Numerical Analysis - Science topic

I have often used this technique when analytical derivatives are not available, such as with experimental data. I have used a similar process (extrapolation) when measurement at a particular location is not physically possible (e.g., near a wall or moving boundary).

a. Why Numerical Techniques are Used for Finding Derivatives?

Numerical techniques are used to find derivatives when the analytical form of a function is either unavailable, difficult to differentiate, or when the function is based on experimental or real-world data points. Numerical differentiation is essential because it provides an approximation of the derivative by using finite differences between values, rather than relying on algebraic methods. This is particularly useful when dealing with complex systems in engineering, physics, or finance, where functions may not have a simple closed-form expression.

Numerical Differentiation is the process of estimating the derivative of a function using discrete data points. In numerical analysis and computation, it is applied to:

b. Analytical vs. Numerical Definition of Derivatives

The analytical definition of a derivative for a function f(x)f(x)f(x) is the limit of the difference quotient as the interval hhh approaches zero:

f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}f′(x)=h→0lim​hf(x+h)−f(x)​

This definition assumes the function is continuous and differentiable in the interval of interest.

In contrast, numerical differentiation approximates the derivative using a small, finite value of hhh rather than taking the limit as h→0h \to 0h→0. The derivative is estimated as:

f′(x)≈f(x+h)−f(x)hf'(x) \approx \frac{f(x+h) - f(x)}{h}f′(x)≈hf(x+h)−f(x)​

c. Forward Difference Approximation

The forward difference approximation estimates the derivative by using the function value at a point xxx and the value at a small step forward, x+hx + hx+h:

f′(x)≈f(x+h)−f(x)hf'(x) \approx \frac{f(x+h) - f(x)}{h}f′(x)≈hf(x+h)−f(x)​

This method approximates the slope of the function at xxx using a small forward step hhh.

d. Forward Difference Example: Approximate Derivative of f(x)=x2f(x) = x^2f(x)=x2

For f(x)=x2f(x) = x^2f(x)=x2, we approximate f′(2)f'(2)f′(2) using the forward difference formula with different step sizes hhh.

f′(2)≈f(2+0.1)−f(2)0.1=(2.1)2−220.1=4.41−40.1=0.410.1=4.1f'(2) \approx \frac{f(2+0.1) - f(2)}{0.1} = \frac{(2.1)^2 - 2^2}{0.1} = \frac{4.41 - 4}{0.1} = \frac{0.41}{0.1} = 4.1f′(2)≈0.1f(2+0.1)−f(2)​=0.1(2.1)2−22​=0.14.41−4​=0.10.41​=4.1

f′(2)≈f(2+0.01)−f(2)0.01=(2.01)2−220.01=4.0401−40.01=0.04010.01=4.01f'(2) \approx \frac{f(2+0.01) - f(2)}{0.01} = \frac{(2.01)^2 - 2^2}{0.01} = \frac{4.0401 - 4}{0.01} = \frac{0.0401}{0.01} = 4.01f′(2)≈0.01f(2+0.01)−f(2)​=0.01(2.01)2−22​=0.014.0401−4​=0.010.0401​=4.01

Exact value: The derivative of f(x)=x2f(x) = x^2f(x)=x2 is f′(x)=2xf'(x) = 2xf′(x)=2x. So, f′(2)=4f'(2) = 4f′(2)=4.

Computational Errors:

e. Backward Difference Approximation

The backward difference approximation estimates the derivative using the function value at xxx and a small step backward, x−hx - hx−h:

f′(x)≈f(x)−f(x−h)hf'(x) \approx \frac{f(x) - f(x-h)}{h}f′(x)≈hf(x)−f(x−h)​

f. Backward Difference Example: Approximate Derivative of f(x)=x2f(x) = x^2f(x)=x2

f′(2)≈f(2)−f(2−0.1)0.1=4−(1.9)20.1=4−3.610.1=0.390.1=3.9f'(2) \approx \frac{f(2) - f(2-0.1)}{0.1} = \frac{4 - (1.9)^2}{0.1} = \frac{4 - 3.61}{0.1} = \frac{0.39}{0.1} = 3.9f′(2)≈0.1f(2)−f(2−0.1)​=0.14−(1.9)2​=0.14−3.61​=0.10.39​=3.9

f′(2)≈f(2)−f(2−0.01)0.01=4−(1.99)20.01=4−3.96010.01=0.03990.01=3.99f'(2) \approx \frac{f(2) - f(2-0.01)}{0.01} = \frac{4 - (1.99)^2}{0.01} = \frac{4 - 3.9601}{0.01} = \frac{0.0399}{0.01} = 3.99f′(2)≈0.01f(2)−f(2−0.01)​=0.014−(1.99)2​=0.014−3.9601​=0.010.0399​=3.99

