Shepard Interpolation Method Research Articles

Simulation of Destructive Test for Arch Reinforcement Area of Tunnel Lining by MSIM

Meshless Shepard interpolation method (MSIM) is used to simulate the destruction process within the arch reinforcement area of tunnel lining. In this method, the shape functions are formed by the partition of unity and the finite cover technology, which is not affected by discontinuous domains, with delta property at any desired node, and applies boundary conditions in an easy and correct way. The MSIM method combines the advantages of both the conventional meshless method and the numerical manifold method. The finite cover technology enables the shape functions to not be affected by the discontinuities in the solution domain, which overcomes the difficulty resulting from the conventional meshless method. The finite covers and the partition of unity functions are formed using the influence domains of a series of nodes, which removes the obstacle from conventional numerical manifold method and has simpler formation of finite covers than numerical manifold method. Virtual crack closure technique is used to calculate the intensity factor of crack-tip stress. The results of progressive destruction process simulation on the cracking patterns within the arch reinforcement area indicate that the method is suitable for tracking the crack propagation in complex stress conditions.

Dynamic crack propagation analysis combined the stable scheme and continuous-discontinuous cellular automaton

A continuous-discontinuous cellular automaton method (CDCA) is developed for dynamic crack propagation. First, a separated time integral scheme is built, in which time difference is divided into two separated time steps: one balances the change of the geometric discretization caused by crack growing, and the other balances the dynamic loading. Second, an adaptive energy conservation strategy is proposed for the CDCA that includes the change in the degree of nodal freedom for the crack tip element, and a new strategy for the inheritance of the degrees of nodal freedom is proposed. This method retains the enrichment scheme for the discontinuity, and the Shepard interpolation method is employed to transform the variables’ values, so as to keeps the energy conservation caused by crack tip location change. Then, the enriched degree of nodal freedom values are easily passed from the crack tip enrichment to the penetrated enrichment and crack tip local coordinate system changing with crack growing. Third, a dynamic crack propagation criterion is developed, which can be used for tensile fractures and compression shear fractures. Based on those theories, a new cellular automaton model is built for dynamic crack propagation. Compared to the extended finite element method (XFEM), the calculation efficiency has been greatly improved, and several numerical examples are given to illustrate accuracy and efficiency of the present method.

Surface for methane combustion: O(3P) +CH4 → OH+CH3**Project supported by the National Natural Science Foundation of China (Grant No. 51574016) and completed while the author was in residence at UNSW, Australia supported by the International Cooperation Training Program for Innovative Talents of USTB.

Kinetic investigations including quasi-classical trajectory and canonical unified statistical theory method calculations are carried out on a potential energy surface for the hydrogen-abstraction reaction from methane by atom O(3P). The surface is constructed using a modified Shepard interpolation method. The ab initio calculations are performed at the CCSD(T) level. Taking account of the contribution of inner core electrons to electronic correlation interaction in ab initio electronic structure calculations, modified optimized aug-cc-pCVQZ basis sets are applied to the all-electrons calculations. On this potential energy surface, the triplet oxygen atom attacks methane in a near-collinear H–CH3 direction to form a saddle point with barrier height of 13.55 kcal/mol, which plays a key role in the kinetics of the title reaction. For the temperature range of 298–2500 K, our calculated thermal rate constants for the O(3P) + CH4 → OH + CH3 reaction show good agreement with relevant experimental data. This work provides detailed mechanism of this gas-phase reaction and a theoretical guidance for methane combustion.

