• Abstract

With the rapid development of computer technology, numerical simulation has become the third scientific research tool besides theoretical analysis and experimental research. As the core of numerical simulation, constructing efficient, accurate and stable numerical methods to simulate complex scientific and engineering problems has become a key issue in computational mechanics. The article outlines the application of singular boundary method to the large-scale and high-frequency acoustic problems. In practical application, the key issue is to construct efficient and accurate numerical methodology to calculate the large-scale and high-frequency sound field. This article focuses on the following two research areas. They are how to discretize partial differential equations into more appropriate linear equations, and how to solve linear equations more efficiently. The bottle neck problems encountered in the computational acoustics are used as the technical routes, i.e., efficient solution of dense linear system composed of ill-conditioned matrix and stable simulation of wave propagation at low sampling frequencies. The article reviews recent advances in emerging applications of the singular boundary method for computational acoustics. This collection can provide a reference for simulating other more complex wave propagation.

• Abstract

A conforming discontinuous Galerkin finite element method was introduced by Ye and Zhang, on simplicial meshes and on polytopal meshes,  which has the flexibility of using discontinuous approximation  and an ultra simple formulation. The main goal of this paper is to improve the above discontinuous Galerkin finite element method so that it  can handle nonhomogeneous Dirichlet boundary conditions effectively. In addition, the method has been generalized in terms of approximation of the weak gradient. Error estimates of optimal order are established for the corresponding discontinuous finite element approximation in both a discrete $H^1$ norm and the $L^2$ norm. Numerical results are presented to confirm the theory.

• Abstract

Semi-discrete and fully discrete mixed finite element methods are considered for  Maxwell-model-based problems of wave propagation in linear viscoelastic solid. This mixed finite element framework allows the use of a large class of existing mixed  conforming finite elements for elasticity   in the spatial discretization. In the fully discrete scheme, a Crank-Nicolson scheme is adopted for the approximation of the temporal derivatives of  stress and velocity variables. Error estimates of the semi-discrete and fully discrete schemes, as well as an unconditional stability result for the fully discrete scheme, are derived. Numerical experiments are provided to verify the theoretical results.

• Abstract

In this paper, we propose a conforming finite element method coupling penalty method for the linearly elastic flexural shell to overcome computational difficulties. We start with discretizing the displacement variable, i.e., the two tangent components of the displacement are discretized by using conforming finite elements (linear element), and the normal component of the displacement is discretized by using conforming Hsieh-Clough-Tocher element (HCT element). Then, the existence, uniqueness, stability, convergence and a priori error estimate of the corresponding analyses are proven and analyzed. Finally, we present numerical experiments with a portion of the conical shell and a portion of the cylindrical shell to verify theoretical convergence results and demonstrate the effectiveness of the numerical scheme.

• Abstract

In this paper, a numerical method for the electroneutral micro-fluids based on the finite element method will be given. In order to deal with the non-linearity of the equation, the modified characteristics  method will be used to deal with the temporal derivates term and the convective term. In this way, the non-linear equation can be linearlized. Then, we will give the unconditional stability and optimal error estimation. At last, some numerical results are given to show the effectiveness of our method. From the stability analysis we can see that the method is unconditionally stable. The numerical results show that our method is robust.

• Abstract

Based on newly developed weight-based smoothness detectors and nonlinear interpolations designed to capture discontinuities for the multiderivative combined dissipative compact scheme (MDCS), hybrid linear and nonlinear interpolations are proposed to form hybrid MDCS. These detectors are derived from the weights used for the nonlinear interpolations and can provide suitable switches between the linear and the nonlinear schemes to realize the characteristics for the hybrid MDCS of capturing discontinuities and maintaining high resolution in the region without large discontinuities. To save computational cost, the nonlinear scheme with characteristic decomposition is only applied in the detected discontinuities region by specially designed hybrid strategy. Typical tests show that the hybrid MDCS is capable of capturing discontinuities and maintaining high resolution power for the smooth region at the same time. With the satisfaction of the geometric conservative law (GCL), the MDCS is further applied on curvilinear mesh to present its promising capability of handling pragmatic simulations.

• Abstract

In this paper, a lattice Boltzmann model with BGK operator (LBGK) for solving time-fractional nonlinear wave equations in Caputo sense is proposed. First, the Caputo fractional derivative is approximated using the fast evolution algorithm based on the sum-of-exponentials approximation.

Then the target equation is transformed into an approximate form, and for which a LBGK model is developed. Through the Chapman-Enskog analysis, the macroscopic equation can be recovered from the present LBGK model.

In addition, the proposed model can be extended to solve the time-fractional Klein-Gordon equation and the time-fractional Sine-Gordon equation. Finally, several numerical examples are performed to show the accuracy and efficiency of the present LBGK model. From the numerical results, the present model has a second-order accuracy in space.

• Abstract

In recent years, various kinds of cloak devices were designed by transformation optics, but these cloak metamaterials are anisotropic and difficult to manufacture. In this paper, we designed the isotropic cloak metamaterials based on the numerical method of the optimization theory to the inverse medium problem. This method has universality, and it is not limited by the shape and type of the cloak device. This isotropic material is easier to manufacture in practice than anisotropic material. A large number of numerical results show the effectiveness of the method.

