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Design sensitivity analysis for transient responses of viscoelastically damped systems using model order reduction techniques

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Abstract

Design sensitivity analysis (DSA) of transient responses, which are indispensable in gradient-based time domain optimization, often requires excessive computational resources for viscoelastically damped systems to directly differentiate and integrate the full-order model (FOM). In this paper, an efficient model-order reduction (MOR)–based DSA framework is developed for capturing the 1st- and 2nd-order derivatives of the transient responses and response functions for viscoelastically damped systems. The damping force is represented by a non-viscous damping model, which depends on the past history of motion via convolution integrals over suitable kernel functions. The direct differentiation method (DDM) is used to derive the DSA. Three robust modal reduction bases, namely multi-model (MM) method, modal strain energy modified by displacement residuals (MSER) method and improved approximation method (IAM) are introduced to reduce the system dimension. Based on a generalized damping model in expression of fraction formula, a reduced state-space formulation without convolution integral term is derived. The 1st- and 2nd-order derivatives of the transient responses and response functions are calculated using a modified precise integration method and the DDM on the reduced stage. The computational efficiency and accuracy of the presented methods are illustrated and compared by two examples. The results indicate that the computational time is significantly reduced by the proposed MOR methods maintaining fairly good accuracy. Among these methods, the MM method represents the most compromise between precise and efficiency and would be the best candidate to be the reduction basis for calculating the time domain DSA of large-scale viscoelastically damped systems.

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Funding

This work was supported by the National Natural Science Foundation of China (Grant Nos. 51805383, 51809201), the Hong Kong Scholars Program (Grant No. XJ2019019) and the Hubei Provincial Natural Science Foundation of China (Grant Nos. 2018CFB345, 2018CFB371).

Author information

Authors and Affiliations

  1. Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin, NT, Hong Kong, China

    Zhe Ding, Qiang Gao & Wei-Hsin Liao

  2. Key Laboratory of Metallurgical Equipment and Control Technology, Ministry of Education, Wuhan University of Science and Technology, Wuhan, 430081, China

    Zhe Ding, Junlei Shi & Qianwen Huang

  3. Hubei Key Laboratory of Mechanical Transmission and Manufacturing Engineering, Wuhan University of Science and Technology, Wuhan, 430081, China

    Zhe Ding, Junlei Shi & Qianwen Huang

Authors
  1. Zhe Ding
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  2. Junlei Shi
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  3. Qiang Gao
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  4. Qianwen Huang
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  5. Wei-Hsin Liao
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Correspondence to Wei-Hsin Liao.

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Responsible Editor: Gengdong Cheng

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The MATLAB codes of the proposed MOR-DSA methods are available upon request to the first and corresponding authors.

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Appendices

Appendix A:: Specific expressions of some matrices in Section 4.1

The expressions of the relative reduced matrices in (37) are

$$ {\bar{\mathbf E}} = diag({{\mathbf{I}}_{{{\bar{r}}_{1}}}} \otimes {{\mathbf{E}}_{1}},{{\mathbf{I}}_{{{\bar{r}}_{2}}}} \otimes {{\mathbf{E}}_{2}}, {\cdots} ,{{\mathbf{I}}_{{{\bar{r}}_{n}}}} \otimes {{\mathbf{E}}_{n}})\in \mathbb{R}^{{\bar{p} \times \bar{p}}}, $$
(66)
$$ {\bar{\mathbf W}} = diag({{\mathbf{I}}_{{{\bar{r}}_{1}}}} \otimes {{\mathbf{W}}_{1}},{{\mathbf{I}}_{{{\bar{r}}_{2}}}} \otimes {{\mathbf{W}}_{2}}, {\cdots} ,{{\mathbf{I}}_{{{\bar{r}}_{n}}}} \otimes {{\mathbf{W}}_{n}})\in \mathbb{R}^{{\bar{p} \times \bar{p}}}, $$
(67)
$$ {\bar{\mathbf{L}}} = \left[ {{{\bar{\mathbf{L}}}_{1}}{{({{\mathbf{I}}_{{{\bar{r}}_{1}}}} \otimes {{\mathbf{P}}_{1}})}^{T}},{{\bar{\mathbf{L}}}_{2}}{{({{\mathbf{I}}_{{{\bar{r}}_{2}}}} \otimes {{\mathbf{P}}_{2}})}^{T}}, {\cdots} ,{{\bar{\mathbf{L}}}_{n}}{{({{\mathbf{I}}_{{{\bar{r}}_{n}}}} \otimes {{\mathbf{P}}_{n}})}^{T}}} \right] \in \mathbb{R}^{{Nm \times \bar{p}}}, $$
(68)
$$ {\bar{\mathbf R}} = \left[ {{{\bar{\mathbf R}}_{1}}{{({{\mathbf{I}}_{{{\bar{r}}_{1}}}} \otimes {{\mathbf{Q}}_{1}})}^{T}},{{\bar{\mathbf R}}_{2}}{{({{\mathbf{I}}_{{{\bar{r}}_{2}}}} \otimes {{\mathbf{Q}}_{2}})}^{T}}, {\cdots} ,{{\bar{\mathbf R}}_{n}}{{({{\mathbf{I}}_{{{\bar{r}}_{n}}}} \otimes {{\mathbf{Q}}_{n}})}^{T}}} \right] \in \mathbb{R}^{{Nm \times \bar{p}}}, $$
(69)
$$ \bar{p} = \sum\limits_{k = 1}^{n} {{{\bar{r}}_{k}}{q_{k}}}. $$
(70)

