Perfectly matched layers for time-harmonic elastodynamics of unbounded domains: theory and finite-element implementation

One approach to the numerical solution of a wave equation on an unbounded domain uses a bounded domain surrounded by an absorbing boundary or layer that absorbs waves propagating outwards from the bounded domain. A perfectly matched layer (PML) is an unphysical absorbing layer model for linear wave equations that absorbs, almost perfectly, outgoing waves of all non-tangential angles-of-incidence and of all non-zero frequencies. This paper develops the PML concept for time-harmonic elastodynamics in Cartesian coordinates, utilising insights obtained with electromagnetics PMLs, and presents a novel displacement-based, symmetric finite-element implementation of the PML for time-harmonic plane-strain or three-dimensional motion. The PML concept is illustrated through the example of a one-dimensional rod on elastic foundation and through the anti-plane motion of a two-dimensional continuum. The concept is explored in detail through analytical and numerical results from a PML model of the semi-infinite rod on elastic foundation, and through numerical results for the anti-plane motion of a semi-infinite layer on a rigid base. Numerical results are presented for the classical soil–structure interaction problems of a rigid strip-footing on a (i) half-plane, (ii) layer on a half-plane, and (iii) layer on a rigid base. The analytical and numerical results obtained for these canonical problems demonstrate the high accuracy achievable by PML models even with small bounded domains.     

On the existence and convergence of the solution of PML equations

In this article we study the mesh termination method in computational scattering theory known as the method of Perfectly Matched Layer (PML). This method is based on the idea of surrounding the scatterer and its immediate vicinity with a fictitious absorbing non-reflecting layer to damp the echoes coming from the mesh termination surface. The method can be formulated equivalently as a complex stretching of the exterior domain. The article is devoted to the existence and convergence questions of the solutions of the resulting equations. We show that with a special choice of the fictitious absorbing coefficient, the PML equations are solvable for all wave numbers, and as the PML layer is made thicker, the PML solution converge exponentially towards the actual scattering solution. The proofs are based on boundary integral methods and a new type of near-field version of the radiation condition, called here the double surface radiation condition. Partly supported by the Finnish Academy, project 37692.     

Inverse problems involving the one-way wave equation: medium function reconstruction

Inverse problems are considered for the linear one-way one-dimensional wave equation or transport equation. In particular the wave speed reconstruction problem for a medium is discussed. Inverse problems for non-stationary, but also non-dispersive, media are examined; this means problems for which the slab medium parameters are both spatially and temporally varying are considered. Both theoretical and numerical results are given for the methods presented. Theoretical results obtained for this equation can be generalised to second order equations.     

Second-order predictor-corrector schemes for nonlinear distributed-order space-fractional differential equations with non-smooth initial data

ABSTRACT

We propose second-order linearly implicit predictor-corrector schemes for diffusion and reaction-diffusion equations of distributed-order. For diffusion equations of distributed order, we propose an analytical solution based on the spectral representation of the fractional Laplacian. Numerically, we approximate the integral term of the equation by the midpoint quadrature rule to obtain a multi-term space-fractional differential equation. The matrix transfer technique is used for spatial discretization of the resulting differential equation and methods based on Padé approximations to the exponential function are used in time. In particular, we discuss the (0,2)- and (1,1)-Padé approximations to the exponential function. The method based on the (1,1)-Padé approximation to the exponential function are seen to produce oscillations for some time steps and we propose a constraint on the choice of the time step to avoid these unwanted oscillations. Stability and convergence of the schemes are discussed. Numerical experiments are performed to support our theoretical observations.     

Second order adjoint sensitivity analysis of multibody systems described by differential–algebraic equations

The optimization strategies employing second order sensitivity information has higher accuracy, but its computation is complex. In this paper, an adjoint variable method applied for the second order design sensitivity analysis of multibody design problems is developed. Based on Lagrange equations of multibody system dynamics, a general objective function, constraint conditions, initial and end conditions, the adjoint variable equations for first order sensitivity analysis and design sensitivity formulations are derived firstly. Then, second order sensitivity analysis formulations, as well as the detailed computation steps, are given based on the previous results. For simplification, the second derivative of the objective function with respect to design variables is translated into an initial value problem of an ordinary differential equation with one variable. Finally, a numerical example of slider–crank mechanism validates the accuracy and efficiency of the method for second order sensitivity analysis.     

