【作者】 张国庆；

【导师】 杨海天；

【作者基本信息】 大连理工大学 ， 固体力学， 2014， 博士

【摘要】 在反问题和优化问题的数值求解过程中,一般需多次数值求解相关的正问题。对于敏度分析困难的反问题和优化问题,若采用非敏度类的算法(如智能类算法)进行求解,将大大增加正问题的求解次数。因此,如何有效地提高求解正问题的计算效率,是提高求解反问题/优化问题的计算效率所必须考虑的。本文针对两个敏度分析困难的反问题：分数阶粘弹性反问题与双模量反问题,提出利用代理模型技术,建立相关正问题的近似求解模型,以降低正问题求解的计算开销。数值验证表明,利用正问题的近似求解模型可显著提高反问题求解的计算效率。本文的研究成果主要包括：1.提出一种利用Kriging代理模型近似求解均质／区域非均质分数阶粘弹性正问题的数值模型。为提高时间域的计算效率,提出了三种建模策略。与原FE/FE-FD模型相比,所提算法单次求解正问题的计算开销明显降低。在反问题求解中,采用基于网格划分策略的连续域蚁群算法,实现了对分数阶粘弹性本构参数的识别。数值验证表明,所提方法可在有效保持计算精度的同时,显著降低反问题的计算成本。2.提出一种利用Kriging代理模型近似求解均质／区域非均质双模量正问题的数值模型。与原FE模型相比,所提算法单次求解正问题的计算开销明显降低。在反问题求解中,采用基于网格划分策略的连续域蚁群算法,实现了对双模量本构参数的识别。数值验证表明,所提方法可在有效保持计算精度的同时,显著降低反问题的计算成本。3.为进一步提高计算效率,提出了一个基于敏度分析求解二维双模量正问题的数值模型,以及一个基于两级敏度分析求解二维双模量反问题的数值模型,并导出了相关的敏度计算公式。采用Newton-Raphson方法求解正问题,采用Gauss-Newton方法求解反问题。数值验证表明,与文中非敏度类算法相比,所提算法使求解双模量正／反问题的计算效率得到更大提高。文中通过多个算例对所提算法进行了数值验证,并分析和讨论了多种因素对计算精度与计算效率的影响。更多还原

【Abstract】 In the process of solving an inverse/optimization problem numerically, its corresponding direct problem usually needs to be solved repeatedly. For the case where sensitivity analysis is difficult to implement, the number of solving direct problems will be greatly increased if a non-sensitivity analysis based algorithm, such as the intelligent algorithm, is adopted. Thereby, how to efficiently solve the direct problem is an important issue taken into account for improving the computational efficiency of solving inverse/optimization problems.This thesis focuses on two problems with the difficulty of sensitivity analysis, i.e. fractional viscoelastic problem and bimodular problem. In order to reduce the computational expense on solving direct problems, two surrogate technique based approximate numerical models to solve direct problems are presented, resulting in a significant increase of computational efficiency of solving inverse problems.The major achievements include1. An approximate numerical model to solve homogeneous/regionally inhomogeneous direct fractional viscoelastic problems is presented using the Kriging surrogate technique. In order to reduce the computational expense on time domain, three strategies are proposed in the modeling process. Compared with the original FE/FE-FD model, the approximate model’s computational expense on a single time to solve the direct problem is drastically reduced. A gridding partition based continuous ant colony algorithm is employed to identify viscoelastic constitutive parameters. Numerical verifications indicate that a great amount of computational cost on solving inverse fractional viscoelastic problems can be saved with sufficient computing accuracy.2. An approximate numerical model to solve homogeneous/regionally inhomogeneous direct bimodular problem is presented using the Kriging surrogate technique. Compared with the original FE model, the approximate model’s computational expense on a single time to solve the direct problem is drastically reduced. A gridding partition based continuous ant colony algorithm is employed to identify bimodular constitutive parameters. Numerical verifications indicate that a great amount of computational cost on solving inverse bimodular problems can be saved with sufficient computing accuracy.3. For further improving the computational efficiency, a sensitivity analysis based numerical model to solve the2-D direct bimudular problem and a two level sensitivity analysis based numerical model to solve the2-D inverse bimudular problem are proposed, respectively. The formulae to calculate sensitivity are derived. The Newton-Raphson method and Gauss-Newton method are employed to solve the direct and inverse bimudular problems, respectively. Numerical verifications indicate that the computational efficiency of solving the direct/inverse bimudular problems is significantly improved using these models in comparison with the non-sensitivity analysis based algorithms presented in this paper.A number of numerical examples are provided to verify the proposed numerical algorithms, and several factors related to the computing accuracy and efficiency are discussed.更多还原