## 1 Introduction  
Electrothermal microactuators are mechanical systems where thermally-induced expansion and contraction of materials is responsible for the induction of motion. A temperature change inside the actuator is caused by Joule heating due to the electrical current flow. In the context of microscale and nanoscale engineering, thermal actuators can be used as microelectromechanical systems (MEMS) capable of translating electrical signals into controllable displacements and forces [1]. Electrothermal microactuators have proven to be very useful MEMS structures, which are commonly used in telecommunication, automotive, aviation and medical industries; e.g., microgrippers, switches, filters and modulators. Compared to the other types (e.g., magnetostatic, piezoelectric and electrostatic), the electrothermal actuators have many advantages. They can generate large forces or displacements per unit volume; require low actuation voltages; have linear material characteristics (strain-stress relationship), despite the high temperatures occurring in them; and moreover, they are relatively easy and cheap to fabricate. The disadvantages of microthermal actuators concern comparatively high-power consumptions and relatively slow response times. The Joule heating effect is possible due to the high value of electrical resistance of the material, which is the polycrystalline silicone. Mainly,two types of such microactuators are used in MEMS: U-beam and V-beam ones. U-beam type actuators were originally demonstrated in the early 1990s, whereas V-beam ones were in the late 1990s, and quickly became very popular for the generation of mechanical motion. U-beam microactuators consist of two beams (arms) which have different cross-section areas. They produce different amounts of Joule heating in the presence of the electric currents, which causes the bending of the apexes of the actuators. For the V-beam microactuator, arms are fabricated with a pre-bending angle. The longitudinally extending arms, also due to the Joule heating effect, move the central shaft. The central shaft aggregates the actuation force of the multiple beams, resulting in a total actuation force [2]. The V-beam actuator, called variously--chevron type, bent-beam or symmetric thermal actuator, is considered in the paper. Figure 1a presents an example shape of the chevron microactuator consisting of four pairs of arms, whereas Figure 1b shows the deformation and map of equivalent stress in the structure. The process of design of such structures has to incorporate many technical aspects and restrictions. Accordingly, optimal design of electrothermal microactuators, requires consideration of many criteria, which depend on the quantities like actuation force, generated displacement, heat or electrical losses, stress concentration, risk of buckling and dynamic characteristics. For such structures, the above criteria strongly depend on the geometry. These criteria are very often contradictory, which leads to the necessity of accepting compromise-solutions. Analytical formulas related to basic electrical, thermal and mechanical quantities have been derived only for the simplified geometry (straight shape arms) of the microactuators. For the arbitrary shape of the geometry in order to calculate objective functions, the quantities derived from numerical simulations of coupled electro-thermo-mechanical analysis have to be used. Functionals expressing assumed criteria, calculated on the basis of multiphysics numerical simulations can be strongly multimodal (have many local minima). It significantly limits the scope of the applicability of classical gradient or non-gradient optimization methods. It can be seen that in order to efficiently optimize such structures, proper multiobjective optimization techniques have to be used [3]. Application of the multiobjective optimization methods leads to a situation where the result is not only a single solution, but a number of optimal solutions. These solutions are called Pareto-optimal solutions.