## 1 Introduction  
The structural complexity of mechanical systems has significantly increased due to the increase of requirements in dynamic performance. Dynamic optimization has also become the most critical issue in structure design, especially for spacecraft structures with strict dynamic requirements [1]. As well known, satellite structures support all the functional components that are housed in it. If the natural frequency of the satellite structure overlaps with the frequency of excitation coming from the engine of launch vehicle, resonance will happen and high-amplitude vibration transmitting through the satellite structure will cause destruction to the payloads inside. Therefore, being one of the most important indicators of dynamic performance, the optimal design of structure with frequency constraint is extremely useful in manipulating the dynamic characteristics in a variety of ways. The ability to manipulate the selected frequency can remarkably improve the performance of the structure [2]. Many studies [3, 4, 5, 6] related to the optimization of structures with frequency constraints are available. [PERSON] and [PERSON] [7] considered different approaches for topology optimization involving simple and multiple eigenfrequencies. [PERSON] and [PERSON] [8] presented an algorithm with relaxation approach to solve the topology optimization with frequency constraints in frame structures. [PERSON] [9] studied the topology optimization with frequency band constraints for resonance avoidance. However, most of these papers consider a single or multiple frequency constraints in the general structural optimization, which use only the information about the global vibration frequency, such as the first-order lateral and the first-order vertical natural frequency. Few studies focus on local frequency requirements, which are sometimes also very important to improve the structural local stiffness for the key payloads installation areas. There are no significant differences between the optimization problems with local and global frequency constraints [10]. One of the most intractable problems in local frequency optimization seems to be the switching of local modes due to structural size modifications [11]. The order of a certain local frequency is usually not constant and is hard to be identified automatically, especially during the structural optimization process, which will cause convergence difficulties to the optimizer [12; 13]. Therefore, the method for structural optimization problems with global frequency constraints could not be directly applied in solving that with local frequency constraints. Mode switching is the main problem in the process of dynamic structure optimization with local frequency constraints. Several survey papers in the field of local modes have been put forward. [PERSON] and [PERSON] [14] restricted the element density to a larger threshold to avoid local mode, which is why distinct gray areas showed in the optimization result of the material layout in the design of frequency maximization of dynamic structures. [PERSON] [15] proposed an improved method of variable penalty factors to avert local mode SIMP. [PERSON] and [PERSON] [16] put forward an approach to punish the element stiffness matrix of low-density areas by using polynomials. However, most of the presented works focus on the topology optimization for continuum structures, few works could be obtained for the sizing optimization problems. In the present work, a new method is proposed to solve the structural optimization problem with local frequency constraints. In order to avoid the obstruction of mode switching, the essential idea of the new method is to transform the original structural dynamic optimization problem with local frequency constraints into an approximate static optimization problem with nodal displacements constraints. The nodal displacement constraints are derived by the initial modal shape of the local area and the ratio between the initial local frequency and the expected local frequency.