We present a simple, yet general, deep neural network representation of the potential energy surface for atomic and molecular systems. It is "first-principle" based, in the sense that no ad hoc approximations or empirical fitting functions are required. When tested on a wide variety of examples, it reproduces the original model within chemical accuracy. This brings us one step closer to carrying out molecular simulations with quantum mechanics accuracy at empirical potential computational cost.

In this paper, an immersed boundary (IB) method for simulating the interfacial flows with insoluble surfactant in three dimensions is developed. We consider a doubly periodic interface separating two fluids where the surfactant exists only along the evolving interface. An equi-arclength parametrization is introduced in order to track the moving interface and maintain good Lagrangian meshes, so stable computations can be performed without remeshing. This surface mesh-control technique is done by adding two artificial tangential velocity components into the Lagrangian marker velocity so that the Lagrangian markers can be equi-arclength distributed during the time evolution. As a result, the surfactant equation on the interface must be modified based on the new parametrization. A conservative scheme for solving the modified surfactant equation has been developed and proved to satisfy the total surfactant mass exactly in discrete level. A series of numerical experiments consisting of the validation of Lagrangian mesh control technique, the convergence study, the study of self-healing dynamics, and the simulations of two-layer fluids under Couette flow have been conducted to test our present numerical scheme.

Particle laden turbulent flows occur in a variety of industrial applications. While the numerical simulation of such flows has seen significant advances in recent years, it still remains a challenging problem. Many studies investigated the rheology of dense suspensions in laminar flows as well as the dynamics of point-particles in turbulence. Here we will present results on the development of numerical algorithms, based on the lattice Boltzmann method, suitable for the study of suspensions of finite-size particles under turbulent flow conditions. The turbulent flow is modeled by the lattice Boltzmann method, and the interaction between particles and carrier fluid is modeled using the bounce-back rule. Direct contact and lubrication force models for particle-particle interactions and particle-wall interaction are taken into account to allow for a full four-way coupled interaction. The accuracy and robustness of the method is discussed by validating the velocity profile in turbulent pipe flow, the sedimentation velocity of spheres in duct flow and the resistance functions of approaching particles. Preliminary results from the turbulent pipe flow simulations with particles show that the angular and axial velocities of the particles are scattered around values of mean axial velocity and shear rate obtained from the Eulerian velocity field.

This paper develops an efficient and accurate numerical method for the computation of electromagnetic waves scattered by random rough surfaces. The method is based upon a combination of the Transformed Field Expansion method, which represents the solution as a provably convergent power series, and the Monte Carlo technique for sampling the probability space. The compelling aspect of the proposed method is that, at each perturbation order and every sample, the governing Transformed Field Expansion equations share the same deterministic Helmholtz operator on a deterministic domain. Thus, an LU factorization of the matrix discretization of this single operator can be employed repeatedly for all orders and every sample. Consequently, the computational complexity of the whole algorithm is significantly reduced as a result. Numerical examples are described which demonstrate the accuracy of the algorithm.

In this paper, a weak Galerkin finite element method (WGFEM) is proposed for solving the Navier-Stokes equations (NSEs). The existence and uniqueness of the WGFEM solution of NSEs are established. The WGFEM provides very accurate numerical approximations for both the velocity field and pressure field, even with very high Reynolds numbers. The salient feature is that the flexibility of the WGFEM for the choice of the order of the velocity and pressure comparing to the standard finite element methods. Optimal order error estimates in both |·|_{$W_h$} and $L^2$ norms are proved for the semi-discretized scheme. Numerical simulations are presented to show the efficiency of the WGFEM.

We design and numerically validate a local discontinuous Galerkin (LDG) method to compute solutions to the initial value problem for a nonlinear variational wave equation originally proposed to model liquid crystals. For the semi-discrete LDG formulation with a class of alternating numerical fluxes, the energy conserving property is verified. A dissipative scheme is also introduced by locally imposing some numerical "damping" in the scheme so as to suppress some numerical oscillations near solution singularities. Extensive numerical experiments are presented to validate and illustrate the effectiveness of the numerical methods. Optimal convergence in $L^2$ is numerically obtained when using alternating numerical fluxes. When using the central numerical flux, only sub-optimal convergence is observed for polynomials of odd degree. Numerical simulations with long time integration indicate that the energy conserving property is crucial for accurately capturing the underlying wave shapes.

In some superconductors the charge density wave (CDW) state is adjacent to the superconducting state in the phase diagram. This CDW phase can be collapsed either pressure or by chemical doping, depending on compound. Among them, in so-called valence skip compounds, a large charge fluctuation with the large electron-phonon interaction is expected. We performed a first-principle study and investigated how the CDW gap is collapsed for several valence-skip compounds, i.e. SnX₃, RbTlX₃(X=F,Cl,Br,I) and CsTlI₃. For all these compounds we found that the CDW gap is rather robust for the uniform volume change, and on the contrary, the magnitude of the CDW gap strongly depends on the position of the anion. We found that this CDW gap is already collapsed at ambient pressure in SnBr₃, SnI₃ and CsTlI₃.

