The Feel++ software

automation, code generation, applications

Christophe Prud'homme -- University of Strasbourg

slides.feelpp.org

Team

Christophe (UNISTRA)

Joubine (UNISTRA)

Vincent (UNISTRA)

Thibaut (INRIA)

Marcela (UPARIS)

Thomas (UNISTRA)

Zohra (UNISTRA)

Abdoulaye (USTTB)

Luca (UNISTRA)

Céline (UNISTRA)

Christophe (CNRS/LNCMI)

Jeremie (CNRS/LNCMI)

Laetitia (INRIA)

Zakaria (INRIA)

Feel++ Suite

What is Feel++ ?

Overview

Framework to solve problems based on ODE and PDE
C++17 (WIP C++20) with a Python layer using Pybind11
Seamless parallelism with default communicator
DevOps: CI/CD (docker + ubuntu/debian packaging)
Tests: Hundreds of tests run in sequential and parallel C++ and Python
Usage: Research, R&D, Teaching, Services

Applications

Health(brain)

Health(Tumor cells)

Industry (ROM,UQ)

Health(Eye/Brain)

Health(Tomography)

Automotive(CFD,ROM)

Health(Rheology)

Physics(High Field Magnets)

Physics(Delectometry)

Health(Micro swimmers)

Engineering (Buildings)

Health(Eye/Brain)

Feel++ Core Library

Laplacian

auto Vh = Pch<4>( mesh, markedelements(mesh, expr("<...>")) );
auto u = Vh->element(), v = Vh->element( g, "g" );
auto l = form1( _test = Vh );
l = integrate( _range = elements( support( Vh ) ),
               _expr = f * id( v ) );
l += integrate( _range = markedfaces( support( Vh ), "Robin" ), _expr = -r_2 * id( v ) );
l += integrate( _range = markedfaces( support( Vh ), "Neumann" ), _expr = -un * id( v ) );

auto a = form2( _trial = Vh, _test = Vh );
a = integrate( _range = elements( support( Vh ) ),
               _expr = inner( k * gradt( u ), grad( v ) ) );
a += integrate( _range = markedfaces( support( Vh ), "Robin" ), _expr = r_1*idt(u)*id(v));
a += on( _range = markedfaces( support(Vh), "Dirichlet" ), _rhs=l, _element=u, _expr = g );
a.solve( _rhs = l, _solution = u );

General Features

A large range of numerical methods to solve PDEs: cG, dG, hdG, rb/mor, …
0D+t, 1D(+t), 2D(+t), 3D(+t)
DSEL for Galerkin methods in C++

Feel++ Recent Developments

Some recent features

// Named Arguments
auto Xh=Pch<1>(_mesh=mesh); (1)
auto u = Xh->element();
// expression handling
auto e3 = expr( "2*x*u*v:x:u:v" ); (2)
auto e3v = expr( e3, symbolExpr( "u", dxv( u ) ), symbolExpr( "v", Nx() ) );
// remeshing in seq and //
auto rm = remesh(_mesh=mesh,_params=<remeshing specs in json>); (3)
// Fast marching : compute distance to range
auto distToBoundary = distanceToRange( _space=Vh, _range=boundaryfaces( mesh ) ); (4)
auto distToBoundaryNarrowBand = (5)
   distanceToRange( _space=Vh, _range=boundaryfaces( mesh ),
                    _max_distance=3.*mesh->hAverage() );

1	Expressivity: using our own named arguments library
2	Expressions handling: based on Ginac, handles automatic differentiation, JIT compilation
3	Remeshing in seq and parallel
4	Versatile levelset/fast marching framework
5	Narrow band FMM

Expression framework and Automatic differentiation

auto mesh = loadMesh<Mesh<Simplex<3,1>>();
auto Vh = Pch<2>( mesh ); (1)
auto u = Vh->element( inner(P()) ); (2)
auto e1 = expr( "u*u:u"); (3)
auto e2 = expr( "2*e1:e1");
auto e3base = expr( "3*e2:e2");
// handle composition
auto e3 = expr( e3base, symbolExpr("e1",e1), symbolExpr("e2",e2), symbolExpr("u",inner(P()) )); (4)
auto grad_e3 = grad<3>( e3 ); (5)
auto diff_e3_x_exact = 3*4*inner(P())*2*Px();
auto diff_e3_y_exact = 3*4*inner(P())*2*Py();
auto diff_e3_z_exact = 3*4*inner(P())*2*Pz();
auto grad_e3_exact = trans(vec(diff_e3_x_exact,diff_e3_y_exact,diff_e3_z_exact));
// compute error (machine error expected)
double error_grad_e3 = normL2(_range=elements(mesh),_expr= grad_e3 - grad_e3_exact ); (6)

