I am performing nonlinear dynamic analysis of a structure following direct integration method using SAP2000. Now, the SAP2000 manual encourages you to set the Hilber-Hughes-Taylor alpha coefficient at a value ranging from 0 to -1/3 in order to attain convergence in analysis. How to decide the value of this coefficient? In a typical case I am not getting convergence using alpha of -0.01 but attaining the same with alpha -1/3. How do I know whether the result that I am obtaining with an alpha value is reliable or not? This is very crucial in my case, since I am trying to identify the collapse point of the structure by doing an incremental dynamic analysis.
Hello dear Sushovan
As you can see in some main papers
1. Hilber, Hans M., Thomas JR Hughes, and Robert L. Taylor. "Improved numerical dissipation for time integration algorithms in structural dynamics." Earthquake Engineering & Structural Dynamics 5.3 (1977): 283-292.
2. Erlicher, Silvano, Luca Bonaventura, and Oreste S. Bursi. "The analysis of the generalized-α method for non-linear dynamic problems." Computational Mechanics 28.2 (2002): 83-104.
it seems to me that the main purpose of the alpha coefficient in the HHT method is elimination of the extra erroneous oscillation of higher modes because of the discretization in space, not providing the convergence, in presence of any type of nonlinearity. Together with some dear colleagues I have done some studies on the nonlinear analyses, I introduce some most relevant reports below:
1. Soroushian, A., P. Wriggers, and J. Farjoodi. "From the notions of nonlinearity tolerances towards a deficiency in commercial Transient Analysis softwares and its solution."Proceedings of 5th ECCOMAS thematic conference on computational methods in structural dynamics and earthquake engineering (COMPDYN 2015), Corfu, Greece Google Scholar. 2015.
2. Soroushian, A. and Wriggers, P. and Farjoodi, J. Time integration of nonlinear equations of motion - numerical instability or numerical inconsistency? Proceedings of the 5th EUROMECH Nonlinear Oscillations Conference, Eindhoven, Netherlands, (2005)
3. Soroushian, Aram, Peter Wriggers, and Jamshid Farjoodi. "Practical integration of semidiscretized nonlinear equations of motion: proper convergence for systems with piecewise linear behavior." Journal of Engineering Mechanics 139.2 (2013): 114-145.
4. Soroushian, Aram, Peter Wriggers, and Jamshid Farjoodi. "On practical integration of semi-discretized nonlinear equations of motion. Part 1: reasons for probable instability and improper convergence." Journal of sound and vibration 284.3-5 (2005): 705-731.
Some special point about the HHT method are in the last reference. (Probably most of these papers can be downloaded from researchgate, other wise you can write to me)
Appreciating your kind attention to the lines above, have nice hours ahead.
Like all implicit schemes, the unconditional stability of this numerical method applies to linear problems. There is no specific result showing stability of this method over the wide range of nonlinear problems that potentially exist. However, a general assumption that you can start with is:
1) the factor "alpha" should be between 0.67 and 1.0. The smaller the value of "alpha" the greater the numerical damping.
2) the factor "beta" can be estimated as (2-alpha)^2/4 and "gamma" as 3/2-alpha
Reference:
Hilber, H.M, Hughes,T.J.R and Talor, R.L. "Improved Numerical Dissipation for Time Integration Algorithms in Structural Dynamics" Earthquake Engineering and Structural Dynamics, 5:282-292, 1977.
Dear All,
Nice question.
(1) As stated in the finite element book of Prof. Bathe (1996), and partly recommended by Tahar Lattrache, the average acceleration method of Newmark can be considered as a first selection method for time integration analysis of nonlinear time integration methods.
(2) Regarding the convergence problems in nonlinear analyses, and specifically when using the HHT method see the paper below by myself and dear colleagues (2005) and specifically its section 4 for methods like HHT.
On practical integration of semi-discretized nonlinear equations of motion. Part 1: Reasons for probable instability and improper convergence , Journal of Sound and Vibration 284 (2005) 705–731
(3) The second part of the above paper returning to 2013 suggests ways to overcome the problems.
The main problem is that at detection of a non-linearity at the end of a time step, in ending the arbitrary iterative non-linearity solution method, some nonzero error remains which is additional to the inaccuracy because of the time integration approximation. Other minor problems exist when time integrating with methods like HHT that do not satisfy the equation of motion at the integration stations.
From the other side when the above-mentioned error can not be smaller than the tolerance after a specific number of iterations the analysis stops, which is undesired and we addressedit as the analysis is not converge. This stop can be simply eliminated from the analysis; see the above references as well as
From the Notions of Nonlinearity Tolerances Towards a Deficiency in Commercial Transient Analysis Softwares and Its Solution, Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COMPDYN 2015), Crete, Greece
from the same authors.
Lots of best wishes for all.
Very sorry for responding twice. I sincerely apologize. Have very nice moments ahead.
- Badges
- Science topic
- Similar topics
- Engineering
- Modeling