Adjoint methods of sensitivity analysis for Lyapunov equation

Kun Yan; Gengdong Cheng; Bo Ping Wang

doi:10.1007/s00158-015-1323-z

Structural and Multidisciplinary Optimization

February 2016, Volume 53, Issue 2, pp 225–237 | Cite as

Adjoint methods of sensitivity analysis for Lyapunov equation

Authors
Authors and affiliations

Kun Yan
Gengdong Cheng
Bo Ping WangEmail author

RESEARCH PAPER

First Online: 29 September 2015

Received: 12 November 2014

Revised: 06 May 2015

Accepted: 20 July 2015

368 Downloads
4 Citations

Abstract

The existing direct sensitivity analysis of optimal structural vibration control based on Lyapunov’s second method is computationally expensive when applied to finite element models with a large number of degree-of-freedom and design variables. A new adjoint sensitivity analysis method is proposed in this paper. Using the new method the sensitivity of the performance index, a time integral of a quadratic function of state variables, with respect to all design variables is calculated by solving two Lyapunov matrix equations. In consideration of computational cost reduction, the new adjoint method is further extended to the reduced order model by Guyan method. This makes the method applicable to large finite element models. Two numerical examples demonstrate the accuracy and efficiency of the proposed method.

Keywords

Adjoint sensitivity analysis Lyapunov’s second method Passive vibration control

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Notes

Acknowledgment

This work is supported by National Natural Science Foundation of China (91216201 and 11372062).

Appendix A Adjoint sensitivity analysis for the reduced model by Guyan reduction

The derivative of objective function with respect to design parameters can be expressed as (42)

where, $ \frac{\partial {\mathbf{X}}_{re}(0)}{\partial {d}_k}={\left(\begin{array}{cc}\hfill \frac{\partial {\mathbf{u}}_{re,0}^T}{\partial {d}_k}\hfill & \hfill \frac{\partial {\mathbf{v}}_{re,0}^T}{\partial {d}_k}\hfill \end{array}\right)}^T $. From the (37), we get that

$$ \frac{\partial {\mathbf{u}}_0}{\partial {d}_k}=\frac{\partial \mathbf{T}}{\partial {d}_k}{\mathbf{u}}_{re,0}+\mathbf{T}\frac{\partial {\mathbf{u}}_{re,0}}{\partial {d}_k},\;\frac{\partial {\mathbf{v}}_0}{\partial {d}_k}=\frac{\partial \mathbf{T}}{\partial {d}_k}{\mathbf{v}}_{re,0}+\mathbf{T}\frac{\partial {\mathbf{v}}_{re,0}}{\partial {d}_k} $$

(54)

T cannot be inverted directly, so premultiply both sides of the equation by matrix T ^T M. Then, we get

$$ {\mathbf{T}}^T\mathbf{M}\frac{\partial {\mathbf{u}}_0}{\partial {d}_k}={\mathbf{T}}^T\mathbf{M}\frac{\partial \mathbf{T}}{\partial {d}_k}{\mathbf{u}}_{re,0}+{\mathbf{T}}^T\mathbf{M}\mathbf{T}\frac{\partial {\mathbf{u}}_{re,0}}{\partial {d}_k} $$

(55)

$$ {\mathbf{T}}^T\mathbf{M}\frac{\partial {\mathbf{v}}_0}{\partial {d}_k}={\mathbf{T}}^T\mathbf{M}\frac{\partial \mathbf{T}}{\partial {d}_k}{\mathbf{v}}_{re,0}+{\mathbf{T}}^T\mathbf{M}\mathbf{T}\frac{\partial {\mathbf{v}}_{re,0}}{\partial {d}_k} $$

(56)

Then

$$ \frac{\partial {\mathbf{u}}_{re,0}}{\partial {d}_k}={\mathbf{M}}_{re}^{-1}{\mathbf{T}}^T\mathbf{M}\frac{\partial \mathbf{T}}{\partial {d}_k}{\mathbf{u}}_{re,0}-{\mathbf{M}}_{re}^{-1}{\mathbf{T}}^T\mathbf{M}\frac{\partial {\mathbf{u}}_0}{\partial {d}_k} $$

