Irregular Domains Research Articles

Hybrid method of space-time and Houbolt methods for solving linear time-dependent problems

A hybrid method, through coupling the space-time method based on the polynomial particular solutions method and the Houbolt method, has been proposed to solve time-dependent problems. The main attractions of the proposed hybrid method are its novelty and high efficiency. Because the Houbolt method is a four level time difference method, the first three time steps initial values have to be calculated in advance. In general, these initial values are computed by Euler method. Consequently, the serious initial bump appears. In the current work, the difficulty of handling the multiple initial values in the solution procedure of the Houbolt method has been alleviated by adopting the space-time method at the beginning procedure, then by following the traditional Houbolt method. Therefore, the proposed hybrid method preserves the advantages of both the space-time method and the Houbolt method. The efficiency of our proposed method is analyzed through three numerical examples in two-dimensional space on regular and irregular domains.

PCT: Point cloud transformer

The irregular domain and lack of ordering make it challenging to design deep neural networks for point cloud processing. This paper presents a novel framework named Point Cloud Transformer (PCT) for point cloud learning. PCT is based on Transformer, which achieves huge success in natural language processing and displays great potential in image processing. It is inherently permutation invariant for processing a sequence of points, making it well-suited for point cloud learning. To better capture local context within the point cloud, we enhance input embedding with the support of farthest point sampling and nearest neighbor search. Extensive experiments demonstrate that the PCT achieves the state-of-the-art performance on shape classification, part segmentation, semantic segmentation, and normal estimation tasks.

Structure-property-processing relationships in extruded liquid crystal polymer film

Liquid crystal polymers (LCPs) derive favorable mechanical, chemical, and electrical behavior from long-range molecular ordering. The microstructure gives rise to anisotropic bulk properties that are problematic for industrial applications, and thus the ability to model the polymer directionality is essential to the design of isotropic material manufacturing processes. This investigation proposes a modeling methodology to simulate the 3D director field in full-scale film extrusion geometries. Wide-angle x-ray scattering (WAXS) is used to validate the predicted orientation for a standard coat-hanger die, and is compared with macroscopic mechanical, thermal, and dielectric testing of LCP film to illustrate the morphological dependence of the polymer properties. The highly anisotropic orientation state resulting from cast film extrusion is both predicted by the model and confirmed experimentally, and this preferred orientation is shown to correlate with observed anisotropy in the bulk properties. Additionally, a practical implementation of the modeling tool is presented to simulate directionality in two alternative die geometries designed to improve bulk isotropy, and it is demonstrated that the model is capable of simulating the resulting order for large, irregular domains typical of industrial processing.

On the Shape-Dependent Problem of Singularity Cancellation Transformations for Weakly Near-Singular Integrals

The singularity cancellation transformations are well-known for calculating weakly singular and near-singular integrals in integral equation solutions. However, some singularity cancellation methods suffer from the shape-dependent problem of inaccuracy and inefficiency for deformed triangles. By the theoretical analysis and numerical verification in this article, the relation between the near-singularity and shape-dependence of the singularity cancellation schemes is discussed. Moreover, a novel framework for devising cancellation transformations of weakly near-singular integrals is presented, which results in fast convergence for both regular and irregular triangular domains. Some numerical results are given to illustrate the validity of the theoretical framework and the efficiency of the proposed transformations.

Mean exit time for diffusion on irregular domains

Many problems in physics, biology, and economics depend upon the duration of time required for a diffusing particle to cross a boundary. As such, calculations of the distribution of first passage time, and in particular the mean first passage time, is an active area of research relevant to many disciplines. Exact results for the mean first passage time for diffusion on simple geometries, such as lines, discs and spheres, are well-known. In contrast, computational methods are often used to study the first passage time for diffusion on more realistic geometries where closed-form solutions of the governing elliptic boundary value problem are not available. Here, we develop a perturbation solution to calculate the mean first passage time on irregular domains formed by perturbing the boundary of a disc or an ellipse. Classical perturbation expansion solutions are then constructed using the exact solutions available on a disc and an ellipse. We apply the perturbation solutions to compute the mean first exit time on two naturally-occurring irregular domains: a map of Tasmania, an island state of Australia, and a map of Taiwan. Comparing the perturbation solutions with numerical solutions of the elliptic boundary value problem on these irregular domains confirms that we obtain a very accurate solution with a few terms in the series only. MATLAB software to implement all calculations is available at https://github.com/ProfMJSimpson/Exit_time.

