Science method

# Numerical Methods - Science method

Explore the latest questions and answers in Numerical Methods, and find Numerical Methods experts.
Questions related to Numerical Methods
Question
I wonder if we can list direct methods to solve the optimal control problem. I would start with:
1. Steepest descent method
Can anyone add more? With appreciations.
I do not think that the answer is so simple.
Direct methods for optimal control are a very broad class of methods that require two ingredients: a transcription approach and an optimisation method.
Virtually all optimisation methods on the market would be applicable if properly paired with a transcription method.
Already among the trancription approaches there are plenty of options:
- Single shooting with polynomial collocation of the controls
- Multiple shooting with polynomial collocation of the controls
- Collocation with pseudo-spectral methods (there are at least 5 variants)
- Collocation with Finite Temporal Elements (see or )
- Transcription with multiple discrete impulses (see Sims and Flanagan)
- Transcription with finite trajectory elements (see https://www.sciencedirect.com/science/article/abs/pii/S0094576511002852)
This is just to list some. There are also direct methods that use neurocontrollers to model the control part.
You can then, mix and match optimiser with transcription method.
Then there are extensions to multi-objective optimal control problems (see ) that use multi-objective optimisers and extensions to problems with uncertainty ( )
Your question seems to be more, can we list all optimisation methods used to solve directly transcribed optimal control problems?
Question
Many Boundary Value problems can be solved by numerical methods ,l,am looking for the possibilty of combining some numerical methods with some integral transforms in order to speed the convergence
I have also applied the principle of superposition to combine multiple analytical solutions (any one of which satisfies the governing partial differential equation) by solving a minimization problem of the residuals between the approximated and desired values. For example, a bunch of exponentially decaying sources in a field (which could be almost anything), adjusting the position and strength of each in order to best match some expectation. I used this to figure out how much of a contaminant was dumped and based on how far it had spread, when and where it was dumped. A guy who was paid to transport 8000 gallons of TCE to a disposal site parked the truck in the vacant lot next to his mistress' house and opened the valve.
Question
Fuzzy boundary broblems have many applications ,so it's very important to compined semi-numericale methods as HPM,VIM,and ADM,......with some artificial algorithms.
Dear Ms kfelati Thank you very much for your useful answer
Question
Hello all,
Source terms have been known to cause reliability issues in numerical methods affecting therein convergence and accuracy. I am currently facing a similar challenge when trying to solve a Poisson equation with a non-zero divergence velocity field. The source term that I am working which is a cavitation source term dependent on local value of pressure.
For the most part, linearizing that source term seems to solve the issue in the literature however even with linearization my Poisson equation does not converge, and even when it does, the solution is inaccurate and often oscillatory.
Any input from the experts would be helpful
Noted professor Stam Nicolis , I will keep that in mind. Thanks for your time and help!
Question
It is believed that:
1- Fully implicit block numerical method performs extremely better than explicit methods and/or partially implicit methods in solving Stiff IVPs of ODEs.
2- Higher ordered methods tend to have better accuracy of the scaled error compared to lesser ordered methods, thurs the later got the advantage in less computational time in most cases.
3- In solving Stiff IVPs of ODE with any of the above methods, the smaller the choices of the step length (say h), the higher the accuracy (lesser error) and the faster the convergence of the method.
In line of the 1-3 above. Iam simulating and comparing a 4-point fully implicit method with 4-point explicit and 4-point diagonally implicit methods. But , iam not getting any advantage in terms of accuracy or less computatinal time. Despite the fact that all the method are zero and A-stables. Please, Sirs is the mistake from the simulation or any other idea ?
I have done considerable research on the subject and published several books. Here is the truth of the matter... You can find many papers claiming that one method or another is quite superior to all others along with an example. The problem is that such authors carefully select one problem out of countless possible ones that illustrates the point they want to make. This is very different from proving that such a method is always or even often superior. While one method might work better for a few problems or a limited class of problems, it might not work well for others. Rather than embarking on a futile search for the perfect universal method, consider several proven algorithms and select the one that works best for the problem you want to solve. Select the method to fit the problem, not the problem to fit the method.
Question
In general, there are numerous methods for estimating battery state in the literature. But, which method is more commonly used in real-world EV applications?
I'd like to give the simple calculation:
SOC of an EV battery is measured by dividing the charge left in your battery by the maximum charge that can be delivered by the battery.
SOH can be calculated by dividing the maximum charge available of the battery by the rated capacity.
Units are in %
Question
Can someone please explain how the time-lag and decrement factor of a building envelope obtained by the admittance method is different from the time lag and decrement factor obtained by any numerical method?
Please refer to the following research artcile:
Question
Some numerical methods (such as FDM and FEM) permit the use of high-order schemes for discretization of computational domain yielding higher-order accurate results. The accuracy attainable by a particular numerical method is a function of many variables and the proposed usage of such numerical solutions. From point of view of a Code Developer or Computational Fluid Dynamist, can it be said that high-order discretization method necessarily produce higher-order numerical solution?
Hello to all, Nice question, maybe could be stated slightly better. In any way, if the question is: does using higher order discretization methods lead to more accurate responses, it seems to me the answer is "not necessarily". Even more if the question is: does using higher order discretization methods lead to responses converging with higher order, again it seems to me the answer is "not necessarily". To explain better, for the first case of the question, using discretization methods of higher order can lead to responses with the potential to converge with higher order. Considering the log-log plot of error with respect to the algorithmic parameter (the convergence plot), this means that close to the exact response, responses produced by methods of higher order can change on a straight line steeper (inclined more) than those associated with methods of lower order. However the location of the steeper line depends on other characteristics of the problem as well as the method, and hence, the accuracy produced when using methods of higher order is not necessarily more than when using methods of lower order. Regarding the second case of the question, when the order of discretization method is higher, the produced responses are of the potential to converge with higher order. However, if there are parameters in the problem that are approximations in convergence towards their exact values with some specific order. This specific order can cause the potential not to be materialized and the resulting responses to converge with an order of accuracy smaller than the order provided by the discretization methods. There are several references in this regard. The oldest I remember now is the paper of Penry and Wood on 1985. With many thanks for your kind attention, have a very nice and healthy day and future.
