Group Preserving Correction Methods for Differential Algebraic Equations

2016 ◽  
Vol 13 (10) ◽  
pp. 7719-7725
Author(s):  
Jianguang Lu ◽  
Yong Feng ◽  
Xiaolin Qin ◽  
Juan Tang
Keyword(s):  
High Accuracy ◽  
Differential Algebraic Equations ◽  
Single Step ◽  
Euler Scheme ◽  
Cayley Transform ◽  
Exponential Mapping ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Time Integrators ◽  
Correction Scheme

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.

scholarly journals Linear differential algebraic equations with constant coefficients

1970 ◽  
Vol 4 (2) ◽  
pp. 28-35 ◽  
Author(s):  
M Sahadet Hossain ◽  
M Mostafizur Rahman
Keyword(s):  
Existence And Uniqueness ◽  
Differential Algebraic Equations ◽  
Canonical Forms ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Constant Coefficients ◽  
Dae Systems ◽  
International University ◽  
Linear Differential ◽  
Modern Mathematics

Differential-algebraic equations (DAEs) arise in a variety of applications. Their analysis and numerical treatment, therefore, plays an important role in modern mathematics. The paper gives an introduction to the topics of DAEs. Examples of DAEs are considered showing their importance for practical problems. Some essential concepts that are really essential for understanding the DAE systems are introduced. The canonical forms of DAEs are discussed widely to make them more efficient and easy for practical use. Also some numerical examples are discussed to clarify the existence and uniqueness of the system's solutions. Keywords: differential-algebraic equations, index concept, canonical forms. DOI: 10.3329/diujst.v4i2.4365 Daffodil International University Journal of Science and Technology Vol.4(2) 2009 pp.28-35

scholarly journals Solving Nonlinear Differential Algebraic Equations by an Implicit Lie-Group Method

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Chein-Shan Liu
Keyword(s):  
Closed Form ◽  
Lie Group ◽  
Iterative Scheme ◽  
Differential Algebraic Equations ◽  
Group Method ◽  
Computational Results ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Closed Form Solutions ◽  
Lie Group Method

We derive an implicit Lie-group algorithm together with the Newton iterative scheme to solve nonlinear differential algebraic equations. Four numerical examples are given to evaluate the efficiency and accuracy of the new method when comparing the computational results with the closed-form solutions.

Component Based Modeling of Electromechanical Systems

2005 ◽  
Author(s):  
Shilpa A. Vaze ◽  
Prakash Krishnaswami ◽  
James DeVault
Keyword(s):  
Design Sensitivity ◽  
Differential Algebraic Equations ◽  
Single Step ◽  
System Response ◽  
Algebraic Equations ◽  
System Behavior ◽  
Governing Equations ◽  
Electromechanical Systems ◽  
System Equations ◽  
Saturation Effects

Simulation methods for electromechanical systems should accommodate their interdisciplinary nature and the fact that these systems often display qualitative changes in system behavior during operation, such as saturation effects and changes in kinematic structure. Current approaches are either based on deriving the system equations by applying a single formulation to all problem domains, or they are based on trying to integrate different software packages/modules to solve the interdisciplinary problem. In this paper, we present a component-based approach which allows the governing equations of each component to be defined in terms of its natural variables. The different component equations are then brought together to form a single system of differential-algebraic equations (DAE’s), which can be numerically solved to obtain the system response. The fact that we have an explicit, unified form of the system governing equations means that this formulation can be easily extended to design sensitivity analysis and optimization of electromechanical systems (EMS). The formulation includes monitor functions which can be used to detect when a qualitative system change has occurred, and to switch to a new set of governing equations to reflect this change. A single step integrator is used to make it easier to switch to a new system behavior, since this will always require a restart of the integrator. There is considerable flexibility in how the components can be defined, and connections between components are themselves modeled as special types of components. Examples of components from the mechanical and electrical side are presented, and two numerical examples are solved to illustrate the efficacy of the proposed method. One example is a link that is driven by a DC motor through a gearbox. The results of this example were verified against Simulink, and good agreement was observed. The second example is a motor driven slider-crank mechanism. The method can be extended to include components from any domain, such as hydraulics, thermal, controls, etc., as long as the governing equations can be written as DAE’s.

Solution Numerical of Stiff ODEs and DAEs in Mechanical and Other Systems

1997 ◽  
Author(s):  
H. Pasic
Keyword(s):  
Formal Solution ◽  
Differential Algebraic Equations ◽  
Single Step ◽  
Algebraic Equations ◽  
Stiff Differential Equations ◽  
Index Zero ◽  
Initial Value ◽  
Polynomial Coefficients ◽  
Multi Stage ◽  
Index 2

Abstract Presented is a formal solution of the initial-value problem of the system of general implicit differential-algebraic equations (DAEs) F(x, y, y’) = 0 of index zero or higher, based on perturbations of the polynomial coefficients of the vector y(x). The equation is linearized with respect to the coefficients and brought into a form suitable for implementation of the weighted residual methods. The solution is advanced by a single-step multi-stage collocation qadrature formula which is stiffly accurate and suitable for solving stiff differential equations and DAEs that arise in many mechanical and other systems. The algorithm is illustrated by two index-2 and index-3 examples — one of which is the well known pendulum problem.

Convergence of Linear Multistep Methods and One-Leg Methods for Index-2 Differential-Algebraic Equations with a Variable Delay

2012 ◽  
Vol 4 (5) ◽  
pp. 636-646 ◽  
Author(s):  
Hongliang Liu ◽  
Aiguo Xiao
Keyword(s):  
Multistep Methods ◽  
Differential Algebraic Equations ◽  
Linear Multistep Methods ◽  
Variable Delay ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Convergence Results ◽  
Index 2

AbstractLinear multistep methods and one-leg methods are applied to a class of index-2 nonlinear differential-algebraic equations with a variable delay. The corresponding convergence results are obtained and successfully confirmed by some numerical examples. The results obtained in this work extend the corresponding ones in literature.

