High-accuracy Solution Research Articles

Low storage explicit Runge-Kutta method

Neste artigo estamos tratando dos métodos explícitos Runge Kutta (LSERK) de alta ordem e baixo armazenamento, que são usados principalmente para a discretização temporal e são estáveis independentemente de sua precisão. O principal objetivo deste trabalho é comparar o RK tradicional com diferentes formas de métodos LSERK. Os experimentos numéricos indicam que tais métodos são altamente precisos e eficazes para propósitos numéricos. Também é mostrado o tempo de CPU e suas implicações na solução. O método é bem adequado para obter uma solução precisa de alta ordem para o problema escalar de segunda ordem do problema de valor inicial (IVP), como é discutido no presente artigo.

A numerical treatment of the coupled viscous Burgers’ equation in the presence of very large Reynolds number

A numerical investigation of the coupled viscous Burgers’ equation for very large Reynolds numbers do not exist in the literature. Coupled viscous Burgers’ equations are solved numerically in the presence of very large Reynolds numbers. For this case, numerical approach to the coupled viscous Burgers’ equation via contributions of two effective methods is used. The first component of the mixed method is finite difference method and the second one is differential quadrature method. For this process, the third order modified cubic B-spline functions are used as base function. To display the effectiveness of the present mixed method four different test problems have been investigated. For various values of the coefficients, more particularly for very large Reynolds numbers, in other words, for the very small value of kinematic viscosity parameters solutions are obtained. Error norms are calculated and compared with analytical results and also with numerical results of the related literature. Present results display that the present mixed method obtains high accurate solutions and in compatibility with both of the analytical and numerical results.

A proper orthogonal decomposition analysis method for transient nonlinear heat conduction problems. Part 2: Advanced algorithm

The present work conducts a systematic and in-depth algorithm investigation for transient nonlinear heat conduction problems solved by using the proper orthogonal decomposition (POD) method. The computational results as presented in Part 1 have revealed some signatures of the basic algorithms, in which poor efficiency is its main shortcoming. In this article, as Part 2, several advanced algorithms are derived to high efficiently deal with the POD-based ROM of transient nonlinear heat conduction problems with temperature-dependent thermal conductivity, including: (i) Element pre-conversion method (EPM), (ii) Multi-level linearization method (MLM), and (iii) EPM–MLM compound method. The accuracy and efficiency are verified by both two-dimensional (2-D) and three-dimensional (3-D) numerical examples with Neumann boundary condition. The results show that these methods can provide high accuracy and high solution quality in the transient nonlinear heat conduction problems. Most importantly, the computational efficiency of these methods has been improved to different degrees when compared to that of the basic algorithm. Among them, the EPM–MLM compound method is the most efficient one. The acceleration effect of POD-based ROM solved by using this compound method is around one order of magnitude higher than that of the basic algorithm, and more pronounced for larger problems.

An Efficient Kriging-Based Constrained Optimization Algorithm by Global and Local Sampling in Feasible Region

Abstract To solve challenging optimization problems with time-consuming objective and constraints, a novel efficient Kriging-based constrained optimization (EKCO) algorithm is proposed in this paper. The EKCO mainly consists of three sampling phases. In phase I of EKCO, considering the significance of constraints, feasible region is constructed via employing a feasible region sampling (FRS) criterion. The FRS criterion can avoid the local clustering phenomenon of sample points. Therefore, phase I is also a global sampling process for the objective function in the feasible region. However, the objective function may be higher-order nonlinear than constraints. In phase II, by maximizing the prediction variance of the surrogate objective, more accurate objective function in the feasible region can be obtained. After global sampling, to accelerate the convergence of EKCO, an objective local sampling criterion is introduced in phase III. The verification of the EKCO algorithm is examined on 18 benchmark problems by several recently published surrogate-based optimization algorithms. The results indicate that the sampling efficiency of EKCO is higher than or comparable with that of the recently published algorithms while maintaining the high accuracy of the optimal solution, and the adaptive ability of the proposed algorithm also be validated. To verify the ability of EKCO to solve practical engineering problems, an optimization design problem of aeronautical structure is presented. The result indicates EKCO can find a better feasible design than the initial design with limited sample points, which demonstrates practicality of EKCO.

