Numerical Methods - Science method

The triple integral I= ∫∫∫(on the domain V) f(x,y,z) dV cannot be calculated either analytically in the general case or even numerically.

Recently we have seen many attempts, some using the Schrödinger equation in QM!?, to evaluate its integral, but unfortunately all ended in failure.

Perhaps this is why Einstein introduced tensor algebra just to estimate the total garavity energy near the sun, as follows:

U= ∫∫∫ G(x,y,z) dx dy dz

.

To our knowledge the only numerical method we know for this triple integral is [1],

i= f(x1,y1,z1) . SW1 +f(x2,y2,z2) . SW2. . . + f(x27,y27,z27) . SW27

Where SW,s are the statistical weights of the triple integral.

Numerical examples,

I=∫∫∫[0,1]^3 f(x,y,z) dx dy dz

for f(x)=x or f(x)=y or f(x)=z

then I is almost the same and is equal to 13.6

for f(x)=x^2 or f(x)=y^2 or f(x)=z^2

then I have almost the same value and equals 11

for f(x)=x+y+z,

I = 40.54

1- A rigorous reform of physics and mathematics.

Determining the governing equations for turbulent combustion and the influence of chemical species and heat generation on the Navier-Stokes equations involves studying:

Recommended Bibliography:

Numerical Methods to Explore:

A small and direct answer is 'No'.

It depends on several parameters. The PDE is linear or nonlinear? The boundary conditions are Dirichlet, Neumann or mixed type? The dimension of the problem? Is it time-depends or no?....

If such algorithm exists, it will be a revolution in the field of numerical methods. 'if'

Best regards

Applying different materials to separate layers of an Eulerian part in Abaqus requires defining the regions within the part and assigning material properties to these regions. The Eulerian mesh in Abaqus is typically used for problems involving fluid flow, multiphase interactions, or highly nonlinear deformations, and it consists of a fixed grid where the material can move.

To apply different materials to several layers in an Eulerian part, follow these steps:

1. Define the Eulerian Domain and Partitioning:

2. Create the Materials:

3. Assign Sections:

4. Assign Sections to the Regions:

5. Define the Initial Conditions for the Eulerian Domain:

6. Set up the Analysis:

7. Verify and Run the Simulation:

Hi,

A sequence ( x_n ) that converges to L is said to have order of convergence q ≥ 1 and rate of convergence μ if

lim n → ∞ |x_{n + 1} − L| / |x_n − L|^q = μ.

Best

A priori the answer is yes, like any mathematical tool. The question is where does it provide "best" help? In those local processes (velocity, acceleration, ...) ordinary calculus, together with the new local differential operators, provides better results; On the other hand, fractional calculus is better in processes in which past history is significant (population growth, epidemic, tumors, ...). I must clarify something: fractional calculus IS NOT A GENERALIZATION of ordinary calculus. On several occasions I have clarified it because it seems to be a recurring error, there is no way in which the fractional derivative is reduced to the ordinary one, therefore it is not a generalization, IT IS AN EXTENSION. Some of the new local differential operators, IF THEY ARE A GENERALIZATION of the ordinary derivative, can be for a value of the order, for example, in the Khalil conformable derivative if α is 1, it reduces to the ordinary derivative, the same thing happens with our derivative N, if the kernel is 1, we have the ordinary derivative.

The best books on computational physics that you can find in my opinion are:

1.Computational Physics Simulation of Classical and Quantum Systems- Philipp O.J. Scherer

2.COMPUTATIONAL PHYSICS-Morten Hjorth-Jensen

3.A Primer on Scientific Programming with Python-Hans Petter Langtangen

First, represent the Zernike polynomial as a complex polynomial in polar coordinates (r, θ) using the Zernike radial polynomials Rl(p) and angular harmonics Pm(θ). Then, evaluate the polynomial at a grid of points on a circular domain (e.g., using a radial and angular resolution). Finally, use the complex values to create a 2D array representing the surface height at each point. You can use libraries like Python's NumPy and SciPy to perform these steps. For example, you can use the `numpy.meshgrid` function to create a grid of (r, θ) values, and then evaluate the Zernike polynomial using NumPy's `polyval` function.

I have solved many heat transfer problems using numerical methods but do not understand your question.

Dear Dr. Jeevan Kafle,

Does this program have funding? If yes, what does the funding include?

Thank you in advance.

The Euler-Lagrange equations for this are simply (d/dt)(M.dq/dt) = -K.q. to find exact solutions all you need are the generalized eigenvalues of (M,K) i.e., the values a where M.v = a.K.v. with v non-zero. If n >= 5 there are symmetric positive definite matrices for which there is no formula for these eigenvalues. But that's the only thing you need numerical methods for.

That's a good question. I rarely do that so not at answering this but you can find some nice tutorials for that from experts in this field.

Tutorials:

This paper: http://cedric.cnam.fr/~bentzc/INITREC/Files/DW10.pdf might be a good starting point.

