Irregular Domains Research Articles

G-Image Segmentation: Similarity-Preserving Fuzzy C-Means With Spatial Information Constraint in Wavelet Space

G-images refer to image data defined on irregular graph domains. This work elaborates a similarity-preserving Fuzzy C-Means (FCM) algorithm for G-image segmentation and aims to develop techniques and tools for segmenting G-images. To preserve the membership similarity between an arbitrary image pixel and its neighbors, a Kullback-Leibler divergence term on membership partition is introduced as a part of FCM. As a result, similarity-preserving FCM is developed by considering spatial information of image pixels for its robustness enhancement. Due to superior characteristics of a wavelet space, the proposed FCM is performed in this space rather than Euclidean one used in conventional FCM to secure its high robustness. Experiments on synthetic and real-world G-images demonstrate that it indeed achieves higher robustness and performance than the state-of-the-art FCM algorithms. Moreover, it requires less computation than most of them.

A meshless method for time fractional nonlinear mixed diffusion and diffusion-wave equation

The paper aims to put forth a radial basis function-based meshless approach for the numerical solution of the time-fractional nonlinear mixed diffusion and diffusion-wave equation. The time-fractional derivative is defined in Caputo's sense and discretized by the finite difference method. The spatial discretization is done using a radial basis function-based local meshless method. Stability of time semi-discretization is rigorously set up. Proposed method's efficiency is validated with different numerical examples on an irregular domain with uniform and nonuniform points. Numerical results obtained demonstrate the ability and accuracy of the present method.

Integral transform analysis of microchannel fluid flow: Irregular geometry estimation using velocimetry data

A hybrid numerical-analytical solution to fluid flow in irregularly-shaped microchannels has been developed. The solution is based on the Generalized Integral Transform Technique (GITT) applied to a trapezoid-like duct to account for fabrication imperfections, as commonly found in commercially manufactured microchannels. With this approach, the eigenseries expansion is done using a 2D eigenfunction basis, constructed from a combination of simple 1D eigenfunctions defined within the original irregular domain. In order to improve convergence, a new filtering scheme based on an analytical solution to the problem in a rectangular channel, which is distorted to match the irregular domain, is employed. The solution is tested on a real microchannel geometry, obtained from a SEM image, and compared with micro-PIV measurements for the flow field in the same channel. In addition, the actual velocimetry data is input to an inverse problem implementation for recovering geometric parameters related to the irregular geometry using the GITT results as the forward solution. The numerical results obtained with the GITT presented a very satisfactory convergence, and the inverse solution was able to recover the channel geometric parameters with very reasonable accuracy.

Multi-Scale Structured Dictionary Learning for 3-D Point Cloud Attribute Compression

Existing 3-D point cloud attribute compression methods are rigid for complex signals defined in spatially irregular domain, as they solely depend on geometric information or adopt pre-defined analytic bases for transform. In this article, we propose a multi-scale structured dictionary learning algorithm for 3-D point cloud attribute compression. The proposed multi-scale dictionary leverages hierarchical sparsity within signals by adapting atoms in an arborescent structure. It supports hierarchical sparse transform for encoding voxelized point cloud attributes with progressive refinement of high-frequency details from coarser to finer scales. To represent spatial blocks with varying amounts of occupied voxels, a non-uniform binary weight matrix is incorporated in dictionary learning to characterize the dimensional irregularity of signals. To guarantee optimal approximation with improved efficiency, alternating optimizations including Alternating Direction Method of Multipliers and Gauss-Seidel iterations are adopted for the weighted mixed $\ell _{2}/\ell _{1}$ -minimization. Furthermore, we develop a lossy attribute compression framework by integrating sparse transform based prediction with dedicated quantization and entropy coding routines. Experimental results demonstrate that the proposed framework outperforms the state-of-the-art transform-based coding methods and is competitive with the most recent MPEG PCC test models in point cloud attribute compression.

Numerical investigation of fractional nonlinear sine-Gordon and Klein-Gordon models arising in relativistic quantum mechanics

This paper presents a method for the approximate solution of the time-fractional nonlinear sine-Gordon and the Klein-Gordon models described in Caputo sense and with the order 1 < γ < 2. This method discretizes the unknown solution in two main steps. First, the temporal discretization of the governing problems is obtained by means of the finite difference scheme. Second, the spatial terms are expanded using local radial basis functions, where each basis function is approximated by a weighted linear summation of function values. The spatial discretization achieved through the local radial basis functions and finite difference (LRBF-FD) is highly accurate, since in local collocation techniques we only consider the nodes located in the subdomain around the collocation node surrounding the local collocation point. In fact, the new approach avoids the ill-conditioning problem resulting from the adoption of the global collocation and leads to sparse systems. The numerical stability and convergence are examined and confirmed numerically to support the theoretical formulation. Numerical experiments assess the effectiveness and capability of the algorithm and demonstrate its good computational performance on irregular domains.

