Interpolation Scheme Research Articles

Lagrange Multivariate Polynomial Interpolation: A Random Algorithmic Approach

The problems of polynomial interpolation with several variables present more difficulties than those of one-dimensional interpolation. The first problem is to study the regularity of the interpolation schemes. In fact, it is well-known that, in contrast to the univariate case, there is no universal space of polynomials which admits unique Lagrange interpolation for all point sets of a given cardinality, and so the interpolation space will depend on the set Z of interpolation points. Techniques of univariate Newton interpolating polynomials are extended to multivariate data points by different generalizations and practical algorithms. The Newton basis format, with divided-difference algorithm for coefficients, generalizes in a straightforward way when interpolating at nodes on a grid within certain schemes. In this work, we propose a random algorithm for computing several interpolating multivariate Lagrange polynomials, called RLMVPIA (Random Lagrange Multivariate Polynomial Interpolation Algorithm), for any finite interpolation set. We will use a Newton-type polynomials basis, and we will introduce a new concept called Z , z -partition. All the given algorithms are tested on examples. RLMVPIA is easy to implement and requires no storage.

A review and experimental evaluation of deep learning methods for MRI reconstruction

Following the success of deep learning in a wide range of applications, neural network-based machine-learning techniques have received significant interest for accelerating magnetic resonance imaging (MRI) acquisition and reconstruction strategies. A number of ideas inspired by deep learning techniques for computer vision and image processing have been successfully applied to nonlinear image reconstruction in the spirit of compressed sensing for accelerated MRI. Given the rapidly growing nature of the field, it is imperative to consolidate and summarize the large number of deep learning methods that have been reported in the literature, to obtain a better understanding of the field in general. This article provides an overview of the recent developments in neural-network based approaches that have been proposed specifically for improving parallel imaging. A general background and introduction to parallel MRI is also given from a classical view of k-space based reconstruction methods. Image domain based techniques that introduce improved regularizers are covered along with k-space based methods which focus on better interpolation strategies using neural networks. While the field is rapidly evolving with plenty of papers published each year, in this review, we attempt to cover broad categories of methods that have shown good performance on publicly available data sets. Limitations and open problems are also discussed and recent efforts for producing open data sets and benchmarks for the community are examined.

Evaluation of Interpolation Scheme Alternatives and Variation of the Number of Slave Cells in a Cut-Cell Methodology

The present work determines numerical solutions applied to flow problems in a cut-cell framework, introducing and evaluating two interpolation alternatives for the treatment of the convective terms and the effect of the variation of the number of slave cells generated near the solid interfaces. Using the upwind, QUICK and WAHYD (TVD) schemes, three benchmark cases were studied in the laminar regime, namely, flow between concentric cylindrical walls, flow in an inclined channel and flow around a cylinder. The numerical results obtained were favorable for the proposed interpolation methodology that prevents velocity over/under-estimations on the finite control volume faces, observing a tendency to produce smaller errors and mid-to-high computational efficiencies when coupled with a smaller number of slave cells generated at the boundaries. Although the magnitude of the errors found were small, improvements are of more significance for quantities that depend on gradient estimations at surfaces.

Optimal design of responsive structures

With recent advances in both responsive materials and fabrication techniques, it is now possible to construct integrated functional structures, composed of both structural and active materials. We investigate the robust design of such structures through topology optimization. By applying a typical interpolation scheme and filtering technique, we prove existence of an optimal design to a class of objective functions which depend on the compliances of the stimulated and unstimulated states. In particular, we consider the actuation work and the blocking load as objectives, both of which may be written in terms of compliances. We study numerical results for the design of a 2D rectangular lifting actuator for both of these objectives, and discuss some intuition behind the features of the converged designs. We formulate the optimal design of these integrated responsive structures with the introduction of voids or holes in the domain, and show that our existence result holds in this setting. We again consider the design of the 2D lifting actuator now with voids. Finally, we investigate the optimal design of an integrated 3D torsional actuator for maximum blocking torque.

