Point Interpolation Research Articles

An efficient meshless numerical method with the error estimate for two-dimensional Schrödinger equation

In this study, we present a capable meshfree numerical approach to numerical solving of the two-dimensional Schrödinger equation (SE). Firstly, the radial point interpolation is obtained to build the shape functions for discretization in the mentioned equation. These meshless shape functions have Kronecker delta attributes. Secondly, the local weak form is constructed over the local sub-domain for the purpose of diminishing the solution of the time-dependent SE in a two-dimensional case and transforming it into a system of algebraic linear equations. Also, semi and fully discrete schemes with their error estimates for this meshless method are obtained. We indicate the proposed method square of error has the order of O(h2m+τ2) in the L2 norm. Moreover, some numerical tests are provided, and their exact and finite difference solutions are compared to verify the efficiency of our algorithm.

Dynamic Analysis of an FGM Beam Undergoing Large Overall Motion in Temperature Field

The dynamic analysis of a functionally gradient material (FGM) beam undergoing large overall motion based on meshless methods is studied in the variable temperature field environment. Considering the high-order terms of nonlinear coupling deformation which were often ignored in previous studies, the high-order rigid–flexible coupled (HOC) dynamic model of a rotating FGM beam with a lumped mass in a temperature field is established by employing Lagrange’s equations of the second kind. Compared with the first-order approximate coupled (FOAC) dynamic model, the HOC dynamic model is more accurate and has a wider scope of application. The validity of the point interpolation (PIM) method and radial point interpolation method (RPIM) in this paper is verified by comparison with simulation results of the assumed mode method (AMM) and the finite element method (FEM). On this basis, the factors affecting the dynamic characteristics of the FGM beam such as the form of temperature field, the functionally gradient index, and the location of added lumped mass are discussed. In this paper, the meshless method with higher computational efficiency and the HOC dynamic model with more accurate results are adopted. This study will provide guidance and reference for dealing with problems in engineering, machinery, aerospace, and other fields in complex environment.

Stability of convergence rates: kernel interpolation on non-Lipschitz domains

Abstract Error estimates for kernel interpolation in Reproducing Kernel Hilbert Spaces usually assume quite restrictive properties on the shape of the domain, especially in the case of infinitely smooth kernels like the popular Gaussian kernel. In this paper we prove that it is possible to obtain convergence results (in the number of interpolation points) for kernel interpolation for arbitrary domains $\varOmega \subset{\mathbb{R}} ^{d}$, thus allowing for non-Lipschitz domains including e.g., cusps and irregular boundaries. Especially we show that, when going to a smaller domain $\tilde{\varOmega } \subset \varOmega \subset{\mathbb{R}} ^{d}$, the convergence rate does not deteriorate—i.e., the convergence rates are stable with respect to going to a subset. We obtain this by leveraging an analysis of greedy kernel algorithms. The impact of this result is explained on the examples of kernels of finite as well as infinite smoothness. A comparison to approximation in Sobolev spaces is drawn, where the shape of the domain $\varOmega $ has an impact on the approximation properties. Numerical experiments illustrate and confirm the analysis.

Artificial intelligence warm-start approach: optimizing the generalization capability of QAOA in complex energy landscapes

To address the issue of the quantum approximate optimization algorithm frequently encountering local minima and the cost of parameter optimization within complex non-convex optimization energy landscapes, we consider a warm-start method. This approach leverages the characteristics of transition states in the enhanced optimizer, specifically descending along unique negative curvature directions, to find smaller local minima. Our research results indicate that with the assistance of an enhanced pre-training structure of the AlphaZero AI model, the initialization generalization ability of the new optimizer is significantly enhanced across various test sets. We train on 2-SAT training sets with clause densities between α ≈ 2.6 and α ≈ 2.89, and transfer to more complex test sets. Additionally, the average residual energy density in transfer learning consistently remains below 0.01, even achieving a high transfer success probability of 98% in hard instances with α ≈ 3.7. The search efficiency, pre-trained by ensemble learning, was significantly enhanced, while only requiring simple interpolation of a few transition points to transfer on the global optimal solutions at higher sample clause densities.

