Abstract
The present manuscript focusses on computational aspects of dispersive computational continua (\(C^2\)) formulation previously introduced by the authors. The dispersive \(C^2\) formulation is a multiscale approach that showed strikingly accurate dispersion curves. However, the seemingly theoretical advantage may be inconsequential due to tremendous computational cost involved. Unlike classical dispersive methods pioneered more than a half a century ago where the unit cell is quasi-static and provides effective mechanical and dispersive properties to the coarse-scale problem, the dispersive \(C^2\) gives rise to transient problems at all scales and for all microphases involved. An efficient block time-integration scheme is proposed that takes advantage of the fact that the transient unit cell problems are not coupled to each other, but rather to a single coarse-scale finite element they are positioned in. We show that the computational cost of the method is comparable to the classical dispersive methods for short load durations.
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Notes
The unit cells geometrical configuration is usually selected as a box-like, such as rectangle in 2D or cuboid in 3D. This imposes a restriction on modeling the physical domain of the problem by spatial periodicity of the unit cell if the problem is solved by direct numerical simulation (DNS). However, in the \(C^2\) approach the unit cell is a computational entity in the same way as the local (classical) Gauss points. Thus, in \(C^2\) the coarse-scale physical domain can be approximated by structured or unstructured meshes and still the computational unit cells being a box-like geometrical entity. Moreover, in \(C^2\) the computational unit cell is finite, but no larger than a coarse-scale element domain. For a more detailed discussion on determining the nonlocal quadrature points please refer to [16, 22]
The unit cells do not necessarily tile the macro-scale domain. Only in special cases of a regular coarse-scale mesh and specific ratios of unit cell size to coarse-scale element the computational unit cells exactly tile the macrostructure. Their behavior is upscaled to the macroscale through the nonlocal quadrature scheme
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The ONR Grant N00014-12-1-0558 to Columbia University is gratefully acknowledged.
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Appendix: Monolithic form of the coupled system
Appendix: Monolithic form of the coupled system
The coupled problem consists of solving at every time step the C-problem (26) and the F-problem (23) for all the unit cells in the domain. In a monolithic matrix form this can be expressed as follows:
For simplicity, the following matrices can been defined:
so that it can be written in a compact monolithic form:
The assembled matrices are of the following form:
\({\mathbf{M }^{C}_{e}}, {\mathbf{K }^{C}_{e}}\) from (27)
where, operator constructs block diagonal matrices, and operator assembles over the coarse-scale and unit cell DOFs.
Similarly for the coupling matrices \({\mathbf{M }^{FC}}, {\mathbf{K }^{FC}}\)
Remark
It is emphasized that fine-scale matrices \(\mathbf{M }^F_I, \mathbf{K }^F_I\) are decoupled to each other. So do fine-scale displacements, which are coupled only to the coarse-scale displacements of the coarse-scale element they belong to. However, coupling matrices \({\mathbf{M }^{FC}_I}, {\mathbf{K }^{FC}_I},{\mathbf{M }^{CF}_I}, {\mathbf{K }^{CF}_I}\) of different unit cells are coupled to each other through the common coarse-scale nodes.
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Fafalis, D., Fish, J. Computational aspects of dispersive computational continua for elastic heterogeneous media. Comput Mech 56, 931–946 (2015). https://doi.org/10.1007/s00466-015-1211-9
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DOI: https://doi.org/10.1007/s00466-015-1211-9