An accelerated Uzawa method for application to frictionless contact problem
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An accelerated Uzawa method for application to frictionless contact problem Yoshihiro Kanno1 Received: 28 November 2016 / Accepted: 11 September 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract The Uzawa method is a method for solving constrained optimization problems, and is often used in computational contact mechanics. The simplicity of this method is an advantage, but its convergence is slow. This paper presents an accelerated variant of the Uzawa method. The proposed method can be viewed as application of an accelerated projected gradient method to the Lagrangian dual problem. Preliminary numerical experiments suggest that the convergence of the proposed method is much faster than the original Uzawa method. Keywords Fast first-order method · Accelerated gradient scheme · Contact mechanics · Nonsmooth mechanics · Uzawa method · Convex optimization
1 Introduction It has been recognized well that contact mechanics has close relation with optimization and variational inequalities [6,28]. The static frictionless contact problem of a linear elastic body, also called Signorini’s problem, is one of the most fundamental problems in contact mechanics. This is a boundary value problem to find the equilibrium configuration of an elastic body, where some portion of the boundary of the body can possibly touch the surface of a rigid obstacle (or the surface of another elastic body). Positive distance between the elastic body and the obstacle surface (i.e., positive gap) implies zero contact pressure (i.e., zero reaction), while nonzero reaction implies zero gap. This disjunction nature can be described by using complementarity conditions. Moreover, the frictionless contact problem can be formulated as a continuous optimization problem under inequality constraints [28]. The Uzawa method is known as a classical method for solving constrained optimization problems [2,5,26]. Due to ease in implementation, the Uzawa method is often
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Yoshihiro Kanno [email protected] Mathematics and Informatics Center, The University of Tokyo, Hongo 7-3-1, Tokyo 113-8656, Japan
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applied to contact problems [10,12,18–20,22,23,27]. Major drawback of the Uzawa method is that its convergence is slow; it exhibits only linear convergence in general. Recently, accelerated, or “optimal” [14,15], first-order methods have received substantial attention, particularly for solving large-scale convex optimization problems; see, e.g., Beck and Teboulle [3], Becker et al. [4], O’Donoghue and Candés [16], and Lee and Sidford [13]. Advantages of most of these methods include ease of implementation, cheap computation per each iteration, and fast local convergence. Application of an accelerated first-order method to computational mechanics can be found in Kanno [11] and Shimizu and Kanno [21]. In this paper, we apply the acceleration scheme in Beck and Teboulle [3] to the Uzawa method. The paper is organized as follows. Section 2 provides an overview of necessary backgrounds of the frictionless contact problem
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