A novel combination of mathematical modelling approaches for simulating nearly hypersonic, viscous, reacting magnetogasdynamic flows

This work focuses on the mathematical combination of magnetogasdynamics (MGD) and the Dunn and Kang chemical kinetic model with a multi-component Harten–Lax–van Leer Contact (HLLC) local Riemann solver using high-order interpolation schemes. Due to the nature of nearly hypersonic, unsteady, viscous, reacting MGD flows, the gas dissociates which is taken into account with 11 species and 26 chemical reactions along with the temperature dependent transport properties through the Wilke model. In the present work, the electromagnetic effects are considered by solving a Poisson equation simultaneously with the chemical reaction equations. The non-linear convective terms are treated numerically with third-, fifth- and seventh-order Weighted Essentially Non-Oscillatory (WENO) interpolation schemes along with second-, fourth- and sixth-order central finite difference schemes for the viscous terms. The combination of these numerical and physical models are analysed for stability and validated for benchmark problems such as (a) Hartmann flow as MGD validation test case and (b) the shock-tube problem for testing the chemical reaction model in which case the reaction rate controlling temperature suggested by Park has also been taken into account. Furthermore, a complex physical problem of an external aerodynamic flow over the cube for the Mach number varying from 1.25 to 3 have also been investigated, because a limited number of studies is available for understanding the flow behaviour of sharp cornered blunt bodies such as cubes. The knowledge of shock structures of these objects has become imperative to predict the atmospheric re-entry trajectories of de-orbiting CubeSats, space debris or fragments of launch vehicles. The main physical finding is that the shock structures are observed to be much more resolved compared to previous works, especially in the wake region. The knowledge gap is bridged by detailing the shock structures and showing the dependence of the shock standoff distance on total enthalpy of the flow.

On the construction of a mode based reduced order model for a moving store

Store separation from aircraft and spacecraft has been a critical issue for the aerospace industry. A potential remedy to reduce resources and to improve turn around time in store separation predictions is the construction of a reduced order model (ROM) that can be utilized for fast and accurate modeling. The objective of this study is to investigate how to construct ROMs for moving bodies and utilize them for both load and subsequently for trajectory predictions of a separating store. To analyze the ROM's capability for load prediction, a preliminary study for modeling distributed pressure loads of pitching and oscillating stores was undertaken. For comparison between modal decomposition and neural networks, the study first used proper orthogonal decomposition (POD), dynamic mode decomposition (DMD), and convolutional neural network (CNN) to reconstruct surface pressures and free stream flow for a pitch up store. Results indicated for pitch up store motion POD would perform most efficiently. The study then used the oscillating store cases to compare two interpolation schemes; namely surface splines and Kriging interpolation. For the oscillating store case, a surrogate ROM-based model was constructed using CFD-simulation data for 12 combinations of reduced frequencies and amplitudes. The resulting ROM-based surrogate model was able to make accurate and fast load predictions for intermediate reduced frequencies and amplitudes. In an attempt to investigate ROM ability to predict store trajectories of intermediate conditions, three cases of store separation at Mach 0.5, 0.8, and 1.2 were considered. It was shown that predictions of trajectories at intermediate Mach numbers can be accomplished using the ROM-based surrogate model constructed from computed store separations at Mach 0.5, 0.8, and 1.2. Results showed a close agreement between CFD and ROM surface loads and as a result trajectory predictions.

Ground-Motion Modeling as an Image Processing Task: Introducing a Neural Network Based, Fully Data-Driven, and Nonergodic Approach

ABSTRACT We construct and examine the prototype of a deep learning-based ground-motion model (GMM) that is both fully data driven and nonergodic. We formulate ground-motion modeling as an image processing task, in which a specific type of neural network, the U-Net, relates continuous, horizontal maps of earthquake predictive parameters to sparse observations of a ground-motion intensity measure (IM). The processing of map-shaped data allows the natural incorporation of absolute earthquake source and observation site coordinates, and is, therefore, well suited to include site-, source-, and path-specific amplification effects in a nonergodic GMM. Data-driven interpolation of the IM between observation points is an inherent feature of the U-Net and requires no a priori assumptions. We evaluate our model using both a synthetic dataset and a subset of observations from the KiK-net strong motion network in the Kanto basin in Japan. We find that the U-Net model is capable of learning the magnitude–distance scaling, as well as site-, source-, and path-specific amplification effects from a strong motion dataset. The interpolation scheme is evaluated using a fivefold cross validation and is found to provide on average unbiased predictions. The magnitude–distance scaling as well as the site amplification of response spectral acceleration at a period of 1 s obtained for the Kanto basin are comparable to previous regional studies.

