Fourth-order Gradient Research Articles

Framework for Laplacian-Level Noninteracting Free-Energy Density Functionals.

A framework for orbital-free Laplacian-level meta-generalized-gradient approximation (meta-GGA) for the noninteracting free-energy-density functionals based upon analysis of the fourth-order gradient expansion is developed. The framework presented here provides a new tool for developing advanced orbital-free thermal functionals at the meta-GGA level of theory. A nonempirical meta-GGA functional, which in the slowly varying density limit correctly reduces to the finite-temperature fourth-order gradient expansion for the noninteracting free energy, is constructed. Application to warm dense helium demonstrates that the developed meta-GGA functional drastically increases the accuracy of orbital-free-density functional theory simulations at temperatures below 40 eV as compared to the lower Thomas-Fermi and GGA rungs.

New Functional Orbital-free Within DFT for Metallic Systems

I present the continuation of a study on Laplacian Level Kinetic Energy (KE) functionals applied to metallic nanosystems. The development of novel Kinetic Energy functionals is an important topic in density functional theory (DFT). The nanoparticles are patterned using gelatin spheres of different sizes, background density and number of electrons. To reproduce the correct kinetic and potential energy density of the various nanoparticles, the use of semilocal descriptors is necessary. Need an explicit density functional expression for the kinetic energy of electrons, including the first e second functional derivative, i.e. the kinetic potential and the kinetic kernel, respectively. The exact explicit form of the non interacting kinetic energy, as a functional of the electron density, is known only for the homogeneous electron gas (HEG), i.e., the Thomas-Fermi (TF) local functional and for 1 and 2 electron systems, i.e., the von Weizsacker (VW) functional. In between these two extreme cases, different semilocal or non local approximations were developed in recent years. Most semilocal KE functionals are based on modifications of the second-order gradient expansion (GE2) or fourth-order gradient expansion (GE4). I find that the Laplacian contribute is fundamental for the description of the energy and the potential of nanoparticles. I propose a new LAP2 semilocal functional which, better than the previous ones, allows us to obtain fewer errors both of energy and potential. More details of the previous calculations can be found in my 2 previous works which will be cited in the text.

Higher-order multi-scale computational approach and its convergence for nonlocal gradient elasticity problems of composite materials

A novel higher-order multi-scale computational approach for efficient nonlocal gradient elasticity simulation of composite materials is proposed in this paper. Based on the use of decoupling technique, raw fourth-order nonlocal gradient equations with periodically oscillatory coefficients are decoupled as the new second-order multi-scale equations for reducing the continuity requirements when implementing multi-scale simulation. Next, a novel macro-micro coupled multi-scale computational model with higher-order corrected terms is rigorously devised for high-resolution multi-scale and nonlocal analysis of composite materials via multi-scale asymptotic analysis. Then, the local error analysis of higher-order multi-scale solutions in the point-wise sense illustrates that establishing higher-order asymptotic corrected terms plays a vital role in investigating the micromechanical mechanisms. Moreover, a rigorous global error estimation with an explicit rate of higher-order multi-scale solutions is first derived in the energy norm sense. In addition, an efficient multi-scale numerical algorithm is presented to effectively simulate nonlocal gradient elasticity problems of composite materials based on finite element method (FEM). And we also derive the convergence estimation of the proposed numerical algorithm. Finally, numerical experiments are conducted to gauge the efficiency and accuracy of the proposed higher-order multi-scale method. This paper offers a unified higher-order two-scale computational framework for enabling nonlocal gradient elasticity behavior simulation of composite materials.

Machine learning of kinetic energy densities with target and feature smoothing: Better results with fewer training data.

Machine learning (ML) of kinetic energy functionals (KEFs), in particular kinetic energy density (KED) functionals, is a promising way to construct KEFs for orbital-free density functional theory (DFT). Neural networks and kernel methods including Gaussian process regression (GPR) have been used to learn Kohn-Sham (KS) KED from density-based descriptors derived from KS DFT calculations. The descriptors are typically expressed as functions of different powers and derivatives of the electron density. This can generate large and extremely unevenly distributed datasets, which complicates effective application of ML techniques. Very uneven data distributions require many training datapoints, can cause overfitting, and can ultimately lower the quality of an ML KED model. We show that one can produce more accurate ML models from fewer data by working with smoothed density-dependent variables and KED. Smoothing palliates the issue of very uneven data distributions and associated difficulties of sampling while retaining enough spatial structure necessary for working within the paradigm of KEDF. We use GPR as a function of smoothed terms of the fourth order gradient expansion and KS effective potential and obtain accurate and stable (with respect to different random choices of training points) kinetic energy models for Al, Mg, and Si simultaneously from as few as 2000 samples (about 0.3% of the total KS DFT data). In particular, accuracies on the order of 1% in a measure of the quality of energy-volume dependence B'=EV0-ΔV-2EV0+E(V0+ΔV)ΔV/V02 (where V0 is the equilibrium volume and ΔV is a deviation from it) are obtained simultaneously for all three materials.

