Large Number Of Grid Points Research Articles

High-Resolution Identification of Sound Sources Based on Sparse Bayesian Learning with Grid Adaptive Split Refinement

Sound source identification technology based on a microphone array has many application scenarios. The compressive beamforming method has attracted much attention due to its high accuracy and high-resolution performance. However, for the far-field measurement problem of large microphone arrays, existing methods based on fixed grids have the defect of basis mismatch. Due to the large number of grid points representing potential sound source locations, the identification accuracy of traditional grid adjustment methods also needs to be improved. To solve this problem, this paper proposes a sound source identification method based on adaptive grid splitting and refinement. First, the initial source locations are obtained through a sparse Bayesian learning framework. Then, higher-weight candidate grids are retained, and local regions near them are split and updated. During the iteration process, Green’s function and the source strength obtained in the previous iteration are multiplied to get the sound pressure matrix. The robust principal component analysis model of the Gaussian mixture separates and replaces the sound pressure matrix with a low-rank matrix. The actual sound source locations are gradually approximated through the dynamically adjusted sound pressure low-rank matrix and optimized grid transfer matrix. The performance of the method is verified through numerical simulations. In addition, experiments on a standard aircraft model are conducted in a wind tunnel and speakers are installed on the model, proving that the proposed method can achieve fast, high-precision imaging of low-frequency sound sources in an extensive dynamic range at long distances.

A numerically robust and stable time–space pseudospectral approach for multidimensional generalized Burgers–Fisher equation

We study the pseudospectral discretization of a nonlinear multidimensional generalized Burgers–Fisher equation. The main objective of this study is to establish that the implementation of the pseudospectral method in time holds equal importance to its application in space when addressing nonlinearity. Typically, the literature recommends using a large number of grid points in the temporal direction to ensure satisfactory outcomes. However, the fact that we obtained excellent accuracy with a relatively small number of temporal grid points is a notable finding in this study. The Chebyshev–Gauss–Lobatto (CGL) points serve as the foundation for the recommended method. By applying the proposed method to the problem, a system of algebraic equations is obtained, which is subsequently solved using the Newton–Raphson technique. We have conducted stability analysis and error estimation for the proposed method. Different researchers’ considerations on test problems have been explored to illustrate the robustness and practicality of the approach presented. The results obtained using the proposed method demonstrate a high level of accuracy, surpassing the existing solutions by a significant margin.

A New Broadening Technique of the Numerically Unresolved Solar Transition Region and Its Effect on the Spectroscopic Synthesis Using Coronal Approximation

Abstract The transition region is a thin layer of the solar atmosphere that controls the energy loss from the solar corona. Large numbers of grid points are required to resolve this thin transition region fully in numerical modeling. In this study, we propose a new numerical treatment, called LTRAC, which can be easily extended to the multidimensional domains. We have tested the proposed method using a one-dimensional hydrodynamic model of a coronal loop in an active region. The LTRAC method enables modeling of the transition region with a numerical grid size of 50–100 km, which is about 1000 times larger than the physically required value. We used the velocity differential emission measure to evaluate the possible effects on the optically thin emission. Lower-temperature emissions were better reproduced by the LTRAC method than by previous methods. Doppler shift and nonthermal width of the synthesized line emission agree with those from a high-resolution reference simulation within an error of several kilometers per second above the formation temperature of 105 K.

A routing and scheduling problem for cross-docking networks with perishable products, heterogeneous vehicles and split delivery

Cross-docking is an effective logistics strategy that has long been employed in supply chains with perishable products, where short processing time of goods is required. In this paper, a new multi-objective mixed-integer programming (MIP) model is proposed for scheduling and routing of heterogeneous vehicles carrying perishable goods across multiple cross-docking systems. Inspired by the actual business operations of a large supermarket chain, multiple cross-docks with simultaneous split pick-up and delivery tasks are considered, each of which processing either general or perishable products. In addition, all pick-up/delivery (P/D) requests for perishable products have a deadline that must be met. The main objectives of the developed model are to reduce distribution costs, accelerate distribution processing time, and maximize the cross-docking network’s capacity utilization rate. To overcome the complexity of the formulated problem, a novel hybrid solution method, namely AUGMECON2-VIKOR is developed and used in the case-study. The results reveal that AUGMECON2-VIKOR outperforms AUGMECON2, particularly when a larger number of grid points are considered in the proposed solution algorithm.

