What is implicit and explicit analysis in FEA? 3

[bongirs]

Hello,

I am trying to understand what it means when a software says that it has an implicit solver as well as explicit solver.
Often I find that people try explain the difference by using quasi-static vs time dependent analysis, but I know that that is not the correct answer. Additionally, how do I as an analyst understand whether the simulation I am doing is explicit or implicit?

Regards,
Sumit

 

[IceBreakerSours]

Partial differential equations can be solved by different methods: Analytical methods such as Laplace transforms, conformal mappings, variational methods, perturbation methods, and numerical analyses. Each of these methods have their own strengths and weaknesses. As it turns out, numerical methods (FEM, CFD, BEM, SPH/DEM/EFG, .. ..) have enjoyed much success in the real world of engineering/research problems.

With clever tweaks of Taylors' series approximation, you can convert the governing equations of equilibrium in to a numerical approximation of those equations. If, in the approximation scheme, the values for a given variable at future time depend on previous values, then the approximation is an explicit numerical scheme (to solve the governing equations of equilibrium). On the other hand, if in the approximation scheme, the values for a given variable at future time depend on the same future time, then a matrix inversion is required (i.e., solving a system of equations). Ultimately, you end up with Ax=B and that is what the whole drama is all about. The other major class of problems in linear algebra is Ax=Lx (L=lambda; a scalar) - the eigenvalue problem.

Note that these are NOT the only kinds of solvers available. Some schemes are a mix of the two schemes, others not so much. Also, note that - from a theoretical standpoint - either scheme MUST give you the exact same response. However, practical considerations end up making a massive difference in deciding which tool (solver) to pick for a given problem.

Also, FEM is just ONE of a variety of numerical methods where implicit/explicit/.. schemes are employed. The key ideas about FEM are in the principle of virtual work, Galerkin method, Raleigh-Ritz approximation, and perhaps, most importantly for an analyst, linear algebra.
Finally, as you must know, mechanics is just ONE area where these numerical methods are applied.

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[btrueblood]

see

https://en.wikipedia.org/wiki/Explicit_and_implicit_methods

...or your college FEA textbook.

 

[bongirs]

Thank you for your elaborate explanation IceBreakerSours! I have a few follow up questions:

1. "Partial differential equations can be solved by different methods" - I studied the direct stiffness method in school and never came across any differential equations in it. So do all FEM have partial differential equations, or it is only available for some FEM's?

2. Does it mean that the explicit analysis does not involve stiffness matrices -> assembly -> solve system of equations?

3. Is explicit analysis same as time-step analysis?

4. So is implicit analysis same as direct stiffness method?

5. Does this imply that implicit cannot be used for dynamic analysis?

6. What could be the reason for choosing either of these solvers if the expected result is the same?

7. You hint towards more solvers than implicit and explicit. What could these be?

 

[bongirs]

Thank you for your response btrueblood!

Somehow Google never showed me that Wiki page. It explains the difference between these 2 methods with clear understanding.
The given example also helps clear this distinction. But the distinction makes sense when you are solving a problem for which you know a differential equation. What about any CAD geometry imported to a FEA software? Let's say an engine that I am analyzing. The solver has no idea about the differential equation for the engine. Then how does it calculate the dy/dt as seen in the example on the Wiki?

 

[IceBreakerSours]

I suggest you purchase a few good books to learn the fundamentals at your own pace. This back-and-forth will address some kinks, not serious gaps.

1) Direct stiffness method has nothing to do with PDEs. DSM was among the earliest tools structural engineers developed to discretize geometries, generate algebraic equations at the back end, and provide an answer to the question that was being addressed. It is taught in introductory FEM courses because it gives you a good foundation. Again, DSM is not FEM.

"..do all FEM have partial differential equations, or it is only available for some FEM's?.."

I think you were trying to suggest numerical methods instead of FEMs. There are no FEMs; there is just one finite element method. Now, to address your question, physics/chemistry/biology/finance/economics/... is where PDEs arise, not in a numerical method. The numerical approximation method is a tool to solve those PDEs.

2) Explicit numerical schemes do not have to solve for equilibrium. Assembly and other pre-processing steps are more or less the same; there are variations, to be sure.

3) I do not know what you mean by time-step analysis but time-marching is essentially what occurs in an explicit scheme.

4) No. There is no interpolation. There is no search for minimum potential energy. DSM is essential to having a visual feel for how domain discretization works, how elements get assembled, how a physical problem ends up as a numerical problem like Ax=B, and how it gets solved.

5) No. Numerical schemes solve the phenomenon of interest, which are typically written out as PDEs. It has nothing to do with the nature of the problem (transient or static, e.g.).

