Data-driven model order reduction with proper symplectic decomposition for flexible multibody system

Abstract

Flexible multibody system plays an important role for the simulation of mechanism system. Due to the requirement of precision or high complexity of the model, the number of the finite elements of flexible multibody system will increase rapidly, which will lead to the decrease in the computational efficiency. In order to save the computational cost for simulating flexible multibody system, a novel model order reduction strategy based on the idea of data-driven model is proposed. In addition, the proposed method which is called symplectic model order reduction is in light of proper symplectic decomposition and symplectic Galerkin projection. At first, the snapshot matrix is obtained by an empirical data ensemble of the full-order model, and the transfer symplectic matrix of high dimension to low dimension is obtained by reduced-order bases using the method of cotangent lift. Then, the discrete governing equations of reduced-order model (ROM) are derived by symplectic discretization. Furthermore, a systematic study of model order reduction in system level and component level is provided in the paper. In addition, for adaption of ROM to parameter variation, a parameter interpolation method is offered to obtain the ROM. Eventually, several examples are used to verify the effectiveness of the proposed method, and the results show that the proposed method has better numerical accuracy and higher computational efficiency with respect to classic proper orthogonal decomposition-based ROM.

Appendices

Appendix A

Definition 1

(Symplectic projection) Let ${{\varvec{\upomega}}} \in {\mathbb{R}}^{2n}$ be a high-dimensional variable, ${\mathbf{z}} \in {\mathbb{R}}^{2m}$ a low-dimensional variable, and the transfer matrix of ${{\varvec{\Pi}}} \in {\mathbb{R}}^{2n \times 2m}$. A linear projection from ${\mathbf{z}}$ to ${{\varvec{\upomega}}}$ has the expression: ${{\varvec{\upomega}}} \mapsto {{\varvec{\Pi}}}\;{\mathbf{z}}$, $n,\;m \in {\mathbb{N}}^{*}$ and $m \ll n$. We call the projection is a symplectic projection, and ${{\varvec{\Pi}}}$ is a symplectic matrix if the following relationship is satisfied:

$$ {{\varvec{\Pi}}}^{{\mathbf{T}}} {\mathbf{J}}_{2n \times 2n} {{\varvec{\Pi}}} = {\mathbf{J}}_{2m \times 2m} , $$

(20)

where in Eq. (20), the formula of ${\mathbf{J}}$ has the following expression:

$$ {\mathbf{J}}_{2i \times 2i} = \left[ {\begin{array}{*{20}c} {\mathbf{0}} & {{\mathbf{I}}_{2i \times 2i} } \\ { - {\mathbf{I}}_{2i \times 2i} } & {\mathbf{0}} \\ \end{array} } \right],\;i = m,n. $$

(21)

${\mathbf{I}}$ is an identity matrix, and the subscript of ${\mathbf{I}}$ represents its dimension.

Definition 2

(Symplectic inverse) According to Definition 1, the symplectic matrix ${{\varvec{\Pi}}}$ belongs to ${\mathbb{R}}^{2n \times 2m}$. For each ${{\varvec{\Pi}}}$, we can define the inverse matrix of ${{\varvec{\Pi}}}$ have the following expression:

$$ {{\varvec{\Pi}}}^{ + } = {\mathbf{J}}_{2m \times 2m}^{{\mathbf{T}}} {{\varvec{\Pi}}}^{{\mathbf{T}}} {\mathbf{J}}_{2n \times 2n} \in {\mathbb{R}}^{2m \times 2n} . $$

(22)

According to Eq. (22), we can obtain:

$$ {{\varvec{\Pi}}}^{ + } {{\varvec{\Pi}}} = {\mathbf{J}}_{2m \times 2m}^{{\mathbf{T}}} {{\varvec{\Pi}}}^{{\mathbf{T}}} {\mathbf{J}}_{2n \times 2n} {{\varvec{\Pi}}} = {\mathbf{J}}_{2m \times 2m}^{{\mathbf{T}}} {\mathbf{J}}_{2m \times 2m} = {\mathbf{I}}_{2m \times 2m} \in {\mathbb{R}}^{2m \times 2m} . $$

(23)

Therefore, Eq. (23) can also meet the characteristic of inverse matrix.

Definition 3

Given an orthogonal matrix ${{\varvec{\Omega}}} \in {\mathbb{R}}^{n \times m}$, then the matrix meet the following relationship:

$$ {{\varvec{\Omega}}}^{{\mathbf{T}}} {{\varvec{\Omega}}} = {\mathbf{I}}_{m \times m} . $$

(24)

Proposition 1

The transfer matrix ${{\varvec{\Pi}}}$ obtained by the above procedure satisfies the properties of symplecticity and orthogonality.

