Physics & special functions

abstract type AbstractPhysics

Abstract type for physics.

struct Physics

Physics data record with the following fields:

• flux::Function: Flux between neighboring control volumes: flux(f,u,edge,data) should return in f[i] the flux of species i along the edge joining circumcenters of neighboring control volumes. u is a 2D array such that for species i, u[i,1] and u[i,2] contain the unknown values at the corresponding ends of the edge.

• storage::Function: Storage term (term under time derivative): storage(f,u,node,data)

  It should return in f[i] the storage term for the i-th equation. u[i] contains the value of the i-th unknown.

• reaction::Function: Reaction term: reaction(f,u,node,data)

  It should return in f[i] the reaction term for the i-th equation. u[i] contains the value of the i-th unknown.

• edgereaction::Function: Edge reaction term: edgereaction(f,u,edge,data)

  It should return in f[i] the reaction term for the i-th equation. u[i] contains the value of the i-th unknown. u is a 2D array such that for species i, u[i,1] and u[i,2] contain the unknown values at the corresponding ends of the edge.

• source::Function: Source term: source(f,node,data).

  It should return the in f[i] the value of the source term for the i-th equation.

• bflux::Function: Flux between neighboring control volumes on the boundary. Called on edges fully adjacent to the boundary: `bflux(f,u,bedge,data)

• breaction::Function: Boundary reaction term: breaction(f,u,node,data) Similar to reaction, but restricted to the inner or outer boundaries.

• bsource::Function: Boundary source term: bsource(f,node,data).

  It should return in f[i] the value of the source term for the i-th equation.

• bstorage::Function: Boundary storage term: bstorage(f,u,node,data) Similar to storage, but restricted to the inner or outer boundaries.

• boutflow::Function: Outflow boundary term boutflow(f,u,edge,data) This function is called for edges (including interior ones) which have at least one ode on one of the outflow boundaries. Within this function, outflownode and isoutflownode can be used to identify that node. There is some ambiguity in the case that both nodes are outflow nodes, in that case it is assumed that the contribution is zero.

• outflowboundaries::Vector{Int64}: List (Vector) of boundary regions which carry outflow bondary conditions. Influences when boutflow is called.

• generic_operator::Function: Generic operator generic_operator(f,u,sys). This operator acts on the full solution u of a system. Sparsity is detected automatically unless generic_operator_sparsity is given.

• generic_operator_sparsity::Function: Function defining the sparsity structure of the generic operator. This should return the sparsity pattern of the generic_operator.

• data::Any: User data (parameters). This allows to pass various parameters to the callback functions.

• num_species::Int8: Number of species including boundary species. Meaningless & deprecated.

Physics(;num_species=0,
         data=nothing,
         flux,
         reaction,
         edgereaction,
         storage,
         source,
         breaction,
         bstorage,
         boutflow,
         outflowboundaries,
         generic,
         generic_sparsity
    )

Constructor for physics data. For the meaning of the optional keyword arguments, see VoronoiFVM.System(grid::ExtendableGrid; kwargs...).

generic_operator_sparsity!(system, sparsematrix)

Set generic operator sparsity, in the case where a generic operator has been defined in physics.

show(io, physics)

Show physics object

abstract type AbstractData{Tv}

Abstract type for user data.

It is possible but not necessary to make user data a subtype of AbstractData and get a prettyprinting show method.

show(
    io::IO,
    _::MIME{Symbol("text/plain")},
    this::VoronoiFVM.AbstractData
)

Pretty print AbstractData

copy!(to::AbstractData, from::AbstractData)

Copy AbstractData.

 boundary_dirichlet!(y,u,bnode,ispec,ireg,val)

Set Dirichlet boundary condition for species ispec at boundary ibc.

 boundary_dirichlet!(y,u,bnode; kwargs...)

Keyword argument version:

• species: species number. Default: 1
• region: boundary region number. By default, all boundary regions.
• value: value

 boundary_neumann!(y,u,bnode,ispec,ireg,val)

Set Neumann boundary condition for species ispec at boundary ibc.

 boundary_neumann!(y,u,bnode; kwargs...)

Keyword argument version:

• species: species number. Default: 1
• region: boundary region number. By default, all boundary regions.
• value: value

 boundary_robin!(y,u,bnode,ispec,ireg,fac,val)

Set Robin boundary condition for species ispec at boundary ibc.

 boundary_robin!(y,u,bnode, args...; kwargs...)

Keyword argument version:

• species: species number. Default: 1
• region: boundary region number. By default, all boundary regions.
• factor: factor
• value: value

   ramp(t; kwargs...)

Ramp function for specifying time dependent boundary conditions

Keyword arguments:

• dt: Tuple: start and end time of ramp. Default: (0,0.1)
• du: Tuple: values at start and end time. Default: (0,0)

hasoutflownode(edge)

Check if one node of the edge is situated on a boundary region listed in outflowboundaries, see [struct Physics].

isoutflownode(edge,inode)

Check if inode (1 or 2) is an outflow node.

isoutflownode(edge,inode,irefgion)

Check if inode (1 or 2) is an outflow node on boundary region iregion.

outflownode(edge)

Return outflow node of edge (1 or 2).

calc_divergences(sys, evelo, bfvelo)

Calculate for each Voronoi cell associated with a node of sys.grid the divergence of the velocity field used to obtain evelo and bfvelo via edgevelocities and bfacevelocities by means of summing all evelos and bfvelos per Voronoi cell.

edgevelocities(grid, velofunc; kwargs...)

