Finite Difference - Science topic

First, notice that there are two reflection effects involving in a PML layer: (a) the reflection produced at the exterior boundary where the PML layer is truncated (which can be controled with the PML absorbing profile) and (b) the reflection phenomena in the inner interface between the physical domain and the PML layer (this kind of reflections are due to the discretization and they are not present in the continuous model).

In your case, if you need to quantify both kind of reflections but you cannot evaluate the computed fields at the interior of the PML layer, eventually your best option could be based on computing two different problems with different physical computational sizes: the first one with a large physical computational domain (where the computed fields are used as reference solution close to the exact one) and the second problem with your actual physical domain and PML setup. Once, you are able to compare both computed fields in these two problems, the difference between them in the physical domain should be the reflection part of the field due to the presence of the PML layer.

you can get help from the following link.

Dear Kulthoum Ismail,

Your PDE is linear and Non-homogeneous.The non- homogeneous function f(x,t) must  be  known real valued continuous function on given domain.It may be any function.There are so many methods for solving this equation under suitable initial and/ or boundary conditions.Some of them are:

I) Method of Separation (Oldest and simplest method)

II) Method of integral Transform

III) Adomian Decomposition method

IV) Perturbation Method

But the non- homogeneous function f  is nonlinear  in U then we have to put certain conditions on f. It is different issue.

Best luck!!!

DNYAN

C ++ is faster , but working with MATLAB is easier. If you have a very large data, that would be better to choose C , however if the data is not that big, MATLAB is better.

Dear Abhishek, regular finite difference scheme computes derivative in P_n using values f(P_n-1), f(P_n), f(P_n+1) of a function f. Splitting scheme computes derivative in Pn using values on other neighbouring points P_n-m, P_n+m with m < n and m = n for a better approximation; so derivative f ' (P_n) value is "splitted" among contributions from a set of points in the computational grid. For an example of splitting method using Mathematica, see the following link.  Gianluca

Dear Dominik

Regarding the time step units its definition its deeply embedden within the units of the different coeficients that you use inside the model such as K, h, rho, and c. To know the specific units of your model you have to perform a dimmensional analysis, however it depends specificaly on k and h, if those coefficients are expresed in a hour dependent unit your time step will be hours. You can also use a dimmenionless form of your model, you can find the proper derivation of dimentionless models in the following book.

I hope you find something useful inside my answer.

Hi, last point: to compute the value of the shooting function you simply have to call ode45 in order to numerically integrate the system of ODEs.

Likewise could a nonstandard frequentist account for irrational probability values given that the sequences is treated in a finite fashion--- if its still considered countable infinite (whatever is that is analogous to in nonstandard analysis) I presume one could not fit all given real number relative frequency value into the collective as there are only countably many positions; 1.one cannot simply re-arrange as in the standard analysis case, so that becomes a different sequences, 2 perhaps treat two numbers very close together as the same number

I would recommend the method presented in the paper

Casulli, Vincenzo, and Paola Zanolli. "A nested Newton-type algorithm for finite volume methods solving Richards' equation in mixed form." SIAM Journal on Scientific Computing 32.4 (2010): 2255-2273.

Speculating in mathematical terms, I think that is easier for a basketball game being played as music (because the 24 second clock play is equal to six musical "tempos") than a football game.

Here in Brazil when a football team like Messi's Barcelona plays, we say that "they play like music", but I can not see it further than a metaphorical term.

Best regards!

Dear Martin Baeker, I really appreciate your help! Your explanation is very clear, but I believe there is something I couldn't get it.

Taking your analogy of the spring as a reference, then you have:

mass * acceleration = force (Newton's second law),

but force = stiffness * displacement

In the elastic wave equations you have:

density (kind of mass) * acceleration = stiffness_type1 * (second order derivative of displacement in the space domain) + other derivatives...

The rhs of the elastic wave equation shouldn't result in a kind of force or stress? Additionally the rhs is formed of 3 parts, including the cross-derivative, which I believe should have, as a multiplier, a kind of combined stiffness.

If what I said is correct, could you give me a tip on how should I think to get the force or stress in the rhs of the equation?

Thanks a lot for your kind attention!

Use compact finite difference schemes, see the work by Lele.

Look at this doc it may be helpful for your topic .Good luck.