On the SAV‐DG method for a class of fourth order gradient flows

AbstractFor a class of fourth order gradient flow problems, integration of the scalar auxiliary variable (SAV) time discretization with the penalty‐free discontinuous Galerkin (DG) spatial discretization leads to SAV‐DG schemes. These schemes are linear and shown unconditionally energy stable. However, the reduced linear systems are rather expensive to solve due to the dense coefficient matrices. In this paper, we provide a procedure to pre‐evaluate the auxiliary variable in the piecewise polynomial space. As a result the computational complexity of reduces to when exploiting the conjugate gradient (CG) solver. This hybrid SAV‐DG method is more efficient and able to deliver satisfactory results of high accuracy. This was also compared with solving the full augmented system of the SAV‐DG schemes.

Scaling of Acceleration Statistics in High Reynolds Number Turbulence

The scaling of acceleration statistics in turbulence is examined by combining data from the literature with new data from well-resolved direct numerical simulations of isotropic turbulence, significantly extending the Reynolds number range. The acceleration variance at higher Reynolds numbers departs from previous predictions based on multifractal models, which characterize Lagrangian intermittency as an extension of Eulerian intermittency. The disagreement is even more prominent for higher-order moments of the acceleration. Instead, starting from a known exact relation, we relate the scaling of acceleration variance to that of Eulerian fourth-order velocity gradient and velocity increment statistics. This prediction is in excellent agreement with the variance data. Our Letter highlights the need for models that consider Lagrangian intermittency independent of the Eulerian counterpart.

Computationally efficient higher-order three-scale method for nonlocal gradient elasticity problems of heterogeneous structures with multiple spatial scales

In this work, an innovative higher-order three-scale (HOTS) computational approach is developed, which can handle and simulate the nonlocal strain-stress gradient elasticity model of heterogeneous structures with multiple spatial scales. The significant characteristics of this study are: (i) the raw fourth-order nonlocal gradient elasticity equations with three-scale structural characteristics are decoupled as the new multi-scale second-order equations for reducing the continuity requirements for multi-scale computational method. (ii) a new micro-meso-macro coupled HOTS computational model for accurately analyzing the nonlocal gradient elasticity behaviors of heterogeneous structures with three-scale structural configurations is rigorously devised by virtue of multi-scale asymptotic analysis. (iii) the small-scale approximate performance of HOTS solutions is derived in the pointwise sense, that illustrates the crucial indispensability of establishing the HOTS solutions for capturing the micro-scale, meso-scale and macro-scale gradient elasticity behaviors of the heterogeneous structures accurately. (iv) the efficient three-scale numerical algorithm on the basis of finite element method (FEM) is presented at length. Finally, several numerical experiments are carried out to validate the applicability of the established HOTS computational methodology, presenting not only the excellent numerical accuracy, but also the advanced computational efficiency. This work provides a unified three-scale computational framework which enables the accurate analysis and efficient simulation of nonlocal strain-stress gradient elasticity problems of heterogeneous structures with multiple spatial scales, and provides a potential application for practical engineering computation.

Unconditionally energy stable C0-virtual element scheme for solving generalized Swift-Hohenberg equation

The very recently introduced Virtual Element Method (VEM) is a numerical method for solving partial differential equations that was created out of the mimetic difference method, but was later reformulated into the Galerkin framework. It is a generalization of the standard Finite Element Method (FEM) to general meshes made up by arbitrary polyhedra. The greatest advantage of VEM is its ability to deal with very complex geometries, i.e., made up by elements of any number of edges not necessarily convex, hanging nodes, flat angles, collapsing nodes, etc., while retaining the same approximation properties of FEM. In this article, the C0-virtual element method is formulated and analyzed to solve generalized Swift-Hohenberg equation on polygonal meshes. The Swift-Hohenberg equation as a central nonlinear model in modern physics has a gradient flow structure. Here, we present the spatial VE discretization based on a mixed formulation for the nonlinear Swift-Hohenberg equation as a class of fourth-order gradient flow problems. For time discretization, we use Crank-Nicolson so that the resulting scheme is unconditionally stable and second-order in time. By following the algebraic implementation of the discrete system, we provide numerical tests validating the theoretical estimates and plotting two-dimensional pattern formation problems.

