Piecewise Linear Interpolation Research Articles

Design and shape optimization of interior permanent magnet synchronous machine based on hybrid cores

By designing stator teeth using grain-oriented sheets (GO) and other parts still with non-grain-oriented sheets (NGO), the electromagnetic performance of the interior permanent magnet synchronous machine (GO-IPMSM) can be improved greatly. As the stator core of the designed GO-IPMSM is a hybrid core that is composed of two different silicon sheets, both the electromagnetic and mechanical performances are affected by the stator joint shape between GO and NGO sheets. Meanwhile, with the adoption of GO, the torque ripple of GO-IPMSM is increased though the average torque is increased as well. For reducing the torque ripple, optimizing the rotor barrier shape is an effective way. In this paper, the piecewise linear interpolation method is employed for establishing the stator joint shape between GO and NGO sheets, and the polynomial method is proposed to establish the rotor barrier shape. Through the analysis, it can be seen that the stator joint shape plays a strong role in affecting the mechanical performance of the GO-IPMSM, while the effect on the electromagnetic performance is weak. The genetic algorithm is used to optimize the rotor barrier shape for achieving high average torque and low torque ripple. Lastly, a quick and accurate efficiency map calculation method based on the Kriging model is proposed for GO-IPMSM with less finite element method (FEM) samples are required.

A rapidly converging domain decomposition algorithm for a time delayed parabolic problem with mixed type boundary conditions exhibiting boundary layers

A rapidly converging domain decomposition algorithm is introduced for a time delayed parabolic problem with mixed type boundary conditions exhibiting boundary layers. Firstly, a space-time decomposition of the original problem is considered. Subsequently, an iterative process is proposed, wherein the exchange of information to neighboring subdomains is accomplished through the utilization of piecewise-linear interpolation. It is shown that the algorithm provides uniformly convergent numerical approximations to the solution. Our analysis utilizes some novel auxiliary problems, barrier functions, and a new maximum principle result. More importantly, we showed that only one iteration is needed for small values of the perturbation parameter. Some numerical results supporting the theory and demonstrating the effectiveness of the algorithm are presented.

Efficient Computation of Geodesics in Color Space.

Although scientists agree that a perceptual color space is not Euclidean and color difference measures, such as CIELAB's ΔE2000, model these aspects of color perception, colormaps are still mostly evaluated through piecewise linear interpolation in a Euclidean color space. In a non-Euclidean setting, the piecewise linear interpolation of a colormap through control points translates to finding shortest paths. Alternatively, a smooth interpolation can be generalized to finding the straightest path. Both approaches are difficult to solve and are compute intensive. We compare the 11 most promising optimization algorithms for the computation of a geodesic either as the shortest or as the straightest path to find the most efficient one to use for colormap interpolation in real-world applications. For two control points, the zero curvature algorithms excelled, especially the 2D relaxation method. For multiple control points, only the mimimal curvature algorithms can produce smooth curves, amongst which the 1D relaxation method performed best.

Asymptotic expansion of the error of the numerical method for solving wave equation with functional delay

A wave equation with functional delay is considered. The problem is discretized. Constructions of the difference method with weights with piecewise linear interpolation are given. A basic method with weights with piecewise cubic interpolation is constructed. The order of the residual is studied without interpolation of the basic method, and the expansion coefficients of the residual with respect to time-steps and space-steps are written out. It is proved that the weighted method with piecewise cubic interpolation converges with order 2 in the energy norm. An equation is written for the main term of the asymptotic expansion of the global error of the basic method. Under certain assumptions, the validity of the application of the Richardson extrapolation procedure is substantiated, and the corresponding numerical method is constructed, that has the fourth order of convergence with respect to time-steps and space-steps. The validity of Runge's formulas for practical estimation of the error is proved. The results of numerical experiments on a test example are presented.

