弹性力学边界条件反演问题的边界元正则化算法及其对比分析-固体力学专业论文.docxVIP

弹性力学边界条件反演问题的边界元正则化算法 及其对比分析 摘 要 针对二维各向同性弹性力学 Cauchy 问题，使用边界元法来进行分析。采用截 断奇异值分解正则化技术来求解该病态问题，并使用 L-曲线法来选择最优正则化参 数，即最优奇异值截断位置，从而得到该问题的解。通过数值算例对求得的边界条 件数值解与解析解进行比较，并进行误差分析，结果表明了截断奇异值分解算法的 有效性和稳定性。通过减少已知数据中的随机偏差和增加边界单元密度可以提高求 解的精确度。 如果部分边界上的边界条件只已知一半，剩余边界上的边界条件全部未知，在 补充内点物理信息的基础上，可求得所有未知的边界条件。对于含内点信息的二维 各向同性弹性力学边界条件识别反问题，本文同样采用边界元法来进行分析。使用 共轭梯度和基于隐式转换的预处理共轭梯度正则化法来求解该病态问题，并采用 Morozov 偏差原理来选择最佳迭代步数。在涉及内点信息的边界条件反问题中，当 已知位移信息的内点很靠近边界时，边界元法存在几乎奇异积分的障碍，常规 Gauss 数值积分失效，首先需要对几乎奇异积分进行正则化，然后再对不适定数理反问题 实施正则化。增加内点的数量和减少已知数据中的随机偏差能够提高结果的精度。 保持一定数量的内点，在边界划分更多单元能够获得更精确的解。当内点沿着整个 边界分布，两种迭代法都对内点的位置不敏感。共轭梯度法迭代步少，计算时间上 更有优势。通过数值算例对求得的边界条件数值解与解析解进行对比，表明了这两 种迭代算法的有效性和稳定性。 关键词：反问题，边界元法，正则化法，边界条件，几乎奇异积分 Boundary Element Regularized Algorithm for Boundary Condition Inverse Problems in Elasticity and Its Comparative Analysis ABSTRACT The boundary element method is developed to analyze the Cauchy boundary condition inverse problems in 2-D isotropic elasticity. Truncated singular value decomposition (TSVD) technique is applied to solving the ill-posed problem. L-Curve method is proposed to select the regularization parameter .i.e. the optimal truncation number, and then the solution of this problem can be obtained. The comparison of numerical and analytical solutions in numerical examples shows that the TSVD algorithm is effective and stable. The regularization errors are also analyzed. The accuracy of the solution can be improved with respect to reducing the amount of noise added into the known data and refining the boundary element mesh size. Displacement or traction boundary conditions are given on a part of boundary, whilst all of displacement and traction vectors are unknown on the rest of the boundary. All the unknown boundary conditions are to be determined with some measurable displacement information at some interior points. The preconditioned conjugate gradient method (PCGM) based on implicit transformation and the conjugate gradient method (CGM) are employed to regularize the ill-pose