Chih-Hao Chen and Siva Nadarajah
This paper aims to present a dynamically adjusted deflated restarting procedure for the generalized conjugate residual method with an inner orthogonalization (GCRO) method.
Abstract
Purpose
This paper aims to present a dynamically adjusted deflated restarting procedure for the generalized conjugate residual method with an inner orthogonalization (GCRO) method.
Design/methodology/approach
The proposed method uses a GCR solver for the outer iteration and the generalized minimal residual (GMRES) with deflated restarting in the inner iteration. Approximate eigenpairs are evaluated at the end of each inner GMRES restart cycle. The approach determines the number of vectors to be deflated from the spectrum based on the number of negative Ritz values, k∗.
Findings
The authors show that the approach restores convergence to cases where GMRES with restart failed and compare the approach against standard GMRES with restarts and deflated restarting. Efficiency is demonstrated for a 2D NACA 0012 airfoil and a 3D common research model wing. In addition, numerical experiments confirm the scalability of the solver.
Originality/value
This paper proposes an extension of dynamic deflated restarting into the traditional GCRO method to improve convergence performance with a significant reduction in the memory usage. The novel deflation strategy involves selecting the number of deflated vectors per restart cycle based on the number of negative harmonic Ritz eigenpairs and defaulting to standard restarted GMRES within the inner loop if none, and restricts the deflated vectors to the smallest eigenvalues present in the modified Hessenberg matrix.
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Mingwu Yuan, Pu Chen, Shanji Xiong, Yuanneng Li and Edward L. Wilson
The advantages of a direct superposition of the Ritz vector in dynamic response analysis (developed by Wilson, Yuan, and Dickens in 1982 and termed the WYD method) are that: no…
Abstract
The advantages of a direct superposition of the Ritz vector in dynamic response analysis (developed by Wilson, Yuan, and Dickens in 1982 and termed the WYD method) are that: no iteration is involved; the method is at least four times faster than the subspace iteration method; and fewer Ritz vectors are necessary for the mode superposition of dynamic response analysis than exact eigenvectors are used. The major purpose of this paper is to illustrate that the WYD method can also be used as a general approximate algorithm to calculate eigenvalues and eigenvectors. The WYD and Lanczos algorithms are very similar and a formula that relates the two is given in this paper. Although the exact algebraic value of only a single eigenvector of a multi‐eigenvalue can be calculated using either the WYD or Lanczos methods, an artificial round‐off is presented that can be used to solve the eigenvalue problem. A method of estimating the error introduced by the WYD method is also developed. A dynamic substructuring technique, based on the WYD method, and which assumes that the connectivities on the interfaces among the substructures need not be considered is also presented.
P. Léger and E.L. Wilson
The evaluation of linear dynamic response analysis of large structures by vector superposition requires, in its traditional formulation, the solution of a large and expensive…
Abstract
The evaluation of linear dynamic response analysis of large structures by vector superposition requires, in its traditional formulation, the solution of a large and expensive eigenvalue problem. A method of solution based on a Ritz transformation to a reduced system of generalized coordinates using load dependent vectors generated from the spatial distribution of the dynamic loads is shown to maintain the high expected accuracy of modern computer analysis and significantly reduces the execution time over eigensolution procedures. New computational variants to generate load dependent vectors are presented and error norms are developed to control the convergence characteristics of load dependent Ritz solutions. Numerical applications on simple structural systems are used to show the relative efficiency of the proposed solution procedures.
