Irregular Domains Research Articles

A study on the numerical solution of the Sobolev equation with a Burgers-type nonlinearity on two-dimensional irregular domains using the local RBF partition of unity method

A study on the numerical solution of the Sobolev equation with a Burgers-type nonlinearity on two-dimensional irregular domains using the local RBF partition of unity method

Local signal detection on irregular domains with generalized varying coefficient models

In spatial analysis, it is essential to understand and quantify spatial or temporal heterogeneity. This paper focuses on the generalized spatially varying coefficient model (GSVCM), a powerful framework to accommodate spatial heterogeneity by allowing regression coefficients to vary in a given spatial domain. We propose a penalized bivariate spline method for detecting local signals in GSVCM. The key idea is to use bivariate splines defined on triangulation to approximate nonparametric varying coefficient functions and impose a local penalty on L 2 norms of spline coefficients for each triangle to identify null regions of zero effects. Moreover, we develop model confidence regions as the inference tool to quantify the uncertainty of the estimated null regions. Our method partitions the region of interest using triangulation and efficiently approximates irregular domains. In addition, we propose an efficient algorithm to obtain the proposed estimator using the local quadratic approximation. We also establish the consistency of estimated nonparametric coefficient functions and the estimated null regions. The numerical performance of the proposed method is evaluated in both simulation cases and real data analysis.

A least-squares Fourier frame method for nonlocal diffusion models on arbitrary domains

We introduce a least-squares Fourier frame method for solving nonlocal diffusion models with Dirichlet volume constraint on arbitrary domains. The mathematical structure of a frame rather than a basis allows using a discrete least-squares approximation on irregular domains and imposing non-periodic boundary conditions. The method has inherited the one-dimensional integral expression of Fourier symbols of the nonlocal diffusion operator from Fourier spectral methods for any d spatial dimensions. High precision of its solution can be achieved via a direct solver such as pivoted QR decomposition even though the corresponding system is extremely ill-conditioned, due to the redundancy in the frame. The extension of AZ algorithm improves the complexity of solving the rectangular linear system to O(Nlog2⁡N) for 1d problems and O(N2log2⁡N) for 2d problems, compared with O(N3) of the direct solvers, where N is the number of degrees of freedom. We present ample numerical experiments to show the flexibility, fast convergence and asymptotical compatibility of the proposed method.

An efficient approach of meshless node placement in three-dimensional subsurface flow modeling

Meshless methods are valuable tools for real-time subsurface flow modeling and are particularly beneficial for adaptively adjusting node positions when incorporating supplementary measurements. However, an algorithm is required for automatically generating high-quality meshless nodes to adapt to irregularly shaped aquifers and arbitrarily distributed wells. This paper introduces a three-dimensional adaptive meshless node placement technique based on advancing front nodes and proposes a sigmoid exclusion circle to ensure adaptability to specific positions of interest. It is crucial for groundwater simulations to have specified computational points at wellbore locations. The generated meshless nodes were used as computational points in the generalized finite difference method. The node quality was assessed by comparing the numerical solutions to the analytical solution, achieving errors of the order of 10–12. The application scenarios of irregularly shaped aquifers demonstrated the high efficiency of the three-dimensional node placement in generating up to 10,000 nodes in <5 s. The effectiveness of the algorithm was verified using three-dimensional irregular computational domains. The algorithm could be easily adapted to two-dimensional problems, and an irregular boundary example is presented. The validation and application cases highlighted the versatility of the proposed method and established its potential for real-world hydrogeological applications.

Elastic least‐squares reverse time migration from topography through anisotropic tensorial elastodynamics

AbstractLeast‐squares reverse time migration is an increasingly popular technique for subsurface imaging, especially in the presence of complex geological structures. However, elastic least‐squares reverse time migration algorithms face significant practical and numerical challenges when migrating multi‐component seismic data acquired from irregular topography. Many associated issues can be avoided by abandoning the Cartesian coordinate system and migrating the data to a generalized topographic coordinate system conformal to surface topology. We introduce a generalized anisotropic elastic least‐squares reverse time migration methodology that uses the numerical solutions of tensorial elastodynamics for propagating wavefields in computational domains influenced by free‐surface topography. We define a coordinate mapping assuming unstretched vertically translated meshes that transform an irregular physical domain to a regular computational domain on which calculating numerical elastodynamics solutions is straightforward. This allows us to obtain numerical solutions of forward and adjoint elastodynamics and generate subsurface images directly in topographic coordinates using a tensorial energy‐norm imaging condition. Numerical examples demonstrate that the proposed generalized elastic least‐squares reverse time migration algorithm is suitable for generating high‐quality images with reduced artefacts and better balanced reflectivity that can accurately explain observed data acquired from topography in a medium characterized by arbitrary heterogeneity and anisotropy. Finally, the computational cost of our method is comparable to that of an equivalent Cartesian elastic least‐squares reverse time migration numerical implementation.

