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The Kohn-Sham model is a powerful, widely used approach for computation of ground state electronic energies and densities in chemistry, materials science, biology, and nanoscience. In this paper, we study adaptive finite element approximations for the Kohn-Sham model. Based on the residual type a posteriori error estimators proposed in this paper, we introduce an adaptive finite element algorithm with a quite general marking strategy and prove the convergence of the adaptive finite element approximations. Using Dörfler's marking strategy, we then get the convergence rate and quasi-optimal complexity. We also carry out several typical numerical experiments that not only support our theory, but also show the robustness and efficiency of the adaptive finite element computations in electronic structure calculations.

This article introduces the uses of transparent synthetic soil for geotechnical problems using optical system, including transparent materials, sample preparation, geotechnical properties, experimental methods, and applications in physical modeling. Four typical kinds of transparent synthetic soil are shown and compared. For amorphous silica powder, normally the consolidated amorphous silica has a higher normalized strength but a lower modulus than the natural clays. For amorphous silica gels, the stressstrain behaviors are consistent with the typical stress-strain behaviors of sand for both dense and loose conditions. For fused silica, it has a higher shearing strength and higher modulus than the natural sand does; the deviatoric stress increases with the confining pressure, but the stress-strain curves of fused silica and the natural sand are particularly similar. For glass sand, with increasing of the relative density, the strainstress relationship varies from strain hardening to stress softening, while its failure form is essentially the same as that of standard sand. According to the geotechnical properties of four typical materials of transparent synthetic soil grain, they are used to simulate different conditions and analyze practical engineering problems in different physical model tests. The process included the generation of a speckle pattern created by the interaction of laser light with transparent particles. Using digital image processing technology, speckle patterns can be obtained and used to calculate the displacement field. By utilizing this optical system, transparent synthetic soil can be used to nonintrusively investigate internal soil deformation, flow problems, and ground movement in physical model tests. Finally, both the advantages and disadvantages of the transparent soil experimental technique are analyzed.

In this paper, we study numerical approximations of a nonlinear eigenvalue problem and consider applications to a density functional model. We prove the convergence of numerical approximations. In particular, we establish several upper bounds of approximation errors and report some numerical results of finite element electronic structure calculations that support our theory.

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