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Many analysis and design problems in engineering and science involve uncertainty to varying degrees. This paper is concerned with the structural vibration problem involving uncertain material or geometric parameters, specified as fuzzy parameters. The requirement is to propagate the parameter uncertainty to the eigenvalues of the structure, specified as fuzzy eigenvalues. However, the usual approach is to transform the fuzzy problem into several interval eigenvalue problems by using the α-cuts method. Solving the interval problem as a generalized interval eigenvalue problem in interval mathematics will produce conservative bounds on the eigenvalues. The purpose of this paper is to investigate strategies to efficiently solve the fuzzy eigenvalue problem.

Based on the fundamental perturbation principle and vertex theory, an efficient perturbation method is proposed, that gives the exact extrema of the first-order deviation of the structural eigenvalue. The fuzzy eigenvalue approach has also been improved by reusing the interval analysis results from previous α-cuts.

The proposed method was demonstrated on a simple cantilever beam with a pinned support, and produced very accurate fuzzy eigenvalues. The approach was also demonstrated on the model of a highway bridge with a large number of degrees of freedom.

This proposed Vertex-Perturbation method is more efficient than the standard perturbation method, and more general than interval arithmetic methods requiring the non-negative decomposition of the mass and stiffness matrices. The new increment method produces highly accurate solutions, even when the membership function for the fuzzy eigenvalues is complex.

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.

A new‐type eigenvalue formulation of the two‐dimensional Helmholtz equation is presented in this paper. A boundary integral equation is derived using the T‐complete functions relevant to the Trefftz method, which is further transformed to the generalized eigenvalue problem. Boundary discretization and a standard eigenvalue computation routine, offered as a black box, are sufficient for the determination of the eigenvalues. The proposed method can reduce the users’ task in preprocessing and initial rough estimation when compared with the existing domain‐type solvers.

In structural mechanics, systems with damping factor get converted to nonlinear eigenvalue problems (NEPs), namely, quadratic eigenvalue problems. Generally, the parameters of NEPs are considered as crisp values but because of errors in measurement, observation or maintenance-induced errors, the parameters may have uncertain bounds of values, and such uncertain bounds may be considered in terms of closed intervals. As such, this paper aims to deal with solving nonlinear interval eigenvalue problems (NIEPs) with respect to damped spring-mass systems having interval parameters.

Two methods, namely, linear sufficient regularity perturbation (LSRP) and direct sufficient regularity perturbation (DSRP), have been proposed for solving NIEPs based on sufficient regularity perturbation method for intervals. LSRP may be used for solving NIEPs by linearizing the eigenvalue problems into generalized interval eigenvalue problems, and DSRP may be considered as a direct solution procedure for solving NIEPs.

LSRP and DSRP methods help in computing the lower and upper eigenvalue and eigenvector bounds for NIEPs which contain the crisp eigenvalues. Further, the DSRP method is computationally efficient compared to LSRP.

The efficiency of the proposed methods has been validated by example problems of NIEPs. Moreover, the procedures may be extended for other nonlinear interval eigenvalue application problems.

The purpose of this paper is to present a numerically adaptive finite element (FE) method for accurate, efficient and reliable eigensolutions of regular second- and fourth-order Sturm–Liouville (SL) problems with variable coefficients.

After the conventional FE solution for an eigenpair (i.e. eigenvalue and eigenfunction) of a particular order has been obtained on a given mesh, a novel strategy is introduced, in which the FE solution of the eigenproblem is equivalently viewed as the FE solution of an associated linear problem. This strategy allows the element energy projection (EEP) technique for linear problems to calculate the super-convergent FE solutions for eigenfunctions anywhere on any element. These EEP super-convergent solutions are used to estimate the FE solution errors and to guide mesh refinements, until the accuracy matches user-preset error tolerance on both eigenvalues and eigenfunctions.

Numerical results for a number of representative and challenging SL problems are presented to demonstrate the effectiveness, efficiency, accuracy and reliability of the proposed method.

The method is limited to regular SL problems, but it can also solve some singular SL problems in an indirect way.

Comprehensive utilization of the EEP technique yields a simple, efficient and reliable adaptive FE procedure that finds sufficiently fine meshes for preset error tolerances on eigenvalues and eigenfunctions to be achieved, even on problems which proved troublesome to competing methods. The method can readily be extended to vector SL problems.