Appendix B:: The first-order sensitivity derivatives in (51)

As defined in (8) and Appendix A, the matrices \({\bar {\mathbf {L}}},{\bar {\mathbf W}},{\bar {\mathbf R}},{\bar {\mathbf E}}\) are related to the viscoelastic damping model involved, which are composed of the relaxation parameters, the order of the damping models and the identity matrix. If the design variable pi is not related to the damping relaxation parameters, the first-order derivatives of \({\bar {\mathbf {L}}},{\bar {\mathbf W}},{\bar {\mathbf R}},{\bar {\mathbf E}}\) are all zero. On the contrary, when the damping model parameters are chosen to be the design variable, the first-order derivatives of \({\bar {\mathbf {L}}},{\bar {\mathbf W}},{\bar {\mathbf R}},{\bar {\mathbf E}}\) should be considered. As defined in (35), \({{\bar {\mathbf {L}}}_k},{{\bar {\mathbf R}}_k}\) are the full column rank of the reduced viscoelastic damping coefficient matrix. It describes the distribution of the damping materials, which is independent of the parameters of the damping models. Besides, Wk,Qk are also independent of any design parameters. Therefore, we have

$$ \frac{{\partial {{{\bar{\mathbf{L}}}}_{k}}}}{{\partial {p_{i}}}} = {\mathbf{0}},{\text{ }}\frac{{\partial {{{\bar{\mathbf R}}}_{k}}}}{{\partial {p_{i}}}} = {\mathbf{0}},\frac{{\partial {{\mathbf{W}}_{k}}}}{{\partial {p_{i}}}} = {\mathbf{0}},\frac{{\partial {{\mathbf{Q}}_{k}}}}{{\partial {p_{i}}}} = {\mathbf{0}},\frac{{\partial {{\bar{\mathbf{C}}}_{0}}}}{{\partial {p_{i}}}} = {\mathbf{0}}. $$
(71)

Substituting (71) into (66)–(69), one can derive the first-order derivatives of the \({\bar {\mathbf {L}}},{\bar {\mathbf W}},{\bar {\mathbf R}},{\bar {\mathbf E}}\):

$$ \begin{array}{l} \frac{{\partial {\bar{\mathbf{L}}}}}{{\partial {p_{i}}}} = \left[ {{{{\bar{\mathbf{L}}}}_{1}}{{({{\mathbf{I}}_{{{\bar r}_{1}}}} \otimes \frac{{\partial {{\mathbf{P}}_{1}}}}{{\partial {p_{i}}}})}^{T}},{{{\bar{\mathbf{L}}}}_{2}}{{({{\mathbf{I}}_{{{\bar r}_{2}}}} \otimes \frac{{\partial {{\mathbf{P}}_{2}}}}{{\partial {p_{i}}}})}^{T}}, {\cdots} ,{{{\bar{\mathbf{L}}}}_{N}}{{({{\mathbf{I}}_{{{\bar r}_{n}}}} \otimes \frac{{\partial {{\mathbf{P}}_{n}}}}{{\partial {p_{i}}}})}^{T}}} \right],\\ \frac{{\partial {\bar{\mathbf E}}}}{{\partial {p_{i}}}} = \left[ {{{\mathbf{I}}_{{{\bar r}_{1}}}} \otimes \frac{{\partial {{\mathbf{E}}_{1}}}}{{\partial {p_{i}}}},{{\mathbf{I}}_{{{\bar r}_{2}}}} \otimes \frac{{\partial {{\mathbf{E}}_{2}}}}{{\partial {p_{i}}}}, {\cdots} ,{{\mathbf{I}}_{{{\bar r}_{n}}}} \otimes \frac{{\partial {{\mathbf{E}}_{n}}}}{{\partial {p_{i}}}}} \right],\\ \frac{{\partial {\bar{\mathbf W}}}}{{\partial {p_{i}}}} = {\mathbf{0}},\frac{{\partial {\bar{\mathbf R}}}}{{\partial {p_{i}}}} = {\mathbf{0}}, \end{array} $$
(72)

