Science topic

Nonlinear Dynamics - Science topic

The study of systems which respond disproportionately (nonlinearly) to initial conditions or perturbing stimuli. Nonlinear systems may exhibit "chaos" which is classically characterized as sensitive dependence on initial conditions. Chaotic systems, while distinguished from more ordered periodic systems, are not random. When their behavior over time is appropriately displayed (in "phase space"), constraints are evident which are described by "strange attractors". Phase space representations of chaotic systems, or strange attractors, usually reveal fractal (FRACTALS) self-similarity across time scales. Natural, including biological, systems often display nonlinear dynamics and chaos.
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I have modelled Railway track bridge System, but while solving it shows Solver was unable to converge the non linear model
Please Help solving this error.
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You may need to linearize the your equations.
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In his book "The Book of Why", Pearl says “Keep in mind also that the regression-based adjustment works only for linear models, which involve a major modeling assumption. With linear models, we lose the ability to model nonlinear interactions, such as when the effect of X on Y depends on the level of M. ” Does this mean that the regression model Y = a + b*X + c*M + d*X*M + e*Z is wrong (where X is the independent variable, M is the moderating variable and Z is the control variable)? I mean, the inclusion of the control variable Z in the regression model does not serve the purpose of blocking the backdoor path (if Z is the only confounder)?
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Holger Steinmetz Thank you for your comment. Although I agree with you, I am still a bit confused, pearl should not be unaware that an equation with a product term is linear in parameters. I have sent an email to pearl about this and with any luck we will get a reply.
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Hi,
I want to ask you: 'Which programming languages do you prefer to use for implementing software in Structural Engineering? (PYTHON, C++, or anything else)'
Recently I have heard about PYTHON and its application in my major, but still, I have doubts about choosing it or C++.
If you think there is a better programming language for implementing software in Structural Engineering, please share your opinion with me regarding the mentioned question. I will be more than happy to receive your suggestions.
Thanks
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My nonlinear model in state space is
xdot= Ax + Bu+ Phi(x), where phi(x) is nonlinearity in the system
I am looking for papers/articles concentrated on indirect MRAC design. Any directions are appreciated.
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x4 is not controllable, but stabilizable if p5 > 0.
x3 and x2 are controllable.
If x4 is stable, then x1 is controllable.
You can run a basic stability test for parameters p1 = p2 = p3 = p4 = p5 = Gb = 1, with the initial condition {x1(0) = 1, x2(0) = 0.5, x3(0) = –0.5, x4(0) = –1}, and u = 0.
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Hello Researchers,
I am solving a nonlinear eigenvalue problem for a plate. The total stiffness matrix is the sum of the linear stiffness matrix and the nonlinear stiffness matrix. The linear stiffness matrix is symmetric but the nonlinear stiffness matrix is asymmetric due to which the total stiffness matrix becomes asymmetric. The mass matrix as usual is symmetric. The 'eig' command in MATLAB is used only for symmetric matrices. Can anyone suggest an eigenvalue solver for asymmetric matrices?
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eig(A,B) is a general eigensolver. Replace A by your stiffness matrix and B by your mass matrix. Depending on the properties of your stiffness matrix the eigenvalues might be real or complex conjugate pairs.
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Hi
I'm working on a research for developing a nonlinear model (e.g. exponential, polynomial and...) between a dependent variable (Y) and 30 independent variables ( X1, X2, ... , X30).
As you know I need to choose the best variables that have most impacts on estimating (Y).
But the question is that can I use Pearson Correlation coefficient matrix to choose the best variables?
I know that Pearson Correlation coefficient calculates the linear correlation between two variables but I want to use the variables for a nonlinear modeling ,and I don't know the other way to choose my best variables.
I used PCA (Principle Component Analysis) for reduce my variables but acceptable results were not obtained.
I used HeuristicLab software to develop Genetic Programming - based regression model and R to develop Support Vector Regression model as well.
Thanks
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Hello Amirhossein Haghighat. The type of univariable pre-screening of candidate predictors you are describing is a recipe for producing an overfitted model. See Frank Harrell's Author Checklist (link below), and look especially under the following headings:
  • Use of stepwise variable selection
  • Lack of insignificant variables in the final model
There are much better alternatives you could take a look at--e.g., LASSO (2nd link below). If you indicate what software you use, someone may be able to give more detailed advice or resources. HTH.
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Two dimensional discrete-time nonlinear dynamical Henon chaotic map generates pseudo-random binary sequence which has been described as below Xn+1 = 1+Yn – aXn^2 and Yn+1 = bXn,where n=0, 1, 2… ..Here, the parameters, a and b are prime importance as the dynamic behaviour of system depends on these values. The system cannot be chaotic unless the value of a and b are 1.4 and 0.3 respectively.
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In attached picture you can see that Henon map shows chaotic characteristics with values of a=1.4 and b=0.3.
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How can I numerically solve coupled polynomial or transcendental equations using MATHEMATICA?
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What do you mean with coupled polynomial? The answer depends con the number of independent variables, the degree and the domain (integers, real or complex). I programm un Maple, Mathematica and Matlab. For nonlinear search of roots, I prefer the last one. It's very fast and accurate.
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Linear stability analysis fails to determine the local stability property of a non-hyperbolic equilibrium point as there is a emergence of a centre subspace (other than stable and unstable subspace) of the linearized system corresponding to the eigenvalue whose real part is zero. Centre manifold of the corresponding nonlinear system may not be unique. So, what is the exact procedure to analyze such kind of situation?
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The short answer is that there is no "exact" procedure for analyzing non-hyperbolic equilibrium points. However, there are some work using blowing up/down methods. You may look at this methods in a differential equation literature. You can check the way we used in one of our paper. "Bifurcations and global dynamics in a predator–prey model with a strong Allee effect on the prey, and a ratio-dependent functional response." P Aguirre, JD Flores, E González-Olivares - Nonlinear Analysis: Real World Applications, 2014. In this paper we analyzed the stability of the origin using blow up/down techniques.
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Hello everyone,
I've implemented a FEM code in matlab, and now I'd like to make it faster. Since I have to perform nonlinear dynamic analysis, I have a lot of iterations, each one requiring to asseble the full stiffness matrix K.
What I would like to do thus is to parallelize the assembling procedure of K. Till now I tried parpool and spmd, the latter with poor results, the first one performing nicely (speedup factor x2... despite using 10 cores...) but only under a certain number of elements. Overcome a certain treshold, parallel computation (14 cores) would take as much as 10 times the single core version.
I understand this may be related to overheads in the comunication between "master" and workers and/or slicing procedures, but it seems I cannot get the hang of it...
Does anyone have suggestions and/or can point me to some useful material on this specific matter?
Thank you all in advance,
Jacopo
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I have been reading papers on asymmetric effects using panel nonlinear ARDL approach but in most cases, I do not see control variables in the model
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If you look at papers on asymmetric effects of a variable(s) on the another variable (say, energy use and financial development on economic growth using panel nonlinear ARDL), you would notice that in most cases, only the decomposed (positive and negative shocks) variables are stated on the right hand side of the model.with no control variable included. My question is: Can a control variable be included in the model? If no, why?
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Hello
Recently I have heard that just SAP should be used for dynamic analysis of building and the results of ETABS are not valid for nonlinear dynamic analysis. I have done nonlinear dynamic analysis with Etabs software. Are the results acceptable?
Is there any evidence for this?