Hello Kumar,

In the presence of laminar-turbulent transition, flow may reattach (e.g. on wings of gliders) as follows: Initially laminar boundary flow may separate because of an adverse pressure gradient. Now the flow becomes strongly unstable and hence turbulent. Hence, it reattaches further downstream because of the stronger resistance of turbulent boundary-layer flow against separation.

Best regards, Johannes

No, not vasp code dmol3 code. A laco-basis DFT code.

If I understand correctly, you want to simulate microwaves being absorbed in a material and also simulate the transient heat in that same material?  If I am correct in what you want to do, this is called a multi-physics simulation.  There are a number of commercial software packages that can do this including Ansys HFSS, CST Microwave Studio, COMSOL, and others.  COMSOL is best known for multiphysics, but other packages can certainly do it.

If you are more interested in developing your own code, I would consider developing a frequency-domain code for the electromagnetic simulation and combine that with a time-domain heat equation simulation.  The thermal response will be so slow enough compared to the electromagnetic waves that I think this is your best approach.

If you want to learn FDTD quickly, I teach a course on the method.  Here is a link to the course website:

If fact, I will be teaching this course again in a few weeks so I am sure I will make some revisions, corrections, and other improvements.  To get a feel for coding FDTD, there is a series of six MATLAB Computer sessions where you watch me type every single line of code as I explain what I am doing and run the code at various stages of development.  The six sessions eventually build a fully functional and rather sophisticated code.  To fully understand the code, also watch the videos through Lecture 10.

If you get through all of this, I think it will be more clear to you how to solve your problem.  If you want to see a frequency-domain version of FDTD, take a look at Lectures 11, 13-15 here:

Hope this helps!

You can see this book., Very helpful in such problems.

You can

Well, block tridiagonal systems arise in many contexts. The solution of Poisson's equation in 2D using finite differences could be encoded that way, although the great majority of people will use iteration, such as S.O.R. or even multigrid. I started solving block tridiagonal systems as a PhD student, and wrote the code in Fortran - I still use that code now, although it has become very considerably more well-written since! It would have been embarrassing to have left it as it was!

I admit, however, that I am not a Matlab user, but people have told me that it works best when one uses matrix manipulations. When one tries to "micro" encode (i.e. using detailed manipulations involving elements of a matrix rather than the whole matrix), then it becomes very slow. Perhaps matlab users could advise about this aspect.

Essentially the block tridiagonal matrix algorithm is identical to that version of Gaussian Elimination that one uses for a simple tridiagonal matrix. The only difference is that, when one divides by a scalar for the TDMA, then its equivalent for the BTDMA is a multiplication by an inverse of a matrix. It may then be possible to organise your Matlab code in an efficient way which extracts blocks from the block tridiagonal matrix and places then into smaller matrices which are then multiplied or inverted as necessary.

I hope that the previous paragraph isn't unclear, but as it is tricky to describe these concepts using only words, I have attached some old teaching notes from 1994. Hopefully these will be of use to you.

Best wishes, Andrew

Dear Elvin,

Please, find attached several papers describing Crank-Nicholson scheme accompanied with its Matlab code, and also these http links:

Yes, 42.

Absolutely.

To be considered also accuracy, CPU time, hardware and effort.

Even among finite difference methods there are different stability criteria. The one you quote is an approximation to that for the use of Euler's method for timestepping when using second order central central differences in space. Backward Euler and Crank Nicholson are unconditionally stable. One may use a fourth order Runge-Kutta scheme for the timestepping and this will be different again.

Hello Jonathan Emil Gundorph Jansen

This problem can be solved by the usual method by discretizing the PDE into FD equations in the usual way. If a uniform grid is created in a zone with rectangular boundaries then the solution will be elementary. However, for your problem, even if you create a uniform grid, near the boundary circle the grid will turn out to be quite non-uniform and special finite difference equations need to be written for such points. That is the reason the FEM is preferred to the FDM. The former can handle non-uniform grid quite easily.

Dear Nauman,

This depends on what kind of stability you look for. One possibility is to prove that the solution remains bounded almost everywhere (stability in the L^\infty), like a maximum principle or that it satisfies a bounded growth in time. For example, often the solution remains positive. If this is a property of the solution of the non-discretised problem, than the same could hold for the discretised one. However, the discretisation should be as such (e.g. Euler implicit in time, and the finite difference discretisation should lead to M-matrices).

In the same spirit, you may prove some monotonic behaviour of the solution. In other words, that two solutions satisfying the same equation, but with ordered boundary and initial conditions, remain bounded as well.

