From d0bab8a3d85f8d7f4d2a6ba390958047d4abb4f0 Mon Sep 17 00:00:00 2001 From: KarsKnook Date: Wed, 14 May 2025 11:49:53 +0100 Subject: [PATCH 01/32] initial commit multicomponent flow demo --- demos/multicomponent/multicomponent.py.rst | 16 ++++++++++++++++ docs/source/advanced_tut.rst | 1 + 2 files changed, 17 insertions(+) create mode 100644 demos/multicomponent/multicomponent.py.rst diff --git a/demos/multicomponent/multicomponent.py.rst b/demos/multicomponent/multicomponent.py.rst new file mode 100644 index 0000000000..961c5a1de5 --- /dev/null +++ b/demos/multicomponent/multicomponent.py.rst @@ -0,0 +1,16 @@ +Multicomponent flow -- microfluidic mixing of hydrocarbons +=================================================== + +.. rst-class:: emphasis + + In this tutorial we demonstrate how Firedrake can be used to + simulate multicomponent flow; specifically the microfluidic mixing of + benzene and cyclohexane. + + The demo was contributed by `Aaron Baier-Reinio + `__ and `Kars Knook + `__. + +We consider a steady, isothermal mixture of benzene and cyclohexane in +a two-dimensional microfluidic container :math:`\Omega`. + diff --git a/docs/source/advanced_tut.rst b/docs/source/advanced_tut.rst index d9fbf252ba..210c14cebf 100644 --- a/docs/source/advanced_tut.rst +++ b/docs/source/advanced_tut.rst @@ -24,3 +24,4 @@ element systems. Rayleigh-Benard convection. Netgen support. Full-waveform inversion: spatial and wave sources parallelism. + Steady multicomponent flow -- microfluidic mixing of hydrocarbons. From 95f8d3829f8e941a47e914c6416ad54c3954faf2 Mon Sep 17 00:00:00 2001 From: KarsKnook Date: Wed, 14 May 2025 13:02:27 +0100 Subject: [PATCH 02/32] more writing --- demos/multicomponent/multicomponent.py.rst | 58 ++++++++++++++++++++-- 1 file changed, 54 insertions(+), 4 deletions(-) diff --git a/demos/multicomponent/multicomponent.py.rst b/demos/multicomponent/multicomponent.py.rst index 961c5a1de5..ba8699bf3a 100644 --- a/demos/multicomponent/multicomponent.py.rst +++ b/demos/multicomponent/multicomponent.py.rst @@ -4,13 +4,63 @@ Multicomponent flow -- microfluidic mixing of hydrocarbons .. rst-class:: emphasis In this tutorial we demonstrate how Firedrake can be used to - simulate multicomponent flow; specifically the microfluidic mixing of - benzene and cyclohexane. + simulate multicomponent flow; specifically the microfluidic + mixing of benzene and cyclohexane. The demo was contributed by `Aaron Baier-Reinio `__ and `Kars Knook `__. -We consider a steady, isothermal mixture of benzene and cyclohexane in -a two-dimensional microfluidic container :math:`\Omega`. +We consider a steady, isothermal, nonreacting mixture of benzene and cyclohexane in +a microfluidic container :math:`\Omega \subset \mathbb{R}^d`. +We model the mixture using the Stokes--Onsager--Stefan--Maxwell equations. +Momentum transport is modelled using the compressible +Stokes momentum equation for a Newtonian fluid, + +.. math:: + + -\nabla \cdot \big\{ + 2 \eta \epsilon(v) + (\zeta - 2\eta / d) (\nabla \cdot v) \mathbb{I} + \big\} + \nabla p = 0 \quad \textrm{in}\ \Omega. + +Here, the unknowns are the :math:`\mathbb{R}^2`-valued velocity :math:`v`, +scalar pressure :math:`p` and density :math:`rho`. +Moreover :math:`\epsilon (v)` denotes the symmetric gradient of :math:`v`, +:math:`\eta, \zeta > 0` are the shear and bulk viscosities, +:math:`d` is the spatial dimension (:math:`d=2` in this demo), +:math:`\mathbb{I}` the :math:`d \times d` identity matrix. + +Let :math:`n` denote the number of chemical species in the mixture; +in this example :math:`n=2` (benzene and cyclohexane). +The continuity equation for molar concentration :math:`c_i` +of species :math:`i \in 1:n` in the abscence of chemical reactions is + +.. math:: + + \partial_t c_i + \frac{1}{M_i} \nabla \cdot J_i = 0 + +where :math:`M_i > 0` is the molar mass of species :math:`i` and +:math:`J_i` its mass flux. + +The mass fluxes must be modelled using a constitutive relation. +For example, in a Fickian model we might use +:math:`J_i = M_i c_i v - D_i \nabla c_i` +where the first term represents advection and the second term Fickian diffusion +(here :math:`D_i > 0` are Fickian diffusion coefficients). +The Fickian approach is appropriate for dilute mixtures (i.e. mixtures where all but +one of the species are present in trace amounts), but it typically is not +thermodynamically consistent in the non-dilute regime and fails to account for +cross-diffusional effects. We therefore model the mass fluxes using the +Onsager--Stefan--Maxwell equations, which capture cross-diffusion in non-dilute +mixtures in a thermodynamically consistent fashion. Sometimes in the literature +these equations are referred to as the generalized Maxwell--Stefan equations [CITE BIRD]. + + + + + ++---------------------------+---------------------------+ +| .. image:: benzene_0.png | .. image:: benzene_4.png | +| :width: 100% | :width: 100% | ++---------------------------+---------------------------+ From 737c860d431ae995bb227c4b13121e8b003b4757 Mon Sep 17 00:00:00 2001 From: KarsKnook Date: Thu, 15 May 2025 18:56:37 +0100 Subject: [PATCH 03/32] text, code, figures --- demos/demo_references.bib | 20 + demos/multicomponent/benzene_0.png | Bin 0 -> 197012 bytes demos/multicomponent/benzene_4.png | Bin 0 -> 217725 bytes .../multicomponent/microfluidic_container.png | Bin 0 -> 399347 bytes demos/multicomponent/multicomponent.py.rst | 444 +++++++++++++++++- 5 files changed, 438 insertions(+), 26 deletions(-) create mode 100644 demos/multicomponent/benzene_0.png create mode 100644 demos/multicomponent/benzene_4.png create mode 100644 demos/multicomponent/microfluidic_container.png diff --git a/demos/demo_references.bib b/demos/demo_references.bib index cb4633b5ef..2329f2e220 100644 --- a/demos/demo_references.bib +++ b/demos/demo_references.bib @@ -34,6 +34,14 @@ @Book{2006:SV year = 2006 } +@book{Perry:2007, + title = {Perry’s Chemical Engineers’ Handbook}, + edition = {8th}, + author = {Green, D. W. and Perry, R. H.}, + year = 2007, + publisher = {McGraw Hill Professional} +} + @InBook{2015:lmscup, author = {O. Bokhove and A. Kalogirou}, title = {Lectures on the Theory of Water Waves}, @@ -308,3 +316,15 @@ @article{Ricker:1953 publisher={Society of Exploration Geophysicists}, doi={https://doi.org/10.1190/1.1437843} } + +@article{Krishna:1997, + title={The {M}axwell--{S}tefan approach to mass transfer}, + author={Krishna, Rajamani and Wesselingh, Johannes A}, + journal={Chemical Engineering Science}, + volume={52}, + number={6}, + pages={861--911}, + year={1997}, + publisher={Elsevier}, + doi={10.1016/S0009-2509(96)00458-7} +} \ No newline at end of file diff --git a/demos/multicomponent/benzene_0.png b/demos/multicomponent/benzene_0.png new file mode 100644 index 0000000000000000000000000000000000000000..d832106c6658455c9310e638e75d25bd5a8721c6 GIT binary patch literal 197012 zcmZsDbwC_Tvu{FhclX6z7I$0R9fB_I?h@QRxVsYw5?q24+$FeMu;7xn=ezg5cg}ft z_mAo6>8a|jnX0b(RqaNpC`qFr5+c5P_YOr?MpEtFJLs5q@1WknLqU4tn*d z7C`;aRp^+45C7Bti_N=t_(l(B5F)zP8ai$|iVFN-M|&0%Ge=W%7B73J|BMjy;)gWt 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zjf(}oAkoy5n-kRk+Cfh5sw6_E)#NwqmQcGJi9u(GhUXvh96}BdiG~ zIA#z!0pvEY*<|{ACL@({!v34ggg&v!n7>I+S zNBRu)C7w%ewllJZ7UPNfI1(qb{Nl=jV&$_xXVbt$Of*v_nox;2`-UFfp%Bj;4Tl1= zG)g>XcTJ-auo8MS{B8|!@}1RtAWhol5h~iDd_Yh8m)0JiMem3Gfr{}9F@gcop%A`E zzG=N*nclVAaL<5H2VXo~;8^>g}#OM4fn3AJ+3j*$OT8LQxLjDyxy%qLmB%-PDm9Utl z9B;Xy2+28lU&gxLpEF;#^grd9-*;Vd@st+*B7eg)6EC-o+JDTgkG$X`f%dKMF;dQK zX!{+}9mg-o>C)yaocm%VucvoXy=ORAtR~OauIv^Zx&_Jobv`__ and `Kars Knook `__. We consider a steady, isothermal, nonreacting mixture of benzene and cyclohexane in -a microfluidic container :math:`\Omega \subset \mathbb{R}^d`. - -We model the mixture using the Stokes--Onsager--Stefan--Maxwell equations. -Momentum transport is modelled using the compressible +a two-dimensional microfluidic container :math:`\Omega \subset \mathbb{R}^2`. +We model the mixture using the Stokes--Onsager--Stefan--Maxwell (SOSM) equations. +Momentum transport is modelled using the steady compressible Stokes momentum equation for a Newtonian fluid, .. math:: @@ -25,40 +23,434 @@ Stokes momentum equation for a Newtonian fluid, \big\} + \nabla p = 0 \quad \textrm{in}\ \Omega. Here, the unknowns are the :math:`\mathbb{R}^2`-valued velocity :math:`v`, -scalar pressure :math:`p` and density :math:`rho`. +scalar pressure :math:`p` and density :math:`\rho`. Moreover :math:`\epsilon (v)` denotes the symmetric gradient of :math:`v`, -:math:`\eta, \zeta > 0` are the shear and bulk viscosities, -:math:`d` is the spatial dimension (:math:`d=2` in this demo), +:math:`\eta, \zeta > 0` are the shear and bulk viscosities respectively, +:math:`d=2` the spatial dimension and :math:`\mathbb{I}` the :math:`d \times d` identity matrix. -Let :math:`n` denote the number of chemical species in the mixture; -in this example :math:`n=2` (benzene and cyclohexane). -The continuity equation for molar concentration :math:`c_i` -of species :math:`i \in 1:n` in the abscence of chemical reactions is +Let :math:`n` denote the number of chemical species in the mixture +(in this example :math:`n=2`, i.e. benzene and cyclohexane). +The continuity equation for the molar concentration :math:`c_i` +of species :math:`i \in \{1:n\}` in the absence of chemical reactions is .. math:: \partial_t c_i + \frac{1}{M_i} \nabla \cdot J_i = 0 + \quad \textrm{in}\ \Omega \quad \forall i \in \{1 : n \}, where :math:`M_i > 0` is the molar mass of species :math:`i` and :math:`J_i` its mass flux. +As we are considering steady flow, the continuity equations simplify to + +.. math:: + + \frac{1}{M_i} \nabla \cdot J_i = 0 + \quad \textrm{in}\ \Omega \quad \forall i \in \{1 : n \}. + +The mass fluxes must be modelled with a constitutive law. +A basic Fickian model may use :math:`J_i = M_i (c_i v - D_i \nabla c_i)` +where :math:`c_i v` represents advection and :math:`-D_i \nabla c_i` Fickian diffusion. +The Fickian approach is appropriate for dilute mixtures +(i.e. mixtures where all of the species but one are present in trace amounts), +but typically is not thermodynamically consistent in the non-dilute regime, +and fails to account for cross-diffusional effects. +We remedy these drawbacks by employing the Onsager--Stefan--Maxwell (OSM) equations +(also known as Maxwell--Stefan equations :cite:`Krishna:1997`), +which, in the present isothermal setting, implicitly determine the mass fluxes through the relations + +.. math:: + + -\frac{1}{M_i} \nabla \mu_i + \frac{1}{\rho} \nabla p &= + \sum_{\substack{j=1 \\ j \neq i}}^n \frac{RT c_j}{\mathscr{D}_{ij} M_i c_T} + \Bigg( \frac{J_i}{M_i c_i} - \frac{J_j}{M_j c_j} \Bigg) + \quad \textrm{in}\ \Omega + \quad \forall i \in \{1 : n \}. + +Here :math:`\mu_i` is the chemical potential of species `i`, +:math:`R` the ideal gas constant, +:math:`T` the temperature, +:math:`\mathscr{D}_{ij}` Stefan--Maxwell diffusion coefficients +and :math:`c_T = \sum_{j=1}^n c_j` the total concentration. +Onsager reciprocal relations imply that +:math:`\mathscr{D}_{ij} = \mathscr{D}_{ji} \ \forall i \neq j` while +:math:`\mathscr{D}_{jj}` is undefined. +Since :math:`n=2` there is only one Stefan--Maxwell diffusion coefficient +:math:`\mathscr{D}_{12} = \mathscr{D}_{21}` +and in the code below we denote it by :code:`D_sm`. + +Only :math:`n-1` of the OSM equations are linearly independent; the :math:`J_i`'s +are fully determined by a mass-average constraint +:math:`v = \frac{1}{\rho} \sum_{j=1}^n J_j`. +We will incorporate this numerically by introducing an augmentation parameter +:math:`\gamma > 0` and reformulating the OSM equations as + +.. math:: + + -\frac{1}{M_i} \nabla \mu_i + \frac{1}{\rho} \nabla p + + \frac{\gamma}{\rho} v &= + \sum_{\substack{j=1 \\ j \neq i}}^n + \frac{\gamma}{\rho^2} J_j + + \frac{RT c_j}{\mathscr{D}_{ij} M_i c_T} + \Bigg( \frac{J_i}{M_i c_i} - \frac{J_j}{M_j c_j} \Bigg) + \quad \textrm{in}\ \Omega + \quad \forall i \in \{1 : n \}. + +In our numerics we will also (weakly) enforce that +:math:`\nabla \cdot v = \nabla \cdot (\frac{1}{\rho} \sum_{j=1}^n J_j )` +as this turns out to yield a well-posed discrete scheme. + +This completes the description of transport phenomena in the mixture. +However, to obtain a closed set of equations, we must also model how the free energy and +volume of the mixture depend on temperature, pressure and composition. +To describe this we must introduce mole fractions :math:`x_i := c_i / c_T`; +note that by definition :math:`\sum_{j=1}^n x_j = 1`. +Thermodynamics requires that the :math:`\mu_i`'s' and :math:`c_T` satisfy + +.. math:: + + \mu_i &= g_i(T, p, x_1, \ldots, x_n) + \quad \textrm{in}\ \Omega + \quad \forall i \in \{1 : n \}, \\ + \frac{1}{c_T} &= \sum_{j=1}^n x_j V_j(T, p, x_1, \ldots, x_n) \quad \textrm{in}\ \Omega, + +where :math:`g_i: \mathbb{R}^{n+2} \rightarrow \mathbb{R}` are partial molar Gibbs functions +and :math:`V_i: \mathbb{R}^{n+2} \rightarrow \mathbb{R}` partial molar volume functions. +These functions are derived from partial derivatives of the Gibbs free energy of the mixture. +In this demo we employ a Margules model :cite:`Perry:2007` for the :math:`g_i`'s. +We also assume that the partial molar volumes are constant +(for liquids this is usually a reasonable assumption), from which it follows that + +.. math:: + + \frac{1}{c_T} = \sum_{j=1}^n \frac{x_j}{c_j^{\textrm{pure}}} \quad \textrm{in}\ \Omega, + +where :math:`c_j^{\textrm{pure}}` is the (known) concentration of pure species `j`. +Note that the above equation is an example of a volumetric equation of state. + + + +Defining parameters for the numerics and nondimensionalised physical quantities:: + + from firedrake import * + + k = 3 # The polynomial degree (for the velocity spaces) + deg_max = 15 # Maximum quadrature degree + gamma = Constant(1e-1) # Augmentation parameter, dimensionless + v_ref_1 = Constant(0.4e-6) # Initialisng reference inlet velocity of benzene (m/s) + + RT = Constant(8.314 * 298.15) # Ideal gas constant times temperature, J / mol + eta = Constant(6e-4) # Shear viscosity, Pa s + zeta = Constant(1e-7) # Bulk viscosity, Pa s + D_sm = Constant(2.1e-9) # Stefan--Maxwell diffusivity, m^2 / s + L_ref = Constant(2e-3) # Reference length, m + + # Constants for the pure species + M_1 = Constant(0.078) # Molar mass of Benzene, kg / mol + M_2 = Constant(0.084) # Molar mass of Cyclohexane, kg / mol + rho_pure_1 = Constant(876) # Density of pure Benzene, kg / m^3 + rho_pure_2 = Constant(773) # Density of pure Cyclohexane, kg / m^3 + c_pure_1 = rho_pure_1 / M_1 # Concentration of pure Benzene, mol / m^3 + c_pure_2 = rho_pure_2 / M_2 # Concentration of pure Cyclohexane, mol / m^3 + + # Constants for the equimolar mixture + c_equi_tot = (c_pure_1 * c_pure_2) / ((0.5 * c_pure_2) + (0.5 * c_pure_1)) # Total equimolar concentration, mol / m^3 + c_equi_1 = 0.5 * c_equi_tot # Equimolar concentration of Benzene, mol / m^3 + c_equi_2 = 0.5 * c_equi_tot # Equimolar concentration of Cyclohexane, mol / m^3 + rho_equi = (M_1 * c_equi_1) + (M_2 * c_equi_2) # Equimolar density, kg / m^3 + + # Reference density and related constants + rho_ref = rho_equi # Reference density, kg / m^3 + c_ref = c_equi_tot # Reference concentration, mol / m^3 + M_ref = rho_ref / c_ref # Reference molar mass, kg / mol + + # Reference convective velocities + v_ref_2 = (c_pure_1 / c_pure_2) * v_ref_1 # Reference inflow velocity of Cyclohexane, m / s + v_ref = 0.5 * (v_ref_1 + v_ref_2) # Reference velocity, m / s + + # Derived quantities + v_sm_ref = D_sm / L_ref # Reference diffusion velocity, m / s + p_ref = eta * v_ref / L_ref # Reference pressure, Pa + lame_ND = (zeta / eta) - 1.0 # Non-dimensionalised Lame parameter, dimensionless + Pe = v_ref / v_sm_ref # Peclet number, dimensionless + Me = p_ref / (RT * c_ref) # Pressure diffusion number, dimensionless + Me_1 = p_ref / (RT * c_pure_1) # Non-dimensionalised partial molar volume of Benzene, dimensionless + Me_2 = p_ref / (RT * c_pure_2) # Non-dimensionalised