Science topic

# Finite Difference Method - Science topic

Explore the latest questions and answers in Finite Difference Method, and find Finite Difference Method experts.
Questions related to Finite Difference Method
Question
Please, can the Newmark-beta method for time integration be used in conjunction with the finite difference method (specifically central difference) for the spatial nodes to accurately solve the beam equation? I am trying to implement this for the 4th-order Euler-Bernoulli beam equation with fixed boundary conditions.
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:
1. Discretize the beam into a series of nodes and use the central difference method to approximate the second and fourth derivatives of the displacement with respect to position, as described earlier.
2. Assemble the stiffness matrix K, the mass matrix M, and the damping matrix C (which will be zero in this case, as damping is not included in the equation).
3. Set up the dynamic equation of motion as follows:
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.
1. Use the Newmark-beta method to solve the dynamic equation of motion for the displacement and velocity at each time step, based on the current state and the previous state. Since the equation now includes the inertial term, the mass matrix M should be used to compute the acceleration at the current time step.
2. Apply the necessary boundary conditions to the solution.
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 ;)
Question
How to access/read the state variable (m_epdev) from inside the Fish UDM in FLAC for various grid zones, like the central zone (2, 2) and adjacent zones (3, 2), (1, 2), (2, 3), and (2, 1), at each iteration step?
(1,3) (2,3) (3,3)
(1,2) (2,2) (3,2)
(1,1) (2,1) (3,1)
Required: state variable (m_epdev) value at each zone at each iteration step inside Fish UDM.
A fruitful reply will be highly appreciated.
Regards,
Dipaloke
Question
I am trying to mathematically model the derivative of the Dirac delta function for the Euler-Bernoulli beam equation using the finite difference method. I have tried using the Gauss and Lorentz delta sequence kernel, but neither could model the Dirac delta accurately for the piezo-beam equation. Any advice would be appreciated.
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:
- - - - -
a = 500;
Delta = a/Pi/(1 + (a*z)^2);
- - - - -
Here is an example onan example on Wolfram Mathematica.
- - - - -
Obviously, the continuous analogue of the delta function admits differentiation. This feature can be used to simulate the moment load.
Sincerely, Sorokin
Question
If there is 2 DOF (Translational, Rotational) Cantilever beam, and it has 7 elements, I want to use System reduction technique to get rid of the rotational Dof, Do I expect to obtain the same first seven natural frequencies?
Because I tried to use SEREP technique and I did not get the first seven natural frequencies, I got the 1,3,5,7,9,11,13 natural frequencies?
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).
Question
• Two-dimensional Steady State Conduction in a Slab Irradiated by a High Energy Laser Beam at The Surface: Use the finite difference method to solve a two-dimensional steady state conduction in a rectangular aluminum (k =200 W/m C) slab subjected to a constant surface heat flux irradiated by a high-energy laser beam at the top surface. For simplicity, assume the heat flux distribution to be to be a constant average value IO = 2 X 108 W/m2 acting over a section of the surface equal to the beam diameter, d=4mm as shown in the figure. The remaining portion of the top surface is subjected to convection with hc=100W/㎡ C. All other surfaces are assumed to be maintained at constant temperature of 𝑇∞=25 C.
Hello Sir
Question
I am analyzing RC beam under elevated temperature using heat transfer approach. The temperature is applied uniformly over the entire surface. I have also defined Amplitude and thermal interaction.
There is no variation in nodal temperature.
Is there any issue in modelling?
#FEM #ABAQUS #heattransfer
Martin Veidt Thankyou for the help. I just found that I was applying temperature as boundary condition instead of load.
Question
Hi
I'm solving nonlinear second order equation by using finite difference method . finally for calculating value at any desired node, knowing three preceding nodes is required however by knowing boundary condition just one of these nodes becomes obvious and still knowing two other values is necessary. it must be noted there are plenty of guesses for values of these nodes which lead to compatible response.
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.
