Solving metamaterial Maxwell’s equations via a vector wave integro-differential equation

In this paper, we discuss the time-domain metamaterial Maxwell’s equations. One major contribution of this paper is that after some effort we find that the metamaterial Maxwell’s equations can be beautifully reduced to a vector wave integro-differential equation involving just one unknown, which is quite similar to that obtained from the standard Maxwell’s equations in vacuum. Then we study the existence and uniqueness of this new modeling equations, and propose a fully-discrete finite element method to solve this model. Numerical results justifying our analysis are presented. This discovery shall make simulation of metamaterials much more efficient than the previous works.     

Approximating thermo-viscoelastic heating of largely strained solid rubber components

Mechanically induced viscoelastic dissipation is difficult to compute when the constitutive model is defined by history integrals. The computation of the viscous energy dissipated is in the form of a double convolution integral. In this study, we present a method to approximate the dissipation for constitutive models in history integral form that represent Maxwell-like materials. The dissipation is obtained without directly computing the double convolution integral. The approximation requires that the total stress can be separated into elastic and viscous components, and that the relaxation form of the constitutive law is defined with a Prony series. A numerical approach often taken to approximate a history integral involves interpolating the history integral’s kernel across a time step. Integration then yields finite difference equations for the evolution of the viscous stresses in time. In the case when the material is modeled with a Prony series, the form of these finite difference equations is similar to the form of the finite difference equations for a Maxwell solid. Since the dissipation rate in a Maxwell solid can be easily computed from knowledge of its viscous stress and the Prony series constants (spring-dashpot constants), we computationally investigated employing a Maxwell solid’s dissipation function to couple thermal and large strain history integral based finite element models of solid rubber components. Numerical data is provided to support this analogy and to help understand its limitations. A rubber cylinder with an imbedded steel disk is dynamically loaded, and the non-uniform heating within the cylinder is computed.     

Exact boundary controllability of the second-order Maxwell system: Theory and numerical simulation

The exact controllability of the second order time-dependent Maxwell equations for the electric field is addressed through the Hilbert Uniqueness Method. A two-grid preconditioned conjugate gradient algorithm is employed to inverse the H.U.M. operator and to construct the numerical control. The underlying initial value problems are discretized by Lagrange finite elements and an implicit Newmark scheme. Two-dimensional numerical experiments illustrate the performance of the method.     

基于策略迭代算法的连续时间线性Markov跳变系统非零和微分反馈Nash控制

针对一类连续时间线性Markov跳变系统,本文提出了一种新的策略迭代算法用于求解系统的非零和微分反馈Nash控制问题.通过求解耦合的数值迭代解,以获得具有线性动力学特性和无限时域二次成本的双层非零和微分策略的Nash均衡解.在每一个策略层,采用策略迭代算法来计算与每一组给定的反馈控制策略相关联的最小无限时域值函数.然后,通过子系统分解将Markov跳变系统分解为N个并行的子系统,并将该算法应用于跳变系统.本文提出的策略迭代算法可以很容易求解非零和微分策略所对应的耦合代数Riccati方程,且对高维系统有效.最后通过仿真示例证明了本文设计方法的有效性和可行性.     

A lattice Boltzmann model for electromagnetic waves propagating in a one-dimensional dispersive medium

A first-order extended lattice Boltzmann (LB) model with special forcing terms for one-dimensional Maxwell equations exerting on a dispersive medium, described either by the Debye or Drude model, is proposed in this study. The time dependent dispersive effect is obtained by the inverse Fourier transform of the frequency-domain permittivity and is incorporated into the LB evolution equations via equivalent forcing effects. The Chapman–Enskog multi-scale analysis is employed to ensure that proposed scheme is mathematically consistent with the targeted Maxwell’s equations at the macroscopic limit. Numerical validations are executed through simulating four representative cases to obtain their LB solutions and compare those with the analytical solutions and existing numerical solutions by finite difference time domain (FDTD). All comparisons show that the differences in numerical values are very small. The present model can thus accurately predict the dispersive effects, and demonstrate first order convergence. In addition to its accuracy, the proposed LB model is also easy to implement. Consequently, this new LB scheme is an effective approach for numerical modeling of EM waves in dispersive media.     

