• Abstract

This paper develops and analyzes interior penalty discontinuous Galerkin (IPDG) method by patch reconstruction technique for Helmholtz problems. The technique achieves high order approximation by locally solving a discrete least-squares over a neighboring element patch. We prove a prior error estimates in the $L^2$ norm and energy norm. For each fixed wave number $k$, the accuracy and efficiency of the method up to order five with high-order polynomials.  Numerical examples are carried out to validate the theoretical results.

• Abstract

In this paper, the finite-time stability and instability are studied for nonlinear impulsive systems. There are mainly four concerns. 1) For the system with stabilizing impulses, a Lyapunov theorem on global finite-time stability is presented. 2) When the system without impulsive effects is globally finite-time stable (GFTS) and the settling time is continuous at the origin, it is proved that  it is still GFTS over any class of impulse sequences, if the mixed impulsive jumps satisfy some mild conditions. 3) For systems with destabilizing impulses, it is shown that to be finite-time stable, the destabilizing impulses should not occur too frequently, otherwise, the origin of the impulsive system is finite-time instable, which are formulated by average dwell time (ADT) conditions respectively. 4) A theorem on finite-time instability is provided for system with stabilizing impulses. For each GFTS theorem of impulsive systems considered in this paper, the upper boundedness of settling time is given, which depends on the initial value and impulsive effects. Some numerical examples are given to illustrate the theoretical analysis.

• Abstract

Stochastic temperature distribution should be carefully inspected in the thermal-failure design of heterogeneous solids with unexpected random energy excitations. Stochastic multiscale modeling for these problems involve multiscale and high-dimensional uncertain thermal parameters, which remains limitation of prohibitive computation. In this paper, we propose a multi-modes based constrained energy minimization generalized multiscale finite element method (MCEM-GMsFEM), which can transform the original stochastic multiscale model into a series of recursive multiscale models sharing the same deterministic material parameters by multiscale analysis. Thus, MCEM-GMsFEM reveals an inherent low-dimensional representation in random space, and is designed to effectively reduce the complexity of repeated computation of discretized multiscale systems. In addition, the convergence analysis is established, and the optimal error estimates are derived. Finally, several typical random fluctuations on multiscale thermal conductivity are considered to validate the theoretical results in the numerical examples. The numerical results indicate that the multi-modes multiscale approach is a robust integrated method with the excellent performance.

• Abstract

Color image segmentation is crucial in image processing and computer vision. Most traditional segmentation methods simply regard an RGB color image as the direct combination of the three monochrome images and ignore the inherent color structures within channels, which contain some key feature information of the image. To better describe the relationship of color channels, we introduce a quaternion-based regularization that can reflect the image characteristics more intuitively. Our model combines the idea of the Mumford-Shah model-based two-stage segmentation method and the Saturation-Value Total Variation regularization for color image segmentation. The new strategy first extracts features from the color image and then subdivides the image in a new color feature space which achieves better performance than methods in RGB color space. Moreover, to accelerate the optimization process, we use a new primal-dual algorithm to solve our novel model. Numerical results demonstrate clearly that the performance of our proposed method is excellent.

• Abstract

In this paper, we develop a direct-forcing immersed boundary projection method for simulating the dynamics in thermal fluid-solid interaction problems. The underlying idea of the method is that we treat the solid as made of fluid and introduce two virtual forcing terms. First, a virtual fluid force distributed only on the solid region is appended to the momentum equation to make the region behave like a real solid body and satisfy the prescribed velocity. Second, a virtual heat source located inside the solid region near the boundary is added to the energy transport equation to impose the thermal boundary condition on the solid boundary. We take the implicit second-order backward differentiation to discretize the time variable and employ the Choi-Moin projection scheme to split the coupled system. As for spatial discretization, second-order centered differences over a staggered Cartesian grid are used on the entire domain. The advantages of this method are its conceptual simplicity and ease of implementation. Numerical experiments are performed to demonstrate the high performance of the proposed method. Convergence tests show that the spatial convergence rates of all unknowns seem to be super-linear in the $1$-norm and $2$-norm while less than linear in the maximum norm.

