建立了柔性压电智能反射面系统的有限元模型和优化控制模型,给出了结构力学建模和形状控制方法以及相应的优化算法。首先,将蜂窝夹层结构的压电智能反射面等效为多层复合板;基于Kirchhoff假设和经典层合板理论,根据虚功原理推导了柔性压电智能反射面的有限元方程;采用蜂窝等效理论计算了反射面蜂窝夹芯等效弹性模量,有限元模型中的单元为四节点四边形压电板单元,每个作动器单元中引入额外的电势自由度。然后,根据建立的有限元方程,推导了反射面变形均方根误差与作动器控制电压的关系式;以均方根误差最小为优化目标,建立了柔性压电智能发射面的静态形状控制优化模型;采用Lagrange乘子法处理了压电作动器工作电压的限制。最后,用提出的方法分析已有模型并验证建模方法。以600 mm口径的平面柔性智能反射面为例,验证了采用压电陶瓷贴片对反射面静态形状控制的可行性及优化算法有效性。仿真结果表明：通过控制压电陶瓷贴片作动器,可以使反射面的静态形状误差减小97%以上,而且作动器的控制电压均在极限电压范围内。

The finite element formulation of a flexible piezoelectric smart reflector was presented based on Kirchhoff classical laminated theory, and its structural mechanic modeling and optimization algorithms were investigated. Firstly, the smart reflector with the honeycomb core was modeled with the equivalent laminate plate theory, and its finite element formulation was derived according to virtual work theory. The honeycomb core equivalent elastic modulus was calculated by using equivalent theory. Then, a simple four-node quadrilateral element was used in the model, and one electric potential degree of freedom was introduced to each active element. Accordingly, the relation between the mean square root error of reflector and the control voltages of actuators was derived, the optimization model for static shape control was created and the voltage limitation for piezoelectric actuator patches was imposed to maintain its control voltage within a practical range. The optimal control voltages were determined by using Lagrange multipliers to minimize the Root Mean Square (RMS) error. Finally, a numerical example of plane smart reflector was given to demonstrate the feasibility of smart mirror concept and the effectiveness of optimization algorithm. Simulation results indicate that the square root error of the smart reflector is reduced by above 90%, and the control voltage of each actuator is in a practical range.