Dear Reader,

Good day to you wherever you are reading this newsletter from. I am back from my short holidays with my family and feeling quite exhausted today.

I will be reflecting on the following:

1. Technical Reflections: Why density is important in Dynamic studies
2. Behind the Scenes at CM Videos: Paid Newsletter - Part 2
3. Quote of the Week: The Seed of Adversity

Technical Reflections

Why density is important in dynamic studies

If you have tried to run an ABAQUS simulation under dynamic conditions (using Dynamic Explicit step for example), you will notice that if you had failed to specify the density within the material model, the simulation will not progress. You cannot progress any dynamic simulations without specifying the density. This is as against a Static General simulation where even if you miss density out in your material model, you will still get a solution.

This is a question that students often ask, as to why this is the case: why is density so important for dynamic studies and less so for static studies within the finite element scheme? It is this that I will try to answer in the following

Finite Element Equation

The best place to start offering an answer is in what is generally regarded as the Finite Element Equation which is written thus:

The above is the basis for every finite element simulation and it is used in the Direct Stiffness Method to determine the values of forces or displacements as part of the finite element solution.

Studying that equation above closely, you find that there is a flagrant ommission of the mass of the element. Also, if the simulation were to progress rapidly so time effects becomes important, that equation is not able to deal with that. So, we make two modifications to the above equation:

• Introduce a mass component usually called a mass matrix to the equation
• Accommodate the effect of time on the simulation.

This leads of to the following broader, more comprehensive Finite Element Equation shown below:

Instantly in Equation (1) above, we notice a mass matrix, Mi for an i-th node in a model. Ci is the damping matrix of that i-th node whilst Ki is the stiffness matrix of the i-th node. Also, the displacement terms have been further differentiated with respect to time to introduce ui-dot (which represents velocity) and ui-dot-dot (which represents acceleration). This later addition enforces a time-effect to the simulation since the partial derivative of the u-terms is with respect to time.

Another important feature of the above equation is the mass matrix, Mi which is further expressed in Equation 2 to represent the dependence of a nodal mass distripution at a point in the model to the overall local mass matrix, Mlocal. It is precisely because of the need to define the mass matrix equation that we must have the density term, rho, as shown in equation 2.

Of course, why is this particularly important for dynamic simulations as against static simulations? This is because during dynamic simulations, the acceleration and velocity terms become significant. Let us say, if a body is displaced along the x-axis by 10 mm so that u = 10 mm. If this displacement happens in a very dynamic initial time of t1 = 0.1 seconds, then the first velocity term will be vi1 = 100 mm/s. And further still, we can calculate the acceleration term. Let us say, this vi1 = 100 mm/s velocity changed to vi2 = 120 mm/s in a time of t2 = 0.01 second (usually associated with an impact event), then the acceleration for such linear translation assessment will become ai = 2000 mm/s^2.

What we are seeing here is that the u-dot and u-dot-dot terms of equation (1) above are no more negligible when multiplied with the mass distribution matrix, Mi and damping matrix, Ci. As a result, all of a sudden, the dynamic force contribution resulting from the velocity and acceleration terms (dynamic forces) developed during the simulation will exceed the static force generated due to Fi = Kiui term.

If the simulation were not to be at dynamic rates, then the acceleration and velocity terms will be vanishingly small and hence needed to be neglected. In which case, even if we do not specify the density, the simulation will still progress to completion without throwing up an execution error.

Behind the Scenes at CM Videos

Paid Newsletter - Part 2

I shared last week of my intention to create a CM Videos Insider Paid Newsletter which will offer more dedicated insights into the issues of computational modelling. It is expected that such discussions as above (with technical contents structured in a lecturing style) will be a regular feature of such newsletter.

I asked for feedback from you and thank you to those who filled the form. I am releasing the same form again this week in case you missed it. Please click the button below to fill the short form to let me know what you think about the plan.

Here are the benefits of such a Paid Newsletter.

• Acess to all the existing features of th free CM Videos newsletter in the format I currently use.
• More indepth lecture-style sharing of technical insights on specific topics within the computational modelling space.
• Priority access to me with direct opportunities for you to reach out to me directly with your questions which I will attempt to answer.
• Priority access and generous discounts to existing and new CM Videos products
• More personal behind the scene stories from my research, work, and business - with behind the scene images and videos to help you get to know me more intimately.

Thank you for filling the feedback form.

Quote of the Week

The Seed of Adversity

I heard someone quote the above in a YouTube podcast that I was watching about growing a YouTube channel and the challenges small YouTube content creators face. The thoughts resonated with me a lot as lately, the YouTube channel has not been growing as fast as I would have liked. I had hoped that by September, I would have crossed the 4000 subscriber mark but today, I am just about a hundred subscribers away from it. Dont get me wrong, I am happy with the growth just wanted more.

As a result, I was feeling rather disappointed in my effort. I keep telling myself, if only I can make better videos, then the numbers will begin to grow but with all effort, this is not the case. I have taken several iterative steps to keep optimizing the content I produce yet, it seems I am not making the progress that should come with my efforts.

So, listening to this YouTube mentor of mine, Sean Cannell, in this video, state that:

Herein lies the benefit of the quote above from Napoleon hill. I am going through the adversity of delayed YouTube growth and often think it is a failure. Yet, within this adversity and seeming failure, there is a seed of equal or greater benefit which I am not aware of. This seed is seen in the 'comments' people leave on the videos. Here is a recent one that really gladdened my heart:

Of course, it is difficult to quantify the benefit people derive from my videos but if I consider the direct emails I get in my inbox with people requesting specific help or stating they actually tried out some of the things I share on the channel, it does indicate that there is benefit to them from my contents.

I should look at more the opportunities of the channel rather than the seeming adversities and failure that one feels occassionally as a YouTube content creator.

I hope you can also use similar approach in your work - focus less on metric but more on the process and what progress and developments you are actually making rather than the outcomes you generate.

Thank you again for reading thus far. I wish you a wonderful weekend and hopefully we will catch up again next week. In the mean time, keep taking care of yourself and God bless you richly.