• Abstract

In this study, the polynomial expansion method (PEM) and the polynomial method of particular solutions (PMPS) are applied to solve a class of linear elliptic partial differential equations (PDEs) in two dimensions with constant coefficients. In the solution procedure, the sought solution is approximated by the Pascal polynomials and their particular solutions for the PEM and PMPS, respectively. The multiple-scale technique is applied to improve the conditioning of the resulted linear equations and the accuracy of numerical results for both of the PEM and PMPS. Some mathematical statements are provided to demonstrate the equivalence of the PEM and PMPS bases as they are both bases of a certain polynomial vector space. Then, some numerical experiments were conducted to validate the implementation of the PEM and PMPS. Numerical results demonstrated that the PEM is more accurate and well-conditioned than the PMPS and the multiple-scale technique is essential in these polynomial methods.

• Abstract

A new algorithm, called symmetric inertial alternating direction method of multipliers (SIADMM), is designed for separable convex optimization problems with linear constraints in this paper. The convergence rate of the SIADMM is proved to be $\mathcal{O}(1/\sqrt{k})$. Two kinds of elliptic equation constrained optimization problems, the unconstrained cases as well as the box-constrained cases of the distributed control and the Robin boundary control, are analyzed theoretically and solved numerically. First, the existence and uniqueness of the solutions to these problems are proved. Second, these continuous optimization problems are transformed into discrete optimization problems by the finite element method, and then the discrete optimization problems are solved by the proposed SIADMM. Numerical experiments with different problems are investigated to demonstrate the efficiency of the SIADMM. And the numerical performance of the SIADMM is better than the performance of the ADMM. Moreover, the numerical results show that the convergence rate of the SIADMM tends to be faster than $\mathcal{O}(1/\sqrt{k})$ in calculation process.

• Abstract

This paper is concerned with the weak Galerkin mixed finite element method (WG-MFEM) for the second-order hyperbolic acoustic wave equation in velocity-pressure formulation. In this formulation, the original second-order differential equation in time and space is reduced to first-order differential equations by introducing the velocity and pressure variables. We employ the usual discontinuous piecewise-polynomials of degree $k\geq 0$ for the pressure and $k+1$ for the velocity. Furthermore, the normal component of the pressure on the interface of elements is enhanced by polynomials of degree $k+1$. The time derivative is approximated by the backward Euler difference. We show the stability of the semi-discrete and fully-discrete schemes, and obtain the suboptimal order error estimates for the velocity and pressure variables. Numerical experiment confirms our theoretical analysis.

• Abstract

In this study, a novel fast-implicit iteration scheme called the alternating cell direction implicit (ACDI) method is combined with the approximate factorization scheme. This application aims to offer a mathematically well-defined version of the ACDI method and to increase the accuracy of the iteration scheme used for the numerical solutions of partial differential equations. The ACDI method is a fast-implicit method that can be used for unstructured grids. The use of fast implicit iteration methods with unstructured grids is not common in the literature. The new ACDI method has been applied to the unsteady diffusion equation to determine its convergence and time-dependent solution ability and character. The numerical tests are conducted for different grid types, such as structured, unstructured quadrilateral, and hybrid polygonal grids. Second, the ACDI was applied to the unsteady advection-diffusion equation to understand the time-dependent and progression capabilities of the presented method. Third, a full potential equation solution is created to understand the complex flow solving ability of the presented method. The results of the numerical study are compared with other fast implicit methods, such as the point Gauss--Seidel (PGS) and line Gauss--Seidel (LGS) methods and the fourth-order Runge-Kutta (RK4) method, which is an explicit scheme, and the Laasonen method, which is a fully implicit scheme. The study increased the abilities of the ACDI method. Due to the new ACDI method, the approximate factorization method, which is used only in structural grids that are known to be advantageous, can be applied to any mesh structure.

• Abstract

Heating or cooling one-dimensional inviscid compressible flow can be modeled by the Euler equations with energy sources. A tricky situation is the sudden appearance of a single-point energy source term. This source is discontinuous in both the time and space directions, and results in multiple discontinuous waves in the solution. We establish a mathematical model of the generalized Riemann problem of the Euler equations with source term. Based on the double CRPs coupling method proposed by the authors, we determine the wave patterns of the solution. Theoretically, we prove the existence and uniqueness of solutions to both "heat removal" problem and "heat addition" problem. Our results provide a theoretical explanation for the effect of instantaneous addition or removal of heat on the fluid.

• Abstract

Precise calculation of the trajectory of store separation is critical in assessing whether the store can be released safely. Store ejection is the initial stage of the releasing process and any uncertainty introduced at this stage will propagate through the whole trajectory. In this work, the impact of the uncertainties in ejector modeling on the simulation of a generic store separation is investigated by using a Monte-Carlo-based approach. To reduce the extremely large computation cost resulted from the direct use CFD in Monte Carlo simulation, the CFD solutions are represented by a time-dependent Kriging model, which is constructed at each time step by using the samples from the URANS simulations. The stochastic outputs, including the distribution of probability density function, expected value and 95\% confidence interval of store separation trajectory, are obtained by the Monte Carlo simulations. The sensitivity analysis is also carried out by using the Monte-Carlo-based method to determine the most significant variables in ejector modeling, which affect the output uncertainty. Our results show that ejector modeling is one of the main uncertainty sources of store separation simulation and the approximation in ejector modeling can cause a significant deviation, especially in the angular displacement.