Thanks in advanced
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Both programs that you mentioned, could able to nonlinear analysis based on plastic Hinge, and both results are accepted. But you must notice that the operation that SAP2000 use to run nonlinear analysis is faster that ETABS and I strongly recommended that if you want to do nonlinear analysis for a frame structure use SAP2000 or higher programs such as PERFORM 3D instead of ETABS, especially if you want to do nonlinear analysis for your articles.
Best Regards
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Specifically, I am using generalized additive models with penalized splines in R - gam() and gamm4(). The smooth term tells me the F-statistic and p-value, but I would like to get the standardized betas so the results can be more comparable to the linear models within my study.
Can someone point me in the right direction? I am coming from a limited mathematical background so resources that go beyond a pure theoretical explanation would be most beneficial for me.
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Hi Robert,
once you leave the linear realm, you pay with the loss of clear interpretability. In this regard, the only thing you get with a GAM is a statistical test of the wiggliness of a relationship. When you use the mgcv package in R, the EDF parameter can be interpreted as a kind of strength of wiggliness but in the end, you have to go with an illustration of the curve.
And by the way: Even in the linear case, the interpretation of standardized coefficients as effect sizes is a bit spicy :)
Greenland, S., Schlesselman, J. J., & Criqui, M. H. (1986). The fallacy of employing standardized regression coefficients and correlations as measures of effect. American Journal of Epidemiology, 123(2), 203-208.
Kim, J. O. J.-O., & Mueller, C. W. (1976). Standardized and unstandardized coefficients in causal analysis: An expository note. Sociological Methods & Research, 4(4), 423-438. doi:10.1177/004912417600400402
King, G. G. (1986). How not to lie with statistics: Avoiding common mistakes in quantitative political science. American Journal of Political Science, 30(3), 666-687. doi:10.2307/2111095
HTH
--Holger
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I am trying to model the shear link in sap2000/ etabs by I do not know how to model the web stiffened beam in this software because for the shear link we have to use web stiffness.
Please suggest how to do it.
I want to perform a nonlinear dynamic analysis of this model.
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you can either use ASCE 41 recommendation in steel chapter or use the following paper which uses a shear spring as the representation of shear link. The web stiffener spacing has been taken into account in this paper. You can use only monotonic parameters for your modelling if you are using sap and aren't interested in deterioration.
I have also shown how you can make a model in OpenSees based on this method which can be also used in SAP.
There are other approaches like modelling the shear link with an elastic element with two shear springs at its ends which has been discussed in the following paper.
I have also talked about and compared all other methods that I have seen in the literature in the following paper:
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In Uncoupled modal response history analysis developed by Prof.  Anil K. Chopra, we get the Nonlinear governing equations of inelastic SDOF systems.
After that, how to solve these Nonlinear governing equations when we know the relationship "force–deformation" (Fsn/Ln − Dn) to get the response-time history of Dn(t) and Fsn(t) ?
And what software can be used?
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I am doing 3pt bending test in LS-Dyna with a glassfibre-epoxy sample. So it is a non-linear dynamic (quasi static) problem. I assume it is a implicit problem since the event is slower and the effects of strain rates are minimal. But my professor used explicit control card to solve this. what difference can it give?
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Explicit solver can be used for modelling slow problems if you use appropriate time scaling or mass scaling.
Without the scaling, the run time will be very long since the load is applied at a very slow rate. And with too much scaling, the results will be incorrect.
When you say he used Explicit, I assume that he used explicit solver with the actually loading rate.
Strain rate effects are related to the rate of displacement or deformation....not whether it is implicit solver or explicit. You still can simulation dynamic problems with implicit analysis and model quasi-static events with explicit, but you need to know what are the suitable keywords to add in both cases.
Here is how I did 3-point bending. I used explicit because I wanted to apply a dynamic load
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Given that the 4th-order non-strict feedback nonlinear dynamics (also, including the transcendental terms)
x_dot=f(x)+g(x).u(x), ..........................................(1)
where x=[x1,x2,x3,x4],
f(x)=[ f11(x2,x4) ; f22(x2,x3,x4); f33(x4); f44(x2,x3,x4) ],
where f22(x2,x3,x4)=atan(x2,x3,x4), and f44(x2,x3,x4)=atan(x2,x3,x4),
g(x)=[0;b1;0;b2].
I am trying to find the solution for above differential equation using the following steps
step 1: convert the dynamics (1) into linear strict feedback using feedback linearization.
z_dot=p(z)+m(z).v(z) ..........................................(2)
where z=[z1;z2;z3;z4], p(z)=[p11(z1,z2); p22(z1,z2,z3);p33(z1,z2,z3,z4); p44(z1,z2,z3,z4)],
m(z)=[0;0;0;v].
step 2: apply the backstepping, v(z), where v is the function of u(x).
Note: elements (in order) in closed-loop system are consist of (a) input; then (b) controller, v(z); (c) original dynamics (1); (d) converted dynamics.
step 3: But the result is showing steady-state error in converted dynamics i.e., especially for z2, z4 (also, very sensitive to initial conditions). but all states of original dynamics are converging very well on zero equilibrium for any initial conditions.
Please help me about some questions as given below
(1) So what should I do for removing this error?
(However I am thinking about applying the Integral backstepping, adaptive backstepping, Dynamic Surface Control (DSC) ).
(2) Should I only concentrate on original dynamics (because converging the original dynamics with the help of converted dynamics)? But my aim is that I want to converge both of the dynamics.
(3) Will it be useful if apply backstepping for converted dynamics (2) or it is better to apply sliding mode directly on original dynamics (1)?
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Thank you so much, Sir Muhammad Shafiq
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Hello everyone!
I am using dlnm package in R (by A. Gasparrini) and I would like to perform qAIC in order to choose the model with the best fit.
Since I am not experienced in R, is there an efficient way to do so?
Thank you in advance!
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Gasparrini himself, provides code that can be used for this purpose; https://github.com/gasparrini/2013_gasparrini_BMCmrm_Rcodedata/blob/master/01.prep.R .
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Can any body suggest me a straightforward way with low computational cost, for obtaining LCO amplitude and frequency for multi-dimensional phase space?
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Dear Mohsen,
an old but useful way is to use the Describing Function method. If I am not mistaken Dr. Ronilson Rocha has used this method with chaotic circuits and systems with good success.
Regards.
Luis
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I tried to solve an equation of Van-der-Pole using linearizations in the current point of the phase space (state-space). This method produced damped oscillations whereas the nonlinear model produced stable oscillations. Is there a method to obtain a solution closed to exact using linear models with some kind of linearizations of something like that? I would like to have a deal with eigenvalues and aigenvectors of the linear equations produced by the former nonlinear equations.
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ode45 matlab can visualize the evolution of the solutions
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I'm working on the FE analysis of nonlinear vibration of circular plate in ansys APDL. I considered the Mooney-Rivlin model of hyperelastic materials. So i want to know that how to find natural frequency of this problem in ansys APDL?
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You can find a relative equilibrium position. First define your equation of motion \ddot{x}=f(x,\dot{x}). Then solve for f(x_eq,0)=0 to get equilibrium point(x_eq).
However, it is an analytical method.
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I have a dataset of 5 variables of quantitative continuous type: 4 independent and 1 dependent (see attached). I tried using linear multiple regression for this (using the standard lm function in R), but no statistical significance was obtained. Then I decided to try to build a nonlinear model using the nls function, but I have relatively little experience in this. Could you help me, please: how to choose the right "equation" for a nonlinear model? Or maybe I'm doing everything wrong at all? So far I have used the standard linear model in the "non-linear" model.