The purpose of this paper is to propose an effective and efficient numerical method that can consider natural frequency in multi-material topology optimization (MMTO) and which is scalable for complex three-dimensional (3D) problems.

The optimization algorithm is developed by combining custom FORTRAN code for MMTO with the open-source software Mystran, which is used as a finite element analysis (FEA) solver. The proposed algorithm allows the designer to shift the fundamental frequency of the design beyond a defined frequency spectrum from the initial designing phase. The methodology is formulated in a smooth and differentiable manner, with the sensitivity expressions, required by gradient-based optimization solvers, presented.

Natural frequency constraint has been successfully implemented into MMTO. The use of open-source software Mystran as an FEA solver in the algorithm provides ability to solve complex problems. Mystran offers powerful built-in functions for eigenvalue extraction using methods like Givens, modified Givens, inverse power and the Lanczos method, which provide the ability to solve complex models. The algorithm is successfully able to solve both two- and three-material MMTO jobs for two-dimensional and 3D geometries.

Natural frequency constraint consideration into topology optimization is very challenging due to three common issues: localized eigenmodes, mode switching and high computational cost. The proposed algorithm addresses these inherent issues, implements natural frequency constraint to MMTO and solves for complex models, which is hardly possible using conventional methods.

A symmetric Craig‐Bampton method of coupled fluid‐structure systems is developed. The reduced matrices of the substructure have the same features as those of conventional substructures. Thus, standard procedures can be applied to assemble and solve the equations of the complete system. To solve the eigenvalue problem related to the substructure a special subspace iteration procedure is presented. The method can be applied to fluid‐structure systems with compressible or incompressible fluids. Constant pressure modes can be dealt with by a convenient separation technique.

The purpose of the article is to construct a new class of higher-order iterative techniques for solving scalar nonlinear problems.

The scheme is generalized by using the power-mean notion. By applying Neville's interpolating technique, the methods are formulated into the derivative-free approaches. Further, to enhance the computational efficiency, the developed iterative methods have been extended to the methods with memory, with the aid of the self-accelerating parameter.

It is found that the presented family is optimal in terms of Kung and Traub conjecture as it evaluates only five functions in each iteration and attains convergence order sixteen. The proposed family is examined on some practical problems by modeling into nonlinear equations, such as chemical equilibrium problems, beam positioning problems, eigenvalue problems and fractional conversion in a chemical reactor. The obtained results confirm that the developed scheme works more adequately as compared to the existing methods from the literature. Furthermore, the basins of attraction of the different methods have been included to check the convergence in the complex plane.

The presented experiments show that the developed schemes are of great benefit to implement on real-life problems.

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.

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.

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.

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.

The purpose of this paper is to develop a discrete element (DE) and multiscale modeling methodology to represent granular media at their particle scale as they interface solid deformable bodies, such as soil‐tool, tire, penetrometer, pile, etc., interfaces.

A three‐dimensional ellipsoidal discrete element method (DEM) is developed to more physically represent particle shape in granular media while retaining the efficiency of smooth contact interface conditions for computation. DE coupling to finite element (FE) facets is presented to demonstrate initially the development of overlapping bridging scale methods for concurrent multiscale modeling of granular media.

A closed‐form solution of ellipsoidal particle contact resolution and stiffness is presented and demonstrated for two particle, and many particle contact simulations, during gravity deposition, and quasi‐static oedometer, triaxial compression, and pile penetration. The DE‐FE facet coupling demonstrates the potential to alleviate artificial boundary effects in the shear deformation region between DEM granular media and deformable solid bodies.

The research is being extended to couple more robustly the ellipsoidal DEM code and a higher order continuum FE code via overlapping bridging scale methods, in order to remove dependence of penetration/shear resistance on the boundary placement for DE simulation.

When concurrent multiscale computational modeling of interface conditions between deformable solid bodies and granular materials reaches maturity, modelers will be able to simulate the mechanical behavior accounting for physical particle sizes and flow in the interface region, and thus design their tool, tire, penetrometer, or pile accordingly.

A closed‐form solution for ellipsoidal particle contact is demonstrated in this paper, and the ability to couple DE to FE facets.