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# Numerical Calculations - Science topic

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Questions related to Numerical Calculations
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I am working on optimal control problems with state/control constraints. My current interest is to use penalty function to convert the constrained optimal control problem into an unconstrained one. Then the unconstrained problem can be solved either via indirect method or direct method. I've been trying to use indirect method to solve the problem.
The numerical difficulties have been mentioned a lot in the literature since the unconstrained optimal control problem contains a small smooth parameter or a penalty parameter. But I have not found any detailed explanation pertaining to this issue. When I use shoothing method to solve the unconstrained optimal control problem (approporiate initial guess is already provided), many issues often arise; one is the constraint may be violated even if I've adjusted the smooth parameter or penalty parameter; another is the singularity problem of the Jacobian.
However, if I use direct method with penalty function, I can get right answers sometimes. Can anyone explain my question? Penalty function works for direct method while not for indirect method? Thanks a lot!
In addition to the above answers, the following points can be helpful for removing these problems or maximizing the error.
1. we can use specific purpose tools [algorithms] or intelligent tools [algorithms] to sense how much the violated errors are, and then we can minimize these errors.
2. A specific software for optimization, for example: "GAMS (General algebraic modeling system)" is software for solving the optimization problem significantly.
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For a 1D system, there is a way to calculate the Zak phase in the discrete form. Suppose C is some closed path in k-space (a 1D BZ). If we suppose the path is discretized into (not necessarily equidistant) ki steps with i=1,…,N and kN+1≡k1, the end result is:
(the formula exists in the appendix)
You can view this as the product (i.e. phase summation) of N small rotations of the eigenvector's phase as it's transported along C; the Im(log)-part merely picks out the phase.
If C is a non-contractible path in the BZ along a reciprocal lattice vector G, it is desirable to enforce a periodic gauge, in which case one would take:
(the formula exists in the appendix)
How we can implement the periodic gauge in numerical calculations? We know it is necessary when we talk about the berry phase. I mean how I can do a numerical calculation for considering periodic gauge mentioned above.
I want to compute the z2 invariant in 2D material by the Fukui-Hatsugai method based on the tight-binding approach not by the Z2FH code included in the OpenMX package. I saw the notes presented at a workshop at Kanazawa University by Dr. Sawahata and the OpenMX manual. I also study the references mentioned by Dr. Sawahata and his coworkers. I saw the Z2FH.C code and it seems any exp(iG.r) coefficients did not consider in the Kramars-pair or Time-reversal subroutines included in this code!; and any Translational symmetry is not included there, but the authors talked about three symmetry included: Translational symmetry, Time-reversal symmetry, and Kramar's degenerate. I know the wavefunctions in any k point derived from the Hamiltonian and then I can compute n-field, but my problem is about fixing gauge, especially about phase coeffect like exp(iG1.r) appears in the relation of symmetry gauges. I can't understand how I should implement these coefficients by a numerical calculation. gratitude to help me
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I did a calculation of the output power of VAWT with the following equation:
[Available output power]=([density of air]*[Swept area of blades]*[wind speed]^3)/2
[Real output power]=[Available output power]*[wind turbine efficiency]
I achieved some values, but I am not sure if it is the right way to calculate the output power.
According to VAWTs on market such as Makemu EOLO 3000, the output power is achieving 2kW at 8m/s which is 10 times bigger than my calculation, on the other hand, the output of EOLO 3000 was achieving more than the [Available output power]. Is there any other way to calculate the output power of VAWT?
It is better to calculate the output power per m^2 of the swept area of the blades.
It may be that the area is different in your case and Makemu EOLO 3000.
May be it is the different area which makes such large difference.
So, you have to specify the area in two cases. The other advise is to use MKS of units.
Best wishes
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A circular plasmid has three different but unique restriction sites for enzymes ‘a’, ‘b’ and ‘c.’ When enzymes ‘a’ and ‘b’ are used together, two fragments of equal size are generated. Enzyme ‘c’ creates fragments of equal size only from one of the fragments generated by those cleaved by ‘a’ and ‘b’. The plasmid is treated with a mixture of ‘a’, ‘b’ and ‘c’ and analysed by agarose gel electrophoresis. The number of bands observed in the gel is __________.
Hi there,
The rule is simple: for circular DNA you get as many fragments as sites present in the molecule. 3 cutting sites here then 3 fragments but 2 will have identical size then 2 bands on gel.
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Hello Researchers,
I am using Gauss Quadrature for numerical integration to obtain the stiffness and mass matrices for a plate element in my FEM code. We know that both these matrices are symmetric. However, I find that due to numerical integration the stiffness and mass matrix turns out to be asymmetric.
Kindly note that the asymmetry is not by any means large. The result of the subtraction of a symmetric matrix from its transpose is a null or zero matrix. If I subtract the stiffness and mass matrix from their respective transposes, the resulting matrix has all the non-diagonal terms of the order 10 to the power of -8 and all diagonal terms are zero (maybe for most cases it can be considered as a zero matrix).
At the point of writing this question, I am suspecting that this discrepancy (i.e asymmetry of the mass and stiffness matrices) is due to the finite precision arithmetic of floating-point numbers. (need your thoughts on whether my suspicions are true)
The end result of not having symmetric stiffness and mass matrices is that the 'eig' function in MATLAB gives incorrect eigenvectors although the eigenvalues are correct.
I would like to know if anyone has encountered such issues and how was it resolved.
I am also attaching a couple of links related to finite precision arithmetic errors below for your reference:
Thank you,
Jatin Poojary
yes, your diagnosis seems correct since the difference is of order of 1e-8. in computers, no floating point number can be represented exactly. therefore, it's common to use some epsilon value is used to avoid it. alternatively, when writing from scratch, only upper or lower diagonal is saved in the memory for a symmetric array to avoid such issues.
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On the basis of the Kaleman function, it is necessary to make numerical calculations for elliptical systems of equations. The results are obtained using computer packages of calculations.
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For the 2D NS equation of the stream-vorticity formula:
$\partial w/ \partial = J(\psi,w) + (1/Re)\Delta w+f(x,y,t); w = -\Delta \psi$.
Using the Fourier-Fourier basis Exp(I(k1*x+k2*y)), I got the evolution in spectral space:
$dW/dt = LW+N(W,\psi_hat)$,
where W, \psi_hat are complex. It has the solution W(t;W0) if the initial values W0 is given.
To search the periodic orbits of this problem with a period T, I tried to solve the equation
$F(W0,T) := W(T;W0)-W0=0$,
Using the Newton search, I can get:
$\partial W/\partial W0 * dW0 + \partial W/\partial T *dT = -F(W0,T)$. (*)
Now the problem I meet is: both dW0 and dT are complex since the coefficient matrix of (*) is complex.
