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[Vanshilar]

So here are some notes on calculating time to strip armor. I think overall your analysis is pretty much on the right track, but there are a number of issues which this post will address.

Warning: math involved, though I'll summarize the conclusions at the bottom.

Assumptions

1. I know it's fairly common to assume the damage as going to a single block of armor, but it turns out, due to the inner/outer armor cell mechanic for Starsector, in practice armor degrades a lot more gracefully than what's implied by assuming a single block. 60% of the DPS goes toward the 9 inner cells, while 40% of the DPS goes toward the 12 outer cells. But the armor rating is based on the inner cells plus half of the outer cells, meaning it'll only "see" half of that 40% of the DPS that went to the outer cells. So the weapon does apply 100% of its damage to the armor cells, but the armor rating will go down by only 80% of the DPS. You can account for this by calculating for the inner and outer cells explicitly, or by setting the DPS to 80% until the armor rating is down to 20% of the base armor. At that point the inner cells are gone and 50% of each of the outer cells are left (which makes the armor rating 20% of the base armor). From there, you have 60% of the DPS go toward hull, while 20% of the DPS goes toward the remaining armor, until the armor is reduced to 0.

Mathematically, when the armor rating goes from 100% to 20%, the weapon has actually already taken off 100% of the base armor rating, just that 20% of it is "missing" due to the armor rating from the outer cells being halved. Then by setting the DPS to 20% when it's really 40% as the armor rating goes from 20% of base armor to 0, the weapon does an additional 40% of the base armor. So in the end the weapon really does 140% of the base armor rating to fully strip off the armor, as it should.

2. It turns out, yes projectile weapons can be approximated as beams, or more accurately, continuous weapons, with the same DPS and hit strength as the base weapon. (Even beams are not calculated continuously, but at 0.1 second intervals IIRC.) This can be seen by using my armor simulation Excel posted here (which does account for inner/outer cells accurately as well as weapon damage type, assuming I coded it correctly). If you use whatever weapon hit strength and armor base value you want, and start with a very small weapon damage (which basically means a small step size), you'll find that every time you double that damage while leaving the weapon hit strength unchanged (i.e. doubling the step size), the number of "0" rows for hull damage (i.e. how long the weapon is hitting armor, until it breaks through and starts hitting hull) will pretty much be halved each time, to the limit of discretization anyway. For example, for 400 hit strength (energy damage), 500 armor rating, what you find for different values of the weapon damage are:

Weapon Damage : Number of steps (number of zeros) before hull is breached
3.125 : 280
6.25 : 140
12.5 : 70
25 : 35
50 : 17
100 : 9
200 : 4
400 : 2

(Side note: it's easy to count the number of zeros before hull is breached in Excel, you just need to type "=COUNTIF(H5:H557,0)" into any of the empty cells.)

In other words, when you calculate for the analytical, continuous case (i.e. step size ~0), as you increase the step size, the location where the formula crosses armor = 0 stays pretty consistent. In this case, at 400 damage, it would've predicted around 2.19 seconds, but of course that's not possible with discrete once-per-second hits so when calculating discrete 400-damage hits it'd say 2 (meaning on the 3rd one you start hitting hull). So it turns out, assuming it's continuous for derivation purposes is a pretty accurate assumption even when looking at projectiles. It's not exact, but it's pretty close for the most part. (Again, I didn't check for what if there are armor bonuses or anything, but this holds true for at least the base case.)

Problem Definition

This is where I think it gets a bit muddy. In this post you describe effective armor as e(A) = uA + v, but your derivation seems to define it the other way around, i.e. x is the effective armor and then you can define the actual armor phi(x) as phi(x) = alpha*x + beta. The definition should stay consistent, and I think this is where it runs into some problems.

