1)}_{\{i_1 \ldots i_{l}\,s\,i_{l+k+1} \ldots i_{p}\}} f_{s\{i_{l+1}\ldots i_{l+k}\}}^{(k)}, \end{aligned}$$

(148a)

$$\begin{aligned} \bar{\varvec{\nu }}_{\mathcal {I}}^{(p)} =&\, \sum _{s=1}^{2\textrm{n}} \sum _{k=2}^{p} \sum _{l=0}^{p-k} \hat{\varvec{\Upsilon }}^{(p-k+1)}_{\{i_1 \ldots i_{l}\,s\,i_{l+k+1} \ldots i_{p}\}} f_{s\{i_{l+1}\ldots i_{l+k}\}}^{(k)}. \end{aligned}$$

(148b)

These new terms have exactly the same structure as those appearing in Eqs. (38b)–(38c) computed for the autonomous part, where one just need to replace the terms of the nonlinear mapping, which has important consequence at the computational level and ease the writing.

The second term involving a gradient and needing a dedicated development involves the gradient of the known autonomous mappings with respect to the normal coordinates, now multiplied with the reduced dynamics of the unknown time-dependent part. It can be expressed as:

(149a)

(149b)

which in turn can be rewritten as:

(150a)

(150b)

The second term on the right-hand side can be explicitly defined as:

(151a)

(151b)

which provides the following compact expression for Eq. (150):

$$\begin{aligned} \left[ \nabla _{\textbf{z}}\varvec{\varPsi }(\textbf{z})\hat{\varvec{f}}(\textbf{z}) \right] =&\, \sum _{\mathcal {I}}\left( \sum _{s=1}^{2\textrm{n}} (\pmb {\phi }_s \hat{f}_{s\mathcal {I}}^{(p)}) + \bar{\bar{\varvec{\mu }}}_{\mathcal {I}}^{(p)} \right) \pi _{\mathcal {I}}^{(p)}, \end{aligned}$$

(152a)

$$\begin{aligned} \left[ \nabla _{\textbf{z}}\varvec{\varUpsilon }(\textbf{z})\hat{\varvec{f}}(\textbf{z}) \right] =&\, \sum _{\mathcal {I}}\left( \sum _{s=1}^{2\textrm{n}} (\pmb {\phi }_s \lambda _s \hat{f}_{s\mathcal {I}}^{(p)}) + \bar{\bar{\varvec{\nu }}}_{\mathcal {I}}^{(p)} \right) \pi _{\mathcal {I}}^{(p)}, \end{aligned}$$

(152b)

where the new terms provided by the above developments are explicitly computed as:

(153a)

(153b)

Again, these coefficients can be directly compared to those reported in Eqs. (38b)–(38c) and in Eqs. (148), since sharing exactly the same structure, where now the non-autonomous part (tilde terms) is on the reduced dynamics coefficients. At the computational level, this symmetric structure can be used to compute easily these new terms.

Explicit expressions for the RHS of low-order ${\varepsilon }^{1}$-homological equations and reduced dynamics coefficients

In the present section, we report explicit expressions of the right-hand sides $\hat{\textbf{E}}_{\mathcal {I}}^{(p)}$ for low-order developments. This serves the purpose of highlighting how different terms in the reduced dynamics are obtained and how they are affected by the forcing and the geometrical parameters of the structure.

For the zero-order homological equations, $\hat{\textbf{E}}^{(0)}$ is simply given by the applied forcing:

$$\begin{aligned} \hat{\textbf{E}}^{(0)} =&\, \hat{\textbf{F}}, \end{aligned}$$

(154a)

$$\begin{aligned} \hat{\varvec{\mu }}^{(0)} =&\, \textbf{0}. \end{aligned}$$

(154b)

Furthermore, since $\hat{\varvec{\mu }}^{(0)}=\textbf{0}$, the resulting mapping is co-linear with the applied forcing $\hat{\textbf{F}}$ and reduced dynamics coefficients $\hat{\textbf{F}}^{0}$ are proportional to the projection of the forcing vector onto the master modal subspace, as detailed in Section 4. This conclusion is obtained by considering that the reduced dynamics coefficient can be computed from the relation:

$$\begin{aligned} \hat{f}^{(p)}_{j\,\mathcal {I}}= & {} \textbf{X}_{j}^{\textrm{T}} \left[ \begin{array}{c} \hat{\textbf{E}}^{(p)}_{\mathcal {I}} \\ -\textbf{M}\hat{\varvec{\mu }}^{(p)}_{\mathcal {I}} \end{array}\right] = \frac{\pmb {\phi }_j^{\textrm{T}}}{\lambda _{j} - \bar{\lambda }_{j}} \hat{\textbf{E}}^{(p)}_{\mathcal {I}} \nonumber \\{} & {} + \frac{\bar{\lambda }_{j} \pmb {\phi }_j^{\textrm{T}}}{\lambda _{j} - \bar{\lambda }_{j}} \textbf{M}\hat{\varvec{\mu }}^{(p)}_{\mathcal {I}}, \end{aligned}$$

