Interpolation Error Research Articles

Tensor interpolation in virtual manufacturing chains for fiber reinforced composites

Preeminent weight-specific mechanical properties predestine fiber-reinforced plastics for application in structural components. Virtual manufacturing chains (CAE-chains) capture a multi-step process sequence by interlinking models with corresponding constitutive laws for each process step. In common numerical solving techniques, spatial discretization of governing equations yields discrete solutions. Since mesh type and fineness are usually problem-specific, mesh-to-mesh mapping must be embedded in the data interfaces between the individual simulation steps. To receive meaningful data, the underlying averaging and interpolation schemes between non-congruent meshes must be mathematically and physically consistent. In engineering applications, interpolation of tensors is usually carried out component-based, treating each component as an independent scalar-field and thus neglecting the tensorial character. The work at hand gives an overview of sophisticated techniques that preserve specific tensor characteristics in interpolation. Compared to available approaches, an enhanced decomposition-based interpolation method is proposed, allowing for a generalization with more than two basic values. In the context of process simulation for short fiber-reinforced injection molding, the influence of the interpolation techniques is evaluated in a three-stage approach with increasing proximity to application. Firstly, mathematical examples demonstrate that component-based interpolation can result in erroneous, non-monotonous tensor characteristics. Secondly, an analytically resolvable problem is derived, and the solution is compared to reconstructed tensors from different interpolation schemes. Thirdly, a numerical CFD-simulation is conducted. Field recovery via interpolation is performed and systematical errors are statistically evaluated. The results reveal significant biases for certain tensor characteristics induced by conventional component-wise interpolation. Overall, a reduction in systematic interpolation errors is achieved by the proposed decomposition-based interpolation.

Back Projection Algorithm for Multi-Receiver Synthetic Aperture Sonar Based on Two Interpolators

The back projection (BP) algorithm is characterized by its high performance for multi-receiver synthetic aperture sonar (SAS). For this reason, it is usually used to evaluate the imaging performance of Fourier-domain methods. However, this algorithm suffers from a large computation load, and the imaging efficiency is seriously lowered. In order to improve the imaging performance, this paper proposes focusing the multi-receiver SAS data using the BP algorithm based on two interpolators, including the linear interpolation and nearest-neighbor interpolation. The former interpolation is used to decrease the interpolation error based on adjacent sampled data; the latter estimates the data at the desired moment by assigning the data value of the nearest sample as estimated data. Then, the imaging performance of the presented method is discussed in detail based on simulations and real-data processing. With the presented method, the imaging performance can be improved without a loss of efficiency compared to nearest-neighbor interpolation without an upsampling operation. In comparison with the traditional BP algorithm, the presented method can be used to improve the imaging efficiency without any loss of performance.

A NEW FRAMEWORK FOR ACCURACY ASSESSMENT OF LIDAR-DERIVED DIGITAL ELEVATION MODELS

Abstract. Existing approaches to accuracy assessment of LiDAR-derived DEM are typically based on statistical methodology and focus on vertical accuracy. This paper presents a new framework which calls for assessment of not only a DEM’s vertical accuracy, but also its ability to preserve a point’s elevation rank in the bare earth topographic surface. New methods to assess each aspect are presented. For DEM’s vertical accuracy, approximation theory from numerical analysis is used to quantify the total error at each DEM point as well as its three error components – sensor error, ground error, and interpolation error. For DEM’s elevation order which is critical to model terrain structure, the concept of isomorphism in set theory is drawn as the mathematical rationale and Kendall’s rank correlation efficient is used to quantify the accuracy. The new framework is illustrated using a DEM derived from LiDAR for sea-level rise vulnerability assessment of a tidal salt marsh. Compared to conventional methods based on statistics, the new framework and methods produce detailed mapping of error distribution thus enable the identification of the main sources of error and where improvement is needed most. Results call for re-evaluation of the current practice of assessing filtering accuracy in LiDAR data processing and further research on relative elevation and isomorphism.

