Radial Basis Function Interpolation Research Articles

A machine learning based approach with an augmented dataset for fatigue life prediction of additively manufactured Ti-6Al-4V samples

The article deals with the prediction of fatigue life using a machine learning (ML) approach. The original dataset is based on the parameters of defects obtained by micro-computed tomography (µ-CT) prior to fatigue tests, stress level and the fatigue life of additively manufactured (AM) Ti-6Al-4V samples. As the original dataset is considered too small to train a comprehensive ML model, the study proposed a novel approach for dataset augmentation. Dataset augmentation is done using inverse transform sampling and multivariate radial basis function (RBF) interpolation with various values of the smoothing parameter (λ). Finally, ML model accuracy is improved up to 0.953 of coefficient of determination (R2).

Consistent C-bar radial-basis Galerkin method for finite strain analysis

For finite-strain problems of quasi-incompressible materials, with hyperelastic and hyperelasto-plastic behavior, we develop a mixed polynomial-enriched radial-basis function (RBF) interpolation. It is characterized by decoupling quadrature and polynomials for the deviatoric and volumetric terms of the right Cauchy–Green tensor. With an enriched interpolation, first and second derivatives of the deformation gradient are continuous. A finite-element mesh is employed for quadrature purposes with the corresponding distribution of Gauss points. In terms of discretization, linear, quadratic and cubic polynomials are combined and function support is established from the number of pre-assigned nodes. Due to the adoption of a Lagrangian kernel, finite strain elasto-plastic constitutive developments are based on the Mandel stress. Combined with the C¯ technique, these developments are found to be convenient from an implementation perspective, as RBF formulations for finite strain plasticity have been limited in terms of generality and amplitude of deformations in the quasi-incompressibility case. Three benchmark tests are successfully solved, and it was found that a combination of reduced integration with a deviatoric cubic basis and volumetric quadratic basis provides superior results in the quasi-incompressible case. Numerical testing shows very good convergence behavior of the results. Besides absence of locking with selective interpolation, outstanding Newton convergence was observed regardless of strain levels.

Quantification of the contribution ratio of relevant input parameters on DEM-based granular flow simulations

Granular flow is affected by multiple parameters, which makes its study challenging. The discrete element method (DEM) is widely employed to simulate granular flow in consideration of particle motion, particularly when the effects of related parameters and particle shape on flow characteristics are being studied. In this study, different combinations of four input parameters (spring coefficient, friction angle between elements, coefficient of restitution, and bottom friction) were first obtained with the help of the Latin hypercube sampling method. Then, a series of simulations were performed using DEM with different sets of input parameters in consideration of different particle shapes and contact models. Radial basis function (RBF) interpolation was then employed to construct a response surface of run-out distance. Monte Carlo simulations were also conducted to obtain the contribution ratio of each input parameter. The result revealed that the bottom friction has a significant influence on the run-out distance, while friction angle between elements and spring coefficient account for a small proportion in the contribution ratio. Moreover, it was confirmed that the coefficient of restitution has a considerable contribution ratio in the front part of elements. The results also revealed that the influence of the particle shape and contact model on the contribution ratio was not as important in comparison.

A fast POD prediction method for hydrogen leakage at different pressures

The complexity of hydrogen leakage at different pressures leads to the challenge of developing a prediction method for both low pressure and supercritical pressure hydrogen leakage. In this paper, a reduced-order model (ROM) based on proper orthogonal decomposition (POD) is developed to rapidly predict the hydrogen dispersion caused by leakage in small holes at different pressures. The subsonic and underexpanded hydrogen jets are simulated and verified with experimental data. Numerical results of some pressures are collected as POD snapshots and POD interpolation methods are performed to predict the velocity and hydrogen concentration distributions at desired unsampled pressures. Different sample plans with different sampling intervals and sample numbers are investigated. POD-linear, POD-spline, and POD-RBF (radial basis function) interpolations are compared. The results show that the POD-linear and POD-spline methods can accurately predict the velocity and hydrogen fraction at different pressures, and both are better than the POD-RBF method. The relative errors in predicting the hydrogen mole fraction between the POD-linear and POD-spline methods and the simulation results are 2.26% and 2.37%, respectively. Best of all, the POD interpolation method proposed in this work significantly reduces the time required for prediction, costing only 3.3 ms.

