Radial Basis Function Interpolation Research Articles

A non-intrusive reduced-order model for finite element analysis of implant positioning in total hip replacements.

Sophisticated high-fidelity simulations can predict bone mass density (BMD) changes around a hip implant after implantation. However, these models currently have high computational demands, rendering them impractical for clinical settings. Model order reduction techniques offer a remedy by enabling fast evaluations. In this work, a non-intrusive reduced-order model, combining proper orthogonal decomposition with radial basis function interpolation (POD-RBF), is established to predict BMD distributions for varying implant positions. A parameterised finite element mesh is morphed using Laplace's equation, which eliminates tedious remeshing and projection of the BMD results on a common mesh in the offline stage. In the online stage, the surrogate model can predict BMD distributions for new implant positions and the results are visualised on the parameterised reference mesh. The computational time for evaluating the final BMD distribution around a new implant position is reduced from minutes to milliseconds by the surrogate model compared to the high-fidelity model. The snapshot data, the surrogate model parameters and the accuracy of the surrogate model are analysed. The presented non-intrusive surrogate model paves the way for on-the-fly evaluations in clinical practice, offering a promising tool for planning and monitoring of total hip replacements.

Study on microcosmic properties and temperature simulation of foamed polypropylene composites.

The injection molding procedure and the compression molding procedure of foamed polypropylene composite are studied. The pore diameter, pore density, and microscopic topography for polypropylene composites was studied by experiments of slicing samples. The foamed temperature is very important for foamed property of the foamed polypropylene composite materials. To analyze the effect of temperature distribution, the virtual boundary meshfree Galerkin method (VBMGM) was employed the radial basis function interpolation method to obtain a detailed discretization formula. The temperature of experiment and numerical calculation from zone a to zone d in different zones of the injection molding procedure is reduced from 406.35 K to 311.63 K; the temperature from Ma zone to Md zone in the compression molding procedure is reduced from 326.35 K to 309.14 K. The experimental observation of the injection molding procedure shows that the foamed property in zone c is ideal, and the average pore diameter and pore density are 26.5 µm and 2.43×109 cells·cm-3 respectively. According to the experimental observation of compression molding process, the foamed property in Md zone is ideal, and the average pore diameter and pore density are 131.2 µm and 6.3×104 cells·cm-3. These results are of great significance to the foam molding of polypropylene composites.

A Novel Interpolation Method for Soil Parameters Combining RBF Neural Network and IDW in the Pearl River Delta

Soil fertility is a critical factor in agricultural production, directly impacting crop growth, yield, and quality. To achieve precise agricultural management, accurate spatial interpolation of soil parameters is essential. This study developed a new interpolation prediction framework that combines Radial Basis Function (RBF) neural networks with Inverse Distance Weighting (IDW), termed the IDW-RBFNN. This framework initially uses the IDW method to apply preliminary weights based on distance to the data points, which are then used as input for the RBF neural network to form a training dataset. Subsequently, the RBF neural network further trains on these data to refine the interpolation results, achieving more precise spatial data interpolation. We compared the interpolation prediction accuracy of the IDW-RBFNN framework with ordinary Kriging (OK) and RBF methods under three different parameter settings. Ultimately, the IDW-RBFNN demonstrated lower error rates in terms of RMSE and MRE compared to direct RBF interpolation methods when adjusting settings based on different power values, even with a fixed number of data samples. As the sample size decreases, the interpolation accuracy of OK and RBF methods is significantly affected, while the error of IDW-RBFNN remains relatively low. Considering both interpolation accuracy and resource limitations, we recommend using the IDW-RBFNN method (p = 2) with at least 60 samples as the minimum sampling density to ensure high interpolation accuracy under resource constraints. Our method overcomes limitations of existing approaches that use fixed steady-state distance decay parameters, providing an effective tool for soil fertility monitoring in delta regions.

Full‐tensor magnetic gradiometry: Comparison with scalar total magnetic intensity, processing and visualization guidelines

AbstractFull‐tensor magnetic gradiometry data have been collected commercially for the last few years. However, to date, there is still no clarity on how to compare these data to scalar total field surveys. Some users display the vertical gradient of the vertical component (Bzz) and compare that to a first vertical derivative of total field with the caveat that ‘they are similar’. Others compute the length of the measured vector and call that total field. We establish the basic formulas to calculate total field from the tensor components and demonstrate this with a real data example from Thompson, Manitoba, Canada. Another key question is whether full‐tensor interpolation is required to obtain total field from tensor data. We compare the results from using a commercial full‐tensor interpolation algorithm with standard minimum curvature of the tensor components individually and with another open‐source code that uses a radial basis function interpolator on the individual tensor components. All three applications produced a total field grid of superior quality to that calculated from a scalar total field survey available for the area of study.

