FDM_1D_verticalコマンドで用いているアルゴリズム
2. 方程式(離散化前)
(Algorithm used in FDM_1D_vertical command;
2. Equations before discretization)
水平方向には物理量が変化しない(\(\PartialDiff{}{x}=\PartialDiff{}{y}=0\))
と仮定すると、運動方程式は
\[\begin{equation}
\rho\PartialDiff{V_x}{t}
=\PartialDiff{\tau_{xx}}{x}
+\PartialDiff{\tau_{xy}}{y}
+\PartialDiff{\tau_{xz}}{z}
=\PartialDiff{\tau_{xz}}{z}
\label{eq.motion.x}
\end{equation}\]
\[\begin{equation}
\rho\PartialDiff{V_y}{t}
=\PartialDiff{\tau_{yx}}{x}
+\PartialDiff{\tau_{yy}}{y}
+\PartialDiff{\tau_{yz}}{z}
=\PartialDiff{\tau_{yz}}{z}
\label{eq.motion.y}
\end{equation}\]
\[\begin{equation}
\rho\PartialDiff{V_z}{t}
=\PartialDiff{\tau_{zx}}{x}
+\PartialDiff{\tau_{zy}}{y}
+\PartialDiff{\tau_{zz}}{z}
=\PartialDiff{\tau_{zz}}{z}
\label{eq.motion.z}
\end{equation}\]
と書ける。これらの方程式の右辺に登場する応力成分は
\(\tau_{xz}\), \(\tau_{yz}\), \(\tau_{zz}\)であり、
それらについて構成則を書き下すと
\[\begin{equation}
\PartialDiff{\tau_{xz}}{t}
=\mu\left(\PartialDiff{V_x}{z}+\PartialDiff{V_z}{x}\right)
=\mu\PartialDiff{V_x}{z}
\label{eq.constitutive.x}
\end{equation}\]
\[\begin{equation}
\PartialDiff{\tau_{yz}}{t}
=\mu\left(\PartialDiff{V_y}{z}+\PartialDiff{V_z}{y}\right)
=\mu\PartialDiff{V_y}{z}
\label{eq.constitutive.y}
\end{equation}\]
\[\begin{equation}
\PartialDiff{\tau_{zz}}{t}
=\lambda\left(
\PartialDiff{V_x}{x}+\PartialDiff{V_y}{y}+\PartialDiff{V_z}{z}
\right)+2\mu\PartialDiff{V_z}{z}
=(\lambda+2\mu)\PartialDiff{V_z}{z}
\label{eq.constitutive.z}
\end{equation}\]
となる。したがって方程式は
- \(V_x\), \(\tau_{xz}\)に関する閉じた方程式系
(\ref{eq.motion.x})(\ref{eq.constitutive.x})
- \(V_y\), \(\tau_{yz}\)に関する閉じた方程式系
(\ref{eq.motion.y})(\ref{eq.constitutive.y})
- \(V_z\), \(\tau_{zz}\)に関する閉じた方程式系
(\ref{eq.motion.z})(\ref{eq.constitutive.z})
に分離できることが分かる。
(\ref{eq.motion.y})(\ref{eq.constitutive.y})式は
(\ref{eq.motion.x})(\ref{eq.constitutive.x})式の
\(x\)を\(y\)で置き換えただけの形になっているので、
以下では(\ref{eq.motion.x})(\ref{eq.constitutive.x})式
および(\ref{eq.motion.z})(\ref{eq.constitutive.z})式のみを用いることにする。
Assuming no horizontal variation of physical quantities
(\(\PartialDiff{}{x}=\PartialDiff{}{y}=0\)),
the equation of motion is written as
Eqs. (\ref{eq.motion.x})-(\ref{eq.motion.z}).
Only the stress components \(\tau_{xz}\), \(\tau_{yz}\), and \(\tau_{zz}\)
appear in the right hand sides of these equations,
and constitutive laws for them are given by
Eqs. (\ref{eq.constitutive.x})-(\ref{eq.constitutive.z}).
Therefore the equations can be separated into
- closed equations (\ref{eq.motion.x}) and (\ref{eq.constitutive.x})
for \(V_x\) and \(\tau_{xz}\),
- closed equations (\ref{eq.motion.y}) and (\ref{eq.constitutive.y})
for \(V_y\) and \(\tau_{yz}\), and
- closed equations (\ref{eq.motion.z}) and (\ref{eq.constitutive.z})
for \(V_z\) and \(\tau_{zz}\).
As Eqs. (\ref{eq.motion.y}) and (\ref{eq.constitutive.y}) have
exactly the same forms as
Eqs. (\ref{eq.motion.x}) and (\ref{eq.constitutive.x})
except for the difference of \(x\) and \(y\),
below we use only Eqs. (\ref{eq.motion.x})(\ref{eq.constitutive.x})
and (\ref{eq.motion.z})(\ref{eq.constitutive.z}).