A Fusion Positioning Method for Indoor Geomagnetic/Light Intensity/Pedestrian Dead Reckoning Based on Dual-Layer Tent-Atom Search Optimization-Back Propagation.

Indoor positioning using smartphones has garnered significant research attention. Geomagnetic and sensor data offer convenient methods for achieving this goal. However, conventional geomagnetic indoor positioning encounters several limitations, including low spatial resolution, poor accuracy, and stability issues. To address these challenges, we propose a fusion positioning approach. This approach integrates geomagnetic data, light intensity measurements, and inertial navigation data, utilizing a hierarchical optimization strategy. We employ a Tent-ASO-BP model that enhances the traditional Back Propagation (BP) algorithm through the integration of chaos mapping and Atom Search Optimization (ASO). In the offline phase, we construct a dual-resolution fingerprint database using Radial Basis Function (RBF) interpolation. This database amalgamates geomagnetic and light intensity data. The fused positioning results are obtained via the first layer of the Tent-ASO-BP model. We add a second Tent-ASO-BP layer and use an improved Pedestrian Dead Reckoning (PDR) method to derive the walking trajectory from smartphone sensors. In PDR, we apply the Biased Kalman Filter-Wavelet Transform (BKF-WT) for optimal heading estimation and set a time threshold to mitigate the effects of false peaks and valleys. The second-layer model combines geomagnetic and light intensity fusion coordinates with PDR coordinates. The experimental results demonstrate that our proposed positioning method not only effectively reduces positioning errors but also improves robustness across different application scenarios.

A reduced-order model for fast predicting ionized flows of hypersonic vehicles along flight trajectory

An improved Reduced-Order Model (ROM) is proposed based on a flow-solution preprocessing operation and a fast sampling strategy to efficiently and accurately predict ionized hypersonic flows. This ROM is generated in low-dimensional space by performing the Proper Orthogonal Decomposition (POD) on snapshots and is coupled with the Radial Basis Function (RBF) to achieve fast prediction speed. However, due to the disparate scales in the ionized flow field, the conventional ROM usually generates spurious negative errors. Here, this issue is addressed by performing flow-solution preprocessing in logarithmic space to improve the conventional ROM. Then, extra orthogonal polynomials are introduced in the RBF interpolation to achieve additional improvement of the prediction accuracy. In addition, to construct high-efficiency snapshots, a trajectory-constrained adaptive sampling strategy based on convex hull optimization is developed. To evaluate the performance of the proposed fast prediction method, two hypersonic vehicles with classic configurations, i.e. a wave-rider and a reentry capsule, are used to validate the proposed method. Both two cases show that the proposed fast prediction method has high accuracy near the vehicle surface and the free-stream region where the flow field is smooth. Compared with the conventional ROM prediction, the prediction results are significantly improved by the proposed method around the discontinuities, e.g. the shock wave and the ionized layer. As a result, the proposed fast prediction method reduces the error of the conventional ROM by at least 45%, with a speedup of approximately 2.0 × 105 compared to the Computational Fluid Dynamic (CFD) simulations. These test cases demonstrate that the method developed here is efficient and accurate for predicting ionized hypersonic flows.

Remeshing and data transfer in the finite cell method for problems with large deformations

AbstractThe simulation of complex structures using standard finite element discretization techniques can be challenging because the creation of the boundary conforming meshes for such structures can be time‐consuming. Therefore, fictitious domain methods are attractive alternatives because the underlying mesh does not have to conform to the boundary. One fictitious domain method is the finite cell method where a structured non‐geometry conforming Cartesian grid is created and the geometry is then described by a simple indicator function. When solving non‐linear problems, for example, if large deformations are taken into account, broken cells are typically heavily distorted and may lead to failure of the overall solution procedure. To tackle this issue remeshing is applied to proceed with the simulation. In previous publications, radial basis functions (RBFs) were applied to map the deformation gradient between the different meshes. However, it is not clear whether this is the best way to transfer data during the remeshing procedure. Therefore, we investigate whether there are alternative methods that can be used instead. This is then compared to the RBF interpolation and applied to different numerical examples with hyperelastic materials.