For example, by defining actual armor phi(x) as alpha*x + beta, we have to then figure out what alpha and beta are (they are not the armor % bonus nor the added bonus, they're the inverse of that). Then in the derivation, you go from:

d(x) = H*B/(H + x)

to:

int (H + x)/alpha*H*B dx

However, you want to be careful with that differential. DPS is damage to the actual armor per second. But in this case, you defined x as the effective armor, meaning that you're integrating over the effective armor dx, rather than the actual armor dphi. If I understand your definitions correctly, then since phi(x) = alpha*x + beta, then dphi = alpha*dx, so:

DPS = dphi/dt (since phi is the actual armor) = alpha*dx/dt = H*B/(H + x)

==> dt = alpha * (H + x)/(H*B) dx

But you ended up with alpha in the denominator rather than the numerator later on. Fortunately since you're comparing ratios later on anyway the alphas will cancel out, so hope it doesn't matter. But it looks like the actual time calculations would be off if done in isolation. (That or I may have misunderstood your definition of variables.)

This is probably more of a writing style, but usually I'm more used to see inequalities written as "(variable) (inequality) (constant)", i.e. "x < 3" or "x > 3". Or, a convention where it's always written in one direction (i.e. always smaller < greater). In this case, 1 - M is a constant, so I would've written it as H/(H + x) <= 1 - M near the bottom of page 1. Also, using "d" as a function is usually discouraged due to confusion with the differential "d". That's more a matter of writing style though.

Also, the armor reduction bonus from Polarized Armor does [/i]not[/i] apply to the residual armor (i.e. the 0.05 after armor is stripped), or at least not in 0.95.1a-RC3 when I tested it. That residual armor is simply 5% of the base armor (including base armor modifying stuff like Heavy Armor) regardless of the hard flux level.

Result Derivation

For the sake of clarity, I think it's better to define x as the current armor. The reason is because then DPS is simply dx/dt, which simplifies a lot of things. So, start by defining a bunch of variables and parameters that'll be needed:

x = current armor
t = current time
T = total time for armor to be removed
W = weapon damage per second
H = weapon hit strength
A = base armor (i.e. initial starting armor when target is still pristine, which includes base armor affecting bonuses like Heavy Armor)
a = bonus % to base armor only for damage reduction purposes (for example, for Polarized Armor)
b = additional additive bonus to base armor only for damage reduction purposes (such as previously a +50 from Elite Impact Mitigation, though I don't think there are any such bonuses right now in the game)
M = maximum armor damage reduction (M = 0.85 by default, or 0.9 with Polarized Armor)
m = minimum damage reduction multiplier, which is 1 - M, i.e. 0.15 by default, or 0.1 with Polarized Armor; I prefer to think of this damage reduction bound in terms of "if H/(H+A) <= 0.15" rather than "if A/(H+A) >= 0.85", since I can then just plug in substitute m when needed, so I prefer to work with m generally speaking
r = minimum effective armor fraction (r = 0.05, i.e. it always counts as the armor having at least 5% of its base armor when armor is (almost) stripped away)
c = (HM/m - b)/a, which is the value of the current armor at which the weapon starts doing more damage than the minimum damage multiplier m

All of these except for x and t are constants, i.e. are fixed numbers which are known at the beginning of the problem (except for T, which is what we're solving for).

So starting from the definition of DPS:

DPS = dx/dt = W * H/(H + ax + b)
==> (H + ax + b) dx = W * H dt
==> dt = (H + ax + b) / (W * H) dx

We can integrate both sides to find t, the time to strip the armor. Deriving it in this way allows us to integrate dx with the bounds (i.e. accounting for m and r, etc.) in terms of the base armor, which is much clearer:

t = a/(2WH) * x^2 + (H + b)/(WH) * x (...plus C harhar)

Now as far as I know, there aren't any additive armor bonuses for damage reduction purposes currently in the game, i.e. b = 0. Thus this simplifies down to:

t = [(a/2H)*x^2 + x]/W (plus C)

(For the sake of completeness, I'll note that DPS is actually a negative number, i.e. it's how much the armor decreases per second, but by convention it's considered a positive quantity. But we can just flip the sign by flipping the integration bounds and being careful about that, so it's not a problem.)

So that's the bulk of it, from here on out it's just accounting for all the messy bounds, inner/outer cells, and all that.