Development of PML absorbing boundary conditions for computational aeroacoustics: A progress review

Recent advances in the development of perfectly matched layer (PML) as absorbing boundary conditions for computational aeroacoustics are reviewed. The PML methodology is presented as a complex change of variables. In this context, the importance of a proper space-time transformation in the PML technique for Euler equations is emphasized. A unified approach for the derivation of PML equations is offered that involves three essential steps. The three-step approach is illustrated in details for the PML of linear and non-linear Euler equations. Numerical examples are also given that include non-reflecting boundary conditions for a ducted channel flow and mixing layer roll-up vortices.     

Absorbing Boundary Conditions of the Second Order for the Pseudospectral Chebyshev Methods for Wave Propagation

In this paper we describe the implementation of one-way wave equations of the second order in conjuction with pseudospectral methods for wave propagation in two space dimensions. These equations are first reformulated as hyperbolic systems of the first order and the absorbing boundaries are implemented by an appropriate modification of the matrix of this system. The resulting matrix corresponding to one-way wave equation based on Padé approximation has all eigenvalues in the complex negative half plane which allows stable integration of the underlying system by any ODE solver in the sense of eigenvalue stability. The obtained numerical scheme is much more accurate than the schemes obtained before which utilized absorbing boundary conditions of the first order, and is also capable of integrating the wave propagation problems on much larger time intervals than was previously possible.     

基于正交轨迹的共形PML数值仿真方法

从材料的观点看,完全匹配层是各向异性的介质.场在完全匹配层遵从麦克斯韦方程.通过设计磁导率和介电常数张量参数来实现任意形状的共形完全匹配层.为匹配层动态稳定,可应用二维正交轨迹网格和FDTD方法实现了数值共形完全匹配层仿真,数值仿真结果表明基于正交轨迹网格的共形完全匹配层为一个共形FDTD或FEM方法提供了一个有效地吸收边界条件.     

Numerical solution to coupled nonlinear Schrödinger equations on unbounded domains

The numerical simulation of coupled nonlinear Schrödinger equations on unbounded domains is considered in this paper. By using the operator splitting technique, the original problem is decomposed into linear and nonlinear subproblems in a small time step. The linear subproblem turns out to be two decoupled linear Schrödinger equations on unbounded domains, where artificial boundaries are introduced to truncate the unbounded physical domains into finite ones. Local absorbing boundary conditions are imposed on the artificial boundaries. On the other hand, the coupled nonlinear subproblem is an ODE system, which can be solved exactly. To demonstrate the effectiveness of our method, some comparisons in terms of accuracy and computational cost are made between the PML approach and our method in numerical examples.     

The numerical solution of a generalized Burgers–Huxley equation through a conditionally bounded and symmetry-preserving method

In this article, we propose a non-standard, finite-difference scheme to approximate the solutions of a generalized Burgers–Huxley equation from fluid dynamics. Our numerical method preserves the skew-symmetry of the partial differential equation under study and, under some analytical constraints of the model constants and the computational parameters involved, it is capable of preserving the boundedness and the positivity of the solutions. In the linear regime, the scheme is consistent to first order in time (due partially to the inclusion of a tuning parameter in the approximation of a temporal derivative), and to second order in space. We compare the results of our computational technique against the exact solutions of some particular initial-boundary-value problems. Our simulations indicate that the method presented in this work approximates well the theoretical solutions and, moreover, that the method preserves the boundedness of solutions within the analytical constraints derived here. In the problem of approximating solitary-wave solutions of the model under consideration, we present numerical evidence on the existence of an optimum value of the tuning parameter of our technique, for which a minimum relative error is achieved. Finally, we linearly perturb a steady-state solution of the partial differential equation under investigation, and show that our simulations still converge to the same constant solution, establishing thus robustness of our method in this sense.     

Numerical solution to coupled nonlinear Schrödinger equations on unbounded domains

The numerical simulation of coupled nonlinear Schrödinger equations on unbounded domains is considered in this paper. By using the operator splitting technique, the original problem is decomposed into linear and nonlinear subproblems in a small time step. The linear subproblem turns out to be two decoupled linear Schrödinger equations on unbounded domains, where artificial boundaries are introduced to truncate the unbounded physical domains into finite ones. Local absorbing boundary conditions are imposed on the artificial boundaries. On the other hand, the coupled nonlinear subproblem is an ODE system, which can be solved exactly. To demonstrate the effectiveness of our method, some comparisons in terms of accuracy and computational cost are made between the PML approach and our method in numerical examples.     

