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In this paper, a superconvergent patch recovery method is proposed for superconvergent solutions of modes in the finite element post-processing stage of variable geometrical Timoshenko beams. The proposed superconvergent patch recovery method improves the solution speed and accuracy of the finite element analysis of a curved beam. The free vibration and natural frequency of the beam were considered for studying forced vibrations and structural resonance. Beam vibration mode analysis was performed for high-precision vibration mode solutions and frequency values. The proposed method can be used to compute beam vibration modes of beams with different shapes and boundary conditions as well as variable cross sections and curvatures. The purpose of this paper is to address these issues.

An adaptive method was proposed to analyse the in-plane and out-of-plane free vibrations of the variable geometrical Timoshenko beams. In the post-processing stage of the displacement-based finite element method, the superconvergent patch recovery method and high-order shape function interpolation technique were used to obtain the superconvergent solution of mode (displacement). The superconvergent solution of mode was used to estimate the error of the finite element solution of mode in the energy form under the current mesh. Furthermore, an adaptive mesh refinement was proposed by mesh subdivision to derive an optimised mesh and accurate finite element solution to meet the preset error tolerance.

The results computed using the proposed algorithm were in good agreement with those computed using other high-precision algorithms, thus validating the accuracy of the proposed algorithm for beam analysis. The numerical analysis of parabolic curved beams, beams with variable cross sections and curvatures, elliptically curved beams and circularly curved beams helped verify that the solutions of frequencies were consistent with the results obtained using other specially developed methods. The proposed method is well suited for the mesh refinement analysis of a curved beam structure for analysing the changes in high-order vibration mode. The parts where the vibration mode changed significantly were locally densified; a relatively fine mesh division was adopted that validated the reliability of the mesh optimisation processing of the proposed algorithm.

The proposed algorithm can obtain high-precision vibration solutions of variable geometrical Timoshenko beams based on more optimized and reasonable meshes than the conventional finite element method. Furthermore, it can be used for vibration problems of parabolic curved beams, beams with variable cross sections and curvatures, elliptically curved beams and circularly curved beams. The proposed algorithm can be extended for application in superconvergent computation and adaptive analysis of finite element solutions of general structures and solid deformation fields and used for adaptive analysis of more complex plates, shells and three-dimensional structures. Additionally, this method can analyse the vibration and stability of curved members with crack damage to obtain high-precision vibration modes and instability modes under damage defects.

In recent years, developments in the field of lightweight armour have been of primary importance to the defence industry. This necessity has led to many organisations adopting composite armours comprising both the traditional heavy armours and new lighter weight ceramic armours. The numerical modelling of metal based armour systems has been well documented over the years using purely continuum based methods; and also the modelling of brittle systems using discrete element methods, therefore it is the objective of this paper to demonstrate how a coupled finite and discrete element approach, can be used in the further understanding of the quantitative response of ceramic systems when subjected to dynamic loadings using a combination of adaptive continuum techniques and discrete element methods. For the class of problems encountered within the defence industry, numerical modelling has suffered from one principal weakness; for many applications the associated deformed finite element mesh can no longer provide an accurate description of the deformed material, whether this is due to large ductile deformation, or for the case of brittle materials, degradation into multiple bodies. Subsequently, two very different approaches have been developed to combat such deficiencies, namely the use of adaptive remeshing for the ductile type materials and a discrete fracture insertion scheme for the modelling of material degradation. Therefore, one of the primary objectives of this paper is to present examples demonstrating the potential benefits of explicitly coupling adaptive remeshing methods to the technique of discrete fracture insertion in order to provide an adaptive discontinuous solution strategy, which is computationally robust and efficient.

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.

An adaptive mesh refinement procedure that uses nodeless variables and quadratic interpolation functions is presented for analysing transient thermal problems. A temperature based finite element scheme with Crank‐Nicolson time marching is used to obtain the thermal solution. The strategies used for mesh adaptation, computing refinement indicators, and time marching are described. Examples in one and two dimensions are presented and comparisons are made with exact solutions. The effectiveness of this procedure for transient thermal analysis is reflected in good solution accuracy, reduction in number of elements used, and computational efficiency.

The purpose of this study is to extend a novel numerical method proposed by the first author, known as the dual mesh control domain method (DMCDM), for the solution of linear differential equations to the solution of nonlinear heat transfer and like problems in one and two dimensions.

