Do you need to know math behind FEA?

Recently I asked a lot of my subscribers what they think is the most difficult thing in FEA.

While “interpreting results” or “too much theory in materials” appeared quite often, I also got several answers like:

My biggest obstacle is actually understanding the mathematical concept behind it. Although I am really interested in the application but I believe understanding the maths behind it will be extremely useful.

As I wrote few emails about it, I figured it will be more efficient to collect the thoughts into a post – so I can simply link to it in future : )

Is understanding “math” behind FEA important?

This is a very important question I think. Whatever you think about it at least a LOT of your time is at stake!

It doesn’t really matter if you are just starting with FEA or you want to go into nonlinear or explicit analysis. If there is something new you want to learn there is a choice you need to make:

Do I want to learn math and theory or not?

Whatever is your answer, learning that theory, understanding match etc. is a very time-consuming task. Which most likely mean it makes sense to consider if it is even needed : )

Below I want to show you how I think about it, and put some arguments forward. Bear in mind that this is not an “ultimate” answer but rather my approach to things. Most likely it comes from my past experiences, philosophy and who I am.

I write this with a bit of courage though, as a few months ago I asked several of very good FEA experts about this. I asked them without showing my position on it (not to suggest anything) and it seems most of them agreed with my opinion.

However, I don’t want to convince you about this – much rather I would love to know your opinion! It would be great if you could share your thoughts on the matter in the comments below!

Cars, calculators, computers and other tools

To start with the topic I think it is best to say that FEA is a tool. A very complex one, but still it serves a purpose and you use it.

What you want from FEA is to model something correctly, and obtain results that interest you. This can be anything from stresses to vibration frequencies. I don’t want to go into specifics here. What is important is the “process”:

FEA works like this:

  • You ask a question by defining a problem, meshing it and supporting it
  • FEA solves your problem and give you an answer
  • You interprete the answer

To this end, it is exactly like using a calculator!

You define the problem by punching it in, calculator solves the problem and then gives you an outcome. It’s up to you what you will do with the outcome.

How does calculator work?

Ach, this is one of my favorite questions… I don’t know!

I’m using one each day, but I would not be able to build it, fix it or even explain to my kids how it operates! I understand that this has something to do with a lot of “zeros” and “ones” but that is about it. I’m a civil engineer and I wouldn’t be able to build a single logic gate if my life depended on it!

The same goes for my car, or the computer I’m using right now… things just work (i hope!).

So the important question arises – is the knowledge of how things work important?

The answer is obvious: of course, it is!

But there should be a second question here which is: to whom?

And this is a bit more difficult : )

Do you want to manufacture calculators?

I think that how things work is a very important knowledge… to people who want to make those things.

To drag the example a bit more, if you like to manufacture calculators it is critical to understand how they work! But is it that important in using them? I dare to disagree… after all, I do use mine and so far nothing bad happened 😛

The same goes for cars (nope, I’m not this kind of guy who can build their own car from few screws and board), computers and the rest of what surrounds us.

I’ve read somewhere, that there is not a single person on the planet who knows how to make a computer mouse. Ranging from plastic injections to optics and electronics the scope of knowledge to gain is simply too big!

And yet… we all use one, don’t we?

Let’s get back to FEA shall we : )

The understanding math behind FEA, all those algorithms and equations is critical… if you want to write your own solver. This is an incredibly difficult field (for complex nonlinear problems at least), but definitely a path to consider. Is there a business there to be made? I have no idea – this is not my path!

If like me you want to use FEA as a tool in design things are a bit different. I think in such case it makes no sense to learn how FEA works with great details. Just as with calculator it is far more important to know what questions to ask and how to interpret outcomes!

Sure I will make few benchmark tests to see if the FEA soft I’m using “actually works” – but this is a completely different thing that checking equations manually…

Just be aware that “not learning how math works” does not mean you can simply ignore the method completely. There are things like “stress averaging” or “convergence” that you need to be aware of to correctly interpret results. You don’t have to solve the math, but being aware of the potential traps is very important!

Why things are as they are?

I think this “quest” to learn theory comes from universities. After all, the theory is something that is easiest to teach. And since no one likes to look bad it is better to say that this is the most important thing there is! (let me just once more say that it really is the most important thing… if you want to create your own solver!). You know organizing laboratories for all students taught by experience designers sounds nice! But since each design takes weeks to do (not to mention teaching in the process!) it is more or less impossible to organize. Even if there would be a will to do so…

I think that “understanding math” and “designing with FEA” are two completely different paths. While creating your own solver must be fun (och, I would love to know that all!) there is also a lot you need to know about design.

