形状関数の微分

形状関数の体積座標

に関する微分 $\frac{ \partial N }{ \partial L0 } _{(In)}$ は、

$\displaystyle \frac{ \partial N }{ \partial L0 } _{(0)}$	$\textstyle =$	$\displaystyle 4 L0 - 1$
$\displaystyle \frac{ \partial N }{ \partial L0 } _{(1)}$	$\textstyle =$	$\displaystyle 0$
$\displaystyle \frac{ \partial N }{ \partial L0 } _{(2)}$	$\textstyle =$	$\displaystyle 0$
$\displaystyle \frac{ \partial N }{ \partial L0 } _{(3)}$	$\textstyle =$	$\displaystyle 0$
$\displaystyle \frac{ \partial N }{ \partial L0 } _{(4)}$	$\textstyle =$	$\displaystyle 4 L1$
$\displaystyle \frac{ \partial N }{ \partial L0 } _{(5)}$	$\textstyle =$	$\displaystyle 4 L2$
$\displaystyle \frac{ \partial N }{ \partial L0 } _{(6)}$	$\textstyle =$	$\displaystyle 4 L3$
$\displaystyle \frac{ \partial N }{ \partial L0 } _{(7)}$	$\textstyle =$	$\displaystyle 0$
$\displaystyle \frac{ \partial N }{ \partial L0 } _{(8)}$	$\textstyle =$	$\displaystyle 0$
$\displaystyle \frac{ \partial N }{ \partial L0 } _{(9)}$	$\textstyle =$	$\displaystyle 0$	(2.57)

体積座標

に関する微分 $\frac{ \partial N }{ \partial L1 } _{(In)}$ については、

$\displaystyle \frac{ \partial N }{ \partial L1 } _{(0)}$	$\textstyle =$	$\displaystyle 0$
$\displaystyle \frac{ \partial N }{ \partial L1 } _{(1)}$	$\textstyle =$	$\displaystyle 4 L1 - 1$
$\displaystyle \frac{ \partial N }{ \partial L1 } _{(2)}$	$\textstyle =$	$\displaystyle 0$
$\displaystyle \frac{ \partial N }{ \partial L1 } _{(3)}$	$\textstyle =$	$\displaystyle 0$
$\displaystyle \frac{ \partial N }{ \partial L1 } _{(4)}$	$\textstyle =$	$\displaystyle 4 L0$
$\displaystyle \frac{ \partial N }{ \partial L1 } _{(5)}$	$\textstyle =$	$\displaystyle 0$
$\displaystyle \frac{ \partial N }{ \partial L1 } _{(6)}$	$\textstyle =$	$\displaystyle 0$
$\displaystyle \frac{ \partial N }{ \partial L1 } _{(7)}$	$\textstyle =$	$\displaystyle 4 L2$
$\displaystyle \frac{ \partial N }{ \partial L1 } _{(8)}$	$\textstyle =$	$\displaystyle 0$
$\displaystyle \frac{ \partial N }{ \partial L1 } _{(9)}$	$\textstyle =$	$\displaystyle 4 L3$	(2.58)

体積座標

に関する微分 $\frac{ \partial N }{ \partial L2 } _{(In)}$ については、

$\displaystyle \frac{ \partial N }{ \partial L2 } _{(0)}$	$\textstyle =$	$\displaystyle 0$
$\displaystyle \frac{ \partial N }{ \partial L2 } _{(1)}$	$\textstyle =$	$\displaystyle 0$
$\displaystyle \frac{ \partial N }{ \partial L2 } _{(2)}$	$\textstyle =$	$\displaystyle 4 L2 - 1$
$\displaystyle \frac{ \partial N }{ \partial L2 } _{(3)}$	$\textstyle =$	$\displaystyle 0$
$\displaystyle \frac{ \partial N }{ \partial L2 } _{(4)}$	$\textstyle =$	$\displaystyle 0$
$\displaystyle \frac{ \partial N }{ \partial L2 } _{(5)}$	$\textstyle =$	$\displaystyle 4 L0$
$\displaystyle \frac{ \partial N }{ \partial L2 } _{(6)}$	$\textstyle =$	$\displaystyle 0$
$\displaystyle \frac{ \partial N }{ \partial L2 } _{(7)}$	$\textstyle =$	$\displaystyle 4 L1$
$\displaystyle \frac{ \partial N }{ \partial L2 } _{(8)}$	$\textstyle =$	$\displaystyle 4 L3$
$\displaystyle \frac{ \partial N }{ \partial L2 } _{(9)}$	$\textstyle =$	$\displaystyle 0$	(2.59)