Finite difference solution of differential equations with stochastic inputs

This paper discusses application of two numerical methods (central difference and predictor corrector) for the solution of differential equations with deterministic as well as stochastic inputs. The methods are applied to a second order linear differential equation representing a series RLC netowrk with step function, sinusoidal and stochastic inputs. It is shown that both methods give correct answers for the step function and sinusoidal inputs. However, the central-difference method of solution is recommended for stochastic inputs. This statement is justified by comparing the auto-correlation and cross-correlation functions of the central-difference solution (with stochastic inputs) with the corresponding theoretical values of a continuous system. It is further shown that the more common predictor-corrector methods, although suitable for solution of differential equations with regular inputs, diverge for stochastic inputs. The reason is that these methods, by the application of several point integral formulas, use a high degree of smoothing on the variable and its derivatives. Inherent in the derivation of these integral formulas is the assumption of the continuity of the variable and its derivatives, a condition which is not satisfied in problems with stochastic inputs.Note that the second order differential equation chosen here for numerical experiments can be solved by classical methods for all of the given inputs, including the probabilistic inputs. The classical methods, however, unlike the numerical solutions, can not be extended to nonlinear differential equations which frequently arise in the digital simulation of engineering problems.     

Computation of optical modes in axisymmetric open cavity resonators

The computation of optical modes inside axisymmetric cavity resonators with a general spatial permittivity profile is a formidable computational task. In order to avoid spurious modes the vector Helmholtz equations are discretised by a mixed finite element approach. We formulate the method for first and second order Nédélec edge and Lagrange nodal elements. We discuss how to accurately compute the element matrices and solve the resulting large sparse complex symmetric eigenvalue problems. We validate our approach by three numerical examples that contain varying material parameters and absorbing boundary conditions (ABC).     

复杂科学与工程问题仿真的隐式微积分建模

针对现代科学与工程仿真遇到愈来愈多难以用经典微积分建模方法描述的复杂问题,在理论研究和工程实践中提出各种含有多个经验参数的唯象偏微分方程模型,或直接采用统计模型来描述.这些模型的物理意义不是很清楚且参数多,其中部分人为参数缺乏物理意义.因此,利用描述问题的基本解或统计分布构造隐式微积分控制方程.这里"隐式"是指可以不需要或难以推导出该控制方程的显式微积分表达式.该方法仅需微积分控制方程的基本解和相应的边界条件就可以进行数值仿真计算.称该方法为隐式微积分方程建模.考虑多相软物质热传导的幂律行为,采用分数阶里斯(Riesz)势核函数为基本解构造稳态问题的隐式分数阶微积分方程模型并进行数值验证.此外,以列维(Lévy)稳态统计分布的概率密度函数为基本解,构造反常扩散现象的隐式分数阶微积分方程模型.该研究的主要数值计算技术基于径向基函数的配点方法.     

Three-Dimensional Lattice Boltzmann Model for the Complex Ginzburg–Landau Equation

In this paper, a lattice Boltzmann model for the three-dimensional complex Ginzburg–Landau equation is proposed. The multi-scale technique and the Chapman–Enskog expansion are used to describe higher-order moments of the complex equilibrium distribution function and a series of complex partial differential equations. The modified partial differential equation of the three-dimensional complex Ginzburg–Landau equation with the third order truncation error is obtained. Based on the complex lattice Boltzmann model, some motions of the stable scroll, such as the scroll wave with a straight filament, scroll ring, and helical scroll are simulated. The comparisons between results of the lattice Boltzmann model with those obtained by the alternative direction implicit scheme are given. The numerical results show that this model can be used to simulate the three-dimensional complex Ginzburg–Landau equation.     