The design is neither easier or more difficult than “math” in FEA… it’s just a different skill set!

You don’t solve equations but instead, you will have other problems. The things like “how to support my model so it makes sense” or “how to interpret outcomes”. Those things don’t get any easier with understanding math I’m afraid. Not to mention all the additional things you need to know to design something with FEA from “what failure modes should I check” to “what are the code requirements”.

Sure that there is an overlap here, but if you want to practically use FEA I would focus far more on practical aspects of it! Sadly there are not a lot of materials and courses teaching that… which is an additional problem (and hopefully a chance for me and my blog!).

I hope this gave you something to think about! As I wrote at the beginning I don’t aim to convince you, but simply to provoke some considerations. I would love to know what you think about it. Share your ideas in the comments below, so we can further discuss this!

Want to learn something practical about FEA?

This is great! Try my Learning FEA course! You can get a free lesson there!

28 Comments

  1. Gaurav December 5, 2017 at 1:52 am - Reply

    As usual Lukasz, you took a problem we face and solved it at grass-roots level. Do you solve engineering problems the same way?

    • Łukasz Skotny December 5, 2017 at 8:16 am - Reply

      Hey!

      Well, I always try to look at things from the basic point first, but since I really enjoy FEA I tend to sometimes overcomplicate what I do simply for the “fun” I guess 😛

      Thanks for writing, I really appreciate that 🙂

      All the best
      Ł

  2. John K. Shebuski December 6, 2017 at 5:13 am - Reply

    I think learning how to perform a basic 3 or 4 element solution from scratch is important, especially if you want to advance and add to your engineering analysis toolbox. It also helps you learn to choose the best elements.

    • Łukasz Skotny December 6, 2017 at 7:33 am - Reply

      Hey John!

      Thanks for the reply!

      I can see where you are coming from – definitely a good point.

      I admit that when it comes to element selection I try to do benchmarks on bigger problems instead… what do you think about such approach?

      All the best!
      Ł

  3. Pavel Manolev December 7, 2017 at 7:30 am - Reply

    Recently we had one particular case, where we needed to go deeply into math behind the software (and to understand it), to find out what input and how it need to be inserted. The main problem was what theory in this area is still not well developed ans, let say, standartised. However it is one special case that, and
    I will agree that ussually we do not need to know how the solver works.

    • Łukasz Skotny December 7, 2017 at 7:48 am - Reply

      Hey Pavel!

      This is a great example! Indeed when something is “new” such approach makes so much sense! I have never been in a situation like this before, but I can totally see the merits!

      Awesome point Mate!

      All the best
      Ł

  4. Peter Bartholomew December 27, 2017 at 10:03 pm - Reply

    Do you need to know the math behind FE? It is possible to work on the basis of a more empirical understanding and there are limits to how much knowledge might be expected of any individual. It may be more important to understand the practical limitations of the manufacturing processes available to you, for example.

    That said, an understanding of the mathematics underpinning FE makes the method far more logical and predicable. There is no harm in understanding convergence rates, variational principles etc. The math merely provide concise descriptions of phenomena you will have to come to terms with anyway.

    • Łukasz Skotny December 27, 2017 at 10:10 pm - Reply

      Hey Peter!

      Thank you for the comment! I think you really neatly wrote something important here!
      It is true that there is a practical limit to what you can learn in a year or 10, and at some point, choices have to be made.

      I agree that knowing “all the math” helps – there should be no doubt about that. But as you pointed out there are other things that may help you even more.

      Thank you for taking the time to write this : )

      All the best
      Ł

  5. Satya January 8, 2018 at 8:13 am - Reply

    Hi there!

    I find it an interesting conversation on FEA. But often when you see FEA analysts with a passion for solving complex FEA problems sometimes fail to understand simple cantilever beam or simply supported beam.

    I support your statement on knowing the math behind the FEA is always helpful. Although learning complex math is difficult, at least its better to have a basic understanding of it.

    • Łukasz Skotny January 8, 2018 at 8:45 am - Reply

      Hey!