Nonlinear dynamic analysis of aircraft CFRP sandwich wings under explosive blast loading: Introducing SVM-DNN algorithm to predict dynamical information

The aircraft wings are a crucial component, particularly during mechanical shock excitation. Thus, for the first time, an inventive adaptable model to simulate the nonlinear dynamics of the wings under explosive blast loading at the moment of the collision is described in this article. It is crucial to increase the structure's stability because of explosive blast loading. The mentioned structure is made of two composite face sheets and a carbon fiber-reinforced polymer (CFRP). Functionally graded (FG) graphene origami (GOri)-enabled auxetic metallic metamaterial (GOEAM) is introduced to improve the stability of this kind of applicable structure, especially in air maneuvers. The analysis is conducted utilizing Reddy's third-order shear deformation of Reddy theory (TSDRT), incorporating Hamilton's principle and Von Kármán nonlinear geometric assumptions. The governing and boundary equations are accurately computed at the domain and boundary edges of the doubly curved structure. The coupled mesh-free radial point interpolation method (MRPIM) and Newton-Raphson method are applied for numerically simulating the current doubly curved structure by employing Newmark's temporal integration method with a constant-average acceleration approach. The present findings are cross-checked against the machine learning method and open-source results from the literature. Based on physical data, machine learning integrates supervised machine learning to predict the nonlinear vibration of the system. In this case, nonlinear transient bending properties are found using hybrid machine learning methods and solutions. The findings demonstrate the significance of GOEAM for nonlinear vibrations of the composite system. The current study's conclusions could be used as a benchmark and useful recommendations for future structural design techniques with enhanced features.

Numerical continued fraction interpolation

UDC 517.524 We show that highly accurate approximations can often be obtained by constructing Thiele interpolating continued fractions by a Greedy selection of the interpolation points together with an early termination condition. The obtained results are comparable with the outcome of state-of-the-art rational interpolation techniques based on the barycentric form.

Dynamic analysis of sandwich conical shell with magneto/electro rheological core using a truly meshfree radial point interpolation method

Dynamic analysis of sandwich conical shell with magneto/electro rheological core using a truly meshfree radial point interpolation method

Point-StyleGAN: Multi-scale point cloud synthesis with style modulation

A point cloud is a set of discrete surface samples. As the simplest 3D representation, it is widely used in 3D reconstruction and perception. Yet developing a generative model for point clouds remains challenging due to the sparsity and irregularity of points. Drawn by StyleGAN, the forefront image generation model, this paper presents Point-StyleGAN, a generator adapted from StyleGAN2 architecture for point cloud synthesis. Specifically, we replace all the 2D convolutions with 1D ones and introduce a series of multi-resolution discriminators to overcome the under-constrained issue caused by the sparsity of points. We further add a metric learning-based loss to improve generation diversity. Besides the generation task, we show several applications based on GAN inversion, among which an inversion encoder Point-pSp is designed and applied to point cloud reconstruction, completion, and interpolation. To our best knowledge, Point-pSp is the first inversion encoder for point cloud embedding in the latent space of GANs. The comparisons to prior work and the applications of GAN inversion demonstrate the advantages of our method. We believe the potential brought by the Point-StyleGAN architecture would further inspire massive follow-up works.

Derivative-free bound-constrained optimization for solving structured problems with surrogate models

We propose and analyze a model-based derivative-free (DFO) algorithm for solving bound-constrained optimization problems where the objective function is the composition of a smooth function and a vector of black-box functions. We assume that the black-box functions are smooth and the evaluation of them is the computational bottleneck of the algorithm. The distinguishing feature of our algorithm is the use of approximate function values at interpolation points which can be obtained by an application-specific surrogate model that is cheap to evaluate. As an example, we consider the situation in which a sequence of related optimization problems is solved and present a regression-based approximation scheme that uses function values that were evaluated when solving prior problem instances. In addition, we propose and analyze a new algorithm for obtaining interpolation points that handles unrelaxable bound constraints. Our numerical results show that our algorithm outperforms a state-of-the-art DFO algorithm for solving a least-squares problem from a chemical engineering application when a history of black-box function evaluations is available.