Large-Eddy Simulation of a Classical Hydraulic Jump: Influence of Modelling Parameters on the Predictive Accuracy

Results from large-eddy simulations of a classical hydraulic jump at inlet Froude number two are reported. The computations were performed using the general-purpose finite-volume-based code OpenFOAM®, and the primary goal was to evaluate the influence of the modelling parameters on the predictive accuracy, as well as establish the associated best-practice guidelines. A benchmark simulation was conducted on a grid with a 1 mm-cell-edge length to validate the solver and provide a reference solution for the parameter influence study. The remaining simulations covered different selections of the modelling parameters: geometric vs. algebraic interface capturing, three mesh resolution levels, and four choices of the convective flux interpolation scheme. Geometric interface capturing led to better accuracy, but deteriorated the numerical stability and increased the simulation times. Interestingly, numerical dissipation was shown to systematically improve the results, both in terms of accuracy and stability. Strong sensitivity to the grid resolution was observed directly downstream of the toe of the jump.

A shape function based interpolation approach for nodal design variable based structural optimization

AbstractIn this article, a novel density interpolation scheme for topological optimization based on nodal density variables is presented. The method relies on the definition of an independent interpolation mesh for the description of the density field. This mesh is used for the interpolation of the density field at the Gauss points of the design domain mesh. Provided that the interpolation mesh is realized by linear elements, the proposed scheme is range restricted and assures physically meaningful values of the interpolated density. Explicit analytical sensitivity expressions can be derived. The utilization of an independent interpolation mesh allows for the utilization of any element type and geometry on the body for the computation of the displacements. It is also shown that the chosen interpolation scheme has interesting properties of volume and material volume conservation with respect to the mesh utilized on the body and the domain of the interpolation mesh can differ from the domain of the mesh for the displacements computation. This allows for the realization of very regular interpolation meshes even in case of very complex shapes of the domain of the body to be optimized. Examples are provided showing the performance of the proposed interpolation scheme.

An Adaptive Interpolation Scheme for Wideband Frequency Sweep in Electromagnetic Simulations

An adaptive interpolation scheme is proposed to accurately calculate the wideband responses in electromagnetic simulations. In the proposed scheme, the sampling points are first carefully divided into several groups based on their responses to avoid the Runge phenomenon and the error fluctuations, and then different interpolation strategies are used to calculate the responses in the whole frequency band. If the relative error does not satisfy the predefined threshold in a specific frequency band, it will be refined until the error criteria is met. The detailed error analysis is also presented to verify the accuracy of the interpolation scheme. At last, two numerical examples including the antenna radiation and the filter simulation are carried out to validate its accuracy and efficiency.

Fast simulation of fully non-stationary wind fields using a new matrix factorization assisted interpolation method

The classical spectral representation method and its improved versions have been widely used to generate non-stationary wind fields. However, they are not efficient for the simulation of non-stationary wind fields with time-varying coherence (i.e., fully non-stationary wind fields) due to the extensive computational demand regarding spectral matrix decomposition or harmonic superposition. To this end, this study develops a new matrix factorization assisted interpolation method, which is composed of an improved interpolation scheme and the monotone projected Barzilai-Borwein based non-negative matrix factorization (MPBB-NMF). Specifically, an improved interpolation scheme is firstly developed to quantify the desired distribution of time-frequency interpolation nodes, and only the decomposed spectra at these nodes are calculated by Cholesky decomposition. Then, MPBB-NMF with a fast global convergence is used to decouple these spectra into products of non-negative time- and frequency-dependent functions. Finally, one-dimensional interpolation is performed for these decoupled functions, which makes it possible to accelerate the harmonic superposition using a small amount of fast Fourier transforms (FFTs). Obviously, the proposed method realizes the decomposition of spectral matrix in the decoupling form. Besides reducing of the number of Cholesky decomposition and invoking FFT, its decoupling only aims at the low dimensional matrix and the interpolation merely involves a few decoupled items. Numerical simulation on the fully non-stationary wind field of a long-span bridge demonstrates the effectiveness and superiority of the proposed method.

Effects of convective schemes and geometric reconstruction scheme on interface of multiple melt flows

Multi-phase polymer melt processing techniques, e.g., multi-layer co-extrusion and multi-melt multi-injection molding (M3IM), are typical methods for preparing multilayer composites. Numerical simulations are now broadly acknowledged to be an advanced and effective method to investigate the evolution of the interface between the melts during these processing techniques. However, how to describe the sharp interface between the melts precisely is still a big challenge. Herein, five different interpolation schemes were used to investigate the melt penetration in M3IM, paying close attention to the effects of different schemes on interface diffusion, wrinkle, and shape. In addition, the morphologies of the interface were characterized experimentally. By comparing the numerical results and experimental results, it can be concluded that the compressive interface capturing scheme for arbitrary meshes (CICSAM) is the most accurate method. Our work is instructive for expounding the variation patterns of interfacial morphologies during the multi-phase polymer melt processing.