partial molar volume of Cyclohexane, dimensionless + M_1_ND = M_1 / M_ref # Non-dimensionalised molar mass of Benzene, dimensionless + M_2_ND = M_2 / M_ref # Non-dimensionalised molar mass of Cyclohexane, dimensionless + + # Margules model parameters + A_12 = Constant(0.4498) # Dimensionless + A_21 = Constant(0.4952) # Dimensionless + +Pure benzene and cyclohexane are piped in through opposing inlets on the left-hand side of the microfluidic container. +These chemicals form a nonideal mixture before flowing out through the outlet on the right-hand side. +The mesh of the two-dimensional microfluidic container is created using :doc:`netgen `. +We use fourth order curved elements to adequately capture the geometry of the container. :: + + import netgen.occ as ngocc + + wp = ngocc.WorkPlane() + wp.MoveTo(0, 1) + wp.Spline([ngocc.Pnt(1, 0), ngocc.Pnt(0, -1)]) + wp.LineTo(1, -1) + wp.Spline([ngocc.Pnt(3, -0.5), ngocc.Pnt(4, -0.5)], tangents={ 1 : ngocc.gp_Vec2d(1, 0) }) + wp.LineTo(4, 0.5) + wp.Spline([ngocc.Pnt(3, 0.5), ngocc.Pnt(1, 1)], tangents={ 0 : ngocc.gp_Vec2d(-1, 0) }) + wp.LineTo(0, 1) + + shape = wp.Face() + shape.edges.Max(ngocc.Y).name = "inlet_1" + shape.edges.Min(ngocc.Y).name = "inlet_2" + shape.edges.Max(ngocc.X).name = "outlet" + + ngmesh = ngocc.OCCGeometry(shape, dim=2).GenerateMesh(maxh=0.055) + mesh = Mesh(Mesh(ngmesh).curve_field(4)) # fourth order curved elements + + inlet_1_ids = [i+1 for i, name in enumerate(ngmesh.GetRegionNames(dim=1)) if name == "inlet_1"] + inlet_2_ids = [i+1 for i, name in enumerate(ngmesh.GetRegionNames(dim=1)) if name == "inlet_2"] + outlet_ids = [i+1 for i, name in enumerate(ngmesh.GetRegionNames(dim=1)) if name == "outlet"] + walls_ids = [i+1 for i, name in enumerate(ngmesh.GetRegionNames(dim=1)) if name == ""] + + ds = ds(mesh, degree=deg_max) + dx = dx(mesh, degree=deg_max) + x_sc, y_sc = SpatialCoordinate(mesh) + +The mesh is displayed below. + +.. image:: microfluidic_container.png + :align: center + :width: 60% + +Thermodynamic constitutive relations:: + + # Margules model for chemical potentials, assuming constant partial molar volumes + def mu_relation(x_1, x_2, p): + mu_1 = (Me_1 * p) + ln(x_1) \ + + (x_2 ** 2) * (A_12 + (2.0 * (A_21 - A_12) * x_1)) + mu_2 = (Me_2 * p) + ln(x_2) \ + + (x_1 ** 2) * (A_21 + (2.0 * (A_12 - A_21) * x_2)) + + return (mu_1, mu_2) + + # Volumetric equation of state, assuming constant partial molar volumes + def conc_relation(x_1, x_2): + x_1_nm = x_1 / (x_1 + x_2) + x_2_nm = x_2 / (x_1 + x_2) + + c_tot = 1.0 / ((x_1_nm * (c_ref / c_pure_1)) + (x_2_nm * (c_ref / c_pure_2))) + c_1 = x_1_nm * c_tot + c_2 = x_2_nm * c_tot + + return (c_tot, c_1, c_2) + +Defining the test and trial functions:: + + W_h = FunctionSpace(mesh, "BDM", k) # Momentum space + V_h = VectorFunctionSpace(mesh, "CG", k) # Bulk velocity space + U_h = FunctionSpace(mesh, "DG", k - 1) # Chemical potential space + P_h = FunctionSpace(mesh, "CG", k - 1) # Pressure space + X_h = FunctionSpace(mesh, "DG", k - 1) # Molar fraction space + R_h = FunctionSpace(mesh, "CG", k - 1) # Density reciprocal space + L_h = FunctionSpace(mesh, "R", 0) # Real space + + Z_h = W_h * W_h * V_h * U_h * U_h * P_h * X_h * X_h * R_h * L_h * L_h + PETSc.Sys.Print("Mesh has %d cells, with %d finite element DOFs" % (mesh.num_cells(), Z_h.dim())) + + sln = Function(Z_h) + mm_1, mm_2, v, mu_aux_1, mu_aux_2, p, x_1, x_2, rho_inv, l_1, l_2 = split(sln) + mu_1 = mu_aux_1 + l_1 + mu_2 = mu_aux_2 + l_2 + u_1, u_2, u, w_1, w_2, q, y_1, y_2, r, s_1, s_2 = TestFunctions(Z_h) + +Defining the variational formulation:: + + c_tot, c_1, c_2 = conc_relation(x_1, x_2) + + # The Stokes viscous terms + A_visc = 2.0 * inner(sym(grad(v)), sym(grad(u))) * dx + A_visc += lame_ND * inner(div(v), div(u)) * dx + + # The augmented Onsager transport matrix terms + A_osm = Pe * (1.0 / c_tot) * ((c_2 / (M_1_ND * M_1_ND * c_1)) * inner(mm_1, u_1) \ + + (c_1 / (M_2_ND * M_2_ND * c_2)) * inner(mm_2, u_2) \ + - (1.0 / (M_1_ND * M_2_ND)) * (inner(mm_1, u_2) + inner(mm_2, u_1))) * dx + A_osm += Pe * gamma * inner(v - (rho_inv * (mm_1 + mm_2)), u - (rho_inv * (u_1 + u_2))) * dx + + # The diffusion driving force terms and Stokes pressure term + B_blf = ((Me * inner(p, div(rho_inv * (u_1 + u_2)))) - inner(p, div(u))) * dx + B_blf -= ((1.0 / M_1_ND) * inner(mu_1, div(u_1)) + (1.0 / M_2_ND) * inner(mu_2, div(u_2))) * dx + + # The div(mass-average constraint) and continuity equation terms + BT_blf = (inner(q, div(rho_inv * (mm_1 + mm_2))) - inner(q, div(v))) * dx + BT_blf -= ((1.0 / M_1_ND) * inner(w_1, div(mm_1)) + (1.0 / M_2_ND) * inner(w_2, div(mm_2))) * dx + + # The total residual + tot_res = A_visc + A_osm + B_blf + BT_blf + + # The thermodynamic constitutive relation and density reciprocal terms + mu_1_cr, mu_2_cr = mu_relation(x_1, x_2, p) + tot_res += (inner(mu_1 - mu_1_cr, y_1) + inner(mu_2 - mu_2_cr, y_2)) * dx + + tot_res += inner(1.0 / rho_inv, r) * dx + tot_res -= inner((M_1_ND * c_1) + (M_2_ND * c_2), r) * dx + + # The density consistency terms + nml = FacetNormal(mesh) + tot_res -= q * inner((rho_inv * (mm_1 + mm_2)) - v, nml) * ds + + # integral constraints + tot_res += inner(x_1 + x_2 - 1, s_1) * dx + tot_res += inner((M_1_ND * c_1) - (M_2_ND * c_2), s_2) * ds(outlet_id) + +We enforce parabolic profiles on :math:`J_i \cdot n` at inflow :math:`i` and on the outflow. +The magnitude of the parabolic profiles are :math:`M_ic_i^\text{ref}v_i^\text{ref}` respectively. +Elsewhere on the boundary :math:`J_i \cdot n = 0` for each :math:`i`. Instead of specifying +the value of the bulk velocity on the inflows and outflow, we enforce :math:`\rho v \cdot n = (J_1 + J_2 )\cdot n` +and :math:`\rho v \times n = 0` in these regions. This means that :math:`v` equals an unknown quantity +so we need to use :code:`EquationBC` instead of :code:`DirichletBC`. :: + + J_1_inflow_bc_func = -M_1_ND * 2.0 * x_sc * (x_sc - 1.0) * (v_ref_1 / v_ref) * (c_pure_1 / c_ref) * as_vector([2.0, -1.0]) + J_2_inflow_bc_func = -M_2_ND * 2.0 * x_sc * (x_sc - 1.0) * (v_ref_2 / v_ref) * (c_pure_2 / c_ref) * as_vector([2.0, 1.0]) + rho_v_inflow_1_bc_func = J_1_inflow_bc_func + rho_v_inflow_2_bc_func = J_2_inflow_bc_func + + J_1_outflow_bc_func = -M_1_ND * 2.0 * (y_sc + 0.5) * (y_sc - 0.5) * (v_ref_1 / v_ref) * (c_pure_1 / c_ref) * as_vector([1.0, 0.0]) + J_2_outflow_bc_func = -M_2_ND * 2.0 * (y_sc + 0.5) * (y_sc - 0.5) * (v_ref_2 / v_ref) * (c_pure_2 / c_ref) * as_vector([1.0, 0.0]) + rho_v_outflow_bc_func = J_1_outflow_bc_func + J_2_outflow_bc_func + + # Boundary conditions on the bulk velocity are enforced via EquationBC + v_inflow_1_bc = EquationBC(inner(v - rho_inv * rho_v_inflow_1_bc_func, u) * ds(inlet_1_id, degree=deg_max) == 0, sln, inlet_1_id, V=Z_h.sub(2)) + v_inflow_2_bc = EquationBC(inner(v - rho_inv * rho_v_inflow_2_bc_func, u) * ds(inlet_2_id, degree=deg_max) == 0, sln, inlet_2_id, V=Z_h.sub(2)) + v_outflow_bc = EquationBC(inner(v - rho_inv * rho_v_outflow_bc_func, u) * ds(outlet_id, degree=deg_max) == 0, sln, outlet_id, V=Z_h.sub(2)) + + # The boundary conditions on the fluxes + flux_bcs = [DirichletBC(Z_h.sub(0), J_1_inflow_bc_func, inlet_1_id), + DirichletBC(Z_h.sub(0), J_1_outflow_bc_func, outlet_id), + DirichletBC(Z_h.sub(0), as_vector([0.0, 0.0]), inlet_2_id), + DirichletBC(Z_h.sub(0), as_vector([0.0, 0.0]), walls_ids), + DirichletBC(Z_h.sub(1), J_2_inflow_bc_func, inlet_2_id), + DirichletBC(Z_h.sub(1), J_2_outflow_bc_func, outlet_id), + DirichletBC(Z_h.sub(1), as_vector([0.0, 0.0]), inlet_1_id), + DirichletBC(Z_h.sub(1), as_vector([0.0, 0.0]), walls_ids), + v_inflow_1_bc, + v_inflow_2_bc, + v_outflow_bc, + DirichletBC(Z_h.sub(2), as_vector([0.0, 0.0]), walls_ids)] + +Furthermore, we use :code:`FixAtPointBC` from :doc:`Steady Boussinesq problem with integral constraints.` +to remove the pressure null space and fix :math:`\mu_i = 0` for each :math:`i`. :: + + import firedrake.utils as firedrake_utils + + class FixAtPointBC(firedrake.DirichletBC): + r'''A special BC object for pinning a function at a point. + + :arg V: the :class:`.FunctionSpace` on which the boundary condition should be applied. + :arg g: the boundary condition value. + :arg bc_point: the point at which to pin the function. + The location of the finite element DOF nearest to bc_point is actually used. + ''' + def __init__(self, V, g, bc_point): + super(FixAtPointBC, self).__init__(V, g, bc_point) + if isinstance(bc_point, tuple): + bc_point = as_vector(bc_point) + self.bc_point = bc_point + + @firedrake_utils.cached_property + def nodes(self): + V = self.function_space() + x = firedrake.SpatialCoordinate(V.mesh()) + xdist = x - self.bc_point + + test = firedrake.TestFunction(V) + trial = firedrake.TrialFunction(V) + xphi = firedrake.assemble(ufl.inner(xdist * test, xdist * trial) * ufl.dx, diagonal=True) + phi = firedrake.assemble(ufl.inner(test, trial) * ufl.dx, diagonal=True) + with xphi.dat.vec as xu, phi.dat.vec as u: + xu.pointwiseDivide(xu, u) + min_index, min_value = xu.min() # Find the index of the DOF closest to bc_point + + nodes = V.dof_dset.lgmap.applyInverse([min_index]) + nodes = nodes[nodes >= 0] + return nodes + + # The auxiliary constraints on the chemical potentials and pressure + aux_point = as_vector([4, 0]) # point on the middle of the outlet + aux_point_bcs = [FixAtPointBC(Z_h.sub(3), Constant(0.0), aux_point), + FixAtPointBC(Z_h.sub(4), Constant(0.0), aux_point), + FixAtPointBC(Z_h.sub(5), Constant(0.0), aux_point)] + +We provide a naive initial guess based on an equimolar constant distribution of benzene and cyclohexane:: + + mm_1, mm_2, v, mu_aux_1, mu_aux_2, p, x_1, x_2, rho_inv, l_1, l_2 = sln.subfunctions + x_1.interpolate(Constant(0.5)) + x_2.interpolate(Constant(0.5)) + rho_inv.interpolate(1.0 / ((M_1_ND * c_1) + (M_2_ND * c_2))) + +and define the nonlinear variational solver object:: + + NLVP = NonlinearVariationalProblem(tot_res, sln, bcs=flux_bcs+aux_point_bcs) + NLVS = NonlinearVariationalSolver(NLVP) + +Newton's method applied directly to the problem with :math:`v_1^\text{ref}=0.4\times 10^{-5}` +with the naive initial guess does not converge. Hence, we apply parameter continuation to :math:`v_1^\text{ref}` +to find a better initial guess. We start by solving the problem for :math:`v_1^\text{ref}=0.4\times 10^{-6}` +with the naive initial guess and use its solution as initial guess for the problem with +:math:`v_1^\text{ref}=0.1\times 10^{-5}`. We repeat this trick with :math:`v_1^\text{ref}=0.2\times 10^{-5}` +and :math:`v_1^\text{ref}=0.3\times 10^{-5}` before solving for :math:`v_1^\text{ref}=0.4\times 10^{-5}`. +We can reuse the nonlinear variational solver object each iteration, but have to reassign :code:`v_ref_1` +to the new value before calling the :code:`solve()` method. Finally, we write each solution to the same +VTK file using the :code:`time` keyword argument.:: + + from firedrake.output import VTKFile + outfile = VTKFile("out/sln.pvd") + cont_vals = [1.0, 2.5, 5, 7.5, 10.0] + n_cont = len(cont_vals) + + for i in range(n_cont): + print(f"Solving for v_ref_1 = {0.4e-6*cont_vals[i]}") + v_ref_1.assign(Constant(0.4e-6*cont_vals[i])) + NLVS.solve() + + p += assemble(-p * dx) / assemble(1 * dx(mesh)) # normalise p to have 0 mean -The mass fluxes must be modelled using a constitutive relation. -For example, in a Fickian model we might use -:math:`J_i = M_i c_i v - D_i \nabla c_i` -where the first term represents advection and the second term Fickian diffusion -(here :math:`D_i > 0` are Fickian diffusion coefficients). -The Fickian approach is appropriate for dilute mixtures (i.e. mixtures where all but -one of the species are present in trace amounts), but it typically is not -thermodynamically consistent in the non-dilute regime and fails to account for -cross-diffusional effects. We therefore model the mass fluxes using the -Onsager--Stefan--Maxwell equations, which capture cross-diffusion in non-dilute -mixtures in a thermodynamically consistent fashion. Sometimes in the literature -these equations are referred to as the generalized Maxwell--Stefan equations [CITE BIRD]. + mu_1_out = Function(U_h) + mu_2_out = Function(U_h) + rho_out = Function(R_h) + c_tot_out = Function(R_h) + c_1_out = Function(R_h) + c_2_out = Function(R_h) + + mu_1_out.interpolate(mu_1) + mu_2_out.interpolate(mu_2) + rho_out.interpolate(1.0 / rho_inv) + c_tot_out.interpolate(c_tot) + c_1_out.interpolate(c_1) + c_2_out.interpolate(c_2) + mm_1.rename("mm_1") + mm_2.rename("mm_2") + v.rename("v") + mu_1_out.rename("mu_1") + mu_2_out.rename("mu_2") + p.rename("p") + x_1.rename("x_1") + x_2.rename("x_2") + rho_inv.rename("rho_inv") + rho_out.rename("rho") + c_tot_out.rename("c_tot") + c_1_out.rename("c_1") + c_2_out.rename("c_2") + outfile.write(mm_1, mm_2, v, mu_1_out, mu_2_out, p, x_1, x_2, rho_inv, \ + rho_out, c_tot_out, c_1_out, c_2_out, time=i) +The mole fraction and stream lines of benzene for :math:`v_1^\text{ref}=0.4\times 10^{-6}` +and :math:`v_1^\text{ref}=0.4\times 10^{-5}` are displayed below on the left and right respectively. +Thanks to parameter continuation and higher-order discretisation methods, we can effectively solve +for low species concentrations and sharp solution gradients. +---------------------------+---------------------------+ | .. image:: benzene_0.png | .. image:: benzene_4.png | From 3ca07ead2b6f7a7133142ccf423c835898ceea89 Mon Sep 17 00:00:00 2001 From: KarsKnook Date: Tue, 20 May 2025 17:45:54 +0100 Subject: [PATCH 04/32] restructure and writing --- demos/multicomponent/multicomponent.py.rst | 623 +++++++++++++-------- 1 file changed, 385 insertions(+), 238 deletions(-) diff --git a/demos/multicomponent/multicomponent.py.rst b/demos/multicomponent/multicomponent.py.rst index 65e4b7c575..7c6531010b 100644 --- a/demos/multicomponent/multicomponent.py.rst +++ b/demos/multicomponent/multicomponent.py.rst @@ -1,10 +1,10 @@ -Multicomponent flow -- microfluidic mixing of hydrocarbons +Multicomponent flow -- microfluidic non-ideal mixing of hydrocarbons =================================================== .. rst-class:: emphasis We show how Firedrake can be used to simulate multicomponent flow; - specifically the microfluidic mixing of benzene and cyclohexane. + specifically the microfluidic non-ideal mixing of benzene and cyclohexane. The demo was contributed by `Aaron Baier-Reinio `__ and `Kars Knook @@ -12,25 +12,150 @@ Multicomponent flow -- microfluidic mixing of hydrocarbons We consider a steady, isothermal, nonreacting mixture of benzene and cyclohexane in a two-dimensional microfluidic container :math:`\Omega \subset \mathbb{R}^2`. -We model the mixture using the Stokes--Onsager--Stefan--Maxwell (SOSM) equations. +Pure benzene and cyclohexane flow in through opposing inlets on the left side of the container. +The chemicals then mix in the center of the container and exit through an outlet on the right. +We use :doc:`netgen ` to build a curved mesh of order :math:`k=3`:: + + from firedrake import * + import netgen.occ as ngocc + + # The polynomial order we will use for our curved mesh and finite element spaces + k = 3 + + # Specify the domain Omega (i.e. the microfluidic container's geometry) + wp = ngocc.WorkPlane() + wp.MoveTo(0, 1) + wp.Spline([ngocc.Pnt(1, 0), ngocc.Pnt(0, -1)]) + wp.LineTo(1, -1) + wp.Spline([ngocc.Pnt(3, -0.5), ngocc.Pnt(4, -0.5)], tangents={ 1 : ngocc.gp_Vec2d(1, 0) }) + wp.LineTo(4, 0.5) + wp.Spline([ngocc.Pnt(3, 0.5), ngocc.Pnt(1, 1)], tangents={ 0 : ngocc.gp_Vec2d(-1, 0) }) + wp.LineTo(0, 1) + + Omega = wp.Face() + + # Label the inlets and outlet; the other walls are unlabelled + Omega.edges.Max(ngocc.Y).name = "inlet_1" + Omega.edges.Min(ngocc.Y).name = "inlet_2" + Omega.edges.Max(ngocc.X).name = "outlet" + + # Construct the mesh + ngmesh = ngocc.OCCGeometry(Omega, dim=2).GenerateMesh(maxh=0.055) + mesh = Mesh(Mesh(ngmesh).curve_field(k)) + + # Get the IDs of the inlets, outlet and walls + inlet_1_id = [i+1 for i, name in enumerate(ngmesh.GetRegionNames(dim=1)) if name == "inlet_1"] + inlet_2_id = [i+1 for i, name in