Alternatively, you may try energy methods. For example, you may show that the L^2, H^1, or similar norm (in space) is decreasing in time, or its growth is bounded. This would be, however, valid for many discretisation schemes and should actually hold for the non-discretised problem.

Good luck,

Sorin

Dear John,

At a quick glance, what you are writing looks fine. The chief problem with using a backward difference in time with a nonlinear problem is that it results in a set of coupled nonlinear equations for the unknowns. Generally this can be solved using an iteration scheme (ad hoc, or multi-dimensional Newton-Raphson, or even Keller-box). This will require some extensive coding. If you were to adopt a forward difference timestepping scheme, then you wouldn't need to iterate at each timestep.

Both types of timestepping have advantages and disadvantages. For forward differences (i.e. Euler) the advantage is rapid encoding and rapid computation per timestep while the disadvantage is the potential need to have quite small time steps in order to maintain numerical stability. For backward differences (e.g. backward Euler) the advantage is that relatively large timesteps can be used (assuming iterative convergence, which isn't guaranteed), while the disadvantages are a much longer code development time, and much more CPU time per time step.

Personally, I would always attempt the explicit scheme first to see roughly how the code behaves and how the physics of the problem behaves. Only then will I proceed to an implicit scheme, and only if it turns out to be necessary to do so.

Regards,

Andrew

This should be sufficient:

and

This

is a very classical approach, whereas this

is a more recent proposal. Be careful, since the Fokker Planck equation is not hyperbolic, it would rather fall in the cathegory of 'advection-diffusion' equations, since it contains both first and second order terms in space.

See the link

Often the simple answer is that depends on the "tradition" of the code.

Then, "better" is too general. We should focus on accuracy and computational cost and hence on the resulting efficiency. FEM is somehow less efficient than FVM for higly non-linear flows but recently DG method received new attention in the CFD community.

Finally, I would stress that FVM is nothing else that a FEM for a specific choice of the test-function.

Tetrahedral elements can fit better complex geometry. However, when you integrate the shape functions with points of Gauss it is less accurate than hexahedral elements. In addition, one of the factors that determines the quality of your mesh is the distortion of your elements. The reason for this lays on the mapping from real to natural space of integration. To sum up, if your geometry is simple, the best option is to mesh it with hexahedral elements. If it is not possible (curved geometries, accute angles or similar) then go with tetrahedal but controlling the distortion of the elements.

Finite volumes ! :)

It really depends on the problem (fluid, structure, electromagnetism, interfaces or not...) the geometry, and the nature of the PDE system

Kind regards,

Thanks a lot for the recommendation. I will try . Best. George

Ivan, see also the attached paper.  Gianluca

You are welcome Ahmed.

On the subject of Bessel functions, their associated addition theorems, etc., you may wish to consult the following section of the Digital Library of Mathematical Functions, at NIST: http://dlmf.nist.gov/10

Hi Romain de Oliveira , did you have any luck with this? I am now in the same position where I am trying to write a matlab code using the finite differences method to solve the light and heavy hole energies of the valance band of a semiconductor Quantum Well.

Do you have any advice you could share?

It is not sure which software you will use.Generally, PML requires approximately a thickness of one lamda (largest wavelength). In COMSOL,the PML is available for frequency domain simulation.However, it does not provide PML for transient simulation based on the elastic wave equations.You can refer to my published work.

I think you could read this paper: Theory of the Lattice Boltzmann Method: Dispersion, Dissipation, Isotropy, Galilean Invariance and Stability

Hello Sudipta Maity, thank you for answering my question

in the first step I use a rectangular waveguide just to test my Matlab code, in the second step I would like use it for analyse a substrat integrated waveguide.

I did not get you. As fast as I know, the procedure is to have a mathematical model to describe a physical process (or model in vague terms). And then one chooses appropriate numerical scheme to solve the mathematical model which is often a form of the differential equation. FDM, FEM, FVM are then picked based on the needs of the problem and one's own choice of expertise!

As others have suggested, if the accuracy of your scheme in time is known, then you can adjust the time step so that the error of the time scheme will be smaller than that of the spatial scheme - for example, if you have a 2nd order scheme in time but a 4th order scheme in space, then the time step should be at least as small as dt = O(dx^2) - this way, the error of the time scheme is O(dt^2) = O(dx^4), and the resulting global convergence rate will allow you to determine the spatial accuracy up to O(dx^4). 

