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The following problem is a prototype of an important class of engineering problems which includes the undamped vibration of elastic structures:

(1.133)

where the primes represent differentiation with respect to x. For example, we may think of an elastic bar of length , cross‐section A, modulus of elasticity E, in which case given in units of Newton (N) or equivalent, the parameter is the coefficient of distributed springs () and the parameter is mass per unit length (kg/m = ). The bar is vibrating in its longitudinal direction.

The boundary conditions are:

and the initial conditions are

where and are given functions in . Here we consider homogeneous Dirichlet boundary conditions. However, the boundary conditions can be homogeneous Neumann or homogeneous Robin conditions, or any combination of those.

The generalized form is obtained by multiplying eq. (1.133) by a test function and integrating by parts:

(1.134)

We now introduce where , . This is known as separation of variables. Therefore we get

(1.135)

which can be written as

(1.136)

Since the functions on the left are independent of t, the function T depends only on t, both expressions must equal some positive constant denoted by . That constant has to be positive because the expression on the left holds for all and if we select then the expression on the left is positive.

The function satisfies the ordinary differential equation

(1.137)

the solution of which is

(1.138)

where ω is the angular velocity (rad/s). Alternatively ω is written as where f is the frequency (Hz).

To find ω and U we have to solve the problem

(1.139)

which will be abbreviated as

(1.140)

There are infinitely many solutions called eigenpairs , . The set of eigenvalues is called the spectrum. If Ui is an eigenfunction and α is a real number then is also an eigenfunction. In the following we assume that the eigenfunctions have been normalized so that

If the eigenvalues are distinct then the corresponding eigenfunctions are orthogonal: Let and be eigenpairs, . Then from eq. (1.140) we have

Subtracting the second equation from the first we see that if then Ui and Uj are orthogonal functions:

(1.141)

and hence .

Importantly, it can be shown that any function can be written as a linear combination of the eigenfunctions:

(1.142)

where

(1.143)

The Rayleigh¹⁵ quotient is defined by

(1.144)

Eigenvalues are usually numbered in ascending order. Following that convention,

(1.145)

that is, the smallest eigenvalue is the minimum of the Rayleigh quotient and the corresponding eigenfunction is the minimizer of on . This follows directly from eq. (1.140). The kth eigenvalue minimizes on the space

(1.146)

where

(1.147)

When the eigenvalues are computed numerically then the minimum of the Rayleigh quotient is sought on the finite‐dimensional space . We see from the definition that the error of approximation in the natural frequencies will depend on how well the eigenfunctions are approximated in energy norm, in the space .

The following example illustrates that in a sequence of numerically computed eigenvalues only the lower eigenvalues will be approximated well. It is possible, however, at least in principle, to obtain good approximation for any eigenvalue by suitably enlarging the space .

Example 1.15 Let us consider the eigenvalue problem

(1.148)

This equation models (among other things) the free vibration (natural frequencies and mode shapes) of a string of length stretched horizontally by the force (N) under the assumptions that the displacements are infinitesimal and confined to one plane, the plane of vibration, and the ends of the string are fixed. The mass per unit length is (kg/m). We assume that κ and are constants. It is left to the reader to verify that the function u defined by

(1.149)

where ai, bi are coefficients determined from the initial conditions and

(1.150)

satisfies eq. (1.148).

If we approximate the eigenfunctions using uniform mesh, and plot the ratio against , where n is the nth eigenvalue, then we get the curves shown in Fig. 1.13. The curves show that somewhat more than 20% of the numerically computed eigenvalues will be accurate. The higher eigenvalues cannot be well approximated in the space . The existence of the jump seen at is a feature of numerically approximated eigenvalues by means of standard finite element spaces using the h‐version [2]. The location of the jump depends on the polynomial degree of elements. There is no jump when .

Figure 1.13 The ratio corresponding to the h version, .

If we approximate the eigenfunctions using a uniform mesh consisting of 5 elements, and increase the polynomial degrees uniformly then we get the curves shown in Fig. 1.14. The curves show that only about 40% of the numerically computed eigenvalues will be accurate. The error increases monotonically for the higher eigenvalues and the size of the error is virtually independent of p.

It is possible to reduce this error by enforcing the continuity of derivatives. Examples are available in [32]. There is a tradeoff, however: Enforcing continuity of derivatives on the basis functions reduces the number of degrees of freedom but entails a substantial programming burden because an adaptive scheme has to be devised for the general case to ensure that the proper degree of continuity is enforced. If, for example, μ would be a piecewise constant function then the continuity of the first and higher derivatives must not be enforced in those points where μ is discontinuous.

From the perspective of designing a finite element software, it is advantageous to design the software in such a way that it will work well for a broad class of problems. In the formulation presented in this chapter continuity is a requirement. Functions that lie in where are also in . In other words, the space is embedded in the space . Symbolically: . The exact eigenfunctions in this example are in .

Figure 1.14 The ratio corresponding to the p version. Uniform mesh, 5 elements.

Table 1.6 Example: p‐Convergence of the 24th eigenvalue in Example 1.16.

p	5	10	15	20
ω 24	194.296	100.787	98.312	98.312

Example 1.16 Let us consider the problem in Example 1.15 modified so that μ is a piecewise constant function defined on a uniform mesh of 5 elements such that on elements 1, 3 and 5, on elements 2 and 4. In this case the exact eigenfunctions are not smooth and the exact eigenvalues are not known explicitly.

At there are 24 degrees of freedom. Suppose that the 24th eigenvalue is of interest. If we increase p uniformly then this eigenvalue converges to 98.312. The results of computation are shown in Table 1.6.

Any eigenvalue can be approximated to an arbitrary degree of precision on a suitably defined mesh and uniform increase in the degrees of freedom. When κ and/or are discontinuous functions then the points of discontinuity must be node points.

Observe that the numerically computed eigenvalues converge monotonically from above. This follows directly from the fact that the eigenfunctions are minimizers of the Rayleigh quotient.

Exercise 1.21 Prove eq. (1.143).

Exercise 1.22 Find the eigenvalues for the problem of Example 1.15 using the generalized formulation and the basis functions , (). Assume that κ and are constants and . Let . Explain what makes this choice of basis functions very special. Hint: Owing to the orthogonality of the basis functions, only hand calculations are involved.