Finite Element Analysis. Barna Szabó

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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.

      1.3.3 Computation of the coefficient matrices

      The coefficient matrices are computed element by element. The numbering of the coefficients is based on the numbering of the standard shape functions, the indices range from 1 through . This numbering will have to be reconciled with the requirement that each basis function must be continuous on I and must have an unique identifying number. This will be discussed separately.

      Computation of the stiffness matrix

The first term of the bilinear form in eq. (1.43) is computed as a sum of integrals over the elements

(1.62)

      We

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