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The moment distribution method, originally introduced by Hardy Cross in his influential publication "Analysis of Continuous Frames by Distributing Fixed‐End Moments" (published in Transactions of the ASCE, 1932;96(1793):1–10), is widely recognized as a fundamental aspect of both structural engineering and global education. This iterative technique aims to improve solution accuracy at each step of distribution and has stood the test of time. In our study, we delve into the mathematical foundation of this method and make a crucial discovery: the terms generated during iterations follow a geometric series pattern. By leveraging this property, we propose a streamlined formula for efficiently calculating the sum of all the terms produced in each step of distribution iteration. Through comparative analysis with alternative methods, we highlight the method's superior precision and robustness in structural analysis. Our investigation primarily focuses on demonstrating the geometric series nature of the terms generated through iterative moment distribution, providing specific theorems as references. Ultimately, our objective is to present concise, closed‐form mathematical solutions that showcase the practical effectiveness of this method in various applications within the field of structural engineering. [ABSTRACT FROM AUTHOR]

In the creation and research of modeling, control and identification systems, the most important stages are the formulation and numerical solution of equations of mathematical models of these systems, which are usually represented in the classes of differential and integral equations. At the same time, the issues of developing and researching computational algorithms that implement methods for numerical solution of equations of mathematical models of systems, in particular, ensuring control over the accuracy of the desired solution, as well as assessing the impact of deviations of parameters of dynamic systems on their movement and quality indicators, are crucial. Both in analyzing the accuracy of numerical study of mathematical models of dynamic systems and in solving problems of synthesis of the latter based on accuracy conditions, the possibility of analytical expression of additional motion of the excited system is of great importance. Mathematical models of non-stationary systems, represented in the form of differential equations in full derivatives, are implemented in the vast majority of applied problems by the Runge-Kutta method of various orders. It is shown that an increase in the productivity of machine calculation can be achieved if it is possible to take into account the difference in the rate of change of different groups of coordinates of the investigated nonstationary system. The problem is formulated and the possibility of rational choice of parameters of the Runge-Kutta method formulas is considered, which allows minimizing the time of integration of equations of the mathematical model of the system. [ABSTRACT FROM AUTHOR]

The problem of estimating the degree of accuracy of the approximate solution of the Cauchy problem for the system of ordinary differential equations is studied. For an approximate solution scheme based on the first two terms in the expansion of the exact solution to the Taylor formula, an error estimator inequality is proved and a definite expression for the corresponding coefficient is derived. [ABSTRACT FROM AUTHOR]

This paper presents a novel and efficient solution procedure to improve 3D solid finite element analysis with an enrichment scheme. To this end, we employ finite elements enriched by polynomial cover functions, which can expand their solution space without requiring mesh refinement or additional nodes. To facilitate this solution procedure, an error estimation method and cover function selection scheme for 3D solid finite element analysis are developed. This enables the identification of nodes with suboptimal solution accuracy, allowing for the adaptive application of cover functions in a systematic and efficient manner. Furthermore, a significant advantage of this procedure is its consistency, achieved by excluding arbitrary coefficients from the formulations employed. The effectiveness of the proposed procedure is demonstrated through several numerical examples. In the majority of the examples, it is observed that the stress prediction error is reduced by more than half after applying the proposed procedure. [ABSTRACT FROM AUTHOR]

In the production and processing of precision shaft-hole class parts, the wear of cutting tools, machine chatter, and insufficient lubrication can lead to changes in their roundness, which in turn affects the overall performance of the relevant products. To improve the accuracy of roundness error assessments, Bat algorithm (BA) is applied to roundness error assessments. An improved bat algorithm (IBA) is proposed to counteract the original lack of variational mechanisms, which can easily lead BA to fall into local extremes and induce premature convergence. First, logistic chaos initialisation is applied to the initial solution generation to enhance the variation mechanism of the population and improve the solution quality; second, a sinusoidal control factor is added to BA to control the nonlinear inertia weights during the iterative process, and the balance between the global search and local search of the algorithm is dynamically adjusted to improve the optimization-seeking accuracy and stability of the algorithm. Finally, the sparrow search algorithm (SSA) is integrated into BA, exploiting the ability of explorer bats to perform a large range search, so that the algorithm can jump out of local extremes and the convergence speed of the algorithm can be improved. The performance of IBA was tested against the classical metaheuristic algorithm on eight benchmark functions, and the results showed that IBA significantly outperformed the other algorithms in terms of solution accuracy, convergence speed, and stability. Simulation and example verification show that IBA can quickly find the centre of a minimum inclusion region when there are many or few sampling points, and the obtained roundness error value is more accurate than that of other algorithms, which verifies the feasibility and effectiveness of IBA in evaluating roundness errors.

