Finite Element Analysis. Barna Szabó

Finite Element Analysis

the software tools available to them, analysts can formulate very efficient discretization schemes.

1.6.1 The exact solution lies in ,

When the solution is smooth then the most efficient finite element discretization scheme is uniform mesh and high polynomial degree. However, all implementations of finite element analysis software have limitations on how high the polynomial degree is allowed to be and therefore it may not be possible to increase the polynomial degree sufficiently to achieve the desired accuracy. In such cases the mesh has to be refined. Uniform refinement may not be optimal in all cases, however. Consider, for example, the following problem:

(1.118) minus epsilon squared u Superscript double-prime Baseline plus c u equals f left-parenthesis x right-parenthesis comma u left-parenthesis 0 right-parenthesis equals u prime left-parenthesis script l right-parenthesis equals 0

where epsilon less-than less-than c , and f is a smooth function. Intuitively, when epsilon squared is small then the solution will be close to u equals f slash c however, because of the boundary condition u left-parenthesis 0 right-parenthesis equals 0 , has to be satisfied, the function u left-parenthesis x right-parenthesis will change sharply over some interval 0 less-than x less-than d left-parenthesis epsilon right-parenthesis less-than less-than script l .

Letting c equals 1 and f left-parenthesis x right-parenthesis equals 1 the exact solution of this problem is

(1.119) u Subscript upper E upper X Baseline left-parenthesis x right-parenthesis equals 1 minus hyperbolic cosine x slash epsilon plus hyperbolic tangent left-parenthesis script l slash epsilon right-parenthesis hyperbolic sine x slash epsilon

which is plotted for various values of epsilon on the interval 0 less-than x slash script l less-than 0.20 in Fig. 1.11. It is seen that the gradient at x equals 0 rapidly increases with respect to decreasing values of epsilon .

Graph depicts the solution uEX(x), given by eq. (1.119), in the neighborhood of x=0 for various values of ε.

Figure 1.11 The solution

, given by eq. (1.119), in the neighborhood of

for various values of epsilon

This is a simple example of boundary layer problems that arise in models of plates, shells and fluid flow. Despite the fact that u Subscript upper E upper X is an analytic function, it may require unrealistically high polynomial degrees to obtain a close approximation to the solution when epsilon is small.

The optimal discretization scheme for problems with boundary layers is discussed in the context of the h p ‐version in [85]. The results of analysis indicate that the size of the element at the boundary is proportional to the product of the polynomial degree p and the parameter epsilon . Specifically, for the problem discussed here, the optimal mesh consists of two elements with the node points located at x 1 equals 0 , x 2 equals d , x 3 equals script l , where d equals upper C p epsilon with 0 less-than upper C less-than 4 slash e .

A practical approach to problems like this is to create an element at the boundary (in higher dimensions a layer of elements) the size of which is controlled by a parameter. The optimal value of that parameter is then selected adaptively.

1.6.2 The exact solution lies in ,

In this section we consider a special case of the problem stated in eq. (1.103):

(1.120) integral Subscript 0 Superscript script l Baseline u prime v Superscript prime Baseline d x equals upper F left-parenthesis v right-parenthesis comma for all v element-of upper E Superscript 0 Baseline left-parenthesis upper I right-parenthesis

with the data u left-parenthesis 0 right-parenthesis equals u left-parenthesis script l right-parenthesis equals 0 , script l equals 1 and upper F left-parenthesis v right-parenthesis defined such that the exact solution is

(1.121) u Subscript upper E upper X Baseline equals x Superscript alpha Baseline left-parenthesis 1 minus x right-parenthesis comma alpha greater-than 1 slash 2 comma 0 less-than x less-than 1

that is,

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