ABSTRACT
IV. A. FSAP AND ASAP FOR RESPONSES AT CRITICAL POINTS: GENERAL THEORY
Consider once again a generic physical system described mathematically by
the following system of K coupled nonlinear equations written in operator form as
represent inhomogeneous source terms; QE∈Q , where QE is also a normed liner space. The components of Q may be operators (rather than just functions) acting on ( )xα and x ,
In view of the above definition, N represents the mapping QEES →⊂:N , where αSSS u ×= , uu ES ⊂ , αα ES ⊂ , and αEEE u ×= . Note that an arbitrary element E∈e is of the form ( )α,ue = . All vectors in this Section are, as usual, column vectors, unless explicitly stated otherwise. If differential operators appear in Eq. (IV.A.1), then a corresponding set of boundary conditions, which is essential to define the domain S of N , must also be given. This set can be represented as
where A and B are operators, and Ω∂ is the boundary of Ω ; the operator ( )αA represents all inhomogeneous boundary terms. Consider that the system response R is a functional of ( )α,ue = defined at a
Ω +== −−= Kδδ ααxue (IV.A.3)
The quantities appearing in the integrand of Eq. (IV.A.3) are defined as follows:
(i) F is the nonlinear function under consideration, (ii) ( )xδ is the customary “Dirac-delta” functional, (iii) IR∈α , i.e., the components ( )Iii ,,1, K=α are restricted throughout
this section to be real numbers; (iv) ( ) ( ) ( )[ ]ααα Myy ,,1 K=y , JM ≤ , is a critical point of F . This critical point can be generally defined in one of the following two ways: If the G-differential of F vanishes at ( )αy , then ( )αy is a critical point
In this case, ( )αy has J components (i.e., JM = ), and ( ) 1
( )αy is a function of α . Occasionally, it may happen that jxF ∂∂ takes on nonzero constant values
sufficiently general to include treatment of extrema (local, relative, or absolute), saddle, and inflexion points of the function F of interest. Note that, in practice, the base-case solution path, and therefore the specific nature and location of the critical point under consideration, are completely known prior to initiating the sensitivity studies.
