Random Green's Function Method for Large-Scale Electronic Structure Calculation

We report a linear-scaling random Green's function(rGF) method for large-scale electronic structure calculation. In this method, the rGF is defined on a set of random states and is efficiently calculated by projecting onto Krylov subspace. With the rGF method, the Fermi–Dirac operator can be obtained directly, avoiding the polynomial expansion to Fermi–Dirac function. To demonstrate the applicability, we implement the rGF method with the density-functional tight-binding method. It is shown that the Krylov subspace can maintain at small size for materials with different gaps at zero temperature, including H_(2)O and Si clusters. We find with a simple deflation technique that the rGF self-consistent calculation of H_(2)O clusters at T = 0 K can reach an error of~ 1 me V per H_(2)O molecule in total energy, compared to deterministic calculations. The rGF method provides an effective stochastic method for large-scale electronic structure simulation.

Dynamic Characteristics of Functionally Graded Timoshenko Beams by Improved Differential Quadrature Method

This study proposes an effective method to enhance the accuracy of the Differential Quadrature Method(DQM)for calculating the dynamic characteristics of functionally graded beams by improving the form of discrete node distribution.Firstly,based on the first-order shear deformation theory,the governing equation of free vibration of a functionally graded beam is transformed into the eigenvalue problem of ordinary differential equations with respect to beam axial displacement,transverse displacement,and cross-sectional rotation angle by considering the effects of shear deformation and rotational inertia of the beam cross-section.Then,ignoring the shear deformation of the beam section and only considering the effect of the rotational inertia of the section,the governing equation of the beam is transformed into the eigenvalue problem of ordinary differential equations with respect to beam transverse displacement.Based on the differential quadrature method theory,the eigenvalue problem of ordinary differential equations is transformed into the eigenvalue problem of standard generalized algebraic equations.Finally,the first several natural frequencies of the beam can be calculated.The feasibility and accuracy of the improved DQM are verified using the finite element method(FEM)and combined with the results of relevant literature.

A Hybrid Level Set Optimization Design Method of Functionally Graded Cellular Structures Considering Connectivity

With the continuous advancement in topology optimization and additive manufacturing(AM)technology,the capability to fabricate functionally graded materials and intricate cellular structures with spatially varying microstructures has grown significantly.However,a critical challenge is encountered in the design of these structures–the absence of robust interface connections between adjacent microstructures,potentially resulting in diminished efficiency or macroscopic failure.A Hybrid Level Set Method(HLSM)is proposed,specifically designed to enhance connectivity among non-uniform microstructures,contributing to the design of functionally graded cellular structures.The HLSM introduces a pioneering algorithm for effectively blending heterogeneous microstructure interfaces.Initially,an interpolation algorithm is presented to construct transition microstructures seamlessly connected on both sides.Subsequently,the algorithm enables the morphing of non-uniform unit cells to seamlessly adapt to interconnected adjacent microstructures.The method,seamlessly integrated into a multi-scale topology optimization framework using the level set method,exhibits its efficacy through numerical examples,showcasing its prowess in optimizing 2D and 3D functionally graded materials(FGM)and multi-scale topology optimization.In essence,the pressing issue of interface connections in complex structure design is not only addressed but also a robust methodology is introduced,substantiated by numerical evidence,advancing optimization capabilities in the realm of functionally graded materials and cellular structures.

Transfer matrix method for free and forced vibrations of multi-level functionally graded material stepped beams with different boundary conditions

Functionally graded materials(FGMs)are a novel class of composite materials that have attracted significant attention in the field of engineering due to their unique mechanical properties.This study aims to explore the dynamic behaviors of an FGM stepped beam with different boundary conditions based on an efficient solving method.Under the assumptions of the Euler-Bernoulli beam theory,the governing differential equations of an individual FGM beam are derived with Hamilton’s principle and decoupled via the separation-of-variable approach.Then,the free and forced vibrations of the FGM stepped beam are solved with the transfer matrix method(TMM).Two models,i.e.,a three-level FGM stepped beam and a five-level FGM stepped beam,are considered,and their natural frequencies and mode shapes are presented.To demonstrate the validity of the method in this paper,the simulation results by ABAQUS are also given.On this basis,the detailed parametric analyses on the frequencies and dynamic responses of the three-level FGM stepped beam are carried out.The results show the accuracy and efficiency of the TMM.

