Non-uniform Mesh Research Articles

Two new optimal and uniform third-order schemes for singular perturbation problems with initial layers

This article presents two numerical methods of the order of three for singular perturbation problems, with a small positive parameter using finite differences. It is a problem with an initial layer in the neighborhood of the initial nodal point whose width is of the order of the small parameter \(\epsilon \). Explicit order three classical methods are modified, and a new scheme is designed for singular perturbation problems. It is a fitted operator method and it is explicit with a variable fitting factor (VFF) evaluated at all nodal points. To reduce the calculation time of the scheme with VFF, the VFF is replaced by a fixed constant fitting factor (CFF). It is implicit with a CFF which is evaluated only one time at the initial nodal point. These two schemes are both optimal concerning the small parameter \(\epsilon \) and uniform of order three. The three order methods presented in this article are superior to the three order methods available in the literature. To view the initial layer when the mesh size is larger than the parameter in the problem, these two fitted operator methods are extended to fitted mesh methods since fitted mesh methods are layer rescaling. The construction of the fitted mesh method is also provided. That is, the uniform mesh is extended to non-uniform mesh. Experimental results are presented to show the optimal and higher-order performance of the two numerical methods using three test problems.

Numerical method for compressible gas-particle flow coupling using adaptive parcel refinement (APR) method on non-uniform mesh

The Eulerian–Lagrangian (EL) approach is usually used for simulating the compressible gas-particle flows. In our previous study, we have developed the compressible multiphase particle-in-cell method (CMP-PIC) and achieved good results for both dilute and dense compressible gas-particle flow. However, the original CMP-PIC was only implemented on the uniform mesh, which limits its resolution, efficiency, and application in the field of engineering. Therefore, in the present study, we have implemented the developed algorithms for coupling terms and the adaptive parcel refinement APR method on the non-uniform mesh. The algorithm of coupling terms is a new method for interpolating between a parcel with definite volume and cells with different volumes; this has not been investigated in previous particle-in-cell (PIC) research. The improved operator of the multiphase coupling terms is put forward to adapt the wide range size ratios since directly using the original operator causes the loss of coupling effects. The APR sets up a numerical “breakup model” of a parcel from the computational perspective, which represents a new approach to a parcel-type numerical method. In the APR method, the parcel resolution can automatically match the resolution of the local Eulerian cell. Therefore, the improved CMP-PIC can handle the simulation when a large parcel enters the fine cell and prevents the abnormal termination due to non-physical packing of particles. Based on conservation and scaling laws, the strategy, split criterion, and split configuration are identified to correlate the parameters of the parent and child parcels. Additionally, the algorithms of the multiphase HLL/HLLC solvers are derived for flux and nozzling terms to adapt the Cartesian grid with varying cell sizes, which is a type of non-uniformity. Several one-dimensional and two-dimensional tests show the correctness and advantages of using the non-uniform mesh, improved operator, and APR method. Results show that the combination of improved interpolation operator and APR method can significantly improve the reliability, resolution of simulation results, and robustness of the algorithm. The simulation time is also reduced due to regional refinement.

Two-phase flow in heterogeneous porous media: A multiscale digital model approach

Pore-scale fluid flow simulation on digital rocks from X-ray CT experiments can predict the physical properties and analyze the distribution characteristics of the fluids, which is critical for many problems in environmental remediation and geo-materials. It is straightforward to numerically simulate the multiphase flow on the digital rocks from CT scanning if they are homogeneous. However, it is of great difficulty to obtain large and high-resolution digital models of heterogeneous samples containing multiscale features only by CT scanning since the CT machines strike a balance between the field of view and resolution. In addition, the current methods of multiphase flow modeling face challenges in dealing with large samples and heavy computations. To address such issues, a unified and integrative modeling approach for constructing multiscale digital images and simulating multiphase flow in complex and heterogeneous pore systems was presented. Firstly, we developed a hybrid modeling method for producing multiscale digital models to represent the heterogeneous pore systems on which two-phase flow simulations are computed. The results show that the constructed models built by the hybrid method contain more tiny pores than the ones from regular CT imaging and exhibit better connectivity and permeability. The pore/throat size distributions, porosity, and permeability of the digital models produced by the hybrid method are closer to the experimental data than the ones from the single-scale models. Moreover, the non-uniform meshes for complex pore systems and parallel computing on a supercomputing cluster are combined to improve the computational efficiency of the two-phase flow simulation. The distribution characteristics of fluids in pore systems at the end of two-phase flow were also analyzed, which improves our understanding of two-phase flow.