A GEOMETRICALLY CONVERGENT PSEUDO–SPECTRAL METHOD FOR MULTI–DIMENSIONAL TWO–SIDED SPACE FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS

<abstract> In this study, we present a geometrically convergent numerical method for solving multi-dimensional two-sided space-time fractional differential equations. The approach represents the solutions of the differential equations in terms of the shifted Chebyshev polynomials. The expansions are evaluated at the shifted Chebyshev-Gauss-Lobatto nodes. We present the approximations for left-sided integration, and the left- and right- sided differentiation. The performance of the method is demonstrated using some two-sided space fractional partial differential equations in one and two dimensions. The numerical results obtained show that the method is accurate and computationally efficient. A theoretical analysis of the convergence of the method is presented, where it is shown that, given a continuously differentiable solution, the numerical solution converges for a sufficiently large number of grid points. </abstract>

High Order Conservative Semi-Lagrangian Scheme for the BGK Model of the Boltzmann Equation

In this paper, we present a conservative semi-Lagrangian finite-difference scheme for the BGK model. Classical semi-Lagrangian finite difference schemes, coupled with an L-stable treatment of the collision term, allow large time steps, for all the range of Knudsen number. Unfortunately, however, such schemes are not conservative. There are two main sources of lack of conservation. First, when using classical continuous Maxwellian, conservation error is negligible only if velocity space is resolved with sufficiently large number of grid points. However, for a small number of grids in velocity space such error is not negligible, because the parameters of the Maxwellian do not coincide with the discrete moments. Secondly, the non-linear reconstruction used to prevent oscillations destroys the translation invariance which is at the basis of the conservation properties of the scheme. As a consequence the schemes show a wrong shock speed in the limit of small Knudsen number. To treat the first problem and ensure machine precision conservation of mass, momentum and energy with a relatively small number of velocity grid points, we replace the continuous Maxwellian with the discrete Maxwellian introduced by Mieussens. The second problem is treated by implementing a conservative correction procedure based on the flux difference form. In this way we can construct a conservative semi-Lagrangian scheme which is Asymptotic Preserving (AP) for the underlying Euler limit, as the Knudsen number vanishes. The effectiveness of the proposed scheme is demonstrated by extensive numerical tests.

A Comparative Study of Cubic B-spline-Based Quasi-interpolation and Differential Quadrature Methods for Solving Fourth-Order Parabolic PDEs

In this work, we present two approaches for simulation of fourth-order parabolic partial differential equations. In the first method, cubic B-spline quasi-interpolation is used to approximate the spatial derivative of the dependent variable and forward difference to approximate the time derivative. In the second method, we have used modified cubic B-spline functions-based differential quadrature method (DQM) for space discretization to get a system of ODEs and then this system is solved by SSP-RK43 method to get the results at knots. The numerical results demonstrate the accuracy of the proposed method. The stability analysis of the methods has also been discussed. It is observed that quasi-interpolation-based method is unconditionally stable, whereas for DQM, the stability has to be checked for a large number of space points. Moreover, for the small number of grid points, DQM gives better results, while for a large number of grid points, quasi-interpolation-based method is better.

Random Force Perturbations: A New Extension of the Cell Perturbation Method for Turbulence Generation in Multiscale Atmospheric Boundary Layer Simulations

AbstractCoupling between mesoscale and large eddy simulation (LES) is critically important for many atmospheric model applications, from predictions of wind energy to fire propagation. The grid‐nesting technique enables bridging between vastly different scales without incurring prohibitive computational costs. However, the transition from coarser to finer resolutions often requires a large number of grid points from inflow boundaries for the development of fine‐scale turbulence features in the LES domain. Recently, the cell perturbation method (CPM) was developed to reduce the turbulence development region with high computational efficiency. Herein, we explore a new method based on the CPM that uses force perturbations in both the horizontal and vertical directions (Force Cell Perturbation Method) instead of the potential temperature perturbations in the original CPM, as an attempt to further explore the performance of the random perturbation techniques. This approach is tested for a neutral and a convective atmospheric boundary layer under idealized conditions. Overall, similar performance is found between the optimal configurations of the CPM and the Force Cell Perturbation Method pointing to the robustness of this family of perturbation methods in accelerating turbulence generation in nested domains. Vertical force perturbations performed better than horizontal force perturbations for both atmospheric stability conditions. The CPM performed best under convective stability conditions. The combination of the force and potential temperature perturbations is found to provide no additional performance improvement over the stand‐alone application of the individual methods.

