Hi Aram,

I am not aware of any general rule regarding the influence of damping on numerical stability. Some years ago I attempted to do a Taylor series expansion for the finite difference 1D wave equation with damping (proportional to velocity) and associated stability analysis, however I came to an impasse when I struck a complex-valued quadratic equation (from memory).

My suspicion is that the type of damping one employs will be important. If the damping term is proportional to net force, it is essentially a penalty factor that causes the (dynamic) equation of motion to mimic the quasi-static solution. This might suggest that the time step increment can be increased somewhat in the presence of damping. However, velocity-dependent damping may not achieve the same result.

I hope this helps and best wishes for your research.

Cheers,

Dion

Dear Dion, many thanks for your informative detailed note.

Yes, the mathematics is in cases complicated. And also yes in fact a main implementation of explicit methods is in wave propagation problems, where the step sizes is generally being selected also such that to provide sufficient accuracy and prevent numerical instability (also taking into account spatial discretization). In these problems, when numerical stability is the governing term, being aware that physical damping can not worsen the stability will ease the selection of step size. Other wise, for each new explicit method, determination of the stability requirements in presence of damping would be essential; the discussion seems also valid when restricting the damping to Rayleigh damping (conventional in practice). Even more, the discussion seems conceptually valid to new unconditionally stable methods and structural dynamic problems, where we would like to guarantee that the physical damping can not ruin the unconditional stability. (The role of step size on computational cost is also important, seemingly highlighting the significance of arriving at an insight on the influence of damping on the stability critical step size.) Again many thanks and best wishes.

Hi Aram,

You can take a look on the following book :

Mechanical Vibrations, Theory and Application to Structural Dynamics

Author : Michel GERADIN and Daniel RIXEN

Editor : Wiley

Regards,

The influence of damping on the stability of the time step integration methods is shown in the paper  by J.T. Katsikadelis "A new direct time integration method for the equation of motion in structural dynamics: ZAMM Vol. 94, pp. 757-774 (2014), Section 2.2. Eq. 26  answers the  question. It shows that for a given time step the increase of damping reduces the spectral radius ρ(Α) of the amplification matrix Α , hence it increases the stability. 

For the other known methods e.g central difference method, mean acceleration method, the answer is given in the book by Katsikadelis J.T., "Dynamic Analysis of Structures, (1012), SYMMETRIA Publications, Athens, Section 4.5, Stability of the numerical methods. 

Dear Jean-Philippe Bournot

Many thanks for your informative message and introducing the book to me. I checked the third edition book, returning to 2014, specifically its seventh chapter, and also checked the preface of the book for the changes compared to the previous editions (which I had seen one of them before). Then I shortly checked the other publications of the authors of the book in Researchgate. However, I could not conclude with the response to the question. Stability analyses on some specific integration methods are addressed in the book but seemingly not a general rule (for arbitrary integration method), addressing the influence of physical damping on the stability critical step size. Maybe I have not reviewed the book in sufficient detail and am incorrect now; if you feel so, kindly please do not hesitate to express your idea and guide me more. Many thanks for your kind attention and contribution towards the response of the question.

With Kind Regards,

Aram

Dear Prof. John T. Katsikadelis

Many sincere thanks for your informative detailed message. I had the paper and checked it. It introduces a nice integration method; and though I already had it, your guidance caused an occasion to review it in further detail. Many sincere thanks. Nevertheless, both in the paper and also seemingly in the book (the book was in Greek and hence I conclude mainly from your explanation above), the influence of physical damping on the numerical stability is discussed considering specific time integration methods. Is the observation addressed for the central difference, linear acceleration method of Newmark, and the method you have introduced in your introduced paper, i.e. physical damping causes larger critical time steps, extendable to all integration methods, including those not introduced yet (and even the specific cases of all parametric methods, e.g. the Generalized-alpha method originally with three parameters). I sincerely appreciate your kind attention and patience, as well as contribution towards the final response of the question.

With Kind Regards,

Aram

The damping factor depends on the parameters used to form the  equation, or the number of intregrating factors.

Dear Okafor Linus.

Many thanks for your contribution. Yes the physical damping is a parameter depending on the system under consideration, as well as, in cases, the spatial discretization method, e.g. finite elements' shape functions. In any way, after the discretization in space the damping matrix is determinate and for linear systems remains so throughout the time integration analysis. The question stated above is how this parameter affects the numerical stability of the responses computed by arbitrary time integration method. To be more precise,  does physical damping in all cases cause stability critical steps larger (than the undamped stability critical steps)? or in other words, is disregarding the physical damping in all cases in the safe side for the numerical stability of the responses computed by time integration. (Of course in view of the existing literature and the ambiguity and complexity of numerical stability in nonlinear problems, the discussion here is limited to linear problems, in consistence with the conventional approach of studying numerical stability, also not forgetting that the requirements for numerical stability in linear analyses is generally essential also for the stability of nonlinear analyses.)

