Non-uniform Point Distributions Research Articles

Comparison of Physics Informed Neural Networks and Finite Element Method Solvers for advection-dominated diffusion problems

We present a comparison of Physics Informed Neural Networks (PINN) and Variational Physics Informed Neural Networks (VPINN) with higher-order and continuity Finite Element Method (FEM). We focus on the one-dimensional advection-dominated diffusion problem and the two-dimensional Eriksson–Johnson model problem. We show that the standard Galerkin method for FEM cannot solve this problem on uniform grid. We discuss the stabilization of the advection-dominated diffusion problem with the Petrov–Galerkin (PG) formulation and present the FEM solution obtained with the PG method. The main benefit of using a stabilization method is that it can deliver a good-quality approximation to the solution on a mesh that is not fully refined towards the singularity. We employ PINN and VPINN methods, defining several strong and weak loss functions. We compare the training and solutions of PINN and VPINN methods with higher-order FEM methods. We consider a case with uniform FEM and uniform distribution of points for PINN, as well as uniform distribution of test functions for VPINN. We also consider adaptive FEM, refined towards edge singularity, and non-uniform distribution of points for PINN, as well as non-uniform distribution of test functions for VPINN.

A high-efficient accurate coupled mesh-free scheme for 2D/3D space-fractional convection-diffusion/Burgers' problems

This paper first develops a high-efficient accurate mesh-free scheme (CSPH-SFCD) for multi-dimensional time-dependent spatial fractional convection-diffusion (SFCD) equations based on the Riemann-Liouville (RL) derivative in convex domain by coupling a corrected smoothed particle hydrodynamics (CSPH) method with a numerical integral of RL. Subsequently, for the first application, the proposed CSPH-SFCD coupled with up-wind (UW) scheme (CSPH-SFCD-UW) is tentatively extended to investigate the nonlinear shock-wave phenomena in 2D space-fractional Burgers' problem (SFBP) with fractional Neumann boundary at high Reynolds number. The proposed mesh-free scheme for time-dependent SFCD/SFBP with fractional boundary is mainly motived by: (a) the SFCD model with RL fractional calculus is discretized by employing a numerical integral formula and a CSPH scheme, and then a conditionally stable CSPH-SFCD method is first derived; (b) the UW scheme is introduced to stabilize the numerical investigation of convection-dominated SFBP; (c) the fractional Neumann boundary (FNB) condition can be easily enforced in the proposed discretized scheme, and the integer-order NB condition is treated by employing virtual particles interpolation technique; (d) the MPI parallel computing technique is adopted to enhance the computational efficiency. Firstly, the numerical convergence and stability of the proposed scheme are discussed and illustrated by solving several multi-dimensional benchmarks. Secondly, the cases of two-sided and left-sided RL fractional derivatives are also discussed and demonstrated in the numerical tests. The advantages of the proposed method are demonstrated by simulating the 2D/3D SFCD equations in convex domains or under non-uniform point distribution. Finally, the 2D two-component SFBP with FNBC are numerically investigated by using the proposed method, and compared with the FDM results. The complex nonlinear non-local properties governed by SFBP are numerically predicted by the proposed scheme. All the numerical results show that the proposed mesh-free method for time-dependent spatial SFCD problems is high-efficient, flexible and accurate. • A high-efficient accurate CSPH-SFCD scheme is first developed to 2D/3D SFCD equations. • The CSPH-SFCD-UW scheme is first proposed to investigate shock-wave behavior in 2D SFBP. • The FNB condition can be easily enforced in the proposed approximated scheme. • The advantages of the present scheme are shown by numerical tests on convex domain.

A Study On the Local Weather Synthesis by The Adjusted Inverse Distance Weighted (AIDW) Interpolation

In this study, a modified spatial interpolation method called Adjusted Inverse Distance Weighted (AIDW) is used to analyze meteorological data around the Islamic University, Bangladesh. For the analysis of meteorological data, data were collected from different areas of Islamic University, Kushtia by the well-known portable weather meter Kestrel 3500 weather meter. For the synthesis of the observed data, an efficient solution to the problem of two dimensional interpolation from irregularly-spaced data points in order to anticipate and calculate the desired value, a non-uniform distribution of sample points is used. This method of research is used on some important parameters related to weather and the weather of the study area. A reasonable agreement with the observed and synthesis data of Adjusted Inverse Distance Weighted (AIDW) interpolation was found in this study.

