(More) Integrals

There are two more options that can be passed to VoronoiFVProblem():

• bulk_integrals: A list of functions following the pattern of rhs_functions
• flux_integrals: A list of functions following the pattern of fluxes but returning only one single value instead of two.

These options are thought to provide the user with the ability to calculate complex integrals even after the VoronoiGeometry has been calculated. The algorithm will sum every member of flux_integrals over all interfaces and sum every member of bulk_integrals over all cells.

To illustrate this, consider the following example:

function surface_int(;para_i,para_j,mass_ij,normal,kwargs...) 
    # kwargs... collects all additional parameters which are not used in the current function.
    weight = mass_ij * sqrt(para_i[:κ]*para_j[:κ])
    return weight
end

b_int(;para_i,mass_i,kwargs...) = mass_i * para_i[:f] * para_i[:κ]^2 

function test_integrals()
    nodes = VoronoiNodes(rand(2,40))
    # calculate Voronoi tessellation and integrate κ(x)=sin(pi*x[1]) and f(x)=x[2]^2 over individual cells and interfaces
    VG_basis = VoronoiGeometry(nodes,cuboid(2,periodic=[]),integrator=HighVoronoi.VI_POLYGON,integrand=x->[sin(pi*x[1]),x[2]^2])

    # set up fluxes and RHS
    vfvp = VoronoiFVProblem(VG_basis,  
                                # note that the exact form of κ and f does not matter since data will be retrieved from VG_basis:
                              integralfunctions = (κ = x->1.0, f = x->1.0, ), 
                              flux_integrals = ( fi = surface_int, ),
                              bulk_integrals = (bi = b_int,) )
    # print the integral of sqrt(κ_i*κ_j) over the interfaces
    println( get_Fluxintegral(vfvp,:fi) )
    # print the integral of f*κ^2 over the bulk
    println( get_Bulkintegral(vfvp,:bi))
end