An Extrapolation Cascadic Multigrid Method Combined with a Fourth-Order Compact Scheme for 3D Poisson Equation

Extrapolation cascadic multigrid (EXCMG) method is an efficient multigrid method which has mainly been used for solving the two-dimensional elliptic boundary value problems with linear finite element discretization in the existing literature. In this paper, we develop an EXCMG method to solve the three-dimensional Poisson equation on rectangular domains by using the compact finite difference (FD) method with unequal meshsizes in different coordinate directions. The resulting linear system from compact FD discretization is solved by the conjugate gradient (CG) method with a relative residual stopping criterion. By combining the Richardson extrapolation and tri-quartic Lagrange interpolation for the numerical solutions from two-level of grids (current and previous grids), we are able to produce an extremely accurate approximation of the actual numerical solution on the next finer grid, which can greatly reduce the number of relaxation sweeps needed. Additionally, a simple method based on the midpoint extrapolation formula is used for the fourth-order FD solutions on two-level of grids to achieve sixth-order accuracy on the entire fine grid cheaply and directly. The gradient of the numerical solution can also be easily obtained through solving a series of tridiagonal linear systems resulting from the fourth-order compact FD discretizations. Numerical results show that our EXCMG method is much more efficient than the classical V-cycle and W-cycle multigrid methods. Moreover, only few CG iterations are required on the finest grid to achieve full fourth-order accuracy in both the \(L^2\)-norm and \(L^{\infty }\)-norm for the solution and its gradient when the exact solution belongs to \(C^6\). Finally, numerical result shows that our EXCMG method is still effective when the exact solution has a lower regularity, which widens the scope of applicability of our EXCMG method.     

On the immersed interface method for solving time-domain Maxwell's equations in materials with curved dielectric interfaces

This paper deals with accurate numerical simulation of two-dimensional time-domain Maxwell's equations in materials with curved dielectric interfaces. The proposed fully second-order scheme is a hybridization between the immersed interface method (IIM), introduced to take into account curved geometries in structured schemes, and the Lax-Wendroff scheme, usually used to improve order of approximations in time for partial differential equations. In particular, the IIM proposed for two-dimensional acoustic wave equations with piecewise constant coefficients [C. Zhang, R.J. LeVeque, The immersed interface method for acoustic wave equations with discontinuous coefficients, Wave Motion 25 (1997) 237-263] is extended through a simple least squares procedure to such Maxwell's equations. Numerical results from the simulation of electromagnetic scattering of a plane incident wave by a dielectric circular cylinder appear to indicate that, compared to the original IIM for the acoustic wave equations, the augmented IIM with the proposed least squares fitting greatly improves the long-time stability of the time-domain solution. Semi-discrete finite difference schemes using the IIM for spatial discretization are also discussed and numerically tested in the paper.     

Application of shifted Jacobi pseudospectral method for solving (in)finite-horizon min–max optimal control problems with uncertainty

The difficulty of solving the min–max optimal control problems (M-MOCPs) with uncertainty using generalised Euler–Lagrange equations is caused by the combination of split boundary conditions, nonlinear differential equations and the manner in which the final time is treated. In this investigation, the shifted Jacobi pseudospectral method (SJPM) as a numerical technique for solving two-point boundary value problems (TPBVPs) in M-MOCPs for several boundary states is proposed. At first, a novel framework of approximate solutions which satisfied the split boundary conditions automatically for various boundary states is presented. Then, by applying the generalised Euler–Lagrange equations and expanding the required approximate solutions as elements of shifted Jacobi polynomials, finding a solution of TPBVPs in nonlinear M-MOCPs with uncertainty is reduced to the solution of a system of algebraic equations. Moreover, the Jacobi polynomials are particularly useful for boundary value problems in unbounded domain, which allow us to solve infinite- as well as finite and free final time problems by domain truncation method. Some numerical examples are given to demonstrate the accuracy and efficiency of the proposed method. A comparative study between the proposed method and other existing methods shows that the SJPM is simple and accurate.     

