Methods For Differential-algebraic Equations Research Articles

Extensions of single-step method for equations of motion from multibody dynamics

• Improved single-step methods are extended to DAEs from multibody dynamics . • Optimal preconditioning strategies are employed in Newton-Raphson iteration. • All the new methods do not suffer from small time-step-size restriction. • Each new method has second-order accuracy and controllable numerical dissipation. Each second-order, unconditionally stable, linear multi-step (LMS) method has its equivalent single-step (SS) method. For the linear two-step method, LMS2, the corresponding SS method was developed as SS2. In this paper, SS2 for first-order ordinary differential equations (ODEs) is designated as SS2-1 and for second-order ODEs is designated as SS2-2. Then SS2-1 and SS2-2 methods are extended to the numerical integration of index 3 differential-algebraic equations (DAEs), index 2 DAEs, and GGL-stabilized index 2 DAEs from multibody dynamics. Optimal preconditioning strategies are employed in the new methods for DAEs. The right and left preconditioners cure the sensitivity to the solution perturbation and the conditioning of the Jacobian matrix for Newton-Raphson iteration. The extended methods are validated by numerical experiments. The effectiveness of the optimal preconditioning strategy and the violation characteristics of constraint equations for the new methods are also verified. The new methods have the advantages of second-order accuracy, and controllable numerical dissipation and they do not suffer from the small time-step-size restriction. Further, there is no acceleration accuracy reduction for equations of motion (EoM) with a non-constant mass matrix. The efficiency of the new methods and the existing generalized-α method for index 3 DAEs are also compared.

Multi-block Generalized Adams-Type Integration Methods for Differential Algebraic Equations

Multi-block Generalized Adams-Type Integration Methods for Differential Algebraic Equations

Asymptotic stability analysis of Runge–Kutta methods for differential-algebraic equations with multiple delays

Asymptotic stability analysis of Runge–Kutta methods for differential-algebraic equations with multiple delays

Linear gradient structures and discrete gradient methods for conservative/dissipative differential-algebraic equations

In this paper, the use of discrete gradients is considered for differential-algebraic equations (DAEs) with a conservation/dissipation law. As one of the most popular numerical methods for conservative/dissipative ordinary differential equations, the framework of the discrete gradient method has been intensively developed over recent decades. Although discrete gradients have been applied to several specific DAEs, no unified framework has yet been constructed. In this paper, the author moves toward the establishment of such a framework, and introduces concepts including an appropriate linear gradient structure for DAEs. Then, it is revealed that the simple use of discrete gradients does not imply the discrete conservation/dissipation laws. Fortunately, however, for the case of index-1 DAEs, an appropriate reformulation and a new discrete gradient enable us to successfully construct a novel scheme, which satisfies both of the discrete conservation/dissipation law and the constraint. This first attempt may provide an indispensable basis for constructing a unified framework of discrete gradient methods for DAEs.

L-Stable Method for Differential-Algebraic Equations of Multibody System Dynamics

An L-stable method over time intervals for differential-algebraic equations (DAEs) of multibody system dynamics is presented in this paper. The solution format is established based on equidistant nodes and nonequidistant nodes such as Chebyshev nodes and Legendre nodes. Based on Ehle’s theorem and conjecture, the unknown matrix and vector in the L-stable solution formula are obtained by comparison with Pade approximation. Newton iteration method is used during the solution process. Taking the planar two-link manipulator system as an example, the results of L-stable method presented are compared for different number of nodes in the time interval and the step size in the simulation, and also compared with the classic Runge-Kutta method, A-stable method, Radau IA, Radau IIA, and Lobatto IIIC methods. The results show that the method has the advantages of good stability and high precision and is suitable for multibody system dynamics simulation under long-term conditions.

Application of the Rosenbrock methods to the solution of unsteady 3D incompressible Navier-Stokes equations

We consider the Rosenbrock methods, namely a family of methods for Differential Algebraic Equations, for the solution of the unsteady three-dimensional Navier-Stokes equations. These multistage schemes are attractive for non-linear problems because they achieve high order in time, ensuring stability properties and linearizing the system to be solved at each timestep. Moreover, as they provide inexpensive ways to estimate the local truncation error, adaptive timestep strategies can be easily devised. In this work we test the Rosenbrock methods for the solution of three-dimensional unsteady incompressible flows. We derive the correct essential boundary conditions to impose at each stage in order to retain the convergence order of the schemes. Then, we consider two benchmark tests: a flow problem with imposed oscillatory pressure gradient whose analytical solution is known and the classical flow past a cylinder. In the latter case, we especially focus on the accuracy in the approximation of the drag and lift coefficients. In both benchmarks we test the performance of a time adaptivity scheme.

