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docs/joss-paper/paper.md

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@@ -124,7 +124,7 @@ The symbolic representation of the strong-form that is encoded in `underworld3`
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Here $\mathrm{H}$ represents sources and sinks of $u$, and ${D u}/{D t}$ is the material time derivative of $u$. The time derivatives of the unknowns are not present in the `PETSc` template but, after time-discretisation, they produce terms that are combinations of fluxes and flux history terms (which combine with $\boldsymbol{\sigma}$ to contribute to $F$) and forces (which combine with $\mathbf{h}$ to contribute to $f$). The explicit time / position dependence in $\sigma$ is to highlight potential changes to boundary conditions or constitutive properties.
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In `underworld3`, the user interacts with the time derivatives explicitly, and provides strong-form expressions for the template \ref{eq:sympy-strong-form}. `Sympy` automatically gathers all the flux-like terms and all the force-like terms into the form required by the `PETSc` template. All evaluations, derivatives and simplifications of functions in the `underworld3` symbolic layer are deferred until final assembly of the `PETSc` template and the compilation of the `C` functions.
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In `underworld3`, the user interacts with the time derivatives explicitly, and provides strong-form expressions for the template (\ref{eq:sympy-strong-form}). `Sympy` automatically gathers all the flux-like terms and all the force-like terms into the form required by the `PETSc` template. All evaluations, derivatives and simplifications of functions in the `underworld3` symbolic layer are deferred until final assembly of the `PETSc` template and the compilation of the `C` functions.
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The main benefits of combining `sympy` with the `PETSc` weak form template is a user environment that 1) provides symbolic, mathematical introspection, particularly in the context of Jupyter notebooks; 2) eliminates much of the `python` or `C` coding required for complex constitutive models; 3) eliminates any need for users to compute derivatives for the Newton solvers in `PETSc`.
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