Proof

• (1) Symplecticity

  According to Eq. (1), then we can have:

  $$ \begin{gathered} {{\varvec{\Pi}}}^{{\mathbf{T}}} {\mathbf{J}}_{2n \times 2n} {{\varvec{\Pi}}} = \left[ {\begin{array}{*{20}c} {{{\varvec{\Psi}}}_{n \times m} } & {\mathbf{0}} \\ {\mathbf{0}} & {{{\varvec{\Psi}}}_{n \times m} } \\ \end{array} } \right]^{{\mathbf{T}}} \left[ {\begin{array}{*{20}c} {\mathbf{0}} & {{\mathbf{I}}_{2n \times 2n} } \\ { - {\mathbf{I}}_{2n \times 2n} } & {\mathbf{0}} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{{\varvec{\Psi}}}_{n \times m} } & {\mathbf{0}} \\ {\mathbf{0}} & {{{\varvec{\Psi}}}_{n \times m} } \\ \end{array} } \right]. \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \left[ {\begin{array}{*{20}c} {\mathbf{0}} & {{{\varvec{\Psi}}}_{n \times m}^{{\mathbf{T}}} {{\varvec{\Psi}}}_{n \times m} } \\ { - {{\varvec{\Psi}}}_{n \times m}^{{\mathbf{T}}} {{\varvec{\Psi}}}_{n \times m} } & {\mathbf{0}} \\ \end{array} } \right]. \hfill \\ \end{gathered} $$

  (25)

  With above results, ${{\varvec{\Psi}}}_{n \times m}$ is an orthogonal matrix, which is obtained by the thin SVD of snapshot matrix. Therefore, Eq. (25) is equal to the following formula:

  $$ {{\varvec{\Pi}}}^{{\mathbf{T}}} {\mathbf{J}}_{2n \times 2n} {{\varvec{\Pi}}} = {\mathbf{J}}_{2m \times 2m} . $$

  (26)

  Then, the transfer matrix ${{\varvec{\Pi}}}$ has the property of symplecticity according to Definition 1.

• (2) Orthogonality

  According to characteristic of the orthogonal matrix expressed in Definition 3, we can have:

  $$ {{\varvec{\Pi}}}^{{\mathbf{T}}} {{\varvec{\Pi}}} = \left[ {\begin{array}{*{20}c} {{{\varvec{\Psi}}}_{n \times m} } & {\mathbf{0}} \\ {\mathbf{0}} & {{{\varvec{\Psi}}}_{n \times m} } \\ \end{array} } \right]^{{\mathbf{T}}} \left[ {\begin{array}{*{20}c} {{{\varvec{\Psi}}}_{n \times m} } & {\mathbf{0}} \\ {\mathbf{0}} & {{{\varvec{\Psi}}}_{n \times m} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{{\varvec{\Psi}}}_{n \times m}^{{\mathbf{T}}} {{\varvec{\Psi}}}_{n \times m} } & {\mathbf{0}} \\ {\mathbf{0}} & {{{\varvec{\Psi}}}_{n \times m}^{{\mathbf{T}}} {{\varvec{\Psi}}}_{n \times m} } \\ \end{array} } \right] $$

  (27)

  Because ${{\varvec{\Psi}}}_{n \times m}$ is orthogonal matrix, so ${{\varvec{\Psi}}}_{n \times m}^{{\mathbf{T}}} {{\varvec{\Psi}}}_{n \times m} = {\mathbf{I}}_{m \times m}$, and Eq. (27) is equal to an identity matrix, and its dimension is ${\mathbb{R}}^{2m \times 2m}$. Therefore, the transfer matrix ${{\varvec{\Pi}}}$ satisfies the characteristic of orthogonality. What’s more, in the light of Definition 2, we can have ${{\varvec{\Pi}}}^{{\mathbf{T}}} = {{\varvec{\Pi}}}^{ + }$.

(1): .

Appendix B

According to Eq. (11), the variation of the action ${\text{S}}$ with respect to ${\mathbf{q}}$ and ${\mathbf{p}}$, then the following formula can be obtained:

$$ \delta {\text{S}} = \delta {\mathbf{q}}^{{\mathbf{T}}} {\mathbf{p}}|_{{t_{n} }}^{{t_{n + 1} }} - \int_{{t_{n} }}^{{t_{n + 1} }} {\delta {\mathbf{q}}\left( {{\mathbf{M}}\left( {\mathbf{q}} \right){\mathbf{a}} + {{\varvec{\Phi}}}\left( {\mathbf{q}} \right)_{{\mathbf{q}}}^{{\mathbf{T}}} {{\varvec{\uplambda}}}_{b} - \frac{1}{2}{\mathbf{v}}^{{\mathbf{T}}} \frac{{\partial {\mathbf{M}}\left( {\mathbf{q}} \right)}}{{\partial {\mathbf{q}}}}{\mathbf{v}} + \frac{{\partial {\text{U}}\left( {\mathbf{q}} \right)}}{{\partial {\mathbf{q}}}} + {\mathbf{Q}}\left( {{\mathbf{q}},{\mathbf{v}},t} \right)} \right){\text{d}}t} , $$

