Piecewise Interpolation Research Articles

Local corner smoothing of linear toolpath based on B-spline fitting and its piecewise interpolation

Line segment toolpaths are commonly utilized in machining of complex parts with curved surfaces. However, the first-order discontinuity of the linear toolpath results in frequent variation of the feedrate, which is bad to the machining quality. To smooth the sharp corners of linear toolpath thus avoiding above problem, this paper presents a local corner smoothing approach. Compared with existing methods, the smoothed toolpath using the proposed the is one integral B-spline, instead of alternative straight-line segments and spline segments, which make the continuity more easily to ensured. Additionally, the integral long B-spline is interpolated through a piecewise feedrate scheduling method, thus the computational burden can be released. Effectiveness of the presented method in corner smoothing and kinematic constraining is demonstrated by illustration test.

Time-variance of 60 GHz vehicular infrastructure-to-infrastructure (I2I) channel

In this paper, we study the time-variance of a roadside infrastructure to infrastructure (I2I) channel operating at 60 GHz millimetre wave (mmWave) band, where the time-variance is caused by moving vehicles acting as scatterers. At first, measurement data is obtained by placing the transmitter (TX) and the receiver (RX) at different heights to emulate a link between two nonidentical roadside units (RSUs), and time-domain channel sounding is performed by sending complementary Golay sequences from the TX to the RX. A linear piece-wise interpolation of the corresponding temporal auto-correlation function (ACF) is used to find the Doppler spread of the I2I channel, where our interpolation method compensates for a slower sampling rate. Next, a framework is presented for time-variant channel impulse response (CIR) simulation which focuses on moving scatterers only and validates the linear piece-wise ACF model. The framework is useful for time-variant vehicular I2I channel simulation and in speed estimation related vehicular applications. Finally, a double-slope curve-fitted analytical model for ACF is proposed as a generalisation to the linear piece-wise model. The proposed model and its Doppler spectrum are found to be in agreement with the analytical results for fixed-to-fixed (F2F) channels with moving scatterers and matches perfectly with the measured data.

Variable-order fractal-fractional time delay equations with power, exponential and Mittag-Leffler laws and their numerical solutions

In this paper, a numerical method based on the Lagrangian piece-wise interpolation is proposed to solve variable-order fractal-fractional time delay equations with power law, exponential decay and Mittag-Leffler memories. These operators permit to describe physical phenomena with variable memory and fractal variable dimension. Numerical methods were applied to simulate the variable-order time delay Mackey–Glass and synaptically coupled FitzHugh–Nagumo models. Our numerical simulations display several new attractors.

An Efficient Analysis Method for LTCC Ridge Waveguide Bandpass Filters via Database Searching

Low temperature cofired ceramic (LTCC) ridge waveguide filters are attractive candidates in implementing microwave filters in modern communication systems because of their competitive features, such as low loss, low cost, compact size, lightweight, high-power handling capability, wide spurious-free out-of-band response, and easy integration. However, optimization of LTCC ridge waveguide filters suffers from low speed or imperfect accuracy when the traditional full-wave simulation or the mode-matching method is utilized for analysis. This article makes research on an alternative analysis method based on the network theory for LTCC inline ridge waveguide filters. The method has high speed since it evaluates filter properties by simply making the cumulative product of transfer matrices of all constitutive laminated ridge waveguide (LRWG) sections and laminated coupling (LCP) sections according to their arranging sequence, and transfer matrices of all these sections are calculated from their characteristic parameters (transmission line or equivalent circuit parameters for LRWG or LCP sections) analytically. Another key feature of the method is that all characteristic parameters of LRWG and LCP sections are obtained by searching precreated database. In order to guarantee the analysis accuracy, two special strategies have been proposed and utilized. First, accurate extraction methods of characteristic parameters have been introduced and applied to the creation of a database for LRWG and LCP sections. Second, a piecewise interpolation scheme has been designed and adopted in the database searching process for both kinds of sections. The efficiency of the promoted analysis method has been verified by two analysis examples of LTCC LRWG filters. One is a three-order narrow-band filter and the other is a nine-order wide-band filter. The curves of scattering parameters calculated using the promoted method agree well with the simulated curves and the time consumed by the method is negligible when compared with the simulation time. The effectiveness of the proposed analysis method in filter design is also briefly introduced in this article.