Data-driven analysis of parametrized acoustic systems in the frequency domain

A data-driven method combined with the formulations of boundary integral equations is developed for the frequency-domain analysis of parametrized acoustic systems, arising from the spatial discretization of the linear Helmholtz equation. The method derives surrogate models for the approximation of frequency response functions at selected field points via the construction of neural networks with radial basis function interpolation. This requires the offline collection of a training database consisting of discrete frequency-parameter samples and their associated response values, whereas the information about system matrices and interference with boundary element kernels are no longer necessary, leading to a matrix-free and non-intrusive nature. To ensure accurate fitting results, a tensor decomposition technique is employed to decouple the naturally formed tensor grid between the frequency and parameter inputs in the database. After establishing two sets of radial basis networks separately, the final reduced-order solutions as a continuous function of the frequency and parameters are represented by the linear combinations of tensor products. In the online phase, the original transfer function at any testing frequency-parameter location can be quickly predicted as direct output from the compact surrogate model. With the analytical solution, the traditional boundary element method and the fast multipole boundary element method as solvers for the generation of training data, two benchmark problems and one complex system are investigated. In all cases, the simplicity, versatility and efficiency of the proposed method are demonstrated through examples of multi-frequency solutions of acoustic systems with parametrized material and geometric properties.

Polyharmonic splines interpolation on scattered data in 2D and 3D with applications

Data interpolation is a fundamental problem in many applied mathematics and scientific computing fields. This paper introduces a modified implicit local radial basis function interpolation method for scattered data using polyharmonic splines (PS) with a low degree of polynomial basis. This is an improvement to the original method proposed in 2015 by Yao et al.. In the original approach, only radial basis functions (RBFs) with shape parameters, such as multiquadrics (MQ), inverse multiquadrics (IMQ), Gaussian, and Matern RBF are used. The authors claimed that the conditionally positive definite RBFs such as polyharmonic splines r2nlnr and r2n+1 “failed to produce acceptable results”.In this paper, we verified that when polyharmonic splines together with a polynomial basis is used on the interpolation scheme, high-order accuracy and excellent conditioning of the global sparse systems are gained without selecting a shape parameter. The scheme predicts functions’ values at a set of discrete evaluation points, through a global sparse linear system. Compared to standard implementation, computational efficiency is achieved by using parallel computing. Applications of the proposed algorithms to 2D and 3D benchmark functions on uniformly distributed random points, the Halton quasi-points on regular or Stanford bunny shape domains, and an image interpolation problem confirmed the effectiveness of the method. We also compared the algorithms with other methods available in the literature to show the superiority of using PS augmented with a polynomial basis. High accuracy can be easily achieved by increasing the order of polyharmonic splines or the number of points in local domains, when small order of polynomials are used in the basis. MATLAB code for the 3D bunny example is shared on MATLAB Central File Exchange (Yao, 2023).

Preserving the positivity of the deformation gradient determinant in intergrid interpolation by combining RBFs and SVD: Application to cardiac electromechanics

The accurate, robust and efficient transfer of the deformation gradient tensor between meshes of different resolution is crucial in cardiac electromechanics simulations. This paper presents a novel method that combines rescaled localized Radial Basis Function (RBF) interpolation with Singular Value Decomposition (SVD) to preserve the positivity of the determinant of the deformation gradient tensor. The method involves decomposing the evaluations of the tensor at the quadrature nodes of the source mesh into rotation matrices and diagonal matrices of singular values; computing the RBF interpolation of the quaternion representation of rotation matrices and the singular value logarithms; reassembling the deformation gradient tensors at quadrature nodes of the destination mesh, to be used in the assembly of the electrophysiology model equations. The proposed method overcomes limitations of existing interpolation methods, including nested intergrid interpolation and RBF interpolation of the displacement field, that may lead to the loss of physical meaningfulness of the mathematical formulation and then to solver failures at the algebraic level, due to negative determinant values. Furthermore, the proposed method enables the transfer of solution variables between finite element spaces of different degrees and shapes and without stringent conformity requirements between different meshes, thus enhancing the flexibility and accuracy of electromechanical simulations. We show numerical results confirming that the proposed method enables the transfer of the deformation gradient tensor, allowing to successfully run simulations in cases where existing methods fail. This work provides an efficient and robust method for the intergrid transfer of the deformation gradient tensor, thus enabling independent tailoring of mesh discretizations to the particular characteristics of the individual physical components concurring to the of the multiphysics model.