(28)

where in Eq. (28), ${\mathbf{a}}$ is the acceleration; the third term, fourth term and fifth term in the integral sign represent the force of FMBS. Without loss of generality, in the following, all the force acted on the FMBS is expressed as F(q,v,t), which includes the external, damping and internal force. In addition, the second term in Eq. (28) is the differential equation, which is popular in the literature of multibody system. Therefore, the second term in Eq. (28) is equal to zero and the variation of action ${\text{S}}$ is:

$$ \delta {\text{S = }}\delta {\mathbf{q}}_{n + 1}^{{\mathbf{T}}} {\mathbf{p}}_{n + 1} - \delta {\mathbf{q}}_{n}^{{\mathbf{T}}} {\mathbf{p}}_{n} $$

(29)

According to Eq. (12), Eq. (29) can be rewritten as:

$$ \delta {\text{S = }}\delta {\mathbf{q}}_{r,n + 1}^{{\mathbf{T}}} {{\varvec{\Psi}}}^{{\mathbf{T}}} {\mathbf{\Psi p}}_{r,n + 1} - \delta {\mathbf{q}}_{r,n}^{{\mathbf{T}}} {{\varvec{\Psi}}}^{{\mathbf{T}}} {\mathbf{\Psi p}}_{r,n} = \delta {\mathbf{q}}_{r,n + 1}^{{\mathbf{T}}} {\mathbf{p}}_{r,n + 1} - \delta {\mathbf{q}}_{r,n}^{{\mathbf{T}}} {\mathbf{p}}_{r,n} . $$

(30)

The state vector is interpolated by the center interpolation method with second-order accuracy [74], then Eq. (30) is approximately equal to:

$$ \delta {\text{S}} \approx \delta \int_{{t_{n} }}^{{t_{n + 1} }} {\left( {\frac{1}{2}{\hat{\mathbf{v}}}^{{\text{T}}} {\mathbf{M}}\left( {{\hat{\mathbf{q}}}} \right){\hat{\mathbf{v}}} - {{\varvec{\Phi}}}\left( {{\hat{\mathbf{q}}}} \right)^{{\mathbf{T}}} {{\varvec{\uplambda}}}_{b} - {\mathbf{U}}\left( {{\hat{\mathbf{q}}}} \right)} \right){\text{d}}t} - \int_{{t_{n} }}^{{t_{n + 1} }} {\delta {\hat{\mathbf{q}}}^{{\text{T}}} {\mathbf{Q}}\left( {{\hat{\mathbf{q}}},{\hat{\mathbf{v}}},t} \right)dt} , $$

(31)

where in Eq. (31), ${\hat{\mathbf{v}}} = \left( {{\mathbf{q}}_{n + 1} - {\mathbf{q}}_{n} } \right)/h$, ${\hat{\mathbf{q}}} = \left( {{\mathbf{q}}_{n + 1} + {\mathbf{q}}_{n} } \right)/2$ and $h = t_{n + 1} - t_{n}$.

Due to Eqs. (30) and (31), the ROM of FMBS can obtained:

$$ \begin{gathered} - {\mathbf{p}}_{r,n} = \int_{{t_{n} }}^{{t_{n + 1} }} {\left( {\frac{{\partial {\mathbf{q}}_{n} }}{{\partial {\mathbf{q}}_{r} }}} \right)^{{\mathbf{T}}} \left( {\frac{{\partial {\text{S}}}}{{\partial {\mathbf{q}}_{n} }}} \right)}^{{\mathbf{T}}} {\text{d}}t = \hfill \\ \;\;\;\;\;\;\;\;{{\varvec{\Psi}}}^{{\mathbf{T}}} \left( { - {\mathbf{M}}\left( {{\hat{\mathbf{q}}}} \right){\hat{\mathbf{v}}} + \frac{1}{4}h{\hat{\mathbf{v}}}^{{\mathbf{T}}} \frac{{\partial {\mathbf{M}}\left( {{\hat{\mathbf{q}}}} \right)}}{{\partial {\hat{\mathbf{q}}}}}{\hat{\mathbf{v}}} - \frac{1}{2}h{{\varvec{\Phi}}}\left( {{\hat{\mathbf{q}}}} \right)_{{\mathbf{q}}}^{{\mathbf{T}}} {{\varvec{\uplambda}}}_{b} - \frac{1}{2}h\frac{{\partial {\text{U}}\left( {{\hat{\mathbf{q}}}} \right)}}{{\partial {\hat{\mathbf{q}}}}} + \frac{1}{2}h{\mathbf{Q}}\left( {{\hat{\mathbf{q}}},{\hat{\mathbf{v}}},t} \right)} \right), \hfill \\ \end{gathered} $$