Question
Can someone please help me with coding a a general expression of n-stencils (where n<m, with m being the number of points) for a second order differential matrix (using finite differences). I have the following code, but it is not giving me the matrices I'm looking for. Thanks in advance.
function D = myfinD()
x = chebpts(6);
%x = linspace(-1,1,6)';
stencils = 3;
x_ = chebpts(stencils);
%x_ = linspace(-1,1,stencils)';
n = numel(x);
d2L = zeros(n,stencils); % Defining an empty matrix
for i = 1:stencils
dL = zeros(n,1);
for k = 1:n
if (x_(i)~=x(k))
for l = 1:n
if (x_(i)~=x(l) && l~=k)
Lprod = ones(n,1);
for j = 1:stencils
if (x_(i)~=x_(j) && j~= k && j~=l)
Lprod = Lprod.*(x-x_(j))/(x_(i)-x_(j));
end
end
dL = dL + Lprod/((x_(i)-x(k))*(x(k)-x(l)));
end
end
end
end
d2L(:,i) = dL;
end
full(d2L)
if (rem(stencils,2)~=0)
if (stencils ~= 1)
m = (stencils-1)/2;
D = spdiags(d2L, -m:m, n, n);
else
D = spdiags(d2L, 0, n, n);
end
full(D)
else
m = stencils/2;
D = spdiags([d2L(:,1:m),zeros(n,1),d2L(:,m+1:end)], -m:m, n, n);
full(D)
end
% confirmation of a three stencil FDS
m = numel(x);
D2 = zeros(m,m);
h = diff(x);
h1 = h(1:end-1); h2 = h(2:end);
d1 = 2./(h1.*(h1+h2)); d2 = -2./(h1.*h2); d3 = 2./(h2.*(h1+h2));
D2(2:end-1,2:end-1) = spdiags([d3 d2 d1], -1:1, m-2, m-2).';
D2(2,1) = d1(1); D2(end-1,end) = d3(end);
full(D2)
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
Question
I am trying to solve a two dimensional elastic problem of following image to find out stress, strain, displacement of the body under the given condition. I intend to use finite difference method for solving the problem
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
Question
I tried phonon calculation using the following tag in INCAR for a Double Perovskite material.
IBRION = -1, EDIFF = 1e-8, IALGO = 38, ISMEAR = 0, SIGMA = 0.05, ADDGRID = .TRUE, ENCUT = 350, LREAL = .FALSE, LWAVE = .FALSE, LCHARG = FALSE, PREC = Accurate. NELM = 150
But convergence is not achieved even after a lot of cycles. (attached screenshot)
Kindly suggest...
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.
Question
Various methods are used to solve and analyze fluid flow problems. One of the methods is the Finite difference method. I wanted to know where can we get some general codes of finite difference methods to learn and implement them.
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 ...
Question
I am trying to solve a time-dependent diffusion equation with finite-difference discretizations using the Newton–Raphson method. However, I encountered some problems in convergence--the solution result changes when changing time step size. Here I want to find some coding examples to refer to. Thank you.
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.
Question
Hi everyone,
I am currently trying to create a contour plot(ABAQUS) of relative micromotions between two surfaces in contact. I have already developed a method for calculation of relative motions. However, I haven't been able to assign my obtained values to the nodes of the geometries that I am evaluating.
Until now, I have read that some people use subroutines to track paired nodes between two surfaces in contact and then they assign the obtained values in a form of a contour plot. Alternatively, developing a MATLAB code and creating a sort of scatter plot with the relative micromotions has also been discussed on some research works.
Nevertheless, I am trying to find a more friendly alternative that doesn't involve a developed code or the use of subroutines.
Creating a field variable from field output is not an option because the first task is to pair closest nodes between surfaces in contact, which is not possible using the operators available here.
If I need to explain myself better, I can provide more details. I will really appreciate any help on this matter.
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.
Question
I'm trying to redo an article (file attached) that uses the hybrid finite difference method - the differential transformation method (DTM FDM) to solve heat conduction problems. I have used Maple but there is something wrong. Can someone help me fix the code?
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
I have the following Laplace's equation on rectangle with length a and width b (picture is attached):
ΔU(x,y)=0
Ux(0,y)=0    : Isolated boundary
Ux(a,y)=f(y) : Current source
U(x,b)=0      : Zero potential
The fourth boundary is quite complex :  Mixed boundary condition (isolated except in two points)
if x=a/3 or 2a/3
U(x,0)=0
else
Uy(x,0)=0
Is there an analytical way to solve this kind of mixed boundary problem? can someone point me to the right direction? I'm a bit lost
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)
Question
Dear all,
What are the factors to be taken into consideration when modeling stone columns in FLAC 3D? Are there differences between the modeling of stone columns and piles?