A PARALLEL DECOUPLING TECHNIQUE TO ACCELERATE CONVERGENCE OF RELAXATION SOLUTION OF INTEGRAL-DIFFERENTIAL-ALGEBRAIC EQUATIONS

2001 ◽  
Vol 02 (03) ◽  
pp. 295-304 ◽  
Author(s):  
ZU-LAN HUANG ◽  
RICHARD M. M. CHEN ◽  
YAO-LIN JIANG
Keyword(s):  
Parallel Processing ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Novel Technique ◽  
Relaxation Solution ◽  
Speed Up ◽  
Accelerate Convergence ◽  
Linear Integral

In this paper, we first study the covergence performance of relaxatio-based algorithms for linear integral differential-algebraic equations (IDAEs), then a parallel decoupling technique to speed up the convergence of the relaxation-based algorithms is derived. This novel technique is suitable for implementation of parallel processing for complicated systems of IDAEs. Factors taking effect on the performance of parallel processing are discussed in detail. Large numerical examples running on a network of IBM RS/6000 SP2 system are given to illustrate how judicious partitionings of matrices can help improve convergence in parallel processing.

scholarly journals Discrete Waveform Relaxation Method for Linear Fractional Delay Differential-Algebraic Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Hongliang Liu ◽  
Yayun Fu ◽  
Bailing Li
Keyword(s):  
Time Lag ◽  
Relaxation Method ◽  
Differential Algebraic Equations ◽  
Waveform Relaxation ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Convergence Results ◽  
Waveform Relaxation Method ◽  
Fractional Delay ◽  
Delay Differential

Fractional order delay differential-algebraic equations have the characteristics of time lag and memory and constraint limit. These yield some difficulties in the theoretical analysis and numerical computation. In this paper, we are devoted to solving them by the waveform relaxation method. The corresponding convergence results are obtained, and some numerical examples show the efficiency of the method.

Ultraspherical Differentiation Method for Solving System of Initial Value Differential Algebraic Equations

2009 ◽  
Vol 9 (3) ◽  
pp. 226-237 ◽  
Author(s):  
M. El-kady ◽  
M.A. Ibrahim
Keyword(s):  
Nonlinear Programming ◽  
Differential Equations ◽  
Spectral Method ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Initial Value ◽  
Numerical Examples ◽  
Ultraspherical Polynomials ◽  
Differentiation Method ◽  
Wide Range

AbstractIn this paper, we introduce a new spectral method based on ultraspherical polynomials for solving systems of initial value differential algebraic equations. Moreover, the suggested method is applicable for a wide range of differential equations. The method is based on a new investigation of the ultraspherical spectral differentiation matrix to approximate the differential expressions in equations. The produced equations lead to algebraic systems and are converted to nonlinear programming. Numerical examples illustrate the robustness, accuracy, and efficiency of the proposed method.

Energy Consistent Time Integration of Non-Conservative Hybrid Multibody Systems

2009 ◽  
Author(s):  
Stefan Uhlar ◽  
Peter Betsch
Keyword(s):  
Multibody Systems ◽  
Time Integration ◽  
Nonholonomic Constraints ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Hybrid Energy ◽  
Joint Friction ◽  
Time Stepping ◽  
Time Integrators ◽  
Coupling Constraints

The contribution at hand deals with the energy-consistent time integration of hybrid multibody systems. The coupling of both rigid and flexible components is facilitated by the introduction of so called coupling constraints, leading to a set of differential algebraic equations governing the motion of the hybrid system. For the modeling of rigid components we rely on the so called rotationless formulation which makes possible the design of mechanical time integrators. In this connection modeling techniques such as the coordinate augmentation, nonholonomic constraints, control issues and modeling of joint friction will be addressed. This leads to a unified approach for the modeling of rigid and flexible bodies, rendering a hybrid-energy-momentum-consistent time stepping scheme. The performance will be demonstrated with the example of a spatial nonholonomic manipulator.

scholarly journals A Direct Eigenanalysis of Multibody System in Equilibrium

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Cheng Yang ◽  
Dazhi Cao ◽  
Zhihua Zhao ◽  
Zhengru Zhang ◽  
Gexue Ren
Keyword(s):  
Eigenvalue Problem ◽  
Condition Number ◽  
Multibody System ◽  
Differential Algebraic Equations ◽  
Direct Application ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Generalized Coordinates ◽  
Lagrange’S Equation ◽  
Matrix Conditioning

This paper presents a direct eigenanalysis procedure for multibody system in equilibrium. The first kind Lagrange’s equation of the dynamics of multibody system is a set of differential algebraic equations, and the equations can be used to solve the equilibrium of the system. The vibration of the system about the equilibrium can be described by the linearization of the governing equation with the generalized coordinates and the multipliers as the perturbed variables. But the multiplier variables and the generalize coordinates are not in the same dimension. As a result, the system matrices in the perturbed vibration equations are badly conditioned, and a direct application of the mature eigensolvers does not guarantee a correct solution to the corresponding eigenvalue problem. This paper discusses the condition number of the problem and proposes a method for preconditioning the system matrices, then the corresponding eigenvalue problem of the multibody system about equilibrium can be smoothly solved with standard eigensolver such as ARPACK. In addition, a necessary frequency shift technology is also presented in the paper. The importance of matrix conditioning and the effectiveness of the presented method for preconditioning are demonstrated with numerical examples.

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