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This paper presents a new approach for estimating the discretization error of finite element analysis of generalized eigenproblems. The method uses smoothed gradients at nodal points to derive improved element‐by‐element interpolation functions. The improved interpolation functions and their gradients are used in the Rayleigh quotient to obtain an improved eigenvalue. The improved eigenvalue is used to estimate the error of the original solution. The proposed method does not require any re‐solution of the eigenproblem. Results for 1‐D and 2‐D C° eigenproblems in acoustics and elastic vibrations are used as examples to demonstrate the accuracy of the proposed method.

The purpose of this paper is to detect the degenerate scale of a 2D bending plate analytically and numerically.

To avoid the time-consuming scheme, the influence matrix of the boundary element method (BEM) is reformulated to an eigenproblem of the 4 by 4 matrix by using the scaling transform instead of the direct-searching scheme to find degenerate scales. Analytical degenerate scales are derived from the boundary integral equation (BIE) by using the degenerate kernel only for the circular case. Numerical results of the direct-searching scheme and the eigen system for the arbitrary shape are also considered.

Results using three methods, namely, analytical derivation, the direct-searching scheme and the 4 by 4 eigen system, are also given for the circular case and arbitrary shapes. Finally, addition of a constant for the kernel function makes original eigenvalues (2 real roots and 2 complex roots) of the 4 by 4 matrix to be all real. This indicates that a degenerate scale depends on the kernel function.

The analytical derivation for the degenerate scale of a 2D bending plate in the BIE is first studied by using the degenerate kernel. Through the reformed eigenproblem of a 4 by 4 matrix, the numerical solution for the plate of an arbitrary shape can be used in the plate analysis using the BEM.

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.

This study aims to develop an adaptive finite element method for structural eigenproblems of cracked Euler–Bernoulli beams via the superconvergent patch recovery displacement technique. This research comprises the numerical algorithm and experimental results for free vibration problems (forward eigenproblems) and damage detection problems (inverse eigenproblems).

The weakened properties analogy is used to describe cracks in this model. The adaptive strategy proposed in this paper provides accurate, efficient and reliable eigensolutions of frequency and mode (i.e. eigenpairs as eigenvalue and eigenfunction) for Euler–Bernoulli beams with multiple cracks. Based on the frequency measurement method for damage detection, using the difference between the actual and computed frequencies of cracked beams, the inverse eigenproblems are solved iteratively for identifying the residuals of locations and sizes of the cracks by the Newton–Raphson iteration technique. In the crack detection, the estimated residuals are added to obtain reliable results, which is an iteration process that will be expedited by more accurate frequency solutions based on the proposed method for free vibration problems.

Numerical results are presented for free vibration problems and damage detection problems of representative non-uniform and geometrically stepped Euler–Bernoulli beams with multiple cracks to demonstrate the effectiveness, efficiency, accuracy and reliability of the proposed method.

The proposed combination of methodologies described in the paper leads to a very powerful approach for free vibration and damage detection of beams with cracks, introducing the mesh refinement, that can be extended to deal with the damage detection of frame structures.

To suggest a polynomial complexity method for determining the range of real eigenvalues in the case of the generalized eigenvalue problem when the elements of the matrices involved are independent intervals.

The basic approach is to make use of approximate interval solutions as regards the right and left eigenvectors of the eigenproblem considered, the so‐called outer solutions, in order to determine the range.

First, a new method for computing the outer solutions has been suggested. The main result of the paper, however, is the development of a simple method for determining the range of the real eigenvalues. Unlike the known general‐purpose methods that have exponential complexity, the present range determination method is much simpler as its complexity is only polynomial.

The method is applicable if certain sufficient conditions reported in the paper are satisfied (an incomplete quadratic system is to have a positive solution and the signs of the outer solutions should satisfy a complete or partial invariance).

The method guarantees reliable numerical results when the original eigenproblems contain interval uncertainties as is, strictly speaking, most often the case in practice.

To the best of the author's knowledge, the present paper suggests, for the first time, a simple method of polynomial complexity for solving the problem considered which is inherently a NP‐hard problem (of exponential complexity).

This article deals with improvement of eigenvalues obtained by finite element analysis of C¹ eigenproblems. The proposed method employs high order gradient smoothing at nodal points to derive improved high order interpolation functions for the single element of each mode. Two different schemes were developed for 1–D C¹ eigenproblems (free vibration of beams) and for 2–D quasi C¹ eigenproblems (transverse vibrations of thin plates). High order Hermitian polynomials are used for the beam problem together with some boundary node corrections, while a combination of high‐order and low‐order approximations are used for the modified formulation of the plate problem. Several smoothing options are proposed for both schemes. Numerical results for both schemes are used as examples to demonstrate the accuracy of the present approach.

The computational efficiency of subspace iteration is addressed relative to the data structures adopted for the very large and generally sparse coefficient matrices. The frequent triangulations and matrix multiplications demand that access to the terms in the coefficient matrices be unbiased. Reliance on virtual memory (paging) operating systems with no special considerations for localized data access is not adequate. Specific data structures must be designed that accommodate the needs of the numerical algorithm yet eliminate unnecessary paging. An implementation of the subspace iteration method using hypermatrix data structures is presented. Use of hypermatrices is shown to provide unbiased and localized data access. The various modifications to the conventional formulation are described and an example problem illustrates the potential benefits of the hypermatrix formulation. Possibilities for adapting hypermatrix data structures to new supercomputer architectures are discussed.

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.

Nodal ordering for the formation of well‐structures stiffness matrices are often performed using graph theory and algebraic graph theory. The purpose of this paper is to present a new method for nodal ordering for profile optimization of finite element models.

In the present method, a combination of graph theory and differential equations is employed. The proposed method transforms the eigenvalue problem involved in optimal ordering of algebraic graph method into a specific initial value problem of an ordinary differential equation.

The transformation of this paper enables many advanced numerical methods for ordinary differential equations to be used in the computation of the eigenproblems.

Combining two different tools, namely graph theory and differential equations, results in a more efficient and accurate method for nodal ordering problem, which is a combinatorial optimization problem. Examples are included to illustrate the efficiency of the present method.

Domain decomposition of finite element models (FEM) for parallel computing are often performed using graph theory and algebraic graph theory. This paper aims to present a new method for such decomposition, where a combination of algebraic graph theory and differential equations is employed.

In the present method, a combination of graph theory and differential equations is employed. The proposed method transforms the eigenvalue problem involved in decomposing FEM by the algebraic graph method, into a specific initial value problem of an ordinary differential equation.

The transformation of this paper enables many advanced numerical methods for ordinary differential equations to be used in the computation of the eigenproblems.

Combining two different tools, namely algebraic graph theory and differential equations, results in an efficient and accurate method for decomposing the FEM which is a combinatorial optimization problem. Examples are included to illustrate the efficiency of the present method.