Advances in Dynamic Relaxation Techniques for Nonlinear Finite Element Analysis

Abstract

Traditionally, the finite element technique has been applied to static and steady-state problems using implicit methods. When nonlinearities exist, equilibrium iterations must be performed using Newton-Raphson or quasi-Newton techniques at each load level. In the presence of complex geometry, nonlinear material behavior, and large relative sliding of material interfaces, solutions using implicit methods often become intractable. A dynamic relaxation algorithm is developed for inclusion in finite element codes. The explicit nature of the method avoids large computer memory requirements and makes possible the solution of large-scale problems. The method described approaches the steady-state solution with no overshoot, a problem which has plagued researchers in the past. The method is included in a general nonlinear finite element code. A description of the method along with a number of new applications involving geometric and material nonlinearities are presented.

Explicit finite deformation analysis of isogeometric membranes

“…In order to obtain solutions of the shell/membrane structures at steady state, a dynamic relaxation method [17,46] is employed and combined with a central difference explicit time integration algorithm [46]. Firstly, one can formulate the discrete equations of motion of a structure as follows:…”

“…Due to the high order k C (k ≥ 1) continuity of NURBS shape functions the Kirchhoff-Love theory can be seamlessly implemented. An explicit time integration scheme [44] is used to compute the transient response of the membrane structures to time-domain excitations, and a dynamic relaxation method [17,45,46] is employed to obtain steady-state solutions. The performance of the current isogeometric formulation and implementation is assessed through a series of examples featuring elastic and hyperelastic materials, and also compared to analytical, experimental and "traditional" (i.e.…”

Incompressible flow strategies for ice creep

“…This section presents a brief overview of the DR algorithm that has been implemented. Further information is given in [34].…”

Adaptive damping for dynamic relaxation problems with non‐monotonic spectral response

“…The method is simple and e ective for non-linear problems, and it has been used for some time in structural applications [1][2][3][4][5][6][7][8]. The DR method was also considered in parallel computing environments [9], and more recently, a comprehensive account of DR has been given with respect to its implementation for single processor [10,11] and parallel processor [12,13] computers.…”