Possible Cases

Case 1. Let's start with the trivial case, where the weapon hit strength is so low that the damage reduction will be equal or greater than the maximum damage reduction, even with stripped armor. Even though it's trivial, it's good at building up intuition for how the armor mechanics work. Take a vulcan cannon (500 frag DPS, hit strength 25 frag, or 6.25) hitting something with 1500 base armor (and thus 75 residual armor). Since its hit strength is so low, it'll always be doing the minimum amount of 15% regardless of the Onslaught's current armor value. This can be written as:

If H/(H + rA) <= m (Equivalently: if H <= rAm/M)
==> DPS = Wm
==> T (from x = A to x = 0.2A) = 0.8A/(Wm)

To account for inner/outer cells, the weapon damage W will really be 80% of it, so this boils down to:

==> T (from x = A to x = 0.2A) = A/(Wm)

At first this might seem a bit odd, that the time still boiled down to just the armor divided by the minimum damage. However, this is how long it takes to get the armor down to 20% of its original base rating. At this point, it'll display as the full base armor rating (i.e. if the ship had 1500 armor, at this point the game will display that a total of 1500 armor has been taken off, because it has), but in reality the armor rating is still at 300 due to the outer cells.

Now for what happens once armor is at 20% of the base value, this is when the inner armor cells are stripped but there are still some outer cells taking damage. At this point, the damage that hits the inner cells will hit hull instead, while the damage that hits the outer cells will continue hitting armor. It is the latter that needs to be reduced to 0 for the armor to be fully stripped. While 40% of the DPS is applied to the outer cells, it'll only count as 20% in terms of the armor rating. So:

T (from x = 0.2A to x = 0) = 0.2A/(0.2Wm) = A/(Wm)

So the total time it takes to strip off all armor is:

T (from x = A to x = 0) = 2A/(Wm)

This can be checked using the armor simulation Excel file. For example, if you set weapon damage to 500, weapon hit strength to 25 (note: since it's frag, the weapon hit strength for damage reduction is 6.25), weapon damage type to "Frag", and armor to 1500, then in this case, A = 1500, W = 125 (frag is only 25% vs armor), m = 0.15, W_h = 500, so:

T (from x = A to x = 0.2A) = A/(Wm) = 80
T (from x = A to x = 0) = 2A/(Wm) = 160

And indeed you'll see that it takes 80 steps for the inner cell armor to get reduced to 0 and for the weapon to start doing hull damage, and another 80 steps before the outer cell armor is reduced to 0.

Case 2. Next, let's take the case where the weapon hit strength is big enough to where it breaks above the minimum damage before the residual armor rA, but is such that at 20% of base armor, it's still doing less than the minimum damage reduction m. This can be written as:

If H/(H + rA) > m (Equivalently: if H > rAm/M)
and if H/(H + 0.2aA + b) <= m (Equivalently: if H <= (0.2aA + b)m/M)

Here, from max armor to 20% armor, the weapon is still doing the minimum damage of m (i.e. usually 15%), so everything is still the same as before, and so:

T (from x = A to x = 0.2A) = (0.8A)/(0.8Wm) = A/(Wm)

At this point, the weapon starts hitting hull, and only 20% of the DPS will go toward decreasing the armor rating. At some point, the DPS will start increasing above the minimum of m. This point is when the current armor x is such that:

H / (H + ax + b) = m
==> x = (HM/m - b)/a

For ease of writing, I'll define c to be this value of (HM/m - b)/a, because it's an important boundary point throughout this post, and I don't want to keep repeating it. It is the value of x below which the damage reduction calculation multiplier H/(H + ax + b) starts increasing above the minimum damage reduction multiplier of m. Note that b is often 0, but I'm including it for the sake of completeness. So:

T (from x = 0.2A to x = c) = (0.2A - c)/(0.2Wm)

Then from x = c to x = rA, the weapon is doing 20% of the quadratic damage to the armor. So (assuming b = 0):

T (from x = c to x = rA) = [(a/2H)*x^2 + x]/(0.2W) (from x = c to x = rA)
  = [(a/2H)*c^2 + c]/(0.2W) - [(a/2H)*(rA)^2 + rA]/(0.2W)

Then from x = rA to x = 0, the weapon is doing 20% of its DPS damage with the damage reduction calculation counted as if the armor value were rA, even though it may be less than that. So the DPS is:

DPS (from x = rA to x = 0) = 0.2WH/(H + rA)
==> T (from x = rA to x = 0) = rA * (H + rA)/(0.2WH)

