A quasi‐interpolation method for solving stiff ordinary differential equations

Based on the idea of quasi‐interpolation and radial basis functions approximation, a numerical method is developed to quasi‐interpolate the forcing term of differential equations by using radial basis functions. A highly accurate approximation for the solution can then be obtained by solving the corresponding fundamental equation and a small size system of equations related to the initial or boundary conditions. This overcomes the ill‐conditioning problem resulting from using the radial basis functions as a global interpolant. Error estimation is given for a particular second‐order stiff differential equation with boundary layer. The result of computations indicates that the method can be applied to solve very stiff problems. With the use of multiquadric, a special class of radial basis functions, it has been shown that a reasonable choice for the optimal shape parameter is obtained by taking the same value of the shape parameter as the perturbed parameter contained in the stiff equation. Copyright © 2000 John Wiley & Sons, Ltd.     

A linearly implicit trapezoidal method for integrating stiff multibody dynamics with contact,joints, and friction

We present a hard constraint, linear complementarity based, method for the simulation of stiff multibody dynamics with contact, joints and friction. The approach uses a linearization of the modified trapezoidal method, incorporates a Poisson restitution model at collision, and solves only one linear complementarity problem per time step when no collisions are encountered. We prove that, under certain assumptions, the method has order two, a fact that is also demonstrated by our numerical simulations. For the unconstrained (ODE) case, the method achieves second‐order convergence and absolute stability while solving only one linear system per step. When we use a special approximation of the Jacobian matrix for the case where the stiff forces originate in springs and dampers attached to two points in the system, the linear complementarity problem can be solved for any value of the time step and numerical simulation demonstrate that the method is stiffly stable. The method was implemented in UMBRA, an industrial‐grade virtual prototyping software. Copyright © 2005 John Wiley & Sons, Ltd.     

求解刚-柔组合结构的隐式-显式混合法

针对整体柔性、局部刚性的组合结构系统的动力反应分析,推出了一种隐式-显式混合算法的逐步积分方法.这种混合求解方法,在一个算法步中,对柔性部分采用显式方法求解,对刚性部分则采用隐式方法求解,充分利用了隐式算法无条件稳定的优点和显式算法计算高效的优点.混合法稳定域大小表达式中的频率为柔性部分的最高频率,而不是整体结构的最高频率,这就避开了局部刚性导致显式算法时间步长需要过于细化的缺点.这种方法可以作为一种高效的逐步积分方法,用来求解土-结构相互作用、液固耦合系统等的动力反应.算例表明了方法的正确性.     

A time‐stepping method for stiff multibody dynamics with contact and friction

We define a time‐stepping procedure to integrate the equations of motion of stiff multibody dynamics with contact and friction. The friction and non‐interpenetration constraints are modelled by complementarity equations. Stiffness is accommodated by a technique motivated by a linearly implicit Euler method. We show that the main subproblem, a linear complementarity problem, is consistent for a sufficiently small time step h. In addition, we prove that for the most common type of stiff forces encountered in rigid body dynamics, where a damping or elastic force is applied between two points of the system, the method is well defined for any time step h. We show that the method is stable in the stiff limit, unconditionally with respect to the damping parameters, near the equilibrium points of the springs. The integration step approaches, in the stiff limit, the integration step for a system where the stiff forces have been replaced by corresponding joint constraints. Simulations for one‐ and two‐dimensional examples demonstrate the stable behaviour of the method. Published in 2002 by John Wiley & Sons, Ltd.     

