The Finite-Element Method—Part II

1989 ◽  
Author(s):  
Shiro Kobayashi ◽  
Soo-Ik Oh ◽  
Taylan Altan
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Integration ◽  
Gaussian Quadrature ◽  
The Finite Element Method ◽  
Gaussian Quadrature Formula ◽  
Stiffness Matrices ◽  
Minimum Number ◽  
Element Technique ◽  
Element Method

Numerical integration is an important part of the finite-element technique. As seen in Section 6.5 of Chap. 6, volume integrations as well as surface integrations should be carried out in order to represent the elemental stiffness equations in a simple matrix form. In deriving the variational principle, it is implicitly assumed that these integrations are exact. However, exact integrations of the terms included in the element matrices are not always possible. In the finite-element method, further approximations are made in the procedure for integration, which is summarized in this section. Numerical integration requires, in general, that the integrand be evaluated at a finite number of points, called Integration points, within the integration limits. The number of integration points can be reduced, while achieving the same degree of accuracy, by determining the locations of integration points selectively. In evaluating integration in the stiffness matrices, it is necessary to use an integration formula that requires the least number of integrand evaluations. Since the Gaussian quadrature is known to require the minimum number of integration points, we use the Gaussian quadrature formula almost exclusively to carry out the numerical integrations.

scholarly journals Gauss-Simpson Quadrature Algorithm for Calcaulting Additional Stress in Foundation Soils

2021 ◽  
Author(s):  
Shuai WANG ◽  
Chao WANG ◽  
Chenliang XING
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Elasticity Theory ◽  
Quadrature Formula ◽  
Gaussian Quadrature ◽  
Additional Stress ◽  
The Finite Element Method ◽  
Integral Sign ◽  
Gaussian Quadrature Formula ◽  
Element Method

Abstract The additional pressure at the bottom of a building’s foundation produces an additional stress in the foundation soils under the building’s foundation. In order to overcome the limitations of traditional elastic theory methods and the finite element method when calculating the additional stress in foundation soils, we use the Gauss-Simpson formula to derive the Gauss-Simpson Quadrature Algorithm based on the elasticity theory. The Gauss-Simpson Quadrature Algorithm is a method designed to calculate the additional stress in foundation soils under an irregularly shaped foundation and an irregular load distribution. This new method is based on the fact that the Gaussian quadrature formula and the Simpson formula are independent of the specific type of integrand. The finite element method with n interpolation points can only achieve an algebraic accuracy of n. The interpolation points of the Gaussian quadrature formula are n zeros of orthogonal polynomials, which can achieve an algebraic accuracy of 2n + 1. Moreover, the weights of the nodes in the quadrature formula are all positive, and thus, it has a high numerical stability. In the proposed method, the Simpson formula is necessary. The Simpson formula is used to transform the implicit additional stress formula with the integral sign into an explicit cumulative integral, which can be considered similar to the rectangular domain case to obtain an explicit analytical algebraic formula for solving the additional stress approximation. In engineering applications, we only need to provide the field engineers with the locations of the interpolation points of the Gauss-Legendre formula, the interpolated weight coefficients, and the specific type of Simpson's formula, and then, the results of the additional stress can be calculated manually, which is nearly impossible using the traditional methods and finite element methods. From the point of view of academic rigor and theoretical completeness, it is possible to use the compound Gauss-Simpson Quadrature Algorithm in conjunction with the looping function in computer programs. Under standard conditions, the proposed Gauss-Simpson Quadrature Algorithm is in good agreement with the results of the traditional elasticity theory.

Rayleigh-Ritz Based Substructure Synthesis for Multiply Supported Structures

1998 ◽  
Vol 122 (1) ◽  
pp. 2-6 ◽  
Author(s):  
C. Morales
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Approximate Solution ◽  
Synthesis Method ◽  
The Finite Element Method ◽  
Compatibility Conditions ◽  
Substructure Synthesis ◽  
Stiffness Matrices ◽  
Element Method

