Direct Solution Algorithm Research Articles

A full operating conditions ejector model for refrigeration systems driven by low-grade heat sources

The ejector refrigeration technology driven by low-grade heat sources constitutes as one of the crucial means to solve the energy shortage in present-day society and to reduce the global environmental pollution. Thermodynamics ejector models are the basis of performance prediction, system control and structural design in ejector refrigeration systems driven by low-grade heat sources. However, the majority of present ejector models are unable to simultaneously meet the general feasibility of full operating conditions and the level of prediction accuracy for improving the energy efficiency of the refrigeration system. In this study, a novel theoretical model of ejectors is successfully derived based on the foundation of compound-choking theory and gas dynamic relations, which easily predicts the entrainment performance under the on-design and off-design conditions in ejector refrigeration systems. Indeed, a direct solution algorithm is presented to quickly and accurately solve this model and a linear correlation is employed between the mixing loss coefficient and the back pressure at the subcritical mode. The developed model is verified with experimental results in R245fa, R134a and R600a ejector refrigeration system. The results show that the model provides a precise estimation of the entrainment performance with the mean relative errors of 2.45 %, 5.49 % and 3.67 % for R245fa, R134a and R600a refrigerants at full working conditions, while the prediction of critical condensing temperatures is considerably approximated by the experimental measurements. In the comparative analysis with the previous model, this model demonstrates a superior prediction performance, especially in the subcritical mode using the linear function mixing loss coefficient. This study will be a valuable aid in the improvement of energy efficiency in ejector refrigeration systems.

Optimizing service design for the intercity demand responsive transit system: Model, algorithm, and comparative analysis

Intercity Demand Responsive Transit (IDRT) systems, providing on-demand “door-to-door” services for intercity travelers, have been recently implemented in a few cities and gained popularity. To devise a theoretical tool for designing efficient IDRT systems, this study proposes a continuous approximation (CA) model to describe the operation metrics of the IDRT. In a two-city system, an optimization model is proposed to study the zoning strategies in the main city, the service area in the satellite city, and headways. The objective is to minimize the average system cost, including the costs of both the operators and the passengers. A two-stage partition algorithm is proposed to generate a near-optimal solution by combining the greedy algorithm and sensitivity-based coordination optimization method. Subsequently, we develop a direct solution algorithm to establish a lower bound for the optimal solution, facilitating a comparison with the near-optimal solution. Throughout all sensitivity analyses, the gaps between the near-optimal solution and the lower-bound solution remain below 3 %. The comparison results demonstrate that the computation time and solution quality of the two-stage partition algorithm are considered acceptable. Simulation is performed to test the validity of the CA model, and the errors in the average system costs are less than 1.16 %. Sensitivity analyses are carried out to evaluate the impacts of several key parameters (e.g., demand densities and line-haul distances) on the optimization results. By comparison with Intercity Fixed Route Transit (IFRT), it can be seen that the choice of IDRT depends on passengers' preference for door-to-door transportation. This study may serve as a valuable reference for the practical implementation of IDRT.

Optimal Identification for Dynamic PV Cell Parameter Based on a Data-Extension-Driven Method

Affected by environmental factors, equipment aging, operating status, etc., the parameters of photovoltaic (PV) models will deviate from the original setting parameters. In order to accurately identify the dynamic parameters of photovoltaics under the general simulation model, traditional parameter identification methods mainly use heuristic intelligent optimization algorithms for direct solution. Due to the limited data collected and the strong randomness of the algorithm, it is easy to make the identification accuracy and stability of photovoltaic parameters difficult to meet the requirements. To this end, this paper proposes an optimal identification method for PV dynamic parameters driven by data expansion. Firstly, the PV external characteristic data is fitted and generalized, which used the generalized regression neural network (GRNN). Then, the extended high-quality data can be used for dynamic parameter identification for PV cell. To confirm the performance of the proposed algorithm in this paper, this paper expands based on the actual external characteristic data of different proportions and uses the general PV simulation model to conduct comparative tests on various commonly used algorithms. The case studies under different scenarios show that the proposed algorithm can provide a more reliable and well-represented fitness function to the metaheuristic algorithms. Therefore, the optimization accuracy and stability of the proposed algorithm for dynamic PV cell parameter identification can be significantly improved simultaneously.