Lithium-Ion Battery State of Health Estimation Using Simple Regression Model Based on Incremental Capacity Analysis Features

Batteries are the core component of electric vehicles (EVs) and energy storage systems (ESSs), being crucial technologies contributing to carbon neutrality, energy security, power system reliability, economic efficiency, etc. The effective operation of batteries requires precise knowledge of the state of health (SOH) of the battery. A lack of proper knowledge of SOH may lead to the improper use of severely aged batteries, which may result in degraded system performance or even a risk of failure. This makes it important to accurately estimate battery SOH using only operational data, and this is the main topic of this study. In this study, we propose a novel method for online SOH estimation for batteries featuring simple online computation and robustness against measurement anomalies while avoiding the need for full cycle discharging and charging operation data. Our proposed method is based on incremental capacity analysis (ICA) to extract battery aging feature parameters and regression using simple piecewise linear interpolation. Our proposed method is compared with back-propagation neural network (BPNN) regression, a popular method for SOH estimation, in case studies involving actual data from battery aging experiments under realistic discharging and temperature conditions. In terms of accuracy, our method is on par with BPNN results (about 5% maximum relative error), while the simplicity of our method leads to better computation efficiency and robustness against data anomalies.

An Approximation Algorithm for Optimal Piecewise Linear Interpolations of Bounded Variable Products

We investigate the optimal piecewise linear interpolation of the bivariate product xy over rectangular domains. More precisely, our aim is to minimize the number of simplices in the triangulation underlying the interpolation, while respecting a prescribed approximation error. First, we show how to construct optimal triangulations consisting of up to five simplices. Using these as building blocks, we construct a triangulation scheme called crossing swords that requires at most - times the number of simplices in any optimal triangulation. In other words, we derive an approximation algorithm for the optimal triangulation problem. We also show that crossing swords yields optimal triangulations in the case that each simplex has at least one axis-parallel edge. Furthermore, we present approximation guarantees for other well-known triangulation schemes, namely for the red refinement and longest-edge bisection strategies as well as for a generalized version of K1-triangulations. Thereby, we are able to show that our novel approach dominates previous triangulation schemes from the literature, which is underlined by illustrative numerical examples.

A new approach to the interpretation of B-type natriuretic peptide concentration in children with congenital heart disease

Abstract Objectives BNP is the unique cardiac marker that reflects not as much as the degree of heart muscle damage, but mostly the severity of hemodynamic disorder, which is important in congenital heart disease. The only disadvantage of this marker is the barely studied reference values for children. It is known that the younger the child is, the higher the BNP value will be. By shifting from interpreting the absolute values towards the application of zlog-transformed data in clinical practice, we can overcome the above problems. Methods We performed an age-adjusted zlog transformation of BNP concentration. The age dependence was accounted for by a piecewise linear interpolation of the logarithms of BNP concentration among healthy children in different age groups from the logarithms of age. Results The concentration of BNP was measured in 351 patients (under 1 year old) with various heart diseases. The median age at the time of testing was 52 days [10; 166]; the median weight was 4.1 kg [3.2; 6.2]. The conditions we investigated included almost all known congenital heart diseases, as well as primary cardiac tumors. After the zlog transformation, we eliminated age-dependence, which was proved by comparing BNP concentrations in two groups of patients with univentricular and biventricular hemodynamics. Conclusions BNP in patients with congenital heart disease reflects the severity of hemodynamic disorders, and zlogBNP is an objective, age-independent and clear mechanism that can be used to interpret this cardiac marker.

Convergence analysis of pressure reconstruction methods from discrete velocities

Magnetic resonance imaging allows the measurement of the three-dimensional velocity field in blood flows. Therefore, several methods have been proposed to reconstruct the pressure field from such measurements using the incompressible Navier–Stokes equations, thereby avoiding the use of invasive technologies. However, those measurements are obtained at limited spatial resolution given by the voxel sizes in the image. In this paper, we propose a strategy for the convergence analysis of state-of-the-art pressure reconstruction methods. The methods analyzed are the so-called Pressure Poisson Estimator (PPE) and Stokes Estimator (STE). In both methods, the right-hand side corresponds to the terms that involving the field velocity in the Navier–Stokes equations, with a piecewise linear interpolation of the exact velocity. In the theoretical error analysis, we show that many terms of different order of convergence appear. These are certainly dominated by the lowest-order term, which in most cases stems from the interpolation of the velocity field. However, the numerical results in academic examples indicate that only the PPE may profit of increasing the polynomial order, and that the STE presents a higher accuracy than the PPE, but the interpolation order of the velocity field always prevails. Furthermore, we compare the pressure estimation methods on real MRI data, assessing the impact of different spatial resolutions and polynomial degrees on each method. Here, the results are consistent with the academic test cases in terms of sensitivity to polynomial order as well as the STE showing to be potentially more accurate when compared to reference pressure measurements.