MINGWU YUAN, SHANJI XIONG and XIAOHONG CHEN
An exact multiple‐level dynamic substructure technique was developed by a combination of WYD algorithm and static multiple‐level substructuring technique. This method is…
Abstract
An exact multiple‐level dynamic substructure technique was developed by a combination of WYD algorithm and static multiple‐level substructuring technique. This method is essentially different from the traditional mode component synthesis. The eigenvalues and eigenvectors created by the method are the eigenpairs for the whole structure and not for the components of structure. On the other hand, the dynamic response by using mode superposition can also be implemented in substructure level. This algorithm actually is an exact substructuring technique which means that substructuring itself did not introduce any additional error except the round‐off when a structure was split into some arbitrary subdomains and the error of WYD or mode superposition themselves. It is no longer necessary to assume any connective condition on the interface between substructures. This method makes the capacity of dynamic analysis of a structural analysis program unlimited. It is especially attractive for the programs on microcomputers. Of course, the method leads to a frequent I/O for a subsequent search of the files from each substructure. It is time consuming compared to the mode component synthesis. But the potential still exists to improve the efficiency by using parallel computation on concurrent computers. In this paper the theory and procedure of the algorithm are presented.
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Wujiu Pan, Heng Ma, Jian Li, Qilong Wu, Junyi Wang, Jianwen Bao, Lele Sun and Peng Gao
Aero-engine casings commonly use composite cylindrical shell structures with excellent properties such as corrosion resistance and fatigue resistance. Still, their vibration…
Abstract
Purpose
Aero-engine casings commonly use composite cylindrical shell structures with excellent properties such as corrosion resistance and fatigue resistance. Still, their vibration behavior is relatively complex and may cause fatigue vibration damage, so it is essential to analyze the vibration characteristics of composite cylindrical shells. The purpose of this paper is to analyze the vibration characteristics of multilayer composite cylindrical shells subjected to external pressures and having different interlayer thickness ratios and provide some theoretical basis for the fatigue damage prediction of cylindrical shell casing to ensure the safety and stability of the engine during flight.
Design/methodology/approach
Firstly, the vibration differential equation with external pressure is established based on Soedel theory considering nonlinear effects, while four symmetric boundary conditions are chosen to constrain the cylindrical shell. Then the Rayleigh–Ritz method, which is more efficient and accurate in calculating large structural systems, is applied to solve the problem, and the theoretical model of three-layer cylindrical shell under external pressure is established. The accuracy of the model is verified by comparing the data with the specialized literature. Subsequently, the effects of different external pressures and different thickness-to-diameter ratios, different length-to-diameter ratios and different interlayer thickness percentages on the natural frequency of multilayer composite cylindrical shells were investigated by control variable analysis.
Findings
The conclusions obtained show that the external pressure increases the natural frequency of the cylindrical shell and that the frequency characteristics of the cylindrical shell vary for different boundary conditions. The effect of length-to-diameter ratio, thickness-to-diameter ratio and the percentage of the thickness of the intermediate layer on the natural frequency of the cylindrical shell are significantly increased under external pressure. Because the presence of external pressure increases the frequency of the cylindrical shell by about 70%, it has almost no effect on the frequency at the minimum number of circumferential waves, and the effect on the frequency at the maximum number of circumferential waves is reduced to about 50%. The frequencies in the SL-SL boundary condition are all in perfect agreement with the S-S boundary condition under the influence of different influencing factors.
Originality/value
In this paper, the effect of external pressure and the natural properties of the cylindrical shell under external pressure on the cylindrical shell’s frequency is considered, emphasizing the effect of different layer thickness ratios on the frequency. This paper aims to summarize the changing law between the natural frequency of the cylindrical shell itself and different design parameters during the flight pressure process. Reliable theoretical predictions are provided for analyzing the vibrational behavior of shells subjected to external pressures in aerospace, as well as a database for the practical production of cylindrical shells.
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A model for a symmetric three‐layer configuration is developed. This model refers to Cu/Invar/Cu (CIC) laminates. Calculated values for the coefficient of thermal expansion (CTE…
Abstract
A model for a symmetric three‐layer configuration is developed. This model refers to Cu/Invar/Cu (CIC) laminates. Calculated values for the coefficient of thermal expansion (CTE) are compared with literature values. The model is then extended to symmetric CIC‐Metalcore boards and its prediction is compared with experimental results. Shearing of FR‐4 is discussed.
