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弹性力学中,把地下洞室外域变换成单位圆外域的映射函数最普遍的形式是Laurent级数,因而求解级数表达式中的系数则成为求解地下洞室解析解问题的关键。而通过搜索边界映射点的方法,可以得出一种求解映射函数表达式系数的新解法。该法首先初始化一组近似的对应关系,通过这一初始映射关系所求解出来的映射函数,得出单位圆上点的相应的映射点及初始映射洞形,由这些相邻映射点之间的距离,根据等距离比的原则,求出对应的在实际地下洞室边界上的点坐标。再根据这一改进的对应关系,求解出第1次迭代的映射函数,如此循环下去,直到近似洞形与实际洞形足够地接近为止。运用该方法可以灵活控制Laurent级数的项数、迭代循环的次数以及映射洞形的精度。该法能快速求解出各种复杂单个单连通区域洞形的映射函数,尤其对目前工程中常见的复杂洞形,都能得出相当精确的映射函数。
In elastic mechanics, Laurent series is the most common form of the mapping function that transforms the underground chamber into the outer circle of unit circle. Therefore, it is the key to solve the analytical solution of underground caverns by solving the coefficients in series expressions. By searching the boundary mapping points, a new solution to the coefficients of the mapping function expression can be obtained. The method first initializes a set of approximate correspondences, and obtains the corresponding mapping points and the initial mapping holes on the unit circle by the mapping function solved by the initial mapping relationship. From these neighboring mapping points Distance, according to the principle of equal distance ratio, find the corresponding point in the actual underground tunnel coordinates. According to the improved correspondence, the mapping function of the first iteration is solved, and the loop continues until the approximate hole shape is sufficiently close to the actual hole shape. The method can be used to flexibly control the number of Laurent series, the number of iteration loops and the accuracy of mapping the hole shape. This method can quickly solve the mapping function of various complicated single-connected-area tunnels. Especially for the complex tunnels commonly found in current engineering, a very accurate mapping function can be obtained.