DOMAIN DECOMPOSITION METHODS APPLIED TO SOLVE FRICTIONLESS-CONTACT PROBLEMS FOR MULTILAYER ELASTIC BODIES A. Ya. Grigorenko 1 , I. I. Dyyak 2 , S. I. Matysyak 2 , and I. I. Prokopyshyn 2 A parallel Dirichlet–Dirichlet domain-decomposition algorithm for solving frictionless-contact problems for elastic bodies made of composite materials is proposed and justified. Numerical results that demonstrate the effectiveness of the approach and its software implementation are presented Keywords: frictionless contact, elastic composite body, parallel Dirichlet–Dirichlet algorithm, domain decomposition method, finite-element method, layered composite Introduction. Solving contact problems of solid mechanics is an important task for many branches such as mechanical engineering and construction. Analytic and numerical solutions of contact problems are outlined in [6, 17]. Numerical approaches based on spline-approximation [13–16] are widely used to solve problems of elasticity. Effective numerical methods for solving contact problems are based on the theory of variational inequalities [1, 4, 8, 18, 19]. The development of domain-decomposition methods (DDMs) in the last decade provided a basis for new research of contact problems for many bodies. Parallel computing is a promising way to solve contact problems. The existing domain-decomposition approaches to solving contact problems are briefly reviewed in [22]. The numerical approaches based on the penalty method and the method of Lagrangian multipliers are most widely used to solve contact problems. The basic shortcoming of the penalty method is that it produces a stiff system of linear algebraic equations. In what follows, we will use Lagrange’s variational principle, the penalty method, and the successive iteration method to develop parallel Neumann and Dirichlet domain-decomposition algorithms for solving the variational equations of frictionless-contact problems for several bodies. In this case, the iterative process involves both iteration in subdomains and finding of contact regions. We will analyze numerically these algorithms for a plane contact problem for two bodies, using the finite-element method with linear and quadratic triangular elements. 1. Formulation of Contact Problem. Consider N contacting elastic bodies W a Ì R 3 with piecewise-smooth boundaries G W a a =¶ , a= 1, N (Fig. 1). Denote W W = = a a 1 N U . The elastic strain state of each body is described by a displacement vector u e a a = = å u i i i 1 3 , a strain tensor $ ,  a a e = = å ij i j ij ee 1 3 , and a stress tensor $ ,  a a s = = å ij i j ij ee 1 3 . The strains are expressed in terms of the displacements by the Cauchy relations: e a a a ij i j j i u x u x = ¶ ¶ + ¶ ¶ æ è ç ç ö ø ÷ ÷ 1 2 (, , ) ij = 13. (1) International Applied Mechanics, Vol. 46, No. 4, 2010 388 1063-7095/10/4604-0388 ©2010 Springer Science+Business Media, Inc. 1 S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, 3 Nesterov St., Kyiv, Ukraine 03057, e-mail: ayagrigorenko@yandex.ru. 2 Ivan Franko Lviv National University; e-mail: ivan_lv@yahoo.com. Translated from Prikladnaya Mekhanika, Vol. 46, No. 4, pp. 25–37, April 2010. Original article submitted May 20, 2009.