   Once again sincerely appreciating your kind attention and contribution, I wish the very best for you and look forward to successful continuation and conclusion of this discussion, towards everyday better transient analyses.

With Kind Regards,

Aram

Dear Aram,

Actually, I omitted to take into acount "general rule" from your request, the book only deal with specific scheme (as you mentioned it). Thus I cannot give you additional information.

Regards,

JPB

Dear Aram,

Stability of numerical time integration is associated with a corresponding energy relation, expressing the development of the energy - or in many cases an equivalent energy - over a time increment. This is a central reason for the importance of 'energy conserving' time integration schemes - which can be augmented with dissipative terms. In the linear case, an energy balance can also be formulated for the Newmark family of integration rules, see CMAME 195 (2006) 6110-6124. However, this algorithmic energy balance contains a re-interpretation of the energy by modifying the elastic energy, and illustrates that the classic stability criteria are actually equivalent to loss of positive definiteness of the algorithmic energy. Similar considerations can be used for fourth order schemes and some non-linear algorithms, but they do not add up to a 'simple general rule' as you seem to want.  

.............. Steen

Dear Jean Phillippe

Many thanks for your kind attention and nice contribution and the book. Hopefully, towards a response soon. Have a nice day, and nicer future.

Dear Steen

Many thanks for your kind attention reply and paper. The paper was really nice, many thanks for it. Though as you have mentioned in the last line, the discussions in the paper are not directly leading to a simple rule, in an energy approach towards the response of the question, it seems to me, also from your note and your paper and also some others, that numerical stability can be considered equivalent to some energy balance equation (or inequality), depending on the nature of problems specially in presence of nonlinearities. Even if this is true, and the stated question can somehow be responded for the linear problems by an energy approach, it seems to me that its later extension to nonlinear analyses and for instance eventually concluding that physical damping is beneficial to numerical stability for all time integration analyses including linear and nonlinear analyses would be complicated. To say better, from some years ago and as addressed in the papers attached to this message, it seems to me that the balance equations or inequalities originated in the physical nature of the problem for the computed responses are the consequence of numerical stability under the shadow of convergence, not the other way around; hence the energy approach to numerical stability analysis though potentially may be able to respond the stated question, it seems not applicable, in presence of nonlinearity, because of the above relation between numerical features and physical behaviors and also the role of the nonlinearity iterations, and residuals, in the nonlinearity analyses, seemingly not taken into account in the balance equations. (To be honest, I am doubtful about the essentiality of studies on numerical stability in nonlinear problems, only in consistence with the existing research trends, try to follow up the studies.) Consequently, it seems to me reasonable that in order to arrive at the response of the question, use the ordinary spectral stability, not the energy balance, to somehow compare the stability in presence and absence of viscous damping; and then, consider the obtained general rule valid also for nonlinear analyses, provided negligible effects and errors from the side of nonlinearity iterations. In the other case, we also might be eventually limited to some restriction on the integration method, e.g. to be energy conserving or decaying. Kindly please accept my sincere apologizes both for the lengthy text and also if not so correct. Once again sincerely appreciating your nice paper and nicer contribution regarding this question, have a great week and marvelous future ahead.

By the way, towards a general rule, the effect of viscous damping on the stability of average and linear acceleration methods of Newmark, central difference method, Fox-Goodwin method, Houbolt method, and different parameters of the Wilson-Theta method, HHT method, CH (optimized) method, and quasi-Wilson-Theta method is tested. The details will hopefully be presented in ECCOMAS2016. In brief, for these methods, viscous damping leads to more numerically stable analyses (compared physically undamped analyses); consequently, any general rule needs to be in consistence.

Dear all, towards the general rule and with attention to the note above, I would be very thankful if you can guide me with any experience you have had in contradiction with the note above, i.e. time integration methods with viscous damping harming the numerical stability and specifically cases at which the region of integration steps causing stable responses ends earlier when affected by viscous damping. Sincerely appreciating your kind attention to the lines above I wish the very best for you and look forward to hearing from you and your guidance, hopefully towards a final response and general rule soon. Have a nice day night and future.

