\chapter{�M�`���V�~��}
\section{�L���v�f�@�ɂ���}
\subsection{�L���v�f�@�̑S�̑�}
�L���v�f�@�̐��藧���C

�ʒu�Â��C

���ށC

�����C

��ʓI�Ȑ���
\section{�M�`�����̃V�~�����[�V����}
%
�܂��C�M�`���̎x�z�������𓱏o����D
���̓��o�ɂ́C����\cite{yagawa1983}�C���т�\cite{idoh1997}���Q�l�ɂ����D
%
�x�z�������́C�ȉ��̃G�l���M�[�ۑ�����(�o�������琶�܂ꂽ)�t�[���G�̖@���𓱓����邱�Ƃœ��o�����(\cite{yagawa1983}pp.2(�u���b�N���})�Cpp.8(�}2.1)�Q��)�D
%
%
\subsubsection{�G�l���M�[�ۑ���}
�Î~���Ă���A���̂̓��̔��������̂̎��P�ʎ��ʓ���̃G�l���M�[$E$�́C
\begin{equation}
E = e
\end{equation}
�������C$e$�͓����G�l���M�[��\���C��M$ c $�C���x$T$��p����ƁC
%
\begin{equation}
e = c T
\end{equation}
�ŕ\����D
%
���������̗v�f���Ɋ܂܂��G�l���M�[�͖��x��$\rho $�Ƃ���ƁC
$\rho Edxdydz = \rho c T dxdydz$�ł���D
�P�ʎ��ԓ�����ω����́C$ \frac{\partial (\rho  c T)}{\partial t}dxdydz�@$�ƂȂ�D
�������C���������̂�$x,y,z$�������̔����������ꂼ��$ dx,dy,dz $�Ƃ���D
%
�G�l���M�[�ۑ������C
\begin{eqnarray}\label{��65}
&& \nonumber \frac{\partial (\rho c T)}{\partial t}dxdydz \\
&& \nonumber =\left( �P�ʎ��ԂɊO������v�f�����ɋ��������M��\dot{Q}dxdydz\right) \\
&& -\left( �P�ʎ��ԂɔM�`���ɂ���ĊO���ɗ��o����M��\dot{q} \right)
\end{eqnarray}
�����ŁC$\dot{Q}$�͒P�ʎ��ԂɒP�ʑ̐ϓ�����ɊO�����狟�������M�ʂŁC�������M�C�t�ˁC���w�����ɂ����̂Ȃǂ�����D
%
�������C�����ŔM���������Ȃ����̂Ƃ���D����������C�z�����ĉ��x�㏸�ȊO�̃G�l���M�[�Ƃ��ĕϐ����������肷��ꍇ�ɂ͏㎮�ɉ�����$ \dot{q} $�ɉ����M�ʂ��l������K�v������D
%
%
\clearpage
\subsubsection{�M�`���ɂ�鍀}
%
\subsubsection{�t�[���G�̖@���F}
�P�ʎ��ԂɒP�ʖʐς��ړ�����M�ʂ́C���̓_�ɂ����鉷�x���z�ɔ�Ⴗ��

%
���C����_�ɂ����鉷�x��T�C�M�`���W�����ɂƂ���΁C���̓_�ɂ�����x,y,z�����ɓ`������M�ʂ̒P�ʎ��ԁC�P�ʖʐϓ�����̒l�́C���ꂼ��
$ -\lambda_{xx}\frac{\partial T}{\partial x} $,
$ -\lambda_{yy}\frac{\partial T}{\partial y} $,
$ -\lambda_{zz}\frac{\partial T}{\partial z} $
�Ə�����D���������āC$dx,dy,dz$���\���ɏ��������C$x_0$��x���ɐ����Ȗʂ��痬������M�ʂ�,
$ \left( -\lambda_{xx}\frac{\partial T}{\partial x} \right) dydz $�ƂȂ�D
��������$dx $���ꂽ�_ $ x_0 + dx $�ł́C�ʒu$x_0 $�ł̉��x���z$ \frac{\partial T}{\partial x} $��p���āC$T_{x+dx}=T +\frac{\partial T}{\partial x} dx $�ƂȂ�̂ŁC���o����M�ʂ́C
$ -\lambda_{xx}\frac{ \partial T_{x+dx}}{\partial x} dydz = \left( -\lambda_{xx} \frac{\partial T}{\partial x} \right) dydz + \frac{\partial }{\partial x} \left( -\lambda_{xx} \frac{\partial T}{\partial x} \right) dxdydz  $�ƂȂ�D
%
%
\begin{figure}[htb]
\center
\includegraphics[width = 14cm]{heatelem.eps}
\caption{���������̂ɏo���肷��M��}
\label{fig:heatelem}
\end{figure}
%
%
����āC�P�ʎ��ԂɒP�ʒ����̂��痬�o����M�ʂ́C
$ \frac{\partial }{\partial x} \left( -\lambda_{xx} \frac{\partial T}{\partial x} \right) dxdydz  $
�ƂȂ�D���l�ɂ��āC$y,z$�������ł����o����ƁC��\ref{��65}��$ \dot{q}$�́C
\begin{equation}
\dot{q} = - \left[ \frac{\partial }{\partial x} \left( -\lambda\frac{\partial T}{\partial x} \right) +\frac{\partial }{\partial y} \left( -\lambda\frac{\partial T}{\partial y} \right) 
+ \frac{\partial }{\partial z} \left( -\lambda\frac{\partial T}{\partial z} \right) \right] dxdydz
\end{equation}
�ƂȂ�D

�@�ȏ���C��\ref{��65}��$ \dot{q} $�������āC���ӂ�dxdxdz�ŏ�����ƁC
%
\begin{equation}\label{�x�z������}
\rho c \frac{\partial T}{\partial t} = \frac{\partial Q}{\partial t} + \frac{\partial }{\partial x}\lambda_{xx}\frac{\partial T}{\partial x} + \frac{\partial }{\partial y}\lambda_{yy}\frac{\partial T}{\partial y} + \frac{\partial }{\partial z}\lambda_{zz}\frac{\partial T}{\partial z}
\end{equation}
�������C��M��$ c $�Ƃ���D
%
%
\subsubsection{�M�`���̎x�z�������̗��U��}\label{KFEM}
%
�{�߂ł́C�����ꂽ�M�`���̎x�z����������ɁC
%�v����
%���b�V���`��̎��R�ȃ��f���ɑ΂���
�ő̓��̂R�����M�`�������K���[�L���@�ŗ��U������D
��\ref{�x�z������}�𐮗�����Ǝ����ɂȂ�D
%
\begin{equation}\label{�x�z������2}
\rho c \frac{\partial T}{\partial t} = \frac{\partial }{\partial x} \left( \lambda \frac{\partial T}{\partial x} \right) + \frac{\partial }{\partial y} \left( \lambda \frac{\partial T}{\partial y} \right) + \frac{\partial }{\partial z} \left( \lambda \frac{\partial T}{\partial z}\right) + \dot{Q}
\end{equation}
�������CT�F���x, �ɁF�M�`����, Q�F�������M��($ \dot{Q}�͒P�ʎ��ԂɒP�ʑ̐ϓ�����ɋ��������M�ʁE���M�� $), �ρF���x, c�F��M, t�F�����@
�ł���D�{�����ł́C�M�`�����镨���͓�����������($ \lambda_{xx} = \lambda_{yy} = \lambda_{zz} = \lambda $)�Ƃ��Čv�Z���s���D
%
�M�`�����̋��E�����́C�M������$ q[W/m^2]$�C���E��ł̊O�����@���x�N�g����$n$
%$ n (= n_x +n_y +n_z) $�H
%
�Ƃ���ƃt�[���G�̖@����莟�̎��ŗ^������D()���̌v�Z���͗v�o�T�D����ȊO�́A\cite{yagawa1983}���D
%
%
\begin{equation}\label{eqn1-2}
	q = - [\lambda] \frac{\partial T}{\partial n}
\begin{pmatrix}
 = -[\lambda](n \cdot ��T) =-
\left[
	\begin{array}{ccc} 
	\lambda_{xx} & 0 & 0 \\
	0 & \lambda_{yy} & 0 \\
	0 & 0 & \lambda_{zz} \\
	\end{array}
	\right]
	\left[
	\begin{array}{c} 
	\frac{\partial T}{\partial x} n_x \\
	\frac{\partial T}{\partial y} n_y \\
	\frac{\partial T}{\partial z} n_z \\
	\end{array}
\right]
\end{pmatrix}
\end{equation}
%
%
%
�@�ȏ���C�M�`�����̎x�z�������ƗL���v�f�̋��E�������^����ꂽ�̂ŁC�x�z���������K���[�L���@�ɂ���ė��U������D
���ɗ��U��������@�ɂ́C���b�c�@�����邪�C�����Ȏ����o�Ă��邽�߁C�Ȃ��D

\subsection{�K���[�L���@�Ƃ́H(��`)}
�ϕ��@�̈��D�ϕ��@���g�������d�݂��c���@����{�Ƃ���ߎ���@


\subsection{�K���[�L���@�Ɋ�Â��L���v�f�@(\cite{yagawa1983}pp.104�`��⊮)}
%
��͑Ώۂ�L���ȗv�f�ɕ������C�v�f���̉��x���z�����̂悤�ɕ\������D
%
\begin{equation}\label{eqn1-9}
T(x,y,z,t)=[N(x,y,z)] \{T(t)\}_{ele}
\end{equation}
//�@(�v�f�����̔C�ӓ_�̉��x��L���v�f�̐ߓ_���xT���g���Ă���킵����)//

%
�������C�L���v�f���ł̔C�ӈʒu(x,y,z)�E�C�ӎ���(t)�̉��x��T�C�v�f�������x�Ƃ����т���d�݊֐�$[ N ](1�~n�̍s�}�g���N�X�Cn:�L���v�f�̐ړ_��)$ �ƁC
���鎞�����ɂ�����L���v�f�̐ߓ_���x�x�N�g��$\{ T \}_{ele} $�ŕ\���D
%
%
$[N]$���d�݊֐��Ƃ��āC��\ref{�x�z������2}�ɃK���[�L���@��K�p����ƁC�ȉ��̎��ɂȂ�D
%
%
\begin{equation}\label{galerkin}
 \int_v [N]^T 
\{
\frac{\partial }{\partial x} \left( \lambda \frac{\partial T}{\partial x} \right) + \frac{\partial }{\partial y} \left( \lambda \frac{\partial T}{\partial y} \right) + \frac{\partial }{\partial z} \left( \lambda \frac{\partial T}{\partial z}\right) + \dot{Q} - \rho c\frac{\partial T}{\partial t} 
\} dv = 0
\end{equation}
%��ڂ�[N]^{T}(�]�u)�̓K���[�L���@�ɂ����́D�������[N]��T=NT�ɂ����́D
%
�������C��t���Y����$T$�͓]�u���C$v$�͗v�f�̈��\���D
%
�㎮�ɂ��āC���̂悤�ɂR�̍��ɕ����čl����D
\begin{equation}\label{2diff}
\int_v [ N ]^T 
\{
\frac{\partial }{\partial x} \left( \lambda \frac{\partial T}{\partial x} \right) + \frac{\partial }{\partial y} \left( \lambda \frac{\partial T}{\partial y} \right) + \frac{\partial }{\partial z} \left( \lambda \frac{\partial T}{\partial z}\right) 
\} dv
\end{equation}
%
\begin{equation}
\int_v [ N ]^T 
\frac{\partial Q}{\partial t}
 dv
\end{equation}
%
\begin{equation}
\int_v [ N ]^T 
\rho c\frac{\partial T}{\partial t}
 dv
\end{equation}
%
�����ŁC��\ref{2diff}�́C�Q�K���������܂ނ̂ŁC�����ϕ���p����1�K�����Ɏ�`�������s���D
%
\begin{equation}\label{bubun}
\int_v N \nabla^2 A dv= \int_S N \nabla A \cdot ndS- \int_v \nabla N \cdot \nabla A dv 
\end{equation}
%
��\ref{bubun}��p���āC��\ref{2diff}��ό`����ƁC�����̂悤�ɂȂ�D
%
%��������
\begin{eqnarray}\label{eqn1-8}
\nonumber & & \int_v [ N ]^T \{ \frac{\partial }{\partial x} \left( \lambda \frac{\partial T}{\partial x} \right) + \frac{\partial }{\partial y} \left( \lambda \frac{\partial T}{\partial y} \right) + \frac{\partial }{\partial z} \left( \lambda \frac{\partial T}{\partial z}\right)  \} dv 	\\
\nonumber & &= \int_S [ N]^T \left( \lambda \frac{\partial T}{\partial x} n_x + \lambda \frac{\partial T}{\partial y} n_y +\lambda \frac{\partial T}{\partial z} n_z \right) dS 			\\
& & - \int_v \{ \frac{\partial [ N]^T }{\partial x} \left( \lambda \frac{\partial T}{\partial x}\right) +\frac{\partial \{ N\}^T }{\partial y} \left( \lambda \frac{\partial T}{\partial y}\right) + \frac{\partial \{ N\}^T }{\partial z} \left( \lambda \frac{\partial T}{\partial z}\right) \}
dv
\end{eqnarray}
%
%
%�i�]�u�s����g���ĕ\���Ă��債�ĈႢ�������낤�����ߎ��D��͓̂]�u�s��ɂȂ�͂�������ˁD�Ƃ����ژ_���Ȃ�ł��傤�ˁD�@���v����\cite{}�j
%

