OilfieldToolsNet/IFEM
4
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

Fixed: Minor corrections in Newmark HHT documentation

Browse Source
This commit is contained in:
Knut Morten Okstad
2018-11-20 15:09:07 +01:00
committed by Knut Morten Okstad
parent dff2eda382
commit 7be71a9a73
2 changed files with 14 additions and 19 deletions
Download Patch File Download Diff File Expand all files Collapse all files

20
src/SIM/HHTSIM.h
View File

@@ -37,7 +37,8 @@ class SystemVector;
the HHT time integration algorithm goes like this:
<H3>Initialisation of time step loop:</H3>
\f[ \beta=\frac{1}{4}(1.0-\alpha_H)^2,\; \gamma=\frac{1}{2}-\alpha_H \f]
\f[ \beta \;=\; \frac{1}{4}(1.0-\alpha_H)^2,\;
\gamma \;=\; \frac{1}{2}-\alpha_H \f]
\f[ \begin{array}{lcll}
t &=& 0 & \mbox{(initial time)} \\
{\bf u}_0 &=& {\bf 0} & \mbox{(initial displacements)} \\
@@ -54,7 +55,8 @@ class SystemVector;
t &=& t + \Delta t_n \\
{\bf u}_n^0 &=& {\bf u}_{n-1} \\
\dot{\bf u}_n^0 &=& \dot{\bf u}_{n-1} \\
\ddot{\bf u}_n^0 &=& \ddot{\bf u}_{n-1}
\ddot{\bf u}_n^0 &=& \ddot{\bf u}_{n-1} \\
\Delta{\bf u}_n &=& {\bf 0} \quad\mbox{(displacement increment)}
\f}
</LI>
@@ -64,12 +66,7 @@ class SystemVector;
\Delta t_n(\frac{\gamma}{2\beta}-1)\ddot{\bf u}_{n-1} \\
{\bf a}_n &=& \frac{1}{2\beta}\ddot{\bf u}_{n-1} +
\frac{1}{\Delta t_n\beta}\dot{\bf u}_{n-1}
\f}\f[ \begin{array}{lcll}
\dot{\bf u}_n^0 &=& {\bf v}_n & \mbox{(predicted velocity)} \\
\ddot{\bf u}_n^0 &=& {\bf a}_n & \mbox{(predicted acceleration)} \\
\Delta{\bf u}_n &=& {\bf 0} & \mbox{(displacement increment)}
\end{array}
\f]
\f}
</LI>
<LI>Assemble FE matrices and right-hand-side force vectors:
@@ -93,10 +90,9 @@ class SystemVector;
<LI>Compute Newton matrix and associated incremental load vector:
\f{eqnarray*}{
{\bf N}_n^0 &=& a_n{\bf M}_n^0 + b_n{\bf C}_n^0 + c_n{\bf K}_n^0 \\
{\bf R}_n^0 &=& (1+\alpha_H)\left[{\bf F}_n^{E,0} -
{\bf F}_{n-1}^{E,0} +
{\bf R}_n^0 &=& (1+\alpha_H)\left[{\bf F}_n^{E,0} - {\bf F}_{n-1}^E +
(\alpha_1{\bf M}_n^0 + \alpha_2{\bf K}_n^0 +
{\bf C}_n^0){\bf v}_n \right] +
{\bf C}_n^0){\bf v}_n\right] +
{\bf M}_n^0{\bf a}_n - {\bf F}_n^{I,0} +
\tilde{\bf R}_{n-1}
\f} where \f{eqnarray*}{
@@ -155,7 +151,7 @@ class SystemVector;
{\bf N}_n^i &=& a_n{\bf M}_n^i + b_n{\bf C}_n^i + c_n{\bf K}_n^i \\
{\bf R}_n^i &=& (1+\alpha_H)\left[{\bf F}_n^{E,i} - {\bf F}_n^{S,i} -
(\alpha_1{\bf M}_n^i + \alpha_2{\bf K}_n^i +
{\bf C}_n^i)\dot{\bf u}_n^i \right] +
{\bf C}_n^i)\dot{\bf u}_n^i\right] +
{\bf F}_n^{I,i} - \alpha_H\tilde{\bf R}_{n-1}
\f}
</LI>

13
src/SIM/NewmarkNLSIM.h
View File

@@ -37,7 +37,8 @@ class SystemVector;
the HHT time integration algorithm goes like this:
<H3>Initialisation of time step loop:</H3>
\f[ \beta=\frac{1}{4}(1.0-\alpha_H)^2,\; \gamma=\frac{1}{2}-\alpha_H \f]
\f[ \beta \;=\; \frac{1}{4}(1.0-\alpha_H)^2,\;
\gamma \;=\; \frac{1}{2}-\alpha_H \f]
\f[ \begin{array}{lcll}
t &=& 0 & \mbox{(initial time)} \\
{\bf u}_0 &=& {\bf 0} & \mbox{(initial displacements)} \\
@@ -53,8 +54,7 @@ class SystemVector;
i &=& 0 \quad\mbox{(iteration counter)}\\
t &=& t + \Delta t_n \\
{\bf u}_n^0 &=& {\bf u}_{n-1} \\
\dot{\bf u}_n^0 &=& \dot{\bf u}_{n-1} \\
\ddot{\bf u}_n^0 &=& \ddot{\bf u}_{n-1}
\Delta{\bf u}_n &=& {\bf 0} \quad\mbox{(displacement increment)}
\f}
</LI>
@@ -66,8 +66,7 @@ class SystemVector;
\frac{1}{\Delta t_n\beta}\dot{\bf u}_{n-1}
\f}\f[ \begin{array}{lcll}
\dot{\bf u}_n^0 &=& {\bf v}_n & \mbox{(predicted velocity)} \\
\ddot{\bf u}_n^0 &=& {\bf a}_n & \mbox{(predicted acceleration)} \\
\Delta{\bf u}_n &=& {\bf 0} & \mbox{(displacement increment)}
\ddot{\bf u}_n^0 &=& {\bf a}_n & \mbox{(predicted acceleration)}
\end{array}
\f]
</LI>
@@ -95,7 +94,7 @@ class SystemVector;
{\bf N}_n^0 &=& a_n{\bf M}_n^0 + b_n{\bf C}_n^0 + c_n{\bf K}_n^0 \\
{\bf R}_n^0 &=& (1+\alpha_H)\left[{\bf F}_n^{E,0} - {\bf F}_n^{S,0} +
(\alpha_1{\bf M}_n^0 + \alpha_2{\bf K}_n^0 +
{\bf C}_n^0){\bf v}_n \right] +
{\bf C}_n^0){\bf v}_n\right] +
{\bf F}_n^{I,0} - \alpha_H\tilde{\bf R}_{n-1}
\f} where \f{eqnarray*}{
a_n &=& \frac{1}{\Delta t_n^2\beta} +
@@ -146,7 +145,7 @@ class SystemVector;
{\bf N}_n^i &=& a_n{\bf M}_n^i + b_n{\bf C}_n^i + c_n{\bf K}_n^i \\
{\bf R}_n^i &=& (1+\alpha_H)\left[{\bf F}_n^{E,i} - {\bf F}_n^{S,i} -
(\alpha_1{\bf M}_n^i + \alpha_2{\bf K}_n^i +
{\bf C}_n^i)\dot{\bf u}_n^i \right] -
{\bf C}_n^i)\dot{\bf u}_n^i\right] -
{\bf F}_n^{I,i} - \alpha_H\tilde{\bf R}_{n-1}
\f}
</LI>