Nonuniform Discretization Research Articles

Non-Uniform Discretization-based Ordinal Regression for Monocular Depth Estimation of an Indoor Drone

At present, the main methods of solving the monocular depth estimation for indoor drones are the simultaneous localization and mapping (SLAM) algorithm and the deep learning algorithm. SLAM requires the construction of a depth map of the unknown environment, which is slow to calculate and generally requires expensive sensors, whereas current deep learning algorithms are mostly based on binary classification or regression. The output of the binary classification model gives the decision algorithm relatively rough control over the unmanned aerial vehicle. The regression model solves the problem of the binary classification, but it carries out the same processing for long and short distances, resulting in a decline in short-range prediction performance. In order to solve the above problems, according to the characteristics of the strong order correlation of the distance value, we propose a non-uniform spacing-increasing discretization-based ordinal regression algorithm (NSIDORA) to solve the monocular depth estimation for indoor drone tasks. According to the security requirements of this task, the distance label of the data set is discretized into three major areas—the dangerous area, decision area, and safety area—and the decision area is discretized based on spacing-increasing discretization. Considering the inconsistency of ordinal regression, a new distance decoder is produced. Experimental evaluation shows that the root-mean-square error (RMSE) of NSIDORA in the decision area is 33.5% lower than that of non-uniform discretization (NUD)-based ordinal regression methods. Although it is higher overall than that of the state-of-the-art two-stream regression algorithm, the RMSE of the NSIDORA in the top 10 categories of the decision area is 21.8% lower than that of the two-stream regression algorithm. The inference speed of NSIDORA is 3.4 times faster than that of two-stream ordinal regression. Furthermore, the effectiveness of the decoder has been proved through ablation experiments.

A new line-by-line methodology based on the spectral contributions of the bands

This study presents a new methodology to solve thermal radiation in participating gases, the spectrally reduced integration (SRI), which employs a non-uniform spectral discretization to considerably reduce the computational cost of the line-by-line (LBL) solution while still maintaining good levels of accuracy. This non-uniform spectral mesh is generated from discretization schemes based on the spectral contributions of the bands, which apply less or more refined spectral resolution depending on the importance of the bands to the radiative transfer. Based on this methodology, several schemes were initially proposed and evaluated for a homogeneous H2O-CO2 mixture test case. Results showed that the most refined schemes were able to generate highly accurate results three to five times faster than the reference LBL solution, while the coarser schemes were still interesting alternatives for faster but still reliable calculations. In fact, a compromise between the accuracy of the SRI solution and the total number of spectral intervals was observed in all the tested schemes. The SRI method also presented similarly promising results when solving non-homogeneous test cases. Finally, the present study also develops two alternatives to address the need of a priori evaluation of the spectral contributions of the bands when using the discretization schemes. These two methodologies produced surprisingly accurate solutions of the non-homogeneous test cases, each with its own advantages and disadvantages, and could be applied together to further simplify the use of the SRI in highly accurate coupled combustion problems.

Derivation of dual-horizon state-based peridynamics formulation based on Euler–Lagrange equation

The numerical solution of peridynamics equations is usually done by using uniform spatial discretisation. Although implementation of uniform discretisation is straightforward, it can increase computational time significantly for certain problems. Instead, non-uniform discretisation can be utilised and different discretisation sizes can be used at different parts of the solution domain. Moreover, the peridynamic length scale parameter, horizon, can also vary throughout the solution domain. Such a scenario requires extra attention since conservation laws must be satisfied. To deal with these issues, dual-horizon peridynamics was introduced so that both non-uniform discretisation and variable horizon sizes can be utilised. In this study, dual-horizon peridynamics formulation is derived by using Euler–Lagrange equation for state-based peridynamics. Moreover, application of boundary conditions and determination of surface correction factors are also explained. Finally, the current formulation is verified by considering two benchmark problems including plate under tension and vibration of a plate.