Applied machine learning: Performance prediction of heat pipe with mesh wick

This study presents an investigation of a heat pipe with a mesh wick, utilizing machine learning (ML) techniques. The model including radial basis function interpolation (RBF), Kriging model (KRG), and the k-nearest neighborhood model (K-NN) were studied and compared. A set of training and validating populations were classified using a k-means clustering technique. The design variable included the geometric shape of heat pipe such as its diameter, the properties and percentage used of working fluid, and the temperature at the evaporator. The prediction case study included the heat transfer rate (q), and total difference temperature between evaporator and condenser section (ΔT). The prediction results found that the ΔT gave the most accurate indicator while the q is passable to applied. The Kriging model proved to be the most accurate, achieving an RMSE of 0.9896 and R2 of 0.9149 for heat transfer rate prediction, and an RMSE of 0.1902 and R2 of 0.9398 for total temperature difference prediction with 90 % training data. The second-best accuracy was achieved by the RBF model, with the linear, thin plate, and cubic spline kernels performing reasonably well.

Multi-body mesh deformation using a multi-level localized dual-restricted radial basis function interpolation

Multi-body dynamic problems with moving boundaries are prevalent in the computational fluid dynamics. In these problems, the original mesh is unsuitable for subsequent solution steps and is needed to be deformed. The radial basis function (RBF) interpolation is a common algorithm for the mesh deformation task. However, the multi-body mesh may contain numerous individuals and volume nodes, which will harm the computational efficiency. In order to optimize this process, we propose a new method for the mesh deformation of the multi-body configuration. In this new approach, each individual is deformed separately. Thus the global problem is decomposed into a series of local mesh deformation problems. As the computational cost to construct the interpolation system in RBF algorithm is proportional to the cube of the number of support points on the surface, converting it into multiple local mesh deformation problems can effectively reduce the CPU cost. To treat each local problem, we employ a dual-restricted RBF interpolation technique which could avoid the influence of moving individual on the other individuals. This new localized approach effectively improves the computational efficiency to construct the interpolation system but sometimes will increase the CPU cost of the mesh updating procedure. To avoid this drawback, the existed multi-level restricted RBF strategy is coupled with the new localized method to further reduce the CPU cost to update the mesh. The combination of the two techniques could enhance their advantages and avoid their drawbacks. Some numerical examples have demonstrated the abilities of the new algorithm. For instance, in the case of the three-dimensional birds flock, the CPU time to construct the interpolation system and deform the volume mesh was respectively reduced by three and two orders of magnitude compared to the global single-level method.

A simple, effective and high-precision boundary meshfree method for solving 2D anisotropic heat conduction problems with complex boundaries

A simple, effective and high-precision boundary meshfree method called virtual boundary meshfree Galerkin method (VBMGM) is used to tackle 2D anisotropic heat conduction problems with complex boundaries. Temperature and heat flux are expressed by virtual boundary element method. The virtual source function is constructed through the utilization of radial basis function interpolation. Calculation model diagram and discrete model diagram of real boundaries, and schematic diagram of VBMGM are demonstrated. Using Galekin method and considering boundary conditions, the integral equation and the discrete formula of VBMGM are given in detail. The benefits of the Galerkin, meshfree, and boundary element methods are all presented in VBMGM. Seven numerical examples of general anisotropic heat conduction problems (including three numerical examples with complex boundaries and four numerical examples with mixed boundary conditions) are computed and contrasted with precise solutions and different numerical methods. The computation time of each example is given. The number of degrees of freedom used in many examples is half or less than that of the numerical method being compared. The suggested method has been demonstrated to be effective and high-precision for solving the 2D anisotropic heat conduction problems with complex boundaries.

A GPU-accelerated Lagrangian method for solving the Liouville equation in random differential equation systems

This work presents and analyzes a numerical approach to efficiently solve the Liouville equation in the context of random ODEs using GPGPUs. Our method combines wavelet compression-based adaptive mesh refinement, Lagrangian particle methods, and radial basis function interpolation to create a versatile algorithm applicable in multiple dimensions. We discuss the advantages and limitations of this algorithm. To demonstrate its effectiveness, we compute the probability density function for various 2D and 3D random ODE systems with applications in physics and epidemiology.

