18 Dec

#### Remarks

• Calculation speed increased 50% for calculating trial functions.

• 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

##### Dec 15, 2020 (Tuesday)
• All boundary conditions implemented (clamped, simply-supported, free).

• Checking whether results on edges satisfy known boundary conditions (unfortunately some of them do not).

• Trying to debug the code. Starting from clamped cases and simply-supported cases.

• One possible reason is the direction of the normal vector $$n$$. The current code assumes the rotation of axis to be counterclockwise by default. This may cause the $$n$$ vector to point opposed to the "inner area". This may be the reason why resulted slope on some edges have negative value. By drawing coordinates to visually / manually verify this theory. I believe this is the cause.

• Trying to implement the correct $$n$$ direction. Not easy. Will be working on this today the entire day I guess.

• By the end of the day, the correct $$n$$ direction has been implemented. Also for clamped and simply-supported edges, it has been verified by checking the corresponding moment / force / slope on edges. For both boundary conditions, visually the results on edges make sense. Free edge is still waiting for verification and possible debugging.

• More testing have been done on free edge. An interesting but disturbing result shows that for some combinations of clamped & free edges, the minimum order of polynomial gives the wrong / unreasonable result. It is known that increasing polynomial order will increase the accuracy, but this is unexpected since, by instinct, the minimum order of polynomial should also work somehow. Trying to fix this tomorrow.
##### Dec 16, 2020 (Wednesday)
• Problem: For rectangle with boundary conditions as C-C-F-C, the minimum solvable polynomial is with order of 10, which gives visually incorrect results. When polynomial order is increased to 11, still the visually incorrect results. When icreased to 12, the results start making sense. But the time it takes to calculate increased significantly.

• To solve the problem, the first thing I want to do is to find a way to make the code run faster, especially when polynomial order is high. My reasoning for this is we probably need to use higher order polynomial to get more accurate results (visually correct ones) eventually. And it actually helps to make the code run faster anyways. So why not do it now.

• One possible way to decrease the time it takes to run the code is to reduce the time it takes to do residual integral. This is tested with various cases that this step (residual integral) is the most time-consuming step, as we increase the polynomial order.

• A side note: function equationsToMatrix gives symbolic expressions (A, B), then using A\B+null(A) really takes a while to solve. Now A and B are converted to double first, then perform the calculation, which is significantly faster in some cases. A problem I had is A and B are symbolic, using null(A) gives me different results as using null(double(A)). After some research and tries, I found null(A)=null(double(A),'r'). The use of 'r' is explained here.

• Also, trial function calculation is put in a loop, the initial polynomial order is small (N=5 for now). The loop checks if N will lead to non-empty trial function. If yes, we get the minimum polynomial order; if not, N is increased until we get non-empty trial function. This may be a temporary solution. It does some calculation until a solution is found, which takes time. But it's acceptable (for now).

• Currently I am calculating residual integral by spliting the plates into slices of rectangles and calculate the area of each, then take the sum. I know there are other ways / built-in functions, but I have symbolic variables in the integral equations. I guess my way of calculating this is the easiest so far (I could be wrong, I will try different ways to do this integral, maybe it will be faster and more accurate). Anyways, spliting the plate into slices of rectangles, the more slices we have, the more accurate the result will be, but the longer it will take. I tried to use one case (rectangle with boundary conditions as C-C-F-C, the minimum solvable polynomial is with order of 10) to calculate the time and integral results. I notice the followings: (1) the time it takes to calculate linearly increases with respect to the amount of slices; (2) the integral results (a coefficient) decrease exponentially with respect to the amount of slices (results closes to plateau after ~50 slices). I will verify this more with more cases, to determine what's the amount of slices we should use.

• The integral of residual function takes the longest time, as stated before. Within the calculation of residual integral, the part that calculate the integral expression (add all slice-integral together) takes the longest time, about >90% of the calculation time. Now, its about to find another / faster way to solve this. I first tried to use different integral limits (y-limit), but it does not have any effect.
##### Dec 17, 2020 (Thursday)
• First tried to write a new method to calculate residual integral. Hoped to be faster. It supposed to take out coefficient C and calculate the rest into matrix A and B, then use A\B to calculate C. Well, the method did somehow worked, but it took longer time to calculate. So I have given up on this method.

• There must be a way to calculate double integral within a contour (ploygon), I am trying to find out how to solve this (I have done this by using small trapizoids, I want to find smarter / faster ways).

