What is the Finite Element Method (FEM)?

In mathematics, partial differential equations (PDEs) can be solved either analytically (exact, continuous) or numerically (approximation, discrete).

Illustrating the position of FEM in the overall picture for solving PDEs

Finite element method (FEM) is a numerical method for solving PDEs. FEM is usually applied to solve for PDEs in continuum mechanics for solid mechanics problems.

Two other popular methods are the finite difference and finite volume methods (FDM FVM), which are frequently used to solve problems in fluid mechanics.

Partial Differential Equations in Continuum Mechanics

Continuum mechanics studies the mechanical behaviour of a material that has been mathematically idealised using partial differential equations (PDEs). Hence, continuum mechanics solves PDEs to understand material behaviours.

Why use idealisation?
It is not economic to fully describe material behaviours over a large range of conditions. Instead, the mathematical expression is idealised such that material behaviour is only represented for a reasonable range of loading conditions. See examples.

How to mathematically idealise/ model a material?
In continuum mechanics, materials are modelled using PDEs. These are equations assembled from (1) physical laws that are universal and (2) constitutive equations that are material specific.

PDEs in continuum mechanics = physical laws + constitutive equations

Examples of PDEs

• Navier equation
• Wave equation as special case of Navier equation
• Heat equation
• Laplace's equation as special case of heat equation
• Navier-Stokes equations

What is Continuum Mechanics?

After generating a continuum (by mathematically representing a real material), the mechanical behaviour of such continuum can then be studied. This is continuum mechanics - the study of materials' mechanical behaviour using mathematical models.

     Solid mechanics and fluid mechanics are two special cases of continuum mechanics.

Physical Laws vs. Constitutive Equations

See also

What are Constitutive Equations?

Physical Laws of Continuum Mechanics

What are Constitutive Equations?

Physical laws of continuum mechanics are valid for any continuum. Constitutive equations are mathematical relationships between kinetics and kinematics quantities of a specific continuum (recall that a continuum is just a mathematical idealisation of a material). In other words, constitutive equations describe the behaviour of a material subjected to certain loading conditions.

Examples
Consider two analyses:

1. With same loading conditions but using two different materials i.e. steel vs. aluminium
2. With same material but test at two different temperatures i.e. room temperature vs. 1000ºC

In the above examples, the constitutive equations are responsible for the the followings

1. Differing response for steel and aluminium under the same loading conditions
2. In practice, a material is only described over a range of conditions it is expected to encounter. Therefore, constitutive equations are formulated to describe different response of the same material over an acceptable range i.e. two different sets of equations for 10-30ºC and 900-1200ºC temperature range for the above analysis

Reynolds' Transport Theorem

The physical laws of continuum mechanics can be generalised using Reynolds' Transport Theorem as follows: