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This study presents a novel hp-version adaptive finite element method (FEM) to investigate the high-precision eigensolutions of the free vibration of moderately thick circular cylindrical shells, involving the issues of variable geometrical factors, such as the thickness, circumferential wave number, radius and length.

An hp-version adaptive finite element (FE) algorithm is proposed for determining the eigensolutions of the free vibration of moderately thick circular cylindrical shells via error homogenisation and higher-order interpolation. This algorithm first develops the established h-version mesh refinement method for detecting the non-uniform distributed optimised meshes, where the error estimation and element subdivision approaches based on the superconvergent patch recovery displacement method are introduced to obtain high-precision solutions. The errors in the vibration mode solutions in the global space domain are homogenised and approximately the same. Subsequently, on the refined meshes, the algorithm uses higher-order shape functions for the interpolation of trial displacement functions to reduce the errors quickly, until the solution meets a pre-specified error tolerance condition. In this algorithm, the non-uniform mesh generation and higher-order interpolation of shape functions are suitable for addressing the problem of complex frequencies and modes caused by variable structural geometries.

Numerical results are presented for moderately thick circular cylindrical shells with different geometrical factors (circumferential wave number, thickness-to-radius ratio, thickness-to-length ratio) to demonstrate the effectiveness, accuracy and reliability of the proposed method. The hp-version refinement uses fewer optimised meshes than h-version mesh refinement, and only one-step interpolation of the higher-order shape function yields the eigensolutions satisfying the accuracy requirement.

The proposed combination of methodologies provides a complete hp-version adaptive FEM for analysing the free vibration of moderately thick circular cylindrical shells. This algorithm can be extended to general eigenproblems and geometric forms of structures to solve for the frequency and mode quickly and efficiently.

This study aims to overcome the involved challenging issues and provide high-precision eigensolutions. General eigenproblems in the system of ordinary differential equations (ODEs) serve as mathematical models for vector Sturm-Liouville (SL) and free vibration problems. High-precision eigenvalue and eigenfunction solutions are crucial bases for the reliable dynamic analysis of structures. However, solutions that meet the error tolerances specified are difficult to obtain for issues such as coefficients of variable matrices, coincident and adjacent approximate eigenvalues, continuous orders of eigenpairs and varying boundary conditions.

This study presents an h-version adaptive finite element method based on the superconvergent patch recovery displacement method for eigenproblems in system of second-order ODEs. The high-order shape function interpolation technique is further introduced to acquire superconvergent solution of eigenfunction, and superconvergent solution of eigenvalue is obtained by computing the Rayleigh quotient. Superconvergent solution of eigenfunction is used to estimate the error of finite element solution in the energy norm. The mesh is then, subdivided to generate an improved mesh, based on the error.

Representative eigenproblems examples, containing typical vector SL and free vibration of beams problems involved the aforementioned challenging issues, are selected to evaluate the accuracy and reliability of the proposed method. Non-uniform refined meshes are established to suit eigenfunctions change, and numerical solutions satisfy the pre-specified error tolerance.

The proposed combination of methodologies described in the paper, leads to a powerful h-version mesh refinement algorithm for eigenproblems in system of second-order ODEs, that can be extended to other classes of applications in damage detection of multiple cracks in structures based on the high-precision eigensolutions.

Moderately thick circular cylindrical shells are widely used as supporting structures or storage cavities in structural engineering, rock engineering, and aerospace engineering. In practical engineering, shells often work with micro-cracks or defects. The existence of micro-crack damage may result in the disturbance of dynamic behaviours and even induce accidental dynamic disasters. The free vibration frequency and mode are important parameters for the dynamic performance and damage identification analysis. In particular, stiffness weakening of the local damage region leads to significant changes in the vibration mode, which makes it difficult for the mesh generated in the conventional finite element method to capture a high-precision solution of the local oscillation.

In response to the above problems, this study developed an adaptive finite element method and a crack damage characterisation method for moderately thick circular cylindrical shells. By introducing the inverse power iteration method, error estimation, and mesh subdivision refinement technique for the analysis of finite element eigenvalue problems, an adaptive computation scheme was constructed for the free vibration problem of moderately thick circular cylindrical shells with circumferential crack damage.

Based on typical numerical examples, the established adaptive finite element solution for the free vibration of moderately thick circular cylindrical shells demonstrated its suitability for solving the high-precision free vibration frequency and mode of cylindrical shell structures. The any order frequency and mode shape of cracked cylindrical shells under the conditions of different ring wave numbers, crack locations, crack depths, and multiple cracks were successfully solved. The influences of the location, depth, and number of cracks on the disturbance of dynamic behaviours were analysed.