Accelerating ADMM for efficient simulation and optimization

The alternating direction method of multipliers (ADMM) is a popular approach for solving optimization problems that are potentially non-smooth and with hard constraints. It has been applied to various computer graphics applications, including physical simulation, geometry processing, and image processing. However, ADMM can take a long time to converge to a solution of high accuracy. Moreover, many computer graphics tasks involve non-convex optimization, and there is often no convergence guarantee for ADMM on such problems since it was originally designed for convex optimization. In this paper, we propose a method to speed up ADMM using Anderson acceleration, an established technique for accelerating fixed-point iterations. We show that in the general case, ADMM is a fixed-point iteration of the second primal variable and the dual variable, and Anderson acceleration can be directly applied. Additionally, when the problem has a separable target function and satisfies certain conditions, ADMM becomes a fixed-point iteration of only one variable, which further reduces the computational overhead of Anderson acceleration. Moreover, we analyze a particular non-convex problem structure that is common in computer graphics, and prove the convergence of ADMM on such problems under mild assumptions. We apply our acceleration technique on a variety of optimization problems in computer graphics, with notable improvement on their convergence speed.

An approximate solution for vibrations of uniform and stepped functionally graded spherical cap based on Ritz method

Based on Ritz method, the vibration approximate solutions of uniform and stepped functionally graded (FG) spherical cap are carried out in this paper. The first-order shear deformation theory (FSDT) is used to derive energy expression. The selections of displacement functions are based on domain decomposition approach, in which the unified Jacobi polynomials are introduced to represent the displacement functions component along axial direction. In addition, the standard Fourier series still denote the displacement functions component along circumferential direction. The various boundary conditions are simulated by applying spring stiffness method. Then the Ritz method is employed to obtain the final results. The solutions of the same condition are compared with those obtained by finite element method (FEM) and published literatures to validate the present method. The results exhibit that the current approach has the advantages of high solution accuracy and fast convergence. On this basis, the numerical results concerning the effects of geometric parameters and boundary conditions on the vibration responses of the structure are also considered.

Study of the Effect of Thermal Dispersion on Internal Natural Convection in Porous Media Using Fourier Series

Natural convection in a porous enclosure in the presence of thermal dispersion is investigated. The Fourier–Galerkin (FG) spectral element method is adapted to solve the coupled equations of Darcy’s flow and heat transfer with a full velocity-dependent dispersion tensor, employing the stream function formulation. A sound implementation of the FG method is developed to obtain accurate solutions within affordable computational costs. In the spectral space, the stream function is expressed analytically in terms of temperature, and the spectral system is solved using temperature as the primary unknown. The FG method is compared to finite element solutions obtained using an in-house code (TRACES), OpenGeoSys and COMSOL Multiphysics®. These comparisons show the high accuracy of the FG solution which avoids numerical artifacts related to time and spatial discretization. Several examples having different dispersion coefficients and Rayleigh numbers are tested to analyze the solution behavior and to gain physical insight into the thermal dispersion processes. The effect of thermal dispersion coefficients on heat transfer and convective flow in a porous square cavity has not been investigated previously. Here, taking advantage of the developed FG solution, a detailed parameter sensitivity analysis is carried out to address this gap. In the presence of thermal dispersion, the Rayleigh number mainly affects the convective velocity and the heat flux to the domain. At high Rayleigh numbers, the temperature distribution is mainly controlled by the longitudinal dispersion coefficient. Longitudinal dispersion flux is important along the adiabatic walls while transverse dispersion dominates the heat flux toward the isothermal walls. Correlations between the average Nusselt number and dispersion coefficients are derived for three Rayleigh number regimes.

