Finite Difference Method - Science topic

You are welcome Uchenna Ogunka,

Yes! the same working principle for the Newmark-beta method in conjunction with the finite difference method can be applied to the dynamic Euler-Bernoulli beam equation that you have provided:

EI * w''''(x) + rhoAw''(t) + q(x) = 0

This equation accounts for the inertial effects of the beam by introducing the term rhoAw''(t), where rho is the density of the material, A is the cross-sectional area, and w''(t) is the second-order time derivative of the deflection w.

To solve this dynamic beam equation using the Newmark-beta method and the finite difference method, you can follow a similar approach as in the previous explanation. Here's a brief summary of the steps:

M * w''(t) + K * w(x) = -q(x)

where w''(t) is the second-order time derivative of the deflection w, and w(x) is the displacement vector at each node.

The main difference between solving the static and dynamic beam equations is the inclusion of the inertial term and the use of the mass matrix M when computing the acceleration at each time step. The overall approach for using the Newmark-beta method in conjunction with the finite difference method remains the same.

Keep in mind that the provided Python code in the previous answer should be modified to account for the dynamic beam equation. Specifically, you will need to adjust the equation of motion to include the inertial term when computing the acceleration at each time step ;)

Rapidly changing functions are incompatible with the finite difference method. It is much better to use numerical integration methods with automatic division of the integration step.

Then the delta-function is replaced by its continuous analog:

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a = 500;

Delta = a/Pi/(1 + (a*z)^2);

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Here is an example onan example on Wolfram Mathematica.

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Obviously, the continuous analogue of the delta function admits differentiation. This feature can be used to simulate the moment load.

Sincerely, Sorokin

In our example, we used planar finite elements with two displacements per node (x and y) and therefore considered only the first 7 bending modes in the xy plane. For the expansion process, the numerical mode shapes were further reduced to 29 displacement DOFs in the y-direction.

From your description, I assume you are trying to use spatial linear finite elements. In this case, bending modes in the xz-plane (and possibly torsional modes) are also present in the numerical solution.

In our experimental model, the excitation was only performed in the y-direction, and the response was also measured in the y-direction using uniaxial accelerometers. Thus, only xy-plane bending was excited and observed.

Due to the lack of observability and controllability for the xz bending and torsional modes, an expansion to a spatial FEM model is unlikely to be successful.

If you are interested in a comparison of different expansion methods (SEREP, SEMM, and M-SEMM), open-source code and an example can be found in the Python library pyFBS (https://gitlab.com/pyFBS/pyFBS/-/blob/master/examples/21_expansion_methods.ipynb).

Hello Sir

Your question is not clear.

Please give a clear problem to understand your question.

Martin Veidt Thankyou for the help. I just found that I was applying temperature as boundary condition instead of load.

Alireza Akbari

please write the ODE, not only the discretization You used.

however, for multi-step methods you need to create the starting values using a single-step method for all required nodes. Use a discretization of the same accuracy.

Note that the second order ODE could be written as system of two first order ODE.

Hello, check something called Kronecker product. I upload my code for 5-point stencil FDM (C is a matrix where every point defines conductivity of the domain at that point):

function U = FDM2D(C)

[nx,ny] = size(C);

dx = 1/(nx-1);

dy = 1/(ny-1);

a = ones(nx,1);

b = -2*ones(nx,1);

c = ones(nx,1);

sq = [a,b,c];

sq = spdiags(sq,-1:1,nx,nx);

sq([nx+1 nx*ny-nx]) = [2 2];

Dxx = kron(sq,eye(ny))/(dx)^2;

Dyy = kron(eye(ny),sq)/(dy)^2;

A = Dxx + Dyy;

A = A.*repmat(U2vek(C)',nx*ny,1);

A(1,:) = [1,zeros(1,nx*ny-1)];

A(end,:) = [zeros(1,nx*ny-1),1];

b = [-100;zeros(nx*ny-2,1);100];

u = A\b;

U = u2mat(u,nx,ny);

% contourf(flip(U),20); axis square; colorbar; title('Pressure')

end

% U2vek(C) function convert conductivity C matrix to a vector form c

function u = U2vek(U)

