力边界法向量在套管应力数值分析中的应用

传统法向量求解方法仅适用于简单几何边界问题。国际上大型有限元软件（如 ABAQUS）虽然能给出一般边界法向量，但非专业人员无法了解其内核理论。因此，深入研究任意边界法向量数值理论十分必要。等参元边界法向量的数值计算方法，是基于石油套管复杂力边界法向量理论研究和工程需要提出的。自然坐标和物理坐标之间映射关系的建立，便于应用等参单元对任意几何形状求解域进行有限元离散。依此将多元函数微分方程求解方法与有限元等参变换方法相结合，建立起等参元边界法向量数学计算模型，并编制出相应的计算软件。该理论的应用，能够更加准确模拟石油套管管内流体、地层蠕变等复杂地质条件下，套管壁上的力边界法向量问题。从而能彻底解决复杂套管力边界法向量问题；能够提高套管强度计算精度、优化套管设计程序。所得等参元力边界法向量数值理论及计算软件，普遍适用于一般边界法向量问题。算例给出两类力边界法向量数值方法，套管外壁法向量数值解与解析解对比分析表明，该方法简单，精度高。

APPLICATION OF FORCE BOUNDARY NORMAL VECTOR IN NUMERICAL ANALYSIS OF CASING STRESSES

The numerical calculation method for normal vector of isoparametric boundary is put forward on the basis of the needs of theoretic study and engineering of normal vectors of complicated force boundary of casings. The conventional normal vector calculation method is only applicable to simple geometric boundaries. Although the large-scale international finite-element software, such as ABAQUS, can be used to get normal vectors with ordinary boundaries, the non-professionals cannot understand the kernel theories. Therefore, an in-depth study of the numeric theory for normal vectors at discretional boundary is essential. By use of the isoparametric cells, to establish the binding or mapping relationship between the natural coordinates and the physical coordinates can facilitate the finite element divergence in a solution domain of a discretional geometric shape, based on which the calculation model for normal vectors at isoparametric boundary can be set up and the related calculation software can be compiled. With this theory, the normal vectors at force boundary on the casing wall can be precisely simulated under the conditions of fluid flow in the casing and complicated geologic conditions like formation creeping, etc. Therefore the complicated force boundary normal vectors can be thoroughly calculated, the accuracy of casing strength calculation can be improved and the casing program design can be optimized. This theory and calculation software for the isoparametric force boundary normal vectors can be commonly used to solve ordinary boundary normal vectors. Calculation examples are given to introduce 2 types of numeric calculation method for force boundary normal vectors. The comparison between the numeric solution and the analytic solution of the normal vectors on the external casing wall proves that this method is simple and highly accurate.