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,,,
Question
I am working on lid driven cavity problem in porous media. I have calculated the fluid flow using finite difference method and average nusselt number by simpson rule varying y from 0 to 1. I don't know how to calculate the entropy generation and bejan number which is described in the attached figure.
THANKS
Question
Can someone explain to me the difference between forward difference method and the backward difference method?
Question
Just to discuss on advantages of CFDMs
Dear Kien Vu
many thanks for the link on CFD!
Question
What are the advantages of Finite volume method (FVM) over Finite difference Method (FDM) for particularly flow simulation (CFD) ?
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.
Question
In Ansys dynamics explicit module, I have modeled a water tank for an earthquake analysis to observe sloshing behavior. I am interested in how Ansys solves the problem using conservation of mass, momentum, and Energy. I have used the eulerian body for water and the lagrangian domain for the water tank. My question Is it using the Finite Element Method (FEM) or Finite Volume Method (FVM) or Finite Difference Method (FDM)? to solve the PDE's
Question
how Silvaco solves the mathematical model. Is it using Finite Element Methods or Finite Difference Methods?
your can use either of them, but FDM is good for your model.
Question
Since the nonlinear term can be discretized nonlocally in several ways, what properties do you need to look at to make the best choice?
Or we have to choose in trial and error method, so the properties of the continuous system are preserved. Basically, I new help in the system of nonlinear odes.
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.)
Question
I want to solve a second-order non-linear and non-homogenous order differential equation using the non-linear shooting method and fourth-order Runk Gutta using Python?
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.
Question
Hi, my name is Christopher and I'm currently doing a research in solving the linear elasticity equations using finite difference scheme as a part of my undergraduate thesis. I have been able to solve for the static linear elasticity (displacement formulation) using the finite difference scheme. However, I'm unable to solve the dynamic linear elasticity (velocity-stress formulation) equations using finite difference (Central Time and Central Space FD). The solution always diverges with time.
Here are the first order hyperbolic governing equation i used:
(∂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)
Central Time Central Space FD scheme:
∂f/∂t=(f^(n+1)-f^(n-1))/2∆t
∂f/∂x=(f_(i+1)^n-f_(i-1)^n)/2∆x
The domain i'm trying to simulate is a 2D cantilever beam with fixed displacement on one end and shear stress on the other end. The rest is traction free boundary.
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.
Question
Attached is an apple-to-apple comparison among Spectral Element Method (SEM), Finite Difference Method (FDM) and Finite Element Method (FEM). The accuracy achieved by FEM or FDM with 100 nodes can be achieved by SEM with 10 nodes.
In my opinion, researchers don't consider SEM just because of complexity in coding when compared with FEM or FDM. But when it comes to accuracy, no way.
Note: here I mean the low order FEM
The difference is in terms of the basis functions used.
Question
I'm looking for a commercial code that solves fluid mechanics problems via the Finite Difference Method.
Check the books by Prof tito toro he used to be in Manchester Metropolitan now in Italy I think they have own FD softwares
Question
hello,
I hope that you are doing well and that the beginning of March is excellent. First of all, I would like to thank the researchers who responded to my last discussion on terrestrial albedo.
Today's discussion is about a problem I found in the numerical modeling of the convection phenomenon (heat transfer)
the goal is to dimension the prototype to be able to dimension it, so I used the finite difference method to discretize the equations of order 1 (I discretized time by a forward scheme and space by a backward scheme).
Despite this and after I wrote a program on python to be able to extract the results, I found values that are not logical for example the water temperature by heating with a radiation value of the order 500 W/m^2 -10 °C.
someone will give me an idea on what I should do or change ... ? the scheme? .........
cordially
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:
Question
Hello everyone, hope all is good.
I want to analyse multi-phase fluid flow through pipe, which software is best to do so ?
Go for Ansys CFX or COMSOL.
Question
During AM process, progressively activation of elements can predict the temperature/crystallinity history, including surroundings heat losses and than this data could be used to determine the deformation as well as the residual stresses within Abaqus AM plugin as one can do from user subroutines like UMATHT -> UMAT ? Are there any differences in term of results or convergence one have analyzed?
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.
Question
Dear Colleagues, I stated a discussion related to this few days before.