We develop an analytical framework designed to solve and analyze heterogeneous‐agent models that endogenously generate fat‐tailed wealth distributions. We exploit the asymptotic linearity of policy functions and the analytical characterization of the Pareto exponent to augment the conventional solution algorithm with a theory of the tail. Our framework allows for a precise understanding of the very top of the wealth distribution (e.g., analytical expressions for top wealth shares, type distribution in the tail, and transition probabilities in and out of the tail) in addition to delivering improved accuracy and speed. [ABSTRACT FROM AUTHOR]

This paper deals with how to solve chemical dissolution-front instability problems, which are nonlinearly coupled by subsurface pore-fluid flow, reactive mass transport and porosity evolution processes in fluid-saturated porous media, through using two different computational schemes. In the first computational scheme, porosity, pressure of the pore-fluid and concentration of the solute are used as fundamental unknown variables to describe a chemical dissolution system, so that it can be named as the PPC scheme. In the second computational scheme, porosity, velocity of the pore-fluid and concentration of the solute are used as fundamental unknown variables to describe a chemical dissolution system, so that it can be named as the PVC scheme. Since the finite element equations of a chemical dissolution-front instability problem on the basis of using the PPC scheme is available, the main focus of this study is on deriving the finite element equations of a chemical dissolution-front instability problem on the basis of using the PVC scheme. In particular, analytical solutions for the property matrices of a four-node rectangular element have been derived and used in both the PVC scheme and the PPC scheme. Through comparing the computational simulation results obtained from using both the PPC scheme and the PVC scheme, it has demonstrated that: (1) if the chemical dissolution system is in a stable state, then the PPC scheme is superior to the PVC scheme because the PPC scheme uses less computational efforts than the PVC scheme; (2) if the chemical dissolution system is in an unstable state, then the PVC scheme is superior to the PPC scheme because the PVC scheme yields more accurate computational simulation results than the PPC scheme. [ABSTRACT FROM AUTHOR]

This paper presents a novel and efficient solution procedure to improve 3D solid finite element analysis with an enrichment scheme. To this end, we employ finite elements enriched by polynomial cover functions, which can expand their solution space without requiring mesh refinement or additional nodes. To facilitate this solution procedure, an error estimation method and cover function selection scheme for 3D solid finite element analysis are developed. This enables the identification of nodes with suboptimal solution accuracy, allowing for the adaptive application of cover functions in a systematic and efficient manner. Furthermore, a significant advantage of this procedure is its consistency, achieved by excluding arbitrary coefficients from the formulations employed. The effectiveness of the proposed procedure is demonstrated through several numerical examples. In the majority of the examples, it is observed that the stress prediction error is reduced by more than half after applying the proposed procedure.

This paper deals with how to implement perturbations in the computational simulations of chemical dissolution‐front instability (CDFI) problems in fluid‐saturated porous media. On the basis of theoretical analysis, it is found that the application of a perturbation to the chemical dissolution front is equivalent to the application of an alternative perturbation to the dimensionless pore‐fluid normal velocity (relative to the planar chemical dissolution front) in the chemical dissolution zone, where the chemical dissolution front is located. This avoids the difficulty to find the spatial coordinates of the chemical dissolution front locations in the computational simulations of CDFI problems. Based on this new finding, a novel algorithm for implementing perturbations in the computational simulations of CDFI problems is proposed. The key point of the proposed algorithm is that the perturbed pore‐fluid normal velocity (relative to the planar chemical dissolution front) is used to directly replace the original pore‐fluid normal velocity (relative to the planar chemical dissolution front) in the related mathematical governing equations (MGEs), so that the proposed algorithm works for the porosity‐velocity‐concentration scheme when it is used to solve CDFI problems in fluid‐saturated porous media. In addition, the related theoretical analysis in this study has answered the previous unanswered question why the application of a perturbation to porosity works in the porosity‐pressure‐concentration scheme but does not work in the porosity‐velocity‐concentration scheme for solving the same CDFI problems in fluid‐saturated porous media. Through solving two illustrative examples with two different distributions of initial porosity, in which one is homogeneous and another is heterogeneous in the chemical dissolution system, the validity and usefulness of the proposed algorithm for implementing perturbations in the computational simulations of CDFI problems have been demonstrated. [ABSTRACT FROM AUTHOR]

