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Computational Science, Engineering & Technology Series
ISSN 17593158 CSETS: 25
DEVELOPMENTS AND APPLICATIONS IN COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping, J.M. Adam, F.J. Pallarés, R. Bru and M.L. Romero
Chapter 2
Recent Advances in GlobalLocal Multiscale Methods for Computational Mechanics S. De and Rahul
Advanced Computational Research Laboratory, Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy NY, United States of America S. De, Rahul, "Recent Advances in GlobalLocal Multiscale Methods for Computational Mechanics", in B.H.V. Topping, J.M. Adam, F.J. Pallarés, R. Bru and M.L. Romero, (Editors), "Developments and Applications in Computational Structures Technology", SaxeCoburg Publications, Stirlingshire, UK, Chapter 2, pp 2547, 2010. doi:10.4203/csets.25.2
Keywords: multiscale modeling, hierarchical multiscale, globallocal method, Jacobianfree NewtonKrylov, preconditioner, massively parallel systems.
Summary
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 sublinear with respect to the full microscale solver. Scale linking is currently performed using hierarchical [1] and concurrent [2] schemes. The globallocal type of multiscale methods fall within the category of hierarchical multiscale methods where the stressstrain relationship at every integration point of the macroscale 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 microscale as no a priori constitutive assumption is made at the macroscale. 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 globallocal 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 globallocal methods. For efficient parallelization, a naïve task decomposition based on distributing individual macroscale 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 coarsegrained parallel algorithm in which groups of macroscale 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 microscale computations. The overlapping groups of processors are shown to achieve optimal concurrency at significantly reduced communication overhead. To accelerate convergence, a multilevel 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. References
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