This paper provides two parallel solutions on the mixed boundary value problem of a unit annulus subjected to a partially fixed outer periphery and an arbitrary traction acting along the inner periphery using the complex variable method. The analytic continuation is applied to turn the mixed boundary value problem into a Riemann-Hilbert problem across the free segment along the outer periphery. Two parallel interpreting methods of the unused traction and displacement boundary condition along the outer periphery together with the traction boundary condition along the inner periphery respectively form two parallel complex linear constraint sets, which are then iteratively solved via a successive approximation method to reach the same stable stress and displacement solutions with the Lanczos filtering technique. Finally, four typical numerical cases coded by \texttt{FORTRAN} are carried out and compared to the same cases performed on \texttt{ABAQUS}. The results indicate that these two parallel solutions are both accurate, stable, robust, and fast, and validate that these two parallel solutions are numerically equivalent.

翻译：本文采用复变函数方法，针对外圆周部分固定且内圆周受任意牵引作用的单位圆环混合边值问题，提出了两种并行求解方案。通过解析延拓将混合边值问题转化为沿外圆周自由段上的黎曼-希尔伯特问题。沿外圆周未利用的牵引边界条件与位移边界条件，以及沿内圆周的牵引边界条件，分别通过两种并行解释方法构成两组并行的复线性约束集，随后采用逐次逼近法结合Lanczos滤波技术迭代求解，最终获得相同的稳定应力与位移解。最后，利用\texttt{FORTRAN}编写代码实现了四个典型数值算例，并与\texttt{ABAQUS}执行相同算例的结果进行对比。结果表明，这两种并行方案均具有精确、稳定、鲁棒且高效的特性，并验证了二者在数值上的等效性。