Numerical solution of two-dimensional stochastic time-fractional Sine–Gordon equation on non-rectangular domains using finite difference and meshfree methods

The nonlinear Sine-Gordon equation is one of the widely used partial differential equations that appears in various sciences and engineering. The main purpose of writing this article is providing an efficient numerical method for solving two-dimensional (2D) time-fractional stochastic Sine–Gordon equation on non-rectangular domains. In this method, radial basis functions (RBFs) and finite difference scheme are used to calculate the approximate solution of the mentioned problem. The complexity of solving this problem arises from its high dimension, irregular area, stochastic and fractional terms. Finite difference technique is applied to overcome on the problem dimension, whereas interpolation method based on RBFs is the best idea for solving problems defined in irregular domains. The stochastic Sine–Gordon equation is transformed into elliptic stochastic differential equations using the finite difference method and meshfree method based on RBFs are used to approximate the obtained stochastic differential equation. Some numerical examples are included to investigate the efficiency and accuracy of the presented method.

3D mixed CNNs with edge-point feature learning

Although deep neural networks have shown good performance on grid data, it is challenging to design deep neural networks for point cloud processing due to the irregular domain and disordering of point cloud. This paper presents a novel convolution block named edge-point features mixed convolution (EPFM-Conv) which effectively integrates the graph-based method and the point-based strategy. The entire EPFM-Conv block constructs the edge branch and point branch to extract the local and global features of point cloud, respectively. In the edge branch, we first construct the local dynamic graphs on point cloud and thereby extract rich detailed features through edge convolution. Next, the features of the edge are further abstracted and aggregated through the grouped residual learning followed by a symmetric function. In the point branch, we directly extract the features of the point by a point-wise multi-layer perceptron. Finally, the features of edge and point are mixed adaptively to obtain the feature representation of point cloud. Compared with the existing methods, the main difference is that the proposed method dexterously constructs the local dynamic graphs on point cloud for extracting rich detailed features, and synchronously designs the edge and point branches to extract the local and global features. Meanwhile, the proposed method uses the grouped residual learning to improve the information flow, thereby effectively deepening the depth of the network. Extensive experiments demonstrate that the proposed EPFM-Conv achieves the excellent performances on shape classification as well as part segmentation tasks.

A study of nonlinear systems arising in the physics of liquid crystals, using MLPG and DMLPG methods

The study of liquid crystals is one of the active areas of physics research. In this paper, the MLPG and direct MLPG (DMLPG) methods are used for the numerical study of the coupled nonlinear sine–Gordon equations in two dimensions arising in the modeling of some phenomena in liquid crystals and superconductors. To approximate numerical integrals in the local weak forms, the MLS and GMLS approximations are used in MLPG and DMLPG methods, respectively. The distribution of regular and scattered points on rectangular and irregular domains has been used to extract the numerical results. By comparing the numerical results, it can be seen that the DMLPG methods are faster, more accurate, and more efficient than the MLPG methods. These are because the GMLS approximation uses the basis polynomials instead of the complex shape functions of the MLS approximation.

A local stabilized approach for approximating the modified time-fractional diffusion problem arising in heat and mass transfer

IntroductionDuring the last years the modeling of dynamical phenomena has been advanced by including concepts borrowed from fractional order differential equations. The diffusion process plays an important role not only in heat transfer and fluid flow problems, but also in the modelling of pattern formation that arises in porous media. The modified time-fractional diffusion equation provides a deeper understanding of several dynamic phenomena. ObjectivesThe purpose of the paper is to develop an efficient meshless technique for approximating the modified time-fractional diffusion problem formulated in the Riemann–Liouville sense. MethodsThe temporal discretization is performed by integrating both sides of the modified time-fractional diffusion model. The unconditional stability of the time discretization scheme and the optimal convergence rate are obtained. Then, the spatial derivatives are discretized through a local hybridization of the cubic and Gaussian radial basis function. This hybrid kernel improves the condition of the system matrix. Therefore, the solution of the linear system can be obtained using direct solvers that reduce significantly computational cost. The main idea of the method is to consider the distribution of data points over the local support domain where the number of points is almost constant. ResultsThree examples show that the numerical procedure has good accuracy and applicable over complex domains with various node distributions. Numerical results on regular and irregular domains illustrate the accuracy, efficiency and validity of the technique. ConclusionThis paper adopts a local hybrid kernel meshless approach to solve the modified time-fractional diffusion problem. The main results of the research is the numerical technique with non-uniform distribution in irregular grids.

A trustable shape parameter in the kernel-based collocation method with application to pricing financial options

In this paper, we focus on the kernel-based solution of high-dimensional elliptic PDEs and propose an efficient algorithm to compute a trustable value of the shape parameter for a given positive definite kernel. The proposed strategy is based on the norm-minimal generalized interpolation problem coming from the optimality of radial basis function interpolation. Also, the effectiveness of the suggested algorithm is assessed by solving parabolic partial integro-differential equations arising in option pricing theory when the dynamic of the underlying asset is driven by a Lévy process. The performance of the proposed algorithm is evaluated by some high-dimensional PDEs on regular and irregular computational domains. Indeed, the numerical results obtained with European put options under Merton’s model show the trustability of the suggested optimal shape parameter.