I would be very grateful for your help.
If you do not have the opportunity to open the code and see the result, I copy it here:
------
library(XLConnect)
wk <- loadWorkbook("base.xlsx")
db <- readWorksheet(wk, sheet=1)
INDEP <- NULL
DEP <- NULL
DEP <- as.numeric(db[,1])
for(i in 1:4){
INDEP[[i]] <- as.numeric(db[,i+1])
}
MODEL <- NULL
SUM <- NULL
MODEL<-nls(DEP ~ k0 + INDEP[[1]]*k1 + INDEP[[2]]*k2 + INDEP[[3]]*k3 + INDEP[[4]]*k4, start=list(k0=0,k1=0,k2=0,k3=0,k4=0))
SUM <- summary(MODEL)
-----
The result is:
-----
Formula: DEP ~ k0 + INDEP[[1]] * k1 + INDEP[[2]] * k2 + INDEP[[3]] * k3 +
INDEP[[4]] * k4
Parameters:
Estimate Std. Error t value Pr(>|t|)
k0 6.04275 1.30085 4.645 6.41e-06 ***
k1 0.03117 0.01922 1.622 0.107
k2 -0.02274 0.01663 -1.367 0.173
k3 -0.01224 0.01717 -0.713 0.477
k4 -0.01435 0.01541 -0.931 0.353
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 1.418 on 186 degrees of freedom
Number of iterations to convergence: 1
Achieved convergence tolerance: 2.898e-08
-----
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It sounds like you already tested your hypothesis with the linear model and used up your p value. So, you can't do more p-value tests.
As far as choosing a model, that isn't a question for the stats folks, but is related to the theory you have (stats folks can say how to implement or approximate the model).
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I'm trying to linearize a non-linear model and write it in state-space form (derive state-space matrices) for control and optimization purposes. The nonlinear model is: x(k+1)=f(x(k),u(k),d(k)), and it contains the minimum of three terms:
f(x(k),u(k),d(k))=x(k)+min(a.u(k),b.x(k),c.d(k))+...
a,b and c are constants and the three terms inside the min() function are all linear. How do I go about linearizing this model? Note that I want to use the linearized model as a prediction model for MPC.
Thank you in advance.
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Linearization essentially considers the small perturbation dynamics about some system state. The linearized system description will depend upon the values of a.x, b.u and c.d. Thus first you need to decide at which point in the state space you want to linearize about. Often the chosen point is the so-called steady state solution where u(k), d(k) are defined constants and x(k+1)=x(k)=constant is the steady state solution. If the min function is the only non-linear term in you equations, then once you have calculated the steady state value of the state the rest is trivial.
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I have a model that consist of two ODEs and one PDE which are all coupled and non-linear. When I linearize the system around the steady state and simulate the system with initial values that are different from steady state I get a stationary deviation in one of the solutions. Due to non-linearities I expect a quite different behavior for the linear system. However, if I multiply some the elements of the system matrix with a constant (relaxation or dampning coefficient?), the linear system now resembles the non-linear system much better in terms of transient behavior and that I no longer have a stationary deviation which I had before. Is this method for fitting the linear model allowed and have some theory that can justify the use of such relaxation constants? Or is it just a 'cheat' which is not valid as the system does change due to the use of such constant?
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It is difficult to give any hint without knowing the equations. However, at a general level linearization around an equilibrium might not provide a correct behaviour if the linearized system is not exponentially stable, i.e., if some of its eigenvalues are zero. For example, the ODE
y'=-y3
is stable around y=0, but its linearization is
y'=0
for which each initial condition is stationary.
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What is the stability of a fixed point (non-isolated), which has zero eigenvalues? Is there any analytical procedure for finding stability of these type of fixed points. If numerical analysis can do the job, then how to do that?
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I suggest you the stability of the iteration processes.
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In the case of fitting nonlinear models, is the COD (R2) sufficient for the selection of the best fitting model or should other criteria be used such as Akaike's information criteria?
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As far as I know, information criteria such as the AIC or BIC can only be used in combination with a likelihood. I can't tell how your fits are being performed, but you mentioning R^2 smacks of least squares.
Note that R^2 is only appropriate for linear models. See e.g.
Maybe this thread can help you further with your problem:
Good luck!
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Dear researchers
As you know, one of the challenges of using nonlinear procedures is to determine the behavior of plastic hinges of members with deformation controlled action that this behavior is assigned to the plastic hinge by a force-deformation curve and its relations using parameters modeling. various researches has shown that the uncertainties in these modeling parameters significantly affect the structural responses.
Also, the acceptance criteria of different performance levels relating to the mentioned force-deformation curve are needed for performance-based design of structures.
There are two questions now:
1- Are force-deformation curves presented in ASCE 41-13 suitable only for nonlinear static analysis (push over)? or also is applicable for nonlinear dynamic analysis?
2- Given that the acceptance criteria presented in ASCE 41-13 are derived based on the mentioned force-deformation relations in this code (a, b and c modeling parameters), what acceptance criteria can be used to evaluate the structure at the IO, LS and CP performance levels if the other force-deformation relations presented in the technical literature (such as Lignos and Hartloper relations for beams and columns of moment frames, respectively) are utilized for concentrated plasticity modeling?
The mentioned curves (Lignos and Hartloper relations) are mostly used in structural modeling to study the structural collapse, in which the collapse is determined by the criteria mentioned in FEMA p-695 and as a result, acceptance criteria in accordance with these behavior curves have not been researched.
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1. Seismic codes suggest simplified force-deformation models in order to estimate the inelastic behavior both in monotonic and cyclic loading. The monotonic F-δ curve is considered as the envelope and the skeleton curve of F-δ loops under cyclic loading. The real inelastic behavior under cyclic loading depends on the material and the dynamic loading, e.g. reinforced concrete under seismic loading. So, stiffness and strength deterioration should be considered under cyclic loading in the concentrated plasticity modeling technique.
2. Uncertainties about deformation capacity are high beyond the point C of the F-δ curve. Even in the Collapse Prevention performance level (before point C), the ultimate deformations shows significant dispersion in experimental cyclic tests (e.g. reinforced concrete). Consequently, appropriate acceptance criteria for different performance levels and for different materials can be found in seismic codes (ASCE 41-13, FEMA, Eurocode, EN 1998-3, etc) or in other technical literature using model safety factors to scale down the proposed mean values to mean plus standard deviation ones.
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Given a system of First Order ODE`s of type y'=Ay+B(y) where B(y) has each of its elements composed by biquadratic polinomials of y.
Do alternatives exist (to the classic Runge-Kutta or Adams-Bashforth / Adams-Moulton methods) to resolve this type of systems improving their computational time and/or their accuracy?
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Dear Yago Isasi,
You can look at also the compact finite difference methods. For example, a sixth-order one is presented in https://doi.org/10.1002/cnm.1349
Best regards,
Tahir Cosgun
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Hello!
I'm currently studying the influence of adding a mass in a cantilever tip, near to a bifurcation.
In order to do it, I simplified a model of the cantilever to a 1DOF mass(m)-spring(k)-damper(b) system, where there is a stiffness linear and nonlinear (koX + k1X^3) so the system can be descrived as a Duffing like behavior ( for large amplitudes).
I wrote the equation of motion of the system as in the Image I annexed and then I made it nondimensional.
Now I wrote the differential equations (velocity, position) just to look at the phase portrait and the poincaré mapping. (For analyse the reponse frequency I have to apply Method of multiple scales and obtain amplitude and phase but for now just want to analyse the system before it).
