Finite Element Analysis. Barna Szabó

1.25 79.49 7.50 2.80 1.63 1.12 4.80 1.15 99.52 4.06 1.63 0.97 0.67 3.92 1.05 29.56 1.24 0.53 0.32 0.22 1.77 0.95 18.89 1.16 0.52 0.32 0.22 2.41 0.85 42.94 3.26 1.52 0.94 0.67 9.84 0.75 60.39 5.14 2.47 1.56 1.11 22.37 0.65 76.07 6.86 3.39 2.16 1.56 42.91 0.55 91.80 8.44 4.28 2.76 2.00 76.22

Table 1.5 Example: Element‐by‐element and total relative errors in energy norm (percent) for selected integer values of α.

	Element number
α	1	2	3	4	5
1	0	0	0	0	0	0
2	11.00	61.24	15.31	7.02	4.55	2.45
3	20.09	4.69	98.83	16.16	9.24	4.62
4	36.98	8.41	17.24	30.13	14.02	8.00

      Errors in numerical integration can be particularly damaging. The reader should be mindful of this when applying the concepts and procedures discussed in this chapter to higher dimensions.

1.7 Eigenvalue problems

      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