Your question about not having an exact solution to compare with has not yet been addressed. The general idea is to compare your numerically obtained solutions on successively finer grids. The simplest case one can imagine is a finite difference problem with equally spaced Cartesian nodes. In this case, say you use a grid with 16 nodes in each spatial direction for your first computation and a grid with 32 nodes in each spatial direction for the second. You will need to be careful that the 32-node grid coincides with the 16-node grid at those 16 nodes, and then you can compute the error between the two numerical solutions at those 16 nodes. In order to get a convergence rate, you will have to take one more refinement - now do the same with 64 nodes in each spatial direction (which again coincides with the previous grids), and compare the numerical solutions on the 32-node grid. From the ratio of the 16-32 error and the 32-64 error, you can get a measure of the convergence - you will probably want to continue this process for another step or two. 

It is also critical that the times that you are comparing at coincide - you may either choose some final time to end all simulations regardless of the space/time steps, or else you may do something similar to the above spatial grid in time and compare the solutions at several different time steps on successive grids, making sure that your time grids coincide as you refine. 

Dear Cosmo, the third derivative with second order accuracy is

where e.g. x_-2 = x - 2h.  Gianluca

Bewteen boundary ponit x_n and internal node x_{n-1}, add a midpoint x_{n+1/2}, and then, apply Taylor formulate, etc,  to the midpoint. Thus you can obtain a difference scheme.

.First of all modify your programs ( or take data from some published work) for the analytical solution and compare Analytical, FDM and FEM solutions. 

. Using that analytical solution, compute RMS error or Maximum error for different degrees of freedom( no. of nodes in the domain) and plot a comparative study on a log log scale.

.Additionally you can show CPU time comparison.

Hope this helps.

Regards,

Pankaj

Dear Yakubu.

my grasp of your problem is smattering.I hope you'll explain this matter  a bit more!

You may use fictitious point method, by defining an point of variable matrix and via discritizing conservation   equation in boundary and use this nes equation for evaluating you fictitious point.

if this is near to your problem , you may find classified methods in GERALD advaned mathematic book

regards

You can get any order you wish.

 Any numerical method that is based on Taylor expansion such as Euler and Runge Kutta solvers can not handle your problem since all these methods are derived under the assumption the solution is smooth and of course bounded in the interval.

   But  there exists some methods based on pade approximations, rational approximation of the solution. These kinds of methods can represent the singularity and go beyond the singularity. I recommend the paper " A numerical methodology for the painleve equation, journal of computational physics, 2011"  by Fornberg and Weideman

I gave an extended answer about being separated by a common tongue on CrossValidates StackExchange website, see the link.

Both econometricians and biostatisticians work with the model

y_ij = x_ij' beta + u_i + e_ij

Typically, in biostatistics, i enumerates individuals, and j, visits/measurements, and in this case the data are called longitudinal. In econometrics, i again may enumerate individuals, and j denotes time (often balanced, e.g., every year); economists call these data panel. Other applications economists work with include countries or states as i, and years as j.

There are several estimators for this model. Of course, OLS provides the benchmark:

beta_OLS = (X'X)^{-1} X'Y, unbiased when E(x_ij u_i)=0, E(x_ij e_ij)=0.

You can construct a between estimator by averaging the system over the faster changing subscript j:

bar y_i = bar x_i' beta + u_i + bar e_ij ~ bar x_i' beta + u_i

beta_between = (bar X' bar X)^{-1} bar X' bar Y

which is unbiased when E( bar x' u) = 0

You can construct the within estimator by forming the deviations from the mean:

y_ij - bar y_i = (x_ij - bar x_i) beta  + e_ij - bar e_i

and construct the GLS estimator on the thus transformed model. The GLS is fully feasible since you know the covariance structure within the individual: Cov[ e_ij - bar e_i, e_ik - bar e_i] = -1/n_i sigma^2_e, and sigma^2_e can be factored in front of the fixed covariance structure matrix.

Finally, the biostatisticians' default approach is the GLS on the whole equations: assigning sigma^2_u and sigma^2_e as variances of the two term, and assuming that u and e are independent of each other and of everything else, you can form a block diagonal covariance matrix of the full residual v_ij = u_i + e_ij, and if Omega = Cov[ v ] across the whole data set, then 

hat beta GLS = (X' Omega^{-1} X)^{-1} X' Omega{-1} Y

which is the efficient estimator under this model.

