# Rapid Reliable Solution of the Parametrized Partial Differential Equations of Continuum Mechanics and Transport

## Pedagogy The Heterogeneous Thermal Block Problem Authors: G Rozza and AT Patera

Q1 Consider an "Upper Bound" model in which we insert Insulator Strips (idealized material of zero conductivity) between each column of blocks. (There will be two Insulator Strips in total.)
(a) Based on this "Upper Bound" model/approximation, develop an analytical expression for the average bottom-wall temperature, sUB, as a function of the block conductivities.
Hint: Treat each column (separately) as three resistors in series; then combine the three resulting column resistances in parallel.

(b) Argue physically (or for those familiar with variational methods, prove mathematically) that sUBs (the true value of the average bottom-wall temperature). (Hint: What do the Insulator Strips do to the flow of heat?)

(c) Consider a random sample of (say) 1000 different parameter values; recall each point in the sample is a 9-vector of block conductivities. (To adequately represent the low values of conductivities, it might be better to sample uniformly in log10 of conductivity.) What is the largest difference between sUB over your sample?; what is the block conductivity 9-vector in your sample that realizes this maximum discrepancy? Can you predict a block conductivity 9-vector that will result in the largest discrepancy between sUB and s over all possible parameters (conductivities) in the admissible range?; do the numerical results support your claim?
Q2 Consider a "Lower Bound" model in which we insert Superconductor Strips (idealized material of infinite conductivity) between each row of blocks. (There will be two Superconductor Strips in total.)
(a) Based on this "Lower Bound" model/approximation, develop an analytical expression for the average bottom-wall temperature, sLB, as a function of block conductivities.
Hint: Treat each row separately as three resistors in parallel; then combine the three resulting row resistances in series.

(b) Argue physically (or for those familiar with variational methods, prove mathematically) that sLBs (the true value of the average bottom-wall temperature). (Hint: What do the Superconductor Strips do to the flow of heat?)

(c) Consider a random sample of (say) 1000 different parameter values; recall each point in the sample is a 9-vector of block conductivities. (To adequately represent the low values of conductivities, it might be better to sample uniformly in log10 of conductivity.) What is the largest difference between sLB and s over your sample?; and what is the block conductivity 9-vector in your sample which realizes this maximum discrepancy? Can you predict a block conductivity 9-vector that will result in the largest discrepancy between sLB and s over all possible parameters (conductivities) in the admissible range?; do the numerical results support your claim?