A positivity preserving adaptive moving mesh method for cancer cell invasion models

In this paper, we present an efficient adaptive moving mesh finite difference method for the modeling of early stage cancer cell invasion of tissue or extracellular matrix (ECM). The cancer cell invasion model is a nonlinear system of reaction–diffusion-taxis partial differential equations (PDEs) describing the time evolution of the cancer cells density and concentrations of proteins of the ECM. The solution of the model exhibits very rapid variations at the boundary of the healthy and cancer cells. Thus, using a uniform grid method to solve the cancer cell invasion model requires a very large number of grid points in order to resolve the steep gradients of the solution precisely. As a result, the computation can become prohibitively expensive and inefficient. In this effort, we propose a positivity preserving scheme for the spatial discretization of the model on an adaptive mesh to accurately capture the sharp structures of the solution. The adaptive mesh is generated by a coordinate transformation obtained as the solution of the optimal mass transfer problem. Several numerical experiments are presented to demonstrate the performance of the adaptive mesh method for solving the cancer cell invasion model. The numerical results show that the proposed adaptive mesh method is effective in capturing and predicting the correct behavior of the cancer cells movement within the ECM.

Physically based visual simulation of the Lattice Boltzmann method on the GPU: a survey

The rapid increase in performance, programmability, and availability of graphics processing units (GPUs) has made them a compelling platform for computationally demanding tasks in a wide variety of application domains. One of these is real-time computational fluid dynamics, which are computationally expensive due to a large number of grid points that require calculations. One commonly used tool to simulate fluid flows is the Lattice Boltzmann method (LBM), mainly due to its simpler formulation when compared to solving the Navier---Stokes equations, and because of its scalability on parallel processing systems. In this paper, we give an up-to-date survey on the research regarding the LBM for fluid simulation using GPUs. We discuss how the method was implemented with different GPU architectures and software frameworks, focusing on optimization techniques and their performance. Additionally, we mention some applications of the method in different areas of study.

Novel Discrete Singular Convolution for High-Frequency Vibration Analysis of Structural Elements

The prediction and control of high-frequency vibrations is of crucial importance to aerostructures. However, developing accurate and numerically stable methods for the analysis of high-frequency vibration is an extremely challenging task. In this paper, a novel discrete singular convolution algorithm is presented for efficient analysis of high-frequency vibration of structural elements. A nonregularized Lagrange’s delta sequence kernel based on harmonic functions is adopted. Taylor series expansion method is used to eliminate the degrees of freedom at fictitious points outside the physical domain, and thus various combinations of boundary conditions can be applied rigorously. Free vibration of beams and rectangular plates with and without two adjacent free edges is solved, and the results are compared with either exact solutions or results obtained by the differential quadrature method using much larger number of grid points. It is demonstrated that the performance of the proposed method for predicting high mode frequencies is excellent and better than existing algorithms. For the beam with both ends simply supported, all mode frequencies obtained by proposed method are very accurate. Such excellent feature of the algorithm was never reported before.

Using the Tensor-Train Approach to Solve the Ground-State Eigenproblem for Hydrogen Molecules

We consider the Born--Oppenheimer approximation of the Schrodinger equation for two hydrogen atoms in the case of large separation distances. We show that the Feschbach--Schur perturbation method can be used to solve the problem for the difference between a given separation and infinite separation. This leads to a simplified problem which can be solved iteratively. We show that this iteration converges for sufficiently large separation distances and we solve the arising sequence of six-dimensional PDEs with a finite element method in combination with low-rank tensor techniques to make the computations tractable. In particular we show how the discretized problems can be represented and solved in the tensor-train format. Since the storage and computational complexity of this format scale linearly in the dimension, a very large number of grid points can be employed, which leads to accurate approximations of the ground-state energy and ground-state wavefunction. Various numerical experiments show the performa...

