Effects Of Truncation Errors Research Articles

A novel structural reliability analysis method combining the improved beluga whale optimization and the arctangent function‐based maximum entropy method

AbstractA novel structural reliability analysis method that combines the improved beluga whale optimization (IBWO) and the arctangent function‐based maximum entropy method (AMEM) is proposed in this paper. It aims to augment the accuracy of failure probability prediction in structural reliability analysis based on the traditional maximum entropy method (MEM). First, the arctangent function is introduced to avoid the effects of truncation error and numerical overflow in the traditional MEM. The arctangent function can nonlinearly transform the structural performance function defined on the infinite interval into a transformed performance function defined on the bounded interval. Subsequently, the undetermined Lagrange multipliers in the maximum entropy probability density function (MEPDF) of the transformed performance function are obtained using IBWO at a swifter convergence speed with heightened convergence accuracy. Finally, the MEPDF of the transformed performance function can be obtained by combining the IBWO and AMEM, and the structural failure probability can be predicted. The analysis of the metro bogie frame as an engineering example reveals that compared with the traditional MEM using the genetic algorithm to solve the Lagrange multipliers, the proposed method diminishes the relative error in failure probability prediction from 20.51% to only 0.09%. This method significantly enhances the prediction accuracy of failure probability.

A Polynomial Chaos Approach to Robust Static Output-Feedback Control With Bounded Truncation Error

This article considers the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_\infty$</tex-math></inline-formula> static output-feedback control for linear time-invariant uncertain systems with polynomial dependence on probabilistic time-invariant parametric uncertainties. By applying polynomial chaos theory, the control synthesis problem is solved using a high-dimensional expanded system, which characterizes stochastic state uncertainty propagation. A closed-loop polynomial chaos transformation is proposed to derive the closed-loop expanded system. The approach explicitly accounts for the closed-loop dynamics and preserves the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {L}_2$</tex-math></inline-formula> -induced gain, which results in smaller transformation errors compared to existing polynomial chaos transformations. The effect of using finite-degree polynomial chaos expansions is first captured by a norm-bounded linear differential inclusion, and then addressed by formulating a robust polynomial chaos based control synthesis problem. This proposed approach avoids the use of high-degree polynomial chaos expansions to alleviate the destabilizing effect of truncation errors, which significantly reduces computational complexity. A numerical example illustrates the effectiveness of the proposed approach.

Polynomial chaos-based [formula omitted] output-feedback control of systems with probabilistic parametric uncertainties

H2 static and dynamic output-feedback control problems are investigated for linear time-invariant uncertain systems. The goal is to minimize the averaged H2 performance in the presence of nonlinear dependence on time-invariant probabilistic parametric uncertainties. By applying the polynomial chaos theory, the control synthesis problem is solved using a high-dimensional expanded system which characterizes stochastic state uncertainty propagation. Compared to existing polynomial chaos-based control designs, the proposed approach addresses the simultaneous presence of parametric uncertainties and white noises. The effect of truncation errors due to using finite-degree polynomial chaos expansions is captured by time-varying norm-bounded uncertainties, and is explicitly taken into account by adopting a guaranteed cost control approach. This feature avoids the use of high-degree polynomial chaos expansions to alleviate the destabilizing effect of expansion truncation errors, thus significantly reducing computational complexity. Moreover, rigorous analysis clarifies the condition under which the stability of the high-dimensional expanded system implies the internal mean square stability of the original system under control. Using iterations between synthesis and post-analysis, a bisection algorithm is proposed to find the smallest bounding parameter on the effect of expansion truncation errors such that robust closed-loop stability is guaranteed. A numerical example is used to illustrate the effectiveness of the proposed approach.

Speed estimation of PMSM using SRUKF algorithm

Speed estimation is a key technology to realize sensorless control for the PMSM. Based on the unscented Kalman filter (UKF) method without linearization of nonlinear system equations, this letter provides square root unscented Kalman filter (SRUKF) algorithm that operates through iterating the square roots of the covariance matrixes obtained by QR decomposition and Cholesky decomposition. The presented method can further improve speed estimation performance through decreasing the effect of truncation error and enhancing the convergence and stability of algorithm. Simulation results of sensorless control system demonstrate the feasibility and effectiveness of the proposed algorithm.

