10. Divergence Behaviour of Polynomial Interpolants on Compact Intervals (The Méray-Runge Phenomenon)

1982 ◽  
pp. 67-72
Author(s):  
Walter Schempp
Keyword(s):  
Runge Phenomenon

On the Development of a Synchronized Harmonic Balance Method for Multiple Frequencies and its Application to LPT Flows

2020 ◽  
Author(s):  
Javier Crespo ◽  
Jesús Contreras
Keyword(s):  
Boundary Conditions ◽  
Time Domain ◽  
Harmonic Balance ◽  
Fourier Transforms ◽  
Harmonic Balance Method ◽  
Single Passage ◽  
Computational Performance ◽  
Runge Phenomenon ◽  
Numerical Instabilities ◽  
The Impact

Abstract The aim of this paper is to describe the development and application of a multi-frequency harmonic balance solver for GPUs, particularly suitable for the simulation of periodic unsteadiness in nonlinear turbomachinery flows comprised of a few dominant frequencies, with an unsteady multistage coupling that bolsters the flow continuity across the rotor/stator interface. The formulation is addressed with the time-domain reinterpretation, where several non-equidistant time instants conveniently selected are solved simultaneously. The set of required frequencies in each row is driven into the governing equations with the help of almost-periodic Fourier transforms for time derivatives and time shifted boundary conditions. The spatial repetitiveness inside each row can be exploited to perform single-passage simulations and the relative circumferential positioning of the rotors or stators and the different blade or vane counts is tackled by means of adding fictitious frequencies referring to non-adjacent rows therefore taking into account clocking and indexing effects. Existing multistage row coupling techniques of harmonic methods rely on the use of non-reflecting boundary conditions, based on linearizations, or time interpolation, which may lead to Runge phenomenon with the resulting numerical instabilities and non-preserving flux exchange. Different sets of time instants might be selected in each row but the interpolation in space and time across their interfaces gives rise to robustness issues due to this phenomenon. The so-called synchronized approach, developed in this work, consist of having the same time instances among the whole ensemble of rows, ensuring that flux transfer at sliding planes is applied more robustly. The combination of a set of shared non-equidistant time instances plus the use of unequal frequencies (real and fictitious) may spoil the Fourier transforms conditioning but this can be dramatically improved with the help of oversampling and instants selection optimization. The resulting multistage coupling naturally addresses typical numerical issues such as flow that might reverse locally across the row interfaces by means of not using boundary conditions but a local flux conservation scheme in the sliding planes. Some examples will be given to illustrate the ability of this new approach to preserve accuracy and robustness while resolving them. A brief analysis of results for a fan stage and a LPT multi-row case is presented to demonstrate the correctness of the method, assessing the impact in the modeling accuracy of the present approach compared with a time-domain conventional analysis. Regarding the computational performance, the speedup compared to a full annulus time-domain unsteady simulation is a factor of order 30 combining the use of single-passage rows and time spectral accuracy.

Derived Nodes Approach for Improving Accuracy of Machining Stability Prediction

2018 ◽  
Vol 140 (3) ◽  
Author(s):  
Le Cao ◽  
Xiao-Ming Zhang ◽  
Tao Huang ◽  
Han Ding
Keyword(s):  
State Space ◽  
Polynomial Interpolation ◽  
Recursive Estimation ◽  
Machining Process ◽  
Stability Prediction ◽  
State Variables ◽  
Piecewise Polynomials ◽  
Discrete Equations ◽  
Runge Phenomenon ◽  
The Stability

Machining process dynamics can be described by state-space delayed differential equations (DDEs). To numerically predict the process stability, diverse piecewise polynomial interpolation is often utilized to discretize the continuous DDEs into a set of linear discrete equations. The accuracy of discrete approximation of the DDEs generally depends on how to deal with the piecewise polynomials. However, the improvement of the stability prediction accuracy cannot be always guaranteed by higher-order polynomials due to the Runge phenomenon. In this study, the piecewise polynomials with derivative-continuous at joint nodes are taken into consideration. We develop a recursive estimation of derived nodes for interpolation approximation of the state variables, so as to improve the discretization accuracy of the DDEs. Two different temporal discretization methods, i.e., second-order full-discretization and state-space temporal finite methods, are taken as demonstrations to illustrate the effectiveness of applying the proposed approach for accuracy improvement. Numerical simulations prove that the proposed approach brings a great improvement on the accuracy of the stability lobes, as well as the rate of convergence, compared to the previous recorded ones with the same order of interpolation polynomials.

Temperature dependence of the optical properties of Bi2O3. A theoretical approach basing on the Kramers-Kronig transformation for polynomial mixed terms models

Open Physics ◽  
2013 ◽  
Vol 11 (1) ◽  
Author(s):  
Gabriel Murariu ◽  
Simona Condurache-Bota
Keyword(s):  
Optical Properties ◽  
Temperature Dependence ◽  
Polynomial Functions ◽  
Thermal Vacuum ◽  
Glass Substrates ◽  
Runge Phenomenon ◽  
Mathematical Expressions ◽  
Transformation Type ◽  
Different Temperatures

AbstractThe Kramers-Kronig transforms (KK) constitute a powerful tool to validate experimental data. The present study is implemented for Bi2O3 thin films deposited by thermal vacuum evaporation at different temperatures of the glass substrates. Since the extraordinary properties of this fabric allow us to consider particular analytical approach as it was previously shown, the reflectance properties of Bi2O3 as a function of temperature could be studied.The novelty of this article is the studying of a global effective analytical representation, based on polynomial functions, in order to obtain a general model that includes temperature dependence of the optical properties, using the Kramers-Kronig transformation type. In the mathematical expressions, were included mix combined term in order to avoid the effects of Runge phenomenon. As a case study was chosen Bi2O3 — a substance less studied in literature. In the last part are the presented and commented the results obtained for a series of eight studied models.

