Finite Element Analysis. Barna Szabó
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Название: Finite Element Analysis

Автор: Barna Szabó

Издательство: John Wiley & Sons Limited

Жанр: Физика

Серия:

isbn: 9781119426462

isbn:

СКАЧАТЬ target="_blank" rel="nofollow" href="#fb3_img_img_d17590df-7b55-5f43-9463-680d0dbb52a0.png" alt="upper N Subscript i Baseline left-parenthesis xi right-parenthesis equals StartRoot StartFraction 2 i minus 3 Over 2 EndFraction EndRoot integral Subscript negative 1 Superscript xi Baseline upper P Subscript i minus 2 Baseline left-parenthesis t right-parenthesis d t i equals 3 comma 4 comma ellipsis comma p plus 1"/>

Graph depicts lagrange shape functions in one dimension, p=2.
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      1 Orthogonality. For :(1.54) This property follows directly from the orthogonality of Legendre polynomials, see eq. (D.13) in the appendix.

      2 The set of shape functions of degree p is a subset of the set of shape functions of degree . Shape functions that have this property are called hierarchic shape functions.

      3 These shape functions vanish at the endpoints of : for .

 Graph depicts legendre shape functions in one dimension, p=4.
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      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.