Gfem Stabilization Techniques Applied to Dynamic Analysis of Non-Uniform Section Bars

Abstract

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.

An enriched formulation of isogeometric analysis applied to the dynamical response of bars and trusses

“…By the precision loss by ill-conditioned systems, the methods are determined with more significant digits, usually at a minimum of 16 plus the power of the higher condition number. The conditioning presented by PUIGA 1 agrees with the results found by Weinhardt et al (2018) for GFEM with a similar enrichment strategy. Comparative computational costs for matrices calculation are shown in Table I, where a little increase in an elapsed time of the enriched models can be observed, mostly by the increase in integration points, necessary to capture the integrals of enrichment functions.…”

“…PUIGA also presented very accurate results for forced vibration analysis compared with all other methods, including GFEM. The highest errors at the end of the spectrum were also observed in GFEM by Weinhardt et al (2018) and do not mean a limitation of the method, as refinement strategies and increase in enrichment level can stabilize numerical accuracy of a desirable set of natural frequencies.…”

“…where l max and l min denote the highest and lowest eigenvalue. In this case, the condition number is extracted from the mass matrix, which dominates the problem's sensitivity (Weinhardt et al, 2018). A detailed discussion about mass matrix dominance in free vibration problem sensitivity is addressed by Petroli et al (2017).…”

“…The extension of the adaptive enriched method applied to thin beams and framed structures was also successful in terms of accuracy and convergence (Arndt et al, 2016). Other forms of GFEM with several enrichment levels were developed by Torii et al (2015) in applications for two-dimensional wave equation, by Shang et al (2016) for vibration of elastoplastic thin beams and recently a set of stabilization techniques of GFEM applied to dynamical response of bars (Weinhardt et al, 2018). It highlights GFEM as an accurate method in structural dynamics.…”

Preface to the special issue of the International Symposium on Solid Mechanics (MecSol 2017)

“…• Composite Materials: distortions in composite wing structures (Makinde et al, 2018), failure in composite pressure vessels (Vignoli and Savi, 2018), and dynamic analysis (Marques et al, 2018). • X-FEM, G-FEM, and BEM: crack propagation and X-FEM modelling (Angelo et al, 2018); G-FEM modelling (Sato et al, 2018) and dynamic analysis (Weinhardt et al, 2018); BEM analysis of buckling of anisotropic plates (Monteiro and Daros, 2018), and 3D frictional contact (Ubessi and Marczak, 2018). • Structural Reliability Methods and Reliability-Based Design Optimization: Multi-objective optimization (Passos and Luersen, 2018) and experimental crack identification (Oliveira Filho et al, 2018).…”