A Radial Basis Function Method with Improved Accuracy for Fourth Order Boundary Value Problems

Accurately approximating higher order derivatives is an inherently difficult problem. It is shown that a random variable shape parameter strategy can improve the accuracy of approximating higher order derivatives with Radial Basis Function methods. The method is used to solve fourth order boundary value problems. The use and location of ghost points are examined in order to enforce the extra boundary conditions that are necessary to make a fourth-order problem well posed. The use of ghost points versus solving an overdetermined linear system via least squares is studied. For a general fourth-order boundary value problem, the recommended approach is to either use one of two novel sets of ghost centers introduced here or else to use a least squares approach. When using either ghost centers or least squares, the random variable shape parameter strategy results in significantly better accuracy than when a constant shape parameter is used.

Study on Characteristics of 3-D Translating-Pulsating Source Green Function of Deep-Water Havelock Form and Its Fast Integration Method

The singularities, oscillatory performances and the contributing factors to the 3-＇D translating-pulsating source Green function of deep-water Havelock form which consists of a local disturbance part and a far-field wave-like part, are analyzed systematically. Relative numerical integral methods about the two parts are presented in this paper. An improved method based on LOBATTO rule is used to eliminate singularities caused respectively by infinite discontinuity and jump discontinuous node from the local disturbance part function, which makes the improvement of calculation efficiency and accuracy possible. And variable substitution is applied to remove the singularity existing at the end of the integral interval of the far-field wave-like part function. Two auxiliary techniques such as valid interval calculation and local refinement of integral steps technique in narrow zones near false singularities are applied so as to avoid unnecessary integration of invalid interval and improve integral accordance. Numerical test results have proved the efficiency and accuracy in these integral methods that thus can be applied to calculate hydrodynamic performance of floating structures moving in waves.

SOME PROBLEMS WITH THE METHOD OF FUNDAMENTAL SOLUTION USING RADIAL BASIS FUNCTIONS

The present work describes the application of the method of fundamental solutions （MFS） along with the analog equation method （AEM） and radial basis function （RBF） approximation for solving the 2D isotropic and anisotropic Helmholtz problems with different wave numbers. The AEM is used to convert the original governing equation into the classical Poisson＇s equation, and the MFS and RBF approximations are used to derive the homogeneous and particular solutions, respectively. Finally, the satisfaction of the solution consisting of the homogeneous and particular parts to the related governing equation and boundary conditions can produce a system of linear equations, which can be solved with the singular value decomposition （SVD） technique. In the computation, such crucial factors related to the MFS-RBF as the location of the virtual boundary, the differential and integrating strategies, and the variation of shape parameters in multi-quadric （MQ） are fully analyzed to provide useful reference.

Function Combined Method for Design Innovation of Children's Bike

As children mature, bike products for children in the market develop at the same time, and the conditions are frequently updated. Certain problems occur when using a bike, such as cycle overlapping, repeating function, and short life cycle, which go against the principles of energy conservation and the environmental protection intensive design concept. In this paper, a rational multi-function method of design through functional superposition, transformation, and technical implementation is proposed. An organic combination of frog-style scooter and children’s tricycle is developed using the multi-function method. From the ergonomic perspective, the paper elaborates on the body size of children aged 5 to 12 and effectively extracts data for a multi-function children’s bike, which can be used for gliding and riding. By inverting the body, parts can be interchanged between the handles and the pedals of the bike. Finally, the paper provides a detailed analysis of the components and structural design, body material, and processing technology of the bike. The study of Industrial Product Innovation Design provides an effective design method to solve the bicycle problems, extends the function problems, improves the product market situation, and enhances the energy saving feature while implementing intensive product development effectively at the same time.

Construction of doubly-periodic solutions to nonlinear partial differential equations using improved Jacobi elliptic function expansion method and symbolic computation

Some doubly-periodic solutions of the Zakharov-Kuznetsov equation are presented. Our approach is to introduce an auxiliary ordinary differential equation and use its Jacobi elliptic function solutions to construct doubly-periodic solutions of the Zakharov-Kuznetsov equation, which has been derived by Gottwald as a two-dimensional model for nonlinear Rossby waves. When the modulus k →1, these solutions reduce to the solitary wave solutions of the equation.