A high-order Lie groups scheme for solving the recovery of external force in nonlinear system

In this paper, we develop the high-order Lie group scheme to estimate a real-time external force exerted on nonlinear system. For estimating a real-time data, the equations of motion and the supplemental data are combined into a set of differential algebraic equations (DAEs), which is solved by an implicit Lie group differential algebraic equations method (LGDAE) and a Newton iterative algorithm. However, an explicit LGDAE cannot work, and an implicit scheme cannot avoid a lot of iterative numbers and overcome numerical instability under a long time span and noisy-level effect. Because the original implicit LGDAE cannot satisfy the constraint of the cone structure, Lie group and Lie algebra at each time step, solutions highly dependent on ones cannot easily converge by a large time step. Therefore, we develop a high-order explicit LGDAE to avoid iterative numbers and numerical instability, and at the same time combined with higher order sliding modes to obtain real-time external force. The accuracy and efficiency of the novel scheme is validated by comparing the estimation results with the previous literature.

Delay-dependent stability of linear multistep methods for DAEs with multiple delays

This paper aims to investigate the asymptotic stability of linear multistep (LM) methods for linear differential-algebraic equations (DAEs) with multiple delays. Based on the argument principle, we first establish the delay-dependent stability criteria of analytic solutions; then, we propose some practically checkable conditions for weak delay-dependent stability of numerical solutions derived by implicit LM methods. Lagrange interpolations are used to compute the delayed terms. Several numerical examples are given to illustrate the theoretical results.

The Signature Method for DAEs arising in the modeling of electrical circuits

We consider the Signature Method ( Σ -method) for the structural analysis of differential–algebraic equations (DAEs) that arise in the modeling and simulation of electrical circuits. Different formulations of the set of model equations are considered. We show that for some formulations the structural approach may fail for certain circuit topologies, while other formulations are better suited for a structural analysis. In particular, we show that for the branch-oriented model equations the Signature Method always succeeds with a structural index that corresponds to the differentiation index of the system. The results are illustrated by a number of examples.

Solving Mechanical Systems with Nonholonomic Constraints by a Lie-Group Differential Algebraic Equations Method

AbstractA Lie-group differential algebraic equations (LGDAE) method, which is developed for solving differential-algebraic equations, is a simple and effective algorithm based on the Lie group GL(n...

Structural Analysis Methods for Differential Algebraic Equations via Fixed-Point Iteration

Motivated by Pryce’s structural analysis method for differential algebraic equations (DAEs), we show the complexity of the fixed-point iteration algorithm (FPIA) and propose a fixed-point iteration method with parameters. It leads to a block fixed-point iteration method (BFPIM) which can be applied to immediately calculate the crucial canonical offsets for large-scale (coupled) DAE systems with block-triangular structure, and its complexity analysis is also given in this paper. Moreover, preliminary numerical experiments show that the time complexity of BFPIM can be reduced by at least O(l) compared to the FPIA.

Group Preserving Correction Methods for Differential Algebraic Equations

The group preserving methods proposed by Liu [Int. J. Non-Linear Mech., 2001 and CMES-Comp. Model. Eng., 2006] for ordinary differential equations or differential algebraic equations (DAEs) adopted the Cayley transform or exponential mapping to formulate the Lie group from its Lie algebra. In this paper, we combine the Euler scheme with the group preserving methods to obtain the high accuracy group preserving techniques. We propose a group preserving correction scheme (GPCS) via exponential mapping and a modified group preserving correction scheme (MGPCS) by considering constraint. The two schemes provide single-step explicit time integrators for systems of DAEs. Some numerical examples are examined, showing that the GPCS and MGPCS work very well and have good computational efficiency and high accuracy.

Block Group Preserving Correction Methods for Differential Algebraic Equations with Multiple Constraints

Block Group Preserving Correction Methods for Differential Algebraic Equations with Multiple Constraints

High-Speed Shielding Current Analysis in High-Temperature Superconducting Film With Cracks

Two types of numerical methods are proposed for calculating the shielding current density in a high-temperature superconducting (HTS) film containing cracks: differential algebraic equation (DAE) method and virtual-voltage method. A numerical code is developed on the basis of both methods and, by using the code, the performance of both methods is numerically investigated. The results of computations show that, if GMRES is implemented as a linear-system solver, the virtual-voltage method is much faster than the DAE method. In addition, the applicability of the H-matrix method to speedup of the virtual-voltage method is numerically accessed. Consequently, it is found that the acceleration by the 'i-I-matrix method is effective especially for a large-sized shielding current analysis in a high-aspect-ratio HTS film.