(57)

$$ \frac{\partial {\mathbf{v}}_{re,0}}{\partial {d}_k}={\mathbf{M}}_{re}^{-1}{\mathbf{T}}^T\mathbf{M}\frac{\partial \mathbf{T}}{\partial {d}_k}{\mathbf{v}}_{re,0}-{\mathbf{M}}_{re}^{-1}{\mathbf{T}}^T\mathbf{M}\frac{\partial {\mathbf{v}}_0}{\partial {d}_k} $$

(58)

In (42), the item $ {\mathbf{X}}_{re}^T(0)\frac{\partial {\mathbf{P}}_{re}}{\partial {d}_k}{\mathbf{X}}_{re}(0) $ can be obtained by adjoint method mentioned in section 3.2.

$$ {\mathbf{X}}_{re}^T(0)\frac{\partial {\mathbf{P}}_{re}}{\partial {d}_k}{\mathbf{X}}_{re}(0)={\displaystyle \sum_{i=1}^{2M}{\displaystyle \sum_{j=1}^{2M}{\lambda}_{ij}^{re}{D}_{ij}^{re,k}}} $$

(59)

where M is number of DOFs in reduced model, $ {\mathbf{D}}^{re,k}=\frac{\partial {\mathbf{A}}_{re}^T}{\partial {d}_k}{\mathbf{P}}_{re}+{\mathbf{P}}_{re}\frac{\partial {\mathbf{A}}_{re}}{\partial {d}_k}+\frac{\partial {\mathbf{Q}}_{re}}{\partial {d}_k} $. Letting

$$ {\mathbf{S}}_{re}={\mathbf{X}}_{re}(0){\mathbf{X}}_{re}{(0)}^T $$

(60)

λ ^re can be obtained by solving one Lyapunov matrix equation

$$ {\mathbf{A}}_{re}{\boldsymbol{\uplambda}}^{re}+{\boldsymbol{\uplambda}}^{re}{\mathbf{A}}_{re}^T+{\mathbf{Z}}_{re}=0 $$

(61)

$ \frac{\partial {\mathbf{A}}_{re}}{\partial {d}_k} $ can be expressed as

$$ \frac{\partial {\mathbf{A}}_{re}}{\partial {d}_k}=\left[\begin{array}{cc}\hfill \mathbf{O}\hfill & \hfill \mathbf{O}\hfill \\ {}\hfill -\frac{\partial {\mathbf{M}}_{re}^{-1}}{\partial {d}_k}{\mathbf{K}}_{re}-{\mathbf{M}}_{re}^{-1}\frac{\partial {\mathbf{K}}_{re}}{\partial {d}_k}\hfill & \hfill -\frac{\partial {\mathbf{M}}_{re}^{-1}}{\partial {d}_k}{\mathbf{C}}_{re}-{\mathbf{M}}_{re}^{-1}\frac{\partial {\mathbf{C}}_{re}}{\partial {d}_k}\hfill \end{array}\right] $$

(62)

where

$$ \frac{\partial {\mathbf{K}}_{\mathrm{re}}}{\partial {d}_k}=\frac{\partial \left({\mathbf{T}}^T\mathbf{K}\mathbf{T}\right)}{\partial {d}_k}=\frac{\partial {\mathbf{T}}^T}{\partial {d}_k}\mathbf{K}\mathbf{T}+{\mathbf{T}}^T\frac{\partial \mathbf{K}}{\partial {d}_k}\mathbf{T}+{\mathbf{T}}^T\mathbf{K}\frac{\partial \mathbf{T}}{\partial {d}_k} $$

(63)

$$ \frac{\partial {\mathbf{C}}_{\mathrm{re}}}{\partial {d}_k}=\frac{\partial \left({\mathbf{T}}^T\mathbf{C}\mathbf{T}\right)}{\partial {d}_k}=\frac{\partial {\mathbf{T}}^T}{\partial {d}_k}\mathbf{C}\mathbf{T}+{\mathbf{T}}^T\frac{\partial \mathbf{C}}{\partial {d}_k}\mathbf{T}+{\mathbf{T}}^T\mathbf{C}\frac{\partial \mathbf{T}}{\partial {d}_k} $$

(64)

