Newton projection method as applied to assembly simulation

Abstract

In this paper, we consider Newton projection method for solving the quadratic programming problem that emerges in simulation of joining process for assembly with compliant parts. This particular class of problems has specific features such as an ill-conditioned Hessian and a sparse matrix of constraints as well as a requirement for the large-scale computations. We use the projected Newton method with a quadratic rate of convergence and suggest some improvements to reduce the solving time: a method for solving the system of linear equations, so-called constraint recalculation method, and compare different approaches for step-size selection. We use the duality principle to formulate alternative forms of the minimization problem that, as a rule, can be solved faster. We describe how to solve the considered nonlinear minimization problem with the nonsmooth objective function by modifying Newton projection method and employing subgradients. In addition, we prove the convergence of the suggested algorithm. Finally, we compare Newton projection method with the other quadratic programming techniques on a number of assembly simulation problems.

Keywords:

• Newton projection method
• quadratic programming
• duality
• nonsmooth objective function
• subgradient

AMS Subject Classifications:

• 90C20
• 90C25
• 90C46
• 65Y20
• 74M15

Acknowledgments

he work was prepared within the scope of joined project of Airbus SAS and St. Petersburg Polytechnic University. The authors would like to thank our colleagues from both Airbus SAS, especially Elodie Bonhomme, and from St. Petersburg Polytechnic University for helpful discussions and comments. This research work was supported by the Academic Excellence Project 5-100 proposed by Peter the Great St. Petersburg Polytechnic University. We are also grateful to Evgeny Shinder for the time invested in reading our manuscript and for several valuable suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

S. Baklanov

S. Baklanov is a PhD student and a researcher in the Virtual Modeling Lab at Peter the Great St. Petersburg Polytechnic University, Russia. He is a C++ Software Engineer in Open Design Alliance. His research interests include convex optimization and its application to simulation the assembly of deformable structures, mathematical modeling and numerical methods.

M. Stefanova

M. Stefanova is a PhD student and a researcher in the Virtual Modeling Lab at Peter the Great St. Petersburg Polytechnic University, Russia. Her research interests include convex optimization and its application to simulation the assembly of deformable structures, mathematical modeling and numerical methods.

S. Lupuleac

S. Lupuleac is an Associate Professor of Applied Mathematics Department and a Head of Virtual Modeling Lab in Peter the Great St. Petersburg Polytechnic University, Russia. He is the Principal Investigator in the series of joint research projects with Airbus Operations SAS under the common name ASRP (Assembly Simulation of Riveting Process), focused on development and application of the methodology for simulation of airframe assembly process. He has co-authored more than 50 scientific publications.