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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.