Loudness stability of binaural sound with spherical harmonic representation of sparse head-related transfer functions

In response to renewed interest in virtual and augmented reality, the need for high-quality spatial audio systems has emerged. The reproduction of immersive and realistic virtual sound requires high resolution individualized head-related transfer function (HRTF) sets. In order to acquire an individualized HRTF, a large number of spatial measurements are needed. However, such a measurement process requires expensive and specialized equipment, which motivates the use of sparsely measured HRTFs. Previous studies have demonstrated that spherical harmonics (SH) can be used to reconstruct the HRTFs from a relatively small number of spatial samples, but reducing the number of samples may produce spatial aliasing error. Furthermore, by measuring the HRTF on a sparse grid the SH representation will be order-limited, leading to constrained spatial resolution. In this paper, the effect of sparse measurement grids on the reproduced binaural signal is studied by analyzing both aliasing and truncation errors. The expected effect of these errors on the perceived loudness stability of the virtual sound source is studied theoretically, as well as perceptually by an experimental investigation. Results indicate a substantial effect of truncation error on the loudness stability, while the added aliasing seems to significantly reduce this effect.

Effect of Numerical Truncation Error on Implicit Finite Difference Methods in Groundwater Transport Models

A correction for truncation errors associated with implicit finite difference method (FDM) of the groundwater transport equation with reaction term, which is generally used in groundwater transport models, is developed here from a Taylor series analysis. An application example is formulated to illustrate the effect of truncation errors on the numerical solution of implicit FDM. The study compares the effect of truncation error on numerical model accuracy of implicit FDM and explicit FDM for groundwater transport equation with reaction term. The explicit FDM constitutes large deviation from analytical solution without truncation error correction. The relative error analysis reveals that error reduces from 75% to 30% after truncation error correction. Therefore, we can conclude that numerical truncation error correction has significant impact on accuracy of groundwater transport models based on explicit FDM than those based on implicit FDM.

A p-adaptation method for compressible flow problems using a goal-based error indicator

An accurate calculation of aerodynamic force coefficients for a given geometry is of fundamental importance for aircraft design. High-order spectral/hp element methods, which use a discontinuous Galerkin discretisation of the compressible Navier–Stokes equations, are now increasingly being used to improve the accuracy of flow simulations and thus the force coefficients. To reduce error in the calculated force coefficients whilst keeping computational cost minimal, we propose a p-adaptation method where the degree of the approximating polynomial is locally increased in the regions of the flow where low resolution is identified using a goal-based error estimator as follows.Given an objective functional such as the aerodynamic force coefficients, we use control theory to derive an adjoint problem which provides the sensitivity of the functional with respect to changes in the flow variables, and assume that these changes are represented by the local truncation error. In its final form, the goal-based error indicator represents the effect of truncation error on the objective functional, suitably weighted by the adjoint solution. Both flow governing and adjoint equations are solved by the same high-order method, where we allow the degree of the polynomial within an element to vary across the mesh.We initially calculate a steady-state solution to the governing equations using a low polynomial order and use the goal-based error indicator to identify parts of the computational domain that require improved solution accuracy which is achieved by increasing the approximation order. We demonstrate the cost-effectiveness of our method across a range of polynomial orders by considering a number of examples in two- and three-dimensions and in subsonic and transonic flow regimes. Reductions in both the number of degrees of freedom required to resolve the force coefficients to a given error, as well as the computational cost, are both observed in using the p-adaptive technique.

Application of generalized flexibility matrix in damage identification using Imperialist Competitive Algorithm

In this article, a new objective function which is formed by using generalized flexibility matrix is presented with the aim of solving the constrained optimization problem in damage detection procedure. The main purpose is to decrease the effects of truncation error which ensue from cutting off the higher order modes. Two different case studies are used to evaluate the proposed approach. These cases include, a 2D-frame and a Howe-Truss structure. In this study, each structure is modeled by finite element method and damages are induced by considering a reduction in the stiffness of the elements. Finally, the inverse problem of damage identification is solved by using the Imperialist Competitive Algorithm (ICA) for each damage scenario. Not only the effect of noise on the obtained results is studied but also a comparison is drawn between the proposed method based on ICA and other conventional optimization algorithms, namely PSO and GA.