Really I am facing the problem .
I want to solve the Radial wave functions for Hydrogen atom for degenerate states like n= 2, 3 ....
For n= 2 , there are two different radial wave functions corresponding to l= 0 and l= 1.
I am using Finite difference method .
I am getting the energy Eigen values but failed to get the degenerate wave functions.
Thanks and Regards
N Das
Respected Durrani,
Thank you for your kind response.
I just try to follow.
Regards
N Das
Question
May I know is there a way to get the Deff of a slab with 3 terms where n=0,1,2. I have looked everywhere and all I found is that I need to use Excel Solver. May I know how Solver can solve this question?
I have encountered a problem where linearizing a n with more than 1 term is almost impossible. After linearizing, I'm planning to graphical method to obtain Deff via gradient.
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.
Question
I want to solve PDE equation using Python. I have used codes of finite difference method for solving.
Could you suggest any solver Partial differential other than FiPy.
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 .
Question
Hello,
My question is concerning the implementation of a parallel-plate waveguide (PPW) using 2D FDFD. I am using the total-field/scattered field method in the paper attached by Rumpf. One way to implement the PPW is to define two PEC objects (by modifying the epsilon matrix on the grid) and specify the total-field between the two objects using the Q matrix to excite the right modes. Is there a way to implement this where the PECs are defined as surfaces or boundary conditions instead of objects?
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.
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:
Hope this helps!!
Question
FEM, FTDT, MOM and BEM which method do you prefer and do use in your field?
Subrat Sahu , you want apply FDTD to what kind of equations (Maxwell, elasticity, Navier-Stokes, ...) ?
Question
Hello Dears
This problem is related to the MATLAB link with LUMERICAL.
I have used the MATLAB link feature with Lumerical software, Also in MATLAB, I have written a script for sweep Jsc according to the grating width of the relevant structure.
There is a FOR loop in MATLAB script, The output of the FOR loop for the first step is a reasonable value, And the outputs after the first stage are all zero,And the output of the last step is irrationally a repetition of the first step.
what is the problem?
I tried hard but I did not succeed.
Finally, I attached the MATLAB script.
Motahari
Question
I am going to solve Richards equation for unsaturated zone using finite difference method. I read in several papers that van Genuchten method is used to estimate hydraulic properties. In van Genuchten formula we can estimate theta and K. what is the difference between the theta  obtained by van Genuchten formula and the theta  obtained by Richards equation. The other question is that in solving Richards equation we assume an initial h. in the next step h should be estimated so that we can calculate  using this equation. How it can be updated in each step.
Shahriar Shahrokhabadi y Zoubair Boulahia , Vahid Moosavi necesito una ayuda en mi simulación por "Simulink" la ecuación de richards ,
Question
Hi dears
I want to do the absorption of a structure for the wavelength range of 280 to 900 with the desired divisions. For example, I want the wavelength distance in this case to be 0.5 nano, in other words, it returns the absorption information every 0.5 nano.
Lumerical(FDTD) apparently does this for random wavelengths. But I want to get the absorption for wavelengths of 280 to 900 nano, at distances of 0.5 nanometers.
In other words, in the image below, I want the value of "value", 0.5 Nano 0.5 Nano change.
Is there a way to do this in Lumerical(FDTD) ?
Motahari
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
Question
I am solving a transinet convection-diffusion equation
u_t + a*u_x - nu*u_xx = 0 in (0,1)
Taking,
Convection coefficient a = 1
Diffusion coefficient nu = 0.005
Number of elements = 100
I got Peclet number Pe = 1
For same case now,
Time step = 0.01
End time = 0.6
I got Courant number C = 1
I need following 4 conditions to fulfill.
1). C =1 and Pe = 5
2). C =1 and Pe = 100
3). C =3 and Pe = 5
4). C =3 and Pe = 100
My question is:
What should be the value of
"Convection coefficient a",
"Number of elements",
"Diffusion coefficient nu",
"Time step" and
"End time"
to get these 4 results separately.
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
Question
Dear researchers
I hope that you are having good and healthy days
I am modeling a direct shear test using FLAC3D V6, the model actually works properly when it assigned to the elastic model, however, the same model does not reach the equilibrium if it switched into Mohr-Coulomb model to model pure sand (without cohesion c=0)
Have anyone of you guys faced the same problem?
