Finite Difference Derivations
Derivations for Finite Difference Schemes and Force Coupling. The current time step, marked by superscript \(n\) is ommitted for clarity where relevenat. Right click equations for output options.
1D Wave
$$\begin{eqnarray}
\require{cancel}
\delta_{tt}w &=& c^{2}\delta_{xx}w \nonumber \\
\frac{2}{k}(\delta_{t\cdot} - \delta_{t-})w &=& c^{2}\delta_{xx}w \nonumber \\
&=& \frac{kc^{2}}{2}\delta_{xx}w + \delta_{t-}w \nonumber \\
&=& \frac{kc^{2}}{2h_{s}}(\delta_{x+}-\delta_{x-})w + \delta_{t-}w \nonumber \\
&=& \frac{kc^{2}}{2h_{s}}(\delta_{x+}w-\delta_{x-}w) + \delta_{t-}w \nonumber \\
\frac{1}{2k}(w^{n+1} - w^{n-1}) &=& \frac{kc^{2}}{2h_{s}}(\delta_{x+}w-\delta_{x-}w) + \delta_{t-}w \nonumber \\
\frac{1}{2k}(w^{n+1} - w^{n-1}) &=& \frac{kc^{2}}{2h_{s}}(\delta_{x+}w-\delta_{x-}w) + \frac{1}{k}(w^{n} - w^{n-1}) \nonumber \\
w^{n+1} - w^{n-1} &=& 2k(\frac{kc^{2}}{2h_{s}}(\delta_{x+}w-\delta_{x-}w) + \frac{1}{k}(w^{n} - w^{n-1})) \nonumber \\
w^{n+1} &=& 2k(\frac{kc^{2}}{2h_{s}}(\delta_{x+}w-\delta_{x-}w) + \frac{1}{k}(w^{n} - w^{n-1})) + w^{n-1} \nonumber \\
w^{n+1} &=& \cancel{2}k(\frac{kc^{2}}{\cancel{2}h_{s}}(\delta_{x+}w-\delta_{x-}w) + \frac{2}{k}(w^{n} - w^{n-1})) + w^{n-1} \nonumber \\
w^{n+1} &=& \cancel{k}(\frac{(kc)^{2}}{h_{s}}(\delta_{x+}w-\delta_{x-}w) + \frac{2}{\cancel{k}}w^{n} - 2w^{n-1} + w^{n-1} \nonumber \\
w^{n+1} &=& \frac{(kc)^{2}}{h_{s}}(\delta_{x+}w-\delta_{x-}w) + 2w^{n} - \cancel{2}w^{n-1} + \cancel{w^{n-1}} \nonumber \\
w^{n+1} &=& \frac{(kc)^{2}}{h_{s}}(\delta_{x+}w-\delta_{x-}w) + 2w^{n} - w^{n-1} \nonumber \\
w^{n+1} &=& \lambda^{2}D_{xx}w + 2w^{n} - w^{n-1},\quad \lambda = \frac{kc}{h_{s}}\nonumber \\
\end{eqnarray}$$
Stiff String
$$\begin{eqnarray}
\require{cancel}
\delta_{tt}w &=& c^{2}\delta_{xx}w - \kappa^{2}\delta_{xxxx}w \nonumber \\
\frac{2}{k}(\delta_{t\cdot} - \delta_{t-})w &=& c^{2}\delta_{xx}w + \kappa^{2}\delta_{xxxx}w \nonumber \\
w^{n+1} &=& (kc)^{2}\delta_{xx}w - (k\kappa)^{2}\delta_{xxxx}w \nonumber \\
w^{n+1} &=& (\lambda)^{2}D_{xx}w - (\mu)^{2}D_{xxxx}w \nonumber,\quad \lambda = \frac{kc}{h_{s}}, \quad \mu = \frac{k\kappa}{h^{2}_{s}} \\
\end{eqnarray}$$
Kirchoff Thin Plate
$$\ddot{u} = -\kappa^{2}\Delta\Delta u, \quad \kappa = \sqrt{\frac{E H^2}{12\rho(1- \nu)} }$$
1D Wave w/ Coupling
Coupling Conditions
$$\begin{eqnarray}
\require{cancel}