On the Richtmyer–Meshkov instability evolving from a deterministic multimode planar interface

AbstractWe investigate the shock-induced turbulent mixing between a light and a heavy gas, where a Richtmyer–Meshkov instability (RMI) is initiated by a shock wave with Mach number $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Ma}= 1.5$. The prescribed initial conditions define a deterministic multimode interface perturbation between the gases, which can be imposed exactly for different simulation codes and resolutions to allow for quantitative comparison. Well-resolved large-eddy simulations are performed using two different and independently developed numerical methods with the objective of assessing turbulence structures, prediction uncertainties and convergence behaviour. The two numerical methods differ fundamentally with respect to the employed subgrid-scale regularisation, each representing state-of-the-art approaches to RMI. Unlike previous studies, the focus of the present investigation is to quantify the uncertainties introduced by the numerical method, as there is strong evidence that subgrid-scale regularisation and truncation errors may have a significant effect on the linear and nonlinear stages of the RMI evolution. Fourier diagnostics reveal that the larger energy-containing scales converge rapidly with increasing mesh resolution and thus are in excellent agreement for the two numerical methods. Spectra of gradient-dependent quantities, such as enstrophy and scalar dissipation rate, show stronger dependences on the small-scale flow field structures as a consequence of truncation error effects, which for one numerical method are dominantly dissipative and for the other dominantly dispersive. Additionally, the study reveals details of various stages of RMI, as the flow transitions from large-scale nonlinear entrainment to fully developed turbulent mixing. The growth rates of the mixing zone widths as obtained by the two numerical methods are ${\sim } t^{7/12}$ before re-shock and ${\sim } (t-t_0)^{2/7}$ long after re-shock. The decay rate of turbulence kinetic energy is consistently ${\sim } (t-t_0)^{-10/7}$ at late times, where the molecular mixing fraction approaches an asymptotic limit $\varTheta \approx 0.85$. The anisotropy measure $\langle a \rangle _{xyz}$ approaches an asymptotic limit of ${\approx }0.04$, implying that no full recovery of isotropy within the mixing zone is obtained, even after re-shock. Spectra of density, turbulence kinetic energy, scalar dissipation rate and enstrophy are presented and show excellent agreement for the resolved scales. The probability density function of the heavy-gas mass fraction and vorticity reveal that the light–heavy gas composition within the mixing zone is accurately predicted, whereas it is more difficult to capture the long-term behaviour of the vorticity.

Effects of Truncation Error of Derivative Approximation for Two-Phase Lattice Boltzmann Method

We verify the accuracy and the truncation errors of approximation to the first derivatives and to Laplacian operator in the lattice Boltzmann method. The truncation errors are calculated by the Taylor series expansion, and the influences are analytically and numerically examined in the simulation of two-phase flow. We propose the 4th-order accurate approximations to derivatives that utilize the property of the tensor unlike the finite difference. The numerical solutions with various approximations agree well with the theoretical ones within the error of 7% in the test of the Laplace's law. Application of the 4th-order accurate approximation scheme to two-phase flow simulation reduces the spurious current around the interface of a stationary droplet about to one-half of the results with the 2nd-order accuracy. The small spurious velocity in the vicinity of the interface of the 4th-order approximation increases the speed of a moving droplet, and distorts the shape of the droplet little. In the simulation of phase separation, the growth rate of the typical domain size calculated with FFT is strongly affected by the truncation error in surface-tension driven regime for short time. It is seen that the 4th-order approximation saturates at 2nd-order accuracy in simulation of a decaying Taylor vortex flow, because the convergence rate of the LBM is 2nd-order in space. It is shown that the approximation method affects the important physical values, such as velocity, moving speed, or domain size in numerical simulations.

Reducing the effect of truncation error in spatial and pointwise models of resonant systems with damping

This paper is concerned with finding analytical solutions to the problems of reducing the effect of truncation error in models of resonant systems that include damping. Pointwise and spatial models of resonant systems are both considered in the paper. It is known that the truncation of an infinite-dimensional model produces errors in the zero locations and zero-frequency content of the system. Our approach is to add a feedthrough term to the truncated model to reduce the uncertainty caused by neglected dynamics. This will improve the accuracy of the model by reducing the errors in the zero locations and zero-frequency content of the system, which is important in ensuring the closed-loop system's robustness when a feedback controller is implemented. The optimal feedthrough terms are determined by minimising the weighted H 2 norm and spatial H 2 norm of the error systems associated with the pointwise and spatial models, respectively. The existence of analytical solutions for optimal feedthrough terms allows models of resonant systems to be compensated in a straightforward manner. Simulation results are presented to demonstrate the effectiveness of our approach in reducing the effect of truncation error.