EquiDistribution Grid Evolution Method for Line Spectrum Estimation

The grid for traditional estimation methods is usually uniform. This is a conservative strategy to form a grid without using any prior information. Estimation results may become better if the true values of parameters are near the artificial grid points. Therefore, a dense grid is usually used to reach expected accuracy, resulting in heavy computation workload. Notice that the performance of estimation is related to the distribution of its grid. Aiming to achieve higher accuracy while use less grid points, a strategy called equidistribution is proposed in this paper. Estimation problem is reformulated by adding the constraint of equidistribution and is solved by a modified grid evolution method, named as EquiDistribution Grid Evolution (EDGE) method. In EDGE, a nonuniform distribution of grid points is evolved according to the distribution of information, and the distribution of information is progressively updated based on the nonuniform grid. The two procedures iterate alternatively. Finally a spectrum based on the adaptively evolved grid is obtained. Compared with previous methods, the proposed method shows advantages on both accuracy and time consumption. The controlling parameter is significantly simplified compared with previous grid evolution method. Furthermore, the resolution and accuracy in parameters’ domain are adaptive and nonuniform. The effectiveness of the EDGE method is demonstrated by numerical simulations.

Real-time remaining useful life prediction based on adaptive kernel window width density

Remaining useful life (RUL) prediction plays an important role in improving the availability and productivity of systems. To improve the accuracy of real-time RUL prediction during system operation, we propose a modeling method for real-time RUL prediction based on adaptive kernel window width density. First, a non-parametric kernel density estimation (KDE) real-time RUL prediction model is proposed, and a window width model with adaptive kernel window width density is established by introducing a local density factor in the window width selection. The local density of sample points is calculated by the k-nearest neighbor distance, and the KDE is performed by adaptively selecting the window width value according to the local density of sample points in the region of non-uniform distribution of sample points. As the monitoring data changes in real time, the kernel density estimates of known samples are used to recursively update the kernel density estimates of new samples. Moreover, the logarithmic transformation of random variables and space mapping are used in the establishment of the RUL prediction model. A model of logarithmic kernel diffeomorphism transformation is established to solve the boundary shift problem of kernel estimation in the prediction to improve the prediction accuracy. Finally, the validity of the method is verified through case studies, and the accuracy of the model is judged using evaluation quasi-measures.

Clustering by centroid drift and boundary shrinkage

Locating the centers before assigning clustering labels is a traditional routine of clustering methods, which also limits the development of new clustering ideas. In this paper, we achieve the clustering task by firstly identifying the boundary points in the feature space, and then we shrink the boundary points to allocate the un-clustered points. Concretely, we propose a Centroid Drift (CD) metric and a Boundary Shrinkage (BS) strategy to detect boundary points in the feature space and allocate labels for un-clustered points, respectively. Both the CD and BS are closely related to the pre-computed k-nearest neighbor matrix, contributing to the decrease of algorithm parameters. Moreover, the common problems of noise points and non-uniform density distribution of data points in clustering task can also be alleviated with our proposed large value suppression and normalization of k-nearest neighbor distance techniques. The experiments on synthetic datasets, real-world face image datasets and hyperspectral images demonstrate the superiorities of our proposed clustering framework.

Modeling acoustic waves in locally enhanced meshes with a staggered-grid finite difference approach

Finite difference is a well-suited technique for modeling acoustic wave propagation in heterogeneous media as well as for imaging and inversion. Typically, the method aims at solving a set of partial differential equations for the unknown pressure field by using a regularly spaced grid. Although finite differences can be fast and cheap to implement, the accuracy of the solution is always restricted by the computational resources. This is a fundamental key point to treat when dealing with large-scale problems. In this work, we present and test a method that uses a non-uniform distribution of grid points to improve on accuracy or to reduce the required computational resources. The applied grid is generated through a coordinate transformation. Differential geometry and generalized coordinates are used to handle and analyze the effect of using a non-uniform grid. Results obtained with the presented method show that the applied transformation as well as the number of points-per-wavelength influences the stability and dispersion in the solution. We exploit this observation to locally improve the accuracy of our simulations. The work presented in this paper allows us to conclude that differential geometry for finite differences can be used to reduce dispersion and hence improve the accuracy when modeling acoustic wave propagation in heterogeneous media. In addition, it can be used to avoid oversampling through the optimization of the number of grid nodes required to have an accurate solution or just honor to the boundaries.