Free vibration and buckling analyses of CNT reinforced laminated non-rectangular plates by discrete singular convolution method

This paper presents the free vibration and buckling analyses of functionally graded carbon nanotube-reinforced (FG-CNTR) laminated non-rectangular plates, i.e., quadrilateral and skew plates, using a four-nodded straight-sided transformation method. At first, the related equations of motion and buckling of quadrilateral plate have been given, and then, these equations are transformed from the irregular physical domain into a square computational domain using the geometric transformation formulation via discrete singular convolution (DSC). The discretization of these equations is obtained via two-different regularized kernel, i.e., regularized Shannon’s delta (RSD) and Lagrange-delta sequence (LDS) kernels in conjunctions with the discrete singular convolution numerical integration. Convergence and accuracy of the present DSC transformation are verified via existing literature results for different cases. Detailed numerical solutions are performed, and obtained parametric results are presented to show the effects of carbon nanotube (CNT) volume fraction, CNT distribution pattern, geometry of skew and quadrilateral plate, lamination layup, skew and corner angle, thickness-to-length ratio on the vibration, and buckling analyses of FG-CNTR-laminated composite non-rectangular plates with different boundary conditions. Some detailed results related to critical buckling and frequency of FG-CNTR non-rectangular plates have been reported which can serve as benchmark solutions for future investigations.

A new numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshes—part 2: numerical simulations and comparison with FEM

A new numerical approach for the time-dependent wave and heat equations as well as for the time-independent Poisson equation developed in Part 1 is applied to the simulation of 1-D and 2-D test problems on regular and irregular domains. Trivial conforming and non-conforming Cartesian meshes with 3-point stencils in the 1-D case and 9-point stencils in the 2-D case are used in calculations. The numerical solutions of the 1-D wave equation as well as the 2-D wave and heat equations for a simple rectangular plate show that the accuracy of the new approach on non-conforming meshes is practically the same as that on conforming meshes for both the Dirichlet and Neumann boundary conditions. Moreover, very small distances ( $$0.1 h - 10^{-9}h$$ where h is the grid size) between the grid points of a Cartesian mesh and the boundary do not decrease the accuracy of the new technique. The application of the new approach to the 2-D problems on an irregular domain shows that the order of accuracy is close to four for the wave and heat equations and is close to five for the Poisson equation. This is in agreement with the theoretical results of Part 1 of the paper. The comparison of the numerical results obtained by the new approach and by FEM shows that at similar 9-point stencils, the accuracy of the new approach on irregular domains is two orders higher for the wave and heat equations and three orders higher for the Poisson equation than that for the linear finite elements. Moreover, the new approach yields even much more accurate results than the quadratic and cubic finite elements with much wider stencils. An example of a problem with a complex irregular domain that requires a prohibitively large computation time with the finite elements but can be easily solved with the new approach is presented.

Fast Spectral Approximation of Structured Graphs with Applications to Graph Filtering

To analyze and synthesize signals on networks or graphs, Fourier theory has been extended to irregular domains, leading to a so-called graph Fourier transform. Unfortunately, different from the traditional Fourier transform, each graph exhibits a different graph Fourier transform. Therefore to analyze the graph-frequency domain properties of a graph signal, the graph Fourier modes and graph frequencies must be computed for the graph under study. Although to find these graph frequencies and modes, a computationally expensive, or even prohibitive, eigendecomposition of the graph is required, there exist families of graphs that have properties that could be exploited for an approximate fast graph spectrum computation. In this work, we aim to identify these families and to provide a divide-and-conquer approach for computing an approximate spectral decomposition of the graph. Using the same decomposition, results on reducing the complexity of graph filtering are derived. These results provide an attempt to leverage the underlying topological properties of graphs in order to devise general computational models for graph signal processing.

