Solution of the two dimensional inverse heat conduction problem using a Maximum Entropy Approach

1. Solution of the two dimensional inverse heat conduction problem using a Maximum Entropy Approach Naveen Nair Krishnan Balasubramaniam Sarit Kumar Das

2. Inverse Heat Conduction Problems • Aim : To determine unknowns such as thermal conditions, unknown geometries or thermo-physical properties from the temperature history and distribution of the sample.

3. Inverse Heat Conduction Problems cont’d… • Most interesting applications involve inversion of surface temperature profiles/ surface temperature histories to determine one unknown boundary condition. • Standard techniques • A • B • C

4. Problem Statement – forward problem • Where • q is the dimensionless temperature which is, in general a function of x, y and t. • kis the thermal conductivity • r the density • Cp the specific heat

5. Problem Statement cont’d • Boundary Conditions • Initial Conditions at x = 0 at x = a at y = b for t <= 0 at y = 0

6. Problem Statement cont’d • The inverse problem will be defined as obtaining the function f (x, t) when given the temperatures (x,b,t) ie. the complete temperature history on the outside edge.

7. Definition of the inverse problem

8. Definition of the inverse problem cont’d… • The aim of the inverse problem reduces now to chose a function f (x, t) that minimizes the J(f). • This problem is fundamentally ill posed • There may exist more than one f (x, t) that reduces J(f) below any arbitrary value. • The mapping between J(f) and f (x, t) need not be unique. • Any solution that provides for |J(f)| <  is called a feasible solution.

9. Choice of a feasible solution – concept of maximum entropy • Entropy of any data is defined as the amount of uncertainty prevalent in the data. • This, therefore is applicable only to statistical / nondeterministic data. (for deterministic data entropy = 0 (by definition) • For an ideal solution to an inverse problem it is intuitive that the solution should be so chosen as to be have the least pre-bias ie. To have the highest amount of inherent uncertainty or highest entropy.

10. Forward Simulation • Input

11. Forward Simulation Output Image Movie Surface Plot Movie