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Chapter 1 Introduction  1.1 Introduction  1.2 Basic procedures and categories of high-performance adaptive finite element method  1.3 Applications of adaptive finite element method in eigenproblems of beam and shell: Vibration, stability, and damage disturbance  1.4 Research aims and contents of the book  References Chapter 2 Adaptive finite element method for vector Sturm-Liouville eigenproblems  2.1 Introduction  2.2 Adaptive approach for eigenproblems in systems of ordinary differential equations  2.2.1 Eigenproblems in system of second order ordinary differential equations  2.2.2 Stop criterion and adaptive procedure  2.3 Finite element solutions of eigenpairs  2.3.1 Finite element discretisation  2.3.2 Inverse iteration for single eigenpairs  2.3.3 Subspace iteration for coincident eigenpairs  2.4 Error estimation for eigenfunctions  2.5 h-version mesh refinement  2.6 Numerical examples  2.6.1 Example 1: Coefficients of variable matrices and adjacent approximate eigenvalues  2.6.2 Example 2: Benchmark eigenproblems in SL12F  2.6.3 Example 3: Coefficients of constant matrices and coincident eigenvalues  2.6.4 Example 4: Free vibration of planar elliptic beams  2.7 Conclusions  References Chapter 3 Adaptive finite element method for vibration of non-uniform and variable curvature beams  3.1 Introduction  3.2 Partial differential governing equations for in-plane and out-of-plane free vibration of variable geometrical Timoshenko beams  3.3 Finite element discretisation  3.4 Superconvergent patch recovery of vibration modes  3.5 Error estimation and mesh refinement  3.6 Finite element mesh refinement strategy and procedure  3.7 Numerical examples  3.7.1 Example 1: In-plane vibration of a parabolic curved beam  3.7.2 Example 2: In-plane vibration of a beam with variable cross-section and curvature  3.7.3 Example 3: In-plane vibration of an elliptically curved beam  3.7.4 Example 4: Out-of-plane vibration of a parabolic curved beam  3.7.5 Example 5: Out-of-plane vibration of a circularly curved beam  3.8 Conclusions  References Chapter 4 Adaptive finite element method for vibration disturbance of cracked beams  4.1 Introduction  4.2 Characterisation method for microcrack damage in circularly curved beams  4.3 Partial differential governing equations and finite element discretisation for free vibration of circularly curved beams  4.4 Local mesh refinement techniques and procedure  4.5 Numerical examples  4.5.1 Example 1: Verification for eigensolutions of free vibration of uncracked curved beam  4.5.2 Example 2: Different depths of single-crack beams  4.5.3 Example 3: Different numbers of multiple-crack damage  4.5.4 Example 4: Different distributions of multiple-crack damage  4.6 Conclusions  References Chapter 5 Adaptive finite element method for damage detection of cracked beams  5.1 Introduction  5.2 Adaptive approach for damage detection of cracked beams  5.2.1 Formulation and analogy of cracked beams  5.2.2 Stop criterion  5.2.3 Analysis strategy  5.3 Adaptive analysis  5.3.1 Finite element solution  5.3.2 Error estimation and mesh refinement  5.4 Newton-Raphson iteration  5.5 Damage refinement  5.6 Algorithms  5.7 Numerical examples  5.7.1 Example 1: Double-clamped uncracked beam with a sinusoidal cross section  5.7.2 Example 2: Stepped cantilever beam with a single crack  5.7.3 Example 3: Cantilever beam with double cracks  5.7.4 Example 4: Cantilever beam with triple cracks  5.7.5 Example 5: Double-clamped uncracked beam with a sinusoidal cross section  5.7.6 Example 6: Stepped cantilever beam with a single crack  5.7.7 Example 7: Cantilever beam with double cracks  5.7.8 Example 8: Cantilever beam with triple cracks  5.8 Conclusions  References Chapter 6 Adaptive finite element method for stability disturbance of cracked beams  6.1 Introduction  6.2 Elastic buckling of circularly curved beams based on Euler-Bernoulli theory  6.3 Characterisation method for micro-crack damage in circularly curved beams  6.4 Local mesh refinement techniques and procedure  6.5 Numerical examples  6.5.1 Example 1: Verification for eigensolutions and refined meshes of elastic buckling of uncracked curved beam  6.5.2 Example 2: Buckling loads under different subtended angles  6.5.3 Example 3: Different locations of single crack damage  6.5.4 Example 4: Different depths of single crack damage  6.5.5 Example 5: Different distributions of multiple crack damage  6.6 Conclusions  References Chapter 7 Adaptive finite element method for vibration of cylindrical shells  7.1 Introduction  7.2 Partial differential governing equations for the free vibration of rotating cylindrical shells  7.3 Mesh refinement procedure of finite element method  7.4 Finite element discretisation  7.5 Error estimation  7.6 Element subdivision and mesh refinement  7.7 Numerical examples  7.7.1 Example 1: Thin-wall thickness  7.7.2 Example 2: Moderately thick-wall thickness  7.7.3 Example 3: Different circumferential wave numbers and thickness-to-length ratios  7.7.4 Example 4: Hinged-hinged and clamped- clamped boundary conditions  7.8 Conclusions  References Chapter 8 Improved hp-version adaptive finite element method for vibration of cylindrical shells  8.1 Introduction  8.2 Differential equations describing the free vibration of moderately thick circular cylindrical shells  8.3 Finite element discretisation and solutions of the differential equations  8.3.1 Higher-order Lagrange shape functions  8.3.2 Finite element discretisation  8.3.3 Inverse iteration technique for eigensolutions  8.4 hp-version adaptive finite element method via error homogenisation and higher-order interpolation  8.4.1 h-version mesh refinement  8.4.2 p-version higher-order interpolation  8.5 Global algorithm and procedure  8.6 Numerical examples  8.6.1 Example 1: h-version and hp-version adaptive finite element analysis  8.6.2 Example 2: Thin-walled circular cylindrical shells  8.6.3 Example 3: Circumferential wave number n  8.6.4 Example 4: Ratio of thickness to radius h/r  8.6.5 Example 5: Ratio of thickness to length h/l  8.7 Conclusions  References Chapter 9 Adaptive finite element method for vibration disturbance of cracked cylindrical shells  9.1 Introduction  9.2 Differential equations describing the free vibration of moderately thick circular cylindrical shell  9.3 Damage characterisation method for circumferential cracks in circular cylindrical shell  9.4 h-version mesh refinement method for eigensolutions of cracked circular cylindrical shell  9.4.1 Finite element solutions  9.4.2 Error estimation  9.4.3 Element subdivision and refinement  9.5 Global algorithm and procedure  9.6 Numerical examples  9.6.1 Example 1: Benchmarks for free vibration of circular cylindrical shell  9.6.2 Example 2: Verification of frequency solutions of cracked circular cylindrical shell under variable circumferential wave numbers  9.6.3 Example 3: Free vibration disturbance by crack location  9.6.4 Example 4: Free vibration disturbance by crack depth  9.6.5 Example 5: Free vibration disturbance by number of multiple cracks  9.7 Conclusions  References Chapter 10 Summary and prospect  10.1 Summary  10.2 Prospect