A Taylor expansion algorithm for integration of 3D near-hypersingular integrals

A new semi-analytic algorithm of nearly singular integrals on higher order element in 3D potential BEM

An enhanced lattice Boltzmann scheme, Taylor series expansion- and least squares-based-lattice Boltzmann method (TLLBM), is adopted to simulate transient thermal behavior in a thin multilayer. The meshless formulation of TLLBM not only handles the changes in material properties from one layer to another, but also facilitates the general geometry handling capabilities. The energy carrier interaction at the interface is governed via diffusive mismatch model (DMM). We simulated transient thermal behavior of a hot-spot in the dual-layer by incorporating a heat source in one of the layers. The sub-continuum effect of anisotropic thermal transport is presented where the heat preferentially flows laterally in the layer with higher conductivity as its thickness is reduced. The insight into the nanoscale thermal behavior acquired via a relatively simple model will be critical for the design and operation of complex data storage and electronic systems, where thermal transport plays an active and critical role

ABSTRACTWe propose new implicit staggered‐grid finite‐difference schemes with optimal coefficients based on the sampling approximation method to improve the numerical solution accuracy for seismic modelling. We first derive the optimized implicit staggered‐grid finite‐difference coefficients of arbitrary even‐order accuracy for the first‐order spatial derivatives using the plane‐wave theory and the direct sampling approximation method. Then, the implicit staggered‐grid finite‐difference coefficients based on sampling approximation, which can widen the range of wavenumber with great accuracy, are used to solve the first‐order spatial derivatives. By comparing the numerical dispersion of the implicit staggered‐grid finite‐difference schemes based on sampling approximation, Taylor series expansion, and least squares, we find that the optimal implicit staggered‐grid finite‐difference scheme based on sampling approximation achieves greater precision than that based on Taylor series expansion over a wider range of wavenumbers, although it has similar accuracy to that based on least squares. Finally, we apply the implicit staggered‐grid finite difference based on sampling approximation to numerical modelling. The modelling results demonstrate that the new optimal method can efficiently suppress numerical dispersion and lead to greater accuracy compared with the implicit staggered‐grid finite difference based on Taylor series expansion. In addition, the results also indicate the computational cost of the implicit staggered‐grid finite difference based on sampling approximation is almost the same as the implicit staggered‐grid finite difference based on Taylor series expansion.

In structural reliability calculation, there are fuzzy uncertainties in the distribution parameters of random variables, which bring the problem of large computation and poor precision. In order to improve accuracy and efficiency of structural reliability, a novel structural reliability calculation method with fuzzy random variables is proposed from the perspective of perturbation principle. First, the fuzzy characteristic parameters of the structural fuzzy random variables are expressed as the sums of their true values and fuzzy perturbations. At the same time, the perturbations are equivalently selected. Second, the probability density function and the performance function are decomposed into a series of intervals under different level cut sets, and the Taylor series is expanded on the upper and lower limits of the intervals. Finally, according to the definition of structural reliability, based on the direct integration method, the sigmoid function is introduced into the reliability function to approximate the step function, and the fuzzy reliability is obtained. The simulation results show the effectiveness and accuracy of the proposed method, and it provides a feasible way for structural reliability analysis with fuzzy variables or fuzzy random variables.

SIMULATION OF NATURAL CONVECTION IN A SQUARE CAVITY BY TAYLOR SERIES EXPANSION- AND LEAST SQUARES-BASED LATTICE BOLTZMANN METHOD

Land surface temperature (LST) responds to land‐use/land‐cover change (LULCC), which modifies surface properties that control the surface energy balance (SEB). Quantifying changes in LST due to individual perturbations caused by LULCC is an attribution problem. Most attribution methods are based on the first‐order Taylor series expansion (FOTSE) of a linearized SEB equation. The accuracy of these methods is affected by the use of FOTSE at two places. The first is to linearize the SEB equation and to obtain an analytical solution for LST (the LST model), and the second is to obtain LST changes as the linear sum of concurrent changes in multiple factors (the attribution model). In this study, we systematically assess the importance of non‐linear effects lost in these linearization processes using the second‐order Taylor series expansion (SOTSE). Results show that while the SOTSE LST model outperforms the FOTSE LST model, the order of Taylor series expansion in the LST model does not significantly influence the attribution of LST changes. However, the SOTSE attribution model is considerably more accurate than the FOTSE attribution model, especially when the magnitude of perturbations is large. Results suggest that contributions from higher‐order and cross‐order terms in the attribution model can be as large as 50%. Sensitivity analysis further shows that non‐linear effects associated with changing surface resistance for LULCC scenarios with large perturbations (e.g., deforestation and urbanization) are particularly strong. In conclusion, we recommend using the FOTSE LST model and the SOTSE attribution model.

