本文選題：摩擦接觸 + 對(duì)偶問(wèn)題　； 參考：《浙江大學(xué)》2016年博士論文

【摘要】：本文主要研究的是彈性力學(xué)問(wèn)題。首先介紹了以形變位移為研究對(duì)象的接觸問(wèn)題,并且分別說(shuō)明了帶摩擦的接觸問(wèn)題和帶損傷的截?cái)嘈蛷椥詥?wèn)題的解存在且唯一的條件。然后我們將帶摩擦接觸問(wèn)題的研究對(duì)象變?yōu)閼?yīng)變張量,即得到原問(wèn)題的對(duì)偶問(wèn)題,證明了原問(wèn)題與對(duì)偶問(wèn)題之間解的互通性,同時(shí)推導(dǎo)得到對(duì)偶問(wèn)題的解存在且唯一的條件。其次,在粘彈性接觸模型下,我們研究了接觸邊界為包含形式的情況,也就是接觸條件由非單調(diào)算子控制,且該算子是具有Clarke次微分形式的單值或多值算子。從無(wú)摩擦粘彈性接觸模型中,我們簡(jiǎn)化得到含偽單調(diào)性橢圓算子的拋物型變分-H半變分不等式,對(duì)該不等式在時(shí)間上的離散格式進(jìn)行分析,即Rothe問(wèn)題。我們證明了Rothe問(wèn)題解的存在性和原問(wèn)題解的唯一性,給出了Rothe問(wèn)題解的收斂性和正則性結(jié)果,并進(jìn)行了二維數(shù)值模擬。最后,本文的研究重點(diǎn)是一類彈性模型中的反問(wèn)題,研究的目標(biāo)是由觀測(cè)數(shù)據(jù)來(lái)反演邊界上的牽引力。我們利用優(yōu)化控制問(wèn)題的思想來(lái)刻畫(huà)這一目標(biāo),并說(shuō)明了反問(wèn)題至少存在一個(gè)解。然后對(duì)目標(biāo)泛函引入Tikhonov正則化方法來(lái)證明正則化后的反問(wèn)題解是唯一存在的,而且當(dāng)正則化參數(shù)趨于0時(shí),正則解是收斂到原反問(wèn)題中L2范數(shù)最小的那個(gè)解。此外,我們推導(dǎo)得到依賴于正則化參數(shù)的數(shù)值解的誤差估計(jì)。特別地,我們分別推導(dǎo)得到了無(wú)摩擦接觸問(wèn)題和帶損傷的截?cái)嘈蛷椥詥?wèn)題的伴隨問(wèn)題,利用伴隨問(wèn)題得到了反問(wèn)題解的約束不等式,并以此構(gòu)造了迭代算法進(jìn)行數(shù)值模擬。
[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.
【學(xué)位授予單位】：浙江大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O241.82;O343

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