Modification of the Time-Effective Moduli of Viscoelastic Bodies

The problem on constructing new time-effective characteristics of a linear viscoelastic body is considered. The initial medium is modeled as a viscoelastic composite. One its part has properties determined by time-effective moduli of the Lagrangian type, but the other part has properties determined by moduli of the Castigliano type. This model makes it possible to employ the methods of mechanics of composite materials, for example, to formulate the effective Voigt and Reuss moduli and to construct iterative transformations narrowing the Voigt and Reuss fork. These transformations are constructed in such a way that, at each iteration, the inequalities following from the minimum total potential energy principle and the theorem of complementary work are satisfied for the effective moduli. It is shown that, at each moment of time t, the sequences of iteratively transformed Voigt and Reuss moduli converge to the same limit, equal to the geometric mean of their initial values. By the example of the problem on bending of a viscoelastic plate, the approximate solutions obtained on the basis of the new time-effective characteristics found, are compared with an analytical solution. Their good agreement points to a high accuracy of the approximate solutions.

CS detection and correction techniques for RTK positioning using single‐frequency GNSS receivers: trends and comparison

Nowadays, high-accuracy satellite-based positioning and navigation solutions are obtained using carrier-phase observations together with a differential correction scheme. A slip of only a few carrier-phase cycles can, however, bias observations causing a reduction in position accuracy. Therefore, reliable detection and handling of cycle slips (CSs) is essential for achieving high-accuracy positioning performance. Generally, it is harder to resolve CS in single-frequency observations, but the area deserves attention as differential positioning using low-cost single-frequency receivers is expected to be the key for many emerging mass-market applications. This study reviews and compares CS detection and correction techniques for single-frequency GNSS receivers with respect to their application for low-cost real-time kinematic (RTK) positioning. No prior performance comparison of CS detection and correction techniques exists; thereby, the authors have proposed four parameters to assess performance quality of CS resolution scheme for RTK positioning. Subsequently, an up-to-date list of existing techniques, classified according to the CS detection strategy, is reviewed with respect to measurement model, CS correction approach and test results. The techniques are also analysed for performance quality and it is observed that they do not conform to all parameters. Finally, conclusions are drawn and future research directions are proposed at the end.

Natural oscillation shapes for a heavy fluid in an elliptic vessel

The problem of constructing of shapes of the natural oscillations in a vertical cylindrical basin (vessel) with an elliptic cross-section in linearized form of an ideal incompressible fluid is considered. Only the gravity force is acting on the fluid. It is required to define the frequencies and shapes of the natural oscillations of the fluid as a function of the system’s parameters. The solution algorithm represented here is based on the method of separation of variables in elliptic coordinates, and the special developed method for the high-accuracy solution of the boundary problem. It used the variational principle and the procedure of continuation in the parameter. This approach shows its effectiveness and obtains the almost complete solution of the problem. For five lower oscillation modes, the natural shapes are determined with high degree of accuracy for a wide range of the eccentricity values.

A whale optimization algorithm based on quadratic interpolation for high-dimensional global optimization problems

Whale Optimization Algorithm (WOA), as a new population-based optimization algorithm, performs well in solving optimization problems. However, when tackling high-dimensional global optimization problems, WOA tends to fall into local optimal solutions and has slow convergence rate and low solution accuracy. To address these problems, a whale optimization algorithm based on quadratic interpolation (QIWOA) is presented. On the one hand, a modified exploration process by introducing a new parameter is proposed to efficiently search the regions and deal with the premature convergence problem. On the other hand, quadratic interpolation around the best search agent helps QIWOA to improve the exploitation ability and the solution accuracy. Moreover, the algorithm tries to make a balance between exploitation and exploration. QIWOA is compared with several state-of-the-art algorithms on 30 high-dimensional benchmark functions with dimensions ranging from 100 to 2000. The experimental results show that QIWOA has faster convergence rate and higher solution accuracy than both WOA and other population-based algorithms. For functions with a flat or sharp bottom, QIWOA is difficult to find the global optimum, but it still performs best compared with other algorithms.

