Second-order adjoint sensitivity analysis methodology (2nd-asam) for large-scale nonlinear systems: II. Application to a nonlinear heat conduction benchmark

TL;DR

This paper demonstrates the application of the second-order adjoint sensitivity analysis methodology (2nd-ASAM) to a nonlinear heat conduction benchmark, efficiently computing multiple derivatives with minimal modifications to the differential equations.

Contribution

It applies 2nd-ASAM to a nonlinear heat conduction problem, showing how to compute all first- and second-order derivatives with limited changes to the existing solver.

Findings

Six adjoint computations yield all derivatives needed.

Only source terms need modification, not the differential operator.

Exact derivatives are obtained for the benchmark problem.

Abstract

This work presents an illustrative application of the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) developed by Cacuci (2015) to a paradigm nonlinear heat conduction benchmark, which models a conceptual experimental test section containing heated rods immersed in liquid lead-bismuth eutectic. This benchmark admits an exact solution, thereby making transparent the underlying mathematical derivations. For this illustrative problem, six large-scale adjoint computations sufficed to compute exactly all five 1st-order and fifteen distinct 2nd-order derivatives of the temperature response to the five model parameters. Very significantly, only the sources on the right-sides of the heat conduction differential operator need to be modified; the left-side of the differential equations (and hence the solver in large-scale practical applications) remains unchanged.