Formal convergence characteristics of elliptically constrained incremental Newton-Raphson algorithms

Abstract

Numerous situations in continuum mechanics, structural stability, optimization and related fields generate problems requiring the solution of nonlinear algebraic equations. To solve such problems, a large assortment of schemes has evolved over the years. This paper will consider the formal numerical properties of the newly developed constrained type incremental Newton-Raphson operator schemes. Specifically the evaluation of the formal behavior of the elliptically constrained version is treated in detail. Note this procedure has the versatility to efficiently handle a wide range of nonlinearities including the possibilities of positive, negative, semi and indefinite tangent properties in an inherently stable manner. The formalism includes such items as determining from both a global as well as local point of view the existence, uniqueness and convergence characteristics. Also included in the developments will be the determination of the occurrence of global safety zones wherein convergence is assured. The approach taken is general enough to provide a framework to enable applications to a wide variety of constrained schemes involving continuous, piecewise continuous, closed or open constraint conditions.