A real-time Lie-group differential algebraic equations method to solve the inverse nonlinear vibration problems

Desiring a real-time recovery of an unknown external force in the nonlinear inverse vibration problem, we transform the nonlinear ordinary differential equation (ODE) of motion into a nonlinear parabolic type partial differential equation (PDE), which can raise the robustness against large noise. Then, we come to a heat source identification problem, of which the numerical method of lines is used to discretize the PDE into a system of differential algebraic equations (DAEs). Consequently, we can develop an implicit Lie-group scheme and a Newton algorithm to stably solve the DAEs for finding the unknown force, damping function, or stiffness function, which is well recovered even under a large noise. Superficially, we seem to transform a simple ODE to a more complex PDE; however, the advantages gained in this transformation will be seen when we test some nonlinear inverse vibration problems with large time span and under large noise.

Regularization of Quasi-Linear Differential-Algebraic Equations

The complete virtual design of dynamical systems, e.g., mechanical systems, electrical circuits, flow problems, or whole production processes, plays a key role in our technological progress. In this context differential-algebraic equations (DAEs) are a very important tool for the analysis and simulation of such dynamical processes. It is well known that the numerical treatment of DAEs is nontrivial in general. The occurrence of hidden constraints (contained in the DAE but only obtainable by differentiating of (parts of) the DAE) impose additional consistency conditions on the initial values and provoke severe difficulties in the direct numerical integration of DAEs as for example drift, instabilities, or convergence problems. Therefore, it is necessary to regularize or remodel the model equations before a robust numerical integration is possible.In this article we present regularization methods for quasi-linear DAEs. In particular, we discuss the projected strangeness-free formulation, the minimally extended formulation, and the overdetermined formulation as regularizations. At the end we give some remarks on the numerical treatment of the presented regularizations.

Recovering A Heat Source and Initial Value by a Lie-Group Differential Algebraic Equations Method

We consider an inverse problem of a nonlinear heat conduction equation for recovering unknown space-dependent heat source and initial condition under Cauchy-type boundary conditions, which is known as a sideways heat equation. With the aid of two extra measurements of temperature and heat flux which are being polluted by noisy disturbances, we can develop a Lie-group differential algebraic equations (LGDAE) method to solve the resulting differential algebraic equations, and to quickly recover the unknown heat source and initial condition simultaneously. Also, we provide a simple LGDAE method, without needing extra measurement of heat flux, to recover the above two unknown functions. The estimated results are quite promising and robust enough against large random noise.

Elastoplastic models and oscillators solved by a Lie-group differential algebraic equations method

In this paper, the viewpoint of non-linear complementarity problem (NCP) is adopted to derive a system of index-one differential algebraic equations (DAEs) for elastoplastic models, by recasting the complementary trio to an algebraic equation through the Fischer–Burmeister NCP-function. Then, we develop a novel algorithm based on the Lie-group GL(n,R) to iteratively solve the resultant DAEs at each time marching step. The Lie-group differential algebraic equations (LGDAE) method is convergent very fast, rendering efficient numerical schemes which can long-term preserve the yield-surface for plasticity models, without resorting on two-phase equations and on-off switching criteria. Several examples, including two non-linear elastoplastic oscillators whose restoring forces are modeled by elastoplastic constitutive equations, are used to assess the performance of the presently developed index-one formulation of elastoplasticity and test the efficiency and accuracy of LGDAE.

Finding unknown heat source in a nonlinear Cauchy problem by the Lie-group differential algebraic equations method

We consider an inverse heat source problem of a nonlinear heat conduction equation, for recovering an unknown space-dependent heat source under the Cauchy type boundary conditions. With the aid of measured initial temperature and initial heat flux, which are disturbanced by random noise causing measurement error, we develop a Lie-group differential algebraic equations (LGDAE) method to solve the resultant differential algebraic equations. The Lie-group numerical method has a stabilizing effect to retain the solution on the associated manifold, which thus naturally has a regularization effect to overcome the ill-posed property of the nonlinear inverse heat source problem. As a consequence, we can quickly recover the unknown heat source under noisy input data only through a few iterations. The initial data used in the recovery of heat source are assumed to be the analytic continuation ones which are not given arbitrarily. Certainly, the measured initial data belong to this type data.

Lie-group differential algebraic equations method to recover heat source in a Cauchy problem with analytic continuation data

The present study recovers a time-dependent heat source H(t) in ut(x,t)=uxx(x,t)+H(t), under measured initial heat flux, and Cauchy boundary conditions. The supplementary initial data are assumed to be analytic continuation ones being obtained by means of measurement, which are not given arbitrarily. We first transform the above problem into an inverse heat conduction problem without having the right-boundary value, and then, further transform it into another inverse heat source problem that we need to find F(t) in Tt(x,t)=Txx(x,t)-xF(t)/ℓ with measured initial and boundary conditions. By using the GL(n,R) Lie-group differential algebraic equations (LGDAE) method to integrate the resultant ordinary differential equations with a priori bound |F(t)|⩽Fmax, we can fast recover H(t) in real time. The accuracy and efficiency are assessed and confirmed by comparing the exact solutions with recovered results, where a large noise up to 10% or 20% is imposed on the input measured data.