��\ref{eqn1-9}����\ref{eqn1-8}�ɓK�p����ƁC�����̂悤�ɂȂ�D
�������C�ߓ_���x�͋�Ԃ̊֐��ł͂Ȃ��̂ŁC�ϕ��̊O�ɏo�����Ƃ��ł���D
%
\begin{eqnarray}\label{eqn1-10}
\nonumber & & \int_v [ N ]^T \{ \frac{\partial }{\partial x} \left( \lambda \frac{\partial T}{\partial x} \right) + \frac{\partial }{\partial y} \left( \lambda \frac{\partial T}{\partial y} \right) 			+ \frac{\partial }{\partial z} \left( \lambda \frac{\partial T}{\partial z}\right)  \} dv 		\\
\nonumber & & = \int_S [ N]^T \left( \lambda \frac{\partial T}{\partial x} n_x + \lambda \frac{\partial T}{\partial y} n_y +\lambda \frac{\partial T}{\partial z} n_z \right) dS 			\\
\nonumber & &	- \int_v \lambda \{ \frac{\partial [ N]^T }{\partial x} \frac{\partial  [N] \{T\}_{ele} }{\partial x} +\frac{\partial [ N]^T }{\partial y}  \frac{\partial [ N] \{T\}_{ele} }{\partial y} + \frac{\partial [ N]^T }{\partial z} \frac{\partial [ N] \{T\}_{ele} }{\partial z} \}
dv \\
\nonumber & & = \int_S [N]^T [\lambda] \frac{\partial T}{\partial n}dS�@\\
& & - \int_v \lambda \{ \frac{\partial [ N]^T }{\partial x}  \frac{\partial  [ N] }{\partial x} + \frac{\partial [ N]^T }{\partial y}  \frac{\partial [ N] }{\partial y} + \frac{\partial [N]^T }{\partial z}  \frac{\partial [N]  }{\partial z} \} dv \cdot \{T\}_{ele}
\end{eqnarray}

//(��ڂ�$[N]^{T}$(�]�u)�̓K���[�L���@�ɂ����́D�������$[N]$��$T=NT$�ɂ�����)//

��\ref{eqn1-10}�̉E�ӑ�ꍀ�Ɏ�\ref{eqn1-2}�̃t�[���G�̖@���̎���K�p�����
%
\begin{equation}
\int_S [N]^T[ \lambda] \frac{\partial T}{\partial n} dS = - \int_S q[N]^T dS
\end{equation}
%yagawa��pp.103�@(3)�����\cite{yagawa1983}
��������D�ȏ���C�x�z�������ɃK���[�L���@��K�p(\ref{galerkin})���邱�ƂŁC���̗��U�����ꂽ����������D
%
%
\begin{eqnarray}\label{galerkin_s}
\nonumber & & - \int_v \lambda \{ \frac{\partial [N]^T }{\partial x}  \frac{\partial  [N] }{\partial x} + \frac{\partial [N]^T }{\partial y}  \frac{\partial [N] }{\partial y} + \frac{\partial [N]^T }{\partial z}  \frac{\partial [N]  }{\partial z} \} dv \cdot \{T\}_{ele} \\
\nonumber & & - \int_S q[N]^T dS + \int_v \frac{\partial Q}{\partial t} dv \\
& & -  \int_v \rho c[N]^T [N] \frac{\partial \{T\}_{ele} }{\partial t} dv = 0
\end{eqnarray}
%
���ӂ�$ -1 $��������
%
%
\begin{eqnarray}\label{galerkin_s2}
\nonumber & &  \int_v \lambda \{ \frac{\partial [N]^T }{\partial x}  \frac{\partial  [N] }{\partial x} + \frac{\partial [N]^T }{\partial y}  \frac{\partial [N] }{\partial y} + \frac{\partial [N]^T }{\partial z}  \frac{\partial [N]  }{\partial z} \} dv \cdot \{T\}_{ele} \\
\nonumber & & + \int_S q[N]^T dS - \int_v \frac{\partial Q}{\partial t} dv \\
& & +  \int_v \rho c[N]^T [N] \frac{\partial \{T\}_{ele} }{\partial t} dv = 0
\end{eqnarray}
%

\subsection{���x���E����}
���ɁC���E�����ɂ��čl����D
���E�����ɂ́C���x�Œ�����C�M���������C�M�`�B�����C�M�t�ˏ�����4�̋��E����������D
%
\subsubsection{���x���E�����F}
���x�Œ苫�E�����́C�L���v�f�@�ł̌Œ苫�E�����ɑ������C
���E$S_1 $��ŌŒ艷�x$ T_fix $�Ƃ���ƁC
\begin{equation}
T = T_{fix}�@�@�i���ES_1��Łj
\end{equation}

�M�������E�����F
�M�������E�����ł́C���E$S_2$��ŔM����$q_0 [W / (m \cdot K)]$�����o������Ƃ������E�����ł���C
\begin{equation}
q =  q_0�@�@�i���ES_2��Łj
\end{equation}
%
���������āC��\ref{galerkin_s2}�ł̋��E�ϕ����́C�����̂悤�ɂȂ�D
%
\begin{equation}
\int_{S2} q[ N]^T dS�@= \int_{S2} q_0[N]^T dS
\end{equation}


�M�`�B���E�����F
�M�`�B���E�����ł́C���E$ S_3 $��Ōő̕\�ʂƎ��͂𗬂�闬�̂Ȃǂ̂Ƃ̊Ԃł̔M�`�B��������D
�M�`�B����$\alpha[W/(m^2 \cdot K)]$�C���̗͂��̂Ȃǂ̉��x��$T_c$�Ƃ���ƁC
%
\begin{equation}
q =  \alpha(T - T_c)�@�@�i���ES_3��Łj
\end{equation}
%
���l�ɂ��āC��\ref{galerkin_s2}�ł̋��E�ϕ����́C�����̂悤�ɂȂ�D(�������C�v�f�S�̂̒l�ł͂Ȃ��C�v�f�̐ߓ_�̒l�����߂Ă��������̂�$[N]^T$���悶�鏇�Ԃ����ւ���D)
%
%
\begin{eqnarray}\label{eqn_s3}
\nonumber \int_{S3} q[N]^T dS�@&=& \int_{S3} \alpha [N]^T (T - T_c) dS \\
\nonumber &=& \int_{S3} \alpha [N]^T( [N] \cdot T_{ele} ) dS - \int_{S3} \alpha T_c [N]^T dS \\
		  &=& \int_{S3} \alpha [ N]^T [N] dS \cdot T_{ele} - \int_{S3} \alpha T_c [ N]^T dS
\end{eqnarray}

�M�t�ˋ��E�����F
�M�t�ˋ��E�����ł́C���E$S_4$��Ŏ��͊��ɑ΂��ĔM�t�˂��s����D
�M�t�˗���$h[W/(m^2 \cdot K]$�C���͊����x��$T_{OUT}$�Ƃ���ƁC�����̂悤�ɂȂ�D
%
%
\begin{equation}
q =  h(T - T_{OUT})�@�@�i���ES_4��Łj
\end{equation}
%
���l�ɂ��āC��\ref{galerkin_s2}�ł̋��E�ϕ����́C�����̂悤�ɂȂ�D
%
\begin{eqnarray}
\nonumber \int_{S4} q[ N]^T dS�@&=& \int_{S3} h(T - T_{OUT}) [N]^T dS \\
\nonumber &=& \int_{S4} h ( [N] \cdot T_{ele} ) [N]^T dS - \int_{S4} h T_{OUT} [N]^T dS \\
		  &=& \int_{S4} h[N] [N]^T dS \cdot T_{ele} - \int_{S4} h T_{OUT} [N]^T dS
\end{eqnarray}
%
%�������C�M�t�˂͉��x���̂̂S��ɔ�Ⴗ��̂ŁC�Ƃ��������C�Ƌߎ����āC������D
%
\clearpage
\subsubsection{���U�����ꂽ��}
%
�ȏ���C�M�`���������̗��U����\ref{galerkin_s2}�ɋ��E�����������������́C���̂悤�ɂȂ�D
%
\begin{equation}\label{risaneq}
	[K] \{T \} + [C]\{\frac{\partial T}{\partial t} \} = \{ F \}
\end{equation}
%
�������C
%
\begin{eqnarray}\label{risanK}
&& \nonumber [K] = \int_v \lambda \{ \frac{\partial [ N]^T }{\partial x}  \frac{\partial  [ N] }{\partial x} + \frac{\partial [N]^T }{\partial y}  \frac{\partial [N] }{\partial y} + \frac{\partial [N]^T }{\partial z}  \frac{\partial [N]}{\partial z} \} dv \\
& & �@�@�@+ \int_{S3} \alpha[N]^T [N] dS + \int_{S4} h[N]^T [N] dS \\ \label{risanC}
&&  [C] = \int_v \rho c[N]^T [N] dv \\ \label{risanF}
&& \nonumber \{F\} = \int_v \frac{\partial Q}{\partial t} dv - \int_{S2} q_0[N]^T dS + \int_{S3} \alpha T_c [N]^T dS \\
& & �@�@�@+ \int_{S4} hT_{OUT} [N]^T dS 
\end{eqnarray}
%
�ł���D
�������C$ [K] $�͔M�`���}�g���N�X�C$ [C] $�͔M�e�ʃ}�g���N�X�C $ \{F\} $�͔M�����x�N�g���ł���D
%
%
\subsubsection{���ԂɊւ��Ă̗��U��}
�K���[�L���@�ł́C��ԂɊւ��ė��U�����s�����D���ɁC���ԂɊւ��ė��U�����s���D
���ԂɊւ��闣�U����@�Ƃ��āC�N�����N�E�j�R���\���̍�������p����(�Q�l����\cite{yagawa1983}(pp.117���))�D
����$ t+\frac{\Delta t }{2}(\Delta �͔������ԑ���) $�ɂ�����ߓ_���x�x�N�g����
\begin{eqnarray} \label{T(t+}
\left\{ T\begin{pmatrix}t+\frac{\Delta  }{2}\end{pmatrix} \right\} = \frac{1}{2}\begin{pmatrix} \left\{T(t + \Delta t) \right\} + \left\{T(t)\right\} \end{pmatrix}
\end{eqnarray}
�ƕ\���D
�܂��C����$ t + \frac{\Delta t}{2} $�ɂ�����ߓ_���x�x�N�g���̎��Ԕ�����
\begin{eqnarray} \label{dT/dt(t+}
\left\{ \frac{\partial T}{\partial t} \begin{pmatrix} t+\frac{\Delta  }{2} \end{pmatrix} \right\}
= 
\frac{ \left\{T(t + \Delta t) \right\} - \left\{T(t)\right\} }{\Delta t}
\end{eqnarray}
�ƕ\���D
�������C$ \{T(t)\} $�͂��鎞��$t$�ɂ�����S�̂̐ߓ_���x�x�N�g����\���D