Piecewise interpolation solution of the Cauchy problem for the transport equation with iterative refinement

This article deals with the method of approximate solution of the Cauchy problem for the transport equation based on the Newton interpolation polynomial of two variables with iterative refinement. In each element of the division of a rectangular domain into subdomains, a polynomial approximation of the partial time derivative is constructed. The interpolation polynomial is transformed into the form of an algebraic polynomial with numerical coefficients. In the integrated form, the polynomial is substituted for the dependent variable in the right side of the expression of the partial time derivative. Iterative renewal of the process at a fixed polynomial degree leads to the analogue of Picard’s method of successive approximations.

Survey of Immersed Boundary Approaches for Finite Element Analysis

AbstractMesh generation for traditional finite element analysis has proven to be very difficult to fully automate especially using hexahedral elements for complex 3D geometry. Several modifications to the finite element method (FEM), such as the meshless methods, have been proposed for avoiding mesh generation. An alternative approach has recently gained popularity where the geometry, created as a solid model in cad software, is embedded or immersed in a nonconforming background mesh for analysis. In this approach, referred to here as the immersed boundary approach, a background mesh that is independent of the geometry is used for piecewise interpolation or approximation of the solution. Therefore, a uniform mesh with regular-shaped or undistorted elements can be used, and such a mesh is easy to generate automatically. When the geometry is immersed in the background mesh, the boundary elements are often only partly inside the geometry and the nodes of the mesh may not be on the boundaries. Many new methods have been developed to integrate over partial elements and to apply boundary and interface conditions when the boundaries of the geometries do not conform to the background mesh. These methods are reviewed in this article with particular emphasis on the implicit boundary method and step boundary method for applying boundary conditions. In addition, B-spline elements and several applications of the immersed boundary approach are surveyed including composite microstructures and structural elements for plates and shells.

3D Curve Generation Using Spherical Polar Piecewise Interpolation in CAD and CAM

In this paper a newly 3D path planning approach and curve generation for design and manufacturing efficiency is considered. The 3D path is generated by a combination of piecewise interpolating curves - along a given number of via-points - created via a spherical coordinate system specified by the polar angles, radial distances and the associated azimuthal angles. Each piecewise interpolating curve is constructed using Hermite polar interpolation in the projective polar plane and the rotating azimuthal plane. To verify the proposed approach, numerical simulations for the generation of a helix design, a 4 and 6 leaf design and a trajectory planning of a picking robot arm are conducted.

Electricity demand forecasting for decentralised energy management

Abstract The world is experiencing a fourth industrial revolution. Rapid development of technologies is advancing smart infrastructure opportunities. Experts observe decarbonisation, digitalisation and decentralisation as the main drivers for change. In electrical power systems a downturn of centralised conventional fossil fuel fired power plants and increased proportion of distributed power generation adds to the already troublesome outlook for operators of low-inertia energy systems. In the absence of reliable real-time demand forecasting measures, effective decentralised demand-side energy planning is often problematic. In this work we formulate a simple yet highly effective lumped model for forecasting the rate at which electricity is consumed. The methodology presented focuses on the potential adoption by a regional electricity network operator with inadequate real-time energy data who requires knowledge of the wider aggregated future rate of energy consumption. Thus, contributing to a reduction in the demand of state-owned generation power plants. The forecasting session is constructed initially through analysis of a chronological sequence of discrete observations. Historical demand data shows behaviour that allows the use of dimensionality reduction techniques. Combined with piecewise interpolation an electricity demand forecasting methodology is formulated. Solutions of short-term forecasting problems provide credible predictions for energy demand. Calculations for medium-term forecasts that extend beyond 6-months are also very promising. The forecasting method provides a way to advance a novel decentralised informatics, optimisation and control framework for small island power systems or distributed grid-edge systems as part of an evolving demand response service.