In order to analyse the phase portrait, etc. I used a matlab program made by Housam Binous, for forced Duffing systems and I tried to adapt to my case ( Basically instead of Force*cos(Wt) and used the electrostatic force as described in the image) but I am not getting the results I expected in the plots, they are very strange).
Could you help me solving this challenge? I'm struggling with it, but It should be simple .
I annexed the matlab files and the image.
Thank you very much !
Kind Regards,
Bruno Silva
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After a long debbuging I found out the voltage AC ( 20 V) and DC ( 4V) values were to big for the analysis for a gap in order of 1um. So by giving large amplitudes values, like 100 um now it is ploting!
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Hi all,
Can you please explain the "Dynamic Pull-in" concept in electrostatic parallel plate MEMS actuators? I would like to know "in detail" about its causes and effects. How is it related to the "Static pull-in"?
Regards,
Raghu
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The pull-in voltage of an electrically actuated MEMS device can be defined statically or dynamically: In the static case, the pull-in voltage is the threshold DC voltage for which the static equilibrium is stable and bounded. In the dynamic case, it is the highest DC voltage, applied as a step, for which the transient step response is bounded. In the static case the equilibrium curve has to be computed to be able to determine pull-in, whereas the dynamic the pull-in voltage is determined using a trial and error approach.
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  • Theory of chaos describes behaviour of dinamic systems evolution which are sensitive on starting conditions. Chaos implies nonlinearity. Nonlinear relationship are a necessary condition for chaotic systems. Existance of nonlinearity alone does not make a chaotic system. All natural processes are nonlinear. Human brain is the most chaotic system on world. Is chaos organisational form of nature?
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I think yes.
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When I am trying code:
AIC(model) an error occurs:
Error in UseMethod("logLik") :
no applicable method for 'logLik' applied to an object of class "c('cesEst', 'list')"
The model is a fitted CES function. So basically I am also unsuccessful in obtaining log likelihood of the model.
Would highly appreciate your help!
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For AIC, if data points are less than 20, use AICc
1st fit the curve using software s such as GraphPad, CurveExpert etc
2nd calculate AICc
based on the residual (software s will do the calc for you or alternatively you can use excel to calculate)
For Durbin-Watson autocorrelation and runs test, I've used excel with good results
see
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Hello,
my question might be more theoretical. The datasets I currently analyse are not normally distributed (according to shapiro-wilk and kolgomorov-smirnov @ p = 0.05). Therefore, I prefer plotting them as boxplots. However, I would actually like to fit a nonlinear model (in that case a hill equation) to the (non-existing) mean in order to extract some parameters. So, is there any way and is it allowed to fit my function to the median instead of the mean values? I haven't found publications where this is done and I assume that this is for a good reason. I just don't know why and what else to do with my data. Thanks for your help! Philipp
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You’ll find an approach involving the use of the L1 and L2-norm solutions in my paper on RG (link below). Unfortunately, the paper may be a bit computationally challenging if your background isn’t math. I’m very occupied these days with research and online supervision. Hence, depending on the nature of your problem, I’ll either send you a suggestion for a solution or link you up with one of my PhD students.
Best regards
Sam
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Eccentrically Braced Frame building of 6 story modeled in SAP 2000. Conducting a non-linear dynamic response to earthquakes.
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I would be grateful for suggestions to solve the following problem.
The task is to fit a mechanistically-motivated nonlinear mathematical model (4-6 parameters, depending on version of assumptions used in the model) to a relatively small and noisy data set (35 observations, some likely to be outliers) with a continuous numerical response variable. The model formula contains integrals that cannot be solved analytically, only numerically. My questions are:
1. What optimization algorithms (probably with stochasticity) would be useful in this case to estimate the parameters?
2. Are there reasonable options for the function to be optimized except sum of squared errors?
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Dear;
You can solve the integrals manually or by a software.
Regards
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Dear Colleagues,
I have recently graduated with a BSc in Mechanical Engineering. During my BSc, I assisted research and projects on a variety of fields ranging from nanomechanics of advanced materials (experimental), predictive analysis of stochastics data input for control (MATLAB), human balance control (theoretical), dynamical modeling of fluid/solid coupling problems, and corresponding CFD in OpenFOAM, computational aerodynamics with HPC. Upon my graduation, I joined a research team at ETH Zurich as a scientific assistant to work on vortex kinematics (theoretical and computational).
My main interest areas are:
  • Nonlinear Dynamics and Chaos, Stochastic Systems, Machine Learning of Dynamical Systems and Fluid Dynamics, Prediction, Nonlinear Control
  • Computational Finance, Financial Analytics
  • Numerical Methods, Computing and Algorithm Development
Clearly, all of the fields mentioned above require a decent knowledge of mathematical modeling, analysis, and computation (mostly by parallel computing over HPCs). One can also argue that these areas are not really far from each other as they can be all classified into an umbrella field of Dynamical Systems Theory.
I will soon start my MSc in Computational Science and Engineering at ETH Zurich. However, I am struggling to decide which specialization area I should choose.
As a part of the program I have to enroll at least in two of the following CORE SUBJECTS:
  • Advanced Numerical Methods for CSE
  • Optimization for Data Science
  • Computational Statistics
  • Advanced Systems Lab (Fast Numerical Codes)
Of this, I am planning to take all as they are rich in content, relevant to my multidisciplinary taste, and beneficial for my future plans. They are also fairly complementary to one another.
I will also have to take two mandatory subjects as a part of the admission requirement:
  • Numerical Methods for CSE
  • High-Performance Computing Lab for CSE
*The program requires me to take 5 courses in my selected specialization area. The rest of the credits necessary to graduate can be chosen freely from any department.
ETH is a top-notch institute for education and research in all three of Control & Robotics, Fluid Dynamics, and Applied/Computational Mathematics. This at least ensures that whatever I choose I will still get a quality education and have a chance to do quality research.
As we all know, modern areas such as robotics, data science, software engineering, neuroscience, computational biology and etc. have rather well-defined career paths. These people would not have as many troubles as a multidisciplinary guy (e.g. my MSc program) to decide what subjects to take and what to focus on.
Now, I lost 2 lost years between the high school and university and I believe this has eliminated some of my flexibility in this kind of decision, especially given that I am in a distance relationship of which I have to also take care of. It is likely that I will prefer to stay at ETH for my Ph.D. or work some time here before my Ph.D. I may also choose to do my Ph.D. in one of the other top schools.
I really appreciate your opinions and advice!
Thank you for your time and patience!
Kind Regards
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CFD is applied to a wide range of research and engineering problems in many fields of study and industries, including aerodynamics and aerospace analysis, weather simulation, natural science and environmental engineering, industrial system design and analysis, biological engineering, fluid flows and heat transfer. You can get the full list by doing google with “CFD meshing software”. Make sure that you are learning the software which is getting used in the company you want to work for. When you learn meshing, it's better to understand fundamentals of grid generation.
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By dynamical systems, I mean systems that can be modeled by ODEs.
For linear ODEs, we can investigate the stability by eigenvalues, and for nonlinear systems as well as linear systems we can use the Lyapunov stability theory.
I want to know is there any other method to investigate the stability of dynamical systems?
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An alternative method of demonstrating stability is given by Vasile Mihai POPOV, a great scientist of Romanian origin, who settled in the USA.
The theory of hyperstability (it has been renamed the theory of stability for positive systems) belongs exclusively to him ... (1965).
See Yakubovic-Kalman-Popov theorem, Popov-Belevitch-Hautus criterion, etc.
