中文摘要：

磁共振系统梯度线圈设计是一个多目标优化问题，在设计时需要综合考虑能耗、磁场能、线性度等设计要求．这些设计要求通常难以同时获得极小解，因此在设计梯度线圈时需要权衡线圈的各方面的设计需求．本文基于柱面可展性和流函数设计方法，结合Pareto优化方法实现了在超椭圆柱设计表面上梯度线圈的多目标设计．分别分析了磁场能、能耗目标对梯度线圈线性度、线圈构型的影响；并在Pareto解空间中分析各目标的相互变化关系，通过数值算例验证了该方法在超椭梯度线圈设计时的有效性与灵活性．优化结果显示，在满足线性度误差小于5％，能耗与磁场能分别小于用户设定值的设计约束下，梯度线圈的多目标设计存在多个局部优化解．该方法可以直观地比较相同目标函数值的情况下各单目标的具体表现，有利于实现不同的设计要求下梯度线圈的最终定型设计．

英文摘要：

The design of gradient coils for a magnetic resonance imaging （lvl~l） ~y~u~m 1~ ~ 1, v J r problem, which usually needs to deal with a couple of conflicting design objectives, such as the stored magnetic energy, power consumption, and target linear gradient distribution. These design requirements usually conflict with each other, and there is no unique optimal solution which is capable of minimizing all objectives simultaneously. Therefore, the design of gradient coils needs to be optimized reasonably with the tradeoff among different design objectives. Based on the developable property of the super-elliptical cylindrical surface and the stream function design method, the multiple objective optimization problem is analyzed by using the Pareto optimization method in this paper. The effect of proposed approach is illustrated by using the stream function method and three aforementioned coil design objectives are analyzed. The influences of the stored magnetic energy and power consumption target on linearity of gradient coil and the configuration of coils are analyzed respectively. The suitable sizes of gradient coils are discussed by analyzing the change of the stored magnetic energy. A weighted sum method is employed to produce the optimal Pareto solutions, in which the multiple objective problem reduces into a single objective function through a weighted sum of all objectives. The quantitative relationship of each design requirement is analyzed in the Pareto solution space, where Pareto optimal solutions can be intuitively found by dealing efficiently with the tradeoff among different coil properties. Numerical examples of super-elliptical gradient coil solutions are provided to demonstrate the effectiveness and versatility of the proposed method to design super-elliptical gradient coils with different coil requirements. The optimization results show that there are multiple available solutions in the convex Pareto solution space under the constraints that the linear gradient deviation is less than 5% an