Numerical solution of static and dynamic problems of imprecisely defined structural systems

Static and dynamic problems with deterministic structural parameters are well studied. In this regard, good number of investigations have been done by many authors. Usually, structural analysis depends upon the system parameters such as mass, geometry, material properties, external loads and boundary conditions which are defined exactly or considered as deterministic. But, rather than the deterministic or exact values we may have only the vague, imprecise and incomplete informations about the variables and parameters being a result of errors in measurements, observations, experiments, applying different operating conditions or it may be due to maintenance induced errors, etc. which are uncertain in nature. Hence, it is an important issue to model these types of uncertainties. Basically these may be modelled through a probabilistic, interval or fuzzy approach. Unfortunately, probabilistic methods may not be able to deliver reliable results at the required precision without sufficient experimental data. It may be due to the probability density functions involved in it. As such, in recent decades, interval analysis and fuzzy theory are becoming powerful tools. In these approaches, the uncertain variables and parameters are represented by interval and fuzzy numbers, vectors or matrices.In general, structural problems for uncertain static analysis with interval or fuzzy parameters simplify to interval or fuzzy system of linear equations whereas interval or fuzzy eigenvalue problem may be obtained for the dynamic analysis. Accordingly, this thesis develops new methods for finding the solution of fuzzy and interval system of linear equations and eigenvalue problems. Various methods based on fuzzy centre, radius, addition, subtraction, linear programming approach and double parametric form of fuzzy numbers have been proposed for the solution of system of linear equations with fuzzy parameters. An algorithm based on fuzzy centre has been proposed for solving the generalized fuzzy eigenvalue problem. Moreover, a fuzzy based iterative scheme with Taylor series expansion has been developed for the identification of structural parameters from uncertain dynamic data. Also, dynamic responses of fractionally damped discrete and continuous structural systems with crisp and fuzzy initial conditions have been obtained using homotopy perturbation method based on the proposed double parametric form of fuzzy numbers. Numerical examples and application problems are solved to demonstrate the efficiency and capabilities of the developed methods. In this regard, imprecisely defined structures such as bar, beam, truss, simplified bridge, rectangular sheet with fuzzy/interval material and geometric properties along with uncertain external forces have been considered for the static analysis. Fuzzy and interval finite element method have been applied to obtain the uncertain static responses. Structural problems viz. multistorey shear building, spring mass mechanical system and stepped beam structures with uncertain structural parameters have been considered for dynamic analysis. In the identification problem, column stiffnesses of a multistorey frame structure have been identified using uncertain dynamic data based on the proposed algorithm. In order to get the dynamic responses, a single degree of freedom fractionally damped spring-mass mechanical system and fractionally damped viscoelastic continuous beam with crisp and fuzzy initial conditions are also investigated.Obtained results are compared in special cases for the validation of proposed methods

Numerical Solution of Some Uncertain Diffusion Problems

Diffusion is an important phenomenon in various fields of science and engineering. These problems depend on various parameters viz. diffusion coefficients, geometry, material properties, initial and boundary conditions etc. Governing differential equations with deterministic parameters have been well studied. But, in real practice these parameters may not be crisp (exact) rather it involves vague, imprecise and incomplete information about the system variables and parameters. Uncertainties occur due to error in measurements, observations, experiments, applying different operating conditions or it may be due to maintenance induced errors, etc. As such, it is an important concern to model these type of uncertainties. Traditionally uncertain problems are modelled through probabilistic approach. But probabilistic methods may not able to deliver reliable results at the required precision without sufficient data. In this context, interval and fuzzy theory may be used to manage such uncertainties. Accordingly, the system parameters and variables are represented here as interval and fuzzy numbers. Generally, we get interval or fuzzy system of equations for uncertain steady state problems with interval or fuzzy parameters whereas interval or fuzzy eigenvalue problems may be obtained for unsteady state. This thesis redefined interval or fuzzy arithmetic in order to handle the uncertain problems. The proposed arithmetic has been used to solve fuzzy and interval system of equations and eigenvalue problems. Various numerical methods viz. Finite Element Method (FEM), Wavelet Method (WM), Euler Maruyama and Milstein Methods are studied by introducing interval or fuzzy theory. The proposed arithmetic has been combined with FEM and WM to develop Interval or Fuzzy Finite Element Method (I/FFEM) and Interval or Fuzzy Wavelet Method (I/FWM). Further, it may be pointed out that sometimes systems may possess uncertainties due to randomness and fuzziness of the parameters. As such, here we have hybridized the concept of fuzziness as well as stochasticity to develop numerical fuzzy stochastic methods viz. interval or Fuzzy Euler Maruyama and Interval/Fuzzy Milstein. These methods are also been used to solve various diffusion problems. Numerical examples and different application problems are solved to demonstrate the efficiency and capabilities of the developed methods. In this respect, imprecisely defined diffusion problems such as heat conduction and conjugate heat transfer in rod, homogeneous and non-homogeneous fin and plate, along with one group, multi group and point kinetic neutron diffusion with interval or fuzzy uncertainties have been investigated. The convergence of the field variables have been investigated with respect to the number of element discretization of the domain in case of I/FEM. Accordingly, convergence of the proposed interval or fuzzy FEM has been studied for unsteady heat conduction in a cylindrical rod. For conjugate heat transfer problems, the convergence of uncertain temperature distributions with respect to the number of element discretizations has also been studied. Further, various combinations of uncertain parameters are considered and the sensitivity of these parameters has been reported. Next, one group and two group problems have been solved and the sensitivity of the uncertain parameters in the context of fast and thermal neutrons are presented. The hybrid fuzzy stochastic methods have also been used to investigate uncertain stochastic point kinetic neutron diffusion problem. Uncertain variation of neutron populations are analysed by considering two random samples. Developed interval or fuzzy WM has also been used to solve uncertain differential equation. Finally obtained results for the said problems are compared in special cases for the validation of proposed methods