If the Liapunov (1892) method involves "guessing the optimal construction" of the Liapunov function to obtain a domain close to the maximum stability domain, Popov's stability criterion provides the maximum stability domain for nonlinearity parameters in the system (see Hurwitz , Aizerman hypothesis, etc.).
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Hello to all friends and researchers.
In the next few weeks I will have a webinar on "Adaptive Fuzzy Sliding mode Control for Nonlinear Dynamic Chemical Processes". But I have trouble presenting content.
If any of you know the information or article or any reference in this field, I will be happy to share it with me and use your comments. Thanks
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This may be unrelated to your problem, but... If by nonlinear and chemical you mean something other than ideal gases, there's a problem. If the fugacity coefficient (indicative of chemical activity) is less than unity, the Gibbs problem (free energy minimization - the driving force and direction of the reaction) is ill-conditioned. The solution is free if you're interested in this aspect of reactions.
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Can we model nonlinear behavior of area elements for in-plane stresses?
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Dear Jay,
One way to define the nonlinearity of materials is to use plastic hinges.
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Wolf's and Rosenstein's algorithms does not seem to include the multidimensional scenario (if I understand them correctely). I want to measure the nonlinear dynamics of the liquid state machine's dynamical system.
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I do not see other way out of this inmense crisis within the European Union. Neither MEDE, nor Eurobonds. From an overlapping generations perspective, with children and young people (who have probability quasi-zero of being infected) being forced to stop their lives and careers, we mid-age and mature people are the ones who must bear the cost of the COVID-crisis. And this means inflation (never debt). Therefore, direct monetization of aid for the shock and partial debt relief. And then, a re-europeization of the investment flows (yes, protectionism) with a strong industrial policy direction in mind.
I am conscious of the asymetric international effects of the shock within the european partners. But, either we together, and in the current generation, bear the whole cost in the form of inflation, or our legacy for future generations (within an already highly leveraged framework) is conmdemned to a Euro-collapse in 15 years. What do you think?
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Well, Europe is better placed with institutional systems and strategies to recover from the COVID-19 (Chinese Virus). Africa with its maximally corrupt politicians are worst of, and in a frightening position. All over the world, efforts must be made for China to pay for this global danger. And Europe and America should be at the front of taming China and recouping compensations from China. This is one major way of protecting European youths and future generations from future bio-terrorism, or techo-science oppression. Africa must learn this.
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Hi All
in phase space reconstruction and with a aim of unfolding a system dynamic (non linear) first we must determine 2 parameter , time lag and number of dimension. my main question is if our dimension more than 3 how can these dimensions be drawn in MATLAB?
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According to its quality and gender
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Can anybody tell me the steps to be fallowed in sap2000 to define and assign to obtain the hysteresis curves for a nonlinear dynamic analysis model.
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hello, Bro ... If you get the answer please share it with me ... thank you so much
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I am trying to model nonlinear, cyclic behaviour of a cantilever beam, I wish to apply a cyclic load at the free end. And I want the midpoint of the beam to be displacement controlled (ie. reverse loading direction once the displacement at midspan reaches predetermined values).
How can I do this?
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Hi Barry,
At a minimum you will need to split your cantilevered beam into two elements at the mid-span. After that, the process is straight forward. Apply the load at the cantilever tip, and use the DisplacementControl integrator:
integrator DisplacementControl $node $dof $dy
$node must be the integer for the node as defined at the midspan. $dof must be 2 if you wish to control the vertical displacement in a 2D model space. $dy is your displacement increment. If you wish to model a cyclic load, you must put this inside a loop:
analysis Static;
foreach peak $peaks {
set steps 100
set dy [expr $peak/$nsteps[]
integrator DisplacementControl $node $dof $dy
analyze $steps;
}
analysis Static;
foreach peak $peaks {
set steps 100
set dy [expr $peak/$nsteps[]
integrator DisplacementControl $node $dof $dy
analyze $steps;
}
The code above takes a list of displacement peaks, and for each peak, it performs 100 static analysis steps until the peak displacement is reached. define a list of peaks accordingly based on what cyclic displacement you want to base your analysis on. For example, if you wish to go back and force between 2 in of vertical displacement, peaks should be:
set $peaks [list 2 -4 4 -4 4]
The first 100 steps will be to 2 inches in 0.02 inch increments, then to -2 inches in 0.04 inch increments (assuming your units are in inches, kip, ksi for example).
You could also define a displacement time series and perform a dynamic analysis, but this tends to be less stable than the simple displacement-controlled static analysis and it also depends on the time step (you will get dynamic effects for a fast loading and a close-to-static response for slow loading, whereas the static analysis does not include any dynamics in the system of equations).
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If we have a nonlinear dynamical system how are we supposed to perform parameter estimation, for different parameters and with respect to which output.
How are we going to come up with an optimization scheme, and choose which set of parameter to optimize the with respect to which output/s.
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Sensitivity Analysis (hinted at by Chakit) will give you some idea of which parameters to prioritise, but the results will only be valid for narrow regions of operation.
Have a look at the linked article for an idea on how to perform parameter estimates over multiple uncertain outputs. The field (rather confusingly) called history matching is fairly recent and dedicated to this sort of work under extremely uncertain and data-scarce problems.
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I have a stochastic mixed-integer non-linear model and it is difficult to be solved. So I want to write the constraints with uniform distribution in the sub-model that is found in the whole model. what are the steps for this? Thanks
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I have been using LINGO for a while, and yes stochastic mixed-integer non-linear model is difficult to be solved, are you sure that writing the constraints with uniform distribution in the sub-model will solve the problem?
There is quite useful lingo library which you can access from the lingo website for free. There are also some examples of writing submodels if you look at SCM planning problem. You might find your exact answer there.
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Consider the famous fractal sets, Mandelbrot and Julia sets. They are based on the idea of choosing two complex numbers Z(0) and C with proper run time and escape-region. They are achieved by repeatedly evaluating the following equation:
Z(n+1) = Z(n)^2+C
For example, in Mandelbrot set, consider a 400×400 mesh when x is in [-2.5 1.5], y is in [-1.5 1.5], run-time is 32 and the escape region is 2.
The final plot is as follows
The yellow part in that figure corresponds to the points in which the value of the function never reaches the escape region. However, the different spectrum of the blue points corresponds to the iterations in which the function crosses the escape region
I have two questions:
a) Is there any study on the transient part of such process (and not steady state)
b) What happens when we don’t consider escape-region?
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Hello Sajad,
There are a number of ways in which fractal structures can appear in dynamical systems. It is important to differentiate here between two situations: the fractal which appears in the border of the Mandelbrot set is in parameter space (since c is a parameter) and the one which appears after you choose a value for c inside the Mandelbrot set and analyse the system's phase space is called the Julia set.
For the Mandelbrot set, the fractal structure appears as the limit between the values of c in which z = 0 diverges or not. Since I have never formally studied dynamics on parameter space, I cannot discuss on it.
For the Julia set, you have what is called a fractal basin boundary. What occurs here is that there are stable invariant manifolds related to one or more unstable periodic orbits (or fixed points), that have zero measure and affect the dynamics of the system by acting as boundaries between different final states. When you set the parameters you described, you are separating the different dynamical behaviors in order to see the boundary between them.
This situation is the mechanism behind final state sensitivity in chaotic systems, that is, an error in initial conditions near the boundary leads to an error in our knowledge of the system's final state which is not linear, but goes with the fractal dimension of the boundary. (See "Grebogi et al, Final state sensitivity: an obstruction to predictability").
