GridapROMs.ParamODEs

module ParamODEs

Parametric FE operators and solutions for transient parametric PDEs.

Extends ParamSteady to the time-dependent setting, building on Gridap's ODEs machinery. Key components:

• Time derivatives — ∂ₚt and ∂ₚtt, parameter-aware analogues of Gridap's ∂t/∂tt, for use inside weak forms.
• Transient trial spaces — TransientTrialParamFESpace and TransientMultiFieldParamFESpace add time-varying Dirichlet conditions to parametric FE spaces.
• ODE operators — ODEParamOperator hierarchy (JointODEParamOperator, SplitODEParamOperator, LinearNonlinearODEParamOperator, …) translate a transient parametric weak form into the residual/Jacobian signature expected by Gridap's time-marching schemes.
• Stage operators — ParamStageOperator wraps a single ODE stage solve, keeping track of the current time, stage coefficients, and parameter realisation.
• Transient FE operators — TransientParamFEOperator and specialisations (SplitTransientParamFEOperator, TransientLinearParamFEOperator, …) combine the spatial FE machinery with time-derivative information.
• ODE solutions — ODEParamSolution iterates over time steps, yielding (realisation, FEFunction) pairs; TransientParamFESolution wraps this iterator into a ParamFEFunction per time step.
• Shifted solvers — ShiftedSolver applies a constant operator shift to a linear solver, enabling efficient generalised-alpha or implicit-explicit schemes.

Modules RBTransient and Extensions depend on the abstractions defined here.

const JointODEParamOperator{O<:ODEParamOperatorType} = ODEParamOperator{O,JointDomains}

const JointTransientParamFEOperator{O<:ODEParamOperatorType} = TransientParamFEOperator{O,JointDomains}

struct LinearNonlinearParamODE <: ODEParamOperatorType end

struct LinearParamODE <: ODEParamOperatorType end

struct NonlinearParamODE <: ODEParamOperatorType end

const ODEParamOperator{T<:ODEParamOperatorType,T<:TriangulationStyle} = ParamOperator{O,T}

Transient extension of the type ParamOperator.

abstract type ODEParamOperatorType <: UnEvalOperatorType end

Parametric extension of the type ODEOperatorType in Gridap.

Subtypes:

• NonlinearParamODE
• LinearParamODE
• LinearNonlinearParamODE

struct ODEParamSolution{V} <: ODESolution
  solver::ODESolver
  odeop::ODEParamOperator
  r::TransientRealisation
  us0::Tuple{Vararg{V}}
end

struct ParamStageOperator{O} <: NonlinearParamOperator
  op::ODEParamOperator{O}
  r::TransientRealisation
  state_update::Function
  ws::Tuple{Vararg{Real}}
  paramcache::AbstractParamCache
end

Stage operator to solve a parametric ODE with a time marching scheme

const SplitODEParamOperator{O<:ODEParamOperatorType} = ODEParamOperator{O,SplitDomains}

const SplitTransientParamFEOperator{O<:ODEParamOperatorType} = TransientParamFEOperator{O,SplitDomains}

struct TransientLinearParamFEOpFromWeakForm{T} <: TransientParamFEOperator{LinearParamODE,T}
  res::Function
  jacs::Tuple{Vararg{Function}}
  constant_forms::Tuple{Vararg{Bool}}
  tpspace::TransientParamSpace
  assem::Assembler
  trial::FESpace
  test::FESpace
  domains::FEDomains
  order::Integer
end

Instance of TransientParamFEOperator, to be used when the transient problem is linear

const TransientMultiFieldParamFESpace = MultiFieldFESpace

struct TransientParamFEOpFromWeakForm{T} <: TransientParamFEOperator{NonlinearParamODE,T}
  res::Function
  jacs::Tuple{Vararg{Function}}
  tpspace::TransientParamSpace
  assem::Assembler
  trial::FESpace
  test::FESpace
  domains::FEDomains
  order::Integer
end

Instance of TransientParamFEOperator, to be used when the transient problem is nonlinear

const TransientParamFEOperator{O<:ODEParamOperatorType,T<:TriangulationStyle} = ParamFEOperator{O,T}

Parametric extension of a TransientFEOperator in Gridap. Compared to a standard TransientFEOperator, there are the following novelties:

• a TransientParamSpace is provided, so that parametric realisations can be extracted directly from the TransientParamFEOperator
• a function representing a norm matrix is provided, so that errors in the desired norm can be automatically computed

Subtypes:

• TransientParamFEOpFromWeakForm
• TransientLinearParamFEOpFromWeakForm

struct TransientParamFESolution{V} <: TransientFESolution
  odesol::ODEParamSolution{V}
  trial
end

Wrapper around a TransientParamFEOperator and ODESolver that represents the parametric solution at a set of time steps. It is an iterator that computes the solution at each time step in a lazy fashion when accessing the solution.

const TransientTrialParamFESpace = UnEvalTrialFESpace

TransientLinearParamFEOperator(res::Function,forms::Tuple{Vararg{Function}}, tpspace,trial,test;kwargs...) -> TransientLinearParamFEOpFromWeakForm{TriangulationStyle}

Returns a linear parametric FE operator