A high-order immersed interface method free of derivative jump conditions for Poisson equations on irregular domains

Immersed Interface Methods (IIM) arise as a very effective tool to solve many interface problems encountered in fluid dynamics, mechanics and other related fields of study. Despite their versatility and potential, IIM-inspired techniques impose as constraints different types of jump conditions in order to be mathematically tractable and usable in practice. To cope with this issue, in this paper we introduce a novel Immersed Interface method for solving Poisson equations with discontinuous coefficients on Cartesian grids. Different from most conventional methods which assume some derivative information at the interface to produce a valid approximation, our approach reduces the number of regular constraints when solving the Poisson problem, requiring to be given only the ordinary jumps of the function. We combine Finite Difference schemes, ghost node strategy, correction formulas, and interpolation rules into a unified and stable numerical model. Moreover, the present method is capable of producing high-order solutions from a unique resource of available data. We attest to the accuracy and robustness of our single jump-based method through a variety of numerical experiments comprising Poisson problems with interfaces that can be now solved from a reduced number of jump conditions.

Uncertainty quantification of bladed disc systems using data driven stochastic reduced order models

This study focusses on the development of stochastic reduced order model for probabilistic characterisation of bladed disc systems with random spatial inhomogeneities. High fidelity finite element modelling is used to mathematically model the system. A two step reduction strategy is applied involving reduction in the state space dimension and reduction in the stochastic dimensions. Information of the spatial inhomogeneities are assumed to be available from limited in situ measurements across the spatial extent and are modelled as non-Gaussian random fields. The stochastic version of the finite element matrices are developed using a polynomial chaos based framework, which optimizes the stochastic dimensionality of the problem. The uncertainties in the input propagates through the system into the response, which are also random. Surrogate models for these response quantities are obtained as PCE and are constructed using the method of stochastic collocations. Challenges involved in application of PCE on complex geometrically irregular spatial domains are addressed. The efficacy of the proposed framework is demonstrated through two numerical examples -an academic bladed disc system and an industrial turbine rotor blade.

A new numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshes—Part 1: the derivations for the wave, heat and Poisson equations in the 1-D and 2-D cases

A new numerical approach for the time-dependent wave and heat equations as well as for the time-independent Poisson equation on irregular domains has been developed. Trivial Cartesian meshes and simple 9-point stencil equations with unknown coefficients are used for 2-D irregular domains. The calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy. The treatment of the Dirichlet and Neumann boundary conditions in the new approach is related to the development of high-order boundary conditions with the stencils that include the same or a smaller number of grid points compared to that for the regular 9-point internal stencils. At similar 9-point stencils, the accuracy of the new approach is two orders higher than that for the linear finite elements. The numerical results for irregular domains in Part 2 of the paper also show that at the same number of degrees of freedom, the new approach is even much more accurate than the quadratic and cubic finite elements with much wider stencils. Similar to our recent results on regular domains, the order of the accuracy of the new approach for the Poisson equation on irregular domains with square Cartesian meshes is higher than that with rectangular Cartesian meshes. The new approach can be directly applied to other partial differential equations.

A numerical solution of time-fractional mixed diffusion and diffusion-wave equation by an RBF-based meshless method

In this paper, we have developed an radial basis function (RBF) based meshless method to solve the time-fractional mixed diffusion and diffusion-wave equation which involves two fractional Caputo derivatives of order $$\alpha \in (0,1)$$ and $$\beta \in (1,2)$$ . The unconditional stability of the proposed numerical scheme is discussed and proved theoretically. The time semi discretization has been done by using the finite difference method and for space discretization, we proposed an RBF-based local collocation method. Some test problems are considered for regular as well as an irregular domain with uniform and non-uniform points to validate the efficiency and accuracy of the method.

An efficient meshless method based on the moving Kriging interpolation for two‐dimensional variable‐order time fractional mobile/immobile advection‐diffusion model

In this work, we introduce an efficient meshless technique for solving the two‐dimensional variable‐order time‐fractional mobile/immobile advection‐diffusion model with Dirichlet boundary conditions. The main advantage of this scheme is to obtain a global approximation for this problem which reduces such problems to a system of algebraic equations. To approximate the first and fractional variable‐order against the time, we use the finite difference relations. The proposed method is based on the moving Kriging (MK) interpolation shape functions. To discretize this model in space variables, we use the MK interpolation. Duo to the fact that the shape functions of MK have Kronecker's delta property, boundary conditions are imposed directly and easily. To illustrate the capability of the proposed technique on regular and irregular domains, several examples are presented in different kinds of domains and with uniform and nonuniform nodes. Also, we use this scheme to simulating anomalous contaminant diffusion in underground reservoirs.