Error modelling method of extended Kalman filter‐based terrain referenced navigation system for integrity assurance under nominal conditions

Navigation safety is further emphasised for terrain referenced navigation (TRN) systems used in middle-range unmanned aerial vehicles (UAVs) than those in cruise missiles flying through mountainous areas, because middle-range UAVs may fly to regions and cities containing both combatants and civilians. For safety assurance of TRN systems, this paper proposes a method to compute protection levels (PLs) to assure the required level of integrity under nominal conditions. For obtaining PLs of TRN systems, all the components of the TRN measurement errors should be identified and overbounded. This study models every error component: the radar altimeter error, the vertical error in the terrain elevation database, the lateral offset induced error, and the interpolation error. Based on the characteristics of TRN systems using terrain information for obtaining navigation solutions, this study developed new models for the lateral offset induced error and the interpolation error as a function of terrain roughness by examining the effect of terrain roughness on the error sources. The simulation results demonstrate that the latitude and longitude PLs obtained by applying the proposed models can sufficiently overbound the navigation solution error for TRN systems under nominal conditions; especially the proposed method shows remarkable performance in regions with low terrain roughness.

Interpolation of spectra from 3D model atmospheres

Context. The use of 3D hydrodynamical simulations of stellar surface convection for model atmospheres is computationally expensive. Although these models have been available for quite some time, their use is limited because of the lack of extensive grids of simulations and associated spectra. Aims. Our goal is to provide a method to interpolate spectra that can be applied to both 1D and 3D models, and implement it in a code available to the community. This tool will enable the routine use of 3D model atmospheres in the analysis of stellar spectra. Methods. We have developed a code that makes use of radial basis functions to interpolate the spectra included in the CIFIST grid of 84 three-dimensional model atmospheres. Spectral synthesis on the hydrodynamical simulations was previously performed with the code ASSϵT. Results. We make a tool for the interpolation of 3D spectra available to the community. The code provides interpolated spectra and interpolation errors for a given wavelength interval, and a combination of effective temperature, surface gravity, and metallicity. In addition, it optionally provides graphical representations of the RMS and mean ratio between 1D and 3D spectra, and maps of the errors in the interpolated spectra across the parameter space.

Rain Station Network Analysis in the Sampean Watershed: Comparison of Variations in Data Aggregation

The lack of rainfall-runoff accuracy is important for some applications. The choice of data aggregation that affects the estimation results is important at the level of accuracy. Some commonly used aggregations are daily, ten days, and monthly rainfall. This study aimed to compare the results of the estimation of the effect of data aggregation and to analyze the density of the rain gauge network in the Sampean watershed. The evaluation of the rain station network is carried out through the Kagan calculation. Rainfall data are from the rainfall data records for 20 years at 33 rain gauge stations. Measurement of the performance of aggregation variations using the relationship between the correlation value of rainfall with the distance between station locations. Station network positioning is assessed from alignment errors and interpolation errors. The results showed differences in the correlation and estimation values ​​in the variation of data aggregation.The greater interval can increase the effectiveness of deployment with minimum error. Based on Kagan's analysis, there is an uneven distribution of gauge stations in the Sampean watershed eventhough the average and interpolation error in the monthly rainfall is less than 5%. It is this inequality that causes gauge stations to be inefficient.
 Keywords : Rain gauge network; correlation; Kagan; data aggregation
 Copyright (c) 2022 Geosfera Indonesia and Department of Geography Education, University of Jember
 This work is licensed under a Creative Commons Attribution-Share A like 4.0 International License

Investigation of the notch depth effect on Charpy fracture energy and fracture surface features of API X65 steel

ABSTRACT Different factors such as notch angle, notch depth, notch root radius, specimen size, hammer geometry and striker velocity affect the value of Charpy impact energy. The aim of present study is to measure the effect of notch depths (in the range from 1.25 to 3 mm) on fracture energy and fracture surface features of API X65 steel plates using Charpy impact tests. For this purpose, 24 specimens were made with 8 different notch depths (3 specimens per notch depth), which were tested to obtain the fracture energy. Furthermore, a computer simulation of the experiment was performed with a three-dimensional model based on Gurson's modified damage law via Abaqus software. The fracture surface morphology was observed using the SEM microscope to evaluate the fracture mechanisms. The experimental results revealed that the relationship between the fracture energy (E) and the notch depth (a) of the Charpy samples for the tested steel was E = 499.65e−0.3501a . Using this equation, the Charpy energy can be determined for this steel for any notch depth with an interpolation error of 3.31%. Fractographic observations indicated that various fracture mechanisms were observed by changing the notch depth, including brittle fracture with the river-like patterns, ductile-brittle fracture, ductile fracture, and fully brittle fracture.

