LATIN: A new view and an extension to wave propagation in nonlinear media

Abstract

The LATIN (acronym of LArge Time INcrement) method was originally devised as a non-incremental procedure for the solution of quasi-static problems in continuum mechanics with material nonlinearity. In contrast to standard incremental methods like Newton and modified Newton, LATIN is an iterative procedure applied to the entire loading path. In each LATIN iteration, two problems are solved: a local problem, which is nonlinear but algebraic and miniature, and a global problem, which involves the entire loading process but is linear. The convergence of these iterations, which has been shown to occur for a large class of nonlinear problems, provides an approximate solution to the original problem. In this paper, the LATIN method is presented from a different viewpoint, taking advantage of the causality principle. In this new view, LATIN is an incremental method, and the LATIN iterations are performed within each load step, similarly to the way that Newton iterations are performed. The advantages of the new approach are discussed. In addition, LATIN is extended for the solution of time-dependent wave problems. As a relatively simple model for illustrating the new formulation, lateral wave propagation in a flat membrane made of a nonlinear material is considered. Numerical examples demonstrate the performance of the scheme, in conjunction with finite element discretization in space and the Newmark trapezoidal algorithm in time. 126D. GIVOLI, R. BHARALI AND L. SLUYS 2. A 'local' problem, which consists of the nonlinear algebraic relations, and an auxiliary equation, which determines a progress direction. This problem is nonlinear but algebraic and miniature.The convergence of these iterations, which has been shown to occur for a large class of nonlinear problems [2,4], provides an approximate solution to the original problem.The terms 'global' and 'local' are used here as in the LATIN literature. Namely, a global problem is a problem involving differential equations, whereas a local problem is a problem involving only algebraic relations. We stress this point, because in other contexts (such as absorbing boundary conditions or certain material models in plasticity that prevent mesh-dependent behavior), the word 'nonlocal' refers to an integral operator, whereas a 'local relation' means that the relation involves only functions and derivatives at a single point.The LATIN method was applied to quasi-static problems in damage mechanics [5], anisotropic plasticity [6], cyclic viscoplasticity [7,8], poroelasticity [9], and rigid plasticity (with a special modified implementation) [10]. It was also applied to sheet cutting problems [11,12], to thermomechanical problems [13], to fatigue crack propagation, in conjunction with XFEM [14], and to tensegrity media [15].Most of the papers on LATIN consider such nonlinearity that after finite element (FE) discretization leads to a single-valued load vector as a function of the displacement vector. In [16], the LATIN method was extended to deal with more general nonlinear behavior, na...

Phase-field fracture irreversibility using the slack variable approach

“…The scope for future studies include the extension towards complex multi-physics problems, large-scale problems, that maybe require special preconditioning techniques. Other future research directions may be towards accelerating the convergence rate using Anderson acceleration [50] or Large time increment method [51,52].…”