Topology Optimization Of Piezoelectric Structures Under Active Control

Structural vibration is ubiquitous in industrial structures,but most of them are harmful,such as they will affect the normal operation of precision instruments and accelerate the fatigue damage of equipments.Piezoelectric materials have the advantages of fast response,large operating bandwidth and efficient energy conversion,and have been widely used in structural active vibration control.In order to improve the vibration control performance of piezoelectric structures,it is significant to optimize the distribution of piezoelectric materials.However,for the optimization design of piezoelectric structures considering the controllability of the system,the design objective is difficult to construct and there is no efficient sensitivity analysis method.For the topological design of piezoelectric smart structures under optimal control,it is difficult to solve the problem of structural steady-state/transient response analysis and the corresponding sensitivity analysis.In order to deal with these difficulties,the following studies are carried out:Piezoelectric material optimization formulations and corresponding sensitivity analysis methods based on singular value decomposition of control matrix and controllability Gramian matrix analysis are established;Topology optimization design methods for piezoelectric smart structures considering harmonic/transient excitation under optimal control are proposed.Main research contents and results include:1.Topology optimization of piezoelectric materials based on singular value decomposition of control matrix.Based on the classical laminated plate theory,the finite element analysis model of piezoelectric structures is established.In order to improve the efficiency of structural response analysis,the mode superposition method is employed to obtain the reduced control equation of the piezoelectric structure.The relationship between the singular value and the control energy of piezoelectric structures is obtained by calculating the inner product of the control force and singular value decomposition of the control matrix.The controllability index which can measure the control energy of piezoelectric structures under certain electric energy input is constructed in the form of singular value multiplication.Using logarithmic form of the controllability index as objective function and relative densities of piezoelectric elements as design variable,the topology optimization model of piezoelectric structures is established.In order to improve the efficiency of optimization,an artificial penalty model on piezoelectric properties is constructed.Based on the adjoint variable method,the sensitivity analysis method for the controllability index with respect to design variables is derived.By comparing with the sensitivity result obtained by the finite difference method,the effectiveness of the proposed method is verified.The topology optimization of piezoelectric materials is solved by gradient based mathematical programming.Numerical results show that the optimization method can effectively reduce the objective function and the optimized design has better vibration control performance than the reference design.Finally,the influence of the volume fraction and the concerned modes in the objective function on the optimized layout of the piezoelectric material are discussed.2.Topological design of piezoelectric smart structures based on Gramian matrix analysis.Considering the active vibration control of piezoelectric laminated plates,the state equation of piezoelectric structures is established with employing the structural modal response as the state variable.By considering the control energy minimization problem,the relationship between the control energy of piezoelectric structures and Gramian matrix is obtained.The relationship shows that Gramian matrix can reflect the degree of controllability of the structure,that is,it can reflect the control energy of the piezoelectric structures.A performance index containing the eigenvalues of Gramian matrix is constructed to reflect the controllability of the concerned modes.By using the relative density of the element as the design variable and the performance index as the design objective,a layout optimization model of piezoelectric structures is established.An adjoint method for sensitivity analysis of the performance index with respect to the design variable is proposed.The numerical examples are solved by mathematical programming method.The results show that the performance index of the optimized design is obviously improved,and the eigenvalue of Gramian matrix corresponding to each mode is also improved.The influence of volume fraction,change of the structure frequency and concerned modes in the objective function on the optimization results are also discussed.The results show that the more the piezoelectric materials is,the higher the performance index is.3.Topological design of thin-shell piezoelectric structure under optimal control.Using LQR optimal control as the feedback control method,the state control equation of piezoelectric smart structure is established.Employing the dynamic compliance of the structure as the objective function,the Riccati equation that the feedback gain matrix needs to meet as the constraint,the optimization model of piezoelectric material is constructed.In the optimization model,the design variable is the relative density of piezoelectric elements.Based on SIMP method,an artificial material penalty model on mechanic and piezoelectric properties of piezoelectric material is constructed to improve the efficiency of the optimization.By introducing adjoint vector,the Lagrangian form of the objective function is established.The numerical optimization problem is solved by the moving asymptotes method.The results show that compared with that of the initial design,the overall vibration of the optimized design has been better controlled.The distribution of the control voltage is consistent with the distribution of piezoelectric materials,which shows the penalty effect of the optimization model on the voltage.The influences of the volume fraction,external excitation frequency and weight matrix coefficients on the optimization results are also considered in the numerical examples.The results show that the external excitation frequency has a significant influence on the optimized layout of piezoelectric materials,while the choice of the weight matrix has no notable effects on the optimized topology.4.Topological design of piezoelectric actuator layer for linear quadratic regulator control of thin-shell structures under transient excitation.Based on the classical plate theory,a finite element model for transient response analysis of piezoelectric structures under optimal control is established.In order to improve the efficiency of response analysis,the control equation of piezoelectric structures is transformed into a reduced one by means of mode superposition method.In the optimization model,the integral of the structural transient response in the specified time interval is taken as the objective function,and the governing equation of the piezoelectric structures under optimal control is taken as the constraint.Based on the adjoint variable method,the sensitivity of the structural transient response with respect to the design variable is derived.Compared with the structural response calculated by commercial software,the validity of the finite element model is verified.The optimization problem is solved by the mathematical programming.The results show that the transient response of the optimized design is effectively reduced.Finally,the influence of volume fraction,time interval in objective function and weight matrix on the optimized layout is discussed.