As far as I understand, biostatisticians typically a rather limited set of variables as x_ij's that may include age, gender, and some other stuff. Economists don't have problems with age and gender, but they may have problems with say education: if your dependent variable is income, then an individual may have taken an intentional decision to reduce their income early in the life and go to college in order to increase their income later in their life. Thus education and income are not unrelated. When that happens, E( hat beta_OLS ) = (X'X)^{-1} X'( X beta + U + E) = beta + (X'X)^{-1} E(X'U) != beta. Likewise, the GLS and the between estimators are biased. However, since u drops out from the within estimator, it remains robust to the violation of the assumption of independence of U and X.

Now, if you rename the within estimator and call it "fixed effects", and rename the GLS estimator and call it "random effects"... well, you just had your Panel Data Econometrics 101.

One of the additional contributions of econometric thinking here is the Hausman test to see whether the beta_GLS/RE and beta_within/FE are more than just the sampling variance away from one another. If they are, we immediately suspect that E(X'U)=0 assumption is violated, and everything except the within estimator is biased.

I would recommend Jeffrey Wooldridge's Econometric Analysis of Cross-Sectional and Panel Data (2nd edition, 2009 or 2010). I think he does start by saying that the model is the same, it's just two different estimators that you apply to it. (Economists have been involved in some deep thought philosophical discussions of fixed effects estimator only applicable when you have a small number of clearly defined, unique units, like the U.S. states, while the random effects estimator is only applicable when you have a random sample of individuals, such as in country-wide samples, but I think this is a superficial debate.)

The answer to your question requires a little of explanation.

There are several different Finite Element Methods, not just one. ALL of these methods can be classified in 2 groups: (1) F.E. methods based on fundamental Lemma of Calculus of Variations (Galerkin method, Petrov-Galerkin, Weighted residual, Galerkin method with weak form) and (2) F.E. method based on a residual functional associated with the differential equation (least-squares method).

There are several types of differential equations. Hyberbolic, Elliptic, Parabolic, First order, Second order, etc. HOWEVER, ALL differential equations have a differential operator associated with them and ALL differential operators can be classified in 3 groups: (1) self-adjoint (2) non-self adjoint (3) non-linear.

For a F.E. method to be used successfully for a specific differential equation, the resulting coefficient matrix in [K]{u}={F} must be positive-definite (This means that [K] can be inverted and the obtained solution {u} will be unique).

The following 3 statements are not opinions but Theorems with proofs:

- When the differential operator is self-adjoint: Galerkin method with weak form and least squares process are the only F.E. methods that guarantee positive-definite coefficient matrix [K].

- When the differential operator is non-self adjoint or non-linear: least square process is the only F.E. method that guarantees a positive-definite coefficient matrix [K].

- When considering an IVP, the differential operator is either non-self adjoint or nonlinear. It is never self-adjoint. Hence, the only F.E. method that can guarantee positive definite coefficient matrices for all IVPs is least squares process.

There is a lot of research done to make the Galerkin method with weak form produce coefficient matrices that are positive-definite. These are known as stabilizing techniques. However, all of these techniques, without exception, end up changing the original IVP.

Many people are under the impression that Galerkin method with weak form is the only F.E. method. This is because ALL commercial F.E. softwares are based on this method. The reason for this being that the differential operators of differential equations describing elastic solid mechanics are always self-adjoint, hence Galerkin method with weak form works great! Least-squares process also works great, however, Least-squares process requires additional resources (in terms of interpolation theory) that didn't exist at the time when commercial F.E. softwares started to appear (1960s). Hence, people didn't consider Least-squares process. Around the late 1990s, those additional resources needed for least-squares were invented, and least-squares started being used in the research area. However, since so many companies were built on Galerkin  method with weak form and they are only marketed towards solid mechanics applications, they still continue to use it.

Other methods to solve differential equations such as: Finite Volume, Finite Difference, Runge-Kutta etc suffer from the same disease than Galerkin method with weak form. Meaning, that they do not guarantee positive-definite coefficient matrices. This does not mean that unique solutions are not possible using these methods, but it means that you will only know it after the [K]{u}={F} is obtained and if [K] is not positive definite, then stabilizing techniques must be used. Runge-Kutta method and Finite Difference methods are very easy and fast to implement. Hence they popularity. However,  when considering complex problems where you might end up having a [K] matrix that is a million-by-million, then they might or might not work and if they don't work, stabilizing techniques might or might not give you the right solution.