The discrete null space method for the energy consistent integration of constrained mechanical systems. Part II: multibody dynamics

In the present work, rigid bodies and multibody systems are regarded as constrained mechanical systems at the outset. The constraints may be divided into two classes: (i) internal constraints which are intimately connected with the assumption of rigidity of the bodies, and (ii) external constraints related to the presence of joints in a multibody framework. Concerning external constraints lower kinematic pairs such as revolute and prismatic pairs are treated in detail. Both internal and external constraints are dealt with on an equal footing. The present approach thus circumvents the use of rotational variables throughout the whole time discretization. After the discretization has been completed a size‐reduction of the discrete system is performed by eliminating the constraint forces. In the wake of the size‐reduction potential conditioning problems are eliminated. The newly proposed methodology facilitates the design of energy–momentum methods for multibody dynamics. The numerical examples deal with a gyro top, cylindrical and planar pairs and a six‐body linkage. Copyright © 2006 John Wiley & Sons, Ltd.     

结构动力方程求解中隐式格式向显式格式的转换

摘 要：以广义加速度隐式算法为例,指出常用隐式算法可转换为显式算法。转换过程主要包括三个步骤：首先是对隐式求解的耦联线性方程组的系数矩阵取逆,然后将此逆矩阵在单位矩阵附近进行级数展开,最后进行级数截断。以常加速度方法的显式化过程为例,详细阐述隐式算法转换为显式算法的过程,着重介绍级数展开的项数对计算精度的影响,以及转换过程中时间步长必须同时满足算法的稳定性和级数收敛条件。揭示隐式算法与显式算法之联系,使可按高精度隐式算法得到与其相伴随的高精度显式算法。     

Comparison of continuous and discontinuous finite element methods for parabolic differential equations employing implicit time integration

This article compares the computational cost, stability, and accuracy of continuous and discontinuous Galerkin Finite Element Methods (GFEM) for various parabolic differential equations including the advection–diffusion equation, viscous Burgers’ equation, and Turing pattern formation equation system. The results show that, for implicit time integration, the continuous GFEM is typically 5–20 times less computationally expensive than the discontinuous GFEM using the same finite element mesh and element order. However, the discontinuous GFEM is significantly more stable than the continuous GFEM for advection dominated problems and is able to obtain accurate approximate solutions for cases where the classic, un-stabilized continuous GFEM fails.     

Using Krylov subspace and spectral methods for solving complementarity problems in many‐body contact dynamics simulation

Many‐body dynamics problems are expected to handle millions of unknowns when, for instance, investigating the three‐dimensional flow of granular material. Unfortunately, the size of the problems tractable by existing numerical solution techniques is severely limited on convergence grounds. This is typically the case when the equations of motion embed a differential variational inequality problem that captures contact and possibly frictional interactions between rigid and/or flexible bodies. As the size of the physical system increases, the speed and/or the quality of the numerical solution decreases. This paper describes three methods – the gradient projected minimum residual method, the preconditioned spectral projected gradient with fallback method, and the modified proportioning with reduced gradient projection method – that demonstrate better scalability than the projected Jacobi and Gauss–Seidel methods commonly used to solve contact problems that draw on a differential‐variational‐inequality‐based modeling approach. Copyright © 2013 John Wiley & Sons, Ltd.     

An efficient high-precision recursive dynamic algorithm for closed-loop multibody systems

As most closed-loop multibody systems do not have independent generalized coordinates, their dynamic equations are differential/algebraic equations (DAEs). In order to accurately solve DAEs, a usual method is using generalized α-class numerical methods to convert DAEs into difference equations by differential discretization and solve them by the Newton iteration method. However, the complexity of this method is O(n²) or more in each iteration, since it requires calculating the complex Jacobian matrix. Therefore, how to improve computational efficiency is an urgent problem. In this paper, we modify this method to make it more efficient. The first change is in the phase of building dynamic equations. We use the spatial vector note and the recursive method to establish dynamic equations (DAEs) of closed-loop multibody systems, which makes the Jacobian matrix have a special sparse structure. The second change is in the phase of solving difference equations. On the basis of the topology information of the system, we simplify this Jacobian matrix by proper matrix processing and solve the difference equations recursively. After these changes, the algorithm complexity can reach O(n) in each iteration. The algorithm proposed in this paper is not only accurate, which can control well the position/velocity constraint errors, but also efficient. It is suitable for chain systems, tree systems, and closed-loop systems.     

