Number Of Grid Points Research Articles

Analysis of the accuracy of calculations using Duo and Level diatomic nuclear motion programs

Analysis of the accuracy of two programs widely-used for computing ro-vibrational spectra of diatomic molecules, namely Duo and Level, is presented. Using model systems for which analytic results are available it is shown that compared to Level, Duo gives similar or usually higher accuracy for line intensities, and is accurate for calculations of bound state energies and corresponding wavefunctions. Furthermore, Duo provides matrix elements accurate to about 10−4–10−5% relative to the analytic values, which is sufficient for developing of accurate methods for experimental determination of some macroscopic gas features, such as pressure, concentration, temperature, and so on; this level of accuracy can only be achieved with Level by significantly increasing the number of grid points in the calculation.

A nonlinear compressible flow disturbance formulation for adaptive mesh refinement wavepacket tracking in hypersonic boundary-layer flows

A new numerical approach for simulating nonlinear wavepackets in hypersonic boundary-layers is presented. The adaptive mesh refinement wavepacket tracking (AMR-WPT) method has been developed as an efficient alternative to conventional direct numerical simulations (DNS). The AMR-WPT method employs the nonlinear disturbances equations (NLDE), an overset dual mesh approach with higher-order interpolation, and adaptive mesh refinement (AMR) to track wavepackets in hypersonic boundary-layer flows. After introducing the numerical details, the method is used to simulate linear and nonlinear wavepackets for an axisymmetric M=9.81 straight cone and 2-D/3-D M=5.35 flat plate boundary-layer. The simulation results are compared against classical stability and transition prediction tools, such as linear stability theory (LST), parabolized stability equations (PSE) and DNS. It is demonstrated that the AMR-WPT method requires only about 10% of the number of grid points when compared to DNS of a nonlinear wavepacket inside a hypersonic flat plate boundary-layer flow.

Multi step structural health monitoring approaches in debonding assessment in a sandwich honeycomb composite structure using ultrasonic guided waves

This paper aims to investigate the use of ultrasonic guided wave (GW) propagation mechanism and the assessment of debonding in a sandwich composite structure (SCS) using a multi-step approach. Towards this, a series of GW propagation-based laboratory experiments and numerical simulations have been carried out on the SCS sample. The debonding regions of variable size and locations were assessed using a pre-defined network of piezoelectric lead zirconate transducers (PZT). Besides, several artificial masses were also placed in the SCS to validate the multi-step structural health monitoring (SHM) strategy. The SHM approach uses a proposed quick damage identification matrix maps and an improved elliptical wave processing (EWP) strategy of the registered GW signals to detect the locations of debonding and other damages in the SCS. The benefit of the proposed damage identification map is to locate the damaged area (sectors) quickly. This identification step is followed by applying the damage localization step using the improved EWP only on the previously identified damage sector region. The proposed EWP has shown the potential to effectively locate the hidden multiple debonding regions and damages in the SCS with a reduced number of calculations using a step-wise approach that uses only a selected number of grid points. The paper shows the effectiveness of the proposed approach based on data gathered from numerical simulations and experimental studies. Thus, using the above-mentioned SHM strategy debondings and damages present within and outside the sensor network are localized. The results were cross verified with nondestructive testing (NDT) methods such as infrared thermography and laser Doppler vibrometry.

2D acoustic wavefield simulation by improved stiffness and mass matrices of bilinear square element

Abstract In the frequency domain, we optimize the entries of the element stiffness matrix computed on a square element with the purpose of approximating the 2D acoustic wave equation with the second spatial order. The optimized matrices are computed from the minimization of the normalized phase velocity as a function of propagation angle and the number of grid points per wavelength, leading to a remarkable reduction in the numerical error. The optimized number of 4.1 sample points in a wavelength, with a maximum error in velocities <0.3%, offering the simulation of large-scale realistic models. The superior performance of the optimized matrices to those from the lumped, consistent and eclectic matrices, is evident from the numerical examples presented, with consequent reduction not only in the numerical dispersion/anisotropy but also in an improved resolution. It is noteworthy that optimized matrices give modeling results with better accuracy than models from the high-order spectral element method. Our methodology is easily extendable to the accurate discretizing of the general 3D elastic wave equation that optimizes the bandwidth of the impedance matrix, availing computational resources for accurately modeling the compressional and shear wave velocities, density, as well as multi-parameter inversion.