Assessment of a localized deterministic spatial interpolation method in generating an indoor radiofrequency radiation map

Picocell antennas are used to improve wireless communication coverage inside an area. More picocell antennas are being used to cover small areas, thus, increasing the radiofrequency radiation (RFR) levels in an indoor environment. One method in estimating the radiofrequency radiation is by using spatial interpolation techniques. This study assessed the robustness of a localized deterministic spatial interpolation method compared to its global interpolation counterpart used in generating a radiofrequency radiation map. This was investigated to speed up the data analysis by taking a smaller number of samples. An omnidirectional picocell antenna located in an office lobby area was chosen. The NARDA Selective Radiation Meter SRM-3000TM with tri-axis antenna was used to check the central frequency and average power density emitted by the antenna. A radiofrequency map was generated using a small sample size in a grid pattern. The local Shepard interpolation method with a radius of influence of 100 cm and a power factor of 1 gives a result with a mean absolute difference lower than the global interpolation method. Thus, error contribution in RFR map generation can be minimized by considering a smaller region of samples in the interpolation.

Dynamics of the CH4 + O(3P) → CH3(ν = 0) + OH(ν′ = 0) reaction**Project supported by the National Natural Science Foundation of China (Grant Nos. 51574016 and 51604018).

The dynamics of the ground-state reaction of CH4 + O(3P) → CH3(ν = 0) + OH(ν′ = 0) have attracted a great deal of attention both theoretically and experimentally. This rapid communication represents extensive quasi-classical trajectory calculations of the vibrational distributions on a unique full-dimensional ab initio potential energy surface for the title reaction, at the collision energy of relevance to previous crossed molecular beam experiments. The surface is constructed using the all electrons coupled-cluster singles and doubles approach plus quasi-perturbative triple excitations with optimized basis sets. A modified Shepard interpolation method is also employed for the construction. Good agreement between our calculations and the available experimental results has been achieved, opening the door for accurate dynamics on this surface.

Fast computation of triangular Shepard interpolants

In this paper, we present an efficient algorithm for the computation of triangular Shepard interpolation method. More precisely, it is well known that the triangular Shepard method reaches an approximation order better than the Shepard one (DellAccio et al., 2016), but it needs to identify useful general triangulation of the node set. Here we propose a searching technique used to detect and select the nearest neighbor points in the interpolation scheme (Cavoretto et al., 2016, 2017). It consists in determining the closest points belonging to the different neighborhoods and subsequently applies to the triangulation-based approach. Numerical experiments and some geological applications show efficiency and accuracy of the interpolation procedure.

Local-metrics error-based Shepard interpolation as surrogate for highly non-linear material models in high dimensions.

Many problems in computational materials science and chemistry require the evaluation of expensive functions with locally rapid changes, such as the turn-over frequency of first principles kinetic Monte Carlo models for heterogeneous catalysis. Because of the high computational cost, it is often desirable to replace the original with a surrogate model, e.g., for use in coupled multiscale simulations. The construction of surrogates becomes particularly challenging in high-dimensions. Here, we present a novel version of the modified Shepard interpolation method which can overcome the curse of dimensionality for such functions to give faithful reconstructions even from very modest numbers of function evaluations. The introduction of local metrics allows us to take advantage of the fact that, on a local scale, rapid variation often occurs only across a small number of directions. Furthermore, we use local error estimates to weigh different local approximations, which helps avoid artificial oscillations. Finally, we test our approach on a number of challenging analytic functions as well as a realistic kinetic Monte Carlo model. Our method not only outperforms existing isotropic metric Shepard methods but also state-of-the-art Gaussian process regression.

Optimization of composite structures with continuous spatial variation of fiber angle through Shepard interpolation

In the present paper, we propose a method for the optimization of composite structures with continuous spatial variation of fiber angle. The key idea is to construct a continuous global function interpolating scattered design points to represent the fiber angle arrangement throughout the design domain. To be more specific, the Shepard interpolation method is employed to construct the interpolation function. The angles at scattered design points are taken as the design variables to be optimized. The Shepard interpolation inherently guarantees spatial continuity of fiber angle. In addition, the design freedom of the proposed method can be readily enlarged by increasing the number of design points or the number of design variables to accommodate complex or rapid local variation of fiber angle. Several examples with in-plane load are investigated. In the simple representative numerical examples, the proposed method shows good performance.