Question
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Hello sirs, as for Matlab code for fractional differential equations, I would recommend FOTF toolbox, FOMCON toolbox and Prof Garrappa's Matlab implementation.
There are also some open source softwares focus on fractional differential equations I am working on:
Any suggestion and advice are welcomed😀
Question
I would like to start a discussion of this specific topic.
Here I would like to discuss the list of possible techniques helpful for performing the simulation of oscillating bodies in quiescent fluid.
This discussion is open to all the students, teachers, and researchers.
I request you to reply here if you are familiar with code development in OpenFoam, IBPM, NEEK1000, lilypad and CFX
Pradyumn Chiwhane I am posting here a previous answer I provided on submerged oscillating bodies in quiescent fluid. The total force exerted by the fluid on the cylinder, you should consider besides the drag the added mass force. Academic references are provided within the following research projects:
Question
Some commonly used numerical methods for resolution of unstructured complex geometry include Finite Volume Method (FVM), Finite Element Method (FEM) and Spectral Element Method (SEM). Complex turbulent flow in a convergent-divergent rocket nozzle involving fluid-structure interactions may present additional challenge with grid resolution near the nozzle wall, hence the relevance of this question.
As a general comment about the FSI methodologies, at present the trend is the Immersed Boundary Method. The genesis of this formulation is in the historical series of Peskin papers. Different formulations have beenrecently developed and presented in the literature. The IBM method can be implemented in FD, FV, FEM numerical formulations.
Question
I have to study the cyclic behaviour of RC Beam-column joints by numerical method. For this study, which software will be the best.
I used ANSYS software but i prefer ABAQUS
Question
Can anyone please help me write a MATLAB program to find the temperature distribution (By numerical method) across a composite building wall subjected to periodic boundary conditions?
Dear Debashish,
You may find the Matlab code for chapter 2 of my book:
A Compendium of Partial Differential Equation Models, CUP
can be adapted to your problem. It is based on the "Method of Lines".
I hope this helps.
Regards,
Graham W Griffiths
Question
Hello everyone, I have a request about solving a system of equations (preferred with "Maple"). please tell me an effective numerically method for solving two nonlinear (higher order) equations with two unknowns. it's about a magneto static instability in dusty plasmas. THANKS
You are most welcome dear Tekle Gemechu . Wish you the best always.
Question
When the value of our Trading Account is PI and our cash is equal PI-qS.
q is number of shares of stock (S).Where q is between one and minus one.(why?)
we know:
1.  dPI=r(PI-qs)ds+qds
2. ds=s(mu)dt+s(sigma)dX
n the theory of evolution and natural selection, the Price equation (also known as Price's equation or Price's theorem) describes how a trait or allele changes in frequency over time. The equation uses a covariance between a trait and fitness, to give a mathematical description of evolution and natural selection. It provides a way to understand the effects that gene transmission and natural selection have on the frequency of alleles within each new generation of a population. The Price equation was derived by George R. Price, working in London to re-derive W.D. Hamilton's work on kin selection. Examples of the Price equation have been constructed for various evolutionary cases. The Price equation also has applications in economics.
It is important to note that the Price equation is not a physical or biological law. It is not a concise or general expression of experimentally validated results. It is rather a purely mathematical relationship between various statistical descriptors of population dynamics. It is mathematically valid, and therefore not subject to experimental verification. In simple terms, it is a mathematical restatement of the expression "survival of the fittest" which is actually self-evident, given the mathematical definitions of "survival" and "fittest".
Question
My field of expertize is in CFD and not in climatology. But I would start a discussion about the relevance of the numerical methods adopted to solve physical models describing the climate change.
I am interested in details in physical as well as mathematical models and the subsequent numerical solution.
Thank you for the very thoughtful remarks. I know the literature mentioned in your note and I admire the style and depth of these books. In many ways, the book "Big Breasts and Wide Hips" connects in my perception with Marquez's "One Hundred Years of Solitude" due to a subtle and mysterious surrealism.
Regarding the Nobel prize examples from your comment, I would like to suggest an article in The Atlantic:
(I disagree with some of the opinions above but the paper is very interesting).
As far as my philosophy is concerned, I see a big role for the models of a continuum which, in the present mathematical form, were introduced by Euler in the 1750s. I also think that we can considerably improve our numerical weather prediction models, but I also share some of your skepticism about very long-term digital projections.
Question
I'm undergoing a Masters degree dissertation on the topic Assessment of concentration and modelling of radionuclide fate-and-transport in groundwater water of Wurno LGA, Sokoto state. I'm facing challenge with the modelling aspect.
If the the transporting materials are ' solute-masses' and if you wish to use continuous models, the mathematical methods are using the PDEs. You need to apply conservation laws (mass, momentum, energy). In particular, based on the kind of diffusion that must take place, CDEs or ADEs may describe the model (in case of contaminant-mass transport). For the numerical methods, you have to specify some conditions on your data (may be ICs and or BCs). Then you can apply CFDMs, FEM or other methods...
Thank you!
Question
I study about numerical solution of fractional partial differential equation using radial basis functions (Multiquadric). According to properties of Multiquadic, if for determined shape parameter c of Multiquadric, numerical method is stability, can I say the numerical method would be stability for smaller values than it?
Dear Maryam,
The parameter 'c' in multiquadratic (and other) RBFs determines the accuracy and/or stability of the numerical scheme being implemented. Currently, no general method has been found for determining the best value for c.
Experience shows that as c becomes smaller, the RBF matrix condition number becomes larger until the equations can no longer be solved due to numerical precision limitations. Some good results for smaller c values are reported by both E J Kansa and Scott Sarra, using extended precision software.
