A Method for the Numerical Integration of the Equations of Motion Arising From a Finite-Element Analysis

1970 ◽  
Vol 37 (3) ◽  
pp. 599-605 ◽  
Author(s):  
C. C. Fu
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Method Of Characteristics ◽  
Truncation Error ◽  
Equations Of Motion ◽  
Step Length ◽  
Integration Step ◽  
Integration Technique ◽  
Element Analysis ◽  
Fifth Order

This paper describes an extended de Vogelaere method which can be used to integrate numerically the equations of motion arising from a finite-element analysis. The method has a truncation error of fifth order in the integration step length, and can be easily programmed for a digital computer. More important is that the method requires a minimum amount of computer storage and computer run time compared with other methods commonly used in numerical analysis and, therefore, is particularly useful in solving a large system of equations. The integration technique has been used to study two-dimensional wave propagation in axisymmetric and plane-strain problems. Results are compared with those of the finite difference and the method of characteristics.

scholarly journals New Analytical Model Used in Finite Element Analysis of Solids Mechanics

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1401 ◽  
Author(s):  
Sorin Vlase ◽  
Adrian Eracle Nicolescu ◽  
Marin Marin
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Three Dimensional ◽  
Elastic Solid ◽  
The Finite Element Method ◽  
Governing Equations ◽  
Element Analysis ◽  
Elastic Multibody System

In classical mechanics, determining the governing equations of motion using finite element analysis (FEA) of an elastic multibody system (MBS) leads to a system of second order differential equations. To integrate this, it must be transformed into a system of first-order equations. However, this can also be achieved directly and naturally if Hamilton’s equations are used. The paper presents this useful alternative formalism used in conjunction with the finite element method for MBSs. The motion equations in the very general case of a three-dimensional motion of an elastic solid are obtained. To illustrate the method, two examples are presented. A comparison between the integration times in the two cases presents another possible advantage of applying this method.

Finite Element Analysis of Freely Rotating Tires

1987 ◽  
Vol 15 (2) ◽  
pp. 134-158 ◽  
Author(s):  
N-T. Tseng
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Failure Modes ◽  
Deformable Body ◽  
Equations Of Motion ◽  
Material Model ◽  
Angular Speed ◽  
Element Analysis ◽  
Automobile Tire ◽  
Truck Tire

Abstract Axisymmetric analysis of an inflated tire rotating with constant angular speed can be used to simulate two loading conditions of a tire during its service life: (1) a freely rotating tire on an automobile that is stuck in snow or mud and (2) the top region of a rolling loaded tire, where footprint loading has little influence on the distribution of its stresses and strains. The equations of motion for a freely rotating deformable body with constant angular speed have been derived and implemented into a finite element code developed in-house. The rotation of a thin disk was used to check the validity of the implemented formulation and coding. Excellent agreement between the numerical and the analytical results was obtained. A cast tire, a radial automobile tire, and a radial truck tire, were then analyzed by the new finite element procedure. The tires were inflated and rotated at speeds up to 241 km/h (150 mph). The elastomers in these tires were simulated by incompressible elements for which the nonlinear mechanical properties were described by the Mooney-Rivlin model. Each ply was simulated by its equivalent orthotropic material model. The finite element predictions agreed well with the available experimental measurements. Significant changes in interply shear strain at the belt edge, the bead load, and the crown cord loads of plies were observed in the finite element analysis. These phenomena suggest three possible failure modes in freely rotating tires, i.e. belt edge separation, bead breakage, and belt failure at crown region.

scholarly journals Exploring the dynamics of flagellar dynein within the axoneme with Fluctuating Finite Element Analysis

2020 ◽  
Vol 53 ◽  
Author(s):  
Robin A. Richardson ◽  
Benjamin S. Hanson ◽  
Daniel J. Read ◽  
Oliver G. Harlen ◽  
Sarah A. Harris
Keyword(s):  
Electron Microscopy ◽  
Finite Element Analysis ◽  
Finite Element ◽  
Molecular Motors ◽  
Electron Tomography ◽  
Protein Complexes ◽  
Step Length ◽  
Experimental Information ◽  
Element Analysis ◽  
Molecular Conformations

Abstract Flagellar dyneins are the molecular motors responsible for producing the propagating bending motions of cilia and flagella. They are located within a densely packed and highly organised super-macromolecular cytoskeletal structure known as the axoneme. Using the mesoscale simulation technique Fluctuating Finite Element Analysis (FFEA), which represents proteins as viscoelastic continuum objects subject to explicit thermal noise, we have quantified the constraints on the range of molecular conformations that can be explored by dynein-c within the crowded architecture of the axoneme. We subsequently assess the influence of crowding on the 3D exploration of microtubule-binding sites, and specifically on the axial step length. Our calculations combine experimental information on the shape, flexibility and environment of dynein-c from three distinct sources; negative stain electron microscopy, cryo-electron microscopy (cryo-EM) and cryo-electron tomography (cryo-ET). Our FFEA simulations show that the super-macromolecular organisation of multiple protein complexes into higher-order structures can have a significant influence on the effective flexibility of the individual molecular components, and may, therefore, play an important role in the physical mechanisms underlying their biological function.

A High-Accuracy Approach to Finite Element Analysis Using the Hexa 27-Node Element

2016 ◽  
Author(s):  
Pedro V. Marcal ◽  
Jeffrey T. Fong ◽  
Robert Rainsberger ◽  
Li Ma
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Truncation Error ◽  
Shell Element ◽  
Pressure Vessels ◽  
Element Formulation ◽  
Element Analysis ◽  
Brick Element ◽  
Similar Solutions ◽  
Selection Of

In most finite-element-analysis codes, accuracy is achieved through the use of the hexahedron hexa-20 elements (a node at each of the 8 corners and 12 edges of a brick element). Unfortunately, without an additional node in the center of each of the element’s 6 faces, nor in the center of the hexa, the hexa-20 elements are not fully quadratic such that its truncation error remains at h2(0), the same as the error of a hexa-8 element formulation. To achieve an accuracy with a truncation error of h3(0), we need the fully-quadratic hexa-27 formulation. A competitor of the hexa-27 element in the early days was the so-called serendipity cubic hexa-32 solid elements (see Ahmad, Irons, and Zienkiewicz, Int. J. Numer. Methods in Eng., 2:419–451 (1970) [1]). The hexa-32 elements, unfortunately, also suffer from the same lack of accuracy syndrome as the hexa20’s. In this paper, we investigate the accuracy of various elements described in the literature including the fully quadratic hexa-27 elements to a shell problem of interest to the pressure vessels and piping community, viz. the shell-element-based analysis of a barrel vault. Significance of the highly accurate hexa-27 formulation and a comparison of its results with similar solutions using ABAQUS hexa-8, and hexa-20 elements, are presented and discussed. Guidelines are proposed for selection of better elements.

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