Geodesic metrics for RBF approximation of some physical quantities measured on sphere

The radial basis function (RBF) approximation is a rapidly developing field of mathematics. In the paper, we are concerned with the measurement of scalar physical quantities at nodes on sphere in the 3D Euclidean space and the spherical RBF interpolation of the data acquired. We employ a multiquadric as the radial basis function and the corresponding trend is a polynomial of degree 2 considered in Cartesian coordinates. Attention is paid to geodesic metrics that define the distance of two points on a sphere. The choice of a particular geodesic metric function is an important part of the construction of interpolation formula.We show the existence of an interpolation formula of the type considered. The approximation formulas of this type can be useful in the interpretation of measurements of various physical quantities. We present an example concerned with the sampling of anisotropy of magnetic susceptibility having extensive applications in geosciences and demonstrate the advantages and drawbacks of the formulas chosen, in particular the strong dependence of interpolation results on condition number of the matrix of the system considered and on round-off errors in general.

Optimising shapes of multiple pin fins in a microchannel using deep reinforcement learning and mesh deformation techniques

The utilisation of advanced pin fin designs in microchannels is useful for enhancing cooling efficiency. Advancements in machine learning and processing power have sparked interest in shape optimisation techniques. This research employs a novel framework that integrates Deep Artificial Neural Networks and Reinforcement Learning with a Computational Fluid Dynamics (CFD) solver to optimise multiple pin fin shapes within a microchannel. By incorporating Radial Basis Function interpolation and Proximal Policy Optimisation alongside FLUENT, acting as the CFD environment, the reinforcement learning agent adeptly explores the design space to enhance the thermohydraulic performance factor (TPF), aiming to maximise Nusselt number while minimising pressure loss. Unlike previous heat transfer optimisation studies, which typically required mesh regeneration at each step, the proposed framework could bypass the meshing step and alter the geometry directly by relying on the RBF interpolation technique to deform the mesh directly. Three distinct scenarios investigated in this study are uniform deformation of all pin fins, deformation of all pin fins arranged in two rows, and individual deformation of each pin fin. Extensive simulations, exceeding 90,000 different cases, demonstrate that although the optimisation process for the individual deformation of each pin fin requires more iterations compared to others, it surpasses them in terms of TPF improvement. Notably, significant improvements are achieved, such as a 49 % enhancement in Nusselt number and a 33 % reduction in pressure drop, culminating in an impressive 63 % increase in TPF compared to the initial geometry.

A comprehensive MRI-based computational model of blood flow in compliant aorta using radial basis function interpolation

BackgroundProperly understanding the origin and progression of the thoracic aortic aneurysm (TAA) can help prevent its growth and rupture. For a better understanding of this pathogenesis, the aortic blood flow has to be studied and interpreted in great detail. We can obtain detailed aortic blood flow information using magnetic resonance imaging (MRI) based computational fluid dynamics (CFD) with a prescribed motion of the aortic wall.MethodsWe performed two different types of simulations—static (rigid wall) and dynamic (moving wall) for healthy control and a patient with a TAA. For the latter, we have developed a novel morphing approach based on the radial basis function (RBF) interpolation of the segmented 4D-flow MRI geometries at different time instants. Additionally, we have applied reconstructed 4D-flow MRI velocity profiles at the inlet with an automatic registration protocol.ResultsThe simulated RBF-based movement of the aorta matched well with the original 4D-flow MRI geometries. The wall movement was most dominant in the ascending aorta, accompanied by the highest variation of the blood flow patterns. The resulting data indicated significant differences between the dynamic and static simulations, with a relative difference for the patient of 7.47±14.18% in time-averaged wall shear stress and 15.97±43.32% in the oscillatory shear index (for the whole domain).ConclusionsIn conclusion, the RBF-based morphing approach proved to be numerically accurate and computationally efficient in capturing complex kinematics of the aorta, as validated by 4D-flow MRI. We recommend this approach for future use in MRI-based CFD simulations in broad population studies. Performing these would bring a better understanding of the onset and growth of TAA.