enumerate(ngmesh.GetRegionNames(dim=1)) if name == "inlet_2"] + outlet_id = [i+1 for i, name in enumerate(ngmesh.GetRegionNames(dim=1)) if name == "outlet"] + walls_ids = [i+1 for i, name in enumerate(ngmesh.GetRegionNames(dim=1)) if name == ""] + + # Define the surface and volume measures using a fixed quadrature degree + deg_max = 15 + ds = ds(mesh, degree=deg_max) + dx = dx(mesh, degree=deg_max) + + # Define the spatial coordinates on the mesh + x_sc, y_sc = SpatialCoordinate(mesh) + +The domain and mesh are visualised below. + +.. image:: microfluidic_container.png + :align: center + :width: 60% + +To model the mixture we employ the Stokes--Onsager--Stefan--Maxwell (SOSM) +partial differential equations and discretise with the method of :cite:`BaierReinio:2025`. +In what follows species 1 refers to benzene and species 2 to cyclohexane. +We shall discretise the following unknowns: + + * :math:`J_1, J_2 \in \textrm{BDM}_k` - species mass fluxes, + * :math:`v \in [\textrm{CG}_k]^2` - barycentric velocity, + * :math:`\mu_1, \mu_2 \in \textrm{DG}_{k-1}` - species chemical potentials, + * :math:`p \in \textrm{CG}_{k-1}` - pressure, + * :math:`x_1, x_2 \in \textrm{DG}_{k-1}` - species mole fractions, + * :math:`\rho^{-1} \in \textrm{CG}_{k-1}` - density reciprocal (i.e. specific volume). + +The equations governing these unknowns are presented below. +We first define the finite element spaces and trial/test functions:: + + # The finite element spaces + J_h = FunctionSpace(mesh, "BDM", k) # Species mass-flux space + V_h = VectorFunctionSpace(mesh, "CG", k) # Velocity space + U_h = FunctionSpace(mesh, "DG", k - 1) # Species chemical potential space + P_h = FunctionSpace(mesh, "CG", k - 1) # Pressure space + X_h = FunctionSpace(mesh, "DG", k - 1) # Species mole fraction space + R_h = FunctionSpace(mesh, "CG", k - 1) # Density reciprocal space + L_h = FunctionSpace(mesh, "R", 0) # Lagrange multiplier space + + # The mixed space + Z_h = J_h * J_h * V_h * U_h * U_h * P_h * X_h * X_h * R_h * L_h * L_h + PETSc.Sys.Print("Mesh has %d cells, with %d finite element DOFs" % (mesh.num_cells(), Z_h.dim())) + + # The trial functions + sln = Function(Z_h) + J_1, J_2, v, mu_aux_1, mu_aux_2, p, x_1, x_2, rho_inv, l_1, l_2 = split(sln) + + # Lagrange multiplier trick for enforcing integral constraints + mu_1 = mu_aux_1 + l_1 + mu_2 = mu_aux_2 + l_2 + + # The test functions + W_1, W_2, u, w_1, w_2, q, y_1, y_2, r, s_1, s_2 = TestFunctions(Z_h) + +Note that we decompose the chemical potentials as :math:`\mu_i = \mu_{i, \textrm{aux}} + l_i` +where :math:`l_i` are Lagrange multipliers. +This will aid in enforcing integral constraints on the solution; see +:doc:`the steady Boussinesq demo ` for an explanation of the process. + +Governing PDEs: SOSM Equations +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + Momentum transport is modelled using the steady compressible Stokes momentum equation for a Newtonian fluid, .. math:: + :name: eq:stokes -\nabla \cdot \big\{ 2 \eta \epsilon(v) + (\zeta - 2\eta / d) (\nabla \cdot v) \mathbb{I} \big\} + \nabla p = 0 \quad \textrm{in}\ \Omega. -Here, the unknowns are the :math:`\mathbb{R}^2`-valued velocity :math:`v`, -scalar pressure :math:`p` and density :math:`\rho`. -Moreover :math:`\epsilon (v)` denotes the symmetric gradient of :math:`v`, -:math:`\eta, \zeta > 0` are the shear and bulk viscosities respectively, -:math:`d=2` the spatial dimension and +Recall that :math:`v` is the barycentric velocity and :math:`p` the pressure. +Moreover :math:`\epsilon (v)` denotes the symmetric gradient of :math:`v` +while :math:`\eta, \zeta > 0` are the shear and bulk viscosities respectively, +:math:`d=2` is the spatial dimension and :math:`\mathbb{I}` the :math:`d \times d` identity matrix. -Let :math:`n` denote the number of chemical species in the mixture -(in this example :math:`n=2`, i.e. benzene and cyclohexane). +We shall non-dimensionalise all of the unknowns in our discretisation. +Hence we introduce a reference velocity :math:`v^{\textrm{ref}}` whose value +will be specified later on when we introduce boundary conditions. +We also choose a reference length of :math:`L^{\textrm{ref}} = 2 \cdot 10^{-3}` m. +It is then natural to define the reference pressure as +:math:`p^{\textrm{ref}} = \eta \cdot v^{\textrm{ref}} / L^{\textrm{ref}}`. :: + + # The (dimensionful) Stokes viscosities + eta = Constant(6e-4) # Shear viscosity, Pa s + zeta = Constant(1e-7) # Bulk viscosity, Pa s + + # Non-dimensionalised Lame parameter, dimensionless + lame_ND = (zeta / eta) - 1.0 + + # Reference quantities used for non-dimensionalisation + v_ref = Constant(0.0) # Reference velocity (to be specified later), m / s + L_ref = Constant(2e-3) # Reference length, m + p_ref = eta * v_ref / L_ref # Reference pressure, Pa + +The non-dimensionalised variational formulation of :ref:`eq. 1 ` is then:: + + # The Stokes viscous terms + stokes_terms = 2.0 * inner(sym(grad(v)), sym(grad(u))) * dx + stokes_terms += lame_ND * inner(div(v), div(u)) * dx + + # The Stokes pressure term + stokes_terms -= inner(p, div(u)) * dx + +Let :math:`n \geq 2` denote the number of chemical species. +Hence :math:`n=2` throughout this demo (benzene and cyclohexane). The continuity equation for the molar concentration :math:`c_i` of species :math:`i \in \{1:n\}` in the absence of chemical reactions is @@ -44,185 +169,119 @@ where :math:`M_i > 0` is the molar mass of species :math:`i` and As we are considering steady flow, the continuity equations simplify to .. math:: + :name: eq:cty - \frac{1}{M_i} \nabla \cdot J_i = 0 - \quad \textrm{in}\ \Omega \quad \forall i \in \{1 : n \}. - -The mass fluxes must be modelled with a constitutive law. -A basic Fickian model may use :math:`J_i = M_i (c_i v - D_i \nabla c_i)` -where :math:`c_i v` represents advection and :math:`-D_i \nabla c_i` Fickian diffusion. -The Fickian approach is appropriate for dilute mixtures -(i.e. mixtures where all of the species but one are present in trace amounts), -but typically is not thermodynamically consistent in the non-dilute regime, -and fails to account for cross-diffusional effects. -We remedy these drawbacks by employing the Onsager--Stefan--Maxwell (OSM) equations -(also known as Maxwell--Stefan equations :cite:`Krishna:1997`), -which, in the present isothermal setting, implicitly determine the mass fluxes through the relations - -.. math:: - - -\frac{1}{M_i} \nabla \mu_i + \frac{1}{\rho} \nabla p &= - \sum_{\substack{j=1 \\ j \neq i}}^n \frac{RT c_j}{\mathscr{D}_{ij} M_i c_T} - \Bigg( \frac{J_i}{M_i c_i} - \frac{J_j}{M_j c_j} \Bigg) - \quad \textrm{in}\ \Omega - \quad \forall i \in \{1 : n \}. - -Here :math:`\mu_i` is the chemical potential of species `i`, -:math:`R` the ideal gas constant, -:math:`T` the temperature, -:math:`\mathscr{D}_{ij}` Stefan--Maxwell diffusion coefficients -and :math:`c_T = \sum_{j=1}^n c_j` the total concentration. -Onsager reciprocal relations imply that -:math:`\mathscr{D}_{ij} = \mathscr{D}_{ji} \ \forall i \neq j` while -:math:`\mathscr{D}_{jj}` is undefined. -Since :math:`n=2` there is only one Stefan--Maxwell diffusion coefficient -:math:`\mathscr{D}_{12} = \mathscr{D}_{21}` -and in the code below we denote it by :code:`D_sm`. - -Only :math:`n-1` of the OSM equations are linearly independent; the :math:`J_i`'s -are fully determined by a mass-average constraint -:math:`v = \frac{1}{\rho} \sum_{j=1}^n J_j`. -We will incorporate this numerically by introducing an augmentation parameter -:math:`\gamma > 0` and reformulating the OSM equations as - -.. math:: - - -\frac{1}{M_i} \nabla \mu_i + \frac{1}{\rho} \nabla p - + \frac{\gamma}{\rho} v &= - \sum_{\substack{j=1 \\ j \neq i}}^n - \frac{\gamma}{\rho^2} J_j + - \frac{RT c_j}{\mathscr{D}_{ij} M_i c_T} - \Bigg( \frac{J_i}{M_i c_i} - \frac{J_j}{M_j c_j} \Bigg) - \quad \textrm{in}\ \Omega - \quad \forall i \in \{1 : n \}. + \nabla \cdot J_i = 0 + \quad \textrm{in}\ \Omega \quad \forall i \in \{1 : n \}, -In our numerics we will also (weakly) enforce that -:math:`\nabla \cdot v = \nabla \cdot (\frac{1}{\rho} \sum_{j=1}^n J_j )` -as this turns out to yield a well-posed discrete scheme. +which are discretised as follows:: -This completes the description of transport phenomena in the mixture. -However, to obtain a closed set of equations, we must also model how the free energy and -volume of the mixture depend on temperature, pressure and composition. -To describe this we must introduce mole fractions :math:`x_i := c_i / c_T`; -note that by definition :math:`\sum_{j=1}^n x_j = 1`. -Thermodynamics requires that the :math:`\mu_i`'s' and :math:`c_T` satisfy + continuity_terms = (inner(w_1, div(J_1)) + inner(w_2, div(J_2))) * dx -.. math:: - - \mu_i &= g_i(T, p, x_1, \ldots, x_n) - \quad \textrm{in}\ \Omega - \quad \forall i \in \{1 : n \}, \\ - \frac{1}{c_T} &= \sum_{j=1}^n x_j V_j(T, p, x_1, \ldots, x_n) \quad \textrm{in}\ \Omega, - -where :math:`g_i: \mathbb{R}^{n+2} \rightarrow \mathbb{R}` are partial molar Gibbs functions -and :math:`V_i: \mathbb{R}^{n+2} \rightarrow \mathbb{R}` partial molar volume functions. -These functions are derived from partial derivatives of the Gibbs free energy of the mixture. -In this demo we employ a Margules model :cite:`Perry:2007` for the :math:`g_i`'s. -We also assume that the partial molar volumes are constant -(for liquids this is usually a reasonable assumption), from which it follows that +Next, we incorporate the volumetric equation of state, which models +how the concentration of the mixture depends on temperature, pressure and composition. +Composition of the mixture is described using mole fractions :math:`x_i := c_i / c_T` where +:math:`c_T = \sum_{j=1}^n c_j` is the total concentration. +Note that by definition :math:`\sum_{j=1}^n x_j = 1`, although at the discrete +level this relation will only hold approximately. +We assume that the mixture is quasi-incompressible in the sense that +its partial molar volumes are constant; this is often a reasonable assumption for liquids. +It follows that the volumetric equation of state is .. math:: + :name: eq:eos \frac{1}{c_T} = \sum_{j=1}^n \frac{x_j}{c_j^{\textrm{pure}}} \quad \textrm{in}\ \Omega, -where :math:`c_j^{\textrm{pure}}` is the (known) concentration of pure species `j`. -Note that the above equation is an example of a volumetric equation of state. - - - -Defining parameters for the numerics and nondimensionalised physical quantities:: - - from firedrake import * - - k = 3 # The polynomial degree (for the velocity spaces) - deg_max = 15 # Maximum quadrature degree - gamma = Constant(1e-1) # Augmentation parameter, dimensionless - v_ref_1 = Constant(0.4e-6) # Initialisng reference inlet velocity of benzene (m/s) - - RT = Constant(8.314 * 298.15) # Ideal gas constant times temperature, J / mol - eta = Constant(6e-4) # Shear viscosity, Pa s - zeta = Constant(1e-7) # Bulk viscosity, Pa s - D_sm = Constant(2.1e-9) # Stefan--Maxwell diffusivity, m^2 / s - L_ref = Constant(2e-3) # Reference length, m - - # Constants for the pure species - M_1 = Constant(0.078) # Molar mass of Benzene, kg / mol - M_2 = Constant(0.084) # Molar mass of Cyclohexane, kg / mol - rho_pure_1 = Constant(876) # Density of pure Benzene, kg / m^3 - rho_pure_2 = Constant(773) # Density of pure Cyclohexane, kg / m^3 - c_pure_1 = rho_pure_1 / M_1 # Concentration of pure Benzene, mol / m^3 - c_pure_2 = rho_pure_2 / M_2 # Concentration of pure Cyclohexane, mol / m^3 +where :math:`c_j^{\textrm{pure}}` is the concentration of pure species `j`. We use values +for :math:`c_j^{\textrm{pure}}` that are measured at room temperature :math:`T=298.15` K +and ambient pressure :math:`10^5` Pa (note that we allow the pressure :math:`p` to vary +in this model but we assume that these variations do not alter :math:`c_j^{\textrm{pure}}`). +We will also make use of the total density of the mixture which is given by :math:`\rho = \sum_{j=1}^n M_j c_j`. +To non-dimensionalise the concentrations and total density, we employ as reference values for these quantities +their values when the mixture is equimolar:: + + # Constants for the pure species (at the ambient temperature and pressure) + M_1 = Constant(0.078) # Molar mass of benzene, kg / mol + M_2 = Constant(0.084) # Molar mass of cyclohexane, kg / mol + rho_pure_1 = Constant(876) # Density of pure benzene, kg / m^3 + rho_pure_2 = Constant(773) # Density of pure cyclohexane, kg / m^3 + c_pure_1 = rho_pure_1 / M_1 # Concentration of pure benzene, mol / m^3 + c_pure_2 = rho_pure_2 / M_2 # Concentration of pure cyclohexane, mol / m^3 # Constants for the equimolar mixture - c_equi_tot = (c_pure_1 * c_pure_2) / ((0.5 * c_pure_2) + (0.5 * c_pure_1)) # Total equimolar concentration, mol / m^3 - c_equi_1 = 0.5 * c_equi_tot # Equimolar concentration of Benzene, mol / m^3 - c_equi_2 = 0.5 * c_equi_tot # Equimolar concentration of Cyclohexane, mol / m^3 + c_equi_tot = 1.0 / ((0.5 / c_pure_2) + (0.5 / c_pure_1)) # Total equimolar concentration, mol / m^3 + c_equi_1 = 0.5 * c_equi_tot # Equimolar concentration of benzene, mol / m^3 + c_equi_2 = 0.5 * c_equi_tot # Equimolar concentration of cyclohexane, mol / m^3 rho_equi = (M_1 * c_equi_1) + (M_2 * c_equi_2) # Equimolar density, kg / m^3 - # Reference density and related constants + # Reference concentration, density and molar mass rho_ref = rho_equi # Reference density, kg / m^3 c_ref = c_equi_tot # Reference concentration, mol / m^3 M_ref = rho_ref / c_ref # Reference molar mass, kg / mol - # Reference convective velocities - v_ref_2 = (c_pure_1 / c_pure_2) * v_ref_1 # Reference inflow velocity of Cyclohexane, m / s - v_ref = 0.5 * (v_ref_1 + v_ref_2) # Reference velocity, m / s + # Non-dimensionalised molar masses + M_1_ND = M_1 / M_ref + M_2_ND = M_2 / M_ref - # Derived quantities - v_sm_ref = D_sm / L_ref # Reference diffusion velocity, m / s - p_ref = eta * v_ref / L_ref # Reference pressure, Pa - lame_ND = (zeta / eta) - 1.0 # Non-dimensionalised Lame parameter, dimensionless - Pe = v_ref / v_sm_ref # Peclet number, dimensionless - Me = p_ref / (RT * c_ref) # Pressure diffusion number, dimensionless - Me_1 = p_ref / (RT * c_pure_1) # Non-dimensionalised partial molar volume of Benzene, dimensionless - Me_2 = p_ref / (RT * c_pure_2) # Non-dimensionalised partial molar volume of Cyclohexane, dimensionless - M_1_ND = M_1 / M_ref # Non-dimensionalised molar mass of Benzene, dimensionless - M_2_ND = M_2 / M_ref # Non-dimensionalised molar mass of Cyclohexane, dimensionless +Our implementation of the non-dimensionalised volumetric equation of state +:ref:`eq. 3 ` is therefore:: - # Margules model parameters - A_12 = Constant(0.4498) # Dimensionless - A_21 = Constant(0.4952) # Dimensionless + # Volumetric equation of state, assuming constant partial molar volumes + def conc_relation(x_1, x_2): -Pure benzene and cyclohexane are piped in through opposing inlets on the left-hand side of the microfluidic container. -These chemicals form a nonideal mixture before flowing out through the outlet on the right-hand side. -The mesh of the two-dimensional microfluidic container is created using :doc:`netgen `. -We use fourth order curved elements to adequately capture the geometry of the container. :: + # Normalise the mole fractions before computing c_tot, + # since they will only sum to one up to discretisation error + x_1_nm = x_1 / (x_1 + x_2) + x_2_nm = x_2 / (x_1 + x_2) - import netgen.occ as ngocc + # Compute c_tot and the species concentrations + c_tot = 1.0 / ((x_1_nm * (c_ref / c_pure_1)) + (x_2_nm * (c_ref / c_pure_2))) + c_1 = x_1_nm * c_tot + c_2 = x_2_nm * c_tot - wp = ngocc.WorkPlane() - wp.MoveTo(0, 1) - wp.Spline([ngocc.Pnt(1, 0), ngocc.Pnt(0, -1)]) - wp.LineTo(1, -1) - wp.Spline([ngocc.Pnt(3, -0.5), ngocc.Pnt(4, -0.5)], tangents={ 1 : ngocc.gp_Vec2d(1, 0) }) - wp.LineTo(4, 0.5) - wp.Spline([ngocc.Pnt(3, 0.5), ngocc.Pnt(1, 1)], tangents={ 0 : ngocc.gp_Vec2d(-1, 0) }) - wp.LineTo(0, 1) + return (c_tot, c_1, c_2) - shape = wp.Face() - shape.edges.Max(ngocc.Y).name = "inlet_1" - shape.edges.Min(ngocc.Y).name = "inlet_2" - shape.edges.Max(ngocc.X).name = "outlet" + c_tot, c_1, c_2 = conc_relation(x_1, x_2) - ngmesh = ngocc.OCCGeometry(shape, dim=2).GenerateMesh(maxh=0.055) - mesh = Mesh(Mesh(ngmesh).curve_field(4)) # fourth order curved elements +Moreover, to express that :math:`\rho^{-1} = \sum_{j=1}^n M_j c_j`, we simply take the +:math:`L^2`-projection of this expression (in non-dimensionalised form):: - inlet_1_ids = [i+1 for i, name in enumerate(ngmesh.GetRegionNames(dim=1)) if name == "inlet_1"] - inlet_2_ids = [i+1 for i, name in enumerate(ngmesh.GetRegionNames(dim=1)) if name == "inlet_2"] - outlet_ids = [i+1 for i, name in enumerate(ngmesh.GetRegionNames(dim=1)) if name == "outlet"] - walls_ids = [i+1 for i, name in enumerate(ngmesh.GetRegionNames(dim=1)) if name == ""] + rho_inv_terms = inner(1.0 / rho_inv, r) * dx + rho_inv_terms -= inner((M_1_ND * c_1) + (M_2_ND * c_2), r) * dx - ds = ds(mesh, degree=deg_max) - dx = dx(mesh, degree=deg_max) - x_sc, y_sc = SpatialCoordinate(mesh) +Next, we must model how the free energy of the mixture depends on +temperature, pressure and composition. +This is accomplished by introducing the species chemical potentials +:math:`\mu_i, \ i \in \{1 : n\}`, which are unknown scalar fields that describe +the local chemical potential energy of the mixture. +Thermodynamics requires that these satisfy -The mesh is displayed below. +.. math:: + :name: eq:gibbs -.. image:: microfluidic_container.png - :align: center - :width: 60% + \mu_i = g_i(T, p, x_1, \ldots, x_n) + \quad \textrm{in}\ \Omega + \quad \forall i \in \{1 : n \}, + +where :math:`g_i: \mathbb{R}^{n+2} \rightarrow \mathbb{R}` are partial molar Gibbs functions. +These functions are derived from partial derivatives of the Gibbs free energy of the mixture. +It is natural to non-dimensionalise the chemical potentials using a reference value of +:math:`\mu^{\textrm{ref}} = RT` where :math:`R` the is ideal gas constant +and :math:`T` the ambient temperature. +In this demo we employ a Margules model :cite:`Perry:2007` +for the :math:`g_i`'s, which in non-dimensionalised form, is implemented as follows:: + + RT = Constant(8.314 * 298.15) # Ideal gas constant times temperature, J / mol -Thermodynamic constitutive relations:: + mu_ref = RT # Reference chemical potential, J / mol + + Me_1 = p_ref / (RT * c_pure_1) # Non-dimensionalised partial molar volume of benzene, dimensionless + Me_2 = p_ref / (RT * c_pure_2) # Non-dimensionalised partial molar volume of cyclohexane, dimensionless + + # Margules model parameters + A_12 = Constant(0.4498) # Dimensionless + A_21 = Constant(0.4952) # Dimensionless # Margules model for chemical potentials, assuming constant partial molar volumes def mu_relation(x_1, x_2, p): @@ -233,98 +292,151 @@ Thermodynamic constitutive relations:: return (mu_1, mu_2) - # Volumetric equation of state, assuming constant partial molar volumes - def conc_relation(x_1, x_2): - x_1_nm = x_1 / (x_1 + x_2) - x_2_nm = x_2 / (x_1 + x_2) +We discretise :ref:`eq. 4 ` through a simple :math:`L^2`-projection:: - c_tot = 1.0 / ((x_1_nm * (c_ref / c_pure_1)) + (x_2_nm * (c_ref / c_pure_2))) - c_1 = x_1_nm * c_tot - c_2 = x_2_nm * c_tot + g_1, g_2 = mu_relation(x_1, x_2, p) + gibbs_terms = (inner(mu_1 - g_1, y_1) + inner(mu_2 - g_2, y_2)) * dx - return (c_tot, c_1, c_2) +It remains to model the mass fluxes (recall the continuity equations in :ref:`eq. 2 `); +this must be done with a constitutive relation. +A basic Fickian constitutive relation may use :math:`J_i = M_i (c_i v - D_i \nabla c_i)` +where :math:`c_i v` represents advection and :math:`-D_i \nabla c_i` Fickian diffusion. +The Fickian approach is appropriate for dilute mixtures +(i.e. mixtures where all of the species but one are present in trace amounts), +but typically is not thermodynamically consistent in the non-dilute regime, +and fails to account for cross-diffusion and thermodynamic non-idealities. +These drawbacks are remedied by employing the Onsager--Stefan--Maxwell (OSM) equations +(also called the Maxwell--Stefan equations :cite:`Krishna:1997`), +which in the present isothermal setting implicitly determine the mass fluxes through the relations -Defining the test and trial functions:: +.. math:: - W_h = FunctionSpace(mesh, "BDM", k) # Momentum space - V_h = VectorFunctionSpace(mesh, "CG", k) # Bulk velocity space - U_h = FunctionSpace(mesh, "DG", k - 1) # Chemical potential space - P_h = FunctionSpace(mesh, "CG", k - 1) # Pressure space - X_h = FunctionSpace(mesh, "DG", k - 1) # Molar fraction space - R_h = FunctionSpace(mesh, "CG", k - 1) # Density reciprocal space - L_h = FunctionSpace(mesh, "R", 0) # Real space + -\frac{1}{M_i} \nabla \mu_i + \frac{1}{\rho} \nabla p &= + \sum_{\substack{j=1 \\ j \neq i}}^n \frac{RT c_j}{\mathscr{D}_{ij} M_i c_T} + \Bigg( \frac{J_i}{M_i c_i} - \frac{J_j}{M_j c_j} \Bigg) + \quad \textrm{in}\ \Omega + \quad \forall i \in \{1 : n \}. - Z_h = W_h * W_h * V_h * U_h * U_h * P_h * X_h * X_h * R_h * L_h * L_h - PETSc.Sys.Print("Mesh has %d cells, with %d finite element DOFs" % (mesh.num_cells(), Z_h.dim())) +Here :math:`\mathscr{D}_{ij} \ \forall i \neq j` are Stefan--Maxwell diffusion +coefficients (note that :math:`\mathscr{D}_{jj}` is undefined). +Onsager reciprocal relations imply that :math:`\mathscr{D}` is symmetric, i.e. +:math:`\mathscr{D}_{ij} = \mathscr{D}_{ji} \ \forall i \neq j`. +Since :math:`n=2` in this demo, we only have one Stefan--Maxwell diffusion coefficient +:math:`\mathscr{D}_{\textrm{sm}} := \mathscr{D}_{12} = \mathscr{D}_{21}`. - sln = Function(Z_h) - mm_1, mm_2, v, mu_aux_1, mu_aux_2, p, x_1, x_2, rho_inv, l_1, l_2 = split(sln) - mu_1 = mu_aux_1 + l_1 - mu_2 = mu_aux_2 + l_2 - u_1, u_2, u, w_1, w_2, q, y_1, y_2, r, s_1, s_2 = TestFunctions(Z_h) +Only :math:`n-1` of the OSM equations are linearly independent. +To uniquely determine the :math:`J_i`'s +one must utilise a mass-average constraint: -Defining the variational formulation:: +.. math:: + v = \frac{1}{\rho} \sum_{j=1}^n J_j. + +This constraint ensures that the continuity equations in :ref:`eq. 2 ` +are consistent with total mass conservation +:math:`\partial_t \rho + \nabla \cdot (\rho v) = 0` +(note that we do not explicitly discretise this equation). +We incorporate the mass-average constraint numerically by +introducing an augmentation parameter +:math:`\gamma > 0` and reformulating the OSM equations as - c_tot, c_1, c_2 = conc_relation(x_1, x_2) +.. math:: + :name: eq:osm - # The Stokes viscous terms - A_visc = 2.0 * inner(sym(grad(v)), sym(grad(u))) * dx - A_visc += lame_ND * inner(div(v), div(u)) * dx + -\frac{1}{M_i} \nabla \mu_i + \frac{1}{\rho} \nabla p + + \frac{\gamma}{\rho} v &= + \sum_{\substack{j=1 \\ j \neq i}}^n + \frac{\gamma}{\rho^2} J_j + + \frac{RT c_j}{\mathscr{D}_{ij} M_i c_T} + \Bigg( \frac{J_i}{M_i c_i} - \frac{J_j}{M_j c_j} \Bigg) + \quad \textrm{in}\ \Omega + \quad \forall i \in \{1 : n \}. - # The augmented Onsager transport matrix terms - A_osm = Pe * (1.0 / c_tot) * ((c_2 / (M_1_ND * M_1_ND * c_1)) * inner(mm_1, u_1) \ - + (c_1 / (M_2_ND * M_2_ND * c_2)) * inner(mm_2, u_2) \ - - (1.0 / (M_1_ND * M_2_ND)) * (inner(mm_1, u_2) + inner(mm_2, u_1))) * dx - A_osm += Pe * gamma * inner(v - (rho_inv * (mm_1 + mm_2)), u - (rho_inv * (u_1 + u_2))) * dx +One can non-dimensionalise :ref:`eq 5. ` by introducing a dimensionless Péclet number +:math:`\mathrm{Pe} = v^{\textrm{ref}} L^{\textrm{ref}} / \mathscr{D}_{\textrm{sm}}` +and pressure diffusion numbers :math:`\mathrm{Me} = p^{\textrm{ref}} / RT c^{\textrm{ref}}`. +Moreover, :ref:`eq 5. ` can be cast into a variational form by +testing against functions :math:`K_i` and integrating by parts the two gradient terms +on the left-hand side (the boundary terms drop out owing to our BCs below). +This leads to the following implementation:: - # The diffusion driving force terms and Stokes pressure term - B_blf = ((Me * inner(p, div(rho_inv * (u_1 + u_2)))) - inner(p, div(u))) * dx - B_blf -= ((1.0 / M_1_ND) * inner(mu_1, div(u_1)) + (1.0 / M_2_ND) * inner(mu_2, div(u_2))) * dx + D_sm = Constant(2.1e-9) # Stefan--Maxwell diffusivity, m^2 / s + Pe = v_ref * L_ref / D_sm # Peclet number, dimensionless + Me = p_ref / (RT * c_ref) # Pressure diffusion number, dimensionless - # The div(mass-average constraint) and continuity equation terms - BT_blf = (inner(q, div(rho_inv * (mm_1 + mm_2))) - inner(q, div(v))) * dx - BT_blf -= ((1.0 / M_1_ND) * inner(w_1, div(mm_1)) + (1.0 / M_2_ND) * inner(w_2, div(mm_2))) * dx + gamma = Constant(1e-1) # Augmentation parameter, dimensionless - # The total residual - tot_res = A_visc + A_osm + B_blf + BT_blf + # The Stefan--Maxwell diffusion terms + osm_terms = Pe * (1.0 / c_tot) * ((c_2 / (M_1_ND * M_1_ND * c_1)) * inner(J_1, W_1) \ + + (c_1 / (M_2_ND * M_2_ND * c_2)) * inner(J_2, W_2) \ + - (1.0 / (M_1_ND * M_2_ND)) * (inner(J_1, W_2) + inner(J_2, W_1))) * dx - # The thermodynamic constitutive relation and density reciprocal terms - mu_1_cr, mu_2_cr = mu_relation(x_1, x_2, p) - tot_res += (inner(mu_1 - mu_1_cr, y_1) + inner(mu_2 - mu_2_cr, y_2)) * dx + # The augmentation terms (for symmetry we also test these terms against u) + osm_terms += Pe * gamma * inner(v - (rho_inv * (J_1 + J_2)), u - (rho_inv * (W_1 + W_2))) * dx - tot_res += inner(1.0 / rho_inv, r) * dx - tot_res -= inner((M_1_ND * c_1) + (M_2_ND * c_2), r) * dx + # The pressure diffusion terms + osm_terms += ((Me * inner(p, div(rho_inv * (W_1 + W_2))))) * dx - # The density consistency terms - nml = FacetNormal(mesh) - tot_res -= q * inner((rho_inv * (mm_1 + mm_2)) - v, nml) * ds + # The chemical potential terms + osm_terms -= ((1.0 / M_1_ND) * inner(mu_1, div(W_1)) + (1.0 / M_2_ND) * inner(mu_2, div(W_2))) * dx - # integral constraints - tot_res += inner(x_1 + x_2 - 1, s_1) * dx - tot_res += inner((M_1_ND * c_1) - (M_2_ND * c_2), s_2) * ds(outlet_id) +Lastly, we weakly enforce that +:math:`\nabla \cdot v = \nabla \cdot (\frac{1}{\rho} \sum_{j=1}^n J_j )`, +using special density consistency terms to handle inhomogeneous BCs:: + + div_mass_avg_terms = inner(q, div(v - (rho_inv * (J_1 + J_2)))) * dx -We enforce parabolic profiles on :math:`J_i \cdot n` at inflow :math:`i` and on the outflow. -The magnitude of the parabolic profiles are :math:`M_ic_i^\text{ref}v_i^\text{ref}` respectively. -Elsewhere on the boundary :math:`J_i \cdot n = 0` for each :math:`i`. Instead of specifying -the value of the bulk velocity on the inflows and outflow, we enforce :math:`\rho v \cdot n = (J_1 + J_2 )\cdot n` -and :math:`\rho v \times n = 0` in these regions. This means that :math:`v` equals an unknown quantity + # The density consistency terms + N = FacetNormal(mesh) + div_mass_avg_terms -= q * inner(v - (rho_inv * (J_1 + J_2)), N) * ds + +This concludes our discussion of the PDE model and its discretisation. +Altogether, our total residual is the sum of forms built above:: + + tot_res = stokes_terms \ + + continuity_terms \ + + rho_inv_terms \ + + gibbs_terms \ + + osm_terms \ + + div_mass_avg_terms + +Boundary conditions +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Let :math:`N` denote the outward unit normal on :math:`\partial \Omega`. +We enforce parabolic profiles on :math:`J_i \cdot N` at inflow :math:`i` and on the outflow. +The magnitudes of the parabolic profiles are :math:`M_i c_i^\text{ref} v_i^\text{ref}` where +:math:`v_i^\text{ref}` are reference velocities that we are free to choose. +Elsewhere on the boundary we enforce :math:`J_i \cdot N = 0`. Finally, instead of specifying +the value of the barycentric velocity :math:`v` on the inflows and outflow, we enforce :math:`\rho v \cdot N = (J_1 + J_2 )\cdot N` +and :math:`\rho v \times N = 0` in these regions. This means that :math:`v` equals an unknown quantity, so we need to use :code:`EquationBC` instead of :code:`DirichletBC`. :: - J_1_inflow_bc_func = -M_1_ND * 2.0 * x_sc * (x_sc - 1.0) * (v_ref_1 / v_ref) * (c_pure_1 / c_ref) * as_vector([2.0, -1.0]) - J_2_inflow_bc_func = -M_2_ND * 2.0 * x_sc * (x_sc - 1.0) * (v_ref_2 / v_ref) * (c_pure_2 / c_ref) * as_vector([2.0, 1.0]) + # Reference species velocities, which we choose to symmetrize so that the molar fluxes agree + v_ref_1 = Constant(0.4e-6) # Reference inflow velocity of benzene, m / s + v_ref_2 = (c_pure_1 / c_pure_2) * v_ref_1 # Reference inflow velocity of cyclohexane, m / s + + parabola_inflow_1 = 2.0 * x_sc * (x_sc - 1.0) * as_vector([2.0, -1.0]) + parabola_inflow_2 = 2.0 * x_sc * (x_sc - 1.0) * as_vector([2.0, 1.0]) + J_1_inflow_bc_func = -M_1_ND * (v_ref_1 / v_ref) * (c_pure_1 / c_ref) * parabola_inflow_1 + J_2_inflow_bc_func = -M_2_ND * (v_ref_2 / v_ref) * (c_pure_2 / c_ref) * parabola_inflow_2 rho_v_inflow_1_bc_func = J_1_inflow_bc_func rho_v_inflow_2_bc_func = J_2_inflow_bc_func - J_1_outflow_bc_func = -M_1_ND * 2.0 * (y_sc + 0.5) * (y_sc - 0.5) * (v_ref_1 / v_ref) * (c_pure_1 / c_ref) * as_vector([1.0, 0.0]) - J_2_outflow_bc_func = -M_2_ND * 2.0 * (y_sc + 0.5) * (y_sc - 0.5) * (v_ref_2 / v_ref) * (c_pure_2 / c_ref) * as_vector([1.0, 0.0]) + parabola_outflow = 2.0 * (y_sc + 0.5) * (y_sc - 0.5) * as_vector([1.0, 0.0]) + J_1_outflow_bc_func = -M_1_ND * (v_ref_1 / v_ref) * (c_pure_1 / c_ref) * parabola_outflow + J_2_outflow_bc_func = -M_2_ND * (v_ref_2 / v_ref) * (c_pure_2 / c_ref) * parabola_outflow rho_v_outflow_bc_func = J_1_outflow_bc_func + J_2_outflow_bc_func # Boundary conditions on the bulk velocity are enforced via EquationBC - v_inflow_1_bc = EquationBC(inner(v - rho_inv * rho_v_inflow_1_bc_func, u) * ds(inlet_1_id, degree=deg_max) == 0, sln, inlet_1_id, V=Z_h.sub(2)) - v_inflow_2_bc = EquationBC(inner(v - rho_inv * rho_v_inflow_2_bc_func, u) * ds(inlet_2_id, degree=deg_max) == 0, sln, inlet_2_id, V=Z_h.sub(2)) - v_outflow_bc = EquationBC(inner(v - rho_inv * rho_v_outflow_bc_func, u) * ds(outlet_id, degree=deg_max) == 0, sln, outlet_id, V=Z_h.sub(2)) - - # The boundary conditions on the fluxes + v_inflow_1_bc = EquationBC(inner(v - rho_inv * rho_v_inflow_1_bc_func, u) * ds(*inlet_1_id) == 0, + sln, inlet_1_id, V=Z_h.sub(2)) + v_inflow_2_bc = EquationBC(inner(v - rho_inv * rho_v_inflow_2_bc_func, u) * ds(*inlet_2_id) == 0, + sln, inlet_2_id, V=Z_h.sub(2)) + v_outflow_bc = EquationBC(inner(v - rho_inv * rho_v_outflow_bc_func, u) * ds(*outlet_id) == 0, sln, + outlet_id, V=Z_h.sub(2)) + + # The boundary conditions on the fluxes and barycentric velocity + # Note that BCs on H(div) spaces only apply to the normal component flux_bcs = [DirichletBC(Z_h.sub(0), J_1_inflow_bc_func, inlet_1_id), DirichletBC(Z_h.sub(0), J_1_outflow_bc_func, outlet_id), DirichletBC(Z_h.sub(0), as_vector([0.0, 0.0]), inlet_2_id), @@ -338,8 +450,40 @@ so we need to use :code:`EquationBC` instead of :code:`DirichletBC`. :: v_outflow_bc, DirichletBC(Z_h.sub(2), as_vector([0.0, 0.0]), walls_ids)] -Furthermore, we use :code:`FixAtPointBC` from :doc:`Steady Boussinesq problem with integral constraints.