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A novel approach to computational homogenization and its application to fully coupled two-scale thermomechanics

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Abstract

The paper introduces a novel approach to computational homogenization by bridging the scales from microscale to macroscale. Whenever the microstructure is in an equilibrium state, the macrostructure needs to be in equilibrium, too. The novel approach is based on the concept of representative volume elements, stating that an assemblage of representative elements should be able to resemble the macrostructure. The resulting key assumption is the continuity of the appropriate kinematic fields across both scales. This assumption motivates the following idea. In contrast to existing approaches, where mostly constitutive quantities are homogenized, the balance equations, that drive the considered field quantities, are homogenized. The approach is applied to the fully coupled partial differential equations of thermomechanics solved by the finite element (FE) method. A novel consistent finite homogenization element is given with respect to discretized residual formulations and linearization terms. The presented FE has no restrictions regarding the thermomechanical constitutive laws that are characterizing the microstructure. A first verification of the presented approach is carried out against semi-analytical and reference solutions within the range of one-dimensional small strain thermoelasticity. Further verification is obtained by a comparison to the classical FE\(^2\) method and its different types of boundary conditions within a finite deformation setting of purely mechanical problems. Furthermore, the efficiency of the novel approach is investigated and compared. Finally, structural examples are shown in order to demonstrate the applicability of the presented homogenization framework in case of finite thermo-inelasticity at different length scales.

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Abbreviations

\(\bigcup \) :

Assemble operator

\(\mid \) :

Condition on the given set

\(\mathrm {D}\) :

Gateaux derivative

\(\wedge \) :

Logical and

\({\overline{\Box }}\) :

Macroscopic quantity, e.g., \({\overline{{\varvec{\sigma }}}}\)

\(\dot{\Box }\) :

Material time derivative

\(\partial _{x}\) :

Partial derivative with respect to x

\(\partial _{xy}^2\) :

Second order partial derivative with respect to x, y

\({ \mathrm div }\) :

Spatial divergence operator

\(\text {grad}\) :

Spatial gradient operator

\(\text {sym}\) :

Symmetry operator

\(\text {d}\) :

Total derivative

\({\Box }^T\) :

Transposition operator

\(\forall \) :

Universal quantifier

\(\theta \) :

Absolute temperature

\({\varvec{\mathrm {K}}}\) :

Assembled stiffness matrix

\({\varvec{\sigma }}\) :

Cauchy stress tensor

\(\varvec{\mathcal {B}}_t\) :

Current configuration

\({{\varvec{g}}}\) :

Current metric tensor

\({{\varvec{x}}}\) :

Current position vector

\({{\varvec{F}}}\) :

Deformation gradient

J :

Determinant of deformation gradient

\({{\varvec{u}}}\) :

Displacement vector

\(w_{\text {ext}}\) :

External power term

r :

Internal heat source

\(w_{\text {int}}\) :

Internal power term

\({{\varvec{q}}}\) :

Spatial heat flux vector

\(q_n\) :

Spatial heat flow

\({{\varvec{t}}}\) :

Spatial surface traction

\({{\varvec{d}}}\) :

Symmetric part of spatial velocity gradient

\(\vartheta \) :

Temperature change with respect to reference temperature

\(q_p\) :

Thermal power

\({\varvec{\mathrm {R}}}\) :

Vector of residuals

\({\varvec{\mathrm {x}}}\) :

Vector of unknowns

BVDH:

Boundary value driven approach to computational homogenization

FE:

Finite element

FEM:

Finite element method

LDBC:

Linear displacement boundary conditions

PDBC:

Periodic displacement boundary conditions

PDE:

Partial differential equation

RVE:

Representative volume element

SST:

Substructure

UTBC:

Uniform traction boundary conditions

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Acknowledgments

This research is financially supported by the Deutsche Forschungsgemeinschaft (DFG) under Contract KA-1163/7, which is gratefully acknowledged by the authors.

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Authors and Affiliations

  1. Institute for Structural Analysis, Technische Universität Dresden, 01062, Dresden, Germany

    Robert Fleischhauer, Marko Božić & Michael Kaliske

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  1. Robert Fleischhauer
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  2. Marko Božić
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  3. Michael Kaliske
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Correspondence to Michael Kaliske.

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Fleischhauer, R., Božić, M. & Kaliske, M. A novel approach to computational homogenization and its application to fully coupled two-scale thermomechanics. Comput Mech 58, 769–796 (2016). https://doi.org/10.1007/s00466-016-1315-x

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  • DOI: https://doi.org/10.1007/s00466-016-1315-x

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