ModelingToolkit: Some questions on usage & best practices

Based on an earlier discussion here on discourse formulated a physical system in Modeling toolkit. The system consists of a discretized 1D PDE, ODEs and an algebraic equation. The physical nature of the system is further described in my earlier posting, if anyone is interested.

I am still in my very early days of using Julia and ModelingToolkit, so I have a few hopefully simple questions about usage of MTK.

My code is as follows:

using OrdinaryDiffEq, ModelingToolkit
using DASSL
using DASKR
using DifferentialEquations
using BenchmarkTools
using Plots
using Sundials

function build_system(n)

	@parameters t # or would @variables t be better!?
    @variables Tgas[1:n](t) Tsol[1:n](t) cR[1:n](t) mflux(t)
	@parameters params[1:3]
	D = Differential(t)
    

    dp, Tin, h = params 

    dx = h / n

	############################################################
	# Build equations ##########################################
	############################################################

	eqns = Array{Equation}(undef, 3 * n + 1)
	count = 1

    # specification as function or intermediate variabel does not matter. I could also make this an 
    # actual @variable, then I could retrieve it later.
    rR = 5.5e6 .* exp.(.-5680 ./(273 .+ Tsol)) .* (1 .- exp.(.-cR./50))
    q = 1e4 * (Tgas - Tsol)
    
    # gas energy balance

    eqns[count] = 0 ~  mflux * 1000 * (Tin - Tgas[1]) - q[1] * dx
    count += 1
	for i in 2:n
	        eqns[count] = 0 ~ mflux * 1000 * (Tgas[i-1] - Tgas[i]) - q[i] * dx
		count += 1
	end

	# solids energy balance
	for i in 1:n
        	eqns[count] = D(Tsol[i]) ~ q[i] / (1000 * 2000)
		count += 1
	end

	# gas momentum balance
	eqns[count] = 0 ~ - mflux[1] + 5e-3 * sqrt(dp)
	count += 1

	# reactant concentration equations
	for i in 1:n
        	eqns[count] = D(cR[i]) ~ -rR[i]
		count += 1
	end

    @named sys = ODESystem(eqns, t, vcat(Tgas, Tsol, cR, [mflux]), params)

    return sys

end

function build_problem(sys; tspan = (0.0, 1.5), param_values = [100e2, 200, 0.5])

    n = Int((length(sys.states) - 1)/3)
    u0 = ones(length(sys.states))
    u0[1:n] .= param_values[2]
    u0[n+1 : 2*n] .= 20
    u0[2*n+1 : 3*n] .= 3000
    u0[3*n+1] = 0.5


    u0map = [
        sys.Tgas => 100.0,
        sys.Tsol => 20.0,
        sys.cR => 3000.0,
        sys.mflux => 0.5
    ]   

    prob = ODEProblem(sys, u0, tspan, param_values, jac=true, sparse=true)
    return prob

end



dp = 100e2
Tin = 200
h = 0.5
parameters = [dp, Tin, h]

N = 100
tspan = (0.0, 1500.0)


sys = build_system(N)
prob = build_problem(sys, tspan=tspan, param_values = parameters)
sol = solve(prob, QBDF())

As you see, my code is structured such that all the @variables are defined in the local scope of function build_system. Since that is the case, how would I later, in another function or in global scope access parts of the solutions, like for example in the plotting recipe?

  • Tutorials typically show something like plot(sol, vars=(Tgas[1])), but that won’t work in my case because of the scoping.

  • Doing something like plot(sol, vars=[sys.Tgas[1]]) does not work either, it gives the error: ArgumentError: sys₊Tgas[1](t) is either an observed nor a state variable. (BTW, there seems to be a typo in the error message, I guess it should say “neither” instead of “either”)

  • What works is: plot(sol, vars=[states(sys)[5]]), but that does not allow me to specify the variable name and is not best practice, I guess

  • in some thread I also found a hint on using the (undocumented?) @nonamespace macro, which works: plot(sol, vars=[@nonamespace sys.Tgas[1]]), but is that really the intended way?

Is there a good way to do it? A question very related to this would be how to define u0map in the function build_problem. Using the definition in my code listing above, building the ODEProblem throws an error, which I guess is again due to namespacing of the MTK variables.

So, what are the best practices here? Is it a non-optimal idea to define the variables, build the ODE problem and plotting the solution in different functions / scopes in the first place?

plot(sol, vars=(sys.Tgas[1])) should work on the latest versions. What version are you using?

Thanks for the hint. I updated MTK to 8.5.1 (sorry, forgot to check which version I had before updating).
But still, I get the following error:

plot(sol, vars=(sys.Tgas[1])) 
# ArgumentError: sys₊Tgas[1](t) is either an observed nor a state variable.
# (should it read "_n_either... nor"?)

Further info:

states(sys)
#301-element Vector{Term{Real, Base.ImmutableDict{DataType, Any}}}:
# Tgas[1](t)
# Tgas[2](t)
# Tgas[3](t)
# ⋮
# cR[100](t)
# mflux(t)
sys.Tgas[1]
# sys₊Tgas[1](t)
Pkg.status()
#  [165a45c3] DASKR v2.8.0
#  [e993076c] DASSL v2.6.0
#  [0c46a032] DifferentialEquations v7.1.0
#  [28b8d3ca] GR v0.64.0
#  [33e6dc65] MKL v0.5.0
#  [961ee093] ModelingToolkit v8.5.1
#  [1dea7af3] OrdinaryDiffEq v6.7.0
#  [91a5bcdd] Plots v1.26.0
#  [d236fae5] PreallocationTools v0.2.4
#  [c3572dad] Sundials v4.9.2
#  [0c5d862f] Symbolics v4.3.0

@YingboMa it should be released right?

I checked the release notes of the past view MTK releases but did not find anything that seems to relate to the namespacing of MTK variables.

Created an issue.