Computational Errors:

g. Central Difference Approximation

The central difference approximation provides a more accurate estimate by using both forward and backward steps:

f′(x)≈f(x+h)−f(x−h)2hf'(x) \approx \frac{f(x+h) - f(x-h)}{2h}f′(x)≈2hf(x+h)−f(x−h)​

This method averages the forward and backward difference approximations, often resulting in a smaller error.

h. Central Difference Example: Approximate Derivative of f(x)=x2f(x) = x^2f(x)=x2

f′(2)≈f(2+0.1)−f(2−0.1)2(0.1)=4.41−3.610.2=0.80.2=4f'(2) \approx \frac{f(2+0.1) - f(2-0.1)}{2(0.1)} = \frac{4.41 - 3.61}{0.2} = \frac{0.8}{0.2} = 4f′(2)≈2(0.1)f(2+0.1)−f(2−0.1)​=0.24.41−3.61​=0.20.8​=4

f′(2)≈f(2+0.01)−f(2−0.01)2(0.01)=4.0401−3.96010.02=0.080.02=4f'(2) \approx \frac{f(2+0.01) - f(2-0.01)}{2(0.01)} = \frac{4.0401 - 3.9601}{0.02} = \frac{0.08}{0.02} = 4f′(2)≈2(0.01)f(2+0.01)−f(2−0.01)​=0.024.0401−3.9601​=0.020.08​=4

Computational Error: 4−4=04 - 4 = 04−4=0

i. Observation: Forward/Backward vs. Central Difference Approximation

j. Second Derivative Approximation

The second derivative approximation using numerical differentiation is often computed using the central difference formula:

f′′(x)≈f(x+h)−2f(x)+f(x−h)h2f''(x) \approx \frac{f(x+h) - 2f(x) + f(x-h)}{h^2}f′′(x)≈h2f(x+h)−2f(x)+f(x−h)​

This estimates the curvature or concavity of the function at xxx.

For f(x)=x2f(x) = x^2f(x)=x2, f′′(x)=2f''(x) = 2f′′(x)=2, and applying the second derivative approximation for f(2)f(2)f(2) with h=0.1h = 0.1h=0.1:

f′′(2)≈(2.1)2−2(2)2+(1.9)2(0.1)2=4.41−8+3.610.01=0.020.01=2f''(2) \approx \frac{(2.1)^2 - 2(2)^2 + (1.9)^2}{(0.1)^2} = \frac{4.41 - 8 + 3.61}{0.01} = \frac{0.02}{0.01} = 2f′′(2)≈(0.1)2(2.1)2−2(2)2+(1.9)2​=0.014.41−8+3.61​=0.010.02​=2

This matches the exact value f′′(2)=2f''(2) = 2f′′(2)=2, showing no computational error.

Thanks for sharing. I wish you every success in your task.

Numerical models can be used to validate the numerical results of back-to-back cold-formed channel sections. For example, the ABAQUS finite element modeling program can be used to develop models that determine the axial strength of cold-formed steel (CFS) back-to-back (BTB) channel sections. The program can also be used to predict the monotonic axial performance of built-up CFS battened columns.

etc.

I think only general advice could be given here:

Strip the problem back to the simplest parts and make sure those work. Specifically, remove contacts and plasticity, does your model solve? (Contact can be substituted with displacement BC to test the overall model behaviour.) If not, review boundary conditions, loads, etc. If so, add in the next 'advanced' feature one by one, try solving until it breaks, and then you will know what caused it and you can fix it. Then onto the next one

Notet1: I haven't investigated what you are doing properly, but it feels like a large forming problem, wouldn't geometric non-linearity be important here?