[nx,ny] = size(U);

u = zeros(nx*ny,1);

for ii=1:ny

c = 1+(ii-1)*nx; % First element of ith U line

l = U(ii,:)'; % Take the ith U line

u(end-ii*nx+1:end-(ii-1)*nx) = l; % Add ith line on top of U

end

end

% u2mat(u,n,m) function convert solution u vector back to its matrix form U

function U = u2mat(u,n,m)

U = [];

for i=1:m

c = 1+(i-1)*n; % First element of ith U line

l = u(c:c+(n-1)).'; % Take the ith U line

U = [l; U]; % Add ith line on top of U

end

end

You can take it a step further from Haswellian and Hyperbola equations and CTCS discretization equations.

(∂V_x) / ∂t = 1 / ρ ((∂σ_xx) / ∂x + (∂σ_xy) / ∂y)

(∂V_y) / ∂t = 1 / ρ ((∂σ_yy) / ∂y + (∂σ_xy) / ∂x)

(∂σ_xx) / ∂t = (λ + 2μ) (∂V_x) / ∂x + λ (∂V_y) / ∂y

(∂σ_yy) / ∂t = (λ + 2μ) (∂V_y) / ∂y + λ (∂V_x) / ∂x

(_Σ_xy) / ∂t = μ ((∂V_x) / ∂y + (∂V_y) / ∂x)

∂f / ∂t = (f ^ (n + 1) -f ^ (n-1)) / 2∆t

∂f / ∂x = (f_ (i + 1) ^ n-f_ (i-1) ^ n) / 2∆x

Dear Sumit,

you can try to start from the WAVECAR that you obtain for your relaxed structure (or the corresponding supercell, i.e. the SPOSCAR file). As you only displace one atom this might help to achieve convergence. Depending on your system size this, however, costs you some memory.

Computational Fluid Dynamics Book by Hoffman Volume 1 can give you an insight into the method. However, for any specific problem, you will find a bunch of codes on CFD websites such as CFD-online, GitHub, and ...

In general, the name diffusion equation (time dependent or not) identifies a linear PDE, whose finite difference discretization leads to a linear algebraic problem that does not require a nonlinear solver like Newton Raphson. This means that your problem is a nonlinear one, for example with nonlinear viscosity (porous media equation and such), and that an implicit method is being used for the time discretization. The correct way to approach these problems is to write the space and time discretization (there is no mention of what time discretization method is used) which leads to a (potentially large) nonlinear system to be solved

at each time step. Then at each time step a nonlinear solver is used to compute the solution of thie algebraic problem. Any of these steps might have been coded incorrectly leading to the problem you mention. One simple way to check what is going on is to compare with the results of a simple explicit method (say forward Euler) employed with a very small time step, such a method would not require NR iterations and therefore any incorrectness in the NR solver would not affect it.

Hi Jose Gaxiola,

Suppose that there is a file of nodal relative motions composed of lines with format X, Y, Z, value. I suggest:

1. Create nodeset from nodes of contact surface.

2. In Initial step create temperature predefined field with type Mapped field using created nodeset.

3. Import XYZ Point data from the mentioned file.

4. Request nodal temperature output.

5. Visualize initial nodal temperatures.

Dear researcher

Follow the link

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

Finite difference methods using a discretization on Cartesian Grid (you can choose which type A, B or C Arakawa Grid) -- I think there are few tutorials on youtube

Finite Element methods (weak form) -- https://www.youtube.com/watch?v=U65GK1vVw4o&list=PLJhG_d-Sp_JHKVRhfTgDqbic_4MHpltXZ (really good tutorial with few examples and coding tutorials)

For using piles in FLAC 3D, alot of parameters of every pile element should be determined such as bending moment and shearing force.

If you use stone columns in FLAC 3D, only some parameters of stone zone should be determined, but bending moment and shearing force of every pile element can not be obtainedobtained and the infill materials of stone columns are totally different with that for RC piles.

Best regards,,,

The following you tube is nice one

see also

Dear Kien Vu

many thanks for the link on CFD!

