Robin boundary conditions across applications¶

Scalar diffusion uses kappa * grad(u) . n + beta*u = g. Solid displacement uses stress(u)*n + beta*u = g. Incompressible flow uses (2*nu*epsilon(u) - p*I)*n + beta*u = g. The coefficient is scalar; the datum has one component for a scalar field and one per displacement or velocity component for a vector field. Robin data act on physical exterior faces, not on immersed constraints or exchange interfaces.

The application entry points below share their physical Problem parameters. Selecting an execution adapter does not remove the Problem boundary terms.

Application	Physical solver or field	Robin path
`coupled_elasticity`	`CoupledElasticityProblem / lib1dsolver CoupledModel1d`	Legacy solid Robin; external FVM native RCR
`coupled_poisson`	`PoissonSolver bulk and embedded + continuity constraint`	Scalar Robin on both fields
`coupled_poisson_elasticity`	`PoissonSolver<1,3> + ElasticStaticProblem<3>`	Scalar and solid Robin; interaction traction stays separate
`elastic_static`	`ElasticStaticProblem`	Vector Robin in native static solve
`elastic_static_adapter`	`ElasticStaticProblem / LinearAdapter`	Vector Robin residual and state Jacobian
`elasticity`	`ElasticityProblem`	Legacy native vector Robin
`elasticity_explicit`	`ElasticityExplicit` (independent equations)	Vector spring and load in explicit dynamics
`elasticity_freq`	`ElasticityFreqProblem`	Complex harmonic vector Robin
`elastodynamics`	`ElastodynamicsSolver`	Native transient Robin
`elastodynamics_adapter`	`ElastodynamicsSolver / IDAAdapter`	Time-dependent Robin residual and Jacobian
`fiber_reinforced_elasticity_freq`	`FiberReinforcedElasticityFreq`	Independent: both fields; shared: matrix faces and physical fiber end sections
`fiber_reinforced_elastodynamics`	`FiberReinforcedElastodynamics`	Matrix/fiber physical boundary contributions
`laplacian`	`PoissonProblem (legacy)`	Legacy scalar Robin
`lifted_elasticity`	`LiftedElasticity` / `static_elasticity_term`	Bulk vector Robin; lifted volume interactions stay distinct
`lifted_poisson`	`run_lifted_poisson`	Bulk scalar Robin
`liver_vascular`	`Vascularization::generate / solve_reference`	Graph pressure/conductance Robin; native resistive territories
`metric_flow_x`	`MetricFlowX::BloodFlowSystem / IDAAdapter`	Native terminal RCR; C=0 is pressure/flow Robin
`metric_flow_x_elastodynamics`	`MetricFlowX::BloodFlowSystem + ElastodynamicsSolver / IDAAdapter`	Solid vector Robin and native terminal RCR, independently
`navier_stokes`	`NavierStokesSolver / IDAAdapter`	Velocity traction Robin, pressure gauge, time and exact Jacobian
`poisson`	`PoissonSolver`	Native scalar Robin
`poisson_adapter`	`PoissonSolver / LinearAdapter`	Scalar Robin assembled operator and load
`pseudocoupling1D`	`CoupledElasticityProblem / lib1dsolver CoupledModel1d`	Legacy solid Robin; external native terminal closures
`reduced_poisson`	`ReducedPoisson`	Scalar Robin on the full bulk boundary
`vascular_network`	`Vascularization::run_application / solve_reference`	Graph pressure/conductance Robin at physical ports or unpaired terminals

Face conditions¶

Use Robin boundary ids, Robin coefficient, and Robin boundary conditions in the selected Problem parameter subsection. Existing function groups retain their own subsection paths; vector data must have the ambient component count. A Robin boundary id must be disjoint from essential and Neumann boundary ids. Coefficients must be finite and nonnegative, and every component of the datum must be finite. Zero coefficient is the prescribed flux or traction limit.