Analysis of efficient preconditioner for solving Poisson equation with Dirichlet boundary condition in irregular three-dimensional domains

This paper analyzes modified ILU (MILU)-type preconditioners for efficiently solving the Poisson equation with Dirichlet boundary conditions on irregular domains. In [1], the second-order accuracy of a finite difference scheme developed by Gibou et al. [2] and the effect of the MILU preconditioner were presented for two-dimensional problems. However, the analyses do not directly extend to three-dimensional problems. In this paper, we first demonstrate that the Gibou method attains second-order convergence for three-dimensional irregular domains, yet the discretized Laplacian exhibits a condition number of O(h−2) for a grid size h. We show that the MILU preconditioner reduces the order of the condition number to O(h−1) in three dimensions. Furthermore, we propose a novel sectored-MILU preconditioner, defined by a sectorized lexicographic ordering along each axis of the domain. We demonstrate that this preconditioner reaches a condition number of order O(h−1) as well. Sectored-MILU not only achieves a similar or better condition number than conventional MILU but also improves parallel computing efficiency, enabling very efficient calculation when the dimensionality or problem size increases significantly. Our findings extend the feasibility of solving large-scale problems across a range of scientific and engineering disciplines.

Gaussian modeling with B-splines for spatial functional data on irregular domains

Functional Data Analysis (FDA) has emerged as a powerful framework for datasets that exhibit continuous variation over specified intervals. Unlike traditional methods, FDA treats data as functions, providing a more comprehensive understanding of their behavior and variability. This approach is particularly beneficial for exploring observations collected over time or space. The Spatial Functional Data Analysis (SFDA) incorporates spatial locations as an additional dimension. This integration enhances the ability to model complex associations between curves in the functional domain. Despite progress in analyzing multivariate spatial data, limited attention has been given to SFDA. This study addresses this gap by proposing a Bayesian Gaussian functional approach based on B-splines. The proposal also addresses challenges posed by irregularly spaced observations in the functional domain, a feature rarely considered in the literature. A simulation study is developed to evaluate performance. Finally, results are explored using a relatively novel real dataset from monitoring stations of temperature in Mexico City.

Refactoring the elastic–viscous–plastic solver from the sea ice model CICE v6.5.1 for improved performance

Abstract. This study focuses on the performance of the elastic–viscous–plastic (EVP) dynamical solver within the sea ice model, CICE v6.5.1. The study has been conducted in two steps. First, the standard EVP solver was extracted from CICE for experiments with refactored versions, which are used for performance testing. Second, one refactored version was integrated and tested in the full CICE model to demonstrate that the new algorithms do not significantly impact the physical results. The study reveals two dominant bottlenecks, namely (1) the number of Message Parsing Interface (MPI) and Open Multi-Processing (OpenMP) synchronization points required for halo exchanges during each time step combined with the irregular domain of active sea ice points and (2) the lack of single-instruction, multiple-data (SIMD) code generation. The standard EVP solver has been refactored based on two generic patterns. The first pattern exposes how general finite differences on masked multi-dimensional arrays can be expressed in order to produce significantly better code generation by changing the memory access pattern from random access to direct access. The second pattern takes an alternative approach to handle static grid properties. The measured single-core performance improvement is more than a factor of 5 compared to the standard implementation. The refactored implementation of strong scales on the Intel® Xeon® Scalable Processors series node until the available bandwidth of the node is used. For the Intel® Xeon® CPU Max series, there is sufficient bandwidth to allow the strong scaling to continue for all the cores on the node, resulting in a single-node improvement factor of 35 over the standard implementation. This study also demonstrates improved performance on GPU processors.