Computing the Riemannian center of mass on meshes

The Riemannian center of mass (a.k.a. Karcher mean or Fréchet mean) provides the equivalent to the Euclidean affine average on manifolds. In spite of its many potential applications in computer graphics and geometric modeling, there exist surprisingly few algorithms to compute it. We present a direct method for computing the Riemannian center of mass on a triangle mesh. Our method works in the polyhedral metric and uses a piecewise-linear interpolation of gradients of the distance fields from a set of control points. We present applications for tracing splines on a surface, comparing to other methods at the state of the art, and showing that we produce quality results while supporting user interaction.

Compact Difference Schemes with Temporal Uniform/Non-Uniform Meshes for Time-Fractional Black–Scholes Equation

In this paper, we are interested in the effective numerical schemes of the time-fractional Black–Scholes equation. We convert the original equation into an equivalent integral-differential equation and then discretize the time-integral term in the equivalent form using the piecewise linear interpolation, while the compact difference formula is applied in the spatial direction. Thus, we derive a fully discrete compact difference scheme with second-order accuracy in time and fourth-order accuracy in space. Rigorous proofs of the corresponding stability and convergence are given. Furthermore, in order to deal effectively with the non-smooth solution, we extend the obtained results to the case of temporal non-uniform meshes and obtain a temporal non-uniform mesh-based compact difference scheme as well as the numerical theory. Finally, extensive numerical examples are included to demonstrate the effectiveness of the proposed compact difference schemes.

Positivity and discretization of Fredholm integral operators

We provide sufficient conditions for vector-valued Fredholm integral operators and their commonly used spatial discretizations to be positive in terms of an order relation induced by a corresponding order cone. It turns out that reasonable Nyström methods preserve positivity. Among the projection methods, persistence is obtained for the simplest ones based on polynomial, piecewise linear or specific cubic interpolation (collocation), as well as for piecewise constant basis functions in a Bubnov-Galerkin approach. However, for semi-discretizations using quadratic splines or sinc-collocation we demonstrate that positivity is violated. Our results are illustrated in terms of eigenpairs for Krein-Rutman operators and form the basis of corresponding investigations for nonlinear integral operators.

Deep Learning for Multigroup Cross-Section Representation in Two-Step Core Calculations

We investigate using deep learning, a type of machine-learning algorithm employing multiple layers of artificial neurons, for the mathematical representation of multigroup cross sections for use in the Griffin reactor multiphysics code for two-step deterministic neutronics calculations. A three-dimensional fuel element typical of a high-temperature gas reactor as well as a two-dimensional sodium-cooled fast reactor lattice are modeled using the Serpent Monte Carlo code, and multigroup macroscopic cross sections are generated for various state parameters to produce a training data set and a separate validation data set. A fully connected, feedforward neural network is trained using the open-source PyTorch machine-learning framework, and its accuracy is compared against the standard piecewise linear interpolation model. Additionally, we provide in this work a generic technique for propagating the cross-section model errors up to the keff using sensitivity coefficients with the first-order uncertainty propagation rule. Quantifying the eigenvalue error due to the cross-section regression errors is especially practical for appropriately selecting the mathematical representation of the cross sections. We demonstrate that the artificial neural network model produces lower errors and therefore enables better accuracy relative to the piecewise linear model when the cross sections exhibit nonlinear dependencies; especially when a coarse grid is employed, where the errors can be halved by the artificial neural network. However, for linearly dependent multigroup cross sections as found for the sodium-cooled fast reactor case, a simpler linear regression outperforms deeper networks.