Best,
Omar
Question
I have used finite difference method to discretise the equation and have written a code for it on mathematica. Anyone knows how to write a code for the truncation error that is the order of accuracy? I am not very familiar with mathematica.
All essentiall elements required to perform a numerical analysis of the finite difference schemes are discussed In the following sites:
Question
I have an one question which is :
What is a difference o relationship between layer-wise and zig-zag theories.
Dear @Ashish thakur
Thank you ,for explination
Question
The finite difference method with Taylor expansion give a good accuracy higher order derivative of normal functions for which the expansion coefficients can be found following this link: https://en.wikipedia.org/wiki/Finite_difference_coefficient
My question is what would be the higher order expansion of the functional derivative knowing that it has a similar Taylor expansion. Please note that I'm talking about the numerical differentiation.
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).
Question
I am writing to inquire if there is a way to extract the inner surface info of a deformed geometry, and build a new part which can exactly fit into the space of the deformed geometry.
I am trying to model a cylindrical tank with fillers (e.g., sand) using the coupled temperature-displacement approach for thermomechanical stress analysis during the heating/cooling cycle (i.e., thermal ratcheting).
The model, solely composed of tank shell, is being heated from 399 ºF to 579 ºF. The result shows that the cylindrical tank shape at 399 ºF (top-left) expanded across the tank wall and a bit buckled towards the bottom of the tank (both top-right and bottom-right).
For now, I would like to fill this deformed tank with a continuum part (without a gap between the tank and the continuum part), and cool down this model from 579 ºF to 399 ºF. To this end, I created a new model and imported the deformed tank geometry at 579 ºF from the previous model. But I am having trouble making a part which would be exactly the same shape as the inner surface of the deformed tank (like a plaster mold casting (bottom-left).
I looked around the ABAQUS documentations/user community, but I couldn’t find any solutions. I would appreciate any input and advice you have on this problem.
Question
I want to solve a 3D wave equations in curved coordinate (for 3D bent rib waveguide) numerically by finite difference method, but I don't know about convenient boundary conditions, which boundary conditions should be applied, Perfect Match Layer or Transparent boundary condition? could anyone help me please?
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?
Question
I have a problem of the form u_xx +a(x)u_x +b(x)u(x-\eta)+c(x)u(x+\gamma)=f(x)
with boundary(interval) conditions u(x)=\phi(x) on -\eta <= x<=0 and u(x)=\psi(x) on 1<= x<= 1+\gamma. where \eta and \gamma are constants. Is that possible to write the matlab code for this problem without using Taylor series expansion for the delay terms? If yes, please show me
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.
Question
Does anybody have experience with the Finite Difference Element Method described in http://www.scc.kit.edu/scc/docs/FDEM/Literatur/FDEM-Survey-Feb09.pdf and would advocate for or against using it? The method works on an unstructured grid, the field quantities are represented by their values at the grid points, and spatial derivatives are evaluated at the grid points by means of fitting polynomials into the gridpoints in the neightborhood of the respective gridpoint.
To me, at first sight, the method seems to have a couple of advantages, compared to FEM:
- The meaning of the state variables of the discretized system is intuitive
- It can be applied as a "black box" to virtually every PDE
- There is no a priori pen and paper work necessary for calculating weak forms, quadrature formulas etc.
- It can readily be extanded to rather high orders
- The PDE can contain arbitrarily high derivatives, in principle
- The error of the derivatives can be estimated
Does anyone know any drawbacks of the method?
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
Question
I have practiced 2nd order FDM discretization technique for 2D Laplace and Poisson Equation but I am totally confused about how to get fourth-order discretization technique. I would be thankful for your spending your valuable time.
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.
Question
I read about the membrane and shear locking appearing in shell element. Most references explain the reasons for shear locking but do not provide sufficient explanations for membrane locking.
I would like to ask about the reasons why membrane locking appears in shell element formulation and how to solve it?
Question
Hi, currently I am looking for technical articles on droplet generation using FDM approach. What are the boundary and initial conditions adopted?
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.
Question
I have read a lot of papers regarding the implementation of the meshless finite difference method and I am also aware as to how to implement the derivative conditions for steady-state (non-time variant) calculations (I will solve the derivative matrix system and the natural boundary equation du/dn = f in order to for a linear algebraic system that can calculate the derivatives and the value of f, then solve for the values of u that would force the value of f to be the desired value).