Question
There are several numerical methods for solving IVPs (of ODEs). There are single step or multi-step methods. The methods may be classified as explicit and implicit methods. When applying numerical methods convergence, consistency and stability are among important points of considerations.
Dear Dudly J B, thanks!
Question
How to find the shape of natural pressure arch formation around underground opening?
kindly suggest the right approach using analytical and numerical methods.
If you perform (2D) UDEC-MC or better UDEC-BB you will see the rotation of the principal stresses out to e.g. about 1 diameter.
Question
Nonlinear algebraic or transcendental scalar equations are usually solved by using iteration methods. Because some of them are so intricate to solve analytically or an approximate numerical solution is required. Nonlinear algebraic equations can occur during several modeling processes or when solving nonlinear BVPs using FDMs......
I cover all of the best methods in this book, which will be free on Thursday (7/15) https://www.amazon.com/dp/B07FL7JR1J The software examples are free here http://dudleybenton.altervista.org/software/Nonlinear_Equations_examples.zip In answer to your question: It depends. Some methods are better than others. Some work well for one class of problems but not another class. In the literature you often find problems carefully selected to prove a point or make one method stand out above all the rest, which might make for an interesting paper but is not really helpful. I try to present a balanced approach.
Question
Dear researchers,
In solving the problem, by using the numerical method, we already understand that the selection of the grid size strongly influences the calculation result, so we must perform a grid independence test. Is the transient case also affected by the time-step size (time delta)? If so, it means that we must also perform a 'time-independent test on problem-solving using numerical methods. What is your opinion?
Thank you very much.
Kind regards.
Thank you so much for your valuable contributions.
Do you agree if, for transient simulation, the result must be "time-step independence", which is similar with the concept of grid-independence?
Thank you.
Have a nice day....
Question
MCMC sampling is often used to produce samples from Bayesian posterior distributions. However, the MCMC method in general associates with computational difficulty and lack of transparency. Specialized computer programs are needed to implement MCMC sampling and the convergence of MCMC calculations needs to be assessed.
A numerical method known as “probability domain simulation (PDS)” (Huang and Fergen 1997) might be an effective alternative to MCMC sampling. A two-dimensional PDS can be easily implemented with Excel spreadsheets (Huang 2020). It outputs the joint posterior distribution of the two unknown parameters in the form of an m×n matrix, from which the marginal posterior distribution of each parameter can be readily obtained. PDS guarantees that the calculation is convergent. Further study of comparing PDS with MCMC is warranted to evaluate the potential of PDS as a general numerical procedure for Bayesian methods.
Huang H 2020 A new Bayesian method for measurement uncertainty analysis and the unification of frequentist and Bayesian inference, preprint,
Huang H and Fergen R E 1995 Probability-domain simulation - A new probabilistic method for water quality modeling. WEF Specialty Conference "Toxic Substances in Water Environments: Assessment and Control" (Cincinnati, Ohio, May 14-17, 1995),
The Excel spreadsheet isn't transparent-it's exactly the opposite, since it provides the result, without showing how it's obtained. There's no problem of principle in programming a Monte Carlo sampling using Excel; just that the code won't be efficient, since an Excel speadsheet isn't designed for such calculations, when they really get interesting.
Question
For explicit numerical numerical methods Courant number should be close to 1. How is the stability criterion defined for transient simulations using implicit numerical methods?
Another issue that has to be taken into account is the convergence of the nonlinear iterations required by the implicit method. For a system y'=f(y), the solution of the nonlinear system derived from an implicit discretization by Newton's method will require to solve linear systems of the form I-dt*Jf, where dt is the time step and Jf the Jacobian of f. For too large dt, these systems may not be invertible any more, or the sufficient condition for the local convergence of Newton's method may be violated.
Question
1. How can I solve the following differential equation system (equations 30 and 35 highlighted in the image) with known initial conditions xp0 and xw0 using the Runge-Kutta method or any other numerical method? HP, HW, GP, GW, gammaP, gammaW, and gammaF are fixed coefficients.
Equations (30) and (35) are two linear relationships between xp, xw and their derivatives. First, you should solve the linear system with respect to the derivatives, so as to put it in the standard form of linear nonhomogeneous ODE systems. Then, once the problem is set in the standard form, you can apply any numerical method.
Question
The Scheme may refer to a FDM or another numerical method.
Higher order can lead to increased variance of a model. i.e. it will fit the training data better but could Be wildly off when used on new data not in the original training data
Question
LBM is described as a powerful tool in treating fluid flows.
We always hear about its advantages as a numerical method.
The question is what are the limitations of LBM when it comes to complex flows?
Below are the list of limitations and scope of further developments in LBM for complex flows"
1. Spurious currents: Unwanted or ghost currents are generated near the interface of the two-phase system due to discretization error in the forcing scheme as well as non-galiliean invariance. Many works are done in order to reduce the spurious currents yet it is challenging in various applications. One way is to use sophisticated collision models e.g. MRT, CM-MRT, cumulative etc. while the other way is to develop appropriate discretization rules for the forcing schemes.
Please refer, Christophe Coreixas , Linlin Fei for CM-MRT and a well picture of why spurious currents and how a well-balanced LBM is done by modifying equilibrium states please refer Zhaoli Guo recently published article in Physics of Fluids. Moreover, an insight into the force disrectization errors can also be understood in this paper.
2. Coupling with physicochemical processes: Complex flows are not encapsulated in mutiphase flows. Thermal and Chemical processes are still need to be coupled for the near imitation of natural processes. E.g. in drying of porous media, we need to couple the fluid flow with a thermal diffusion-advection method to show the non-isothermal effects. However, the ooccurrencef spurious currents hinder the process. One may use another set of PDFs for thermal process or use straight forward temperature equation with entropy balance but in the end it is again a query of how good your model is in respect to noisy currents.
Please refer, Feifei Qin for proper understanding of entropic LBM, using multirange forcing scheme to suppress spurious currents and then coupling with thermal effects.