High Density Salt and Pepper Noise Filter based on Shepard Interpolation Method

This paper has proposed the technique in order to remove salt-and-pepper noise from the corrupted image. The proposed scheme consists of two steps. The first, the adaptive median filter is applied to noisy image for detecting noisy and noise-free pixels. Finally, the output image of previous step will be divided into many non-overlapping windows of which the noise-free pixel values are used to compute the new pixel value by the Shepard method. The experiment results of our approach had shown the better performance than the existing method.

Theoretical study on photoexcitation dynamics of a bis-diimine Cu(I) complex in solutions

We investigate the photoexcitation dynamics of [Cu(phen)2] + (phen=1,10-phenanthroline) in solutions by using the molecular mechanics with Shepard interpolation method, which enables us to generate potential energy surfaces of the Cu complex efficiently. The calculated flattening rates and coherent vibration motions are in good agreement with the experimental results. We find that the photoexcitation dynamics in the gas phase is much different from those in solutions, which indicates the importance of the solvent effects on the photoexcitation dynamics.

A new dual reciprocity hybrid boundary node method based on Shepard and Taylor interpolation method and Chebyshev polynomials

A new dual reciprocity hybrid boundary node method (DHBNM) is proposed in this paper, in which the Shepard and Taylor interpolation method (STIM) and Chebyshev polynomials interpolation are proposed. Firstly, the Shepard interpolation is used to construct zero level shape function, and the high-power shape functions are constructed through the Taylor expansion, and through those two methods, no inversion is needed in the whole process of the shape function construction. Besides, Chebyshev polynomials are used as the basis functions for particular solution interpolation instead of the conical function, radial basis functions, and the analytical solutions of the basic form of particular solutions related to Chebyshev polynomials for elasticity are obtained, by means of this method, no internal node is needed, and interpolation coefficients can be given as explicit functions, so no inversion is needed for particular solution interpolation, which costs a large amount of computational expense for the traditional method. Based on those two methods, a new dual reciprocity hybrid boundary node method is developed, compared to the traditional DHBNM, no inversion is needed for both shape function construction and particular solution interpolation, which greatly improves the computational efficiency, and no internal node is needed for particular solution interpolation. Numerical examples are given to illustrate that the present method is accurate and effective.

A new hybrid boundary node method based on Taylor expansion and the Shepard interpolation method

SummaryA novel meshless method based on the Shepard and Taylor interpolation method (STIM) and the hybrid boundary node method (HBNM) is proposed. Based on the Shepard interpolation method and Taylor expansion, the STIM is developed to construct the shape function of the HBNM. In the STIM, the Shepard shape function is used as the basic function, which is the zero‐level shape function, and the high‐power basic functions are constructed through Taylor expansion. Four advantages of the STIM are the interpolation property, the arbitrarily high‐order consistency, the absence of inversion for the whole process of shape function construction, and the low computational expense. These properties are desirable in the implementation of meshless methods. By combining the STIM and the HBNM, a much more effective meshless method is proposed to solve the elasticity problems. Compared with the traditional HBNM, the STIM can improve accuracy because of the use of high‐power basic functions and can also improve the computational efficiency because there is no inversion for the shape function construction process. Numerical examples are given to demonstrate the accuracy and efficiency of the proposed method. Copyright © 2015 John Wiley & Sons, Ltd.

Hybrid many-body-expansion/Shepard-interpolation method for constructing ab initio potential energy surfaces for quantum dynamics calculations

A hybrid many-body-expansion/Shepard-interpolation method for constructing globally accurate potential energy surfaces (PESs) is presented. Incorporation of the many-body expansion strategy into the conventional Shepard interpolation method, greatly reduced the configuration space need to be interpolated, and furthermore, all the asymptotic channels can be represented accurately. A total of 30 “ab initio” points (taking LSTH PES as the source of pseudo ab inito calculations) are found to be sufficient for constructing chemically accurate H3 PES appropriate for uses in dynamics calculations. The excellence of the constructed PES is demonstrated by accurate quantum dynamics calculations.

