Equation scaling
When working with SI units in SEAS models numbers might get very large.
Rescaling the equations might be advantageous to avoid large round-off errors in finite precision.
In this section, we show how to properly scale the elasticity equations.
The linear elasticity equations in first order form are given by
\[\begin{aligned}
\sigma_{ij} &= \lambda \delta_{ij}\frac{\partial u_k}{\partial x_k} +
\mu\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right) \\
- \frac{\partial \sigma_{ij}}{\partial x_j} &= f_i
\end{aligned}\]
We define scaled quantities
\[\bar{x}_i = \frac{x_i}{L}, \quad \bar{u}_i = \frac{u_i}{u_c}, \quad
\bar{\sigma}_{ij} = \frac{\sigma_{ij}}{\sigma_c},\]
where \(L, u_c, \sigma_c\) are scaling constants.
Inserting these into the linear elasticity equations gives
\[\begin{aligned}
\sigma_c\bar{\sigma}_{ij} &= L^{-1}u_c\lambda
\delta_{ij}\frac{\partial \bar{u}_k}{\partial \bar{x}_k} +
L^{-1}u_c\mu\left(\frac{\partial \bar{u}_i}{\partial \bar{x}_j} +
\frac{\partial \bar{u}_j}{\partial \bar{x}_i}\right) \\
- L^{-1}\sigma_c\frac{\partial \bar{\sigma}_{ij}}{\partial \bar{x}_j} &= f_i
\end{aligned}\]
Multiplying the first equation with \(\sigma_c^{-1}\),
multiplying the second equation with \(L\sigma_c^{-1}\), and defining
\[\bar{\lambda} = \sigma_c^{-1}u_cL^{-1}\lambda, \quad
\bar{\mu} = \sigma_c^{-1}u_cL^{-1}\mu, \quad
\bar{f}_i = L\sigma_c^{-1} f_i\]
leads to
\[\begin{aligned}
\bar{\sigma}_{ij} &= \bar{\lambda}
\delta_{ij}\frac{\partial \bar{u}_k}{\partial \bar{x}_k} +
\bar{\mu}\left(\frac{\partial \bar{u}_i}{\partial \bar{x}_j} +
\frac{\partial \bar{u}_j}{\partial \bar{x}_i}\right) \\
- \frac{\partial \bar{\sigma}_{ij}}{\partial \bar{x}_j} &= \bar{f}_i
\end{aligned}\]
That is, we recovered the original equations and we only need to scale
the mesh and the parameters.
Example
We change units with the scaling constants
\[L = 10^3, \quad u_c = 1, \quad \sigma_c = 10^6
\]
In the rescaled equations, the spatial dimension of the mesh is [km],
velocities are in [m/s], and stresses are in [MPa].
Parameters and source terms are scaled with
\[\bar{\lambda} = 10^{-9}\lambda, \quad
\bar{\mu} = 10^{-9}\mu, \quad
\bar{f}_i = 10^{-3} f_i\]
i.e. the Lamé parameters are given in [GPa] and force in [10-3 N/m-3].