Transient analyses of underwater acoustic propagation with the modified meshfree radial point interpolation method and newmark time integration techniques

Underwater acoustic propagation plays a very important role in underwater communication and ocean engineering field. The radial point interpolation method (RPIM) is an effective numerical approach to solve underwater acoustic propagations though considerable numerical errors can always be caused due to the discontinuously differentiable nodal interpolation function in one integration cell. The origin of this numerical error primarily comes from the mismatch between the background integration cell and the nodal interpolation function support. To address this issue, a modified meshfree RPIM is developed in this work for transient analyses of underwater acoustic propagations. In the proposed modified RPIM, the construction of the interpolation function supports are dependent on the integration cells. In consequence, in one background cell all the quadrature points share one mutual support domain and the continuously differentiable interpolation functions can be obtained. Therefore, the numerical integration errors can be evidently decreased and the related numerical dispersion effects in wave analysis can be largely suppressed. Owing to the adequately small spatial dispersion errors from the proposed modified RPIM, it is observed that the modified RPIM with the Newmark time domain discretization scheme can monotonically improve the solution precision by directly reducing the temporal discretization interval. This property makes the modified RPIM clearly outperform the original RPIM and is especially suitable for handling very complex underwater acoustic propagations.

A novel train–bridge interaction computational framework based on a meshless box girder model

In traditional train–bridge coupled system (TBCS), simply supported box girder bridges are often modeled using Euler beam elements, neglecting their spatial structure. This simplification may yield inaccurate results, impacting the running safety analysis of high-speed railway. To address this issue, a novel train–bridge interaction computational framework based on first-order shear deformation theory (FSDT) and radial point interpolation method (RPIM) for the box girder bridge model is proposed. In this model, the displacement fields of top, bottom, and web plates are represented using FSDT and numerically discretized by RPIM. The traditional TBCS is upgraded by replacing the Euler beam model with the novel model. This is the first time that the framework has been applied to TBCS field. Several numerical examples are presented to highlight the accuracy of this novel model, and illustrate the differences, and advantages of it over the traditional model. The results indicate that the proposed model closely matches the accuracy of the solid element (C3D20) model; it can provide a comprehensive bridge response compared to traditional TBCS, and the latter may underestimate the dynamic response of the structure. The proposed model holds significant potential for the simulation of box structures and widespread application in the field of TBCS.

A boundary point interpolation method for acoustic problems with particular emphasis on the calculation of Cauchy principal value integrals

A boundary point interpolation method (BPIM) is presented in this paper for numerical solving acoustic problems. The BPIM is a boundary-type meshless method that combines the point interpolation technique for constructing approximate functions with boundary integral equations for reformulating boundary value problems. Since effective numerical integration techniques are crucial in the BPIM and boundary integral equation methods for calculating regular and singular integrals, two Clenshaw-Curtis integration techniques (CCITs) for regular and weakly singular integrals are derived in a unified manner with the same integration points on quadratic integration cells. In addition, a unilateral CCIT, a power series expansion technique, and an improved CCIT are proposed to effectively calculate strongly singular integrals or Cauchy principal value integrals in the BPIM. These integration techniques provide direct and efficient formulas for calculating boundary integrals on quadratic integration cells, and can be directly applied to the calculation of regular and singular integrals in the boundary element method and other meshless boundary integral equation methods. Explicit expressions of integration weights in all CCITs are derived, and the values of integration points and weights are listed. Numerical results are finally provided to validate the effectiveness of the present meshless method and integration techniques.

Iteration complexity and finite-time efficiency of adaptive sampling trust-region methods for stochastic derivative-free optimization

ASTRO-DF is a prominent trust-region method using adaptive sampling for stochastic derivative-free optimization of nonconvex problems. Its salient feature is an easy-to-understand-and-implement concept of maintaining “just enough” replications when evaluating points throughout the search to guarantee almost-sure convergence to a first-order critical point. To reduce the dependence of ASTRO-DF on the problem dimension and boost its performance in finite time, we present two key refinements, namely: (i) local models with diagonal Hessians constructed on interpolation points based on a coordinate basis; and (ii) direct search using the interpolation points whenever possible. We demonstrate that the refinements in (i) and (ii) retain the convergence guarantees while matching existing results on iteration complexity. Uniquely, our iteration complexity results match the canonical rates without placing assumptions on iterative models’ quality and their independence from function estimates. Numerical experimentation on a testbed of problems and comparison against existing popular algorithms reveals the computational advantage of ASTRO-DF due to the proposed refinements.