      I agree that math helps. But as you pointed out complex match is well… complex : )

      I think that as you pointed out it is better to spend this time understanding and learning how things work – in the long run it should help you more : )

      All the best!
      Ł

  6. Darinel V. Mata April 13, 2018 at 2:38 am - Reply

    We use tools to make our time efficient if we know the use of the tool and what can it do.

    For example, the calculator: we know the why is it always has 7 output if we add 3 and 4 or if we integrate 2xdx from 3 to 4. But the circuit board is unknown to us. We know the concept behind the “Math” not the “Program” or “Algorithm” behind for the software to work.

    Learning the “Math” or the language it use to interpret the behavior of structure is a very important aspect for us to know how to communicate with them. This should be learn first to make use this tool – FEA – more effective.

    • Łukasz Skotny April 13, 2018 at 6:33 am - Reply

      By all means… but I would say that “math” (like real equations and stuff) is rather the “circuit board” of FEA. Understanding how the model behaves and how to interpret outcomes is critical without a doubt!

      All the best
      Ł

  7. Cormac April 13, 2018 at 2:04 pm - Reply

    There are really two branches of math required to understand FEA. The equations defining the relationships between elements mentioned here are obvious, but the methods used by the software to implement these equations and to generate a solution are a whole other branch of math that can contribute in a different but equally significant way to the issues users have with FE solvers.

    In an ideal situation FE models are solved perfectly just as in the theory you find in FEA books, but in reality speed vs. accuracy tradeoffs are made in how the equations are actually solved in software. Computers also bring a whole new set of problems that have nothing to do with FEA theory such as floating point errors and even bits being flipped due to quantum effects/cosmic rays. There’s enough error trapping in most systems to avoid this, but there’s a reason error correcting RAM is available.

    When FE theory is discussed it glosses over the solving step saying a ‘matrix is inverted’, when in fact different choices of inversion algorithm can give different results (Or take much much longer). This is why even an expert in FEA theory will still not be able to explain everything going on in a solver unless they also know their computer science.

    • Łukasz Skotny April 13, 2018 at 8:11 pm - Reply

      Wow Cormac!

      I admit that I’m not sure if you are joking about the cosmic rays or not O.o

      But I admit that without a doubt everything is much more complicated than it seems 🙂

      All the best mate!
      Ł

  8. jeremy theler April 20, 2018 at 1:39 pm - Reply

    You can con through life not knowing the details of what you use. What actually happens is that we all know things in different levels. So I am a Nuclear Engineer and know (almost) all the details about how electricity comes to my socket down from Uranium. But I do not know all the details of the capitalist economic system I live in.
    Nevertheless, those who actually know math and understand everything that happens from the real physical system down to the computed results, will have far (far far far) more opportunities and changes to better solve the FEM problem than those who do not.
    As a side-detail, I should point out that in order to completely understand what happens behind a FEM code, it must be open source (and if it is free-as-in-free-speech, even better). Freedom is a necessary condition to understand FEM.

    • Łukasz Skotny April 20, 2018 at 2:53 pm - Reply

      Hey!

      I completely agree that we know stuff on different levels, and economic example is a good one. However, I think that it’s not “knowing something” or “not knowing something” type of a choice. That would be too simplistic – it is always better to know (just as it is better to be young, healthy and rich than old, sick and poor!).

      The dilemma is knowing THIS or knowing THAT, as everything takes time to learn, and mastery is reached after years not weeks of study. It took me around 10 years to learn what I know about shell stability (and stability in general). Now I’m fairly confident I can design pretty weird stuff… and yet I have no idea how mathematics behind riks algorithm works. I’m just using it as a tool to reach my goals (the design).

      If instead, I would spend last 10 years on FEA math I’m fairly certain I would know a lot in that field… but I wouldn’t be able to design anything… as I would be lacking 10 years of experience in design. I could, of course, make my own algorithm or make the existing one better… but that is not what I wanted to do (I’m interested in design).

      We as humanity already know way too much about stuff for one person to know it all. This means that we have to specialize in order to achieve something. I decided to specialize in practical aspects and design (and teaching/preaching that). I consider myself quite successful at it, and not a single time in my 10 years lack of mathematics was a problem/got me into trouble. Sure it would be better to know it all… but I would have to forsake what I know how to get that… and since I’m designing stuff not writing my own solvers… I prefer my way 🙂

      I don’t think I agree with you about the free FEM argument. I mean if I would be math-solver guy making my own – sure this would be important. Also, some Ph.D. researchers in similar fields definitely would enjoy it. But as it is right now I won’t have time (nor skill) to actually take a look at the FEM code. I just did few benchmarks at the start to verify if it works (and if I understand stuff correctly) and went with it. Also if a company does develop soft for 30+ years and I’m using algorithms that were created in the 70’… I’m pretty certain they already found all the mistakes : ) In that regard I would be more concerned with new solutions, as “simple” errors may not be found yet… and the fact that I can “check the code” myself doesn’t help me one bit… after all I’m not qualified to understand it!