My problem is how to implement such conditions should I desire to integrate the PDE with respect to time. How do I exactly do this?
As far as I understand, I cannot openly solve for dependent variable u as there is an initial condition for u already, which means that the values of the derivatives are fixed already - and in return, there is a fixed value of f already.
If I intend to force f to be equal to the desired natural boundary condition, it would require me to change the values of u, which is again, not permissible as there are initial conditions. Thoughts?
State the governing boundary value problem explicitly.
Question
Which is the latest method to solve non linear equation using finite difference method.
Can you tell me some methods for solving non linear equations using finite difference method.
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?
Question
Dear researchers
Whether material exists or can exist that is not elastic but has a linear behavior?
brittle materials, such as glass and most rocks , are described as linear and non elastic
Question
i can solve analytically using method of separation of variables. but is finite difference method applicable to reach the numerical solution?
Where are you?
Question
Dear my colleagues,
In recent days, I searched a lot to know that which cells size is most appropriate for numerical groundwater modeling (finite difference method). But I could find any reference. Could any help me in this regard. Thank 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
Question
I'm looking for a numerical method to solve the space-fractional reaction-diffusion equation.
What is the best numerical method to solve 2D space-fractional partial differential equation?
Numerical Methods
Spectral element method
Finite element method
Finite Difference Method
Space-fractional Partial Differential Equations
finite element, b-spline, finite difference methods
Question
In FE some elements are overlapping. I read a paper says thst such overlapping causes higher in plane stiffness only. I wonder does the impact on accuracy of solutions very big?
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.
Question
When using Return Mapping algorithm, I implement Newton–Raphson iteration to reach the value of plastic multiplier. I know that the total value of plastic multiplier should not be negative. But, what should I do if one increment of plastic multiplier in an iteration is negative. Should I stop iteration or continue or what?
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.
Question
Can any body suggest me any source code/software that can generate synthetic seismogram using using finite difference method.
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.
With Best Wishes,
Samir G. Pandya
Question
I know about mesh density, skewness, wrap angles, and aspect ratio.
Per literature, aspect ratio is a quick tool to define quality of mesh.
Is there a rule of thumb for acceptable values of elements aspect ratio?
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
Question
Is it possible to have a mixed mesh with plane elements(plane stress/strain) and bernoulli beams?For example if we have a CST element connecting with a beam at one node.In that case,in this node beam will have only axial displacements?(cause lateral displacements aquire C1 continuity and CST offers C0)Is that possible?
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.
Question
Dear friends
I have a serious problem in order to regenerate the results of the attached paper. I do exactly mentioned method (FEM) in the paper but I don't get correct results.
This paper about gas face seal and compressible zeroth and first order perturbed Reynolds equation. I Solved zeroth order with FEM and FDM correctly and I got same results that bring out in the paper (Opening force for various pressure ratio) and I solve first order equation with 1 pressure ratio (figure 5 and 6) (I am thankful to you if look at the paper), but in other figure I can't get the paper's results. I try many method to solve linear system of equation (Ax=b) such as direct method, iterative method (i.e. gauss-sidel, cgs, gmres, bicg, pcg ...) but I failed, also I try many network grid and there is no achievement.
So what should I do?
I really know my question is general, but I really don't know about the errors.
Thanks dear Ryan Vogt and Debopam Ghosh
Question
I have developed a honeycomb sandwiched structure in Salome platform, meshed with netgen 2D, then calculated the compression stresses on it using Calculix open source FE.
I tried to optimise the element offset in order to eliminate such problem.
Attached is the photo of the honeycomb core.
by digraph
Question
Hello RG users,
The rocks are recrystallized limestones and dolomites and physically in various conditions such as in some areas the outcrop is fresh and intact, in some areas it is lightly or highly decomposed and weathered, even somewhere there are residual soils. Also, some areas contain discontinuities and some not.
Assume that I have the plain 3D geometry. I simply cannot assume that the whole area is homogeneous and has same characteristics.
My question is How can I efficiently represent such varying conditions (heterogeneity) in a 3D model based on FEM, DEM, FDEM etc. ? Is there any software capable of doing it?
Thank you!
Best Regards,
Bekir SF
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.
Question