High order unconditionally energy stable RKDG schemes for the Swift–Hohenberg equation

We propose unconditionally energy stable Runge–Kutta (RK) discontinuous Galerkin (DG) schemes for solving a class of fourth order gradient flows including the Swift–Hohenberg equation. Our algorithm is geared toward arbitrarily high order approximations in both space and time, while energy dissipation remains preserved for arbitrary time steps and spatial meshes. The method integrates a penalty free DG method for spatial discretization with a multi-stage algebraically stable RK method for temporal discretization by the energy quadratiztion (EQ) strategy. The resulting fully discrete DG method is proven to be unconditionally energy stable. By numerical tests on several benchmark problems we demonstrate the high order accuracy, energy stability, and simplicity of the proposed algorithm.

FOURTH-ORDER STRAIN GRADIENT BAR-SUBSTRATE MODEL WITH NONLOCAL AND SURFACE EFFECTS FOR THE ANALYSIS OF NANOWIRES EMBEDDED IN SUBSTRATE MEDIA

This paper presents a new analytical bar-substrate model for the analysis of an isotropic and homogeneous nanowire embedded in an elastic substrate. A fourth-order strain gradient model based on a thermodynamic approach is employed to represent the small-scale effect (nonlocal effect) while the Gurtin-Murdoch continuum model based on the surface elastic theory is used to account for the size-dependent effect (surface energy effect). The proposed model is derived from the virtual displacement principle, leading to the governing differential equations and its associated natural boundary conditions. The analytical solutions of the sixth-order governing differential equation for the nanowire-substrate element are provided, and were employed in numerical simulations. Two numerical simulations are used to demonstrate the performance and to investigate the characteristics of the fourth-order strain gradient model on nanowire responses, when compared to the classical model and the second-order strain gradient model. The first simulation investigates the influences of nonlocal and surface effects on the responses of a nanowire embedded in an elastic substrate, while the second simulation study assessed the sensitivity of system stiffness on parameters in the nanowire-substrate model.

A Systematic Way to Extend the Debye-Hückel Theory beyond Dilute Electrolyte Solutions.

An extended Debye-Hückel theory with fourth order gradient term is developed for electrolyte solutions; namely, the electric potential φ(r) of the bulk electrolyte solution can be described by ∇2φ(r) = κ2φ(r) + LQ2∇4φ(r), where the parameters κ and LQ are chosen to reproduce the first two roots of the dielectric response function of the bulk solution. Three boundary conditions for solving the electric potential problem are proposed based upon the continuity conditions of involving functions at the dielectric boundary, with which a boundary element method for the electric potential of a solute with a general geometrical shape and charge distribution is derived. Solutions for the electric potential of a spherical ion and a diatomic molecule are found and used to calculate their electrostatic solvation energies. The validity of the theory is successfully demonstrated when applied to binary as well as multicomponent primitive models of electrolyte solutions.

Влияние градиентных поправок при расчете энергии электронного газа на поверхности металлов

The paper describes the results of studying the effect of gradient corrections to the kinetic and exchange-correlation energy functional in calculating the surface energy of a metal surface; the calculations are performed within the framework of the density functional theory. The electron density distribution profile near the metal surface was calculated by the variational method for two test functions, which differ by taking into account the electron density oscillations. The exact form of the kinetic and exchange-correlation energy functional is unknown; therefore, to calculate the surface energy of the selected metals, various gradient corrections for the second and fourth order electron gas inhomogeneity are used. The effect of the discreteness of the ionic lattice and the orientation of the crystallographic planes on the spatial distribution of the electron gas is taken into account within the framework of perturbation theory; the Ashcroft pseudopotential is taken as the one to describe the electron-ion interaction. The use of a fourth-order gradient correction for the exchange-correlation and kinetic energies has little effect on the calculated values of the surface energy of alkali metals. The calculation results do not always agree well with the experimental values of the selected metals. This may be due to the fact that the relaxation of the metal surface is not taken into consideration and because of the large error in obtaining the experimental values of the surface energy

Adaptive higher-order phase-field modeling of anisotropic brittle fracture in 3D polycrystalline materials