In the DMCDM, a mesh of finite elements is used for the approximation of the variables and another mesh of control domains for the satisfaction of the governing equation. Both meshes fully cover the domain but the nodes of the finite element mesh are inside the mesh of control domains. The salient feature of the DMCDM is that the concept of duality (i.e. cause and effect) is used to impose boundary conditions. The method possesses some desirable attributes of the finite element method (FEM) and the finite volume method (FVM).

Numerical results show that he DMCDM is more accurate than the FVM for the same meshes used. Also, the DMCDM does not require the use of any ad hoc approaches that are routinely used in the FVM.

To the best of the authors’ knowledge, the idea presented in this work is original and novel that exploits the best features of the best competing methods (FEM and FVM). The concept of duality is used to apply gradient and mixed boundary conditions that FVM and its variant do not.

Numerical simulation of the single point incremental forming (SPIF) processes can be very demanding and time consuming due to the constantly changing contact conditions between the tool and the sheet surface, as well as the nonlinear material behaviour combined with non-monotonic strain paths. The purpose of this paper is to propose an adaptive remeshing technique implemented in the in-house implicit finite element code LAGAMINE, to reduce the simulation time. This remeshing technique automatically refines only a portion of the sheet mesh in vicinity of the tool, therefore following the tool motion. As a result, refined meshes are avoided and consequently the total CPU time can be drastically reduced.

SPIF is a dieless manufacturing process in which a sheet is deformed by using a tool with a spherical tip. This dieless feature makes the process appropriate for rapid-prototyping and allows for an innovative possibility to reduce overall costs for small batches, since the process can be performed in a rapid and economic way without expensive tooling. As a consequence, research interest related to SPIF process has been growing over the last years.

In this work, the proposed automatic refinement technique is applied within a reduced enhanced solid-shell framework to further improve numerical efficiency. In this sense, the use of a hexahedral finite element allows the possibility to use general 3D constitutive laws. Additionally, a direct consideration of thickness variations, double-sided contact conditions and evaluation of all components of the stress field are available with solid-shell and not with shell elements. Additionally, validations by means of benchmarks are carried out, with comparisons against experimental results.

It is worth noting that no previous work has been carried out using remeshing strategies combined with hexahedral elements in order to improve the computational efficiency resorting to an implicit scheme, which makes this work innovative. Finally, it has been shown that it is possible to perform accurate and efficient finite element simulations of SPIF process, resorting to implicit analysis and continuum elements. This is definitively a step-forward on the state-of-art in this field.

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.

The concepts of solution error and optimal mesh in adaptive finite element analysis are revisited. It is shown that the correct evaluation of the convergence rate of the error norms involved in the error measure and the optimal mesh criteria chosen are essential to avoid oscillations in the refinement process. Two mesh optimality criteria based on: (a) the equal distribution of global error, and (b) the specific error over the elements are studied and compared in detail through some examples of application.

In modeling an aircraft wing, structural idealizations are often employed in hand calculations to simplify the structural analysis. In real applications of structural design, analysis and optimization, finite element methods are used because of the complexity of the geometry, combined and complex loading conditions. The purpose of this paper is to give a comprehensive study on the effect of using different structural idealizations on the design, analysis and optimization of thin walled semi-monocoque wing structures in the preliminary design phase.

In the design part of the paper, wing structures are designed by employing two different structural idealizations that are typically used in the preliminary design phase. In the structural analysis part, finite element analysis of one of the designed wing configurations is performed using six different one and two dimensional finite element pairs which are typically used to model the sub-elements of semi-monocoque wing structures. Finally in the optimization part, wing structure is optimized for minimum weight by using finite element models which have the same six different finite element pairs used in the analysis phase.

Based on the results presented in the paper, it is concluded that with the simplified methods, preliminary sizing of the wing configurations can be performed with enough confidence as long as the simplified method based designs are also optimized iteratively, which is what is practiced in the design phase of this study.

This research aims at investigating the effect of using different one and two dimensional element pairs on the final analyzed and optimized configurations of the wing structure, and conclusions are inferred with regard to the sensitivity of the optimized wing configurations with respect to the choice of different element types in the finite element model.

The paper presents a new technique for a compatible transition from a h‐refined to a p‐refined finite element mesh. At one or more faces of particularly designed pNh‐transition elements a low order h‐discretization may be combined with a usual p‐mesh in the other parts of the elements. The pNh‐elements are conform finite elements which can be applied in an adaptive scheme controlled by a residue based error estimate. Typical applications which require strongly a local mesh refinement within a p‐finite element mesh are, e.g. the approximation of high gradients and the determination of contact areas. Numerical examples demonstrate the efficiency of the pNh‐element technique for such problems.