So to answer your question, F.E. method can be used to solve IVP. Either using Galerkin method with weak form combined with stabilizing techniques (see works of T. Hughes) or using least-squares process. If you are interested on using F.E. method to solve ODEs in time, then you should real this article:

- K. S. Surana, L. Euler, J. N. Reddy, A. Romkes: Methods of Approximation in hpk Framework for ODEs in Time Resulting from Decoupling of Space and Time in IVPs. American J. Computational Mathematics 1(2): 83-103 (2011)

This article explains into details how to handle IVPs by decoupling space and time and as a result, obtaining a system of ODEs in time. Then it compares F.E. methods and other methods such as Newmark, Wilson method. I hope that gives you a better idea of the subject.

Do you mean the Navier-Stokes equation?

Fully agree with Amaechi J. Anyaegbunam,

there many books with explicitly written FEM matrices for Timoshenko beam eg. Finite Element Procedures: K.J. Bathe etc etc It is more efficent to use FEM rather than FD - you will not have problems with Boundary conditions (like difficulties within FD method)

If you are using classical finite elements on triangles, then the gradient will be discontinuous along edges. You may first define the ''trianglegradient''  for the inner of each triangle using the appropriate interpolation scheme and evaluating the gradient of the interpolating polynomial  in the center of gravity of the triangle, e.g. for piecewise linear elements by piecewise linear interpolation. for the nodes you may use the arithmetic mean of the triangle gradients for all triangles to which the node belongs.

These the following three parameters

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 =( epsilon/ gama ). h

Your normal media conductivity has nothing to do with the PML one. Especially in a stretched space formulation this is much easier to see why. The PML "conductivity" is just a loss term that is used to attenuate the right components in the PML. For a correction based PML implementation you can try this http://dx.doi.org/10.1109/TAP.2011.2180344

but I have never used it in cylindrical coordinates and obviously as I have developed it I should not be the one to promote it ... Good luck.

Hi Ghaffar! Once you have descritized and your numerical scheme is obtained. I suppose you have a two level scheme. Taking initial condition as Un , compute your rhs. Apply boundary conditions to obtain a system of equations whose size depends on the points taken along space direction. This system will be linear if given PDE is linear else it is non-linear. Solve it to obtain solution Un+1.

Now Un+1 is obtained , transfer it to Un and get solution at new time step 2Dt as explained above.

Dear Muhammad,

Which equation are you trying to solve ?

First order or secund order or hyperbolic ?

Best regards,

MH

All the numbers determine a stability of numerical scheme. As common, if they reach some critical limit, the numerical solution begins oscillate in space and time with grid period. That oscillatory solution is non-physical and grows rapidly providing the numerical overflow.

I think, the detailes of numerical scheme stability analysis and recommendations about the scheme choosing can be found in any book about the CFD or heat and mass transfer simulation, e.g.:

"Computational Fluid Dynamics" by K.A. Hoffmann;

"Computational techniques for fluid dynamics" by C.A.J. Fletcher;

"Numerical Heat Transfer and Fluid Flow" by S.V. Patankar;

"The Theory of Difference Schemes" or "Computational heat transfer" by A.A. Samarskii;

"Essential Computational Fluid Dynamics" by O. Zikanov.

I can highly recommend the following book which are relatively new published in 2008:

Title: COMPUTATIONAL METHODS FOR PLASTICITY - THEORY AND APPLICATIONS

Authors: EA de Souza Neto, D Peri´c and DRJ Owen

Publisher: Wiley

ISBN: 978-0-470-69452-7

The book is very easy to read ( considering the topic ) and covers almost every aspect of computation plasticity (800 pages), small and large strain formulations, isotropic and an-isotropic formulations, elastoplastic formulation, flow rules and plastic multipliers, viscoplasticity, etc

Best of all each topic is started out in one dimension and then expanded to a full 3D formulation. Furthermore, the book is very easy to implement from, as each constitutive formulation is summarized in pseudo code - documenting the implementation step by step.

The book give a general introduction to constitutive modeling and is by far the best book I ever read on this very interesting and challenging topic.

