Search Results (48)

Developing improved mathematical and numerical models of groundwater flow is crucial for monitoring and forecasting water resources. Most existing models are linear and fail to capture the complex physical processes involved in groundwater dynamics. This study aims to develop a nonlinear mathematical model for observing and forecasting changes in groundwater levels influenced by water intake wells, evaporation, and precipitation. The proposed mathematical model is formulated as a nonlinear differential equation. To solve this model, it was reduced to a dimensionless form, and the quasi-linearization method was employed to simplify the calculations. The finite difference method was then used to obtain a numerical solution. An algorithm and software were developed to demonstrate the results of the calculations. Numerical calculations performed using the developed software provide insights into the effects of water intake wells, surface evaporation, and precipitation on groundwater levels. The findings of this study hold practical significance for optimizing the sustainable use of water resources and highlighting how the location and flow rate of water intake wells impact groundwater levels. Full article

IpFlux is a cost-free software developed to provide a simplified, accessible, and accurate solution for hydraulic analysis in open-channel flows. It addresses the need for tools that support rapid decision-making during early design stages, especially when conventional software may be too complex, resource-intensive, or costly. Written in Python, IpFlux features an intuitive interface and implements both explicit and implicit formulations to compute normal and critical depths, hydraulic jumps, flow through weirs and gates, backwater curves, and compound cross-sections. Thanks to its focused interface and direct data entry, IpFlux enables significantly faster estimations than traditional tools used for similar hydraulic calculations, particularly in early project stages. The software’s accuracy and applicability are demonstrated by comparing its outputs against classical references and selected results from established tools such as HEC-RAS and ANSYS Fluent. While IpFlux is not intended to replace advanced simulation software, it offers a reliable and user-friendly alternative for preliminary analyses in engineering projects, as well as for educational purposes in hydraulic engineering. Full article

Structural design in Europe should strongly follow EN 1998-1 or so called Eurocode 8 (EC8), for a seismic resistance assessment of structures. Eurocode 8 recommends two linear methods and two nonlinear methods. The nonlinear methods require some knowledge about the nonlinear behavior of beams and joints in the structure, which makes the linear methods preferable. An alternative method of the seismic loading representation is to use artificial accelerograms with the same or similar spectra as the response spectrum used for modal spectrum analysis. Using an artificial diagram, three approaches in finite element methods exist: explicit time integration, implicit time integration, and modal dynamics. A typical six-story steel structure is modeled using the finite element method, and all linear methods are examined in both horizontal directions. The structure is examined by the modal response spectrum method using sufficient modes, as well as with and without the residual mode. The results are compared, and conclusions concerning the efficiency and precision of methods are deduced. Time history loading by accelerograms reveals higher dynamics and stress in the structural response than the modal response spectrum and lateral forces methods. The time history analysis methods have almost no difference in accuracy, and the modal dynamics method is the cheapest one. Full article

This paper presents a Finite Element Method (FEM) framework for solving the screened Poisson equation with a Dirac delta function as the forcing term. The singularity introduced by the delta function poses challenges for standard numerical methods, particularly in higher dimensions. To address this, we employ integrated Legendre basis functions, which yield sparse and structured system matrices characterized by a Banded-Block-Banded-Arrowhead ($B^3$-Arrowhead) form. In one dimension, the resulting linear system can be solved directly. In two and three dimensions, the equation can be efficiently solved using a generalized Alternating Direction Implicit (ADI) method combined with reverse Cholesky factorization. Numerical results in 1D, 2D, and 3D confirm that the method accurately captures the localized impulse response and reproduces the expected Green’s function behavior. The proposed approach offers a robust and scalable solution framework for partial differential equations with singular source terms and has potential applications in physics, engineering, and computational science. Full article

In this paper, a numerical method of a two-dimensional (2D) integro-differential equation with two fractional Riemann–Liouville (R-L) integral kernels is investigated. The compact difference method is employed in the spatial direction. The integral terms are approximated by a second-order convolution quadrature formula. The alternating direction implicit (ADI) compact difference scheme reduces the CPU time for two-dimensional problems. Simultaneously, the stability and convergence of the proposed ADI compact difference scheme are demonstrated. Finally, two numerical examples are provided to verify the established ADI compact difference scheme. Full article

In this paper, we will introduce a compact alternating direction implicit (ADI) difference scheme for solving the two-dimensional (2D) time fractional nonlinear Schrödinger equation. The difference scheme is constructed by using the $L1-2-3$ formula to approximate the Caputo fractional derivative in time and the fourth-order compact difference scheme is adopted in the space direction. The proposed difference scheme with a convergence accuracy of $O({\tau }^{1+\alpha }+h_x^4+h_y^4)(\alpha \in (0,1))$ is obtained by adding a small term, where $\tau$, $h_x$, $h_y$ are the temporal and spatial step sizes, respectively. The convergence and unconditional stability of the difference scheme are obtained. Moreover, numerical experiments are given to verify the accuracy and efficiency of the difference scheme. Full article