%
��\ref{risaneq}�Ɏ�\ref{T(t+},\ref{dT/dt(t+}��������ƁC
\begin{eqnarray}\label{crankeq}
\begin{pmatrix} \frac{1}{2}[K]+\frac{1}{\Delta t}[C] \end{pmatrix} \left\{ T(t+ \Delta t) \right\}
&=& 
\begin{pmatrix}-\frac{1}{2}[K] + \frac{1}{\Delta t}[C] \end{pmatrix} \{ T(t) \} + \{ F \}
\end{eqnarray}
����𒀎������ċ��߂�Ηǂ��D
�������C�N�����N�E�j�R���\���̍������ł́C$ \Delta t $�̑傫����������x(1sec�`)�傫���Ȃ�Ɖ����U������Ƃ̕񍐂�����D
($http://computation.cside.com/TA/time_integration_04.html$�Q��)
���̂��߁C�V�~�����[�V�������s���̎��Ԃɂ̐ݒ�ɂ͒��ӂ��K�v�ł���\��������D

\subsubsection{�N�����N�E�j�R���\���̍�������p���Ȃ����U����@(�v�����E�o�T)}
(�ȉ��C�ؑ�����ɂ�����)
$ n+1 $���Ԍ�̐ߓ_���x�̃}�g���N�X$ \{ T \}_{n+1} $ ��p���āC�����̂悤�ɕό`����D
%
%
\begin{eqnarray}\label{���U����}
&& \nonumber	[K] \{T \} + [C]\left\{\frac{\partial T}{\partial t} \right\} = \{ F \} \\
&& \nonumber	[K] \{T \} + [C] \frac{ \{ T\}_{n+1} - \{ T \}_{n}}{h} = \{ F \} \\
&&  \begin{pmatrix} [K] + \frac{1}{h} [C] \end{pmatrix} \{ T \}_{n+1} = \{ F \} +\frac{1}{h} [C] \{ T \}_{n}
\end{eqnarray}
�������C$ h $�͒P�ʔ������ԂƂ���D
%
%
\subsubsection{�L���v�f�`��ɍ��킹�����U��}
%
\subsubsection{2�������̏ꍇ}
(1��)�O�p�`�v�f�̏ꍇ
%
������u���C
\begin{equation}
u = a x +b y + c
\end{equation}
�Ƃ���ƁC
%
%
\subsubsection{3�������̏ꍇ}
(1��)�l�ʑ̗v�f
%
3�����̃��f�����l�ʑ̗̂L���v�f�̃��b�V���ŋ�؂�ꍇ���l����D
���}�̂悤�ɁC�P�˂Q�˂R�˂P�ƉE�˂������߂�悤�Ɏl�ʑ̗v�f�̐ߓ_�ԍ������߂�ƁC���_�S�����܂�D
%
\begin{figure}[htb]
\center
\includegraphics[width = 5cm]{tetrahedral.eps}
\caption{�l�ʑ̌`��̗L���v�f}
\label{fig:tetrahedral_screw_}
\end{figure}
%
�v�f�̊e�ߓ_
���̗v�f���̔C�ӓ_(x,y,z)�ɂ����鉷�x��$ T $�Ƃ���ƁC�W��$ a,b,c,d $�����߂邱�Ƃ�$ T $�͎����̂悤�ɕ\�킷���Ƃ��ł���D
%
\begin{equation}
T(x,y,z) = a x +b y + c z +d
\end{equation}
%
%
�W��$ a,b,c,d $�́C�e�ߓ_�ł̉��x$ T_1,T_2,T_3,T_4 $�Ɗe���_�̂R�������W($i��1�`4$�̂Ƃ���$x_i,y_i,z_i$���\���ʒu���W)�����̎��̂悤�ɑ���������̕������������ċ��߂�D
\begin{eqnarray}
\nonumber T_1 = a x_1 +b y_1 + c z_1 +d \\
\nonumber T_2 = a x_2 +b y_2 + c z_2 +d \\
\nonumber T_3 = a x_3 +b y_3 + c z_3 +d \\
T_4 = a x_4 +b y_4 + c z_4 +d
\end{eqnarray}
%
%�������C$x_i,y_i,z_i$��$i���P�`�S$�Ŋe�ߓ_�̔ԍ��ɑΉ����C���ӂ�$ T_i $�͊e���_�ł̉��x�ɑΉ�����D
%
����������ċ��߂�$ a,b,c,d $���܂Ƃ߂�ƁC�v�f���̔C�ӓ_�̉��x$T$�́C�`��֐�$N$��p���Ď����̂悤�ɋߎ����邱�Ƃ��ł���D
%
\begin{equation}
T = N_1 u_1 +N_2 u_2 + N_3 u_3 + N_4 u_4
\end{equation}
%
�������C$ N_1, N_2, N_3, N_4 $�͎����Ɏ����`��֐��ł���C
%
\begin{equation}
N_i = \frac{1}{6V} ( a_i x + b_i y + c_i z +d_i )�@�@�@(i=1,2,3,4)
\end{equation}
�ƂȂ�D
%
�܂��C$ V $�͎l�ʑ̂̑̐ς�\���C�}\ref{fig:tetrahedral_screw}�̋Ǐ��ߓ_1,2,3�𒸓_�Ƃ����O�p�`�̖ʐςƓ��O�p�`����Ǐ��ߓ_�S�܂ł̍���L�Ƃ̐ς̂R���̂P�Ɠ������D
�܂��C���_O����ߓ_1,2,3,4�ւ̃x�N�g����p����Ǝ�\ref{v=1/6det}�̂悤�ɂ�����D

%
%������V�́C�����̂悤�ɍs�񎮂ɂ��l�ʑ̗v�f�̑̐ς��v�Z�������̂ł���D
%
%
\begin{equation}\label{v=1/6det}
V = \frac{1}{6} det
	\begin{array}{|cccc|}
	1 & x_1 & y_1 & z_1 \\
	1 & x_2 & y_2 & z_2 \\
	1 & x_3 & y_3 & z_3 \\
	1 & x_4 & y_4 & z_4 \\
	\end{array}
\end{equation}
%
%
�܂��C$a_i,b_i,c_i,d_i,$�́C���Ɏ����s�񎮂̌v�Z�l�ł���D
%
\begin{eqnarray}
\nonumber a_i &=& (-1)^i (y_k z_l + y_l z_j + y_j z_k - y_k z_j - y_l z_k -y_j z_l) \\
 &=& (-1)^i det
	\begin{array}{|ccc|}
	1 & y_j & z_j \\
	1 & y_k & z_k \\
	1 & y_l & z_l \\
	\end{array} \\
\nonumber b_i &=& (-1)^i (x_j z_l + x_k z_j + x_l z_k - x_l z_j - x_j z_k -x_k z_l) \\
 &=& (-1)^i det
	\begin{array}{|ccc|}
	x_j & 1 & z_j \\
	x_k & 1 & z_k \\
	x_l & 1 & z_l \\
	\end{array} \\
\nonumber c_i &=& (-1)^i (x_j y_k + x_k y_l + x_l y_j - x_l y_k - x_j y_l -x_k y_j) \\
 &=& (-1)^i det
	\begin{array}{|ccc|}
	x_j & y_j & 1 \\
	x_k & y_k & 1 \\
	x_l & y_l & 1 \\
	\end{array} \\
\nonumber d_i &=& (-1)^i (x_j y_k z_l + x_k y_l z_j + x_l y_j z_k - x_l y_k z_j - x_j y_l z_k -x_k  y_j z_l) \\
 &=& (-1)^i det
	\begin{array}{|ccc|}
	x_j & y_j & z_j \\
	x_k & y_k & z_k \\
	x_l & y_l & z_l \\
	\end{array} \\
\end{eqnarray}
%
�������C
\begin{eqnarray}
\nonumber i=1�ɑ΂��āCj=2,k=3,l=4 \\
\nonumber i=2�ɑ΂��āCj=3,k=4,l=1 \\
\nonumber i=3�ɑ΂��āCj=4,k=1,l=2 \\
\nonumber i=4�ɑ΂��āCj=1,k=2,l=3
\end{eqnarray}
%
�̑Ή��֌W�ƂȂ��Ă���D
%
%
�܂��C��Ɏ������`��֐��́C���̐����������D
%
\begin{eqnarray}
&�@& x,y,z�̑������ł���.\\
&�A& N_i(x_j,y_j,z_j) = \left \{
\begin{array}{l}
0�@(i \neq j�̂Ƃ�) \\
1�@(i=j�̂Ƃ�)
\end{array}
\right.
�@(i=1,2,3,4, j=1,2,3,4)) \\
&�B& N_1 + N_2 + N_3 + N_4 = 1�@\label{�`��֐��̘a}
\end{eqnarray} 
%
%
�l�ʑ̗v�f���̔C�ӓ_$ (x,y,z) $�ł̌`��֐�$ N_1 $�̒l�́C�ߓ_$ 2,3,4 $�ň͂܂��ʂƐߓ_1�̊Ԃɂ����镨���ʂ��ߎ�������̂ł���C�ߓ_$ 2,3,4 $�Ɠ_$ (x,y,x) $�𒸓_�Ƃ���l�ʑ̂̑̐ςƐߓ_$ 1,2,3,4 $�ň͂܂ꂽ�l�ʑ̗v�f�S�̂̑̐ς̔�ɓ������D
%
\begin{figure}[htb]
\center
\includegraphics[width = 4cm]{tetrahedral.eps}
\caption{�l�ʑ̌`��̗L���v�f}
\label{fig:tetrahedral_screw}
\end{figure}
%
���l�ɁC�`��֐�$ N_2 $�́C�ߓ_$ 1,3,4 $�Ɨv�f���̔C�ӓ_$ x,y,z $�𒸓_�Ƃ���l�ʑ̂̑̐ςƗL���v�f�S�̂̑̐ςƂ̔�ɓ������D

�܂��C���Ƃ��΁C�ߓ_$ 2,3,4 $�ň͂܂��O�p�`���ʏ�ł́C$ N_1 = 0 $�Ȃ̂�(�A�̐������)�C�����ʂ́C$ u_2,u_3,u_4 $�Ɉˑ����邪�C$ u_1 $�Ɩ��֌W�ƂȂ�D

�ߓ_$ 2,3,4 $�ň͂܂ꂽ�O�p�`�̖ʏ�̓_�C�ӓ_$ P(x,y,z,T) $�̉��xT�́C$ u_1 $�̉e���͎󂯂��C������$ u_2,u_3,u_4 $�ɂ݂̂Ɉˑ�����D
%