A New Method of Distribution of Measurement Points on Curvilinear Surfaces of Products.

The article presents the method of selecting scanning lines along which coordinate measurements, performed by using, e.g., a coordinate measuring machine working in the single point probing mode, of free-form surfaces of measured workpieces may be conducted. Additionally, the proposed method supports the user of a coordinate measuring system during defining the number of measurement points distributed along selected scanning lines, thus deciding on the final distribution of measurement points on a measured surface of a product. The proposed method enables distributing measurement points in the parts of a measured product characterized by the worst quality of manufacturing. Moreover, the new method is very automated, therefore it affects the increase in the efficiency of coordinate measurements. The effect of using the new method is the non-uniform distribution of measurement points located on free-form surfaces. The presented algorithm takes into account the lengths of selected cross-sections of a measured curvilinear surface of an object, its geometrical complexity and the accuracy of the probe radius correction process. The decision regarding the number of measurement points is made on the basis of the accuracy analysis of the calculations of the corrected measurement points obtained during the probe radius correction process and the accuracy of the substitute model representing a measured curvilinear surface. Two methods of the correction process were used. The accuracy of the applied methods of the probe radius compensation process was estimated on the basis of the deviations calculated between corrected measurement points and scanning lines. The selection of scanning lines and the number of measurement points was realized by using the expert system based on the fuzzy logic. The paper presents the results of both simulation and experimental investigations. The numerical calculations were performed for two selected free-form surfaces. The verification of the developed algorithm was carried out during experimental investigations based on a measurement of a selected free-form surface. The conducted research was aimed at verifying the correctness of the distribution of measurement points generated using the proposed method. In the case of real measurements, measurement points should be located in the places of surfaces of products characterized by the largest deviations of manufacturing. The results of the conducted investigations confirm the usefulness of the developed algorithm for defining the distribution of measurement points on curvilinear surfaces in the coordinate measuring technique. Moreover, the way of implementation of the developed method of the distribution of measurement points in selected commercial measurement software is presented, thus enabling the use of the new method in the industry.

Temporal connectivity in finite networks with nonuniform measures

Soft Random Geometric Graphs (SRGGs) have been widely applied to various models including those of wireless sensor, communication, social and neural networks. SRGGs are constructed by randomly placing nodes in some space and making pairwise links probabilistically using a connection function that is system specific and usually decays with distance. In this paper we focus on the application of SRGGs to wireless communication networks where information is relayed in a multi hop fashion, although the analysis is more general and can be applied elsewhere by using different distributions of nodes and/or connection functions. We adopt a general non-uniform density which can model the stationary distribution of different mobility models, with the interesting case being when the density goes to zero along the boundaries. The global connectivity properties of these non-uniform networks are likely to be determined by highly isolated nodes, where isolation can be caused by the spatial distribution or the local geometry (boundaries). We extend the analysis to temporal-spatial networks where we fix the underlying non-uniform distribution of points and the dynamics are caused by the temporal variations in the link set, and explore the probability a node near the corner is isolated at time $T$. This work allows for insight into how non-uniformity (caused by mobility) and boundaries impact the connectivity features of temporal-spatial networks. We provide a simple method for approximating these probabilities for a range of different connection functions and verify them against simulations. Boundary nodes are numerically shown to dominate the connectivity properties of these finite networks with non-uniform measure.