Numerical Methods for Solving the Hartree-Fock Equations of Diatomic Molecules II

AbstractIn order to solve the partial differential equations that arise in the Hartree- Fock theory for diatomicmolecules and inmolecular theories that include electron correlation, one needs efficient methods for solving partial differential equations. In this article, we present numerical results for a two-variablemodel problem of the kind that arises when one solves the Hartree-Fock equations for a diatomic molecule. We compare results obtained using the spline collocation and domain decomposition methods with third-order Hermite splines to results obtained using the more-established finite difference approximation and the successive over-relaxation method. The theory of domain decomposition presented earlier is extended to treat regions that are divided into an arbitrary number of subregions by families of lines parallel to the two coordinate axes. While the domain decomposition method and the finite difference approach both yield results at the micro-Hartree level, the finite difference approach with a 9- point difference formula produces the same level of accuracy with fewer points. The domain decompositionmethod has the strength that it can be applied to problemswith a large number of grid points. The time required to solve a partial differential equation for a fine grid with a large number of points goes down as the number of partitions increases. The reason for this is that the length of time necessary for solving a set of linear equations in each subregion is very much dependent upon the number of equations. Even though a finer partition of the region has more subregions, the time for solving the set of linear equations in each subregion is very much smaller. This feature of the theory may well prove to be a decisive factor for solving the two-electron pair equation, which – for a diatomic molecule – involves solving partial differential equations with five independent variables. The domain decomposition theory also makes it possible to study complex molecules by dividing them into smaller fragments that are calculated independently. Since the domain decomposition approachmakes it possible to decompose the variable space into separate regions in which the equations are solved independently, this approach is well-suited to parallel computing.

An efficient and positivity‐preserving layer method for modeling radiation belt diffusion processes

AbstractAn efficient and positivity‐preserving layer method is introduced to solve the radiation belt diffusion equation and is applied to study the bounce resonance interaction between relativistic electrons and magnetosonic waves. The layer method with linear interpolation, denoted by LM‐L (layer method‐linear), requires the use of a large number of grid points to ensure accurate solutions. We introduce a monotonicity‐ and positivity‐preserving cubic interpolation method to be used with the Milstein‐Tretyakov layer method. The resulting method, called LM‐MC (layer method‐monotone cubic), can be used to solve the radiation belt diffusion equation with a much smaller number of grid points than LM‐L while still being able to preserve the positivity of the solution. We suggest that LM‐MC can be used to study long‐term dynamics of radiation belts. We then develop a 2‐D LM‐MC code and use it to investigate the bounce resonance diffusion of radiation belt electrons by magnetosonic waves. Using a previously published magnetosonic wave model, we demonstrate that bounce resonance with magnetosonic waves is as important as gyroresonance; both can cause several orders of magnitude increase of MeV electron fluxes within 1 day. We conclude that bounce resonance with magnetosonic waves should be taken into consideration together with gyroresonance.

Hybrid OpenMP/MPI programs for solving the time-dependent Gross–Pitaevskii equation in a fully anisotropic trap

We present hybrid OpenMP/MPI (Open Multi-Processing/Message Passing Interface) parallelized versions of earlier published C programs (D. Vudragovic et al., Comput. Phys. Commun. 183, 2021 (2012)) for calculating both stationary and non-stationary solutions of the time-dependent Gross-Pitaevskii (GP) equation in three spatial dimensions. The GP equation describes the properties of dilute Bose-Einstein condensates at ultra-cold temperatures. Hybrid versions of programs use the same algorithms as the C ones, involving real- and imaginary-time propagation based on a split-step Crank-Nicolson method, but consider only a fully-anisotropic three-dimensional GP equation, where algorithmic complexity for large grid sizes necessitates parallelization in order to reduce execution time and/or memory requirements per node. Since distributed memory approach is required to address the latter, we combine MPI programing paradigm with existing OpenMP codes, thus creating fully flexible parallelism within a combined distributed/shared memory model, suitable for different modern computer architectures. The two presented C/OpenMP/MPI programs for real- and imaginary-time propagation are optimized and accompanied by a customizable makefile. We present typical scalability results for the provided OpenMP/MPI codes and demonstrate almost linear speedup until inter-process communication time starts to dominate over calculation time per iteration. Such a scalability study is necessary for large grid sizes in order to determine optimal number of MPI nodes and OpenMP threads per node.

Block-structured grids for Eulerian gyrokinetic simulations

In order to predict the turbulent transport in magnetic fusion experiments, global (i.e., full-torus) gyrokinetic simulations are often carried out. In this context, one frequently encounters situations in which the plasma temperature varies by large factors across the radial simulation domain. In grid-based Eulerian codes, this enforces the use of a very large number of grid points in the two-dimensional velocity space, and, thus, an enormous computational effort. To minimize the computational requirements, one may employ block-structured grids, adapted to the radial changes of the temperature. As the block-structured grids rely on a general approach, they can be applied to different Eulerian gyrokinetic implementations. In this paper, we explain the construction and implementation of such grids in the gyrokinetic code GENE, F. Jenko et al. (2000), and present corresponding simulation results.