` -to remove the pressure null space and fix :math:`\mu_i = 0` for each :math:`i`. :: +It is now natural to assign :math:`v^\textrm{ref}` +to be the average of the species reference velocities:: + + v_ref.assign(0.5 * (v_ref_1 + v_ref_2)) + +Integral constraints +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +At the continuous level the OSM equations imply that + +.. math:: + + \nabla (x_1 + \ldots + x_n) = 0 \quad \textrm{in}\ \Omega. + +Hence, at the discrete level, we expect :math:`x_1 + \ldots + x_n` to approximately be a constant. +However, we have not yet incoporated any equations to make this constant be one. +We accomplish this by enforcing that :math:`\int_{\Omega} (x_1 + \ldots + x_n - 1) \mathrm{d} x = 0`:: + + tot_res += inner(x_1 + x_2 - 1, s_1) * dx + +Moreover, the steady SOSM problem still does not have a unique solution +since we have not specified how much mass of fluid is present in :math:`\Omega`. +For uniqueness we must pin this down by imposing one more constraint. +Instead of directly imposing the value of :math:`\int_{\Omega} \rho \mathrm{d} x`, +to demonstrate the flexibility of our approach we enforce that, on the outflow, +the species have equal average densities:: + + tot_res += inner((M_1_ND * c_1) - (M_2_ND * c_2), s_2) * ds(*outlet_id) + +Analogously to :doc:`the steady Boussinesq demo` we use +:code:`FixAtPointBC` to remove the pressure nullspace and pin the +:math:`\mu_{i, \textrm{aux}}` at a DOF (by carefully studying which rows in the +discretised Jacobian are linearly dependent, one checks that it is +mathematically valid to do this):: import firedrake.utils as firedrake_utils @@ -375,20 +519,23 @@ to remove the pressure null space and fix :math:`\mu_i = 0` for each :math:`i`. nodes = nodes[nodes >= 0] return nodes - # The auxiliary constraints on the chemical potentials and pressure - aux_point = as_vector([4, 0]) # point on the middle of the outlet + # Fix the auxiliary chemical potentials and pressure at a point + aux_point = as_vector([4, 0]) # A point on the middle of the outlet aux_point_bcs = [FixAtPointBC(Z_h.sub(3), Constant(0.0), aux_point), FixAtPointBC(Z_h.sub(4), Constant(0.0), aux_point), FixAtPointBC(Z_h.sub(5), Constant(0.0), aux_point)] +Solving the system using Newton's method +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + We provide a naive initial guess based on an equimolar constant distribution of benzene and cyclohexane:: - mm_1, mm_2, v, mu_aux_1, mu_aux_2, p, x_1, x_2, rho_inv, l_1, l_2 = sln.subfunctions + J_1, J_2, v, mu_aux_1, mu_aux_2, p, x_1, x_2, rho_inv, l_1, l_2 = sln.subfunctions x_1.interpolate(Constant(0.5)) x_2.interpolate(Constant(0.5)) rho_inv.interpolate(1.0 / ((M_1_ND * c_1) + (M_2_ND * c_2))) -and define the nonlinear variational solver object:: +and define the nonlinear variational solver object, which by defualt uses Newton's method:: NLVP = NonlinearVariationalProblem(tot_res, sln, bcs=flux_bcs+aux_point_bcs) NLVS = NonlinearVariationalSolver(NLVP) @@ -401,7 +548,7 @@ with the naive initial guess and use its solution as initial guess for the probl and :math:`v_1^\text{ref}=0.3\times 10^{-5}` before solving for :math:`v_1^\text{ref}=0.4\times 10^{-5}`. We can reuse the nonlinear variational solver object each iteration, but have to reassign :code:`v_ref_1` to the new value before calling the :code:`solve()` method. Finally, we write each solution to the same -VTK file using the :code:`time` keyword argument.:: +VTK file using the :code:`time` keyword argument. :: from firedrake.output import VTKFile outfile = VTKFile("out/sln.pvd") @@ -411,9 +558,10 @@ VTK file using the :code:`time` keyword argument.:: for i in range(n_cont): print(f"Solving for v_ref_1 = {0.4e-6*cont_vals[i]}") v_ref_1.assign(Constant(0.4e-6*cont_vals[i])) + v_ref.assign(0.5 * (v_ref_1 + v_ref_2)) NLVS.solve() - p += assemble(-p * dx) / assemble(1 * dx(mesh)) # normalise p to have 0 mean + p += assemble(-p * dx) / assemble(1 * dx(mesh)) # Normalise p to have 0 mean mu_1_out = Function(U_h) mu_2_out = Function(U_h) @@ -429,8 +577,8 @@ VTK file using the :code:`time` keyword argument.:: c_1_out.interpolate(c_1) c_2_out.interpolate(c_2) - mm_1.rename("mm_1") - mm_2.rename("mm_2") + J_1.rename("J_1") + J_2.rename("J_2") v.rename("v") mu_1_out.rename("mu_1") mu_2_out.rename("mu_2") @@ -443,10 +591,9 @@ VTK file using the :code:`time` keyword argument.:: c_1_out.rename("c_1") c_2_out.rename("c_2") - outfile.write(mm_1, mm_2, v, mu_1_out, mu_2_out, p, x_1, x_2, rho_inv, \ + outfile.write(J_1, J_2, v, mu_1_out, mu_2_out, p, x_1, x_2, rho_inv, \ rho_out, c_tot_out, c_1_out, c_2_out, time=i) - The mole fraction and stream lines of benzene for :math:`v_1^\text{ref}=0.4\times 10^{-6}` and :math:`v_1^\text{ref}=0.4\times 10^{-5}` are displayed below on the left and right respectively. Thanks to parameter continuation and higher-order discretisation methods, we can effectively solve From c8aa070891158b5280ab94182cef209e17ac678b Mon Sep 17 00:00:00 2001 From: KarsKnook Date: Tue, 20 May 2025 17:57:36 +0100 Subject: [PATCH 05/32] reference --- demos/demo_references.bib | 7 +++++++ 1 file changed, 7 insertions(+) diff --git a/demos/demo_references.bib b/demos/demo_references.bib index 2329f2e220..f799191642 100644 --- a/demos/demo_references.bib +++ b/demos/demo_references.bib @@ -327,4 +327,11 @@ @article{Krishna:1997 year={1997}, publisher={Elsevier}, doi={10.1016/S0009-2509(96)00458-7} +} + +@article{BaierReinio:2025, + title={High-order finite element methods for three-dimensional multicomponent convection-diffusion}, + author={Baier-Reinio, Aaron and Farrell, Patrick E}, + journal={arXiv preprint arXiv:2408.17390}, + year={2025} } \ No newline at end of file From 5145f3ca5c8570ba44e737cd54ce2924db66b839 Mon Sep 17 00:00:00 2001 From: KarsKnook Date: Tue, 20 May 2025 18:42:28 +0100 Subject: [PATCH 06/32] bibliography --- demos/multicomponent/multicomponent.py.rst | 5 +++++ 1 file changed, 5 insertions(+) diff --git a/demos/multicomponent/multicomponent.py.rst b/demos/multicomponent/multicomponent.py.rst index 7c6531010b..22afb2b6ea 100644 --- a/demos/multicomponent/multicomponent.py.rst +++ b/demos/multicomponent/multicomponent.py.rst @@ -603,3 +603,8 @@ for low species concentrations and sharp solution gradients. | .. image:: benzene_0.png | .. image:: benzene_4.png | | :width: 100% | :width: 100% | +---------------------------+---------------------------+ + +.. rubric:: References + +.. bibliography:: demo_references.bib + :filter: docname in docnames From c8e58a940e0a2ca94f351551df2fc626106f5e28 Mon Sep 17 00:00:00 2001 From: "Patrick E. Farrell" Date: Wed, 21 May 2025 11:31:08 +0100 Subject: [PATCH 07/32] Small suggestions (#4335) * Small suggestions * Don't mention Gibbs-Duhem without explaining later * Change initial paragraph --- demos/demo_references.bib | 4 ++-- demos/multicomponent/multicomponent.py.rst | 15 ++++++++++----- 2 files changed, 12 insertions(+), 7 deletions(-) diff --git a/demos/demo_references.bib b/demos/demo_references.bib index f799191642..2c32ee3cfb 100644 --- a/demos/demo_references.bib +++ b/demos/demo_references.bib @@ -37,7 +37,7 @@ @Book{2006:SV @book{Perry:2007, title = {Perry’s Chemical Engineers’ Handbook}, edition = {8th}, - author = {Green, D. W. and Perry, R. H.}, + author = {Green, Don W. and Perry, Robert H.}, year = 2007, publisher = {McGraw Hill Professional} } @@ -334,4 +334,4 @@ @article{BaierReinio:2025 author={Baier-Reinio, Aaron and Farrell, Patrick E}, journal={arXiv preprint arXiv:2408.17390}, year={2025} -} \ No newline at end of file +} diff --git a/demos/multicomponent/multicomponent.py.rst b/demos/multicomponent/multicomponent.py.rst index 22afb2b6ea..9afeb92a7d 100644 --- a/demos/multicomponent/multicomponent.py.rst +++ b/demos/multicomponent/multicomponent.py.rst @@ -3,13 +3,18 @@ Multicomponent flow -- microfluidic non-ideal mixing of hydrocarbons .. rst-class:: emphasis - We show how Firedrake can be used to simulate multicomponent flow; + We show how Firedrake can be used to simulate multicomponent flow, specifically the microfluidic non-ideal mixing of benzene and cyclohexane. The demo was contributed by `Aaron Baier-Reinio `__ and `Kars Knook `__. +Multicomponent fluids are those composed of two or more species. Solving the equations +describing such fluids is challenging, because there are many variables to solve for, the +equations are nonlinear, and because the system possesses subtle properties like the +mass-average constraint and the mole fraction sum. + We consider a steady, isothermal, nonreacting mixture of benzene and cyclohexane in a two-dimensional microfluidic container :math:`\Omega \subset \mathbb{R}^2`. Pure benzene and cyclohexane flow in through opposing inlets on the left side of the container. @@ -133,11 +138,11 @@ We also choose a reference length of :math:`L^{\textrm{ref}} = 2 \cdot 10^{-3}` It is then natural to define the reference pressure as :math:`p^{\textrm{ref}} = \eta \cdot v^{\textrm{ref}} / L^{\textrm{ref}}`. :: - # The (dimensionful) Stokes viscosities + # The (dimensional) Stokes viscosities eta = Constant(6e-4) # Shear viscosity, Pa s zeta = Constant(1e-7) # Bulk viscosity, Pa s - # Non-dimensionalised Lame parameter, dimensionless + # Non-dimensionalised Lamé parameter, dimensionless lame_ND = (zeta / eta) - 1.0 # Reference quantities used for non-dimensionalisation @@ -465,7 +470,7 @@ At the continuous level the OSM equations imply that \nabla (x_1 + \ldots + x_n) = 0 \quad \textrm{in}\ \Omega. Hence, at the discrete level, we expect :math:`x_1 + \ldots + x_n` to approximately be a constant. -However, we have not yet incoporated any equations to make this constant be one. +However, we have not yet incorporated any equations to make this constant be one. We accomplish this by enforcing that :math:`\int_{\Omega} (x_1 + \ldots + x_n - 1) \mathrm{d} x = 0`:: tot_res += inner(x_1 + x_2 - 1, s_1) * dx @@ -535,7 +540,7 @@ We provide a naive initial guess based on an equimolar constant distribution of x_2.interpolate(Constant(0.5)) rho_inv.interpolate(1.0 / ((M_1_ND * c_1) + (M_2_ND * c_2))) -and define the nonlinear variational solver object, which by defualt uses Newton's method:: +and define the nonlinear variational solver object, which by default uses Newton's method:: NLVP = NonlinearVariationalProblem(tot_res, sln, bcs=flux_bcs+aux_point_bcs) NLVS = NonlinearVariationalSolver(NLVP) From 11062d88b5f338a6026d08b76cbd9239ceee929a Mon Sep 17 00:00:00 2001 From: KarsKnook Date: Wed, 21 May 2025 13:39:08 +0100 Subject: [PATCH 08/32] typos --- demos/multicomponent/multicomponent.py.rst | 30 +++++++++++----------- 1 file changed, 15 insertions(+), 15 deletions(-) diff --git a/demos/multicomponent/multicomponent.py.rst b/demos/multicomponent/multicomponent.py.rst index 9afeb92a7d..47ae3c4bce 100644 --- a/demos/multicomponent/multicomponent.py.rst +++ b/demos/multicomponent/multicomponent.py.rst @@ -249,7 +249,7 @@ Our implementation of the non-dimensionalised volumetric equation of state c_tot, c_1, c_2 = conc_relation(x_1, x_2) -Moreover, to express that :math:`\rho^{-1} = \sum_{j=1}^n M_j c_j`, we simply take the +Moreover, to express that :math:`1 / \rho^{-1} = \sum_{j=1}^n M_j c_j`, we simply take the :math:`L^2`-projection of this expression (in non-dimensionalised form):: rho_inv_terms = inner(1.0 / rho_inv, r) * dx @@ -358,14 +358,14 @@ introducing an augmentation parameter One can non-dimensionalise :ref:`eq 5. ` by introducing a dimensionless Péclet number :math:`\mathrm{Pe} = v^{\textrm{ref}} L^{\textrm{ref}} / \mathscr{D}_{\textrm{sm}}` -and pressure diffusion numbers :math:`\mathrm{Me} = p^{\textrm{ref}} / RT c^{\textrm{ref}}`. +and pressure diffusion number :math:`\mathrm{Me} = p^{\textrm{ref}} / RT c^{\textrm{ref}}`. Moreover, :ref:`eq 5. ` can be cast into a variational form by testing against functions :math:`K_i` and integrating by parts the two gradient terms on the left-hand side (the boundary terms drop out owing to our BCs below). This leads to the following implementation:: D_sm = Constant(2.1e-9) # Stefan--Maxwell diffusivity, m^2 / s - Pe = v_ref * L_ref / D_sm # Peclet number, dimensionless + Pe = v_ref * L_ref / D_sm # Péclet number, dimensionless Me = p_ref / (RT * c_ref) # Pressure diffusion number, dimensionless gamma = Constant(1e-1) # Augmentation parameter, dimensionless @@ -413,8 +413,8 @@ The magnitudes of the parabolic profiles are :math:`M_i c_i^\text{ref} v_i^\text :math:`v_i^\text{ref}` are reference velocities that we are free to choose. Elsewhere on the boundary we enforce :math:`J_i \cdot N = 0`. Finally, instead of specifying the value of the barycentric velocity :math:`v` on the inflows and outflow, we enforce :math:`\rho v \cdot N = (J_1 + J_2 )\cdot N` -and :math:`\rho v \times N = 0` in these regions. This means that :math:`v` equals an unknown quantity, -so we need to use :code:`EquationBC` instead of :code:`DirichletBC`. :: +and :math:`\rho v \times N = 0` in these regions. To enforce that :math:`v` is equal to an unknown quantity, +we must use :class:`~.EquationBC` instead of :class:`~.DirichletBC`. :: # Reference species velocities, which we choose to symmetrize so that the molar fluxes agree v_ref_1 = Constant(0.4e-6) # Reference inflow velocity of benzene, m / s @@ -432,7 +432,7 @@ so we need to use :code:`EquationBC` instead of :code:`DirichletBC`. :: J_2_outflow_bc_func = -M_2_ND * (v_ref_2 / v_ref) * (c_pure_2 / c_ref) * parabola_outflow rho_v_outflow_bc_func = J_1_outflow_bc_func + J_2_outflow_bc_func - # Boundary conditions on the bulk velocity are enforced via EquationBC + # Boundary conditions on the barycentric velocity are enforced via EquationBC v_inflow_1_bc = EquationBC(inner(v - rho_inv * rho_v_inflow_1_bc_func, u) * ds(*inlet_1_id) == 0, sln, inlet_1_id, V=Z_h.sub(2)) v_inflow_2_bc = EquationBC(inner(v - rho_inv * rho_v_inflow_2_bc_func, u) * ds(*inlet_2_id) == 0, @@ -471,20 +471,20 @@ At the continuous level the OSM equations imply that Hence, at the discrete level, we expect :math:`x_1 + \ldots + x_n` to approximately be a constant. However, we have not yet incorporated any equations to make this constant be one. -We accomplish this by enforcing that :math:`\int_{\Omega} (x_1 + \ldots + x_n - 1) \mathrm{d} x = 0`:: +We accomplish this by enforcing that :math:`\int_{\Omega} (x_1 + \ldots + x_n - 1) \ {\rm d} x = 0`:: tot_res += inner(x_1 + x_2 - 1, s_1) * dx Moreover, the steady SOSM problem still does not have a unique solution since we have not specified how much mass of fluid is present in :math:`\Omega`. For uniqueness we must pin this down by imposing one more constraint. -Instead of directly imposing the value of :math:`\int_{\Omega} \rho \mathrm{d} x`, +Instead of directly imposing the value of :math:`\int_{\Omega} \rho \ {\rm d} x`, to demonstrate the flexibility of our approach we enforce that, on the outflow, the species have equal average densities:: tot_res += inner((M_1_ND * c_1) - (M_2_ND * c_2), s_2) * ds(*outlet_id) -Analogously to :doc:`the steady Boussinesq demo` we use +Analogously to :doc:`the steady Boussinesq demo ` we use :code:`FixAtPointBC` to remove the pressure nullspace and pin the :math:`\mu_{i, \textrm{aux}}` at a DOF (by carefully studying which rows in the discretised Jacobian are linearly dependent, one checks that it is @@ -533,7 +533,7 @@ mathematically valid to do this):: Solving the system using Newton's method ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -We provide a naive initial guess based on an equimolar constant distribution of benzene and cyclohexane:: +We provide a naive initial guess based on an equimolar spatially uniform distribution of benzene and cyclohexane:: J_1, J_2, v, mu_aux_1, mu_aux_2, p, x_1, x_2, rho_inv, l_1, l_2 = sln.subfunctions x_1.interpolate(Constant(0.5)) @@ -551,8 +551,8 @@ to find a better initial guess. We start by solving the problem for :math:`v_1^\ with the naive initial guess and use its solution as initial guess for the problem with :math:`v_1^\text{ref}=0.1\times 10^{-5}`. We repeat this trick with :math:`v_1^\text{ref}=0.2\times 10^{-5}` and :math:`v_1^\text{ref}=0.3\times 10^{-5}` before solving for :math:`v_1^\text{ref}=0.4\times 10^{-5}`. -We can reuse the nonlinear variational solver object each iteration, but have to reassign :code:`v_ref_1` -to the new value before calling the :code:`solve()` method. Finally, we write each solution to the same +We can reuse the nonlinear variational solver object each iteration, but have to reassign :code:`v_ref_1` +and :code:`v_ref` before calling the :code:`solve()` method. Finally, we write each solution to the same VTK file using the :code:`time` keyword argument. :: from firedrake.output import VTKFile @@ -599,10 +599,10 @@ VTK file using the :code:`time` keyword argument. :: outfile.write(J_1, J_2, v, mu_1_out, mu_2_out, p, x_1, x_2, rho_inv, \ rho_out, c_tot_out, c_1_out, c_2_out, time=i) -The mole fraction and stream lines of benzene for :math:`v_1^\text{ref}=0.4\times 10^{-6}` +The mole fraction and streamlines of benzene for :math:`v_1^\text{ref}=0.4\times 