SO how can I get a REAL dT?
THANK YOU VERY MUCH!
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Hello everyone,
I do not know the dynamic viscosity of my material (ZrB2-SiC). It's ceramic.
Is there any way to calculate it numerically?
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I have not found out any such curves by searching internet. If I try to solve it myself, then what I have is a functional differential equation, which I cannot solve. if (x,y) is a point on that curve, then (x+y''/y', y+1/y'') must be yet another point on curve (the ' indicates differentiation).
would that be a closed curve or some sort of spiral? I have not found any way to numerically solve it, for I cannot jump from (x,y) to (x+\del x, x+\del y) in small intervals. How can I, at least , numerically solve it?
Also, is there generalization of curve as 3D surface, 4D hypersurface, or in general, an (n-1) hypersurface (differential manifold)embedded in n-dimensional space?
What is seeked for a twice-differentiable curve which is expressble analytically
The locus of centers of curvature of a logarithmic spiral is a logarithmic spiral. Try to choose the constants in the curvature equation so that it is the same logarithmic spiral.
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Hi all,
I'm interested in simulating CdTe solar cells with drift-diffusion modelling (coupled with transfer-matrix modelling). I have been able to find old papers where they explain the roll-over effect observed in J-V curves (near the Voc) due to a bad back contacts (such as the papers by Burgelman), but I haven't been able to find anything recent. I'm assuming the way CdTe layers are processed have changed a lot since back in the late 90's early 2000s.
Any pointers towards a recent 2016+ paper where they manage to model the roll-over effect due to the back contact would be greatly appreciated.
Thank you!
Jason
i work on this you can see my work
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Hello everyone,
Using the spectral collocation method (SCM), I have successfully obtained the complex wavenumber k, displacement U and stress field S of different LAMB modes.
However, when I used the obtained k/U/S form the SCM method to solve the LAMB wave edge reflection problem, I had trouble with the calculation of energy reflection coefficients.
The sum of all the non-zero energy reflection coefficients (corresponding to the propagating modes) should be equal to 1, but I failed to get that right and the sum of those coefficients turned to be variable.
Power flux has been calculated as described in Mode-exciting method for Lamb wave-scattering analysis (JASA, 2004) by Arief Gunawan, and Sohichi Hirose in the form:
<P> = Real ( -iw/2*intgrate(sigma*·(du/dt), -h,h) )
If anyone has ever met this problem or knows how to solve it? Or, if there is any program available online for us to use and study?
I have been checking my program over and over again for half a month, but I still can't find the causes. Thank you so much for spending your time reading my question. I would really appreciate it if you can help me with this problem.
My program was written with reference to the article of Prof. Pagneux and has been attached as NromalIncident-Vincent Pagneux way.zip, in which the LambDispersionValidation.m can be run to check the basic results of my program (compared with the online program from https://www.mathworks.com/matlabcentral/fileexchange/73050-lamb-wave-dispersion-curve, as shown in the figure below).
1. I can get the dispersion curves (as shown in the figure below). So I think the results of complex wavenumber are right.
2. For LAMB waves propagating in +x direction, the displacement field was assumed as u = U*exp(i(kx-wt)). The numerically obtained k may have the form a + bi, a - bi, -a + bi, -a - bi (a and b are positive real numbers). In the GetkInOrder.m, I have kept only those k with positive real part (+x propagating) and positive imaginary part (physically decaying with propagation).
3. For LAMB waves propagating in -x direction, the displacement field was assumed as u = U*exp(i(-kx-wt)), so the k will be still kept k = a+bi.
Best regards,
Hao Qiu
Hi, I am actually working on the problem of energy reflection/transmission coefficient in case of piezoelectric material using the Poynting vector formula.
Can you check the work of Zaitsev please, he give the formula of power flow.
flow of pure mechanical power:
PM = 1/2 Re (ω2/v) Cijkl nk Ul Uj ,
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I have been trying to get the spectrum for HHG based on Lewenstein model but I am not able to get the correct spectrum. Are there any matlab code available to compare?
What's your method for calculating the ground state?
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I am using finite difference method to discretize the parabolic equation (explicit method), but confused about the chemical reaction term. Solving the generic equation for i specie involves a couple of species in the kinetic reaction equation since it is reversible ( Ex: A+B <-->C ). What is the best way to conquer the complexity in this manner? should I go with separation method and solve diffusion and reaction term separately. I am not quit sure if it is applicable. Thanks in advance.
Kafia Oulmi Thank you for the recommendations.
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I’m using COMSOL for circuit breaker PSS-1 2D time-dependent simulation. I have coupled all moduls: Heat Transfer in Fluids, CFD-Modul, Electric Currents, Magnetic Fields, Moving Mesh
in HT-Modul I use 3 HeatSources for Simulation of radiation, Joule-Heat(ec volumetric los), enthalpy transport.
in CFD Modul I use Force for Lorenz-Force consideration
EC and MF moduls are coupled with external current density.
MF:full field, ”in plane“ field, gauge A-fixing
boundary conditions: walls
initial values: T=293K, P=1atm(with hydrostatic correction), DC terminal 100A and ground.
between electrodes T=10000K or I tryed to use Gauss-Pulse function (quasi same effect)
Solver: fully coupled
Method: High-nonlinear Newton
Goal: to simulate plasma in switching device correct and watch internal thermo-hydrodynamics and V-A characteristics of circuit breaker.
when I start to calculate my modell, I receive no convergence in both Stationary and Time-Dependent cases.
when somebody need a comsol-file, just ask it
Agreed with the link provided by dear Ijaz Durrani
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Nowadays, many researchers are devoting their research work to the non-singular kernel. Most of them have mentioned that "the main advantage of this kind of operators is that the singular power-law kernel is now replaced by a non-singular kernel," which is easier to use in theoretical analysis, numerical calculations, and real-world applications. But in my opinion, the singular power-law kernel is very easy to use in the mentioned above calculations and applications. Kindly share your thoughts.
I prefer, instead of discussing the advantages of one operator over another, to consider that both are tools that can be useful and effective in problems of different nature.
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I have evaluated the adsorption and desorption of two pollutants from aqueous solutions in singular and binary conditions with three different adsorbents.
In my adsorption dataset, there are four inputs, which include Initial concentration, adsorbents dosage, contact time, and temperature. Among them, only the Initial concentration of pollutants is variable and the rest of the environmental conditions are constant for all the samples.
Regarding the explained condition, is it possible and rational run a linear or non-linear regression model for my dataset?