A mixed three-node triangular element with continuous nodal stress for fully dynamic consolidation of porous media

A mixed three-node triangular element with continuous nodal stresses (mixed T3-MINI-CNS) is presented for dynamics of porous media based on the three-variable u-w-p Biot model and the numerical manifold method (NMM). The displacement and velocity approximations are derived using the constrained and orthonormalized least-square (CO-LS) scheme, which possess continuous derivatives and Delta property at nodes. The pressure approximation takes linear piecewise interpolation. Relative to higher-order element, the mixed quadratic six-node triangle, mixed T3-MINI-CNS can model dynamic response of porous media more precisely with fewer unknowns, especially the short-term transient response. Moreover, mixed T3-MINI-CNS is immune from locking in both undrained and rigid skeleton limits and achieves more accurate pressure results than other characteristic locking-free elements. Based on NMM and u-w-p formulation, the most versatile three-node triangular mesh can always be used, avoiding difficulties in mesh generation. As fluid acceleration is involved, mixed T3-MINI-CNS is capable of totally predicting dynamic response of porous mixture, especially under rapid loading condition. In addition, reliability and precision of the time integration are assessed in terms of the energy balance condition. Through calculating benchmark problems, convergence, accuracy, and reliability of the mixed T3-MINI-CNS are thoroughly investigated and validated.

Comparison of Preisach and Congruency-Based Static Hysteresis Models Applied to Non-Oriented Steels

Two hysteresis models are investigated for modeling static hysteresis phenomenon in nonoriented magnetic steels. First, the static Preisach model is studied, and two types of experimental data, i.e., first-order reversal curves (FORCs) and concentric hysteresis loops, are employed to identify the weight function. Due to negative values in the obtained experimental weight functions, the resulting Preisach models show nonphysical behavior. To eliminate this nonphysical behavior, an optimization procedure is applied to remove these negative values. The nonphysical behavior is resolved in the modified Preisach model; however, the model still has poor accuracy for asymmetric loops. Alternatively, a new congruency-based static hysteresis model is proposed. In this model, third- or higher-order reversal curves are approximated using measured FORCs and second-order reversal curves (SORCs) by means of a congruency regularity. According to this regularity, any third- or higher-order reversal curve is congruent to SORCs with the same reversal point. A 3-D piecewise interpolation function is employed to interpolate the measured major loop, FORCs, and SORCs. The proposed model results are compared with quasi-static measurements. The comparison reveals that the model is capable of approximating the magnetization curves for the measurements containing complex-shaped $B$ waveforms with a relative energy error of less than 4%.

A Time-Domain Approach to Channel Estimation and Equalization for the SC-FDM System

Similar to orthogonal frequency division multiplexing receiver, the frequency-domain (FD) channel estimation (CE) and equalization are indispensable parts in the coherent single carrier frequency division multiplexing (SC-FDM) receiver. When the channel varies slowly, the FD processing is cost effective. For the applications with high mobility, the traditional implementation of the SC-FDM receiver causes significant performance degradation. In this paper, we propose to estimate time-varying pieces of channel taps for pilot symbols based on basis expansion model (BEM), and subsequently to reconstruct time-domain (TD) channel response for data symbols by utilizing the Slepian sequences-based piece-wise interpolation. Furthermore, two simplified schemes, i.e., the Slepian sequences-based multiple-point interpolation and the segmented BEM, are developed to reduce the computational complexity of the TD-CE. The Cramer-Rao lower bound (CRLB) of channel impulse response (CIR) estimation is also analyzed. In the light of the sparsity of TD channel gain matrix, we design an iterative least-squares QR decomposition algorithm to equalize the SC-FDM symbols. In the simulated SC-FDM system, when considering the carrier frequency of 5.9 GHz and the velocity of about 510 km/h, we observe that the traditional FD methods cause the demodulation failure, while the proposed TD processing schemes achieve ideal error-probability performance and preserve a relatively low complexity.

Quasi-classical trajectory-based non-equilibrium chemical reaction models for hypersonic air flows

Phenomenological models, such as Park’s widely used two temperature model, overpredict the reaction rate coefficients at vibrationally cold conditions and underpredict it at vibrationally hot conditions. To this end, two new chemical reaction models, the nonequilibrium total temperature (NETT) and nonequilibrium piecewise interpolation models for the continuum framework are presented. The focus is on matching the reaction rate coefficients calculated using a quasiclassical trajectory based dissociation cross section database. The NETT model is an intuitive model based on physical understanding of the reaction at a molecular level. A new nonequilibrium parameter and the use of total temperature in the exponential term of the Arrhenius fit ensure the NETT model has a simple and straightforward implementation. The efficacy of the new model was investigated for several equilibrium and nonequilibrium conditions in the form of heat bath simulations. Additionally, two-dimensional hypersonic flows around a flat blunt-body were simulated by employing various chemical reaction models to validate the new models using experimental shock tube data. Park’s two temperature model predicted higher dissociation rates and a higher degree of dissociation leading to lower peak vibrational temperatures compared to those predicted by the new nonequilibrium models. Overall, the present work demonstrates that the new nonequilibrium models perform better than Park’s two temperature model, especially in simulations with a high degree of nonequilibrium, particularly as observed in re-entry flows.