Dear all, yesterday I tried to also respond to the question during a key talk for MS 506 in ECCOMAS 2016, in Crete, Greece.

It seems we can expect the numerically stable analyses when the damping is viscous, no physical instability exist, and the corresponding undamped analysis is numerically stable.

The question was not the main issue of the talk, I have not prepared the corresponding full paper and meanwhile due to the questions of attendees there exist some unclear details to be finalized (for which I sincerely appreciate the attendees of the session). In any way, hopefully the issue will be finalized in very near future; it can help simpler study of stability for newer methods. Have a great rest of day and wonderful Sunday ahead.

I understand that the query of Dr. Aram Soruoushian is related to viscous damping, which is an approximations of the real life damping. There are other types of damping which model better the actual dissipative forces, including those described by a fractional derivative. See for example the papers

Katsikadelis J.T. (2009). Numerical Solution of Multi-term Fractional Differential Equations, ZAMM, Zeitschrift für Angewandte Mathematik und Mechanik, 89, (7), 593 – 608 (2009) / DOI 10.1002/zamm.200900252

Katsikadelis, J.T. (2012). The Fractional Distributed Order Oscillator. A Numerical Solution, Journal of the Serbian Society for Computational Mechanics Vol. 6, No.1, pp. 148-149

Katsikadelis, J.T. (2013). Numerical solution of distributed order fractional differential equations, Journal of Computational Physics, 259 (2014) 11–22.

Katsikadelis, J.T. (2014). Generalized Fractional Derivatives and their Applications to Mechanical Systems, Special Issue, Archive of Applied Mechanics, Special Issue, DOI 10.1007/s00419-014-0969-0.

In all these investigations convergence exists for the developed time step integration schemes, even the damping is not viscous.

Therefore, an answer to the posed query may not be meaningful.

Dear Prof. John T. Katsikadelis, 

Many thanks for your guidance and the nice papers you introduced. Regarding the discussion, however, it seems even if we can prove that viscous dampings of the Rayliegh type (not all viscously damped systems) can not change an stable analysis in the undamped case to unstable analyses, first we have arrived at a general rule for all time integration methods, practical for many civil engineering applications (or probably also many other areas), and secondly, the proof can be considered as a basis to be later extended also to other types of dampings including those mentioned by you. (And even such a approach is in agreement with the existing tradition of using Rayliegh viscous dampings in the study of time integration methods stability, i.e spectral stability.) Hopelessly the full paper is not finalized yet; an attempt regarding such a proof was presented in the talk on Friday morning in ECCOMAS 2016, and will be talked about also elsewhere this summer, for which some details need to be finalized and I will write about them also here. In any way, with many sincere thanks for your kind and informative guidance and also the nice literature, have a nice day, great week, and wonderful future ahead.

With Kind Regards,

Aram

Dear all interested and/or contributors in the answers to this question,

In this starting hours of 2017, I thought it is nice to once again sincerely appreciate your kind attentions to this question and wish all of you and your families and beloveds moments all full of joy success and health throughout 2017. Hopefully this question will be finalized much before the end of the year. . . . . . 

With Kind Regards,

Aram Soroushian

Dear All,

Hello again, with may sincere thanks for all your kind attention, it seems to me that based on the typology of the contributions of the excitation and the damping in the integration formulation, we may be able to guarantee better stability conditions when the physical damping is increased. I have published (still e-published) a paper regarding this idea and its details, and will add its e-copy here when having the permission, hopefully soon and hopefully towards receiving your nice comments about it and the answer of the question then. Have very nice moments and future ahead, and again many thanks for your time and kind attention.

With Kind Regards,

Aram

Hello again,

Besides, few hours ago I also found a weaker general rule for the effect of physical damping on numerical stability in the short and about old paper below:

Wood, W.L., 1987. On the effect of natural damping on the stability of a time‐stepping scheme. Communications in applied numerical methods, 3(2), pp.141-144.

where the most important message seems to be "natural damping does not always ‘help’ the stability of the algorithm"

The contents of this paper is in complete consistence with my new paper (addressed in my previous post above), and addressing a special case of it. Hopelessly I have just met the above-mentioned paper of W.L. Wood and therefore will try to add it in the references of my paper in the proofreading stage.

With many thanks for your kind attention, have very nice moments and future ahead.

Hello again,

With many thanks for all your kind attention ad comments, the paper that seems responding the question is now attached to this post. It relates the response to the presence of damping and excitation in the integration scheme and also extends the claim to other damping. Kindly please let me have your any comment about the paper. It would be a great honor for me and I would be sincerely thankful. Have a very nice day.