A Matrix-Free Newton–Krylov Parallel Implicit Implementation of the Absolute Nodal Coordinate Formulation

Upper part: beam with three elements characterized by the generalized coordinates e₁ through e₄. In the approach adopted herein, the elements in a beam are separated; each element has a set of 12 generalized coordinates associated with it (lower part), which leads to the presence of some simple kinematic constraints (Eq. (11)). This decouples the problem and increases opportunities for parallelism at the price of a larger albeit sparser problem.

Displacement in the x-direction of the pendulum tip (ANCF, abaqus, and FEAP comparison)

The y-position of the tip of the cantilever beam subject to a concentrated force of 60 N down the negative y-direction plotted against the number of elements used to discretize the beam. Equilibrium is reached by the numerical solution (continuous line) at −0.1016 m and by the analytical result (broken line) at −0.1019 m. Closer agreement can be obtained by redefining the curvature in Eq. (8) according to the work performed in Ref. [26].

Advancing the numerical solution in time draws on three loops: the outer one marches forward in time with step size h; at each time step t_n+1, the second loop solves the algebraic discretization problem in Eq. (18) using a stopping test controlled by the parameter εNS. Each iteration in this second loop launches a third loop whose role is that of producing the correction δn+1(k) in the value of the acceleration and Lagrange multipliers. The corrections are computed at various degrees of accuracy by controlling the value of the residual εNS(k) in the stopping test of the BICGSTAB iterative solver [17] applied for the linear system in Eq. (22).

Displacement in the y-direction of the pendulum tip (ANCF, abaqus, and FEAP comparison)

Displacement in the z-direction of the pendulum tip (ANCF, abaqus, and FEAP comparison)

Conservation of total energy for the time evolution of a flexible pendulum using the Newmark integration algorithm

The number of iterations in the nonlinear Newton solver is shown as a function of time. The solver typically converges within 2–4 iterations.

A system containing 101,025 beam elements constrained together with 640,146 constraints to form a net configuration. The net was constrained along one of the edges and subjected to a gravitational acceleration of 9.81 m/s² in the negative y direction. An animation of the simulation can be found at http://sbel.wisc.edu/Animations/#47

The computational time for net models of several sizes were recorded using several different GPUs. The time it took to simulate ten time steps of the simulation was plotted as a function of the number of beams required to create the net. The scaling analysis went up to 0.5 × 10⁶ ANCF elements.