(\ref{eq.motion.x})式の両辺を\(\rho\)で割ると
\[\begin{eqnarray}
\PartialDiff{V_x}{t}
&=& \frac{1}{\rho}\PartialDiff{\tau_{xz}}{z}
\nonumber \\
&=& \PartialDiff{}{z}\left(\frac{1}{\rho}\tau_{xz}\right)
-\PartialDiff{}{z}\left(\frac{1}{\rho}\right)\tau_{xz}
\nonumber \\
&=& \PartialDiff{}{z}\left(\frac{\tau_{xz}}{\rho}\right)
+\frac{1}{\rho^2}\PartialDiff{\rho}{z}\tau_{xz}
\nonumber \\
&=& \PartialDiff{}{z}\left(\frac{\tau_{xz}}{\rho}\right)
+\frac{1}{\rho}\PartialDiff{\rho}{z}\frac{\tau_{xz}}{\rho}
\label{eq.motion.x.divided_by_rho}
\end{eqnarray}\]
となり、
\[\begin{equation}
\tau’_{xz}\equiv\tau_{xz}/\rho
\label{eq.tau_xz_dash.definition}
\end{equation}\]
とおけば
\[\begin{equation}
\PartialDiff{V_x}{t}
=\PartialDiff{\tau’_{xz}}{z}
+\frac{1}{\rho}\PartialDiff{\rho}{z}\tau’_{xz}
\label{eq.motion.x.use_tau_dash}
\end{equation}\]
が得られる。一方(\ref{eq.constitutive.x})式より
(密度\(\rho\)が時刻\(t\)に依存しないことを用いて)
\[\begin{equation}
\PartialDiff{\tau’_{xz}}{t}
=\frac{1}{\rho}\PartialDiff{\tau_xz}{t}
=\frac{\mu}{\rho}\PartialDiff{V_x}{z}
=V_s^2\PartialDiff{V_x}{z}
\label{eq.constitutive.x.use_tau_dash}
\end{equation}\]
が得られる。
(\ref{eq.motion.x.use_tau_dash})(\ref{eq.constitutive.x.use_tau_dash})
式は\(V_x\)と\(\tau’_{xz}\)に関する閉じた方程式系になっている。
FDM_1D_verticalコマンドでは
密度の空間変化\(\PartialDiff{\rho}{z}\)が十分に小さいものとし、
(\ref{eq.motion.x.use_tau_dash})式の右辺第2項を無視して
\[\begin{equation}
\PartialDiff{V_x}{t}
=\PartialDiff{\tau’_{xz}}{z}
\label{eq.motion.x.use_tau_dash.approx}
\end{equation}\]
とする。
(\ref{eq.constitutive.x.use_tau_dash})(\ref{eq.motion.x.use_tau_dash.approx})
式は密度\(\rho\)を明示的に含まない形になっており、
S波速度\(V_s(z)\)のみを与えれば計算を実行できる。
そのため密度変化\(\PartialDiff{\rho}{z}\)が
十分に小さいと仮定したことが矛盾を生じたり、
速度構造の与え方に対する制約になったりすることは無い。
Dividing Eq. (\ref{eq.motion.x}) by \(\rho\) reduces to
Eq. (\ref{eq.motion.x.divided_by_rho}),
which can be rewritten as (\ref{eq.motion.x.use_tau_dash}),
where \(\tau’_{xz}\) is defined by
Eq. (\ref{eq.tau_xz_dash.definition}).
From Eq. (\ref{eq.constitutive.x}),
we obtain Eq. (\ref{eq.constitutive.x.use_tau_dash})
because the density \(\rho\) is independent of time \(t\).
Eqs. (\ref{eq.motion.x.use_tau_dash})
and (\ref{eq.constitutive.x.use_tau_dash})
constitute a closed equation system for \(V_x\) and \(\tau’_{xz}\).
The FDM_1D_vertical command assumes that
the spatial variation of the density (\(\PartialDiff{\rho}{z}\))
is small enough to ignore
the 2nd term of the right hand side of
Eq. (\ref{eq.motion.x.use_tau_dash}),
giving Eq. (\ref{eq.motion.x.use_tau_dash.approx}).
Eqs. (\ref{eq.constitutive.x.use_tau_dash})
and (\ref{eq.motion.x.use_tau_dash.approx})
do not explicitly consist of the density \(\rho\),
meaning that the calculation requires
only the S-wave velocity \(V_s(z)\).
Therefore the assumption of small density variation
\(\PartialDiff{\rho}{z}\) does not cause an inconsistency,
nor results in a constraint for the velocity stcuture.
同様にして(\ref{eq.motion.z})(\ref{eq.constitutive.z})式から
\[\begin{equation}
\PartialDiff{V_z}{t}
=\PartialDiff{\tau’_{zz}}{z}
+\frac{1}{\rho}\PartialDiff{\rho}{z}\tau’_{zz}
\approx \PartialDiff{\tau’_{zz}}{z}
\label{eq.motion.z.use_tau_dash}
\end{equation}\]
\[\begin{equation}
\PartialDiff{\tau’_{zz}}{t}=V_p^2\PartialDiff{V_z}{z}
\label{eq.constitutive.z.use_tau_dash}
\end{equation}\]
が得られる。ここで
\[\begin{equation}
\tau’_{zz}\equiv\tau_{zz}/\rho
\label{eq.tau_zz_dash.definition}
\end{equation}\]
とおいた。
In the same way,
Eqs. (\ref{eq.motion.z}) and (\ref{eq.constitutive.z})
can be arranged to Eqs. (\ref{eq.motion.z.use_tau_dash})
and (\ref{eq.constitutive.z.use_tau_dash}),
where \(\tau’_{zz}\) is defined by
Eq. (\ref{eq.tau_zz_dash.definition}).