Finite Element Analysis. Barna Szabó

Finite Element Analysis - Barna Szabó


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upper N equals 0.5"/> 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 p equals 1.

Graph depicts the ratio (ωFE/ωEX)n corresponding to the h version, p=2.
corresponding to the h version,
.

      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.

Graph depicts the ratio (ωFE/ωEX)n corresponding to the p version. Uniform mesh, 5 elements.
corresponding to the p version. Uniform mesh, 5 elements.

p 5 10 15 20
ω 24 194.296 100.787 98.312 98.312

      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 mu 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 1.22 Find the eigenvalues for the problem of Example 1.15 using the generalized formulation and the basis functions phi Subscript n Baseline left-parenthesis x right-parenthesis equals sine left-parenthesis n pi x slash script l right-parenthesis, (n equals 1 comma 2 comma ellipsis comma upper N). Assume that κ and mu are constants and mu slash kappa equals 1. Let script l equals 10. Explain what makes this choice of basis functions very special. Hint: Owing to the orthogonality of the basis functions, only hand calculations are involved.

      Up to this point we have been concerned with the finite element method based on the generalized formulation, called the principle of virtual work. There are many other finite element methods. All finite element methods share the following attributes:

      1 Formulation. A bilinear form is defined on the normed linear spaces X, Y (i.e. , ) and the functional


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