Quality Evaluation Method of Anchor Chain Flash Butt Welding Based on Deep Learning

Flash butt welding, a mainstream welding method employed in producing anchor chains, is a critical manufacturing process affecting the quality of anchor chains. Ultrasonic and load testing are used to evaluate the welding quality of anchor chains, but the cost of checking and replacing unqualified chain links is high. A deep learning-based quality evaluation method for flash butt welding is proposed to reduce the cost of detecting and replacing substandard chain links. First, displacement and current sensors collect electrode position and current signals during welding. Second, since the number of qualified anchor links is much larger than that of unqualified ones, a new data synthesis method is proposed: nearest-neighbor splicing sampling, which achieves the enhancement of minority samples by segmenting and combining existing data samples according to the features of anchor chain welding. Then, a piecewise linear interpolation method is used to handle the varying data length problem, thus satisfying the input requirements of the convolutional neural network (CNN). Finally, a CNN model is established, and dropout is used to reduce the over-fitting phenomenon. The experimental results show that the accuracy of the under-sampling method, over-sampling method, and nearest-neighbor splicing sampling method are 93.8%, 95.9%, and 96.3%, respectively, and the sensitivity, specificity, and accuracy of the CNN model are 95.7%, 93%, and 94.3%, respectively, which are better than those of the support vector machine (SVM).

Energy stability of a temporal variable-step difference scheme for time-fractional nonlinear fourth-order reaction-diffusion equation

In this paper, we propose a nonuniform numerical formula for the Caputo fractional derivative at the half-grid based on the piecewise linear interpolation and construct a difference scheme for the time-fractional nonlinear fourth-order reaction-diffusion equation. By virtue of two discrete tools: the discrete orthogonal convolution kernels and the discrete complementary convolution kernels, we obtain the positive definiteness of the discrete time-fractional derivative. Then a discrete variational energy dissipation law of the proposed difference scheme is established for the time-fractional nonlinear fourth-order reaction-diffusion equation, which is asymptotically compatible with the associated energy dissipation law for the classical equation as the value of fractional order approaches to one. Numerical experiments demonstrate the effectiveness and the energy dissipation of the proposed difference scheme with an adaptive time-stepping strategy.

A real-time data acquisition method of industrial production line based on OPC technology

In view of the low timeliness and accuracy of traditional production line data acquisition methods, this study proposed a real-time data acquisition method of industrial production line based on OPC technology. Considering the heterogeneity of industrial production line data sources, a data transmission scheme is generated based on OPC technology and multi-point communication with Modbus protocol after data preprocessing by piecewise linear interpolation method. Then, the improved decision tree algorithm is used to design the data processing mode. Finally, the current state of industrial production line data is judged by comparing the threshold value, and the real-time data collection of industrial production line is completed after the collection interval is determined. Experimental results show that the acquisition accuracy of this method is up to 96.8%, the maximum acquisition time is only 12.5 s, and the highest packet loss rate is only 1.89%, indicating that this method improves the acquisition quality.

Parameter-uniform convergence analysis of a domain decomposition method for singularly perturbed parabolic problems with Robin boundary conditions

We construct and analyze a domain decomposition method to solve a class of singularly perturbed parabolic problems of reaction-diffusion type having Robin boundary conditions. The method considers three subdomains, of which two are finely meshed, and the other is coarsely meshed. The partial differential equation associated with the problem is discretized using the finite difference scheme on each subdomain, while the Robin boundary conditions associated with the problem are approximated using a special finite difference scheme to maintain the accuracy. Then, an iterative algorithm is introduced, where the transmission of information to the neighbours is done using a piecewise linear interpolation. It is proved that the resulting numerical approximations are parameter-uniform and, more interestingly, that the convergence of the iterates is optimal for small values of the perturbation parameters. The numerical results support the theoretical results about convergence.

