基于δ法求解高阶时变非线性微分方程

该文基于二阶线性微分方程δ法绘制相平面原理,提出一种新颖而简单计算圆弧圆心和半径的方法实现高阶时变非线性微分方程相平面的作图,从而得到求解高阶时变非线性微分方程时域解的算法,并与龙格-库塔法等解析法相比具有计算简单、结果精度高的特点。     

A high-order integral algorithm for highly singular PDE solutions in Lipschitz domains

We present a new algorithm, based on integral equation formulations, for the solution of constant-coefficient elliptic partial differential equations (PDE) in closed two-dimensional domains with non-smooth boundaries; we focus on cases in which the integral-equation solutions as well as physically meaningful quantities (such as, stresses, electric/magnetic fields, etc.) tend to infinity at singular boundary points (corners). While, for simplicity, we restrict our discussion to integral equations associated with the Neumann problem for the Laplace equation, the proposed methodology applies to integral equations arising from other types of PDEs, including the Helmholtz, Maxwell, and linear elasticity equations. Our numerical results demonstrate excellent convergence as discretizations are refined, even around singular points at which solutions tend to infinity. We demonstrate the efficacy of this algorithm through applications to solution of Neumann problems for the Laplace operator over a variety of domains—including domains containing extremely sharp concave and convex corners, with angles as small as π/100 and as large as 199π/100.     

A new energy harvester using a cross-ply cylindrical membrane shell integrated with PVDF layers

In this paper, we proposed a smart cylindrical membrane shell panel (SCMSP) model for vibration-based energy harvester. The SCMSP is made of an orthotropic elastic core covered by outer PVDF layers with transverse polarization vector. Electrodynamics governing equations of motion are derived by applying extended Hamilton’s principle. The governing equations are based on Donnell’s linear thin shell theory. The SCMSP displacement fields are expanded by means of double Fourier series satisfying immovable edges with free rotation boundary conditions and coupled system of linear partial differential equations are obtained. The discretized linear ordinary differential equations of motion are obtained using Galerkin method. The output power is taken as an indicating criterion for the generator. A parametric study for MEMS applications is conducted to predict the power generated due to radial harmonic ambient vibration. Optimal resistance value is also obtained for the particular electrode distribution that gives maximum output power. A low vibration amplitude (5?Pa), and a low-frequency (471.79?Hz) vibration source is targeted for the resonance operation, in which the output power of 0.4111?μW and peak-to-peak voltage of 0.2952?V are predicted.     

Application of the Haar functions to solution of differential equations

In this paper, it is proposed that Haar functions should be used for solving ordinary differential equations of a time variable in facility. This is because integrated forms of Haar functions of any degree can be illustrated by linear- and linear segment-functions like as triangles. Fortunately, since they are placed where Haar functions are defined in a specified form respectively, these functions are computable by algebraic operations of quasi binary numbers. Therefore, when a given function is approximated in a form of stairsteps on a Haar function system their integration can be termwise executed by shift and add operations of coefficients of the approximation. The use of this system is comparable with an application using the midpoint rule in numerical integration. In this line, nonlinear differential equations can be solved like as linear differential equations.     

基于等倾线法实现高阶非线性微分方程求解

该文提出基于二阶线性系统等倾线法绘制相平面原理，实现高阶非线性微分方程的求解，并与龙格-库塔法等解析法相比具有计算简单、结果精度高的特点。     

A symplectic FDTD algorithm for the simulations of lossy dispersive materials

A high-order symplectic finite-difference time-domain (SFDTD) algorithm, based on the matrix splitting, the symplectic integrator propagator and the auxiliary differential equation (ADE) technique, is presented. The algorithm is applied to the lossy Lorentz–Drude dispersive model. With the rigorous and artful formula derivation, the detailed formulations are provided. Moreover, the present algorithm can also be applicable to the lossy Drude and Lorentz dispersive model in a straightforward manner. An excellent agreement is achieved between the SFDTD-calculated and exact theoretical results when calculating the transmission coefficient in simulation of metal films. Focusing action of the matched left-handed materials (LHMs) slab is also achieved as the second example in the two-dimensional space. Numerical results for a more realistic structure, the simulation of periodic arrays of silver split-ring resonators (SRRs) using the Drude dispersion model, are also included, and the results agree well with those obtained by the finite element method (FEM).     