1	create \(P^2_{c,h}\)
2	create an element of Vh such that \(u(x)=x^T x\)
3	create expressions
4	define symbols in expressions
5	differentiate the expressions
6	compute \(L^2\) error norm

Feel++ Recent Developments:: Python

Feel++ in Python

import feelpp
from feelpp.operators import *

mesh= feelpp.load(m, mesh_name, 0.1)
Xh = feelpp.functionSpace(mesh=mesh)
v=Xh.element()
v.on(range=feelpp.elements(mesh), expr=feelpp.expr("1"))
e_meas = mesh.measure()
M=mass(test=Xh,trial=Xh,range=feelpp.elements(mesh))
assert(abs(M.energy(v,v)-e_meas)<1e-10)
S=stiffness(test=Xh,trial=Xh,range=feelpp.elements(mesh))
assert(S.energy(v,v)<1e-10)
v.on(range=feelpp.elements(mesh), expr=feelpp.expr("x+y:x:y"))
assert(abs(S.energy(v,v)-2*e_meas)<1e-10)

Other Features

WIP: DSEL for Galerkin methods in Python like in C++
WIP: MOR integration in complex workflows
Parallel execution of toolboxes
- Advanced parametric studies including UQ (see MS57)
- Reinforcement learning for micro-swimming (see MS96)
- Ensemble kalman filter in applications with sensors

Feel++ Toolboxes in Python

import feelpp
from feelpp.toolboxes.core import *
from feelpp.toolboxes.fluid import *
from feelpp.toolboxes.heat import *

def simulate(toolbox, export=True, buildModelAlgebraicFactory=True,data=None):
    toolbox.init(buildModelAlgebraicFactory)
    #toolbox.printAndSaveInfo()
    if toolbox.isStationary():
        toolbox.solve()
        if export:
            toolbox.exportResults()
    else:
        if not toolbox.doRestart():
            toolbox.exportResults(toolbox.timeInitial())
        toolbox.startTimeStep()
        while not toolbox.timeStepBase().isFinished():
            if feelpp.Environment.isMasterRank():
                print("time simulation: {}s/{}s with step: {}".format(toolbox.time(),toolbox.timeFinal(),toolbox.timeStep()))
            toolbox.solve()
            if export:
                toolbox.exportResults()
            toolbox.updateTimeStep()
    return not toolbox.checkResults()

ht = heat(dim=3,order=1)
ok=simulate(ht)
meas = ht.postProcessMeasures().values()
df=pd.DataFrame([meas])
# do something with the pandas dataframe
....

cfd=fluid(dim=3,order=2)
ok=simulate(cfd)
....

Toolboxes Configuration and Recent Developments

Toolboxes :: Setup

JSON file structure

Models and physical properties
Parameters
Mesh
Material properties
Boundary/Initial conditions
Post-processing
Checks (testsuite,benchmarks,…)

HifiMagnets (MS57)

Design High Field Magnets up to \(37T\)
Automated generation linked to material database using templating
Full design simulation toolchain generation (2DAxi and 3D)

Plane channel turbulence testcase (Spalart-Allmaras) - Github

{ "Name": "turbulence-plane-channel",
  "ShortName": "turbulence-plane-channel",
  "Models": {
    "fluid": {
      "equations": "Navier-Stokes",
      "turbulence": {
        "enable": 1,
        "model": "Spalart-Allmaras"}}},
  "Parameters": {
    "umax": 15.6,
    "tfix": 0.01,
    "chi": "t<tfix:t:tfix",
    "umax_inlet": "umax:umax",
    "H": 0.0635,
    "Re": 32000},
  "Materials": {
    "Omega": {
      "rho": "1",
      "mu": "(rho*umax*(H/2))/Re:rho:umax:H:Re"}},
  "BoundaryConditions": {
    "velocity": {
      "Dirichlet": {
        "Gamma4": {
          "expr": "{ umax_inlet*(2*fluid_dist2wall/H)^(1/7),0 }:umax_inlet:fluid_dist2wall:H",
          "turbulence_bc": "inlet"},
        "walls": {
          "markers": ["Gamma1","Gamma3"],
          "expr": "{0,0}",
          "turbulence_bc": "wall"}}},
    "pressure": {
      "Dirichlet": {
        "Gamma2": {
          "expr": "1.2"   }}},
    "AAfluid": {
      "outlet": {
        "outlet": {
          "expr": "0"}}}},
  "PostProcess": {
    "Exports": {
      "fields": ["velocity","pressure"],
      "expr": {
        "dist2wall": "fluid_dist2wall:fluid_dist2wall",
        "mu_t": "materials_mu_t:materials_mu_t",
        "curl_U_magnitude": "fluid_curl_U_magnitude:fluid_curl_U_magnitude",
        "SA_nu": "fluid_turbulence_SA_nu:fluid_turbulence_SA_nu" }}}}