For example, within a two-dimensional Navier–Stokes configuration:

subsection Navier-Stokes
  set Dirichlet boundary ids = 0,1,2
  set Robin boundary ids = 3
  subsection Domain
    set Grid generator = hyper_cube
    set Grid generator arguments = 0:1:true
  end
  subsection Robin coefficient
    set Variable names = x,y,t
    set Function expression = 2+t
  end
  subsection Robin boundary conditions
    set Variable names = x,y,t
    set Function expression = 0;1+t
  end
end

The Navier–Stokes boundary datum is a traction. It is not multiplied by the fluid density used for body acceleration. A physical Robin traction face fixes the absolute pressure reference, including the zero-coefficient limit; the solver then imposes no additional pressure pin. Both the native backward Euler path and the IDA contributor evaluate the boundary data at their current time. The IDA state Jacobian includes the boundary velocity operator.

In shared harmonic reinforcement, matrix faces use the matrix boundary data. Fiber Robin data act on the matrix displacement trace at the physical fiber end sections; they create no independent fiber unknown. In independent formulations, each physical field has its own boundary conditions. Harmonic problems apply the spring operator and traction to the real and imaginary field components with their existing phase convention.

The legacy scalar applications register their functions under Immersed Problem/Functions (laplacian) and Reduced Poisson/Functions (reduced_poisson). For example, on a colorized unit square, this configuration prescribes the constant solution u=1 with zero volume forcing:

subsection Immersed Problem
  set Dirichlet boundary ids = 0
  set Robin boundary ids = 1,2,3
  subsection Dirichlet boundary conditions
    set Function expression = 1
  end
  subsection Functions
    subsection Robin coefficient
      set Function expression = 2
    end
    subsection Robin boundary conditions
      set Function expression = 2
    end
    subsection Robin coefficient 1
      set Function expression = 3
    end
    subsection Robin boundary conditions 1
      set Function expression = 3
    end
  end
end

Replace Immersed Problem with Reduced Poisson for the reduced solver. Per-id subsections override individual function entries; omitted entries retain the global configuration, including after a reparse. A changed coefficient does not automatically change the prescribed datum. The experimental legacy MATRIX_FREE_PATH preprocessor branch is not exposed by the CMake applications; these boundary conditions are implemented in their assembled solvers.

Pressure and flow at network terminals¶

The vascular network and liver applications solve a Poiseuille pressure system as part of their physical reference calculation. They are not only geometric generators. Their terminal Robin relation is

outward_flow = beta * pressure - g

Vascularization/Physiology/Reference Robin conditions maps a physical node id to its conductance and source. Conductance has units m^3/(Pa s), and the source has units m^3/s. A resistor to a reservoir satisfies beta = 1/R and g = P_reservoir/R. A positive conductance anchors the pressure of its connected component. Zero conductance prescribes a flux and still requires another pressure anchor. These terminal conditions do not alter physical node ids, vessel resistances, or the resistive links between tissue territories.

Native flow terminal models¶

The standalone MetricFlowX application reads native MetricFlowX parameters. The coupled MetricFlowX–elastodynamics application keeps those terminal data separate from the solid Robin faces. Its flow contributor delegates residual and both Jacobian blocks to BloodFlowSystem; it does not replace the native terminal equations.

For a purely resistive terminal, select Outlet boundary condition type = RCR and Terminal parameter source = Parameters under MetricFlowSystem<1, 3>. Set Default terminal C = 0, nonnegative Default terminal R1 and Default terminal R2 with positive sum, and Default terminal P_out. The native condition then reads

p_terminal - P_out = (R1 + R2) * Q_out

With positive compliance, the RCR capacitor pressure is a differential unknown. Terminal parameters CSV file supplies per-boundary overrides. The generated coupled-network path supplies its Generated terminal resistance as native R2, with R1 = C = 0, and uses Generated outlet pressure as P_out.

The optional coupled_elasticity and pseudocoupling1D applications pass the one-dimensional case file to CoupledModel1d::init. That backend retains its native terminal resistance and RCR models. The ImmersX mechanical Robin data belong to the tissue problem and do not overwrite those flow terminal data. These applications require the optional 1dsolver library; the solid Problem can also be used and tested independently of that library.

Both legacy coupled drivers pass the physical flow time to the tissue solve. Changing the time refreshes the mechanical boundary coefficient, applied load and essential lifting together. This preserves time-dependent Robin data even when the tissue is solved as a sequence of static problems.

See Elasticity and Elastodynamics for the solid formulations.