A second-order L2-1σ Crank-Nicolson difference method for two-dimensional time-fractional wave equations with variable coefficients

Based on quadratic and linear polynomial interpolation approximations and the Crank-Nicolson technique, a new second-order difference approximation method of the Caputo fractional derivative of order α∈(1,2) is proposed. This method is different from the known second-order methods, which is called the L2-1σ Crank-Nicolson method in this paper. Using the L2-1σ Crank-Nicolson method of the Caputo fractional derivative, a fully discrete L2-1σ Crank-Nicolson difference method for two-dimensional time-fractional wave equations with variable coefficients is developed. The unconditional stability and convergence of the method are rigorously proved. The optimal error estimates in the discrete L2-norm and H1-norm are obtained under a relatively weak regularity condition. The error estimates show that the method has the second-order convergence in both time and space for all α∈(1,2). In order to overcome the loss of the temporal convergence order caused by the stronger singularity of the exact solution at the initial time, a nonuniform L2-1σ Crank-Nicolson difference method is also developed on a class of general nonuniform time meshes. Numerical results confirm the theoretical convergence result. The effectiveness of the nonuniform method for non-smooth solutions with the stronger singularity at the initial time is tested on a class of initially graded time meshes.

The Variable-Step L1 Scheme Preserving a Compatible Energy Law for Time-Fractional Allen-Cahn Equation

In this work, we revisit the adaptive L1 time-stepping scheme for solving the time-fractional Allen-Cahn equation in the Caputo's form. The L1 implicit scheme is shown to preserve a variational energy dissipation law on arbitrary nonuniform time meshes by using the recent discrete analysis tools, i.e., the discrete orthogonal convolution kernels and discrete complementary convolution kernels. Then the discrete embedding techniques and the fractional Gr\"onwall inequality were applied to establish an $L^2$ norm error estimate on nonuniform time meshes. An adaptive time-stepping strategy according to the dynamical feature of the system is presented to capture the multi-scale behaviors and to improve the computational performance.

A second-order scheme with nonuniform time grids for Caputo–Hadamard fractional sub-diffusion equations

In this paper, a second-order scheme with nonuniform time meshes for Caputo–Hadamard fractional sub-diffusion equations with initial singularity is investigated. Firstly, a Taylor-like formula with integral remainder is proposed, which is crucial to studying the discrete convolution kernels and error estimate. Secondly, we come up with an error convolution structure (ECS) analysis for Llog,2−1σ interpolation approximation to the Caputo–Hadamard fractional derivative. The core result in this paper is an ECS bound and a global consistency analysis established at an offset point. By virtue of this result, we obtain a sharp L2-norm error estimate of a second-order Crank–Nicolson-like scheme for Caputo–Hadamard fractional differential equations. Ultimately, an example is presented to show the sharpness of our analysis.

Three-dimensional Haar wavelet method for singularly perturbed elliptic boundary value problems on non-uniform meshes

Three-dimensional Haar wavelet method for singularly perturbed elliptic boundary value problems on non-uniform meshes

A fourth-order numerical method for solving a class of derivative-dependent nonlinear singular boundary value problems

In this paper, a high order numerical scheme based on quartic B-spline functions is proposed to solve derivative-dependent nonlinear singular boundary value problems. Convergence of the method is analyzed. Five test problems are considered to illustrate the accuracy and efficiency of the method. The results obtained by present method are compared with those obtained by the uniform mesh cubic B-spline collocation (UCS) method [1] and non-uniform mesh cubic B-spline collocation (NCS) method [1]. The CPU time of proposed method is compared with that of the NCS method.