Toolboxes :: PostProcessing

Postprocessing Features

Compute quantities based on expressions
Export data/results to visualisation software
Statistics : mean, max, min, integrals
Norms (w/ range support) : \(\|\cdot\|_{L^2}, \|\cdot\|_{H^1}, \|\nabla(\cdot)\|_{L^2}\)
Evaluation at points, cut lines and cut planes
Specific toolbox outputs: eg. flow rates, stresses, fluxes…

PostProcessing Example

"PostProcess": { "heat" {
    "Exports": {
        "fields":["temperature","pid"]
    },
    "Measures": {
        "Normal-Heat-Flux": {
            "%1%": {
                "markers":"%1%",
                "direction":"outward",//"inward",
                "index1":["Interior_wall","Exterior_wall"] } },
        "Statistics": {
            "temperature_%1%": {
                "type":["min","max"],
                "field":"temperature",
                "markers":"%1%",
            "index1":["Interior_wall","Exterior_wall"] }}
        },
    "tolerance":1e-1
    }
}

Falling disk in parallel

Recent developments and features

Core: Mesh adaptation (event based)
Core: Repartitioning
Fluid: Rigid and Elastic moving bodies
All Toolboxes: Plugin system in C++ and Python

WIP

Time discretisation: parareal with possibility to accelerate with MOR
Scenarii: setup complex workflows in C++ (already doable in Python)

Generic Toolbox :: Coefficient Form PDEs

Find scalar or vector fields \(u_i,i=1...,N\) such that

\[\begin{aligned} d_i \frac{\partial u_i}{\partial t} + \nabla \cdot \left( -c_i \nabla u_i - \alpha u_i + \gamma_i \right)& \\ + \beta_i \cdot \nabla u_i + a u_i &= f \quad \text{ in } \Omega \end{aligned}\]

Coefficients can depend on the unknowns \(u_j,j=1 \ldots N\).
Automatic differentiation built on top of Feel++ expression handling
Various schemes based on Newton or Picard

Predator-Prey coupled model [Github]

"Models": {
    "cfpdes":{ "equations":["equation1","equation2"] },
    "equation1":{
        "setup":{
            "unknown":{"basis":"Pch1","name":"prey","symbol":"u"},
            "coefficients":{
                "c":"1", // diffusion
                "a":"-( (1-equation1_u) - equation2_v/(equation1_u+thealpha) ):thealpha:equation1_u:equation2_v", // life and death
                "d":"1" // time derivative
            } } },
    "equation2":{
        "setup":{
            "unknown":{ "basis":"Pch1", "name":"predator", "symbol":"v" },
            "coefficients":{
                "c":"thedelta:thedelta", // diffusion
                "a":"-( (thebeta*equation1_u)/(equation1_u+thealpha) - thegamma ):thebeta:thealpha:thegamma:equation1_u", // life and death
                "d":"1"// time derivative
            } } } }
    ...

Cahn Hilliard

Cahn-Hilliard in L-Shape domain \(6e-3s\)

ch2d

\[\begin{aligned} f(u)=u^2(u^2-1),\, \lambda=10^{-4} \end{aligned}\]

Phase separation modeling

\[\begin{aligned} \partial_t u-\nabla \cdot M\left(\nabla\left(f'(u)-\lambda \Delta u\right)\right)&=0 \quad \text { in } \Omega \text {, }\\ M\nabla\left(f'(u)-\lambda \Delta u\right)&=0 \text { on } \partial \Omega,\\ M \lambda \nabla u \cdot n&=0 \quad \text { on } \partial \Omega . \end{aligned}\]