3. The enigma of porous media: Complex flows need a medium like porous media for example. The intricate geometry of a porous media of widest pore spaces to narrowest channel of meniscus invasion often comes with a hurdle. For example, the Haines jumps. Haines jumps occur when a near pore-space is suddenly invaded due to contact angle instability. However, the invasion is so rapid that accurate tracking of the meniscus as well as having a mesh independent lattice domain is a challenge. To know more about Haines jumps in drying please refer my works where we overcame the challenge of mesh dependence as well as showcase the effect of haines jumps to drying kinetics.
Below are the list of papers worh reading to understand more about LBM
Question
How can optical memories based photonic crystal create a hysteresis loop in a structure?
(In the analysis of these structures using the numerical method, how can we change the power from minimum to maximum and vice versa)
Please introduce reference in the field
In all numerical analysis software, power is increased from minimum to maximum. But in the same conditions we can not minimize the power from the maximum value.
Please know how to do this process.
Question
I need the basics for including energy loss in Hamilton formulation for Finite element analysis for vibration of viscoelastic materials. The papers I read use complex modulus to represent viscoelastic losses or convolution integrals. Can someone give me a link where the formulation starts from Hamilton's principle?
Question
I want to integrate two carbon fiber materials together and want to model it mathematically considering the processes of joining the materials and the possible stress around the joints due to external loads.
I will appreciate if someone can recommend and the numerical methods which one is better suited for this purpose.
Thank you all in advance
Assuming you have a strong join (no delamination issues etc.) then you have continuity of the displacement and the stresses across the join. However, strains will *not* be continuous across the join. The weak form of the elasticity equations will still hold, and energy minimization (for static problems) also holds. Using FEM, make the element boundaries conform with the join to avoid the problems of trying to approximate discontinuous functions (e.g., strains) with the discontinuity inside an element.
Question
A lot of time i found different papers in which numerical methods execute to find the solution of given model these numerical methods apply with initial condition but in analytical methods we used wave transformation to convert PDE into ODE. Here my question is raised in any case there is numerical method in applied and wave transformation is not given and we want to apply the analytical method, analytical method need wave transformation so, how can we make a wave transformation?
If you are on a d-dimensional interval, you can expand the solution in the Fourier basis, then rewrite the PDE in this basis. For example, the heat equation would turn into a system of decoupled linear ODEs, with coefficient given by the eigenvalues of the Laplacian. This idea is the basis for the so-called spectral methods (see for example the book by Canuto, Hussaini, Quarteroni and Zhang).
Numerically, the initial condition is transformed into the Fourier domain by Fast Fourier Transform (FFT), and the final solution is transformed back into the original domain by Inverse FFT.
Question
I am reading this article to understand Richardson's extrapolation for NSFD. With MATLAB, I did the simulations and got the same result with the step size they used. But when I increase the step size, I will get negative results in the Richardson Extrapolation (I have used the Aitken-Neville algorithm). What is the problem? Is it the formula used? Or the code that I have used? that result in negative values.
Thank you!
Hi!
To be honest, I have no idea why you have these issues. Does your method have a formal order of accuracy? If it doesn't, then Richardson's Extrapolation won't do anything. Is your set of equations a stiff set? If so, then explicit methods can and will diverge irrespective of their order of accuracy. Direct methods, where the ODEs are converted into a matrix/vector system will often yield good solutions. Even if they are nonlinear, then a Newton-Raphson approach can work well.
Question
Please, how can I output global displacement (U) in Comsol? I want to calculate the compliance of my model. (Please, see the attached Comsol code).
compliance = U transpose x Global stiffness matrix x U
I have used these lines of code to obtain the global stiffness matrix (Kf).
MA = mphmatrix(model, 'sol1', 'out', {'K','E'})
K = MA.K; % stiffness vector
Kf = full(K); % global stiffness matrix
Thank you.
Hello all,
is there anybody who tried to extract the system matrices from COMSOL and tried to solve it using any of the numerical methods (central difference, Newmark's method)? If yes, could you please share your Matlab code for any dummy problem?
Thank you
Question
Lagrange multipliers might become tricky for large number of variables and most of the examples on the internet are for small systems where it is possible to try every state of multipliers (either relative inequality constraints being active or not). Is there any suggestion for systematic numerical algorithm (iterative one) to solve optimization problems with inequality constraints by Lagrange Multipliers?
Question
Considering the rate of contaminant removal as pseudo 2nd order from the batch study, column study is being performed. I would like to predict the removal pattern of the contaminant in a non-equilibrium condition. A partial differential equation is coming as advection-dispersion-reaction equation(mentioned in the attached doc file). How can I get the concentration distribution (c vs t) using this equation? I found that it can be solved in Matlab using a numerical method. But somehow I couldn't figure it out using Matlab till now. Any help will be highly appreciable. Thanks in advance.
Importantly I need the solution to that equation. Any method or process will definitely welcome. If you have any idea then kindly guide me to the solution. Thank you.
Question
I am developing a solver to simulate compression of oil in shock absorber filled with viscoelastic compressible oil. The model will consider the loss due to viscosity which will decide the response of damper to the shock it experiences. Please include any link or document you think curucial for this.
I have made models for incompressible flows but I am having a hard time to find resources for this case.
Thank You!
Thank you A.N.M. Mominul Islam Mukut ! This document gives serious insight to my problem as well as implementation.
Question
It's a very tedious job to write the charatcteristic equation and then find the eigen values, especially in large dense matrices.
Is there any general (so called) method by which one can find the eigen values of matrix?
There are many methods that can do what you mentioned.
If you care about time and work, you can use some algorithms or software codes that can do this job. For methods and codes in MATLAB, see the attached link below (last pages)
Question
1. Due to the influ­ence of university education and papers published in leading international scientific journals, young scientists, in particular, believe that this is the sole way to advance knowledge. Experienced and practically oriented hydrologists do not always share their enthusiasm.
