Ritz Method

The variational functional I corresponding to the boundary value problem (2.1a), (2.1b) and (2.1c) is given by equation (2.32). The solution to the boundary value problem is obtained by extremising the functional I, that is, by setting the first variation of δI to zero. Using the expression (2.37) for δI, substituting δu/α for
u and multiplying all the terms by α , we get

(3.1)

Note that, using the above equation is equivalent to using the integral form corresponding to the virtual work formulation (equation 2.6) or the weighted residual (weak) formulation (equation 2.5b). While using the weighted residual formulations, however, one needs to choose an appropriate form for the weight function w also. This point is made clear in section 3.6

As stated in the introduction, in Ritz method, an approximate solution is assumed as a series of N terms of the following form :

(3.2)

where are unknown coefficients and are a set of linearly independent functions of x defined over the interval [0,L]. The term "linear independence" means as follows. Consider a linear combination of involving scalar coefficients . If the only solution of the equation

for x  [0,L]

(3.3)

is that all the coefficient are trivially zero, i.e.

for i = 0,1...N,

(3.4)

then, the set of functions is said to be linearly independent over the interval [0,L]. The functions are called as the basis functions . Here, the basis functions are defined over the whole bar, i.e. for the interval [0,L]. Thus, they are called as the global basis functions. Later on, in finite element formulation, the (local) basis functions will be defined over a part of the element, called as an element.

When the weighted residual formulation is used, in general, the weight functions need not be the same as the basis functions. However, when both the sets are identical, this special case of the weighted residual formulation is called as the Galerkin formulation. This point is discussed in more details in section 3.6.

Note that the function is required to satisfy the three admissibility conditions stated in section 2.6. These conditions impose certain restrictions on the basis functions . The second condition implies that must be unconstrained at . The third condition requires that the derivatives must be finite over the whole interval . The third restriction on the functions is imposed by the first admissibility condition that the form (expression 3.2) must satisfy the Dirichlet boundary condition (equation 2.1b). Then we get

(3.5)

If we assume that all except satisfy the condition

at x = 0

(3.6)

then, from equation (3.5), we get

(3.7)

Then, the series in expression (3.2) starts from i = 1 rather than form i = 0. Thus,

(3.8)

where are linearly independent and satisfy the condition (3.6).

Taking the variation of equation (3.2), we get

	(property vii of section 2.5),
	(property ii of section 2.5). (3.9)	(3.9)

Substituting the expression (3.9) for in equation (1), we get

(3.10)

Note that, in the above expression, each of the coefficients can be varied independently. That is, for given j, we can choose to be non zero and all the remaining to be zero. Then, the above equation leads to the following set of equations:

(3.11)

Substituting the expression (3.8) for in equation (3.11) and changing the summation index from i to j, we get

( )-

(3.12)

Now define the following quantities :

	(3.13)
	(3.14)

Then, equation (3.12) becomes

(3.15)

In matrix form, this can be written as

(3.16)

Note that, if the functions are non-dimensional, then the dimensions of the vectors and turn out to be that of the displacement and force respectively. Then, the dimension of becomes that of stiffness (i.e. force per unit length). Therefore, is called as the stiffness matrix, is called the displacement vector and is called as the force vector.

The matrix has the following properties.

• Expression (3.13) shows that

(3.17)

Thus, the matrix is symmetric.

• Note that

(3.18)

Thus,

for any non-trivial vector .

(3.19)

Therefore, is a positive definite matrix. This implies that is invertible or exists. Hence,

(3.20)

Thus, Ritz method converts the differential equation of a boundary value problem into a set of algebraic equations using the corresponding integral form. Solution of the algebraic equations (equation 3.16) gives the coefficients of the assumed form of the solution (expression 3.8). Examples in the next section illustrate this procedure. These examples also show that, as the number of terms is increased, the solution becomes more accurate. It also illustrates that at the points of singularity in loading or geometric property (like A) or material property (like E), the accuracy is poor if only a few terms are chosen.

Amongst the many sets of linearly independent functions available, the set of polynomials offers a very simple and easy choice as far as the mathematical operations of Ritz method are concerned. Therefore, we choose as the following set :

(3.21)

But, are supposed to satisfy the condition = 0 at x = 0. The first member of this set does not satisfy this condition. Therefore, we choose the set without the first function. Therefore

(3.22)

Note that these functions satisfy all the constraints arising out of the three admissibility conditions required to be satisfied by .