Problem description

This set of tutorials define a custom equation to model the conventional projectile motion problem:

\begin{align} \dot{\boldsymbol{x}} & = \boldsymbol{v}, \label{1} \\ \dot{\boldsymbol{v}} & = \boldsymbol{a} = \boldsymbol{g} - \mu \boldsymbol{v}, \label{2} \end{align}

where \(\boldsymbol{x}\) and \(\boldsymbol{v}\) are the position and velocity of the projectile, respectively. \(\boldsymbol{g}\) is the gravitational acceleration, and \(\mu\) is the dynamic viscosity. The projectile's trajectory can be numerically integrated using the backward-Euler method:

\begin{align} \mathbf{r} = \begin{Bmatrix} r_x \\ r_v \end{Bmatrix} & = \begin{Bmatrix} \tilde{\boldsymbol{x}} - \boldsymbol{x}_n - \left(t - t_n\right) \dot{\boldsymbol{x}} \\ \tilde{\boldsymbol{v}} - \boldsymbol{v}_n - \left(t - t_n\right) \dot{\boldsymbol{v}} \\ \end{Bmatrix}, \label{3} \\ \begin{Bmatrix} \boldsymbol{x} \\ \boldsymbol{v} \end{Bmatrix} & = \mathop{\mathrm{root}}\limits_{\tilde{\boldsymbol{x}}, \tilde{\boldsymbol{v}}} \left( \mathbf{r} \right), \label{4} \end{align}

subject to appropriate initial conditions \(\boldsymbol{x}_0\) and \(\boldsymbol{v}_0\)

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Among these equations:

\(\eqref{1}\) and \(\eqref{3}\) can be defined using VecBackwardEulerTimeIntegration.
\(\eqref{2}\) is not available in NEML2, and is the focus of this set of tutorials. We will name it ProjectileAcceleration.
\(\eqref{4}\) is the ImplicitUpdate.

Outline

Each tutorial builds on top of the previous tutorials, introduces and explains one small piece in model development. It is therefore recommended to follow these tutorials in order.

The final model is then used in another set of tutorials on parameter calibration and inference.

Table of Contents

Problem description

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Outline