High-Precision Modeling for Aerospace TT&C Channels Using Piecewise Osculating Interpolation

Polynomial interpolation is widely used in channel modeling for aerospace tracking, telemetry, and command channel simulators, to accurately regenerate the satellite motion parameters with a high sampling rate from the available low sampling rate motion data. In this paper, a novel piecewise osculating interpolation approach is proposed to improve modeling precision. It uses the available distance and its higher-order derivatives at each pair of adjacent knots to construct the piecewise interpolation polynomial, thus achieving a better precision and continuity of the resulting motion parameters. Moreover, it is efficiently implemented with an interpolation filter based on a modified Farrow structure, and the designed filter can interpolate the motion data obtained both through uniform or nonuniform sampling. Additionally, a sampling optimization algorithm is proposed. It optimizes the motion data sampling by minimizing the maximum interpolation error given the average sampling interval and interpolation polynomial order, and therefore, improves the precision further without increasing the costs. The validity of the proposed approach is verified by numerical simulation, and its advantages are demonstrated by comparison with existing methods.

A highly efficient time-space-domain optimized method with Lax-Wendroff type time discretization for the scalar wave equation

In this paper, we propose an efficient time-space-domain optimized (OptTS) finite difference scheme to model 2D and 3D scalar wave propagation. It adopts piecewise constant interpolation coefficients for several consecutive Courant number ranges, which avoids the extra time costs caused by loading the coefficients consecutively according to different wave velocities in heterogeneous media. The new scheme is capable of suppressing numerical dispersion errors using a large time interval because we introduce a Lax-Wendroff type time discretization, which utilizes the wave-field information of O(M1M2) neighboring grid points in each step, with merely O(M1+M2) computation, where M1 and M2 are the stencil radii to generate the 2nd-order and 4th-order spatial derivatives. The improved computational efficiency and accuracy of OptTS are validated by substantial experiments using both theoretical analysis and numerical modeling. They show that in heterogeneous models, the maximal Courant numbers reach 1.0 and 0.85 in 2D and 3D cases respectively without severe temporal error, and the scheme is computationally nearly three times faster than its high-order counterpart, providing a powerful tool for large-scale modeling and high-resolution imaging.

Capacity Curve Estimation for High-Rise Buildings Using Limited Number of Sensors

ABSTRACT Capacity curve estimated from seismic floor accelerations can provide an informative description of the structural behavior soon after an earthquake. Detailed information on instrumented buildings is not always available. Therefore, this paper proposes a strategy based on the piece-wise cubic polynomial interpolation procedure to estimate the capacity curve for high-rise buildings from accelerations using limited sensors placed at regular intervals along the height of the building. The validity of the method is assessed by numerical simulations and shaking table tests.

A novel method for numerical simulation of sand motion model in beach formation based on fractional Taylor–Jumarie series expansion and piecewise interpolation technique

In this paper, for the first time, fractional Taylor–Jumarie series expansion is used to solve a glass of time-fractional delay partial differential equation by piecewise interpolation reproducing kernel method (RKM), this class of equations describe sand motion model in beach formation. The aim of this work is to obtain more accurate numerical solution by fractional Taylor–Jumarie series expansion and piecewise interpolation technique. Three numerical experiments are provided to show the advantage of this method, the results show the characteristics of sand motion. The research in this paper is the theoretical basis for further sand motion study.

Convex Properties of Center-of-Mass Trajectories for Locomotion Based on Divergent Component of Motion

This letter presents an in-depth analysis of the convex properties of center-of-mass (CoM) trajectories for legged robot locomotion based on the concept of Divergent Component of Motion (DCM). In particular, we show that the union of all possible trajectories forms a bounded convex set under appropriate boundary conditions. Additionally, we describe in detail our approach of generating closed-form CoM trajectories through piecewise interpolation over a sequence of waypoints and show how to compute the CoM trajectory efficiently through equations given in a matrix form. Applying the convex properties to our trajectory-generation approach, we present an algorithm for computing convex overapproximations of the CoM waypoints. Finally, we provide an example of usage in placing waypoints that lead to feasible CoM trajectories with respect to kinematic and dynamic constraints. The approach is validated with a multi-contact scenario in simulation with the humanoid robot TORO.