So, taking this all together:

T (from x = A to x = 0) = A/(Wm) + (0.2A - c)/(0.2Wm) + [(a/2H)*c^2 + c]/(0.2W) - [(a/2H)*(rA)^2 + rA]/(0.2W) + rA * (H + rA)/(0.2WH)

If you'd like to simplify, this ends up being:

T (from x = A to x = 0) = (5/W) * [0.4A/m - c/m + (a/2H)*c^2 + c  + (1 - a/2)*(rA)^2 / H]

Case 3. Now, let's take the case where the weapon hit strength is big enough to where the weapon will be doing more than the minimum damage before the base armor reaches 20%, i.e. it starts doing more than the minimum damage m (usually 0.15) prior to the weapon hitting hull. But the weapon is still doing minimum damage at full base armor. This can be written as:

If H/(H + 0.2aA + b) > m (Equivalently: if H > (0.2aA + b)m/M)
and if H/(H + aA + b) <= m (Equivalently: if H <= (aA + b)m/M)

As with before, define the point c as the value of x below which the damage reduction calculation multiplier H/(H + ax + b) starts increasing above the minimum damage reduction multiplier of m. Then, from the beginning until x = c, the weapon is still doing the minimum damage:

T (from x = A to x = c) = (A - c)/(0.8Wm)

Then, we have the quadratic part from x = c to x = 0.2A, where, assuming b = 0:

T (from x = c to x = 0.2A) = [(a/2H)*x^2 + x]/W (from x = c to x = 0.2A)
  = [(a/2H)*c^2 + c]/(0.8W) - [(a/2H)*0.2A*0.2A + 0.2A]/(0.8W)

At this point the weapon will start hitting hull, while still in the quadratic part. So, from x = 0.2A to x = rA (assuming b = 0):

T (from x = 0.2A to x = rA) = [(a/2H)*x^2 + x]/(0.2W) (from x = 0.2A to x = rA)
  = [(a/2H)*0.2A*0.2A + 0.2A]/(0.2W) - [(a/2H)*(rA)^2 + rA]/(0.2W)

Then, as with before, from x = rA to x = 0, the weapon is doing damage as if x = rA, so as with before:

DPS (from x = rA to x = 0) = 0.2WH/(H + rA)
==> T (from x = rA to x = 0) = rA * (H + rA)/(0.2WH)

So, taking all this together:

T (from x = A to x = 0) = (A - c)/(0.8Wm) + [(a/2H)*c^2 + c]/(0.8W) - [(a/2H)*0.2A*0.2A + 0.2A]/(0.8W) + [(a/2H)*0.2A*0.2A + 0.2A]/(0.2W) - [(a/2H)*(rA)^2 + rA]/(0.2W) + rA * (H + rA)/(0.2WH)

You can probably simplify it (particularly since r = 0.05, and far as I know there's nothing that modifies that in the game), but I won't bother to here.

Case 4. This is the last case, where the weapon hit strength is so big that from the get-go, it's never as low as the minimum damage reduction multiplier m. This can be written as:

If H/(H + aA + b) > m (equivalently: if H > (aA + b)m/M)

Then we can proceed very similarly as Case 3, except now we don't have to worry about the beginning minimum damage section. Thus we start off directly with the quadratic case (assuming b = 0):

T (from x = A to x = 0.2A) = [(a/2H)*x^2 + x]/W (from x = A to x = 0.2A)
  = [(a/2H)*A^2 + A]/(0.8W) - [(a/2H)*0.2A*0.2A + 0.2A]/(0.8W)
  = [(a/2H)*0.96A^2 + 0.8A]/(0.8W)
  = [(a/H)*0.6A^2 + A]/W

The rest is the same as in case 3. So, taking all this together:

T (from x = A to x = 0) = [(a/H)*0.6A^2 + A]/W + [(a/2H)*0.2A*0.2A + 0.2A]/(0.2W) - [(a/2H)*(rA)^2 + rA]/(0.2W) + rA * (H + rA)/(0.2WH)

Summary

So, to summarize, assuming my math is correct (which is always a dangerous assumption and is never guaranteed):

Case 1: If H <= rAm/M
then T = 2A/(Wm)

