Convex optimization techniques in compliant assembly simulation
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Convex optimization techniques in compliant assembly simulation Maria Stefanova1 · Olga Minevich2 · Stanislav Baklanov1 · Margarita Petukhova1 · Sergey Lupuleac1 · Boris Grigor’ev1 · Michael Kokkolaras3 Received: 13 June 2019 / Revised: 19 February 2020 / Accepted: 19 February 2020 © The Author(s) 2020
Abstract A special class of quadratic programming (QP) problems is considered in this paper. This class emerges in simulation of assembly of large-scale compliant parts, which involves the formulation and solution of contact problems. The considered QP problems can have up to 20,000 unknowns, the Hessian matrix is fully populated and ill-conditioned, while the matrix of constraints is sparse. Variation analysis and optimization of assembly process usually require massive computations of QP problems with slightly different input data. The following optimization methods are adapted to account for the particular features of the assembly problem: an interior point method, an active-set method, a Newton projection method, and a pivotal algorithm for the linear complementarity problems. Equivalent formulations of the QP problem are proposed with the intent of them being more amenable to the considered methods. The methods are tested and results are compared for a number of aircraft assembly simulation problems. Keywords Quadratic programming · Aircraft assembly · Contact problem · Variation simulation analysis · Massive computations
* Maria Stefanova [email protected] * Olga Minevich [email protected] 1
Peter the Great St. Petersburg Polytechnic University, Saint Petersburg, Russia
2
Technische Universität München, Munich, Germany
3
McGill University, Montreal, Canada
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1 Introduction In the last decade a new modeling approach has been developed and applied to variation simulation and assembly optimization in aerospace and automotive industry (see Lupuleac et al. 2010, 2011, 2019b; Dahlström and Lindkvist 2007; Lindau et al. 2016; Yang et al. 2016). This approach considers the contact interaction between compliant parts. By using the variational formulation (Galin 1961; Tu and Gazis 1964; Lions and Stampacchia 1967; Kinderlehrer and Stampacchia 1980) and the substructuring (Turner et al. 1956; Guyan 1965; Wriggers 2006; Petukhova et al. 2014), contact detection is reduced to a quadratic programming problem that allows for large-scale computations of contact problems during variation simulation and assembly optimization. Since the accuracy of the assembly model influences the simulation results, it is important to use fine FEM meshes that can reflect the most essential features of assemblies. Implementation of highly refined computational meshes and need for massive computations of problems with slightly different input data require choosing appropriately tailored solvers for the QP problem; that is the primary objective of the present paper. The paper is organized as follows: in Sect. 2 the contact problem arising in assembly simulation is presented and its
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