Statistical higher-order multi-scale method for nonlinear thermo-mechanical simulation of random composite materials with temperature-dependent properties

Stochastic multi-scale modeling and simulation for nonlinear thermo-mechanical problems of composite materials with complicated random microstructures remains a challenging issue. In this paper, we develop a novel statistical higher-order multi-scale (SHOMS) method for nonlinear thermo-mechanical simulation of random composite materials, which is designed to overcome limitations of prohibitive computation involving the macro-scale and micro-scale. By virtue of statistical multi-scale asymptotic analysis and Taylor series method, the SHOMS computational model is rigorously derived for accurately analyzing nonlinear thermo-mechanical responses of random composite materials both in the macro-scale and micro-scale. Moreover, the local error analysis of SHOMS solutions in the point-wise sense clearly illustrates the crucial indispensability of establishing the higher-order asymptotic corrected terms in SHOMS computational model for keeping the conservation of local energy and momentum. Then, the corresponding space-time multi-scale numerical algorithm with off-line and on-line stages is designed to efficiently simulate nonlinear thermo-mechanical behaviors of random composite materials. Finally, extensive numerical experiments are presented to gauge the efficiency and accuracy of the proposed SHOMS approach

Homogenization with quasistatic Tresca's friction law: qualitative and quantitative results

The problems of frictional contacts are the key to the investigation of mechanical performances of composite materials under varying service environments. The paper considers a linear elasticity system with strongly heterogeneous coefficients and quasistatic Tresca's friction law, and we study the homogenization theories under the frameworks of H-convergence and small $\epsilon$-periodicity. The qualitative result is based on H-convergence, which shows the original oscillating solutions will converge weakly to the homogenized solution, while our quantitative result provides an estimate of asymptotic errors in the $H^1$ norm for the periodic homogenization. We also design several numerical experiments to validate the convergence rates in the quantitative analysis

Higher-order multi-scale deep Ritz method for multi-scale problems of authentic composite materials

The direct deep learning simulation for multi-scale problems remains a challenging issue. In this work, a novel higher-order multi-scale deep Ritz method (HOMS-DRM) is developed for thermal transfer equation of authentic composite materials with highly oscillatory and discontinuous coefficients. In this novel HOMS-DRM, higher-order multi-scale analysis and modeling are first employed to overcome limitations of prohibitive computation and Frequency Principle when direct deep learning simulation. Then, improved deep Ritz method are designed to high-accuracy and mesh-free simulation for macroscopic homogenized equation without multi-scale property and microscopic lower-order and higher-order cell problems with highly discontinuous coefficients. Moreover, the theoretical convergence of the proposed HOMS-DRM is rigorously demonstrated under appropriate assumptions. Finally, extensive numerical experiments are presented to show the computational accuracy of the proposed HOMS-DRM. This study offers a robust and high-accuracy multi-scale deep learning framework that enables the effective simulation and analysis of multi-scale problems of authentic composite materials