Browsing by Subject "Cauchy problem"

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An optimization approach for determining interfacial interactions
(2017-08) Palvadi, Naga Sundeep; Liechti, K. M.; Lu, Nanshu; Landis, Chad M.; Bonnecaze, Roger; Ravi-Chandar, Krishnaswamy
Traction separation relations are fundamentally important to describe the constitutive response of both adhesive and cohesive interfaces. Departing from the heuristic nature of existing methods, in this doctoral work a framework is proposed to independently solve for the tractions and displacements by redefining the problem at hand as a Cauchy inverse boundary value problem. The proposed framework combines the philosophy of reciprocity theorems and iterative solution techniques of Cauchy problems to develop an optimization scheme which can be further utilized to measure the traction separation relationships. An attractive feature of this method is that no a priori assumption on the functional form relating interfacial tractions and displacement quantities is made which, in turn, enables the user to apply this framework to characterize rate dependent traction separation relations as well. Numerical experiments were carried out to demonstrate the efficiency of this optimization-finite element method for double cantilever beam type geometry with surrogate traction separation relationships. Guidelines regarding the amount and type of physical measurements required as input were further developed and refined based on these numerical experiments. Two loading devices were built and physical experiments were performed separately to characterize the interfacial behavior of polydimethylsiloxane (PDMS) and polystyrene. Both the devices produce a mode-I delamination and the distinguishing feature was the type of the data generated and used as input in the optimization scheme. Geometric shadow moiré was used to measure, full field, out of plane deformation of the top adherand of a double cantilever beam and this displacement data was the input used to measure interfacial response in one type of tests. A double cantilever beam with electromechanical sensors combined with a microscope constituted the other test. This setup generated minimal data consisting of load, displacement, beam end rotation and crack length which were used as input data to measure the same interfacial response. Results obtained from both the tests performed at different loading rates highlighted that the interfacial response between the elastic bodies exhibited rate dependent delamination behavior. Realizing the inherent sensitivity of inverse problems to experimental errors, procedures pertaining to experimental data analysis and practical error elimination are also proposed in order to achieve robust results. In that vein, additional work done on measuring out of plane deformation from geometric shadow moiré intensity profiles and its evolution as a potent non-contact measurement technique will be further discussed.
The Boltzmann equation : sharp Povzner inequalities applied to regularity theory and Kaniel & Shinbrot techniques applied to inelastic existence
(2008-08) Alonso, Ricardo Jose, 1972-; Martínez Gamba, Irene, 1957-
This work consists of three chapters. In the first chapter, a brief overview is made on the history of the modern kinetic theory of elastic and dilute gases since the early stages of Maxwell and Boltzmann. In addition, I short exposition on the complexities of the theory of granular media is presented. This chapter has the objectives of contextualize the problems that will be studied in the remainder of the document and, somehow, to exhibit the mathematical complications that may arise in the inelastic gases (not present in the elastic theory of gases). The rest of the work presents two self-contained chapters on different topics in the study of the Boltzmann equation. Chapter 2 focuses in studying and extending the propagation of regularity properties of solutions for the elastic and homogeneous Boltzmann equation following the techniques introduced by A. Bobylev in 1997 and Bobylev, Gamba and Panferov in 2002. Meanwhile, chapter 3 studies the existence and uniqueness of the inelastic and inhomogeneous Cauchy problem of the Boltzmann equation for small initial data. A new set of global in time estimates, proved for the gain part of the inelastic collision operator, are used to implement the scheme introduced by Kaniel and Shinbrot in the late 70’s. This scheme, known as Kaniel and Shinbrot iteration, produces a rather simple and beautiful proof of existence and uniqueness of global solutions for the Boltzmann equation with small initial data.
Well-posedness for the space-time monopole equation and Ward wave map
(2008-05) Czubak, Magdalena, 1977-; Uhlenbeck, Karen K.
We study local well-posedness of the Cauchy problem for two geometric wave equations that can be derived from Anti-Self-Dual Yang Mills equations on R2+2. These are the space-time Monopole Equation and the Ward Wave Map. The equations can be formulated in different ways. For the formulations we use, we establish local well-posedness results, which are sharp using the iteration methods.