Enhancing non-intrusive reduced-order models with space-dependent aggregation methods

AbstractIn this manuscript, we combine non-intrusive reduced-order models (ROMs) with space-dependent aggregation techniques to build a mixed-ROM, able to accurately capture the flow dynamics in different physical settings. The flow prediction obtained using the mixed formulation is derived from a convex combination of the predictions of several previously trained reduced-order models (ROMs), with each model assigned a space-dependent weight. The ROMs incorporated in the mixed model utilize different reduction methods, such as proper orthogonal decomposition and autoencoders, and various approximation techniques, including radial basis function interpolation (RBF), Gaussian process regression, and feed-forward artificial neural networks. Each model’s contribution is given higher weights in regions where it performs best and lower weights where its accuracy is lower compared to the other models. Additionally, a random forest regression technique is used to determine the weights for previously unseen conditions. The performance of the aggregated model is assessed through two test cases: the 2D flow past a NACA 4412 airfoil at a 5-degree angle of attack, with the Reynolds number ranging between $$1 \times 10^{5}$$ 1 × 10 5 and $$1 \times 10^{6}$$ 1 × 10 6 , and a transonic flow over a NACA 0012 airfoil, with the angle of attack as the varying parameter. In both scenarios, the mixed-ROM demonstrated improved accuracy compared to each individual ROM technique, while providing an estimate for the predictive uncertainty.

Accurately identifying the defects of bubbles and foreign objects under the protective films of electric vehicle batteries by using 3D point clouds

Abstract For the defects of bubbles and foreign objects under the protective film of electric vehicle batteries, it is difficult to accurately identify them over traditional 2D optical images. In this paper, we first propose a supervoxel-based region growing algorithm for pre-segmentation of point clouds. Secondly, we utilize radial basis function interpolation and threshold segmentation methods to accurately segment defect point clouds from the entire point cloud. Finally, we develop a feature descriptor and combine it with support vector machine to classify bubbles and foreign objects under the film. This paper achieves the identification of bubbles and foreign objects under the film through two steps: point cloud segmentation and point cloud classification. Experimental results demonstrate that the proposed point cloud segmentation method exhibits high robustness to noise and the intrinsic curvature of the workpiece. Additionally, in the classification scenario presented in this paper, the proposed feature descriptor outperforms classical feature descriptors. Compared to image-based deep learning methods, the defect recognition algorithm proposed in this paper has clear principles and superior performance, with precision and recall of 95.63% and 96.95%, and an intersection over union metric of 0.926.

3D spatial distribution of soil pollutants based on geo-shadowing anisotropic RBF-PCA

Research on soil contamination has become increasingly important, but there is limited information about where to sample for pollutants. Thus, the use of three-dimensional (3D) spatial interpolation techniques has been promoted in this area of study. However, the application of traditional interpolation methods is limited in geography, especially in the expression of anisotropy, and it is not associated with geographical properties. To address this issue, we used a test site (a factory in Nanjing) to develop a new research method based on the geographical shading radial basis function (RBF) interpolation method, which considers 3D anisotropy and geographical attribute expression. Drilling and uniform sampling were used to sample the contaminated area at this test site. This approach included two steps: i) An ellipsoid with anisotropic properties was constructed. Thus, the first step was to determine the shape of the ellipsoid using principal component analysis (PCA) to determine the main orientations and construct a rotational and stretched matrix. The second step was determining the ellipsoid size by computing the range using the variogram method for orientations. ii) During field measurement, the geospatial direction influences soil attribute values, so a shadowing calculation method was derived for quadratic weight determination. Then, the weight of the attribute value of known points can be assigned to meet the field conditions. Lastly, the model was evaluated using the root mean square error (RMSE). For the 2D space, the RMSE values of Kriging, RBF, and the proposed method are 6.09, 7.12, and 5.02, respectively. The R2 values of Kriging, RBF, and the proposed method are 0.871, 0.832, and 0.946, respectively. For the 3D space, the RMSE values of Kriging, RBF, and the proposed method are 2.65, 2.23, and 2.58, respectively. The R2 values of Kriging, RBF, and the proposed method are 0.934, 0.912, and 0.953, respectively. The resulting fitted model was relatively smooth and met experimental needs. Thus, we believe that the interpolation method can be applied as a new method to predict the distribution of soil pollutants.

