Читать книгу Finite Element Analysis - Barna Szabó - Страница 26

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

The domain is partitioned into M non‐overlapping intervals called finite elements. A partition, called finite element mesh, is denoted by . Thus . The boundary points of the elements are the node points. The coordinates of the node points, sorted in ascending order, are denoted by xi, () where and . The kth element Ik has the boundary points xk and , that is, .

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 , and 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 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)

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 is defined by the mapping function

(1.60)

A finite element space S is a set of functions characterized by , the assigned polynomial degrees and the mapping functions , . 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 if and only if u satisfies the conditions on the right of the vertical bar (). The first condition is that u must lie in the energy space. In one dimension this implies that u must be continuous on I. The expression indicates that on element Ik the function is mapped from the standard polynomial space .

The finite element test space, denoted by , is defined by the intersection , that is, is zero in those boundary points where essential boundary conditions are prescribed. The number of basis functions that span is called the number of degrees of freedom.

The process by which the number of degrees of freedom is progressively increased by mesh refinement, with the polynomial degree fixed, is called h‐extension and its implementation the h‐version of the finite element method. The process by which the number of degrees of freedom is progressively increased by increasing the polynomial degree of elements, while keeping the mesh fixed, is called p‐extension and its implementation the p‐version of the finite element method. The process by which the number of degrees of freedom is progressively increased by concurrently refining the mesh and increasing the polynomial degrees of elements is called hp‐extension and its implementation the hp‐version of the finite element method.

Remark 1.4 It will be explained in Chapter 5 that the separate naming of the h, p and hp versions is related to the evolution of the finite element method rather than its theoretical foundations.