I would like to share my thoughts. First of all, hydrology is not a science, it is engineering. Considering hydrology a science has no basis in the current educational system. Second, we need to focus on the physical understanding of hydrological process, not on meaningless mathematical refinement. Asking the right question is more important than trying to find answers of irrelevant questions. Third, there is no forum for practicing hydrologists to share knowledge and experience. All journals are in the hands of academic professors, who have very little experience or interest in the practical application of hydrology in real world; their primary emphasis is in obtaining grant money and producing papers. It appears to me that spending all grant money on real world problems for a few years will make a big difference and a solid advance in hydrology. What about giving it a try?
Question
Hello All,
I am a novice to numerical methods. I am trying to numerically solve the following two coupled equations using Matlab; any suggestions are greatly appreciated.
Flux balance in spherical coordinates:
d/dr(r^2*N(r)) = r^2*[Ko*Exp(-E/R/T(r))*(Vmax - V(r))].....................(1)
The term in [ ] is a source term.
N is zero at r = 0 (symmetric boundary condition);
Flux, N, is here is convective in nature and is due to pressure gradient as given below.
N = (K/u)*(P(r)/r/T(r))*dP(r)/dr..................................................(2)
P is Ps at r = R (surface).
I have already worked out the T(r) and V(r) profiles separately.
Thanks!
Question
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”.
For the review please see
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, , 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.
Question
In compressible flow , five equation should be solved simultaneously .I need to solve these equations by numerical methods. Is there any references that explains this algorithms??
An important preliminary step to numerical solution of compressible flow equations is to understand which kinds of flow regimes have to be studied. The most appropriate numerical methods for the high Mach number case (where Mach number = typical velocity of the flow / speed of sound and high means any value from about 1 upwards) are rather different from those in the low Mach number regime (where low means below approximately 0.1). If diabatic heating or external forces are also to be considered, other non dimensional parameters also may play a role in determining the most appropriate method. As remarked above, the literature is huge and it is impossible to give suggestions without at least an idea of the kind of regime one is interested in.
Question
I am working in Fractional Optimal Control problem of Epidemic models, I am trying to solve the problem numerically but I can not do it, as there is no toolbox and any sample MATLAB code is available to solve fractional optimal control problem. Please share the MATLAB code to solve Fractional Optimal Control Problem. Please help me.
Thank You in advance.
Thank you very much Sir.
Question
I am studying mass-spring-damper systems with coulombs friction. There are multiple discussions on simulating such systems using numerical methods and the problems that arise due the discontinuous excitation but I wanted to know if an analytical solution exists. To be mathematically clear about the problem, I am trying to analytically solve the following.
m*(d2x/dt2) + c*(dx/dt) + k*x = F*sign(dx/dt)
where the sign function is defined as:
sign(var) = 0 if var = 0
sign(var) = 1 if var > 0
sign(var) = -1 if var < 0
Note: I am aware of treating such systems as piece-wise linear nonlinear systems but I want to know whether a general solution exists that is capable of solving the problem without breaking it to a number of mini-problems.
Dear Amir, this equation has the following ananlitics. It can be glue for v more than zero exact solution and for v less than zero. In both cases they are spirals and exactly solvable, but tends to different stationary points. You can even draw its phase space picture: you draw two spirals(exact solutions)each on proper halfspace and jump from one to another, when v=0.
Question
I am interested in collaborating with any researcher working on modelling corona virus using fractional derivatives. If you are a researcher or you have a related project, please feel free to let me know if you need someone to collaborate with you on this research study. If you know someone else working on this research project, please share my collaboration interest with him.her. I would be very happy to collaborate on this research project with other researchers worldwide.
Thank you very much Dr. Abdelkader Mohamed Abdelkader Elsayed and Dr., Rashid Nasrolahpour ! I greatly appreciate that!
Question
If an Eigenfunction expansion method is used to transform a nonlinear PDE (in space and time) into a system of truncated nonlinear ODEs in time which are then solved numerically (using Runge-Kutta method or any other numerical method), can we then say Eigenfunction expansion method is a total analytic method, a semi-analytic method or just a method of transformation of PDE to ODE ?
Question
I'm reviewing a lot of papers where the authors take a 3-D autonomous chaotic system (think Lorenz) and add a fourth variable bidirectionally coupled to the other three and then report its unusual properties which typically include lines of equilibria, initial conditions behaving like bifurcation parameters, and sometimes hyperchaos. Usually these systems have two identical Lyapunov exponents (often two zeros) and a Kaplan-Yorke dimension ~1.0 greater than the dimension determined by other methods. Thus it seems clear that the system has a constant of the motion such that it is actually 3-dimensional with an extraneous variable nonlinearly dependent on the other three. Are there algebraic or numerical methods for demonstrating this by finding a constant of the motion?
Dear Professor Sprott,
x = 3u^2+4u+5y-z
Question
Dear all,
I have a differential equation which encompasses a multiplication two unknowns (u &w) at the above manner (both are functions of time and space). I do not understand how it should be discretized? I follow the procedures as space discretization, time discretization, Newton-Raphson method and variatoinal derivative, but does not look like to be correct.
Thank you.
There are plent of FEM books in literature. Please check the following paper. It precisely deals with what you are asking
Akin, J. E., A Least Squares Finite Element Solution of Nonlinear Equations, in The Mathematics of Finite Elements and Applications, J. R. Whiteman (Ed.),Academic Press, London, 153—62, 1973.
Question
There are numerous methods I have been looking in the building energy benchmark field. However, there are different lookouts for every method applied. It is very difficult to decide what's the best method for benchmark residential buildings energy.
My approach is towards the in-house appliances energy for benchamrking.
I have already gone thought some of the related papers also I attached, please suggest any best method/ methods or any means by which I got insight.
Thank you in advance
Nice Contribution Luis E. Ortiz
Question
I've used a code to calculate flow rate in DFN. Now I want to calculate permeability using the Darcy law. The flow rate is calculated in m3/m.s.
