Keywords: multiscale modeling, hierarchical multiscale, global-local method, Jacobian-free Newton-Krylov, preconditioner, massively parallel systems.

The objective of multiscale modeling is to predict the response of complex systems at all relevant spatial and temporal scales at a cost that is sub-linear with respect to the full micro-scale solver. Scale linking is currently performed using hierarchical [1] and concurrent [2] schemes. The global-local type of multiscale methods fall within the category of hierarchical multiscale methods where the stress-strain relationship at every integration point of the macro-scale is computed by suitably deforming an associated representative volume element (RVE). The major advantage of this class of methods is the ability to model arbitrary nonlinearities at the micro-scale as no a priori constitutive assumption is made at the macro-scale. Of particular interest are matrix free methods to minimize memory requirements for a large scale computational problems. In this chapter we will discuss both implicit and explicit global-local multiscale methods, their current developments, challenges and applications to computational mechanics.

For these methods to be widely used in the solution of practical problems it is important to ensure both efficiency and reliability. While there has been excellent progress in the development of multiscale methods, the issue of efficiency has not received sufficient attention. In this chapter we will present current advances in parallelization strategies as well as efficient preconditioners to accelerate global-local methods. For efficient parallelization, a naïve task decomposition based on distributing individual macro-scale integration points to a single group of processors is not optimal and leads to communication overheads and the idling of processors. To overcome this problem, we have developed a coarse-grained parallel algorithm in which groups of macro-scale integration points are distributed to a layer of processors. Each processor in this layer communicates locally with a group of processors that are responsible for the micro-scale computations. The overlapping groups of processors are shown to achieve optimal concurrency at significantly reduced communication overhead.

To accelerate convergence, a multi-level preconditioning strategy based on [3] has also been developed in which at any given Newton step, the information regarding the invariant subspaces in all previous Newton steps as well as the restriction of the Jacobian matrix to those spaces is utilized to effectively deflate the spectrum of the current Jacobian matrix. This preconditioning strategy is powerful as each invariant subspace results in an eigenvalue of the preconditioned Jacobian matrix with a multiplicity which is at least equal to the dimension of that space. This approach neither incurs additional expensive microscale computations for designing the preconditioner nor requires an explicit Jacobian formation.

[1]

S. Yip, "Handbook of materials modeling", XXIX ed. Springer, Berlin, 2005.

[2]

S. Kolhoff, P. Gumbsch, H.F. Fishmeister, "Crack propagation in BCC crystals studied with a combined finite-element and atomistic model", Philosophical Magazine A, 64, 851-878, 1991. doi:10.1080/01418619108213953

[3]

K. Burrage, J. Erhel, B. Pohl, "Restarted GMRES preconditioned by deflation", Journal of Computational and Applied Mathematics, 69, 303–318, 1996. doi:10.1016/0377-0427(95)00047-X

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