Biofuel Combustion. Energetics and Kinetics of Hydrogen Abstraction from Carbon-1 inn-Butanol by the Hydroperoxyl Radical Calculated by Coupled Cluster and Density Functional Theories and Multistructural Variational Transition-State Theory with Multidimensional Tunneling

Multistructural canonical variational transition-state theory with small-curvature multidimensional tunneling (MS-CVT/SCT) is employed to calculate thermal rate constants for hydrogen-atom abstraction from carbon-1 of n-butanol by the hydroperoxyl radical over the temperature range 250-2000 K. The M08-SO hybrid meta-GGA density functional was validated against CCSD(T)-F12a explicitly correlated wave function calculations with the jul-cc-pVTZ basis set. It was then used to compute the properties of all stationary points and the energies and Hessians of a few nonstationary points along the reaction path, which were then used to generate a potential energy surface by the multiconfiguration Shepard interpolation (MCSI) method. The internal rotations in the transition state for this reaction (like those in the reactant alcohol) are strongly coupled to each other and generate multiple stable conformations, which make important contributions to the partition functions. It is shown that neglecting to account for the multiple-structure effects and torsional potential anharmonicity effects that arise from the torsional modes would lead to order-of-magnitude errors in the calculated rate constants at temperatures of interest in combustion.

Shadow Detecting and Shadow Interpolation Algorithm for InSAS

The shadow detecting algorithm based on the coherence and the Sigma filter is used to pick up the shadow of interferometric synthetic aperture sonar (InSAS), which can eliminate small separated shadow areas. To solve the problems such as great computer complexity of traditional Shepard interpolation method and large fluctuant of linear interpolation method for the large shadow area, an improved Shepard interpolation method is proposed. Interpolation boundary is picked up by using diffuse search, and interpolation source is adaptively chosen according to the size of shadow area. The method carries out a perfect tradeoff between performance and computer speed. Lake trial dataset is used to validate the performance of proposed method. The results indicate that the proposed method can eliminate the fluctuant from the linear interpolation method and can process in real time in the InSAS system.

Multipath Variational Transition State Theory: Rate Constant of the 1,4-Hydrogen Shift Isomerization of the 2-Cyclohexylethyl Radical

We propose a new formulation of variational transition state theory called multipath variational transition state theory (MP-VTST). We employ this new formulation to calculate the forward and reverse thermal rate constant of the 1,4-hydrogen shift isomerization of the 2-cyclohexylethyl radical in the gas phase. First, we find and optimize all the local-minimum-energy structures of the reaction, product, and transition state. Then, for the lowest-energy transition state structures, we calculate the reaction path by using multiconfiguration Shepard interpolation (MSCI) method to represent the potential energy surface, and, from this representation, we also calculate the ground-state vibrationally adiabatic potential energy curve, the reaction-path curvature vector, and the generalized free energy of activation profile. With this information, the path-averaged generalized transmission coefficients <γ> are evaluated. Then, thermal rate constant containing the multiple-structure anharmonicity and torsional anharmonicity effects is calculated using multistructural transition state theory (MS-TST). The final MP-VTST thermal rate constant is obtained by multiplying k(MS-T)(MS-TST) by <γ>. In these calculations, the M06 density functional is utilized to compute the energy, gradient, and Hessian at the Shepard points, and the M06-2X density functional is used to obtain the structures (conformers) of the reactant, product, and the saddle point for computing the multistructural anharmonicity factors.

Direct Simulation of Excited-State Intramolecular Proton Transfer and Vibrational Coherence of 10-Hydroxybenzo[h]quinoline in Solution

We investigate an ultrafast excited-state intramolecular proton transfer (ESIPT) reaction and the subsequent coherent vibrational motion of 10-hydroxybenzo[h]quinoline in cyclohexane by the electronically embedded multiconfiguration Shepard interpolation method, which enables us to generate the potential energy surface of the reaction effectively and thus carry out a direct excited-state dynamics simulation with low computational costs. The calculated time scale of the ESIPT and the frequencies and lifetimes of coherent motions are in good agreement with the experimental results. The present study reveals that the coherent motions are caused by not only the proton transfer itself but also the backbone displacement induced by the ESIPT. We also discuss the effects of the solvent on the dynamics of the coherent vibrational modes.