This paper is concerned with the convergence characteristics and application of the Rayleigh-Ritz based substructure synthesis method to structures for which the use of a kinematical procedure taking into account all the compatibility conditions, is not possible. It is demonstrated that the synthesis in this case is characterized by the fact that the mass and stiffness matrices have the embedding property. Consequently, the estimated eigenvalues comply with the inclusion principle, which in turn can be utilized to prove convergence of the approximate solution. The method is applied to a frame and is compared with the finite element method. [S0739-3717(00)00201-4]

A MODIFIED TRIANGULATION ALGORITHM TAILORED FOR THE SMOOTHED FINITE ELEMENT METHOD (S-FEM)

2013 ◽  
Vol 11 (01) ◽  
pp. 1350069 ◽  
Author(s):  
Y. LI ◽  
M. LI ◽  
G. R. LIU
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
The Finite Element Method ◽  
Advancing Front ◽  
Stiffness Matrices ◽  
Smoothed Finite Element Method ◽  
Complicated Geometry ◽  
Discrete Domains ◽  
Volume Method ◽  
Element Method

Meshing is one of the key tasks in using the finite element method (FEM), the smoothed finite element method (S-FEM), finite volume method (FVM), and many other discrete numerical methods. Linear triangular (T3) mesh is one of the most widely used mesh, because it can be generated and refined automatically for discrete domains of complicated geometry, and hence save significantly the time for model creation. This paper presents a modified triangulation algorithm based on the advancing front technique to provide a comprehensive linear triangular mesh generator with six connectivity lists, including element–node (Ele–N) connectivity, element–edge (Ele–Eg) connectivity, edge–node (Eg–N) connectivity, edge–element (Eg–Ele) connectivity, node–edge (N–Eg) connectivity and node–element (N–Ele) connectivity. These six connectivity lists are generated along the way when the T3 elements are created, and hence it is done in a most efficient fashion. The connectivity is recorded in the usual counter-clockwise convention for convenient utilization in various S-FEM models for effective analyses. In addition, an algorithm is developed for renumbering the nodes in the T3 mesh to obtain a minimized bandwidth of stiffness matrices for both FEM and S-FEM models.

A Finite-Element Method for Through Flow Calculations in Turbomachines

1976 ◽  
Vol 98 (3) ◽  
pp. 403-414 ◽  
Author(s):  
Ch. Hirsch ◽  
G. Warzee
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Axisymmetric Flow ◽  
Axial Flow ◽  
The Finite Element Method ◽  
Flow Equations ◽  
Through Flow ◽  
Method Of Solution ◽  
Element Technique ◽  
Element Method

A new method for the numerical solution of the meridional through-flow equations in an axial flow machine is presented based on the finite-element method. A rigorous derivation of the pitch-averaged flow equations is presented and the assumption of axisymmetric flow leads, with the introduction of a stream function, to the equation to be solved. A description is given of the finite-element technique which is applied in this problem. The method of solution allows the calculation of transonic stages. Numerical results are compared with experimental data and show very satisfactory agreement. This method appears, therefore, to compare very favorably with the other methods used up to now. Although the present results pertain to axial flow machines, the method is easily applicable to radial flow machines as well and the way of solution for this case is indicated.

scholarly journals Modal analysis of damaged structures by the modified finite element method

1998 ◽  
Vol 20 (1) ◽  
pp. 29-46 ◽  
Author(s):  
Nguyen Cao Menh ◽  
Nguyen Tien Khiem ◽  
Dao Nhu Mai ◽  
Nguyen Viet Khoa
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Standard Procedure ◽  
Frame Structures ◽  
The Finite Element Method ◽  
Shape Functions ◽  
Stiffness Matrices ◽  
Flexible Supports ◽  
Cubic Polynomial ◽  
Element Method

The classical 3D beam element has been modified and developed as a new finite element for vibration analysis of frame structures with flexible connections and cracked members. The mass and stiffness matrices of the modified elements are established basing on a new form of shape functions, which are obtained in investigating a beam with flexible supports and crack modeled through equivalent springs. These shape functions remain the cubic polynomial form and contain flexible connection (or crack) parameters. They do not change standard procedure of the finite element method (FEM). Therefore, the presented method is easy for engineers in application and allows to analyze Eigen-parameters of structures as functions of the connection (or crack) parameters. The proposed approach has been applied to calculate natural frequencies and mode shape of typical frame structures in presented examples.

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