Multiple Boundary Value Problem Research Articles

Trust‐region based return mapping algorithm for implicit integration of elastic‐plastic constitutive models

SummaryA new method for the solution of the non‐linear equations forming the core of constitutive model integration is proposed. Specifically, the trust‐region method that has been developed in the numerical optimization community is successfully modified for use in implicit integration of elastic‐plastic models. Although attention here is restricted to these rate‐independent formulations, the proposed approach holds substantial promise for adoption with models incorporating complex physics, multiple inelastic mechanisms, and/or multiphysics. As a first step, the non‐quadratic Hosford yield surface is used as a representative case to investigate computationally challenging constitutive models. The theory and implementation are presented, discussed, and compared with other common integration schemes. Multiple boundary value problems are studied and used to verify the proposed algorithm and demonstrate the capabilities of this approach over more common methodologies. Robustness and speed are then investigated and compared with existing algorithms. Through these efforts, it is shown that the utilization of a trust‐region approach leads to superior performance versus a traditional closest‐point projection Newton–Raphson method and comparable speed and robustness to a line search augmented scheme. Copyright © 2017 John Wiley & Sons, Ltd.

The Use of Transformation and Superposition in the Solution of Multiply-connected Boundary Value Problems

The development of two general solution methods for linear multiple boundary value problems was considered in terms of the elastic torsion of multiply-connected inhomogenious prisms. Two mathematical techniques, the transformation and the superposition, are introduced, and, in conjunction with the finite element procedure, they were applied to several example problems for the purpose of verification of the accuracy/reliability. The results obtained indicated that the two methods are versatile, easily adaptable to any arbitrary multiply-connected linear boundary value problem, and provide reliable results.

On a Multiple Boundary Value Problem in Generalized Aplanatic Lens Design and its Solution in Series

Series solutions are presented for the problem of the numerical design of generalized aplanatic lenses zoned, or stepped in thickness, and extremized in the sense of having maximum aperture or minimum volume for fixed values of the other lens parameters, including the zone angular apertures or f/numbers, some of which also may be required to be minimal or nearly minimal in some sense. The solutions are developed by the method of limits or Cauchy-Kovalevsky process which in turn leads to the establishment of an existence and convergence theory for the problem. Applications to both classical optics and contemporary microwave and acoustic lens antenna problems are illustrated.