Case 2: If H > rAm/M and H <= (0.2aA + b)m/M (and assuming b = 0 due to quadratic section)
then T = (5/W) * [0.4A/m - c/m + (a/2H)*c^2 + c  + (1 - a/2)*(rA)^2 / H]

Case 3: If H > (0.2aA + b)m/M and H <= (aA + b)m/M (and assuming b = 0 due to quadratic section)
then T = (A - c)/(0.8Wm) + [(a/2H)*c^2 + c]/(0.8W) - [(a/2H)*0.2A*0.2A + 0.2A]/(0.8W) + [(a/2H)*0.2A*0.2A + 0.2A]/(0.2W) - [(a/2H)*(rA)^2 + rA]/(0.2W) + rA * (H + rA)/(0.2WH)

Case 4: If H > (aA + b)m/M (and assuming b = 0 due to quadratic section)
then T = [(a/H)*0.6A^2 + A]/W + [(a/2H)*0.2A*0.2A + 0.2A]/(0.2W) - [(a/2H)*(rA)^2 + rA]/(0.2W) + rA * (H + rA)/(0.2WH)

These 4 cases are demonstrated in the attached graph, for 500 DPS, 1500 base armor, with minimum damage reduction multiplier 0.15. Note that the 4 cases are basically where the DPS starts going above the minimum damage reduction multiplier m. Case 1 = never, Case 2 = between 5% of base armor and 20% of base armor, Case 3 = between 20% of base armor and 100% of base amor, and Case 4 = above 100% of base armor.

Note that these cases are all defined in terms of the hit strength H relative to the base armor A. That makes it easier to get an intuitive feel for which case is applicable in a given situation.

-----

Now, regarding the hull damage while the armor rating goes from 20% to 0, you can follow the above analysis but now plugging in for hull damage during T from x = 0.2*A to 0 all the way through, or you can make a simple observation: during this period, the weapon does 20% of its DPS to armor, and 60% of its DPS to hull. Both are equally affected by the damage reduction calculation, so this proportion stays constant. So, this means that the weapon will do 60% of the base armor to hull during this time, subject to the difference between damage to armor and damage to hull as a multiplier (0.5 for HE, 1 for energy, 2 for kinetic, 4 for frag).

Taking the frag case, it means that the weapon will do 2.4x the base armor to hull, i.e. if the base armor were 1500, the weapon would do 3600 damage to hull during this time. Looking through the ship_data.csv, for all the ships, even if you add on Armored Weapon Mounts (+10% base armor) and Shield Shunt (+15% base armor), along with Heavy Armor (+150/300/400/500 base armor based on size), all of the ships will still have hull remaining after armor is stripped, except for the Dram and the Ox. So basically, this isn't something to be concerned about; you can almost always assume that the ship has enough hull to last through until the armor is gone. (In practice, this isn't true, because you could be hitting a fresh patch of armor when the ship has already taken hull damage through a hole in the armor on another part of the ship, but a ship taking damage from multiple directions is beyond the scope of this analysis.)

So this means that to calculate total time to kill, you can start with the time to fully strip armor above, and then start off the hull with already having taken some damage (60% of armor, modified by damage type), and then work from there. That makes the calculation a lot easier to do since you'll be working purely with damage to hull, assuming residual armor. (Until somebody wants to incorporate the effect of Combat Endurance's hull regen, anyway.)

[Warnoise]

I hope Alex is checking this thread because I wouldn't want to see all this effort going to waste

[float]

Thanks Vanshilar for the detailed writeup. I knew from the start that not accounting for the grid would be the greatest source of inaccuracy. However, I don't think there is any issue with alpha. I wrote that phi "maps actual armor to effective armor," so in phi(x) = alpha x + beta, x is the actual armor and phi(x) is the effective armor (as you pointed out, doing it in reverse would be silly as alpha and beta would be meaningless). The integral I have is initially from A (total actual armor) to 0, where du is the actual armor differential, but I change to dv (differential in effective armor) since d takes effective armor as its argument (hence switching from d(phi(u)) to d(v)). This is where the 1/alpha factor comes from, and its purpose is to make the interval intersections at the end meaningful (it wouldn't make sense to intersect two intervals where one is over effective armor and the other is over actual armor). I could have done this the other way too, making the damage function take actual armor as its argument.