Numerical solutions for blood flow in elastic vessels

We consider the differential–algebraic system for the blood flow and pressure in the systemic arteries. By the operator splitting method, we transform the system into the hyperbolic one, introduce the bicharacteristics, and perform the time–space nonuniform discretization, obtaining the innovative difference scheme. Our results are illustrated with numerical experiments.

Determination of horizon size in state-based peridynamics

Peridynamics is based on integro-differential equations and has a length scale parameter called horizon which gives peridynamics a non-local character. Currently, there are three main peridynamic formulations available in the literature including bond-based peridynamics, ordinary state-based peridynamics and non-ordinary state-based peridynamics. In this study, the optimum horizon size is determined for ordinary state-based peridynamics and non-ordinary state-based peridynamics formulations by using uniform and non-uniform discretisation under dynamic and static conditions. It is shown that the horizon sizes selected as optimum sizes for uniform discretisation can also be used for non-uniform discretisation without introducing significant error to the system. Moreover, a smaller horizon size can be selected for non-ordinary state-based formulation which can yield significant computational advantage. It is also shown that same horizon size can be used for both static and dynamic problems.

A new type of peridynamics: Element-based peridynamics

The general peridynamic theories are constructed using the interactions of particles in a horizon. The interactions of particles can be connected by using spring, rod and beam, etc. Element-based peridynamics (EBPD) is proposed in the present study, in which the interactions of particles are expressed by using elements in a horizon, and the basic element concepts in the continuum mechanics are reserved exactly. The force density of EBPD is described using 2-node rod elements for one-dimensional (1D) problems, 3-node triangle elements for two-dimensional (2D) problems and 4-node tetrahedron elements for three-dimensional (3D) problems. The EBPD equations are derived from the principle of the minimum potential energy. The treatments of surface effects, boundary conditions, failure criterion and solving process of the EBPD are provided in detail. The examples of the 1D singular bar, 2D stationary crack near-tip solution, 2D plate crack propagation and 3D cubic demonstrate the effectivity of the proposed EBPD. The proposed EBPD model is used for elasticity, and defined new nonlocal stress and strain expressions. In addition, no restriction of Poisson’s ratio and no zero energy modes are included in EBPD. Furthermore, EBPD has the potential to study the nonuniform discretization, varying horizons, plasticity, thermodynamic and anisotropic material. It is also convenient to couple with FEM to reduce the computational burden.

Physical Cauer circuits in nonlinear eddy-current modeling

The paper re-examines the use of physical Cauer circuits (CCs) for modeling magnetization processes in a ferromagnetic lamination and in a hollow thick-wall cylinder taking into account magnetic nonlinearity of their materials. The saturation is shown to be the main reason for the absence of conventional skin effect inherited from the linear eddy-current theory. It is shown that flux densities in the outer and inner layers of a lamination become comparable in magnitude when the material is operated at moderate and high average flux densities. In this typical case and contrary to common practice, there is no need to use CCs with non-uniformly discretized sections. It is shown that simple uniform CCs can be more accurate than circuits obtained using non-uniform discretization.

Stress analysis of sandwich plates with functionally graded cores using peridynamic differential operator and refined zigzag theory

Abstract This study presents a novel non-local model for the stress analysis of sandwich plates with a functionally graded core using Peridynamic Differential Operator (PDDO) and Refined Zigzag Theory (RZT). The through-thickness material properties of the functionally graded cores were tailored by means of mixing rules. The PDDO converts the equilibrium equations of the RZT from the differential form into the integral form. This makes the PDDO capable of solving the local differential equations accurately. The RZT is very suitable for the stress analysis, especially for thick and moderately thick plates. It contains only seven kinematic variables and eliminates the use of the shear correction factors. A typical sandwich structure consists of a soft core and stiff orthotropic face-sheets. The mismatch of the stiffness at the core and face sheet interfaces results in an increase in the interfacial shear stresses, leading to the core-face sheet delamination. The interfacial stresses can be mitigated by functionally grading the material properties of the core through the thickness. The PD-RZT stress and displacement predictions were compared with the analytical solutions by using the uniform and non-uniform mesh discretizations and good agreements were achieved. It was observed that the functionally graded cores offered some advantages with respect to the classical cores and minimized the stress concentrations at the interface of the core and the face sheets.