Mixed Form

\[\begin{array}{rl} \frac{\partial u}{\partial t}-\nabla \cdot M \nabla \mu &=0 \quad \text { in } \Omega, \\ \mu-f'(u)+\lambda \Delta u&=0 \text { in } \Omega . \end{array}\]

Phase separation modeling

"Models":
{
  "cfpdes":{ "equations":["equation1","equation2"] },
  "equation1":{
    "setup":{
        "unknown":{"basis":"Pch1","name":"c","symbol":"c"},
        "coefficients":{
           "d": "1",
           "gamma": "{-equation2_grad_mu_0,-equation2_grad_mu_1,-equation2_grad_mu_2}"
  }}},
  "equation2":{
    "setup":{
        "unknown":{"basis":"Pch1","name":"mu","symbol":"mu"},
        "coefficients":{
           "gamma":"{lambda*equation1_grad_c_0,lambda*equation1_grad_c_1, lambda*equation1_grad_c_2}",
           "a":"1",
           "f": "equation1_c^2*(equation1_c^2 - 1)"
  } } }
}

Model equations

\[\begin{aligned} -\frac{1}{\mu}\Delta a_\theta + \frac{1}{\mu r^2}a_\theta + \sigma \frac{\partial a_\theta}{\partial t} =0 &\text{ in } \Omega \\ a_\theta = \frac{r}{2}\text{B}_\text{applied} &\text{ on } \partial\Omega \end{aligned}\]

Non-Linear Part (e-j power law):

\[\sigma=\begin{cases}\frac{j_c}{e_c}\left(\frac{||-\partial a_\theta / \partial t||}{e_c}\right)^{(1-n)/n} &\text{in Superconductor}\\ 0 &\text{in Air} \end{cases}\]

feelpp example supra

Bulk Superconductor Model

"Models":{
    "cfpdes":{ "equations":"magnetic" },
    "magnetic":{
        "common":{"setup":{"unknown":{
            "basis":"Pch1",
            "name":"Atheta","
            symbol":"Atheta"
        }}},
        "models":[{
            "name":"magnetic_Conductor",
            "materials":"Conductor",
            "setup":{"coefficients":{
                    "c":"x/mu:x:mu",
                    "a":"1/mu/x:mu:x",
                    "d":"materials_Conductor_sigma*x
                         :materials_Conductor_sigma:x"}}
        },{
            "name":"magnetic_Air",
            "materials":"Air",
            "setup":{"coefficients":{
                    "c":"x/mu:x:mu",
                    "a":"1/mu/x:mu:x"}}}
        ]
    }
}
...

Toolbox:: Hybridized Discontinuous Galerkin

Second order elliptic problems

Darcy

\[\begin{aligned} \mathbf{j} + \mathcal{K} \nabla p &= 0 & \text{ in } \Omega \\ \nabla \cdot \mathbf{j} &= f & \text{ in } \Omega \\ p &= 0 & \text{ on } \Gamma_D \\ \mathbf{j} \cdot \mathbf{n} &= g_N & \text{ on } \Gamma_N \end{aligned}\]

\(p\) is the potential
\(\mathbf{j}\) is the flux

Elasticity, \(d=2,3\)

\[\begin{aligned} \mathcal{A} \underline{\boldsymbol{\sigma}}-\underline{\boldsymbol{\epsilon}}(\boldsymbol{u})&=0 \quad \text { in } \Omega \subset \mathbb{R}^{d},\\ \nabla \cdot \underline{\boldsymbol{\sigma}}&=\boldsymbol{f} \quad \text { in } \Omega,\\ \boldsymbol{u}&=\boldsymbol{g} \text { on } \partial \Omega . \end{aligned}\]

\(\boldsymbol{u}: \Omega \rightarrow \mathbb{R}^{3}\) displacement
\(\underline{\boldsymbol{\epsilon}}(\boldsymbol{u}):=\frac{1}{2}\left(\nabla \boldsymbol{u}+(\nabla \boldsymbol{u})^{\top}\right)\) strain tensor
\(\underline{\boldsymbol{\sigma}}: \Omega \rightarrow S\) stress tensor where \(S\) is the set of all symmetric matrices in \(\mathbb{R}^{d\times d}\)

Methodology: HDG

Provides optimal approximation of both primal and flux/stress variables \(p/\boldsymbol{u}\) and \(\mathbf{j}/\underline{\boldsymbol{\sigma}}\) respectively;
Requires less globally coupled degrees of freedom than DG methods of comparable accuracy;
Allows local element-by-element postprocessing to obtain new approximations with enhanced accuracy and conservation properties

