Adjoint methods of sensitivity analysis for Lyapunov equation

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.

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 dum** 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.