���ɁC�M�`���������̍����̎��ɂ́C���x$ T $��$ x,y,z $�����̕Δ������܂܂�邽�߁C��������߂邱�Ƃɂ���D
%
��ɁC���߂���
%
\begin{equation}
\nonumber T = N_1 T_1 +N_2 T_2 + N_3 T_3 + N_4 T_4
\end{equation}
%
���C�e�ߓ_�ł̉��x$ T_i $(i=1,2,3,4)�́C�ߓ_�ŌŒ�̒萔�ł���̂ŁC
%
\begin{eqnarray}
\nonumber \frac{\partial T }{\partial x}  &=& \frac{\partial N_1 }{\partial x} T_1 + \frac{\partial N_2 }{\partial x} T_2 + \frac{\partial N_3 }{\partial x} T_3 + \frac{\partial N_4 }{\partial x} T_4   \\
  &=& \frac{a_1 T_1 + a_2 T_2 + a_3 T_3 + a_4 T_4 }{6V}   \\  
\frac{\partial T }{\partial y}  &=& \frac{b_1 T_1 + b_2 T_2 + b_3 T_3 + b_4 T_4 }{6V}  \\
\frac{\partial T }{\partial z}  &=& \frac{c_1 T_1 + c_2 T_2 + c_3 T_3 + c_4 T_4 }{6V} 
\end{eqnarray}
%
�����ŁC$ a_i�`c_i,T_i�@(i=1,2,3,4),V $�͒萔�Ȃ̂ŁC$ \frac{\partial T }{\partial x},\frac{\partial T }{\partial y},\frac{\partial T }{\partial z} $�͒萔�ƂȂ�D
%
%


�ȏ���C$ \frac{\partial [N] }{\partial x} , \frac{\partial [N]^T }{\partial x} $���v�Z���邱�Ƃ��ł���D
������L���v�f���Ƃɋ��߂āC�ϕ����C�S�̍����s������D



\subsubsection{�v�f���̐ߓ_�Ԃ̌v�Z���ɕϊ�����}
%
���U��������\ref{risanK},\ref{risanC},\ref{risanF}��v�f���̐ߓ_�Ԃ̊֌W�ɒ����C�v�Z�\�Ȏ��ɕό`����D
%
\subsection{$ (x,y,z) $���W�n���� $ (\xi,\eta,\zeta) $���W�n�ւ̍��W�ϊ�}
\subsubsection{�`��֐��̍��W�ϊ�}
�����ł܂��C�l���₷�����邽�߂ɁC��\ref{risanK}�̉E�ӑ�1,2,3����$ K_1,K_2,K_3 $�Ƃ����C $ (\xi ,\eta,\zeta ) $���W�n�ɕϊ�����ߒ��ɂ��čl����D
%
\begin{eqnarray}\label{risanK1}
&& \nonumber (��\ref{risanK}�̉E�ӑ�ꍀ���Čf)�F \\
&&K_1 =  \int_v \lambda \left\{ \frac{\partial [ N]^T }{\partial x}  \frac{\partial  [ N] }{\partial x} + \frac{\partial [N]^T }{\partial y}  \frac{\partial [N] }{\partial y} + \frac{\partial [N]^T }{\partial z}  \frac{\partial [N]}{\partial z} \right\} dv
\end{eqnarray}
%
�ȉ��C��\ref{risanK1}��$ (\xi ,\eta,\zeta ) $���W�ɕϊ����Čv�Z����ߒ��������D
%
�l�ʑ̗v�f���̔C�ӓ_$ (x,y,z) $�͌`��֐�$ N_1,N_2,N_3,N_4 $��p���Ď����̂悤�ɕ\����D
%
��\ref{xyzN}�̏ؖ��́C��Ɏ����D
%
\begin{eqnarray}\label{xyzN}
\nonumber x &=& N_1 x_1 + N_2 x_2 + N_3 x_3 + N_4 x_4 \\
y &=& N_1 y_1 + N_2 y_2 + N_3 y_3 + N_4 y_4 \\
\nonumber z &=& N_1 z_1 + N_2 z_2 + N_3 z_3 + N_4 z_4
\end{eqnarray}
%
������C�}\ref{fig:zahyo_trans}�̂悤�ȍ��W�n�ɕϊ�����D
%
\begin{figure}[htb]
\center
\includegraphics[width = 10cm]{zahyo_trans.eps}
\caption{$ (x,y,z) $���W�n����C$ (\xi ,\eta,\zeta ) $���W�n�ւ̕ϊ�}
\label{fig:zahyo_trans}
\end{figure}
%
�f�J���g���W�n$ (x,y,z) $�ŕ\���ꂽ�l�ʑ̓��̔C�ӓ_���C�l�ʑ̓��̍��W�n$ (\xi ,\eta,\zeta ) $�ŕ\���āC�����o����铙�������W�ϊ����s���D

��\ref{xyzN}���l�ʑ̓��̍��W�n�ŕ\���ƁC����\ref{xyz��}�̗l�ɂȂ�D

%
\begin{eqnarray}\label{xyz��}
\nonumber x &=& (1-\xi -\eta -\zeta ) x_1 + \xi  x_2 +  \eta  x_3 + \zeta  x_4 \\
y &=& (1-\xi - \eta -\zeta ) y_1 + \xi  y_2 +  \eta  y_3 + \zeta  y_4 \\
\nonumber z &=& (1-\xi - \eta -\zeta ) z_1 + \xi  z_2 +  \eta  z_3 + \zeta  z_4
\end{eqnarray}
%
%
�����̎��͈ȉ���蓱�o����D
\begin{eqnarray}
x=N 
\left[\begin{array}{c}
x_1 \\
x_2 \\
x_3 \\
x_4 \\
\end{array}
\right]
,
y=N 
\left[\begin{array}{c}
y_1 \\
y_2 \\
y_3 \\
y_4 \\
\end{array}
\right]
,
z=N
\left[\begin{array}{c}
z_1 \\
z_2 \\
z_3 \\
z_4 \\
\end{array}
\right]
,
N=
\left[\begin{array}{cccc}
1-\xi - \eta -\zeta & \xi & \eta & \zeta
\end{array}
\right]
\end{eqnarray}
%
�Ƃ���ƁC
%
\begin{eqnarray}
\nonumber
x
&=&
N 
\left[\begin{array}{c}
 x_1 \\
 x_2 \\
 x_3 \\ 
 x_4
\end{array}
\right]
\\
\nonumber
&=&
\left[\begin{array}{cccc}
 1-\xi - \eta -\zeta & \xi & \eta & \zeta
\end{array}
\right]
\left[\begin{array}{c}
 x_1 \\
 x_2 \\
 x_3 \\
 x_4
\end{array}
\right]
\\
\nonumber
&=&(1-\xi - \eta -\zeta )x_1 + \xi x_2 +  \eta x_3 + \zeta x_4
\end{eqnarray}
%
���l�ɂ��āC
%
\begin{eqnarray}
y=(1-\xi - \eta -\zeta )y_1 + \xi y_2 +  \eta y_3 + \zeta y_4 \\
\nonumber�@z=(1-\xi - \eta -\zeta )z_1 + \xi z_2 +  \eta z_3 + \zeta z_4
\end{eqnarray}
%
���ꂼ���ό`���āC
%
\begin{eqnarray}\label{xyzforJ}
\nonumber�@x &=& (x_2 - x_1)\xi +(x_3-x_1) \eta +(x_4-x_1)\zeta +x_1 \\
 y &=& (y_2 - y_1)\xi +(y_3-y_1) \eta +(y_4-y_1)\zeta +y_1  \\
\nonumber�@z &=& (z_2 - z_1)\xi +(z_3-z_1) \eta +(z_4-z_1)\zeta + z_1 
\end{eqnarray}
%
�ƂȂ�D

�v�Z�ɗp����̈�́C$ 0��\xi ��1, �@0�� \eta ��1, �@0��\zeta ��1, �@0��\xi + \eta +\zeta ��1 $�ƂȂ�悤�Ȏl�ʑ̓����ɂ��čl���C�`��֐�$ N(1�~4�̍s�}�g���N�X) $�͎����̂悤�ɒu�����������̂�p����D
%
\begin{eqnarray}
N&=&
\left[
\begin{array}{cccc}
N_{1} & N_{2} & N_{3} & N_{4}
\end{array}
\right]
\\
&=&
\left[
\begin{array}{cccc}
1-\frac{V_{2}}{V}-\frac{V_{3}}{V}-\frac{V_{4}}{V} & \frac{V_{2}}{V} & \frac{V_{3}}{V} & \frac{V_{4}}{V}
\end{array}
\right]
\\
&=&
\left[
\begin{array}{cccc}
1-\xi - \eta -\zeta & \xi & \eta & \zeta
\end{array}
\right]
\end{eqnarray}

�������C$ V $�͎l�ʑ̗v�f�̑̐ρC$ V_{2},V_{3},V_{4} $�͂��ꂼ��C�ߓ_$1,3,4$�Ɨv�f�����̔C�ӓ_$ (x,y,z) $�𒸓_�Ƃ���l�ʑ̂̑̐ρC�ߓ_$1,2,4$�Ɨv�f�����̔C�ӓ_$ (x,y,z) $�𒸓_�Ƃ���l�ʑ̂̑̐ρC�ߓ_$1,2,3$�Ɨv�f�����̔C�ӓ_$ (x,y,z) $�𒸓_�Ƃ���l�ʑ̂̑̐ς�\���D�܂��C\ref{�`��֐��̘a}���C�`��֐�$ N_{1} $��\������D

\subsubsection{
$
\frac{\partial N}{\partial x} ,
\frac{\partial N}{\partial y} ,
\frac{\partial N}{\partial z} 
$
��$ (\xi, \eta, \zeta) $���W�n�ւ̕ϊ�}