• Abstract

Multivariate time series segmentation is an important problem in data mining and it has arisen in more and more practical applications  in recent years. The task of time series segmentation is to partition a time series into segments by detecting the abrupt changes or anomalies in the time series. Multivariate time series segmentation can provide meaningful information for further data analysis, prediction and policy decision. A time series can be considered as a piecewise continuous function, it is natural to take its total variation  norm as a prior information of this time series. In this paper, by minimizing the negative log-likelihood function of a time series, we propose a total variation based  model  for multivariate time series segmentation. An iterative process is applied to solve the proposed model and a search combined the dynamic programming method  is designed  to determine the  breakpoints. The experimental results show that the  proposed method is efficient for   multivariate time series segmentation and it is competitive to the existing methods for multivariate time series segmentation.

• Abstract

In this paper, the Peng-Robinson equation of state with dynamic boundary conditions is discussed, which considers the interactions with solid walls. At first, the model is introduced and the regularization method on the nonlinear term is adopted. Next, The scalar auxiliary variable (SAV) method in temporal and finite element method in spatial are used to handle the Peng-Robinson equation of state. Then, the energy dissipation law of the numerical method is obtained. Also, we acquire the convergence of the discrete SAV finite element method (FEM). Finally, a numerical example is provided to confirm the theoretical result.

• Abstract

This paper documents the first attempt to apply a localized method of fundamental solutions (LMFS) to the acoustic analysis of car cavity containing sound-absorbing materials. The LMFS is a recently developed meshless approach with the merits of being mathematically simple, numerically accurate, and requiring less computer time and storage. Compared with the traditional method of fundamental solutions (MFS) with a full interpolation matrix, the LMFS can obtain a sparse banded linear algebraic system, and can circumvent the perplexing issue of fictitious boundary encountered in the MFS for complex solution domains. In the LMFS, only circular or spherical fictitious boundary is involved. Based on these advantages, the method can be regarded as a competitive alternative to the standard method, especially for high-dimensional and large-scale problems. Three benchmark numerical examples are provided to verify the effectiveness and performance of the present method for the solution of car cavity acoustic problems with impedance conditions.

• Abstract

The main contents of this paper are to establish a finite element fully-discrete approximate scheme for multi-term time-fractional mixed sub-diffusion and diffusion-wave equation with spatial variable coefficient, which contains a time-space coupled derivative. The nonconforming $EQ_1^{rot}$ element and Raviart-Thomas element are employed for spatial discretization, and L1 time-stepping method combined with the Crank-Nicolson scheme are applied for temporal discretization. Firstly, based on some significant lemmas, the unconditional stability analysis of the fully-discrete scheme is acquired. With the assistance of the interpolation operator $I_h$ and projection operator $R_h$, superclose and convergence results of the variable $u$ in $H^1$-norm and the flux $\vec{p}=\kappa_5(\textbf{x})\nabla u(\textbf{x},t)$ in $L^2$-norm are obtained, respectively. Furthermore, the global superconvergence results are derived by applying the interpolation postprocessing technique. Finally, the availability and accuracy of the theoretical analysis are corroborated by experimental results of numerical examples on anisotropic meshes.

• Abstract

This paper proposes a deep-learning-based Robin-Robin domain decomposition method (DeepDDM) for Helmholtz equations. We first present the plane wave activation-based neural network (PWNN), which is more efficient for solving Helmholtz equations with constant coefficients and wavenumber $k$ than finite difference methods (FDM). On this basis, we use PWNN to discretize the subproblems divided by domain decomposition methods (DDM), which is the main idea of DeepDDM. This paper will investigate the number of iterations of using DeepDDM for continuous and discontinuous Helmholtz equations. The results demonstrate that: DeepDDM exhibits behaviors consistent with conventional robust FDM-based domain decomposition method (FDM-DDM) under the same Robin parameters, i.e., the number of iterations by DeepDDM is almost the same as that of FDM-DDM. By choosing suitable Robin parameters on different subdomains, the convergence rate is almost constant with the rise of wavenumber in both continuous and discontinuous cases. The performance of DeepDDM on Helmholtz equations may provide new insights for improving the PDE solver by deep learning.