Dragan D. Milašinović, Petar Marić, Žarko Živanov and Miroslav Hajduković
The problems of inelastic instability (buckling) and dynamic instability (resonance) have been the subject of extensive investigation and have received wide attention from the…
Abstract
Purpose
The problems of inelastic instability (buckling) and dynamic instability (resonance) have been the subject of extensive investigation and have received wide attention from the structural mechanics community. This paper aims to tackle these problems in thin-walled structures, taking into account geometrical and/or material non-linearity.
Design/methodology/approach
The inelastic buckling mode interactions and resonance instabilities of prismatic thin-walled columns are analysed by implementing the semi-analytical finite strip method (FSM). A scalar damage parameter is implemented in conjunction with a material modelling named rheological-dynamical analogy to address stiffness reduction induced by the fatigue damage.
Findings
Inelastic buckling stresses lag behind the elastic buckling stresses across all modes, which is a consequence of the viscoelastic behaviour of materials. Because of the lag, the same column length does not always correspond to the same mode at the elastic and inelastic critical stress.
Originality/value
This paper presents the influence of mode interactions on the effective stresses and resonance instabilities in thin-walled columns due to the fatigue damage. These mode interactions have a great influence on damage variables because of the fatigue and effective stresses around mode transitions. In its usual semi-analytical form, the FSM cannot be used to solve the mode interaction problem explained in this paper, because this technique ignores the important influence of interaction of the buckling modes when applied only for undamaged state of structure
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A method is shown which allows evaluation of production tolerances for high speed PCBs (microstrip, stripline) in order to maintain the impedance Z within a specified range.
Xuanhua Fan, Keying Wang and Shifu Xiao
As a practical engineering method, earthquake response spectra play an important role in seismic hazard assessment and in seismic design of structures. However, the computing…
Abstract
Purpose
As a practical engineering method, earthquake response spectra play an important role in seismic hazard assessment and in seismic design of structures. However, the computing scale and the efficiency of commercial software restricted the solution of complex structures. There is a clear need of developing large-scale and highly efficient finite element procedures for response spectrum analysis.
Design/methodology/approach
In this paper, the kernel theories for earthquake response spectra are deduced and the corresponding parallel solution flow via the modal superposition method is presented. Based on the algorithm and the parallel data structure of JAUMIN framework, a parallel finite element (FE) solution module is established. Using the solution procedure on a supercomputer equipped with up to thousands of processors, the correctness and parallel scalability of the algorithm are evaluated via numerical experiments of typical engineering examples.
Findings
The results show that the solution module has the same precision as the commercial FE software ANSYS; the maximum solution scale achieves 154 million degrees of freedom (DOFs) with a favorable parallel computing efficiency, going far beyond the computing ability of the commercial FE software.
Originality/value
The solution scale in this paper is very challenging for the large-scale parallel computing of structural dynamics and will promote the dynamic analysis ability of complex facilities greatly.
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Successful teams tend to be highly cohesive and team cohesion to be particularly helpful in allowing teams and their members to sustain their success even in the most challenging…
Abstract
Successful teams tend to be highly cohesive and team cohesion to be particularly helpful in allowing teams and their members to sustain their success even in the most challenging times. One disillusioning consequence of this reciprocity between cohesion and performance would suggest that failures made by teams and/or their members likely jeopardize their success by preventing them from capitalizing on such virtuous circles associated with team cohesion. Yet, many teams uphold their performance despite the failures they have to cope with, suggesting that the potential vicious circles can be overcome. This chapter aims at illuminating the vicious and virtuous circles associated with team cohesion that are induced by either collective failures of teams or individual failures of their members. It therefore offers a multilevel perspective not only on the emergence and diffusion of failures at the individual and team levels, but also on the critical role that team cohesion plays for a team’s (dys)functional coping across these levels. It is theorized that collective failures triggered exogenously can help build team cohesion, and that whether endogenously-triggered collective failures bring about the vicious or the virtuous circles of team cohesion depends on whether the individual failures developing into collective failures are triggered endogenously or exogenously. The implications of this conceptual work are discussed in light of the literatures on error/failure management and group cohesiveness.