scholarly journals Survey of appropriate matching algorithms for large scale systems of differential algebraic equations

2012 ◽  
Author(s):  
Jens Frenkel ◽  
Günter Kunze ◽  
Peter Fritzson
Keyword(s):  
Large Scale ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Large Scale Systems

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 A New Block Structural Index Reduction Approach for Large-scale Differential Algebraic Equations

2020 ◽  
Author(s):  
Juan Tang ◽  
Yongsheng Rao
Keyword(s):  
Large Scale ◽  
Differential Algebraic Equations ◽  
Coupled Systems ◽  
Algebraic Equations ◽  
Structural Index ◽  
Triangular Form ◽  
Large Scale Systems ◽  
Index Reduction ◽  
New Generation ◽  
Block Structures

A new generation of universal tools and languages for modeling and simulation multi-physical domain applications emerged and became widely accepted, which generate large-scale systems of differential algebraic equations (DAEs) automatically. Motivated by the characteristics of DAEs systems with large dimension, high index or block structures, we first propose a modified Pantelides’ algorithm (MPA) for any high order DAEs based on its Σ matrix, which is similar to Pryce’s Σ method. By introducing a vital parameter vector, a modified Pantelides’ algorithm with parameter 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 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 Approximation of potential-driven flow dynamics in large-scale self-similar tree networks

2011 ◽  
Vol 467 (2134) ◽  
pp. 2810-2824 ◽  
Author(s):  
Jason Mayes ◽  
Mihir Sen
Keyword(s):  
Mathematical Model ◽  
Analytical Solutions ◽  
Large Scale ◽  
Differential Algebraic Equations ◽  
Flow Dynamics ◽  
Algebraic Equations ◽  
Tree Networks ◽  
Self Similar ◽  
Large Scale Flow ◽  
The Mathematical Model

Dynamic analysis of large-scale flow networks is made difficult by the large system of differential-algebraic equations resulting from its modelling. To simplify analysis, the mathematical model must be sufficiently reduced in complexity. For self-similar tree networks, this reduction can be made using the network’s structure in way that can allow simple, analytical solutions. For very large, but finite, networks, analytical solutions are more difficult to obtain. In the infinite limit, however, analysis is sometimes greatly simplified. It is shown that approximating large finite networks as infinite not only simplifies the analysis, but also provides an excellent approximate solution.

scholarly journals Automated Linearization of Nonlinear Coupled Differential and Algebraic Equations

1994 ◽  
Vol 116 (2) ◽  
pp. 429-436 ◽  
Author(s):  
J. D. Trom ◽  
M. J. Vanderploeg
Keyword(s):  
Finite Difference ◽  
Dynamic Systems ◽  
Large Scale ◽  
Algebraic Equations ◽  
New Approach ◽  
Large Scale Systems ◽  
Equation Formulation ◽  
Relative Coordinate ◽  
Equilibrium Configurations ◽  
Multibody Dynamic

This paper presents a new approach for linearization of large multibody dynamic systems. The approach uses an analytical differentiation of terms evaluated in a numerical equation formulation. This technique is more efficient than finite difference and eliminates the need to determine finite difference pertubation values. Because the method is based on a relative coordinate formalism, linearizations can be obtained for equilibrium configurations with non-zero Cartesian accelerations. Examples illustrate the accuracy and efficiency of the algorithm, and its ability to compute linearizations for large-scale systems that were previously impossible.

scholarly journals Automated Linearization of Nonlinear Coupled Differential and Algebraic Equations

1991 ◽  
Author(s):  
Martin J. Vanderploeg ◽  
Jeff D. Trom
Keyword(s):  
Finite Difference ◽  
Dynamic Systems ◽  
Large Scale ◽  
Algebraic Equations ◽  
New Approach ◽  
Large Scale Systems ◽  
Equation Formulation ◽  
Equilibrium Configurations ◽  
Multi Body ◽  
Body Dynamic

Abstract This paper presents a new approach for linearization of large multi-body dynamic systems. The approach uses an analytical differentiation of terms evaluated in a numerical equation formulation. This technique is more efficient than finite difference and eliminates the need to determine finite difference pertubation values. Because the method is based on a relative coordinate formalism, linearizations can be obtained for equilibrium configurations with non-zero Cartesian accelerations. Examples illustrate the accuracy and efficiency of the algorithm, and its ability to compute linearizations for large-scale systems that were previously impossible.

scholarly journals A graph-theoretic approach to sparse matrix inversion for implicit differential algebraic equations

2013 ◽  
Vol 4 (1) ◽  
pp. 243-250
Author(s):  
H. Yoshimura
Keyword(s):  
Bipartite Graph ◽  
Large Scale ◽  
Jacobian Matrix ◽  
Sparse Matrix ◽  
Mechanical Systems ◽  
Differential Algebraic Equations ◽  
Matrix Inversion ◽  
Theoretic Approach ◽  
Algebraic Equations ◽  
Input Output

Abstract. In this paper, we propose an efficient numerical scheme to compute sparse matrix inversions for Implicit Differential Algebraic Equations of large-scale nonlinear mechanical systems. We first formulate mechanical systems with constraints by Dirac structures and associated Lagrangian systems. Second, we show how to allocate input-output relations to the variables in kinematical and dynamical relations appearing in DAEs by introducing an oriented bipartite graph. Then, we also show that the matrix inversion of Jacobian matrix associated to the kinematical and dynamical relations can be carried out by using the input-output relations and we explain solvability of the sparse Jacobian matrix inversion by using the bipartite graph. Finally, we propose an efficient symbolic generation algorithm to compute the sparse matrix inversion of the Jacobian matrix, and we demonstrate the validity in numerical efficiency by an example of the stanford manipulator.

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

2016 ◽  
Vol 13 (10) ◽  
pp. 7705-7711 ◽  
Author(s):  
Juan Tang ◽  
Wenyuan Wu ◽  
Xiaolin Qin ◽  
Yong Feng
Keyword(s):  
Fixed Point ◽  
Structural Analysis ◽  
Iteration Method ◽  
Large Scale ◽  
Complexity Analysis ◽  
Differential Algebraic Equations ◽  
Iteration Algorithm ◽  
Algebraic Equations ◽  
Fixed Point Iteration ◽  
Dae Systems

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.

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