Prediction–correction projection immersed interface method for interfacial stress-induced flows and applications to electrohydrodynamics

We propose a robust, high-resolution prediction–correction projection immersed interface method (IIM) for solving the unsteady incompressible Navier–Stokes equations with traction boundary conditions, which arise from free surface flows driven by capillary, electric, and elastic forces. This method combines the advantages of traditional body-fitted moving mesh methods and immersed boundary methods (IBM), allowing for the accurate imposition of boundary conditions on free surfaces using Cartesian grids and providing detailed interface information that is typically smoothed out in traditional IBM. The irregular liquid-phase domain is embedded within a square region and discretized using a dynamically adaptive Cartesian mesh. The free surface is captured using a narrow-band level set with hybrid reinitialization. A prediction–correction projection scheme is constructed, incorporating additional pressure predictions and corrections to enhance robustness. The resulting Helmholtz/Poisson equations are solved using the augmented IIM for boundary value problems. Grid refinement analysis demonstrates second-order convergence of the L∞ error, even in challenging cases such as the oscillating drop test with low viscosity. We further apply this method to electrohydrodynamic (EHD) problems, constructing an implicit augmented IIM to solve the equations governing the electric field and charge conservation. Numerical experiments demonstrate that this method accurately addresses highly intense EHD phenomena, such as the formation of Taylor cones, highlighting its robustness.

An efficient and accurate mapping method for elliptic equations in irregular annular domains

In this paper, we introduce a coordinate transformation, which transforms the irregular annular domain to a unit disk. We present its basic properties. As examples, we consider Poisson type equation and Cauchy–Navier elastic equations with variable coefficients in two-dimensional irregular annular domains, and prove the existence and uniqueness of weak solutions. We also construct the mixed Fourier–Legendre spectral schemes, and derive the optimal convergence of numerical solutions under the H1-norm. The numerical results indicate that the suggested method achieves high-order accuracy.

Two-dimensional conjugated metal–organic frameworks grown on a MoS2 surface

Molybdenum disulfide (MoS2) features an atomically flat surface without dangling bonds. Molecular self-assembly on this surface provides an effective route to constructing heterostructure devices. In this work, we show the successful synthesis of M3(1,4,5,8,9,12-hexaazatriphenylene, HAT)2 (M = Ni, Co) conjugated metal–organic frameworks (c-MOFs) on a MoS2 surface. In the frameworks, HAT molecules constitute a honeycomb lattice while the metal atoms constitute a Kagome lattice. The random orientations of the frameworks with respect to the substrate and irregular domain shapes indicate that the frameworks interact weakly with the MoS2. The successful synthesis of 2D c-MOFs on inert substrates opens a door for the construction of advanced 2D van der Waals heterojunctions.

Irregular domains: Special coordinates for a pseudospectral method

Working with a special coordinate system, this study demonstrates how to obtain numerical solutions with geometric convergence for the eigenstates of a Laplacian operator in irregular prismatic domains (both annular and single) that are simply connected. An appropriate coordinate system, which defines a tightly bounded domain, allows for a fair mesh for series approximation nodes. Three independent criteria were used to verify the consistency of the solutions: the Rayleigh quotient, the divergence theorem, and a partial derivative equation (PDE) transformed from an eigenvalue problem to a boundary value problem with Robin conditions. Supporting the proposed method, examples show a few hundred eigenstates obtained in a single computation, with at least 10 significant figures and a low computational cost.

The localized RBF interpolation with its modifications for solving the incompressible two-phase fluid flows: A conservative Allen–Cahn–Navier–Stokes system

In this research work, we apply a numerical scheme based on the first-order time integration approach combined with the modifications of the meshless approximation for solving the conservative Allen–Cahn–Navier–Stokes equations. More precisely, we first utilize a first-order time discretization for the Navier–Stokes equations and the time-splitting technique of order one for the dynamics of the phase-field variable. Besides, we use the local interpolation based on the Matérn radial function for spatial discretization. We should solve a Poisson equation with the proper boundary conditions to have the divergence-free property during the numerical algorithm. Accordingly, the applied numerical procedure could not give a stable and accurate solution. Instead, we solve a regularization system in a discrete form. To prevent the instability of the numerical solution concerning the convection term, a biharmonic term with a small coefficient based on the high-order hyperviscosity formulation has been added, which has been approximated by a scalable interpolation based on the combination of polyharmonic spline with polynomials (known as the PHS+poly). The obtained full-discrete problem is solved using the biconjugate gradient stabilized method considering a proper preconditioner. We investigate the potency of the numerical scheme by presenting some simulations via uniform, hexagonal, and quasi-uniform nodes on rectangular and irregular domains. Besides, we have compared the proposed meshless method with the standard finite element method due to the used CPU time.