A distributed dynamic load identification method based on the hierarchical-clustering-oriented radial basis function framework using acceleration signals under convex-fuzzy hybrid uncertainties

Load identification is a hotly studied topic due to the widespread recognition of its importance in structural design and health monitoring. This paper explores an effective identification method for the distributed dynamic load (DDL) varying in both time progress and space dimensions using limited acceleration responses. As for the reconstruction of spatial distribution, the radial basis function (RBF) interpolation strategy, whose hyper-parameters are determined by a hierarchical clustering algorithm, is applied to approximate the DDL and then transform the continuous function into finite dimensions. In the time domain, based on the inverse Newmark iteration, the RBF coefficients at each discrete instant are obtained by the least square solution of the modal forces. Considering the multi-source uncertainties lacking exact probability distributions, a multi-dimensional interval model is developed to quantify convex parameters and fuzzy parameters uniformly. Further, a Chebyshev-interval surrogate model with different orders is constructed to obtain the fuzzy-interval boundaries of DDLs. Eventually, three examples are discussed to demonstrate the feasibility of the developed DDL identification approach considering hybrid uncertainties. The results suggest its promising applications in different structures and loading conditions.

Real-time optical time interpolation using spectral interferometry.

A simple scheme for all-optical time interpolation using spectral interferometry is put forward that is, in principle, capable of single-shot measurements. In this method, the arrival time of optical timing pulses is encoded into the spectrum of a time-stretched supercontinuum via cross phase modulation. The proof-of-concept test setup points toward femtosecond-level absolute timing capabilities with only minor additions to modern optical clockwork.

LAMMPS lb/fluid fix version 2: Improved hydrodynamic forces implemented into LAMMPS through a lattice-Boltzmann fluid

The first version of this code (Mackay et al., 2013) [10] implemented long-range hydrodynamic interactions into the open-source molecular dynamics package LAMMPS. This was done through the creation of a fix, lb/fluid which was subsequently included as a user-package in the main LAMMPS distribution. Here we substantially update this package by making improvements to its accuracy, adding significant new features, and by simplifying the use of the package. A new two-pass interpolation and spreading scheme is introduced which results in the improved accuracy and numerical stability. New features include new output options, several added computes, and mesh geometry option suitable for micro- and nano-fluidic device simulations. The original package could require fairly careful calibration to obtain accurate thermostating and accurate reproduction of properties related to the hydrodynamic size of objects such as colloids. This process has now been largely automated so that the default settings should suffice for most applications. Program summaryProgram title: fix lb/fluidCPC Library link to program files:https://doi.org/10.17632/2289cnrdtz.1Licensing provisions: GPLv3Programming language: C++Journal reference of previous version: Comput. Phys. Commun. 184 (2013) 2021–2031.Does the new version supersede the previous version?: YesReasons for the new version: The new version improves accuracy, adds new features, and simplifies the use of the package.Summary of revisions: A new two-pass interpolation and spreading scheme is introduced to relate properties on the fluid mesh to off-lattice particle properties. New features include output options, several added computes, and mesh geometry suitable for micro- and nano-fluidic device simulations. Calibration processes have been largely automated so that the default settings should suffice for most applications.Nature of problem: The inclusion of long-range hydrodynamic effects into molecular dynamics simulations requires the presence of an explicit solvent. Prior to the implementation of this fix, the only option for incorporating such a solvent into a LAMMPS [1] simulation is the explicit inclusion of each of the individual solvent molecules. This is obviously quite computationally intensive, and for large system sizes can quickly become impractical.Solution method: As an alternative, we have implemented a coarse-grained model for the fluid, simplifying the problem, while retaining the solvent degrees of freedom. We use a thermal lattice-Boltzmann model for the fluid, which is coupled to the molecular dynamics particles at each fluid time step.

Five-Axis Trajectory Generation Considering Synchronization and Nonlinear Interpolation Errors

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

Volume‐of‐fluid based two‐phase flow methods on structured multiblock and overset grids

AbstractA geometric volume‐of‐fluid (VOF) method is implemented on the static/dynamic overset grids with a new VOF interpolation algorithm developed. The VOF value cannot be directly interpolated through the overset grid boundaries since the VOF function is a discontinuous step function. In the present study, the interface is reconstructed in the fringe cells of the overset grids using the interpolated distance function and normal vector. The VOF value in the fringe cells is calculated geometrically based on the reconstructed interface on the physical domain. The gravity and surface tension forces are implemented using a VOF based ghost fluid method, where a ghost pressure interpolation scheme is developed to handle the pressure jump across the interface due to the density changes and surface tension effect. Numerical tests and application examples have been performed for the verification and validation of the numerical methods developed in the present study. The advection tests show that the effect of the VOF overset interpolation on the accuracy of the VOF method is small. The numerical test for the static pressure shows that the present VOF based ghost fluid method can greatly reduce the spurious velocity observed in the body force method. The sharp pressure profile due to the surface tension effects can be preserved across the interface. Application examples demonstrate the capability and superiority of the VOF based numerical methods for the prediction of the violent ship flows involving strong air water interface interactions and complex geometries.