Due to its ability to simulate complex microstructure evolution, the phase-field modeling has been extensively developed to investigate brittle fracture in recent years. However, low computational efficiency still imposes substantial difficulties in the development of phase-field models. In this work, we develop a novel adaptive phase-field approach based on the isogeometric meshfree collocation method (IMCM) to simulate the crack propagation in 2D and 3D polycrystalline materials. The concept of IMCM is based upon the correspondence between the isogeometric collocation and reproducing kernel meshfree method to facilitate a robust mesh adaptivity in isogeometric collocation. The strong form collocation formulation further enhances the computational efficiency of phase-field modeling by reducing the number of point evaluations. The present numerical framework is utilized for the adaptive phase-field modeling which introduces the anisotropy of fracture resistance for each grain in polycrystals. Furthermore, the discrete displacement and phase-field equations are generalized to enable the calculation of both second- and fourth-order gradients, which are required to solve the phase-field models using IMCM. The smoothness and higher-order continuity of IMCM enable the fourth-order phase-field equation to be solved directly without splitting it into two second-order differential equations. The fourth-order model can capture the crack surface accurately with fewer nodes than the second-order model. Several numerical examples of isotropic and anisotropic brittle fracture in polycrystalline materials are investigated to demonstrate the effectiveness and robustness of the proposed approach.

Vibration energy flow analysis of periodic nanoplate structures under thermal load using fourth-order strain gradient theory

A nonlocal Kirchhoff plate model with fourth-order strain gradient theory is firstly proposed to study variations with band gap frequencies and vibration energy distribution. The temperature rise is supposed to vary linearly through the thickness of the periodic nanoplate structures. The dynamic equations of finite periodic nanoplate structures under thermal load with the small-scale effect and the nonlinear membrane strain taken into consideration are derived based on the finite element method. The structural intensity approach is developed to predict the vibration energy flow of the periodic nanoplates. Effects of the temperature rise on band gaps and the structural intensity are divided into two parts. One is nonlinear effects of temperature rise, and the other is effects of the thermal load. In the numerical calculation, the natural frequencies of single-layer graphene sheets computed by the nonlocal finite element method with fourth-order strain gradient agree well with analytical results, which validate the effectiveness of the present method. The influences of nonlocal parameters and thermal load on the band gap and structural intensity are considered, respectively. The boundary value has been achieved to determine the critical mechanical load or thermal load for analyzing in depth the effects of the mechanical load and thermal load on the structural intensity. The proposed method shows that the vibration energy flow pattern may be controlled by adjusting the magnitude of the mechanical or thermal load.

FFT-based solver for higher-order and multi-phase-field fracture models applied to strongly anisotropic brittle materials

This paper presents the application of a fast Fourier transform (FFT) based method to solve two phase field models designed to simulate crack growth of strongly anisotropic materials in the brittle regime. By leveraging the ability of the FFT-based solver to generate solutions with higher-order and global continuities, we design two simple algorithms to capture the complex fracture patterns (e.g. sawtooth, and curved crack growth) common in materials with strongly anisotropic surface energy via the multi-phase-field and high-order phase-field frameworks. A staggered operator-split solver is used where both the balance of linear momentum and the phase field governing equations are formulated in the periodic domain. The unit phase field of the initial failure region is prescribed by the penalty method to alleviate the sharp material contrast between the initial failure region and the base material. The discrete frequency vectors are generalized to estimate the second and fourth order gradients such that the Gibbs effect near shape interfaces or jump conditions can be suppressed. Furthermore, a preconditioner is adopted to improve the convergence rate of the iterative linear solver. Three numerical experiments are used to systematically compare the performance of the FFT-based method in the multi-phase-field and high-order phase-field frameworks.

A fourth-order gauge-invariant gradient plasticity model for polycrystals based on Kröner’s incompatibility tensor

In this paper we derive a novel fourth-order gauge-invariant phenomenological model of infinitesimal rate-independent gradient plasticity with isotropic hardening and Kröner’s incompatibility tensor inc ( ε p ) : = Curl [ ( Curl ε p ) T ] , where ε p is the symmetric plastic strain tensor. Here, gauge-invariance denotes invariance under diffeomorphic reparametrizations of the reference configuration, suitably adapted to the geometrically linear setting. The model features a defect energy contribution that is quadratic in the tensor inc ( ε p ) and it contains isotropic hardening based on the rate of the plastic strain tensor ε · p . We motivate the new model by introducing a novel rotational invariance requirement in gradient plasticity, which we call micro-randomness, suitable for the description of polycrystalline aggregates on a mesoscopic scale and not coinciding with classical isotropy requirements. This new condition effectively reduces the increments of the non-symmetric plastic distortion p · to their symmetric counterpart ε · p = sym p · . In the polycrystalline case, this condition is a statement about insensitivity to arbitrary superposed grain rotations. We formulate a mathematical existence result for a suitably regularized non-gauge-invariant model. The regularized model is rather invariant under reparametrizations of the reference configuration including infinitesimal conformal mappings.