Interpolation techniques for building a continuous map from discrete wireless sensor network data

ABSTRACTWireless sensor networks (WSNs) typically gather data at a discrete number of locations. However, it is desirable to be able to design applications and reason about the data in more abstract forms than in points of data. By bestowing the ability to predict inter‐node values upon the network, it is proposed that it will become possible to build applications that are unaware of the concrete reality of sparse data. This interpolation capability is realised as a service of the network. In this paper, the ‘map’ style of presentation has been identified as a suitable sense data visualisation format. Although map generation is essentially a problem of interpolation between points, a new WSN service, called the map generation service, which is based on a Shepard interpolation method, is presented. A modified Shepard method that aims to deal with the special characteristics of WSNs is proposed. It requires small storage, can be localised and integrates the information about the application domain to further reduce the map generation cost and improve the mapping accuracy. Empirical analysis has shown that the map generation service is an accurate, a flexible and an efficient method. Copyright © 2011 John Wiley &amp; Sons, Ltd.

Multi-structural variational transition state theory. Kinetics of the 1,4-hydrogen shift isomerization of the pentyl radical with torsional anharmonicity

We present a new formulation of variational transition state theory (VTST) called multi-structural VTST (MS-VTST) and the use of this to calculate the rate constant for the 1,4-hydrogen shift isomerization reaction of 1-pentyl radical and that for the reverse reaction. MS-VTST uses a multi-faceted dividing surface and provides a convenient way to include the contributions of many structures (typically conformers) of the reactant and the transition state in rate constant calculations. In this particular application, we also account for the torsional anharmonicity. We used the multi-configuration Shepard interpolation method to efficiently generate a semi-global portion of the potential energy surface from a small number of high-level electronic structure calculations using the M06 density functional in order to compute the energies and Hessians of Shepard points along a reaction path. The M06-2X density functional was used to calculate the multi-structural anharmonicity effect, including all of the structures of the reactant, product and transition state. To predict the thermal rate constant, VTST calculations were performed to obtain the canonical variational rate constant over the temperature range 200–2000 K. A transmission coefficient is calculated by the multidimensional small-curvature tunneling (SCT) approximation. The final MS-CVT/SCT thermal rate constant was determined by combining a reaction rate calculation in the single-structural harmonic oscillator approximation (including tunneling) with the multi-structural anharmonicity torsional factor. The calculated forward rate constant agrees very well with experimentally-based evaluations of the high-pressure limit for the temperature range 300–1300 K, although it is a factor of 2.5–3.0 lower than the single-structural harmonic oscillator approximation over this temperature range. We anticipate that MS-VTST will be generally useful for calculating the reaction rates of complex molecules with multiple torsions.

An Improved Bubble Packing Method for Unstructured Grid Generation with Application to Computational Fluid Dynamics

An improved bubble packing method (BPM) is proposed to generate high-quality unstructured grids for prediction of the flow field in a domain with complex geometry. For a curved-boundary domain, bubble departure from the curved boundaries during the dynamic movement of bubbles can be avoided by using the mapping and the arc-length parameterization methods. Furthermore, the grid density of the whole region can be controlled effectively. Local mesh refinement is achieved by adding bubbles with different sizes to the real and artificial vertices of the domain, and vertex information is transferred to the inner nodes of the domain using the Shepard interpolation method. In order to validate the proposed BPM, a finite-volume solver on an unstructured collocated grid is developed to simulate both square and polar lid-driven cavity flows. The numerical simulation results agree well with the experimental data under different Reynolds numbers.