Consistency Problems In Multiscale Modeling And Computation

In this thesis we discuss on two kinds of consistency problems in multiscale modeling,including the coupling consistency between differente regions and modeling consistency between different models.For the modeling consistency we investigate how to calculate the macro stress tensor based on atomistic model.More specifically,we study the following two stress formulation:· Stress calculation based on finite temperature Cauchy-Born rule(FTCB).In this part we would first derive the Virial stress for alloy system,starting from classical Cauchy-Born rule.Then we further simplify the ensemble average computation by using quasi-harmonic approximation and Fourier transformation.For complex lattice,we also take the inner displacement and effect of zero phonon into consideration.As numerical examples,we apply the FTCB method into the computation of five different systems and compare the result with MD simulation.· Stress calculation based on Irving-Kirkwood formulation.We first generalize the IK formulation for the Cauchy stress in Eulerian coordinate.We showed that quantities,such as density and momentum,need to be correctly defined at atomistic level for the conservation relations hold at macroscopic level.Secondly by comparing the two time average procedures we demonstrate that stress in Eulerian system must be evaluated spatially and temporally at the same time,even the kernel function is sparable.Thirdly,the relation between Cauchy and first Piola-Kirchhoff stress was investigated.As to the coupling consistency,we focus on the designing of interface condition(boundary conditions for inner region).Taking the simulation of molecular dynamics and wave equation as examples,we studied how to build the absorbing boundary condition based on outside information:· Boundary condition for MD simulation based on Krylov subspace approximation.In this part we would propose an effective and easy-to-implement boundary condition based on linearization and Krylov subspace.The boundary information was projected into a lower dimensional subspace,which can avoid the reflections at boundary.We would apply this new boundary condition in the simulation of crack and dislocations.· Boundary condition for wave equation based on neural network.Starting from the classical absorbing boundary conditions(ABC)theory,we construct linear model and nonlinear model to learn the complex kernel function in the ABC theory.We use the deep network as our approximate function space,while the training data are driven by simulation of wave equation in a larger region.According to the numerical test,the learning boundary condition can well absorbed waves at both edge and corner.