Another famous example in which fractal basin boundaries appear and which can be more understandable is the "Hénon map". (See, for example, "Chaos", by Alligood et al, chapter 4).
Formally answering your questions:
a) In this case, chaos manifests itself as a transient behavior and its relevance depends on your application. I haven't seen transient analysis in the Julia set (although I am certain it exists), but I know it was studied in other systems. (See, for example, "Transient Chaos" by Lai and Tel).
b) If you don't consider the escape region, you cannot distinguish between the orbits that diverge and the ones that do not. Therefore, you will not be able to highlight the boundary between these sets.
Kind regards
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ANN, which has the ability to model nonlinear systems, could be used to predict BW from age.
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Dear Ananta Kumar Das,
Yeah, the MLPNN (multilayer perceptron neural networks) can easily be used to predict the body weight from age.
Especially, the LM (Levenberg Marquardt) or BR (Bayesian regularisation) based MLPNN can be more productive to estimate the body weight from age. However, it's interesting to see how training model accuracy varies for different age inputs having constant body weight output. To make a system more robust, you have to take huge dataset (may be 1000 or more).
You can visit these links for more clarification:
Furthermore, the training of network models are implicitly dependent on many parameters like number of epochs, number of layers, number of hidden neurons in the hidden layer and most importantly the division ratio of training dataset.
The procedure to apply MLPNN models are given in following link:
Regards
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One of the salient phenomena of nonlinear dynamics called antimonotonicity (creation followed by destruction of bifurcation bubbles) is widely observed in the literature. However, a crucial question remains unanswered: that of knowing its real practical utility. Answers and suggestions will be welcome.
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For answer to your query, I would like to suggest you : https://www.sciencedirect.com/topics/computer-science/antimonotonicity
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I am studying the rheological behaviour of some pastes at different temperatures. A common approach to modelling the influence of temperature is to consider its effect on the apparent viscosity of the material. Other studies have separately modelled the parameters of the Hershel-Bulkley model as a function of temperature. This was achieved by fitting the data obtained for each parameter (σ0, k, and n) of the Hershel-Bulkley model to the Arrhenius equation separately. From this, estimates for the pre-exponential factor and activation energy, Ea, were obtained. Since the Arrhenius equation is a non-linear model, I assume these estimates were obtained through non-linear regression.
Questions:
1. Is my assumption correct?
2. If yes, should I retain values of 8.314 J/mol.K for R and 273.15 K for absolute temperature?
Suggestions would be greatly appreciated.
Thank you!
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is VIF also suitable to check multicollinearity for non-linear models? is there other methods to measure the nonlinear dependency between the predictors ?
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Hi,
I think I might have something useful.
To strictly answer your question, from an engineering standpoint, I would suggest calculating the feature importance of predictors using either supervised or unsupervised techniques. Features with the same score are the ones with high multicollinearity and hence you can eliminate them. I have been using this technique for my research and so far, it has given me excellent result for selecting optimum set of features.
Here is the link for the github code, which I use for feature selection. (Note: slight modifications are important if you are targeting classification or regression.) Please check it out.
Hope it helps you
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In the case of storage at uncontrolled temperatures, in order to establish shelf life.
I am working with the Weibullian model, any other that I can use?
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You can use Matrix Variate Bingham antipodal symmetric distribution model or for a simple generalized Matrix Variate Angular Central Gaussian distribution, which is antipodal symmetric distribution model. For a vector case you may use Von Mises Fisher distribution defined on unit hyper sphere.
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How is linear algebra used to represents nonlinear models in 3d game programming or in real coordinate space applications?
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You need more advanced topics than linear algebra.
Different techniques in computer science are useful for doing the job. See, for example, the following article:
https://www.sciencedirect.com › topics › computer-science › nonlinear-transformation...
Best regards
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I ran an abaqus model of nonlinear static analysis and there is no error message. The message file actually said the analysis completed in 54 increments. However when I loaded odb to abaqus for visualization, there are no increments to show for the step. Any reason why?
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Mr LI,
Your question is puzzling. Would you please share either the. CAE file or the input file?
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Hi guys,
Reading my question title surely has given at least some of you a flashback on their experiences during the estimation of a nonlinear system model. I hope to get some tips, tricks and useful critic on my proceedings with the model estimation. The project I’m working on is part of my bachelor’s thesis. I am thankful for every useful input form you!
What am I identifying?
My task is to identify a fully-fledged driving simulator capable of movements in nine degrees of freedom in total. The main goal is to obtain a good system model which fits the estimation and validation data and can be used for further investigation (if needed). Most importantly, as I’m conducting the thesis with an automotive OEM, not only the identification per se but also the whole process from generating measurement data, selecting a suitable model and optimizing the parameters shall be worked out in order to have a reference for future research purposes.
The driving simulator has been shown to by nonlinear and dynamic and shall be investigated as a MIMO system.
What has been done so far?
Measurement: All nine degrees of freedom have been excited with suitable position signals (Sine sweeps, discrete sine excitations, white and pink noise, Amplitude modulated pseudo binary signals) and the output has been measured as acceleration. Not only have the individual degrees of freedom (longitudinal, lateral, …) been excited (which would be a SISO case) but also a multidimensional excitation (by exciting all degrees of freedom) has been performed to identify the MIMO system.
Model selection: I am working with Matlab’s built-in toolbox from Prof. Ljung as well as a Lolimot identification toolbox from Prof. Nelles. So far, I have gained deep insight in both toolboxes and examined the different approaches in more or less full detail. In the beginning, I have played around with the GUI to get a feeling for the system models. Now, I’m exclusively working with the toolbox functions in Matlab scripts to change the model and estimation parameters arbitrarily. I want to concentrate my thesis on the estimation of a Lolimot, Narx, Hammerstein-Wiener and a linear model. This way, I want to compare the different models and I want to show that a linear model for example is not sufficient for the underlying driving simulator. In conclusion, I want to find the model that performs best for my system.
What am I planning to do next?
In the next steps of my bachelor’s thesis, I want to examine the above mentioned system models and thus have to perform a parameter optimization. The models rely on a different set of parameters (e.g. time delays, nonlinearity estimator parameters, …). As testing out all parameter combinations does not seem to be a viable option w.r.t. computing time, I have defined a DoE and want to perform a subset selection which will be representative of all parameter combinations. Using this subset (which is noticeably smaller than the huge amount of parameter possibilities of the DoE) the models shall be estimated and compared using their respective loss function values. This allows me to assign a unique value to every parameter set of the obtained subset which reflects whether the model is better or worse. Next, I want to build a response surface model and find its global minimum to find the best parameter combination of the whole subset and consequently of all parameter variations.
What questions do I have?
Before I work on the above mentioned parameter optimization, I want to make sure that I have understood everything this far and that my data is suitable for an identification. I have gained quite some understanding reading various system identification publications, however I still am not sure on two things.
Excitation signals:
The above mentioned excitations have been measured with a set of acceleration sensors all around the vehicle mockup. The measurement output has shown some pretty good results, which I used to identify other system properties like latency, phase lag, etc. I am sure that the measured signals themselves are pretty good and show minor noise in the relevant frequencies and obviously a bit more noise for lower frequencies where the noise characteristics of the sensor itself takes over. However, I am not sure whether the type of excitation is right. For dynamic systems sine sweeps and APRBS signals have yielded good results in the literature. However, an APRBS signal (step excitations with different amplitudes) shows steep peaks in the measured output of the simulator. The vehicle moves (for vertical signals) up, idles a few seconds and moves down again. The peaks result in the steep movement up and back down again. Between that is just dead time. Thus, I am not sure, whether the system dynamic has been excited strong enough. A sine sweep seems to be better and the system models estimated with both toolboxes seem to confirm that or at least manage to obtain a fit to the estimation sweep data, whereas the APRBS data is very hard to fit.