Solution of Combined Heat and Power Economic Dispatch Problem Using Direct Optimization Algorithm

This paper presents the solution to the combined heat and power economic dispatch problem using a direct solution algorithm for constrained optimization problems. With the potential of Combined Heat and Power (CHP) production to increase the efficiency of power and heat generation simultaneously having been researched and established, the increasing penetration of CHP systems, and determination of economic dispatch of power and heat assumes higher relevance. The Combined Heat and Power Economic Dispatch (CHPED) problem is a demanding optimization problem as both constraints and objective functions can be non-linear and non-convex. This paper presents an explicit formula developed for computing the system-wide incremental costs corresponding with optimal dispatch. The circumvention of the use of iterative search schemes for this crucial step is the innovation inherent in the proposed dispatch procedure. The feasible operating region of the CHP unit three is taken into account in the proposed CHPED problem model, whereas the optimal dispatch of power/heat outputs of CHP unit is determined using the direct Lagrange multiplier solution algorithm. The proposed algorithm is applied to a test system with four units and results are provided.

Solution of Combined Heat and Power Economic Dispatch Problem Using Genetic Algorithm

This research proposes a synergistic meta-heuristic algorithm for solving the extreme operational complications of combined heat and power economic dispatch problem towards the advantageous economic outcomes on the cost of generation. The combined heat and power (CHP) is a system that provides electricity and thermal energy concurrently. For its extraordinary efficiency and significant emission reduction, it is considered a promising energy prospect. The broad application of combined heat and power units requires the joint dispatch of power and heating systems, in which the modelling of combined heat and power units plays a vital role. The present research employs the genetic optimization algorithm to evaluate the cost function, heat and power dispatch values encountered in a system with simple cycle cogeneration unit and quadratic cost function. The system was first modeled to determine the various parameters of combined heat and power units towards solving its economic dispatch problem directly. In order for modelling to be done, a general structure of combined heat and power must be defined. The test system considered consists of four units: two conventional power units, one combined heat and power unit and one heat-only unit. The algorithm was applied to test system while taking into account the power and heat units, bounds of the units and feasible operation region of cogeneration unit. Output decision variables of 4-unit test systems plus cost function from Genetic Algorithm (GA), was determined using appropriate codes. The proposed algorithm produced a well spread and diverse optimal solution and also converged reasonably to the actual optimal solution in 51 iterations. The result obtained compared favourably with that obtained with the direct solution algorithm discussed in a previous paper. We conclude that the genetic algorithm is quite efficient in dealing with non-convex and constrained combined heat and power economic dispatch problem.

A Fast Tomographic Reconstruction Method for Flame Temperature Distribution Measurement Based on Direct Solution Algorithm

The rapid and accurate measurement of the flame temperature distribution is of great significance to the structural design and health diagnosis of the engine. Aiming at the low reconstruction efficiency of traditional flame temperature distribution reconstruction algorithms, a Direct Solution algorithm for flame temperature distribution reconstruction is proposed in this paper based on the structural characteristics of the reconstruction equations. By setting several numerical cases, the performance of the Direct Solution algorithm and some commonly used traditional algorithms, such as Simultaneous Algebraic Reconstruction Technique (SART), Least Squares QR-factorization (LSQR) algorithm, Non-Negative Least Squares QR-factorization (NNLS) algorithm, is compared in the reconstruction of the flame temperature distribution. The results show that the efficiency of the Direct Solution method is 169.4, 7.4, and 3483.3 times higher than that of the SART, LSQR, and NNLS algorithms under the condition of 40 × 40 grids. In addition, with the increase of the number of grids, the growth rate of the reconstruction time of the Direct Solution algorithm is much lower than that of other algorithms. The overall reconstruction accuracy of the Direct Solution algorithm is better than that of SART and LSQR algorithms. This shows that it has an excellent comprehensive performance and has a great application prospect in the rapid reconstruction of the temperature distribution.

Research on GPU parallel algorithm for direct numerical solution of two-dimensional compressible flows

In this paper, a novel parallel algorithm is proposed to solve the problems of heavy computation and long simulation time in the field of compressible flows. In this algorithm, a third-order upwind scheme and a fourth-order central difference scheme are employed, with a third-order Runge-Kutta method for time stepping. Considering the powerful floating-point computing ability of the Graphics Processing Unit (GPU), this paper establishes the algorithm on the basis of GPU. Moreover, the direct numerical simulation method is adopted in this algorithm to improve the solution accuracy of the simulation results. To further enhance the efficiency of the algorithm, several optimization strategies are explored in the design of the algorithm as well. Both accuracy and feasibility of the algorithm are verified by a classical two-dimensional example. Compared with solving this example on the Central Processing Unit platform, the experimental results demonstrate that the maximum speedup ratio achieved by our approach is 18.03 times.