This article presents an accurate porosity‐velocity‐concentration approach, in which porosity, pore‐fluid velocity and the concentration of dissolvable substances in the pore fluid are selected as four primary unknown variables for solving reactive mass transport problems involving chemical dissolution in fluid‐saturated porous media with arbitrarily initial porosity distributions. The first advantage of using the proposed approach is that since pore‐fluid velocity, instead of pore‐fluid pressure, is selected as the primary unknown variable to describe the pore‐fluid flow process, the pore‐fluid velocity obtained from the proposed approach is more accurate than that obtained from the numerical simulation, in which pore‐fluid pressure is selected as the primary unknown variable to describe the pore‐fluid flow. The second advantage of using the proposed approach is that because the property matrices of a four‐node rectangular element are precisely calculated in a purely mathematical way, the overall accuracy of numerical solutions can be ensured. After the proposed approach is verified by a benchmark problem, it has been applied for solving reactive mass transport problems involving chemical dissolution in fluid‐saturated porous media with three different kinds of initial porosity distributions. It has been demonstrated that: (1) the proposed approach can produce highly‐accurate numerical solutions for solving reactive mass transport problems involving chemical dissolution in fluid‐saturated porous media with arbitrarily initial porosity distributions; (2) the initial porosity distribution in a porous medium can have remarkable effects on the reactive mass transport process in the porous medium; (3) the porosity and dimensionless concentration fronts propagate from the entrance to the exit of the problem domain, which is identical to the pore‐fluid flow direction. [ABSTRACT FROM AUTHOR]

In this paper, we present an automatic procedure that enhances the solution accuracy of edge-based smoothed 2D solid finite elements (three-node triangular and four-node quadrilateral elements). To obtain an enhanced solution, an adaptive enrichment scheme that uses enriched 2D solid finite elements and can effectively improve solution accuracy by applying cover functions adaptively without mesh-refinement is adopted in this procedure. First, the error of the edge-based finite element solution is estimated using a devised error estimation method, and appropriate cover functions are assigned for each node. While the edge-based smoothed finite elements provide piecewise constant strain fields, the proposed enrichment scheme uses the enriched finite elements to obtain a higher order strain field within the finite elements. Through various numerical examples, we demonstrate the accuracy improvement and efficiency achieved. [ABSTRACT FROM AUTHOR]

In this paper, we present an automatic procedure that enhances the solution accuracy of edge-based smoothed 2D solid finite elements (three-node triangular and four-node quadrilateral elements). To obtain an enhanced solution, an adaptive enrichment scheme that uses enriched 2D solid finite elements and can effectively improve solution accuracy by applying cover functions adaptively without mesh-refinement is adopted in this procedure. First, the error of the edge-based finite element solution is estimated using a devised error estimation method, and appropriate cover functions are assigned for each node. While the edge-based smoothed finite elements provide piecewise constant strain fields, the proposed enrichment scheme uses the enriched finite elements to obtain a higher order strain field within the finite elements. Through various numerical examples, we demonstrate the accuracy improvement and efficiency achieved.

Permanent magnet synchronous motors (PMSMs) have recently gained increasing industrial and research interest. The PMSM design and especially its performance evaluation through two-dimensional (2D) and three-dimensional (3D) magnetic field analysis using finite element method (FEM) is not trivial, as it is a quite demanding and time consuming procedure. Furthermore, there has always been doubt on the accuracy of 2D analysis results and on which type of analysis is the most preferable one for the specific motor. This work aims to emphasize the difficulties and the advantages that arise from the 3D analysis. The aforementioned simulations were performed for the case of a four-pole PMSM. Several magnetic and electromechanical quantities have been determined. Additionally, various materials were utilized for the construction of stator and rotor cores and their effect on the motor's behavior was investigated. Finally, the derived results are compared and useful conclusions regarding the solution accuracy are given. [ABSTRACT FROM AUTHOR]