An affine generalized optimal scheme with improved free-surface expression using adaptive strategy for frequency-domain elastic wave equation

Effective frequency-domain numerical schemes are central for forward modeling and inversion of the elastic wave equation. The rotated optimal nine-point scheme is a highly utilized finite-difference (FD) numerical scheme. This scheme makes a weighted average of the derivative terms of the elastic wave equations in both the original and the rotated coordinate systems. In comparison with the classical nine-point scheme, it can simulate S-waves better and has higher accuracy at nearly the same computational cost. Nevertheless, this scheme limits the rotation angle to 45°; thus, the grid sampling intervals in the x- and z-directions need to be equal. Otherwise, the grid points will not lie on the axes, which dramatically complicates the scheme. Affine coordinate systems do not constrain axes to be perpendicular to each other, providing enhanced flexibility. Based on the affine coordinate transformations, we developed a new affine generalized optimal nine-point scheme. At the free surface, we applied the improved free-surface expression with an adaptive parameter-modified strategy. The new optimal scheme had no restriction that the rotation angle must be 45°. Dispersion analysis shows that our scheme can effectively reduce the required number of grid points per shear wavelength for both equal and unequal sampling intervals compared to the classical nine-point scheme. Moreover, this reduction improves with the increase of Poisson’s ratio. Three numerical examples demonstrate that our scheme can provide more accurate results than the classical nine-point scheme in terms of both the internal and free-surface grid points.

A simple and efficient numerical method for pricing discretely monitored early-exercise options

We present a simple, fast, and accurate method for pricing a variety of discretely monitored options in the Black-Scholes framework, including autocallable structured products, single and double barrier options, and Bermudan options. The method is based on a quadrature technique, and it employs only elementary calculations and a fixed one-dimensional uniform grid. The convergence rate is O(1/N4) and the complexity is O(MNlogN), where N is the number of grid points and M is the number of observation dates. Besides Black-Scholes, our method is also applicable to more general frameworks such as Merton’s jump diffusion model.

Out-of-Plane Free Vibration Analysis of Continuous Curved Girders with Combined Linetypes Using Differential Quadrature Element Method

In this study, the out-of-plane free vibration equations of circular curved and clothoid transition curved girders with various boundary conditions were derived. The continuous curved girder with combined linetypes was discretized into separate curved girder elements. Based on the differential quadrature element method (DQEM), the Chebyshev–Gauss–Labatto unequal mesh division was applied to discretize the vibration equations of the continuous curved girder into those of the curved girder elements, along with the boundary conditions, internal geometric compatibility conditions, and force balance conditions. The out-of-plane natural frequency equations were derived considering the boundary conditions by the substitution method. The free vibration was solved for continuous curved girders with combined linetypes of circular and clothoid transition curves. The effectiveness of the DQEM was verified, and the effect of the number of grid points on the accuracy of the solution was evaluated. In addition, the effects of the bending/torsional stiffness ratio, warping coefficient, and boundary conditions on the natural frequencies of continuous curved girders with combined linetypes were investigated. The results showed that the DQEM can be efficiently used to solve the free vibration of multispan continuous curved girders with various linetype combinations. The frequencies of the continuous curved girder with combined linetypes decrease with the weakening of boundary constraints, but increase with increasing warping coefficient and bending/torsional stiffness ratio.

Moving boundary truncated grid method for electronic nonadiabatic dynamics.