Abstract 13566: Predictors of Progressive QRS Duration in Left Bundle Branch Block

Introduction: QRS duration in patients with left bundle branch block (LBBB) predicts adverse clinical outcomes and response to cardiac resynchronization therapy (CRT). Changes in LBBB QRS duration over time have not been reported. Hypothesis: QRS duration in patients with LBBB changes over time and may be predicted by patient demographics and comorbidities. Methods: We retrospectively identified 450 patients with LBBB on an index ECG (QRS ≥120 msec) recorded between 1992 and 2000 and examined any follow-up ECGs recorded after approximately 1, 2, 3, 5, and 10 years. Piecewise linear interpolations were performed, and separate generalized linear models (GLM) with Schwartz Bayesian Criterion effect selection were used to relate baseline clinical variables to changes in QRS duration at follow-up. Results: The cohort was 53.1% male, average age 66.7 yr, and mean index ECG QRS duration 144.7 msec. Although there was significant variability, the duration of the mean QRS increased as a tapering exponential function of time (p<0.0001) with progressively less increase at later follow-up. Three additional variables strongly and independently predicted progressive lengthening of the QRS within the overall GLM: heart failure with reduced ejection fraction (HFrEF) (p<0.0001), index ECG QRS duration (neg) (p<0.0001), and male gender (p=0.0063). In a subgroup of patients with an index ECG QRS <150 msec, these four GLM model variables also predicted the increase in QRS: time (p=0.0129), index QRS duration (p=0.0042), HFrEF (p=0.0001), and male gender (p=0.0084). (Equation, Figure below) ΔQRS(t;i) = 50.22 - 9.71*e -0.29*t - 0.344*QRS(0;i) + 7.62* 1 HFrEF (i) +5.12* 1 Male (i) Adjusted R 2 = 0.103 Conclusions: QRS duration progressively prolongs with time in patients with LBBB especially in patients with relatively narrow QRS and can be predicted by the presence of HFrEF and male gender. These predictors may be useful in better informing clinical prognosis and decisions regarding CRT.

Discrete Algorithm for a Disk Morphological Filter Based on Piecewise Linear Interpolation

Discrete Algorithm for a Disk Morphological Filter Based on Piecewise Linear Interpolation

A variable gain impulsive observer for Lipschitz nonlinear systems with measurement noises

This paper investigates the variable gain impulsive observer design problem for Lipschitz nonlinear systems. It is assumed that the measurements are contaminated by noise and received by observer at aperiodic instants. To establish a tractable design condition for impulsive observers, the piecewise linear interpolation method is used to construct the variable gain function. To quantify the impact of the measurement noises and exogenous disturbance on the estimation error, a Lyapunov-based condition for establishing exponential input-to-state stability (EISS) property of the observation error dynamics is presented. Then it is shown that the EISS condition can be expressed as a set of linear matrix inequalities (LMIs) by introducing a piecewise quadratic Lyapunov function. A convex optimization problem is proposed in which the EISS gain is minimized. Comparisons with the existing methods show the effectiveness of the proposed design technique.

A Shortcut Method to Solve for a 1D Heat Conduction Model under Complicated Boundary Conditions

The function of boundary temperature variation with time, f(t) is generally defined according to measured data. For f(t), which has a complicated expression, a corresponding one-dimensional heat conduction model was constructed under the first type of boundary conditions (Dirichlet conditions) in a semi-infinite domain. By taking advantage of the Fourier transform properties, a theoretical solution was given for the model, under the condition that f(t) does not directly participate in the transformation process. The solution consists of the product of erfc(t) and f(0) and the convolution of erfc(t) and the derivative of f(t). The piecewise linear interpolation equation of f(t), based on the measured data of temperature, was substituted into the theoretical solution, thus quickly solving the model and deriving a corresponding analytical solution. Based on the analytical solution under the linear decay function boundary condition, the inflection point method and curve fitting method for calculating the thermal diffusivity were introduced and exemplified, and the variation laws of the appearance moment of the inflection point were discussed. The obtained results show that the values of thermal diffusivity calculated by the two methods are basically consistent, and that the inflection point values rise with the increasing values of the initial temperature variation of the boundary, the decrease in boundary temperature velocity, and the distance from the boundary, respectively.