Constructing Connectome Atlas by Graph Laplacian Learning.

The recent development of neuroimaging technology and network theory allows us to visualize and characterize the whole-brain functional connectivity in vivo. The importance of conventional structural image atlas widely used in population-based neuroimaging studies has been well verified. Similarly, a "common" brain connectivity map (also called connectome atlas) across individuals can open a new pathway to interpreting disorder-related brain cognition and behaviors. However, the main obstacle of applying the classic image atlas construction approaches to the connectome data is that a regular data structure (such as a grid) in such methods breaks down the intrinsic geometry of the network connectivity derived from the irregular data domain (in the setting of a graph). To tackle this hurdle, we first embed the brain network into a set of graph signals in the Euclidean space via the diffusion mapping technique. Furthermore, we cast the problem of connectome atlas construction into a novel learning-based graph inference model. It can be constructed by iterating the following processes: (1) align all individual brain networks to a common space spanned by the graph spectrum bases of the latent common network, and (2) learn graph Laplacian of the common network that is in consensus with all aligned brain networks. We have evaluated our novel method for connectome atlas construction in comparison with non-learning-based counterparts. Based on experiments using network connectivity data from populations with neurodegenerative and neuropediatric disorders, our approach has demonstrated statistically meaningful improvement over existing methods.

Multivariate approximation of functions on irregular domains by weighted least-squares methods

AbstractWe propose and analyse numerical algorithms based on weighted least squares for the approximation of a bounded real-valued function on a general bounded domain $\varOmega \subset \mathbb{R}^d$. Given any $n$-dimensional approximation space $V_n \subset L^2(\varOmega )$, the analysis in Cohen and Migliorati (2017, Optimal weighted least-squares methods. SMAI J. Comput. Math., 3, 181–203) shows the existence of stable and optimally converging weighted least-squares estimators, using a number of function evaluations $m$ of the order $n \ln n$. When an $L^2(\varOmega )$-orthonormal basis of $V_n$ is available in analytic form, such estimators can be constructed using the algorithms described in Cohen and Migliorati (2017, Optimal weighted least-squares methods. SMAI J. Comput. Math., 3, 181–203, Section 5). If the basis also has product form, then these algorithms have computational complexity linear in $d$ and $m$. In this paper we show that when $\varOmega $ is an irregular domain such that the analytic form of an $L^2(\varOmega )$-orthonormal basis is not available, stable and quasi-optimally weighted least-squares estimators can still be constructed from $V_n$, again with $m$ of the order $n \ln n$, but using a suitable surrogate basis of $V_n$ orthonormal in a discrete sense. The computational cost for the calculation of the surrogate basis depends on the Christoffel function of $\varOmega $ and $V_n$. Numerical results validating our analysis are presented.

The couple of Hermite-based approach and Crank-Nicolson scheme to approximate the solution of two dimensional stochastic diffusion-wave equation of fractional order

The main aim of this study is presenting a semi-discretization scheme to find the numerical solution of two dimensional (2D) stochastic time fractional diffusion-wave equation, which obtains from classical 2D diffusion-wave equation by replacing integer time derivative with Caputo fractional time derivative of order α (1 < α ≤ 2) and inserting some stochastic factors. In this scheme, first Crank-Nicolson and linear spline techniques are used to discrete mentioned problem in the time direction and then Hermite-based approach is applied to obtain the approximate solution in each time step. It is not required any discretization in the spatial directions and therefore this approach is an efficient tool to solve various problems which have been defined on irregular domains. Finally, to confirm this claim that obtained numerical results are accurate and in reliable agreement with the theoretical discussion, some test problems are included in the numerical example section. The values of maximum error, error associated with norm 2, and RMS–error are reported to demonstrate accuracy and reliability of the proposed method. Also, the domain of last example is considered in a long range of the time interval to study the stability of our method on time variable.

