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The interpolation finite difference method (IFDM) allows free numerical analysis of elliptic partial differential equations over arbitrary domains. Conventionally, in the finite difference method (FDM), the calculation is performed using the second-order accuracy central difference. For engineering problems, second-order accuracy calculations are often sufficient. On the other hand, much research has been carried out to improve the accuracy of numerical calculations. Although there is much research in the FDM field, the development of numerical calculations by the spectral method is decisive in improving the calculation accuracy. Numerical calculations are usually performed by double precision calculations. If double precision calculations ensure 15 significant digits in floating point computing, such numerical calculations will be the ultimate goal to reach. A numerical calculation that does not seem to have an error even though it originally has an error is defined as a virtual error-zero (VE0) calculation. In this paper, we will examine an overall picture of high-accuracy numerical calculation by the IFDM in the numerical calculation of the 1D Poisson equation. It becomes clear that a VE0 calculation is always possible in the numerical calculation method, defined as the compact interpolation finite difference scheme [(m)]. [ABSTRACT FROM AUTHOR]

In a previous paper, a calculation system for a high-accuracy, high-speed calculation of a one-dimensional (1D) Poisson equation based on the interpolation finite difference method was shown. Spatial high-order finite difference (FD) schemes, including a usual second-order accurate centered space FD scheme, are instantaneously derived on the equally spaced/unequally spaced grid points based on the definition of the Lagrange polynomial function. The upper limit of the higher order FD scheme is not theoretically limited but is studied up to the tenth order, following the previous paper. In the numerical analyses of the 1D Poisson equation published in the previous paper, the FD scheme setting method, SAPI (m), m = 2, 4, ..., 10, was defined. Due to specifying the value of m, the setting of FD schemes is uniquely defined. This concept is extended to the numerical analysis of two-dimensional Poisson equations. In this paper, we focus on Poiseuille flows passing through arbitrary cross sections as numerical calculation examples. Over regular and irregular domains, three types of FD methods—(i) forward time explicit method, (ii) time marching successive displacement method, and (iii) alternative direction implicit method—are formulated, and their characteristics of convergence and numerical calculation errors are investigated. The numerical calculation system of the 2D Poisson equation formulated in this paper enables high-accuracy and high-speed calculation by the high-order difference in an arbitrary domain. Especially in the alternative direction implicit method using the band diagonal matrix algorithm, convergence is remarkably accelerated, and high-speed calculation becomes possible. [ABSTRACT FROM AUTHOR]

Among the methods of the numerical analysis of the physical phenomena of the continuum, the finite difference method (FDM) is the first examined method and has been established as a full numerical calculation system over the regular domain. However, there is a general perception that generality in numerical calculations cannot be expected over complex irregular domains. As using the FDM, the development of computational methods that are applicable over any irregular domain is considered to be a very important contemporary problem. In the FDM, there is a marked characteristic that the theory developed by the (spatial) one-dimensional (1D) problem is naturally applied to the 2D and 3D problems. The calculation method is called the interpolation FDM (IFDM). In this paper, attention is paid to 1D Laplace and Poisson equations, and the whole image of the IFDM using the algebraic polynomial interpolation method (APIM), the IFDM-APIM, is described. Based on the Lagrange interpolation function, the spatial difference schemes from 2nd order to 10th order including odd order are calculated and defined instantaneously over equally/unequally spaced grid points, then, high-order accurate and high-speed computations become possible. [ABSTRACT FROM AUTHOR]

To ensure the safe and stable operation of wind turbines, it is essential that the blades have sufficient bending stiffness to prevent excessive deflection and tower impact. A method combining static loading test and numerical analysis is proposed to solve the problem of identifying the cross-sectional bending stiffness of actual blades. Two working conditions, namely multi-point static test and single-point static calibration, are considered. A mathematical model of a cantilever beam with an initial small curvature and variable cross section is established. By fitting the deflection measurement data obtained from the static loading test, the deflection differential equation is solved by the finite difference method to obtain expressions for the cross-sectional bending stiffness under large and small deflection. The analysis results of several blade models show that the bending stiffness identification error for most blade sections is less than 10%, confirming the effectiveness of this method. This provides a practical and theoretical basis for the design, analysis, and refinement of test parameters for large-scale blades. [ABSTRACT FROM AUTHOR]

The finite difference method (FDM) based on Cartesian coordinate systems can be applied to numerical analyses over any complex domain. A complex domain is usually taken to mean that the geometry of an immersed body in a fluid is complex; here, it means simply an analytical domain of arbitrary configuration. In such an approach, we do not need to treat the outer and inner boundaries differently in numerical calculations; both are treated in the same way. Using a method that adopts algebraic polynomial interpolations in the calculation around near-wall elements, all the calculations over irregular domains reduce to those over regular domains. Discretization of the space differential in the FDM is usually derived using the Taylor series expansion; however, if we use the polynomial interpolation systematically, exceptional advantages are gained in deriving high-order differences. In using the polynomial interpolations, we can numerically solve the Poisson equation freely over any complex domain. Only a particular type of partial differential equation, Poisson's equations, is treated; however, the arguments put forward have wider generality in numerical calculations using the FDM. [ABSTRACT FROM AUTHOR]