Point Interpolation Research Articles

Fractional Boundary Element Solution for Nonlinear Nonlocal Thermoelastic Problems of Anisotropic Fibrous Polymer Nanomaterials

This paper provides a new fractional boundary element method (BEM) solution for nonlinear nonlocal thermoelastic problems with anisotropic fibrous polymer nanoparticles. This comprehensive BEM solution comprises two solutions: the anisotropic fibrous polymer nanoparticles problem solution and the nonlinear nonlocal thermoelasticity problem. The nonlinear nonlocal thermoelasticity problem solution separates the displacement field into complimentary and specific components. The overall displacement is obtained using the boundary element methodology, which solves a Navier-type problem, and the specific displacement is derived using the local radial point interpolation method (LRPIM). The new modified shift-splitting (NMSS) technique, which minimizes memory and processing time requirements, was utilized to solve BEM-created linear systems. The performance of NMSS was evaluated. The numerical results show how fractional and graded parameters influence the thermal stresses of nonlinear nonlocal thermoelastic issues involving anisotropic fibrous polymer nanoparticles. The numerical findings further reveal that the BEM results correlate very well with the finite element method (FEM) and analytical results, demonstrating the validity and correctness of the proposed methodology.

A meshless computational framework for a modified dynamic system of vehicle coupled with plate structure

In previous simulations of train–bridge interaction systems (TBIS), the supporting system for the train are commonly treated as beam structures, leading to less accurate results, particularly for small-span cases. To address this limitation, a modified vertical TBIS is proposed. In the presented TBIS, the supporting system is considered as a Reissner–Mindlin plate, and the displacement field is described by first-order shear deformation theory (FSDT). To establish the model, radial point interpolation method (RPIM), a meshless method, is employed. Finally, a coupled dynamic equation is established to calculate various responses of the system. Several numerical examples are presented to illustrate the disparities between the system based on plate model and traditional beam model. The results indicate that the beam model yields higher estimates of the mid-span vertical displacement of the bridge, while the peak of the mid-span vertical acceleration is smaller compared to the plate model; additionally, it is observed that the carbody is primarily influenced by rail irregularities. Consequently, the proposed plate model offers distinct advantages over the beam model in providing comprehensive structural response information, thereby offering novel insights into bridge design and analysis. Additionally, this marks the inaugural application of the meshless method in the field of TBIS, which further extends the application scope of meshless methods.

Numerical Solution of Natural Convection Problems Using Radial Point Interpolation Meshless (RPIM) Method Combined with Artificial-Compressibility Model

A numerical method is used to solve the thermal analysis of natural convection in enclosures. This paper proposes the use of an implicit artificial-compressibility model in conjunction with the Radial Point Interpolation Meshless (RPIM) method to mimic laminar natural convective heat transport. The technique couples the pressure with the velocity components using an artificial compressibility model. The RPIM is used to discretize the spatial terms of the governing equation. We solve the semi-algebraic system implicitly in backward Euler pseudo-time. The proposed method solves two test problems—natural convection in the annulus of concentric circular cylinders and trapezoidal cavity. Additionally, the results are validated using experimental and numerical data available in the literature. Excellent agreement was seen between the numerical results acquired with the suggested method and those obtained through the standard techniques found in the literature.

Identification of dominant subspaces for model reduction of structured parametric systems

SummaryIn this paper, we discuss a novel model reduction framework for linear structured dynamical systems. The transfer functions of these systems are assumed to have a special structure, for example, coming from second‐order linear systems or time‐delay systems, and they may also have parameter dependencies. Firstly, we investigate the connection between classic interpolation‐based model reduction methods with the reachability and observability subspaces of linear structured parametric systems. We show that if enough interpolation points are taken, the projection matrices of interpolation‐based model reduction encode these subspaces. Consequently, we are able to identify the dominant reachable and observable subspaces of the underlying system. Based on this, we propose a new model reduction algorithm combining these features and leading to reduced‐order systems. Furthermore, we discuss computational aspects of the approach and its applicability to a large‐scale setting. We illustrate the efficiency of the proposed approach with several numerical large‐scale benchmark examples.

Radial basis point interpolation for strain field calculation in digital image correlation

In order to extract smooth and accurate strain fields from the noisy displacement fields obtained by digital image correlation (DIC), a point interpolation meshless (PIM) method with a radial basis function (RBF) is introduced for full-field strain calculation, which overcomes the problems of slow calculation speed and unstable matrix inverse calculation of the element-free Galerkin method (EFG). The radial basis point interpolation method (RPIM) with three different radial basis functions and the moving least squares (MLS) and pointwise least squares (PLS) methods are compared by analyzing and validating the strain fields with high-strain gradients in simulation experiments. The results indicate that the RPIM is nearly 80% more computationally efficient than the MLS method when a larger support domain is used, and the efficiency of the RPIM is nearly 26% higher than that of the MLS method when a smaller support domain is used; the strain calculation accuracy is slightly lower than that of the MLS method by 0.3–0.5%, but the stability of the calculation is significantly improved. In contrast with the PLS method, which is easily affected by the noise and the size of the strain calculation window, the RPIM is insensitive to the displacement noise and the size of the support domain and can obtain a similar calculation accuracy. The RPIM with multiquadric (MQ) radial basis functions performs well in balancing the computational accuracy and efficiency and is insensitive to shape parameters. The application cases show that the method can effectively compute the strain field at the crack tip, validating its applicability to the study of the plastic region at the crack tip. In conclusion, the proposed RPIM-based method provides an accurate, practical, and robust approach for full-field strain measurements.

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.