A high-order numerical approach with Cartesian meshes for modeling of wave propagation and heat transfer on irregular domains with inhomogeneous materials

Recently we have developed a new numerical approach for PDEs with constant coefficients on irregular domains and Cartesian meshes. In this paper we extend it to a much more general case of PDEs with variable coefficients that have a lot of applications; e.g., the modeling of functionally graded materials, the inhomogeneous materials obtained by 3-D printing and many others. Here, we consider the 2-D wave and heat equations for isotropic and anisotropic inhomogeneous materials. The idea of the extension to the case of PDEs with variable coefficients is based on the representation of the stencil coefficients as functions of the mesh size. This leads to the increase in the size of the local system of algebraic equations solved for each grid point of the new approach; however, this does not change the size of the global system of semidiscrete equations and practically does not increase the computational costs of the proposed technique. Similar to our previous technique, the new 2-D approach with compact 9-point stencils uses trivial Cartesian meshes for complex irregular domains and provides the fourth order of accuracy for the wave and heat equations with variable coefficients. The calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of the new technique. At similar 9-point stencils, the accuracy of the new approach is much higher than that for the linear finite elements. The numerical results for irregular domains show that at the same number of degrees of freedom, the new approach is even much more accurate than the high-order (up to the third order) finite elements with much wider stencils. The wave and heat equations are uniformly treated with the new approach.

An effective pure meshfree method for 1D/2D time fractional convection-diffusion problems on irregular geometry

An easy implemented and effective pure meshfree method is first developed to solve the 1D/2D constant/variable-order time fractional convection-diffusion equation (TF-CDE) on non-regular domain with two boundary conditions in this paper. The proposed method (CSPH-FDM) is derived from that the finite difference scheme (FDM) for Caputo time fractional derivative and a corrected smoothed particle hydrodynamics (CSPH) without kernel derivative for spatial derivatives. In the proposed CSPH-FDM, the high-order spatial derivative is divided into multi first-order derivatives and solved continuously by the CSPH, the Neumann boundary condition can be accurately treated, the two distribution cases of the local refinement and irregular particles or the arbitrary irregular shape domain can be easily and effectively implemented by the CSPH. To demonstrate the validity and numerical convergent order of the proposed method, several 1D/2D analytical examples with local refinement and irregular particles distributions or on complex geometries are first investigated, in which a four-order derivate problem with Neumann boundary is also considered. Subsequently, the CSPH-FDM is extended to predict the solute moving process versus time by two TF-CDEs on different irregular domains and compared with other numerical results. All the numerical results show the flexible application ability and reliability of the present method.

A MILP model for hydro unit commitment with irregular vibration zones based on the constrained Delaunay triangulation method

Irregular Vibration Zones are common in the giant hydropower stations built during the past decade in southwest China. They significantly affect the safe operation of the hydropower station, and thus became a key issue for hydropower operations in southwest China. The irregular Vibration Zone (VZ) constraint leads to nonconvex, discrete safe operating zones even with holes, which exacerbates the difficulty of solving the Hydro Unit Commitment (HUC) problem. This paper presents a novel Mixed Integer Linear Programming (MILP) model for the HUC problem with the irregular VZ constraint. The method firstly aggregates the power generation performance function and the irregular VZ constraint into a single function, which is highly nonconvex and discrete. The constrained Delaunay triangulation method is then utilized to partition the irregular domain of the function and construct the MILP compatible constraints with the disaggregation convex combination method. The practical behavior of the model is then analyzed through two cases for a giant hydropower station located in southwest China, which suffers from irregular VZs. The results show the high efficiency and the high accuracy of our model. The major advantage of the proposed method is that it can automatically model the general irregular VZs with fewer integer variables.

Natural Convection of Dusty Hybrid Nanofluids in Diverging–Converging Cavities Including Volumetric Heat Sources

Abstract Two Semi-Implicit-Method for Pressure-Linked-Equations (SIMPLE) techniques have been performed to solve the dimensionless equations governing a dusty hybrid nanofluid flow in wavy enclosures contain a volumetric heat source. These techniques are applied to evaluate the pressure terms for both the hybrid nanofluid and the dusty particles based on the control volume solver. Two systems of equations are proposed to simulate the hybrid nanofluid phase and the dusty particles phase. In addition, a body-fitted method is applied to map the irregular domain into a rectangular domain, and an inverse map technique is used to present the obtained data inside the given wavy domain. The hybrid mixture, here, is consisting of water as a base fluid, and the nanoparticles are Al2O3 and Cu. The controlling parameters are the Rayleigh numbers RaE and Ral, the ratio of the densities of the mixture Ds, the dusty parameter αs, and the total nanoparticles volume fraction. The results revealed that the absolute values of the stream function are reduced by 54.5% when the heating modes are switched from (Ral/RaE) &amp;lt; 1 to (Ral /RaE) &amp;gt; 1. Also, the average Nusselt is enhanced by 5.2% at a = 0.9, 6.74% at a = 0.95, and 11.36% at a = 1.1 when the nanoparticles volume fraction is increased from 1% to 5%.