Should I multiply the flow across the mesh or divide the model width and then use Darcy law ?
Can I have your phone or email address?
Question
I am getting higher natural frequency value of a sandwich plate when computed on the basis of HSDT as compared to that of FSDT. But in general it is expected that the higher order kinematics (cubic in-plane displacement) should give lower frequency values than that computed using first order kinematics (linear displacement). I know that my result is correct, but I am unable to justify this point. kindly help
You may check your report correctly once again.
Question
Dear Colleagues
Can anyone suggest me a good numerical method for solving a system of fractional partial differential equations?
Check the following paper
Numerical scheme for solving system of fractional partial differential equations with Volterra‐type integral term through two‐dimensional block‐pulse functions
Xie, Jiaquan, Ren, Zhongkai, Li, Yugui, Wang, Xiaogang, Wang, TaoJournal:Numerical Methods for Partial Differential EquationsYear:2019
Question
I just started using gmsh for 2D triangulations and I'm observing that both the nodes and elements are being outputted in ascending order (consecutively). This is good as it makes reading the data easier but I want to know if this is always the case.
Secondly, I also observed that the physical-line boundaries are outputted as elements with two nodes  (line-elements in a 2D triangular mesh). I have also figured out that the actual elements (triangles) are listed after these line-elements. This then means that the true number of triangles (elements) in the mesh is the number of elements oputputted in the mesh file less the number of line-elements. Kindly let me know if this is the correct idea.
Thanks in anticipation.
The documentation on the MSH file format states:
"All the node, element and entity tags (their global identification numbers) should be strictly positive. (Tag 0 is reserved for internal use.) Important note about efficiency: tags can be "sparse", i.e., do not have to constitute a continuous list of numbers (the format even allows them to not be ordered)"
So better not make that assumption. I would be surprised if nodes and elements in MSH files generated by gmsh weren't in ascending order. However, gmsh can still read and modify MSH files for which this is not the case - this could for example be the case when you're manually editing or creating MSH files.
Question
All,
I have two non-linear coupled differential equations. Is not a problem to solve them with Euler’s numerical method. The system is very stable and converge to a better solution every time you make h smaller.
But how do you prove the stability for this Euler’s numerical method for the two non-linear coupled differential equations?
Thanks
Hi Jim,
Without knowing more about your particular problem, I offer the following:
First linearize the system to form, du/dt = A u, where A is a 2x2 matrix and u is a 2x1 vector. Then calculate the eigenvalues. If they are all contained within the "Euler stability contour" in the left-half plane, then the system will be stable for small perturbations. However, if any of the eigenvalues are exterior to the stability contour, then the system is likely to be unstable.
The above also applies to nonlinear coupled ODE systems of any order. To find examples of this process, do an Internet search on: ODE stability. The wikipedia entry: https://en.wikipedia.org/wiki/Euler_method is a good starting point.
I hope this helps.
Kind regards,
Graham W Griffiths
Question
A Research on roundness of a circle: An investigation of pie π. Since π is a long digits irrational number can π actually be constant? Or is it gotten from iteration of closed figure using numerical method? We will be using a 2d circles from 3 d circular objects, we will check the ratio of it diameter to it perimeter after cut neglecting the reduction in length due to cutting chips.
π is use in the determination of some parameters and quantities in mechanical engineering design and fluid mechanism e.t.c. and most importantly in force distribution of rotation of earth F=2πf Alare's Formula. So exactness is important to know if the forces are evenly distributed or turbulent .
Do you have any reference on π ? Or have you done any research on investigation of π? Help me with solution. We need a team to join I and Kehinde Alare.
Thanks sir
Question
Hi everyone,
I have been working on a laminar compressible microtube flow analysis (supercritical fluid). Mass flow inlet, pressure outlet and constant wall heat flux boundary conditions are applied to the circular microtube (0.5 mm diameter). I get fluid properties by using CFD Post and there is a mistake at CFD Post total pressure results (it was used areaAverage at related cross-section plane) as can be seen on below figure. In addition, theoretical total pressures are calculated by using momentum equation between two control surfaces (the related values (density, velocity etc.) were taken from numerical solution). We are also suspicious about getting this mistaken result due to operating pressure condition which is 101.325 kPa. I would like to grateful if you could share your recommendations.
Actually, it was carried out lots of validation for this study (based on the change of enthalpy and mass flow rate) and everything seems quite logical. Also, as Oğuzhan mentioned, the change of pressure towards axial direction was calculated by using momentum equation (pressure drop was calculated depending on friction resulted from wall shear and thermal acceleration) and the result can be seen in figure entitled as calculated, so as the change of pressure make sense for this theoretical approach calculated by using the Ansys data (density, velocity, etc.), the change of pressure taken from CFD-post (it is stated "Ansys Results") by using areaAve(related cross-section plane) definition is not logical. Is there any missed point about compressible pressure drop calculation in Fluent? (For example, selection of compressible flow etc.). Thank you for helping.
Question
What are the possible methods to solve non-linear system of ordinary differential equations of n equations containing n variables?
List possible analytical method through which we could solve the nonlinear system. Also list possible numerical methods as well.
Share any material or article or book or even videos you know worth related about solving nonlinear equations.
Thanks.
Finding a closed form solution is difficult to say the least. However, there is a rich theory on the behavior of such solutions. There is plenty of work on numerical analysis of this subject. However, there is a wealth of work on the behavior of such systems - particularly their asymptotic properties. For autonomous systems, i.e., x_dot=f(x), where f is independent of t, at a point where f(x0)=|= 0, locally a solution curve through x0 behaves like the vector field f. The questions arises about the solution near a equilibrium point. f(x0)=0. This gives rise to the stable, unstable and central manifolds. That is locally the equation is approximated by the linear equation x_dot= Df*x. The eigenvalues of Df(x0) determine the behavior of the solutions and define the stable manifold ( associated with eigenvalues with negative real part), unstable manifold (associated with eigenvalues with positive real part) and central manifold (associated with eigenvalues with zero real part). A point x0 is a hyperbolic equilibrium if f(x0)=0 and none of the eigenvalues of Df(x0) have zero real part.