Exact Taylor series coefficients for variable-feedrate CNC curve interpolators

Recent papers [Yang DCH, Kong T. Computer Aided Design 1994;26:225–34; Yeh S-S, Hsu P-L. Computer Aided Design 1999;31:349–57] formulate real-time CNC interpolators for variable feedrates along parametric curves. These interpolators employ truncated Taylor series to compute successive reference-point parameter values, but in both papers an erroneous coefficient for the highest (quadratic) term is cited. The derivation of the proper coefficients is a straightforward, although somewhat convoluted, exercise in chain-rule differentiation. Compact recursive formulae are presented here to compute the correct coefficients, up to the cubic term, in cases where the feedrate depends on: (i) elapsed time; (ii) curve arc length; or (iii) local path curvature. The local and cumulative effects of truncation errors on the accuracy of such interpolator schemes are also assessed, and compared with the essentially exact interpolators for Pythagorean-hodograph curves.

Stability analysis of single-phase thermosyphon loops by finite-difference numerical methods

In this paper, examples of the application of finite-difference numerical methods in the analysis of stability of single-phase natural circulation loops are reported. The problem is addressed here for its relevance to thermal-hydraulic system code applications, with the aim to point out the effects of truncation error on stability prediction. The methodology adopted for analyzing in a systematic way the effect of various finite-difference discretization can be considered the numerical analogue of the usual techniques adopted for PDE stability analysis. Three different single-phase loop configurations are considered involving various kinds of boundary conditions. In one of these cases, an original dimensionless form of the governing equations is proposed, adopting the Reynolds number as a flow variable. This allows for an appropriate consideration of transition between laminar and turbulent regimes, which is not possible with other dimensionless forms, thus enlarging the range of validity of model assumptions.

Effects of the Order of Accuracy of Finite-Difference Method and Grid Resolution on LFS of Wall Turbulence.

In the large eddy simulation (LES) of turbulent flow, grid scale (GS) flow field is described by solving discretized governing equations of flow motion with a subgrid scale (SGS) model. Hence, the reliability of the LES data depends strongly not only on the reliability of the SGS model but also on the accuracy of the numerical method. In the LES a finite difference method (FDM) is widely used for practical problems. However, the truncation error of FDM is usually not so small compared with the SGS stress. The problem can be removed by increasing grid resolution, but it is still difficult to keep enough grid resolution in the LES of high Reynolds number flows. In this study, the effects of the order of accuracy of FDM and grid resolution on the LES of plane channel turbulent flow are estimated by the fully conservative higher order staggered FDM scheme proposed recently. The simulations of the flow at Rer=100 are submitted without SGS model with various grid resolution and several order of FDM up to 12th-order accuracy, and the fact that the FDM discretization has the effects of both truncation error and sharp cutoff is revealed. The effect of truncation error decreases the wall friction loss, while the effect of sharp cutoff increases it. Then, the flow at Rer=180 is simulated with the dynamic SGS model, and it is pointed out that we have to take the numerical error into consideration in constructing the SGS model for LES of practical problems.

The effect of truncation error on the numerical prediction of linear stability boundaries in a natural circulation single-phase loop

This paper discusses the effects of truncation error on the stability of natural circulation in a single-phase loop as predicted by finite difference numerical methods. A short description of the characteristics of the selected problem is firstly given, also showing results obtained by a modal solution. Different numerical schemes are applied and the related effect on predicted stability is shown. The results obtained are then discussed in relation to the applicability of thermal–hydraulic system codes in the analysis of fluid-dynamic instabilities in real systems.