The moving boundary truncated grid method is developed to study the wave packet dynamics of electronic nonadiabatic transitions between a pair of diabatic potential energy surfaces. The coupled time-dependent Schrödinger equations (TDSEs) in the diabatic representation are integrated using adaptive truncated grids for both the surfaces. As time evolves, a variable number of grid points fixed in space are activated and deactivated without any advance information of the wave packet dynamics. Essential features of the truncated grid method are first illustrated through applications to three one-dimensional model problems, including the systems of single avoided crossing, dual avoided crossing, and extended coupling region with reflection. As a demonstration for chemical applications, the truncated grid method is then employed to study the dynamics of photoisomerization of retinal in rhodopsin described by a two-electronic-state two-dimensional model. To demonstrate the capability of the truncated grid method to deal with the electronic nonadiabatic problem in high dimensionality, we consider a multidimensional electronic nonadiabatic system in two, three, and four dimensions. The results indicate that the correct grid points are automatically activated to capture the growth and decay of the wave packets on both of the surfaces. Therefore, the truncated grid method greatly decreases the computational effort to integrate the coupled TDSEs for multidimensional electronic nonadiabatic systems.

A strongly coupled immersed boundary method for fluid-structure interaction that mimics the efficiency of stationary body methods

Strongly coupled immersed boundary (IB) methods solve the nonlinear fluid and structural equations of motion simultaneously for strongly enforcing the no-slip constraint on the body. Handling this constraint requires solving several large dimensional systems that scale by the number of grid points in the flow domain even though the nonlinear constraints scale only by the small number of points used to represent the fluid-structure interface. These costly large scale operations for determining only a small number of unknowns at the interface creates a bottleneck to efficiently time-advancing strongly coupled IB methods. In this manuscript, we present a remedy for this bottleneck that is motivated by the efficient strategy employed in stationary-body IB methods while preserving the favorable stability properties of strongly coupled algorithms—we precompute a matrix that encapsulates the large dimensional system so that the prohibitive large scale operations need not be performed at every time step. This precomputation process yields a modified system of small-dimensional constraint equations that is solved at minimal computational cost while time advancing the equations. We also present a parallel implementation that scales favorably across multiple processors. The accuracy, computational efficiency and scalability of our approach are demonstrated on several two dimensional flow problems. Although the demonstration problems consist of a combination of rigid and torsionally mounted bodies, the formulation is derived in a more general setting involving an arbitrary number of rigid, torsionally mounted, and continuously deformable bodies.

Staggered Mesh Method for Correlation Energy Calculations of Solids: Random Phase Approximation in Direct Ring Coupled Cluster Doubles and Adiabatic Connection Formalisms

We propose a staggered mesh method for correlation energy calculations of periodic systems under the random phase approximation (RPA), which generalizes the recently developed staggered mesh method for periodic second order Møller-Plesset perturbation theory (MP2) calculations [Xing; Li; Lin J. Chem. Theory Comput. 2021]. Compared to standard RPA calculations, the staggered mesh method introduces negligible additional computational cost. It avoids a significant portion of the finite-size error and can be asymptotically advantageous for quasi-1D systems and certain quasi-2D and 3D systems with high symmetries. We demonstrate the applicability of the method using two different formalisms: the direct ring coupled cluster doubles (drCCD) theory, and the adiabatic-connection (AC) fluctuation-dissipation theory. In the drCCD formalism, the second order screened exchange (SOSEX) correction can also be readily obtained using the staggered mesh method. In the AC formalism, the staggered mesh method naturally avoids the need of performing "head/wing" corrections to the dielectric operator. The effectiveness of the staggered mesh method for insulating systems is theoretically justified by investigating the finite-size error of each individual perturbative term in the RPA correlation energy, expanded as an infinite series of terms associated with ring diagrams. As a side contribution, our analysis provides proof that the finite-size error of each perturbative term of standard RPA and SOSEX calculations scales as , where Nk is the number of grid points in a Monkhorst-Pack mesh.