• Abstract

We have analyzed the kinetics of solid circular particles interacting with fluid, outer boundary and internal square shaped obstacles tilted at a $ 45^{\circ} $ angle. The effects on the motion of particle due to collision with obstacles and wall are inspected. An Eulerian approach is used to study the behavior of particle in the fixed computational mesh. The interactions between fluid, particles and obstacles have been carried out in the whole domain by using fictitious boundary method (FBM). In this work, the particulate flow simulations are computed by using finite element solver FEATFLOW. Numerical results are presented by assigning different alignments to the obstacles and varying their positions in the domain. Particle-wall, particle-particle and particle-obstacle collisions are treated by applying a modified collision model proposed by Glowinski et al. The rapid change in drag forces acting on obstacles due to nearby passing particles and its effect on the fluid motion has been investigated.

• Abstract

In the present study, we extend the order-preserving (OP) criterion proposed in our latest studies to the WENO-Z-type schemes. Firstly, we innovatively present the concept of the generalized mapped WENO schemes by rewriting the Z-type weights in a uniform formula from the perspective of the mapping relation. Then, we naturally introduce the OP criterion to improve the WENO-Z-type schemes, and the resultant schemes are denoted as MOP-GMWENO-X, where the notation ``X'' is used to identify the version of the existing WENO-Z-type scheme in this paper. Finally, extensive numerical experiments have been conducted to demonstrate the benefits of these new schemes. We draw the conclusion that, the convergence properties of the proposed schemes are equivalent to the corresponding WENO-X schemes. The major benefit of the new schemes is that they have the capacity to achieve high resolutions and simultaneously remove spurious oscillations for long simulations. The new schemes have the additional benefit that they can greatly decrease the post-shock oscillations on solving 2D Euler problems with strong shock waves.

• Abstract

We propose a class of up to fourth-order maximum-principle-preserving and mass-conserving schemes for the conservative Allen-Cahn equation equipped with a non-local Lagrange multiplier. Based on the second-order finite-difference semi-discretization in the spatial direction, the integrating factor Runge-Kutta schemes are applied in the temporal direction. Theoretical analysis indicates that the proposed schemes conserve mass and preserve the maximum principle under reasonable time step-size restriction, which is independent of the space step size. Finally, the theoretical analysis is verified by several numerical examples.

• Abstract

This study is focused on the forest edge flow by using numerical method. To model the effects of a forest canopy on airflow, source terms are introduced into the governing equations. The lattice Boltzmann method in conjunction with the standard $k-\varepsilon$ model is applied to solve the turbulent wind field. In order to perform the simulation on non-uniform grids, the Taylor series expansion and least square based lattice Boltzmann method (TLLBM) is adopted to improve the accuracy and computational efficiency. The present method and code are verified with an earlier forest edge simulation. A series of forest canopies are established to explore the impacts of canopy morphology on wind field. These canopies cover 3 canopy architectures and the Leaf Area Index (LAI) ranges from 2.0 to 4.0. The further study is carried out by adjusting the canopy foliage amount and the canopy architecture. The present study demonstrates the potential of lattice Boltzmann method to simulate the high Re number forest edge flow. The impacts of canopy morphology on zero plane displacement, aerodynamic roughness length, friction wind velocity, permeability coefficient, wall-shear stress are illustrated in detail. The results show that the canopy sub-layer wind field, especially the wind velocity profiles within and above the forest canopy, are mainly determined by canopy morphology.

• Abstract

In this paper, we construct a new cell-centered nonlinear finite volume scheme that preserves  the extremum principle for heterogeneous anisotropic diffusion equation on distorted meshes. We introduce a new nonlinear approach to construct the conservative flux, that is,  a linear second order flux is firstly given and a nonlinear conservative flux is then constructed by using an adaptive method and a nonlinear weighted method.  Our new scheme does not need to use the convex combination of the cell-center unknowns  to approximate the auxiliary unknowns, so it can deal with the problem with general discontinuous coefficients. Numerical results show that our new  scheme performs more robust  than some existing schemes on highly distorted meshes.