An expanded particle swarm optimization based on multi-exemplar and forgetting ability

There are two phenomena in human society and biological systems. One is that people prefer to extract knowledge from multiple exemplars to obtain better learning ability. The other one is the forgetting ability that helps the encoding and consolidation of new information by removing unused or unwanted memories. Inspired by these phenomena, this paper transplants the multi-exemplar and forgetting ability to particle swarm optimization (PSO), and proposes an eXpanded PSO, called XPSO. Firstly, XPSO expands the “social-learning” part of each particle from one exemplar to two exemplars, learning from both the locally and the globally best exemplars. Secondly, XPSO assigns different forgetting abilities to different particles, simulating the forgetting phenomenon in the human society. Under the multi-exemplar learning model with forgetting ability, XPSO further adopts an adaptive scheme to update the acceleration coefficients and selects a reselection mechanism to update the population topology. The effectiveness of these additional proposed strategies is verified by extensive experiments. Moreover, comparison results among XPSO and other 9 popular PSO as well as 3 non-PSO algorithms on CEC’13 test suite suggest that XPSO attains a very promising performance for solving different types of functions, contributing to both higher solution accuracy and faster convergence speed.

Free vibration characteristics of functionally graded porous spherical shell with general boundary conditions by using first-order shear deformation theory

The paper analyzed the free vibration of functionally graded porous spherical shell (FGPSS) based on Ritz method. The energy method and first-order shear deformation theory (FSDT) are adopted to derive the formulas. In this paper, the displacement functions are improved on basis of domain decomposition method, in which the unified Jacobi polynomials are introduced to represent the displacement functions component along meridional direction, and the displacement functions component along circumferential direction is still Fourier series. In addition, the spring stiffness method is formed a unified format to deal with various complex boundary conditions and continuity conditions. Then the final solutions can be obtained based on Ritz method. To prove the validity of proposed method, the results of the same condition are compared with those obtained by FEM, published literatures and experiment. The results show that the proposed method has advantages of fast convergence, high calculation efficiency, high solution accuracy and simple boundary simulation.

A novel technique based on Bernoulli wavelets for numerical solutions of two-dimensional Fredholm integral equation of second kind

PurposeA novel technique based on Bernoulli wavelets has been proposed to solve two-dimensional Fredholm integral equation of second kind. Bernoulli wavelets have been created by dilation and translation of Bernoulli polynomials. This paper aims to introduce properties of Bernoulli wavelets and Bernoulli polynomials.Design/methodology/approachTo solve the two-dimensional Fredholm integral equation of second kind, the proposed method has been used to transform the integral equation into a system of algebraic equations.FindingsNumerical experiments shows that the proposed two-dimensional wavelets technique can give high-accurate solutions and good convergence rate.Originality/valueThe efficiency of newly developed two-dimensional wavelets technique has been validated by different illustrative numerical examples to solve two-dimensional Fredholm integral equations.

ПРОГРАММНАЯ СИСТЕМА МНОГОКРИТЕРИАЛЬНОЙ ОПТИМИЗАЦИИ МЕТОДОМ ЭВОЛЮЦИОННОГО МОДЕЛИРОВАНИЯ ИНВЕСТИЦИОННОГО ПОРТФЕЛЯ

A financial market changing state requires an automation to determine investment strategies in combination with the high accuracy of solutions found. There is offered a solution for the problem mentioned with the use of methods of an evolutionary modeling, in particular, a genetic algorithm developed. It allows defining optimum Pareto options of an investment portfolio (IP) differing with the balance between its profitability and a risk. The algorithm mentioned is used in the developed software program with a module structure. It contains the collection and analysis subsystems of data on security characteristics, IP structure optimization, and also IP choice the most corresponding to investor’s requirements. The software system working capacity was analyzed at IP formation on the basis of actual data during the time period chosen the data obtained correspond to the indices of possible investment strategies.