Aerodynamic and Structural Assessment of Floating Wind Turbine Rotor Under Varying Tilt Angle

Predicting the aerodynamic performance of floating offshore wind turbines (FOWTs) proves challenging due to platform motion induced by waves. The effect of wind and waves results in a six-degree-of-freedom motion of the platform, directly influencing turbine performance. Understanding the impact of specific degrees of freedom (DOF) motions on aerodynamics and structural response is crucial for effective wind turbine design. This research examines the impact of rotor tilt on both aerodynamic performance and structural response. The investigation employs computational fluid dynamics (CFD) analysis and mapping aerodynamic loads onto the finite element (FE) mesh for structural analysis. The study employs a comprehensive 3D simulation, utilizing the moving reference frame (MRF) method for the NREL 5 MW reference wind turbine CFD simulations. It explores different rotor tilt angles (5°, 10°, 15°, and 20°) encountered by offshore structures during their operation and examines their impact on aerodynamic performance. Predicted aerodynamic loads were mapped onto the blade FE mesh using the radial basis function (RBF) interpolation technique and solved using the open-source FE solver CalculiX. The analysis shows that the turbine performance is relatively unaffected up to a tilt angle of 10°. However, further increase in rotor tilt angle adversely impacts turbine performance, leading to notable reductions in thrust and power output. The fluid-structure coupled analysis provided insights into the deformations and stresses experienced by the turbine blade, indicating a notable increase in flap-wise displacement for larger tilt angles, while edge-wise displacement is not as significantly affected. The maximum stress location on the blade generally correlates well with actual observations.

Enhancing OpenEP: atrial conduction velocity and conduction velocity heterogeneity quantification through EP Workbench

Abstract Background Alterations in atrial conduction velocity have been implicated in the pathogenesis of atrial arrhythmias, with conduction velocity thought to be slower and more heterogeneous in areas promoting atrial fibrillation. However, clinical studies of conduction velocity are challenging since there are no available platforms that provide researchers and clinicians with the ability to quantify conduction velocity and analyse conduction heterogeneity from electroanatomic mapping data. Purpose In this paper we sought to address these limitations by enhancing the OpenEP Python library by (1) automating calculations using previously-published conduction velocity estimation algorithms; (2) providing an interactive graphical representing of conduction velocity and conduction heterogeneity and (3) implementing a histogram analysis tool to quantify conduction velocity and enable the identification of slow conduction regions. Method Two well-known and previously published methods for calculating cardiac conduction velocity, Triangulation and polynomial surface fitting, were implemented. Conduction velocity estimation was improved further by excluding regions with wave collisions and focal discharges. These regions were identified using the divergence of conduction velocity vector fields, with positive divergence representing focal sources and negative divergence representing areas of collision. Finally, calculated conduction velocities using electrogram data were interpolated over the surface mesh via Radial Basis Function (RBF) interpolation method. All algorithms were implemented in the OpenEP Python library (openep-py), with visualisation tools provided in the complementary graphical interface, EP Workbench. Results An extensible "Analysis" feature was created in openep-py, to provide conduction velocity and divergence calculations which are fully customisable through the graphical interface based on user preferences. The EP Workbench desktop application displays resulting cardiac surface maps to depict calculated conduction velocity and divergence fields, together with conventional activation and voltage maps (Figure 1). The histogram analysis tool can be used to identify regions of slow conduction velocity from electroanatomic mapping data. Using a threshold of <0.3m/s, two slow conducting channels are visualised on the posterior wall of a test case (Figure 2). The tool is not specific to conduction velocity and may also be used to quantify other data types in EP Workbench. Finally, computed conduction velocity and divergence values and surface maps are stored in the standard OpenEP data structure, allowing further analysis following data export. Conclusion We have extended OpenEP to provide a comprehensive set of tools for conduction velocity calculation and analysis which will facilitate investigation of the structural and functional basis of conduction velocity alterations in disease states.

Quasi-Interpolation on Chebyshev Grids with Boundary Corrections

Quasi-interpolation is a powerful tool for approximating functions using radial basis functions (RBFs) such as Gaussian kernels. This avoids solving large systems of equations as in RBF interpolation. However, quasi-interpolation with Gaussian kernels on compact intervals can have significant errors near the boundaries. This paper proposes a quasi-interpolation method with Gaussian kernels using Chebyshev points and boundary corrections to improve the approximation near the boundaries. The boundary corrections use a linear approximation of the function beyond the interval to estimate the truncation error and add correction terms. Numerical studies on test functions show that the proposed method reduces errors significantly near boundaries compared to quasi-interpolation without corrections, for both equally spaced and Chebyshev points. The convergence and accuracy with the boundary corrections are generally better with Chebyshev points compared to equally spaced points. The proposed method provides an efficient way to perform quasi-interpolation on compact intervals while controlling the boundary errors. This study introduces a novel approach to quasi-interpolation modification, which significantly enhances convergence rates and minimizes errors at boundary points, thereby advancing the methods for boundary approximation.