• Abstract

Imposing appropriate numerical boundary conditions at the symmetrical center $r = 0$ is vital when computing compressible fluids with radial symmetry. Extrapolation and other traditional techniques are often employed, but spurious numerical oscillations or wall-heating phenomena can occur. In this paper, we emphasize that because of the conservation property, the updating formula of the boundary cell average can coincide with the one for interior cell averages. To achieve second-order accuracy both in time and space, we associate obtaining the inner boundary value at $r=0$ with the resolution of the corresponding one-sided generalized Riemann problem (GRP). Acoustic approximation is applied in this process. It creates conditions to avoid the singularity of type $1/ r$ and aids in obtaining the value of the singular quantity using L'Hospital's rule. Several challenging scenarios are tested to demonstrate the effectiveness and robustness of our approach.

• Abstract

A discontinuous Galerkin (DG) scheme for solving semilinear elliptic problem is developed and analyzed in this paper. The DG finite element discretization is first established, then the corresponding well-posedness is provided by using Brouwer's fixed point method. Some optimal priori error estimates under both DG norm and $L^2$ norm are presented,  respectively. Numerical results are given to illustrate the efficiency of the proposed approach.

• Abstract

In this work, a non-classical analytical approach for buckling analysis of partially cracked generally orthotropic plate is proposed under the thermal domain. The derivation for the governing equation is based on the non-classical approach using Kirchhoff's thin plate theory and the modified couple stress theory. The effect of fibre orientation on critical buckling temperature is incorporated by considering the coefficients of mutual influence. Line spring model is applied with some modifications to formulate all the crack terms while the thermal effects are introduced in form of thermal in-plane moments and forces. The final governing equation is solved using Galerkin's method and the relation for critical buckling temperature as affected by fibre orientation is obtained. The variation of critical buckling temperature as affected by fibre orientation for different values of crack length, crack location and length scale parameter is presented. Also, the effect of fibre orientation on fundamental frequency under the thermal domain is analysed.

• Abstract

The main aim of this paper is to present an analytical approach on the postbuckling for torsionally loaded sandwich carbon nanotube (CNT) reinforced cylindrical shells with the auxetic core. The considered shells consist of three layers, external and internal CNT reinforced layers, and the auxetic lattice core made by isotropic material. The homogenization model for honeycomb auxetic lattice core is utilized, and the equilibrium equations are formulated based on the nonlinear Donnell's thin shell theory with von K\'{a}rm\'{a}n geometrical nonlinearities. The three terms of deflection are considered, the Airy's stress function and Galerkin's method are utilized, the explicit expression of critical buckling of torsionally loaded shells and load-deflection expression of postbuckling states are achieved. The effects of two carbon nanotube reinforced layers, the auxetic core layer, the volume fraction of carbon nanotube on the torsional buckling behavior are examined and remarked.

• Abstract

This paper presents an Eulerian diffuse-interface method using a high-order compact difference scheme for simulating elastic-plastic flows with the Mie--Gruneisen (MG) equation of state (EoS). For simulations of multimaterial problems, numerical errors were generated in the material discontinuities owing to inconsistent treatment of the convective terms. Based on the normal-stress-based mechanical equilibrium assumption for elastic-plastic solids, we introduce an improved form of the consistent localized artificial diffusivity (LAD) method to ensure an oscillation-free interface for velocity and normal stress. The proposed algorithm uses a hyperelastic model. A mixture type of the model system was formed by combining the conservation equations for the basic conserved variables, an equation of a unified deviatoric tensor describing solid deformation, and an additional set of equations for solving the material quantities in the MG EoS. Several one- and two-dimensional problems with various discontinuities, including the elastic-plastic Richtmyer--Meshkov instability, were considered for testing the proposed method.