I agree with Rabin Thapa, you can run a linear regression predicting adsorption and desorption (are these two variables?) by the use of initial dosage as predictor. I wondered however, why you did not want to include the adsorbent (you said you had three different) as additional predictor.
Best
Sven
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Hello, the software I use is ANSYS Fluent. The subject of my research is the analysis of temperature distribution in the oven. At present, I have a very simple model to test, just how to try the heat of the heater to pass the dance. Or problems such as inability to converge during calculations, floating-point changes, etc. are also repeated.
1.
In the current model, the solid part has a heater, box, and door; and the fluid part has internal air, external air, and a simple cylindrical fan. There is a 0.5mm gap between the door and the box so that the internal air can be transmitted to the external air.
2.
In the cell zone condition, there is an energy parameter of 3570000 (W/m3) for setting the heater.
3.
In the boundary conditions, the six sides of the outside air, the side where the inside air connects to the outside air, and the fan exhaust are set as pressure-out. The fan inlet is set to velocity-intel, and the parameter is 1m/s. The heater, cabinet, and door surface are all set to convection, and the value of heat transfer coefficient is only 500W/m2-K for the heater, and 25W/m2-K for other parameters.
4.
In solution methods, after changing the gradient mode to green-gauss cell-based, numerical calculations are performed.
The above is a rough setting. I wonder if there are any more experienced or professional people who can help me with the problem?
When you are modelling something complex, it is always good to take an incremental approach as opposed to setting up everything in one and then get stuck when it doesn't solve.
So in this case, try building up the model piece-by-piece. First include only the air inside the oven. Does it run? If so, include the walls with heat transfer, then include the heat generation, then the outside air, then the connection between inside and outside, etc. Every time you modify the model, run it, debug as required, evaluate the results and only go forward when you are happy with the state of the model so far.
Otherwise it is very difficult to tell what is the exact issue. It can be something trivial, like a few poor quality elements, non-physical boundary conditions, mistyped material properties, and so on, but you wouldn't know. If you develop the model step-by-step, when it stops solving you will know that it is very likely due to the latest change you made to the model.
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I already familiar with Wolf and Sandri papers which suggest method to calculate the full spectrum of Lyapunov exponents by calculation of the Jacobin matrix, but it is too expensive for large (>1000 nodes) networks.
- Determining Lyapunov exponents from a time series, A Wolf, JB Swift, HL Swinney, JA Vastano - Physica D: Nonlinear Phenomena, 1985
- Numerical calculation of Lyapunov exponents, M Sandri - Mathematica Journal, 1996
I appreciate any guide in advance.
Time complexity is nearly the same as for integration of two individual trajectories. All other calculations are not very time-consuming.
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Hello all,
I am looking for an method / algorithm/ or logic which can help to figure out numerically whether the function is differentiable at a given point.
To give a more clear perspective, let's say while solving a fluid flow problem using CFD, I obtain some scalar field along some line with graph similar to y = |x|, ( assume x axis to be the line along which scalar field is drawn and origin is grid point, say P)
So I know that at grid point P, the function is not differentiable. But how can I check it using numeric. I thought of using directional derivative but couldn't get along which direction to compare ( the line given in example is just for explaining).
Ideally when surrounded by 8 grid points , i may be differentiable along certain direction and may not be along other. Any suggestions?
Thanks
The answer to a question about the numerical algorithms for resolving the issue of differentiability of a function is typically provided by the textbooks on experimental mathematics.
I recommend in particular: Chapter 5: “Exploring Strange Functions on the Computer” in the book: “Experimental Mathematic in Action”.
You can also get a copy of the text in a form of a preprint from
Judging by the quote placed in the beginning of Chapter 5, the issue of investigation of the “strange functions” was equally challenging i 1850s as it is 170 years later:
“It appears to me that the Metaphysics of Weierstrass’s function
still hides many riddles and I cannot help thinking that enter-
ing deeper into the matter will finally lead us to a limit of our
intellect, similar to the bound drawn by the concepts of force
and matter in Mechanics. These functions seem to me, to say
it briefly, to impose separations, not, like the rational numbers”
(Paul du Bois-Reymond, [129], 1875)
The situation described in your question is even more complicated because the function is represented only by a few values on a rectangular grid and it is additionally assumed that the function is not differentiable at a certain point. In this situation I can suggest to use the techniques employed in the theory of generalized functions (distributions).
For a very practical example you can consult a blog: “How to differentiate a non-differentiable function”:
In order to answer your question completely I would like to know what is the equation, boundary conditions and the numerical scheme used to obtain a set of the grid point values mentioned in the question.
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With the limited ability of personal computers, it is often impossible to do numerical calculations because it involves a very large number of cells, for example, with 10 million cells. This problem is usually experienced when facing problems with large geometric sizes. What do we do to reduce the number of cells in a numerical simulation, but give good results?
Kind regards,
Nazar
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Dear Colleagues!
I ask you for cooperation in the implementation of the project
"Numerical calculation of the counterintuitive behavior of an underwater cylinder of infinite length under hydrodynamic loading"
Detailed description in attached file
While no rotation is considered. Thank.
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I want to calculate natural convection in the horizontal rectangular enclosure (heated from below) and in the vertical rectangular enclosure by using the Nusselt correlation. However, I can not find the Nusselt correlation for the high aspect ratio and high Rayleigh number. Does anyone of you know these correlations and if yes, can you please share the references with me. Thank you.
Best Regards.
You can find the following articles useful
International Journal of Heat and Mass Transfer
Volume 139, August 2019, Pages 121-129📷
Experimental investigation on very-high-Rayleigh-number thermal convection in tilted rectangular enclosures
You can change the tilting angle to get vertical or horizontal enclosure
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How we can compute eigenvalues of a 2*2 block matrix when each block is a square matrix?
First, you need to understand the method, and then you can use Wolfram Mathematica, Maple, Mathlab, or any other software.
For block matrices, follow the Silvester method, as shown in the attached article.
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I am modeling tailing dams, specifically studying liquefaction. I have imputed a seismographic record that has a duration of 200 seconds. When i configure the "seismic phase" using dynamic analysis, it is when my question appears. There is an option to configure the Dynamic time interval but i wanna know what specifically it is and how can i calibrate it correctly consideding that i have a Seismographic record of 200 seconds duration, or it doesnt affect??? However i want to know the meaning of that and and what is his influence in the Dynamic calculation.
The time interval is used in the Newmark implicit time history integration scheme to solve the dynamic finite element equation in PLAXIS code. The time interval should be not exceeded the critical time interval which depends on the the maximum frequency of input motion and the size of the finite element mesh in order to obtain reliable solution and ensure that the wave does not move through a single step distance larger than the minimum dimension of tan element. I suggest if you could review the PLAXIS manual – scientific part- section 7: Dynamic and you will find how to compute the critical time interval.
