Stabilized Implicit Cosimulation Method: Solver Coupling With Algebraic Constraints for Multibody Systems

2015 ◽  
Vol 11 (2) ◽  
Author(s):  
Bernhard Schweizer ◽  
Pu Li ◽  
Daixing Lu ◽  
Tobias Meyer
Keyword(s):  
Numerical Stability ◽  
Time Integration ◽  
Equation System ◽  
Stability And Convergence ◽  
Linear Recurrence ◽  
Test Model ◽  
Integration Schemes ◽  
Coupling Equations ◽  
Coupling Variables ◽  
The Stability

In this manuscript, an implicit cosimulation method is analyzed, where the solvers are coupled by algebraic constraint equations. We discuss cosimulation approaches on index-2 and on index-1 level and investigate constant, linear and quadratic approximation functions for the coupling variables. The key idea of the method presented here is to discretize the Lagrange multipliers between the macrotime points (extended multiplier approach) so that the coupling equations and their time derivatives can simultaneously be fulfilled at the macrotime points. Stability and convergence of the method are investigated in detail. Following the stability analysis for time integration schemes based on Dahlquist's test equation, an appropriate cosimulation test model is used to examine the numerical stability of the presented cosimulation method. Discretizing the cosimulation test model by means of a linear cosimulation approach yields a system of linear recurrence equations. The spectral radius of the recurrence equation system characterizes the numerical stability of the underlying cosimulation method. As for time integration methods, 2D stability plots are used to graphically illustrate the stability behavior of the coupling approach.

Explicit and Implicit Cosimulation Methods: Stability and Convergence Analysis for Different Solver Coupling Approaches

2015 ◽  
Vol 10 (5) ◽  
Author(s):  
Bernhard Schweizer ◽  
Pu Li ◽  
Daixing Lu
Keyword(s):  
Numerical Stability ◽  
Time Integration ◽  
Stability And Convergence ◽  
Test Model ◽  
Recurrence Equations ◽  
Integration Schemes ◽  
Solver Coupling ◽  
Applied Forces ◽  
The Stability ◽  
Stability Definition

The numerical stability and the convergence behavior of cosimulation methods are analyzed in this manuscript. We investigate explicit and implicit coupling schemes with different approximation orders and discuss three decomposition techniques, namely, force/force-, force/displacement-, and displacement/displacement-decomposition. Here, we only consider cosimulation methods where the coupling is realized by applied forces/torques, i.e., the case that the coupling between the subsystems is described by constitutive laws. Solver coupling with algebraic constraint equations is not investigated. For the stability analysis, a test model has to be defined. Following the stability definition for numerical time integration schemes (Dahlquist's stability theory), a linear test model is used. The cosimulation test model applied here is a two-mass oscillator, which may be interpreted as two Dahlquist equations coupled by a linear spring/damper system. Discretizing the test model with a cosimulation method, recurrence equations can be derived, which describe the time discrete cosimulation solution. The stability of the recurrence equations system represents the numerical stability of the cosimulation approach and can easily be determined by an eigenvalue analysis.

Dynamic Response of Shear-Deformable Axisymmetric Orthotropic Circular Plate Structures Solved by the DQEM and EDQ Based Time Integration Schemes

2003 ◽  
Author(s):  
Chang-New Chen
Keyword(s):  
Dynamic Response ◽  
Dynamic Equilibrium ◽  
Time Integration ◽  
Equation System ◽  
Plate Structures ◽  
Integration Schemes ◽  
Time Element ◽  
Time Integration Schemes ◽  
Orthotropic Circular Plate ◽  
Shear Deformable

The dynamic response of shear-deformable axisymmetric orthotropic circular plate structures is solved by using the DQEM to the spacial discretization and EDQ to the temporal discretization. In the DQEM discretization, DQ is used to define the discrete element model. Discrete dynamic equilibrium equations defined at interior nodes in all elements, transition conditions defined on the inter-element boundary of two adjacent elements and boundary conditions at the structural boundary form a dynamic equation system at a specified time stage. The dynamic equilibrium equation system is solved by the direct time integration schemes of time-element by time-element method and stages by stages method which are developed by using EDQ and DQ. Numerical results obtained by the developed numerical algorithms are presented. They demonstrate the developed numerical solution procedure.