Free vibration of non-uniform axially functionally graded beams using the asymptotic development method

The asymptotic development method is applied to analyze the free vibration of non-uniform axially functionally graded(AFG) beams, of which the governing equations are differential equations with variable coefficients. By decomposing the variable flexural stiffness and mass per unit length into reference invariant and variant parts, the perturbation theory is introduced to obtain an approximate analytical formula of the natural frequencies of the non-uniform AFG beams with different boundary conditions.Furthermore, assuming polynomial distributions of Young's modulus and the mass density, the numerical results of the AFG beams with various taper ratios are obtained and compared with the published literature results. The discussion results illustrate that the proposed method yields an effective estimate of the first three order natural frequencies for the AFG tapered beams. However, the errors increase with the increase in the mode orders especially for the cases with variable heights. In brief, the asymptotic development method is verified to be simple and efficient to analytically study the free vibration of non-uniform AFG beams, and it could be used to analyze any tapered beams with an arbitrary varying cross width.

A STUDY ON THE WEIGHT FUNCTION OF THE MOVING LEAST SQUARE APPROXIMATION IN THE LOCAL BOUNDARY INTEGRAL EQUATION METHOD

The meshless method is a new numerical technique presented in recent years.It uses the moving least square(MLS)approximation as a shape function.The smoothness of the MLS approximation is determined by that of the basic function and of the weight function,and is mainly determined by that of the weight function.Therefore,the weight function greatly affects the accuracy of results obtained.Different kinds of weight functions,such as the spline function, the Gauss function and so on,are proposed recently by many researchers.In the present work,the features of various weight functions are illustrated through solving elasto-static problems using the local boundary integral equation method.The effect of various weight functions on the accuracy, convergence and stability of results obtained is also discussed.Examples show that the weight function proposed by Zhou Weiyuan and Gauss and the quartic spline weight function are better than the others if parameters c and α in Gauss and exponential weight functions are in the range of reasonable values,respectively,and the higher the smoothness of the weight function,the better the features of the solutions.

An improved influence function method for predicting subsidence caused by longwall mining operations in inclined coal seams

Prediction of surface subsidence caused by longwall mining operation in inclined coal seams is often very challenging. The existing empirical prediction methods are inflexible for varying geological and mining conditions. An improved influence function method has been developed to take the advantage of its fundamentally sound nature and flexibility. In developing this method, the original Knothe function has been transformed to produce a continuous and asymmetrical subsidence influence function. The empirical equations for final subsidence parameters derived from col- lected longwall subsidence data have been incorporated into the mathematical models to improve the prediction accuracy. A number of demonstration cases for longwall mining operations in coal seams with varying inclination angles, depths and panel widths have been used to verify the applicability of the new subsidence prediction model.

Uncertainty analysis of strain modal parameters by Bayesian method using frequency response function

Structural strain modes are able to detect changes in local structural performance, but errors are inevitably intermixed in the measured data. In this paper, strain modal parameters are considered as random variables, and their uncertainty is analyzed by a Bayesian method based on the structural frequency response function （FRF）. The estimates of strain modal parameters with maximal posterior probability are determined. Several independent measurements of the FRF of a four-story reinforced concrete flame structural model were performed in the laboratory. The ability to identify the stiffness change in a concrete column using the strain mode was verified. It is shown that the uncertainty of the natural frequency is very small. Compared with the displacement mode shape, the variations of strain mode shapes at each point are quite different. The damping ratios are more affected by the types of test systems. Except for the case where a high order strain mode does not identify local damage, the first order strain mode can provide an exact indication of the damage location.

An influence function method based subsidence prediction program for longwall mining operations in inclined coal seams

The distribution of the final surface subsidence basin induced by longwall operations in inclined coal seam could be significantly different from that in flat coal seam and demands special prediction methods. Though many empirical prediction methods have been developed, these methods are inflexible for varying geological and mining conditions. An influence function method has been developed to take the advantage of its fundamentally sound nature and flexibility. In developing this method, significant modifications have been made to the original Knothe function to produce an asymmetrical influence function. The empirical equations for final subsidence parameters derived from US subsidence data and Chinese empirical values have been incorpo- rated into the mathematical models to improve the prediction accuracy. A corresponding computer program is developed. A number of subsidence cases for longwall mining operations in coal seams with varying inclination angles have been used to demonstrate the applicability of the developed subsidence prediction model.

