Abstract
This paper presents a computationally efficient homogenization method for transient heat conduction problems. The notion of relaxed separation of scales is introduced and the homogenization framework is derived. Under the assumptions of linearity and relaxed separation of scales, the microscopic solution is decomposed into a steady-state and a transient part. Static condensation is performed to obtain the global basis for the steady-state response and an eigenvalue problem is solved to obtain a global basis for the transient response. The macroscopic quantities are then extracted by averaging and expressed in terms of the coefficients of the reduced basis. Proof-of-principle simulations are conducted with materials exhibiting high contrast material properties. The proposed homogenization method is compared with the conventional steady-state homogenization and transient computational homogenization methods. Within its applicability limits, the proposed homogenization method is able to accurately capture the microscopic thermal inertial effects with significant computational efficiency.
Highlights
With the advent of micro-fabrication technologies [1], the demand for miniature devices utilizing heterogeneous materials is steadily increasing
To capture the micro inertia effects, the transient balance of energy has to be solved at the microscale and both, the macroscopic heat flux and rate of change of macroscopic internal energy, must be computed/upscaled using computational homogenization
To ensure consistent scale transition in computational homogenization, specific types of boundary conditions are required at the microscale, which will be defined through the downscaling procedure
Summary
With the advent of micro-fabrication technologies [1], the demand for miniature devices utilizing heterogeneous materials is steadily increasing. A full separation of scales indicates that the characteristic macroscopic loading time scale is much larger than all the microscopic characteristic diffusion times, independently of the ratios between the characteristic diffusion times of the different microstructural constituents In such regimes, transient effects are negligible at the microscale and it is appropriate to use a steady-state energy balance equation at the microscale. To benefit from the model reduction, the microscopic problem is projected on the reduced basis subspace, which yields an evolution equation for the microscale thermal inertia in terms of the coefficients of the reduced basis These evolution equations for the amplitudes of the reduced variables, together with the macroscopic energy balance and the effective homogenized constitutive equations give rise to an enriched continuum description at the macroscale. Introduction of the relaxed separation of scales. A model reduction technique for the microscale which leads to an enriched continuum formulation at the macroscale
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