(155)

which for the zero-order development yields:

$$\begin{aligned} \hat{f}^{(0)}_{j} = \frac{\pmb {\phi }_j^{\textrm{T}}}{\lambda _{j} - \bar{\lambda }_{j}} \hat{\textbf{F}}. \end{aligned}$$

(156)

The first-order non-autonomous asymptotic development yields:

$$\begin{aligned} \hat{\textbf{E}}_{\mathcal {I}}^{(1)} = \hat{\textbf{E}}_{\{i_1\}}^{(1)} =&\, -\textbf{M}\sum _{s=1}^{2\textrm{n}}\left[ {\varvec{\Upsilon }}^{(2)}_{\{s \,i_1\}} \hat{f}^{(0)}_{s} + {\varvec{\Upsilon }}^{(2)}_{\{i_1 s\}} \hat{f}^{(0)}_{s} \right] \nonumber \\&\quad - 2\varvec{G}(\varvec{\Psi }^{(1)}_{\{i_1\}}, \hat{\varvec{\Psi }}^{(0)}), \end{aligned}$$

(157a)

$$\begin{aligned} \hat{\varvec{\mu }}^{(1)}_{\{i_1\}} =&\, \sum _{s=1}^{2\textrm{n}} \left[ {\varvec{\Psi }}^{(2)}_{\{s\,i_1\}} \hat{f}^{(0)}_{s} + {\varvec{\Psi }}^{(2)}_{\{i_1 s\}} \hat{f}^{(0)}_{s} \right] , \end{aligned}$$

(157b)

thus providing an explicit expression for the reduced dynamics coefficients:

$$\begin{aligned}{} & {} \hat{f}^{(1)}_{j\{i_1\}} = \frac{\pmb {\phi }_j^{\textrm{T}}}{\lambda _{j} - \bar{\lambda }_{j}} \left[ -\textbf{M}\sum _{s=1}^{2\textrm{n}}\left[ {\varvec{\Upsilon }}^{(2)}_{\{s\, i_1\}} \hat{f}^{(0)}_{s} +{\varvec{\Upsilon }}^{(2)}_{\{i_1 s\}} \hat{f}^{(0)}_{s} \right] \right. \nonumber \\{} & {} \left. - 2\varvec{G}(\varvec{\Psi }^{(1)}_{\{i_1\}},\hat{\varvec{\Psi }}^{(0)}) \right] + \frac{\bar{\lambda }_{j} \pmb {\phi }_j^{\textrm{T}}}{\lambda _{j} - \bar{\lambda }_{j}} \left[ \sum _{s=1}^{2\textrm{n}} \textbf{M}\left[ {\varvec{\Psi }}^{(2)}_{\{s\,i_1\}} \hat{f}^{(0)}_{s}\right. \right. \nonumber \\{} & {} \left. \left. + \varvec{\Psi }^{(2)}_{\{i_1 s\}} \hat{f}^{(0)}_{s} \right] \right] , \end{aligned}$$

(158)

which highlights how the reduced dynamics coefficients at first order are heavily affected by the ${{\varvec{G}}}(\varvec{\Psi }^{(1)}_{\{i_1\}},\hat{\varvec{\Psi }}^{(0)})$ vector. Finally, the second-order expansions yields:

$$\begin{aligned} \hat{\textbf{E}}_{\mathcal {I}}^{(2)} = \hat{\textbf{E}}_{\{i_1i_2\}}^{(2)} =&\, -\textbf{M}\sum _{s=1}^{2\textrm{n}}\left[ {\varvec{\Upsilon }}^{(3)}_{\{s\, i_1i_2\}} \hat{f}^{(0)}_{s} +{\varvec{\Upsilon }}^{(3)}_{\{i_1 s \,i_2\}} \hat{f}^{(0)}_{s}\right. \nonumber \\&\quad \left. + {\varvec{\Upsilon }}^{(3)}_{\{i_1 i_2 s\}} \hat{f}^{(0)}_{s} \right] \nonumber \\&\, - \textbf{M}\sum _{s=1}^{2\textrm{n}}\left[ {\varvec{\Upsilon }}^{(2)}_{\{i_1 s\}} \hat{f}^{(1)}_{s\, \{i_2\}} + {\varvec{\Upsilon }}^{(2)}_{\{s\, i_2\}} \hat{f}^{(1)}_{s \{i_1\}} \right] \nonumber \\&\quad - \textbf{M}\sum _{s=1}^{2\textrm{n}} \left[ \hat{\varvec{\Upsilon }}^{(1)}_{\{s\}} f^{(2)}_{s\{ i_1 i_2\}} \right] \nonumber \\&\, + 2\varvec{ G}(\varvec{\Psi }^{(2)}_{\{i_1 i_2\}}, \hat{\varvec{\Psi }}^{(0)} ) + 2\varvec{ G}(\varvec{\Psi }^{(1)}_{\{i_1\}}, \hat{\varvec{\Psi }}^{(1)}_{\{i_2\}} ) \nonumber \\&\quad + 3\varvec{ H}(\varvec{\Psi }^{(1)}_{\{i_1\}}, \varvec{\Psi }^{(1)}_{\{i_2\}}, \hat{\varvec{\Psi }}^{(0)}) \; , \end{aligned}$$