Rolling normal filtering for point clouds

3D geometric features represent rich details of 3D models, whose scale is much larger than noise. Traditional point cloud denoising methods cannot handle the task of processing and analyzing these features. Rolling guidance normal filtering is proved to be a useful tool in image and mesh small features removing. However, its direct extension to point cloud processing will lead to artifacts such as shape shrinkage and non-uniform distribution of points. To address these issues, we propose a new point position updating formulation and adopt a multi-normal strategy to overcome sharp edge shrinkage. Compared with other state-of-the-art denoising methods, our approach is more robust in removing small-scale geometric features while retaining large-scale structures. Even compared to its mesh counterpart, our method exhibits superiority in preventing large-scale sharp structures from severe distortion. Finally, a variety of experiments demonstrate that our approach shows its advantages in geometric feature removal against previous methods.

Meshfree finite difference approximations for functions of the eigenvalues of the Hessian

We introduce meshfree finite difference methods for approximating nonlinear elliptic operators that depend on second directional derivatives or the eigenvalues of the Hessian. Approximations are defined on unstructured point clouds, which allows for very complicated domains and a non-uniform distribution of discretisation points. The schemes are monotone, which ensures that they converge to the viscosity solution of the underlying PDE as long as the equation has a comparison principle. Numerical experiments demonstrate convergence for a variety of equations including problems posed on random point clouds, complex domains, degenerate equations, and singular solutions.

Optimal Non-Uniform Deployments in Ultra-Dense Finite-Area Cellular Networks

Network densification and heterogenisation through the deployment of small cellular access points (picocells and femtocells) are seen as key mechanisms in handling the exponential increase in cellular data traffic. Modelling such networks by leveraging tools from Stochastic Geometry has proven particularly useful in understanding the fundamental limits imposed on network coverage and capacity by co-channel interference. Most of these works however assume infinite sized and uniformly distributed networks on the Euclidean plane. In contrast, we study finite sized non-uniformly distributed networks, and find the optimal non-uniform distribution of access points which maximises network coverage for a given non-uniform distribution of mobile users, and vice versa.

HASEonGPU—An adaptive, load-balanced MPI/GPU-code for calculating the amplified spontaneous emission in high power laser media

We present an adaptive Monte Carlo algorithm for computing the amplified spontaneous emission (ASE) flux in laser gain media pumped by pulsed lasers. With the design of high power lasers in mind, which require large size gain media, we have developed the open source code HASEonGPU that is capable of utilizing multiple graphic processing units (GPUs). With HASEonGPU, time to solution is reduced to minutes on a medium size GPU cluster of 64 NVIDIA Tesla K20m GPUs and excellent speedup is achieved when scaling to multiple GPUs. Comparison of simulation results to measurements of ASE in Y b3+:Y AG ceramics show perfect agreement. Program summaryProgram title: HASEonGPUCatalogue identifier: AFAM_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AFAM_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public License, version 3No. of lines in distributed program, including test data, etc.: 84610No. of bytes in distributed program, including test data, etc.: 3791861Distribution format: tar.gzProgramming language: C++, Matlab.Computer: GPU cluster or workstation with CUDA-capable GPUs (compute capability ≥2.0).Operating system: Linux.Has the code been vectorized or parallelized?: Yes, can utilize 1 CPU core per compatible GPU.RAM: Several Gb, depending on input size and number of GPUs. 4000000000 bytes (4 GB) per GPU is recommended.Classification: 4.13, 6.5, 15.External routines: CUDA, Boost Program Options, OpenMPINature of problem:The algorithm described by D. Albach in [1, 2] uses ray-tracing techniques and Monte Carlo integration to calculate Amplified Spontaneous Emission (ASE) with high precision. It requires a high number of sampling points as well as a high number of rays to reach the desired results. Additionally, reflections on the upper and lower surfaces of the medium increase the workload by an order of magnitude. On traditional CPU-based systems the computation is time-consuming, which limits the number of simulations that can be performed.Solution method:HASEonGPU uses a non-uniform distribution of sampling points within the gain medium to focus computation on areas of interest. This is further improved by combining the Monte Carlo integration with importance sampling [3]. To improve execution time further, the algorithm is highly parallelized to run on a GPU and supports adaptive sampling resolutions and random restarts. It can also be executed in a GPU cluster, where linear scaling is achieved by a coarse-granular load balancing that distributes the workload among all GPUs in a master–worker-scheme over MPI.Restrictions:Presently, the number of rays used for the Monte Carlo integration of a single sampling point within the gain medium is limited by the available memory on the GPU (about 108 rays per GB of GPU memory). Furthermore, when using MPI as a workload distribution mechanism, one of the MPI processes will act as a scheduling master and its GPU cannot participate in the computation.Unusual features:The software can run on a workstation (threaded) as well as on a large-scale GPU cluster (MPI) that provides the required GPU hardware. The simulation parameters include polychromatic laser pulses as well as surface coatings, cladding, and refractive indices of the gain medium. This also allows the simulation of reflections on the upper and lower surfaces of the medium. If a desired mean square error metric is not met with a set number of rays, the algorithm can automatically increase the number of rays to improve the results.Additional comments:The source code also includes a MATLAB script that can be used to call HASEonGPU directly from MATLAB code to integrate it into existing simulation setups. There are also examples included on how to execute HASEonGPU from the command line as well as an example experiment that uses MATLAB and the provided script. More detailed information can be found in the README file.Running time:Depending on the number of sampling points, desired sampling resolution for each point, and number of GPUs, the execution time can vary strongly. A typical cylindrical gain medium of 6 cm diameter simulated with 4210 non-uniformly distributed sampling points can be simulated with a sufficient precision in 1 min on a single NVIDIA Tesla K20m GPU. Running time as well as precision can be further optimized through various parameters.