Regards

Benny Endelt

Hi Dimitris, when we say that time updates of magnetic field at (n+1/2), we mean that the magnetic field values at (n-1/2) and electric field values at n are used to calculate the magnetic field value at time (n+1/2) [the electric and magnetic fields are interleaved in space also, but I am only specifying the time indices for the sake of understanding]. So, when you say, "magnetic field time updates are about n+3/2 and n+1/2 timesteps", it probably means you are trying to get the magnetic field values at (n+3/2) in terms of the magnetic field values at (n+1/2) and electric field values at (n+1). So they seem to be consistent, only with the difference that in one case by present time step we mean (n+3/2) and in the other case the present time-step refers to (n+1/2). I had a look at chapter 3 and 4 of Taflove once again, the update is the same as that of Inan and Marshall. Can you please specify the particular portion of Taflove's book? I might get a clear picture.

Dear Sir,

1- First of all you have to make inflation around the cylinder about 5 layer and

2- after get the run and have the answers, you have to select' area weighted average' and snap to select the parameter and get the 'Pressure' then select ' coefficient of pressure'.

good luck

To have some idea whether the finite scheme is working fine or not choose grid points N, 2N and 4N. Assume solutions for 4N points as accurate, compute Max. Absolute Errors in N and 2N grid points.Find order of accuracy from here by using formula       log( E_N / E_2N) / log(2).

Viscous dissipation in general  is the irreversible conversion of mechanical energy to internal energy (heating) due to viscosity. See this link for more details: http://rheology.tripod.com/z07.13.pdf

When I calculated rotary heat exchangers i used to calculate a heat balance error.

But note if you make a very accurate solution of the model, it does not mean it  is accurate if you compare with practical tests.

Explicit methods is where you are only using grid points at the current time step to approximate the solution at the next time step. Implicit methods use information from the next time step to set the solution at the next time step. Really the major different is explicit schemes just require you to update, implicit require you to either solve a linear system, or solve a root finding problem, based on if the PDE is linear or non-linear. 

You can use fos command or use the strength reduction factor rechnique

At least on uniform grids, finite difference operators become Toeplitz matrices.

Convolutions, too, are Toeplitz matrices on finite sequences.

Now, the fact is that Toeplitz operators, (i.e. infinite dimensional Toeplitz matrices) commute, as they share the same (Fourier-type) eigenvectors. This means that for infinite-dimensional systems, you can apply convolutions and finite difference operators in arbitrary order.

For finite dimensions this is not true, but the "commutativity" still holds at interior points. Thus it's only on the boundary that non-commutatiivity shows up. You will have to create a small boundary correction, but if you have very long data sequences, this is not such a big problem, as convolutions and finite difference operators are "almost commutative."

So the answer is that you may apply the operators in any way you prefer, if you only pay attention to what happens at the boundaries.

Hello Madhup,

The problem with boundary conditions described in your question is quite well known. I will start from the simple scheme in which the derivatives inside of the domain are calculated using the central differences. On the lower and upper boundaries I will apply forward and backward differences respectively. This solution is usually acceptable for most of the problems including convection in the Cartesian and spherical geometry.

If you are looking for more accurate solution I would like to suggest to read

Strict Stability of High-Order Compact Implicit

Finite-Difference Schemes: The Role of

Boundary Conditions for Hyperbolic PDEs, I

Saul S. Abarbanel and Alina E. Chertock

Journal of Computational Physics

160, 42–66 (2000)

Based on my recent experience it looks that the both solutions can be used to solve the boundary value problems.

In the past I was using often the book:

Fundamentals of Computational Fluid Dynamics

by Patrick J. Roache

where all problems of computational fluid dynamics are discussed very clearly and in accessible manner.

All the best in your calculations,

Janusz

hi ,Muhammad Abdulrasool

   I come from China, Xi'an jiaotong university , we share the same subject , I read you program .also I run it ,thus I get the good static characteristics which match the lund's result but I can not get the good dynamic characteristics. particularly,for the cross damping coefficient (Cxy,Cyx),I can not get the Cxy=Cyx, so I think there must be some 

errors when we caculate the parameters (A1,A2....A11). now, I can not locate this error. i think we can figure it out together. 

1) I agree with Andrew. There are rules of thumb to decide the size of the supercell, and in general the larger the better. If you are interested in a properties a good choice (if computationally available) is to start from 1x1x1, to 2x2x2, 3x3x3 and so on, until the properties converges.