Note2: If it is truly a contact related problem, consider trying general contact, a lot of research goes into that function from DS, there are some features that should be superior to the contact pair implementation.

Good luck!

Maria-Cristina Ioncica A universal solution will be to find articles with a high-resolution graph and then read at least a dozen points precisely and fit the curve in programs such as Matlab. In the attachment, I am sending links to 2 of my articles, where you can find the cardiac-cycle chart you are looking for. However, I would like to point out that the variability of blood speed/flow in blood vessels depends on many factors, including physical activity, diseases, and others.

The best software for impact simulation is just ls-dyna. Don' waste your time with abaqus.

hi Nayak,

did you find a solution to your problem? i am also trying to define solid wood in abaqus cae. i have elastic data but plastic data and i can't find how to define this data in abaqus. I have limited time left for my master thesis. I need serious help from you and other friends.

Regards.

I may write on "How to simulate Physical-Based Numerical Models with Meshless Methods"

The Reynolds number UD/nu=2300 is based on a smooth circular pipe of diameter D, with STEADY mean flow velocity U and fluid of kinematic viscosity nu. Below this critical Reynolds number any perturbation due to an obstacle will not cause persistent turbulence to occur far downstream of the obstacle. Of course locally the wake of a blunt body placed in the pipe can be turbulent, but soon the flow will relaminarize if we travel further downstream. Above Re=2300 the flow does not need to be turbulent. It can be turbulent if there is a sufficiently large initial upstream perturbation. In principle the flow can remain laminar if the inlet is very smooth and care is taken to avoid vibrations. Experimentally fully developed laminar pipe flows have been achieved for Re=500. 000. It is important to realize that this critical Reynolds number does depend on the geometry of the cross-section of the channel. For a rectangular channel of height h and width w >>h, usually one considers a Reynolds number Re=Uh/nu based on the channel heigth. The critical Reynolds number for allowing turbulence is around Re=hU/nu=1100. There is however much less literature on flows through slit shaped channels than circular pipes. If you consider an open channel flow, clearly the critical Reynolds number will be quite different from Re=2300 and of course it does depend on the length scale used in the definition of this Reynolds number!

Are you asking how to make your particles a helix/spiral shape? First, before doing that, I recommend running simulations that will just use the effective properties that your helices should give. That will let you explore device ideas without needed the complexity and inefficiency of having to resolve the spirals in your grid. Second, you will want to create a small simulation of a helix to retrieve the effective properties. This is usually called homogenization or parameter retrieval. Third, if you need to, move on to your more complicated 3D simulation.

That still leaves the question of how to build a helix in a 3D grid. One way you can do this is to create the helix in a CAD software such as SolidWorks or Blender. You can export that model as an STL file, which is just a surface mesh that MATLAB (or whatever software you are using) can import. From there, there are codes available that can import those STL files and "voxelize" them into a 3D array. If you are using MATLAB, search the MathWorks website for "voxelize" and you will find multiple solutions. Once you have that, you can create copies of the spiral in your FDTD grid. This approach will let you import even more complicated shapes relatively easily. Alternatively, you can create the spiral directly in your grid. I would do this by creating a array that has a list of points along the center of your spiral. From there, you can calculate what points your FDTD grid should be assigned to that spiral by calculating their distance to the points on the spiral. If they are within a certain distance, assign the material properties of your spiral. Otherwise, assign the material properties of whatever medium the spirals are residing in.

I am sure there are plenty of other ways to do this. If you are interested, I dedicated almost all of Chapter 1 to describing techniques for building geometries into arrays for simulation using the finite-difference method. I do not specifcally talk about spirals, but you may find some of that chapter very helpful. Here is a link to the book website:

Hope this helps!!

It would be better if you give an example that covers your problem more precisely. Can you share the code you used it for solving the problem?

Yes, you have to define a multi-phase model, e.g., VOF or Euler-Euler.

Mass weighted averages are more meaningful. For combustion simulations, local density and mass depend on temperature... Area weighted average is based on the mesh...

Yah. No issues.