Mitigating the integration error in numerical simulations of Newtonian systems

We introduce a method for mitigating the numerical integration errors of linear, second‐order initial value problems. We propose a methodology for constructing an optimal state‐space representation that gives minimum numerical truncation error, and in this sense, is the optimal state‐space representation for modelling given phase‐space dynamics. To that end, we utilize a simple transformation of the state‐space equations into their variational form. This process introduces an inherent freedom, similar to the gauge freedom in electromagnetism. We then utilize the gauge function to reduce the numerical integration error. We show that by choosing an appropriate gauge function the numerical integration error dramatically decreases and one can achieve much better accuracy compared to the standard state variables for a given time‐step. Moreover, we derive general expressions yielding the optimal gauge functions given a Newtonian one degree‐of‐freedom ODE. For the n degrees‐of‐freedom case we describe MATLAB^® code capable of finding the optimal gauge functions and integrating the given system using the gauge‐optimized integration algorithm. In all of our illustrating examples, the gauge‐optimized integration outperforms the integration using standard state variables by a few orders of magnitude. Copyright © 2006 John Wiley & Sons, Ltd.     

Comparative study of six explicit and two implicit finite difference schemes for solving one-dimensional parabolic partial differential equations

Eight finite difference schemes used in solving parabolic partial differential equations are compared with respect to accuracy, execution time and programming effort. The analysis presented is useful in selecting the appropriate numerical scheme depending on the emphasis placed upon accuracy, execution time or programming effort.     

An implicit unitless error and step‐size control method in integrating unified viscoplastic/creep ODE‐type constitutive equations

A series of numerical analyses are carried out to investigate the difficulties in numerical integration of unified viscoplastic/creep constitutive equations, which are normally represented as a system of ordinary differential equations (ODEs). The problems of numerically integrating the constitutive equations are identified and analysed. To overcome the stiffness problems, implicit methods are used for the numerical integration and a generic technique is introduced to calculate the Jacobian matrix. A normalization technique is introduced in the paper to convert the integration errors for each time increment to unitless errors. Thus, a single tolerance can be used to control the accuracy and step size in integrating a set of unified viscoplastic/creep constitutive equations. In addition, an implicit step‐size control method is proposed and used in the integrations. This method reduces the possibility of rejection of an integration increment due to poor accuracy. Copyright © 2007 John Wiley & Sons, Ltd.     

An efficient reduced-order modeling approach for non-linear parametrized partial differential equations

For general non-linear parametrized partial differential equations (PDEs), the standard Galerkin projection is no longer efficient to generate reduced-order models. This is because the evaluation of the integrals involving the non-linear terms has a high computational complexity and cannot be pre-computed. This situation also occurs for linear equations when the parametric dependence is nonaffine. In this paper, we propose an efficient approach to generate reduced-order models for large-scale systems derived from PDEs, which may involve non-linear terms and nonaffine parametric dependence. The main idea is to replace the non-linear and nonaffine terms with a coefficient-function approximation consisting of a linear combination of pre-computed basis functions with parameter-dependent coefficients. The coefficients are determined efficiently by an inexpensive and stable interpolation at some pre-computed points. The efficiency and accuracy of this method are demonstrated on several test cases, which show significant computational savings relative to the standard Galerkin projection reduced-order approach. Copyright © 2008 John Wiley & Sons, Ltd.     

