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A unified methodology to solve problems of frictionless unilateral contact as well as adhesive contact between linear elastic solids is presented. This methodology is based on energetic principles and is casted to a minimization problem of the total potential energy. Appropriate boundary integral forms of the energy are defined and the quadratic problem form of the contact problem is proposed. The problem is solved by the collocation boundary element method (BEM). To solve the quadratic problem two algorithms are developed, both being variants of the well-known conjugate gradient algorithm. The difference between them is given by the explicit construction or not of the quadratic-problem matrix. This matrix has the same physical meaning as the stiffness matrix commonly used in the context of the finite element method (FEM). Both symmetric and non-symmetric formulations of this matrix are presented and discussed, showing that the non-symmetric one provides more accurate results. The present procedure, in addition to its interest by itself, can also be extended to problems where dissipative phenomena take place such as friction, damage and plasticity. Essential steps of the numerical implementation are briefly presented and the numerical solutions of some standard problemsof frictionless contact are given and compared to those obtained by other well-known BEM and FEM procedures for contact problems.

The purpose of this review article is to push amplitude equations as far as possible from threshold. We focus on the Stuart–Landau amplitude equation describing the supercritical Hopf bifurcation of the flow in the wake of a cylinder for critical Reynolds number . After having reviewed Stuart's weakly nonlinear multiple-scale expansion method, we first demonstrate the crucial importance of the choice of the critical parameter. For the wake behind a cylinder considered in this paper, choosing instead of considerably improves the prediction of the Landau equation. Although Sipp and Lebedev (2007 J. Fluid Mech593 333–58) correctly identified the adequate bifurcation parameter ϵ, they have plotted their results adding an additional linearization, which amounts to using as approximation to ϵ. We then illustrate the risks of calculating ‘running’ Landau constants by projection formulas at arbitrary values of the control parameter. For the cylinder wake case, this scheme breaks down and diverges close to . We propose an interpretation based on the progressive loss of the non-resonant compatibility condition, which is the cornerstone of Stuart's multiple-scale expansion method. We then briefly review a self-consistent model recently introduced in the literature and demonstrate a link between its properties and the above-mentioned failure. [ABSTRACT FROM AUTHOR]