Summary: In the conventional discrete velocity method (DVM), the local solution of collisionless Boltzmann equation with a piecewise constant distribution for the distribution function is utilized to reconstruct distribution function at the cell interface and then calculate numerical flux of Boltzmann equation for updating the distribution function at cell center. In this process, a numerical dissipation will be introduced into the solution due to neglecting of the collision effect at the cell interface. This numerical dissipation may deteriorate the solution accuracy of conventional DVM in the continuum flow regime, in which the particle collision happens frequently. To overcome this defect, two improved schemes are first presented in this work, in which the local discrete solution of Boltzmann equation with Shakhov model is adopted to evaluate the distribution function at the cell interface, while the equilibrium state of the local solution is computed by different ways. One of the improved schemes evaluates the equilibrium state exactly by the moments of distribution functions according to the compatibility condition, while the other computes the equilibrium state approximately by a simple average at the cell interface. Since the collision effect is incorporated in evaluation of numerical flux, the improved schemes can provide reasonable solutions in all flow regimes. On the other hand, they introduce some extra computational efforts for determining the collision term at the cell interface as compared with the conventional DVM. To assess the performance of different methods for simulation of flows in all flow regimes, a comprehensive study is then carried out in this work. [ABSTRACT FROM AUTHOR]

Insight has been investigated under the assumption that participants solve insight problems with insight processes and/or experiences. A recent trend has involved presenting participants with the solution and analysing the resultant experience as if insight has taken place. We examined self-reports of the aha experience, a defining aspect of insight, before and after feedback, along with additional affective components of insight (e.g., pleasure, surprise, impasse). Classic insight problems, compound remote associates, and non-insight problems were randomly interleaved and presented to participants. Solution feedback increased ratings of aha experience in both insight and non-insight problems, with this result being driven by responses to solutions that were initially incorrectly generated. Ratings of aha for correctly generated solutions decreased after the correct solution was presented. These findings have implications for insight research paradigms as well as informing teaching methods. [ABSTRACT FROM AUTHOR]

• A lattice Boltzmann flux solver (LBFS) is presented for simulation of conjugate heat transfer problems. • The concept of thermal resistance is introduced to evaluate the thermal conductivity at the cell interface. • The energy equation is discretized by FVM and the numerical flux is evaluated by the local solution of LBE. • The present scheme provides accurate results for conjugate heat transfer problems on non-uniform mesh and curved boundary. In this work, a lattice Boltzmann flux solver (LBFS) is presented for simulation of conjugate heat transfer problems. Specifically, the mass and momentum equations used in the flow domain and the energy equation applied in both the flow and solid domains are discretized by finite volume method (FVM) and the numerical fluxes at the cell interface are reconstructed by the local solution of lattice Boltzmann equation (LBE) truncated to the Navier-Stokes level. To calculate the numerical fluxes, the Chapman-Enskog analysis is carried out first to reveal the connection between the macroscopic fluxes and the solution of LBE. With this relationship, the macroscopic fluxes can then be calculated locally and independently at each cell interface by the solution of LBE, which makes the developed scheme be very easy for application on non-uniform mesh and curved boundary as compared with lattice Boltzmann method (LBM). Overall, the present method well combines the advantages of simplicity and kinetic nature of LBM and geometric flexibility of Navier-Stokes solver discretized by FVM. In addition, in the calculation of numerical flux of energy equation, the concept of thermal resistance is innovatively introduced to evaluate the thermal conductivity at the cell interface. As compared with simple average technique, the introduced strategy can provide a more accurate prediction for temperature field. Numerical results showed that the solid-solid and fluid-solid conjugate heat transfer problems can be well simulated by the developed scheme and the second-order accuracy is achieved in space. [ABSTRACT FROM AUTHOR]