A Spectral Stochastic Semi-Lagrangian Method for Convection-Diffusion Equations with Uncertainty

Real life convection-diffusion problems are characterized by their inherent or externally induced uncertainties in the design parameters. This paper presents a spectral stochastic finite element semi-Lagrangian method for numerical solution of convection-diffusion equations with uncertainty. Using the spectral decomposition, the stochastic variational problem is reformulated to a set of deterministic variational problems to be solved for each Wiener polynomial chaos. To obtain the chaos coefficients in the corresponding deterministic convection-diffusion equations, we implement a semi-Lagrangian method in the finite element framework. Once this representation is computed, statistics of the numerical solution can be easily evaluated. These numerical techniques associate the geometrical flexibility of the finite element method with the ability offered by the semi-Lagrangian method to solve convection-dominated problems using time steps larger than its Eulerian counterpart. Numerical results are shown for a convection-diffusion problem driven with stochastic velocity and for an incompressible viscous flow problem with a random force. In both examples, the proposed method demonstrates its ability to better maintain the shape of the solution in the presence of uncertainties and steep gradients.     

A highly accurate adaptive finite difference solver for the Black–Scholes equation

In this paper, we develop a highly accurate adaptive finite difference (FD) discretization for the Black–Scholes equation. The final condition is discontinuous in the first derivative yielding that the effective rate of convergence in space is two, both for low-order and high-order standard FD schemes. To obtain a method that gives higher accuracy, we use an extra grid in a limited space- and time-domain. This new method is called FD6G2. The FD6G2 method is combined with space- and time-adaptivity to further enhance the method. To obtain solutions of high accuracy, the adaptive FD6G2 method is superior to both a standard and an adaptive second-order FD method.     

An Adaptive P 1 Finite Element Method for Two-Dimensional Maxwell’s Equations

Recently a new numerical approach for two-dimensional Maxwell’s equations based on the Hodge decomposition for divergence-free vector fields was introduced by Brenner et al. In this paper we present an adaptive P ₁ finite element method for two-dimensional Maxwell’s equations that is based on this new approach. The reliability and efficiency of a posteriori error estimators based on the residual and the dual weighted-residual are verified numerically. The performance of the new approach is shown to be competitive with the lowest order edge element of Nédélec’s first family.     

High Spatial Order Energy Stable FDTD Methods for Maxwell’s Equations in Nonlinear Optical Media in One Dimension

In this paper, we consider electromagnetic (EM) wave propagation in nonlinear optical media in one spatial dimension. We model the EM wave propagation by the time-dependent Maxwell’s equations coupled with a system of nonlinear ordinary differential equations (ODEs) for the response of the medium to the EM waves. The nonlinearity in the ODEs describes the instantaneous electronic Kerr response and the residual Raman molecular vibrational response. The ODEs also include the single resonance linear Lorentz dispersion. For such model, we will design and analyze fully discrete finite difference time domain (FDTD) methods that have arbitrary (even) order in space and second order in time. It is challenging to achieve provable stability for fully discrete methods, and this depends on the choices of temporal discretizations of the nonlinear terms. In Bokil et al. (J Comput Phys 350:420–452, 2017), we proposed novel modifications of second-order leap-frog and trapezoidal temporal schemes in the context of discontinuous Galerkin methods to discretize the nonlinear terms in this Maxwell model. Here, we continue this work by developing similar time discretizations within the framework of FDTD methods. More specifically, we design fully discrete modified leap-frog FDTD methods which are proved to be stable under appropriate CFL conditions. These method can be viewed as an extension of the Yee-FDTD scheme to this nonlinear Maxwell model. We also design fully discrete trapezoidal FDTD methods which are proved to be unconditionally stable. The performance of the fully discrete FDTD methods are demonstrated through numerical experiments involving kink, antikink waves and third harmonic generation in soliton propagation.     

Shape optimization of continua using NURBS as basis functions

The present paper introduces a numerical solution to shape optimization problems of domains in which boundary value problems of partial differential equations are defined. In the present paper, the finite element method using NURBS as basis functions in the Galerkin method is applied to solve the boundary value problems and to solve a reshaping problem generated by the H1 gradient method for shape optimization, which has been developed as a general solution to shape optimization problems. Numerical examples of linear elastic continua illustrate that this solution works as well as using the conventional finite element method.     

A Legendre Pseudospectral Penalty Scheme for Solving Time-Domain Maxwell’s Equations

In this paper we present a Legendre pseudospectral algorithm based on a tensor product formulation for solving the time-domain Maxwell equations. Our approach starts by conducting an analysis for finding well-posed boundary operators for the Maxwell equations. We then discuss equivalent characteristic boundary conditions for common physical boundary constraints. These theoretical results are then employed to construct a pseudospectral penalty scheme which is asymptotically stable at the semidiscrete level. Numerical computations based on the proposed scheme are also provided for different cases where exact solutions exist. By measuring the differences between the computed and exact solutions, we observe the expected convergence patterns of the scheme. This work is supported by National Science Council grant No. NSC 95-2120-M-001-003.     