Finite element formulation for the dynamic analysis of shear deformable thin-walled beams

The present study develops a finite element formulation for the natural vibration and dynamic analysis of thin-walled members with asymmetric cross-sections. The formulation captures shear deformation within the middle surface, global and local warping, viscous damping, translational and rotatory inertial effects, and inertial coupling. The formulation starts with the extended variational form of Hamilton’s principle to account for non-conservative damping effects, in conjunction with a special interpolation scheme aimed at circumventing shear locking. The resulting finite element has two nodes with seven degrees of freedom per node. The discretized equations of motion are integrated using the unconditionally stable Newmark integration scheme with constant average acceleration. For members with doubly symmetric sections, comparisons with shell solutions show that the formulation reliably predicts the natural frequencies and vibration modes associated with flexure/shear and torsion/warping responses. Further comparisons with other non-shear deformable thin-walled beam solutions show that shear deformation effects gain significance in shorter spans, higher vibration modes, and in flexural vibration about the major axis. For monosymmetric and asymmetric sections, comparisons with benchmark solutions showcase the ability of the formulation to capture the inertial coupling arising between the flexure/shear and torsion/warping responses, while keeping the computational cost to a minimum. Comparisons to shell-based solutions demonstrate the ability of the solution to accurately capture the damped time history response of members with doubly symmetric, monosymmetric, and asymmetric sections subjected to harmonic, ramped, and blast-like forces. A separate treatment is provided for degenerate cross-sections with a vanishing global warping function.

A Modified Local Distance-weighted (MLD) Method of Interpolation and Its Numerical Performances for Large Scattered Datasets

The purpose of this study was to propose a new interpolation scheme that was designed to remedy the shortcomings encountered in two popular interpolation methods; the triangle-based blending (TBB) method and the inverse distance weighted (IDW) method. At the same time, the proposed method combines their desirable aspects, which are the local nature and non-use of quadratic surface construction, making it comparatively less time-consuming and more independent of the global effect. Because of these properties, the new scheme was named the ‘Modified Local Distance-Weighted (MLD)’ method, and it was tested in detail with different sizes of datasets. The datasets involved in this investigation were of two types; uniformly and non-uniformly distributed. The performances were carefully monitored and assessed via several criteria; accuracy, sensitivity to parameters, CPU-time, storage requirement and ease of implementation. For comparative purposes, three alternative interpolation methods were simultaneously carried out, i.e. TBB, IDW and Radial Basis Function-Based (RBF) method. The investigation clearly revealed promising aspects of the proposed scheme where a good quality of results was anticipated, and CPU-time and storage were seen to have been significantly reduced. The research strongly indicates the benefits of the proposed method for larger sized datasets for real practical scientific and engineering uses in the future.

Eigenvectors-guided topology optimization to control the mode shape and suppress the vibration of the multi-material plate

Although the structural vibration can be suppressed by adjusting eigenvalues outside the excitation frequency band, it becomes increasingly difficult to suppress vibration as the width of the excitation frequency band increases. Besides, for some high-precision equipment, e.g. space telescope mirror substrate and rocket motor casings, their complex and diverse modals directly affect the performance. Thus, a novel optimization algorithm was proposed, which applies to achieve eigenvector-based modal control and vibration suppression. The eigenvectors were defined as the objective function, and Nelson’s method without truncation error was used to calculate the sensitivity information. Due to the introduction of dynamic equations, the non-linearity of the objective function is prominent. Then, an improved solver that can handle this non-linear topology optimization problem was proposed. The optimization was performed under a multi-material framework, and the extended multi-material interpolation scheme was proposed to readily realize the optimization with three or more materials. In addition, the consistent mass matrix constructed by multi-material interpolation was used to describe the mass matrix without concentrated mass in the dynamic equation. Moreover, the modal assurance criterion was used to track jumping modals. Finally, the modal controllability was achieved through several numerical examples, which verify the effectiveness of the proposed method.