Some Applications

Electrostatic
Heat transfer
Flow in porous media
Elasticity and Poro-Elasticity

The integral boundary condition

Electric potential in a nanoscale floating gate nMOS

Magnetic field in a high field magnet (\(36T\))

Tissue perfusion in the Lamina Cribrosa

Feature: Integral Boundary Condition (IBC)

\[\begin{aligned} \int_{\Gamma_{ibc}} \mathbf{j} \cdot \mathbf{n} &= I_{target}\\ p&=\text{constant but unknown} \end{aligned}\]

A HDG method for elliptic problems with integral boundary condition: Theory and Applications, Silvia Bertoluzza,Giovanna Guidoboni,Romain Hild,Daniele Prada,Christophe Prud’homme,Riccardo Sacco,Lorenzo Sala,Marcela Szopos, 2021 submitted

HDG Laplacian: formulation

Notations

\(\mathcal{T}_h\) the collection of elements \(K\) such that \(\Omega = \bigcup_{K\in \mathcal{T}_h} K\).
\(h:= max_{K \in \mathcal{T}_h} h_K\), \(\partial K\) the boundary of \(K\) with its measure \(|F|\)
\(\mathbf{n}_{\partial K}\) is the associated unit outward normal vector.
The skeleton of \(\mathcal{T}_h\) is the collection of all the faces of \(\mathcal{T}_h\) into the set \(\mathcal{F}_h\).
\(\mathcal{F}_h = \mathcal{F}_h^\Gamma \cup \mathcal{F}_h^0, \; \mathcal{F}_h^\Gamma = \mathcal{F}_h^D \cup \mathcal{F}_h^N \cup \mathcal{F}_h^{ibc}\)

Function Spaces

\(V_h = \Pi_{K \in \mathcal{T}_h} V(K), \qquad V(K) = \left[ P_k (K) \right]^n\)
\(W_h = \Pi_{K\in\mathcal{T}_h} W(K), \qquad W(K) = \left[ P_k (K) \right]\)
\(\widetilde M_h = \{ \mu \in L^2(\mathcal{F}_h): \mu\rvert_F \in P_k(F) \; \forall F \in \mathcal{F}_h \setminus \mathcal{F}_h^{ibc} \},\)
\(M^*_h = \{ \mu \in C^0(\mathcal{F}^{ibc}_h): \mu\rvert_F \in P_0(F) \; \forall F \in \mathcal{F}_h^{ibc} \} \cong \mathbb{R}\)
\(M_h=\widetilde M_h \oplus M^*_h\)

HDG Laplacian: formulation

Discrete formulation Find \(\boldsymbol{j}_h \in V_h, \; p_h \in W_h\) and \(\hat{p}_h \in M_h\) such that:

\[\begin{aligned} \sum_{K\in\mathcal{T}_h} \left[ \left( \mathcal{K}^{-1} \boldsymbol{j}^K_h, \boldsymbol{v}_h^K \right)_K - \left( p_h^K, \nabla\cdot\boldsymbol{v}_h^K \right)_K + \langle \hat{p}_h, \boldsymbol{v}_h^K \cdot {\boldsymbol{n}}_{\partial K}\rangle_{\partial K} \right] = 0 && \forall \boldsymbol{v}_h \in V_h \\ \sum_{K\in\mathcal{T}_h} \left[ -\left(\boldsymbol{j}_h^K,\nabla w_h^K \right)_K + \langle \hat{\boldsymbol{j}}_h^{\partial K}\cdot {\boldsymbol{n}}_{\partial K}, w_k^K \rangle_{\partial K} \right] = \sum_{K\in\mathcal{T}_h} \left( f, w_h^K \right)_K && \forall w_h \in W_h \\ \sum_{K\in\mathcal{T}_h} \langle \hat{\boldsymbol{j}}_h^{\partial K} \cdot {\boldsymbol{n}}_{\partial K}, \mu_h \rangle_{\partial K} = \langle g_N, \mu_h \rangle_{\Gamma_N} + I_{target} |\Gamma_{ibc}|^{-1} \langle\mu_h, 1\rangle_{\Gamma_{ibc}} && \forall \mu_h \in M_h\\ \hat{\mathbf j}_h^{\partial K} = \mathbf j_h^K + \tau_K(p_h^K - \hat{p_h}^{\partial K})\mathbf n && \forall\partial K, \end{aligned}\]