Surrogate modeling for the homogenization of elastoplastic composites based on RBF interpolation

We propose a new framework for creating a surrogate model of computational homogenization for elastoplastic composite materials that serves as a homogenized constitutive law for decoupled two-scale analysis. Two key ingredients of the proposed surrogate modeling are the introduction of a variable representing the macroscopic strain history along with the macroscopic strain and the application of radial basis function (RBF) interpolation. Not only the coefficients of the RBF but also the type of its function form are considered hyperparameters and are determined by applying an optimization algorithm. In the offline process, numerical material tests (NMTs) are performed on a unit cell consisting of multiple elastoplastic materials subjected to various patterns of macroscopic strain to create a database that stores the discrete relationships between variables representing macroscopic deformation states with the macroscopic stresses. Then, RBF interpolation is applied to represent the macroscopic stress in a continuous function with these input data as independent variables. In the online process, this function is used as a macroscopic constitutive law in the macroscopic analysis, which can be followed by localization analysis using an arbitrary macroscopic strain history if necessary.

Instantaneous Physics‐Based Ground Motion Maps Using Reduced‐Order Modeling

AbstractPhysics‐based simulations of earthquake ground motion are useful to complement recorded ground motions. However, the computational expense of performing numerical simulations hinders their applicability to tasks that require real‐time solutions or ensembles of solutions for different earthquake sources. To enable rapid physics‐based solutions, we present a reduced‐order modeling approach based on interpolated proper orthogonal decomposition (POD) to predict peak ground velocities (PGVs). As a demonstrator, we consider PGVs from regional 3D wave propagation simulations at the location of the 2008 MW 5.4 Chino Hills earthquake using double‐couple sources with varying depth and focal mechanisms. These simulations resolve frequencies ≤1.0 Hz and include topography, viscoelastic attenuation, and S‐wave speeds ≥500 m/s. We evaluate the accuracy of the interpolated POD reduced‐order model (ROM) as a function of the approximation method. Comparing the radial basis function (RBF), multilayer perceptron neural network, random forest, and k‐nearest neighbor, we find that the RBF interpolation gives the lowest error (≈0.1 cm/s) when tested against an independent data set. We also find that evaluating the ROM is 107–108 times faster than the wave propagation simulations. We use the ROM to generate PGV maps for 1 million different focal mechanisms, in which we identify potentially damaging ground motions and quantify correlations between focal mechanism, depth, and accuracy of the predicted PGV. Our results demonstrate that the ROM can rapidly and accurately approximate the PGV from wave propagation simulations with variable source properties, topography, and complex subsurface structure.

A new variable shape parameter strategy for RBF approximation using neural networks

The choice of the shape parameter highly effects the behaviour of radial basis function (RBF) approximations, as it needs to be selected to balance between the ill-conditioning of the interpolation matrix and high accuracy. In this paper, we demonstrate how to use neural networks to determine the shape parameters in RBFs. In particular, we construct a multilayer perceptron (MLP) trained using an unsupervised learning strategy, and use it to predict shape parameters for inverse multiquadric and Gaussian kernels. We test the neural network approach in RBF interpolation tasks and in a RBF-finite difference method in one and two-space dimensions, demonstrating promising results.

A framework for simulating snow accumulation and ice accretion on high-speed trains

A simulation framework for three-dimensional particle trajectory tracking, snow accumulation and ice accretion modelling on solid bodies in high-speed air flow is presented. The framework, named HADICE, solves the aerodynamic flow field as Reynolds averaged Navier-Stokes (RANS), with optional hybrid RANS/LES capabilities for modelling complex turbulent flows. Particle trajectory tracking is performed in an Eulerian-Lagrangian approach and wall collision is modelled using a hard collision model with reflect condition and a momentum based trap condition. A novel accurate and robust iterative evaluation method for the local collection efficiency is proposed for arbitrary 3D geometry and flow. Ice accretion is modelled in a multi-step approach and iced geometry is updated through mesh deformation using a radial basis function (RBF) interpolation method. Snow accumulation and ice accretion predictions based on the framework are validated against climatic wind tunnel experimental measurements. A full 3D simulation is demonstrated for snow accumulation and ice accretion on a 1:8 scaled high-speed train model. Using the presented framework, snow accumulation and icing simulation for high-speed trains can be conducted in an accurate and efficient manner, which is of great importance for physical investigations and the design of anti-snow and ice protection systems.