      This, of course, doesn’t mean I’m into the most expensive soft you can find on the market… to be honest, I wouldn’t be able to afford it I think. But some testing is needed so it’s either a “rich company” (that can afford them) or popular product on the market for few years (people did design, caught and reported errors).

      I have found few errors in software in my time and reported them… luckily I found them before the design I was making was submitted… not after. The other way around would be pretty bad!

      Let me know what do you think about it 🙂

      All the best
      Ł

  9. Krzysztof April 24, 2018 at 12:00 pm - Reply

    It is very important to know theory and the math (also advanced). This knowledge give to the engineer self confidence in his possibility of work. The most important is to be smart and recognize the difference between two sides. One is the best engeneering practice and engeneering methodology and second the scientific approach and research. Between is gap but the sorce of knowlage basied at the same theory. It is my point of view.

    • Łukasz Skotny April 24, 2018 at 3:04 pm - Reply

      Hey Krzysztof!

      Without a doubt, there is a huge gap between science and “engineering”! There is a big movement of “bridging the gap” by teaching people complex math. I don’t know if this would work or not. I prefer the completely different approach. I’d rather teach people “deep understanding” of things. So they simply see how stuff really work. If someone would need the math (truth be told I usually don’t) they will learn it… but the math itself is not the way… not to mention it’s very tiring for me as a practicing engineer. I would rather see this “beautiful engineering magic” in everything I design rather than the constant flow of differential equations.

      However, I’m fully aware that there are people preferring this the other way around 🙂

      I don’t think that mathematics is necessary to be confident in one’s design. I’m really confident in what I do… but I know next to nothing about how math work in nonlinear solvers I use almost every day! Just as I don’t know how the processor work on a PC I use to run the calculations. It’s just another layers, and as humanity develops new things it will be layer, after layer after layer. One’s life will be too short to get to the core of “all” the knowledge in one field!

      Thank you for writing. It’s so awesome to know other peoples opinions on this!

      All the best
      Ł

      • Krzysztof April 25, 2018 at 6:25 am - Reply

        I agree and disagree ;). Some aspects doesn’t not require deep knowledge of mathematics but if you want to be expert in some are of calculations ( no science) it is very helpfully. You can keep control at the equations and you are not “black box” user. In typical daily engineer work much more important is to understood what, why and were and for what. the simple solutions are the best, the life not require complicate yourself work 😀 :D. Have a nice day!!!!

        • Łukasz Skotny April 25, 2018 at 7:26 am - Reply

          Hey!

          Well, as you said, “I agree and disagree” 😉

          Complicating your life is never a good thing for sure 🙂

          I’m curious about your experience. Could you please share an example where knowing the “deep” mathematics actually help in the design? Because in all honesty, I don’t see that…

          All the best
          Ł

  10. Joe June 1, 2018 at 1:37 pm - Reply

    Hi Lukasz
    Another great post. I totally agree. It is much more important to be able to do a simplified hand calculation to check your results than to understand the inner workings of the software. (Unless you are writing the software). Even if you were writing the software you would need to use software libraries. How many programmers know how the computer calculates a square root for example.

    • Łukasz Skotny June 2, 2018 at 4:20 am - Reply

      Hey Joe!

      Ha! I never thought about that 🙂 Libraries are indeed a good argument here. As far as I know, there may even be entire libraries doing parts of FEA available 🙂

      I’m really happy that you like the post, and thanks for dropping a comment 🙂

      All the best
      Ł

  11. Devanshi Handa June 19, 2018 at 8:53 am - Reply

    Hi Lukasz,

    As an undergrad student wishing to further my career in the field of CFD, I feel I am in a constant dilemna over not knowing where to draw the line while studying math. As you said, it is not about the choice of “knowing something” or “not knowing something” as the former is always better.