Numerical Analysis of Local Discontinuous Galerkin Method for the Time-Fractional Fourth-Order Equation with Initial Singularity

In this paper, efficient methods seeking the numerical solution of a time-fractional fourth-order differential equation with Caputo’s derivative are derived. The solution of such a problem has a weak singularity near the initial time t=0. The Caputo time-fractional derivative with derivative order α∈(0,1) is discretized by the well-known L1 formula on nonuniform meshes; for the spatial derivative, the local discontinuous Galerkin (LDG) finite element method is used. Based on the discrete fractional Gronwall’s inequality, we prove the stability of the proposed scheme and the optimal error estimate for the solution, i.e., (2−α)-order accurate in time and (k+1)-order accurate in space, when piece-wise polynomials of degree at most k are used. Moreover, a second-order and nonuniform time-stepping scheme is developed for the fractional model. The scheme uses the L2-1σ formula for the time fractional derivative and the LDG method for the space approximation. The stability and temporal optimal second-order convergence of the scheme are also shown. Finally, some numerical experiments are presented to confirm the theoretical results.

High-order numerical algorithms for the time-fractional convection–diffusion equation

In this paper, efficient methods are derived for seeking a numerical solution to the time-fractional convection–diffusion equation whose solution very likely exhibits a weak regularity at the starting time. Here, the time-fractional derivative in the Caputo sense with order in is discretized by the - methods with uniform and non-uniform meshes and the spatial derivative is approximated by the local discontinuous Galerkin (LDG) finite element methods. The fully discrete schemes for both situations are established and analyzed. It is shown that the derived schemes are numerically stable and convergent. Finally, some numerical examples are performed to testify the effectiveness of the obtained algorithm.

A higher-order uniformly convergent defect correction method for singularly perturbed convection-diffusion problems on an adaptive mesh

This paper presents a defect correction scheme based on finite-difference discretization for a singularly perturbed convection-dominated diffusion problem. The solution of this class of problems exhibits a multiscale character. There are narrow regions in which the solution grows exponentially and reveals layer behaviour. The defect correction method that we propose improves the efficiency of a numerical solution through iterative improvement and generates a stable higher-order method over an adaptive non-uniform polynomial-Shishkin mesh. An extensive theoretical analysis is presented, which establishes that the method is second-order uniformly convergent and highly stable. The convergence obtained is optimal because it is free from any logarithmic term. The numerical result for two model problems is presented, agreeing with the theoretical estimates. Furthermore, we compare the results with those of other non-uniform mesh found in the literature.

Solving Fokker–Planck equations using deep KD-tree with a small amount of data

The Fokker–Planck (FP) equation can deterministically describe the evolution of the probability density function, which plays an extremely significant role in the fields of stochastic dynamics. Unfortunately, the limited samples that arise from the consideration of engineering practice are inevitable, which restricts the solving of the FP equation. Accordingly, in the present study, a super-DL-FP framework is established to solve the steady-state FP equation with a small amount of data, through combining the deep KD-tree and the DL-FP approach proposed in [Chaos 30, 013133 (2020)]. It should be emphasized that the normalization condition is of great importance and has to be considered in solving the steady-state FP equation. An appropriate integral estimation for the normalization condition under non-uniform meshing can effectively improve the precision of the solution, but it is still a challenging problem, especially for the case of small data. Thus, the so-called deep KD-tree method is innovatively proposed to estimate the normalized integral with a small random dataset. The main target is to obtain the appropriate discrete integral points and corresponding integral volumes by executing multiple KD-tree segmentation based on random data on the integral region. Several numerical experiments and comparisons are implemented to illustrate the superior performance of the super-DL-FP method. The obtained results indicate that the proposed algorithm can accomplish higher accuracy in the sense of lower cost than the well-known algorithms like center difference scheme, Chebyshev spectrum algorithm, and normalized flow approach.