Influence of nanostructure geometry on light trapping in solar cells

The aim of this research was to detect the influence of the silicon nanowires geometry and silver nanoparticles on the wafer surface on light trapping for application in solar cells with radial p–n junction. Si-NWs arrays were fabricated on p-type (100) silicon wafers with specific resistance of 10 Ω × cm. We used MacEtch method to form Si nanowires. In our study to deposit the additional silver nanoparticles on the surface of the nanowires, the flushing time was reduced to 10 min. The structural and optical properties of silicon nanowires were studied. The absorption, transmittance and reflectance spectra for periodic nanostructures with different parameters were calculated by FDTD (Finite-Difference Time-Domain) method. The FDTD method is a powerful numerical algorithm for direct solution of Maxwell’s equations. The realization of this method is discrete both in space and in time. The presented structures are described on a discrete mesh consisting of Yee cells and the Maxwell equations. They are solved discretely over time on this grid. As a result, through the method of the catalytic etching, we obtained nanowires of 0.5–2 μm in length and diameter in range of 50–100 nm. The diameter of silver nanoparticles was in the range of 20–50 nm. The reflection coefficient of light in wide spectral range was less than 1%. Reflection spectra peaked at the wavelength of 850 nm that points out on the influence of Ag nanoparticles.

Accuracy analysis and verification of the method for calculation of geodetic problem on earth ellipsoid surface

The method for calculation of geodetic problem on earth ellipsoid surface is the basic premise for the realization of high-precision marine delimitation technology. Based on the analysis of the construction principle of the existing solution of geodetic problem method, the algorithm for direct geodetic problem solution(DGPS) based on nested coefficient method and Bessel’s inverse geodetic problem solution(IGPS), which are suitable for various distance solutions, are optimized. Then, the experimental verification scheme of the solution of geodetic problem is designed. Finally, the crossing and reducing accuracy of different calculation examples are verified by different geodetic problem solution methods, and the experiment shows that: the verified accuracy of point-position plotting, distance calculation and azimuth calculation meet the requirements of the indexes of high-precision marine delimitation.

Numerical Algorithms for Direct Solution of Fourth Order Ordinary Differential Equations

This paper examines the derivation of hybrid numerical algorithms with step length(k) of five for solving fourth order initial value problems of ordinary differential equations directly. In developing the methods, interpolation and collocation techniques are considered. Approximated power series is used as interpolating polynomial and its fourth derivative as the collocating equation. These equations are solved using Gaussian-elimination approach in finding the unknown variables aj, j=0,...,10 which are substituted into basis function to give continuous implicit scheme. The discrete schemes and its derivatives that form the block are obtainedby evaluating continuous implicit scheme at non-interpolating points. The developed methods are of order seven and the results generated when the methods were applied to fourth order initial value problems compared favourably with existing methods.order initial value problems compared favourably with existing methods.

Infrared attitude measurement method for spinning projectile under snow background

In order to obtain the attitude of spinning projectile over snow surface, an infrared attitude measurement method for spinning projectile under snow background is proposed. First of all, combining the radiation characteristics of the snow surface and the propagation law of the atmospheric infrared radiation, the relationship between the infrared radiation over snow and the inclination angle to the ground is deduced. Secondly, the infrared attitude measurement model is built on the basis of the rotation characteristics of the projectile. At last, the attitude solution algorithm of spinning projectile is designed. On the basis of the error transfer coefficient, the effects of direct solution algorithm and indirect solution algorithm are analyzed, and the feasibility of the designed solution method is verified by experiment. The results indicate that the pitch angle solution error is within ±1.5° and the roll angle solution error is within −1.5° to 1°, which can provide a good angles estimation for projectile guidance over snow.