One of the things I wasn't sure about was how the effective armor interacts with the residual armor. I thought I read somewhere that IM's elite effect did work on residual armor, which would suggest that the effective residual armor is phi(mA) = alpha m A + B. Of course there is also the question of whether the m applies before or after the phi, i.e. is it phi(mA) or m phi(A). But this assumes that polarized armor would also work with residual armor. If that doesn't work, but IM did, then we'd need two different versions of effective armor maps, one that works on residual armor and one that doesn't. I'd rather not have to deal with that.

Your derivations look correct -- I can't say anything about the actual equations themselves as they're rather difficult to read (please consider typesetting -- even uploading images of typeset equations would make a huge difference), but the casework seems solid. Note that you do make some implicit assumptions -- in particular, the case structure implies that rA < 0.2aA + b, i.e. the residual armor is less than effective armor at 80% damage dealt, which is the case when r=0.05 (default) but may fail if r >= 0.2. Also, your analysis assumes that residual armor will never have any effective armor modifiers (I don't actually know if it does, used to, or ever will, but it's something to keep in mind).

I think beyond approximation errors (assuming continuous weapons, etc), the biggest thing we're missing with the armor analysis is an account of the ship's exposed armor area as well as how scattered shots are. It's unrealistic to assume that every shot hits the exact same spot every time, and severely downplays the benefit that having a lot of attackable area has for a ship's armor tankiness. I know that it would be easier to simulate at this point, so I might try and do both -- knowing that simulation is going to be slow means I probably wouldn't be able to get nice graphs out of it.

[Vanshilar]

However, I don't think there is any issue with alpha. I wrote that phi "maps actual armor to effective armor," so in phi(x) = alpha x + beta, x is the actual armor and phi(x) is the effective armor (as you pointed out, doing it in reverse would be silly as alpha and beta would be meaningless).

Ahh, I see. Yeah, you might want to think about renaming some of the variables then, because later you talk about "at x effective armor" etc. which makes it more confusing, since x is sometimes the actual armor and sometimes the effective armor. Granted, I know in that context you're just talking about x in terms of, as an input to the d() function for DPS, and you have it defined in terms of effective armor rather than actual armor, but that's why I prefer defining it directly in terms of actual armor.

Also, "effective armor" just means the value A that's used in the H/(H + A) part of the damage reduction formula. The actual damage taken off is directly to the actual armor, and there is no "effective armor" there. Hence why I just kept everything in terms of the actual armor.

(As a side note for those who might be unaware, stuff like Heavy Armor, Armored Weapon Mounts, and Shield Shunt directly change the base armor, so that in combat, that modified armor value is what the game uses for the ship's base armor for all intents and purposes. Effective armor is because stuff like Polarized Armor modifies the A in the H/(H+A) part of the damage reduction formula while not changing the actual armor.)

One of the things I wasn't sure about was how the effective armor interacts with the residual armor. I thought I read somewhere that IM's elite effect did work on residual armor, which would suggest that the effective residual armor is phi(mA) = alpha m A + B. Of course there is also the question of whether the m applies before or after the phi, i.e. is it phi(mA) or m phi(A).

It's not that it works on residual armor -- it's that Impact Mitigation's effect is actually to decrease the incoming weapon's damage by 25%, when going up against armor, and when going up against hull for the damage reduction calculation H/(H+A) (since that's always as if the weapon is going up against armor), though not the damage itself. Granted, it's true that 0.75*H/(0.75*H + A) is mathematically equivalent to H/(H + 1.33*A), but it's not actually applied to the armor itself. Thus it's always active, so to speak, regardless of residual armor or not.

As far as I know, however, at least for version 0.95.1a-RC3, Polarized Armor has no effect on residual armor. For example, for a ship with base armor 1000, the residual armor value used for damage reduction when hitting hull will always be 50, regardless of the Polarized Armor ship's hard flux level. This means that it's true that there's a brief moment when the actual armor might be say 45 and Polarized Armor makes it say 55, and then the mathematical modeling would be incorrect, but to me that's so small as to not worry about. (Chances are, the continuous approximation probably affects things more than that anyway, even though its effect is actually quite small.)