The main objective of the article is to investigate thermal-hydraulic performance of a corrugated cooling fin with louvered surfaces. The investigation is carried out using the fin geometry of one most commonly used liquid-to-air heat exchangers. The investigation was carried out by numerically simulating the airflow with louvered fin geometry. The simulation model was verified by comparing simulated j- and f-factors with the corresponding values of several experimental correlations. The j-factors deviated less than 10.7 % from two of the experimental correlations, whereas deviations ranging between -45.5 % to 86.4 % were reported for the f-factor. The thermal part of the model was validated with good confidence, while the frictional part of the model was validated with a smaller degree of certainty. [ABSTRACT FROM AUTHOR]

Abstract: Many methods are available for the solution of radiative heat transfer problems in participating media. Among these, the discrete ordinates method (DOM) and the finite volume method (FVM) are among the most widely used ones. They provide a good compromise between accuracy and computational requirements, and they are relatively easy to integrate in CFD codes. This paper surveys recent advances on these numerical methods. Developments concerning the grid structure (e.g., new formulations for axisymmetrical geometries, body-fitted structured and unstructured meshes, embedded boundaries, multi-block grids, local grid refinement), the spatial discretization scheme, and the angular discretization scheme are described. Progress related to the solution accuracy, solution algorithm, alternative formulations, such as the modified DOM and FVM, even-parity formulation, discrete-ordinates interpolation method and method of lines, and parallelization strategies is addressed. The application to non-gray media, variable refractive index media, and transient problems is also reviewed. [Copyright &y& Elsevier]

In the production and processing of precision shaft-hole class parts, the wear of cutting tools, machine chatter, and insufficient lubrication can lead to changes in their roundness, which in turn affects the overall performance of the relevant products. To improve the accuracy of roundness error assessments, Bat algorithm (BA) is applied to roundness error assessments. An improved bat algorithm (IBA) is proposed to counteract the original lack of variational mechanisms, which can easily lead BA to fall into local extremes and induce premature convergence. First, logistic chaos initialisation is applied to the initial solution generation to enhance the variation mechanism of the population and improve the solution quality; second, a sinusoidal control factor is added to BA to control the nonlinear inertia weights during the iterative process, and the balance between the global search and local search of the algorithm is dynamically adjusted to improve the optimization-seeking accuracy and stability of the algorithm. Finally, the sparrow search algorithm (SSA) is integrated into BA, exploiting the ability of explorer bats to perform a large range search, so that the algorithm can jump out of local extremes and the convergence speed of the algorithm can be improved. The performance of IBA was tested against the classical metaheuristic algorithm on eight benchmark functions, and the results showed that IBA significantly outperformed the other algorithms in terms of solution accuracy, convergence speed, and stability. Simulation and example verification show that IBA can quickly find the centre of a minimum inclusion region when there are many or few sampling points, and the obtained roundness error value is more accurate than that of other algorithms, which verifies the feasibility and effectiveness of IBA in evaluating roundness errors.

Abstract: Computational fluid dynamics (CFD) has become a cutting-edge tool for investigating hemodynamic dysfunctions in the body. It has the potential to help physicians quantify in more detail the phenomena difficult to capture with in vivo imaging techniques. CFD simulations in anatomically realistic geometries pose challenges in generating accurate solutions due to the grid distortion that may occur when the grid is aligned with complex geometries. In addition, results obtained with computational methods should be trusted only after the solution has been verified on multiple high-quality grids. The objective of this study was to present a comprehensive solution verification of the intra-aneurysmal flow results obtained on different morphologies of patient-specific cerebral aneurysms. We chose five patient-specific brain aneurysm models with different dome morphologies and estimated the grid convergence errors for each model. The grid convergence errors were estimated with respect to an extrapolated solution based on the Richardson extrapolation method, which accounts for the degree of grid refinement. For four of the five models, calculated velocity, pressure, and wall shear stress values at six different spatial locations converged monotonically, with maximum uncertainty magnitudes ranging from 12% to 16% on the finest grids. Due to the geometric complexity of the fifth model, the grid convergence errors showed oscillatory behavior; therefore, each patient-specific model required its own grid convergence study to establish the accuracy of the analysis. [Copyright &y& Elsevier]

Different data types (single-, double-, and extended precision floating point numbers) used in computations substantially affects the following main efficiency estimation criteria: performance, accuracy, and amount of computations. The influence of different variants in the organization of computations on the efficiency of basis-matrix algorithms is studied. The concept of an expert support system is proposed for the organization of computations on linear models to obtain specified parameter values with respect to the accuracy and performance criteria. [ABSTRACT FROM AUTHOR]