Numerical methods for the two-dimensional Fokker-Planck equation governing the probability density function of the tempered fractional Brownian motion

In this paper, we study the numerical schemes for the two-dimensional Fokker-Planck equation governing the probability density function of the tempered fractional Brownian motion. The main challenges of the numerical schemes come from the singularity in the time direction. When $0<H<0.5$, a change of variables $\partial \left(t^{2H}\right)=2Ht^{2H-1}\partial t$ avoids the singularity of numerical computation at $t=0$, which naturally results in nonuniform time discretization and greatly improves the computational efficiency. For $0.5<H<1$, the time span dependent numerical scheme and nonuniform time discretization are introduced to ensure the effectiveness of the calculation and the computational efficiency. By numerically solving the corresponding Fokker-Planck equation, we obtain the mean squared displacement of stochastic processes, which conforms to the characteristics of the tempered fractional Brownian motion.

Weak form of bond-associated non-ordinary state-based peridynamics free of zero energy modes with uniform or non-uniform discretization

The non-ordinary state-based peridynamics (NOSB PD) is attractive because of its ability to employ existing constitutive relations for material models. The deformation gradient tensor and the force density vector appearing in the equilibrium equations are expressed in terms of nonlocal integrals. The definitions of these nonlocal integrals affect the accuracy and stability of PD predictions. Therefore, this study introduces a more accurate representation of the deformation gradient and the bond associated (BA) force density vector by using the peridynamic differential operator (PDDO). Also, it presents the weak form of BA-NOSB PD governing equations in order to impose natural and essential boundary conditions without the use of Lagrange multipliers for implicit and explicit analysis. By considering a two-dimensional rectangular plate with and without a hole under tension, the numerical results demonstrate the accuracy of BA-NOSB PD with no oscillations and zero energy modes.

A geometric full multigrid method and fourth-order compact scheme for the 3D Helmholtz equation on nonuniform grid discretization

A full multigrid method and fourth-order compact difference scheme is designed to solve the 3D Helmholtz equation on unequal mesh size. Three dimensional restriction and prolongation operators of the multigrid method on unequal grids could be constructed based on volume law. Two numerical experiments are implemented, and the results show the computational efficiency and accuracy of the full multigrid method. The study also illustrates that the full multigrid method with fourth-order compact difference scheme has great advantages in computation, which has been time consuming, and in iterative convergence efficiency.

Analytical sensitivity computation using collocation method with non-uniform mesh discretisation for numerical solutions of optimal control problems

ABSTRACT A numerical computational method based on the orthogonal collocation on finite element with sparse variable time points is proposed for the optimal control problems (OCPs). This approach generates a non-uniform mesh in the whole time horizon to obtain a better approximation than the uniform discretisation methods. The original problem is converted to a nonlinear programming problem, where only the discretised control parameters and the interval parameters used for the jumps of control are treated as optimisation variables. To improve the convergence rate and the efficiency, the analytical first-order and second-order sensitivities of state at collocation points with respect to the control, as well as the variable interval parameters are derived, respectively. The proposed method is illustrated by testing two OCPs with different complexities. The detailed comparisons between the proposed method and other methods are also carried out. The research results reveal the effectiveness of proposed approach.