Dear Sourangshu Ghosh ,

It's a good question to be clearly described. I agree with all the above comments. I have used FVM since 1995, with so many practical and theoretical case. So far. the results are in good agreement with most experiment results I have tried. In my opinion, the best results of computation are strongly depend on the grid topology and the cell structure, the cell number, physical models, spacial discretization, solution methods, transient formulation (if the problem is unsteady), solution controls, etc.

Thank you.

Perhaps you may check this post:

your can use either of them, but FDM is good for your model.

I assume the (best) discretization of the nonlinear terms depends on the associated physics. For example, if you are solving the conservation laws, one might want to use a scheme which (numerically) conserves the quantities and may be other physics (2nd law consistency, for example in the gas dynamics, curl preserving schemes in MHD, etc.)

This is a typical two-point boundary values ​​problem. It's best to replace the second-order equation with two first-order equations. Then you just need to use any method of minimizing the function and integrating the equations. The possible stiffness of the equations can make numerical calculations difficult. In the case of strong non-linearity, multiple solutions can also be expected.

A good choice here would be use of staggered grid. You can associate different quantities with different grid elements, e.g. diagonal stress components go at cell centers, off-diagonal stress components go at vertices, and velocity components go to cell edges (Vy sits at horizontal edges and so on). This way you'll have, for example, dVx/dx approximated at a cell center, exactly where sigma_xx is attached.

Dear Chennakesava Kadapa,

The difference is in terms of the basis functions used.

Check the books by Prof tito toro he used to be in Manchester Metropolitan now in Italy I think they have own FD softwares

Hi Firyal Latrache,

I wrote a lot of such programs. Usually such a problem comes from incorrect derivation of equations or some small typo in the script.

Filippo Maria Denaro idea is very good, start with something very simple, e.g. 1D time dependent problem with analytical solution - compare your results.

Such problem can be caused also by the stability of the schema. Have you set a stability condition of the solution (range for dx and dt)?

You can look at this article, just assume that eps or u is equal to zero:

Go for Ansys CFX or COMSOL.

There are very few people working with AM Plugin. Perhaps, in my opinion, results would be more reliable than the traditional method. However, I don't have sufficient data to support this claim.

Respected Durrani,

Thank you for your kind response.

I just try to follow.

Regards

N Das

I suspect we are not making good progress in this effort because the problem has not been clearly stated. The objectives and expectations as presented are vague. First, you say n=0, then you speak of higher order terms. You begin with "linearize" then proceed to discuss nonlinearities. If we are not sure where you are going, how can we forge a path or know if we have arrived at the destination? I think you need to step back and consider exactly what you hope to accomplish and what constraints must be considered in doing so.

Hello Parthasarathi Subramanian,

You can use fenics. In this package, there are also some demo to solve some pdes. You can reformulate your pde to use that codes .

I think the method depends on how you are using the waveguide and what you want to learn. Do you want to calculate the modes that this waveguide would support? If so, the link pointed out by Ahmed is perfect. However, that is a rigorous analysis. If the PPW is to be used as a microwave transmission line, it may be perfectly valid to analyze it using an electrostatic approximation. Under Topic 10 at the following link I provide a MATLAB function tlcalc() to analyze many different types of transmission lines.

https://empossible.net/academics/emp3302/

You might also want to simulate waveguide circuits and look at different waveguide discontinuities. This is actually a large 3D problem. Consider using finite-element or finite-difference time-domain. Here are some links to start learning FDTD:

https://empossible.thinkific.com/courses/1D-FDTD

https://empossible.net/academics/emp5304/

Hope this helps!!

Subrat Sahu , you want apply FDTD to what kind of equations (Maxwell, elasticity, Navier-Stokes, ...) ?