Abstract: A software package of realization of mathematical models of pollutants transport in rivers is offered. This package is designed as a up-to-date convenient, reliable tool for specialists of various areas of knowledge such as ecology, hydrology, building, agriculture, biology, ichthyology and so on. It allows us to calculate pollutant concentrations at any point of the river depending on the quantity and the conditions of discharging from several pollution sources. One-, two-, and three-dimensional advection–diffusion mathematical models of river water quality formation both under classical and new, original boundary conditions are realized in the package. New finite-difference schemes of calculation have been developed and the known ones have been improved for these mathematical models. At the same time, a number of important problems which provide practical realization, high accuracy and short time of obtaining the solution by computer have been solved. In particular: (a) the analytical description of plane or spatial region for which the diffusion equations and boundary conditions are investigated, i.e. the analytical description of the bank line and the bottom of the river; (b) the analytical description of dependence of coefficients of the equation on the spatial coordinates; (c) the analytical description of dependence of non-homogeneous parts of the diffusion equation (i.e. the capacities of pollution sources) on the spatial coordinates and on the time; (d) the correct choice of ratios between spatial steps of the grid, and also between them and the step of digitations of time in the difference scheme. [Copyright &y& Elsevier]

The problem-solving time and the solution accuracy are expressed as functions of a parameter. An optimization problem of choosing the parameter using a “time cost–solution accuracy” criterion is considered. A Pareto-optimal set of solutions is obtained. The best value of the parameter is chosen by the ideal-point method. [ABSTRACT FROM AUTHOR]

Dynamic economic dispatch with valve-point effects (DED_ vpe) is a high-dimensional constrained optimization problem with non-convex and non-smooth characteristics. Hybrid methods are one of the most advanced methods to solve the problem. However, most of these methods improve the solution accuracy at the expense of algorithm robustness. This paper proposes an adaptive hybrid backtracking search optimization algorithm (AHBSA) for solving the DED_ vpe. The core idea of AHBSA lies in designing a suitable coupling structure based on the current best individual (called optimal partial coupling). The structure hybridizes an improved BSA mutation operator and the DE/best/1 operator with equal probability. The improved BSA mutation operator uses the current best individual and the historical population to update individual position, called BSA/best/old. It is also the first research work of extending BSA to the problem. In addition, an adaptive parameter control mechanism is proposed to select an appropriate 'mixrate' value for achieving better coupling. The performance of AHBSA is validated on six DED test cases of three systems. Experimental results demonstrate that, compared with some representative methods, AHBSA not only reduces the fuel cost but also ensures the robustness of the algorithm. • A hybrid method AHBSA is proposed to solve DED problem with valve-point effects. • An optimal partial coupling structure is designed in AHBSA. • BSA is the first attempt at solving DED problem. • Results on six benchmark DED test cases show AHBSA is competitive. [ABSTRACT FROM AUTHOR]

Abstract: A new method based on a hydrid finite volume/finite element discretization of the radiative transfer equation is described and applied to multi-dimensional rectangular enclosures with gray absorbing-emitting-scattering media. The spatial discretization is carried out using the finite volume method and the angular discretization is performed using basis functions commonly employed in the finite element method. The predicted results for two- and three-dimensional enclosures are compared with analytical solutions and show that the numerical solution converges to the analytical one as the discretization is refined. [Copyright &y& Elsevier]

We study the effect of the Galerkin/Least-Squares (GLS) stabilization on the finite element discretization of optimal control problems governed by the linear Oseen equations. Control is applied in the form of suction or blowing on part of the boundary. Two ways of including the GLS stabilization into the discretization of the optimal control problem are discussed. In one case the optimal control problem is first discretized and the resulting finite-dimensional problem is then solved. In the other case, the optimality conditions are first formulated on the differential equation level and are then discretized. Both approaches lead to different discrete adjoint equations and, depending on the choice of the stabilization parameters and grid size, may significantly affect the computed control. The effect of the order in which the discretization is applied and the choice of the stabilization parameters are illustrated using two test problems. The cause of the differences in the computed controls are explored numerically. Diagnostics are introduced that may guide the selection of sensible stabilization parameters. [Copyright &y& Elsevier]