We study the spin-dependent electron transport by using ab initio calculations for a molecular junction consisting of a phenylene capped by two carbon chains which is sandwiched between two zigzag-edged graphene nanoribbon (ZGNR) electrodes modulated by external magnetic fields or ferromagnets. It is shown that the spin-charge transport can be adjusted by the phenylene rotation angles in respect to ZGNR electrodes and spin orientation in electrodes. Specifically, we demonstrate that the proposed molecular device exhibits switching, (dual) spin-filtering, rectifying, negative differential resistance (NDR) effects. Interestingly, when the phenylene rotation angle is θ=60^◦, the maximum value of the peak-to-valley ratio of NDR is up to 26, the spin-resolved rectification ratio can reach up to 2.47×10⁴, and the spin filtering efficiency reaches up to 100%. The physical mechanisms of these effects are analyzed via the spin-resolved transmission spectrum associated with local density of states (LDOS) and molecular projected self-consistent Hamiltonian (MPSH) eigenvalues.

Numerical dispersion error is inevitable when the finite element method is employed to simulate acoustic problems. Studies have shown that the dispersion error is essentially rooted at the "overly-stiff" property of the standard FEM model. To reduce the dispersion error effectively, a discrete model that provides a proper softening effects is needed. Thus, the stable node-based smoothed finite element method (SNS-FEM) which contains a stable item is presented. In this paper, the SNS-FEM is investigated in details with respect to the pollution effect. Different kinds of meshes are employed to analyze the relationship between the dispersion error and the parameter involved in the stable item. To ensure the SNS-FEM can be applied to the practical engineering problems effectively, the relationship is finally constructed based on the hexagonal patch for the commonly used unstructured mesh is very similar to it. By minimizing the discretization error, an optimal parameter equipped in this novel SNSFEM is formulated. Both theoretical analysis and numerical examples demonstrate that the SNS-FEM with the optimal parameter reduces the dispersion error significantly compared with the FEM and the well performed GLS, especially for mid- and high-wave number problems.

In this paper, we first compute the multiple non-trivial solutions of the Schrödinger equation on a square, by using the Liapunov-Schmidt reduction and symmetry-breaking bifurcation theory, combined with Legendre pseudospectral methods. Then, starting from the non-trivial solution branches of the corresponding nonlinear problem, we further obtain the whole positive solution branch with $D_4$ symmetry of the Schrödinger equation numerically by pseudo-arclength continuation algorithm. Next, we propose the extended systems, which can detect the fold and symmetry-breaking bifurcation points on the branch of the positive solutions with $D_4$ symmetry. We also compute the multiple positive solutions with various symmetries of the Schrödinger equation by the branch switching method based on the Liapunov-Schmidt reduction. Finally, the bifurcation diagrams are constructed, showing the symmetry/peak breaking phenomena of the Schrödinger equation. Numerical results demonstrate the effectiveness of these approaches.

This paper presents an analysis of the stability and accuracy of different Lattice Boltzmann schemes when employed for direct numerical simulations of turbulent flows. The Single-Relaxation-Time scheme of Bhatnagar, Gross and Krook (BGK), the Multi-Relaxation-Time scheme (MRT) and the Regularized Lattice Boltzmann scheme (RLB) are considered. The stability and accuracy properties of these schemes are investigated by computing three-dimensional Taylor-Green vortices representing homogeneous isotropic turbulent flows. Varying Reynolds numbers and grid resolutions were considered. As expected, the BGK scheme requires sufficiently high grid resolutions for stable and accurate simulations. Surprisingly, the MRT scheme when used without any turbulence model fails to obtain mesh convergence for the type of flow considered here. The RLB scheme allows for stable simulations but exhibits a strong dissipative behavior. A similar behavior was found when employing the mentioned LBM schemes for numerical simulations of turbulent channel flows at varying Reynolds numbers and resolutions. The obtained insights on accuracy and stability of the considered Lattice Boltzmann methods can become useful especially for the design of effective turbulence models to be used for high Reynolds number flows.

We present a robust approach to perform 3D nonrigid image registration suitable for large deformation and topology change, and develop a software package named DTHB3D Reg (Dynamic Truncated Hierarchical B-spline based 3D Image Registration). The optimum spatial transformation, defined using truncated hierarchical B-splines, is obtained through the minimization of an energy functional. The optimization process minimizes sum of squared difference in the intensity values of the grayscale images. Control points are dynamically updated without constructing large matrices as in finite element method. To improve the computational efficiency, an adaptive strategy carries out refinement only in the regions with large deformation. The proposed method is demonstrated on 3D synthetic and medical images to show robustness on topology change as compared to other image registration methods.