Carlsson, kristoffer

Larsson, Fredrik

Runesson, Kenneth

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On the use of flux boundary conditions in FE2 procedures withseamless scale-bridgingKristoffer 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 and study the possibilitiesto obtain upper and lower bounds on the the overall (two-scale) solution. The strategy presented byLarsson 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, and thereby studying the solution for the lowerbound effective conductivity, while satisfying the conformity requirement, we propose a novel mixeddual variational format. In this format, the finite element multiscale problem, sometimes denoted”Finite Element squared” (FE2), (i) incorporates the flux boundary conditions when computationalhomogenization is adopted, and (ii) is a conforming approximation when the subscale fluctuation isfully 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 (upper bound) or the dual format and flux boundaryconditions on SVE’s (lower bound) while allowing for the transition from homogenization tosingle-scale analysis.We compare the numerical performance of the two different variational settings (and consequentboundary conditions) in the FE2 context where the scale separation is successively diminished. Inparticular, we compare the convergence properties for FE-refinement on both ”scales” and discussthe possibilities to utilize the two approximations in a strategy 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).

On the Use of Flux Boundary Conditions in FE2 Procedures with Seamless Scale-bridging

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On the Use of Flux Boundary Conditions in FE2 Procedures with Seamless Scale-bridging

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