Analysis of 3D frame problems by DQEM using EDQ

The differential quadrature element method (DQEM) and extended differential quadrature (EDQ) have been proposed by the author. The development of a differential quadrature element analysis model of three-dimensional shear-undeformable frame problems adopting the EDQ is carried out. The element can be a nonprismatic beam. The EDQ technique is used to discretize the element-based governing differential equations, the transition conditions at joints and the boundary conditions on domain boundaries. An overall algebraic system can be obtained by assembling all of the discretized equations. A numerically rigorous solution can be obtained by solving the overall algebraic system. Mathematical formulations for the EDQ-based DQEM frame analysis are carried out. By using this DQEM model, accurate results of frame problems can efficiently be obtained. Numerical results demonstrate this DQEM model.     

Field equations for incompressible non-viscous fluids using artificial intelligence

In this article, we introduce new field equations for incompressible non-viscous fluids, which can be treated similarly to Maxwell’s electromagnetic equations based on artificial intelligence algorithms. Lagrangian and Hamiltonian formulations are used to arrive at field equations that are solved using convolutional neural networks. Four linear differential equations, which describe the two fields, namely, the dynamic pressure and the vortex fields, are derived, and these can be used in place of Euler’s equation. The only assumption while deriving this equation is that the dynamic pressure and vortex fields obey the superposition principle. The important finding to be noted is that Euler’s fluid equations can be converted into field equations analogous to Maxwell’s electromagnetic equations. We solve the flow problem for laminar flow past a cylinder, sphere, and cone in two dimensions similar to the conduction in a uniform electric field and arrive at closed-form expressions. These closed-form expressions, which are obtained for the potentials of fluid flow, are similar to the streamline potential functions in the case of fluid dynamics.

     

A method for solving systems of linear interval equations applied to the Leontief input–output model of economics

A new approach to solving systems of linear interval equations based on the generalized procedure of interval extension is proposed. This procedure is based on the treatment of interval zero as an interval centered around zero, and for this reason it is called the “interval extended zero” method. Since the “interval extended zero” method provides a fuzzy solution to interval equations, its interval representations are proposed. It is shown that they may be naturally treated as modified operations of interval division. These operations are used for the modified interval extensions of known numerical methods for solving systems of linear equations and finally for solving systems of linear interval equations. Using a well known example, it is shown that the solution obtained by the proposed method can be treated as an inner interval approximation of the united solution and an outer interval approximation of the tolerable solution, and lies within the range of possible AE-solutions between the extreme tolerable and united solutions. Generally, we can say that the proposed method provides the results which can be treated as approximate formal solutions sometimes referred to as algebraic solutions. Seven known examples are used to illustrate the method’s efficacy and advantages in comparison with known methods providing formal (algebraic) solutions to systems of linear interval equations. It is shown that a new method provides results which are close to the so-called maximal inner solutions (the corresponding method was developed by Kupriyanova, Zyuzin and Markov) and the algebraic solutions obtained by the subdifferential Newton method proposed by Shary. It is important that the proposed method makes it possible to avoid inverted interval solutions. The influence of the system’s size and number of zero entries on the results is analyzed by applying the proposed method to the Leontief input–output model of economics.     

Parallel methods for ordinary differential equations

Remarkably few methods have been proposed for the parallel integration of ordinary differential equations (ODEs). In part this is because the problems do not have much natural parallelism (unless they are virtually uncoupled systems of equations, in which case the method is obvious). In part it is because the subproblems arising in the solution of ODEs (for example, the solution of linear equations) are the ones that have provided the challenges for parallelism. This paper surveys some of the methods that have been proposed, and suggests some additional methods that are suitable for special cases, such as linear problems. It then looks at the possible application of large-scale parallelism, particularly across the method. If efficiency is of no concern (that is, if there is an arbitrary number of proceessors) there are some ways in which the solution of stiff equations can be done more rapidly; in fact, a speed up from a parallel time of 0(N ²) to 0(logN) forN equations might be possible if communication time is ignored. This is obtained by trying to perform as much as possible of the matrix arithmetic associated with the solution of the linear equations at each step in advance of that step and in parallel with the integration of earlier steps.     