where \(\frac {{\partial {{\mathbf {P}}_k}}}{{\partial {p_i}}},\frac {{\partial {{\mathbf {E}}_k}}}{{\partial {p_i}}}\) can be easily obtained from (8). Therefore, the specific expressions of the terms in (51) are

$$ \begin{array}{@{}rcl@{}} \frac{{\partial \left( { - {{{\bar{\mathbf{M}}}}^{- 1}}{{\bar{\mathbf{C}}}_{0}}} \right)}}{{\partial {p_{i}}}} & = {{\bar{\mathbf{M}}}^{- 1}}\frac{{\partial {\bar{\mathbf{M}}}}}{{\partial {p_{i}}}}{{\bar{\mathbf{M}}}^{- 1}}{{\bar{\mathbf{C}}}_{0}} - {{\bar{\mathbf{M}}}^{- 1}}\frac{{\partial {{\bar{\mathbf{C}}}_{0}}}}{{\partial {p_{i}}}}{\text{ }} \to \\& {\text{ }}\frac{{\partial \left( { - {{{\bar{\mathbf{M}}}}^{- 1}}{{\bar{\mathbf{C}}}_{0}}} \right)}}{{\partial {p_{i}}}} = {{\bar{\mathbf{M}}}^{- 1}}\frac{{\partial {\bar{\mathbf{M}}}}}{{\partial {p_{i}}}}{{\bar{\mathbf{M}}}^{- 1}}{{\bar{\mathbf{C}}}_{0}}, \end{array} $$
(73)
$$ \frac{{\partial \left( { - {{{\bar{\mathbf{M}}}}^{{\text{ - 1}}}}{\bar{\mathbf{L}}}} \right)}}{{\partial {p_{i}}}} = {{\bar{\mathbf{M}}}^{{\text{ - 1}}}}\frac{{\partial {\bar{\mathbf{M}}}}}{{\partial {p_{i}}}}{{\bar{\mathbf{M}}}^{{\text{ - 1}}}}{\bar{\mathbf{L}}} - {{\bar{\mathbf{M}}}^{{\text{ - 1}}}}\frac{{\partial {\bar{\mathbf{L}}}}}{{\partial {p_{i}}}}, $$
(74)
$$ \begin{array}{@{}rcl@{}} \frac{{\partial \left( { - {{{\bar{\mathbf W}}}^{- 1}}{{{\bar{\mathbf R}}}^{T}}} \right)}}{{\partial {p_{i}}}} &= {{\bar{\mathbf W}}^{- 1}}\frac{{\partial {\bar{\mathbf W}}}}{{\partial {p_{i}}}}{{\bar{\mathbf W}}^{- 1}}{{\bar{\mathbf R}}^{T}} - {{\bar{\mathbf W}}^{- 1}}\frac{{\partial {{{\bar{\mathbf R}}}^{T}}}}{{\partial {p_{i}}}}{\text{ }} \to \\& {\text{ }}\frac{{\partial \left( { - {{{\bar{\mathbf W}}}^{- 1}}{{{\bar{\mathbf R}}}^{T}}} \right)}}{{\partial {p_{i}}}} = {\mathbf{0}}, \end{array} $$
(75)
$$ \begin{array}{@{}rcl@{}} \frac{{\partial \left( {{{{\bar{\mathbf W}}}^{- 1}}{\bar{\mathbf E}}} \right)}}{{\partial {p_{i}}}} & = - {{\bar{\mathbf W}}^{- 1}}\frac{{\partial {\bar{\mathbf W}}}}{{\partial {p_{i}}}}{{\bar{\mathbf W}}^{- 1}}{\bar{\mathbf E}} + {{\bar{\mathbf W}}^{- 1}}\frac{{\partial {\bar{\mathbf E}}}}{{\partial {p_{i}}}}{\text{ }} \to \\& {\text{ }}\frac{{\partial \left( {{{{\bar{\mathbf W}}}^{- 1}}{\bar{\mathbf E}}} \right)}}{{\partial {p_{i}}}} = {{\bar{\mathbf W}}^{- 1}}\frac{{\partial {\bar{\mathbf E}}}}{{\partial {p_{i}}}}. \end{array} $$
(76)

Appendix C:: The second-order sensitivity derivatives in (58)

