Finite Element Analysis. Barna Szabó

Finite Element Analysis

left-parenthesis xi right-parenthesis equals StartFraction 1 Over StartRoot 2 left-parenthesis 2 i minus 3 right-parenthesis EndRoot EndFraction left-parenthesis upper P Subscript i minus 1 Baseline left-parenthesis xi right-parenthesis minus upper P Subscript i minus 3 Baseline left-parenthesis xi right-parenthesis right-parenthesis i equals 3 comma 4 comma ellipsis"/>

Hint: note that upper P Subscript n Baseline left-parenthesis 1 right-parenthesis equals 1 for all n and use equations (D.10) and (D.12) in Appendix D.

1.3.2 Finite element spaces in one dimension

We are now in a position to provide a precise definition of finite element spaces in one dimension.

The domain upper I equals StartSet x vertical-bar 0 less-than x less-than script l EndSet is partitioned into M non‐overlapping intervals called finite elements. A partition, called finite element mesh, is denoted by normal upper Delta . Thus upper M equals upper M left-parenthesis normal upper Delta right-parenthesis . The boundary points of the elements are the node points. The coordinates of the node points, sorted in ascending order, are denoted by xi, ( i equals 1 comma 2 comma ellipsis comma upper M plus 1 ) where x 1 equals 0 and x Subscript upper M plus 1 Baseline equals script l . The kth element Ik has the boundary points xk and x Subscript k plus 1 , that is, upper I Subscript k Baseline equals StartSet x vertical-bar x Subscript k Baseline less-than x less-than x Subscript k plus 1 Baseline EndSet .

Various approaches are used for the construction of sequences of finite element mesh. We will consider four types of mesh design:

1 A mesh is uniform if all elements have the same size. On the interval the node points are located as follows:

2 A sequence of meshes () is quasiuniform if there exist positive constants C1, C2, independent of K, such that(1.56) where (resp. ) is the length of the largest (resp. smallest) element in mesh . In two and three dimensions ℓk is defined as the diameter of the kth element, meaning the diameter of the smallest circle or sphere that envelopes the element. For example, a sequence of quasiuniform meshes would be generated in one dimension if, starting from an arbitrary mesh, the elements would be successively halved.

3 A mesh is geometrically graded toward the point on the interval if the node points are located as follows:(1.57) where is called grading factor or common factor. These are called geometric meshes.

4 A mesh is a radical mesh if on the interval the node points are located by(1.58)

The question of which of these schemes is to be preferred in a particular application can be answered on the basis of a priori information concerning the regularity of the exact solution and aspects of implementation. Practical considerations that should guide the choice of the finite element mesh will be discussed in Section 1.5.2.

When the exact solution has one or more terms like StartAbsoluteValue x minus x 0 EndAbsoluteValue Superscript alpha , and alpha greater-than 1 slash 2 is a fractional number, then the ideal mesh is a geometrically graded mesh and the polynomial degrees are assigned in such a way that the smallest elements are assigned the lowest polynomial degree, the largest elements the highest. The optimal grading factor is q equals left-parenthesis StartRoot 2 EndRoot minus 1 right-parenthesis squared almost-equals 0.17 which is independent of α. The assigned polynomial degrees should increase at a rate of approximately 0.4 [45].

The ideal meshes are radical meshes when the same polynomial degree is assigned to each element. The optimal value of θ depends on p and α:

(1.59) theta equals StartFraction p plus 1 slash 2 Over alpha minus 1 slash 2 plus left-parenthesis n minus 1 right-parenthesis slash 2 EndFraction

where n is the number of spatial dimensions. For a detailed analysis of discretization schemes in one dimension see reference [45].

The relationship between the kth element of the mesh and the standard element upper I Subscript st is defined by the mapping function

(1.60) x equals upper Q Subscript k Baseline left-parenthesis xi right-parenthesis equals StartFraction 1 minus xi Over 2 EndFraction x Subscript k Baseline plus StartFraction 1 plus xi Over 2 EndFraction x Subscript k plus 1 Baseline comma xi element-of upper I Subscript st Baseline period

A finite element space S is a set of functions characterized by normal upper Delta , the assigned polynomial degrees p Subscript k Baseline greater-than-or-equal-to 1 and the mapping functions upper Q Subscript k Baseline left-parenthesis xi right-parenthesis , k equals 1 comma 2 comma ellipsis comma upper M left-parenthesis normal upper Delta right-parenthesis . Specifically;

(1.61)

where p and Q represent, respectively, the arrays of the assigned polynomial degrees and the mapping functions. This should be understood to mean that u element-of upper S if and only if u satisfies the conditions on the right of the vertical bar ( vertical-bar ). The first condition u element-of upper E left-parenthesis upper I right-parenthesis is that u must lie in the energy space. In one dimension this implies that u must be continuous on I. The expression Скачать книгу