Carlsson, Kristoffer

Larsson, Fedrik

Runesson, Kenneth

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Comparing FE2 procedures with seamless scale-bridging using aprimal and dual formulationKristoffer Carlsson∗, Fredrik Larsson∗, and Kenneth Runesson∗∗ Chalmers University of TechnologyDepartment of Industrial and Material SciencesSE-412 96 Gothenburg, Swedenweb page: https://www.chalmers.se/en/departments/ims/ABSTRACTWe consider discretization-based homogenization of micro-heterogeneous stationary heat conductionproblems within the wider framework of Variationally Multiscale methods. The strategy presentedby Larsson and Runesson[1] is adopted, whereby the solution procedure for the fine-scale fluctuationsis based on the local mesh-size in the macroscopic finite element problem. This allows for bridgingthe extremes – from single scale analysis to computational homogenization – seamlessly within onefinite element simulation.One crucial component in two-scale analysis, often denoted ”Finite Element squared” (FE2) is theprolongation of the macroscopic fields onto the subscale problem. In computational homogenizationon Statistical Volume Elements (SVEs), it is well known that temperature (Dirichlet) and flux(Neumann) boundary conditions result in upper and lower bounds, respectively, on the effective heatconductivity. However, in the standard (primal) variational format, it is only the Dirichlet boundaryconditions that satisfy the conformity requirement in the standard (primal) variational form. Hence,other choices of boundary conditions (e.g. flux or periodic boundary conditions) are not directlyapplicable if one wants to consider connected adjacent SVE’s without assuming (infinite) separationof scales.In order to allow for flux boundary conditions, while satisfying the conformity requirement, wepropose a novel mixed dual variational format. In this format, the finite element multiscale problem,sometimes denoted ”Finite Element squared” (FE2), (i) incorporates the flux boundary conditionswhen computational homogenization is adopted, and (ii) is a conforming approximation when thesubscale fluctuation is fully resolved within each macroscopic element.We are thus able to compute the FE2 approximation using either the (standard) primal format andtemperature boundary conditions on SVE’s or the dual format and flux boundary conditions onSVE’s while allowing for the transition from homogenization to single-scale analysis.We compare the numerical performance of the two different variational settings (and consequentboundary conditions) in the FE2 context with different choices of discretization on the macroscale and choices of ”SVE-quadrature”. In particular, we compare the convergence properties forFE-refinement on both ”scales” and discuss the possibilities to utilize the two approximations in astrategy to control the approximation error.REFERENCES[1] Larsson F. and Runesson K. Adaptive bridging of scales in continuum modeling based on errorcontrol. International Journal for Multiscale Computational Engineering, pp. 371-392, (2008).

Comparing FE2 procedures with seamless scale-bridging using a primal and dual formulation

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Comparing FE2 procedures with seamless scale-bridging using a primal and dual formulation

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