In this paper, the numerical solution for two-dimensional nonlinear parabolic equations is studied using an alternating-direction implicit (ADI) Crank–Nicolson (CN) difference scheme. Firstly, we use the CN format in the time direction, and then use the CN format in the space direction before discretizing the second-order center difference quotient. In addition, we strictly prove that the proposed ADI difference scheme has unique solvability and is unconditionally stable and convergent. The extrapolation method is further applied to improve the numerical solution accuracy. Finally, two numerical examples are given to verify our theoretical results. Full article

This paper presents the parallelization of two widely used implicit numerical solvers for the solution of partial differential equations on structured meshes, namely, the ADI (Alternating-Direction Implicit) solver for tridiagonal linear systems and the SIP (Strongly Implicit Procedure) solver for the penta-diagonal systems. Both solvers were parallelized using CUDA (Computer Unified Device Architecture) Fortran on GPGPUs (General-Purpose Graphics Processing Units). The parallel ADI solver (P-ADI) is based on the Parallel Cyclic Reduction (PCR) algorithm, while the parallel SIP solver (P-SIP) uses the wave front method (WF) following a diagonal line calculation strategy. To map the solution schemes onto the hierarchical block-threads framework of the CUDA on the GPU, the P-ADI solver adopted two mapping methods, one block thread with iterations (OBM-it) and multi-block threads (MBMs), while the P-SIP solver also used two mappings, one conventional mapping using effective WF lines (WF-e) with matrix coefficients and solution variables defined on original computational mesh, and a newly proposed mapping using all WF mesh (WF-all), on which matrix coefficients and solution variables are defined. Both the P-ADI and the P-SIP have been integrated into a two-dimensional (2D) hydrodynamic model, the CCHE2D (Center of Computational Hydroscience and Engineering) model, developed by the National Center for Computational Hydroscience and Engineering at the University of Mississippi. This study for the first time compared these two parallel solvers and their efficiency using examples and applications in complex geometries, which can provide valuable guidance for future uses of these two parallel implicit solvers in computational fluids dynamics (CFD). Both parallel solvers demonstrated higher efficiency than their serial counterparts on the CPU (Central Processing Unit): 3.73~4.98 speedup ratio for flow simulations, and 2.166~3.648 speedup ratio for sediment transport simulations. In general, the P-ADI solver is faster than but not as stable as the P-SIP solver; and for the P-SIP solver, the newly developed mapping method WF-all significantly improved the conventional mapping method WF-e. Full article

A point cloud is a simple and concise 3D representation, but point cloud generation is a long-term challenging task in 3D vision. However, most existing methods only focus on their effectiveness of generation and auto-encoding separately. Furthermore, both generative adversarial networks (GANs) and auto-encoders (AEs) are the most popular generative models. But there is a lack of related research that investigates the implicit connections between them in the field of point cloud generation. Thus, we propose a new bidirectional network (BI-Net) trained with collaborative learning, introducing more priors through the alternate parameter optimizations of a GAN and AE combination, which is different from the way of combining them at the network structure and loss function level. Specifically, BI-Net acts as a GAN and AE in different data processing directions, where their network structures can be reused. If optimizing only the GAN without the AE, there is no direct constraint of ground truth on the generator’s parameter optimization. This unique approach enables better network optimization and leads to superior generation results. Moreover, we propose a nearest neighbor mutual exclusion (NNME) loss to further homogenize the spatial distribution of generated points during the reverse direction. Extensive experiments were conducted, and the results show that the BI-Net produces competitive and high-quality results on reasonable structure and uniform distributions compared to existing state-of-the-art methods. We believe that our network structure (BI-Net) with collaborative learning could provide a new promising method for future point cloud generation tasks. Full article

In this paper, we consider the numerical solution of large-scale discrete-time projected Lyapunov equations. We provide some reasonable extensions of the most frequently used low-rank iterative methods for linear matrix equations, such as the low-rank Smith method and the low-rank alternating-direction implicit (ADI) method. We also consider how to reduce complex arithmetic operations and storage when shift parameters are complex and propose a partially real version of the low-rank ADI method. Through two standard numerical examples from discrete-time descriptor systems, we will show that the proposed low-rank alternating-direction implicit method is efficient. Full article

