Abstract: The theory of the Generalized J-integral (GJ-integral) provides a rigorous mathematical framework for the shape sensitivity analysis of singular points in boundary value problems. In this context, singularities encompass geometric discontinuities, domain boundaries, and junctions where distinct boundary conditions meet. The framework of the GJ-integral J!.Se; / is presented, where Se is the “seed” function and is the perturbation vector field. This framework consists of: (1) Construction of J! using a function Se of the saddle point derived from the Lagrangian; and (2) Extension of the domain integral R! to an arbitrary domain !. As techniques for application, the following three methods are developed, starting from R˝ within the defining domain˝: (1) Separating singularities from regular regions; (2) Focusing on specific target singularities; and (3) the GJ-scan, which numerically computes the vector field such that R˝ >0. While the mathematical foundation for problems possessing a minimizer has been previously established, this paper extends the framework to saddle-point problems. We establish the GJ-integral’s connection with the saddle-point structure, providing a rigorous proof that identifies it as the Fréchet derivative of the Lagrangian under incompressibility constraints. Finally, by employing the specialized techniques of Separating, Focusing, and the GJ-scan, we explore Stokes flow and evaluate the shape sensitivity of its boundaries.

Cite: Ohtsuka K. , Kovtunenko V.
Generalized J-integral in saddle point problems
Computational Mathematics and Modeling. 2026. P.1-24. DOI: 10.1007/s10598-026-09753-0 OpenAlex

Dates:

Published print:	Oct 9, 2026
Published online:	Oct 9, 2026

Identifiers:

≡ OpenAlex:

W7221113869

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