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

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

Chinedu Nwaigwe Great, Interested, In fact, 'm pleased to learn you're working on open-source programs to solve integral equations. Publish your code to make your study more publicly available and to allow others to build on your work.

A partner with skills in code development and numerical analysis would be ideal. As you indicated, knowledge of Git and Python would be a must. However, further knowledge of numerical techniques for solving integral equations would be advantageous.

Thanks for your response dear

Yes I've used this technique to get the results. But, as I mentioned i want to know why Floquet ports can't model lossy (as well as dispersive) half-space medium without using the Open add space boundary condition (BC).

Because, by considering a quarter wavelength space between the medium and Floquet port (via using Open add space BC ), we don't model a half-space anymore. Accordingly, we can obtain results approximately by considering sufficient thickness for the lossy medium.

To stack the plies of a composite material in numerical analysis, you will need to use a software that is capable of modeling composite materials, such as finite element analysis (FEA) software. The specific steps will depend on the software you are using, but in general, you will need to follow these steps:

To stack the plies in the specific stacking sequence of [(±45)/(0,90)]4s, you will need to define the orientation of each ply relative to the previous ply. For example, if the first ply is oriented at 45 degrees, the second ply would be oriented at -45 degrees, the third ply would be oriented at 0 degrees, and so on. The software will automatically calculate the resulting ply orientations based on the specified stacking sequence.

I found this solution of the Lotka-Volterra system of first-order ODEs via VIM:

And here is a good overview of how to leverage a Laplace Transform when solving ODEs with VIM:

I hope you can put these together for LVIM.

Coefficient alpha is based on correlations, so you be having a problem with missing data for "complete" cases. But you should also check to see if your problem is structural in the sense that you are including items from two (or more) different surveys in the same analysis, because it would be impossible to calculate correlations when the variables are drawn from different data sets.

Akila Dulanjalee Wijethunge Can you share your Abaqus model (.inp format)?

Hi dear Kourosh Vaferi,

You can import your 2D model from CATIA software into AutoCAD software and write output DXF.

Good luck!

Thank you all for the clarifications

I do not think such a general question can have a definite answer. A particular question to classify the initial data leading to closed periodic trajectories (without collisions) for 3 body case makes sense and is of importance. I guess the most appropriate person to discuss all that is R Montgomery from UC Santa Cruz.

Dear Dipaloke Majumder ,

So your input file looks like something like this ? (CLOAD statements for each node ?) :

*CLOAD

node number, direction, magnitude

*CLOAD

node number, direction, magnitude) or

*CLOAD

node number, direction, magnitude) or

...

Aditya Kumar Mishra replication is tough though i do agree with the expert comments above that physical replication like visualization is a must one of the important criteria I feel is to To assess applicability one must always specify the requirements along with exact what attribute of a previous results of interest.

Bonjour, one of the reasons is the type of mesh implemented and the computer processor you use. To solve these types of equations your computer must be powerful. I recommend you to work with as few meshes as possible.

Dear Akash Meena,

as I explained in my previous answers, all these effects are considered, at least in the manuscript you refer to, as volume forces and their magnitude is obtained from empirical expressions. Both phenomena, the recoil pressure, and the vapor-induced shear stress occur due to the same physical effect, namely the rapid evaporation of material. Thereby, the recoil pressure is simply a pressure acting on the keyhole surface and directed in the inward normal direction of the free surface. The shear stress, however, acts in the tangential direction, which can be obtained with/from the normal inward vector.

Best,

Antoni Artinov

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.

The answer is simply no. Additionally, considering "realist (platonic)" and "non-realist (non-platonic)" doesn't actually help with the answer I am going to provide, and the article also begs the question. It's like asking why you like music, is it because it sounds good, or is it because it makes you feel good? Well, that depends on what you mean! Equally, asking a working mathematician about the independence of math, or the construction of math will get you very confused looks. They way one treats math, is ever which is the most convenient, or the most sensible to the person. As such, the article in question does not particularly respect nor delineate the historical and functional differences between these two foundations of mathematics very well. Mathematics is a very broad, messy, overlapping subject. In fact, most of the math I regularly use does not really involve calculations, or functions per se. But, as the article is a pre-print, I assume it simply represents a scribbling of his thoughts.