• Abstract

The superconvergence of a two-dimensional time-independent nonlinear Schr\"{o}dinger equation are analyzed with the rectangular Lagrange type finite element of order $k$. Firstly, the error estimate and superclose property are given in $H^1$-norm with order $\mathcal{O}(h^{k+1})$ between the finite element solution $u_h$ and the interpolation function $u_I$ by use of the elliptic projection operator. Then, the global superconvergence is obtained by the interpolation post-processing technique. In addition, some numerical examples with the order $k=1$ and $k=2$ are provided to demonstrate the theoretical analysis.

• Abstract

The aim of this paper is to present the backward substitution method for solving a class of fractional dual-phase-lag models of  heat transfer. The proposed method is based on the Fourier series expansion along the spatial coordinate over the orthonormal basis formed by the eigenfunctions of the corresponding Sturm-Liouville problem. This Fourier expansion of the solution  transforms the original fractional partial differential equation into a sequence of multi-term fractional ordinary differential equations. These fractional equations are solved by the use of the backward substitution method. The numerical examples with temperature-jump boundary condition  and parameters of the tissue confirm the high accuracy and efficiency of the proposed numerical scheme.

• Abstract

A cell-centered Lagrangian scheme is developed for the numerical simulation of wave propagations in one dimensional (1D) elastic-plastic flow. The classical elastic-plastic material model initially proposed by Wilkins is adopted. The linear elastic model (Hooke's Law), perfectly plastic model and von Mises yield criterion are used to describe the constitutive relationship of elastic-plastic solid. The second-order extension of this scheme is achieved by a linear reconstruction method. Various numerical tests are simulated to check the capability of this scheme in capturing nonlinear elastic-plastic waves. Compared with the well-developed operator splitting method used in simulating elastic-plastic flow, this scheme is more accurate due to the consideration of a list of 64 different types of the nonlinear elastic-plastic waves when constructing the elastic-perfectly plastic Riemann solver. The numerical simulations of typical examples show competitive results.

• Abstract

We employ the Riemann-Hilbert (RH) method to study the Hirota equation with arbitrary order zero poles under zero boundary conditions. Through the spectral analysis, the asymptoticity, symmetry, and analysis of the Jost functions are obtained, which play a key role in constructing the RH problem. Then we successfully established the exact solution of the equation without reflection potential by solving the RH problem. Choosing some appropriate parameters of the resulting solutions, we further derive the soliton solutions with different  order poles, including four cases of a fourth-order pole, two second-order poles, a third-order pole and a first-order pole, and four first-order points. Finally,  the  dynamical behavior of these solutions are analyzed via graphic analysis.

• Abstract

The Peng-Robison equation of state, one of the most extensively applied equations of state in the petroleum industry and chemical engineering, has an excellent appearance in predicting the thermodynamic properties of a wide variety of materials. It has been a  great challenge on how to design  numerical schemes  with preservation of mass conservation and energy dissipation law. Based on the exponential time difference combined with the stabilizing technique and added Lagrange multiplier enforcing the mass conservation, we develop the efficient first- and second-order numerical schemes with preservation of maximum bound principle (MBP) to solve the single-component two-phase  diffuse interface model with Peng-Robison equation of state.  Convergence analyses as well as energy stability are also proven. Several two-dimensional and three-dimensional experiments are performed to verify these theoretical results.

• Abstract

A well-balanced Runge-Kutta discontinuous Galerkin method is presented for the numerical solution of multilayer shallow water equations with mass exchange and non-flat bottom topography. The governing equations are reformulated as a nonlinear system of conservation laws with differential source forces and reaction terms. Coupling between the flow layers is accounted for in the system using a set of exchange relations. The considered well-balanced Runge-Kutta discontinuous Galerkin method is a locally conservative finite element method whose approximate solutions are discontinuous across the inter-element boundaries. The well-balanced property is achieved using a special discretization of source terms that depends on the nature of hydrostatic solutions along with the Gauss-Lobatto-Legendre nodes for the quadrature used in the approximation of source terms. The method can also be viewed as a high-order version of upwind finite volume solvers and it offers attractive features for the numerical solution of conservation laws for which standard finite element methods fail. To deal with the source terms we also implement a high-order splitting operator for the time integration. The accuracy of the proposed Runge-Kutta discontinuous Galerkin method is examined for several examples of multilayer free-surface flows over both flat and non-flat beds. The performance of the method is also demonstrated by comparing the results obtained using the proposed method to those obtained using the incompressible hydrostatic Navier-Stokes equations and a well-established kinetic method. The proposed method is also applied to solve a recirculation flow problem in the Strait of Gibraltar.