FREE VIBRATION ISOGEOMETRIC ANALYSIS OF FRAMED STRUCTURES

Abstract

Isogeometric analysis (IGA) represents a recently developed technology in computational mechanics that offers the possibility of integrating methods for analysis and Computer Aided Design (CAD) into a single, unified process. The implications to practical engineering design scenarios are profound, since the time taken from design to analysis is greatly reduced, leading to dramatic gains in efficiency. In this study an introduction to Isogeometric finite element analysis on linear elasticity problems in 2D has been given using non uniform rational B-splines (NURBS) as basis functions. Theory of B-Splines and FEM have been studied and derived the equations needed to perform linear elasticity analysis. An Isogeometric finite element solver has been programmed in Python. The IGA solver was used to compute the free vibration frequencies of bar and beam element. Both the elements formulated are 1D elements with beam element based on Euler-Bernoulli beam theory. The free vibration frequencies of both the elements were computed in the framework of NURBS based IGA and compared with analytical frequencies. The numerical frequency matched with analytical frequency for first 40 modes for both the elements. Since the higher modes frequencies are irrelevant in context of structural engineering, the results obtained can be termed as sufficiently good for further analysis. The IGA solver was then used to analyse plate element based on Reissner-Mindlin theory of plates. Reissner-Mindlin theory was preferred instead of Kirchhoff-Love theory to account for transverse shear deformations which are necessary to be considered for analysis of thick plates. The plate element was analysed for different boundary conditions: All sides simply supported and all sides clamped, and mesh sizes: 25, 36, 49, 64, 81, 100, 121, 256, 441. The good behaviour of the method was verified and compared with analytical results. Plot of relative error percentage and the high rate of convergence in every case considered can be seen in the results. The relative percentage error reduced to 0.00896% from an initial error of 11.3% with mesh refinement for simply supported plate subjected to UDL. This reduction in error was even more drastic in case of Clamped plate subjected to UDL with error reducing from 92.12% to 1.18% for same amount of mesh refinement. The simply supported plate produced convergent solutions in fewer mesh density and without the need for selective integration. A good solver must be capable of automatically meshing the problem domain without consuming much computation power. Hence another IGA framework was prepared based on Polynomial over Hierarchical T-splines (PHT-splines) which is capable of being locally refined. NURBS based IGA is not suited for local refinement because of its global tensor product structure. Aquad tree structure was used to construct PHT-spline elements since they are better suited to track connectivity between elements across different refinement levels. For adaptivity, Zienkiewicz-Zhu error estimator was formulated which is a recovery based a-posteriori error estimator. Dörfler marking scheme was used to mark the elements to be refined after computing the error for each element at the current refinement level. To exploit the good nature of automatic adaptive refinement, a cantilever beam was analysed, because of the presence of re-entrant corners and stress concentrations. The cantilever beam is based on Euler-Bernoulli beam theory. The promised results were obtained with the convergence plot showing good convergence. A total of 12 refinement steps were needed to reduce the error in problem domain within the prescribed limits. The mesh structure as well as the contour plots of stresses are shown in the results. A cluster of elements can be seen at the re-entrant corners as expected, increasing at each refinement level until the error reduced below the limit prescribed.