6) To give you a flavor, most (not all) implicit schemes are unconditionally stable - which has nothing to do with a particular model's convergence. Explicit schemes, on the other hand, run in to a stability limit known as the Courant-Friedrich-Levy (CFL) condition. Modeling fully incompressible behavior is rather straightforward for an implicit scheme, not so with an explicit scheme. When there is way too much going on in the phenomenon of interest in a very short span of time (think car crash), explicit schemes handle them much better than implicit schemes.

I do not wish to add more because it gets overwhelming pretty fast.

7) Simplest example is an implicit-explicit scheme

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[bongirs]

Firstly, Thank you very much for such a descriptive reply! It surely helps resolve some of my doubts. However, it raises even more questions than it answers . Anyway, I do not want to take a lot of your time, I guess you need to have a PhD to understand the complexity of FEM/numerical methods/PDE. Still here is a gist of what I understood

1. You need numerical methods for solving PDE is what I understand, and a numerical method may not necessarily mean FEM. I used to believe that FEM is the part where you convert PDE to a set of simultaneous equations for getting a solution set, because computationally you can solve simultaneous equations but not PDE directly. But I guess what you are saying is that you don't always have to bring the PDE's to a system of simultaneous equations? Also, what is DSM then if it is not FEM, and what is FEM then? (I know this one got pretty confusing )

2. Explicit does involve stiffness matrices, but somehow you don't need them to be assembled into a system of simultaneous equations?

3. I meant like in some software, if you are doing non-linear analysis, you get to define the time steps for the non-linearity iterations, like t=1, 2, 3.5 ... 10sec.

4. OK so what exactly is implicit analysis? I cannot find any good video YouTube nor Google is providing any help. The problem with FEM books is that either they talk only at the basic level or they talk in such an advanced level that a novice cannot understand. Its becoming difficult to understand the "big picture" for me....

5. So you mean to say that there is a different PDE for every different type of simulation? So are these PDE's defining the stiffness matrix (or whatever you call it depending on scenario) of each element? If yes then why don't they solve these PDE by hand, get the element stiffness matrices and then code these matrices into the software, why make the software solve the PDE for a know unit element? If everything I said is incorrect and the PDE does not relate to elements then what does it relate to? Obviously if the software can import any CAD geometry and solve PDE for that, then the PDE has to define the elements which discretize any given geometry.

6. So, implicit or explicit has no particular advantage or disadvantage over one another. Its just that each has its own ability to simulate realistically a given phenomenon. So deformations that occur in a small amount of time can be solved better with explicit. Then which type of simulations is explicit not able to solve as good as implicit?

7. You could not have me confounded any more lol

 

[IceBreakerSours]

You really do have to spend time studying this.

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[ThomasH]

@bongirs

I your first post you mentioned, as I understood it, working as an analyst. Do you work as an analyst today?

Thomas

 

[bongirs]

Thomas, I am a graduate student whose research project is a drop test simulation using RADIOSS. As I am seeking full time position, I was asked this question in an interview and I could not answer it.

 

[IceBreakerSours]

Perhaps the fastest way of gaining a necessary (yet insufficient) understanding of the method is to go through the documentation that comes with your favorite code. Out of the documentations I have seen for three different solvers, Abaqus wins hands down. If you have access to Abaqus, I can recommend which sections/books to go through.

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[bongirs]

I am using RADIOSS and the documentation is incompetent to say the least.
I can check the Abaqus documentation as my university has licenses for all major FEA software's.

 

[bongirs]

I found an answer in a presentation.
Basically, implicit solver considers a linear transformation of acceleration between two time steps. This requires inversion of stiffness matrix for each step.
Explicit assumes a constant acceleration between two time steps. This requires inversion of mass matrix.
It is said that mass matrix inversion is easier than that for stiffness matrix, I don't know why.

 

[IceBreakerSours]

FYI: While many use the term "inversion", matrix inverse is not computed. As it turns out, matrix inversion is, in fact, a massive waste of resources (for real-world applications).

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[bongirs]

If you are talking about solving the final system of simutaneous equations, then I know that computationally methods like Gauss Jordan are better than finding the inverse of co-efficient matrix, which is the part where I agree with you. However, I read that it necessary to find inverse of mass matrix and stiffness matrix to come towards these simultaneous equations. In other words, you have to find inverse of these matrices to be able to come up with a final system of equations F = Kd. Do you disagree on this as well?

 

[ThomasH]

Hi
The mass matrix is often assumed as lumped (only diagonal terms). It makes the inversion very easy.

After reading your posts I think you need to seriously study the theory behind your research project. Not just asking questions in a forum and perhaps read some manuals but actually study the basics. There is so much that can't be covered with this Q and A approach.

Good Luck

Thomas

 

[IceBreakerSours]

No, I do not disagree. And ThomasH has provided the answer to your question.

If I were signing off on someone's FE skills, besides their basic engineering skills, ideally, I'd want them to have implemented a simple element, simple solver, and a non-trivial material. Or, some home-brew advanced coding that gets used by a commercial code.

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