Design of large-angle distortion-free and spot adjustable LED spot array projector based on microlens array and eyepiece

The spot array projector has long been a challenging and intriguing research focus in the fields of projection and lighting applications. In the current literature, there is a lack of a comprehensive and detailed presentation on the design and analysis method for a spot array generator based on the structure of imaging eyepiece and MLA. We present a novel design and optimization method for a large-angle, distortion-free and spot adjustable LED spot array projector that is composed of an eyepiece, two microlens arrays (MLAs), and a micro-LED array (MLEDA). The eyepiece system is optimized using imaging optical methods to project sub-beams to the target plane with a large angle. The sub-lens of condenser MLA is also optimized using imaging optical methods to refocus the collimating beam and match the numerical aperture (NA) with the eyepiece, and the sub-lens of the collimating MLA is acquired by using simulated annealing (SA) global illumination optics optimization method to achieve collimation and far-field homogenization. The predistortion MLEDA and the MLAs are proposed and implemented by the radial basis function (RBF) interpolation method, which correct the large-angle distortion introduced by the eyepiece. Both near-field and far-field applications can be realized by the proposed system. In the near-field applications, different spot geometries at the near-field target plane can be achieved. In the far-field applications, the power matching of the MLEDA is used to improve far-field uniformity of spot array. Moreover, the predefined-geometry arrangement spot array can be realized in both near and far fields. Two design examples with full field of view (FOV) projection of 80° and 100° are provided to validate the proposed method. Overall, the proposed system offers a promising solution for various applications requiring target identification or 3D calibration.

Bayesian inversion for rapid multiwell interpretation of well logs and core data: Applications to unconventional organic-shale formations

Multimineral analysis is widely used to calculate in situ porosity, fluid saturation, and mineralogy of rocks penetrated by a well. It delivers weight/volumetric concentrations of rock solid/fluid constituents by combining multiple borehole geophysical measurements and often is referred to as petrophysical joint inversion. Recently, a probabilistic method was developed for improved petrophysical estimation of rock constituents and their uncertainty from well logs. This method mitigates borehole and instrument-related environmental effects present in the measurements and efficiently propagates the uncertainty from measurement noise and rock-physics models (RPMs) to compositional/petrophysical estimations. The probabilistic estimation method is extended to the challenging conditions of multiple neighboring wells penetrating similar rock formations where borehole/drilling environmental conditions, borehole instruments, and measurement noise may vary from well to well. A calibration step is performed in a few key wells with core data and/or advanced borehole measurements; it enables the same RPMs and prior models to be implemented in nearby wells but with limited measurements. In addition, a precomputed surrogate model constructed with radial basis function interpolation is implemented for accurate and efficient nuclear-property calculations. The multiwell interpretation method is verified using synthetic and field examples of organic-rich shale formations. Results find that the probabilistic method (1) improves rock petrophysical/compositional estimations by mitigating borehole environmental effects and incorporating a priori knowledge, (2) yields regionally consistent compositions among wells, and (3) quantifies the uncertainty of the estimations. As a result, the probabilistic approach is especially suitable for assessing petrophysical/compositional properties in multiwell settings with complex rock constituents and/or limited borehole measurements.

A 2D prediction step using multiquadric local interpolation with adaptive parameter estimation for image compression

We present and analyze several prediction strategies in the 2D setting based on multiquadric radial basis function interpolation with either linear or Weighted Essentially Non Oscillatory (WENO) shape parameter approximation. When considered within Harten’s framework for Multiresolution, these prediction operators give rise to sparse multi-scale representations of 2D signals, whose compression capabilities are demonstrated through numerical experiments. It is well know that the accuracy of multiquadric interpolation depends on the choice of the shape parameter. In addition, in [6], it was shown that the use of data-dependent strategies in the selection of the shape parameter leads to more accurate reconstructions. We shall show that our local adaptive estimates of the shape parameters lead to non-separable, fully 2D, reconstruction strategies that lead, in turn, to efficient compression algorithms.

