diff --git a/docs/source/_static/references.bib b/docs/source/_static/references.bib new file mode 100644 index 0000000..4fb720e --- /dev/null +++ b/docs/source/_static/references.bib @@ -0,0 +1,37 @@ +@article{Taylor1973, + title={A numerical solution of the {N}avier-{S}tokes equations using the finite element technique}, + author={Taylor, Cedric and Hood, Paul}, + journal={Computers \& Fluids}, + volume={1}, + number={1}, + pages={73--100}, + year={1973}, + publisher={Elsevier} +} + +@misc{Ladyzhenskaya1963, + title={The mathematical theory of incompressible viscous flows}, + author={Ladyzhenskaya, O}, + year={1963}, + publisher={Gordon and Breach, New York} +} + +@article{Brezzi1974, + title={On the existence, uniqueness and approximation of saddle-point problems arising from {L}agrangian multipliers}, + author={Brezzi, Franco}, + journal={Publications des s{\'e}minaires de math{\'e}matiques et informatique de Rennes}, + number={S4}, + pages={1--26}, + year={1974} +} + +@article{Babuvska1971, + title={Error-bounds for finite element method}, + author={Babu{\v{s}}ka, Ivo}, + journal={Numerische Mathematik}, + volume={16}, + number={4}, + pages={322--333}, + year={1971}, + publisher={Springer} +} diff --git a/docs/source/conf.py b/docs/source/conf.py index c80c962..691f5ac 100644 --- a/docs/source/conf.py +++ b/docs/source/conf.py @@ -38,7 +38,7 @@ templates_path = ['_templates'] # -- Options for sphinxcontrib.bibtex ------------------------------------ -bibtex_bibfiles = ['demos/demo_references.bib'] +bibtex_bibfiles = ['demos/demo_references.bib', '_static/references.bib'] # -- Options for HTML output diff --git a/docs/source/user_guide/fluid_solver.rst b/docs/source/user_guide/fluid_solver.rst index 60fead5..c3cff4a 100644 --- a/docs/source/user_guide/fluid_solver.rst +++ b/docs/source/user_guide/fluid_solver.rst @@ -66,13 +66,13 @@ The dimensional version of the momentum equation is where :math:`\nu_\mathrm{eff}` is the effective viscosity in the porous medium, and :math:`K`, its permeability. It is also common to use the effective dynamic -viscosity :math:`\mu_\mathrm{eff} = \nu_\mathrm{eff} \rho` +viscosity :math:`\mu_\mathrm{eff} = \nu_\mathrm{eff} \rho`. The inverse permeability :math:`K^{-1}` can be provided directly in cases where it is zero in some regions, i.e. liquid-only regions. It is common to take :math:`\nu_\mathrm{eff}=\nu` for simplicity (the default here). -The nondimensional implementation currently assume :math:`\nu_\mathrm{eff}=\nu` +The nondimensional implementation currently assumes :math:`\nu_\mathrm{eff}=\nu` and :math:`K^{-1}>0`, such that @@ -91,7 +91,7 @@ For the dimensional version, the user must also pass the following keys: * Optional: ``"effective kinematic viscosity"`` or ``"effective dynamic viscosity"``: :math:`\nu_\mathrm{eff}` and :math:`\mu_\mathrm{eff}`, respectively -To use the nondimensional version, the user must also pass the following key: +To use the nondimensional version, the user must pass the following key: * ``"Darcy number"``: :math:`\mathrm{Da}` @@ -120,10 +120,11 @@ keys. Numerical considerations ------------------------ -The discretization is done using Taylor-Hood elements (piecewise linear -pressure, piecewise quadratic velocity), which satisfy the inf-sup condition. +The discretization is done using Taylor-Hood elements :cite:`Taylor1973` +(piecewise linear pressure, piecewise quadratic velocity), which satisfy the +inf-sup condition :cite:`Ladyzhenskaya1963,Brezzi1974,Babuvska1971`. -To achieve convergence other than at low a Reynolds number, it may be required +To achieve convergence other than at a low Reynolds number, it may be required to do continuation on the parameter so that Newton's method has better initial guesses. This can be done for example, by passing the Reynolds number as a :class:`firedrake.constant.Constant`, and assigning a larger value after each @@ -140,3 +141,8 @@ For the Darcy number, using a very small value (:math:`\mathrm{Da}\sim done in topology optimization. For smaller values, convergence issues can arise. Reversly, as :math:`\mathrm{Da}\to\infty`, we simply recover Navier-Stokes. + +.. rubric:: References + +.. bibliography:: ../_static/references.bib + :filter: docname in docnames