Comparison of Implicit Time Integration Schemes for Nonlinear Dynamic Problems

2010 ◽  
Author(s):  
Murat Demiral
Keyword(s):  
Dynamic Equilibrium ◽  
Time Integration ◽  
Nonlinear Problems ◽  
Implicit Time ◽  
Implicit Methods ◽  
Equilibrium Equations ◽  
Integration Schemes ◽  
Implicit Time Integration ◽  
Time Integration Schemes ◽  
The Stability

Implicit time integration schemes are used to obtain stable and accurate transient solutions of nonlinear problems. Methods that are unconditionally stable in linear analysis are sometimes observed to have convergence problems as in the case of solutions obtained with a trapezoidal method. On the other hand, a composite time integration method employing a trapezoidal rule and a three-point backward rule sequentially in two half steps can be used to obtain accurate results and enhance the stability of the system by means of a numerical damping introduced in the formulation. To have a better understanding of the differences in the numerical implementation of the algorithms of these two methods, a mathematical analysis of dynamic equilibrium equations is performed. Several practical problems are studied to compare the implicit methods.

scholarly journals Numerical Stability of Partitioned Approach in Fluid-Structure Interaction for a Deformable Thin-Walled Vessel

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Kelvin K. L. Wong ◽  
Pongpat Thavornpattanapong ◽  
Sherman C. P. Cheung ◽  
Jiyuan Tu
Keyword(s):  
Fluid Structure Interaction ◽  
Time Integration ◽  
Computational Time ◽  
Fluid Structure ◽  
Structure Interaction ◽  
Integration Schemes ◽  
Partitioned Approach ◽  
Time Integration Schemes ◽  
Implicit Approach ◽  
The Stability

Added-mass instability is known to be an important issue in the partitioned approach for fluid-structure interaction (FSI) solvers. Despite the implementation of the implicit approach, convergence of solution can be difficult to achieve. Relaxation may be applied to improve this implicitness of the partitioned algorithm, but this commonly leads to a significant increase in computational time. This is because the critical relaxation factor that allows stability of the coupling tends to be impractically small. In this study, a mathematical analysis for optimizing numerical performance based on different time integration schemes that pertain to both the fluid and solid accelerations is presented. The aim is to determine the most efficient configuration for the FSI architecture. Both theoretical and numerical results suggest that the choice of time integration schemes has a significant influence on the stability of FSI coupling. This concludes that, in addition to material and its geometric properties, the choice of time integration schemes is important in determining the stability of the numerical computation. A proper selection of the associated parameters can improve performance considerably by influencing the condition of coupling stability.

A Mixed and Multi-Step Higher-Order Implicit Time Integration Family

2010 ◽  
Vol 224 (10) ◽  
pp. 2097-2108 ◽  
Author(s):  
M Rezaiee-Pajand ◽  
S R Sarafrazi
Keyword(s):  
Time Integration ◽  
Higher Order ◽  
Trapezoidal Rule ◽  
Implicit Time ◽  
Benchmark Problems ◽  
Integration Schemes ◽  
Implicit Time Integration ◽  
Time Integration Schemes ◽  
The Stability ◽  
Stability And Accuracy

This article develops a new time integration family for second-order dynamic equations. A combination of the trapezoidal rule and higher-order Newton backward extrapolation functions are utilized in the formulation. Five members of the suggested family are extensively studied in this article. Most members of the presented time integration family are new. The stability and accuracy of the proposed time integration schemes are investigated by solving some benchmark problems. Numerical results are checked and compared with well-known strategies. The findings of the article show the efficiency, accuracy and robustness of the suggested techniques.

Stability of Sequential Modular Time Integration Methods for Coupled Multibody System Models

2010 ◽  
Vol 5 (3) ◽  
Author(s):  
Martin Arnold
Keyword(s):  
Differential Equations ◽  
Numerical Stability ◽  
Multibody System ◽  
Time Integration ◽  
Coupled Systems ◽  
Stability And Convergence ◽  
Integration Methods ◽  
System Models ◽  
Time Integration Methods ◽  
Numerical Stability And Convergence

The interacting components of complex technical systems are often described by coupled systems of differential equations. In dynamical simulation, these coupled differential equations have to be solved numerically. Cosimulation techniques, multirate methods, and other approaches that exploit the modular structure of coupled systems are frequently used as alternatives to classical time integration methods. The numerical stability and convergence of such modular time integration methods is studied for a class of sequential modular methods for coupled multibody system models. Theoretical investigations and numerical test results show that the stability of these sequential modular methods may be characterized by a contractivity condition. A linearly implicit stabilization of coupling terms is proposed to guarantee numerical stability and convergence.

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