Ultrasound Imaging of Bottom Opening Crack Based on Frequency-Domain Reverse Time Migration

In 2-dimension (2D), a frequency-domain reverse time migration (FRTM) based on the optimal 9-point finite difference method (FDM) is introduced into the field of ultrasonic non-destructive testing (NDT). This study solves the challenge that conventional ultrasound imaging methods are difficult to image defects completely. In numerical investigations, the use of multiple reflection waves and the necessity of bottom identification are explained. In actual inspection, the method of bottom identification is given, and then the bottom opening crack (BOC) of different lengths in the aluminum block can be completely imaged. Furthermore, the conventional total focusing method (TFM) cannot image the sides of the BOCs using the same data as FRTM. Therefore, the quantitative analysis of BOC based on the complete imaging results of FRTM is more convincing. On this basis, the optimization methods for the number of grid points placed at each wavelength and the frequency calculation range are also given. On the premise of ensuring high image quality, the imaging computation time is reduced to 1/18 of the original, which improves the computing efficiency of the FRTM.

Numerical analysis of the effect of attack angle on the boundary layer flow along an archery bare shaft arrow

The velocity boundary layer, formed on an archery bare-shaft arrow, is computed by using two kinds of finite difference codes; an axisymmetric code and a three-dimensional code. Two types of arrow point, i.e., bullet point and streamlined point, are attached to a cylindrical arrow shaft. The Reynolds number, based on the shaft diameter, is varied in a range 10, 000 ≤ Re ≤ 20, 000, and the attack angle, α, is fixed at α = 0.0°. Critical assessment is performed on the grid dependency of velocity profiles, the difference in velocity profiles between axisymmetric and 3D computations, the effect of interpolation method on the interpolants by comparing the spline and the tri(bi)-cubic interpolations. The results of assessment indicates that the effect of varying the number of grid points, simulation codes and the interpolation method, considered in the present study, is reasonably small for the steady state boundary layers.

Concurrent calculation of radiative transfer in the atmospheric simulation in ECHAM-6.3.05p2

Abstract. ​​​​​​​The scalability of the atmospheric model ECHAM6 at low resolution, as used in palaeoclimate simulations, suffers from the limited number of grid points. As a consequence, the potential of current high-performance computing architectures cannot be used at full scale for such experiments, particularly within the available domain decomposition approach. Radiation calculations are a relatively expensive part of the atmospheric simulations, taking up to approximately 50 % or more of the total runtime. This current level of cost is achieved by calculating the radiative transfer only once in every 2 h of simulation. In response, we propose extending the available concurrency within the model further by running the radiation component in parallel with other atmospheric processes to improve scalability and performance. This paper introduces the concurrent radiation scheme in ECHAM6 and presents a thorough analysis of its impact on the performance of the model. It also evaluates the scientific results from such simulations. Our experiments show that ECHAM6 can achieve a speedup of over 1.9× using the concurrent radiation scheme. By performing a suite of stand-alone atmospheric experiments, we evaluate the influence of the concurrent radiation scheme on the scientific results. The simulated mean climate and internal climate variability by the concurrent radiation generally agree well with the classical radiation scheme, with minor improvements in the mean atmospheric circulation in the Southern Hemisphere and the atmospheric teleconnection to the Southern Annular Mode. This empirical study serves as a successful example that can stimulate research on other concurrent components in atmospheric modelling whenever scalability becomes challenging.

