Nonlinear finite elements/Functions

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There are certain terms involving relationships between functions that you will often encounter in papers dealing with finite elements and continuum mechanics. We list some of the basic terms that you will see. More details can be found in books on advanced calculus and functional analysis.

FunctionsEdit

Let   and   be two sets. A function is a rule that assigns to each   an element of  . A function is usually denoted by

 

Sometimes, one also writes

 

For example, if the function is  , then we may write  .

Domain and RangeEdit

For a function  , the set   is called the domain of  . See Figure 1.

 
Figure 1. Domain and range of a function.

The range of   is the set

 

Therefore,  .

One-to-one mapping (injection)Edit

A function   is called one-to-one (or an injection) if no two distinct elements of   are mapped to the same element of  .

Onto mapping (surjection)Edit

A function   is called onto if for every   there is an   such that  .

If that case,  .

One-to-one and onto mapping (bijection)Edit

When a mapping is both one-to-one and onto it is called a bijection. For example, if   and   with

 

the map is one-to-one and onto.

On the other hand if

 

the map is one-to-one but not onto.

If we choose

 

the map is neither one-to-one nor onto.

Image, pre-image, and inverse functionsEdit

Suppose we have a function  . Let   be a subset of   (see Figure~1). Let us define

 

Then,   is called the image of  .

On the other hand, if   is a subset of   and we define

 

Then,   is called the inverse image or pre-image of  .

If   is one-to-one and onto, then there is a unique function   such that

 

The function   is called an inverse function of  .

Identity mapEdit

The map   such that   for all   is called the identity map. This map is one-to-one and onto.

CompositionEdit

A notation that you will commonly see in papers on nonlinear solid mechanics is the composition of two functions. See Figure 2.

 
Figure 2. Composition of functions.

Let   and   be two functions such that   and  . The composition   is defined as

 

Let us consider a stretch ( ) followed by a translation ( ). Then we can write   and  .

The composition   is given by

 

The inverse composition   is given by

 

Isomorphism and HomeomorphismEdit

You will also come across the terms isomorphism and homeomorphism in the literature on nonlinear solid mechanics.

Isomorphism is a very general concept that appears in several areas of mathematics. The word means, roughly, "equal shapes". It usually refers to one-to-one and onto maps that preserve relations among elements.

A homeomorphism is a continuous transformation between two geometric figures that is continuous in both directions. The map has to be one-to-one to be homeomorphic. It also has to satisfy the requirements on an equivalence relation.

Continuously differentiable functionsEdit

A function

 

is said to be  -times continuous differentiable or of class   if its derivatives of order   (where  ) exist and are continuous functions.

Figure 3 shows three functions ( ,  ,  ) and their derivatives.

 
Figure 3: Continuity of functions.

  FunctionsEdit

The function   is called the Heaviside step function (usually written  ) which is defined as

 

The derivative of the Heaviside function is the Dirac delta function (written  ) which has the defining property that

 

for any function   and any constant  . The delta function is singular and discontinuous. Hence, the Heaviside function is not continuously differentiable. Sometimes the Heaviside function is said to belong to the class of   functions.

  FunctionsEdit

The function   in Figure 3 (also called a hat function) is continuous but has discontinuous derivatives. In this particular case, the function has the form

 

Such functions that are differentiable only once are called   functions.

  FunctionsEdit

The function   in Figure 3 is infinitely differentiable and has continuous derivatives every time it is differentiable. Such functions are called   functions. Since the function can be differentiated once to give a continuous derivative, it also falls into the category of   functions.

Sobolev spaces of functionsEdit

You will find Sobolev spaces being mentioned when you read the finite element literature. A clear understanding of these function spaces needs a knowledge of functional analysis. The book Introduction to Functional Analysis with Applications to Boundary Value Problems and Finite Elements by B. Daya Reddy is a good starting point that is just right for engineers. We will not get into the details here.

Of particular interest in finite element analysis are Sobolev spaces of functions such as

 

where

 

The function space   is the space of square integrable functions.

Of interest to us is an outcome of Sobolev's theorem which says that if a function is of class   then it is actually a bounded   function. If we choose our basis functions from the set of square integrable functions with continuous derivatives, certain singularities are automatically precluded.