An Efficient Computing Explicit Method for Structural Dynamics

2009 ◽  
Vol 131 (2) ◽  
Author(s):  
Shuenn-Yih Chang
Keyword(s):  
Structural Dynamics ◽  
Unconditional Stability ◽  
Explicit Methods ◽  
Explicit Method ◽  
Implicit Methods ◽  
Time Step ◽  
Integration Algorithm ◽  
Structural Dynamic ◽  
Linear Elastic ◽  
Linearized Stability

An integration algorithm, which integrates the most important advantage of explicit methods of the explicitness of each time step and that of implicit methods of the possibility of unconditional stability, is presented herein. This algorithm is analytically shown to be unconditionally stable for any linear elastic and nonlinear systems except for the instantaneous stiffness hardening systems with the instantaneous degree of nonlinearity larger than 43 based on a linearized stability analysis. Hence, its stability property is better than the previously published algorithm (Chang, 2007, “Improved Explicit Method for Structural Dynamics,” J. Eng. Mech., 133(7), pp. 748–760), which is only conditionally stable for instantaneous stiffness hardening systems although it also possesses unconditional stability for linear elastic and any instantaneous stiffness softening systems. Due to the explicitness of each time step, the possibility of unconditional stability, and comparable accuracy, the proposed algorithm is very promising for a general structural dynamic problem, where only the low frequency responses are of interest since it consumes much less computational efforts when compared with explicit methods, such as the Newmark explicit method, and implicit methods, such as the constant average acceleration method.

Comparisons of Structure-Dependent Explicit Methods for Time Integration

2015 ◽  
Vol 15 (03) ◽  
pp. 1450055 ◽  
Author(s):  
Shuenn-Yih Chang
Keyword(s):  
Time Integration ◽  
Low Frequency ◽  
Unconditional Stability ◽  
Explicit Methods ◽  
Explicit Method ◽  
Computationally Efficient ◽  
Total Response ◽  
Time Step ◽  
Order Accuracy ◽  
Practical Applications

Chang explicit method (CEM)1,2 and CR explicit method3 (CRM) are two structure-dependent explicit methods that have been successfully developed for structural dynamics. The most important property of both integration methods is that they involve no nonlinear iterations in addition to unconditional stability and second-order accuracy. Thus, they are very computationally efficient for solving inertial problems, where the total response is dominated by low frequency modes. However, an unusual overshooting behavior for CR explicit method is identified herein and thus its practical applications might be largely limited although its velocity computing for each time step is much easier than for the CEM.

An Improved Structure-Dependent Explicit Method for Structural Dynamics

2008 ◽  
Author(s):  
Shuenn-Yih Chang ◽  
Chiu-Li Huang
Keyword(s):  
Low Frequency ◽  
Conditional Stability ◽  
Explicit Method ◽  
Time Step ◽  
Structural Dynamic ◽  
Linear Elastic ◽  
Acceleration Method ◽  
The Stability ◽  
Stiffness Softening ◽  
Hardening System

An explicit method is presented herein whose coefficients are determined from the initial structural properties of the analyzed system. Thus, it is structure-dependent. This method has a great stability property when compared to the previously published method [6], which is unconditionally stable for linear elastic and any instantaneous stiffness softening systems while it only has conditional stability for an instantaneous stiffness hardening system. The most important improvement of this method is that it has unconditional stability for general instantaneous stiffness hardening systems in addition to linear elastic and instantaneous stiffness softening systems. This implies that a time step may be selected base on accuracy consideration only and the stability problem might be neglected. Hence, many computational efforts can be saved in the step-by-step solution of a general structural dynamic problem, where the response is dominated by the low frequency modes and the high frequency responses are of no great interest, when compared to an explicit method, such as the Newmark explicit method, and an implicit method, such as the constant average acceleration method.

scholarly journals One-Parameter Controlled Non-Dissipative Unconditionally Stable Explicit Structure-Dependent Integration Methods with No Overshoot

2021 ◽  
Vol 11 (24) ◽  
pp. 12109
Author(s):  
Veerarajan Selvakumar ◽  
Shuenn-Yih Chang
Keyword(s):  
Time Integration ◽  
Unconditional Stability ◽  
Second Order ◽  
Numerical Dissipation ◽  
Implicit Methods ◽  
Time Step ◽  
Order Accuracy ◽  
Structural Dynamic ◽  
Integration Methods ◽  
New Family