High-order continuous Galerkin methods for multi-dimensional advection–reaction–diffusion problems

We construct in this paper two efficient spectral algorithms in the frequency space for solving unsteady advection–reaction–diffusion equations with constant and variable coefficients. We first consider a Jacobi–Galerkin method for solving linear equations with constant coefficients. We then develop the direct solution algorithm for the linear advection–reaction–diffusion equations with variable coefficients using the Jacobi–Galerkin method with numerical integration. The proposed Jacobi–Galerkin methods, both in temporal and spatial discretizations, are successfully developed to handle the two-dimensional unsteady advection–reaction–diffusion equations with constant and variable coefficients and with fractional orders. In these methods, the model solution is expanded in both space and time in terms of polynomials bases built upon a linear combination of Jacobi polynomials. The homogeneous initial and Dirichlet boundary conditions are satisfied exactly by expanding the model solution in terms of these polynomials. The proposed Jacobi–Galerkin methods yield an exponential rate of convergence when the solution is smooth and allow a great flexibility to handle multi-dimensional time fractional advection–reaction–diffusion equations. Finally, a series of numerical examples are presented to demonstrate the efficiency and flexibility of the methods.

A computationally efficient numerical model for heat transfer simulation of deep borehole heat exchangers

Being a promising technique for seasonal thermal energy storage, the deep borehole heat exchanger (DBHE) has also provided a new variance of ground-coupled heat pump systems especially for applications in cold-climate regions. Numerical simulations are recognized as the appropriate means in the study on heat transfer around the deep boreholes, which often features time-consuming computations. A model has been established in this study for DBHEs with coaxial tubes, considering coupled heat transfer in the tubes and surrounding subsurface. A software package is developed based on the finite difference method for thermal analysis of the DBHEs, and an algorithm for direct solution of resulted algebraic equation set is used so as to achieve very efficient computation. The model and the numerical algorithm are validated by comparison with simulation results from a reference using the finite element method. Performance of DBHEs is then assessed with parameter analyses. A diagram is presented to show relations of the nominal DBHE capacity to its key parameters such as the borehole depth, geothermal heat flux and thermal conductivity of subsurface, which is helpful in engineering practice. Studies also indicate that a high-resistance inner pipe helps in improving the heat transfer capacity of DBHEs.

Accuracy Directly Controlled Fast Direct Solution of General ${\mathcal{ H}}^{2}$ -Matrices and Its Application to Solving Electrodynamic Volume Integral Equations

The dense matrix resulting from an integral equation (IE) based solution of Maxwell's equations can be compactly represented by an ${\cal H}^2$-matrix. Given a general dense ${\cal H}^2$-matrix, prevailing fast direct solutions involve approximations whose accuracy can only be indirectly controlled. In this work, we propose new accuracy-controlled direct solution algorithms, including both factorization and inversion, for solving general ${\cal H}^2$-matrices, which does not exist prior to this work. Different from existing direct solutions, where the cluster bases are kept unchanged in the solution procedure thus lacking explicit accuracy control, the proposed new algorithms update the cluster bases and their rank level by level based on prescribed accuracy, without increasing computational complexity. Zeros are also introduced level by level such that the size of the matrix blocks computed at each tree level is the rank at that level, and hence being small. The proposed new direct solution has been applied to solve electrically large volume IEs whose rank linearly grows with electric size. A complexity of $O(NlogN)$ in factorization and inversion time, and a complexity of $O(N)$ in storage and solution time are both theoretically proven and numerically demonstrated. For constant-rank cases, the proposed direct solution has a strict $O(N)$ complexity in both time and memory. Rapid direct solutions of millions of unknowns can be obtained on a single CPU core with directly controlled accuracy.

Development of a direct-solution algorithm for determining the optimal crop planning of farms using deficit irrigation

The irrigation farms placed in areas of scare water demand methodologies that can increase their profitability via more efficient use of their resources. Determining the combination of factors that maximizes the profitability of any productive process requires the use of optimization methodologies. Traditionally, these types of problems were solved using heuristic methods. However, a direct-solution algorithm would produce faster and more accurate solutions. The aim of this work was to develop a direct-solution algorithm capable of determining the crop planning (area and volume of water per crop) that maximizes the profitability of an irrigation farm. The data required by the algorithm include the total cultivable area of the farm and the amount of available irrigation water as well as the “gross margin vs. irrigation depth” functions of the considered crops. Cultivating one or two crops is the way to reach higher profitability, but this strategy is not suitable from an agricultural point of view (i.e., crop rotation, diseases, weather risks, regulations of agricultural policies, etc.). Due to this algorithm must be compatible with the MOPECO model, a methodology has been developed to allow its implementation in this model. The objective of this software is to maximize the profitability of irrigation farms by incorporating a more efficient use of irrigation water using regulated deficit irrigation techniques. The current version of this model uses genetic algorithms for determining optimal crop planning, which are time consuming. For a hypothetical 100ha farm, considering 10 different crops and 11 scenarios of water availability, the developed algorithm adapted to MOPECO achieved gross margins around 0.5% lower than LINGO, and 1.1% higher than genetic algorithms, decreasing the calculation time requirements by between 50 and 100 and approximately 2000 times, respectively. Another relevant result is the fact that the algorithm may be used manually, by drawing the tangent lines between the gross margin curves, for reaching the optimal combinations of irrigation depth and, indirectly, the cultivable area of each crop. Moreover, the algorithm allows to understand the relationships among crops, which may advise users in the determination of the optimal solution under real conditions. This methodology also highlights the importance of using regulated deficit irrigation techniques when managing irrigation farms with a low supply of irrigation water. The developed algorithm may also be useful in the optimization of other production processes.