���ɁC�����֐��̕Δ����@���C
%
\begin{eqnarray}
\nonumber
\left[
\begin{array}{c}
\frac{\partial N}{\partial \xi } \\
\frac{\partial N}{\partial  \eta } \\
\frac{\partial N}{\partial \zeta } \\
\end{array}
\right]
&=&
\left[
\begin{array}{ccc}
\frac{\partial x}{\partial \xi }\cdot \frac{\partial N}{\partial x} & \frac{\partial y}{\partial \xi }\cdot\frac{\partial N}{\partial y} & \frac{\partial z}{\partial \xi }\cdot\frac{\partial N}{\partial z} \\
\frac{\partial x}{\partial  \eta }\cdot \frac{\partial N}{\partial x} & \frac{\partial y}{\partial  \eta }\cdot\frac{\partial N}{\partial y} & \frac{\partial z}{\partial  \eta }\cdot\frac{\partial N}{\partial z} \\
\frac{\partial x}{\partial \zeta }\cdot \frac{\partial N}{\partial x} & \frac{\partial y}{\partial \zeta }\cdot\frac{\partial N}{\partial y} & \frac{\partial z}{\partial \zeta }\cdot\frac{\partial N}{\partial z} \\
\end{array}
\right] \\ \label{[J]}
&=&
\left[
\begin{array}{ccc}
\frac{\partial x}{\partial \xi } & \frac{\partial y}{\partial \xi } & \frac{\partial z}{\partial \xi } \\
\frac{\partial x}{\partial  \eta } & \frac{\partial y}{\partial  \eta } & \frac{\partial z}{\partial  \eta } \\
\frac{\partial x}{\partial \zeta } & \frac{\partial y}{\partial \zeta } & \frac{\partial z}{\partial \zeta } \\
\end{array}
\right]
\left[
\begin{array}{c}
\frac{\partial N}{\partial x} \\
\frac{\partial N}{\partial y} \\
\frac{\partial N}{\partial z} \\
\end{array}
\right] \\
&=&
\left[
\begin{array}{c}\label{NJN}
J
\end{array}
\right]
\left[
\begin{array}{c}
\frac{\partial N}{\partial x} \\
\frac{\partial N}{\partial y} \\
\frac{\partial N}{\partial z} \\
\end{array}
\right]
\end{eqnarray}
%
%
������[J]�̓��R�r�s��ł���D���ӂ�$ [J]^{-1} $���悶�邱�Ƃɂ�莟���𓾂�D
%
\begin{eqnarray}
\left[
\begin{array}{c}
\frac{\partial N}{\partial x} \\
\frac{\partial N}{\partial y} \\
\frac{\partial N}{\partial z} \\
\end{array}
\right]
=
\left[
\begin{array}{c}
J
\end{array}
\right]
^{-1}
\left[
\begin{array}{c}
\frac{\partial N}{\partial \xi } \\
\frac{\partial N}{\partial \eta } \\
\frac{\partial N}{\partial \zeta } \\
\end{array}
\right]
\end{eqnarray}
%
%
\begin{eqnarray}
\nonumber
[J]^{-1}
&=&
\frac{1}{det|J|}
\left[
\begin{array}{ccc}
\frac{\partial y}{\partial  \eta }\cdot\frac{\partial z}{\partial \zeta } - \frac{\partial z}{\partial  \eta }\cdot\frac{\partial y}{\partial \zeta } & \frac{\partial z}{\partial \xi }\cdot\frac{\partial y}{\partial \zeta } - \frac{\partial y}{\partial \xi }\cdot\frac{\partial z}{\partial \zeta } & \frac{\partial y}{\partial \xi }\cdot\frac{\partial z}{\partial  \eta } - \frac{\partial z}{\partial \xi }\cdot\frac{\partial y}{\partial  \eta }
\\
\frac{\partial z}{\partial  \eta }\cdot\frac{\partial x}{\partial \zeta } - \frac{\partial x}{\partial  \eta }\cdot\frac{\partial z}{\partial \zeta } & \frac{\partial x}{\partial \xi }\cdot\frac{\partial z}{\partial \zeta } - \frac{\partial z}{\partial \xi }\cdot\frac{\partial x}{\partial \zeta } & \frac{\partial z}{\partial \xi }\cdot\frac{\partial x}{\partial  \eta }
 - \frac{\partial x}{\partial \xi }\cdot\frac{\partial z}{\partial  \eta }
\\
\frac{\partial x}{\partial  \eta }\cdot\frac{\partial y}{\partial \zeta } - \frac{\partial y}{\partial  \eta }\cdot\frac{\partial x}{\partial \zeta } & \frac{\partial y}{\partial \xi }\cdot\frac{\partial x}{\partial \zeta } - \frac{\partial x}{\partial \xi }\cdot\frac{\partial y}{\partial \zeta } & \frac{\partial x}{\partial \xi }\cdot\frac{\partial y}{\partial  \eta } - \frac{\partial y}{\partial \xi }\cdot\frac{\partial x}{\partial  \eta }
\end{array}
\right]
\end{eqnarray}
%
��\ref{xyzforJ}��������
%
\tiny{
\begin{eqnarray}
[J]^{-1}
&=&
%{\scriptstyle
\frac{1}{det|J|}
\left[
\begin{array}{ccc}
\nonumber (y_3-y_1)(z_4-z_1)-(z_3-z_1)(y_4-y_1) & (z_2-z_1)(y_4-y_1)-(y_2-y_1)(z_4-z_1) &
(y_2-y_1)(z_3-z_1)- (z_2-z_1)(y_3-y_1) \\
\nonumber (z_3-z_1)(x_4-x_1)-(x_3-x_1)(z_4-z_1) & (x_2-x_1)(z_4 -z_1) - (z_2-z_1)(x_4-x_1) & (z_2-z_1)(x_3-x_1)- (x_2-x_1)(z_3-z_1) \\
\nonumber (x_3-x_1)(y_4-y_1)-(y_3-y_1)(x_4-x_1) & (y_2-y_1)(x_4-x_1) - (x_2-x_1)(y_4-y_1) & (x_2-x_1)(y_3-y_1) - (y_2-y_1)(x_3-x_1)
\end{array}
\right]
\end{eqnarray}
}
\normalsize{}	%�t�H���g�T�C�Y���ʏ�̎��ԍ��̕\���̂���
\begin{eqnarray}
\end{eqnarray}
%
%
�ƂȂ�D������C
\begin{eqnarray}
\nonumber
[J]^{-1}
%\left[
%}
&=&
\frac{1}{det|J|}
[A]
\\
&=&
\frac{1}{det|J|}
\left[
\begin{array}{ccc}\label{detJA}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33} \\
\end{array}
\right]
\end{eqnarray}
%
�Ƃ����D������C$ [A] $�̊e�v�f�͎l�ʑ̂̐ߓ_�̍��W�l($ \in $����)�ō\�����ꂽ�����l�s��ł��邱�Ƃ�������D

�܂��C����\ref{Jdef}���C
\begin{eqnarray}\label{Jdef}
[J] = \left[
\begin{array}{ccc}
\frac{\partial x}{\partial \xi } & \frac{\partial y}{\partial \xi } & \frac{\partial z}{\partial \xi } \\
\frac{\partial x}{\partial  \eta } & \frac{\partial y}{\partial  \eta } & \frac{\partial z}{\partial  \eta } \\
\frac{\partial x}{\partial \zeta } & \frac{\partial y}{\partial \zeta } & \frac{\partial z}{\partial \zeta } \\
\end{array}
\right]
\end{eqnarray}
%
%
\begin{eqnarray}
\nonumber det|J| &=& (x_2-x_1)(y_3-y_1)(z_4-z_1) + (y_2-y_1)(z_3-z_1)(x_4-x_1) +(z_2-z_1)(x_3-x_1)(y_4-y_1) \\
\nonumber & & -(x_4-x_1)(y_3-y_1)(z_2-z_1) - (x_2-x_1)(y_4-y_1)(z_3-z_1) - (x_3-x_1)(y_2-y_1)(z_4-z_1) 
\\
\nonumber &=& (x_2-x_1)\{ (y_3-y_1)(z_4-z_1)-(y_4-y_1)(z_3-z_1) \} \\
&&+ (y_2-y_1)\{ (z_3-z_1)(x_4-x_1)-(x_3-x_1)(z_4-z_1) \} \\
\nonumber && + (z_2-z_1)\{ (x_3-x_1)(y_4-y_1)-(y_3-y_1)(x_4-x_1) \}
\end{eqnarray}
�Ə���������D

%
%
%
��\ref{NJN}��ό`���C��\ref{detJA}���������
\begin{eqnarray}
\left[
\begin{array}{c}
\frac{\partial N}{\partial x} \\
\frac{\partial N}{\partial y} \\
\frac{\partial N}{\partial z} \\
\end{array}
\right]
&=&
\left[
\begin{array}{c}
J
\end{array}
\right]
^{-1}
\left[
\begin{array}{c}
\frac{\partial N}{\partial \xi } \\
\frac{\partial N}{\partial  \eta } \\
\frac{\partial N}{\partial \zeta } \\
\end{array}
\right] \\
&=&
\frac{1}{det|J|}
\left[
\begin{array}{ccc}\label{detJA}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33} \\
\end{array}
\right]
%
\left[
\begin{array}{c}
\frac{\partial N}{\partial \xi } \\
\frac{\partial N}{\partial  \eta } \\
\frac{\partial N}{\partial \zeta } \\
\end{array}
\right]
\\
\nonumber
&=&
\frac{1}{det|J|}
\left[
\begin{array}{c}
a_{11}\frac{\partial N}{\partial \xi } + a_{12}\frac{\partial N}{\partial  \eta } + a_{13}\frac{\partial N}{\partial \zeta } \\
a_{21}\frac{\partial N}{\partial \xi } + a_{22}\frac{\partial N}{\partial  \eta } + a_{23}\frac{\partial N}{\partial \zeta } \\
a_{31}\frac{\partial N}{\partial \xi } + a_{32}\frac{\partial N}{\partial  \eta } + a_{33}\frac{\partial N}{\partial \zeta } \\
\end{array}
\right]
\\
\nonumber
&=&
\frac{1}{det|J|}
\left[
\begin{array}{c}
a_{11}\begin{array}{cccc}[-1 & 1 & 0 & 0]\end{array} + a_{12}\begin{array}{cccc}[-1 & 0 & 1 & 0]\end{array} + a_{13}\begin{array}{cccc}[-1 & 0 & 0 & 1]\end{array} \\
a_{21}\begin{array}{cccc}[-1 & 1 & 0 & 0]\end{array} + a_{22}\begin{array}{cccc}[-1 & 0 & 1 & 0]\end{array} + a_{23}\begin{array}{cccc}[-1 & 0 & 0 & 1]\end{array} \\
a_{31}\begin{array}{cccc}[-1 & 1 & 0 & 0]\end{array} + a_{32}\begin{array}{cccc}[-1 & 0 & 1 & 0]\end{array} + a_{33}\begin{array}{cccc}[-1 & 0 & 0 & 1]\end{array} \\
\end{array}
\right]
\\
&=& \label{a_11}
\frac{1}{det|J|}
\left[
\begin{array}{cccc}
-a_{11} - a_{12} - a_{13} & a_{11} & a_{12} & a_{13} \\
-a_{21} - a_{22} - a_{23} & a_{21} & a_{22} & a_{23} \\
-a_{31} - a_{32} - a_{33} & a_{31} & a_{32} & a_{33} \\
\end{array}
\right]
\\
&=& \label{N_xN_yN_z}
\frac{1}{det|J|}
\left[
\begin{array}{c}
\left[ N_x \right] \\
\left[ N_y \right] \\
\left[ N_z \right] \\
\end{array}
\right]
\end{eqnarray}
�Ƃ����B

%�ƂȂ�D
%��\ref{detJA}���C
%$ \frac{\partial N}{\partial x} ,\frac{\partial N}{\partial y },\frac{\partial N}{\partial z } $
%�̊e�X�ɂ��ċ��߂�D
%
%\begin{eqnarray}
%\nonumber
%\frac{\partial N}{\partial x}
%
%&=&
%\frac{1}{det|J|}\left[ a_{11}\frac{\partial N}{\partial \xi } + a_{12}\frac{\partial N}{\partial  \eta } + a_{13}\frac{\partial N}{\partial \zeta }  \right] \\
%\nonumber
%&=&
%\frac{1}{det|J|}\left[a_{11}\left[\begin{array}{c}-1 \\ 1 \\ 0 \\ 0 \\ \end{array}\right] + a_{12}\left[\begin{array}{c}-1 \\ 0 \\ 1 \\ 0 \\ \end{array}\right] + a_{13}\left[\begin{array}{c}-1 \\ 0 \\ 0 \\ 1 \\ \end{array}\right]
%\right] \\
%\nonumber
%&=&
%\frac{1}{det|J|}\left[\begin{array}{c}-a_{11} - a_{12} - a_{13} \\ a_{11} \\ a_{12} \\ a_{13} \\ \end{array}
%\right] \\
%&=& \label{Nx}
%\frac{1}{det|J|}\left[ N_x \right]�@(�Ƃ���)
%\end{eqnarray}
%
%\begin{eqnarray}
%\nonumber
%\frac{\partial N}{\partial y}
%&=&
%\frac{1}{det|J|}\left[\begin{array}{c}-a_{21} - a_{22} - a_{23} \\ a_{21} \\ a_{22} \\ a_{23} \\ \end{array}
%\right] \\
%&=& \label{Ny}
%\frac{1}{det|J|}\left[ N_y \right]�@(�Ƃ���)
%\\
%\nonumber
%\frac{\partial N}{\partial z}
%&=&
%\frac{1}{det|J|}\left[\begin{array}{c}-a_{31} - a_{32} - a_{33} \\ a_{31} \\ a_{32} \\ a_{33} \\ \end{array}
%\right] \\
%&=& \label{Nz}
%\frac{1}{det|J|}\left[ N_z \right]�@(�Ƃ���)
%\end{eqnarray}
%
%�ƂȂ�D

