**1. Introduction**

Topology optimization method is firstly developed by Bendsøe and Kikuchi [1] and Bendsøe [2], which is originally used for an elastic material distribution problem in 1988. Compared with sizing and shape optimization, topology optimization is a more powerful tool, which allows new holes and connections to be generated in the design domain with a prescribed amount of material. Recently, the topology optimization method has been extended to many other fields, such as materials science [3, 4], micro-machines [5, 6], precision and ultra-precision machining [7, 8], optical focusing [9, 10], and so on. With the development of computer technology, topology optimization is developing very fast, and the calculation process of topology optimization problem has been significantly simplified.

In the research of piezoelectric actuators, piezoelectric stacks are usually used in the driving field, which requires precise but minute motion [11, 12]. Flexure hinge type piezoelectric actuators are widely applied in precision machining, in-situ mechanical measurement, biomedicine and other fields because of its compact structure, long stroke and fast response speed [13–15]. The piezoelectric actuators are usually composed of piezoelectric stack, compliant mechanism and slider [16–20], here the compliant mechanisms are used as transmission and amplifying mechanisms. In the field of piezoelectric actuators, the modeling and design of compliant mechanisms are key issues. For example, bridge-type [21] and parallelogram-type [22] compliant mechanisms are applied to the reach of precision engineering. Sigmund [23, 24] has designed an inverting displacement amplifier, which has been used to obtain the maximum mechanical advantage. In addition, topology optimization methods have been assembled in commercial software [25], which makes the applications of topology optimization methods much easier.

There mainly three different types of topology optimization method are used to handle the design objectives and constraints of piezoelectric actuators: density-based methods, boundary variation methods, hard-kill methods, respectively [25, 26]. (1) Density-based methods, which include Solid Isotropic Material with Penalization (SIMP) and Rational Approximation of Materials Properties (RAMP) technique. (2) Boundary variation methods (level set and phase field). (3) Hard-kill methods, typically Evolutionary Structural Optimization (ESO) method and Bidirectional Evolutionary Structural Optimization (BESO) method. Many scholars have conducted indepth research on the above three types of topology optimization method, the design theories and methods have developed rapidly. Topology optimization has becoming one of the important methods of piezoelectric actuator design.

However, topology optimization of piezoelectric actuators is a complicated problem, which contains the process of determining the connectivity, shape, and location of voids inside a given design domain. Traditional researches have given some topological structure for the design of piezoelectric actuators [27–34], the not solved problem is that which kind of compliant mechanisms with the flexure hinges makes the output performances best. Yang et al. [35] developed a static topology optimization method to solve the problem, and some topology optimization methods have also been extended to design the compliant mechanism of piezoelectric actuators [36–40]. At present, there is little research on using topology optimization method to design flexible hinges, and its theories and methods are very scarce, which has a lot of exploration space. Therefore, it is particularly important to explore the theories and methods of topology optimization for the design of flexure hinge type piezoelectric actuator.

We hope that this chapter will provide a summary of the recent advances and novel applications of topology optimization methods, provide exposure to lesser known, yet promising, techniques, and serve as a resource for those new to the field. The presentation of each method focuses on new developments and novel applications. So in Section 2, the operating principle of piezoelectric actuators is introduced, the problems are described and the topology optimization model is established, then the simulation analysis is performed. The prototype is fabricated and the experimental system is built in Section 3. Systematic experimental test is conducted to study the actual performance of the actuator in Section 4, and the conclusion and discussion of this chapter are in Section 5.