体積座標

に関する微分 $\frac{ \partial N }{ \partial L3 } _{(In)}$ については、

$\displaystyle \frac{ \partial N }{ \partial L3 } _{(0)}$	$\textstyle =$	$\displaystyle 0$
$\displaystyle \frac{ \partial N }{ \partial L3 } _{(1)}$	$\textstyle =$	$\displaystyle 0$
$\displaystyle \frac{ \partial N }{ \partial L3 } _{(2)}$	$\textstyle =$	$\displaystyle 0$
$\displaystyle \frac{ \partial N }{ \partial L3 } _{(3)}$	$\textstyle =$	$\displaystyle 4 L3 - 1$
$\displaystyle \frac{ \partial N }{ \partial L3 } _{(4)}$	$\textstyle =$	$\displaystyle 0$
$\displaystyle \frac{ \partial N }{ \partial L3 } _{(5)}$	$\textstyle =$	$\displaystyle 0$
$\displaystyle \frac{ \partial N }{ \partial L3 } _{(6)}$	$\textstyle =$	$\displaystyle 4 L0$
$\displaystyle \frac{ \partial N }{ \partial L3 } _{(7)}$	$\textstyle =$	$\displaystyle 0$
$\displaystyle \frac{ \partial N }{ \partial L3 } _{(8)}$	$\textstyle =$	$\displaystyle 4 L2$
$\displaystyle \frac{ \partial N }{ \partial L3 } _{(9)}$	$\textstyle =$	$\displaystyle 4 L1$	(2.60)

したがって、形状関数の自然座標 $\{ \xi \}$ に関する微分 $\frac{ \partial N }{ \partial \xi } _{(In)}$ は、式 2.42 より、

$\displaystyle \frac{ \partial N }{ \partial \xi } _{(0)}$	$\textstyle =$	$\displaystyle 1 - 4 L0$
$\displaystyle \frac{ \partial N }{ \partial \xi } _{(1)}$	$\textstyle =$	$\displaystyle 4 L1 - 1$
$\displaystyle \frac{ \partial N }{ \partial \xi } _{(2)}$	$\textstyle =$	$\displaystyle 0$
$\displaystyle \frac{ \partial N }{ \partial \xi } _{(3)}$	$\textstyle =$	$\displaystyle 0$
$\displaystyle \frac{ \partial N }{ \partial \xi } _{(4)}$	$\textstyle =$	$\displaystyle 4 (L0 - L1)$
$\displaystyle \frac{ \partial N }{ \partial \xi } _{(5)}$	$\textstyle =$	$\displaystyle -4 L2$
$\displaystyle \frac{ \partial N }{ \partial \xi } _{(6)}$	$\textstyle =$	$\displaystyle -4 L3$
$\displaystyle \frac{ \partial N }{ \partial \xi } _{(7)}$	$\textstyle =$	$\displaystyle 4 L2$
$\displaystyle \frac{ \partial N }{ \partial \xi } _{(8)}$	$\textstyle =$	$\displaystyle 0$
$\displaystyle \frac{ \partial N }{ \partial \xi } _{(9)}$	$\textstyle =$	$\displaystyle 4 L3$	(2.61)

$\displaystyle \frac{ \partial N }{ \partial \eta } _{(0)}$	$\textstyle =$	$\displaystyle 1 - 4 L0$
$\displaystyle \frac{ \partial N }{ \partial \eta } _{(1)}$	$\textstyle =$	$\displaystyle 0$
$\displaystyle \frac{ \partial N }{ \partial \eta } _{(2)}$	$\textstyle =$	$\displaystyle 4 L2 - 1$
$\displaystyle \frac{ \partial N }{ \partial \eta } _{(3)}$	$\textstyle =$	$\displaystyle 0$
$\displaystyle \frac{ \partial N }{ \partial \eta } _{(4)}$	$\textstyle =$	$\displaystyle -4 L1$
$\displaystyle \frac{ \partial N }{ \partial \eta } _{(5)}$	$\textstyle =$	$\displaystyle 4 (L0 - L2)$
$\displaystyle \frac{ \partial N }{ \partial \eta } _{(6)}$	$\textstyle =$	$\displaystyle -4 L3$
$\displaystyle \frac{ \partial N }{ \partial \eta } _{(7)}$	$\textstyle =$	$\displaystyle 4 L1$
$\displaystyle \frac{ \partial N }{ \partial \eta } _{(8)}$	$\textstyle =$	$\displaystyle 4 L3$
$\displaystyle \frac{ \partial N }{ \partial \eta } _{(9)}$	$\textstyle =$	$\displaystyle 0$	(2.62)

$\displaystyle \frac{ \partial N }{ \partial \zeta } _{(0)}$	$\textstyle =$	$\displaystyle 1 - 4 L0$
$\displaystyle \frac{ \partial N }{ \partial \zeta } _{(1)}$	$\textstyle =$	$\displaystyle 0$
$\displaystyle \frac{ \partial N }{ \partial \zeta } _{(2)}$	$\textstyle =$	$\displaystyle 0$
$\displaystyle \frac{ \partial N }{ \partial \zeta } _{(3)}$	$\textstyle =$	$\displaystyle 4 L3 - 1$
$\displaystyle \frac{ \partial N }{ \partial \zeta } _{(4)}$	$\textstyle =$	$\displaystyle -4 L1$
$\displaystyle \frac{ \partial N }{ \partial \zeta } _{(5)}$	$\textstyle =$	$\displaystyle -4 L2$
$\displaystyle \frac{ \partial N }{ \partial \zeta } _{(6)}$	$\textstyle =$	$\displaystyle 4 (L0 - L3)$
$\displaystyle \frac{ \partial N }{ \partial \zeta } _{(7)}$	$\textstyle =$	$\displaystyle 0$
$\displaystyle \frac{ \partial N }{ \partial \zeta } _{(8)}$	$\textstyle =$	$\displaystyle 4 L2$
$\displaystyle \frac{ \partial N }{ \partial \zeta } _{(9)}$	$\textstyle =$	$\displaystyle 4 L1$	(2.63)