Numerical study of the multi-dimensional Galilei invariant fractional advection–diffusion equation using direct mesh-less local Petrov–Galerkin method

This article presents a local mesh-less procedure for simulating the Galilei invariant fractional advection–diffusion (GI-FAD) equations in one, two, and three-dimensional spaces. The proposed method combines a second-order Crank–Nicolson scheme for time discretization and the second-order weighted and shifted Grünwald difference (WSGD) formula. This time discretization scheme ensures unconditional stability and convergence with an order of O(τ2). In the spatial domain, a mesh-less weak form is employed based on the direct mesh-less local Petrov–Galerkin (DMLPG) method. The DMLPG method employs the generalized moving least-square (GMLS) approximation in conjunction with the local weak form of the equation. By utilizing simple polynomials as shape functions in the GMLS approximation, the necessity for complex shape function construction in the MLS approximation is eliminated. To validate and demonstrate the effectiveness of the proposed algorithm, a variety of problems in one, two, and three dimensions are investigated on both regular and irregular computational domains. The numerical results obtained from these investigations confirm the accuracy and reliability of the developed approach in simulating GI-FAD equations.

Uniqueness, regularity, continuity, and stability of solutions of integral equations in irregular domains

This study explores the uniqueness, regularity, continuity, and stability of solutions to integral equations related to elliptical boundary value problems in irregular domains. Traditional methods usually consider soft boundaries, but this work extends these results to include domains with rough boundaries. By employing Sobolev spaces, especially fractional Sobolev spaces, we develop a comprehensive theoretical framework. Under appropriate conditions, the studied integral equation has a unique solution in fractional Sobolev spaces, maintaining the regularity properties of the input function. Two new theorems are introduced to strengthen the findings. The first theorem establishes that solutions are continuous with respect to perturbations in the input data. This means small changes in the input do not cause significant deviations in the solution, ensuring robustness in practical applications. The second theorem demonstrates the stability of the solutions when the domain geometry undergoes small changes. This stability is crucial for real-world scenarios where the exact shapes of domains may not be perfectly known or may vary slightly. These results significantly improve the mathematical understanding of boundary value problems in non-smooth domains. The proposed extended framework allows for more generalized applications, addressing challenges in various fields of physics and engineering. The ability to handle irregular domains with rough boundaries opens up new possibilities for modeling and solving complex problems where traditional soft boundary assumptions are not valid. This research provides a robust basis for future studies and applications, highlighting the importance of considering fractional Sobolev spaces in the analysis of integral equations.

Large-scale flood modeling and forecasting with FloodCast

Large-scale hydrodynamic models generally rely on fixed-resolution spatial grids and model parameters as well as incurring a high computational cost. This limits their ability to accurately forecast flood crests and issue time-critical hazard warnings. In this work, we build a fast, stable, accurate, resolution-invariant, and geometry-adaptive flood modeling and forecasting framework that can perform at large scales, namely FloodCast. The framework comprises two main modules: multi-satellite observation and hydrodynamic modeling. In the multi-satellite observation module, a real-time unsupervised change detection method and a rainfall processing and analysis tool are proposed to harness the full potential of multi-satellite observations in large-scale flood prediction. In the hydrodynamic modeling module, a geometry-adaptive physics-informed neural solver (GeoPINS) is introduced, benefiting from the absence of a requirement for training data in physics-informed neural networks (PINNs) and featuring a fast, accurate, and resolution-invariant architecture with Fourier neural operators. To adapt to complex river geometries, we reformulate PINNs in a geometry-adaptive space. GeoPINS demonstrates impressive performance on popular partial differential equations across regular and irregular domains. Building upon GeoPINS, we propose a sequence-to-sequence GeoPINS model to handle long-term temporal series and extensive spatial domains in large-scale flood modeling. This model employs sequence-to-sequence learning and hard-encoding of boundary conditions. Next, we establish a benchmark dataset in the 2022 Pakistan flood using a widely accepted finite difference numerical solution to assess various flood simulation methods. Finally, we validate the model in three dimensions - flood inundation range, depth, and transferability of spatiotemporal downscaling - utilizing SAR-based flood data, traditional hydrodynamic benchmarks, and concurrent optical remote sensing images. Traditional hydrodynamics and sequence-to-sequence GeoPINS exhibit exceptional agreement during high water levels, while comparative assessments with SAR-based flood depth data show that sequence-to-sequence GeoPINS outperforms traditional hydrodynamics, with smaller simulation errors. The experimental results for the 2022 Pakistan flood demonstrate that the proposed method enables high-precision, large-scale flood modeling with an average MAPE of 14.93 % and an average Mean Absolute Error (MAE) of 0.0610 m for 14-day water depth simulations while facilitating real-time flood hazard forecasting using reliable precipitation data.