On the Euler-Imshenetskii-Darboux transformation of linear second-order equations

It is shown how the linear Euler-Imshenetskii-Darboux (EID) differential transformation can be used for generating infinite sequences of linear second-order ordinary differential equations starting from certain standard equations. In so doing, the method of factorization of differential operators and operator identities obtained by means of this method are used. Generalizations of some well-known integrable cases of the Schrödinger equation are found. An example of an integrable equation with the Liouville coefficients, which apparently cannot be solved by the well-known Kovacic and Singer algorithms and their modifications, is constructed. An algorithm for solving the constructed class of equations has been created and implemented in the computer algebra system REDUCE. The corresponding procedure GENERATE is a supplement to the ODESOLVE procedure available in REDUCE. Solutions of some equations by means of the GENERATE procedure in REDUCE 3.8, as well as those obtained by means of DSOLVE in Maple 10, are presented. Although the algorithm based on the Euler-Imshenetskii-Darboux transformation is not an alternative to the existing algorithms for solving linear second-order ordinary differential equations, it is rather efficient within the limits of its applicability.     

A program for perspective views of three-dimensional surfaces

A numerical method is proposed for solving linear differential equations of second order without first derivatives. The new method is superior to de Vogelaere's for this class of equations, and for non-linear equations it becomes an implicit extension of de Vogelaere's method. The global truncation error at a fixed steplength h is bounded by a term of order h⁴, and the interval of absolute stability is [?2.4, 0]. The work of Coleman and Mohamed (1978) is readily adapted to provide truncation error estimates which can be used for automatic error control. It is suggested that the new method should be used in preference to de Vogelaere's for linear equations, and in particular to solve the radial Schrödinger equation. the radial Schrödinger equation.     

NICE—An explicit numerical scheme for efficient integration of nonlinear constitutive equations

The paper presents a simple but efficient new numerical scheme for the integration of nonlinear constitutive equations. Although it can be used for the integration of a system of algebraic and differential equations in general, the scheme is primarily developed for use with the direct solution methods for solving boundary value problems, e.g. explicit dynamic analysis in ABAQUS/Explicit. In the developed explicit scheme, where no iteration is required, the implementation simplicity of the forward-Euler scheme and the accuracy of the backward-Euler scheme are successfully combined. The properties of the proposed NICE scheme, which was also implemented into ABAQUS/Explicit via User Material Subroutine (VUMAT) interface platform, are compared with the properties of the classical forward-Euler scheme and backward-Euler scheme. For this purpose two highly nonlinear examples, with the von Mises and GTN material model considered, have been studied. The accuracy of the new scheme is demonstrated to be at least of the same level as experienced by the backward-Euler scheme, if we compare them on the condition of the same CPU time consumption. Besides, the simplicity of the NICE scheme, which is due to implementation similarity with the classical forward-Euler scheme, is its great Advantage.     

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.

     

Controlling the level of sparsity in MPC

In optimization algorithms used for on-line Model Predictive Control (MPC), linear systems of equations are often solved in each iteration. This is true both for Active Set methods as well as for Interior Point methods, and for linear MPC as well as for nonlinear MPC and hybrid MPC. The main computational effort is spent while solving these linear systems of equations, and hence, it is of greatest interest to solve them efficiently. Classically, the optimization problem has been formulated in either of two ways. One leading to a sparse linear system of equations involving relatively many variables to compute in each iteration and another one leading to a dense linear system of equations involving relatively few variables. In this work, it is shown that it is possible not only to consider these two distinct choices of formulations. Instead it is shown that it is possible to create an entire family of formulations with different levels of sparsity and number of variables, and that this extra degree of freedom can be exploited to obtain even better performance with the software and hardware at hand. This result also provides a better answer to a recurring question in MPC; should the sparse or dense formulation be used.     