Multi-domain integrative Swin transformer network for sparse-view tomographic reconstruction

SummaryDecreasing projection views to a lower X-ray radiation dose usually leads to severe streak artifacts. To improve image quality from sparse-view data, a multi-domain integrative Swin transformer network (MIST-net) was developed and is reported in this article. First, MIST-net incorporated lavish domain features from data, residual data, image, and residual image using flexible network architectures, where a residual data and residual image sub-network was considered as a data consistency module to eliminate interpolation and reconstruction errors. Second, a trainable edge enhancement filter was incorporated to detect and protect image edges. Third, a high-quality reconstruction Swin transformer (i.e., Recformer) was designed to capture image global features. The experimental results on numerical and real cardiac clinical datasets with 48 views demonstrated that our proposed MIST-net provided better image quality with more small features and sharp edges than other competitors.

Local wavefield refinement using Fourier interpolation and boundary extrapolation for finite-element method based on domain reduction method

The domain reduction method based on the finite-element method (FEM) is promising for multiscale seismic wave simulations for local complex structures and rough topography. However, a transition region is required to refine the wavefields from coarse to fine grids, which may lead to strong artifacts. Traditional wavefield interpolation methods can only handle a small upsampling ratio due to their low accuracy. Here, we propose a high accuracy upsampling method using Fourier interpolation and boundary extrapolation. It requires that the wavefield in the transition region is evenly sampled, which may slightly reduce the flexibility of FEMs. Fortunately, this constraint only exists in two layers of nodes, in which the coarse and fine grids are collocated. To further apply our interpolation method to the wavefield upsampling on the nonuniform grid, we adopt cubic interpolation within the local region of interest, which is adequate in accuracy for regularizing the wavefield on the fine nonuniform grid. For a topographic surface, we perform wavefield extrapolation on the ground surface to avoid abnormal amplitude variations below and beyond the surface, which can greatly reduce potential artifacts caused by the Fourier interpolation. Numerical experiments find that our method is superior to traditional interpolation methods in accuracy, especially for large upsampling ratios. Its relative error is less than 2% even for a 40-time upsampling; in contrast, the relative error of linear and cubic interpolation is up to 90% and 41% in the same situation, respectively. We further verify the feasibility of our method for 3D heterogeneous models with rough topography, multilayer heterogeneity, and random perturbations of background models. The proposed method yields a significant speeding of the multiscale seismic simulation using the FEM for 3D models with complex local structures and topographies while being highly accurate.

Semi-endogenous Versus Schumpeterian Growth Models: A Critical Review of the Literature and New Evidence

Abstract Several studies have tested semi-endogenous versus Schumpeterian growth models using different methodological approaches. This paper critically reviews these studies including their approaches and provides new evidence on this issue, by analyzing both time-series data from the United States and panel data from 19 OECD countries over the period 1980–2014. The review finds much support for Schumpeterian growth theory, but shows that all studies reviewed have several limitations. These limitations include conceptual problems associated with the use of the number/stock of patents as a measure of the flow/stock of knowledge, the possibility of spurious regressions due to non-stationary data, potential mismeasurement of R&D inputs due to possible interpolation and deflation errors, misspecification problems that can arise in difference models when variables are cointegrated, and potential spurious rejections of the unit root hypothesis for R&D intensity when the lag length in unit root tests is too small. The present study avoids these limitations and finds strong evidence in favor of semi-endogenous growth.

