# Design and Optimization of Modular Solid Rocket Grain Matching Multi-Thrust Performance Curve
## 1 Introduction  
The solid rocket grain, a component of the solid rocket motor, plays a crucial role in determining the rocket's internal ballistic performance (thrust-time curve or pressure-time curve). In order to meet the requirements of some flight missions, designers need to develop solid rocket grains that exhibit a multi-thrust performance curve. Consequently, one of the key responsibilities of designers is to design a grain shape or configuration that aligns with the desired multi-thrust profiles. Researchers and engineers have embraced the concept of combined grain to achieve complex performance curves [1, 2, 3, 4], such as the utilization of star grain and tube grain to generate a dual-thrust performance curve [5, 6]. Moreover, the combination of star grain and slot grain are found in the boost stage of a multi-thrust motor [7]. Designing combined grain configurations necessitates a substantial amount of practical knowledge, and most of the combination patterns may not produce ideal and steady thrust platforms. During every stage, large pressure fluctuations may occur. To address this issue, one may use several segments of end-burning grain with varying burning rates to make the thrust platform assteady as possible. However, the burning surface area is constrained by the end-burning grain radius. To achieve larger thrust, a larger grain radius would be required, which may not be feasible in practical applications [7]. To find the most promising grain shape (or grain configuration) matching a given multi-thrust performance curve, an optimization method should be introduced. Two main optimization methods are commonly used: size optimization and shape optimization. Size optimization involves the selection of a grain type, such as star grain or finocyl grain, by designers. The detailed parameters of the grain are then optimized using global optimization algorithms like genetic algorithms. This method is straightforward and has been extensively studied over the years [8; 9; 10]. While it can achieve a performance curve that matches the given requirements, there is no mathematical guarantee of finding the best performance-matching grain. Additionally, since the grain shape is pre-selected, the results may be influenced by the designers' experience. On the other hand, the grain reverse design, realized by the shape optimization, does not require the pre-selection of a grain shape. Instead, the grain shape is represented using techniques like splines [11] or neural networks [12; 13; 14], and it gradually evolves during the iteration process. This approach allows for the exploration of complex and irregular grain shapes, resulting in excellent matches to the desired performance curve. Moreover, it may even lead to the discovery of novel grain shapes during the optimization process. However, shape optimization is computationally intensive, with 3D grain shape optimizations taking days and 2D grain shape optimizations taking hours [14]. Furthermore, it may present new challenges in manufacturing irregular grain shapes. Both size optimization and shape optimization have their respective limitations and shortcomings. Therefore, it is necessary to introduce the concept of modular grain. Modular grain represents a unique category within the realm of combined grain, wherein two or three distinct grain shapes are employed as fundamental templates. These templates can be flexibly combined to achieve a wide range of performance curves, encompassing single-thrust, dual-thrust, and triple-thrust configurations. [PERSON] [15] discusses the modulation of solid rocket motor grains and suggests it might reduce the cost and cycles during the development. X-Bow Systems, the pioneer in modular solid motors, has designed and built a suite of modular solid rocket motors and small launch vehicles for both orbital and suborbital launch services [16]. The concept of modular grain may enable rapid and responsive motor design, prototyping, testing, and production, making the product more competitive in the market [17].