Generalized Adjoint for Physical Processes with Parameterized Discontinuities. Part III: Multiple Threshold Conditions

Abstract

The generalized adjoint is extended to special situations in which the concerned switches are triggered simultaneously by more than one threshold condition. It is shown that the involved threshold conditions can be combined into a single threshold condition represented by the envelope of the involved threshold surfaces in the space constituted by model variables and time. This envelope is piecewise smooth, and the concerned switch point is a nonsmooth point on the envelope at the intersection of the involved threshold surfaces. When the concerned switch point is perturbed along different sectors (formed by the involved threshold surfaces) on the envelope surface, different threshold conditions should be used in the tangent linear and adjoint matching conditions. This complicates the tangent linearization and the backward adjoint integration. Four basic types of situations are examined and illustrated by using simple examples, showing how the generalized adjoint can be extended to these complicated situations.