Modular Modeling Using Lagrangian DAEs

2000 ◽  
Author(s):  
Robert Piché ◽  
Mikko Palmroth
Keyword(s):  
Software Design ◽  
Initial Conditions ◽  
Differential Algebraic Equations ◽  
Simulation Software ◽  
Algebraic Equations ◽  
Feedback Control System ◽  
Physical Systems ◽  
Modular Modeling ◽  
Index Reduction ◽  
Analytical System

Abstract Layton’s Analytical System Dynamics theory for modeling multidisciplinary physical systems with Lagrangian differential algebraic equations (DAEs) is extended by introducing a technique for using hierarchical reusable modules. Connections between submodels are represented in a systematic manner using kinematic constraints and Lagrange multipliers. Simulation software design issues are discussed: data structures, consistent initial conditions, index reduction, and DAE solvers. An example of an electromechanical feedback control system is presented in detail.

scholarly journals Numerical solution for consistent initial conditions of constrained mechanical systems

2003 ◽  
Vol 25 (3) ◽  
pp. 170-185
Author(s):  
Dinh Van Phong
Keyword(s):  
Numerical Solution ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Differential Algebraic Equations ◽  
System Of Equations ◽  
Algebraic Equations ◽  
Flexible Approach ◽  
Constrained Mechanical Systems ◽  
Consistent Initial Values

The article deals with the problem of consistent initial values of the system of equations of motion which has the form of the system of differential-algebraic equations. Direct treating the equations of mechanical systems with particular properties enables to study the system of DAE in a more flexible approach. Algorithms and examples are shown in order to illustrate the considered technique.

Towards an Energy-Based Linear Analysis of Nonholonomic Systems

1999 ◽  
Author(s):  
Marwan Bikdash ◽  
Richard A. Layton
Keyword(s):  
Steady State ◽  
Singular Systems ◽  
Nonholonomic Constraints ◽  
Nonholonomic Systems ◽  
Linear Analysis ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Indirect Approach ◽  
Physical Systems ◽  
Future Work

Abstract Guidelines toward an energy-based, linear analysis of discrete physical systems are presented, based on previous work in systematic modeling using Lagrangian differential-algebraic equations (DAEs). Recent work in this area is extended by accommodating nonholonomic constraints and explicit inputs. An equilibrium postulate is proposed and equilibrium is characterized for static and steady-state conditions. Lagrangian DAEs are linearized using a local, indirect approach. Alternate descriptor formulations leading to different linear singular systems are compared and one formulation is determined to be a good foundation for future work in linear analysis using Lagrangian DAEs.

Index reduction of differential algebraic equations by differential Dixon resultant

2018 ◽  
Vol 328 ◽  
pp. 189-202 ◽  
Author(s):  
Xiaolin Qin ◽  
Lu Yang ◽  
Yong Feng ◽  
Bernhard Bachmann ◽  
Peter Fritzson
Keyword(s):  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Index Reduction

Projection Method With Minimal Correction Procedure for Numerical Simulation of Constrained Dynamics

2017 ◽  
Author(s):  
Patrick S. Heaney ◽  
Gene Hou
Keyword(s):  
Projection Method ◽  
Numerical Technique ◽  
Correction Method ◽  
Differential Algebraic Equations ◽  
Constrained Systems ◽  
Correction Procedure ◽  
Algebraic Equations ◽  
Differential Algebraic Equation ◽  
Index Reduction ◽  
Algebraic Constraints

This paper describes a numerical technique for simulating the dynamics of constrained systems, which are described generally by differential-algebraic equations. The Projection Method for index reduction of a differential-algebraic equation and a minimal correction procedure are described. This procedure ensures algebraic constraints are satisfied during the numerical integration of the reduced index system of differential equations. Two examples illustrate how the method can be utilized to solve constrained multibody and rotational dynamics problems. The efficiency and accuracy of the proposed index-reduction and minimal correction method are then evaluated.

Simple Recipe for Accurate Solution of Fractional Order Equations

2012 ◽  
Vol 8 (3) ◽  
Author(s):  
Sambit Das ◽  
Anindya Chatterjee
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Initial Conditions ◽  
Special Functions ◽  
Differential Algebraic Equations ◽  
Accurate Solution ◽  
Algebraic Equations ◽  
Matlab Program ◽  
Fractional Differential

Fractional order integrodifferential equations cannot be directly solved like ordinary differential equations. Numerical methods for such equations have additional algorithmic complexities. We present a particularly simple recipe for solving such equations using a Galerkin scheme developed in prior work. In particular, matrices needed for that method have here been precisely evaluated in closed form using special functions, and a small Matlab program is provided for the same. For equations where the highest order of the derivative is fractional, differential algebraic equations arise; however, it is demonstrated that there is a simple regularization scheme that works for these systems, such that accurate solutions can be easily obtained using standard solvers for stiff differential equations. Finally, the role of nonzero initial conditions is discussed in the context of the present approximation method.