10^{-6}` and :math:`v_1^\text{ref}=0.4\times 10^{-5}` are displayed below on the left and right respectively. -Thanks to parameter continuation and higher-order discretisation methods, we can effectively solve -for low species concentrations and sharp solution gradients. +Owing to parameter continuation and the high-order discretisation, we can robustly solve the problem +even in the presence of low species concentrations and sharp solution gradients. +---------------------------+---------------------------+ | .. image:: benzene_0.png | .. image:: benzene_4.png | From 68a72b7f920fdee5d90de796f4aa3743ba2188a8 Mon Sep 17 00:00:00 2001 From: KarsKnook <57411502+KarsKnook@users.noreply.github.com> Date: Wed, 21 May 2025 13:42:42 +0100 Subject: [PATCH 09/32] swap order --- docs/source/advanced_tut.rst | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/docs/source/advanced_tut.rst b/docs/source/advanced_tut.rst index cc96162941..ea508f23db 100644 --- a/docs/source/advanced_tut.rst +++ b/docs/source/advanced_tut.rst @@ -24,5 +24,5 @@ element systems. Rayleigh-Benard convection. Netgen support. Full-waveform inversion: spatial and wave sources parallelism. - Steady multicomponent flow -- microfluidic mixing of hydrocarbons. Steady Boussinesq problem with integral constraints. + Steady multicomponent flow -- microfluidic mixing of hydrocarbons. From 9dfc2964a61cd55e6963eef9c20bc8fa3a287f03 Mon Sep 17 00:00:00 2001 From: KarsKnook Date: Mon, 26 May 2025 11:30:31 +0100 Subject: [PATCH 10/32] resolving suggestions --- demos/multicomponent/multicomponent.py.rst | 24 +++++++++++----------- 1 file changed, 12 insertions(+), 12 deletions(-) diff --git a/demos/multicomponent/multicomponent.py.rst b/demos/multicomponent/multicomponent.py.rst index 47ae3c4bce..0800e2a84f 100644 --- a/demos/multicomponent/multicomponent.py.rst +++ b/demos/multicomponent/multicomponent.py.rst @@ -1,5 +1,5 @@ Multicomponent flow -- microfluidic non-ideal mixing of hydrocarbons -=================================================== +==================================================================== .. rst-class:: emphasis @@ -94,11 +94,11 @@ We first define the finite element spaces and trial/test functions:: # The mixed space Z_h = J_h * J_h * V_h * U_h * U_h * P_h * X_h * X_h * R_h * L_h * L_h - PETSc.Sys.Print("Mesh has %d cells, with %d finite element DOFs" % (mesh.num_cells(), Z_h.dim())) + PETSc.Sys.Print(f"Mesh has {mesh.num_cells()} cells, with {Z_h.dim()} finite element DOFs") # The trial functions - sln = Function(Z_h) - J_1, J_2, v, mu_aux_1, mu_aux_2, p, x_1, x_2, rho_inv, l_1, l_2 = split(sln) + solution = Function(Z_h) + J_1, J_2, v, mu_aux_1, mu_aux_2, p, x_1, x_2, rho_inv, l_1, l_2 = split(solution) # Lagrange multiplier trick for enforcing integral constraints mu_1 = mu_aux_1 + l_1 @@ -434,10 +434,10 @@ we must use :class:`~.EquationBC` instead of :class:`~.DirichletBC`. :: # Boundary conditions on the barycentric velocity are enforced via EquationBC v_inflow_1_bc = EquationBC(inner(v - rho_inv * rho_v_inflow_1_bc_func, u) * ds(*inlet_1_id) == 0, - sln, inlet_1_id, V=Z_h.sub(2)) + solution, inlet_1_id, V=Z_h.sub(2)) v_inflow_2_bc = EquationBC(inner(v - rho_inv * rho_v_inflow_2_bc_func, u) * ds(*inlet_2_id) == 0, - sln, inlet_2_id, V=Z_h.sub(2)) - v_outflow_bc = EquationBC(inner(v - rho_inv * rho_v_outflow_bc_func, u) * ds(*outlet_id) == 0, sln, + solution, inlet_2_id, V=Z_h.sub(2)) + v_outflow_bc = EquationBC(inner(v - rho_inv * rho_v_outflow_bc_func, u) * ds(*outlet_id) == 0, solution, outlet_id, V=Z_h.sub(2)) # The boundary conditions on the fluxes and barycentric velocity @@ -490,7 +490,7 @@ Analogously to :doc:`the steady Boussinesq demo ` we use discretised Jacobian are linearly dependent, one checks that it is mathematically valid to do this):: - import firedrake.utils as firedrake_utils + import functools class FixAtPointBC(firedrake.DirichletBC): r'''A special BC object for pinning a function at a point. @@ -506,7 +506,7 @@ mathematically valid to do this):: bc_point = as_vector(bc_point) self.bc_point = bc_point - @firedrake_utils.cached_property + @functools.cached_property def nodes(self): V = self.function_space() x = firedrake.SpatialCoordinate(V.mesh()) @@ -535,14 +535,14 @@ Solving the system using Newton's method We provide a naive initial guess based on an equimolar spatially uniform distribution of benzene and cyclohexane:: - J_1, J_2, v, mu_aux_1, mu_aux_2, p, x_1, x_2, rho_inv, l_1, l_2 = sln.subfunctions + J_1, J_2, v, mu_aux_1, mu_aux_2, p, x_1, x_2, rho_inv, l_1, l_2 = solution.subfunctions x_1.interpolate(Constant(0.5)) x_2.interpolate(Constant(0.5)) rho_inv.interpolate(1.0 / ((M_1_ND * c_1) + (M_2_ND * c_2))) and define the nonlinear variational solver object, which by default uses Newton's method:: - NLVP = NonlinearVariationalProblem(tot_res, sln, bcs=flux_bcs+aux_point_bcs) + NLVP = NonlinearVariationalProblem(tot_res, solution, bcs=flux_bcs+aux_point_bcs) NLVS = NonlinearVariationalSolver(NLVP) Newton's method applied directly to the problem with :math:`v_1^\text{ref}=0.4\times 10^{-5}` @@ -556,7 +556,7 @@ and :code:`v_ref` before calling the :code:`solve()` method. Finally, we write e VTK file using the :code:`time` keyword argument. :: from firedrake.output import VTKFile - outfile = VTKFile("out/sln.pvd") + outfile = VTKFile("out/solution.pvd") cont_vals = [1.0, 2.5, 5, 7.5, 10.0] n_cont = len(cont_vals) From 4a4e1f75a834b6691b6b25030a67433cb7d25a5d Mon Sep 17 00:00:00 2001 From: Connor Ward Date: Wed, 4 Jun 2025 11:27:15 +0100 Subject: [PATCH 11/32] Fix underlines --- demos/multicomponent/multicomponent.py.rst | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/demos/multicomponent/multicomponent.py.rst b/demos/multicomponent/multicomponent.py.rst index 0800e2a84f..38d2a1d92c 100644 --- a/demos/multicomponent/multicomponent.py.rst +++ b/demos/multicomponent/multicomponent.py.rst @@ -405,7 +405,7 @@ Altogether, our total residual is the sum of forms built above:: + div_mass_avg_terms Boundary conditions -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +~~~~~~~~~~~~~~~~~~~ Let :math:`N` denote the outward unit normal on :math:`\partial \Omega`. We enforce parabolic profiles on :math:`J_i \cdot N` at inflow :math:`i` and on the outflow. @@ -531,7 +531,7 @@ mathematically valid to do this):: FixAtPointBC(Z_h.sub(5), Constant(0.0), aux_point)] Solving the system using Newton's method -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ We provide a naive initial guess based on an equimolar spatially uniform distribution of benzene and cyclohexane:: From 4042260dd6539a7c2f21e3d3519bbc2b8958a147 Mon Sep 17 00:00:00 2001 From: KarsKnook Date: Thu, 5 Jun 2025 17:36:10 +0100 Subject: [PATCH 12/32] smaller problem for CI --- demos/multicomponent/multicomponent.py.rst | 12 ++++++++++-- tests/firedrake/demos/test_demos_run.py | 1 + 2 files changed, 11 insertions(+), 2 deletions(-) diff --git a/demos/multicomponent/multicomponent.py.rst b/demos/multicomponent/multicomponent.py.rst index 38d2a1d92c..5d4dfb5fd5 100644 --- a/demos/multicomponent/multicomponent.py.rst +++ b/demos/multicomponent/multicomponent.py.rst @@ -43,9 +43,17 @@ We use :doc:`netgen ` to build a curved mesh of order :math:`k=3 Omega.edges.Max(ngocc.Y).name = "inlet_1" Omega.edges.Min(ngocc.Y).name = "inlet_2" Omega.edges.Max(ngocc.X).name = "outlet" + + # Smaller problem when running CI tests + import os + if os.getenv("FIREDRAKE_CI") == "1": + maxh = 0.14 + k = 2 + else: + maxh = 0.055 # Construct the mesh - ngmesh = ngocc.OCCGeometry(Omega, dim=2).GenerateMesh(maxh=0.055) + ngmesh = ngocc.OCCGeometry(Omega, dim=2).GenerateMesh(maxh=maxh) mesh = Mesh(Mesh(ngmesh).curve_field(k)) # Get the IDs of the inlets, outlet and walls @@ -55,7 +63,7 @@ We use :doc:`netgen ` to build a curved mesh of order :math:`k=3 walls_ids = [i+1 for i, name in enumerate(ngmesh.GetRegionNames(dim=1)) if name == ""] # Define the surface and volume measures using a fixed quadrature degree - deg_max = 15 + deg_max = 3*k ds = ds(mesh, degree=deg_max) dx = dx(mesh, degree=deg_max) diff --git a/tests/firedrake/demos/test_demos_run.py b/tests/firedrake/demos/test_demos_run.py index 4d8eb623d8..7086dce967 100644 --- a/tests/firedrake/demos/test_demos_run.py +++ b/tests/firedrake/demos/test_demos_run.py @@ -35,6 +35,7 @@ Demo(("matrix_free", "poisson"), []), Demo(("matrix_free", "rayleigh-benard"), ["hypre", "mumps", "vtk"]), Demo(("matrix_free", "stokes"), ["hypre", "mumps", "vtk"]), + Demo(("multicomponent", "multicomponent"), ["vtk, netgen"]), Demo(("multigrid", "geometric_multigrid"), ["vtk"]), Demo(("netgen", "netgen_mesh"), ["mumps", "netgen", "slepc", "vtk"]), Demo(("nonlinear_QG_winddrivengyre", "qg_winddrivengyre"), ["vtk"]), From 133f07a80f17d1a9ae9f6c8ab342d2321c3dec7d Mon Sep 17 00:00:00 2001 From: KarsKnook Date: Mon, 9 Jun 2025 12:28:07 +0100 Subject: [PATCH 13/32] fixes --- demos/multicomponent/multicomponent.py.rst | 50 +++++++++++----------- 1 file changed, 25 insertions(+), 25 deletions(-) diff --git a/demos/multicomponent/multicomponent.py.rst b/demos/multicomponent/multicomponent.py.rst index 5d4dfb5fd5..d4a006949f 100644 --- a/demos/multicomponent/multicomponent.py.rst +++ b/demos/multicomponent/multicomponent.py.rst @@ -137,7 +137,11 @@ Recall that :math:`v` is the barycentric velocity and :math:`p` the pressure. Moreover :math:`\epsilon (v)` denotes the symmetric gradient of :math:`v` while :math:`\eta, \zeta > 0` are the shear and bulk viscosities respectively, :math:`d=2` is the spatial dimension and -:math:`\mathbb{I}` the :math:`d \times d` identity matrix. +:math:`\mathbb{I}` the :math:`d \times d` identity matrix. Despite the fact +that we are considering compressible flow, we still need +a constraint on :math:`\nabla \cdot v` (as is the case for incompressible flow). +However, we postpone the discussion of this constraint to the end of this section as it +involves other quantities that we have not yet described. We shall non-dimensionalise all of the unknowns in our discretisation. Hence we introduce a reference velocity :math:`v^{\textrm{ref}}` whose value @@ -189,7 +193,7 @@ As we are considering steady flow, the continuity equations simplify to which are discretised as follows:: - continuity_terms = (inner(w_1, div(J_1)) + inner(w_2, div(J_2))) * dx + continuity_terms = (inner(div(J_1), w_1) + inner(div(J_2), w_2)) * dx Next, we incorporate the volumetric equation of state, which models how the concentration of the mixture depends on temperature, pressure and composition. @@ -396,11 +400,11 @@ Lastly, we weakly enforce that :math:`\nabla \cdot v = \nabla \cdot (\frac{1}{\rho} \sum_{j=1}^n J_j )`, using special density consistency terms to handle inhomogeneous BCs:: - div_mass_avg_terms = inner(q, div(v - (rho_inv * (J_1 + J_2)))) * dx + div_mass_avg_terms = inner(div(v - (rho_inv * (J_1 + J_2))), q) * dx # The density consistency terms N = FacetNormal(mesh) - div_mass_avg_terms -= q * inner(v - (rho_inv * (J_1 + J_2)), N) * ds + div_mass_avg_terms -= inner(dot(v - (rho_inv * (J_1 + J_2)), N), q) * ds This concludes our discussion of the PDE model and its discretisation. Altogether, our total residual is the sum of forms built above:: @@ -421,8 +425,8 @@ The magnitudes of the parabolic profiles are :math:`M_i c_i^\text{ref} v_i^\text :math:`v_i^\text{ref}` are reference velocities that we are free to choose. Elsewhere on the boundary we enforce :math:`J_i \cdot N = 0`. Finally, instead of specifying the value of the barycentric velocity :math:`v` on the inflows and outflow, we enforce :math:`\rho v \cdot N = (J_1 + J_2 )\cdot N` -and :math:`\rho v \times N = 0` in these regions. To enforce that :math:`v` is equal to an unknown quantity, -we must use :class:`~.EquationBC` instead of :class:`~.DirichletBC`. :: +and :math:`\rho v \times N = 0` in these regions. Boundary conditions that couple +unknowns and/or are nonlinear must be implemented with :class:`~.EquationBC` instead of :class:`~.DirichletBC`. :: # Reference species velocities, which we choose to symmetrize so that the molar fluxes agree v_ref_1 = Constant(0.4e-6) # Reference inflow velocity of benzene, m / s @@ -452,16 +456,16 @@ we must use :class:`~.EquationBC` instead of :class:`~.DirichletBC`. :: # Note that BCs on H(div) spaces only apply to the normal component flux_bcs = [DirichletBC(Z_h.sub(0), J_1_inflow_bc_func, inlet_1_id), DirichletBC(Z_h.sub(0), J_1_outflow_bc_func, outlet_id), - DirichletBC(Z_h.sub(0), as_vector([0.0, 0.0]), inlet_2_id), - DirichletBC(Z_h.sub(0), as_vector([0.0, 0.0]), walls_ids), + DirichletBC(Z_h.sub(0), 0, inlet_2_id), + DirichletBC(Z_h.sub(0), 0, walls_ids), DirichletBC(Z_h.sub(1), J_2_inflow_bc_func, inlet_2_id), DirichletBC(Z_h.sub(1), J_2_outflow_bc_func, outlet_id), - DirichletBC(Z_h.sub(1), as_vector([0.0, 0.0]), inlet_1_id), - DirichletBC(Z_h.sub(1), as_vector([0.0, 0.0]), walls_ids), + DirichletBC(Z_h.sub(1), 0, inlet_1_id), + DirichletBC(Z_h.sub(1), 0, walls_ids), v_inflow_1_bc, v_inflow_2_bc, v_outflow_bc, - DirichletBC(Z_h.sub(2), as_vector([0.0, 0.0]), walls_ids)] + DirichletBC(Z_h.sub(2), 0, walls_ids)] It is now natural to assign :math:`v^\textrm{ref}` to be the average of the species reference velocities:: @@ -534,9 +538,9 @@ mathematically valid to do this):: # Fix the auxiliary chemical potentials and pressure at a point aux_point = as_vector([4, 0]) # A point on the middle of the outlet - aux_point_bcs = [FixAtPointBC(Z_h.sub(3), Constant(0.0), aux_point), - FixAtPointBC(Z_h.sub(4), Constant(0.0), aux_point), - FixAtPointBC(Z_h.sub(5), Constant(0.0), aux_point)] + aux_point_bcs = [FixAtPointBC(Z_h.sub(3), 0, aux_point), + FixAtPointBC(Z_h.sub(4), 0, aux_point), + FixAtPointBC(Z_h.sub(5), 0, aux_point)] Solving the system using Newton's method ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -563,7 +567,6 @@ We can reuse the nonlinear variational solver object each iteration, but have to and :code:`v_ref` before calling the :code:`solve()` method. Finally, we write each solution to the same VTK file using the :code:`time` keyword argument. :: - from firedrake.output import VTKFile outfile = VTKFile("out/solution.pvd") cont_vals = [1.0, 2.5, 5, 7.5, 10.0] n_cont = len(cont_vals) @@ -590,22 +593,19 @@ VTK file using the :code:`time` keyword argument. :: c_1_out.interpolate(c_1) c_2_out.interpolate(c_2) - J_1.rename("J_1") - J_2.rename("J_2") - v.rename("v") mu_1_out.rename("mu_1") mu_2_out.rename("mu_2") - p.rename("p") - x_1.rename("x_1") - x_2.rename("x_2") - rho_inv.rename("rho_inv") rho_out.rename("rho") c_tot_out.rename("c_tot") c_1_out.rename("c_1") c_2_out.rename("c_2") - - outfile.write(J_1, J_2, v, mu_1_out, mu_2_out, p, x_1, x_2, rho_inv, \ - rho_out, c_tot_out, c_1_out, c_2_out, time=i) + + names = ["J_1", "J_2", "v", "mu_aux_1", "mu_aux_2", "p", "x_1", "x_2", + "rho_inv", "l_1", "l_2"] + for field, name in enumerate(names): + solution.subfunctions[field].rename(name) + + outfile.write(*solution.subfunctions, mu_1_out, mu_2_out, rho_out, c_tot_out, c_1_out, c_2_out, time=i) The mole fraction and streamlines of benzene for :math:`v_1^\text{ref}=0.4\times 10^{-6}` and :math:`v_1^\text{ref}=0.4\times 10^{-5}` are displayed below on the left and right respectively. From cd61bf3642c0977325655dfb22f59192765c05ad Mon Sep 17 00:00:00 2001 From: KarsKnook Date: Mon, 9 Jun 2025 14:23:53 +0100 Subject: [PATCH 14/32] naming fields outside continuation loop --- demos/multicomponent/multicomponent.py.rst | 38 +++++++++++----------- 1 file changed, 19 insertions(+), 19 deletions(-) diff --git a/demos/multicomponent/multicomponent.py.rst b/demos/multicomponent/multicomponent.py.rst index d4a006949f..cf545c11fa 100644 --- a/demos/multicomponent/multicomponent.py.rst +++ b/demos/multicomponent/multicomponent.py.rst @@ -571,6 +571,25 @@ VTK file using the :code:`time` keyword argument. :: cont_vals = [1.0, 2.5, 5, 7.5, 10.0] n_cont = len(cont_vals) + names = ["J_1", "J_2", "v", "mu_aux_1", "mu_aux_2", "p", "x_1", "x_2", + "rho_inv", "l_1", "l_2"] + for