Explicit efficient constrained model predictive control

A more efficient model predictive control algorithm is algebraic model predictive control. The efficiency is improved by forming the prediction horizon with a non-uniform distribution of points. The most efficient case reduces the problem down to a single prediction point, defining the horizon length as well as removing any redundant control calculations. This paper shows that for this single prediction point case, the constrained control can be easily evaluated explicitly, thus making the control calculation deterministic and more efficient. We also show that because an explicit control can be determined for the system, the closed-loop constrained stability bounds can be evaluated a priori. Using the key constraint of the peak location of the response also allows the system to meet the constraints while still being explicit and deterministic. The proposed explicit algorithm is applied to two numerical simulation examples with the performance, computational efficiency and stability analysed.

Steric effects on electrokinetic flow of non-linear biofluids

Abstract Some of electrokinetic-based biomicrofluidic devices work at zeta potentials that are sufficiently high for ionic size (steric) effects to show up. In the present effort, consideration is given to the steric effects on the hydrodynamics of electroosmotic flow in a rectangular microchannel. The distinction between this research and the previous ones is that we account for non-linear rheology of the fluids encountered in biomicrofluidic systems by means of the power-law viscosity model. The method of analysis consists of a finite-difference-based numerical procedure for a non-uniform distribution of grid points, which is applied to the dimensionless form of the governing equations including the modified Poisson-Boltzmann equation. Our findings show that, unlike the electroosmotic flow of Newtonian fluids, the inclusion of the ionic size may enhance the velocity of the power-law fluids. We demonstrate that this phenomenon is due to the reduction of the fluid effective viscosity which is another consequence of the steric effects, besides the already explored feature of lowering the electroosmotic body force. Consequently, the ultimate influence of the ionic finite volumes on the electroosmotic flow of the power-law fluids is determined by the relative importance of the above-mentioned phenomena. Generally, the reduction of the body force is dominant; nevertheless, the opposite is true for shear-thickening fluids of very high flow behavior index, resulting in higher fluid velocities in the presence of the steric effects.

The localized method of approximated particular solutions for solving two-dimensional incompressible viscous flow field

The purpose of this paper is to demonstrate that the localized method of approximated particular solutions (LMAPS) is a stable, accurate tool for simulating two-dimensional incompressible viscous flow fields with Chorin׳s projection method. Totally there are two numerical experiments conducted: the two-dimensional lid-driven cavity flow problem, and the two-dimensional backward facing step problem. Throughout this study, the LMAPS has been tested by non-uniform point distribution, extremely narrow rectangular domain, internal flow, velocity or pressure driven flow and high velocity or pressure gradient, etc. All results are similar to results of finite element method (FEM) or other literature, and it is concluded that the LMAPS has high potential to be applied to more complicated engineering applications.

Lattice Boltzmann simulation for forced two-dimensional turbulence.