You may also follow https://au.mathworks.com/matlabcentral/answers/3779-for-loops-and-zero-matlab-experience

https://au.mathworks.com/matlabcentral/answers/142015-how-to-use-a-for-loop-to-display-zeros-of-a-function

Shahriar Shahrokhabadi y Zoubair Boulahia , Vahid Moosavi necesito una ayuda en mi simulación por "Simulink" la ecuación de richards ,

Hello again

To solve the above problem,

First, enable the (''use linear wavelength spacing'') option as shown below, then pay attention to the rest of the description:

Determine the number of frequency points according to the type of divisions of your wavelength range. For example, if you want the distance to be 0.5 nano, then subtract the beginning of the interval from the end of it, then multiply by 2, and add 1 at the end.

Finally, the number of frequency points is determined.

For example, above is the number of frequency points:

(900-280)*2+1=1241

Thanks

Motahari

Dear Muhammad Bilal Ghaffar,

Peclet number relates mainly to spatial discretization, while Courant number to temporal discretization.

Courant number: The Courant number is defined as Cr= epsilon Dt/h

Diffusion number: The diffusion number is defined as S=gama Dt/(h*h)

Grid Péclet number: The Péclet number is defined as Pe = Cr/S .

They are used in Advection Diffusion Equation.When the Péclet number is high, the convection term dominates and when the Péclet number is low the diffusion term.

The Courant number reflects the importance of the convection process, the grid Peclet and Courant numbers can be respectively defined as t 1 do Pe(h) = BA. The values of these parameters are generally higher for coarser materials. For 3D transport models, the main problem is the large CPU time demand because all known finite difference and finite element algorithms have to take into account the grid Peclet number and Courant number limits.

Co = J^ _ < (0.5... 1) Limit up to 1

Ashish

Hello dear jasim

use FLAC 2D software and x velocity, n step, instead shear force.....

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.

.

.

https://civil808.com/shop/products/s23-34/23267/%D9%81%DB%8C%D9%84%D9%85-%D8%A2%D9%85%D9%88%D8%B2%D8%B4%DB%8C-%D9%85%D8%AF%D9%84%D8%B3%D8%A7%D8%B2%DB%8C-%D8%B9%D8%AF%D8%AF%DB%8C-%D8%AA%D8%B3%D8%AA-%D8%A8%D8%B1%D8%B4-%D9%85%D8%B3%D8%AA%D9%82%DB%8C%D9%85-%D8%AE%D8%A7%DA%A9-%D9%88-%D8%AA%D8%A7%D8%AB%DB%8C%D8%B1-%D8%AA%D8%B2%D8%B1%DB%8C%D9%82-%D8%AF%D9%88%D8%BA%D8%A7%D8%A8-%D8%A8%D9%88%D8%B3%DB%8C%D9%84%D9%87

All essentiall elements required to perform a numerical analysis of the finite difference schemes are discussed In the following sites:

Dear @Ashish thakur

Thank you ,for explination

Dario Fasino ,

Thank you for the clarification, however I'm a bit not very sure of whether to choose F(f + 2h𝜂) or F(f + h𝜂1 + h𝜂2) in analogy to f(i+2) (for normal functions).

Hi So-Bin Cho,

The next links may be of interest for you:

Mozhgan Mahdizadeh Rokhi

the definition of the BCs for a well posed problem for hyperbolic equations is a standard topic of Mathematics textbook about PDE. Could you describe the set of equations for your problem?

Dear Mesfin, to write a code as such, you may construct a standard finite difference scheme, and enforce the condition that x-\eta and x+\gamma are nodes of the discretization, so that u(x-\eta) and u(x+\gamma) correspond to suitable node values. You only need the space step to be a submultiple of both the delays \eta and \gamma.

Finite element method (FEM) is a powerful and popular numerical method on solving partial differential equations (PDEs), with flexibility in dealing with complex geometric domains and various boundary conditions. So it has a wide range of applications in Mechanical Engineering, Thermal and Fluid flows, Electromagnetic, Biomathematics, Geo Mechanics, etc.

Please refer the attachments for FEM

My earlier link gives a derivation of the 4th order 9-point compact stencil for a 2D Cartesian Poisson equation with a non-constant inhomogeneous term.

thanks Nariman Abbasnezhad Mohammad Amin Motamedi

No FDM is not helpful in droplet evaporation or any multiphase flow and heat transfer related problem. The answer given by Mr. Denaro is well enough to understand that you will further need some good schemes and algorithms which are only available in FVM or FEM analysis. However there are some papers which talk about the moisture transfer and cooling with the help of FVM/FEM and FDM simultaneously.

