Abstract
A framework for the homogenization of nonlinear problems is discussed with respect to block LU factorization of the micro–macro coupled equation, and based on the relation between the characteristic deformation and the Schur-Complement as the homogenized tangent stiffness. In addition, a couple of approximation methods are introduced to reduce the computational cost, i.e., a simple scheme to reuse the old characteristic deformation and a sophisticated method based on the mode-superposition method developed by our group. Note that these approximation methods satisfy the equilibrium conditions in both scales. Then, using a simplified FE model, the conventional algorithm, a relative algorithm originating from the block LU factorization, and the above-mentioned algorithms with the approximated Schur-Complement are compared and discussed. Finally, a large-scale heart simulation using parallel computation is presented, based on the proposed method.
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Abbreviations
- Y, y:
-
Position vector around the deformation in the microstructure
- X, x:
-
Position vector around the deformation in the macrostructure
- u :
-
Macroscopic displacement vector
- {u}:
-
Macroscopic structure nodal displacement vector
- {ue}:
-
Macroscopic structure nodal displacement vector per element
- w :
-
Periodic component of the microscopic displacement vector
- {w}:
-
Periodic component of the nodal displacement vectors of all microstructures
- {wQ}:
-
Periodic component of the nodal displacement vector of a single microstructure
- {we}:
-
Periodic component of the nodal displacement vector per element
- F :
-
The deformation gradient tensor
- Z :
-
The displacement gradient tensor
- C :
-
The right Cauchy–Green tensor
- E :
-
The Green–Lagrange strain tensor
- Π:
-
The first Piola–Kirchhoff stress tensor
- I :
-
The identity tensor
- I c , II c , III c :
-
Principal invariants
- J :
-
Determinant F
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Acknowledgments
Our current research on homogenization method stems from Reference [10] coauthored by Hirohisa Noguchi. We again recognize his enthusiasm and contribution in a broad range of computational mechanics. This work was supported by Core Research for Evolutional Science and Technology, Japan Science and Technology Agency.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Okada, Ji., Washio, T. & Hisada, T. Study of efficient homogenization algorithms for nonlinear problems. Comput Mech 46, 247–258 (2010). https://doi.org/10.1007/s00466-009-0432-1
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DOI: https://doi.org/10.1007/s00466-009-0432-1