This study can be used as a reference for the adaptive finite element solution of free vibration of moderately thick circular cylindrical shells with cracks and lays the foundation for further development of a high-performance computation method suitable for the dynamic disturbance and damage identification analysis of general cracked structures.

This paper aims to propose a new adaptive strategy for two-dimensional (2D) nonlinear finite element (FE) analysis of the minimal surface problem (MSP) based on the element energy projection (EEP) technique.

By linearizing nonlinear problems into a series of linear problems via the Newton method, the EEP technique, which is an effective and reliable point-wise super-convergent displacement recovery strategy for linear FE analysis, can be directly incorporated into the solution procedure. Accordingly, a posteriori error estimate in maximum norm was established and an adaptive 2D nonlinear FE strategy of h-version mesh refinement was developed.

Three classical known surfaces, including a singularity problem, were analysed. Moreover, an example whose analytic solution is unavailable was considered and a comparison was made between present results and those computed by the MATLAB PDE toolbox. The results show that the adaptively-generated meshes reflect the difficulties inherent in the problems and the proposed adaptive analysis can produce FE solutions satisfying the user-preset error tolerance in maximum norm with a fair adaptive convergence rate.

The EEP technique for linear FE analysis was extended to the nonlinear procedure of MSP and can be expected to apply to other 2D nonlinear problems. The employment of the maximum norm makes point-wisely error control on the sought surfaces possible and makes the proposed method distinguished from other adaptive FE analyses.

A general strategy is developed for adaptive finite element (FE) analysis of free vibration of elastic membranes based on the element energy projection (EEP) technique.

By linearizing the free vibration problem of elastic membranes into a series of linear equivalent problems, reliable a posteriori point-wise error estimator is constructed via EEP super-convergent technique. Hierarchical local mesh refinement is incorporated to better deal with tough problems.

Several classical examples were analyzed, confirming the effectiveness of the EEP-based error estimation and overall adaptive procedure equipped with a local mesh refinement scheme. The computational results show that the adaptively-generated meshes reasonably catch the difficulties inherent in the problems and the procedure yields both eigenvalues with required accuracy and mode functions satisfying user-preset error tolerance in maximum norm.

By reasonable linearization, the linear-problem-based EEP technique is successfully transferred to two-dimensional eigenproblems with local mesh refinement incorporated to effectively and flexibly deal with singularity problems. The corresponding adaptive strategy can produce both eigenvalues with required accuracy and mode functions satisfying user-preset error tolerance in maximum norm and thus can be expected to apply to other types of eigenproblems.

A knowledge‐based system for forming quadrilateral finite elements, XFORMQ, was developed at the Centre of Flexible Manufacturing Research and Development of McMaster University, Canada. It automatically forms quadrilateral elements of good quality in conjunction with existing triangular mesh generators. XFORMQ can model geometries as complicated as those handled by triangular mesh generators. It allows for pre‐specified element sizes and rapid transition of element density. The concepts of ‘layer’ and ‘polygon patterns’, which considerably simplify the mesh generation rules and ensure the quality of formed elements, are introduced. Several test cases with different degrees of difficulties were used to evaluate XFORMQ's capabilities with satisfactory results. XFORMQ has the potential of generating meshes arising from the adaptive finite element analysis with quadrilateral elements.

This study aimed to overcome the challenging issues involved in providing high-precision eigensolutions. The accurate prediction of the buckling load bearing capacity under different crack damage locations, sizes and numbers, and analysing the influence mechanism of crack damage on buckling instability have become the needs of theoretical research and engineering practice. Accordingly, a finite element method was developed and applied to solve the elastic buckling load and buckling mode of curved beams with crack damage. However, the accuracy of the solution depends on the quality of mesh, and the solution inevitably introduces errors due to mesh. Therefore, the adaptive mesh refinement method can effectively optimise the mesh distribution and obtain high-precision solutions.

For the elastic buckling of circular curved beams with cracks, the section damage defect analogy scheme of a circular arc curved beam crack was established to simulate the crack size (depth), position and number. The h-version finite element mesh adaptive analysis method of the variable section Euler–Bernoulli beam was introduced to solve the elastic buckling problem of circular arc curved beams with crack damage. The optimised mesh and high-precision buckling load and buckling mode solutions satisfying the preset error tolerance were obtained.

The results of testing typical examples show that (1) the established section damage defect analogy scheme of circular arc curved beam crack can effectively realise the simulation of crack size (depth), position and number. The solution strictly satisfies the preset error tolerance; (2) the non-uniform mesh refinement in the algorithm can be adapted to solve the arbitrary order frequencies and modes of cracked cylindrical shells under the conditions of different ring wave numbers, crack positions and crack depths; and (3) the change in the buckling mode caused by crack damage is applicable to the study of elastic buckling under various curved beam angles and crack damage distribution conditions.