2) K-points mesh has to be reduced if larger direct space cell is considered ( smaller k-space). But again you need to accurately covnerge the k-points. FDM requires well converged parameters.

Also you need to geometrically converge the cell below the usual threshold (10meV/A). I personally used 1meV/A, but it depends on the system.

3) I would suggest Atsushi Togo phonopy software to perform those calculations.

Thanks Nasir Haniffa, yes the independent variables are real and the dependent also are real except (Q_E10). so I'll separating the real and imaginary parts and solve them. I'll let you know about the results.

Ory. WENO uses divided differences  to determine smoothest interpolation stencil (smallest total variation TV), TVD schemes apply upwind when oscillations are detected smoothing the solution by numerical diffusion. At the discontinuity TVD becomes first order accurate whereas WENO tries to remain non-oscillatory and higher-order by selection of the proper interpolation stencil(s). WENO (weighted ENO) takes some kind of the average from all discretization stencils for this reason it is less non-TVD then original ENO scheme.

hi

if you use FDM and have specified differential equation and boundary conditions it is easy to implement boundary condition to FDM. So the most difficult form of boundary is mixed or robin boundary condition that you could implement it into your domain by ghost node method ( fictional node method). check it on the net.

regards 

I am not sure if you mean only the use of second and higher differences in numerical approximations, but some older texts have very thorough expositions.

three  are: Mathematical Physics Jeffreys and Jeffreys Chapter 9  CUP 1956

 Methods of Mathematical Computation Herriot, Wiley 1963

Modern Computing Methods UK National Physical Lab Philos Library NY 1961

Start by linear izing the adsorption term.

Hi Chris,

the SOR method can diverge if the matrix is not self-adjoint positive definite. However the convergence can be very slow even for self-adjoint matrices.

The first thing that you should try is one of the Krylov subspace method, i.e., the Conjugate Gradient method (for self-adjoint positive definite matrices) or the GMRES method (for general matrices). These methods only require the implementation of the matrix-vector product and the scalar product. Thus are easy to implement.

If these methods do not yield a desirable speed of convergence you can still go for more elaborated methods like preconditioned GMRES, Multigrid, AMG, AMLI, etc. But try the simple methods first. ;)

You can find the methods e.g. in Saads book (see link below).

Regards

These are classic Navier Stokes equations and heat transfer or energy equation. I do not believe these are of 3rd order, however they are highly nonlinear and extremely difficult to solve with your own codes. It is recommended to use already available commercial software like ANSYS, COMSOL, or ABAQUS to solve such type of problems as they have the required solvers for such problems. A large number of example problems are also available with these software. They are mostly finite element or finite volume based software.

P.S. MATLAB have a very elementary PDE toolbox for solving PDEs. These equations can not be solved with that. 

In general, no. The anti-difference operator applied to exp(x²) is a finite algebraic combination of anti-derivatives that cannot be expressed in terms of elementary functions. 

Is this a time-domain finite-difference scheme or a frequency-domain finite-difference scheme?  The implementation of these two is VERY different.  I don't see any time parameters so I assume this is some sort of steady-state or frequency-domain analysis.  There appears to be some terms missing.  If the equations are describing finite-differences, they need to be divided by something.  I would suggest providing the original analytical equation.

Anyway, if you want a brief introduction for steady-state (using linear algebra), check out Lecture 9 here:

For a time-domain scheme, check out

Dear sir

thank you very much.

best regards

The phase shift is associated with the properties of waves on a 2-D membrane.  Morse and Feshbach, 1953 (Methods of Theoretical Physics, Part I, Section 7.3, and especially Figure 7.11 and the summary on page 893-894) give the Green's functions for 1-D, 2-D and 3-D media.  In the time domain in 2-D the pulse shape has a "tail" that is not present in the 1-D and 3-D solutions.  This tail corresponds to a Hankel function in the frequency domain (page 891) which has a frequency independent phase shift of 90degrees (-i).  Also see Chapman, 1978, GJRAS v54, 481-518. 

I am not sure wheter  you use the Chebsheve method to discritize the spatial domain and a marching method to slove the wave equation. If so, the stability connect closely with the stepsize in time domian. One resolution is decreasing the stepsize, the other is use implicit marching scheme.

Hi,

SEISMIC_CPML of Computational Infrastructure for Geodynamics (CIG) is one of the options.

In the github link, there is one which supports 3D viscoelastic media and MPI at the sametime.

Best regards,