A Condensation Method for the Dynamic Analysis of Vertical Vehicle–Track Interaction Considering Vehicle Flexibility

This paper is aimed at developing a computationally efficient approach to simulate the vertical dynamic behavior of vehicle–track coupled system. With the finite element method, the car body, bogies, and rail are modeled as Euler beams supported by springs and dashpots, which can investigate the influence of flexibility of the vehicle on structural dynamic response. By a variant of component-mode synthesis (CMS), the degrees-of-freedom (DOFs) within the substructures are condensed and the two substructures are coupled through nonlinear Hertzian theory. Although the system matrix is updated and factorized during the calculation, the total computational efficiency is significantly improved due to the much smaller size of the equations of motion and direct solution algorithm instead of iterative procedure. Compared with an existing model, the accuracy and efficiency of the method are investigated. Application of the model is shown by numerical example.

Research of Design Synthesis for MEMS Device of Serpentine Spring

The design synthesis of MEMS device is an inverse problem of the MEMS analysis. The performance analysis equations are transformed to the performance design equations according to the predetermined performance. The Direct Solution Algorithm derived by the Newton iterative method is used to solve the nonlinear performance design equations, and then the MEMS device will possess predetermined properties. The serpentine spring was synthesized with this method. The computational efficiency and accuracy is greatly improved by comparison with the foreign natural simulation optimization algorithms of the genetic algorithms. The genetic algorithms needs tens of thousands of times analyses, the relative error of the calculation is 1%, but this method only needs dozens of analyses and the relative error of the calculation is 0.01%.

New spectral-Galerkin algorithms for direct solution of high even-order differential equations using symmetric generalized Jacobi polynomials

A family of symmetric generalized Jacobi polynomials (GJPs) is introduced and used for solving high even-order linear elliptic differential equations in one variable by the Galerkin method. Some efficient and accurate algorithms are developed and implemented for solving such equations. The use of GJPs leads to a simplified analysis, more precise error estimates and very efficient numerical algorithms. The methods lead to linear systems with specially structured matrices that can be efficiently inverted. Two numerical examples are presented aiming to demonstrate the accuracy and the efficiency of the proposed algorithms.

An efficient direct solver for multidimensional elliptic Robin boundary value problems using a Legendre spectral-Galerkin method

In this paper, a Legendre–Galerkin method for solving second-order elliptic differential equations subject to the most general nonhomogeneous Robin boundary conditions is presented. The homogeneous Robin boundary conditions are satisfied exactly by expanding the unknown variable using a polynomial basis of functions which are built upon the Legendre polynomials. The direct solution algorithm here developed for the homogeneous Robin problem in two-dimensions relies upon a tensor product process. Nonhomogeneous Robin data are taken into account by means of a lifting. Such a lifting is performed in two successive steps, the first one to account for the data specified at the corners and the second one to account for the boundary values prescribed in the interior of the sides. Numerical results indicating the high accuracy and effectiveness of these algorithms are presented.

A More Effective Technique of Design Synthesis for MEMS with Expected Performance

A design synthesis technique based on sensitivity for Micro-Electro-Mechanical Systems (MEMS) proposed. This new technique can be called Sensitivity-Based Direct Solution Algorithm (DSA) of design synthesis for MEMS with expected performance. Design synthesis with expected performance is regarded as a reverse problem of MEMS analysis. Behavior equation group can be educed from analysis equations. This behavior equation group can be solved using any solution algorithm of non-linear equation group. Newton Iteration Method based on sensitivity is adopted. Comparing with Genetic Optimization Algorithm (GA) , computational workload of DSA is greatly decreased.