A second-order difference scheme for the singularly perturbed Sobolev problems with third type boundary conditions on Bakhvalov mesh

We consider a singularly perturbed initial-third boundary value Sobolev problem. Firstly, the asymptotic behaviour of the exact solution is analysed. Then, a second-order finite difference scheme is constructed on the special non-uniform mesh. By using energy estimate, the stability and convergence of the proposed scheme are investigated in the discrete energy norm. Finally, three numerical examples are solved to validate the theory.

An Extended Finite Difference Method for Singular Perturbation Problems on a Non-Uniform Mesh

An extended second order finite difference method on a variable mesh is proposed for the solution of a singularly perturbed boundary value problem. A discrete equation is achieved on the non uniform mesh by extending the first and second order derivatives to the higher order finite differences. This equation is solved efficiently using a tridiagonal solver. The proposed method is analysed for convergence, and second order convergence is derived. Model examples are solved by the proposed scheme and compared with available methods in the literature to uphold the method.

Phase-Only Synthesis for Large Planar Arrays via Zernike Polynomials and Invasive Weed Optimization

A synthesis method for the design of large planar array antennas with phase-only control is here presented. The synthesis is based on the use of Zernike polynomials, as global basis function for the phase, to reduce the number of optimization variables with respect to the number of elements of the array. Invasive weed optimization (IWO) is applied to polynomial coefficients’ optimization to circumvent nonlinearity and local trapping issues typical of phase-only problems. The periodicity of the array factor is exploited to reduce the optimization to a Voronoi cell of the grating-lobes’ lattice and nonuniform meshing is introduced to best adapt the control stations to the beam shape requirements. The technique is applied to the design of shaped beams for continental coverage from geostationary satellites.

A three-dimensional off-lattice Boltzmann method for the simulation of blood flow through a model irregular stenosis

In the present work, a three-dimensional characteristic-based off-lattice Boltzmann method is developed in general cylindrical curvilinear coordinates to handle body-fitted non-uniform meshes that typically arise in blood flow simulation of stenosed arteries. To handle the singularity point at r = 0, the azimuthal mapping approach together with a special periodic boundary condition is developed. The numerical solver is validated, using reference data from literature, for steady flow through a stenosed lumen and for pulsatile flow through an abdominal aortic aneurysm. Thereafter, the solver is applied to study pulsatile blood flow through a model irregular arterial stenosis with an aerial occlusion of 75%. The surface irregularity of the stenosis is modeled using a sine function while keeping the cosine-shaped occlusion. The degree of irregularity is controlled by the amplitude and frequency of the sine function. Flow characteristics such as wall shear stress (WSS), divergence of WSS, oscillatory shear index, relative residence time (RRT), the turbulence kinetic energy (TKE), and power spectral density are used to investigate the near-wall vascular remodeling caused by the resulting disturbed flow. The present study demonstrates that for a given areal occlusion, an increase in the amplitude and frequency of the surface irregularity increases the number of locations susceptible for perfusion of low-density lipoproteins and promotes flow disturbances in the stenotic and post-stenotic regions. An increase in the Reynolds number not only increases the peaks of the WSS but also brings the peaks together causing alternating stress in the stenosed region. Furthermore, a positive correlation is found between the Reynolds number and the TKE production. On the other hand, an increase in the Womersley number reduces the TKE production, curtails the disturbed flow, and reduces the RRT of the solutes, all of which further reduce the risk factors. Overall, the developed solver demonstrates the importance of shape of the stenosis on the blood flow dynamics for physiological inflow conditions.