The weighted residual method (WRM) by itself is an analytical method that is generally used to obtain an approximate solution. In principal, if you consider a polynomial basis of infinite order, you can always get the exact solution. However, we try to use lower order basis functions to capture the major properties of the solution. In simpler problems this may actually be the same as the exact analytical solution, however, in general, it's mostly an approximation.

This method is also used in numerical solution schemes such as FEM, where the solution domain is discretized into finite-dimensional small elements and then WRM is applied to each element.

You can use MATLAB or Mathematica code for solving the system of ODE of higher order by any numerical method.

Hi

It may say that you do the simulation with your own code, or with Fluent or CFX?

HEC-RAS can solve the 2-D Shallow water equation using a finite volume approach.

Louis Nguyen To obtain Non-equilibrium (NE)-SAFT from an existing equilibrium SAFT code, various changes must be made to the code.

To begin, you would need to adjust the equations used in the SAFT model to account for the polymer volume (or polymer density) dependency on temperature and pressure. This would very certainly necessitate some adjustments to the mathematical form of the equations.

Second, you'd need to change the code's input such that temperature and volume-dependent variables may be provided as independent variables rather than temperature and pressure. This would very certainly necessitate changes to the input procedures and input storage.

Finally, the output routines would need to be modified to allow for the computation of thermodynamic parameters important to the NE-SAFT model, such as the equation of state, heat capacity, and internal energy.

It may be feasible to discover a faster workaround, but it is probable that making these changes to the current code would be easier and easier than trying to find a workaround. Changing the code directly ensures that the result is compatible with the NE-SAFT model.

It's also worth noting that these revisions will very certainly necessitate a solid grasp of the mathematical underpinnings of the SAFT model, as well as the alterations required to achieve the NE-SAFT model.

Well it's a good idea to find some of them, first. The first equation implies that y=0 is an equilibrium, so a class of equilibria is of the form (x,0,z). That reduces the problem. From the last equation it then should be possible to solve for z and, from the second, for x.

Then look at the other factor of the first equation; and so on.

Hereby attached is a system of three Ordinary Differential Equations (ODE). We have to use numerical methods such as RK4 and Euler to obtain the results using Java. I think RK4 is better for this kind of problem. In this thread, Dudley J Benton has already mentioned C programming for this. It is really amazing.

For the parameter values, you can use your own values for your convenience.

High-order or low-order scheme is based on a Tailor expansion of the local linear equation, however not all the equations are linear, such as Euler equations, so we need first to consider how to approximate the problem to a locally linearization process. This is done by Riemann solver or flux splitting. However, even we can linearize equations locally, but not every where can we improve the accurate order as high as we want. When discontinuity exists, as proved by Godunov, the linear scheme must no more than 1st-order if stability computing shock waves. Break works are done by TVD scheme, ENO and then WENO schemes, if we construct nonlinear schemes we can bypass the restrict of linear schemes and gain high-order at smooth regions, but in fact, near the shock waves, all schemes must automatically reduce the order if they want get a stable result.

More to say, the linearization process of the equations are also with orders, most of them are first order based on a constant assumption of the Riemann problem. If you want to improve it order, you can use generalized RIemann solver or use ADER method or UGKS.... But they are really more expensive and may cause other problems. First-order linearilization is still the most common used in the study of PDEs.

So just to my knowledge, higher-order may not lead to higher-accuracy.

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?

Good luck.

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.

The former have been more extensively tested than the latter, for the time being.

Dear Ms kfelati Thank you very much for your useful answer

Noted professor Stam Nicolis , I will keep that in mind. Thanks for your time and help!

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.

Dear Mohammad Alipour

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 %

Dear Debasish Mahapatra ,

Please refer to the following research artcile:

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.

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😀

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:

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.

I used ANSYS software but i prefer ABAQUS

Dear Debashish,

You may find the Matlab code for chapter 2 of my book:

can be adapted to your problem. It is based on the "Method of Lines".

The code can be downloaded from:

I hope this helps.

Regards,

Graham W Griffiths

You are most welcome dear Tekle Gemechu . Wish you the best always.

Dear Y.C. Zhong

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.

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!

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.

Dear Dudly J B, thanks!

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.

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.

Dear Aram Soroushian ,

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?

Please give your comment.

Thank you.

Have a nice day....

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.

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.

see

Equations (30) and (35) are two linear relationships between x_p, x_w 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.

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

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.

Have you got answer about it?

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.

Dear Waseem

The Lie symmetry method can be applied to construct such transformations.

Regards,

Kamyar

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

Perhaps you can check this article:

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.

Thank you A N M Mominul Islam Mukut ! This document gives serious insight to my problem as well as implementation.

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)

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?

Have a look at via MATLAB :: https://au.mathworks.com/help/matlab/math/partial-differential-equations.html

ode45 : : https://au.mathworks.com/help/matlab/ref/ode45.html

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, [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.

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.

Thank you very much Sir.

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.

Thank you very much Dr. Abdelkader Mohamed Elsayed and Dr., Rashid Nasrolahpour ! I greatly appreciate that!

Dear Professor Sprott,

x = 3u^2+4u+5y-z

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.

Nice Contribution Luis E. Ortiz

Can I have your phone or email address?

You may check your report correctly once again.