Convergence of the classical Rayleigh-Ritz method and the finite element method

AIAA Journal ◽  
1990 ◽  
Vol 28 (8) ◽  
pp. 1509-1516 ◽  
Author(s):  
Leonard Meirovitch ◽  
Moon K. Kwak
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Ritz Method ◽  
The Finite Element Method ◽  
Element Method

The Finite-Element Method—Part I

1989 ◽  
Author(s):  
Shiro Kobayashi ◽  
Soo-Ik Oh ◽  
Taylan Altan
Keyword(s):  
Heat Transfer ◽  
Finite Element Method ◽  
Finite Element ◽  
Ritz Method ◽  
Approximate Solutions ◽  
Plane Elasticity ◽  
Heat Transfer Analysis ◽  
Deformation Analysis ◽  
The Finite Element Method ◽  
Element Method

The concept of the finite-element procedure may be dated back to 1943 when Courant approximated the warping function linearly in each of an assemblage of triangular elements to the St. Venant torsion problem and proceeded to formulate the problem using the principle of minimum potential energy. Similar ideas were used later by several investigators to obtain the approximate solutions to certain boundary-value problems. It was Clough who first introduced the term “finite elements” in the study of plane elasticity problems. The equivalence of this method with the well-known Ritz method was established at a later date, which made it possible to extend the applications to a broad spectrum of problems for which a variational formulation is possible. Since then numerous studies have been reported on the theory and applications of the finite-element method. In this and next chapters the finite-element formulations necessary for the deformation analysis of metal-forming processes are presented. For hot forming processes, heat transfer analysis should also be carried out as well as deformation analysis. Discretization for temperature calculations and coupling of heat transfer and deformation are discussed in Chap. 12. More detailed descriptions of the method in general and the solution techniques can be found in References [3-5], in addition to the books on the finite-element method listed in Chap. 1. The path to the solution of a problem formulated in finite-element form is described in Chap. 1 (Section 1.2). Discretization of a problem consists of the following steps: (1) describing the element, (2) setting up the element equation, and (3) assembling the element equations. Numerical analysis techniques are then applied for obtaining the solution of the global equations. The basis of the element equations and the assembling into global equations is derived in Chap. 5. The solution satisfying eq. (5.20) is obtained from the admissible velocity fields that are constructed by introducing the shape function in such a way that a continuous velocity field over each element can be denned uniquely in terms of velocities of associated nodal points.

A Comparison of Computed Deflections of Symmetric Angle-Ply Laminate Plates by the Ritz Method and the Finite Element Method

2014 ◽  
Author(s):  
Kenneth Carroll ◽  
Ernesto Gutierrez-Miravete
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Composite Laminates ◽  
Ritz Method ◽  
The Finite Element Method ◽  
Symbolic Manipulation ◽  
Transverse Loading ◽  
Simply Supported ◽  
Element Method

When a simply supported composite plate is subjected to a lateral load, the presence of the twist coupling stiffnesses in the governing differential equations of equilibrium does not allow the determination of an exact solution for the deflection and numerical methods must be used. This paper describes a comparison of computed approximations to the deflection of composite laminates subjected to transverse loading obtained using the Ritz method and the finite element method. The Ritz method is implemented with the symbolic manipulation program Maple and ANSYS is used to perform the finite element calculations. Reliable results are obtained using both methods.

The Frequencies and Damping of a Cantilever Composite Plate With a Viscoelastic Layer

1995 ◽  
Author(s):  
P. Cupiał ◽  
J. Nizioł
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Ritz Method ◽  
Method Comparison ◽  
Viscoelastic Layer ◽  
The Finite Element Method ◽  
Complex Eigenvalues ◽  
Geometric Boundary ◽  
Loss Factors ◽  
Element Method

Abstract In the paper the natural frequencies and modal loss factors of a three-layered rectangular cantilever plate with high-modulus composite face layers and a viscoelastic mid-layer are discussed. Two analysis approaches are considered, the Ritz method and the finite element method. The Ritz analysis is based on an appropriate 2-D plate model. Algebraic polynomials are used which satisfy the necessary geometric boundary conditions. The discretized eigenvalue problem is then solved for complex eigenvalues, from which damped frequencies and modal loss factors are extracted. Both frequencies and loss factors are shown to converge with an increasing number of the polynomials used. Apart from the Ritz method, frequencies and modal loss factors are calculated by the finite element method based on the so-called strain energy method. Comparison between the two approaches is then given.

scholarly journals Numerical Solution to One-dimensional Consolidation by the Finite Element Method

2019 ◽  
pp. 1-15
Author(s):  
E. Nwankwo ◽  
O. D. Aluko ◽  
S. Iyeke ◽  
A. Aladenika
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Solution ◽  
Water Pressure ◽  
Ritz Method ◽  
The Finite Element Method ◽  
One Dimensional ◽  
Matlab Program ◽  
Consolidation Behavior ◽  
Element Method

Adequate prediction of structures settlement is of utmost importance in order to prevent future failure of civil engineering structures due to excessive settlement resulting from an inadequate settlement prediction. In this paper, laboratory consolidation test was performed on five different clay samples from different locations to determine the soil consolidation in terms of pore water pressure. A formulation of Finite Element (FE) method was also developed for solving one-dimensional consolidation problem and its validity checked out. The one-dimensional consolidation differential equation was solved using finite element analysis by Rayleigh-Ritz method to obtain an approximate solution and ten elements were used to discretize the domain. MATLAB program was used to write the finite element codes. Considering the graphs generated from the MATLAB program which compares the consolidation behavior of the soil sample from analytical and numerical point of view, it is seen that there is a good agreement between Terzaghi’s exact solution to consolidation behavior of soils and numerical solution using the finite element method.

Simulation of tuning graphene plasmonic behaviors by ferroelectric domains for self-driven infrared photodetector applications

Nanoscale ◽  
2019 ◽  
Vol 11 (43) ◽  
pp. 20868-20875 ◽  
Author(s):  
Junxiong Guo ◽  
Yu Liu ◽  
Yuan Lin ◽  
Yu Tian ◽  
Jinxing Zhang ◽  
...  
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Interfacial Effect ◽  
Infrared Photodetector ◽  
The Finite Element Method ◽  
Ferroelectric Domains ◽  
Element Method

We propose a graphene plasmonic infrared photodetector tuned by ferroelectric domains and investigate the interfacial effect using the finite element method.

Sign in / Sign up

Export Citation Format

Share Document