Numerical approximation of 2D multi-term time and space fractional Bloch–Torrey equations involving the fractional Laplacian

In this paper, we propose a novel numerical technique to 2D multi-term time and space fractional Bloch–Torrey equations defined on an irregular convex domain. First, we consider the problem with space integral Laplacian operator. We present the semi-discrete and fully-discrete schemes by using the L1 formula on a temporal graded mesh and an unstructured-mesh Galerkin finite element method (FEM) based on the matrix transfer technique (MTT). Moreover, the unconditional stability and convergence of the schemes are discussed and theoretically proved. Then we extend the technique and develop a numerical scheme for the case of space fractional Laplacian operator. In addition, the recent L1+ formula is utilized to improving the temporal accuracy. Finally, three numerical examples in rectangular, elliptical and human brain-like irregular domains are given to demonstrate the accuracy and efficiency of the proposed numerical scheme.

PARALLEL CUDA IMPLEMENTATION OF THE ALGORITHM FOR SOLVING THE NAVIER-STOKES EQUATIONS USING THE FICTITIOUS DOMAIN METHOD

An important direction in the development of numerical simulation methods is the study of approximate methods for solving problems of mathematical physics in complex multidimensional fields. To solve many applied problems in irregular areas, the method of fictitious areas is widely used, which is characterized by a high degree of automation of programming. The main idea of the method of fictitious domains is that the problem is solved not in the original complex domain, but in some other, simpler domain. This allows you to create software immediately for a fairly wide class of problems with arbitrary computational domains. The possibilities of applying the method of fictitious domains to the problems of hydrodynamics in the variables “stream function, velocity vortex” have been considered in many works. In this paper, we study a numerical method for solving the Navier-Stokes equations in doubly connected domains. To solve the two-dimensional Navier-Stokes equations in irregular domains, an approximate method based on the method of fictitious domains is proposed. A computational finite difference algorithm for solving an auxiliary problem of the method of fictitious domains has been developed. The results of numerical simulation of two-dimensional Navier-Stokes equations by the method of fictitious domains with continuation by the lowest coefficient are presented. For this problem, a parallel algorithm was developed using CUDA technologies, which was tested on various mesh dimensions

Multi-windowed vertex-frequency analysis for signals on undirected graphs

The recent emerging graph signal processing technologies have been widely applied to analyze signals defined on irregular domains, e.g., data collected from social networks, sensor networks, or transportation systems. Vertex frequency analysis, especially the windowed graph Fourier transform, is one of the most important tools for graph signal analysis and representations. Nevertheless, with a selected window function, it is rather challenging to construct tight frames via the windowed graph Fourier transform. To facilitate the construction of tight frames, in this paper, we consider multi-windowed graph Fourier transforms to develop novel vertex frequency analysis methods. Firstly, under the multi-windowed setting, tight graph Fourier frames are elaborately constructed to fulfill technical demands in different application scenarios. The canonical dual frames of the multi-windowed graph Fourier frames are investigated to establish the reconstruction formulas of graph signals. Additionally, we propose shift multi-windowed graph Fourier frames by directly using the shift operators, e.g., the adjacency matrix. The related tight frames, dual frames and their constructions are also discussed. Experimental results show that the proposed two types of frames can efficiently extract vertex-frequency features of synthetic graph signals. Furthermore, anomaly data can also be detected by these frames.

Influence of surface pinning in the domain on the magnetization dynamics in permalloy striped domain films

Abstract We studied the influence of surface pinning on the domain structure and magnetization dynamics in permalloy stripe domain films. The effect of the pinning in the film was controlled by introducing the defects via the interface roughness of the silicon substrate. The results show that the pinning, induced by these defects, leads to the irregularity of the stripe domain and change of their local distributions. The irregular stripe domain pattern results in the weakening of magnetization resonant- frequency and intensity of the film. The results further were verified by micromagnetic simulation, confirming that the regularity and homogeneity of the stripe domain are important for the excitation of spin waves in stripe domain films. Our results show the important significance of the domain structure for the application of spin-wave propagation in magnetic films.

VoroCNN: deep convolutional neural network built on 3D Voronoi tessellation of protein structures.

Effective use of evolutionary information has recently led to tremendous progress in computational prediction of three-dimensional (3D) structures of proteins and their complexes. Despite the progress, the accuracy of predicted structures tends to vary considerably from case to case. Since the utility of computational models depends on their accuracy, reliable estimates of deviation between predicted and native structures are of utmost importance. For the first time, we present a deep convolutional neural network (CNN) constructed on a Voronoi tessellation of 3D molecular structures. Despite the irregular data domain, our data representation allows us to efficiently introduce both convolution and pooling operations and train the network in an end-to-end fashion without precomputed descriptors. The resultant model, VoroCNN, predicts local qualities of 3D protein folds. The prediction results are competitive to state of the art and superior to the previous 3D CNN architectures built for the same task. We also discuss practical applications of VoroCNN, for example, in recognition of protein binding interfaces. The model, data and evaluation tests are available at https://team.inria.fr/nano-d/software/vorocnn/. Supplementary data are available at Bioinformatics online.

