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rGOOSEFEM/docs/examples/dynamics_inertia-damping.rst
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dynamics_inertia-damping.rst
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*****************
Inertia + damping
*****************
Velocity Verlet
===============
.. code-block:: cpp
// position, velocity, acceleration (and history: last increment)
xt::xtensor<double,2> u = xt::zeros<double>({nnode, ndim});
xt::xtensor<double,2> v = xt::zeros<double>({nnode, ndim});
xt::xtensor<double,2> a = xt::zeros<double>({nnode, ndim});
xt::xtensor<double,2> v_n = xt::zeros<double>({nnode, ndim});
xt::xtensor<double,2> a_n = xt::zeros<double>({nnode, ndim});
// residual force
xt::xtensor<double,2> fr = xt::zeros<double>({nnode, ndim});
// compute mass matrix
// (often assumed constant & diagonal, remove either assumption if needed)
GooseFEM::MatrixDiagonal M(...);
...
// time increments
for ( ... )
{
// store history
xt::noalias(v_n) = v;
xt::noalias(a_n) = a;
// new displacement
xt::noalias(u) = u + dt * v + 0.5 * std::pow(dt,2.) * a;
// update residual force (and mass matrix if needed)
...
// estimate 1: new velocity
xt::noalias(v) = v_n + dt * a_n;
// estimate 1: new residual force (and mass matrix if needed)
...
// estimate 1: new acceleration
M.solve(fr, a);
// estimate 2: new velocity
xt::noalias(v) = v_n + .5 * dt * ( a_n + a );
// estimate 2: new residual force (and mass matrix if needed)
...
// estimate 2: new acceleration
M.solve(fr, a);
// new velocity
xt::noalias(v) = v_n + .5 * dt * ( a_n + a );
// new residual force (and mass matrix if needed)
...
// new acceleration
M.solve(fr, a);
}
Example
=======
.. todo::
Compile, run, and view instructions
| :download:`main.cpp <dynamics/Elastic-VelocityVerlet/main.cpp>`
| :download:`CMakeLists.txt <dynamics/Elastic-VelocityVerlet/CMakeLists.txt>`
| :download:`plot.py <dynamics/Elastic-VelocityVerlet/plot.py>`
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