Extended Group Foliation Method and Functional Separation of Variables to Nonlinear Wave Equations

Generalized functional separation of variables to nonlinear evolution equations is studied in terms of the extended group foliation method, which is based on the Lie point symmetry method. The approach is applied to nonlinear wave equations with variable speed and external force. A complete classification for the wave equation which admits functional separable solutions is presented. Some known results can be recovered by this approach.

Exactness of penalization for exact minimax penalty function method in nonconvex programming

The exact minimax penalty function method is used to solve a noncon- vex differentiable optimization problem with both inequality and equality constraints. The conditions for exactness of the penalization for the exact minimax penalty function method are established by assuming that the functions constituting the considered con- strained optimization problem are invex with respect to the same function η （with the exception of those equality constraints for which the associated Lagrange multipliers are negative these functions should be assumed to be incave with respect to η）. Thus, a threshold of the penalty parameter is given such that, for all penalty parameters exceeding this threshold, equivalence holds between the set of optimal solutions in the considered constrained optimization problem and the set of minimizer in its associated penalized problem with an exact minimax penalty function. It is shown that coercivity is not suf- ficient to prove the results.

Buckling analysis of functionally graded plates partially resting on elastic foundation using the differential quadrature element method

We extend the differential quadrature element method (DQEM) to the buckling analysis of uniformly in-plane loaded functionally graded (FG) plates fully or partially resting on the Pasternak model of elastic support. Material properties of the FG plate are graded in the thickness direction and assumed to obey a power law distribution of the volume fraction of the constituents. To set up the global eigenvalue equation, the plate is divided into sub-domains or elements and the generalized differential quadrature procedure is applied to discretize the governing, boundary and compatibility equations. By assembling discrete equations at all nodal points, the weighting coefficient and force matrices are derived. To validate the accuracy of this method, the results are compared with those of the exact solution and the finite element method. At the end, the effects of different variables and local elastic support arrangements on the buckling load factor are investigated.

Intercalation Assembly Method and Intercalation Process Control of Layered Intercalated Functional Materials

Layered intercalated functional materials of layered double hydroxide type are an important class of functional materials developed in recent years. Based on long term studies on these materials in the State Key Laboratory of Chemical Resource Engineering in Beiiing University of Chemical Technology, the orinciole for the design of controlled intercalation processes in the light of tuture production processing requirements has been developed. Intercalation assembly methods and technologies have been invented to control the intercalation process for preparing layered intercalated materials with various structures and functions.

Large deformation analysis of a cantilever beam made of axially functionally graded material by homotopy analysis method

Large deformation of a cantilever axially functionally graded (AFG) beam subject to a tip load is analytically studied using the homotopy analysis method (HAM). It is assumed that its Young’s modulus varies along the longitudinal direction according to a power law. Taking the solution of the corresponding homogeneous beam as the initial guess and obtaining a convergence region by adjusting an auxiliary parameter, the analytical expressions for large deformation of the AFG beam are provided. Results obtained by the HAM are compared with those obtained by the finite element method and those in the previous works to verify its validity. Good agreement is observed. A detailed parametric study is carried out. The results show that the axial material variation can greatly change the deformed configuration, which provides an approach to control and manage the deformation of beams. By tailoring the axial material distribution, a desired deformed configuration can be obtained for a specific load. The analytical solution presented herein can be a helpful tool for this procedure.

Trial function method and exact solutions to the generalized nonlinear Schrdinger equation with time-dependent coefficient

In this paper, the trial function method is extended to study the generalized nonlinear Schrodinger equation with time- dependent coefficients. On the basis of a generalized traveling wave transformation and a trial function, we investigate the exact envelope traveling wave solutions of the generalized nonlinear Schrodinger equation with time-dependent coefficients. Taking advantage of solutions to trial function, we successfully obtain exact solutions for the generalized nonlinear Schrodinger equation with time-dependent coefficients under constraint conditions.