Finite Element Analysis. Barna Szabó

Finite Element Analysis

Finite Element Analysis - Barna Szabó

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S Superscript p Super Subscript k Baseline left-parenthesis upper I Subscript st Baseline right-parenthesis"/> indicates that on element Ik the function u left-parenthesis x right-parenthesis is mapped from the standard polynomial space script upper S Superscript p Super Subscript k Baseline left-parenthesis upper I Subscript st Baseline right-parenthesis.

      The finite element test space, denoted by upper S Superscript 0 Baseline left-parenthesis upper I right-parenthesis, is defined by the intersection upper S Superscript 0 Baseline left-parenthesis upper I right-parenthesis equals upper S left-parenthesis upper I right-parenthesis intersection upper E left-parenthesis upper I right-parenthesis, that is, u element-of upper S Superscript 0 Baseline left-parenthesis upper I right-parenthesis is zero in those boundary points where essential boundary conditions are prescribed. The number of basis functions that span upper S Superscript 0 Baseline left-parenthesis upper I right-parenthesis 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 p Subscript k plus 1. 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

      We will be concerned with the evaluation of the integral on the kth element:

integral Subscript x Subscript k Baseline Superscript x Subscript k plus 1 Baseline Baseline kappa left-parenthesis x right-parenthesis u prime Subscript n Baseline v Subscript n Superscript prime Baseline d x equals integral Subscript x Subscript k Baseline Superscript x Subscript k plus 1 Baseline Baseline kappa left-parenthesis x right-parenthesis left-parenthesis sigma-summation Underscript j equals 1 Overscript p Subscript k Baseline plus 1 Endscripts a Subscript j Baseline StartFraction d upper N Subscript j Baseline Over d x EndFraction right-parenthesis left-parenthesis sigma-summation Underscript i equals 1 Overscript p Subscript k Baseline plus 1 Endscripts b Subscript i Baseline StartFraction d upper N Subscript i Baseline Over d x EndFraction right-parenthesis d x period

      (1.63)d x equals StartFraction x Subscript k plus 1 Baseline minus x Subscript k Baseline Over 2 EndFraction d xi identical-to StartFraction script l Subscript k Baseline Over 2 EndFraction d xi

      where script l Subscript k Baseline equals Overscript def Endscripts x Subscript k plus 1 Baseline minus x Subscript k is the length of the kth element. Also,

StartFraction d Over d x EndFraction equals StartFraction d Over d xi EndFraction StartFraction d xi Over d x EndFraction equals StartFraction 2 Over x Subscript k plus 1 Baseline minus x Subscript k Baseline EndFraction StartFraction d Over d xi EndFraction identical-to StartFraction 2 Over script l Subscript k Baseline EndFraction StartFraction d Over d xi EndFraction dot

      Therefore

integral Subscript x Subscript k Baseline Superscript x Subscript k plus 1 Baseline Baseline kappa left-parenthesis x right-parenthesis u prime Subscript n Baseline v Subscript n Superscript prime Baseline d x equals StartFraction 2 Over script l Subscript k Baseline EndFraction integral Subscript negative 1 Superscript plus 1 Baseline kappa left-parenthesis upper Q Subscript k Baseline left-parenthesis xi right-parenthesis right-parenthesis left-parenthesis sigma-summation Underscript j equals 1 Overscript p Subscript k Baseline plus 1 Endscripts a Subscript j Baseline StartFraction d upper N Subscript j Baseline Over d xi EndFraction right-parenthesis left-parenthesis sigma-summation Underscript i equals 1 Overscript p Subscript k Baseline plus 1 Endscripts b Subscript i Baseline StartFraction d upper N Subscript i Baseline Over d xi EndFraction right-parenthesis d xi period

      We define

      (1.64)k Subscript i j Superscript left-parenthesis k right-parenthesis Baseline equals StartFraction 2 Over script l Subscript k Baseline EndFraction integral Subscript negative 1 Superscript plus 1 Baseline kappa left-parenthesis upper Q Subscript k Baseline left-parenthesis xi right-parenthesis right-parenthesis StartFraction d upper N Subscript i Baseline Over d xi EndFraction StartFraction d upper N Subscript j Baseline Over d xi EndFraction d xi

      and write

      (1.65)integral Subscript x Subscript k Baseline Superscript x Subscript k plus 1 Baseline Baseline kappa left-parenthesis x right-parenthesis u prime Subscript n Baseline v Subscript n Superscript prime Baseline d x equals sigma-summation Underscript i equals 1 Overscript p Subscript k Baseline plus 1 Endscripts sigma-summation Underscript j equals 1 Overscript p Subscript k Baseline plus 1 Endscripts k Subscript i j Superscript left-parenthesis k <hr><noindex><a href=Скачать книгу