Multi-scale method for the quasi-periodic structures of composite materials

Exactness of formal asymptotic solutions of a Dirichlet problem modeling the steady state of functionally-graded microperiodic nonlinear rods

“…where the value of C 0 is obtained by substituting the boundary condition v 0 (1) = 1 into (18). Observe that v 0 (x) → x as n → ∞, so the macroscopic trend v 0 seems to approximate the exact solution u ε with sufficient accuracy for weak nonlinearities.…”

“…Such first-order approximations are formed by superposing a macroscopic trend and a local perturbation, which are related to the effective behavior and the influence of the microstructure, respectively. However, there is a need to consider approximations containing higher-order terms 4 when knowledge of the details of the local behavior of the exact solutions is required and the traditional first-order approximations fail to reproduce such local details [18][19][20]. To the best of our knowledge, only asymptotic and twospace homogenization methods are capable of providing such higher-order terms.…”

Asymptotic computation for transient heat conduction performance of periodic porous materials in curvilinear coordinates by the second‐order two‐scale method

“…On the right hand of the heat equation, the source term f ε ( ξ , t ) is often dependent on the density ρ ε ( ξ ) , then we also assume there is periodicity in f ε ( ξ , t ), that is, It is remarked that the model problem can simulate the heat conduction process in any proper curvilinear coordinates and the quasi‐periodicity of the problem is presented by the metric function with obvious physical and geometric meanings. Although the authors of developed the SOTS analysis method of the heat conduction problem with quasi‐periodic structure, the expression of the heat conductivity was only given by , which is not very clear, and for complex structures, cannot be determined directly. So the model proposed in this paper is more effective and practical.…”

“…It is remarked that the model problem (3) can simulate the heat conduction process in any proper curvilinear coordinates and the quasi-periodicity of the problem is presented by the metric function with obvious physical and geometric meanings. Although the authors of [18,19] developed the SOTS analysis method of the heat conduction problem with quasi-periodic structure, the expression of the heat conductivity was only given by a "…”

“…In practical computation, it has been validated that the SOTS solutions are much better than the first-order solutions and homogenization solutions. After that, the SOTS analysis method has been developed and extended to study the linear elasticity and thermoelastic problems for quasi-periodic structures [18][19][20], the integrated heat transfer problems with conduction, convection and radiation in periodic or porous materials [21][22][23][24], and the dynamic thermoelastic problems of periodic composite or random particulate composites [25][26][27][28][29][30].As is known that in practical applications, various structures made of porous materials are designed and created to satisfy the industrial demand in many particular situations, such as the curved thermal protection layer in the thermal protection system of the spacecraft, hollow cylinder, ball, and the honeycomb structure, most of which are not periodically arranged along the Cartesian axis. However, the SOTS asymptotic expansion is based on the periodicity of the composite structures.…”