In order to elucidate my answer better, some background in the cartography of mathematics is needed. There are many different universes (formal distinct foundations of mathematics as unique fields) of mathematics that have their own level of reasoning, and focuses. To name a few, category theory, abstract group theory, analysis, proof theory, many-valued logic, and the list just keeps going. All of which are employed at different levels to ascertain certain properties of math, or even to articulate certain questions. For instance, if one wants to study the different universes of mathematics, category theory is generally involved, and the object considered is called a topos. Or if one wishes to study how numbers work, one can employ number theory to study them as unique things, or you can employ analysis and study them as functions, or you can study them with group theory and consider them as action as well. In this view, no field of mathematics has a primacy over other mathematics, only advantages to the inquiries at hand.

Here is a simple question that I think illustrates the point I am making: is two an element of four, or not? That is to ask, in the construction of numbers, are they considered logically unique (aka type theory), or as informal primitives so that numbers are just simply numbers (set theory)? It is in fact this very question that helps separate type theory and set theory. This question, is akin to asking is meaning found in words or what the words represent? However both are true to a certain degree, and from different perspectives. If we are partial to the former, we are essentially asking, does the construction of words form the meaning they express? Yes, but only if we consider meaning as inherent to language alone (intensional). That is language makes meaning, not the world outside of our minds. If we are partial to the latter however, then words denote things, they are analogs to events, and point to common descriptions that we see (extensional). In the same manner, type theory considers numbers as things in themselves, to say "there are two dogs" is to say two dogs. Because the number two is different then dogs. Equally, computer scientists often employ type theory because it logically constructs things, whereas, mathematicians like set theory because its very good at describing things, and there relationships. It would be very burdensome for a mathematician if we had to logically construct everything from the bottom up. Instead of saying, let us consider a sequence of integers. The computer scientist would have to define every part of that sentence.

I hope this helps clarify the question.

For a function describing some physical property, when complex arguments and complex results are physically meaningful, then often the physics requires the function to be analytic. But if the only physically valid arguments and results are real values, then the physics only requires (infinitely) smooth functions.

For example, exp(-1/z^2) is not analytic at z=0, but exp(-1/x^2) is infinitely smooth everywhere on the real line (and so may be valid physically).

One place where this happens is in using centre manifolds to rigorously construct low-D model of high-D dynamical systems. One may start with an analytic high-D system (e.g., dx/dt=-xy, dy/dt=-y+x^2) and find that the (slow) centre manifold typically is only locally infinitely smooth described by the divergent series (e.g., y=x^2+2x^4+12x^4+112x^6+1360x^8+... from section 4.5.2 in http://bookstore.siam.org/mm20/). Other examples show a low-D centre manifold model is often only finitely smooth in some finite domain, again despite the analyticity of the original system.

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

Nesrine El Houari Thank you but already check the manual and did not mention it

Hello,

Proceed according to the attachment.

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,,,

My advice is to find better starting point using simple approaches (golden rate etc) and to read the theory of Newton's methods. Regards

Paolo Lampitella

Paolo I know that paper and the answer is yes and not... The local truncation error is expressed in terms of pointwise derivatives but I mean an expression of the LTE wherein the integrals appear explicitly, congruently to the initial integral equation.

Ok, let me say that we could think to start from any MDE derived in a classic way for a PDE and integrate its LHS and RHS to generate the integral modified equation.

But if we start from the original idea that the weak equation does not require the differentiability, we should not introduce any Taylor expansion in space. Otherwise we should regularize the variable and define the Taylor expansion for the smooth averaged function.

Hi dear Bartosz Grzeszykowski

The following articles could be helpful:

if you have a time-dept simulation, you can create a multiplyiting function with LIST_REEL and DEFI_FONC.

Franco

Dear Alireza Akbari looks like your equation is a nonlinear PDE, there are tables for those:

However I could not find yours, but don't worry, I tell you a trick we use in MHD.

1. You linearized it, i.e., you solve the PDE as a function of e^{i(k.r - omega t)}

2. You get a complex polinom, but I don't see any parameters in your equation.

3. Anyway you can try an algebraic manipulator such as math or maple and find the roots. However, I find it strange that there is not a parameter, you need it to scan the complex solution.