Isogeometric collocation method to simulate phase-field crystal model

PurposeThe purpose of this study is to develop a new numerical algorithm to simulate the phase-field model.Design/methodology/approachFirst, the derivative of the temporal direction is discretized by a second-order linearized finite difference scheme where it conserves the energy stability of the mathematical model. Then, the isogeometric collocation (IGC) method is used to approximate the derivative of spacial direction. The IGC procedure can be applied on irregular physical domains. The IGC method is constructed based upon the nonuniform rational B-splines (NURBS). Each curve and surface can be approximated by the NURBS. Also, a map will be defined to project the physical domain to a simple computational domain. In this procedure, the partial derivatives will be transformed to the new domain by the Jacobian and Hessian matrices. According to the mentioned procedure, the first- and second-order differential matrices are built. Furthermore, the pseudo-spectral algorithm is used to derive the first- and second-order nodal differential matrices. In the end, the Greville Abscissae points are used to the collocation method.FindingsIn the numerical experiments, the efficiency and accuracy of the proposed method are assessed through two examples, demonstrating its performance on both rectangular and nonrectangular domains.Originality/valueThis research work introduces the IGC method as a simulation technique for the phase-field crystal model.

Visibility graph-based covariance functions for scalable spatial analysis in non-convex partially Euclidean domains.

We present a new method for constructing valid covariance functions of Gaussian processes for spatial analysis in irregular, non-convex domains such as bodies of water. Standard covariance functions based on geodesic distances are not guaranteed to be positive definite on such domains, while existing non-Euclidean approaches fail to respect the partially Euclidean nature of these domains where the geodesic distance agrees with the Euclidean distances for some pairs of points. Using a visibility graph on the domain, we propose a class of covariance functions that preserve Euclidean-based covariances between points that are connected in the domain while incorporating the non-convex geometry of the domain via conditional independence relationships. We show that the proposed method preserves the partially Euclidean nature of the intrinsic geometry on the domain while maintaining validity (positive definiteness) and marginal stationarity of the covariance function over the entire parameter space, properties which are not always fulfilled by existing approaches to construct covariance functions on non-convex domains. We provide useful approximations to improve computational efficiency, resulting in a scalable algorithm. We compare the performance of our method with those of competing state-of-the-art methods using simulation studies on synthetic non-convex domains. The method is applied to data regarding acidity levels in the Chesapeake Bay, showing its potential for ecological monitoring in real-world spatial applications on irregular domains.

An essay on the foundations of variational methods: Exploring Sobolev Spaces for boundary integral equations

This work addresses the uniqueness and regularity of solutions to integral equations associated with elliptic boundary value problems in irregular domains. Traditional results often assume smooth (Lipschitz) boundaries, but this study extends these results to more general domains with irregular boundaries. By leveraging Sobolev spaces, particularly fractional Sobolev spaces , and the properties of the Slobodetskii norm, we develop a robust theoretical framework. Our main theorem demonstrates that, under suitable conditions, has a unique solution in , and this solution inherits the regularity properties from the function . The results provide significant advancements in the mathematical understanding of boundary value problems in non-smooth domains, with potential applications in various fields of physics and engineering.

A novel semi-implicit WLS scheme for time-memory nonlinear behavior in 2D variable-order TF-NLSEs

In this paper, a novel hybrid semi-implicit meshless weighted least-squares (WLS) scheme H-SIFPM is developed for the first time by coupling the semi-implicit finite point-set method and finite difference method (FDM), and then applied to predict the time-memory nonlinear behavior dominated by a 2D variable-order time-fractional nonlinear Schrödinger equation (TF-NLSE). The proposed H-SIFPM for TF-NLSE is mainly derived from that: firstly, the variable-order time-fractional with Caputo derivative is approximated by a FDM scheme and then imposed to the approximation of WLS as a boundary condition; secondly, the hybrid process is implicitly discretized in temporal level, and an implicit meshless FPM scheme is developed; thirdly, an iterative technique is adopted to treat the implicit format. Subsequently, the estimated error and stability of the proposed scheme is tentatively proved. Moreover, the accuracy and capacity of the proposed H-SIFPM are also tested by solving several examples with constant/variable-order cases, in which a second-order numerical convergent rate is illustrated. We also demonstrate the advantages of the meshless method by solving the problem on complex irregular domain with local refinement. Finally, the nonlinear dispersion behavior dominated by 2D TF-NLSE/TF-GPE with different boundary conditions are predicted, and the FDM results are also presented for comparison. All numerical results show that the proposed H-SIFPM scheme for TF-NLSEs is stable, accurate and flexible.