Tensor-structured algorithm for reduced-order scaling large-scale Kohn–Sham density functional theory calculations

We present a tensor-structured algorithm for efficient large-scale density functional theory (DFT) calculations by constructing a Tucker tensor basis that is adapted to the Kohn–Sham Hamiltonian and localized in real-space. The proposed approach uses an additive separable approximation to the Kohn–Sham Hamiltonian and an L1 localization technique to generate the 1-D localized functions that constitute the Tucker tensor basis. Numerical results show that the resulting Tucker tensor basis exhibits exponential convergence in the ground-state energy with increasing Tucker rank. Further, the proposed tensor-structured algorithm demonstrated sub-quadratic scaling with system-size for both systems with and without a gap, and involving many thousands of atoms. This reduced-order scaling has also resulted in the proposed approach outperforming plane-wave DFT implementation for systems beyond 2000 electrons.

Reconstruction of RANS Model and Cross-Validation of Flow Field Based on Tensor Basis Neural Network

The solution of the Reynolds-averaged Navier-Stokes (RANS) equation has been widely used in engineering problems. However, this model does not provide satisfactory prediction accuracy. Because the widely used eddy viscosity model assumes a linear relationship between the Reynolds stress and the average strain rate tensor and these linear models cannot capture the anisotropic characteristics of the actual flow. In this paper, two kinds of flow field structures of two-dimensional cylindrical flow and circular tube jet are calculated by using the RANS model. Secondly, in order to improve the prediction accuracy of the RANS model, the Reynolds stress of the RANS model is reconstructed by the tensor basis neural network algorithm based on nonlinear eddy viscosity model. Finally, the model trained by neural network is cross-validated, and compare the cross-test results with the traditional RANS k-eps model. The results show that the multi-layer neural network method has achieved good results in turbulence model reconstruction.

Spectral methods for Langevin dynamics and associated error estimates

We prove the consistency of Galerkin methods to solve Poisson equations where the differential operator under consideration is hypocoercive. We show in particular how the hypocoercive nature of the generator associated with Langevin dynamics can be used at the discrete level to first prove the invertibility of the rigidity matrix, and next provide error bounds on the approximation of the solution of the Poisson equation. We present general convergence results in an abstract setting, as well as explicit convergence rates for a simple example discretized using a tensor basis. Our theoretical findings are illustrated by numerical simulations.

The Measurement of Overall Elastic Stiffness and Bulk Modulus in Anisotropic Materials: Semiconductors

In this study, the correlation between macroscopic and microscopic properties of the II-IV semiconductor compounds ZnX (X = S, Se, Te) is investigated. Based on constructing orthonormal tensor basis elements using the form-invariant expressions, the elastic stiffness for cubic system materials is decomposed into two parts; isotropic (two terms) and anisotropic parts. A scale for measuring the overall elastic stiffness of these compounds is introduced and its correlation with the calculated bulk modulus and lattice constants is analyzed. The overall elastic stiffness is calculated and found to be directly proportional to bulk modulus and inversely proportional to lattice constants. A scale quantitative comparison of the contribution of the anisotropy to the elastic stiffness and to measure the degree of anisotropy in an anisotropic material is proposed using the Norm Ratio Criteria (NRC). It is found that ZnS is the nearest to isotropy (or least anisotropic) while ZnTe is the least isotropic (or nearest to anisotropic) among these compounds. The norm and norm ratios are found to be very useful for selecting suitable materials for electro-optic devices, transducers, modulators, acousto-optic devices.

The Calculation of Stiffness for Semiconductor Components

In this study, the correlation between macroscopic and microscopic properties of the II-IV semiconductor compounds CdX (X = S, Se, Te) is investigated. Based on constructing orthonormal tensor basis elements using the form-invariant expressions, the elastic stiffness for cubic system materials is decomposed into two parts; isotropic (two terms) and anisotropic parts. A new scale for measuring the overall elastic stiffness of these compounds is introduced and its correlation with the calculated bulk modulus and lattice constants is analyzed. The overall elastic stiffness is calculated and found to be directly proportional to bulk modulus and inversely proportional to lattice constants. A scale quantitative comparison of the contribution of the anisotropy to the elastic stiffness and to measure the anisotropy degree in an anisotropic material is proposed using the Norm Ratio Criteria (NRC). It is found that CdS is the nearest to isotropy (or least anisotropic) while CdTe is the least near to isotropy (or nearest to anisotropic) among these compounds. The norm and norm ratios are found to be very useful for selecting suitable materials for electro-optic devices, transducers, modulators, acousto-optic devices.