$ \frac{\partial {\mathbf{M}}_{\mathrm{re}}^{-1}}{\partial {d}_k} $ can be obtained by

$$ \frac{\partial {\mathbf{M}}_{\mathrm{re}}^{-1}}{\partial {d}_k}=-{\mathbf{M}}_{\mathrm{re}}^{-1}\frac{\partial {\mathbf{M}}_{re}}{\partial {d}_k}{\mathbf{M}}_{\mathrm{re}}^{-1} $$

(65)

where

$$ \frac{\partial {\mathbf{M}}_{\mathrm{re}}}{\partial {d}_k}=\frac{\partial \left({\mathbf{T}}^T\mathbf{M}\mathbf{T}\right)}{\partial {d}_k}=\frac{\partial {\mathbf{T}}^T}{\partial {d}_k}\mathbf{M}\mathbf{T}+{\mathbf{T}}^T\frac{\partial \mathbf{M}}{\partial {d}_k}\mathbf{T}+{\mathbf{T}}^T\mathbf{M}\frac{\partial \mathbf{T}}{\partial {d}_k} $$

(66)

The matrix $ \frac{\partial {\mathbf{Q}}_{re}}{\partial {d}_k} $ can be expressed as

$$ \frac{\partial {\mathbf{Q}}_{re}}{\partial {d}_k}=\frac{\partial {\mathbf{t}}^T\mathbf{Q}\mathbf{t}}{\partial {d}_k}=\frac{\partial {\mathbf{t}}^T}{\partial {d}_k}\mathbf{Q}\mathbf{t}+{\mathbf{t}}^T\frac{\partial \mathbf{Q}}{\partial {d}_k}\mathbf{t}+{\mathbf{t}}^T\mathbf{Q}\frac{\partial \mathbf{t}}{\partial {d}_k} $$

(67)

$$ \frac{\partial \mathbf{t}}{\partial {d}_k}=\left[\begin{array}{cc}\hfill \frac{\partial \mathbf{T}}{\partial {d}_k}\hfill & \hfill \mathbf{O}\hfill \\ {}\hfill \mathbf{O}\hfill & \hfill \frac{\partial \mathbf{T}}{\partial {d}_k}\hfill \end{array}\right] $$

(68)

And now, it remains to derive the solution of $ \frac{\partial \mathbf{T}}{\partial {d}_k} $ in terms of sensitivities of stiffness matrix. From (34), $ \frac{\partial \mathbf{T}}{\partial {d}_k} $ can be expressed as

$$ \frac{\partial \mathbf{T}}{\partial {d}_k}=\left\{\begin{array}{c}\hfill \mathbf{O}\hfill \\ {}\hfill -\frac{\partial {\mathbf{K}}_{ss}^{-1}}{\partial {d}_k}{\mathbf{K}}_{sm}-{\mathbf{K}}_{ss}^{-1}\frac{\partial {\mathbf{K}}_{sm}}{\partial {d}_k}\hfill \end{array}\right\} $$

(69)

where

$$ \frac{\partial {\mathbf{K}}_{ss}^{-1}}{\partial {d}_k}=-{\mathbf{K}}_{ss}^{-1}\frac{\partial {\mathbf{K}}_{ss}}{\partial {d}_k}{\mathbf{K}}_{ss}^{-1} $$

(70)

Note that $ \frac{\partial {\mathbf{K}}_{ss}}{\partial {d}_k} $ and $ \frac{\partial {\mathbf{K}}_{sm}}{\partial {d}_k} $ are part of $ \frac{\partial \mathbf{K}}{\partial {d}_k} $. For the case the design variable is dumping coefficient of the dumped spring, $ \frac{\partial {\mathbf{K}}_{ss}}{\partial {d}_k} $ and $ \frac{\partial {\mathbf{K}}_{sm}}{\partial {d}_k} $ is zero, the computational cost can be further reduced.

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Copyright information

Authors and Affiliations

Kun Yan
- 1
Gengdong Cheng
- 1
Bo Ping Wang
- 2
Email author

1.State Key Laboratory of Structural Analysis for Industrial Equipment, Faculty of Vehicle Engineering and MechanicsDalian University of TechnologyDalianPeople’s Republic of China
2.Department of Mechanical and Aerospace EngineeringUniversity of Texas at ArlingtonArlingtonUSA

Adjoint methods of sensitivity analysis for Lyapunov equation

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Acknowledgment

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