Unconditionally Energy Stable DG Schemes for the Swift–Hohenberg Equation

The Swift-Hohenberg equation as a central nonlinear model in modern physics has a gradient flow structure. Here we introduce fully discrete discontinuous Galerkin (DG) schemes for a class of fourth order gradient flow problems, including the nonlinear Swift-Hohenberg equation, to produce free-energy-decaying discrete solutions, irrespective of the time step and the mesh size. We exploit and extend the mixed DG method introduced in [H. Liu and P. Yin, J. Sci. Comput., 77: 467--501, 2018] for the spatial discretization, and the "Invariant Energy Quadratization" method for the time discretization. The resulting IEQ-DG algorithms are linear, thus they can be efficiently solved without resorting to any iteration method. We actually prove that these schemes are unconditionally energy stable. We present several numerical examples that support our theoretical results and illustrate the efficiency, accuracy and energy stability of our new algorithm. The numerical results on two dimensional pattern formation problems indicate that the method is able to deliver comparable patterns of high accuracy.

Kinetic energy densities based on the fourth order gradient expansion: performance in different classes of materials and improvement via machine learning.

We study the performance of fourth-order gradient expansions of the kinetic energy density (KED) in semi-local kinetic energy functionals depending on the density-dependent variables. The formal fourth-order expansion is convergent for periodic systems and small molecules but does not improve over the second-order expansion (the Thomas-Fermi term plus one-ninth of the von Weizsäcker term). Linear fitting of the expansion coefficients somewhat improves on the formal expansion. The tuning of the fourth order expansion coefficients allows for better reproducibility of the Kohn-Sham kinetic energy density than the tuning of the second-order expansion coefficients alone. The possibility of a much more accurate match with the Kohn-Sham kinetic energy density by using neural networks (NN) trained using the terms of the 4th order expansion as density-dependent variables is demonstrated. We obtain ultra-low fitting errors without overfitting of NN parameters. Small single hidden layer neural networks can provide good accuracy in separate KED fits of each compound, while for joint fitting of KEDs of multiple compounds multiple hidden layers were required to achieve good fit quality. The critical issue of data distribution is highlighted. We also show the critical role of pseudopotentials in the performance of the expansion, where in the case of a too rapid decay of the valence density at the nucleus with some pseudopotentials, numeric instabilities can arise.

Assessing the performance of the Tao-Mo semilocal density functional in the projector-augmented-wave method.

We assess the performance of the recently proposed Tao-Mo (TM) semilocal exchange-correlation functional [J. Tao and Y. Mo, Phys. Rev. Lett. 117, 073001 (2016)] using the projector-augmented-wave method with the plane wave basis set. The meta-generalized gradient approximation level semilocal functional constructed by Tao-Mo is an all-purpose exchange-correlation functional for the quantum chemistry and solid-state physics. The exchange of the TM functional is based on the density matrix expansion technique together with the slowly varying fourth order gradient expansion. The correlation functional corresponding to the exchange is based on the one-electron self-interaction-free Tao-Perdew-Staroverov-Scuseria functional. Our test includes solid-state lattice constants, bulk moduli, bandgaps, cohesive energies, magnetic moments and vacancy-formation energies of transition metals. It is observed that in the plane wave basis, the TM functional performs accurately in predicting all the solid state properties at the semilocal level.

From hyperbolic diffusion scheme to gradient method: Implicit Green–Gauss gradients for unstructured grids

This paper introduces a new approach to constructing algorithms for gradient computations on unstructured grids. The proposed approach is to derive a gradient algorithm from a hyperbolic diffusion scheme, which solves the diffusion equation and equations for the solution gradient, by ignoring the residual component for the diffusion equation. The resulting scheme forms a globally coupled linear system of equations for the gradients with the Green–Gauss gradient formula on the right hand side, which can be solved efficiently by iterative techniques. Accuracy and iterative performance can be controlled by parameters inherited from the generating hyperbolic diffusion scheme. Second- or fourth-order gradient accuracy can be obtained on regular grids, and first-order accuracy, which is sufficient for second-order finite-volume schemes, can be obtained on irregular grids. Numerical results indicate that the method gives highly accurate gradients especially for highly-curved high-aspect-ratio grids typical in high-Reynolds-number boundary layer flows over complex geometries, and can be employed in finite-volume solvers as an efficient alternative to least-squares or Green–Gauss gradient methods. It is also shown that one of the parameters has a role of dissipation, and can be tuned to yield a smooth gradient distribution over a discontinuous function, thereby allowing an implicit finite-volume solver to converge for a discontinuous solution with a mild overshoot. The paper concludes with a discussion for future developments.