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Which turbulence model is most suitable for numerical calculations of flow and heat transfer in annular finned tubes geometry?
Thank you very much Abderazak Bennia
So, there are 2 turbulence models that suitable for computation of finned annular tubes with RANS, i.e. k-w SST and k-e RNG.
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Solving Schrödinger equation numerically we will have finite size matrix, which has finite number of eigenvalues, that is - discrete spectrum. But we know, that atoms also have continuous spectrum. Is it possible somehow modify algorithm for quantum mechanic calculations to account continuous spectrum?
Well, correct, but then you talk about conduction electrons
free from the atoms, this does not usually need detailed calculation. You use the well known theory of nearly free electrons, as in metals. Im talking about solids, not single atoms.
I cannot imagine numerical calcultions for electrons free from individual atoms either. The spectrum would be continuous, size would not be enough and eigenvalues too many.
something like collision theory might hold.
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The wall maybe any kind of wall or shell. I mean the numerical calculation by codes and software to predict sound level and pressure. please tell me the name of the codes and the way that used to predict acoustic properties
If you are just looking to do basic transmission loss calculations, you can calculate the sound transmission loss (STL) by using STL = 10log(1/τ) where τ is the transmission loss (you can look up tables of values for common materials).
If you are looking for transmission through more complex structures in terms of geometry or material, I agree that Comsol would be a viable option.
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Hello Everyone !
I want to perform FFT for the case when a rectangular wings is oscillating in direction of roll . The vertices, being generated at the tips are also changing their locations as the wings moves. In Tecplot 360 we can only specify the coordinates of the spatial position through probe or by entering the coordinates values .
So lets say if I put a point on the vortex at certain times step , in the next step the point will remain on its position but the vortex will be shifted to another position . Is there any way to move the position of probe point along with vortex at the time passes by .
You nare not mentioning, which simulation software you are using. But as a side remark: in ANSYS CFD-Post Point objects and thereby Probed Values can be made dependent on an CCL expression value and thereby easily made be dependent on time, if that functional dependency can be described by a formula.
Regarding Tecplot I do not know.
Best regards,
Dr. Th. Frank.
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Hi dear researchers,
I am currently implementing direct collocation NMPC,
However, I have a doubt on the control value for the first control u_0
that should be applied to the system once the control trajectory is found as collocation downsamples the system dynamics before solving the dynamic NLP problem.
will the control value u_0 be (u*(0)+u*(1))/2?
x_k+1 = f(x_k,u_0);
Should this be the case, will that not have a detrimental effect in the accuracy of the true control?
Kind regards
Dear Yves, I suggest you to see links and attached files on topic.
Dynamic Optimization in JModelica.org - Semantic Scholar
An Introduction to Trajectory Optimization: How to Do Your Own Direct ...
Introduction to Nonlinear Model Predictive Control and Moving ...
Hybrid Optimal Theory and Predictive Control for Power Management ...
Lifted collocation integrators for direct optimal control in ACADO toolkit
Real-Time Optimization
Fast Numerical Methods for Mixed-Integer Nonlinear Model-Predictive ...
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Hello! Please help to find a simple cross-shore sandy beach evolution (erosion+accretion) model for using in day-week scale (i had initial and intermediate profiles (once at month), and 3 hr forcing factors - wave height, period, wave length etc.
I need to estimate beach dynamics between my surveys (primary for getting beach width and slope).
If I plotted that right, your location is on the South eastern section of the Baltic. That location has a fairly long fetch to the west. If you don't have access to any wave buoy or had an ADCP deployed on the bottom for information then you'd have to derive that from a Wind sensor that might have been located near your site. I have some clients who have had some of our Axys Wave buoys deployed in the Baltic, but they were more central or western, closer to Sweden and Poland. I'd refer you to the DHI (Danish Hydraulic Institute) in Copenhagen. Also, Van Oord, whose office is in the Netherlands, may have done some work in your region. Nortek AS (Norway) who makes ADCP for waves and currents also may have a client who has been working your area.
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General algebraic equation cannot be solved CALSSICALLY after degree 4 (Galois theory). Some case seems tracktable for the quintic using hypergeometric function and other technics. BUT I would know what is actually the best algorithm for solving numerically any algebraic equation (technic + calgorithmic complexity) ? for the two case Real and Real+Complex. Thx in advance for any answer.
HP
Hi, there is an interesting way to compute simultaneously all the roots of an univariate polynomial: the Weierstrass method. You can have a look for example to the following paper:
SIMULTANEOUS COMPUTATION OF ALL THE ZERO-CLUSTERS OF A UNIVARIATE POLYNOMIAL , Jean-Claude YAKOUBSOHN
in the file attached (proceeding of a relatively old conference). Best
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There are various methods that have been used in solving the fractional differential equations, but I am wondering what are the most powerful and efficient ones that can be applied effectively in solving the fractional differential equations?
HAM and HPM coupled with the integral transformations like Laplace, Fourier, etc. Especially if you apply them to linear/nonlinear problems by considering the fractional operators without a singular kernel, I hope you will obtain great results.
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When dealing with Boson system, we often meet Bogoliubov transformation, which is different from the unitary transformation. The matrix may be large that needs the numerical calculation. My question is how to perform the Bogoliubov transformation numerically?
see the following interesting papers
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I used differential evolution to determine the interatomic potential parameters of a model by minimizing a objective function that is the sum of the squared difference of experimental and calculated values of a set of properties. Now I'm trying to calculate the covariance matrix (and correlation matrix) for the fitted parameters. Is there a method for numerically calculate this matrix by probing the objective function? I've been able to numerically calculate the Hessian at the minimum and tried to use it's inverse as the covariance matrix, but the resulting correlation matrix has values that are outside the range (-1,1) so I'm no confident it is correct.
It you need to compute the covariance, you can use Matlab or use the simple covariance model of
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Hi,
Maxwell Stress Tensor (MST) method is usually used to calculate the optical force exerted on an object. However, my question is about the applicability range of this method. When do we use MST method and when we prefer to use other methods. Is there any reference that explains about different methods of optical force calculation?
As long as the objects are in vacuum/air, there is no problem to use MST to compute the optical forces. In fact, MST, Abraham, Minkowski, and Einstein-Laub tensors all give the same result of optical force.
However, when it comes to objects in a medium, different methods predict different results and there are issues such as how to separate field momentum and material momentum. As far as I know, there is no stress tensor that can predict forces which agree with all experimental results yet.