A Comparison of Parallel Algorithms for Numerical Solution of Parabolic Problems with Fractional Power Elliptic Operators

In this article we construct parallel solvers analyze the efficiency and accuracy of general parallel solvers for three dimensional parabolic problems with the fractional power of elliptic operators. The proposed discrete method are targeted for general non-constant elliptic operators, the second motivation for the usage of such schemes arises when non-uniform space meshes are essential. Parallel solvers are required to solve the obtained large size systems of linear equations. The detailed scalability analysis is done in order to compare the efficiency of prposed parallel algorithms. Results of computational experiments are presented and analyzed.

A Fitted Mesh Cubic Spline in Tension Method for Singularly Perturbed Problems with Two Parameters

A numerical treatment via a difference scheme constructed by the Crank–Nicolson scheme for the time derivative and cubic spline in tension for the spatial derivatives on a layer resolving nonuniform Bakhvalov-type mesh for a singularly perturbed unsteady-state initial-boundary-value problem with two small parameters is presented. Error analysis of the constructed scheme is discussed and shown to be parameter-uniformly convergent with second-order convergence. Numerical experimentation is taken to confirm the theoretical findings.

Efficient and robust quadratures for isogeometric analysis: Reduced Gauss and Gauss–Greville rules

This work proposes two efficient quadrature rules, reduced Gauss quadrature and Gauss–Greville quadrature, for isogeometric analysis. The rules are constructed to exactly integrate one-dimensional B-spline basis functions of degree p, and continuity class Cp−k, where k is the highest order of derivatives appearing in the Galerkin formulation of the problem under consideration. This is the same idea we utilized in Zou et al. (2021), but the rules therein produced negative weights for certain non-uniform meshes. The present work improves upon Zou et al. (2021) in that the weights are guaranteed to be positive for all meshes. The reduced Gauss quadrature rule is built element-wise according to the element basis degree and smoothness. The Gauss–Greville quadrature rule combines the proposed reduced Gauss quadrature and Greville quadrature Zou et al. (2021). Both quadrature rules involve many fewer quadrature points than the full Gauss quadrature rule and avoid negative quadrature weights for arbitrary knot vectors. The proposed quadrature rules are stable and accurate, and they can be constructed without solving nonlinear equations, therefore providing efficient and easy-to-use alternatives to full Gauss quadrature. Various numerical examples, including curved shells, demonstrate that they achieve good accuracy, and for p=5 and 6 eliminate locking.

Mesh manipulation for local structural property tailoring of medical warp-knitted textiles

Medical meshes are used as structural reinforcement both in clinical surgery and tissue engineering. However, complex loading conditions often found in such applications result in a non-homogenous stress distribution, for which the uniform reinforcement provided by the meshes is not optimal. This study aims to design a textile reinforcement with a spatially heterogeneous, load-tailored fiber architecture. To this end, we developed a simple method of manipulating a standard uniform mesh by stretching in warp and weft directions to various extents in order to control fiber orientation and fiber volume fraction. Subsequent thermal treatment locked the manipulated configurations allowing further use of the meshes. Firstly, samples in five uniform configurations (two manipulated longitudinally (warp direction), two manipulated transversely (weft direction), one non-manipulated) were obtained and analyzed regarding their morphology as well as their mechanical properties under cyclic uniaxial loading. Significant effects of the manipulation on key characteristics of the pores such as angles, side lengths, aspect ratios, and fiber volume fraction were shown. Tensile testing demonstrated the range of tensile properties achievable with the simple manipulation of the mesh, not only in magnitude but also in the shape of the stress-strain response curve. Finally, local manipulation combining different mesh configurations was exemplarily applied to create a spatially heterogeneous load-tailored reinforcement to match local strain directions in tissue-engineered tubular heart valves. The proposed method enables the use of well-established uniform medical meshes to produce load-tailored non-uniform mesh reinforcement for many applications in an easy-to-implement manner.