\delta_{t\cdot}w &=& J^{T}\delta_{t\cdot}u \nonumber \\
f &=& T\delta_{x-}w_{0} \nonumber \\
\end{eqnarray}$$
From 1D Wave Equation above
$$\begin{eqnarray}
\require{cancel}
w^{n+1} &=& \frac{(kc)^{2}}{h_{s}}(\delta_{x+}w-\delta_{x-}w) + 2w^{n} - w^{n-1} \nonumber \\
\end{eqnarray}$$
Translates to
$$\begin{eqnarray}
\require{cancel}
w_{0}^{n+1} &=& \frac{(kc)^{2}}{h_{s}}(\delta_{x+}w_{0}-\delta_{x-}w_{0}) + 2w_{0}^{n} - w_{0}^{n-1}, \quad w_{0}^{n} &=& F \nonumber \\
w_{0}^{n+1} &=& \frac{(kc)^{2}}{h_{s}}(\delta_{x+}w_{0}-\frac{f}{T}) + 2w_{0}^{n} - w_{0}^{n-1} \nonumber \\
w_{0}^{n+1} &=& \lambda\delta_{x+}w_{0} - \frac{(kc)^{2}}{h_{s}T}f) + 2w_{0}^{n} - w_{0}^{n-1} \nonumber \\
\end{eqnarray}$$
Stiff String w/ Coupling
Coupling Conditions
$$\begin{eqnarray}
\require{cancel}
f &=& T\delta_{x-}w_{0} - EI\delta_{xxx}w_{0} \nonumber \\
\frac{dE_{s}}{dt} &=& \dot{w_0}(Tw^{\prime} - EIw^{\prime\prime\prime}) + EI\dot{w^{\prime}}w^{\prime\prime} \nonumber \\
f &=& \dot{w_0}(Tw^{\prime} - EIw^{\prime\prime\prime}) \nonumber \\
\delta_{t\cdot}w_{0} &=& J^{T}\delta_{t\cdot}u, \quad \delta_{xx}w_0 = 0 \nonumber \\
\end{eqnarray}$$
Definitions
$$\begin{eqnarray}
\delta_{xxxx} &=& \delta_{xx}\delta_{xx} \nonumber \\
&=& \frac{2}{h}(\delta_{x\cdot} - \delta_{x-})\delta_{xx} \nonumber \\
\end{eqnarray}$$
Stiff String
$$\delta_{tt}w = c^{2}\delta_{xx}w - \kappa^{2}\delta_{xxxx}w, \quad c = \sqrt{\frac{T}{\rho A}}, \quad \kappa = \sqrt{\frac{EI}{\rho AL^{4}}}$$
Coupling, omitting \(w_{0}\) for clarity
$$\begin{eqnarray}
\require{cancel}
\rho H \delta_{tt} &=& T\delta_{xx} - EI\delta_{xxxx} \nonumber \\
&=& \frac{2T}{h}(\delta_{x\cdot}-\delta_{x-}) - EI\frac{2}{h}(\delta_{x\cdot}-\delta_{x-})\delta_{xx} \nonumber \\
&=& \frac{2T}{h}\delta_{x\cdot}-\frac{2T}{h}\delta_{x-} - \frac{2EI}{h}(\delta_{x\cdot}\delta_{xx})-\frac{2EI}{h}\delta_{x-}\delta_{xx} \nonumber \\
&=& \frac{2}{h}(T\delta_{x\cdot}- EI(\delta_{x\cdot}\delta_{xx})- (T\delta_{x-} - EI\delta_{x-}\delta_{xx})), \quad f &=& T\delta_{x-}w_{0} - EI\delta_{xxx}w_{0} \nonumber \\
&=& \frac{2}{h}(T\delta_{x\cdot}- EI(\delta_{x\cdot}\delta_{xx})- (T\delta_{x-} - EI\delta_{x-}\delta_{xx})) \nonumber \\
&=& \frac{2}{h}(T\delta_{x\cdot}- EI(\delta_{x\cdot}\delta_{xx}) - f) \nonumber \\
\end{eqnarray}$$
Kirchoff Thin Plate w/ Coupling
$$\ddot{u} = -\kappa^{2}\Delta\Delta u, \quad \kappa = \sqrt{\frac{E H^2}{12\rho(1- \nu)} }$$