Multiscale Algorithms Research For Performance Analysis Of Random Heterogeneous Materials

Random heterogeneous material(RHM) is one kind of common heterogeneous material in engineering. Prediction of physical and mechanical properties for RHMs is multiscale problem essentially. Since it is difficult to solve the multiscale problem efficiently by utilizing classical continuum models and corresponding numerical methods, the development of more efficient multiscale models and algorithms has been one of important problems in scientific and engineering computing. Till now, some homogenization methods and multiscale methods have been proposed to predict the performance of RHMs. However, difficulties still exist in the precision and efficiency of those methods for complex RHMs. For RHMs with complex microstructures and a high contrast of constituent properties, some effective numerical methods are developed to predict the effective properties in this dissertation. With those methods, the computational cost of multiscale methods could be reduced, while the accuracy of effective properties could be improved.Firstly, four typical multiscale methods are compared, and those methods are divided broadly into two types according to the manner of obtaining multiscale solution, i.e., up-down method and uncoupling method. In the up-down method, the multiscale problem is solved by two steps. Macroscopic solution is solved at first and multiscale solution is resolved locally then if it is needed. The computational cost of the method depends on the need for the multiscale solution. In contrast, the uncoupling method focuses on the multiscale solution in the global domain directly. The computational cost of the method is comparable to that of classical numerical methods. Based on the analysis of two types of methods, an appropriate multiscale method could be selected or designed for a certain multiscale problem. The computational complexity of four methods is compared, and the equivalence of multiscale solutions computed by two methods is proved.Secondly, Richardson extrapolation method is introduced to improve the accuracy of approximate effective coefficients(AECs) for RHMs and the efficiency of multiscale methods. AEC is the parameter which should be computed at first in the up-down method. The accuracy of AEC depends on the size of representative volume element(RVE). Enlarging RVE will improve the accuracy of AEC, however it will lead to remarkable increase of computational cost. It is verified numerically that AECs provided by Dirichlet boundary condition(DBC) and Neumann boundary condition(NBC) converge with first-order accuracy. And then Richardson extrapolation method is introduced to improve the convergence rate of AEC. More accurate AEC could be obtained by extrapolation of some AECs in small RVEs. So it is unnecessary to solve auxiliary problems within large RVE, and a lot of cost could be saved. Efficiency of Richardson extrapolation method is validated by numerical examples. And it is proved that under DBC and NBC, AECs converge monotonously as RVE size goes to infinity. Moreover, multiparameter mean ergodic theorems, which are useful for the proof of convergence rate for AEC, are proved here.Thirdly, for RHM with a high contrast of constituent properties, a new Robin boundary condition(RBC) is proposed for the auxiliary problem. For those materials, DBC and NBC cannot offer accurate AEC. Two continuous properties for solution of auxiliary problem on the boundary of RVE are deduced based on the ergodic theorem. And then RBC is constructed. The convergence of AEC under RBC as RVE size goes to infinity is proved. Compared to DBC and NBC, RBC provides more accurate AEC. Though Dirichlet-Neumann mixed boundary condition(DNBC) gives more accurate AEC than DBC and NBC as well, the adjusting factor in RBC makes it more flexible, and much better than DNBC. With RBC, an accurate AEC could be obtained within small RVE so that there is no need to solve auxiliary problems within large RVE and the cost could be reduced significantly.Finally, effective mechanical properties of a polymer ternary blend with core-shell structure are predicted by statistical second-order two-scale(SSOTS) method. Core-shell particles are dispersed randomly in the matrix in the ternary blend. Richardson extrapolation method is integrated into SSOTS method, in order to improve the accuracy of approximate effective stiffness tensor. And unified strength theory is integrated into SSOTS method to select more appropriate yield criterion and to provide more accurate yield strength. SSOTS method is validated by comparison with experiment and homogenization methods. Moreover, the relation between shell thickness in core-shell structures and effective mechanical properties of the ternary blend is discussed, which could provide reference to new materials research.