Combining differential quadrature method with simulation technique to solve non‐linear differential equations

Nowadays, most of the ordinary differential equations (ODEs) can be solved by modelica‐based approaches, such as Matlab/Simulink, Dymola and LabView, which use simulation technique (ST). However, these kinds of approaches restrict the users in the enforcement of conditions at any instant of the time domain. This limitation is one of the most important drawbacks of the ST. Another method of solution, differential quadrature method (DQM), leads to very accurate results using only a few grids on the domain. On the other hand, DQM is not flexible for the solution of non‐linear ODEs and it is not so easy to impose multiple conditions on the same location. For these reasons, the author aims to eliminate the mentioned disadvantages of the simulation technique (ST) and DQM using favorable characteristics of each method in the other. This work aims to show how the combining method (CM) works simply by solving some non‐linear problems and how the CM gives more accurate results compared with those of other methods. Copyright © 2008 John Wiley & Sons, Ltd.     

Multiple positive solutions of integral BVPs for high-order nonlinear fractional differential equations with impulses and distributed delays

In this article, we study the existence of multiple solutions of the integral boundary value problems for high-order nonlinear fractional differential equations with impulses and distributed delays. Some sufficient criteria will be established by the fixed point index theorem in cones. As application, one example is given to demonstrate the validity of our main results.     

Uniformly convergent non‐standard finite difference methods for singularly perturbed differential‐difference equations with delay and advance

A new class of fitted operator finite difference methods are constructed via non‐standard finite difference methods ((NSFDM)s) for the numerical solution of singularly perturbed differential difference equations having both delay and advance arguments. The main idea behind the construction of our method(s) is to replace the denominator function of the classical second‐order derivative with a positive function derived systematically in such a way that it captures significant properties of the governing differential equation and thus provides the reliable numerical results. Unlike other FOFDMs constructed in standard ways, the methods that we present in this paper are fairly simple to construct (and thus enrich the class of fitted operator methods by adding these new methods). These methods are shown to be ε‐uniformly convergent with order two which is the highest possible order of convergence obtained via any fitted operator method for the problems under consideration. This paper further clarifies several doubts, e.g. why a particular scheme is not suitable for the whole range of values of the associated parameters and what could be the possible remedies. Finally, we provide some numerical examples which illustrate the theoretical findings. Copyright © 2005 John Wiley & Sons, Ltd.     

An algorithm for computing a companion block diagonal form for a system of linear differential equations

In this paper we describe a new algorithm which reduces in a finite number of steps a linear system of differential equations to a companion block diagonal form. This form is particularly convenient if one wishes to compute invariants at singularities.     

An exponentially fitted method for solving Burgers' equation

In this paper, an exponentially fitted method is used to numerically solve the one‐dimensional Burgers' equation. The performance of the method is tested on the model involving moderately large Reynolds numbers. The obtained numerical results show that the method is efficient, stable and reliable for solving Burgers' equation accurately even involving high Reynolds numbers for which the exact solution fails. Copyright © 2009 John Wiley & Sons, Ltd.     

Novel partitioned time integration methods for DAE systems based on L‐stable linearly implicit algorithms

Real‐time applications based on the principle of Dynamic Substructuring require integration methods that can deal with constraints without exceeding an a priori fixed number of steps. For these applications, first we introduce novel partitioned algorithms able to solve DAEs arising from transient structural dynamics. In particular, the spatial domain is partitioned into a set of disconnected subdomains and continuity conditions of acceleration at the interface are modeled using a dual Schur formulation. Interface equations along with subdomain equations lead to a system of DAEs for which both staggered and parallel procedures are developed. Moreover under the framework of projection methods, also a parallel partitioned method is conceived. The proposed partitioned algorithms enable a Rosenbrock‐based linearly implicit LSRT2 method, to be strongly coupled with different time steps in each subdomain. Thus, user‐defined algorithmic damping and subcycling strategies are allowed. Secondly, the paper presents the convergence analysis of the novel schemes for linear single‐Degree‐of‐Freedom (DoF) systems. The algorithms are generally A‐stable and preserve the accuracy order as the original monolithic method. Successively, these results are validated via simulations on single‐ and three‐DoFs systems. Finally, the insight gained from previous analyses is confirmed by means of numerical experiments on a coupled spring–pendulum system. Copyright © 2011 John Wiley & Sons, Ltd.