Nearest neighbour distance matching Leave‐One‐Out Cross‐Validation for map validation

Abstract Several spatial and non‐spatial Cross‐Validation (CV) methods have been used to perform map validation when additional sampling for validation purposes is not possible, yet it is unclear in which situations one CV method might be preferred over the other. Three factors have been identified as determinants of the performance of CV methods for map validation: the prediction area (geographical interpolation vs. extrapolation), the sampling pattern and the landscape spatial autocorrelation. In this study, we propose a new CV strategy that takes the geographical prediction space into account, and test how the new method compares with other established CV methods under different configurations of these three factors. We propose a variation of Leave‐One‐Out (LOO) CV for map validation, called Nearest Neighbour Distance Matching (NNDM) LOO CV, in which the nearest neighbour distance distribution function between the test and training data during the CV process is matched to the nearest neighbour distance distribution function between the target prediction and training points. Using random forest as a machine learning algorithm, we then examine the suitability of NNDM LOO CV as well as the established LOO (non‐spatial) and buffered‐LOO (bLOO, spatial) CV methods in two simulations with varying prediction areas, landscape autocorrelation and sampling distributions. LOO CV provided good map accuracy estimates in landscapes with short autocorrelation ranges, or when estimating geographical interpolation map accuracy with randomly distributed samples. bLOO CV yielded realistic error estimates when estimating map accuracy in new prediction areas, but generally overestimated geographical interpolation errors. NNDM LOO CV returned reliable estimates in all scenarios we considered. While LOO and bLOO CV provided reliable map accuracy estimates only in certain situations, our newly proposed NNDM LOO CV method returned robust estimates and generalised to LOO and bLOO CV whenever these methods were the most appropriate approach. Our work recognises the necessity of considering the geographical prediction space when designing CV‐based methods for map validation.

On the rational approximation of Markov functions, with applications to the computation of Markov functions of Toeplitz matrices

We investigate the problem of approximating the matrix function f(A) by r(A), with f a Markov function, r a rational interpolant of f, and A a symmetric Toeplitz matrix. In a first step, we obtain a new upper bound for the relative interpolation error 1 − r/f on the spectral interval of A. By minimizing this upper bound over all interpolation points, we obtain a new, simple, and sharp a priori bound for the relative interpolation error. We then consider three different approaches of representing and computing the rational interpolant r. Theoretical and numerical evidence is given that any of these methods for a scalar argument allows to achieve high precision, even in the presence of finite precision arithmetic. We finally investigate the problem of efficiently evaluating r(A), where it turns out that the relative error for a matrix argument is only small if we use a partial fraction decomposition for r following Antoulas and Mayo. An important role is played by a new stopping criterion which ensures to automatically find the degree of r leading to a small error, even in presence of finite precision arithmetic.

A Fast Mean-Preserving Spline for Interpolating Interval Data

Abstract Interpolation of interval data where the mean is preserved, e.g., estimating smoothed, pseudodaily meteorological variables based on monthly means, is a common problem in the geosciences. Existing methods for mean-preserving interpolation are computationally intensive and/or do not readily accommodate bounded interpolation, where the interpolated data cannot exceed a threshold value. Here we present a mean-preserving, continuous, easily implementable, and computationally efficient method for interpolating one-dimensional interval data. Our new algorithm provides a straightforward solution to the interpolation problem by utilizing Hermite cubic splines and midinterval control points to interpolate interval data into smaller partitions. We further include adjustment schemes to restrict the interpolated result to user-specified minimum and maximum bounds. Our method is fast, portable, and broadly applicable to a range of geoscientific data, including interpolating unbounded time series such as mean temperature, and bounded data including mean wind speed or cloud-cover fraction. Significance Statement Interpolation is often utilized to mathematically estimate smaller time step values when such data are not readily available, for example, the estimation of daily temperature when only monthly temperature values are available. We propose a novel interpolation method based on linking segments of flexible continuous curves that ensures the average of interpolated result will be the same as the original value, which is important for minimizing interpolation errors. We find that our new method takes significantly less computational time when compared with other existing methods, while retaining a similar degree of precision. Furthermore, we outline an additional procedure for users to specify the minimum and maximum bounds of interpolated results if applicable.

