06 Jan

#### Remarks, Week 12/18/2020

• Calculation speed increased 50% for calculating trial functions (Turns out more work need to be done for this part during the next two weeks).

• Automatically determine minimum polynomial order.

• Calculation speed increased >90% during calculation of residual integral. The way to do this is by implementing a new integration method. The new method and the old one show matched results.

• Details can be found in my previous blog post here.

#### Remarks, Week 12/23/2020

• Solving trial function calculation problem. Visually reasonable results achieved with reasonable / faster calculation time. (The problem is when changing symbolic expression to double numerical value, numbers are cut to 17 digits, which is not accurate enough and cause more trouble. The current solution is to change number of digits to 100 and results are more reasonable. The calculation time is longer then using 17 digits, but significantly faster than using null(sym), I would say >95% increase.)

• Check free boundary conditions, using the new / faster residual calculation method.

• Stress implemented.

• Details can be found in my previous blog post here.

#### Remarks, Week 12/30/2020

• MWR for plate bending, code clean up. Code can be found here

• Plate buckling research.

• Try solving buckling equation using MWR. (Result: Not how it works. So I need to try something else.)

• Details can be found in my previous blog post here.

#### Remarks for the Current Week

• Starting from the end of last year, I have been studying plate buckling.

• There are tons of research materials, I am working on a book, Theory of Elastic Stability by Timoshenko & Gere, which is very classic and cited by most research articles.

• The general buckling governing equation, for rectangular plates, is $$\frac{\partial^4 \omega_{(x, y)}}{\partial x^4} + 2 \frac{\partial^4 \omega_{(x, y)}}{\partial x^2 \partial y^2} + \frac{\partial^4 \omega_{(x, y)}}{\partial y^4} + N_{xx} \frac{\partial^2 \omega_{(x, y)}}{\partial x^2} + N_{yy} \frac{\partial^2 \omega_{(x, y)}}{\partial y^2} = 0$$

• In the book & in literature research, special cases (e.g. rectangular plates, simply-supported edges, etc) have been widely used. I will start learning how to handle these special cases first, and see if we should approximate some other arbitrary cases as those special cases with existing solutions, or we should develop approximate solutions by ourselves, like using Method of Weighted Residual in dealing with plate bending.

• Special Case 1: Rectangular plate, simply-supported on all 4 edges, force along one direction only. Study note can be found here: SS-Rectangular-1DirectionForce.

• Special Case 2-n: Other cases are summarized here: Other_cases. There aren't many useful information here. The analytical solutions are too difficult to calculate and they work for rectangular case only, which is very limited.

• I am currently working on this paper On Solving the Irregularly-Shaped Plate Buckling Problem.

#### Plans for Next Week / Next Few Weeks

• Installing ANSYS / Solidworks on lab server. (I personally prefer Solidworks, since we can also use it for 3D design, and it is said to be more direct to use than ANSYS. However, it is not free. The cost of Solidworks license can be found here). How to do it has been summarized in my previous blog post (here). I guess the first thing we need to do is to install Windows system (64 bit) on the server. I can do it, but it may take longer time since I have no previous experience. So if anyone can help with that, it would be great. Thanks.

• After installing ANSYS / Solidworks, I need to learn how to use it. There are tutorials online and our case study is not complicated. We will see how long this will take.

• Validation of plate bending. This is necessary since our approach gives approximated solutions, they need validation. However, depending on what method I eventually choose to use for calculating buckling, we may or may not need to validate plate buckling. If we use some existing (approximated/analytical) solutions, they have been validated through-out the years, so we do not need to repeat the work.

• After the above steps, I will move on to dealing with decoupling of given foldable structures, i.e. loading plates & connected plates, and figuring out how loads are transmitted to each other.