(159a)

$$\begin{aligned} \hat{\varvec{\mu }}_{\{i_1i_2\}}^{(2)} =&\, - \sum _{s=1}^{2\textrm{n}}\left[ \varvec{\Psi }^{(3)}_{\{s\,i_1i_2\}} \hat{f}^{(0)}_{s} \right. \nonumber \\&\quad \left. + \varvec{\Psi }^{(3)}_{\{i_1 s\,i_2\}} \hat{f}^{(0)}_{s} +\varvec{\Psi }^{(3)}_{\{i_1 i_2 s\}} \hat{f}^{(0)}_{s} \right] \nonumber \\&\, - \sum _{s=1}^{2\textrm{n}}\left[ \varvec{\Psi }^{(2)}_{\{i_1 s\}} \hat{f}^{(1)}_{s\{i_2\}} + \varvec{\Psi }^{(2)}_{\{s \,i_2\}} \hat{f}^{(1)}_{s \{i_1\}} \right] \nonumber \\&\quad - \sum _{s=1}^{2\textrm{n}} \left[ \hat{\varvec{\Psi }}^{(1)}_{\{s\}} f^{(2)}_{s\, \{i_1 i_2\}} \right] \; , \end{aligned}$$

(159b)

and the second-order reduced dynamics coefficients:

$$\begin{aligned} \hat{f}^{(2)}_{j\,\{i_1 i_2\}} =&\,\frac{\pmb {\phi }_j^{\textrm{T}}}{\lambda _{j} - \bar{\lambda }_{j}} \,\biggl [ \biggl . -\textbf{M}\sum _{s=1}^{2\textrm{n}}\left[ \varvec{\Upsilon }^{(3)}_{\{s\,i_1i_2\}} \hat{f}^{(0)}_{s} \right. \nonumber \\&\left. + \varvec{\Upsilon }^{(3)}_{\{i_1 s\,i_2\}} \hat{f}^{(0)}_{s} + \varvec{\Upsilon }^{(3)}_{\{i_1 i_2 s\}} \hat{f}^{(0)}_{s} \right] \nonumber \\&\, - \textbf{M}\sum _{s=1}^{2\textrm{n}}\left[ \varvec{\Upsilon }^{(2)}_{\{i_1 s\}} \hat{f}^{(1)}_{s\{i_2\}}\right. \nonumber \\&\left. + \varvec{\Upsilon }^{(2)}_{\{s\,i_2\}} \hat{f}^{(1)}_{s\{i_1\}} \right] - \textbf{M}\sum _{s=1}^{2\textrm{n}} \left[ \hat{\varvec{\Upsilon }}^{(1)}_{\{s\}} f^{(2)}_{s\{i_1 i_2\}} \right] \nonumber \\&\, + 2\varvec{ G}(\varvec{\Psi }^{(2)}_{\{i_1 i_2\}}, \hat{\varvec{\Psi }}^{(0)} ) + 2\varvec{ G}(\varvec{\Psi }^{(1)}_{\{i_1\}}, \hat{\varvec{\Psi }}^{(1)}_{\{i_2\}} ) \nonumber \\&\,+ 3\varvec{ H}(\varvec{\Psi }^{(1)}_{\{i_1\}}, \varvec{\Psi }^{(1)}_{\{i_2\}}, \hat{\varvec{\Psi }}^{(0)}) \biggl .\biggl ] \nonumber \\&+ \frac{\bar{\lambda }_{j} \pmb {\phi }_j^{\textrm{T}}}{\lambda _{j} - \bar{\lambda }_{j}} \, \biggl [ \biggl . - \textbf{M}\sum _{s=1}^{2\textrm{n}}\left[ \varvec{\Psi }^{(3)}_{\{s\, i_1i_2\}} \hat{f}^{(0)}_{s}\right. \nonumber \\&\left. + \varvec{\Psi }^{(3)}_{\{i_1 s\, i_2\}} \hat{f}^{(0)}_{s} + \varvec{\Psi }^{(3)}_{\{i_1 i_2 s\}} \hat{f}^{(0)}_{s} \right] \nonumber \\&\, - \sum _{s=1}^{2\textrm{n}} \textbf{M}\left[ \varvec{\Psi }^{(2)}_{\{i_1 s\}} \hat{f}^{(1)}_{s\, i_2} + \varvec{\Psi }^{(2)}_{\{s\,i_2\}} \hat{f}^{(1)}_{s\{i_1\}} \right] \nonumber \\&- \sum _{s=1}^{2\textrm{n}} \textbf{M}\left[ \hat{\varvec{\Psi }}^{(1)}_{\{s\}} f^{(2)}_{s\{i_1 i_2\}} \right] \biggl .\biggl ]. \end{aligned}$$

(160)