    I feel its the work of the mathematicians to bring in the “new methods” which are turned into solving codes by “computer science engineers”. But then, am I not just pressing buttons if I don’t know the math behind it? If I do chose to study the numerical techniques of solving the system of linear equations for quenching my thirst of knowledge, which would take not just a few weeks but a considerable amount of time, what would i gain from it If I am just gonna use the software anyway?

    • Łukasz Skotny June 19, 2018 at 6:28 pm - Reply

      Hey Devanshi!

      This is a tricky question – mostly because apart from few animations I saw I know literally nothing about CFD!

      If the assumption that it is more or less similar to FEA is correct (if not… just ignore my comment) I would say this:

      There is a difference in understanding how to solve a set of matrix operations and various equations and in knowing what stuff does and why. Sadly it is actually difficult to gain an understanding of parameters meaning without digging into math. It requires effort and also a chance to work on real-life problems at the same time (best if someone who understands stuff supervises you, or at least check what you do!). Studying math is a way to get there I think – but it will take years more I believe. If you get a chance work with more experienced people and learn from them. Somehow I doubt that practicing CFD engineer would start coaching you by asking to learn math… but maybe I misunderstand the field, as I don’t know it at all!

      Good luck!
      Ł

  12. Ben Gregory June 27, 2018 at 9:57 pm - Reply

    Łukasz,

    You’ve presented a neat thought experiment with this post and I was excited to see so much participation!
    Is theory important to practice? (In this case, are the mathematics important to practicing FEA?)

    I would say it depends; on intention and risk.

    Let’s start with a car and look at the typical intent of driving one, transportation. We understand the theory of our intention, to go from point A to point B. We learn to operate the car, are trained in and obey the traffic laws (I hope), and use the car to get to our destination. We understand the theory our intention rests upon.

    Now let’s look at FEA. What is the intention? Take for instance product design and performance of a new cell phone case, pretty low risk. The intention is to improve and optimize the material and geometry of the case to minimize damage to the consumer’s cell phone, as well as to remain functional and attractive to the consumer. (Thanks a lot marketing!)

    Now take for instance the FEA on a 150 story skyscraper. The intention of the FEA is to understand and predict the behavior of the structure under loaded conditions which are directly tied to the safety of the structure and its future occupants. To say you fully understand and can anticipate its behavior implies confidence and is the basis for trust with the owners and its occupants. Needless to say, the risk to human life is much higher than the cell phone case example.

    I wouldn’t care if the guy (or gal) designing my Samsung Note 8’s case understood the math behind FEA, but I’d be pretty upset if the gal (or guy) running the FEA on the Burj Kalifia didn’t have a DEEP understanding of how the consistent – mass matrix works in structural dynamics. This applies to the entire spectrum of math behind the study. They should understand the math behind it and use the FEA as I use a calculator; to more efficiently compute operations and mathematics I already understand.

    At first glance your calculator analogy is attractive; but I don’t think it quite maps out to FEA, because we actually understand the mathematical theories behind the operations a calculator performs. We know how to add, subtract, multiple, take exponents, logarithms, etc (I hope)… We simply use the calculator to become more efficient at executing these operations. What we don’t fully know (or need to know) are the minute details on how these theoretical operations are processed in our tool or how the tool was manufactured, as long as we understand the theory and can verify the result makes sense.

    For example take multiplication. We all know how to perform and execute at least one algorithm to compute the multiplication of two numbers. (I’m a big fan of the long multiplication method I learned in 3rd grade!). However, there are indeed several algorithms that will compute the exact same result. Whether our calculators use long multiplication, lattice multiplication, or binary multiplication, who cares. As long as we understand what multiplication is and can verify it.

    Back to the Burj Khalifa, regardless of the math, I would be upset if there were not a strong team of competent designers, specifiers, contractors, materials specialists, and overall sky scraper specialists involved with the project who have a proven track record of success with these tall, complicated, beasty buildings. Success breeds success, but keep that success and confidence rooted in truth and understanding, not ego.

    Objectively understanding where our personal talents, aspirations, and commitments contribute most to our passions are far, far more important than understanding partial differential equations. I would expect a person with a deep mathematical understanding of the practical and theoretical nature of structures to perform the FEA on such tall buildings and I would expect everyone practicing FEA to continue to improve the quantitative, abstract, and creative aspects of their brain.

    If we’re going to use FEA, why not just go ahead and learn the math behind it. Engineering degrees require the advanced math, just learn it.