Here are several papers of mine and my collaborators discussing the applicability of different methods. We found that, in a medium, optical force actually depends on lattice symmetry of the microstructures, and radiation pressure on a surface can become radiation tension.
• "Electromagnetic stress at the boundary: Photon pressure or tension?", S. Wang, J. Ng, M. Xiao, and C.T. Chan, Science Advances 2, e1501485 (2016).
• "Analytic derivation of electrostrictive tensors and their application to optical force density calculations", W. Sun, S.B. Wang, J. Ng, L. Zhou, and C.T. Chan, Physical Review B 91, 235439 (2015).
• "Closed-form expressions for effective constitutive parameters: Electrostrictive and magnetostrictive tensors for bianisotropic metamaterials and their use in optical force density calculations", N. Wang, S. Wang, Z.Q. Zhang, and C.T. Chan, Physical Review B 98, 045426 (2018).
• "Electromagnetic stress tensor for an amorphous metamaterial medium", N. Wang, S. Wang, and J. Ng, Physical Review A 97, 033839 (2018).
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I'm a little bit confused in regard of implementing MMA method. Should I transform the objective function into a Taylor series in order to use this method?
Most of tutorials explain the general theme of the technique but not how to implement it.
Hi, an implementation of this method can be found (software pyOpt) :
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Hi, can we get the steady state solution directly of TPCT in Ansys / Fluent ?
Or, we should use transient solution till reach the final steady state ?
certainly.
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Hello all,
I want to implement Interval Newton Generalized Bisection method to find the root of a non-linear equation.
Though I have understood the overall idea ( start with an interval, evaluate new interval, if they intersect : unique solution). But I am not unable to understand when and how to bisect. I understand it's probably when there are more than 1 root.
May some one provide simple example/reference where this may have been explained.
Thanks
Validated Numerics by Warwick Tucker discusses the Interval Newton Method including the automatic bisection (which occurs when there is a root of the derivative inside the interval iterate): https://press.princeton.edu/titles/9488.html
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Up to my knowledge, I find that the simulation with regard to the simulate of two-dimensional mixed Volterra-Fredholm equatio is confined to be in a closed subset of R^n and there are several wonderful methods. However, I happen to get a integral equation in the following form (please see the attached picture). The only difference with the existing ones is that the integration with respect to x is from 0 to infinity and the boundary condition available is f(x,T)=h(x), with h() given, which is also quite weird. I desire to do a rough simulation in the last part to make a brief illustration but I get no idea and have no inspirations from the literature. Also, since I get little knowledge and experience in simulation of solutions to equations like that, I find it's hard for me to think out a practical method to do the simulation. I'm asking if I can get some useful inspirations from you, who may be experienced in numerical simulation of integral equations. Thanks a lot for your generouse help, your attention and your precious time.
Of course you can compactify: just change variables in x! And if you know h(x), you know how it behaves at infinity, in particular, so you know how f(x,T) behaves.
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It is said that approximate solutions are found where there is difficulty in finding exact solution or analytical solution. But still we calculate approximate solution for problems with exact solution or analytical solution. Being a student of computational mathematics I wanted to know the importance of finding approximate solution or numerical solutions.
You can flip your question around and ask: Why compute exact or analytic solutions?
Exact and analytic solutions are of great educational, academic, theoretical, comprehension and validation value, but their quantitative results for a given practical problem at hand are at best approximate due to an inevitable mismatch between ideal theory and observable reality.
In practice, when quantitative results for a given real-world problem are required, the numerically approximate estimation may often be demonstrably better (more accurate, faster, comprehensive, correct, exhaustive, efficient, cost-effective, etc.) than the analytical.
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Anscombe's quartet comprises four datasets that have nearly identical simple descriptive statistics, yet appear very different when graphed. Each dataset consists of eleven (x,y) points. They were constructed in 1973 by the statistician Francis Anscombe to demonstrate both the importance of graphing data before analyzing it and the effect of outliers on statistical properties. He described the article as being intended to counter the impression among statisticians that "numerical calculations are exact, but graphs are rough.
Yes, and I expect most statisticians and data analysts have seen Anscombe's Quartet. Anscombe's original paper "Graphs in Statistical Analysis" has been cited 1,243 times according to Google Scholar. It is a standard teaching tool for basic statistics and is included in textbooks from a range of research fields, including these:
• Quinn & Keough, Experimental design and data analysis for biologists (2002)
• Schensul, Schensul, & LeCompte, Essential Ethnographic Methods: Observations, Interviews, and Questionnaires (1999)
• Helsel & Hirsh, Statistical methods in Water Resources (2002)
• Cromley & McLafferty, GIS and Public Health (2011)
• Pearce, Clarke, Dyke, & Kempson, Manual of Crop Experimentation (1988)
• Gad & Weil, Statistics for toxicologists (1982)
• Moses & Knutsen, Ways of Knowing: Competing Methodologies in Social and Political Research (2012)
It also appears in manuals or supplementary data analysis texts for R, SPSS, SAS, and STATA.
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Which programming language is the best to use in CFD?
FORTRAN, MATLAB or any other?
Is there difference between the performance of these codes and applications such as OpenFOAM?
You may like to use an open source automated PDE solver such as Fenics:
You need to do a minimal coding (either in C++ or Python ) for writing a variational form for your problem and defining the input functions.
.
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I am to numerically calculate the value of cost, for a certain optimal quadratic regulator. Cost in quadratic regulator is J=J(t)= int((x^2+u^2).dt, 0...inf). I have the numerical values of x and u as data signals. I want to calculate the total cost. Am I allowed to use Matlab syntax: sum(x.^2+u.^2), to have a nonscaled evaluation of cost for a fixed-step solver. Is it correct to evaluate J as J= sum((x.^2+u.^2).dt), where dt is the fixed-step size. The model has been simulated by Matlab syntax ode45 or similar solvers. What happens to the calculation if I don't use a fixed-step solver? And how to calculate cost for a variable-step solver?.
Dear Harald, Thanks. With regard to your recent comment, when I calculated the cost through two different formulations as:
J=sum (dt.*(x.^2+u.^2)) and J=sum(dt.*x.^2)+sum(dt.*u.^2), I got two different values for J.
As I know, integral and sigma operators are linear in the arguments of function under integral. For example int ((f+g).dt)=int(f.dt)+int(g.dt); also, sigma (f(i)+g(i))=sigma (f(i))+ sigma (g(i)).
I've used ode45 as the solver. And in my problem, we could take u=x^3 and u^2=x^6, hence the cost could be calculated as J=sum(d.*(x.^2+x.^6)). But why I got a different value when I calculated J=sum(d.*x.^2)+sum(d.*x.^6) for a fixed d signal, d=diff (T).