Although many families of integration methods have been successfully developed with desired numerical properties, such as second order accuracy, unconditional stability and numerical dissipation, they are generally implicit methods. Thus, an iterative procedure is often involved for each time step in conducting time integration. Many computational efforts will be consumed by implicit methods when compared to explicit methods. In general, the structure-dependent integration methods (SDIMs) are very computationally efficient for solving a general structural dynamic problem. A new family of SDIM is proposed. It exhibits the desired numerical properties of second order accuracy, unconditional stability, explicit formulation and no overshoot. The numerical properties are controlled by a single free parameter. The proposed family method generally has no adverse disadvantage of unusual overshoot in high frequency transient responses that have been found in the currently available implicit integration methods, such as the WBZ-α method, HHT-α method and generalized-α method. Although this family method has unconditional stability for the linear elastic and stiffness softening systems, it becomes conditionally stable for stiffness hardening systems. This can be controlled by a stability amplification factor and its unconditional stability is successfully extended to stiffness hardening systems. The computational efficiency of the proposed method proves that engineers can do the accurate nonlinear analysis very quickly.

Numerical Investigation on the Vibration of Steam Turbine Inlet Valves and the Feedback to the Dynamic Flow Field

2015 ◽  
Author(s):  
Clemens Bernhard Domnick ◽  
Friedrich-Karl Benra ◽  
Dieter Brillert ◽  
Hans Josef Dohmen ◽  
Christian Musch
Keyword(s):  
Steam Turbine ◽  
Structural Dynamics ◽  
Fluid Dynamic ◽  
Dynamic Flow ◽  
Piston Rings ◽  
Time Step ◽  
Structural Dynamic ◽  
Turbine Inlet ◽  
Valve Stem ◽  
Highly Nonlinear

The power output of steam turbines is controlled by steam turbine inlet valves. These valves have a large flow capacity and dissipate in throttled operation a huge amount of energy. Due to that, high dynamic forces occur in the valve which can cause undesired valve vibrations. In this paper, the structural dynamics of a valve are analysed. The dynamic steam forces obtained by previous computational fluid dynamic (CFD) calculations at different operating points are impressed on the structural dynamic finite element model (FEM) of the valve. Due to frictional forces at the piston rings and contact effects at the bushings of the valve plug and the valve stem the structural dynamic FEM is highly nonlinear and has to be solved in the time domain. Prior to the actual investigation grid and time step studies are carried out. Also the effect of the temperature distribution within the valve stem is discussed and the influence of the valve actuator on the vibrations is analysed. In the first step, the vibrations generated by the fluid forces are investigated. The effects of the piston rings on the structural dynamics are discussed. It is found, that the piston rings are able to reduce the vibration significantly by frictional damping. In the second step, the effect of the moving valve plug on the dynamic flow in the valve is analysed. The time dependent displacement of the valve is transferred to CFD calculations using deformable meshes. With this one way coupling method the response of the flow to the vibrations is analysed.

scholarly journals Contrast-Independent, Partially-Explicit Time Discretizations for Nonlinear Multiscale Problems

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3000
Author(s):  
Eric T. Chung ◽  
Yalchin Efendiev ◽  
Wing Tat Leung ◽  
Wenyuan Li
Keyword(s):  
Linear Equations ◽  
Nonlinear Problems ◽  
Time Discretization ◽  
Numerical Results ◽  
Explicit Methods ◽  
Implicit Methods ◽  
Time Step ◽  
Multiscale Problems ◽  
Time Stepping ◽  
Space Decomposition

This work continues a line of work on developing partially explicit methods for multiscale problems. In our previous works, we considered linear multiscale problems where the spatial heterogeneities are at the subgrid level and are not resolved. In these works, we have introduced contrast-independent, partially explicit time discretizations for linear equations. The contrast-independent, partially explicit time discretization divides the spatial space into two components: contrast dependent (fast) and contrast independent (slow) spaces defined via multiscale space decomposition. Following this decomposition, temporal splitting was proposed, which treats fast components implicitly and slow components explicitly. The space decomposition and temporal splitting are chosen such that they guarantees stability, and we formulated a condition for the time stepping. This condition was formulated as a condition on slow spaces. In this paper, we extend this approach to nonlinear problems. We propose a splitting approach and derive a condition that guarantees stability. This condition requires some type of contrast-independent spaces for slow components of the solution. We present numerical results and show that the proposed methods provide results similar to implicit methods with a time step that is independent of the contrast.