Some aspects in Kelvin-Helmholtz instability with and without Boussinesq approximation

The topic of this paper is the Kelvin-Helmholtz instability, a phenomenon which occurs on the interface of a stratified fluid, in the presence of a parallel shear flow, when there is a velocity and density difference across the interface of two adjacent layers. This paper focuses on a numerical simulation modelled by the Taylor-Goldstein equation, which represents a more realistic case compared to the basic Kelvin-Helmholtz shear flow. The Euler system is solved with new modelled smooth velocity and density profiles at the interface. The flux at cell boundaries is reconstructed by implementing a third order WENO (Weighted Essentially Non-Oscillatory) method. Next, a Riemann solver builds the fluxes at cell interfaces. The use of both Rusanov and HLLC solvers is investigated. Temporal discretization is done by applying the second order TVD (total variation diminishing) Runge-Kutta method on a uniform grid. Numerical simulations are performed comparatively for both Kelvin-Helmholtz and Taylor-Goldstein instabilities, on the same simulation domains. We find that increasing the number of grid points leads to a better accuracy in shear layer vortices visualization. Thus, we can conclude that applying the Taylor-Goldstein equation improves the realism in the general fluid instability modelling.

A fast estimator for quantifying the shape dependence of the 3D bispectrum

The dependence of the bispectrum on the size and shape of the triangle contains a wealth of cosmological information. Here we consider a triangle parameterization which allows us to separate the size and shape dependence. We have implemented an FFT based fast estimator for the three dimensional (3D) bin averaged bispectrum, and we demonstrate that it allows us to study the variation of the bispectrum across triangles of all possible shapes (and also sizes). The computational requirement is shown to scale as ∼ N g 3 log N g 3 where N g is the number of grid points along each side of the volume. We have validated the estimator using a non-Gaussian field for which the bispectrum can be analytically calculated. The estimated bispectrum values are found to be in good agreement (< 10 % deviation) with the analytical predictions across much of the triangle-shape parameter space. We also introduce linear redshift space distortion, a situation where also the bispectrum can be analytically calculated. Here the estimated bispectrum is found to be in close agreement with the analytical prediction for the monopole of the redshift space bispectrum.

Numerical Simulation of the Deflagration to Detonation Transition in a Tube with Repeated Obstacles: Experimental Scale Simulation Using the Artificial Thickened Flame Method

Abstract The Artificial Thickened Flame (ATF) method, which involves artificially increasing the flame thickness so as to simulate with a coarse grid resolution, is applied to reduce the computational cost of predicting the Deflagration to Detonation Transition (DDT) in a tube with repeated obstacles. While simulation results depended on the parameter N (the number of grid points in laminar flame thickness), it was found that N values of more than 10 may be excessive. The results show that the chosen simulation method predicts the flame speed as compared to a reference experiment and captures the detail of the strong ignitions near the corner between the obstacle and the sidewall. The present simulation also captures the wrinkle flame front structure during the acceleration of flame.

Spatio-temporal dependency of vegetation dynamics on climatic variables during 1982–2015 over India

The abnormally changing climate has a direct or indirect impact on the vegetation dynamics. The main objective here is to understand how the vegetation dynamics respond to climate change during the historical period, which could act as a clarion call for the future assessment as well. To accomplish the same, role of potential climatic variables viz., solar radiation, precipitation, maximum and minimum temperatures on changing normalized difference vegetation index (NDVI) is analysed. Vegetation dynamics largely depend on geographical conditions. This study is applied to whole India, comprising 1139 grid points over the period 1982–2015. This study made use of satellite derived longest available data record from Global Inventory Monitoring and Modelling System (GIMMS) and Climate data records (CRU) from climate research unit to study the spatio-temporal changes and assess the driving forces for change in vegetation dynamics. The trends of NDVI and all the climatic variables are analysed using modified Mann–Kendall approaches. Significant climatic trends along with NDVI trends are evidenced over India during period of analysis. Linear association between NDVI and each climatic variable are analysed based on Pearson correlation coefficient at monthly and seasonal scales. In this study, we have proposed spatio-temporal dependency modelling of vegetation dynamics (i.e., NDVI) on climatic variables. The stepwise linear regression analysis is performed for dependency modelling at grid level covering whole of India. Concurrent and Lag-1 climatic variables are used to consider temporal dependency, and for spatial dependency both average- and max-pooling-based climatic features are extracted. It is evident from the present study that consideration of spatio-temporal dependency could improve the modelling performance (of NDVI-climatic variable) to a great extent in more number of grid points. Spatio-temporal dependency modelling could achieve higher coefficient of determination (R 2 ) values (≥0.5) in 314 grids, which was possible only in 157 grids in case of simple dependency modelling.