Sliding Modes for the Simulation of Mechanical and Electrical Systems Defined by Differential-Algebraic Equations

2010 ◽  
Vol 5 (4) ◽  
Author(s):  
Sachit Rao ◽  
Vadim Utkin ◽  
Martin Buss
Keyword(s):  
Sliding Mode Control ◽  
Sliding Mode ◽  
Initial Conditions ◽  
Differential Algebraic Equations ◽  
Engineering Systems ◽  
Sliding Modes ◽  
Algebraic Equations ◽  
Electric Networks ◽  
Electrical Systems ◽  
Mode Control

We offer a technique, motivated by feedback control and specifically sliding mode control, for the simulation of differential-algebraic equations (DAEs) that describe common engineering systems such as constrained multibody mechanical structures and electric networks. Our algorithm exploits the basic results from sliding mode control theory to establish a simulation environment that then requires only the most primitive of numerical solvers. We circumvent the most important requisite for the conventional simulation of DAEs: the calculation of a set of consistent initial conditions. Our algorithm, which relies on the enforcement and occurrence of sliding mode, will ensure that the algebraic equation is satisfied by the dynamic system even for inconsistent initial conditions and for all time thereafter.

Unique solvability of a system of ordinary differential equations modeling a warm cloud parcel and avoiding saturation adjustment

2020 ◽  
Author(s):  
Nikolas Porz
Keyword(s):  
Climate Models ◽  
Weather Forecast ◽  
Forecast Model ◽  
Differential Algebraic Equations ◽  
Simulation Software ◽  
Integration Scheme ◽  
Recent Effort ◽  
Algebraic Equations ◽  
Macroscopic Theory ◽  
Consistent Solution

<p>The representation of cloud processes in weather and climate models is crucial for their feedback on atmospheric flows. Since there is no general macroscopic theory of clouds, the parameterization of clouds in corresponding simulation software depends fundamentally on the underlying modeling assumptions. We present a new model of intermediate complexity (a one-and-a-half moment scheme) for warm clouds, which is derived from physical principles. Our model consists of a system of differential-algebraic equations which allows for supersaturation and thus avoids the commonly used but somewhat outdated concept of so called 'saturation adjustment'. This is made possible by a non-Lipschitz right-hand side, which allows for nontrivial solutions. In a recent effort we have proved under mild assumptions on the external forcing that this system of equations has a unique physically consistent solution, i.e., a solution with a nonzero droplet population in the supersaturated regime. For the numerical solution of this system we have developed a semi-implicit integration scheme, with efficient solvers for the implicit parts. The model conserves air and water (if one accounts for the precipitation), and it comes with eight parameters that cannot be measured since they describe simplified processes, so they need to be fitted to the data. For further studies we implemented our cloud micro physics model into ICON, the weather forecast model operated by the German forecast center DWD.</p>

scholarly journals A New Block Structural Index Reduction Approach for Large-Scale Differential Algebraic Equations

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2057
Author(s):  
Juan Tang ◽  
Yongsheng Rao
Keyword(s):  
Large Scale ◽  
Differential Algebraic Equations ◽  
Coupled Systems ◽  
Algebraic Equations ◽  
Triangular Form ◽  
Large Scale Systems ◽  
Index Reduction ◽  
Dae Systems ◽  
New Generation ◽  
Block Structures

A new generation of universal tools and languages for modeling and simulation multi-physical domain applications has emerged and became widely accepted; they generate large-scale systems of differential algebraic equations (DAEs) automatically. Motivated by the characteristics of DAE systems with large dimensions, high index or block structures, we first propose a modified Pantelides’ algorithm (MPA) for any high order DAEs based on the Σ matrix, which is similar to Pryce’s Σ method. By introducing a vital parameter vector, a modified Pantelides’ algorithm with parameters has been presented. It leads to a block Pantelides’ algorithm (BPA) naturally which can immediately compute the crucial canonical offsets for whole (coupled) systems with block-triangular form. We illustrate these algorithms by some examples, and preliminary numerical experiments show that the time complexity of BPA can be reduced by at least O(ℓ) compared to the MPA, which is mainly consistent with the results of our analysis.

scholarly journals Index Reduction for Differential-algebraic Equations with Mixed Matrices

2019 ◽  
Vol 66 (5) ◽  
pp. 1-34 ◽  
Author(s):  
Satoru Iwata ◽  
Taihei Oki ◽  
Mizuyo Takamatsu
Keyword(s):  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Index Reduction
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