I think I am ignoring something, yeah, right, there was an error, I caught. I think for a signal, if d is not fixed, for example d=[.1 .2] then [.1 .2].*[3^2 4^3] differs from d.*(3^2+4^3). It is a bit tricky but fathomable.
Please comment if you've gotten another idea, thank you. Saeb.
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I have a dataset with categorical and numerical data. I want to apply clustering algorithm (e.g. K-means). As far as I know, the data should be numerical to calculate the distance between points (samples).
Any advice how to deal with the categorical data in this case? Is it allowed to change the categorical data to numerical data? If yes, how this can be done?
Thanks
Hi Lucy,
K-means deals with categorical attributes using its distance function, where a distance of zero is assigned if the values are identical; otherwise, the distance is one. Say, you have the values red, green, and blue; then the distance between red and red is zero but the distance between red and green is one.
HTH.
Samer, PhD.
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I want to do some complicated numerical calculations in my research problems. Please suggest me any software to do this.
fortran est l'un des programmes scientifiques les plus utilisés
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Hi,
How is it possible to calculate the DDSDDE numerically when dealing with finite strain plasticity problem.
The formula which I am using currently is:
DDSDDE= 2*(dS/dC)
Where: S- Second Piola Kirchoff stresses
C- Right Cauchy-Green deformation tensor
Is this formula correct?
Best Regards,
Thank you guys for the references
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Could anybody share the source code of mathematical model of biofiltration, biosoption or adsoprtion? Usually, model is represented by system of PDE that includes mass transfer kinetic and dynamic equations, sorption isotherm and in the case of biological treatment - equation of biofilm degradation and microbiological growth.
I will appreciate any help provided. Thank you!
Dr Sheth, Yes, if I achieve some results, I will make it open source.
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We live in a world of computations, there is no doubt about it. Computations that increase in terms of computarional demand every day. What do you think is the future in solving large scale numerical models? Some say OMP, others MPI and the romantics say cloud computing. What is your opinion and why?
PS: Any other methods used to solve in parallel or other techniques are more than welcome to be presented.
Well, the most common model used today is MPI+X. MPI is used for distributed memory communication and X is one of the shared-memory methods (OpenMP, OpenACC, CUDA or OpenCL).
I would recommend a recent presentation by Bill Gropp on "MPI+X on The Way to Exascale"
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Take the viscous flow over a cylinder as an example. We know that for a specific range of Re number the Karman vortex appears behind the cylinder. This is due to the inequality in roughness of the lower and upper surfaces of the cylinder.
Also, in a numerical simulation in which the flow and the cylinder are 100% symmetric the Karman vortex appears.
One can say this is due to the round off error in the numerical calculations. But I'm not sure about that.
Does anybody have another theory or answer?
Thanks
Vortex shedding is a natural phoenomenon we can experiment constantly in our life. The symmetrical solution appears at very low Re number but it is not stable when Re is greater than a threshold. Such kind of bifurcation is very typical of the non-linear dynamics, that is when the convective part of the momentum transport has a magnitude order greater than the diffusion counterpart. Vortex shedding is still laminar and deterministic up to a certain Re number, then becomes turbulent and a wider range of characteristic scales appear.
A numerical solution, wherein the scheme has no relevant artificial viscosity and the grid is fine to solve correctely the boundary layer, simply reproduces the instabily of the flow problem.
Note that using half domain and symmetric conditions forces the solution to be symmetric, a fact that is not physical, at least if you do not solve the statistically averaged NS equations.
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Hi,
I am looking to solve an equation in three variables like
s == x*tanh(x), s == y*tanh(y), s == -sqrt(x^2+y^2)*tan(sqrt(x^2+y^2)).
Specifically, I'm only looking for the smallest non-zero value of s that satisfies this equation (there are infinite solutions in s, x, y).
I know that there should be a solution at s = 1.7089319, and that this is indeed the smallest non-zero value of s. I deduced this heuristically from an implicit plot using a program called GrafEq. (I tried the plot in Matlab but apparently it's too complicated for Matlab to handle).
However, using vpasolve can be rather problematic. Here is my code:
syms s x y
[sol_s, sol_x, sol_y] = vpasolve([s == x*tanh(x), s == y*tanh(y), s == -sqrt(x^2+y^2)*tan(sqrt(x^2+y^2))], [s, x, y], [1 2; -Inf Inf; -Inf Inf])
The output is
sol_s = Empty sym: 0-by-1
and similarly with sol_x and sol_y.
In other words, Matlab can't even find any solution for s (let alone show it is the smallest).
Apart from altering the search range for x and y (from -Inf Inf to NaN NaN or to some finite range), is there any other method or tool (in Matlab or elsewhere) I could use to solve this kind of equation numerically?
Thanks.
Hello Arnold, Matlab fslove function is very suitable for solving a set of nonlinear equations.
Regards,
Ashraf
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Can anyone please help me with Conference papers / Articles on analysis of beams  using Timoshenko beam theory, effect of geometrical properties on analysis and also papers with numerical calculations to compare the results obtained using my procedure.
I’ll be very grateful for any help you can give me with this.
Vamshi Reddy,
In the attachment You can find the dynamic solution of Timoshenko's and Bernoulli's simply supported beams subjected to force moving with velocity v0 with rotational inertia Jb taken into account.
I dont have the file in English, but "mathematics is international".
Best Regards,
Bartosz Grzeszykowski
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I am modelling a shallow foundation with midas gts nx and ı want to compare result with traditional bearing capacity method.Is there anyone know is it neccesary to apply pressure into the foundation etc.For ex after making pressure or displacement on foundation how can ı found bearing capacity?
By increasing the load on the foundation, you can monitor the settlement. Then, you can calculate bearing capacity by load-settlement curve easily.
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I want to start learn CFD, I am good enough with Hoffman Computational Fluid Dynamics book but I need an starter maybe multimedia course to get my mind ready, so if you have any good free online course for start learning CFD please let me know about that. plus I'm good at programming and numerical calculation methods
Have a look at this free course: CFD Master Class (link attached).
If you submit all the assignments you can also get certified.
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hi dear all; consider 5 complex nonlinear equations with 5 variables. what is the simplest method for solution?
Hi. The easiest way is to use  fmincon() function in Matlab. Note that, It may not give you the global optima.
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e.g. I calculate 4 combinations(i.e.nCr(10,1)=10,nCr(10,2)=45,(10,3)=120,(10,4)=210 using for loop).Can I get these four answers as four entries of a matrix?