This study can provide a novel strategy for the adaptive mesh refinement for finite element analysis of elastic buckling of circular arc curved beams with crack damage. The adaptive mesh refinement method established in this study is fundamentally different from the conventional finite element method which employs the user experience to densify the meshes near the crack. It can automatically and flexibly generate a set of optimised local meshes by iteratively dividing the fine mesh near the crack, which can ensure the high accuracy of the buckling loads and modes. The micro-crack in curved beams is also characterised by weakening the cross-sectional stiffness to realise the characterisation of locations, depths and distributions of multiple crack damage, which can effectively analyse the disturbance behaviour of different forms of micro-cracks on the dynamic behaviour of beams.

Gives a bibliographical review of the error estimates and adaptive finite element methods from the theoretical as well as the application point of view. The bibliography at the end contains 2,177 references to papers, conference proceedings and theses/dissertations dealing with the subjects that were published in 1990‐2000.

This study aims to propose a general and efficient adaptive strategy with local mesh refinement for two-dimensional (2D) finite element (FE) analysis based on the element energy projection (EEP) technique.

In view of the inflexibility of the existing global dimension-by-dimension (D-by-D) recovery method via EEP technique, in which displacements are recovered through element strips, an improved element D-by-D recovery strategy was proposed, which enables the EEP recovery of super-convergent displacements to be implemented mostly on a single element. Accordingly, a posteriori error estimate in maximum norm was established and an EEP-based adaptive FE strategy of h-version with local mesh refinement was developed.

Representative numerical examples, including stress concentration and singularity problems, were analyzed; the results of which show that the adaptively generated meshes reasonably reflect the local difficulties inherent in the physical problems and the proposed adaptive analysis can produce FE displacement solutions satisfying the user-specified tolerances in maximum norm with an almost optimal adaptive convergence rate.

The proposed element D-by-D recovery method is a more efficient and flexible displacement recovery method, which is implemented mostly on a single element. The EEP-based adaptive FE analysis can produce displacement solutions satisfying the specified tolerances in maximum norm with an almost optimal convergence rate and thus can be expected to apply to other 2D problems.

This study aimed to solve the engineering problem of free vibration disturbance and local mesh refinement induced by microcrack damage in circularly curved beams. The accurate identification of the crack damage depth, number and location depends on high-precision frequency and vibration mode solutions; therefore, it is critical to obtain these reliable solutions. The high-precision finite element method for the free vibration of cracked beams needs to be developed to grasp and control error information in the conventional solutions and the non-uniform mesh generation near the cracks. Moreover, the influence of multi-crack damage on the natural frequency and vibration mode of a circularly curved beam needs to be detected.

A scheme for cross-sectional damage defects in a circularly curved beam was established to simulate the depth, location and the number of multiple cracks by implementing cross-section reduction induced by microcrack damage. In addition, the h-version finite element mesh adaptive analysis method of the Timoshenko beam was developed. The superconvergent solution of the vibration mode of the cracked curved beam was obtained using the superconvergent patch recovery displacement method to determine the finite element solution. The superconvergent solution of the frequency was obtained by computing the Rayleigh quotient. The superconvergent solution of the eigenfunction was used to estimate the error of the finite element solution in the energy norm. The mesh was then subdivided to generate an improved mesh based on the error. Accordingly, the final optimised meshes and high-precision solution of natural frequency and mode shape satisfying the preset error tolerance can be obtained. Lastly, the disturbance behaviour of multi-crack damage on the vibration mode of a circularly curved beam was also studied.

Numerical results of the free vibration and damage disturbance of cracked curved beams with cracks were obtained. The influences of crack damage depth, crack damage number and crack damage distribution on the natural frequency and mode of vibration of a circularly curved beam were quantitatively analysed. Numerical examples indicate that the vibration mode and frequency of the beam would be disturbed in the region close to the crack damage, and a greater crack depth translates to a larger frequency change. For multi-crack beams, the number and distribution of cracks also affect the vibration mode and natural frequency. The adaptive method can use a relatively dense mesh near the crack to adapt to the change in the vibration mode near the crack, thus verifying the efficacy, accuracy and reliability of the method.

The proposed combination of methodologies provides an extremely robust approach for free vibration of beams with cracks. The non-uniform mesh refinement in the adaptive method can adapt to changes in the vibration mode caused by crack damage. Moreover, the proposed method can adaptively divide a relatively fine mesh at the crack, which is applied to investigating free vibration under various curved beam angles and crack damage distribution conditions. The proposed method can be extended to crack damage detection of 2D plate and shell structures and three-dimensional structures with cracks.