In this study, the representation of discretization error using Taylor series in finite difference solutions is investigated as well as the behavior of the exact solutions to the finite difference equations as a function of the grid size and grid refinement factor. The results are compared to the classical Richardson Extrapolation method whereby the numerical solution (or the error) is explicitly expressed as a Taylor series expansion. The exact finite difference solutions are used to demonstrate that oscillatory convergence is a common occurrence. The expansion of the numerical solutions in Taylor series is based on the exact finite difference solutions that are obtained using different discretization schemes. It is shown that in some cases the numerical solution exhibited a singular behavior which can not be remedied easily. Some exact finite difference solutions also exhibited oscillatory behavior which was not due to the use of mixed order terms as is usually believed by the Computational Fluid Dynamics community. Moreover, representation of the numerical solution using Taylor series is not always satisfactory even in case of relatively simple one-dimensional problems.

Data interpolation is critical in the analysis of geophysical data when some data are missing or inaccessible. We interpolate irregular or missing potential field data using the relation between adjacent data points inspired by the Taylor series expansion (TSE). The TSE method first finds the derivatives of a given point near the query point using data from neighboring points and then uses the Taylor series to obtain the value at the query point. The TSE method works by extracting local features represented as derivatives from the original data for interpolation in the area of data vacancy. Compared with other interpolation methods, the TSE provides a complete description of potential field data. Specifically, the remainder in TSE can measure local fitting errors and help obtain accurate results. Implementation of the TSE method involves two critical parameters — the order of the Taylor series and the number of neighbors used in the calculation of derivatives. We find that the first parameter must be carefully chosen to balance between the accuracy and numerical stability when data contain noise. The second parameter can help us build an over-determined system for improved robustness against noise. Methods of selecting neighbors around the given point using an azimuthally uniform distribution or the nearest-distance principle are also presented. Our approach is first illustrated by a synthetic gravity data set from a single survey line, and then it is generalized to the case over a survey grid. In both numerical experiments, the TSE method demonstrates an improved interpolation accuracy in comparison with the minimum curvature method. Finally, we apply the TSE method to a ground gravity data set from the Abitibi Greenstone Belt, Canada, and an airborne gravity gradient data set from the Vinton Dome, Louisiana, USA.

This paper presents an effective approach for lower-order (LO) modeling of an electric vehicle–integrated off-grid microgrid (OMG) system. The seventh-order system (SOS) of the OMG is reduced to a second-order model (SOM) while preserving the original system’s dynamic characteristics and ensuring computational efficiency. The Taylor series (TS) and Laurent series (LS) expansions are employed to simplify the complex system that plays a significant role in the reduction process. The expansion parameters of the higher-order system (HOS) of OMG and its lower-order model (LOM) are exploited to construct the fitness function. The proposed approach constructs three sub-objective functions based on TS and LS. These sub-objective functions are then combined into a single fitness function to obtain an improved LOM by enhancing the transient and steady-state responses with respect to the HOS of OMG. To minimize the error, the resultant fitness function is optimized using the brown bear optimization (BBO) algorithm. The optimization is performed under two key constraints: (i) ensuring zero steady-state error, and (ii) satisfying the Hurwitz stability criterion. To demonstrate the efficacy of the proposed LOM, it is compared with other LOMs obtained from different approximation techniques. The proposed LOM and other LOMs are graphically validated through step, impulse, Bode, Nichols, and Nyquist response comparisons with the HOS. Additionally, the performance error criteria (PEC), time-domain specifications (TDSs) and frequency domain specifications (FDSs) of the proposed LOM are compared with other LOMs using the HOS to establish the validation and applicability of the proposed method.