Supersonic Transitional Airfoil Shapes of Minimum Drag

Feasibility of a procedure incorporating transition considerations in optimizing total drag has been demonstrated. The Schittkowski algorithm was modie ed to demonstrate the approach on two-dimensional airfoils. Cubic spline basis functions were used to describe the airfoils, and total drag (wave 1 friction) was minimized under the constraint of a e xed airfoil area. Reynolds numbers were assumed such that the airfoil was transitional, generally with the forward portion laminar and the aft turbulent. The transition locus was calculated using the fast transition prediction module, which provides rapid computation of the Tollmien ‐ Schlichting wave amplie cation factor N and estimates transition point by the e N method. The laminar friction drag was evaluated using self-similar solutions of the boundary-layer equations. The friction drag for the turbulent portion was computed assuming a one-seventh velocity proe le. With this framework, the total drag was a function of the functional xtr, the streamwise location of transition. As a validation of the method, the algorithm gave the correct global optimum for the inviscid case, which is the parabolic arc proe le. For the viscous case, signie cantly different locations of the maximum thickness led to only small differences in the minimum total drag. In addition, the drag reductions from the optimal inviscid parabolic proe le were about 10%. Convergence with respect to the number of spline knots was achieved. Important challenges were met to reduce the effect of truncation errors in the numerical approximation of differentiations such as those used in the evaluation of the Hessian matrix. Despite these dife culties, generalization of the approach to treat ine nite yawed and swept wings accounting for suction, crosse ow instabilities, and vortex drag appears feasible.

Short‐range order in some polymer melts from X‐ray diffraction

Wide-angle X-ray diffraction measurements were performed on polymer melts of isotactic and syndiotactic polypropylene (IPP and SPP), poly(ethylenepropylene) (PEP), polystyrene (PS), polyisobutylene (PIB), and polyethylene (PE), to study the dependence of the short-range structure of polymer liquids on chain architecture. Total structure functions, which comprise intra- and intermolecular contributions, were derived from the scattering data. The trivial Fourier components of the intramolecular structure (C(SINGLE BOND)>C ≃ 1.54 Å and C(SINGLE BOND)C(SINGLE BOND)C ≃ 2.55 Å) were subtracted from the total structure functions. The remaining functions contain only those intramolecular contributions dependent on the chain's conformational degrees of freedom, plus the intramolecular contributions. The structural differences between the polymers in momentum space are discerned only when the trivial components are subtracted. This subtraction also reduces the effects of truncation errors on Fourier transformation to real space. The short-range structure of PIB appears very different compared to all the others, which correlates with anomalies in a number of physical properties for this polymer. © 1996 John Wiley & Sons, Inc.

Short-range order in some polymer melts from X-ray diffraction

Wide-angle X-ray diffraction measurements were performed on polymer melts of isotactic and syndiotactic polypropylene (IPP and SPP), poly(ethylenepropylene) (PEP), polystyrene (PS), polyisobutylene (PIB), and polyethylene (PE), to study the dependence of the short-range structure of polymer liquids on chain architecture. Total structure functions, which comprise intra- and intermolecular contributions, were derived from the scattering data. The trivial Fourier components of the intramolecular structure (C(SINGLE BOND)>C ≃ 1.54 Å and C(SINGLE BOND)C(SINGLE BOND)C ≃ 2.55 Å) were subtracted from the total structure functions. The remaining functions contain only those intramolecular contributions dependent on the chain's conformational degrees of freedom, plus the intramolecular contributions. The structural differences between the polymers in momentum space are discerned only when the trivial components are subtracted. This subtraction also reduces the effects of truncation errors on Fourier transformation to real space. The short-range structure of PIB appears very different compared to all the others, which correlates with anomalies in a number of physical properties for this polymer. © 1996 John Wiley & Sons, Inc.

Truncation error in grid generation: A case study

AbstractTheoretical proofs state that the planar Winslow or homogenous Thompson–Thames–Mastin (hTTM) map is a diffeomorphism, yet numerical solutions to the hTTM equations produce folded grids on the so‐called “horseshoe” domain. A quasi‐analytic solution to the horseshoe problem is constructed to demonstrate that folding is due to truncation error effects. Higher‐order difference methods are also explored. © 1995 John Wiley &amp; Sons, Inc.

Energy dependent relaxation and the theory of resonance Raman scattering

We present a ‘‘transform’’ theory for analyzing resonance Raman scattering under the condition of energy or state dependent dampibng. The energy dependence of the damping is taken to be linear in the Raman quantum but arbitrary in the remaining subspace of vibrations. The theory is tested using exact sum-over-states model calculations. The effects of truncation error are explicitly assessed. The method is applied to a Raman excitation profile of cytochrome P450, which is ‘‘red shifted’’ with respect to the resonant absorption band.