In this paper, we develop a finite difference method for solving the wave equation with fractional damping in 1D and 2D cases, where the fractional damping is given based on the Caputo fractional derivative. Firstly, based on the weighted method, we propose a new numerical approximation for the Caputo fractional derivative and apply it for the 1D case to obtain a time-stepping method. We then develop an alternating direction implicit (ADI) scheme for the 2D case. Using the discrete energy method, we prove that the proposed difference schemes are unconditionally stable and convergent in both 1D and 2D cases. Finally, several numerical examples are given to verify the theoretical results. Full article

A novel dimension splitting method is proposed for the efficient numerical simulation of a biochemotaxis model, which is a coupled system of chemotaxis–fluid equations and incompressible Navier–Stokes equations. A second-order pressure correction method is employed to decouple the velocity and pressure for the Navier–Stokes equations. Then, the alternating direction implicit scheme is used to solve the velocity equation, and the operator with dimension splitting effect is used instead of the traditional elliptic operator to solve the pressure equation. For the chemotactic equation, the operator splitting method and extrapolation technique are used to solve oxygen and cell density to achieve second-order time accuracy. The proposed dimension splitting method splits the two-dimensional problem into a one-dimensional problem by splitting the spatial derivative, which reduces the computation and storage costs. Finally, through interesting experiments, we show the evolution of the cell plume shape during the descent process. The effect of changing specific parameters on the velocity and plume shape during the descent process is also studied. Full article

In this study, a novel gas-extruded membrane bearing was developed, and an optimization algorithm was applied to solve a Reynolds equation that describes the load-bearing characteristics of this bearing. This was effective in improving the solution rate of the Reynolds equation, significantly reducing the difficulty of obtaining a solution, avoiding high programming difficulty, and achieving a high solution accuracy. Through a comparative analysis, the error in the accuracy of the alternating implicit difference method was addressed, and the traditional finite element method for solving the same model was verified, with an average error of 2% reached to verify its applicability. The algorithm was also used to analyze the load-bearing capacity of the gas-extruded membrane bearing. This revealed not only a positive correlation of the average load-bearing capacity of the gas-extruded membrane bearing with the frequency and amplitude of vibration but also a negative correlation with radial clearance, with the cut-off frequency reaching 19 Khz. The load-bearing capacity of the gas-extruded membrane bearing proposed in this paper reached 1.28 N, which indicated an error of 3.28% with the theoretical approach. To sum up, this research provides an important reference for the design and manufacture of novel gas-extruded membrane bearings. Full article

In this study, a two-dimensional contaminant transport model with time-varying axial input sources subject to non-linear sorption, decay, and production is numerically solved to find the concentration distribution profile in a heterogeneous, finite soil medium. The axial input sources are assigned on the coordinate axes of the soil medium, with background sources varying sinusoidally with space. The groundwater velocities are considered space-dependent in the longitudinal and transversal directions. Various forms of axial input sources are considered to study their transport patterns in the medium. The alternating direction implicit (ADI) and Crank-Nicolson (CN) methods are applied to approximate the two-dimensional governing equation, and the obtained algebraic system of equations in each case is further solved by MATLAB scripts. Both approximate solutions are illustrated graphically for various hydrological input data. The influence of various hydrogeological input parameters, such as the medium’s porosity, density, sorption conditions, dispersion coefficients, etc., on the contaminant distribution is analyzed. Further, the influence of constant and varying velocity parameters on groundwater contaminant transport is studied. The stability of the proposed model is tested using the Peclet and Courant numbers. Substantial similarity is observed when the approximate solution obtained using the CN method is compared with the finite element method in a special case. The proposed approximate solution is compared with the existing numerical solutions, and an overall agreement of 98–99% is observed between them. Finally, the stability analysis reveals that the model is stable and robust. Full article

Leak 2D is a new two-dimensional dual permeability mathematical model for the simulation of the preferential flow in the vadose zone. In this model, water flow in the soil matrix domain is described by the two-dimensional h-based Richards’ equation. Water flow in the fracture domain is estimated using the kinematic wave approach. Richards’ equation is solved by a combination of the alternating direction implicit (A.D.I.) method and the Douglas and Jones predictor−corrector method. The kinematic wave equation is solved explicitly. In the present paper, Leak 2D is calibrated and validated with data obtained in a Hele–Shaw apparatus filled with sand. Preferential flow is achieved by inserting four artificial macropores of various sizes into the soil. Six irrigations of various intensities and durations were used for the calibration and validation process. The water content at various depths was recorded by five sensors that were inserted into the soil. A comparison of the simulated water content with the measured profiles shows that Leak 2D can sufficiently describe preferential flow into the unsaturated zone of the soil, even under extreme irrigation conditions. Full article