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The purpose of this paper is devoted to present an accurate assessment for determine natural frequencies for uniform and non-uniform Euler-Bernoulli beams and frames by an adaptive generalized finite element method (GFEM). The present paper concentrates on developing the C1 element of the adaptive GFEM for vibration analysis of Euler-Bernoulli beams and frames.

The variational problem of free vibration is formulated and the main aspects of the adaptive GFEM are presented and discussed. The efficiency and convergence of the proposed method in vibration analysis of uniform and non-uniform Euler-Bernoulli beams are checked. The application of this technique in a frame is also presented.

The present paper concentrates on developing the C1 element of the adaptive GFEM for vibration analysis of Euler-Bernoulli beams and frames. The GFEM, which was conceived on the basis of the partition of unity method, allows the inclusion of enrichment functions that contain a priori knowledge about the fundamental solution of the governing differential equation. The proposed enrichment functions are dependent on the geometric and mechanical properties of the element. This approach converges very fast and is able to approximate the frequency related to any vibration mode.

The main contribution of the present study consisted in proposing an adaptive GFEM for vibration analysis of Euler-Bernoulli uniform and non-uniform beams and frames. The GFEM results were compared with those obtained by the h and p-versions of FEM and the c-version of the CEM. The adaptive GFEM has shown to be efficient in the vibration analysis of beams and has indicated that it can be applied even for a coarse discretization scheme in complex practical problems.

This paper aims to propose a generalized finite element technique that can accurately approximate the solution of the flexural-shear cantilever model of wall-frame structures proposed by Heidebrecht and Stafford Smith.

This approach adopts scaled monomials as enrichment functions, and they are highly effective in accurately capturing the solution of the problem, as it consists of smooth functions such as polynomials, hyperbolic and trigonometric functions. Several numerical experiments are performed on the static and modal analyses of the flexural-shear cantilever wall-frame structures using the proposed generalized finite element method (GFEM), and their accuracies are compared with those obtained using the standard finite element method.

The proposed GFEM is able to achieve theoretical convergence rates of the static and modal analyses, which are, in principle, identical to those of the standard FEM, for various polynomial orders of its shape functions such as quadratic, cubic and quartic orders. The proposed GFEM with quartic enrichment functions can provide more accurate solutions than the standard FEM, and thus can be effectively used at the initial design stage of high-rise wall-frame structures.

This work is the first paper where the GFEM is applied to the analysis of high-rise wall-frame structures, and the developed technique can be used as a good analysis tool at the initial design stage.

The purpose of this paper is to evaluate some numerical integration strategies used in generalized (G)/extended finite element method (XFEM) to solve linear elastic fracture mechanics problems. A range of parameters are here analyzed, evidencing how the numerical integration error and the computational efficiency are improved when particularities from these examples are properly considered.

Numerical integration strategies were implemented in an existing computational environment that provides a finite element method and G/XFEM tools. The main parameters of the analysis are considered and the performance using such strategies is compared with standard integration results.

Known numerical integration strategies suitable for fracture mechanics analysis are studied and implemented. Results from different crack configurations are presented and discussed, highlighting the necessity of alternative integration techniques for problems with singularities and/or discontinuities.

This study presents a variety of fracture mechanics examples solved by G/XFEM in which the use of standard numerical integration with Gauss quadratures results in loss of precision. It is discussed the behaviour of subdivision of elements and mapping of integration points strategies for a range of meshes and cracks geometries, also featuring distorted elements and how they affect strain energy and stress intensity factors evaluation for both strategies.

The purpose of this paper is to evaluate the capabilities of the generalized finite element method (GFEM) under the context of the geometrically nonlinear analysis. The effect of large displacements and deformations, typical of such analysis, induces a significant distortion of the element mesh, penalizing the quality of the standard finite element method approximation. The main concern here is to identify how the enrichment strategy from GFEM, that usually makes this method less susceptible to the mesh distortion, may be used under the total and updated Lagrangian formulations.

An existing computational environment that allows linear and nonlinear analysis, has been used to implement the analysis with geometric nonlinearity by GFEM, using different polynomial enrichments.

The geometrically nonlinear analysis using total and updated Lagrangian formulations are considered in GFEM. Classical problems are numerically simulated and the accuracy and robustness of the GFEM are highlighted.

This study shows a novel study about GFEM analysis using a complete polynomial space to enrich the approximation of the geometrically nonlinear analysis adopting the total and updated Lagrangian formulations. This strategy guarantees the good precision of the analysis for higher level of mesh distortion in the case of the total Lagrangian formulation. On the other hand, in the updated Lagrangian approach, the need of updating the degrees of freedom during the incremental and iterative solution are for the first time identified and discussed here.

This paper aims to present a computational framework to generate numeric enrichment functions for two-dimensional problems dealing with single/multiple local phenomenon/phenomena. The two-scale generalized/extended finite element method (G/XFEM) approach used here is based on the solution decomposition, having global- and local-scale components. This strategy allows the use of a coarse mesh even when the problem produces complex local phenomena. For this purpose, local problems can be defined where these local phenomena are observed and are solved separately by using fine meshes. The results of the local problems are used to enrich the global one improving the approximate solution.