The second-order derivatives of (58) are as follows:

$$ \begin{array}{@{}rcl@{}} \frac{{{\partial^{2}}\left( { - {{{\bar{\mathbf{M}}}}^{- 1}}{\bar{\mathbf{K}}}} \right)}}{{\partial {p_{i}}\partial {p_{j}}}} &=& \frac{{\partial {{{\bar{\mathbf{M}}}}^{- 1}}}}{{\partial {p_{j}}}}\frac{{\partial {\bar{\mathbf{M}}}}}{{\partial {p_{i}}}}{{\bar{\mathbf{M}}}^{- 1}}{\bar{\mathbf{K}}} + {{\bar{\mathbf{M}}}^{- 1}}\frac{{{\partial^{2}}{\bar{\mathbf{M}}}}}{{\partial {p_{i}}\partial {p_{j}}}}{{\bar{\mathbf{M}}}^{- 1}}{\bar{\mathbf{K}}} \\&& +{{\bar{\mathbf{M}}}^{- 1}}\frac{{\partial {\bar{\mathbf{M}}}}}{{\partial {p_{i}}}}\frac{{\partial {{{\bar{\mathbf{M}}}}^{- 1}}}}{{\partial {p_{j}}}}{\bar{\mathbf{K}}} + {{\bar{\mathbf{M}}}^{- 1}}\frac{{\partial {\bar{\mathbf{M}}}}}{{\partial {p_{i}}}}{{\bar{\mathbf{M}}}^{- 1}}\frac{{\partial {\bar{\mathbf{K}}}}}{{\partial {p_{j}}}}\\&&- \frac{{\partial {{{\bar{\mathbf{M}}}}^{- 1}}}}{{\partial {p_{j}}}}\frac{{\partial {\bar{\mathbf{K}}}}}{{\partial {p_{i}}}} - {{\bar{\mathbf{M}}}^{- 1}}\frac{{{\partial^{2}}{\bar{\mathbf{K}}}}}{{\partial {p_{i}}\partial {p_{j}}}}, \end{array} $$
(77)
$$ \begin{array}{@{}rcl@{}} \frac{{{\partial^{2}}\left( { - {{{\bar{\mathbf{M}}}}^{- 1}}{{\bar{\mathbf{C}}}_{0}}} \right)}}{{\partial {p_{i}}\partial {p_{j}}}}&=& \frac{{\partial {{{\bar{\mathbf{M}}}}^{- 1}}}}{{\partial {p_{j}}}}\frac{{\partial {\bar{\mathbf{M}}}}}{{\partial {p_{i}}}}{{\bar{\mathbf{M}}}^{- 1}}{{\bar{\mathbf{C}}}_{0}} + {{\bar{\mathbf{M}}}^{- 1}}\frac{{{\partial^{2}}{\bar{\mathbf{M}}}}}{{\partial {p_{i}}\partial {p_{j}}}}{{\bar{\mathbf{M}}}^{- 1}}{{\bar{\mathbf{C}}}_{0}} \\&& + {{\bar{\mathbf{M}}}^{- 1}}\frac{{\partial {\bar{\mathbf{M}}}}}{{\partial {p_{i}}}}\frac{{\partial {{{\bar{\mathbf{M}}}}^{- 1}}}}{{\partial {p_{j}}}}{{\bar{\mathbf{C}}}_{0}}, \end{array} $$
(78)
$$ \begin{array}{@{}rcl@{}} \frac{{{\partial^{2}}\left( { - {{{\bar{\mathbf{M}}}}^{- 1}}{\bar{\mathbf{L}}}} \right)}}{{\partial {p_{i}}\partial {p_{j}}}} &= &\frac{{\partial {{{\bar{\mathbf{M}}}}^{- 1}}}}{{\partial {p_{j}}}}{{\bar{\mathbf{M}}}^{- 1}}\frac{{\partial {\bar{\mathbf{M}}}}}{{\partial {p_{i}}}}{{\bar{\mathbf{M}}}^{- 1}}{\bar{\mathbf{L}}} + {{\bar{\mathbf{M}}}^{- 1}}\frac{{{\partial^{2}}{\bar{\mathbf{M}}}}}{{\partial {p_{i}}\partial {p_{j}}}}{{\bar{\mathbf{M}}}^{- 1}}{\bar{\mathbf{L}}} \\&&+ {{\bar{\mathbf{M}}}^{- 1}}\frac{{\partial {\bar{\mathbf{M}}}}}{{\partial {p_{i}}}}\frac{{\partial {{{\bar{\mathbf{M}}}}^{- 1}}}}{{\partial {p_{j}}}}{\bar{\mathbf{L}}} + {{\bar{\mathbf{M}}}^{- 1}}\frac{{\partial {\bar{\mathbf{M}}}}}{{\partial {p_{i}}}}{{\bar{\mathbf{M}}}^{- 1}}\frac{{\partial {\bar{\mathbf{L}}}}}{{\partial {p_{j}}}} \\&&- \frac{{\partial {{{\bar{\mathbf{M}}}}^{- 1}}}}{{\partial {p_{j}}}}\frac{{\partial {\bar{\mathbf{L}}}}}{{\partial {p_{i}}}} - {{\bar{\mathbf{M}}}^{- 1}}\frac{{{\partial^{2}}{\bar{\mathbf{L}}}}}{{\partial {p_{i}}\partial {p_{j}}}}, \end{array} $$
(79)
$$ \frac{{{\partial^{2}}\left( { - {{{\bar{\mathbf W}}}^{- 1}}{{{\bar{\mathbf R}}}^{T}}} \right)}}{{\partial {p_{i}}\partial {p_{j}}}} = {\mathbf{0}},\frac{{{\partial^{2}}\left( { - {{{\bar{\mathbf W}}}^{- 1}}{\bar{\mathbf E}}} \right)}}{{\partial {p_{i}}\partial {p_{j}}}} = {{\bar{\mathbf W}}^{- 1}}\frac{{{\partial^{2}}{\bar{\mathbf E}}}}{{\partial {p_{i}}\partial {p_{j}}}}, $$
(80)