Best Regards.

PDE's can be solved with the software Mathematica. They have a free version online.

Dear Parthiv Chandra K

The literature that may be useful are,

Due to different factors

Dear friend,

This problem you describe has been known for a long time and can be due to several causes.

Firstly, the use of FEM for the calculation of mesh cages is not always recommended. This is due to the fact that the shape changes of the netlike structure can be relatively large. This is due to the fact that the hydrodynamic loads are relatively large compared to the tensile forces in the mesh bars.

A second reason may be the discretisation of the netlike structure. A discretisation that is too fine-meshed can lead to numerical problems, a discretisation that is too coarse may be physically impermissible. Here you could try a little - also in conjunction with the hydrodynamic loads.

And last but not least, you should allow a slight stretching of the net material. You will see that this has a positive effect on the peak loads in the mesh bars (they become smaller), which are ultimately responsible for the structural vibrations.

If all this does not help, you need to test what frequencies of the waves are allowed, that means also in the hydrodynamic loads (you mentioned the Morison equation, which is known to consist of the Froude-Krylov force, the hydrodynamic inertia force and the viscous load).

Good luck, Mathias Paschen

for rectangular footing if your conducting plane strain condition it will become strip footing?

Ramarajan J. If the motion is prescribed or can be described by formula, than the approach will be no FSI simulation needed.for purely the rotational motion it would be sufficient to assign the calculated surface velocity as a boundary condition to the walls of the rotational spherical body. For the additional axial displacement, the body real needs to be displaced with regard to the originally generated mesh and this could be accomplished by mesh morphing (in CFX) or the so-called "moving-deforming mesh" algorithm in Fluent.In contrary, if the rotational spherical body can freely move in the fluid under the influence of acting hydrodynamic forces, then a description of the body motion is not possible by simple formula. In this case not a FSI, but a 6DOF (6-degree-of-freedom) simulation would be required. The body itself is not deformable, I assume. So its motion can be described as the motion of a rigid body under the influence of the acting hydrodynamical force and moment on its center of mass. Combining 6DOF with moving-deforming mesh would then do the work.

take this video as an example: https://youtu.be/pn0J_XSOkds

There are two behavior values that you need to set, one for the "reference" and the other one is for the "mobile" that needs to be "Rigid" in your case. Attached please find a sample analysis that I did long time ago. Two joints are applied in this model. The top joint is Revolute joint and the bottom one is Planar that enables applying side displacement while the part is rotating.

By the way, in addition to rotational velocity, rotational acceleration is available with joint load that can simulate the impact part of your simulation (sudden start).

In my opinion, the stationary solution you got right now is good enough and performing a transient analysis as I said earlier is more expensive and time consuming but it is still an option.

Dear Hamidreza Javidrad

I council you this article:

I hope you can find what you are searching for concerning "deltas".

Your output is a result of FEM theory, which ensures only force equilibrium at nodes and displacement compatibility across adjacent elements, but not equilibrium of stress. Given a deformed mesh, the stress in a node changes depending on whether that node belongs to one element or to the adjacent one.

In addition, it seems that yor model has a geometrical singularity (zero notch radius), where the theoretical stress is clearly infinite (whilst the FE stress is finite, no matter how fine is the mesh).

Finally, it is not clear to me the purpose of your analysis and what result you expect from it.

Dear Nate Tran ,

It is a bit difficult to understand how you setup your numerical model. I mean you did not say anything regarding how the material is setup (it is setup with Stress-Strain curve or ...) and what nonlinear model is used for this simulation (e.g., Multilinear Kinematic Hardening, Bilinear Kinematic Hardening and etc. ). For that reason I can suggest general guidelines for this type of simulation.

Also, if you haven't considered any fracture concept (e.g., XFEM, etc.) you can't visually see the breakage happening in your model. Therefore, you only see that the analysis is not converged and consequently terminated.

Another important thing is that, you are trying to see the effect of different strain rate in your analysis. If this is the case, then I suggest you use transient structural analysis instead of static structural analysis. Static structural analysis doesn't consider time integration in the numerical analysis and you won't be able to see the effect of different strain rate.

Hope this answer helps!!