�܂��C�l�ʑ̂̑̐�$ V $�́C
%
\begin{eqnarray}
\nonumber
6V
&=&
\left|
\begin{array}{cccc}
1   & 1   & 1   &   1 \\
x_1 & x_2 & x_3 & x_4 \\
y_1 & y_2 & y_3 & y_4 \\
z_1 & z_2 & z_3 & z_4
\end{array}
\right|
\\
\nonumber
&=&
\left|
\begin{array}{cccc}
1 & x_1 & y_1 & z_1 \\
1 & x_2 & y_2 & z_2 \\
1 & x_3 & y_3 & z_3 \\
1 & x_4 & y_4 & z_4
\end{array}
\right|
\label{6V}
\\
&=&
\left|
\begin{array}{cccc}
1 & x & y & z \\
0 & x_2 - x_1 & y_2 - y_1 & z_2 - z_1 \\
0 & x_3 - x_1 & y_3 - y_1 & z_3 - z_1 \\
0 & x_4 - x_1 & y_4 - y_1 & z_4 - z_1 \\
\end{array}
\right|
(��2�C3�C4�s����1�s������)
\end{eqnarray}
%

\tiny{
\begin{eqnarray}
\nonumber
\begin{pmatrix}
=
1
\times
\left|
\begin{array}{ccc}
x_2 - x_1 & y_2 - y_1 & z_2 - z_1 \\
x_3 - x_1 & y_3 - y_1 & z_3 - z_1 \\
x_4 - x_1 & y_4 - y_1 & z_4 - z_1 \\
\end{array}
\right|
+
0
\times
\left|
\begin{array}{ccc}
x & y & z \\
x_3 - x_1 & y_3 - y_1 & z_3 - z_1 \\
x_4 - x_1 & y_4 - y_1 & z_4 - z_1 \\
\end{array}
\right|
&+&
0
\times
\left|
\begin{array}{ccc}
x & y & z \\
x_2 - x_1 & y_2 - y_1 & z_2 - z_1 \\
x_4 - x_1 & y_4 - y_1 & z_4 - z_1 \\
\end{array}
\right|
\\
\nonumber
&+&
0
\times
\left|
\begin{array}{ccc}
x & y & z \\
x_2 - x_1 & y_2 - y_1 & z_2 - z_1 \\
x_3 - x_1 & y_3 - y_1 & z_3 - z_1 \\
\end{array}
\right|
\end{pmatrix}
\end{eqnarray}
}

\normalsize{}

%
%
\begin{eqnarray}
\label{6V_}
&=&
\left|
\begin{array}{ccc}
x_2 - x_1 & y_2 - y_1 & z_2 - z_1 \\
x_3 - x_1 & y_3 - y_1 & z_3 - z_1 \\
x_4 - x_1 & y_4 - y_1 & z_4 - z_1 \\
\end{array}
\right|
(��]���q�W�J�ő��̍���0�{)
\end{eqnarray}
%
%
��\ref{xyzforJ}����\ref{[J]}�ɑ������,\\
%
\begin{eqnarray}
\nonumber
J
&=&
\left[
\begin{array}{ccc}
\frac{\partial x}{\partial \xi } & \frac{\partial y}{\partial \xi } & \frac{\partial z}{\partial \xi } \\
\frac{\partial x}{\partial  \eta } & \frac{\partial y}{\partial  \eta } & \frac{\partial z}{\partial  \eta } \\
\frac{\partial x}{\partial \zeta } & \frac{\partial y}{\partial \zeta } & \frac{\partial z}{\partial \zeta } \\
\end{array}
\right]
\\
\label{detJ}
&=&
\left[
\begin{array}{ccc}
x_2 - x_1 & y_2 - y_1 & z_2 - z_1 \\
x_3 - x_1 & y_3 - y_1 & z_3 - z_1 \\
x_4 - x_1 & y_4 - y_1 & z_4 - z_1 \\
\end{array}
\right]
\end{eqnarray}
%
%
��\ref{detJ}�͎�\ref{6V}�Ɠ������̂ŁC
%
\begin{eqnarray}\label{6V}
det|J|=6V
\end{eqnarray}
�ƂȂ�D

�܂��C
%	
%//�ȉ��C�v����//$dxdydz = det|J|d\xi d\eta d\zeta $ �ł��鍪����m�肽��
���R�r�A���̒�`\cite{��������ϕ�}���C
\begin{eqnarray}
\frac{\partial(x,y,z)}{\partial(\xi,\eta,\zeta)}
&=& det
\left[
\begin{array}{ccc}
\frac{\partial x}{\partial \xi } & \frac{\partial y}{\partial \xi } & \frac{\partial z}{\partial \xi } \\
\frac{\partial x}{\partial  \eta } & \frac{\partial y}{\partial  \eta } & \frac{\partial z}{\partial  \eta } \\
\frac{\partial x}{\partial \zeta } & \frac{\partial y}{\partial \zeta } & \frac{\partial z}{\partial \zeta }
\end{array}
\right]
\end{eqnarray}
�����ό`���āC
\begin{eqnarray}
dxdydz &=& det[J] d\xi d\eta d\zeta
%
\end{eqnarray}
������p����$dV$�����W�ϊ�����D
%
%
\begin{eqnarray}\label{dV=1/6V}
\nonumber dV &=& dxdydz \\
\nonumber &=& det|J|d\xi d \eta d\zeta \\
&=& 6V d\xi d \eta d\zeta
\end{eqnarray}
�ȏ���C
��\ref{risanK1}�ɁC��\ref{N_xN_yN_z}�C��\ref{6V}�����D
�܂��A��\ref{dV=1/6V}��p���āC�̐ϗv�f$ dV $��$(x,y,z)$���W����$(\xi ,\eta ,\zeta)$���W�ւ̕ϊ�����p����
%
%
\begin{eqnarray}
\nonumber
&&
\int \int \int_{V_{\xi  \eta \zeta }}\lambda \left\{ \frac{1}{6V}[ N_x]^T  \frac{1}{6V}[ N_x] + \frac{1}{6V}[ N_y]^T  \frac{1}{6V}[ N_y] + \frac{1}{6V}[ N_z]^T  \frac{1}{6V}[ N_z] \right\} 6Vd\xi d \eta d\zeta 
\\
&=&\label{K_m_}
\frac{6V}{36V^2}
\lambda
\int \int \int_{V_{\xi  \eta \zeta }}  \left\{ [ N_x]^T [ N_x] + [ N_y]^T [ N_y] + [ N_z]^T [ N_z] \right\} d\xi d \eta d\zeta
\end{eqnarray}
%
������$ \left\{ [ N_x]^T [ N_x] + [ N_y]^T [ N_y] + [ N_z]^T [ N_z] \right\} = [K_m]$�Ƃ����(������$ [K_m] $ �͎����l�s��D

%(�掮(\ref{Nx}�C\ref{Ny}�C\ref{Nz})���))�C
\begin{eqnarray}
\nonumber
K_m &=&
[ N_x]^T [ N_x] + [ N_y]^T [ N_y] + [ N_z]^T [ N_z]
\\
\nonumber
&=&
\left[
\begin{array}{c}
-a_{11} -a_{12} -a_{13} \\
a_{11} \\
a_{12} \\
a_{13}
\end{array}
\right]
\left[
\begin{array}{cccc}
-a_{11} -a_{12} -a_{13} & a_{11} & a_{12} & a_{13}
\end{array}
\right]
\\
\nonumber
&&+
\left[
\begin{array}{c}
-a_{21} -a_{22} -a_{23} \\
a_{21} \\
a_{22} \\
a_{23}
\end{array}
\right]
\left[
\begin{array}{cccc}
-a_{21} -a_{22} -a_{23} & a_{21} & a_{22} & a_{23}
\end{array}
\right]
\\
\nonumber
&&+
\left[
\begin{array}{c}
-a_{31} -a_{32} -a_{33} \\
a_{31} \\
a_{32} \\
a_{33}
\end{array}
\right]
\left[
\begin{array}{cccc}
-a_{31} -a_{32} -a_{33} & a_{31} & a_{32} & a_{33}
\end{array}
\right]
\\
\nonumber
&=&
\left[
\begin{array}{cccc}
(a_{11} +a_{12} +a_{13})^2 & \cdot & \cdot & -(a_{11} +a_{12} +a_{13})a_{13} \\
-a_{11}(a_{11} + a_{12} +a_{13}) & {a_{11}}^2 & \cdot & \cdot \\
-a_{12}(a_{11} +a_{12} +a_{13}) & a_{12}a_{11} & {a_{12}}^2 & \cdot \\
-a_{13}(a_{11} +a_{12} +a_{13}) & a_{13}a_{11} & a{13}a_{12} & {a_{13}}^2
\end{array}
\right]
\\
\nonumber
&& +
\left[
\begin{array}{cccc}
(a_{21} +a_{22} +a_{23})^2 & \cdot & \cdot & -(a_{21} +a_{22} +a_{23})a_{23} \\
-a_{21}(a_{21} + a_{22} +a_{23}) & {a_{21}}^2 & \cdot & \cdot \\
-a_{22}(a_{21} +a_{22} +a_{23}) & a_{22}a_{11} & {a_{22}}^2 & \cdot \\
-a_{23}(a_{21} +a_{22} +a_{23}) & a_{23}a_{11} & a{23}a_{22} & {a_{23}}^2
\end{array}
\right]
\\
\label{K_m__}
&& +
\left[
\begin{array}{cccc}
(a_{31} +a_{32} +a_{33})^2 & \cdot & \cdot & -(a_{31} +a_{32} +a_{33})a_{33} \\
-a_{31}(a_{31} + a_{32} +a_{33}) & {a_{31}}^2 & \cdot & \cdot \\
-a_{32}(a_{31} +a_{32} +a_{33}) & a_{32}a_{11} & {a_{32}}^2 & \cdot \\
-a_{33}(a_{31} +a_{32} +a_{33}) & a_{33}a_{11} & a{33}a_{12} & {a_{33}}^2
\end{array}
\right]
\end{eqnarray}
%
%
\begin{eqnarray}
\nonumber
(K_1(=��\ref{K_m_}))
&=& 
\frac{\lambda}{6V}[K_m]
\int \int \int_{V_{\xi  \eta \zeta }}  d\xi d \eta d\zeta \\
\nonumber
&=&
\frac{\lambda}{6V}[K_m]
\int_{0}^{1} \int_{0}^{1-\zeta }  \left[\xi \right]_{0}^{1- \eta -\zeta }  d \eta d\zeta \\
\nonumber
&=&
\frac{\lambda}{6V}[K_m]
\int_{0}^{1} \int_{0}^{1-\zeta } (1- \eta -\zeta ) d \eta d\zeta \\
\nonumber
&=&
\frac{\lambda}{6V}[K_m]
\int_{0}^{1} \left[ (1-\zeta )��-\frac{1}{2}��^2 \right]_0^{1-\zeta } d��d\zeta \\
\nonumber
&=&
\frac{\lambda}{6V}[K_m]
\int_{0}^{1} \frac{1}{2} (1-\zeta )^2  d\zeta \\
\nonumber
&=&
\frac{\lambda}{6V}[K_m]
\int_{0}^{1} \frac{1}{2} \left[ \frac{1}{3}\zeta ^3 - \zeta ^2 + \zeta  \right]_0^1  d\zeta \\
&=&
\frac{\lambda}{6V}[K_m]\frac{1}{6}
=
\frac{\lambda}{36V}[K_m]
\end{eqnarray}
�ƌv�Z�ł���D
�������A$ K_m = $( �� $ \ref{K_m__} $ )