where

$$ \frac{{\partial {{{\bar{\mathbf{M}}}}^{- 1}}}}{{\partial {p_{i}}}} = - {{\bar{\mathbf{M}}}^{- 1}}\frac{{\partial {\bar{\mathbf{M}}}}}{{\partial {p_{i}}}}{{\bar{\mathbf{M}}}^{- 1}}, $$
(81)
$$ \frac{{{\partial^{2}}{\bar{\mathbf{M}}}}}{{\partial {p_{i}}\partial {p_{j}}}} = {{\mathbf{T}}^{T}}\frac{{{\partial^{2}}{\mathbf{M}}}}{{\partial {p_{i}}\partial {p_{j}}}}{\mathbf{T}},\quad\frac{{{\partial^{2}}{\bar{\mathbf{K}}}}}{{\partial {p_{i}}\partial {p_{j}}}} = {{\mathbf{T}}^{T}}\frac{{{\partial^{2}}{\mathbf{K}}}}{{\partial {p_{i}}\partial {p_{j}}}}{\mathbf{T}}, $$
(82)
$$ \frac{{{\partial^{2}}{\bar{\mathbf{L}}}}}{{\partial {p_{i}}\partial {p_{j}}}} = \left[ {{{{\bar{\mathbf{L}}}}_{1}}{{({{\mathbf{I}}_{{{\bar r}_{1}}}} \otimes \frac{{{\partial^{2}}{{\mathbf{P}}_{1}}}}{{\partial {p_{i}}\partial {p_{j}}}})}^{T}}, {\cdots} , {{{\bar{\mathbf{L}}}}_{N}}{{({{\mathbf{I}}_{{{\bar r}_{n}}}} \otimes \frac{{{\partial^{2}}{{\mathbf{P}}_{n}}}}{{\partial {p_{i}}\partial {p_{j}}}})}^{T}}} \right], $$
(83)
$$ \frac{{{\partial^{2}}{\bar{\mathbf E}}}}{{\partial {p_{i}}\partial {p_{j}}}} = \left[ {{{\mathbf{I}}_{{{\bar r}_{1}}}} \otimes \frac{{{\partial^{2}}{{\mathbf{E}}_{1}}}}{{\partial {p_{i}}\partial {p_{j}}}},{{\mathbf{I}}_{{{\bar r}_{2}}}} \otimes \frac{{{\partial^{2}}{{\mathbf{E}}_{2}}}}{{\partial {p_{i}}\partial {p_{j}}}}, {\cdots} ,{{\mathbf{I}}_{{{\bar r}_{n}}}} \otimes \frac{{{\partial^{2}}{{\mathbf{E}}_{n}}}}{{\partial {p_{i}}\partial {p_{j}}}}} \right]. $$
(84)

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Ding, Z., Shi, J., Gao, Q. et al. Design sensitivity analysis for transient responses of viscoelastically damped systems using model order reduction techniques. Struct Multidisc Optim 64, 1501–1526 (2021). https://doi.org/10.1007/s00158-021-02937-9

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  • DOI: https://doi.org/10.1007/s00158-021-02937-9

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