HDG Laplacian

Meshes

auto mesh=loadMesh(_mesh=new Mesh<Simplex<3>>); (1)
auto complement_integral_bdy = complement(faces(mesh), (2)
  [&mesh]( auto const& e ) {
    if ( e.hasMarker() &&
         e.marker().matches(mesh->markerName("Ibc*") ) )
      return true;
    return false;
   });
auto face_mesh = createSubmesh( mesh, complement_integral_bdy); (3)
auto ibc_mesh = createSubmesh( mesh, markedfaces(mesh,"Ibc*")); (4)

1	load mesh
2	build set of non ibc facets
3	\(\mathcal{F}_h^{ibc}\) and
4	\(\mathcal{F}_h\setminus\mathcal{F}_h^{ibc}\).

Function Spaces

Vh_ptr_t Vh = Pdhv<OrderP>( mesh); (1)
Wh_ptr_t Wh = Pdh<OrderP>( mesh );
Mh_ptr_t Mh = Pdh<OrderP>( face_mesh );
// only one degree of freedom
Ch_ptr_t Ch = Pch<0>(ibc_mesh );
// $n$ IBC
auto ibcSpaces = product( nb_ibc, Ch); (2)
auto Xh = product( Vh, Wh, Mh. ibcSpaces  ); (3)

1	create the spaces \(V_h,W_h,\tilde{M}_h\) and \(M_h^*\).
2	handle arbirary number of IBCs
3	initialize spaces and product space

\[V_h\times W_h\times \tilde{M}_h\times (M_h^*)^n\]

HDG laplacian

Construction of algebraic system

auto a = blockform2( Xh )
auto rhs = blockform1( Xh );

. . .
// Assembling the right hand side
rhs(1_c) += integrate(_range=elements(mesh),_expr=-f*id(w));
. . .
// Assembling the main matrix
a(0_c,0_c) += integrate(_range=elements(mesh),
                        _expr=(trans(lambda*idt(u))*id(v)) );
. . .
//$\langle \hat{p}_h\rvert_{\tilde{M}_h}, \boldsymbol{v}_h^K \cdot {\boldsymbol{n}}_{\partial K}\rangle$ $$
a(0_c,2_c) += integrate(_range=internalfaces(mesh),
         _expr=( idt(phat)*(leftface(trans(id(v))*N())+
                rightface(trans(id(v))*N()))));

solving and postprocessing

a( 3_c, 0_c, i, 0 ) +=
   integrate( _range=markedfaces(mesh,"Ibc"),
             _expr=(trans(idt(u))*N()) * id(nu) );
auto U = Xh.element();
a.solve(_solution=U, _rhs=rhs, _name="hdg");
auto up = U(0_c);
auto pp = U(1_c);
auto phat = U(2_c);
auto ip = U(3_c,0);

// postprocessing
auto Whp = Pdh<OrderP+1>( mesh );
auto pps = product( Whp );
auto PP = pps.element();
auto ppp = PP(0_c);
auto b = blockform2( pps, solve::strategy::local, backend() );
b( 0_c, 0_c ) = integrate( _range=elements(mesh), _expr=inner(gradt(ppp),grad(ppp)));
auto ell = blockform1( pps, solve::strategy::local, backend() );
ell(0_c) = integrate( _range=elements(mesh), _expr=-lambda*grad(ppp)*idv(up));
b.solve( _solution=PP, _rhs=ell, _name="sc.post", _local=true);
ppp=PP(0_c);
ppp += -ppp.ewiseMean(P0dh)+pp.ewiseMean(P0dh);

Toolbox HDG

Similar to CFPDEs, except that only one equation for now
- time dependence
- Other terms in the PDEs
- non-linear coefficients
IBCs
- arbitrary number
- Coupling with 0D+t models using FMU
  - time splitting approach to avoid iterating
Extended to PoroElasticity
WIP: HHO support
WIP: Order reduction (RB and NiRB)
Future: merge with CFPDEs

High Field Magnets resistive / superconductors magnets

Ocular Mathematical Virtual Simulator (OMVS)

Final Words

Develop Feel++ as a service
Streamline complex workflows for advanced usage/studies
- data assimilation,
- machine learning,
- UQ on complex models
- …
Go Digital twins
- Building energy modeling
- High field magnets
- Eye

Links