On the Geometric Stabilization for Discrete Hamiltonian Systems With Holonomic Constraints

2009 ◽  
Author(s):  
Hiroaki Yoshimura ◽  
Kenji Soya
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Geometric Constraint ◽  
Differentiation Formula ◽  
Holonomic Constraints ◽  
Constrained Mechanical Systems ◽  
Constraint Stabilization ◽  
Backward Differentiation Formula ◽  
External Forces ◽  
Baumgarte Stabilization

This paper develops a discrete Hamiltonian system with holonomic constraints with Geometric Constraint Stabilization. It is first shown that constrained mechanical systems with nonconservative external forces can be formulated by using canonical symplectic structures in the context of Hamiltonian systems. Second, it is shown that discrete holonomic Hamiltonian systems can be developed via the discretization based on the Backward Differentiation Formula and also that geometric constraint stabilization can be incorporated into the discrete Hamiltonian systems. It is demonstrated that the proposed method enables one to stabilize constraint violations effectively in comparison with conventional methods such as Baumgarte Stabilization and Gear–Gupta–Leimkuhler Stabilization, together with an illustrative example of linkage mechanisms.

Backward Differentiation Formula and Newmark-Type Index-2 and Index-1 Integration Schemes for Constrained Mechanical Systems

2019 ◽  
Vol 15 (2) ◽  
Author(s):  
T. Meyer ◽  
P. Li ◽  
B. Schweizer
Keyword(s):  
Mechanical Systems ◽  
Differential Algebraic Equations ◽  
Differentiation Formula ◽  
Constrained Mechanical Systems ◽  
Time Points ◽  
Backward Differentiation Formula ◽  
Integration Schemes ◽  
Intermediate Time ◽  
Dae Systems ◽  
Index 2

Abstract Various methods for solving systems of differential-algebraic equations (DAE systems) are known from literature. Here, an alternative approach is suggested, which is based on a collocated constraints approach (CCA). The basic idea of the method is to introduce intermediate time points. The approach is rather general and may basically be applied for solving arbitrary DAE systems. Here, the approach is discussed for constrained mechanical systems of index-3. Application of the presented formulations for nonmechanical higher index DAE systems is also possible. We discuss index-2 formulations with one intermediate time point and index-1 implementations with two intermediate time points. The presented technique is principally independent of the time discretization method and may be applied in connection with different time integration schemes. Here, implementations are investigated for backward differentiation formula (BDF) and Newmark-type integrator schemes. A direct application of the presented approach yields a system of discretized equations with larger dimensions. The increased dimension of the discretized system of equations may be considered as the main drawback of the presented technique. The main advantage is that the approach may be used in a very straightforward manner for solving rather arbitrary multiphysical DAE systems with arbitrary index. Hence, the method might, for instance, be attractive for general purpose DAE integrators, since the approach is not tailored for special DAE systems (e.g., constrained mechanical systems). Numerical examples will demonstrate the straightforward application of the approach.

Deterministic Mechanism of Irreversibility

2018 ◽  
Vol 14 (3) ◽  
pp. 5708-5733 ◽  
Author(s):  
Vyacheslav Michailovich Somsikov
Keyword(s):  
Internal Energy ◽  
Hamiltonian Systems ◽  
Classical Mechanics ◽  
Motion Equation ◽  
Holonomic Constraints ◽  
Dissipative Processes ◽  
Motion Energy ◽  
Analytical Review ◽  
History Of ◽  
Material Points

The analytical review of the papers devoted to the deterministic mechanism of irreversibility (DMI) is presented. The history of solving of the irreversibility problem is briefly described. It is shown, how the DMI was found basing on the motion equation for a structured body. The structured body was given by a set of potentially interacting material points. The taking into account of the body’s structure led to the possibility of describing dissipative processes. This possibility caused by the transformation of the body’s motion energy into internal energy. It is shown, that the condition of holonomic constraints, which used for obtaining of the canonical formalisms of classical mechanics, is excluding the DMI in Hamiltonian systems. The concepts of D-entropy and evolutionary non-linearity are discussed. The connection between thermodynamics and the laws of classical mechanics is shown. Extended forms of the Lagrange, Hamilton, Liouville, and Schrödinger equations, which describe dissipative processes, are presented.

scholarly journals A NEW SUPERCLASS OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR STIFF ORDINARY DIFFERENTIAL EQUATIONS

2014 ◽  
Vol 07 (01) ◽  
pp. 1350034 ◽  
Author(s):  
M. B. Suleiman ◽  
H. Musa ◽  
F. Ismail ◽  
N. Senu ◽  
Z. B. Ibrahim
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Results ◽  
New Method ◽  
Differentiation Formula ◽  
Stiff Ordinary Differential Equations ◽  
Backward Differentiation Formula ◽  
Error Constant

A superclass of block backward differentiation formula (BBDF) suitable for solving stiff ordinary differential equations is developed. The method is of order 3, with smaller error constant than the conventional BBDF. It is A-stable and generates two points at each step of the integration. A comparison is made between the new method, the 2-point block backward differentiation formula (2BBDF) and 1-point backward differentiation formula (1BDF). The numerical results show that the method developed outperformed the 2BBDF and 1BDF methods in terms of accuracy. It also reduces the integration steps when compared with the 1BDF method.

DERIVATION OF 2-POINT ZERO STABLENUMERICAL ALGORITHM OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR SOLVING FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS

2021 ◽  
Vol 5 (2) ◽  
pp. 579-583
Author(s):  
Muhammad Abdullahi ◽  
Bashir Sule ◽  
Mustapha Isyaku
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Taylor Series ◽  
Taylor Series Expansion ◽  
Order Ordinary Differential Equation ◽  
Differentiation Formula ◽  
First Order ◽  
Backward Differentiation Formula

This paper is aimed at deriving a 2-point zero stable numerical algorithm of block backward differentiation formula using Taylor series expansion, for solving first order ordinary differential equation. The order and zero stability of the method are investigated and the derived method is found to be zero stable and of order 3. Hence, the method is suitable for solving first order ordinary differential equation. Implementation of the method has been considered

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