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MULTIGRID PRECONDITIONED CONJUGATE GRADIENT METHOD IN THE MECHANICAL ANALYSIS OF HETEROGENEOUS SOLIDS

• A fast solver method called the multigrid preconditioned conjugate gradient method is proposed for the mechanical analysis of heterogeneous materials on the mesoscale. Even small samples of a heterogeneous material such as concrete show a complex geometry of different phases. These materials can be modelled by projection onto a uniform, orthogonal grid of elements. As one major problem theA fast solver method called the multigrid preconditioned conjugate gradient method is proposed for the mechanical analysis of heterogeneous materials on the mesoscale. Even small samples of a heterogeneous material such as concrete show a complex geometry of different phases. These materials can be modelled by projection onto a uniform, orthogonal grid of elements. As one major problem the possible resolution of the concrete specimen is generally restricted due to (a) computation times and even more critical (b) memory demand. Iterative solvers can be based on a local element-based formulation while orthogonal grids consist of geometrical identical elements. The element-based formulation is short and transparent, and therefore efficient in implementation. A variation of the material properties in elements or integration points is possible. The multigrid method is a fast iterative solver method, where ideally the computational effort only increases linear with problem size. This is an optimal property which is almost reached in the implementation presented here. In fact no other method is known which scales better than linear. Therefore the multigrid method gains in importance the larger the problem becomes. But for heterogeneous models with very large ratios of Young's moduli the multigrid method considerably slows down by a constant factor. Such large ratios occur in certain heterogeneous solids, as well as in the damage analysis of solids. As solution to this problem the multigrid preconditioned conjugate gradient method is proposed. A benchmark highlights the multigrid preconditioned conjugate gradient method as the method of choice for very large ratio's of Young's modulus. A proposed modified multigrid cycle shows good results, in the application as stand-alone solver or as preconditioner.

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Document Type: Conference Proceeding Stefan Häfner, Carsten KönkeORCiDGND https://doi.org/10.25643/bauhaus-universitaet.2962Cite-Link https://nbn-resolving.org/urn:nbn:de:gbv:wim2-20170327-29626Cite-Link http://euklid.bauing.uni-weimar.de/ikm2006/index.php_lang=de&what=papers.html Klaus GürlebeckGND, Carsten KönkeORCiDGND English 2017/03/24 2006/07/14 2017/03/27 Bauhaus-Universität Weimar Bauhaus-Universität Weimar Fakultät Bauingenieurwesen / Institut für Strukturmechanik 29 Architektur ; CAD; Computerunterstütztes Verfahren 500 Naturwissenschaften und Mathematik / 510 Mathematik 56 Bauwesen / 56.03 Methoden im Bauingenieurwesen Bauhaus-Universität Weimar / Internationales Kolloquium über Anwendungen der Informatik und Mathematik in Architektur und Bauwesen, IKM, Weimar / Internationales Kolloquium über Anwendungen der Informatik und Mathematik in Architektur und Bauwesen, IKM, Weimar, 17. 2006 Creative Commons 4.0 - Namensnennung-Nicht kommerziell (CC BY-NC 4.0)