The numerical solution of fractional differential equations using the Volterra integral equation method based on thin plate splines

Finding the approximate solution of differential equations, including non-integer order derivatives, is one of the most important problems in numerical fractional calculus. The main idea of the current paper is to obtain a numerical scheme for solving fractional differential equations of the second order. To handle the method, we first convert these types of differential equations to linear fractional Volterra integral equations of the second kind. Afterward, the solutions of the mentioned Volterra integral equations are estimated using the discrete collocation method together with thin plate splines as a type of free-shape parameter radial basis functions. Since the scheme does not need any background meshes, it can be recognized as a meshless method. The proposed approach has a simple and computationally attractive algorithm. Error analysis is also studied for the presented method. Finally, the reliability and efficiency of the new technique are tested over several fractional differential equations and obtained results confirm the theoretical error estimates.

     

High Order Weighted Essentially Non-oscillation Schemes for Two-Dimensional Detonation Wave Simulations

In this paper, we demonstrate the detailed numerical studies of three classical two dimensional detonation waves by solving the two dimensional reactive Euler equations with species with the fifth order WENO-Z finite difference scheme (Borges et al. in J. Comput. Phys. 227:3101?C3211, 2008) with various grid resolutions. To reduce the computational cost and to avoid wave reflection from the artificial computational boundary of a truncated physical domain, we derive an efficient and easily implemented one dimensional Perfectly Matched Layer (PML) absorbing boundary condition (ABC) for the two dimensional unsteady reactive Euler equation when one of the directions of domain is periodical and inflow/outflow in the other direction. The numerical comparison among characteristic, free stream, extrapolation and PML boundary conditions are conducted for the detonation wave simulations. The accuracy and efficiency of four mentioned boundary conditions are verified against the reference solutions which are obtained from using a large computational domain. Numerical schemes for solving the system of hyperbolic conversation laws with a single-mode sinusoidal perturbed ZND analytical solution as initial conditions are presented. Regular rectangular combustion cell, pockets of unburned gas and bubbles and spikes are generated and resolved in the simulations. It is shown that large amplitude of perturbation wave generates more fine scale structures within the detonation waves and the number of cell structures depends on the wave number of sinusoidal perturbation.     

Some variants of the Chebyshev–Halley family of methods with fifth order of convergence

In this paper we present some techniques for constructing high-order iterative methods in order to approximate the zeros of a non-linear equation f(x)=0, starting from a well-known family of cubic iterative processes. The first technique is based on an additional functional evaluation that allows us to increase the order of convergence from three to five. With the second technique, we make some changes aimed at minimizing the calculus of inverses. Finally, looking for a better efficiency, we eliminate terms that contribute to the error equation from sixth order onwards.

The paper contains a comparative study of the asymptotic error constants of the methods and some theoretical and numerical examples that illustrate the given results. We also analyse the efficiency of the aforementioned methods, by showing some numerical examples with a set of test functions and by using adaptive multi-precision arithmetic in the computation.     

Long-Time Performance of Unsplit PMLs with Explicit Second Order Schemes

A gradual long-time growth of the solution in perfectly matched layers (PMLs) has been previously reported in the literature. This undesirable phenomenon may hamper the performance of the layer, which is designed to truncate the computational domain for unsteady wave propagation problems. For unsplit PMLs, prior studies have attributed the growth to the presence of multiple eigenvalues in the amplification matrix of the governing system of differential equations. In the current paper, we analyze the temporal behavior of unsplit PMLs for some commonly used second order explicit finite-difference schemes that approximate the Maxwell’s equations. Our conclusion is that on top of having the PML as a potential source of long-time growth, the type of the layer and the choice of the scheme play a major role in how rapidly this growth may manifest itself and whether or not it manifests itself at all.     

Solving biharmonic equation using the localized method of approximate particular solutions

Some localized numerical methods, such as finite element and finite difference methods (FDMs), have encountered difficulties when solving fourth or higher order differential equations. Localized methods, which use radial basis functions, are considered the generalized FDMs and, thus, inherit the similar difficulties when solving higher order differential equations. In this paper, we deal with the use of the localized method of approximate particular solutions (LMAPS), a recently developed localized radial basis function collocation method, in solving two-dimensional biharmonic equation in a bounded region. The technique is based on decoupling the biharmonic problem into two Poisson equations, and then the LMAPS is applied to each Poisson's problem to compute numerical solutions. Furthermore, the influence of the shape parameter and different radial basis functions on the numerical solution is discussed. The effectiveness of the proposed method is demonstrated by solving three examples in both regular and irregular domains.