Reconstruction of γ Dose Rate Field and Algorithm Validation Based on Inverse Distance Weight Interpolation

The inverse distance weighting (IDW) interpolation algorithm is used to reconstruct the γ radiation field. The traditional IDW interpolation algorithm is improved. The power exponent of distance P in the IDW for each interpolation point is not fixed and varies from one point to the other point. A fitting expression of P is obtained, which is a function of the coordinates of each point and can minimize the interpolation error when the number of sampling points is specified. Afterward, the improved algorithm is used to reconstruct a γ radiation field of a single source, and the theoretic results are compared with the results from Geant4, yielding an average relative error of 7.50%. The interpolated results from the experimental measurements align well with the actual data, with an average relative error of only 0.12%. The P derived from the interpolated experimental measurement data shows an error of 2.0% compared to the power exponent obtained from the Geant4 data interpolation. Then we set up a double-source γ radiation scene experiment and measured the count rate data at the grid points. At the same time, the experiment scene was simulated by Geant4. The improved IDW algorithm could not reconstruct the double-source γ radiation field well, thus further improvement is needed.

Augmented-Resolution Digital image correlation algorithm for vibration measurements

Digital Image Correlation (DIC) is a widespread technique, and many different implementations were developed in the literature to optimize the algorithm performance. In this paper, a new DIC algorithm is optimized for vibration measurements, where the displacement amplitude drastically decreases with the increase in frequency. Even if high-resolution cameras are used, it is mandatory to use a DIC algorithm with reliable sub-pixel performance. DIC algorithms are generally optimized for large complex displacement fields, and they solve the sub-pixel problem by fitting correlation data through a second-order polynomial function. An innovative solution is proposed to push the sub-pixel accuracy and increase the signal-to-noise ratio of the measurement, by exploiting image resizing to achieve virtual higher resolutions and by adopting a quartic polynomial function to interpolate the displacement map with an increased number of interpolation points. The proposed algorithm is validated through synthetic and experimental data and compared to literature DIC algorithms.

Fractal Interpolation Function based Thin Wire Antennas

This paper presents an approach for the design of wire antennas based on fractal interpolation functions (FIFs). The interpolation points and the contraction factors of the FIFs are chosen as free parameters to modify the antenna geometry. The proposed structures’ gain and radiation pattern can be optimized using FIF parameters. Producible prefractal antennas obtained in the intermediate iterations of fractal generation have compact sizes compared to classical counterparts. The error in prefractal geometry and the original fractal is bounded, and can be determined in terms of the finest producible detail’s dimensions. The emerging structures have multiband behavior due to their self-similar and symmetric nature. To illustrate the approach, we have provided finite element based simulations for several prefractal antennas. |S11|, the gain, the radiation efficiency, the radiation patterns, and feed point impedances for the demonstrated antennas are calculated numerically. The results indicate that produced antennas can be used in applications that require limited mechanical size, multiple operating bands, and controlled radiation patterns.

A Symmetric Non-Stationary Loop Subdivision with Applications in Initial Point Interpolation

Loop subdivision is a significant surface scheme with wide applications in fields like computer graphics and wavelet. As a type of stationary scheme, Loop subdivision cannot adjust the limit surface directly. In this paper, we present a new way to solve this problem by proposing a symmetric non-stationary Loop subdivision based on a suitable iteration. This new scheme can be used to adjust the limit surfaces freely and thus can generate surfaces with different shapes. For this new scheme, we show that it is C2 convergent in the regular part of mesh and is at least tangent plane continuous at the limit positions of the extraordinary points. Additionally, we present a non-uniform generalization of this new symmetric non-stationary subdivision so as to locally control the shape of the limit surfaces. More interestingly, we present the limit positions of the initial points, both for the symmetric non-stationary Loop subdivision and its non-uniform generalization. Such limit positions can be used to interpolate the initial points with different valences, generalizing the existing result. Several numerical examples are given to illustrate the performance of the new schemes.

