ON APPROXIMATION OF PERIODIC ATEB-FUNCTIONS

Two versions of approximation formulae for periodic Ateb-sine and Ateb-cosine in the first quarter of their common period are proposed. The first version is a Pade type approximation derived when constructing analytical solution of corresponding integral equation by iteration method with transforming the power series into a closed sum by Shanks’ formula. Two iteration approximations are considered. The first one is more concise but of worse approximation accuracy which deteriorates with increasing the argument value. To improve the approximation accuracy a hybrid approximation is proposed when the values of the Ateb-functions in the beginning (for the cosine) and in the end (for the sine) of the quarter period are computed by a separate formula obtained a priory by the asymptotic method. The comparison analysis of the approximate and exact values of the special functions indicates the error of the approximation proposed to be less than one per cent. The second variant of approximation is by replacing the periodic Ateb-functions by trigonometric functions of specific argument. The arguments are chosen so that the values of the special functions are exact at specific points of the quarter period. Five such collocation points are introduced in the paper. To implement this version of approximation a separate table of the values of the periodic Ateb-functions at the collocation points is compiled. The computational examples presented in the paper show the approximate values of the special functions obtained by the second version of approximation to have a good accuracy.

Efficiency-House Optimization to Widen the Operation Range of the Double-Suction Centrifugal Pump

Most pumping machineries have a problem of obtaining a higher efficiency over a wide range of operating conditions. To solve that problem, an optimization strategy has been designed to widen the high-efficiency range of the double-suction centrifugal pump at design (Qd) and nondesign flow conditions. An orthogonal experimental scheme is therefore designed with the impeller hub and shroud angles as the decision variables. Then, the “efficiency-house” theory is introduced to convert the multiple objectives into a single optimization target. A two-layer feedforward artificial neural network (ANN) and the Kriging model were combine based on a hybrid approximate model and solved with swarm intelligence for global best parameters that would maximize the pump efficiency. The pump performance is predicted using three-dimensional Reynolds-averaged Navier–Stokes equations which is validated by the experimental test. With ANN, Kriging, and a hybrid approximate model, an optimization strategy is built to widen the high-efficiency range of the double-suction centrifugal pump at overload conditions by 1.63%, 1.95%, and 4.94% for flow conditions 0.8Qd, 1.0Qd, and 1.2Qd, respectively. A higher fitting accuracy is achieved for the hybrid approximation model compared with the single approximation model. A complete optimization platform based on efficiency-house and the hybrid approximation model is built to optimize the model double-suction centrifugal pump, and the results are satisfactory.

Hybrid Approximation Hierarchical Boundary Element Methods for Acoustic Problems

A multipole expansion approximation boundary element method (MEA BEM) based on the hierarchical matrices (H-matrices) and the multipole expansion theory was proposed previously. Though the MEA BEM can obtain higher accuracy than the adaptive cross-approximation BEM (ACA BEM), it demands more CPU time and memory than the ACA BEM does. To alleviate this problem, in this paper, two hybrid BEMs are developed taking advantage of the high efficiency and low memory consumption property of the ACA BEM and the high accuracy advantage of the MEA BEM. Numerical examples are elaborately set up to compare the accuracy, efficiency and memory consumption of the ACA BEM, MEA BEM and hybrid methods. It is indicated that the hybrid BEMs can reach the same level of accuracy as the ACA BEM and MEA BEM. The efficiency of each hybrid BEM is higher than that of the MEA BEM but lower than that of the ACA BEM. The memory consumptions of the hybrid BEMs are larger than that of the ACA BEM but less than that of the MEA BEM. The algorithm used to approximate the far-field submatrices corresponding to the cells and their nearest interactional cells determines the accuracy, efficiency and memory consumption of the hybrid BEMs. The proposed hybrid BEMs have both operation and storage logarithmic-linear complexity. They are feasible.