Project velocity onto grid edges. That is, we compute the path integrals of the given velofunc along the Voronoi cell edges as provided by integrate.

edgevelocities(grid, vel; kwargs...)

Compute VoronoiFVM.edgevelocities for a finite element flow field computed by ExtendableFEM.

bfacevelocities(grid, velofunc; kwargs...)

Similar to edgevelocities, but for boundary faces.

bfacevelocities(grid, vel; kwargs...)

Compute VoronoiFVM.bfacevelocities for a finite element flow field computed by ExtendableFEM.

bfacenodefactors(sys)

Node form factors per boundary face

mutable struct Node{Tc, Tp, Ti} <: VoronoiFVM.AbstractNode{Tc, Tp, Ti}

Structure holding local node information.

• index::Any: Index in grid

• region::Any: Inner region number

• nspec::Any: Number of species defined in node

• icell::Any: Number of discretization cell the node is invoked from

• coord::Matrix: Grid coordinates

• cellnodes::Matrix: Grid cell nodes

• cellregions::Vector: Grid cell regions

• time::Float64: System time

• embedparam::Float64: Current value of embedding parameter

• params::Vector: parameters (deprecated)

• fac::Float64: Form factor

• _idx::Any: Local loop index

mutable struct BNode{Tv, Tc, Tp, Ti} <: VoronoiFVM.AbstractNode{Tc, Tp, Ti}

Structure holding local boundary node information.

• index::Any: Index in grid

• ibface::Any: BFace number it is called from

• ibnode::Any: local node number

• region::Any: Boundary region number

• cellregions::Vector

• nspec::Any: Number of species defined in node

• coord::Matrix: Grid coordinates

• bfacenodes::Matrix

• bfaceregions::Vector

• allcellregions::Vector

• bfacecells::Adjacency

• Dirichlet::Any

• time::Float64: System time

• embedparam::Float64: Current value of embedding parameter

• params::Vector: Parameters (deprecated)

• dirichlet_value::Vector

• fac::Float64

mutable struct Edge{Tc, Tp, Ti} <: VoronoiFVM.AbstractEdge{Tc, Tp, Ti}

Structure holding local edge information.

• index::Any: Index in grid

• node::Vector: Index

• region::Any: Inner region number corresponding to edge

• nspec::Any: Number of species defined in edge

• icell::Any: Number of discretization cell the edge is invoked from

• coord::Matrix: Grid coordinates

• cellx::Matrix

• edgenodes::Matrix

• cellregions::Vector

• has_celledges::Bool

• time::Float64: System time

• embedparam::Float64: Current value of embedding parameter

• params::Vector: Parameters (deprecated)

• fac::Float64: Form factor

• _idx::Any: Local loop index

• outflownoderegions::Union{Nothing, SparseArrays.SparseMatrixCSC{Bool, Int64}}

• outflownode::Int64: Outflow node

mutable struct BEdge{Tc, Tp, Ti} <: VoronoiFVM.AbstractEdge{Tc, Tp, Ti}

Structure holding local edge information.

• index::Any: Index in grid

• node::Vector: Index

• region::Any: Inner region number corresponding to edge

• nspec::Any: Number of species defined in edge

• icell::Any: Number of discretization cell the edge is invoked from

• coord::Matrix: Grid coordinates

• bedgenodes::Matrix

• bfaceedges::Matrix

• bfaceregions::Vector

• time::Float64: System time

• embedparam::Float64: Current value of embedding parameter

• params::Vector: Parameters (deprecated)

• fac::Float64

time(edge_or_node)

Return actual simulation time stored in node or edge

region(edgeornode)

Return region number node or edge is belonging to

parameters(edge_or_node)

Return abstract vector of parameters passed via vector of unknonws. This allows differentiation with respect to these parameters.

embedparam(edge_or_node)

Return embedding parameter stored in node or edge

project(edge, vector)

Project d-vector onto d-dimensional vector, i.e. calculate the dot product of vector with the difference of the edge end coordinates.

edgelength(edge)

Return length of edge

meas(edge::VoronoiFVM.AbstractEdge) -> Any

Calculate the length of an edge.

fbernoulli(x)

Bernoulli function $B(x)=\frac{x}{e^x-1}$ for exponentially fitted upwinding.

The name fbernoulli has been chosen to avoid confusion with Bernoulli from JuliaStats/Distributions.jl

Returns a real number containing the result.

While x/expm1(x) appears to be sufficiently accurate for all x!=0, it's derivative calculated via ForwardDiff.jl is not, so we use the polynomial approximation in the interval (-bernoulli_small_threshold, bernoulli_small_threshold).

Also, see the discussion in #117.

fbernoulli_pm(x)

Bernoulli function $B(x)=\frac{x}{e^x-1}$ for exponentially fitted upwind, joint evaluation for positive and negative argument

Usually, we need $B(x), B(-x)$ together, and it is cheaper to calculate them together.

Returns two real numbers containing the result for argument x and argument -x.

For the general approach to the implementation, see the discussion in fbernoulli.

inplace_linsolve!(A, b)

Non-allocating, non-pivoting inplace solution of square linear system of equations A*x=b using Doolittle's method.

After solution, A will contain the LU factorization, and b the result.

A pivoting version is available with Julia v1.9.

inplace_linsolve!(A, b, ipiv)

Non-allocating, pivoting, inplace solution of square linear system of equations A*x=b using LU factorization from RecursiveFactorizations.jl.

After solution, A will contain the LU factorization, and b the result. ipiv must be an Int64 vector of the same length as b.