Audio-lossless robust watermarking against desynchronization attacks

Audio-lossless robust watermarking (ALRW) is a new research issue in the field of information hiding, which can resist those content-preserving manipulations while can restore the cover audio files completely in case of no attacks. Most ALRW countermeasures proposed in the literature usually focus on additive noise-like manipulations such as MP3 compression, low-pass filtering and Gaussian additive noise, but few are resistant to challenging desynchronization attacks such as random cropping and time-scale modification (TSM). The main reason is that the exploited robust features in the existing ALRW algorithms are related to the sample position. In this paper, we propose an ALRW method by efficiently embedding a watermark into the histogram shape of the low-frequency component after the discrete wavelet transform (DWT) and reversibly hiding the distortion due to the robust watermark embedding as the compensation information for restoration of the cover audio file. The use of the histogram shape in the DWT low-frequency component as the exploited robust feature is: 1) mathematically invariant to TSM; 2) robust to random cropping attacks since an audio clip before and after the cropping often has similar distribution, and 3) resistant to interpolation errors during time scaling and those common audio processing operations. To reduce the compensation information, the robust watermarking process is elaborately and luminously designed to represent the difference between the original and the robust watermarked audio. As a result, an ALRW system against desynchronization attacks is achieved with satisfactory performance. Experimental results show that the proposed ALRW scheme can effectively reduce the compensation information and provide strong robustness to those common manipulations including TSM, random cropping, MP3 lossy compression and other noise-like manipulations. In case of no attacks, the cover audio signal can be recovered without any loss.

Motion corrected fetal body magnetic resonance imaging provides reliable 3D lung volumes in normal and abnormal fetuses.

ObjectivesTo calculate 3D‐segmented total lung volume (TLV) in fetuses with thoracic anomalies using deformable slice‐to‐volume registration (DSVR) with comparison to 2D‐manual segmentation. To establish a normogram of TLV calculated by DSVR in healthy control fetuses.MethodsA pilot study at a single regional fetal medicine referral centre included 16 magnetic resonance imaging (MRI) datasets of fetuses (22–32 weeks gestational age). Diagnosis was CDH (n = 6), CPAM (n = 2), and healthy controls (n = 8). Deformable slice‐to‐volume registration was used for reconstruction of 3D isotropic (0.85 mm) volumes of the fetal body followed by semi‐automated lung segmentation. 3D TLV were compared to traditional 2D‐based volumetry. Abnormal cases referenced to a normogram produced from 100 normal fetuses whose TLV was calculated by DSVR only.ResultsDeformable slice‐to‐volume registration‐derived TLV values have high correlation with the 2D‐based measurements but with a consistently lower volume; bias −1.44 cm3 [95% limits: −2.6 to −0.3] with improved resolution to exclude hilar structures even in cases of motion corruption or very low lung volumes.ConclusionsDeformable slice‐to‐volume registration for fetal lung MRI aids analysis of motion corrupted scans and does not suffer from the interpolation error inherent to 2D‐segmentation. It increases information content of acquired data in terms of visualising organs in 3D space and quantification of volumes, which may improve counselling and surgical planning.

Importance of Weather Conditions in a Flight Corridor

Current research initiatives, such as the Single European Sky Air Traffic Management Research Program, call for an air traffic system with improved safety and efficiency records and environmental compatibility. The resulting multi-criteria system optimization and individual flight trajectories require, in particular, reliable three-dimensional meteorological information. The Global (Weather) Forecast System only provides data at a resolution of around 100 km. We postulate a reliable interpolation at high resolution to compute these trajectories accurately and in due time to comply with operational requirements. We investigate different interpolation methods for aerodynamic crucial weather variables such as temperature, wind speed, and wind direction. These methods, including Ordinary Kriging, the radial basis function method, neural networks, and decision trees, are compared concerning cross-validation interpolation errors. We show that using the interpolated data in a flight performance model emphasizes the effect of weather data accuracy on trajectory optimization. Considering a trajectory from Prague to Tunis, a Monte Carlo simulation is applied to examine the effect of errors on input (GFS data) and output (i.e., Ordinary Kriging) on the optimized trajectory.