So the question here is: Is such an excitation with dead time between measurement output peaks even suitable for an identification?
Another question is: The discrete sinusoidal excitations have been designed to excite the system with one sinusoidal signal which is faded in and out, then there is 2 seconds of dead time and then the next sinusoid follows. The measured output follows suit and shows excitation with dead time between the sinusoids. Is this critical as well?
The final question here is: I have also conducted measurements with white and pink noise inputs. The statistical character makes this kind of input especially useful. Though, the signals had to be manipulated in amplitude and smoothened to not overexcite the simulator dynamics (and eventually to crash the simulator). This means, that the frequency band is not as wide as a ‘normal’ white noise, but should be in the relevant are of the simulator. Is an identification with that kind of estimation signal suitable?
Estimation and validation data:
When estimating the system models, estimation, validation and test data can be assigned. The system is being estimated based on the estimation data (training data) and can be validated by plotting the system output for the validation data. What I fundamentally do not seem to understand, or have not read yet, is whether the estimation and validation data can be fundamentally different. In most examples I have seen that the system has been trained with e.g. step inputs and validated with a different independent set of step inputs. It was then tested again with a third independent step input. What I am trying to do however, is to estimate the system based on e.g. the Sweep data, to and to validate it on the white noise signals. The question thus is: Is that even a good approach? The signals are fundamentally different.
As far as I understand or want to understand is that a successful identification of a system should be capable to represent all input-output combinations possible for the system. It is very clear to me that this will never be the case. But the underlying system in my case is able to perform sine and step excitations and many more. Should I have measured an input-output combination that contained all kinds of excitations?
In other words: What is the best way to estimate and validate the model in my case? Ljung’s toolbox does not even take validation data in consideration during estimation, it much rather relies on the user to evaluate the fit to the validation data. This is very understandable, since in most cases the evaluation is a mere decision of the user.
I am thanking all of you for input to my problems!
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Hi Ziad. I recommand you read this book
"The Volterra And Wiener Theories of Nonlinear Systems" by Dr. MARTIN SCHETZEN. This book, like all his other books, is very easy to understand .
I believe Dr. Schetzen's was Norbert Wiener's student at MIT and is among the best experts of this subject. I strongly recommend this book to anyone interested in the subject
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Hello everyone,
Can anyone explain me about effective stiffness to be used in SAP2000 to model nonlinear boundary conditions. If the behavior of boundary is different in tension and compression (having different stiffnesses in tension and compression).
I am very much greatful to all you.
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For your valuable inputs in helping to solve the problem. I recently read a paper suggested by my friend and I am attaching here its title.
Influence of isolator characteristics on the response of base-isolated structures
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Actually, I was reading Shabana’s Book: Computational dynamics, John Wiley & Sons. 2010. As I'm investigating the dynamics of a multi-body system (having rigid and flexible structures). This code was introduced in chapter 9 of Shabana's book with some demonstrations of different parts of the code and it's usage. Yet, to my best knowledge, it was not mentioned from where to get this code. I would like to know where may I find this code, and/or other alternatives with similar or more capabilities?
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To obtain SAMS/2000 you have to reach out to Professor Shabana directly, I would suggest you getting his contact information from University of Illinois at Chicago's website. I strongly believe that SAMS/2000 should be the reference for all other commercial codes. SAMS/2000 is carefully developed by a pioneer professor in the field of computational multi body dynamics for rigid and flexible bodies. It took more than 35 years of research by professor Shabana and hundreds of his Phd students to develop the code!
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Hello,
I am in the process of making a predictive nonlinear model for my data with one dependent variable and 3 independent variables. I am currently using SPSS and R but the best model i have at the moment is not descriptive enough. My R2 is about 0.8 and my goal is above 0.9. I know getting a model means a lot of trials which i have attempted but i havent gotten better fit. is there a script or code that i can use to make a more effective model?
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Hello, one would have liked to know the types of your dependent variables. However, since you have access SPSS softwares package, you really can use Discriminant Analysis to obtain your desired model. It theoretically requires one categorical dependent variable and three interval data independent variables.
Regression analysis is ruled out because it requires interval data dependent and independent variables. If your dependent variable is interval type and the independent variables are categorical type, you are now thinking about Analysis of Variance which deals with classification and not modelling.
Indeed, Discriminant Analysis does both classification and modelling. It is really a powerful but uncommon technique.
I hope my input is helpful ?
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Hi there,
I am running a latent growth model (LGM) with an outcome variable that was measured across four time periods.
The goal is to establish the shape of the unconditional LGM before I use covariates to predict the intercept and growth factors of the LGM.
In assessing the shape of the unconditional LGM, the quadratic slope mean is NOT significant, but the quadratic slope variance IS significant. Does this pattern of results suggest that the overall shape of the unconditional LGM is non-linear? Or, given the non-significant quadratic slope mean, should I only be assessing the linear LGM?
Thank you!
Navio
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Navio,
1. The parameter tests are known to have low power. So, rather than testing the significance of the mean and variance parameters of the latent slope factors, you may compare models (e.g. linear vs quadratic latent growth) based on chi-squared difference test or even better, based on likelihood-based fit indices, such as AIC and BIC, in order to determine whether you need a latent quadratic slope factor.
2. "Having a statistically non-significant mean but significant variance for the latent quadratic slope factor" does not suggest that the overall shape of the trajectories is linear (or non-linear) because there is no "overall" shape of the trajectories in Latent Growth Modeling. You are interested in modeling the individual trajectories and that is why you are using LGM instead of Repeated Measures Anova/GLM.
3. Assuming the model is correct, the pattern you ended up with (i.e. significant variance but non-significant mean for the quadratic slope factor) indicates that there is significant variability in individuals’ quadratic growth rates (i.e. the amount of acceleration/deceleration in the rate of change). Non-significant mean only suggests that the estimated quadratic slope coefficient is probably negative for some individuals and positive for others but they cancel out and their mean is not significantly different than zero. If the models fit indices also indicate a better fit for the quadratic growth model, the statistically significant variance can be interpreted as an additional proof that there is a need for the quadratics growth model because individuals seem to grow at the different quadratic rates.
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Many autorities of complex systems theory claim that where the chaos theory begins, classical science ends.The theory of the chaos provides a different approach to solving problems, first of all, in natural sciences.It shows that centuries interpretations of physical phenomena(mechanic paradigm) still do not apply to a significant extent. Will the chaos theory overcomes the centuries mechanic paradigm in economics, leadership and management?
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Nonlinear dynamics can open the path to enclosure-overcoming mathematics that allow science to become explicative, thus can help to overcome the traditional, mere descriptive since enclosure-confined rank of science: cf. Number form theory linked to Structure wave theory.
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actually I want to know the features of linear and nonlinear model and also appropriate data for nonlinear models
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Yes, there is a new method which is called Piecewise Analytic Method (PAM). It does more than Runge-Kutta.
1. PAM gives a general analytic formula that can be used in differentiation and integration.
2. PAM can solve highly non-linear differential equation.
3. The accuracy and error can be controlled according to our needs very easily.
4. PAM can solve problems which other famous techniques can’t solve.
5. In some cases, PAM gives the exact solution.
6. ....