薄膜体声波谐振器的FDTD数值分析

提出了基于时域有限差分方法对薄膜体声波谐振器进行数值分析的新方法。利用时域有限差分法理论对压电材料的控制方程,牛顿方程和电学方程在空间和时间进行了离散化,通过得到的差分方程直接得出了声场传播的时域数值解。使用该数值方法对薄膜体声波谐振器的电学特性阻抗进行了分析,并将结果与一维Mason模型的解析解进行了比较验证。     

A Runge-Kutta discontinuous Galerkin method for the Euler equations

This paper presents a Runge-Kutta discontinuous Galerkin (RKDG) method for the Euler equations of gas dynamics from the viewpoint of kinetic theory. Like the traditional gas-kinetic schemes, our proposed RKDG method does not need to use the characteristic decomposition or the Riemann solver in computing the numerical flux at the surface of the finite elements. The integral term containing the non-linear flux can be computed exactly at the microscopic level. A limiting procedure is carefully designed to suppress numerical oscillations. It is demonstrated by the numerical experiments that the proposed RKDG methods give higher resolution in solving problems with smooth solutions. Moreover, shock and contact discontinuities can be well captured by using the proposed methods.     

A novel time-marching scheme using numerical Green’s functions: A comparative study for the scalar wave equation

The present paper presents the formulation of a novel time-marching method based on the Explicit Green’s Approach (ExGA) to solve scalar wave propagation problems. By means of the weighted residual method in both time and space, the time integral expression concerning the ExGA is readily established. The arising ExGA time integral expression is spatially discretized in a finite element sense and a recursive scheme that employs time-domain numerical Green’s function matrices is adopted to evaluate the displacement and the velocity vectors. These Green’s matrices are computed by the time discontinuous Galerkin finite element method only at the first time step. The system of coupled equations originated from the time discontinuous Galerkin method is then solved by an iterative predictor–multicorrector algorithm. Once the Green’s matrices are computed, no iterative process is required to obtain the displacement and the velocity vectors at any time level. At the end of the paper, numerical examples are presented in order to compare the proposed approach with other approaches.     

Improving accuracy of high-order discontinuous Galerkin method for time-domain electromagnetics on curvilinear domains

The paper discusses high-order geometrical mapping for handling curvilinear geometries in high-accuracy discontinuous Galerkin simulations for time-domain Maxwell problems. The proposed geometrical mapping is based on a quadratic representation of the curved boundary and on the adaptation of the nodal points inside each curved element. With high-order mapping, numerical fluxes along curved boundaries are computed much more accurately due to the accurate representation of the computational domain. Numerical experiments for two-dimensional and three-dimensional propagation problems demonstrate the applicability and benefits of the proposed high-order geometrical mapping for simulations involving curved domains.     

Asymptotic initial-value method for a system of singularly perturbed second-order ordinary differential equations of convection–diffusion type

In this paper, a numerical method is suggested to solve a class of boundary value problems (BVPs) for a weakly coupled system of singularly perturbed second-order ordinary differential equations of convection–diffusion type. First, in this method, an asymptotic expansion approximation of the solution of the BVP is constructed by using the basic ideas of a well known perturbation method namely Wentzal, Kramers and Brillouin (WKB). Then, some initial value problems (IVPs) are constructed such that their solutions are the terms of this asymptotic expansion. These problems happen to be singularly perturbed problems and, therefore, exponentially fitted finite difference schemes are used to solve these problems. As the BVP is converted into a set of IVPs and an asymptotic expansion approximation is used, the present method is termed as asymptotic initial-value method. The necessary error estimates are derived and examples provided to illustrate the method.     

Application of lower order integration schemes in linear shell problems

A method is presented by which steady flow solutions may be obtained to problems which involve non-Newtonian memory fluids. The finite element method is used in conjunction with a Galerkin form of the equations of motion and continuity. Integral constitutive laws are directly employed without extra-stress differential equations. The stress is computed by construction of the portion of the streamline lying upstream of element quadrature points. This construction is shown to be quite simple, owing to the special form of finite element trial velocity fields. Two test problems are analyzed which use the integral form of the Maxwell constitutive law. The interaction between the fluid elasticity and solution procedures for the discrete equations is discussed.