field, name in enumerate(names): + solution.subfunctions[field].rename(name) + + mu_1_out = Function(U_h) + mu_2_out = Function(U_h) + rho_out = Function(R_h) + c_tot_out = Function(R_h) + c_1_out = Function(R_h) + c_2_out = Function(R_h) + + mu_1_out.rename("mu_1") + mu_2_out.rename("mu_2") + rho_out.rename("rho") + c_tot_out.rename("c_tot") + c_1_out.rename("c_1") + c_2_out.rename("c_2") + for i in range(n_cont): print(f"Solving for v_ref_1 = {0.4e-6*cont_vals[i]}") v_ref_1.assign(Constant(0.4e-6*cont_vals[i])) @@ -579,31 +598,12 @@ VTK file using the :code:`time` keyword argument. :: p += assemble(-p * dx) / assemble(1 * dx(mesh)) # Normalise p to have 0 mean - mu_1_out = Function(U_h) - mu_2_out = Function(U_h) - rho_out = Function(R_h) - c_tot_out = Function(R_h) - c_1_out = Function(R_h) - c_2_out = Function(R_h) - mu_1_out.interpolate(mu_1) mu_2_out.interpolate(mu_2) rho_out.interpolate(1.0 / rho_inv) c_tot_out.interpolate(c_tot) c_1_out.interpolate(c_1) c_2_out.interpolate(c_2) - - mu_1_out.rename("mu_1") - mu_2_out.rename("mu_2") - rho_out.rename("rho") - c_tot_out.rename("c_tot") - c_1_out.rename("c_1") - c_2_out.rename("c_2") - - names = ["J_1", "J_2", "v", "mu_aux_1", "mu_aux_2", "p", "x_1", "x_2", - "rho_inv", "l_1", "l_2"] - for field, name in enumerate(names): - solution.subfunctions[field].rename(name) outfile.write(*solution.subfunctions, mu_1_out, mu_2_out, rho_out, c_tot_out, c_1_out, c_2_out, time=i) From fe14543a67297322db7ca3b09fc023f706934e87 Mon Sep 17 00:00:00 2001 From: KarsKnook Date: Tue, 10 Jun 2025 10:49:28 +0100 Subject: [PATCH 15/32] clarifying equationBC --- demos/multicomponent/multicomponent.py.rst | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/demos/multicomponent/multicomponent.py.rst b/demos/multicomponent/multicomponent.py.rst index cf545c11fa..65c4d0d9a0 100644 --- a/demos/multicomponent/multicomponent.py.rst +++ b/demos/multicomponent/multicomponent.py.rst @@ -424,8 +424,8 @@ We enforce parabolic profiles on :math:`J_i \cdot N` at inflow :math:`i` and on The magnitudes of the parabolic profiles are :math:`M_i c_i^\text{ref} v_i^\text{ref}` where :math:`v_i^\text{ref}` are reference velocities that we are free to choose. Elsewhere on the boundary we enforce :math:`J_i \cdot N = 0`. Finally, instead of specifying -the value of the barycentric velocity :math:`v` on the inflows and outflow, we enforce :math:`\rho v \cdot N = (J_1 + J_2 )\cdot N` -and :math:`\rho v \times N = 0` in these regions. Boundary conditions that couple +the value of the barycentric velocity :math:`v` on the inflows and outflows, +we enforce :math:`v = \rho^{-1}((J_1 + J_2)\cdot N)N`. Boundary conditions that couple unknowns and/or are nonlinear must be implemented with :class:`~.EquationBC` instead of :class:`~.DirichletBC`. :: # Reference species velocities, which we choose to symmetrize so that the molar fluxes agree From a6f0e1ba57b5c548721423c42152d402fd0668ba Mon Sep 17 00:00:00 2001 From: KarsKnook Date: Wed, 11 Jun 2025 12:22:18 +0100 Subject: [PATCH 16/32] better FixAtPointBC --- demos/boussinesq/boussinesq.py.rst | 71 +++++++++++----------- demos/multicomponent/multicomponent.py.rst | 65 ++++++++++---------- 2 files changed, 71 insertions(+), 65 deletions(-) diff --git a/demos/boussinesq/boussinesq.py.rst b/demos/boussinesq/boussinesq.py.rst index ac093bd556..e10eed70c7 100644 --- a/demos/boussinesq/boussinesq.py.rst +++ b/demos/boussinesq/boussinesq.py.rst @@ -160,40 +160,43 @@ cannot always be sure that the linear solver at hand is correctly utilising the To directly eliminate the nullspace we introduce a class :code:`FixAtPointBC` which implements a boundary condition that fixes a field at a single point. :: - import firedrake.utils as firedrake_utils - - class FixAtPointBC(firedrake.DirichletBC): - r'''A special BC object for pinning a function at a point. - - :arg V: the :class:`.FunctionSpace` on which the boundary condition should be applied. - :arg g: the boundary condition value. - :arg bc_point: the point at which to pin the function. - The location of the finite element DOF nearest to bc_point is actually used. - ''' - def __init__(self, V, g, bc_point): - super(FixAtPointBC, self).__init__(V, g, bc_point) - if isinstance(bc_point, tuple): - bc_point = as_vector(bc_point) - self.bc_point = bc_point - - @firedrake_utils.cached_property - def nodes(self): - V = self.function_space() - x = firedrake.SpatialCoordinate(V.mesh()) - xdist = x - self.bc_point - - test = firedrake.TestFunction(V) - trial = firedrake.TrialFunction(V) - xphi = firedrake.assemble(ufl.inner(xdist * test, xdist * trial) * ufl.dx, diagonal=True) - phi = firedrake.assemble(ufl.inner(test, trial) * ufl.dx, diagonal=True) - with xphi.dat.vec as xu, phi.dat.vec as u: - xu.pointwiseDivide(xu, u) - min_index, min_value = xu.min() # Find the index of the DOF closest to bc_point - - nodes = V.dof_dset.lgmap.applyInverse([min_index]) - nodes = nodes[nodes >= 0] - return nodes - + import functools + + class FixAtPointBC(DirichletBC): + r'''A special BC object for pinning a function at a point. + + :arg V: the :class:`.FunctionSpace` on which the boundary condition should be applied. + :arg g: the boundary condition value. + :arg bc_point: the point at which to pin the function. + The location of the finite element DOF nearest to bc_point is actually used. + ''' + def __init__(self, V, g, bc_points): + super().__init__(V, g, bc_points) + + @functools.cached_property + def nodes(self): + V = self.function_space() + if V.mesh().ufl_coordinate_element().degree() != 1: + # Ensure a P1 mesh + coordinates = V.mesh().coordinates + P1 = coordinates.function_space().reconstruct(degree=1) + P1_mesh = Mesh(Function(P1).interpolate(coordinates)) + V = V.reconstruct(mesh=P1_mesh) + + points = [tuple(self.sub_domain)] + vom = VertexOnlyMesh(V.mesh(), points) + P0 = FunctionSpace(vom, "DG", 0) + Fvom = Cofunction(P0.dual()).assign(1) + + # Take the basis function with the largest abs value at bc_point + v = TestFunction(V) + F = assemble(Interpolate(inner(v, v), Fvom)) + with F.dat.vec as Fvec: + max_index, _ = Fvec.max() + nodes = V.dof_dset.lgmap.applyInverse([max_index]) + nodes = nodes[nodes >= 0] + return nodes + We use this to fix the pressure and auxiliary temperature at the origin:: aux_bcs = [FixAtPointBC(Z.sub(1), 0, as_vector([0, 0])), diff --git a/demos/multicomponent/multicomponent.py.rst b/demos/multicomponent/multicomponent.py.rst index 65c4d0d9a0..23d6a80c3a 100644 --- a/demos/multicomponent/multicomponent.py.rst +++ b/demos/multicomponent/multicomponent.py.rst @@ -504,37 +504,40 @@ mathematically valid to do this):: import functools - class FixAtPointBC(firedrake.DirichletBC): - r'''A special BC object for pinning a function at a point. - - :arg V: the :class:`.FunctionSpace` on which the boundary condition should be applied. - :arg g: the boundary condition value. - :arg bc_point: the point at which to pin the function. - The location of the finite element DOF nearest to bc_point is actually used. - ''' - def __init__(self, V, g, bc_point): - super(FixAtPointBC, self).__init__(V, g, bc_point) - if isinstance(bc_point, tuple): - bc_point = as_vector(bc_point) - self.bc_point = bc_point - - @functools.cached_property - def nodes(self): - V = self.function_space() - x = firedrake.SpatialCoordinate(V.mesh()) - xdist = x - self.bc_point - - test = firedrake.TestFunction(V) - trial = firedrake.TrialFunction(V) - xphi = firedrake.assemble(ufl.inner(xdist * test, xdist * trial) * ufl.dx, diagonal=True) - phi = firedrake.assemble(ufl.inner(test, trial) * ufl.dx, diagonal=True) - with xphi.dat.vec as xu, phi.dat.vec as u: - xu.pointwiseDivide(xu, u) - min_index, min_value = xu.min() # Find the index of the DOF closest to bc_point - - nodes = V.dof_dset.lgmap.applyInverse([min_index]) - nodes = nodes[nodes >= 0] - return nodes + class FixAtPointBC(DirichletBC): + r'''A special BC object for pinning a function at a point. + + :arg V: the :class:`.FunctionSpace` on which the boundary condition should be applied. + :arg g: the boundary condition value. + :arg bc_point: the point at which to pin the function. + The location of the finite element DOF nearest to bc_point is actually used. + ''' + def __init__(self, V, g, bc_points): + super().__init__(V, g, bc_points) + + @functools.cached_property + def nodes(self): + V = self.function_space() + if V.mesh().ufl_coordinate_element().degree() != 1: + # Ensure a P1 mesh + coordinates = V.mesh().coordinates + P1 = coordinates.function_space().reconstruct(degree=1) + P1_mesh = Mesh(Function(P1).interpolate(coordinates)) + V = V.reconstruct(mesh=P1_mesh) + + points = [tuple(self.sub_domain)] + vom = VertexOnlyMesh(V.mesh(), points) + P0 = FunctionSpace(vom, "DG", 0) + Fvom = Cofunction(P0.dual()).assign(1) + + # Take the basis function with the largest abs value at bc_point + v = TestFunction(V) + F = assemble(Interpolate(inner(v, v), Fvom)) + with F.dat.vec as Fvec: + max_index, _ = Fvec.max() + nodes = V.dof_dset.lgmap.applyInverse([max_index]) + nodes = nodes[nodes >= 0] + return nodes # Fix the auxiliary chemical potentials and pressure at a point aux_point = as_vector([4, 0]) # A point on the middle of the outlet From 08026f568e50c0f764379e51bca009a43d3ce735 Mon Sep 17 00:00:00 2001 From: KarsKnook Date: Wed, 11 Jun 2025 13:44:25 +0100 Subject: [PATCH 17/32] typo --- demos/boussinesq/boussinesq.py.rst | 8 ++++---- demos/multicomponent/multicomponent.py.rst | 8 ++++---- 2 files changed, 8 insertions(+), 8 deletions(-) diff --git a/demos/boussinesq/boussinesq.py.rst b/demos/boussinesq/boussinesq.py.rst index e10eed70c7..35170045ff 100644 --- a/demos/boussinesq/boussinesq.py.rst +++ b/demos/boussinesq/boussinesq.py.rst @@ -170,8 +170,8 @@ implements a boundary condition that fixes a field at a single point. :: :arg bc_point: the point at which to pin the function. The location of the finite element DOF nearest to bc_point is actually used. ''' - def __init__(self, V, g, bc_points): - super().__init__(V, g, bc_points) + def __init__(self, V, g, bc_point): + super().__init__(V, g, bc_point) @functools.cached_property def nodes(self): @@ -183,8 +183,8 @@ implements a boundary condition that fixes a field at a single point. :: P1_mesh = Mesh(Function(P1).interpolate(coordinates)) V = V.reconstruct(mesh=P1_mesh) - points = [tuple(self.sub_domain)] - vom = VertexOnlyMesh(V.mesh(), points) + point = [tuple(self.sub_domain)] + vom = VertexOnlyMesh(V.mesh(), point) P0 = FunctionSpace(vom, "DG", 0) Fvom = Cofunction(P0.dual()).assign(1) diff --git a/demos/multicomponent/multicomponent.py.rst b/demos/multicomponent/multicomponent.py.rst index 23d6a80c3a..21cb9abcfc 100644 --- a/demos/multicomponent/multicomponent.py.rst +++ b/demos/multicomponent/multicomponent.py.rst @@ -512,8 +512,8 @@ mathematically valid to do this):: :arg bc_point: the point at which to pin the function. The location of the finite element DOF nearest to bc_point is actually used. ''' - def __init__(self, V, g, bc_points): - super().__init__(V, g, bc_points) + def __init__(self, V, g, bc_point): + super().__init__(V, g, bc_point) @functools.cached_property def nodes(self): @@ -525,8 +525,8 @@ mathematically valid to do this):: P1_mesh = Mesh(Function(P1).interpolate(coordinates)) V = V.reconstruct(mesh=P1_mesh) - points = [tuple(self.sub_domain)] - vom = VertexOnlyMesh(V.mesh(), points) + point = [tuple(self.sub_domain)] + vom = VertexOnlyMesh(V.mesh(), point) P0 = FunctionSpace(vom, "DG", 0) Fvom = Cofunction(P0.dual()).assign(1) From 37738c567c9a7523f349a6cdd29f1b68bcf5732e Mon Sep 17 00:00:00 2001 From: Aaron Baier-Reinio <130484015+ABaierReinio@users.noreply.github.com> Date: Sun, 22 Jun 2025 12:46:03 +0100 Subject: [PATCH 18/32] Update demos/multicomponent/multicomponent.py.rst Clarify that (1 / specific volume) = density. Co-authored-by: Pablo Brubeck --- demos/multicomponent/multicomponent.py.rst | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demos/multicomponent/multicomponent.py.rst b/demos/multicomponent/multicomponent.py.rst index 21cb9abcfc..9c4109d332 100644 --- a/demos/multicomponent/multicomponent.py.rst +++ b/demos/multicomponent/multicomponent.py.rst @@ -261,7 +261,7 @@ Our implementation of the non-dimensionalised volumetric equation of state c_tot, c_1, c_2 = conc_relation(x_1, x_2) -Moreover, to express that :math:`1 / \rho^{-1} = \sum_{j=1}^n M_j c_j`, we simply take the +Moreover, to express that :math:`1 / \rho^{-1} = \rho = \sum_{j=1}^n M_j c_j`, we simply take the :math:`L^2`-projection of this expression (in non-dimensionalised form):: rho_inv_terms = inner(1.0 / rho_inv, r) * dx From 46ee4c1be4bbf9fde14152c80a160f4dca4d2424 Mon Sep 17 00:00:00 2001 From: Aaron Baier-Reinio <130484015+ABaierReinio@users.noreply.github.com> Date: Sun, 22 Jun 2025 15:03:13 +0100 Subject: [PATCH 19/32] Update demos/multicomponent/multicomponent.py.rst Minor formatting changes Co-authored-by: Pablo Brubeck --- demos/multicomponent/multicomponent.py.rst | 6 ++---- 1 file changed, 2 insertions(+), 4 deletions(-) diff --git a/demos/multicomponent/multicomponent.py.rst b/demos/multicomponent/multicomponent.py.rst index 9c4109d332..31ace5c339 100644 --- a/demos/multicomponent/multicomponent.py.rst +++ b/demos/multicomponent/multicomponent.py.rst @@ -302,10 +302,8 @@ for the :math:`g_i`'s, which in non-dimensionalised form, is implemented as foll # Margules model for chemical potentials, assuming constant partial molar volumes def mu_relation(x_1, x_2, p): - mu_1 = (Me_1 * p) + ln(x_1) \ - + (x_2 ** 2) * (A_12 + (2.0 * (A_21 - A_12) * x_1)) - mu_2 = (Me_2 * p) + ln(x_2) \ - + (x_1 ** 2) * (A_21 + (2.0 * (A_12 - A_21) * x_2)) + mu_1 = (Me_1 * p) + ln(x_1) + (x_2 ** 2) * (A_12 + (2.0 * (A_21 - A_12) * x_1)) + mu_2 = (Me_2 * p) + ln(x_2) + (x_1 ** 2) * (A_21 + (2.0 * (A_12 - A_21) * x_2)) return (mu_1, mu_2) From b53d79f434b2ebdff5e794b81fc1e4a8266dcfdc Mon Sep 17 00:00:00 2001 From: Aaron Baier-Reinio <130484015+ABaierReinio@users.noreply.github.com> Date: Sun, 22 Jun 2025 15:03:36 +0100 Subject: [PATCH 20/32] Update demos/multicomponent/multicomponent.py.rst Minor formatting changes Co-authored-by: Pablo Brubeck --- demos/multicomponent/multicomponent.py.rst | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/demos/multicomponent/multicomponent.py.rst b/demos/multicomponent/multicomponent.py.rst index 31ace5c339..d75ebaace2 100644 --- a/demos/multicomponent/multicomponent.py.rst +++ b/demos/multicomponent/multicomponent.py.rst @@ -381,9 +381,9 @@ This leads to the following implementation:: gamma = Constant(1e-1) # Augmentation parameter, dimensionless # The Stefan--Maxwell diffusion terms - osm_terms = Pe * (1.0 / c_tot) * ((c_2 / (M_1_ND * M_1_ND * c_1)) * inner(J_1, W_1) \ - + (c_1 / (M_2_ND * M_2_ND * c_2)) * inner(J_2, W_2) \ - - (1.0 / (M_1_ND * M_2_ND)) * (inner(J_1, W_2) + inner(J_2, W_1))) * dx + osm_terms = (Pe / c_tot) * ((c_2 / (M_1_ND * M_1_ND * c_1)) * inner(J_1, W_1) + + (c_1 / (M_2_ND * M_2_ND * c_2)) * inner(J_2, W_2) + - (1 / (M_1_ND * M_2_ND)) * (inner(J_1, W_2) + inner(J_2, W_1))) * dx # The augmentation terms (for symmetry we also test these terms against u) osm_terms += Pe * gamma * inner(v - (rho_inv * (J_1 + J_2)), u - (rho_inv * (W_1 + W_2))) * dx From b168413b71f4514a415468513f75a9d3210180cf Mon Sep 17 00:00:00 2001 From: Aaron Baier-Reinio <130484015+ABaierReinio@users.noreply.github.com> Date: Sun, 22 Jun 2025 15:05:29 +0100 Subject: [PATCH 21/32] Update demos/multicomponent/multicomponent.py.rst Minor formatting changes Co-authored-by: Pablo Brubeck --- demos/multicomponent/multicomponent.py.rst | 19 ++++++------------- 1 file changed, 6 insertions(+), 13 deletions(-) diff --git a/demos/multicomponent/multicomponent.py.rst b/demos/multicomponent/multicomponent.py.rst index d75ebaace2..44b54b5134 100644 --- a/demos/multicomponent/multicomponent.py.rst +++ b/demos/multicomponent/multicomponent.py.rst @@ -577,19 +577,12 @@ VTK file using the :code:`time` keyword argument. :: for