Weighted reproducing kernel collocation method based on error analysis for solving inverse elasticity problems

For inverse problems equipped with incomplete boundary conditions, a simple solution strategy to obtain approximations remains a challenge in the fields of engineering and science. Based on our previous study, the weighted reproducing kernel collocation method (W-RKCM) shows optimal convergence in solving inverse Cauchy problems. As such, this work further introduces the W-RKCM to solve inverse problems in elasticity. From mathematical error estimate and numerical convergence study, it is shown that the weighted least-squares formulation can properly balance the errors in the domain and on the boundary. By comparing the approximations obtained by W-RKCM with those obtained by the direct collocation method, the reproducing kernel shape function can retain the locality without using a large support size, and the corresponding approximations exhibit extremely high solution accuracy. The stability of the W-RKCM is demonstrated by adding noise on the boundary conditions. This work shows the efficacy of the proposed W-RKCM in solving inverse elasticity problems as no additional technique is involved to reach the desired solution accuracy in comparison with the existing methods in the literature.

Analysis for center deviation of circular target under perspective projection

PurposeAccurate feature localization is a fundamental problem in computer vision and visual measurement. In a perspective projection model of the camera, the projected center of a spatial circle and the center of the projection ellipse are not identical. This paper aims to show how to locate the real projection center precisely in the perspective projection of a space circle target.Design/methodology/approachBy analyzing the center deviation caused by projection transformation, a novel method is presented to precisely locate the real projection center of a space circle using projective geometry. Solution distribution of the center deviation is analyzed, and the quadratic equation for determining the deviation is derived by locating vanishing points. Finally, the actual projected center of the circular target is achieved by solving the deviation quadratic equations.FindingsThe procedures of the author’s method are simple and easy to implement. Experimental data calculated that maximum deviation occurs at approximately between 3π/10 and 2π/5 of the angle between the projection surface and the space target plane. The absolute reduction in error is about 0.03 pixels; hence, it is very significant for a high-accuracy solution of the position of the space circle target by minimizing systematic measurement error of the perspective projection.Originality/valueThe center deviation caused by the space circle projection transformation is analyzed, and the detailed algorithm steps to locate the real projection center precisely are described.

A review and verification of analytical weight function methods in fracture mechanics

AbstractThe weight function method is a powerful method for the determination of key parameters of stress intensity factors and crack opening displacements that are fundamental for crack analyses in fracture mechanics. The method has a remarkable computational efficiency without compromising solution accuracy and is especially suited for cracks in real‐world components with complex stress gradients. A critical issue for successful development and reliable application of the weight function method is how to achieve verifiable and high solution accuracy. This paper provides several reflections on the current state of the art in weight function methods, including a brief historical perspective and description of several analytical and numerical weight methods, with main focus on the two‐dimensional (2D) analytical approaches. A variety of typical application examples are presented. A platform for point‐by‐point accuracy assessment by using the Green's functions is established on a completely independent numerical approach with validated accuracy. Rigorous and systematic verification of the accuracy levels of different 2D analytical weight function approaches are conducted using several benchmark crack geometries. The sources of inaccuracy of different analytical weight function methods are analysed. A brief account on weight function methods for analysing three‐dimensional (3D) crack problems with arbitrary univariant and bivariant stress distributions is provided.

Quadrilateral overlapping elements and their use in the AMORE paradigm

We give the formulation of new 4-node overlapping quadrilateral finite elements and demonstrate their performance in several numerical examples. The elements are insensitive to mesh distortions and lead overall to accurate numerical solutions. We show how the quadrilateral elements are utilized in the AMORE scheme of “automatic meshing with overlapping and regular elements”. Since in this paradigm the finite elements interior to the domain of analysis are undistorted regular elements and the distortion-tolerant overlapping elements are used for discretizations near the boundaries, high solution accuracy is achieved. This accuracy is obtained with much less effort for meshing than in traditional finite element analysis and with a reasonable computational expense.

Electrodynamic analysis of the director antenna by the method of integral equations with a reduction in the number of partitioning segments of passive conductors

It has been solved the problem task for the numerical electrodynamic analysis of a Yagi-antenna using a mathematical model developed with the Poklington integral equation and the Krylov-Bogolyubov method with a reduction in the number of partitioning segments of passive conductors. The obtained results demonstrate the high efficiency of the proposed model, due to the combination of high accuracy solutions and saving computational resources.