%gouse
���l�ɂ��āC$ K_2,K_3 $�̓��o���s���D
%
%
\subsubsection{K2�̓��o}
%
\ref{eqn_s3}�����C�e�ߓ_�̒l�����߂����̂ŁC���Ԃ����ւ��āC
\begin{eqnarray}\label{eqn_s3_2}
\nonumber \int_{S3} q[N]^T dS�@&=& \int_{S3} \alpha(T - T_c) [N]^T dS \\
\nonumber &=& \int_{S3} [N]^T \alpha(T - T_c)  dS\\
\nonumber &=& \int_{S3} \alpha [N]^T ( [N] \cdot T_{ele} ) dS - \int_{S3} \alpha [N]^T T_c  dS \\
		  &=& \int_{S3} \alpha [N]^T [N] dS \cdot T_{ele} - \int_{S3} \alpha [N]^T T_c  dS
\end{eqnarray}
%
%
\subsubsection{$dS$�̓��o}
$(x,y,z)$��ԓ��̃x�N�g��$u,v$�̔����ω��C$du,dv$�́C�P�ʃx�N�g��$e(i,j,k)$��p���āC
%
\begin{eqnarray}
du
&=&
\left[
\begin{array}{ccc}
a_1 & a_2 & a_3
\end{array}
\right]
\left[
\begin{array}{c}
 i \\
 j \\
 k
\end{array}
\right]
\end{eqnarray}

\begin{eqnarray}
dv
&=&
\left[
\begin{array}{ccc}
b_1 & b_2 & b_3
\end{array}
\right]
\left[
\begin{array}{c}
 i \\
 j \\
 k
\end{array}
\right]
\end{eqnarray}
%
�ƕ\����D

�܂��C��Ԓ��̎O�p�`�����v�f$dS$��
$dS=\frac{1}{2} | du \times dv | $
�ƕ\����D
%
\begin{figure}[htb]
\center
\includegraphics[width = 4cm]{ds.eps}
\caption{�O�p�`�����v�f$dS$}
\label{fig:ds}
\end{figure}
%

�e�[���[�̒藝���C
\begin{eqnarray}
x( \xi + d \xi,\eta,\zeta) &=& x(\xi,\eta,\zeta) + \frac{\partial}{\partial \xi} x(\xi,\eta,\zeta) d \xi \\
\nonumber \frac{\partial}{\partial \xi} x(\xi,\eta,\zeta) d \xi &=& x( \xi + d \xi,\eta,\zeta) - x(\xi,\eta,\zeta) \\
\nonumber &=& (x_2 - x_1)(\xi +d\xi)+(x_3-x_1)\eta + (x_4-x_1)\zeta \\ 
\nonumber && - \{(x_2-x_1)\xi +(x_3-x_1)\eta +(x_4-x_1)\zeta\} \\
&=& (x_2-x_1)d\xi
\end{eqnarray}

�����́C$ [J] $�̊e������$ d\xi , d\eta, d\zeta $���悶�����̂ł���D
�䂦�ɁC�����ɂȂ�D
\begin{eqnarray}
\nonumber
[J]
\begin{array}{ccc}
[d\xi & d\eta & d\zeta]
\end{array}
%
&=&
%
\left[
\begin{array}{ccc}
\frac{\partial x}{\partial \xi } & \frac{\partial y}{\partial \xi } & \frac{\partial z}{\partial \xi } \\
\frac{\partial x}{\partial  \eta } & \frac{\partial y}{\partial  \eta } & \frac{\partial z}{\partial  \eta } \\
\frac{\partial x}{\partial \zeta } & \frac{\partial y}{\partial \zeta } & \frac{\partial z}{\partial \zeta } \\
\end{array}
\right]
%
\begin{array}{ccc}
[d\xi & d\eta & d\zeta]
\end{array}
\\
&=& 
\left[
\begin{array}{ccc}
x_2 - x_1 & y_2 - y_1 & z_2 - z_1 \\
x_3 - x_1 & y_3 - y_1 & z_3 - z_1 \\
x_4 - x_1 & y_4 - y_1 & z_4 - z_1 \\
\end{array}
\right]
\begin{array}{ccc}
[d\xi & d\eta & d\zeta]
\end{array}
\end{eqnarray}
%
%
��\ref{xyzforJ}���Čf����D
\begin{eqnarray}\label{xyz��_2}
\nonumber x &=& (1-\xi -\eta -\zeta ) x_1 + \xi  x_2 +  \eta  x_3 + \zeta  x_4 \\
\nonumber &=& (x_2 - x_1)\xi + (x_3 - x_1)\eta + (x_4 - x_1)\zeta + x_1\\
 y &=& (y_2 - y_1)\xi +(y_3-y_1) \eta +(y_4-y_1)\zeta +y_1  \\
\nonumber�@z &=& (z_2 - z_1)\xi +(z_3-z_1) \eta +(z_4-z_1)\zeta +z_1 
\end{eqnarray}
������C$u,v$�̔����ω�$ du, dv$���ȉ��̎��ŕ\���Ƃ���D
$\eta =0$ ��$ \xi - \zeta $���ʂōl����ƁC

\begin{eqnarray}
d u &=& ( \frac{\partial x}{\partial \xi}d\xi) i + ( \frac{\partial y}{\partial \xi}d\xi) j +( \frac{\partial z}{\partial \xi}d\xi) k
\\
%
d v &=& ( \frac{\partial x}{\partial \zeta}d\zeta) i + ( \frac{\partial y}{\partial \zeta}d\zeta) j +( \frac{\partial z}{\partial \zeta}d\zeta) k
%
\end{eqnarray}
�@�@//���Ɋւ��āC�Ł�0�̎��̉��̌W���̌v�Z�͂ǂ��ɂ��z���Čv�Z�ł���悤�Ɏ����������D


%
�ȏ���C$ du \times dv $�͒P�ʃx�N�g��$e$��p���āC�����̂悤�ɏ�����D
%
\begin{eqnarray}
\nonumber e=
\left[
\begin{array}{ccc}
e_1 & e_2 & e_3
\end{array}
\right]
\end{eqnarray}
��p����
%
%
\begin{eqnarray}\label{duxdv}
\nonumber du \times dv &=&
\left|
\begin{array}{ccc}
e_1 & e_2 & e_3\\
\frac{\partial x}{\partial  \xi} d\xi & \frac{\partial y}{\partial  \xi} d\xi & \frac{\partial z}{\partial  \xi} d\xi \\
\frac{\partial x}{\partial  \zeta} d\zeta & \frac{\partial y}{\partial  \zeta} d\zeta & \frac{\partial z}{\partial  \zeta} d\zeta
\end{array}
\right|
\\
\nonumber &=& \left|
\begin{array}{ccc}
e_1 & e_2 & e_3\\
\frac{\partial x}{\partial  \xi} & \frac{\partial y}{\partial  \xi}& \frac{\partial z}{\partial  \xi} \\
\frac{\partial x}{\partial  \zeta} & \frac{\partial y}{\partial  \zeta} & \frac{\partial z}{\partial  \zeta}
\end{array}
\right| d \xi d \zeta
\\
\nonumber &=& \left|
\begin{array}{ccc}
e_1 & e_2 & e_3\\
x_2 - x_1 & y_2 - y_1& z_2 -z_1 \\
x_4 - x_1 & y_4 - y_1 & z_4 -z_1
\end{array}
\right| d \xi d \zeta
\\
\nonumber &=& \left[
\begin{array}{c}
(y_2 - y_1)(z_4 - z_1)e_1 - (z_2 - z_1)(y_4 - y_1)e_1\\
+(x_4 - x_1)(z_2 - z_1)e_2 - (x_2 - x_1)(z_4 - z_1)e_2\\
+(x_2 - x_1)(y_4 - y_1)e_3 - (x_4 - x_1)(y_2 - y_1)e_3\\
\end{array}
\right]
d \xi d \zeta
\\
% 3x1 * 1x3 �Ȃ�s�񎮂̒l�Ɠ������͂Ȃ�D
&=& \left[
\begin{array}{ccc}
M_1 & M_2 & M_3
\end{array}
\right]
\left[
\begin{array}{c}
e_1\\
e_2\\
e_3
\end{array}
\right]
d \xi d \zeta
\end{eqnarray}
%
�������C
\begin{eqnarray}\label{M1M2M3}
\begin{array}{c}
M_1 = (y_2 - y_1)(z_4 - z_1) - (z_2 - z_1)(y_4 - y_1)\\
M_2 = (x_4 - x_1)(z_2 - z_1) - (x_2 - x_1)(z_4 - z_1)\\
M_3 = (x_2 - x_1)(y_4 - y_1) - (x_4 - x_1)(y_2 - y_1)
\end{array}
\end{eqnarray}
�Ƃ���D
%
��\ref{duxdv},\ref{M1M2M3},�s�^�S���X�̒藝���C�����ʐς͎��̗l�Ɍv�Z�ł���D
\begin{eqnarray}\label{ds=}
\nonumber dS &=& \frac{1}{2}
|
du \times dv
|
\\
&=&
\frac{1}{2} \sqrt{ M_1^2 + M_2^2 + M_3^2} d \xi d \zeta
\end{eqnarray}
%
�������C$ 0 \leqq \zeta \leqq 1- \xi ,  0 \leqq \xi \leqq 1 $�Ƃ���D
%
%
%
%
\subsubsection{$K_2, K_3$�����߂�}
�Ƃ���ŁCN�̒�`�́C�P�~N�̍s�}�g���N�X�ł������D
�䂦�ɁC
\begin{eqnarray}\label{NTN}
\nonumber[N]^T \cdot [N]
&=&
\left[
\begin{array}{c}
N_1\\
N_2\\
N_3\\
N_4
\end{array}
\right]
\left[
\begin{array}{cccc}
N_1 & N_2 & N_3 & N_4
\end{array}
\right]\\
&=&
\left[
\begin{array}{cccc}
N_1^2 & N_1 N_2 & N_1 N_3 & N_1 N_4\\
N_2 N_1 & N_2^2 & N_2 N_3 & N_2 N_4\\
N_3 N_1 & N_3 N_2 & N_3^2 & N_3 N_4\\
N_4 N_1 & N_4 N_2 & N_4 N_3 & N_4^2
\end{array}
\right]
\end{eqnarray}
�ƂȂ�C�Ώ̍s��ƂȂ�D
$N_1,N_2,N_3,N_4$�͂��ꂼ��s��ł͂Ȃ��C�s��̐����Ȃ̂ŁC�Ίp�����Ə㔼���i�܂��͉������j�����߂�΂悢�D

��\ref{risanK}�̉E�ӑ��C�O�����v�Z����ɂ�����C��\ref{NTN}�̍s��̊e�v�f�ɂ��āC�ϕ��v�Z���s���D
�����v�f���Ƃɂ܂Ƃ߁C�l�ʑ̗v�f�S�̂̌v�Z�Ƃ��Ă܂Ƃ߂邱�Ƃɂ���D
%
%
\begin{eqnarray}\label{k2_23}
&& \nonumber ([K]�̑��C�O��) = \int_{S3} \alpha[N]^T [N] dS + \int_{S4} h[N]^T [N] dS \\
\end{eqnarray}
%
%\begin{eqnarray}
%\nonumber ([K]�̑��) = \alpha \int_{S3} [N] [N]^T dS\\
%\nonumber ([K]�̑�O��) = h \int_{S4} [N] [N]^T dS\\
%\end{eqnarray}
%
�����ŁC�{�v�Z�ł́C��������$ (x,y,z)$��ԓ��̎l�ʑ̂�$ (\xi,\eta,\zeta) $��ԓ��ɍ��W�ϊ������l�ʑ̓��ōl����D
�ȉ��C$ \eta = 0 $�Ƃ���$ \xi - \zeta $���ʂ̖ʐϕ����l����D
%
\begin{figure}[htb]
\center
\includegraphics[width = 8cm]{xi_zeta_integra.eps}
\caption{$ \xi - \zeta $���ʂł̖ʐϕ�$ \int_{0}^{1} d \xi \int_{0}^{1-\xi} d\zeta $}
\label{fig:xi_zeta_integra}
\end{figure}
%
%
\begin{eqnarray}
\nonumber \int_{0}^{1} [N]^T [N] dS &=& \int_{0}^{1} d\xi\int_{0}^{1-\xi} [N]^T [N] d\zeta \\
\end{eqnarray}
�����ŁC$ [N] [N]^T  $�̍s��(��\ref{NTN})�Ɋ܂܂��v�f���Ƃɐϕ����s���D