Edit: I forgot to address the elite part. Impact Mitigation's old elite effect was to add 50 armor for damage reduction calculation purposes. It did this when armor was full, I'm not sure how it was applied as armor decreased (i.e. I'm not sure if it continued adding 50, or if it decreased at the same rate as armor). However, it did not affect the residual armor, which would still be at 5% of the base armor. But as an unstated bonus, if the target's residual armor were less than 50, then it would use 50 for residual armor. So it basically had a hidden effect where the residual armor was floored at 50. But if the target's base armor were 1000 or greater, it had no effect on the residual armor.

Your derivations look correct -- I can't say anything about the actual equations themselves as they're rather difficult to read (please consider typesetting -- even uploading images of typeset equations would make a huge difference), but the casework seems solid.

Yeah I may make it into a more formal document in the future, but I didn't want to mess around with that. I did try to keep the parentheses different though (i.e. () [] {}) to hopefully keep things more clear, but yeah it's hard to read. On the plus side, much easier to put into Excel this way, haha. I never bothered to learn LaTeX, I'd be using Word's equation editor, sigh.

Note that you do make some implicit assumptions -- in particular, the case structure implies that rA < 0.2aA + b, i.e. the residual armor is less than effective armor at 80% damage dealt, which is the case when r=0.05 (default) but may fail if r >= 0.2. Also, your analysis assumes that residual armor will never have any effective armor modifiers (I don't actually know if it does, used to, or ever will, but it's something to keep in mind).

Yeah I probably should have set some bounds on the modifiers the way you did, i.e. I assumed that r <= 0.2, and there are some "edge cases" that I think I may have to correct (for example, I'm not sure if I accounted for the additive bonus b correctly -- for example I think I should modify the "If" statement for Case 1 for things like if b = 1000 or something; but fortunately, there are no such modifiers in the game currently, as far as I know, so b is basically always 0 right now). It can be hard to know which game mechanics we can assume won't change in the future; but realistically the equations would need to be updated when that point comes, and having the equations account for all sorts of hypotheticals makes it harder to understand and harder to debug.

Yeah I assumed that residual armor doesn't have any effective armor modifiers, because I'm not aware of any in the game right now. (Again, this is because Impact Mitigation actually affects the hit strength and not the armor, even though it's mathematically equivalent, and Polarized Armor has no effect on residual armor.) If Alex added something to the game that affects it, I'd have to modify the equations.

I think beyond approximation errors (assuming continuous weapons, etc), the biggest thing we're missing with the armor analysis is an account of the ship's exposed armor area as well as how scattered shots are. It's unrealistic to assume that every shot hits the exact same spot every time, and severely downplays the benefit that having a lot of attackable area has for a ship's armor tankiness. I know that it would be easier to simulate at this point, so I might try and do both -- knowing that simulation is going to be slow means I probably wouldn't be able to get nice graphs out of it.

That might be something to consider later on, but it might be one of those things where it might be better for the player to just get a "feel" for how things work, rather than trying to be mathematically precise. For example, we commonly just look at a weapon's DPS as stated on the stat card, even though in practice some shots miss, etc. We also aren't including things like the effect of CR, shipsystem bonuses, and so forth. In this case the reason for doing the mathematical modeling is to get a feel for the effect of the various skills and hullmods, so it just needs to be close enough to that.

Yes a bigger ship is "flatter" to the weapon, and thus has more armor cells to chew through compared with a smaller ship. However, smaller ships are harder to hit (so, the chance that there is no hit at all increases). The spread of a weapon also depends on the weapon's distance to the target, how much the target is maneuvering, etc. So it's hard to define the problem properly to get meaningful numbers.

Having said that, I think a Monte Carlo simulation of that would actually be rather straightforward to program in Matlab or whatever is your program of choice. Unfortunately I don't have access to those programs anymore, so it's basically just Excel for me now.

[intrinsic_parity]

Personally, I don't see the point in modeling stuff as a continuous system when it's actually discrete, and you can very easily just numerically evaluate the discrete system to get the exact solution with no modeling error.