Forced vibrations of a cantilever beam using radial point interpolation methods: A comparison study

Meshless methods are a type of numerical method used to simulate continuum mechanics problems. These methods have been applied to several types of problems, there are a few works using meshless method focused on dynamic problems, but most works study static loading conditions. The current work aims at using two different meshless methods, the Radial Point Interpolation Method (RPIM) and the Natural Neighbours RPIM (NNRPIM), in a dynamic problem, specifically forced vibrations. This problem requires time integration, therefore three different time integration methods have also been tested, namely: the Central Difference Method (CDM), the Wilson method, and the Newmark method. The CDM is an explicit method, while the other two are implicit. A discretization study was performed to assess the ideal nodal discretization before the numerical and time integration methods are validated. For the implicit methods, different time step lengths were also tested. In the final example damping was introduced. The results prove the validity of the two meshless methods by having similar results to the Finite Element Method (FEM) using the three distinct time integration methods, different loading conditions, and damping.

The Biomechanical Analysis of Tibial Implants Using Meshless Methods: Stress and Bone Tissue Remodeling Analysis

Total knee arthroplasty (TKA) stands out as one of the most widely employed surgical procedures, establishing itself as the preferred method for addressing advanced osteoarthritis of the knee. However, current knee prostheses require refined design solutions. This research work focuses on a computational analysis of both the mechanical behavior of a knee joint implant and the bone remodeling process in the tibia following implantation. This research study delves into how specific design parameters, particularly the stem geometry, impact the prosthesis’s performance. Utilizing a computed tomography scan of a tibia, various TKA configurations were simulated to conduct analyses employing advanced discretization techniques, such as the finite element method (FEM) and the radial point interpolation method (RPIM). The findings reveal that the introduction of the implant leads to a marginal increase in the stress values within the tibia, accompanied by a reduction in the displacement field values. The insertion of the longest tested implant increased the maximum stress from 5.0705 MPa to 6.1584 MPa, leading to a displacement reduction from 0.016 mm to 0.0142 mm. Finally, by combining the FEM with a bone remodeling algorithm, the bone remodeling process of the tibia due to an implant insertion was simulated.

Automated segmentation of the left-ventricle from MRI with a fully convolutional network to investigate CTRCD in breast cancer patients.

Purpose: The goal of this study was to develop a fully convolutional network (FCN) tool to automatedly segment the left-ventricular (LV) myocardium in displacement encoding with stimulated echoes MRI. The segmentation results are used for LV chamber quantification and strain analyses in breast cancer patients susceptible to cancer therapy-related cardiac dysfunction (CTRCD). Approach: A DeepLabV3+ FCN with a ResNet-101 backbone was custom-designed to conduct chamber quantification on 45 female breast cancer datasets (23 training, 11 validation, and 11 test sets). LV structural parameters and LV ejection fraction (LVEF) were measured, and myocardial strains estimated with the radial point interpolation method. Myocardial classification validation was against quantization-based ground-truth with computations of accuracy, Dice score, average perpendicular distance (APD), Hausdorff-distance, and others. Additional validations were conducted with equivalence tests and Cronbach's alpha () intraclass correlation coefficients between the FCN and a vendor tool on chamber quantification and myocardial strain computations. Results: Myocardial classification results against ground-truth were , , and for the validation set and , , and for the test set. The confidence intervals (CI) and two one-sided t-test results of equivalence tests between the FCN and vendor-tool were to 2.42%, p-value < 0.001 for LVEF ( versus ), and to 0.63%, p-value < 0.001 for longitudinal strain ( versus ). Conclusions: The validation results were found equivalent to the vendor tool-based parameter estimates, which show that accurate LV chamber quantification followed by strain analysis for CTRCD investigation can be achieved with our proposed FCN methodology.