An efficient preconditioner for adaptive Fast Multipole accelerated Boundary Element Methods to model time-harmonic 3D wave propagation

This paper presents an efficient algebraic preconditioner to speed up the convergence of Fast Multipole accelerated Boundary Element Methods (FM-BEMs) in the context of time-harmonic 3D wave propagation problems and in particular the case of highly non-uniform discretizations. Such configurations are produced by a recently-developed anisotropic mesh adaptation procedure that is independent of partial differential equation and integral equation. The new preconditioning methodology exploits a complement between fast BEMs by using two nested GMRES algorithms and rapid matrix–vector calculations. The fast inner iterations are evaluated by a coarse hierarchical matrix (H-matrix) representation of the BEM system. These inner iterations produce a preconditioner for FM-BEM solvers. It drastically reduces the number of outer GMRES iterations. Numerical experiments demonstrate significant speedups over non-preconditioned solvers for complex geometries and meshes specifically adapted to capture anisotropic features of a solution, including discontinuities arising from corners and edges.

An accurate and efficient numerical method for solving linear peridynamic models

• An accurate and efficient numerical method for linear peridynamic models. • Coupling of time-splitting technique and localized radial basis function method. • Valid for any dimension, irregular domain, large time step, and long execution time. • More accurate, efficient and robust than existing numerical methods. • Performance of the method demonstrated with some representative examples. In this paper, we combine the recently developed localized radial basis functions-based pseudo-spectral method with the time-splitting technique to solve a linear wave equation arising from modelling the wave dynamics using peridynamic formulation in continuum mechanics . Specifically, we adopt this combined method for solving a Hamiltonian ordinary differential equation system, which is equivalent to the original linear peridynamic equation after introducing a new variable. The proposed approach inherits advantages of these two related methods in space and time: (1) offering high accuracy and efficiency in the solution of the problem under irregular domains for both uniform and non-uniform discretizations ; (2) extending the applicability of the approach to multi-dimensions; and (3) maintaining a good approximation for problems at large time-step and long time integration. Numerical results indicate that the proposed method is simple, accurate, efficient, and stable for solving various linear peridynamic problems.

Calibration of Dupire local volatility model using genetic algorithm of optimization

The problem of calibration of local volatility model of Dupire has been formalized. It uses genetic algorithm as alternative to regularization approach with further application of gradient descent algorithm. Components that solve Dupire’s partial differential equation that represents dynamics of underlying asset’s price within Dupire model have been built. This price depends in particular on values of volatility parameters. Local volatility is parametrized in two dimensions (by Dupire model): time to maturity of the option and strike price (execution price). On maturity axis linear interpolation is used while on strike axis we use B-Splines. Genetic operators of mutation and selection are then applied to parameters of B-Splines. Resulting parameters allow us to obtain the values of local volatility both in knot points and intermediate points using interpolation techniques. Then we solve Dupire equation and calculate model values of option prices. To calculate cost function we simulate market values of option prices using classic Black-Scholes model. An experimental research to compare simulated market volatility and volatility obtained by means of calibration of Dupire model has been conducted. The goal is to estimate the precision of the approach and its usability in practice. To estimate the precision of obtained results we use a measure based on average deviation of modeled local volatility from values used to simulate market prices of the options. The research has shown that the approach to calibration using genetic algorithm of optimization requires some additional manipulations to achieve convergence. In particular it requires non-uniform discretization of the space of model parameters as well as usage of de Boor interpolation. Value 0.07 turns out to be the most efficient mutation parameter. Using this parameter leads to quicker convergence. It has been proved that the algorithm allows precise calibration of local volatility surface from option prices.

Peridynamic least squares minimization

This study presents the peridynamic approximation of a field variable and its temporal and spatial derivatives based on least squares minimization. Such capability permits the conversion of local form of differentiation to its nonlocal analytical integral form. It enables the analysis of discrete and scattered data for numerical differentiation and approximation of the field variable without employing any special techniques. Also, it enables the numerical solution of differential field equations with single and multiple variables in a computational domain of either uniform or nonuniform discretization. The implicit solution to the discrete form of the differential equations can be achieved by employing standard techniques for solving sparse non-symmetric systems. The accuracy of this approach is demonstrated by considering numerical differentiation of discrete data, and solving ordinary and partial differential equations with particular characteristics.

