Solution methods for eigenvalue problems in structural mechanics

Bathe, Klaus‐Jürgen; Wilson, Edward L.

doi:10.1002/nme.1620060207

Cited by 177 publications

(87 citation statements)

References 9 publications

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“…Eigenvalue problems occur frequently in problems arising in many branches of science, such as computational fluid mechanics [Cliffe et al, 1994], statistics [Rapcsak, 2004], engineering [Roger et al, 1986;Bathe and Wilson, 1973;Andy and Nair, 2005;Auckenthaler et al, 2011;Misrikhanov and Ryabchenko, 2006], quantum physics [Scott et al, 1990], and meteorologic modelling [Cullum and Willoughby, 1986]. Such problems are usually solved using numerical methods.…”

Section: Introductionmentioning

confidence: 99%

An Arnoldi-Frontal Approach for the Stability Analysis of Flows in a Collapsible Channel

Hao

Cai

Roper

et al. 2016

Int. J. Appl. Mechanics

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In this paper, we have developed a new approach based on a combination of the Arnoldi and frontal methods, which is suitable for solving large sparse asymmetric and generalized complex eigenvalue problems. The new eigensolver seeks the most unstable eigen-solution in the Krylov subspace and makes use of the efficiency of the frontal solver developed for the finite element methods. The approach is used for a stability analysis of flows in a collapsible channel and is found to significantly improve the computational efficiency compared to the traditionally used QZ solver or a standard Arnoldi method. With the new approach, we are able to validate the previous results obtained either on a much coarser mesh or estimated from unsteady simulations. New neutral stability solutions of the system are also obtained which are beyond the limit of previously used methods.

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Section: Introductionmentioning

confidence: 99%

An Arnoldi-Frontal Approach for the Stability Analysis of Flows in a Collapsible Channel

Hao

Cai

Roper

et al. 2016

Int. J. Appl. Mechanics

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“…However, as the model size increases, the solution to the eigenvalue problem becomes increasingly expensive. The symmetric, banded nature of elemental matrices makes subspace iteration (Bathe and Wilson 1972) or the Block Lanczos method (Lanczos 1950) suitable for large FE eigenvalue problems. Today, structural dynamics calculation methods are tailored to reduce the computational requirements of the FE analysis, for example see Guyan Reduction (Guyan 1965).…”

Section: Finite Element Analysismentioning

confidence: 99%

Calibration under uncertainty for finite element models of masonry monuments

Atamturktur¹,

Hemez²,

Unal³

2010

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About the Cover:The figures on the cover illustrate three aspects of the validation of a computational model developed to predict the vibration response of historic masonry monuments. Top right: The National Cathedral, Washington, DC, where vibration response of vaults is experimentally measured to assess the effect of structural damage. The measurements provide acceleration time series used to verify and validate predictions of the numerical simulation. Top left: The computational finite element model is shown to illustrate the mesh discretization of a vault. The model is analyzed to simulate the vibration response. Colors represent different material properties whose uncertainties are propagated through the calculation. Bottom: The measurement of a vibration mode shape of the structure (solid line) is compared to the prediction of the finite element model (dashed line). Crosses illustrate the uncertainty of predictions due to variability of the boundary conditions and material properties. Statistical tests are implemented to quantify the accuracy of the finite element model, given measurement variability and prediction uncertainty. Tables Table 3- Calibration Under Uncertainty for Finite List of Figures List of ABSTRACTHistorical unreinforced masonry buildings often include features such as load bearing unreinforced masonry vaults and their supporting framework of piers, fill, buttresses, and walls. The masonry vaults of such buildings are among the most vulnerable structural components and certainly among the most challenging to analyze. The versatility of finite element (FE) analyses in incorporating various constitutive laws, as well as practically all geometric configurations, has resulted in the widespread use of the FE method for the analysis of complex unreinforced masonry structures over the last three decades. However, an FE model is only as accurate as its input parameters, and there are two fundamental challenges while defining FE model input parameters: (1) material properties and (2) support conditions. The difficulties in defining these two aspects of the FE model arise from the lack of knowledge in the common engineering understanding of masonry behavior. As a result, engineers are unable to define these FE model input parameters with certainty, and, inevitably, uncertainties are introduced to the FE model.As the complexity of the building increases, as is the case for historical unreinforced masonry buildings, the errors and uncertainties in the analysis also increase. In the presence of high and numerous uncertainties originating from multiple sources, deterministic approaches in which parameters are defined as constant values assumed to be known with certainty cannot be implemented reliably. Probabilistic methods, however, provide a rigorous and rational means in treating the uncertainty present in the FE analysis of historical unreinforced masonry buildings. The way in which uncertainty in historical unreinforced masonry construction is treated is one of the novel and main contributions...

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“…4 was earlier analyzed in [3]. We analyzed the same structure in this study to demonstrate some important features of the accelerated subspace iteration method.…”

Section: Analysis Of a Building Framementioning

confidence: 99%

An accelerated subspace iteration method

1981

Computer-Aided Design

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Solution methods for eigenvalue problems in structural mechanics

Cited by 177 publications

References 9 publications

An Arnoldi-Frontal Approach for the Stability Analysis of Flows in a Collapsible Channel

An Arnoldi-Frontal Approach for the Stability Analysis of Flows in a Collapsible Channel

Calibration under uncertainty for finite element models of masonry monuments

An accelerated subspace iteration method

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