The command I used is  >> n=10;
>> for k=1:4
nchoosek(n,k)
end  (which gives above four answers)
How to get these answers as a matrix of four entries?(1st answer as 1st entry, 2nd answer as second entry of the matrix and others accordingly).So that we can further work with that matrix
Maybe something like
n=10:13;k=2:4;
nCr=zeros(length(n),length(k));
for i=1:length(n)
for j=1:length(k)
nCr(i,j)=nchoosek(n(i),k(j));
end
end
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If I have to calculate 10C4(i.e. nchoosek(10,4)),10C3,10C2,10C1, Can I calculate all these four values using one command?
.e.g. in case of factorial If I have to calculate 1! 2! 3! 4! 5!, My command will be n=1:5 followed by factorial(n), which gives required factorials in one command.But in case of combination nchoosek(n,k) does not work the same way.It  gives single answer for single values of n and k.If anyone has an alternative solution to my problem, kindly guide me
.In short, I want to calculate more than one combination using single code or single program or single command as in case of factorial above.
In the link is a small vectorized function I wrote that will do the job for one 'n' and a vector of 'k'.
So, for your example you will need to write:
nCk(10,4:-1:1)
And you will get:
ans =
210
120
45
10
For further details take a look here:
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I want to solve the laplace equation on rectangle geometry.
My problem is that one axisc is much bigger than the other.
x=100 nm & y~1cm
One edge with Dirichlet boundary
Other two edges with Neumann boundaries equal zero
And the fourth edge with Neumann boundary equal to same non zero function
How can I overcome this badly scale ratio?
Which methods should I use?
One more thing. I goofed. My memory failed me when I cited the second example in the book. There is not a discontinuous potential in that example but I remembered it wrong. However, the first example in the book is still good for your application. Also, note that if you make one of your boundaries of interest the vertical center line in the first example, the solution will satisfy homogeneous Neumann conditions there. So this example does not force you to have two discontinuities if your example of interest only has one. But if this still doesn't get the job done (maybe you want a Dirichlet boundary condition instead) you can find a lot more examples of analytical solutions by consulting any textbook, or the internet, for solutions to electrostatics problems.
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Hey!!
I need to multiply viscous dissipation/heating term with a factor of 0.4( or say is as efficiency factor) and the resulting term should be further used as effective viscous dissipation term in further numerical calculations in fluent.
I searched a lot but could not find a way to implement this in fluent whereas several research papers that I am referring have implemented this in fluent.
Kindly help me in implementing this in fluent. Help will be highly appreciated.
Thanks
Thank you so much for your swift response.
With best regards
Buchibabu
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I'm trying to measure the pitch length of a non-perfect helix. I have some data points on that non-perfect helix for which I know (estimate) the axis and radius.
Currently, I'm sorting all the points (N points) along the axis and then measure the axial distance and also the angle of rotation for each couple of successive points (N-1 couples). So, each couple has its own pitch. Then I average all these calculated pitches, to get a single pitch for the whole helix.
The problem is that, as the helix is not perfect, so the pitch variation is significant which makes the measured pitch unreliable.
I was wondering if there is any solid and robust approach for pitch calculation of a non-perfect helix for which the axis is known?
Dear Alireza,
What about applying the method of least squares by fitting your data points onto the equation of a perfect helix, where the pitch would be one of the parameters to be optimized? The direction of the helical axis could also be optimized, if the axis is not necessarily exactly fixed.
You would need to form the equation for the helix, and then devise a way of computing the spatial distance of a given point from the helix; the latter could be done numerically by minimizing the distance along the helix, which would be a quick one-parameter minimization, or analytically if it is feasible, keeping in mind that the minimum distance is not necessarily perpendicular to the helix axis. After this, you could find the optimal parameters of the helix equation (including the pitch) by minimizing the sum of squares of the distances.
I think this would be the standard, robust way of doing it. You could use your current estimate as an initial guess for the optimization problem, which itself would be numerically quick to carry out.
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Dear colleagues, I'm solving Poisson's equation with Neumann boundary conditions in rectangular area as you can see at the pic 1. I realized fully explicit algorithm, but it costs to much operations per time. Now I'm looking for economy algorithm in 2D.
Can you advice any? If you are using c++, maybe it's any open source librarys with poisonn's solver?
Fast Fourier Transformations are not allowed.
I think you want to solve Poisson's equation in a rectangle with Neumann boundary conditions  on two sides or all four sides. Any way apply 5-point finite difference scheme and form a block tri-diagonal system and solve it by Gauss Seidel method. Since it is not time dependent problem, so nothing like explicit method. For detail see, G D Smith or Greenspan or L Collatz or M K Jain book.
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In some numeric calculations which concerned distillation process I found out that it can be higher than 1 so I wonder in which specific conditions this can occur. Thank you for any reply.
In thermodynamics, activity coefficient is used to determine the non-ideal behaviour of a mixture or departure from ideality as predicted by the Raoult's law. In more physical sense, it defines the escaping tendency of the constitutive molecules from the mixture.
So, if activity coefficient is less than 1.0, it indicates that molecules have strong attractive force and therefore more energy is required to separate them. This is also called the negative deviation from Raoult's law since the actual vapour pressure of the mixture is less than what is predicted by the Raoult's law. Example: acetone and chloroform. In distillation application, such scenario forms a maximum boiling point azeotrope. Example: water (20.2%) - hydrochloric acid (79.2%) which boils at 110 oC, higher than its constituent's boiling point.
Conversely, when activity coefficient is greater than 1.0, it implies that molecules have strong repelling force and exhibits positive deviation from Raoult's law. In this case, lesser energy is needed to separate the constituent molecules. Example: isopropyl alcohol and isopropyl ether. In distillation application, such scenario forms a minimum boiling point azeotrope system. Example: water (4.4%) - ethanol (95.6%) which boils at  78.2 oC, lower than its constituent's boiling point.
Hope it helps,
SM
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We define numerical reproducibility as an ability to obtain a bit-wise identical floating-point result from multiple runs of the same code. Indeed, the non-reproducibility of results often happens. But, we search for those applications where this non-reproducibility impacts significantly results or leads to completely wrong results. The accuracy of results is also of our interest, but not for very ill-conditional problems.
From my experience, non-reproducible results are the standard case and not the exception. Typically, as soon as one uses parallel code or optimization (I'm referring to Fortran code here) the results will differ from run to run. Differences are larger if one uses different computers and different set-ups (e.g. compiler vendors or versions).
For the actual use, i.e. when generating results, this is not a problem for me. However, when I try to write automated tests that check if the code is fine after an update, its often hard for me to find meaningful thresholds for deviations from the recorded results.