Локальная интерполяция и аппроксимация в задачах эвристического синтеза цифровых фильтров

The article has an overview character with elements of generalization, but at the same time it includes information brought to the level of immediate practical implementation. Approaches to heuristic and functional synthesis of digital filters based on local nterpolation and approximation are considered. Digital differentiators based on local polynomial interpolation, digital integrators based on local polynomial and spline interpolation and digital filters with a U-shaped amplitude-frequency characteristic based on polynomial OLS approximation are obtained. The obtained results can be used in practice according to their functional purpose, taking into account the recommendations mentioned. The idea of numerical differentiation based on interpolation is not new, but the deep connection between this approach and digital filtering is not often reflected in literature due to the fact that this concept is limited to numerical differentiation based on finite differences. The gap is filled in this work. Numerical integration is usually considered on the basis of stepwise or piecewise linear or piecewise parabolic interpolation (method of rectangles, trapezoids, Simpson). However, regardless of the choice of the interpolation method, the digital integrator has a certain generalized structure, and approaches to numerical integration can be based on more effective methods of interpolation, which is shown in the article. The OLS filters (Savitzky-Golay) are described in the literature. However, the steps for their practical implementation are clearly insufficient. Traditionally the general description of filters does not develop further 4th degree of the approximating polynomial and is often limited to the moving average filters. The frequency properties of the filters are also not fully described. In this work the description of the OLS filters obtained at the degree of the approximating polynomial 0-8 was given, their frequency properties were studied, and sufficient information was given to obtain digital filters based on them with a U-shaped amplitude-frequency characteristic without pulsations in the passband. Thus, the field of filters application is extended beyond the OLS smoothing.

Interpretation of Retrospective BG Measurements.

This study investigates blood glucose (BG) measurement interpolation techniques to represent intermediate BG dynamics, and the effect resampling of retrospective BG data has on key glycemic control (GC) performance results. GC protocols in the ICU have varying BG measurement intervals ranging from 0.5 to 4 hours. Sparse data pose problems, particularly in comparing GC performance or model fitting, and thus interpolation is required. Retrospective data from SPRINT in Christchurch Hospital Intensive Care Unit (ICU) (2005-2007) were used to analyze several interpolation techniques. Piecewise linear, spline, and cubic interpolation functions, which force interpolation through measured data, as well as 1st and 2nd Order B-spline basis functions, are used to identify the interpolated trace. Dense data were thinned to increase sparsity and obtain measurements (Hidden Measurements) for comparison after interpolation. Performance is assessed based on error in capturing hidden measurements. Finally, the effect of minutely versus hourly sampling of the interpolated trace on key GC performance statistics was investigated using retrospective data received from STAR GC in Christchurch Hospital ICU, New Zealand (2011-2015). All of the piecewise functions performed considerably better than the fitted interpolation functions. Linear piecewise interpolation performed the best having a mean RMSE 0.39 mmol/L, within 2 standard deviations of the BG sensor error. Minutely sampled BG resulted in significantly different key GC performance values when compared to raw sparse BG measurements. Linear piecewise interpolation provides the best estimate of intermediate BG dynamics and all analyses comparing GC protocol performance should use minutely linearly interpolated BG data.

Parameter calibration for a shear modified GTN model and its application to forming limit prediction

Damage evolution is an unavoidable problem in the forming process of sheet metal, and an improper forming technology may cause ductile fracture of the final parts. The damage mechanics theory can give great insight to the failure mechanism and the numerical simulation using damage model can provide a new way to predict the fracture behaviour for sheet metal. The shear modified GTN model by Nahshon and Hutchinson with a piecewise interpolation function was applied, which can capture the damage evolution under both low and high stress triaxiality. Then, a method combined numerical simulation and optimization algorithm was adopted to calibrate the damage-related parameters. Furthermore, the shear modified model was used to predict the forming limit of a high strength steel 22MnB5.With the help of digital image correlation technique (DIC), both the strain field and forming limit curve obtained by the numerical simulation were compared with the experimental ones, and satisfactory results are obtained.