Minimum energy control of time delay systems via piecewise linear polynomial functions

The piecewise linear polynomial function approach to the minimum energy control of linear systems with time delay, is presented in this paper. The concepts of a delay shift matrix and an operational matrix for integration are employed in solving the related state and costate equations containing terms with advanced and delayed arguments. An attractive feature of the present method is its ultimate simplicity and convenience. The differential equations with delay and advance terms are converted into a set of linear algebraic equations using a recurrence algorithm. An example demonstrates the accuracy of the method.     

Unconditionally stable split-step finite difference time domain formulations for double-dispersive electromagnetic materials

Systematic split-step finite difference time domain (SS-FDTD) formulations, based on the general Lie–Trotter–Suzuki product formula, are presented for solving the time-dependent Maxwell equations in double-dispersive electromagnetic materials. The proposed formulations provide a unified tool for constructing a family of unconditionally stable algorithms such as the first order split-step FDTD (SS1-FDTD), the second order split-step FDTD (SS2-FDTD), and the second order alternating direction implicit FDTD (ADI-FDTD) schemes. The theoretical stability of the formulations is included and it has been demonstrated that the formulations are unconditionally stable by construction. Furthermore, the dispersion relation of the formulations is derived and it has been found that the proposed formulations are best suited for those applications where a high space resolution is needed. Two-dimensional (2-D) and 3-D numerical examples are included and it has been observed that the SS1-FDTD scheme is computationally more efficient than the ADI-FDTD counterpart, while maintaining approximately the same numerical accuracy. Moreover, the SS2-FDTD scheme allows using larger time step than the SS1-FDTD or ADI-FDTD and therefore necessitates less CPU time, while giving approximately the same numerical accuracy.     

Computation of analytical expressions for transfer functions

A new method for computing transfer functions of linear time-invariant multi-variable control systems is presented. This method has the following attractive features: (i) it produces analytical expressions for transfer functions, (ii) computations are inherently parallel and can be implemented, for instance, on parallel processors with a fine-grain architecture, (iii) calculations of transfer functions are reduced to two well-studied computational problems: solution of simultaneous linear equations and the fast Fourier transform (FFT), and (iv) state-variable formulations can be avoided. Theoretical conclusions are demonstrated with numerical examples.     

Exponential difference schemes for solving boundary-value problems for convection-diffusion type equations

The paper considers the numerical solution of boundary-value problems for multidimensional convection-diffusion type equations (CDEs). Such equations are useful for various physical processes in solids, liquids and gases. A new approach to the spatial approximation for such equations is proposed. This approach is based on an integral transformation of second-order one-dimensional differential operators. A linear version of CDE was chosen for simplicity of the analysis. In this setting, exponential difference schemes were constructed, algorithms for their implementation were developed, a brief analysis of the stability and convergence was made. This approach was numerically tested for a two-dimensional problem of motion of metallic particles in water flow subject to a constant magnetic field.     

Frame element with lateral deformable supports: Formulations and numerical validation

This paper presents the theory and the numerical validation of three different formulations of nonlinear frame elements with nonlinear lateral deformable supports. The governing differential equations of the problem are derived first and the three different finite element formulations are then presented. The first model follows a displacement-based formulation, which is based on the virtual displacement principle. The second one follows the force-based formulation, which is based on the virtual force principle. The third model follows the Hellinger–Reissner mixed formulation, which is based on the two-field mixed variational principle. The selection of the displacement and force interpolation functions for the different formulations is discussed. Tonti’s diagrams are used to conveniently represent the equations governing both the strong and the weak forms of the problem. The general matrix equations of the three formulations are presented, with some details on the issues regarding the elements’ implementations in a general-purpose finite element program. The convergence, accuracy, and computational times of the three elements are studied through a numerical example. The distinctive element characteristics in terms of force and deformation discontinuities between adjacent elements are discussed. The capability of the proposed frame models to trace the softening response due to softening of the foundation is also investigated. Overall, the force-based and the mixed models are much more accurate than the displacement-based model and require very few elements to reach the converged solution. The force-based element is slightly more accurate than the mixed model, but it is more prone to numerical instabilities as it involves inverting the element flexibility matrix.