    When told it’s too difficult or too complex, think of this. In 1800 only 12% of the world’s population was literate, today over 85% of the world is literate. In 1800 when asking nobility or clergy what percent of the world’s population they thought were even capable of reading and writing, the response was that 20%, mayyybe 30% were even capable of comprehending and producing the written symbols and combinations we know as words.

    Think of the freedom and realization that came with learning to read and write and can we gain a similar realization by having more people like you and me thoroughly understand and apply advanced mathematics.

    Thanks,

    Ben Gregory, P.E.

    • Łukasz Skotny June 28, 2018 at 9:39 pm - Reply

      Wow Ben!

      This is definitely longest comment so far 🙂

      A lot of great ideas, but I don’t agree with all. I will start with what I don’t agree on.

      You said that the scope of what you do is important – and I agree… I mean I would be angry if someone designing my smartphone was “weak”… not because I care for my phone, but I care about engineering (BTW didn’t some phones caused fires due to overheating like a 1-2 years ago?). I would love everyone to be competent as it would serve engineers well… but I understand it’s a neverending fight… bah maybe even a naive dream. Now there are a lot of uneducated engineers that tell their customers stuff can be done with 25% of the budget in half the time! This influence work environment of us all… and is actually dangerous I think!

      I dislike your comparison, as I’ve learned at school that death of a one person is a tragedy, and hundreds are just a statistic. If someone is designing a small house and fails it is just as bad as if someone designs a skyscraper and fails. “Statistically” speaking this is of course not the case… but try to explain that to the family of the house owner if things go south… This is why I believe everyone should be competent, and distinctions that you make don’t support that I think. Good engineering is important – and it’s important that we all do it properly!

      To drag this a bit more, it would seem from the argument you are making that if I want to drive my car faster I should learn more about how the engine is built and how it works… I believe such nuances are influential only in the extremely limited amount of cases (like being a super F1 driver or something… or at least I guess those folks know how these things operate). Afer all you haven’t provided me with a clear example where knowing high-math was critical in any design. I would rather say that it is important to know the traffic rules (like “how engineering works, what boundary conditions are etc”) and then simply do work (hopefully with the supervision of more experienced people). And this is where I completely agree with you. It is obvious that one person cannot know everything. That in a super big project it is critical to assemble a team that can handle it. Of course that there will be people understanding stuff we don’t work with us… this makes engineering beautiful after all doesn’t it? The question is how many super math gurus are really needed in those teams.

      If I would design a super big project I would like one of those on board. Not because of accuracy to the slightest, however. Let’s face it, I do my best to learn few hours each day about FEA and what I do. I learn about practical aspects of things, I do tests and fun stuff and I try to expand. I already know I won’t know everything there is to know from the practical stuff – not to even start about theory learning and advanced math. If someone is a math expert that is as talented as I and work as much in the complex math domain she/he will know things I won’t even hear about in my life… but I can tell the same about me! I wouldn’t trust a guy who can write his own super FEA nonlinear algorithm to design my house… he simply can’t do it… it’s not his job! But on a big project his knowledge would help to cut on computing time… you know which algorithm is the best, tweaking data I/O maybe even hardware setup (although this may be a different field altogether). So in such a big job having a “math guru” on the team saves you time… engineering (as I understand it) happens somewhere else.

      About the calculator – this is a tricky parabole (I like the car much more!) – sure you know how the multiplication works… but I would say that this is the “traffic rules” – you know WHAT you want to multiply, not how calculator does it. I don’t know which algorithm is implemented in mine… and I don’t care. All I know is that I need to multiply this with that and that the outcomes should be more or less this much… and to me, this has nothing to do with the “higher math” of FEA.

      As for learning trust me I completely understand you. In fact I would love to know the complex math in and out – this would at least feel great! But I also know this is at least a 10+ year study project (to the level I would consieder sufficient…) and my “to learn list” is full of things I consieder higher priority… and I think this is the biggest problem I have with high math – percived value Vs learning time. Sadly so far I haven’t heard arguments that would shift how much value I place on the math side of things…

      At the end – thank you so much for writing this – clearly, this was an effort and I really appreciate this! Also if you would ever want to chat about it just drop me an email so we can set up a date – I would love to! (enterfea@enterfea.com).

      I really see that you are as passionate about this as I am, and I would love to discuss this further!
      All the best Ben!
      Ł

Leave A Comment

Do NOT follow this link or you will be banned from the site!