A comparison of the finite difference and multiresolution method for the elliptic equations with Dirichlet boundary conditions on irregular domains

We make a comparison of the finite difference and multiresolution method for solving the elliptic equations on irregular domains. The Dirichlet boundary condition is treated by the ghost fluid method (GFM) for the finite difference method and the Lagrange multiplier for the multiresolution method. Numerical results illustrate the improved convergence rate of errors and their gradients with the multiresolution method, up to eighth-order. Moreover, the wavelet-based multiresolution has an advantage of preconditioning of the linear system arising from the discretization of elliptic equation over the finite difference method.

Monomial Augmentation Guidelines for RBF-FD from Accuracy Versus Computational Time Perspective

Local meshless methods using RBFs augmented with monomials have become increasingly popular, due to the fact that they can be used to solve PDEs on scattered node sets in a dimension-independent way, with the ability to easily control the order of the method, but at a greater cost to execution time. We analyze this ability on a Poisson problem with mixed boundary conditions in 1D, 2D and 3D, and reproduce theoretical convergence orders practically, also in a dimension-independent manner, as demonstrated with a solution of Poisson's equation in an irregular 4D domain. The results are further combined with theoretical complexity analyses and with conforming execution time measurements, into a study of accuracy vs. execution time trade-off for each dimension. Optimal regimes of order for given target accuracy ranges are extracted and presented, along with guidelines for generalization.

Hyperspectral Image Classification with Localized Graph Convolutional Filtering

The nascent graph representation learning has shown superiority for resolving graph data. Compared to conventional convolutional neural networks, graph-based deep learning has the advantages of illustrating class boundaries and modeling feature relationships. Faced with hyperspectral image (HSI) classification, the priority problem might be how to convert hyperspectral data into irregular domains from regular grids. In this regard, we present a novel method that performs the localized graph convolutional filtering on HSIs based on spectral graph theory. First, we conducted principal component analysis (PCA) preprocessing to create localized hyperspectral data cubes with unsupervised feature reduction. These feature cubes combined with localized adjacent matrices were fed into the popular graph convolution network in a standard supervised learning paradigm. Finally, we succeeded in analyzing diversified land covers by considering local graph structure with graph convolutional filtering. Experiments on real hyperspectral datasets demonstrated that the presented method offers promising classification performance compared with other popular competitors.

A point-cloud deep learning framework for prediction of fluid flow fields on irregular geometries

We present a novel deep learning framework for flow field predictions in irregular domains when the solution is a function of the geometry of either the domain or objects inside the domain. Grid vertices in a computational fluid dynamics (CFD) domain are viewed as point clouds and used as inputs to a neural network based on the PointNet architecture, which learns an end-to-end mapping between spatial positions and CFD quantities. Using our approach, (i) the network inherits desirable features of unstructured meshes (e.g., fine and coarse point spacing near the object surface and in the far field, respectively), which minimizes network training cost; (ii) object geometry is accurately represented through vertices located on object boundaries, which maintains boundary smoothness and allows the network to detect small changes between geometries and (iii) no data interpolation is utilized for creating training data; thus accuracy of the CFD data is preserved. None of these features are achievable by extant methods based on projecting scattered CFD data into Cartesian grids and then using regular convolutional neural networks. Incompressible laminar steady flow past a cylinder with various shapes for its cross section is considered. The mass and momentum of predicted fields are conserved. We test the generalizability of our network by predicting the flow around multiple objects as well as an airfoil, even though only single objects and no airfoils are observed during training. The network predicts the flow fields hundreds of times faster than our conventional CFD solver, while maintaining excellent to reasonable accuracy.

A meshless radial basis function based method for modeling dual-phase-lag heat transfer in irregular domains

In this paper we present a new numerical technique for the dual-phase-lag model appearing in the heating tissue. The problems are described as general linear partial differential equations with mixed temporal and spatial derivatives defined in irregular 2D domains. We consider heat transfer problems in the inhomogeneous medium, i.e., the coefficients of the governing partial differential equations depend on the spatial coordinates. To solve the problem, we use the combination of the time-stepping method for temporal variables with the meshless semi-analytical technique for the spatial approximation. The effectiveness of the proposed scheme is demonstrated by solving dual-phase-lag heat conduction problems including the generalized bio-heat problem in single and multi-connected irregular domains.