You can see :
Also, You can write your comments and follow the update of PAM in the discussion
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I stopped working in this field nearly 20 years ago. But even than there existed several methods that outperformed RK4 by lengths, the comparison is done on the estimated accuracy obtained by the same number of calls to the ODE's right hand side function. Have a look in the following books
Hairer, E. and Nørsett, S. P. and Wanner, G.: Solving Ordinary Differential Equations, Part I - Nonstiff Problems, Springer Verlag, 1987.
Hairer, E. and Wanner, G.: Solving Ordinary Differential Equations, Part II - Stiff and Differential-Algebraic Problems, Springer Verlag, 1991.
There the methods (for non-stiff ODE) of Dormand/Prince are highly recommended (I used one in my time). Somewhere in the middle of the first book is a twosided graphics where the orbits of a specific problem are shown obtained by different methods (and the same maximum number of rhs calls). The orbits of the chosen ODE are known from theory to return to the starting point. The impressive fact of the graphics is to show how good this feature is reproduced by the methods applied: the orbit of 1-step Euler metod leaves the pages and does not return, the orbit of the RK4 method leaves a large gap between start and end points, the orbit of the DP method without stepsize control leaves a small gap, and the orbit of DP with stepsize control leaves no visible gap; even more, the last method needed much less rhs calls!
This was state of the art 20 years ago. Maybe that meanwhile better methods have been developed, maybe the initially discussed PAM is one. Anyway, I recommend that you replace the RK4 method in your code at least by the DP4/5 method. If you use Mathlab then apply ode45 as ODE solver, if your code is in Fortran or C then search the internet for DOPRI5.
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The assortment of an appropriate input variables among them is one of the involved processes during the pre-processing of data in non-linear modeling system. It is neighboring by few basic questions regarding this issue including which inputs, what arrangement of input variable and exactly how much the training data should be used for improvement of model.
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It is case dependent. However, PCA techniques result good in most of the cases.
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When plotting a bifurcation diagram in nonlinear dynamics, the axis x displays a given phase parameter. Are there examples in which the phase parameter stands for time passing (for example, from the value T0 to the value T200 seconds, or months, or years)?
Thanks!
To make an example, I was thinking to something like the one in the Figure below, concerning the phase transitions among liquids, solids and gases: if you leave, e.g,., that the temperature raises of one degree every second, can we say that the axis x displays time (apart temperature values)?
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see my theory
In which I discuss the possibility of rapid movement given specific situations arise in my theory... You speak of some in your question...
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Dear all,
I am currently dealing with a problem of spurious oscillations in wave propagation in elastoplastic bodies.
The problem in question is a 1D column subjected to a 10Hz sine wave acceleration input motion at the bottom of the mesh. Abaqus/Standard is used for the dynamic analysis.
Generally, the stress strain behavior looks fine, the same with displacements and velocities. The oscillations are most pronouncedly experienced in the calculated accelerations. They appear in time when the element/Gauss point is subjected to a sharp change in stiffness, e.g. from elastic branch into plastic one when no smooth hyperbolic relationship is used. The oscillations are of much higher frequency than the input motion [say around 100Hz].
I am wondering where these oscillations come from. I have tried to include numerical damping in the HHT direct time integration scheme however this did not influence the observed oscillations. I am wondering if further “play” with alpha, beta, gamma parameters in the HHT method could result in damping out the oscillation I experience. So far I have only used set of parameters as suggested in Abaqus Manual for “application=moderate dissipation” option.
I have also tried the effects of time and space discretization, non of the two was effective [for space discretization going for much finer solution than the minimum of 10 nodes per wave length as typically advised]. The problem remains and is insensitive to mesh or time step refinement. The only way of removing the oscillations is applying the Rayleigh damping, however this seems to be artificial way of removing the problem since the constitutive model is elastoplastic and deemed to be capable of accounting for material damping.
Generally, Abaqus manual says “The principal advantage of these operators is that they are unconditionally stable for linear systems; there is no mathematical limit on the size of the time increment that can be used to integrate a linear system” so I understand that maybe this scheme is not stable or inaccurate for the nonlinear dynamic problems or a family of nonlinear dynamic problems. Would some another commonly used direct time integration scheme, such as lets say Bathe be more accurate here?
Anyone has experienced maybe a similar problem? Where is it I can look for the reason of the oscillations?
Thanks in advance for any help and advice on that.
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Hey Fabian,
Thanks for your answer and ideas!
Regarding the element type, I have not mentioned that, is it a 1D column but the one built of continuum elements. I use quadratic full integration elements but I also tried quadratic reduced integration and linear elements. No change in the observed oscillations.
Regarding the input, I have also tried various options here, such as a sin function in a tabular form, sin/cos function by defining *Amplitude, definition=periodic. Indeed, the insufficient sampling, for example for tabular definition of sin function will cause introduction of higher frequencies in the input motion. However, I did ensure this is not a problem in my analysis.
If I run my model with elastic material, no oscillations occur. Thus, I would assume both, the input motion definition and the element type [here not sure, maybe there is some coupling between element type and elastoplasticity?] should be fine. The problem of the spurious oscillations appears when "non-linearity" is material behavior is introduced.
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I am trying to determine the equilibrium points in the astrodynamics system, but the equilibrium condition is a highly nonlinear system of equations. I have tried the 'fsolve' in Matlab, but it is very sensitive to the initial guess of the solution and is lack of robustness. So I am wondering whether there is any better solver in Matlab or any other software package.
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Yue Wang - thanks for the complement, but the title should be just "Dr."
For the homotopy method, I suggest you check out these packages: PHCpack, PHClab, and Hom4PS-3.
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Hye all,
Previously, i have developed MPC using linear ARX model (from system identification toolbox). <-- here, I do not use MPC toolbox available in Matlab, I have designed the MPC based on this book: MPC using Matlab, by Liuping Wang.
My question is, how do I use NARX equation (from system identification toolbox) into the MPC? 
Any opinions, references from all very much appreciated.
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Hi,
Here is a complete workshop on how to implement MPC and MHE in MATALB.
The workshop shows a complete explanation of the implementation with coding examples. the codes are also provided.
I thought of sharing it here as it might be helpful.
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I am working on shunt active power filtering. I am devising a model on simulink that can suggest compensation currents for real world non linear loads (e.g. microwave, energy savers etc). So I have measured the 3phase voltage and currents for these loads and have converted in csv file to import in matlab. now I want to generate a non linear model that can relate with these voltages and currents. Can anyone suggest a mechanism to do that?
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You can use the command
ident
and select time domain data then import your data and continue with system identification procedure.
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Why are horizontal degrees of freedom (DOF) mainly considered in the dynamic equation of motion rather than other DOFs such as vertical and rotational for control of structures?
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In some situations, the horizontal components of earthquake ground motions can excite the vertical degrees of freedom (cause vibration vertical deformation/vibration). This may happen depending on the structural configuration.
For example, consider a fictitious structure, which looks like the letter “F” with fixed support. In this fictitious example, there are two cantilever beams supported on a column.
If this structure is subjected to only the horizontal component of an earthquake, the column deforms and vibrates as a simple vertical cantilever. This initiates the vertical deformation and vibration of the cantilever beams.
With best regards,
Daniel.
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Lagrange’s equation, as shown in your write-up, is a PDE; can you convert the PDE into a system of ODEs? Most use the Method of Lines to do such conversions. Then try the FortranCalculus (FC) language that got its start by NASA's Apollo Space program and got us to the moon!