The famous theorem of Hartman and Grobman ( developed at the same time, Hartman in the US and Grobman in the former Soviet Union unbeknown to one another) give a complete characterization of hyperbolic fixed points. There has been much work on analyzing the central manifold since the work of G. D. Birkhoff and still on going today and has led the concept of a dynamical system and is extremely active.
For the case where the system is non-automous, i.e., f = f(t, x), there is a wealth of literature examining the asymptotic properties of the solutions of the equation. This field of endeavor goes by the name "asymptotic integration." In this case often the methods used establish existence of a solution and to derive proprieties come from functional analysis, fixed point theorems, topological arguments, etc.
A good place to start is Phil Hartman's, classic reference, known by many as the "tomb", "Ordinary Differential Equations." Also a good place to start with dynamic systems is the work of Steve Smale, Charlie Pugh and the references there in their work.
Question
I need to simulate the transition of Flow through Coarse Porous Media such as Rock-fill dams, to investigation of Water level profile in each distance from up-stream and determination of discharge of fluid seepage from the body of these media. Notice, I want to simulate a Single-Phase Flow.
How can I simulate this project? Please suggest and introduce a useful software for this issue to me...
What is your idea about Flow 3D, Fluent, ABAQUS,...
Dear Dr. Majid Heydari ,
It would be appreciated if you explain more about the method you used to simulate the POROUS MEDIA for your Ph.D. thesis.
Best regards,
Mehdi
Question
Please help me to solve numerically of the following 4th order nonlinear ODE
f''' + ff'' +1 -f'*f' +k(ff''''-2f'f'''+f''*f'')=0
boundary conditions
f(0)=0, f'(0)=0, f'(infinity)=1, f'''(0)= -[1+kf''(0)]
where k is a parameter and the solution exists for k<0.325786 and one may choose infinity= a finite number, say 10. I am unable to solve it through 4th order Runge-Kutta with shooting technique and Matlab solver 'bvp4c'. Here I have also the attached the article in which I found this equation.
The difficulty is the condition f'''(0+kf''(0)=-1.
Question
Sometimes we hear somethings on stiffness problem of solution for chemical reaction modeling.
There is an option in Fluent software to figure it out.
Have you guys used it so far? tell us when and why did you use it?
As Pierre Mandel kindly mentioned, when characteristic scales of your problem are very far (orders of magnitude) you see Stiff Differential Equations (ODE or PDE), It means you have small stability region or it means you have second or third order "Bifurcation" which needs a very high order numerical solution schemes and small time (discretization in general) steps.
There are two types of stiff problems in modeling advection-diffusion-reaction (ADR):
- If you model single specie, when time scale of Peclet number and Damkohler number are very different, you have an stiff ADE (or ADR, or transport equation), you can solve it either by implicit solver or small time steps, or dual time step, or other numerical tricks.
- If you have multi-rate mass transport and your species are internally related (reactive transport), you cannot solve this problem with small time step. Mathematically you have to get integral of infinity and you are always unstable. Those problem can only be solved with implicit solvers (all species in one shot!)
If you need more information on this topic, a very good and classic source for learning is this book chapter:
Approaches to modeling of reactive transport in porous media
Carl I Steefel, Kerry TB MacQuarrie Publication date: 1996/1/1 Reviews in Mineralogy and Geochemistry Volume 34 Issue 1 Pages 85-129 Publisher GeoScienceWorld
Question
Guys, Sometimes after initializing the problem in Fluent and triggering the calculate button, a residual/residuals start from a big non-zero magnitude. For instance, to me, it in many cases happens for some convection-diffusion equation when I'm simulating combustion. Interesting point is that the results looks ok and have good agreement with experimental results. In this page:
It's said that:
"For most problems, the default convergence criterion in ANSYS FLUENT is sufficient. This criterion requires that the scaled residuals defined by Equation 26.13-4 or 26.13-9 decrease to 10-3 for all equations except the energy and P-1 equations, for which the criterion is 10-6."
As it is said, the residuals need DECREASE TO 10-3. Then it means that it doesn't matter from where the residual starts, it should finally decrease to 10-3.
1. Am I right?
2. What's your opinion?
Another question:
3. Why does it happans? (I mean why does the residual start from a non-zero magnitude?) and How to figure it out?
I've attached two screenshots of the residual of a project for instance.
In my honest opinion there are many situations where this statement from the ANSYS Fluent manual is rather misleading, if not even wrong. In particular in CFD simulations with strong heat and mass transfer almost the entire physics, material parameters and thermodynamics is bound to a veery good convergence of the energy equation. In my opinion a scaled residual (RMS Res) of 10^-3 for the energy equation usually leads to rather physically wrong fluid temperatures (i.e. large errors in comparison to the real physics) with the corresponding consequences for the accuracy of the overall simulation.
The CFD best practice is:
1) define meaningful, local and sensitive physical target quantities (e.g. a temperature profile or a distribution of a certain sensitive species of your mixture).
2) run a series of CFD simulations on a reasonable resolved mesh (not the coarsest one) with systematically refined (and reached !!) convergence criterion, i.e. RMS Res<10^-3, <10^-4, <10^-5. And so forth, if still there is not yet a convergent trend.
3) Make a comparison of your sensitive target quantities from this series of investigations. If the target quantities do NOT change anymore significantly (in the range of your wanted error level of the simulation), than you have found your convergence criterion to which you need to converge own all of your following simulations.
Any other explanation or strategy is not leading to realiable CFD results i my opinion. You can read this strategy in the ERCOFTAC Best Practice Guidelines for Industrial CFD and in the ASME Standard for CFD Simulations and Heat Transfer. And once it is an European and American standard approach, I think we should strive to fulfill it, doesn't we as responsible engineers?
Best regards,
Dr. Th. Frank.