Multilevel Techniques for the Solution of HJB Minimum-Time Control Problems

The solution of minimum-time feedback optimal control problems is generally achieved using the dynamic programming approach, in which the value function must be computed on numerical grids with a very large number of points. Classical numerical strategies, such as value iteration (VI) or policy iteration (PI) methods, become very inefficient if the number of grid points is large. This is a strong limitation to their use in real-world applications. To address this problem, the authors present a novel multilevel framework, where classical VI and PI are embedded in a full-approximation storage (FAS) scheme. In fact, the authors will show that VI and PI have excellent smoothing properties, a fact that makes them very suitable for use in multilevel frameworks. Moreover, a new smoother is developed by accelerating VI using Anderson’s extrapolation technique. The effectiveness of our new scheme is demonstrated by several numerical experiments.

Automated Temporal Tracking of Coherently Evolving Density Fronts in Numerical Models

AbstractOceanic density fronts can evolve, be advected, or propagate as gravity currents. Frontal evolution studies require methods to temporally track evolving density fronts. We present an automated method to temporally track these fronts from numerical model solutions. First, at all time steps contiguous density fronts are detected using an edge detection algorithm. A front event, defined as a set of sequential-in-time fronts representing a single time-evolving front, is then identified. At time stepi, a front is compared to each front at time stepi+ 1 to determine if the two fronts are matched. Anifront grid point is trackable if the minimum distance to thei+ 1 front falls within a range. Theifront is forward matched to thei+ 1 front when a sufficient number of grid points are trackable and the front moves onshore. A front event is obtained by forward tracking a front for multiple time steps. Within an event, the times that a grid point can be tracked is its connectivity and a pruning algorithm using a connectivity cutoff is applied to extract only the coherently evolving components. This tracking method is applied to a realistic 3-month San Diego Bight model solution yielding 81 front events with duration ≥ 7 h, allowing analyses of front event properties including occurrence frequency and propagation velocity. Sensitivity tests for the method’s parameters support that this method can be straightforwardly adapted to track evolving fronts of many types in other regions from both models and observations.

René 41 & waspaloy based comparative study for high pressure turbine blades used in turboshaft engines

This study aims to present the comparison between 2 different nickel-based superalloy materials, rené 41 and waspaloy through their application in designing a high-pressure (HP) turbine blade for a turboshaft engine. NACA 4-digit airfoil profile has been used to create the airfoil for the HP turbine blade. MATLAB software is used to create a graphical user interface for the selection of various parameters used to define the airfoil profile including the NACA airfoil number, angle of attack, number of grid points, grid type, trailing edge type, chord length, and the plot style. For this study, NACA 9618 airfoil is used to model a single HP turbine blade. Numerical flow analysis using vortex panel method (VPM) is carried out on the airfoil to obtain the flow characteristics. Further, several types of failure methodologies occurring in the turbine blades and standard industrial methods used to reduce failure cycles are discussed. Comparisons are carried out showing precipitation-hardened rené 41 exhibiting relatively larger Von-Mises stress values in comparison with age-hardened waspaloy at elevated temperature for thermo-mechanical loading in the blade structure. Steady-state simulation at 39,000 rpm yields maximum temperature values of 1056 °C and 1220 °C for rené 41 and waspaloy respectively, also yielding relatively close values for critical displacement at the shroud section of the blade. Analysis models are proposed to determine the effect of mechanical and thermal loads on the high-pressure turbine blade design.