Nonlinear finite elements/Calculus of variations

Ideas from the calculus of variations are commonly found in papers dealing with the finite element method. This handout discusses some of the basic notations and concepts of variational calculus. Most of the examples are from Variational Methods in Mechanics by T. Mura and T. Koya, Oxford University Press, 1992.

The calculus of variations is a sort of generalization of the calculus that you all know. The goal of variational calculus is to find the curve or surface that minimizes a given function. This function is usually a function of other functions and is also called a functional.

Maxima and minima of functionsEdit

The calculus of variations extends the ideas of maxima and minima of functions to functionals.

For a function of one variable  , the minimum occurs at some point  . For a functional, instead of a point minimum, we think in terms of a function that minimizes the functional. Thus, for a functional   we can have a minimizing function  .

The problem of finding extrema (minima and maxima) or points of inflection (saddle points) can either be constrained or unconstrained.

The unconstrained problem.Edit

Suppose   is a function of one variable. We want to find the maxima, minima, and points of inflection for this function. No additional constraints are imposed on the function. Then, from elementary calculus, the function   has

  • a minimum if   and  .
  • a maximum if   and  .
  • a point of inflection if  .

Any point where the condition   is satisfied is called a stationary point and we say that the function is stationary at that point.

A similar concept is used when the function is of the form  . Then, the function   is stationary if

 

Since  ,  ,  , and   are independent variables, we can write the stationarity condition as

 

The constrained problem - Lagrange multipliers.Edit

Suppose we have a function  . We want to find the minimum (or maximum) of the function   with the added constraint that

 

The added constraint is equivalent to saying that the variables  ,  , and   are not independent and we can write one of the variables in terms of the other two.

The stationarity condition for   is

 

Since the variables  ,  , and   are not independent, the coefficients of  ,  , and   are not zero.

At this stage we could express   in terms of   and   using the constraint equation (1), form another stationarity condition involving only   and  , and set the coefficients of   and   to zero. However, it is usually impossible to solve equation (1) analytically for  . Hence, we use a more convenient approach called the Lagrange multiplier method.

Lagrange multiplier method.Edit

From equation (1) we have

 

We introduce a parameter   called the Lagrange multiplier and using equation (2) we get

 

Then we have,

 

We choose the parameter   such that

 

Then, because   and   are independent, we must have

 

We can now use equations (1), (3), and (4) to solve for the extremum point and the Lagrange multiplier. The constraint is satisfied in the process.

Notice that equations (1), (3) and (4) can also be written as

 

where

 

Minima of functionalsEdit

Consider the functional

 

We wish to minimize the functional   with the constraints (prescribed boundary conditions)

 

Let the function   minimize  . Let us also choose a trial function (that is not quite equal to the solution  )

 

where   is a parameter, and   is an arbitrary continuous function that has the property that

 

(See Figure 1 for a geometric interpretation.)

 
Figure 1. Minimizing function   and trial functions.

Plug (6) into (5) to get

 

You can show that equation (8) can be written as (show this)

 

where

 

and

 

The quantity   is called the first variation of   and the quantity   is called the second variation of  . Notice that   consists only of terms containing   while   consists only of terms containing  .

The necessary condition for   to be a minimum is

 
Remark.Edit

The first variation of the functional   in the direction   is defined as

 

To find which function makes   zero, we first integrate the first term of equation (9) by parts. We have,

 

Since   at   and  , we have

 

Plugging equation (12) into (9) and applying the minimizing condition (11), we get

 

or,

 

The fundamental lemma of variational calculus states that if   is a piecewise continuous function of   and   is a continuous function that vanishes on the boundary, then

 

Applying (14) to (13) we get

 

Equation (15) is called the Euler equation of the functional  . The solution of the Euler equation is the minimizing function that we seek.

Of course, we cannot be sure that the solution represents and minimum unless we check the second variation  . From equation (10) we can see that   if   and   and in that case the problem is guaranteed to be a minimization problem.

We often define

 

where   is called a variation of  .

In this notation, equation (9) can be written as

 

You see this notation in the principle of virtual work in the mechanics of materials.

An exampleEdit

Consider the string of length   under a tension   (see Figure 2). When a vertical load   is applied, the string deforms by an amount   in the  -direction. The deformed length of an element   of the string is

 

If the deformation is small, we can expand the relation into a Taylor series and ignore the higher order terms to get

 
 
Figure 2. An elastic string under a transverse load.

The force T in the string moves a distance

 

Therefore, the work done by the force   (per unit original length of the string) (the stored elastic energy) is

 

The work done by the forces   (per unit original length of string) is

 

We want to minimize the total energy. Therefore, the functional to be minimized is

 

The Euler equation is

 

The solution is