A displacement-controlled arc-length solution scheme

Abstract

Tracing load-displacement paths in structural mechanics problems is complicated in the presence of critical points of instability where conventional load-or displacement control fails. To deal with this, arc-length methods have been developed since the 1970s, where control is taken over increments of load at these critical points, to allow full transit of the load-displacement path. However, despite their wide use and incorporation into commercial finite element software, conventional arc-length methods still struggle to cope with non-zero displacement constraints. In this paper we present a new displacement-controlled arc-length method that overcomes these shortcomings through a novel scheme of constraints on displacements and reaction forces. The new method is presented in a variety of serving suggestions, and is validated here on six very challenging problems involving truss and continuum finite elements. Despite this paper's focus on structural mechanics, the new procedure can be applied to any problems that involve nonhomogeneous Dirichlet constraints and challenging equilibrium paths.

On the implementation of a material point‐based arc‐length method

“…As is the case for many arc‐length papers in the current literature, the arc‐length method has only been presented for the problems with load control. The case of displacement controlled problems is much more involved with an alternative arc‐length constraint equation needed, please see Pretti et al 16 for more details on the formulation of a displacement controlled arc‐length method.…”

“…, this ensures that if the required Newton-Raphson process continues to exceed the desired number of iterations, the arc length will not continue to decrease so much that the tracing of the equilibrium path begins to halt as the allowed distance becomes tiny. There is no theory on the choice of value of 𝛾, therefore 𝛾 = 4 is chosen here based on Pretti et al 16 In the first iteration of each load step, there is clearly no information as to the change in nodal displacements, Δu (0) = 0. In this case, solving (4) produces the solutions for the initial load factor increment of…”

“…The value of $$$ \gamma $$$ controls the minimum allowed arc length through the entire simulation such that $$$ \Delta {l}_i $$$ is a value in the interval $$$ \left[\frac{\Delta {l}_0}{\gamma },\Delta {l}_0\right] $$$, this ensures that if the required Newton‐Raphson process continues to exceed the desired number of iterations, the arc length will not continue to decrease so much that the tracing of the equilibrium path begins to halt as the allowed distance becomes tiny. There is no theory on the choice of value of $$$ \gamma $$$, therefore $$$ \gamma =4 $$$ is chosen here based on Pretti et al 16 …”

“…These tractions and body forces result in a Cauchy stress field, 𝜎 ij , through the body. Discretising (16) with background mesh elements, E, and imposing Galerkin's weak form produces…”

“…6 Crisfield 7 adapted the method proposed by Riks such that the arc-length method is more suited for use in the FEM when used with the modified Newton-Raphson method. Since its conception, many variations of the arc-length method have been proposed to solve complex equilibrium paths including (but not limited to) Bergan et al, 8 Ramm, 9 Powell and Simons, 10 Lam and Morley, 11 Fafard and Massicotte, 12 Krenk, 13 Nicholson, 14 May 15 and Pretti et al 16 The essence of the arc-length method comes from the constraint equation, or arc-length equation, which is included in the non-linear equations of the problem and solved to give an incremental load factor and corresponding incremental displacements. By solving for both load and displacement throughout the simulation, the issues which may arise in traditional load/displacement controlled schemes due to snap-through or snap-back responses are mitigated.…”