Investigation of radial basis function dynamic mesh method with rotation correction based on adaptive background mesh

Strategies to improve the quality, robustness, and efficiency of dynamic mesh deformation are investigated. Radial basis function (RBF) interpolation is used for the background mesh, and volume-weighted interpolation is utilized for the computational mesh. An adaptive background mesh with an octree structure is automatically generated. It is refined according to the resolution near the boundaries and the wall's interior is also covered by mesh which is of sufficient resolution and prevents a negative volume due to large deformations of complex configurations. Secondly, the RBF method based on adaptive background mesh, the RBF interpolation angle method based on adaptive background mesh, and the damping function-RBF interpolation angle method based on adaptive background mesh have been developed. This method is applicable to complex shapes with multiple gaps, ensuring a high-quality mesh and high computational efficiency for large deformations. The numerical validation of the interpolation rotation angles for different RBFs indicates that the CTPS C0 function is suitable for large gradients in the boundary deformation. The RBF interpolation angle method ensures a smoother deformation for rigid and large deformations. The damping function-RBF interpolation angle method ensures mesh orthogonality near the boundary during elastic deformation, improving the boundary layer's quality and deformation robustness. Several examples show that the proposed method improves the mesh quality by about 30% and increases the time consumption by about 10%. In addition, the methods are applicable to rigid/elastic deformation of objects with multiple boundaries.

A new contact and road model for multi-body dynamic simulation of wheeled vehicles on soft-soil terrain

AbstractIn this paper, a new high-performance and memory-efficient contact and road model is developed. Specifically, the road is modeled as a rectangular structured grid of deformable springs in the vertical direction, thus enabling fast execution. The new road model stands out due to its ability to handle large road scenarios by allocating computer memory dynamically for each spring, resulting in efficient memory utilization. Furthermore, each spring represents a small road patch that entails various information, such as the soil elevation, the soil properties, and the soil compaction, allowing for complicated simulations incorporating spatially varying soil properties and phenomena related to the multi-pass effect. In addition, using the new contact model, complex terrain geometries are handled in a computationally efficient way by approximating locally the irregular road profile with a suitable equivalent plane. For this, two different strategies are proposed, namely the radial basis function (RBF) interpolation method and the 3D enveloping contact model. Finally, the proposed techniques are implemented in Altair MotionSolve, a comprehensive multi-body simulation software for complex mechanical systems. In particular, a single-wheel test bed is initially examined followed by a four-wheeled rover model and the next-generation NATO reference mobility model (NG-NRMM). In all cases, the proposed model is validated by using available experimental data. Lastly, a case involving both wheeled and tracked vehicles is also examined by using a shared road model.

Extending error bounds for radial basis function interpolation to measuring the error in higher order Sobolev norms

Radial basis functions (RBFs) are prominent examples for reproducing kernels with associated reproducing kernel Hilbert spaces (RKHSs). The convergence theory for the kernel-based interpolation in that space is well understood and optimal rates for the whole RKHS are often known. Schaback added the doubling trick [Math. Comp. 68 (1999), pp. 201–216], which shows that functions having double the smoothness required by the RKHS (along with specific, albeit complicated boundary behavior) can be approximated with higher convergence rates than the optimal rates for the whole space. Other advances allowed interpolation of target functions which are less smooth, and different norms which measure interpolation error. The current state of the art of error analysis for RBF interpolation treats target functions having smoothness up to twice that of the native space, but error measured in norms which are weaker than that required for membership in the RKHS. Motivated by the fact that the kernels and the approximants they generate are smoother than required by the native space, this article extends the doubling trick to error which measures higher smoothness. This extension holds for a family of kernels satisfying easily checked hypotheses which we describe in this article, and includes many prominent RBFs. In the course of the proof, new convergence rates are obtained for the abstract operator considered by Devore and Ron in [Trans. Amer. Math. Soc. 362 (2010), pp. 6205–6229], and new Bernstein estimates are obtained relating high order smoothness norms to the native space norm.