Also, if you're looking for free/open source programming languages, Python is pretty close to MATLAB syntax wise, and should be easy to pick up if you were comfortable in MATLAB. I would also highly recommend the programming language Julia. I've been transitioning a lot my MATLAB code to Julia for performance reasons, but it has very similar syntax to MATLAB, and should be a similarly easy transition.

Also, it's 100% worth learning latex! Overleaf is a really nice website that handles all the compiling stuff for you.

[float]

Edit: I forgot to address the elite part. Impact Mitigation's old elite effect was to add 50 armor for damage reduction calculation purposes. It did this when armor was full, I'm not sure how it was applied as armor decreased (i.e. I'm not sure if it continued adding 50, or if it decreased at the same rate as armor). However, it did not affect the residual armor, which would still be at 5% of the base armor. But as an unstated bonus, if the target's residual armor were less than 50, then it would use 50 for residual armor. So it basically had a hidden effect where the residual armor was floored at 50. But if the target's base armor were 1000 or greater, it had no effect on the residual armor.

This is interesting, I had no idea it worked like that. So it does seem like the effective armor map doesn’t apply to residual armor. This is slightly odd, since it means direct comparison of effective and actual armor values (i.e if actual residual armor is less than or greater than effective current armor).

Personally, I don't see the point in modeling stuff as a continuous system when it's actually discrete, and you can very easily just numerically evaluate the discrete system to get the exact solution with no modeling error.

The biggest reason for using math (where a continuous approximation is much easier to deal with) instead of simulation is efficiency. If you can find the symbolically calculate the time to strip/destroy a ship as a function of armor, then you just need to plug in values for the armor and constants to get a number, which takes constant O(1) time.

Consider how long a simulation would take. You’d need to simulate every time step basically, which means that the simulation time is proportional to the armor strip time. Since we know the latter is O(A^2), the simulation time is also O(A^2). The constant factor is also something to consider if including the grid — since every simulation step involves reading and modifying 21 armor values, the constant factor would be at least that.

This is fine if we just want to get the value at one point, but for something like a graph where we’re conparing strip time at many different armor values, efficiency is definitely something to keep in mind.

Also, just a small suggestion for your tool (which is very cool, I played around with it): it would be great if there were text boxes in addition to sliders so you could enter exact values. I'm not familiar with matplotlibs GUI tools, so if that's a lot of work, I understand not wanting to.

I noticed that too — the sliders are a bit annoying. I’m trying to update the tool to work with an armor grid, but I doubt simulation is fast enough for a dynamically updating graph (see above). So I’ll probably have to make even more continuous assumptions, like pretending the armor grid is actually a density function over R^2 and that the damage falloff surrounding an impact point is some radial distribution centered at that point.

[intrinsic_parity]

Personally, I don't see the point in modeling stuff as a continuous system when it's actually discrete, and you can very easily just numerically evaluate the discrete system to get the exact solution with no modeling error.

The biggest reason for using math (where a continuous approximation is much easier to deal with) instead of simulation is efficiency. If you can find the symbolically calculate the time to strip/destroy a ship as a function of armor, then you just need to plug in values for the armor and constants to get a number, which takes constant O(1) time.

Consider how long a simulation would take. You’d need to simulate every time step basically, which means that the simulation time is proportional to the armor strip time. Since we know the latter is O(A^2), the simulation time is also O(A^2). The constant factor is also something to consider if including the grid — since every simulation step involves reading and modifying 21 armor values, the constant factor would be at least that.

This is fine if we just want to get the value at one point, but for something like a graph where we’re conparing strip time at many different armor values, efficiency is definitely something to keep in mind.

Nah you can just use 'shots' as your independant variable if you are dealing with only one weapon i.e. work with damage/shot rather than damage/sec and then transform to time after you simulate. I've done it many times, it's not hard, and you will virtually never need more than ~100 shots so pretty much any computer should be able to run the program in like a few ms lol. The computer probably spends more time updating the GUI than actually running the program. Even if you simulated time directly, with time steps of like 10 ms, it should still only take a few seconds to run for any modern computer. I do lots of simulating dynamic systems for my research, this is really not close to any sort of computational limits, and I guarantee you spent way more time doing all those derivations than it would take to write and run a numerical sim.