Spatially-variant image deconvolution for photoacoustic tomography.

Photoacoustic tomography (PAT) system can reconstruct images of biological tissues with high resolution and contrast. However, in practice, the PAT images are usually degraded by spatially variant blur and streak artifacts due to the non-ideal imaging conditions and chosen reconstruction algorithms. Therefore, in this paper, we propose a two-phase restoration method to progressively improve the image quality. In the first phase, we design a precise device and measuring method to obtain spatially variant point spread function samples at preset positions of the PAT system in image domain, then we adopt principal component analysis and radial basis function interpolation to model the entire spatially variant point spread function. Afterwards, we propose a sparse logarithmic gradient regularized Richardson-Lucy (SLG-RL) algorithm to deblur the reconstructed PAT images. In the second phase, we present a novel method called deringing which is also based on SLG-RL to remove the streak artifacts. Finally, we evaluate our method with simulation, phantom and in vivo experiments, respectively. All the results show that our method can significantly improve the quality of PAT images.

Scaling of radial basis functions

Abstract This paper studies the influence of scaling on the behavior of radial basis function interpolation. It focuses on certain central aspects, but does not try to be exhaustive. The most important questions are: How does the error of a kernel-based interpolant vary with the scale of the kernel chosen? How does the standard error bound vary? And since fixed functions may be in spaces that allow scalings, like global Sobolev spaces, is there a scale of the space that matches the function best? The last question is answered in the affirmative for Sobolev spaces, but the required scale may be hard to estimate. Scalability of functions turns out to be restricted for spaces generated by analytic kernels, unless the functions are band-limited. In contrast to other papers, polynomials and polyharmonics are included as flat limits when checking scales experimentally, with an independent computation. The numerical results show that the hunt for near-flat scales is questionable, if users include the flat limit cases right from the start. When there are not enough data to evaluate errors directly, the scale of the standard error bound can be varied, up to replacing the norm of the unknown function by the norm of the interpolant. This follows the behavior of the actual error qualitatively well, but is only of limited value for estimating error-optimal scales. For kernels and functions with unlimited smoothness, the given interpolation data are proven to be insufficient for determining useful scales.

Fast strain mapping in abdominal aortic aneurysm wall reveals heterogeneous patterns.

Abdominal aortic aneurysm patients are regularly monitored to assess aneurysm development and risk of rupture. A preventive surgical procedure is recommended when the maximum aortic antero-posterior diameter, periodically assessed on two-dimensional abdominal ultrasound scans, reaches 5.5mm. Although the maximum diameter criterion has limited ability to predict aneurysm rupture, no clinically relevant tool that could complement the current guidelines has emerged so far. In vivo cyclic strains in the aneurysm wall are related to the wall response to blood pressure pulse, and therefore, they can be linked to wall mechanical properties, which in turn contribute to determining the risk of rupture. This work aimed to enable biomechanical estimations in the aneurysm wall by providing a fast and semi-automatic method to post-process dynamic clinical ultrasound sequences and by mapping the cross-sectional strains on the B-mode image. Specifically, the Sparse Demons algorithm was employed to track the wall motion throughout multiple cardiac cycles. Then, the cyclic strains were mapped by means of radial basis function interpolation and differentiation. We applied our method to two-dimensional sequences from eight patients. The automatic part of the analysis took under 1.5minper cardiac cycle. The tracking method was validated against simulated ultrasound sequences, and a maximum root mean square error of 0.22mm was found. The strain was calculated both with our method and with the established finite-element method, and a very good agreement was found, with mean differences of one order of magnitude smaller than the image spatial resolution. Most patients exhibited a strain pattern that suggests interaction with the spine. To conclude, our method is a promising tool for investigating abdominal aortic aneurysm wall biomechanics as it can provide a fast and accurate measurement of the cyclic wall strains from clinical ultrasound sequences.