Question
I need to use matlab to get the displacement and velocity time history of the 3 degree of freedom system (image attached). i can use any numerical method or ode45 but one of the element in third equation has sigma of udot which cannot be taken as constant. Kindly help how to solve these type of equations?
Any help will be greatly appreciated.. Thanks in advance.
Yew-Chung Chak .. Now i got it. Thanks very much for your help.
Question
Yanai (1973) fluxes computation methods for atmospheric data at grid points available at equal time interval.
And Dennis Shea points out an example at:
Question
As we are aware of the importance of eigen values of matrix and also the mathematical efforts needed to solve for the same. But can we reduce those efforts or can we at-least generalize the method to compute all the eigen values. As we know:
1. power method is mostly used numerical method to compute eigen value , but as the size of matrix increases it's not smart approach to use this method.
2. Using characteristic polynomial , computationally, is even cumbersome approach because one needs numerical method to find the roots of characteristic polynomial.
Can there exist any algorithm which can be used on general matrix to find the eigen value??
In addition to the helpful answers, I have a little bit to add. Mathematicians failed to find exact roots of polynomials with degree n > 4 ( Galois theory). Numerical approximations are a must. The study of autonomous dynamical systems or similar problems is necessarily based on the eigenvalues. But exact solutions are far-fetched, so they turn to study the qualitative behavior. Bounds of the eigenvalues are enough to predict the solution behavior of the whole system under discussion.
Question
When the buoyancy effect is significant in supercritical fluids passing in the tube, some eddies can form in any location of tube, especially at the vicinity of supercritical temperature. So, what is the numerical method to solve like the compressible problem? As my experience and literature, there is no way to calculate this type of problem by using RANS models, so I have just started using LES models. Although it was used a very low time step size (about E-5) and tried all of the subgrid models, I couldn't solve this problem. I would like to be grateful if you could share your recommendations. Thanks
Are you looking at density variation (compressible) and phase change at super critical temperature ?
Gravitational effects in horizontal ducts are observed at very low velocities such as free-forced convection situations .
i would advise dense phase model in cfd to assess such situations.
please elaborate the problem so that we can discuss in detail.
i would suggest to look into physics so that we can choose appropriate model for analysis
Question
Computational costs are important in numerical methods, as if they were obtained by counting the number of operations(flops). The order to count this number has been asked by MATLAB that I have recently encountered.
Zahra,
If you have a mathematical expression in a form of a matlab code, then you can count the number of the multiplications to count manually the multiplication operations to execute this mathematical expression.
A practical way to measure the complexity is by means of the cpu time. As the cpu time increases it means more complexity. The matlab can output the spent time by the computer to run the code.
Best wishes
Question
We know that , In applied mathematics, discontinuous Galerkin methods (DG methods) form a class of numerical methods for solving differential equations. They combine features of the finite element and the finite volume framework and have been successfully applied to hyperbolic, elliptic, parabolic and mixed form problems arising from a wide range of applications . so i want to know ,What is the difference between the discontinuous Galerkin method(DGM) and discontinuous Galerkin finite elelment(DGFEMs) method?
Give examples.
Question
Dear Professor Baines,
We need the DOI number for you book edited after the Reading Conference in 1985 :
Conference on Numerical Methods for Fluid Dynamics,
Reading, U.K., april 1985.
J.M. VANEL, R. PEYRET, P. BONTOUX, “A pseudospectral solution of vorticity-streamfunction
equations using the influence matrix technique”, Num. Meth. Fluid Dyn., II, Clarendon, ed. Morton &
Baines, 463-475, 1986.
Would you be kind to give us Roger Peyret and myself the DOI reference number requested by ISI Web Knowledge and Publons.
Sincerely, with our best regards,
Patrick Bontoux and Roger Peyret
Dear Doctor,
I thank very much for the information concerning an URL reference ! I will inquire with Publons which asked me the DOI information referring to this publication.
Great thanks,
With my best regards, Patrick
Question
How can one find a vector field B that satisfies the following equation for a known solar function phi which operates on 3D real space?
What kind of equation is this, are there known methods to solve this equation for B (either analytic or numerical methods)?
Thanks
Yes, B is divergence free in this example, and so we do have that del^2(B) = 0. So does this problem become a Laplace equation with boundary conditions given by the original equation curl(B) = grad(phi)?
Question
Dear colleagues
Hi,
As you know, for determining the order of accuracy of a numerical solution on a uniform mesh one normally needs to successively double the number of grid points in each direction (for a rectangular solution domain) and calculate the error at a specific location, say x ,which is common to all the grids (i.e. coarse, fine, and finer, if two levels of mesh refinement is employed) then use Log(E2/E1)/Log2=m, where m is the point wise order of accuracy, E2=ABS(Fine solution(x)-Coarse solution(x)), and E1=ABS(Finer solution(x)-Fine solution(x))
However, when a structured boundary layer-type non-uniform mesh is employed the situation is much more complicated. For example, doubling the grid points in each direction (in a rectangular domain) may not guarantee that at a particular location, say x, the grid spacing will also be halved. Not to mention that doubling the grid points may also not guarantee that the reference point (x) coincide with one of the grids in the fine mesh.
My question is how can we calculate the point wise order of accuracy of the numerical method in this case?
To elaborate, I should say that I am using the LBM to simulate a natural convection problem.
Thanks Dear Prof. Filippo Maria Denaro
I followed your suggestion and studied the text by Peric & Ferziger it was very useful. I generated structured non-uniform meshes at three levels (fine, medium, and coarse) by first producing a fine mesh and then extracting coarser mesh from it by skipping grid points. This way I could get a grid spacing ratio of about two for all of the media. As Peric & Ferziger suggested, if the non-uniform mesh is smooth enough and the grid spacing ratio among the successive non-uniform meshes (fine, medium, and coarse) remain constant, the order of accuracy can be predicted by:
Ln(E2/E1)/Ln2 for each grid point. where 2 is the grid spacing ratio in my case.
Question