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Dear Prof.,
We have an integral in which we have singularity. We try to use Matlab for numerical calculation. However, We have not found the correct way to calculate the integral. We use trapz or cumtrapz. However we can not remove the infinities in the integral. Please let you suggest me about this problem.
use quadgk . the gauss kronrod method doesn't evaluate the function at the
boudaries. If the singularity is inside the interval, you must of cut the interval there
and integrate on the parts separately.
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i have one more question,( i am new to this field of study sorry for asking very fundamental question)
i am doing the experiment on exciting plasmon with kretschmann method.
using a 40nm gold film deposited on a slide glass.as for now im trying out water as the index matching layer between the prism and the slide gass which contains the layer.
my question is how do i calculate the plasmon excitaiton angle(with numerical calculation)  for this specific setup ?
thank you
thank you ..
do you know if there are any  specific papers which i can relate to ?
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Recently I am stuty on the Numeric Calculation of 3-D translating-pulsating source(Haskind source) Green's function in Havelock form.LOBATTO rule could be used to eliminate the singulatity of of the Green's function expressed in single integration form and the Hess &Smith Polynomial Method could be adopted to calculated the complex exponentila integral.But the complex exponential integral for derivatives of Green's function oscillates with high frequency when theta=pi due to its tending to intinity.And how to deal with this singularity of partial derivative for 3-D translating-pulsating source Green's function????????
If you need a more particular knowledge of this problem ,you could read the attachment named the singularity of partial derivative.pdf
If anyone willing to tell me any method to deal with singularity of partial derivative for 3-D translating-pulsating source Green's function in Havelock form,or recommend me any material about this problem,it will be much appreciated.
Your response will be highly appreciated.
Dear Yu,
I hadn't time to check your math carefully, but from a physical point of view there should be no problems with 'tau'->0. In general you should have two poles corresponding to a system of two Kelvin-like waves splinted due to oscillations. They turn into one standard Kelvin wave with 'omega'->0. For slow moving source the intensity of ring waves is much stronger than K-wave, but it's not important for the math because ring waves are not related to any pole, if I'm not mistaken. It means that there should no be any math divergences at 'tau'->0.
I would recommend you to check carefully the asymptotic of k*exp(k*alpha)*E1(k*alpha) - 1/alpha for 'k'->Inf (in your first document). I suspect that it should compensate the divergence. Again, if I'm not mistaken, E1(x) has step-like discontinuities around arg(x)->+-pi, so all Heviside functions should be compensated too.
Regards,
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I have a set of measured data points that empirically defines a function of two variables. I need to fit this data set with a nonlinear analytical function that is given but it contains five adjustable fitting parameters in addition to the two variables. Let us define the error measure between the fit and the data using the traditional sum-square-error (with an optional user-selected weighting function) definition. The goal is to numerically calculate the fitting parameters that minimize this error measure, and some numerical routine is needed to do that. My problem is that all of the publicly available routines that I have tried look for relative minimums in the error measure instead of the global minimum. The routine declares success when a relative minimum is found. For my specific application there is an enormous number of relative minimums, so that almost any change in the initial guess produces an entirely different returned result. I have to manually keep track of which relative minimum (out of many because of many initial guesses) is the smallest. A smarter routine would not declare success each time a relative minimum in the error measure is found. Instead, it would remember results from prior initial guesses and recognize that if the most recent calculated relative minimum is not smaller than all prior calculations, then it needs to keep working. Does anyone know of a publicly available routine (e.g., a package in MATLAB) that does this?
Did you think about any member of numerous family of genetic algorithms, simulated annealing, particle swarm optimization and alike?  Well, they do not guarantee a global solution either but at least automatcally 'think' for you what other starting point might be better.  The only way to find global optimum I know is interval arithmetics.  See Wikipedia for that.
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I am trying to update a thermal finite element model with experimental thermography data. I use a gradient based nonlinear least square optimization algorithm using the trust-region reflective algorithm.
The objective function of my results is simplified explained the difference between the experimental measured temperature and the numerical calculated temperature. The optimized parameters are the experimental parameters as the heat pulse length and the orientation of the heat sources.
Out of my results I found out that if I optimize more than 4 different parameters, the results converge to non-unique solutions which are dependent of the initial parameter values.
I already checked the independancy of the parameters and received covariance values of less than 0.4 between the parameters.
I also checked the routine using a CMA evolution strategy based on the methodology of Hansen which delivers the same problem.
The tolerance of the optimization routine is set on 10-12.
Does anybody has an idea what the problem could be?
Thank you very much for your input.
As far as I know this problem is not solved, i.e. there is no algorithm which guarantees to find unique global solution (provided that the global solution is unique indeed).  You may try non-gradient based algorithm, such as so called "amoeba algorithm". By controling the "stretching parameter" there is a hope that the barycentric coordinates would be always large enough to avoid all local minima and settle down on the global minimum. Once found, the global minimum can be refined by gradient based algorithm.
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The question is about a imaginary cubic R^3 pointcoordinate-representation in which each corepoint (monade) is a basment for a graph which orientation can be defined using two angles (phi and psi).
This could have the advantage to be able to integrate functions based on tangents within the two angles defining a single line in the ordinary coordinatesystem and can solve problems like send and return values in communication systems (VSWR - r).
Once implemented - which is a very difficult quest, there could be a way to simplify functionality that can be integrated seperately - isn't it ?
The two angled representation in sort of cubes of the R^3 can then be used for magnetic and electric waves or for example building a function for calculating the evolutionvelocity of pointamout-mathematical functions for numerical sciences.
Using the Taylor-rows with the bernoulli-numbers for tangential calculations an amount of points or mathematical elements could be paralellized and the precision could be adapted fluently!
sorry I do not have the expertise to answer this question.
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Vertical electric field is using to tune the electrical properties of channel in FETs. In new 2D material this is referred to the shifting of electrochemical potential. In contrast in simple Nonequilibrium Green function formulism applying voltage to the gate or gates (DG/double gate channels) does not influenced Fermi level position in numerical calculation. Instead, it will change the shape of conduction band.
Is there any mathematical relation or physics formula that can explain the displacement or shifting of the Fermi level with respect to Fermi level on the time that no external voltage applied to the gate or gates, (considering martial properties)?
Anybody can explain this paradox?  Is there any good literatures on the shifting of Fermi level due to applying electric field?
Dear Vahid, there is no paradox. The Fermi level shift is a consequence of applying the bias. Example: if you apply a negative bias to the gate, the channel region will become less attractive to electrons (the conduction band shifts upwards). This will change the carrier balance in the channel (less electrons compared to holes), and the Fermi level will shift towards the valence band.
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And what are the benefits of calculation r squared for linear regression ?