Finite Element Analysis. Barna Szabó

Finite Element Analysis - Barna Szabó


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of implementation, keeping the condition number of the coefficient matrices small, and personal preferences. For the symmetric positive‐definite matrices considered here the condition number C is the largest eigenvalue divided by the smallest. The number of digits lost in solving a linear problem is roughly equal to log Subscript 10 Baseline upper C. Characterizing the condition number as being large or small should be understood in this context. In the finite element method the condition number depends on the choice of the basis functions and the mesh.

      Lagrange shape functions

      Lagrange shape functions of degree p are constructed by partitioning upper I Subscript st into p sub‐intervals. The length of the sub‐intervals is typically 2 slash p but the lengths may vary. The node points are xi 1 equals negative 1, xi 2 equals 1 and negative 1 less-than xi 3 less-than xi 4 less-than midline-horizontal-ellipsis less-than xi Subscript p plus 1 Baseline less-than 1. The ith shape function is unity in the ith node point and is zero in the other node points:

      (1.50)upper N Subscript i Baseline left-parenthesis xi right-parenthesis equals product Underscript StartLayout 1st Row k equals 1 2nd Row k not-equals i EndLayout Overscript p plus 1 Endscripts StartFraction xi minus xi Subscript k Baseline Over xi Subscript i Baseline minus xi Subscript k Baseline EndFraction comma i equals 1 comma 2 comma ellipsis comma p plus 1 comma xi element-of upper I Subscript st Baseline dot

      These shape functions have the following important properties:

      (1.51)upper N Subscript i Baseline left-parenthesis xi Subscript j Baseline right-parenthesis equals Start 2 By 2 Matrix 1st Row 1st Column 1 2nd Column if i equals j 2nd Row 1st Column 0 2nd Column if i not-equals j EndMatrix and sigma-summation Underscript i equals 1 Overscript p plus 1 Endscripts upper N Subscript i Baseline left-parenthesis xi right-parenthesis equals 1 period

      Exercise 1.5 Sketch the Lagrange shape functions for p equals 3.

      Legendre shape functions

      For p equals 1 we have

      (1.52)upper N 1 equals StartFraction 1 minus xi Over 2 EndFraction comma upper N 2 equals StartFraction 1 plus xi Over 2 EndFraction dot

      For p greater-than-or-equal-to 2 we define the shape functions as follows:

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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