关于悬索桥中经典Melan方程的适定性与高效逼近
On the well-posedness and efficient approximation for the classical Melan equation in suspension bridges
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中文总结 AI 辅助
本文研究悬索桥经典Melan方程的适定性与高效逼近，通过简化模型推导先验估计并证明唯一性，提出迭代方法并验证其几何收敛，实际桥梁计算展示效果。
中文摘要 AI 辅助
本文考虑用于模拟悬索桥的经典Melan方程。我们首先研究了简化“低刚度”模型解析解的显式表达式和全局性质，在此基础上推导了原始经典Melan方程的先验估计，并在显式荷载条件下建立了其在下向活载下具有非负积分的唯一解，从而在工程背景下证明了相应挠度曲线的唯一性。我们还开发了一种高效的迭代逼近方法，以简化“低刚度”模型的解作为首次迭代，并证明了其具有显式误差估计的几何收敛性。通过对两座实际桥梁的计算，验证了该方法的适用性和计算效率，同时量化了非线性非局部项对解的影响，并阐明了简化模型与原始模型之间的关系。文中验证并解释了几项工程观察结果，并提出了一些相关的开放问题。
英文摘要
The classical Melan equation modeling suspension bridges is considered. We first study the explicit expression and global properties of the analytical solution for the simplified ``less stiff'' model, based on which we derive an a priori estimate for the original classical Melan equation and establish, under an explicit load condition, that it has a unique solution with nonnegative integral under downward live loads, thereby showing the uniqueness of the corresponding deflection curve in the engineering setting. We also develop an efficient iterative approximation method by taking the solution of the simplified ``less stiff'' model as the first iterate, and prove its geometric convergence with explicit error estimates. The applicability and computational efficiency of the method are demonstrated through calculations for two actual bridges, which also quantify the influence of the nonlinear nonlocal term on the solution and clarify the relationship between the simplified and original models. Several engineering observations are verified and explained, and some related open problems are suggested.
发表机构
- Lanzhou University of Technology（兰州理工大学）
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