An Adaptive Parallel Arc-Length Method

TL;DR

The paper addresses the serial bottleneck of quasi-static arc-length continuations by introducing the Adaptive Parallel Arc-Length Method (APALM), which reparameterizes the solution path with arc-length and partitions it into parallelizable sub-intervals. A multi-level framework, driven by error measures that compare coarse and fine arc-lengths, guides refinement where the load-displacement path is highly nonlinear. Three implementations are described—ASALM (serial), ASPALM (serial initialization with parallel corrections), and APALM (fully parallel corrections)—and demonstrated on isogeometric Kirchhoff-Love shells exhibiting snap-through and bifurcations. Results show APALM can reproduce reference paths with adaptive refinements and achieve meaningful speedups with modest numbers of workers, validating its potential for large-scale quasi-static analyses and future extensions to path exploration and space-time refinement.

Abstract

Parallel computing is omnipresent in today's scientific computer landscape, starting at multicore processors in desktop computers up to massively parallel clusters. While domain decomposition methods have a long tradition in computational mechanics to decompose spatial problems into multiple subproblems that can be solved in parallel, advancing solution schemes for dynamics or quasi-statics are inherently serial processes. For quasi-static simulations, however, there is no accumulating 'time' discretization error, hence an alternative approach is required. In this paper, we present an Adaptive Parallel Arc-Length Method (APALM). By using a domain parametrization of the arc-length instead of time, the multi-level error for the arc-length parametrization is formed by the load parameter and the solution norm. By applying local refinements in the arc-length parameter, the APALM refines solutions where the non-linearity in the load-response space is maximal. The concept is easily extended for bifurcation problems. The performance of the method is demonstrated using isogeometric Kirchhoff-Love shells on problems with snap-through and pitch-fork instabilities. It can be concluded that the adaptivity of the method works as expected and that a relatively coarse approximation of the serial initialization can already be used to produce a good approximation in parallel.

Figures (13)

• Figure 1: Load (left), displacement (middle), and arc-length control (right) for structural analysis problems. The question mark (?) indicates the iteration where load and displacement control encounter a limit point. In these situations, the next point obtained is typically difficult to find.
• Figure 2: Concept of the APALM. The large open circles represent reference solutions from a previously computed level. The small solid circles represent new data on the interval between two reference solutions, computed by the arc-length method (here the large dashed circle). The black dashed line indicates the curve estimation for which the sum is equal to the total curve length.
• Figure 3: Error measures on a nearly straight interval (a) and a curved interval (b). For the nearly straight interval, the distance $\delta L$ (see b) is sufficiently small, whereas for the curved interval, it is too big. The measures $\Delta L$ and $\Delta L'$, $\overline{\Delta L}$, and $\delta L$ are, respectively, the coarse arc length, the fine arc length, the lower distance, and the absolute error.
• Figure 4: Domain parameterizations on the curve-length domain $s$ and the parameter domain $\xi$ with levels $\ell$ and $\ell+1$. \ref{['fig:APALM_domain_ori']} illustrates the original domain, \ref{['fig:APALM_domain_interior']} illustrates the insertion of interior points in the case of a sufficiently close approximation of the end-point of the domain and \ref{['fig:APALM_domain_stretch']} illustrates the full insertion of all sub-domain solutions combined with the stretching of the curve length domain.
• Figure 5: The data structure behind the APALM. The axes represent data sets, which are monotonically increasing when the axis has an arrow. Solid arrows represent mappers from one axis to another, and dashed arrows represent data references. The mappers $\Xi(s_i)$ and $\mathcal{S}(\xi_i)$ map between the curve parametrisation and the curve length axes. The former takes a curve length $s_i$ and returns the curve parameter $\xi_i$, and the latter maps the inverse. The mappers $\mathcal{U}(\xi_k)$ and $\mathcal{U}'(\xi_k)$ return the solution $\vb*{w}_j$ and the previous solution $(\vb*{w}')_j$, respectively, given a parametric coordinate $\xi_j$, and the mapper $\mathcal{L}(\xi_j)$ returns the level on which the coordinate $\xi_j$ was computed. The guess is a data reference to the previous solution. The thick solid intervals represent running jobs assigned with an ID, and the thick dashed intervals represent queued intervals. Each interval is represented by a start-point and an end-point tuple $(\xi_l,\xi_{l+1})$. The red lines, squares, and arros represent the submit operation when solutions are added to the data structure.

Theorems & Definitions (2)

• Remark 1: Difference with parallel-in-time methods
• Remark 2: Path-dependency