• Trying another way to do residual integral. Using variable expressions for upper / lower limits. In this way, there will be a lot fewer slices (a lot fewer iterations of integral needed, save times), and more accurate (upper and lower bounds are now the exact boundary, not approximated rectangles). Results show much faster (80% improve).

• Results validuated using rectangle with all edge simply-supported (existing analytical solution). Extremely good match, with error within 5%.

• Results also validated using other shapes and boundary conditions. Results on boundaries match the requested boundary conditions. However, results at other points can not be validated now, since there does not exist analytical solution. FEA is needed now, and this will be next step.
##### Dec 18, 2020 (Friday)
• One problem in the code that I noticed this morning is that the new residual calculation does not make sense when polygon shape becomes random or the polynomial order increases. After degugging, I noticed that the way the code slice the polygon may give wrong solutions. The reason is the interation point of lines give much more accuracy (e.g. to the 50th decimal point) but visually they look the same. Therefore, the slices give wrong results for some random polygon. I tried to round up the intersection points to 10 decimal point and it works fine now. However, there must be another way to solve this problem (to neglect the high / redundent accuracy).

• One thing I noticed during these trials: If the boundary conditions along all edges are the same, we only need minimum polynomial order to get visually correct results. However, if the boundary conditions are mixed, higher polynomial order may be necessary to get visually correct results. One more thing to deal with later.

• Using the new residual integral calculation, the corfficient c is very close to the old residual integral calculation. (note: old residual integral calculation gives approximate c)

• Checked new residual integral calculation v.s. old way. The time they take is as follows:
Ploygon Shape Boundary Conditions Polynomial Order Old Method Calculation Time (s) New Method Calculation Time (s) Speed Increase
Right triangle SSS 7(minimum) 17.7071 0.5290 97.01%
Right triangle SSS 8 104.3383 2.6930 97.42%
Right triangle SSS 9 434.9891 17.4255 95.99%
Right triangle SSS 10 1341.2622 94.3956 92.96%
Random triangle SSS 7(minimum) 9.7053 0.5142 94.7%
Random triangle SSS 8 47.6480 2.0723 95.65%
Random triangle SSS 9 309.8624 14.8110 95.22%
Random triangle SSS 10 1008.6408 51.5274 94.89%
Random triangle SCC 7(minimum) 16.5692 0.7590 95.42%
Random triangle SCC 8 96.7196 4.0388 95.82%
Random triangle SCC 9 413.7540 19.0235 95.4%
Rectangle SSSS 8(minimum) 11.0170 0.3224 97.07%
Rectangle SSSS 9 60.1881 1.0085 98.32%
Rectangle SSSS 10 276.4640 3.2389 98.83%
Rectangle SSSS 11 1032.9852 19.2665 98.13%
Rectangle CCCC 8(minimum) 10.1211 0.2797 97.24%
Rectangle CCCC 9 62.4365 0.8610 98.62%
Rectangle CCCC 10 286.7006 3.5423 98.76%
Rectangle SCSC 8 (minimum) 9.5832 0.3042 96.83%
Rectangle SCSC 9 62.0498 0.9841 98.41%
Rectangle SCSC 10 276.7912 3.4167 98.77%
Rectangle CSSC 8(minimum) 10.3034 0.2880 97.21%
Rectangle CSSC 9 61.3386 0.9279 98.49%
Rectangle CSSC 10 274.8270 3.4119 98.75%
Random 4 edges CSFS 11(minimum) 194.0299 10.4531 94.61%
Random 4 edges CSFS 12 1128.4976 67.9565 93.98%
Random 4 edges CSFS 13 3217.7059 217.5573 93.24%
Random 5 edges CCCCC 10(minimum) 18.3101 1.4104 92.3%
Random 5 edges SSSSS 13(minimum) 36.8810 3.2176 91.28%
Random 5 edges CSCSC 12(minimum) 28.7605 2.5393 91.17%
Random 6 edges CCCCCC 12(minimum) 26.0530 2.7281 89.53%
Random 6 edges SSSSSS 16(minimum) 1083.1359 132.9720 87.72%
Random 6 edges CSCSSC 15(minimum) 322.4051 38.8201 87.96%

#### To-Do List

• Free edge validation

• Comparison of two plates.

• Stress and strain calculation.

• The current implement works for convex polygons, and concave polygons with concaves vertically, but not horizontally.