field, name in enumerate(names): solution.subfunctions[field].rename(name) - mu_1_out = Function(U_h) - mu_2_out = Function(U_h) - rho_out = Function(R_h) - c_tot_out = Function(R_h) - c_1_out = Function(R_h) - c_2_out = Function(R_h) - - mu_1_out.rename("mu_1") - mu_2_out.rename("mu_2") - rho_out.rename("rho") - c_tot_out.rename("c_tot") - c_1_out.rename("c_1") - c_2_out.rename("c_2") + mu_1_out = Function(U_h, name="mu_1") + mu_2_out = Function(U_h, name="mu_2") + rho_out = Function(R_h, name="rho") + c_tot_out = Function(R_h, name="c_tot") + c_1_out = Function(R_h, name="c_1") + c_2_out = Function(R_h, name="c_2") for i in range(n_cont): print(f"Solving for v_ref_1 = {0.4e-6*cont_vals[i]}") From 65a4bccadbaf314fd8bcb4446d022389666535f3 Mon Sep 17 00:00:00 2001 From: Aaron Baier-Reinio <130484015+ABaierReinio@users.noreply.github.com> Date: Sun, 22 Jun 2025 15:06:14 +0100 Subject: [PATCH 22/32] Update demos/multicomponent/multicomponent.py.rst Minor formatting changes Co-authored-by: Pablo Brubeck --- demos/multicomponent/multicomponent.py.rst | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demos/multicomponent/multicomponent.py.rst b/demos/multicomponent/multicomponent.py.rst index 44b54b5134..f665e86951 100644 --- a/demos/multicomponent/multicomponent.py.rst +++ b/demos/multicomponent/multicomponent.py.rst @@ -538,7 +538,7 @@ mathematically valid to do this):: return nodes # Fix the auxiliary chemical potentials and pressure at a point - aux_point = as_vector([4, 0]) # A point on the middle of the outlet + aux_point = (4, 0) # A point on the middle of the outlet aux_point_bcs = [FixAtPointBC(Z_h.sub(3), 0, aux_point), FixAtPointBC(Z_h.sub(4), 0, aux_point), FixAtPointBC(Z_h.sub(5), 0, aux_point)] From b9928c655bc2235ccf6139c9cc2294b6efac1728 Mon Sep 17 00:00:00 2001 From: Aaron Baier-Reinio <130484015+ABaierReinio@users.noreply.github.com> Date: Sun, 22 Jun 2025 15:06:37 +0100 Subject: [PATCH 23/32] Update demos/boussinesq/boussinesq.py.rst Minor formatting changes Co-authored-by: Pablo Brubeck --- demos/boussinesq/boussinesq.py.rst | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demos/boussinesq/boussinesq.py.rst b/demos/boussinesq/boussinesq.py.rst index 35170045ff..ae6cf751b7 100644 --- a/demos/boussinesq/boussinesq.py.rst +++ b/demos/boussinesq/boussinesq.py.rst @@ -199,7 +199,7 @@ implements a boundary condition that fixes a field at a single point. :: We use this to fix the pressure and auxiliary temperature at the origin:: - aux_bcs = [FixAtPointBC(Z.sub(1), 0, as_vector([0, 0])), + aux_bcs = [FixAtPointBC(Z.sub(1), 0, (0, 0)), FixAtPointBC(Z.sub(2), 0, as_vector([0, 0]))] :code:`FixAtPointBC` takes three arguments: the function space to fix, the value with which it From 0414465e3aaf7bd60fb8a625eb580c7289b582d8 Mon Sep 17 00:00:00 2001 From: Aaron Baier-Reinio <130484015+ABaierReinio@users.noreply.github.com> Date: Sun, 22 Jun 2025 15:07:16 +0100 Subject: [PATCH 24/32] Update demos/multicomponent/multicomponent.py.rst Minor wording change Co-authored-by: Pablo Brubeck --- demos/multicomponent/multicomponent.py.rst | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demos/multicomponent/multicomponent.py.rst b/demos/multicomponent/multicomponent.py.rst index f665e86951..1e8ed7aa17 100644 --- a/demos/multicomponent/multicomponent.py.rst +++ b/demos/multicomponent/multicomponent.py.rst @@ -104,7 +104,7 @@ We first define the finite element spaces and trial/test functions:: Z_h = J_h * J_h * V_h * U_h * U_h * P_h * X_h * X_h * R_h * L_h * L_h PETSc.Sys.Print(f"Mesh has {mesh.num_cells()} cells, with {Z_h.dim()} finite element DOFs") - # The trial functions + # The discrete solution solution = Function(Z_h) J_1, J_2, v, mu_aux_1, mu_aux_2, p, x_1, x_2, rho_inv, l_1, l_2 = split(solution) From f7cfe89fa9a0f6e4c32aebef41285125c77e2fb8 Mon Sep 17 00:00:00 2001 From: Aaron Baier-Reinio <130484015+ABaierReinio@users.noreply.github.com> Date: Sun, 22 Jun 2025 15:11:23 +0100 Subject: [PATCH 25/32] Update demos/multicomponent/multicomponent.py.rst Minor formatting changes Co-authored-by: Pablo Brubeck --- demos/multicomponent/multicomponent.py.rst | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/demos/multicomponent/multicomponent.py.rst b/demos/multicomponent/multicomponent.py.rst index 1e8ed7aa17..0faf5321ea 100644 --- a/demos/multicomponent/multicomponent.py.rst +++ b/demos/multicomponent/multicomponent.py.rst @@ -585,8 +585,8 @@ VTK file using the :code:`time` keyword argument. :: c_2_out = Function(R_h, name="c_2") for i in range(n_cont): - print(f"Solving for v_ref_1 = {0.4e-6*cont_vals[i]}") - v_ref_1.assign(Constant(0.4e-6*cont_vals[i])) + v_ref_1.assign(cont_vals[i]) + print(f"Solving for v_ref_1 = {float(v_ref_1)}") v_ref.assign(0.5 * (v_ref_1 + v_ref_2)) NLVS.solve() From 45422acbf6f6224a8c7ca41033e3beabb9908f08 Mon Sep 17 00:00:00 2001 From: Aaron Baier-Reinio <130484015+ABaierReinio@users.noreply.github.com> Date: Sun, 22 Jun 2025 15:11:34 +0100 Subject: [PATCH 26/32] Update demos/multicomponent/multicomponent.py.rst Minor formatting changes Co-authored-by: Pablo Brubeck --- demos/multicomponent/multicomponent.py.rst | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/demos/multicomponent/multicomponent.py.rst b/demos/multicomponent/multicomponent.py.rst index 0faf5321ea..0d6d82004c 100644 --- a/demos/multicomponent/multicomponent.py.rst +++ b/demos/multicomponent/multicomponent.py.rst @@ -569,7 +569,8 @@ and :code:`v_ref` before calling the :code:`solve()` method. Finally, we write e VTK file using the :code:`time` keyword argument. :: outfile = VTKFile("out/solution.pvd") - cont_vals = [1.0, 2.5, 5, 7.5, 10.0] + vmax = 0.4e-5 + cont_vals = [0.1 * vmax, 0.25 * vmax, 0.5 * vmax, 0.75 * vmax, vmax] n_cont = len(cont_vals) names = ["J_1", "J_2", "v", "mu_aux_1", "mu_aux_2", "p", "x_1", "x_2", From e73f614a7a0c17de1f0c56f2c18d595fe4ee23d2 Mon Sep 17 00:00:00 2001 From: Aaron Baier-Reinio <130484015+ABaierReinio@users.noreply.github.com> Date: Sun, 22 Jun 2025 15:14:25 +0100 Subject: [PATCH 27/32] Update demos/multicomponent/multicomponent.py.rst Minor formatting changes Co-authored-by: Pablo Brubeck --- demos/multicomponent/multicomponent.py.rst | 9 +++------ 1 file changed, 3 insertions(+), 6 deletions(-) diff --git a/demos/multicomponent/multicomponent.py.rst b/demos/multicomponent/multicomponent.py.rst index 0d6d82004c..cbbaa93843 100644 --- a/demos/multicomponent/multicomponent.py.rst +++ b/demos/multicomponent/multicomponent.py.rst @@ -443,12 +443,9 @@ unknowns and/or are nonlinear must be implemented with :class:`~.EquationBC` ins rho_v_outflow_bc_func = J_1_outflow_bc_func + J_2_outflow_bc_func # Boundary conditions on the barycentric velocity are enforced via EquationBC - v_inflow_1_bc = EquationBC(inner(v - rho_inv * rho_v_inflow_1_bc_func, u) * ds(*inlet_1_id) == 0, - solution, inlet_1_id, V=Z_h.sub(2)) - v_inflow_2_bc = EquationBC(inner(v - rho_inv * rho_v_inflow_2_bc_func, u) * ds(*inlet_2_id) == 0, - solution, inlet_2_id, V=Z_h.sub(2)) - v_outflow_bc = EquationBC(inner(v - rho_inv * rho_v_outflow_bc_func, u) * ds(*outlet_id) == 0, solution, - outlet_id, V=Z_h.sub(2)) + bc_data = {inlet_1_id: rho_v_inflow_1_bc_func, inlet_2_id: rho_v_inflow_2_bc_func, outlet_id: rho_v_outflow_bc_func} + F_bc = sum(inner(v - rho_inv * flux, u) * ds(*subdomain) for subdomain, flux in bc_data.items()) + v_bc = EquationBC(F_bc == 0, solution, (*inlet_1_id, *inlet_2_id, *outlet_id), V=Z_h.sub(2)) # The boundary conditions on the fluxes and barycentric velocity # Note that BCs on H(div) spaces only apply to the normal component From 6a1a3b2b224fa29c8c66c96c6ca816047707e5d7 Mon Sep 17 00:00:00 2001 From: Aaron Baier-Reinio <130484015+ABaierReinio@users.noreply.github.com> Date: Sun, 22 Jun 2025 15:14:39 +0100 Subject: [PATCH 28/32] Update demos/multicomponent/multicomponent.py.rst Minor formatting changes Co-authored-by: Pablo Brubeck --- demos/multicomponent/multicomponent.py.rst | 4 +--- 1 file changed, 1 insertion(+), 3 deletions(-) diff --git a/demos/multicomponent/multicomponent.py.rst b/demos/multicomponent/multicomponent.py.rst index cbbaa93843..43b5dfb93e 100644 --- a/demos/multicomponent/multicomponent.py.rst +++ b/demos/multicomponent/multicomponent.py.rst @@ -457,9 +457,7 @@ unknowns and/or are nonlinear must be implemented with :class:`~.EquationBC` ins DirichletBC(Z_h.sub(1), J_2_outflow_bc_func, outlet_id), DirichletBC(Z_h.sub(1), 0, inlet_1_id), DirichletBC(Z_h.sub(1), 0, walls_ids), - v_inflow_1_bc, - v_inflow_2_bc, - v_outflow_bc, + v_bc, DirichletBC(Z_h.sub(2), 0, walls_ids)] It is now natural to assign :math:`v^\textrm{ref}` From 38a6ef9f1461a850c9dccd7cfd4fa52857a9ad3b Mon Sep 17 00:00:00 2001 From: ABaierReinio Date: Sun, 22 Jun 2025 16:32:20 +0100 Subject: [PATCH 29/32] Fix description of BCs --- demos/multicomponent/multicomponent.py.rst | 32 ++++++++++++---------- 1 file changed, 17 insertions(+), 15 deletions(-) diff --git a/demos/multicomponent/multicomponent.py.rst b/demos/multicomponent/multicomponent.py.rst index 43b5dfb93e..383462f667 100644 --- a/demos/multicomponent/multicomponent.py.rst +++ b/demos/multicomponent/multicomponent.py.rst @@ -418,12 +418,14 @@ Boundary conditions ~~~~~~~~~~~~~~~~~~~ Let :math:`N` denote the outward unit normal on :math:`\partial \Omega`. -We enforce parabolic profiles on :math:`J_i \cdot N` at inflow :math:`i` and on the outflow. -The magnitudes of the parabolic profiles are :math:`M_i c_i^\text{ref} v_i^\text{ref}` where +On inflow :math:`i` and on the outflow we introduce parabolic mass flux profiles :math:`G_i`, +with magnitudes :math:`M_i c_i^\text{ref} v_i^\text{ref}` where :math:`v_i^\text{ref}` are reference velocities that we are free to choose. +We then strongly enforce :math:`J_i \cdot N = G_i \cdot N` as Dirichlet boundary conditions +only apply to the normal component of H(div) functions. Elsewhere on the boundary we enforce :math:`J_i \cdot N = 0`. Finally, instead of specifying -the value of the barycentric velocity :math:`v` on the inflows and outflows, -we enforce :math:`v = \rho^{-1}((J_1 + J_2)\cdot N)N`. Boundary conditions that couple +the value of the barycentric velocity :math:`v` on the inflows and outflows, +we enforce :math:`v = \rho^{-1}(G_1 + G_2)`. Boundary conditions that couple unknowns and/or are nonlinear must be implemented with :class:`~.EquationBC` instead of :class:`~.DirichletBC`. :: # Reference species velocities, which we choose to symmetrize so that the molar fluxes agree @@ -432,15 +434,15 @@ unknowns and/or are nonlinear must be implemented with :class:`~.EquationBC` ins parabola_inflow_1 = 2.0 * x_sc * (x_sc - 1.0) * as_vector([2.0, -1.0]) parabola_inflow_2 = 2.0 * x_sc * (x_sc - 1.0) * as_vector([2.0, 1.0]) - J_1_inflow_bc_func = -M_1_ND * (v_ref_1 / v_ref) * (c_pure_1 / c_ref) * parabola_inflow_1 - J_2_inflow_bc_func = -M_2_ND * (v_ref_2 / v_ref) * (c_pure_2 / c_ref) * parabola_inflow_2 - rho_v_inflow_1_bc_func = J_1_inflow_bc_func - rho_v_inflow_2_bc_func = J_2_inflow_bc_func + G_1_inflow_bc_func = -M_1_ND * (v_ref_1 / v_ref) * (c_pure_1 / c_ref) * parabola_inflow_1 + G_2_inflow_bc_func = -M_2_ND * (v_ref_2 / v_ref) * (c_pure_2 / c_ref) * parabola_inflow_2 + rho_v_inflow_1_bc_func = G_1_inflow_bc_func + rho_v_inflow_2_bc_func = G_2_inflow_bc_func parabola_outflow = 2.0 * (y_sc + 0.5) * (y_sc - 0.5) * as_vector([1.0, 0.0]) - J_1_outflow_bc_func = -M_1_ND * (v_ref_1 / v_ref) * (c_pure_1 / c_ref) * parabola_outflow - J_2_outflow_bc_func = -M_2_ND * (v_ref_2 / v_ref) * (c_pure_2 / c_ref) * parabola_outflow - rho_v_outflow_bc_func = J_1_outflow_bc_func + J_2_outflow_bc_func + G_1_outflow_bc_func = -M_1_ND * (v_ref_1 / v_ref) * (c_pure_1 / c_ref) * parabola_outflow + G_2_outflow_bc_func = -M_2_ND * (v_ref_2 / v_ref) * (c_pure_2 / c_ref) * parabola_outflow + rho_v_outflow_bc_func = G_1_outflow_bc_func + G_2_outflow_bc_func # Boundary conditions on the barycentric velocity are enforced via EquationBC bc_data = {inlet_1_id: rho_v_inflow_1_bc_func, inlet_2_id: rho_v_inflow_2_bc_func, outlet_id: rho_v_outflow_bc_func} @@ -449,12 +451,12 @@ unknowns and/or are nonlinear must be implemented with :class:`~.EquationBC` ins # The boundary conditions on the fluxes and barycentric velocity # Note that BCs on H(div) spaces only apply to the normal component - flux_bcs = [DirichletBC(Z_h.sub(0), J_1_inflow_bc_func, inlet_1_id), - DirichletBC(Z_h.sub(0), J_1_outflow_bc_func, outlet_id), + flux_bcs = [DirichletBC(Z_h.sub(0), G_1_inflow_bc_func, inlet_1_id), + DirichletBC(Z_h.sub(0), G_1_outflow_bc_func, outlet_id), DirichletBC(Z_h.sub(0), 0, inlet_2_id), DirichletBC(Z_h.sub(0), 0, walls_ids), - DirichletBC(Z_h.sub(1), J_2_inflow_bc_func, inlet_2_id), - DirichletBC(Z_h.sub(1), J_2_outflow_bc_func, outlet_id), + DirichletBC(Z_h.sub(1), G_2_inflow_bc_func, inlet_2_id), + DirichletBC(Z_h.sub(1), G_2_outflow_bc_func, outlet_id), DirichletBC(Z_h.sub(1), 0, inlet_1_id), DirichletBC(Z_h.sub(1), 0, walls_ids), v_bc, From 24c356d9952c24d02ea080e5b56f7eb3096b4fbf Mon Sep 17 00:00:00 2001 From: ABaierReinio Date: Sun, 29 Jun 2025 13:35:44 +0100 Subject: [PATCH 30/32] Use mathrsfs when building doc PDF --- docs/source/conf.py | 1 + 1 file changed, 1 insertion(+) diff --git a/docs/source/conf.py b/docs/source/conf.py index dfb14aba20..c421aa8a8e 100644 --- a/docs/source/conf.py +++ b/docs/source/conf.py @@ -274,6 +274,7 @@ # Additional stuff for the LaTeX preamble. 'preamble': r'''\newcommand{\require}[1]{} \usepackage{cancel} +\usepackage{mathrsfs} % \sphinxDUC{2730}{\ding{80}} \sphinxDUC{27F6}{$\rightarrow$} From 7278e918aecdeb697ebaa131084b440407d5c150 Mon Sep 17 00:00:00 2001 From: ABaierReinio Date: Sun, 29 Jun 2025 13:37:03 +0100 Subject: [PATCH 31/32] Store boundary ids as tuples so they can be used as dictionary keys --- demos/multicomponent/multicomponent.py.rst | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/demos/multicomponent/multicomponent.py.rst b/demos/multicomponent/multicomponent.py.rst index 383462f667..241dfaca9c 100644 --- a/demos/multicomponent/multicomponent.py.rst +++ b/demos/multicomponent/multicomponent.py.rst @@ -57,10 +57,10 @@ We use :doc:`netgen ` to build a curved mesh of order :math:`k=3 mesh = Mesh(Mesh(ngmesh).curve_field(k)) # Get the IDs of the inlets, outlet and walls - inlet_1_id = [i+1 for i, name in enumerate(ngmesh.GetRegionNames(dim=1)) if name == "inlet_1"] - inlet_2_id = [i+1 for i, name in enumerate(ngmesh.GetRegionNames(dim=1)) if name == "inlet_2"] - outlet_id = [i+1 for i, name in enumerate(ngmesh.GetRegionNames(dim=1)) if name == "outlet"] - walls_ids = [i+1 for i, name in enumerate(ngmesh.GetRegionNames(dim=1)) if name == ""] + inlet_1_id = tuple([i+1 for i, name in enumerate(ngmesh.GetRegionNames(dim=1)) if name == "inlet_1"]) + inlet_2_id = tuple([i+1 for i, name in enumerate(ngmesh.GetRegionNames(dim=1)) if name == "inlet_2"]) + outlet_id = tuple([i+1 for i, name in enumerate(ngmesh.GetRegionNames(dim=1)) if name == "outlet"]) + walls_ids = tuple([i+1 for i, name in enumerate(ngmesh.GetRegionNames(dim=1)) if name == ""]) # Define the surface and volume measures using a fixed quadrature degree deg_max = 3*k From 73645304cd0c88fe75bc2406137ccefdaf19fd27 Mon Sep 17 00:00:00 2001 From: ABaierReinio Date: Sun, 29 Jun 2025 13:37:15 +0100 Subject: [PATCH 32/32] Include runnable python script --- demos/multicomponent/multicomponent.py.rst | 3 +++ 1 file changed, 3 insertions(+) diff --git a/demos/multicomponent/multicomponent.py.rst b/demos/multicomponent/multicomponent.py.rst index 241dfaca9c..eaf7a28e26 100644 --- a/demos/multicomponent/multicomponent.py.rst +++ b/demos/multicomponent/multicomponent.py.rst @@ -609,6 +609,9 @@ even in the presence of low species concentrations and sharp solution gradients. | :width: 100% | :width: 100% | +---------------------------+---------------------------+ +A Python script version of this demo can be found :demo:`here +`. + .. rubric:: References .. bibliography:: demo_references.bib

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