Rearch of Data Interpolation in Airlevel Posture Sensor

2011 ◽  
Vol 271-273 ◽  
pp. 225-228
Author(s):  
Bai Hua Li ◽  
Lin Hua Piao ◽  
Ming Ming Ji
Keyword(s):  
Lagrange Interpolation ◽  
Polynomial Interpolation ◽  
Piecewise Linear ◽  
Linear Interpolation ◽  
Piecewise Linear Interpolation ◽  
Runge Phenomenon ◽  
Software Compensation ◽  
The Given ◽  
The Relationship ◽  
Zero Voltage

The method of interpolation was used in the process of software compensation of temperature in order to improve the environmental performance of air level posture sensor. Two algorithms including Lagrange interpolation and piecewise linear interpolation were calculated and compared in Matlab software, and then the optimization scheme could be achieved. The result showed that although the curse of Lagrange interpolation included all the given data positions, the Runge phenomenon in polynomial interpolation made the accuracy of interpolation lower. Piecewise linear interpolation reflected the relationship between environment and zero voltage more accurately. Piecewise linear interpolation not only can be used to improve the accuracy of software compensation of air level posture sensor.

scholarly journals A new approach for constructing mock-Chebyshev grids

2021 ◽  
Author(s):  
Ali IBRAHIMOGLU
Keyword(s):  
Numerical Experiments ◽  
Polynomial Interpolation ◽  
Numerical Results ◽  
Uniform Grid ◽  
New Approach ◽  
Chebyshev Nodes ◽  
Chebyshev Points ◽  
Runge Phenomenon ◽  
Equidistant Nodes

Polynomial interpolation with equidistant nodes is notoriously unreliable due to the Runge phenomenon, and is also numerically ill-conditioned. By taking advantage of the optimality of the interpolation processes on Chebyshev nodes, one of the best strategies to defeat the Runge phenomenon is to use the mock-Chebyshev points, which are selected from a satisfactory uniform grid, for polynomial interpolation. Yet, little literature exists on the computation of these points. In this study, we investigate the properties of the mock-Chebyshev nodes and propose a subsetting method for constructing mock-Chebyshev grids. Moreover, we provide a precise formula for the cardinality of a satisfactory uniform grid. Some numerical experiments using the points obtained by the method are given to show the effectiveness of the proposed method and numerical results are also provided.

scholarly journals Physically Consistent Scar Tissue Dynamics from Scattered Set of Data: A Novel Computational Approach to Avoid the Onset of the Runge Phenomenon

2021 ◽  
Vol 11 (18) ◽  
pp. 8568
Author(s):  
Pier Nicola Sergi ◽  
Natalia De la Oliva ◽  
Jaume del Valle ◽  
Xavier Navarro ◽  
Silvestro Micera
Keyword(s):  
Foreign Body Reaction ◽  
Scar Tissue ◽  
Nerve Fibers ◽  
Artificial Materials ◽  
Complex Biological Process ◽  
Runge Phenomenon ◽  
Standard Procedures ◽  
Low Degree ◽  
Before And After ◽  
Physically Based

The foreign body reaction is a complex biological process leading to the insulation of implanted artificial materials through a capsule of scar tissue. In particular, in chronic implantations of neural electrodes, the prediction of the scar tissue evolution is crucial to assess the implant reliability over time. Indeed, the capsule behaves like an increasing insulating barrier between electrodes and nerve fibers. However, no explicit and physically based rules are available to computationally reproduce the capsule evolution. In addition, standard approaches to this problem (i.e., Vandermonde-based and Lagrange interpolation) fail for the onset of the Runge phenomenon. More specifically, numerical oscillations arise, thus standard procedures are only able to reproduce experimental detections while they result in non physical values for inter-interval times (i.e., times before and after experimental detections). As a consequence, in this work, a novel framework is described to model the evolution of the scar tissue thickness, avoiding the onset of the Runge phenomenon. This approach is able to provide novel approximating functions correctly reproducing experimental data (R2≃0.92) and effectively predicting inter-interval detections. In this way, the overall performances of previous approaches, based on phenomenological fitting polynomials of low degree, are improved.

A New Algebra Interpolation Polynomial without Runge Phenomenon

2013 ◽  
Vol 303-306 ◽  
pp. 1085-1088 ◽  
Author(s):  
You Jun Chen ◽  
Hong Ying He ◽  
Shi Lu Zhang
Keyword(s):  
Computational Complexity ◽  
Continuous Functions ◽  
Interpolation Polynomial ◽  
High Order ◽  
Order Derivative ◽  
High Order Derivative ◽  
Runge Phenomenon

Proposed a new algebra interpolation polynomial with preferable stability, analyzed the related properties as well as stability and computational complexity, etc. Proved the new algebra interpolation polynomial can approximate any continuous functions, and it can be used to calculate the high order derivative without Runge phenomenon.

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