Finite Element Analysis. Barna Szabó

Finite Element Analysis

left-parenthesis e Subscript r Baseline right-parenthesis Subscript upper E

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) minus left-parenthesis kappa u Superscript prime Baseline right-parenthesis Superscript prime Baseline plus c u equals minus mu StartFraction partial-differential squared u Over partial-differential t squared EndFraction comma x element-of left-parenthesis 0 comma script l right-parenthesis comma t element-of left-parenthesis 0 comma infinity right-parenthesis

where the primes represent differentiation with respect to x. For example, we may think of an elastic bar of length script l , cross‐section A, modulus of elasticity E, in which case kappa identical-to upper A upper E greater-than 0 given in units of Newton (N) or equivalent, the parameter c greater-than-or-equal-to 0 is the coefficient of distributed springs ( normal upper N slash normal m normal m squared ) and the parameter mu greater-than 0 is mass per unit length (kg/m = 1 0 Superscript negative 6 Baseline normal upper N normal s squared slash normal m normal m squared ). The bar is vibrating in its longitudinal direction.

The boundary conditions are:

u left-parenthesis 0 comma t right-parenthesis equals 0 comma u left-parenthesis script l comma t right-parenthesis equals 0

and the initial conditions are

u left-parenthesis x comma 0 right-parenthesis equals f left-parenthesis x right-parenthesis comma StartFraction partial-differential u Over partial-differential t EndFraction vertical-bar Subscript left-parenthesis x comma 0 right-parenthesis Baseline equals g left-parenthesis x right-parenthesis

where f left-parenthesis x right-parenthesis and g left-parenthesis x right-parenthesis are given functions in upper L squared left-parenthesis upper I right-parenthesis . 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

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