Marcos Arndt

GFEM for modal analysis of 2D wave equation

Purpose – The purpose of this paper is to present an application of the Generalized Finite Element Method (GFEM) for modal analysis of 2D wave equation. Design/methodology/approach – 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. Findings – The results indicate that the proposed approach is able to obtain more accurate results for higher vibration modes than standard polynomial FEM. Originality/value – 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 concer...

The Generalized Finite Element Method Applied to Free Vibration of Framed Structures

The vibration analysis is an important stage in the design of mechanical systems and buildings subject to dynamic loads like wind and earthquake. The dynamic characteristics of these structures are obtained by the free vibration analysis. The Finite Element Method (FEM) is commonly used in vibration analysis and its approximated solution can be improved using two refinement techniques: h and p-versions. The h-version consists of the refinement of element mesh; the p-version may be understood as the increase in the number of shape functions in the element domain without any change in the mesh. The conventional p-version of FEM consists of increasing the polynomial degree in the solution. The h-version of FEM gives good results for the lowest frequencies but demands great computational cost to work up the accuracy for the higher frequencies. The accuracy of the FEM can be improved applying the polynomial p refinement. Some enriched methods based on the FEM have been developed in last 20 years seeking to increase the accuracy of the solutions for the higher frequencies with lower computational cost. Engels (1992) and Ganesan & Engels (1992) present the Assumed Mode Method (AMM) which is obtained adding to the FEM shape functions set some interface restrained assumed modes. The Composite Element Method (CEM) (Zeng, 1998a and 1998b) is obtained by enrichment of the conventional FEM local solution space with non-polynomial functions obtained from analytical solutions of simple vibration problems. A modified CEM applied to analysis of beams is proposed by Lu & Law (2007). The use of products between polynomials and Fourier series instead of polynomials alone in the element shape functions is recommended by Leung & Chan (1998). They develop the Fourier p-element applied to the vibration analysis of bars, beams and plates. These three methods have the same characteristics and they will be called enriched methods in this chapter. The main features of the enriched methods are: (a) the introduction of boundary conditions follows the standard finite element procedure; (b) hierarchical p refinements are easily implemented and (c) they are more accurate than conventional h version of FEM. At the same time, the Generalized Finite Element Method (GFEM) was independently proposed by Babuska and colleagues (Melenk & Babuska, 1996; Babuska et al., 2004; Duarte et al., 2000) and by Duarte & Oden (Duarte & Oden, 1996; Oden et al., 1998) under the following names: Special Finite Element Method, Generalized Finite Element Method, Finite Element Partition of Unity Method, hp Clouds and Cloud-Based hp Finite Element Method.

The composite element method applied to free vibration analysis of trusses and beams

This work deals with an enrichment technique of the finite element solution to the free vibration problems, called the composite element method [J. Sound Vibration 218 (1998) 619, 659, Key Engrg. Mater. 145-149 (1998) 773]. The enrichment of the solution space is obtained combining the FEM and the high accuracy of closed form solutions from the classical theory. The analytical solutions must be in accordance with some special boundary conditions in such a way they do not change the nodal values of FEM and, also, they must be the solutions to the frequency equation. The CEM can be improved using two types of approach: h and c-versions. The h-version, the same as FEM, consists of the refinement of the element mesh. The c-version is the increase of degrees of freedom related to the classical theory. Truss and beams elements are used in order to verify the numerical efficiency of the CEM. Some examples are presented and the frequencies and mode shapes of vibration obtained by CEM are compared with the FEM solution, and also the classical theory. The numerical results have shown that CEM is more accurate than FEM with the same number of total degrees of freedom. The CEM is also more accurate to determine higher frequencies than the FEM, except for the last ones. The results suggest that, for higher frequencies, numerical instabilities are presented.

An Introduction to the Composite Element Method Applied to the Vibration Analysis of Trusses

This paper introduces a new type of Finite Element Method (FEM), called Composite Element Method (CEM). The CEM was developed by combining the versatility of the FEM and the high accuracy of closed form solutions from the classical analytical theory. Analytical solutions, which fulfil some special boundary conditions, are added to FEM shape functions forming a new group of shape functions. CEM results can be improved using two types of approach: h-version and c-version. The h-version, as in FEM, is the refinement of the element mesh. On the other hand, in the c-version there is an increase of degrees of freedom related to the classical theory (c-dof). The application of CEM in vibration analysis is thus investigated and a rod element is developed. Some samples which present frequencies and vibration mode shapes obtained by CEM are compared to those obtained by FEM and by the classical theory. The numerical results show that CEM is more accurate than FEM for the same number of total degrees of freedom employed. It is observed in the examples that the c-version of CEM leads to a super convergent solution.

Accurate assessment of natural frequencies for uniform and non-uniform Euler-Bernoulli beams and frames by adaptive generalized finite element method

Purpose – 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. Design/methodology/approach – 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. Findings – 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 ...

GFEM STABILIZATION TECHNIQUES APPLIED TO DYNAMIC ANALYSIS OF NON-UNIFORM SECTION BARS

The Finite Element Method FEM , although widely used as an approximate solution method, has some limitations when applied in dynamic analysis. As the loads excite the high frequency and modes, the method may lose precision and accuracy. To improve the representation of these highfrequency modes, we can use the Generalized Finite Element Method GFEM to enrich the approach space with appropriate functions according to the problem under study. However, there are still some aspects that limit the GFEM applicability in problems of dynamics of structures, as numerical instability associated with the process of enrichment. Due to numerical instability, the GFEM may lose precision and even result in numerically singular matrices. In this context, this paper presents the application of two proposals to minimize the problem of sensitivity of the GFEM: an adaptation of the Stable Generalized Finite Element Method for dynamic analysis and a stabilization strategy based on preconditioning of enrichment. Examples of one-dimensional modal and transient analysis are presented as bars with cross section area variation. Numerical results obtained are discussed analyzing the effects of the adoption of preconditioning techniques on the approximation and the stability of GFEM in dynamic analysis.

Isogeometric analysis of free vibration of framed structures: comparative problems

Purpose 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. Design/methodology/approach 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. Findings The results show advantages of IGA over FEM and GFEM in the frequency error spectra, mostly in the higher frequencies. Originality/value Isogeometric analysis shows a feasible tool in structural analysis, with emphasis for problems that requires a high amount of vibration modes.

ISOGEOMETRIC ANALYSIS OF FREE VIBRATION OF TRUSSES AND PLANE STRESS

Actually Isogeometric Analysis (IGA) have proven high accuracy and efficacy in dynamical problems, openning possibilities to improve the traditional FEM models. At the same time of IGA development, some steps forward in GFEM were done concerning also applications for dynamical problems. The aim of this paper is to test the response of IGA for free vibration problems of trusses and plane stress. Based on numerical applications, IGA models have their convergence and accuracy checked and compared with those developed in FEM and GFEM. The results shows high accuracy for IGA models, and reinforce its way as a promising tool.