A Dissipative Momentum-Conserving Time Integration Algorithm for Nonlinear Structural Dynamics

2021 ◽  
pp. 2150116
Author(s):  
Salvatore Lopez
Keyword(s):  
Angular Momentum ◽  
Structural Dynamics ◽  
Time Integration ◽  
Integration Scheme ◽  
Second Phase ◽  
Time Step ◽  
Integration Algorithm ◽  
Nonlinear Structural Dynamics ◽  
Nonlinear Structural ◽  
Solution Point

A second-order accurate single-step time integration method for nonlinear structural dynamics is developed. The method combines algorithmic dissipation of higher modes and conservation of linear and angular momentum and is composed of two phases. In the first phase, a solution point is computed by a basic integration scheme, the generalized-[Formula: see text] method being adopted due to its higher level of high-frequency dissipation. In the second phase, a correction is hypothesized as a linear combination of the solution in the basic step and the gradient of vector components of the incremental linear and angular momentum. By solving a system composed of six linear equations, the searched for corrected solution in the time step is then provided. The novelty in the presented integration scheme lies in the way of imposing the conservation of linear and angular momentum. In fact, this imposition is carried out as a correction of the computed solution point in the time step and not through an enlarged system of equations of motion. To perform tests on plane and spatial motion of three-dimensional structural models, a small strains — finite rotations corotational formulation is also described.

Analysis of Newmark Explicit Integration Method for Nonlinear Systems

2006 ◽  
Vol 22 (4) ◽  
pp. 321-329 ◽  
Author(s):  
S.-Y. Chang ◽  
Y.-C. Huang ◽  
C.-H. Wang
Keyword(s):  
Nonlinear Systems ◽  
Integration Method ◽  
Stability Limit ◽  
Explicit Method ◽  
Initial Stiffness ◽  
Explicit Integration ◽  
Time Step ◽  
Linear Elastic ◽  
Stiffness Softening ◽  
Theoretical Results

AbstractNumerical properties of the Newmark explicit method in the solution of nonlinear systems are explored. It is found that the upper stability limit is no longer equal to 2 for the Newmark explicit method for nonlinear systems. In fact, it is enlarged for stiffness softening and is reduced for stiffness hardening. Furthermore, its relative period error increases with the increase of the step degree of nonlinearity for a given value of the product of the natural frequency and the time step. It is also verified that the viscous damping determined from an initial stiffness is effective to reduce displacement response in the solution of a nonlinear system as that for solving a linear elastic system. All the theoretical results are confirmed with numerical examples.

Accurate Integration of Nonlinear Systems Using Newmark Explicit Method

2009 ◽  
Vol 25 (3) ◽  
pp. 289-297 ◽  
Author(s):  
S.-Y. Chang
Keyword(s):  
Nonlinear System ◽  
Elastic System ◽  
Stability Limit ◽  
Explicit Method ◽  
Analytical Evaluation ◽  
Time Step ◽  
Linear Elastic ◽  
Instantaneous Degree Of Nonlinearity ◽  
Stiffness Softening ◽  
Relative Period

AbstractIn the step-by-step solution of a linear elastic system, an appropriate time step can be selected based on analytical evaluation resultsHowever, there is no way to select an appropriate time step for accurate integration of a nonlinear system. In this study, numerical properties of the Newmark explicit method are analytically evaluated after introducing the instantaneous degree of nonlinearity. It is found that the upper stability limit is equal to 2 only for a linear elastic system. In general, it reduces for instantaneous stiffness hardening and it is enlarged for instantaneous stiffness softening. Furthermore, the absolute relative period error increases with the increase of instantaneous degree of nonlinearity for a given product of the natural frequency and the time step. The rough guidelines for accurate integration of a nonlinear system are also proposed in this paper based on the analytical evaluation results. Analytical evaluation results and the feasibility of the rough guidelines proposed for accurate integration of a nonlinear system are confirmed with numerical examples.

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