An Improved Imaging Algorithm for High-Resolution Spotlight SAR with Continuous PRI Variation Based on Modified Sinc Interpolation.

This paper focuses on an improved imaging algorithm for spotlight synthetic aperture radar (SAR) with continuous Pulse Repetition Interval (PRI) variation in extremely high-resolution. Conventional SAR systems are limited in that a wide swath cannot be achieved with a high azimuth resolution in the meantime. This limitation can be overcome by Pulse Repetition Frequency (PRF) variation in a SAR system. However, there are problems such as the ambiguities of point targets or extended targets caused by nonuniform sampling. A reconstructive method, Nonuniform Discrete Fourier Transform (NUDFT) has been presented in the current literature, but it is rather computationally expensive. In this paper, a modified sinc interpolation based on NUDFT is proposed, which is used to reconstruct the uniformly sampled echo in time domain. Since the interpolation kernel length is relatively short, it is more computationally efficient. Then, the two-step processing approach combined with the modified sinc interpolation is further presented, which has much better accuracy than that combined with the conventional sinc interpolation. Both the simulated data and the extracted GF-3 data experiment demonstrate the validity and accuracy of the proposed approach.

Improved Algorithm of Boundary Integral Equation Approximation in 2D Vortex Method for Flow Simulation Around Curvilinear Airfoil

The problem of the accuracy improving is considered for vortex method. The general Galerkin-type approach is considered for the numerical solution of the boundary integral equation. It is shown that the airfoil surface line representation as a polygon consisting of rectilinear panels can lead to incorrect behavior of the numerical solution of the boundary integral equation with respect to unknown vortex sheet intensity, especially in case of considerably different lengths of the neighboring panels. However, in this case, the exact analytical formulae are derived for the coefficients of the linear algebraic system, which approximate the integral equation. The approach is developed which makes it possible to take into account explicitly the curvature of the airfoil surface line. The approximate analytical formulae are derived for the coefficients of the linear system, which are expressed through the integrals of the fractions, which denominators can be equal to zero. Numerical example is considered for the elliptic airfoil in the incident flow at its essentially non-uniform discretization. The suggested algorithm has acceptable numerical complexity, however it permits to obtain rather good numerical results.

A comparison study on peridynamic models using irregular non-uniform spatial discretization

The applicability of peridynamic models to problems with irregularly non-uniformly discretized solution domain is critical. In this study, a systematic comparison study on results predicted using eight different peridynamic models, including bond-based, ordinary state-based and non-ordinary state-based mechanics and heat conduction models, for three different types of mechanical problems, including thermal, mechanics and coupled thermo-mechanics, with irregular non-uniform spatial discretization is performed. It is found that for the case of irregular but semi-uniform spatial discretization, all these models yield good predictions compared to analytical local solutions. For the case of irregular and non-uniform spatial discretization, models formulated specifically for this configuration give much better results than the conventional formulations which do not consider the neighborhood difference among material points in the spatial discretization. For either cases of spatial discretization, the bond-associated correspondence material model predicts the most accurate results.

Markov Chain Investigation of Discretization Schemes and Computational Cost Reduction in Modeling Photon Multiple Scattering

Establishing fast and reversible photon multiple scattering algorithms remains a modeling challenge for optical diagnostics and noise reduction purposes, especially when the scattering happens within the intermediate scattering regime. Previous work has proposed and verified a Markov chain approach for modeling photon multiple scattering phenomena through turbid slabs. The fidelity of the Markov chain method has been verified through detailed comparison with Monte Carlo models. However, further improvement to the Markov chain method is still required to improve its performance in studying multiple scattering. The present research discussed the efficacy of non-uniform discretization schemes and analyzed errors induced by different schemes. The current work also proposed an iterative approach as an alternative to directly carrying out matrix inversion manipulations, which would significantly reduce the computational costs. The benefits of utilizing non-uniform discretization schemes and the iterative approach were confirmed and verified by comparing the results to a Monte Carlo simulation.