For me, too, it is a bit unclear if your question is about divergence in numerical time integration, or about determination of stability boundaries.

Adding to what the Gabriel Thomas wrote, if the question is about stability boundaries, then as an alternative to a brute force grid algorithm you could also consider numerical continuation and bifurcation analysis. There exist a number of freely available software implementations. One of these is MatCont, co-developed by Yuri Kuznetsov from Utrecht. (If you need pointers, feel free to let me know.)

Please, share with me any report on Springer method for derivation of the diffusion coefficient of composites from the diffusion coefficient of matrix...

Malcolm White :

I see, I did find the imput for the Volumetric loss density, electromagnetic (ec)

But, when I fix the Heat Source with this source, I got no Temperature (see attached image please)

Dear Franklin Uriel Parás Hernández ,

Since I got more experience with scripting, I want to share my opinion from scripting point of view.

I found ANSYS more interesting than COMSOL because of its outstanding scripting capabilities. Depending on the module you want to use, ANSYS either uses JavaScript (e.g., DesignModeler and Mechanical) or IronPython (Workbench interface, Fluent etc.) in the background. In addition, Ansys provided APIs in IronPython programing language for almost all its modules (e.g., DesignModeler and Mechanical) which makes automation very easy. I mean you are not going to be limited to go with only one programming option and different gateways are provided for programing.

whereas, in COMSOL you will be limited to APIs provided in Java only, which is fine too.

This is my two cents and hope other people can share more great opinions from other perspectives.

Thanks,

me too, i need to know, How can estimate the Abaqus chaboche parameters, please ?

see

Sir Szuhsien:

You are welcome, and i wish you good health and happiness....

Hi Md Wasi Uddin

I think this article might helpful for your research:

Wish you all the best luck!

KHOA

Maybe you wan to consider the recursive least squares algorithm (RLS). RLS is the recursive application of the well-known least squares (LS) regression algorithm, so that each new data point is taken in account to modify (correct) a previous estimate of the parameters from some linear (or linearized) correlation thought to model the observed system. The method allows for the dynamical application of LS to time series acquired in real-time. As with LS, there may be several correlation equations with the corresponding set of dependent (observed) variables. For the recursive least squares algorithm with forgetting factor (RLS-FF), adquired data is weighted according to its age, with increased weight given to the most recent data.

Years ago, while investigating adaptive control and energetic optimization of aerobic fermenters, I have applied the RLS-FF algorithm to estimate the parameters from the K_La correlation, used to predict the O₂ gas-liquid mass-transfer, hence giving increased weight to most recent data. Estimates were improved by imposing sinusoidal disturbance to air flow and agitation speed (manipulated variables). The proposed (adaptive) control algorithm compared favourably with PID. Simulations assessed the effect of numerically generated white Gaussian noise (2-sigma truncated) and of first order delay. This investigation was reported at (MSc Thesis):

Dear Md Wasi Uddin

To understand how to compare your results with an experiment or literature results, please learn the theory of Dimensional Analysis. According to this theory, a phenomenon is governed by dimensionless numbers. For example, in one case, the Nusselt number is affected by the Reynolds and Prandtl numbers. Therefore, when you compare the Nusselt numbers you get with the Nusselt numbers from the literature, you must compare them to the same Reynolds and Prandtl numbers. For the case you are comparing, you need to equate the definitions of Re, Pr, and Nu between your case and the case in the literature. For example, if the Re and Nu numbers are defined by involving the hydraulic diameter in the literature, then you calculate your results at the same Re and Pr.

Good luck....

Explicit solver can be used for modelling slow problems if you use appropriate time scaling or mass scaling.

Without the scaling, the run time will be very long since the load is applied at a very slow rate. And with too much scaling, the results will be incorrect.

When you say he used Explicit, I assume that he used explicit solver with the actually loading rate.

Strain rate effects are related to the rate of displacement or deformation....not whether it is implicit solver or explicit. You still can simulation dynamic problems with implicit analysis and model quasi-static events with explicit, but you need to know what are the suitable keywords to add in both cases.

Here is how I did 3-point bending. I used explicit because I wanted to apply a dynamic load

Simon Smith Thank you so much for your explanation 🙏