This paper extends the probabilistic finite element method (PFEM) to vector-valued and matrix-valued functions using techniques from matrix calculus and Kronecker algebra, and presents the PFEM in Kronecker notation for linear mad nonlinear continua. The results derived are easily amenable to computa-tional procedures.

This paper proposes the application of reliability-based optimization and robust design methods of a coil tube-spring. The perturbation method, the Edgeworth series, the reliability-based optimization, the reliability sensitivity technique, and the robust design method are employed to present practical and effective approaches of reliability-based optimization and robust design for coil tube-spring with non-normal distribution parameters, on the condition of the known first four moments of the original random variables. Theoretical formulae for reliability-based optimization and robust design are obtained. The respective programmes can be used to obtain the reliability-based optimization and robust design parameters of a coil tube-spring with non-normal distribution parameters accurately and quickly.

将可靠性设计理论与灵敏度分析方法相结合，讨论了具有任意分布参数的机械零件的可靠性灵敏度设计问题，提出了可靠性灵敏度设计的计算方法，给出了可靠性灵敏度的变化规律，研究了设计参数的改变对机械零件的可靠性的影响，为机械零件的可靠性设计提供了理论依据。

将可靠性优化设计理论、可靠性灵敏度技术与稳健设计方法相结合，讨论了具有非正态分布参数的车辆零件的可靠性稳健设计问题，提出了可靠性稳健设计的计算方法。把可靠性灵敏度溶入可靠性优化设计模型之中，将可靠性稳健设计归结为满足可靠性要求的多目标优化问题。在基本随机参数的前四阶矩已知的情况下，通过计算机程序可以实现具有非正态分布参数的车辆零件的可靠性稳健设计，迅速准确地得到具有非正态分布参数的车辆零件的可靠性稳健设计信息。数值算例表明所提出的方法是一种非常方便和实用的可靠性稳健设计方法。

The reliability-based optimization, the reliability sensitivity technique, and therobust design are employed to present a practical and effective approach of robust reliabilitydesign for mechanical components on the condition of known probabilistic characteristics oforiginal random variables. The theoretical formula of robust reliability design for mechanicalcomponents is obtained. The respective program can be used to obtain the robust reliabilitydesign parameters accurately and quickly. According to the numerical results, the methodproposed is a convenient and practical robust reliability design method.