Computation of nonlinear balanced realization and model reduction based on Taylor series expansion

In the paper we consider the systems of delay differential equations, which are mathematical models of many technical processes with delay in time. An explicit fifth-order hybrid method for systems of delay differential equations for the variable step of numerical integration is constructed. This algorithm is based on the most commonly used explicit fifth-order Runge-Kutta methods for ordinary systems of differential equations and the construction of Newton polynomials for the prehistory of the model and Taylor formula. The basic principles of construction of such explicit hybrid methods of higher orders of convergence are given. An accurate estimate of the local error of numerical integration by this method is obtained. This algorithm allows the use of numerical integration steps greater than the magnitude of the delay or procedures for adjusting the magnitude of the step depending on the calculation error. Such a problem of numerical integration of time-delay dynamical systems occurs when the time interval is large enough compared to the delay. Implicit continuous Runge-Kutta methods were previously used for numerical integration such problems, which complicated the numerical algorithm, because nonlinear systems of equations have to be solved at each step of numerical integration. The explicit hybrid method constructed by us for systems of delay differential equations is convenient for programming, has a high speed of calculation of numerical solution, in comparison with implicit methods for such models of problems. Also, this method has no restrictions on the magnitude of the delay, in contrast to the hybrid methods with decomposition in Taylor series for delay. The method obtained by us can be used to construct maps of dynamic modes in the study of regular and chaotic behavior of time-delay dynamical systems.

This paper focuses on estimating the variable stiffness of the Series Elastic Actuator(SEA) through torque and relative position between the motor and the payload measurements. Although position and torque sensors are effective in control schemes, the exorbitant price are not convenient for the application of robotics. Through the mathematical modeling, the variable stiffness of the series elastic actuator sometimes is difficult to obtain accurately. It should be noticed that, an accurate estimation of the variable stiffness is important. In this paper, a Kalman filter based on Taylor expansion is designed to estimate the variable stiffness. Specially, in order to improve the accuracy of the model, the process is divided into two steps for solving the problem of hysteresis, and the recursive least squares algorithm is used to estimate the spring stiffness. Then, according to the accurate spring stiffness value, an estimator of the variable stiffness is presented. The effectiveness of the proposed method on variable stiffness identification is proved through experiments.

In this paper an outline is presented of historical and current developments in the application of recurrent Taylor series to the integration of systems of ordinary differential equations. The idea of an extremely accurate and fast method for numerical solutions of differential equations is presented in the paper. In general Taylor series are not included or not even mentioned in surveys on numerical integration techniques as the programs were written by mathematicians with the main objective of demonstrating the feasibility of the concept and with the goal of providing integration algorithms of very high accuracy. For this reason the programs should be looked upon as a stimulus for writing more advanced software employing Taylor series better able to compete with programs using other methods. An attempt in this direction is TKSL, a program the results of which will be dealt with.

A method is presented to determine aquifer parameters simultaneously and uniquely based on time‐drawdown data collected from a single observation well in the vicinity of a pumping well partially penetrating a confined leaky aquifer. The Hantush solution is analyzed inversely for the identification of hydraulic properties using the Taylor series and a nonlinear least squares method. Aquifer parameters investigated include vertical and horizontal hydraulic conductivities, storage coefficient, and the leakage factor of the overlying aquitard. The developed method is verified using a hypothetical example and is applied to a field case. Error limits between the determined and true parameter values can be less than 0.1% depending on the quality of the field data.

Addition theorems can be constructed by doing three‐dimensional Taylor expansions according tof(r+r′)=exp (r′⋅∇)f(r). Since, however, one is normally interested in addition theorems of irreducible spherical tensors, the application of the translation operator in its Cartesian form exp (x′∂/∂x)exp (y′∂/∂y)exp (z′∂/∂z) would lead to enormous technical problems. A better alternative consists in using a series expansion for the translation operator exp (r′⋅∇) involving powers of the Laplacian ∇2and spherical tensor gradient operators\documentclass{article}\pagestyle{empty}\begin{document}$\mathcal{Y}^{m}_{\ell}(\nabla)$\end{document}, which are irreducible spherical tensors of ranks zero and ℓ, respectively (Santos, F. D. Nucl Phys A 1973, 212, 341). In this way, it is indeed possible to derive addition theorems by doing three‐dimensional Taylor expansions (Weniger, E. J. Int J Quantum Chem 2000, 76, 280). The application of the translation operator in its spherical form is particularly simple in the case ofBfunctions and leads to an addition theorem with a comparatively compact structure. Since other exponentially decaying functions like Slater‐type functions, bound‐state hydrogenic eigenfunctions, and other functions based on generalized Laguerre polynomials can be expressed by simple finite sums ofBfunctions, the addition theorems for these functions can be written down immediately. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002