$i)$ $ N_1^2 $�̂Ƃ�
\begin{eqnarray}
\nonumber \int_0^1 \int_0^{1-\xi} N_1^2 d\zeta d\xi &=& \int_0^1 \int_0^{1-\xi} (1-\xi -\zeta )^2 d\zeta d\xi \\
\nonumber &=& \int_0^1 -\frac{1}{3} \left[ (1 - \xi - \zeta)^3 \right]_0^{1-\xi} d\xi \\
\nonumber &=& \int_0^1 \frac{1}{3} (1 - \xi )^3 d\xi \\
\nonumber &=& -\frac{1}{12} \left[ (1 - \xi)^4 \right]_0^{1} d\xi \\
&=& \frac{1}{12}
\end{eqnarray}

$ii)$ $N_1 N_2 $�̂Ƃ�

$ \cdot $

$ \cdot $

$ \cdot $

���l�ɂ��āC$ N_2^2,N_3^2,...N_2 N_4,N_3 N_4 $�����߂�D
�܂��C$ N_1 N_2 $ �C $ N_2 N_1 $�͐ς̏��Ԃ����ւ��\�Ȃ��߁C���l�ƂȂ�D

���߂����ʁC�ȉ��̂悤�ɂȂ�D
%
\begin{eqnarray}
\nonumber \int_{0}^{1} [N]^T [N] dS &=& \int_{0}^{1} d\xi\int_{0}^{1-\xi} [N]^T [N] d\zeta \\
 &=& 
\left[
\begin{array}{cccc}
\frac{1}{12} & \frac{1}{24} & 0 & \frac{1}{24} \\
\frac{1}{24} & \frac{1}{12} & 0 & \frac{1}{24} \\
0 & 0 & 0 & 0 \\
\frac{1}{24} & \frac{1}{24} & 0 & \frac{1}{12}
\end{array}
\right]
\end{eqnarray}
%
�䂦�ɁC��\ref{ds=}�C\ref{k2_23}���C
\begin{eqnarray}
\nonumber ([K]�̑�2,3������\alpha , h������������) &=& \int_{Sx} [N]^T [N] dS \\
\nonumber &=& \frac{1}{2}  \int_{Sx} [N]^T [N]  \sqrt{ M_1^2 + M_2^2 + M_3^2} d \xi d \zeta \\
\nonumber &=& \frac{1}{2} \sqrt{ M_1^2 + M_2^2 + M_3^2}
\left[
\begin{array}{cccc}
\frac{1}{12} & \frac{1}{24} & 0 & \frac{1}{24} \\
\frac{1}{24} & \frac{1}{12} & 0 & \frac{1}{24} \\
0 & 0 & 0 & 0 \\
\frac{1}{24} & \frac{1}{24} & 0 & \frac{1}{12}
\end{array}
\right]
\end{eqnarray}
%

���ɁC�M�e�ʃ}�g���N�X�ɂ��Ă̎��̓��o���s���D
K�Ɠ��l�ɂ��čŏI�I��$ (x,y,z) $��Ԃɕϊ�����ƁC�����̂悤�ɂȂ�(��\ref{dV=1/6V}��p����dV�����W�ϊ�����)�D
\begin{eqnarray}
(��\ref{risanC}�̉E��) &=& \int_v \rho c [N]^T[N]dv \\
\nonumber &=& \rho c\int_0^1\int_0^{1-\xi} \int_0^{1-\xi - \eta}[N]^T[N] det|J| d\xi d\eta d\zeta \\
&=& \rho c \cdot 6V
\left[
\begin{array}{cccc}
\frac{1}{60} & \frac{1}{120} & \frac{1}{120} & \frac{1}{120} \\
\frac{1}{120} & \frac{1}{60} & \frac{1}{120} & \frac{1}{120} \\
\frac{1}{120} & \frac{1}{120} & \frac{1}{60} & \frac{1}{120} \\
\frac{1}{120} & \frac{1}{120} & \frac{1}{120} & \frac{1}{60}
\end{array}
\right]
\end{eqnarray}
%
�܂��C�M�����x�N�g���ɂ��Ă��C
K�Ɠ��l�ɂ��čŏI�I��$ (x,y,z) $��Ԃɕϊ�����ƁC�����̂悤�ɂȂ�D
\begin{eqnarray}
(��\ref{risanF}�̉E�ӂ��) &=& \int_s [N]^T ds \\
&=&
\frac{1}{2} \sqrt{ M_1^2 + M_2^2 + M_3^2}
\left[
\begin{array}{c}
\frac{1}{6} \\
\frac{1}{6} \\
0 \\
\frac{1}{6} 
\end{array}
\right]
\end{eqnarray}
%
%
\begin{equation}\label{risaneq_2}
	[K] \{T \} + [C]\{\frac{\partial T}{\partial t} \} = \{ F \}
\end{equation}
%
�������C
\begin{eqnarray}\label{risanK_2}
\nonumber
[K] &=& \int_v \lambda \{ \frac{\partial [ N]^T }{\partial x}  \frac{\partial  [ N] }{\partial x} + \frac{\partial [N]^T }{\partial y}  \frac{\partial [N] }{\partial y} + \frac{\partial [N]^T }{\partial z}  \frac{\partial [N]}{\partial z} \} dv \\
& & �@�@�@+ \int_{S3} \alpha[N]^T [N] dS + \int_{S4} h[N]^T [N] dS \\ 
\nonumber
&=&  \frac{\lambda}{36V}[K_m]\\
\nonumber
  & &  +  
\frac{1}{2} \alpha \sqrt{ M_1^2 + M_2^2 + M_3^2}
\left[
\begin{array}{cccc}
\frac{1}{12} & \frac{1}{24} & 0 & \frac{1}{24} \\
\frac{1}{24} & \frac{1}{12} & 0 & \frac{1}{24} \\
0 & 0 & 0 & 0 \\
\frac{1}{24} & \frac{1}{24} & 0 & \frac{1}{12}
\end{array}
\right]
 +  
\frac{1}{2} h \sqrt{ M_1^2 + M_2^2 + M_3^2}
\left[
\begin{array}{cccc}
\frac{1}{12} & \frac{1}{24} & 0 & \frac{1}{24} \\
\frac{1}{24} & \frac{1}{12} & 0 & \frac{1}{24} \\
0 & 0 & 0 & 0 \\
\frac{1}{24} & \frac{1}{24} & 0 & \frac{1}{12}
\end{array}
\right]
\\
\nonumber && (�������C\eta=0�̎�)
\\
%
%
\label{risanC_2}
[C] &=& \int_v \rho c[N]^T [N] dv \\
&=& \rho c \cdot 6V
\left[
\begin{array}{cccc}
\frac{1}{60} & \frac{1}{120} & \frac{1}{120} & \frac{1}{120} \\
\frac{1}{120} & \frac{1}{60} & \frac{1}{120} & \frac{1}{120} \\
\frac{1}{120} & \frac{1}{120} & \frac{1}{60} & \frac{1}{120} \\
\frac{1}{120} & \frac{1}{120} & \frac{1}{120} & \frac{1}{60}
\end{array}
\right]
\\
%
%
\label{risan_F}
\nonumber
 \{F\} &=& \int_v \frac{\partial Q}{\partial t} dv - \int_{S2} q_0[N]^T dS + \int_{S3} \alpha T_c [N]^T dS \\
& & �@�@�@+ \int_{S4} hT_{OUT} [N]^T dS \\
&=& \int_v \frac{\partial Q}{\partial t} dv - q_0 \cdot \frac{1}{2} \sqrt{ M_1^2 + M_2^2 + M_3^2}
\left[
\begin{array}{c}
\frac{1}{6} \\
\frac{1}{6} \\
0 \\
\frac{1}{6} 
\end{array}
\right] \\
\nonumber &&+
\alpha T_c \cdot \frac{1}{2} \sqrt{ M_1^2 + M_2^2 + M_3^2}
\left[
\begin{array}{c}
\frac{1}{6} \\
\frac{1}{6} \\
0 \\
\frac{1}{6} 
\end{array}
\right]
+ hT_{OUT} \cdot
\frac{1}{2} \sqrt{ M_1^2 + M_2^2 + M_3^2}
\left[
\begin{array}{c}
\frac{1}{6} \\
\frac{1}{6} \\
0 \\
\frac{1}{6} 
\end{array}
\right]
\\
&& (�������C\eta=0�̎�)
\end{eqnarray}
%
%
\subsection{���l�v�Z�@�ɍ��킹�����̕ό`}
\subsubsection{�K�E�X�U�C�f���@}
�{�����ł́A���l�v�Z�̑��x���d�����A�v�Z���x���J��Ԃ��v�Z�񐔂ł̒��߂����҂ł���K�E�X�U�C�f���@��p���Čv�Z���s���B�܂��A����v�Z���\�ł���ƍl�����AGPU�Ȃǂ��g����������v�Z�̎��������҂��Ă���B

\begin{eqnarray}\label{Gaus}
A x(n) + b &=& 0 \\
(D-F)x(n) &=& -b \\
x(n) &=& D^{-1}Fx(n) - D^{-1}b 
\end{eqnarray}

�������AD�́AA�̑Ίp�����̂悤�ȍs��B
F�́AA����D�����������̂ƂȂ�B
��\ref{crankeq}�𓖂Ă͂߂�ƁA�����ɂȂ�B
\begin{eqnarray}
A &=& \begin{pmatrix} \frac{1}{2}[K]+\frac{1}{\Delta t}[C] \end{pmatrix} \\
x[n] &=& \left\{ T(t+ \Delta t) \right\} \\
-b &=& \begin{pmatrix}-\frac{1}{2}[K] + \frac{1}{\Delta t}[C] \end{pmatrix} \{ T(t) \} + \{ F \}
\end{eqnarray}
�܂��A$ A $�����߁A�Ίp���������ɂȂ�悤�ȍs���T���A$F,D $���`���āA�v�Z��i�߂�B
�㎮���A$ x[n] $�͐ߓ_�̉��x��\��$ n�~1 $�̃x�N�g���ɂȂ�B
���R�r�@�ł́A$x$����x�ɍX�V���邪�A�K�E�X�E�U�C�f���@�ł�1�s���X�V����B
���̂��߁A���v�Z�X�e�b�v��1�̐ߓ_�����X�ɍX�V����Ă����B



\section{�����̏���}
%
\subsection{���_���x�����̏ꍇ}
�H�i�Ɛ��̉Ȋw\cite{foodandwater}��pp.201�ɂ��΁C
�����́C���̉��x�ɂ�����O�a�����C��$ H_s $�Ɗ����͋C���̐����C��$ H_c $�Ƃ̍�$ H_s - H_c $�Ɏx�z����C�P�ʕ\�ʐς�����̊������x$ R_c $�͎��̂悤�ɕ\����D
\begin{eqnarray}
R_c = - \frac{W_0}{A} \cdot \frac{dW}{d\theta} = K(H_s - H_c)
\end{eqnarray}
%
�������C$ W_0 $�F�������̖��������ʁC$ A $�F�������̊����\�ʐρC$ \frac{dW}{d\theta} $�F�P�ʎ��Ԃɂ�����P�ʍޗ�����������̐��̎��ʂ̕ω��C$ K $�F���������ړ��W���D���͋C�̗����Ƌ��ɑ��傷��D

