A CA approach for estimating transient surface heating

(applying NKS to inverse problems in engineering)

Vallorie J. Peridier

College of Engineering, Temple University

Model Problem

Schematic:

Study context:

(1) NKS/2005 Summer School project objectives:
	•   obtain quantitative results for a basic inverse problem using an NKS approach.
	•  the basic inverse problem: 	— we know (Subscript[T, M])(t),  and need to estimate   q''(t)
						— model is 1-d in space & medium behaves as a "heat sink."
(2)  Results reported in Complex Systems 16.

The CA Model

Steve Wolfram's suggestion:

The CA Model

The energy propagation CA model: 
                   (Rule 226 with right shift & column 1,  "(Subscript[q, CA])(i),"  prescribed)

...But will (Subscript[q, CA])(i) be a satisfactory approximation for q''?

Model Physics: Generating the data

Variables and governing equations: Dimensioned form

Variables:	x	[m]		(distance into the medium);
		\(\(q\^''\)\)    [(W/(m^2))]		(surface heating);
		t	[sec] 		(time);
		T	\(\([\^o C]\)\)		(temperature);

Parameters:	k	[(W/(((m )^o) C))]		(thermal conductivity);
		α	[((m^2)/sec)]		(thermal diffusion coefficient of the medium; α=(k/ρ)c);
		r	[m]		(distance of temperature sensor from surface);

Problem statement:	
		((∂T)/(∂t)) = α(((∂^2)T)/(∂(x^2)));	T(x->∞) = T(t=0) = 0;   –k ((∂T)/(∂x))(x=0) = (q^(''))(t)

To compute input data for the CA/inverse solver, one could solve above problem numerically, and then:
			(Subscript[T, m])(t) = T(\(x = r, \ t\))

Model Physics: Generating the data

Variables and governing equations: Dimensionless O(1) quantities

Variables:	ξ     =(x/r)			(distance into the medium);
		 φ  =((\(q\^''\))/(     \(\(q\^''\)\_MAX\)))	  		(surface heating);
		  
		τ    =t (α/(r^2))			(time);
		
		 θ    =(T/Tscale) 	 	(temperature);        Tscale = \(\(\((\(q\^''\)\_MAX\ )\)\ r\)\/k\)

Problem statement:	
		∂(θ/∂)τ = (∂^2) (θ/∂)(ξ^2);	 θ(ξ->∞) = θ(τ=0)= 0;     – ∂(θ/∂)ξ(ξ=0) = φ(τ)

To compute (Subscript[T, m])(t) , the   input data for the CA/inverse solver,  we solve above problem numerically; then:
			(Subscript[T, m])(t) = Tscale· θ(ξ=1,τ)

Model Physics: Generating the data

φ(τ)  vs.  (Subscript[q, CA]) (i)

Can the thermocouple distinquish whether the (dimensionless) surface heating is:
	– a continuous O(1) function  φ(τ), or 
	– a integer {0,1} approximation, (Subscript[q, CA])(i)?      	REALLY, NO ...

Inverse solution procedure

Estimating (Subscript[θ, CA])at the thermocouple depth

Say that in the CA evolution:	
	 i denotes the row (the timestep),  with Δτ the CA row spacing,
	 j denotes the column corresponding to the thermocouple depth, with Δξ the CA column spacing.

The "quantum of heat" {0,1}  which arrives at the thermocouple at the ith timestep is:  	q(i,j).	

Taking the initial temperature to be: 	(Subscript[θ, CA])(τ=0) = 0;	

We can use a (basic) "lumped capacitance" model to estimate the dimensionless temperature at the thermocouple:
	
	(Subscript[θ, CA])(τ=i·Δτ)   	= θ(i,j)           = θ(i-1,j) (e^(-(Δτ)(Δξ))) + κ·q(i,j)·(Δτ)(Δξ)

Inverse solution procedure

Estimating (Subscript[θ, CA])at the thermocouple depth (cont.)

A quantitative estimate for each of the above element:

	• Δτ	Say the sampling timestep of the thermocouple is Δt.	
	           Then:
			Δτ = (Δt) (α/(r^2))

	• Δξ	Diffusion processes in semi-∞ media scale as (((ξ^2))/τ)∼O(1).  
	          So to first order, (more on this later):
	          
			Δξ ∼ (((Δτ))^(1/2)) 


	• j	Which column?  Since the thermocouple is location at ξ=1,    j = (1/((Δξ))):

			j ∼ (((Δτ))^(-1/2)),  (rounded to nearest integer)

Inverse solution procedure

Estimating (Subscript[θ, CA])at the thermocouple depth (cont.)

A quantitative estimate for each of the above element (cont.):
	• κ	A constant, set once to give good agreement with a the exact solution for a test case.

For example, for (Subscript[q, CA])  shown in above slide, and Δτ=(π/50), κ=2 gives this result:

Inverse solution procedure

The "search" procedure, for an input (Subscript[T, M]),  generates (Subscript[q, CA]) and  \(TraditionalForm\`\(q\^''\)\_MAX\)

The inverse search method steps:
	(1.)	Algorithm gets an input (Subscript[T, M])(i) vector and creates an initial guess for (Subscript[q, CA])(i).

	(2.)	Algorithm re-arranged the (Subscript[q, CA])(i) vector so that the error in the least-squares fit 
	      	of (Subscript[T, M])(i) as. (Subscript[θ, CA])(i) is minimized.

	(3.)	A "1" is substituted for a "0" at the (suitably lagged) position in (Subscript[q, CA])(i)  
		where the difference between (Subscript[T, M])(i) and (Subscript[θ, CA])(i)  is greatest.

	(4.)     Steps 2,3 are repeated until there is no improvement in the fit. Now we have:

(5.) 	Lastly,  the least squares procedure yields the multiplicative scale factor between  (Subscript[T, M])(i), (Subscript[θ, CA])(i)  so:
		 		 (Subscript[\(q\^''\), MAX])  = (((Tscale · k))/r)

Inverse solution procedure

Sample result:

Inverse solution procedure

More on the spatial discretization Δξ:

•  If this "CA diffusion model in semi-∞ medium" were used to study spatial variations in ξ, 	
	    note that, since ξ∼(Sqrt[τ])·O (1):   
		
			             Δξ = ((Δτ)/(2(Sqrt[τ])))·O(1)  	=((Δτ)/(2(Sqrt[j·Δτ])))·O(1)
    
	•  Consequently, the scale Δξ varies as the square root of the column index j:   
			
				(Subscript[Δξ, j])  ∼  ((Sqrt[Δτ])/(Sqrt[j]))·

CONCLUSIONS: Lessons I learned in NKS Summer School:

(1.)  Use CA evolutions to represent the propagating quantity.

(2.)  Rule searches with graphical assessments are preferred over direct modelling.	

(3.)  Integer-variable representation turns out to be both adequate and an asset.
	
(4.)  Dimensionless  analysis facilitates: 
	– the reconciliation of the NKS/engineering problem descriptions;
	– obtaining quantitative results.

(5.)  NKS Summer School is confidently recommended!