A frozen Gaussian approximation-based multi-level particle swarm optimization for seismic inversion

In this paper, we propose a frozen Gaussian approximation (FGA)-based multi-level particle swarm optimization (MLPSO) method for seismic inversion of high-frequency wave data. The method addresses two challenges in it: First, the optimization problem is highly non-convex, which makes hard for gradient-based methods to reach global minima. This is tackled by MLPSO which can escape from undesired local minima. Second, the character of high-frequency of seismic waves requires a large number of grid points in direct computational methods, and thus renders an extremely high computational demand on the simulation of each sample in MLPSO. We overcome this difficulty by three steps: First, we use FGA to compute high-frequency wave propagation based on asymptotic analysis on phase plane; Then we design a constrained full waveform inversion problem to prevent the optimization search getting into regions of velocity where FGA is not accurate; Last, we solve the constrained optimization problem by MLPSO that employs FGA solvers with different fidelity. The performance of the proposed method is demonstrated by a two-dimensional full-waveform inversion example of the smoothed Marmousi model.

MPI + OpenCL implementation of a phase-field method incorporating CALPHAD description of Gibbs energies on heterogeneous computing platforms

Phase-field method uses a non-conserved order parameter to define the phase state of a system and is a versatile method for moving boundary problems. It is a method of choice for simulating microstructure evolution in the domain of materials engineering. Solution of phase-field evolution equations avoids explicit tracking of interfaces and is often implemented on a structured grid to capture microstructure evolution in a simple and elegant manner. Restrictions on the grid size to accurately capture the interface curvature effects lead to large number of grid points in the computational domain and render the simulation computationally intensive for realistic simulations in 3D. However, the availability of powerful heterogeneous computing platforms and super clusters provides the advantage to perform large scale phase-field simulations efficiently. This paper discusses a portable implementation to extend simulations across multiple CPUs using MPI to include use of GPUs using OpenCL. The solution scheme adapts an isotropic stencil that avoids grid-induced anisotropy. Use of separate OpenCL kernels for problem specific portions of the code ensure that the approach can be extended to different problems. Performance analysis of parallel strategies used in the study illustrate the massively parallel computing possibility for phase-field simulations across heterogeneous platforms.

The local GDQ method applied to general higher-order theories of doubly-curved laminated composite shells and panels: The free vibration analysis

This paper presents a general two-dimensional approach for solving doubly-curved laminated composite shells using different kinematic expansions along the three orthogonal directions of the curvilinear shell model. The Carrera Unified Formulation (CUF) with different thickness functions along the three orthogonal curvilinear directions is applied to completely doubly-curved shells and panels, different from spherical and cylindrical shells and plates. Furthermore, the fundamental nuclei for doubly-curved structures are presented in their explicit form for the first time by the authors. These fundamental nuclei also allow to consider doubly-curved structures with variable thickness. In addition, the theoretical model includes the Murakami’s function (also known as zig-zag effect). For some problems it is useful to have an in-plane kinematic expansion which is different from the normal one. The 2D free vibration problem is numerically solved through the Local Generalized Differential Quadrature (LGDQ) method, which is an advanced version of the well-known Generalized Differential Quadrature (GDQ) method. The main advantage of the LGDQ method compared to the GDQ method is that the former can consider a large number of grid points without losing accuracy and keeping the very good stability features of GDQ method as already demonstrated in literature by the authors.

Ab initio calculation of the deformation potential and photoelastic coefficients of silicon with a non-uniform finite-difference solver based on the local density approximation

The band diagram, deformation potential and photoelastic tensor of silicon are calculated self-consistently under uniaxial and shear strain by solving for the electronic wavefunctions with a finite-difference method. Many-body effects are accounted for by the local density approximation. In order to accommodate the large number of grid points required due to the diverging electrostatic potential near the atomic nuclei in an all-electron calculation, a non-uniform meshing is adopted. Internal displacements are taken into account by adding an additional coordinate transform to the method of Bir and Pikus. Good consistency of the calculated deformation potential and photoelastic coefficients is obtained with prior experimental and theoretical results, validating the numerical methods. Furthermore, it is shown that a slight correction of the multiplicative coefficient of the Xα approximation for conduction bands results in good agreement with experiment for both the direct and indirect bandgaps.