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 This thesis presented a finite element method based on Chebyshev polynomials (CFE) for the analysis of plates, shell and fluid problems. Chebyshev polynomials are sequence of orthogonal polynomials that are defined recursively. The values of the polynomials belong to the interval [-1,1] and vanish at the Gauss points. Therefore, high-order shape functions satisfied the interpolation condition at the points, can be performed with Chebyshev polynomials. The full gauss quadrature rule was used for approximation of integral calculations.For plate problems, static and free vibration analyses of thick and thin square, rhombic and circular plates, modeled by Reissner-Mindlin plate theory, subjected to different boundary conditions were carried out. Both regular and irregular meshes were considered. Analysis results show that the CFE automatically overcomes shear locking without the formation of spurious zero energy modes by increasing the order of the shape functions.For shell problems, static and free vibration analyses of cylindrical and spherical shells based on flat shell theory were considered. The obtained results show that CFE can overcome membrane locking that arises in curved structures.For fluid problems, the CFE was applied to solve the nonhomogeneous wave equation expressed in acoustic pressure. The obtained results showed the CFE is a robust and efficient method.
 Acknowledgements iAbstract iiContents iiiList of Figures vList of Tables viiNomenclature ixList of Acronyms xChapter 1: Introduction 11.1 Plate bending theory 11.2 Shell theory 31.3 Fluid mechanics 41.4 Finite element method 51.5 Shear locking, hourglass and membrane locking phenomena 61.6 Novel methods to overcome the phenomena 71.7 A finite element method based on Chebyshev polynomials 8Chapter 2: Introduction to mechanics for structures and fluid 102.1 Solid mechanics 102.2.1 Plane stress problem 132.2.2 Plate problem 142.2.3 Shell problems 152.2 Fluid mechanics 172.3 Coupling of fluid-structure 19Chapter 3: The construction of a finite element method based on Chebyshev polynomials 213.1 Chebyshev polynomials 213.2 Lagrangian approximation based on Chebyshev polynomials 243.3 Construction shape functions based on Chebyshev polynomials 253.4 Apply CFE for structural and compressible fluid analyses 273.4.1 Finite element formulation for the plane stress using CFE 273.4.2 Finite element formulation for the plate using CFE 283.4.3 Finite element formulation for the shell using CFE 303.4.4 Finite element formulation for the fluid dynamic using CFE 333.4.5 Finite element formulation for coupling of fluid-structure using CFE 33Chapter 4: Numerical results 354.1 Plate analyses 354.1.1 Static analysis of plates 354.1.2 Free vibration 454.2 Shell analyses 524.2.1 Cylindrical shell 534.2.2 Spherical shell 564.3 Fluid-structure coupling 59Chapter 5. Conclusion 635.1. Conclusion 635.2. Limitations and future works 63References 64Biography 69
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