dynamics_inertia.rst
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dynamics_inertia.rst
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			*******
			Inertia
			*******

			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;

			// new residual force (and mass matrix if needed)
			...

			// new acceleration
			M.solve(fr, a);

			// new velocity
			xt::noalias(v) = v_n + .5 * dt * ( a_n + a );
			}

			Example
			=======

			.. todo::

			Compile, run, and view instructions:

			.. code-block:: bash

			mkdir build
			cd build
			make
			./main
			python3 plot.py

			\| :download:`main.cpp <dynamics/Elastic-Verlet/main.cpp>`
			\| :download:`CMakeLists.txt <dynamics/Elastic-Verlet/CMakeLists.txt>`
			\| :download:`plot.py <dynamics/Elastic-Verlet/plot.py>`