A revisit to smoothness preserving fractal perturbation of a bivariate function: Self-Referential counterpart to bicubic splines

Construction of fractal interpolation surfaces has recently been considered in the standpoint of a parameterized class of fractal (self-referential) functions corresponding to a given bivariate continuous function. In this paper, we describe a procedure so that the elements in this parameterized class preserve smoothness (C(2,2)-regularity) of the original bivariate function defined on a rectangle. As a consequence, we generalize the bicubic spline by means of a two-parameter family of fractal functions, which we call bicubic fractal splines. Under certain hypotheses, upper bounds for the interpolation error for the bicubic fractal spline and its derivatives are obtained. A detailed exposition of C(2,2)-regular self-referential functions is provided not only as a prelude to the bicubic fractal splines, but also to elucidate the study of smoothness preserving bivariate self-referential functions appeared recently in the fractal literature.

A lowest-order free-stabilization Virtual Element Method for the Laplacian eigenvalue problem

In this paper, we propose a Virtual Element Method (VEM) for the Laplacian eigenvalue problem, which is designed to avoid the requirement of the stabilization terms in standard VEM bilinear forms. In the present method, the constructions of the bilinear forms depend on higher order polynomial projection. To exactly compute the bilinear forms, we need to modify the virtual element space associated to the higher order polynomial projection. Meanwhile, the continuity and coercivity of the discrete VEM bilinear forms depend on the number of vertices of the polygon. By the spectral approximation theory of compact operator and the projection and interpolation error estimates, we prove correct spectral approximation and error estimates for the VEM discrete scheme. Finally, we show numerical examples to verify the theoretical results, including the Laplace eigenvalue problem and the Steklov eigenvalue problem.

Quadratic serendipity element shape functions on general planar polygons

This paper proposes a method for the construction of quadratic serendipity element (QSE) shape functions on planar convex and concave polygons. Existing approaches for constructing QSE shape functions are linear combinations of the pair-wise products of generalized barycentric coordinates with linear precision, restricted to the convex polygonal domain or resort to numerical optimization. We extend the construction to general polygons with no more than three collinear consecutive vertices. This is done by defining coefficients of the linear combination as the oriented area of triangles with vertices from the polygonal domain, which can be either convex or concave. The proposed shape functions possess linear to quadratic precision. We prove the interpolation error estimates for mean value coordinate-based QSE shape functions on convex and concave polygonal domains satisfying a set of geometric constraints for standard finite element analysis. We also tailor a polygonal mesh generation scheme that improves the uniformity and avoids short edges of Voronoi diagrams for their use in the QSE-based polygonal finite element computation. Numerical tests for the 2D Poisson equations on various domains are presented, demonstrating the optimal convergence rates in both the L2-norm and the H1-seminorm.

Five-Axis Trajectory Generation Considering Synchronization and Nonlinear Interpolation Errors

Abstract This paper presents a novel real-time interpolation technique for five-axis machine tools to attain higher speed and accuracy. To realize computationally efficient real-time interpolation of 6-DOF tool motion, a joint workpiece–machine coordinate system interpolation scheme is proposed. Cartesian motion of the tool center point (TCP) is interpolated in the workpiece coordinate system (WCS), whereas tool orientation is interpolated in the machine coordinate system (MCS) based on the finite impulse response filtering. Such an approach provides several advantages: (i) it eliminates the need for complex real-time spherical interpolation techniques, (ii) facilitates efficient use of slower rotary drive kinematics to compensate for the dynamic mismatch between Cartesian and rotary axes and achieve higher tool acceleration, and (iii) mitigates feed fluctuations while interpolating near kinematic singularities. To take advantage of such benefits and realize accurate joint WCS–MCS interpolation scheme, tool orientation interpolation errors are analyzed. A novel approach is developed to adaptively discretize long linear tool moves and confine interpolation errors within user set tolerances. Synchronization errors between TCP and tool orientation are also characterized, and peak synchronization error level is determined to guide the interpolation parameter selection. Finally, blending errors during non-stop continuous interpolation of linear toolpaths are modeled and confined. Advantages of the proposed interpolation scheme are demonstrated through simulation studies and validated experimentally. Overall, proposed technique can improve cycle times up to 10% while providing smooth and accurate non-stop real-time interpolation of tool motion in five-axis machining.