The direct numerical simulations of forced two-dimensional turbulent flow are presented by using the lattice Boltzmann method. The development of an energy-enstrophy double cascade is investigated in the two cases of external force of two-dimensional turbulence, Gaussian force and Kolmogorov force. It is found that the friction force is a necessary condition of the occurrence of a double cascade. The energy spectrum k(-3) in the enstrophy inertial range is in accord with the classical Kraichnan theory for both external forces. The energy spectrum of the Gaussian force case in an inverse cascade is k(-2); however, the Kolmogorov force drives the k(-5/3) energy in a backscatter cascade. The result agrees with Scott's standpoint, which describes nonrobustness of the two-dimensional turbulent inverse cascade. Also, intermittency is found for the enstrophy cascade in two cases of the external force form. Intermittency refers to the nonuniform distribution of saddle points in the two-dimensional turbulent flow.

3D Delaunay triangulation of non-uniform point distributions

In view of the simplicity and the linearity of regular grid insertion, a multi-grid insertion scheme is proposed for the three-dimensional Delaunay triangulation of non-uniform point distributions by recursive application of the regular grid insertion to an arbitrary subset of the original point set. The fundamentals and difficulties of three-dimensional Delaunay triangulation of highly non-uniformly distributed points by the insertion method are reviewed. Current strategies and methods of point insertions for non-uniformly distributed spatial points are discussed. An enhanced kd-tree insertion algorithm with a specified number of points in a cell and its natural sequence derived from a sandwich insertion scheme is also presented.The regular grid insertion, the enhanced kd-tree insertion and the multi-grid insertion have been rigorously studied with benchmark non-uniform distributions of 0.4–20 million points. It is found that the kd-tree insertion is more efficient in locating the base tetrahedron, but it is also more sensitive to the triangulation of non-uniform point distributions with a large amount of conflicting elongated tetrahedra. Including the grid construction time, multi-grid insertion is the most stable and efficient for all the uniform and non-uniform point distributions tested.

An immersed boundary–lattice Boltzmann approach to study the dynamics of elastic membranes in viscous shear flows

A combined immersed boundary–lattice Boltzmann approach is used to simulate the dynamics of elastic membrane immersed in a viscous incompressible flow. The lattice Boltzmann method is utilized to solve the flow field on a regular Eulerian grid, while the immersed boundary method is employed to incorporate the fluid–membrane interaction with a Lagrangian representation of the deformable immersed boundary. The distinct feature of the method used here is to employ the combination of simple Peskin's IBM and standard LBM. In order to obtain more accurate and truthful solutions, however, a non-uniform distribution of Lagrangian points and a modified Dirac delta function are used. Two test cases are presented. In the first case, we consider a vesicle suspended in a simple shear flow commonly known as tank-treading motion. The computed results were compared with experiments, which showed reasonably good agreement. For the second test case, we consider individual healthy (soft) and sick (stiff) RBCs suspended in a shear flow. The simulation results demonstrated that elastic deformation plays an important role in overall RBC motions characterized as tank-treading and tumbling motions, in which the natural state of the elastic membrane is an essential consideration. In addition, the results confirm that the combination of the immersed boundary and lattice Boltzmann methods permits the simulation of the complex biological phenomena.

A Parallel Adaptive Treecode Algorithm for Evolution of Elastically Stressed Solids

AbstractThe evolution of precipitates in stressed solids is modeled by coupling a quasi-steady diffusion equation and a linear elasticity equation with dynamic boundary conditions. The governing equations are solved numerically using a boundary integral method (BIM). A critical step in applying BIM is to develop fast algorithms to reduce the arithmetic operation count of matrix-vector multiplications. In this paper, we develop a fast adaptive treecode algorithm for the diffusion and elasticity problems in two dimensions (2D). We present a novel source dividing strategy to parallelize the treecode. Numerical results show that the speedup factor is nearly perfect up to a moderate number of processors. This approach of parallelization can be readily implemented in other treecodes using either uniform or non-uniform point distribution. We demonstrate the effectiveness of the treecode by computing the long-time evolution of a complicated microstructure in elastic media, which would be extremely difficult with a direct summation method due to CPU time constraint. The treecode speeds up computations dramatically while fulfilling the stringent precision requirement dictated by the spectrally accurate BIM.