(32)

$$ \begin{gathered} {\mathbf{p}}_{r,n + 1} = \int_{{t_{n} }}^{{t_{n + 1} }} {\left( {\frac{{\partial {\mathbf{q}}_{n + 1} }}{{\partial {\mathbf{q}}_{r,n + 1} }}} \right)^{{\mathbf{T}}} \left( {\frac{{\partial {\text{S}}}}{{\partial {\mathbf{q}}_{n + 1} }}} \right)}^{{\mathbf{T}}} {\text{d}}t = \hfill \\ \;\;\;\;\;\;\;\;{{\varvec{\Psi}}}^{{\mathbf{T}}} \left( {{\mathbf{M}}\left( {{\hat{\mathbf{q}}}} \right){\hat{\mathbf{v}}} + \frac{1}{4}h{\hat{\mathbf{v}}}^{{\mathbf{T}}} \frac{{\partial {\mathbf{M}}\left( {{\hat{\mathbf{q}}}} \right)}}{{\partial {\hat{\mathbf{q}}}}}{\hat{\mathbf{v}}} + \frac{1}{2}h{{\varvec{\Phi}}}\left( {{\hat{\mathbf{q}}}} \right)_{{\mathbf{q}}}^{{\mathbf{T}}} {{\varvec{\uplambda}}}_{b} - \frac{1}{2}h\frac{{\partial {\text{U}}\left( {{\hat{\mathbf{q}}}} \right)}}{{\partial {\hat{\mathbf{q}}}}} + \frac{1}{2}h{\mathbf{Q}}\left( {{\hat{\mathbf{q}}},{\hat{\mathbf{v}}},t} \right)} \right). \hfill \\ \end{gathered} $$

(33)

Appendix C

The expressions of some new symbols in the Jacobian matrix of Eq. (16) have the following forms:

$$ {\mathbf{M}}2{\mathbf{q}} = - \frac{1}{h}{\mathbf{M}}_{r} - \frac{1}{2}\frac{{\partial \left( {{\mathbf{M}}_{r} {\hat{\mathbf{v}}}_{r} } \right)}}{{\partial {\hat{\mathbf{q}}}}}{{\varvec{\Psi}}}_{n \times m} , $$

$$ {\mathbf{\Phi \lambda }}_{b} 2{\mathbf{q}} = - \frac{1}{4}h\frac{{\partial \left( {{{\varvec{\Psi}}}_{n \times m}^{{\mathbf{T}}} {{\varvec{\Phi}}}\left( {{{\varvec{\Psi}}}_{n \times m} {\hat{\mathbf{q}}}_{r} } \right)_{{\mathbf{q}}}^{{\mathbf{T}}} } \right)_{r} {{\varvec{\uplambda}}}_{r,b} }}{{\partial {\hat{\mathbf{q}}}}}{{\varvec{\Psi}}}_{n \times m} , $$

$$ {\mathbf{F}}2{\mathbf{q}} = \frac{1}{4}h\frac{{\partial \left( {{{\varvec{\Psi}}}_{n \times m}^{{\mathbf{T}}} {\mathbf{F}}\left( {{{\varvec{\Psi}}}_{n \times m} {\hat{\mathbf{q}}}_{r} ,{{\varvec{\Psi}}}_{n \times m} {\hat{\mathbf{v}}}_{r} ,t} \right)} \right)}}{{\partial {\hat{\mathbf{q}}}}}{{\varvec{\Psi}}}_{n \times m} + \frac{1}{2}\frac{{\partial \left( {{{\varvec{\Psi}}}_{n \times m}^{{\mathbf{T}}} {\mathbf{F}}\left( {{{\varvec{\Psi}}}_{n \times m} {\hat{\mathbf{q}}}_{r} ,{{\varvec{\Psi}}}_{n \times m} {\hat{\mathbf{v}}}_{r} ,t} \right)} \right)}}{{\partial {\hat{\mathbf{v}}}}}{{\varvec{\Psi}}}_{n \times m} . $$

Appendix D

See Tables 9 , 10, 11, 12, 13 and 14.

Table 9 Max ϵ_rms for different number of ROBs with Young’s modulus E ∈ [0.01,1]GPa

Table 10 Max ϵ_rms of position at different truncated error based on system level and component level

Table 11 Max ϵ_rms of velocity at different truncated error based on system level and component level

Table 12 Max ϵ_rms of position at different truncated error based on system level and component level

Table 13 Max ϵ_rms of velocity at different truncated error based on system-level and component-level

Table 14 Max ϵ_rel of position and velocity at different truncated error ε