Introduction to FEM

The finite element method is a numerical procedure that can be used to obtain solutions to a large class of engineering problems involving stress analysis, heat transfer, electro­magnetism, and fluid flow. Having a clear understanding of the basic concepts will enable you to use a general-purpose finite element software, such as ANSYS.

In general, engineering problems are mathematical models of physical situations. Mathematical models of many engineering problems are differential equations with a set of corresponding boundary and/or initial conditions. The differential equations are derived by applying the fundamental laws and principles of nature to a system or a control volume. These governing equations represent balance of mass, force, or energy. When possible, the exact solution of these equations renders detailed behavior of asystem under a given set of conditions.

The analytical solutions are composed of two parts:

a homogenous part and

a particular part

In any given engineering problem, there are two sets of design parameters that influence the way in which a system behaves. First, there are those parameters that provide information regarding the natural behavior of a given system. These parameters include material and geometric properties such as modulus of elasticity, thermal conductivity, viscosity, and area, and second moment of area.

On the other hand, there are parameters that produce disturbances in a system.

Problem Type	Examples of Parameters that Produce Disturbances in a System
Solid Mechanics	external forces and moments; support reaction
Heat Transfer	temperature difference; heat input
Fluid Flow and Pipe Networks	pressure difference; rate of flow
Electrical Network	voltage difference

TABLE Parameters causing disturbances in various engineering systems

It is important to understand the role of these parameters infinite element modeling in terms of their respective appearances in stiffness or conductance matrices and load or forcing matrices. The system characteristics will always show up in the stiffness matrix, conductance matrix, or resistance matrix, whereas the disturbance parameters will always appear in the load matrix.

There are many practical engineering problems for which we cannot obtain exact solutions. This inability to obtain an exact solution may be attributed to either the complex nature of governing differential equations or the difficulties that arise from dealing with the boundary and initial conditions. To deal with such problems, we resort to numerical approximations. In contrast to analytical solutions, which show the exact behavior of a system at any point within the system, numerical solutions approximate exact solutions only at discrete points, called nodes. The first step of any numerical procedure is discretization.

This process divides the medium of interest into a number of small sub regions (elements) and nodes. There are two common classes of numerical methods:

Finite difference methods and

Finite element methods.

With finite difference methods, the differential equation is written for each node, and the derivatives are replaced by differential equations. This approach results in a set of simultaneous linear equations. Although finite difference methods are easy to understand and employ in simple problems, they become difficult to apply to problems with complex geometries or complex boundary conditions. This situation is also true for problems with non-isotropic material properties.

In contrast, the finite element method uses integral formulations rather than difference equations to create a system of algebraic equations. Moreover, a continuous function is assumed to represent the approximate solution for each element. The complete solution is then generated by connecting or assembling the individual solutions, allowing for continuity at the interelemental boundaries.