Finite Element Analysis Essay
Finite Element Analysis
Finite Element Analysis (FEA) was first introduced by Alexander Hrennikoff in 1941 through his works and by Richard Courant in 1943 through the utilization of the Ritz method of numerical analysis and minimization of variational calculus. The need for solving complex elasticity and structural analysis problems in civil and aeronautical engineering sparked the birth of FEA (Weaver et al, 1973). Hrennikoff and Courant, who used different approaches in their works, shared a common characteristic, which discretizes the mesh of a continuous domain into a set of discrete sub-domains called elements.
Hrennikoff used a lattice analogy to discretizes the domain while Courant divided the domain into finite triangular subregions to solve second order elliptic partial differential equations involving problems in the torsion of a cylinder (Solin et al 2003). John Argyris (University of Stuttgart) and Ray W. Clough (Berkeley) began to thoroughly develop the FEA in mid to late 1950s for airframe and structural analysis and in civil engineering. A description of the study, which is wider in scope, and centers on the “stiffness and deflection of complex structures,” got published in 1956 by M.
J. Turner, R. W. Clough, H. C. Martin and L. J. Topp. In 1965, National Aeronautics and Space Administration (NASA) requested for the development of NASTRAN, a software for finite element. The publication of Strang and Fix’s An Analysis of the Finite Element Method in 1973 provided the method with a rigorous mathematical foundation. Since then, it has been established as a branch of applied mathematics for numerical modeling of physical system in various disciplines such as engineering and aeronautics.
Finite Element Analysis has been developed to an astounding precision in the early 1970s with the advent of expensive mainframe computers owned usually by aeronautics, automotive and nuclear industries. Highly accurate results for all kinds of parameters can now be produced by modern supercomputers. Through this method, the distribution of stresses and displacements and the areas where structures bend and twist can be viewed in detail.
The complexity of both modeling and analysis of a system can also be controlled through a wide array of simulation options offered by FEA software (Strang et al, 1973). To address most engineering applications, the software helps in managing a desired level of accuracy needed and associated computational time requirements. It also allows entire designs to be constructed, refined, and optimized before manufacturing. Several industrial applications such as standard of engineering designs and the design process have been greatly improved by this tool.
Not only that, the time it takes for a product to be manufactured has also been remarkably reduced with the use of the method. Introduction: A numerical technique known as Finite Element Analysis is used for finding approximate solutions of partial differential equations (PDE) and of integral equations, the approach being based upon complete elimination of differential equation or approximating the system of ordinary differential equations (Solin et al, 2003). It consists of a material so stressed and analyzed to produce new product design or to refine existing products.
The primary challenge in solving partial differential equations is to create an equation that approximates the equation to be studied while the errors in the input and intermediate calculations do not accumulate, causing the results to be meaningless (Weaver et al, 1973). One way of solving partial differential equations is to use the Finite Element Method (FEM). This method is the best choice when the domain changes, when the desired precision varies over the entire domain, or when the solution lacks smoothness.
This method helps a company, for instance, to determine whether a proposed design can perform to the client’s specifications prior to manufacturing or not. Otherwise, an existing product or structure can be modified to qualify for a new service condition using FEA. The same can also be done in cases of structural failure. Two types of analysis are in use today in the industry. First, the 2-D modeling conserves simplicity and allows the analysis to be run on a relatively normal computer, yielding a less accurate result. The other is 3-D, which produces more accurate results. However, it runs effectively only on the fastest computers.
The programmer can insert various algorithms within each of these schemes which may make the system function linearly or otherwise. Linear systems do not take into account plastic deformation, unlike non-linear systems, and are less complex. Nonlinear system is capable of testing a material all the way to fracture. Finite Element Analysis uses a complex system of points called nodes which makes for the grid called mesh. This mesh is programmed to contain the material and structural properties which determines how the structure will react to certain loading conditions (Solin et al, 2003).
Nodes are assigned at a certain density throughout the material depending on the anticipated stress levels of a particular area. Higher nodes are usually found on regions that are likely to receive large amounts of stress (Solin et al, 2003). References: Solin, P. and Segeth, D. (2003). Higher-Order Finite Element Methods. Chapman & Hall Strang, Gilbert and Fix, George (1973). An Analysis of Finite Element Method. Prentice Hall Weaver, William and Gere, James (1973). Matrix Analysis of Framed Structures (3rd ed). Springer-Verlag New York, LLC