State the governing boundary value problem explicitly.

Primitive variables, streamfunction/vorticity or velocity potential? Laminar/turbulent? Steady/unsteady?

All of this could have been stated in the original question!

Doesn't ADI give you good performance? It is fairly easy. You could use multigrid, which can be very fast but it isn't easy. Why do you wish to go for an easy method?

brittle materials, such as glass and most rocks , are described as linear and non elastic

Hello Mekashaw Ali

Where are you?

Dear Mr. Martijn and Andras,

I hope things are going well with you.

First of all I highly appreciate for your help. I would like to know from you about the criteria of CHANGING IN SIZE OF ADJACENT CELLS. As you mentioned that the cell size should be smaller around or along stresses such as wells. So is there any criteria for decreasing cell size?

Thank you

finite element, b-spline, finite difference methods

In my opinion, overlapping elements can dramatically change the accuracy of the results. Mesh distribution has to be as much as regular to obtain accurate results.

Dear Megahed.

Let me introduce first. I work about 14 y with plastic zone analysis of plane framed structures. My software PPLANAV run based on NR, which mean solve some very simple problems, like columns, beams and portals.

After all this time, I didn't worked with bigger structures because of time consummed processing, which becomes higher when you have many numerical data involved and many NR iteration too.

Saying this, I may ask you how numerically the computer arrived to this negative number? I guess that you must check it precisely, to guarantee that there is no mistake in your matrices, in the answers produced first. Therefore, it recommends to slow down and analyze.

How the computer program produced this answer? There is a possibility of any kind of buckling is involved or coupled? It is there any numerical bug? It is some interdependency between the tensors that lead to an negative value?

Actually, in a case of a simple supported steel beam with a single vertical load at the mid-length, after the plasticity has formed a potential zone where will be the collapse, but there is some little elastic part yet, hundreds of iterations happen till reach the end. Making a plot of its deformed shape, between two consecutive iterations, you will see it like a string, downward and upward. Therefore, the iterations of NR, trying to null the residual force vector, make this change of direction because the structure is very sensible at this time.

So, you must certify yourself that the unstability phenomena or the resistance limit are occurring at that point, that there is a sound explanation for this numerical correction. If this is the case, everything is fine. Otherwise, look to your code and search if there is no other thing producing the unespected answer.

Good luck.

Dear Anupam Patel,

Attached herewith Software/ Research Article links that will help you regarding source code/software that can generate synthetic seismogram using using finite difference method.

01. http://www.spice-rtn.org/library/software.1.html

02.

03. http://sw3d.mff.cuni.cz/software/index.htm

04.

I hope the information will help you.

With Best Wishes,

Samir G. Pandya

in ansys workbench for static analysis, ERROR response show the released energy between proximity elements, so if your error response is minimum, your mesh, so your mesh quality is good and then your other response have accuracy

Spyros: Let’s talk about plane stress and Bernoulli beam. We use numerical techniques to simulate physical reality. So you will have to decide what is the reality in this case. You have a wall and one end of a beam is supported by the wall. If you want the beam to transfer moment to the wall, you will need to embed the beam into the wall. The amount of embedment will affect the stresses. So even if the 2d finite element had in-plane rotational dof, I would still recommend embedding beam inside 2d mesh.

Thanks dear Ryan Vogt and Debopam Ghosh

by digraph

Dear Bakir,

You have distinct weathering zones in your bench slope. So, you can define these zones in your model as a starting point. You can use GSI for estimating material properties of these zones. Since you do not have any defined joint sets, as you said, you can apply RS2 or RS3, both are FEM analysis program. Your slope dimension seems greather than REV size of your rock mass. Therefore, you can model your slope using continuum approach. Your bench slope is more prone to circular or non-circular failure mechanisms rather than discontunity controlled slip failures, such as plane, wedge or toppling.

Dear S.M. Iqbal S. Zainal

Please see the attached figure.