弹性问题及其边界反演的数值分析

发布时间：2018-05-30 11:57

本文选题：摩擦接触 + 对偶问题　；参考：《浙江大学》2016年博士论文

【摘要】：本文主要研究的是弹性力学问题。首先介绍了以形变位移为研究对象的接触问题,并且分别说明了带摩擦的接触问题和带损伤的截断型弹性问题的解存在且唯一的条件。然后我们将带摩擦接触问题的研究对象变为应变张量,即得到原问题的对偶问题,证明了原问题与对偶问题之间解的互通性,同时推导得到对偶问题的解存在且唯一的条件。其次,在粘弹性接触模型下,我们研究了接触边界为包含形式的情况,也就是接触条件由非单调算子控制,且该算子是具有Clarke次微分形式的单值或多值算子。从无摩擦粘弹性接触模型中,我们简化得到含伪单调性椭圆算子的抛物型变分-H半变分不等式,对该不等式在时间上的离散格式进行分析,即Rothe问题。我们证明了Rothe问题解的存在性和原问题解的唯一性,给出了Rothe问题解的收敛性和正则性结果,并进行了二维数值模拟。最后,本文的研究重点是一类弹性模型中的反问题,研究的目标是由观测数据来反演边界上的牵引力。我们利用优化控制问题的思想来刻画这一目标,并说明了反问题至少存在一个解。然后对目标泛函引入Tikhonov正则化方法来证明正则化后的反问题解是唯一存在的,而且当正则化参数趋于0时,正则解是收敛到原反问题中L2范数最小的那个解。此外,我们推导得到依赖于正则化参数的数值解的误差估计。特别地,我们分别推导得到了无摩擦接触问题和带损伤的截断型弹性问题的伴随问题,利用伴随问题得到了反问题解的约束不等式,并以此构造了迭代算法进行数值模拟。
[Abstract]:In this paper, the problem of elasticity is studied. First, the contact problem of deformation displacement is introduced, and the existence and unique conditions of the contact problem with friction and the truncated elastic problem with damage are discussed respectively. Then we change the object with friction contact into strain Zhang Liang, that is, we obtain the dual problem of the original problem, prove the interworking between the solution of the original problem and the dual problem, and derive the existence and unique condition of the solution of the dual problem. Secondly, in the viscoelastic contact model, we study the case where the contact boundary is a form of inclusion, that is, the contact condition is controlled by a non-monotone operator, and the operator is a single-valued or multi-valued operator with Clarke subdifferential form. From the frictionless viscoelastic contact model, we simplified the parabolic variational -H semi-variational inequality with pseudo-monotonic elliptic operators, and analyzed the discrete scheme of the inequality in time, that is, the Rothe problem. We prove the existence of the solution of the Rothe problem and the uniqueness of the solution of the original problem, give the convergence and regularity results of the solution of the Rothe problem, and carry out the two-dimensional numerical simulation. Finally, this paper focuses on the inverse problem in a class of elastic models. The objective of the study is to invert the tractive force on the boundary from the observational data. We use the idea of optimal control problem to characterize this objective and show that there is at least one solution to the inverse problem. Then the Tikhonov regularization method is introduced to the target functional to prove that the regularized inverse problem solution is unique, and when the regularization parameter approaches 0, the regular solution converges to the solution with the smallest L 2 norm in the original inverse problem. In addition, the error estimates of numerical solutions dependent on regularization parameters are derived. In particular, we derive the adjoint problems of the frictionless contact problem and the truncated elastic problem with damage respectively. By using the adjoint problem, we obtain the constrained inequalities of the inverse problem solution, and construct an iterative algorithm for numerical simulation.
【学位授予单位】：浙江大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.82;O343

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