The implementation of the two-scale G/XFEM formulation follows the object-oriented approach presented by the authors in a previous work, where it is possible to combine different kinds of elements and analyses models with the partition of unity enrichment scheme. Beside the extension of the G/XFEM implementation to enclose the global–local strategy, the imposition of different boundary conditions is also generalized.

The generalization done for boundary conditions is very important, as the global–local approach relies on the boundary information transferring process between the two scales of the analysis. The flexibility for the numerical analysis of the proposed framework is illustrated by several examples. Different analysis models, element formulations and enrichment functions are used, and the accuracy, robustness and computational efficiency are demonstrated.

This work shows a generalize imposition of different boundary conditions for global–local G/XFEM analysis through an object-oriented implementation. This generalization is very important, as the global–local approach relies on the boundary information transferring process between the two scales of the analysis. Also, solving multiple local problems simultaneously and solving plate problems using global–local G/XFEM are other contributions of this work.

This paper aims to present an adaptive approach of the generalized finite element method (GFEM) for transient dynamic analysis of bars and trusses. The adaptive GFEM, previously proposed for free vibration analysis, is used with the modal superposition method to obtain precise time-history responses.

The adaptive GFEM is applied to the transient analysis of bars and trusses. To increase the precision of the results and computational efficiency, the modal matrix is responsible for the decoupling of the dynamic equilibrium equations in the modal superposition method, which is used with only the presence of the problem’s most preponderant modes of vibration. These modes of vibration are identified by a proposed coefficient capable of indicating the influence of each mode on the transient response.

The proposed approach leads to more accurate results of displacement, velocity and acceleration when compared to the traditional finite element method.

In this paper, the application of the adaptive GFEM to the transient analysis of bars and trusses is presented for the first time. A methodology of identification of the preponderant modes to be retained in the modal matrix is proposed to improve the quality of the solution. The examples showed that the method has a strong potential to solve dynamic analysis problems, as the approach had already proved to be efficient in the modal analysis of different framed structures. A simple way to perform h-refinement of truss elements to obtain reference solutions for dynamic problems is also proposed.

The purpose of this paper is to present a homogenization approach to model mechanical structures with multiple scales and periodicity, as they occur, e.g. in power transformer windings, subjected to magnetic forces.

The idea is based on the framework of generalized finite element methods (GFEM), where the normal polynomial finite element basis functions are enriched by problem dependent basis functions, which are, in this case, the eigenmodes of a quasi‐periodic unit cell setup. These eigenmodes are used to enrich the standard polynomial basis functions of higher order on a coarse grid modeling the whole periodic structure.

It is shown that heterogeneous magnetomechanical structures can be homogenized with the developed method, as demonstrated by homogenization of a transformer coil setup.

An efficient homogenization procedure is proposed on the basis of the GFEM, which is extended using a special set of enrichment functions, i.e. the mechanic eigenmodes of a generalized eigenvalue problem.

The purpose of this paper is to present an application of the Generalized Finite Element Method (GFEM) for modal analysis of 2D wave equation.

The GFEM can be viewed as an extension of the standard Finite Element Method (FEM) that allows non-polynomial enrichment of the approximation space. In this paper the authors enrich the approximation space with sine e cosine functions, since these functions frequently appear in the analytical solution of the problem under study. The results are compared with the ones obtained with the polynomial FEM using higher order elements.

The results indicate that the proposed approach is able to obtain more accurate results for higher vibration modes than standard polynomial FEM.

The examples studied in this paper indicate a strong potential of the GFEM for the approximation of higher vibration modes of structures, analysis of structures subject to high frequency excitations and other problems that concern high frequency oscillatory phenomena.

The purpose of this paper is to check the efficiency of isogeometric analysis (IGA) by comparing its results with classical finite element method (FEM), generalized finite element method (GFEM) and other enriched versions of FEM through numerical examples of free vibration problems.

Since its conception, IGA was widely applied in several problems. In this paper, IGA is applied for free vibration of elastic rods, beams and trusses. The results are compared with FEM, GFEM and the enriched methods, concerning frequency spectra and convergence rates.

The results show advantages of IGA over FEM and GFEM in the frequency error spectra, mostly in the higher frequencies.

Isogeometric analysis shows a feasible tool in structural analysis, with emphasis for problems that requires a high amount of vibration modes.

The purpose of this study is to evaluate the accuracy and numerical stability of the Generalized Finite Element Method (GFEM) for solving structural dynamic problems.

The GFEM is a numerical method based on the Partition of Unity (PU) concept. The method can be understood as an extension of the conventional Finite Element Method (FEM) for which the local approximation provided by the shape functions can be improved by means of enrichment functions. Polynomial enrichment functions are hereby used combined with an implicit time-stepping integration technique for improving the dynamical response of the models. Both consistent and lumped mass matrices techniques are tested. The method accuracy and stability are investigated through linear and nonlinear elastic problems.

The results indicate that the adopted strategies can provide stable and accurate responses for GFEM in dynamic analyses. Furthermore, the mass lumping technique provided remarkable reductions of the system of equation condition number, therefore leading to more stable numerical models.

The evaluated features of GFEM models for implicit time-stepping integration schemes represent new information of great deal of interest regarding linear and nonlinear dynamic analyses using such a method.