How to make a circular heat map or diagram in Mathematica?

Is there any way to make a circular heat map in Mathematica?

Thanks for your help!

Here is a toy example for regular heat map. Can anyone help me make it to a circular one?

data1={{9., 1., 6., 7., 6., 3., 1., 3., 10., 2., 2., 5., 2., 5., 3.,    1.}, {5., 5., 5., 4., 4., 6., 4., 6., 9., 1., 2., 10., 2., 1., 1.,    6.}, {2., 7., 6., 2., 8., 10., 8., 9., 2., 5., 3., 9., 7., 8., 7.,    5.}, {6., 6., 2., 1., 8., 2., 8., 3., 8., 5., 5., 4., 6., 2., 3.,    6.}, {8., 1., 8., 2., 5., 8., 5., 3., 5., 3., 4., 2., 2., 4., 4.,    1.}, {10., 2., 8., 10., 3., 6., 1., 9., 3., 5., 2., 5., 1., 3., 7.,    9.}};  ArrayPlot[data1, ColorFunction -> ColorData["LightTerrain"],   Frame -> True,   FrameTicks -> {{{{1, "r1"}, {2, "r2"}, {3, "r3"}, {4, "r4"}, {5,        "r5"}, {6, "r6"}}, None},    {None, {{1, "c1"}, {2, "c2"}, {3, "c3"}, {4, "c4"}, {5, "c5"}, {6,        "c6"}, {7, "c7"}, {8, "c8"}, {9, "c9"}, {10, "c10"}, {11,        "c11"}, {12, "c12"}, {13, "c13"}, {14, "c14"}, {15, "c15"}, {16,        "c16"}}}},   Epilog -> {Text["Sector1", {2, 8}], Text["Sector2", {6, 8}],     Text["Sector3", {10, 8}], Text["Sector4", {14, 8}]},   ImagePadding -> {{20, 20}, {20, 80}}, ImageSize -> Large] 

enter image description here

I want to make a circular diagram like the following one.

enter image description here


Is it possible to make a one like this?

enter image description here

Thanks a lot!

Heat balance multilayer with Neumann condition

I am solving the heat equation (diffusion only) over a 3-layer system (physical properties varying from layer to layer).

km = 0.128; rhom = 925; Cpm = 1550;  ka = 0.024; rhoa = 1.292; Cpa = 1003;  thickness = 2*0.136;    L1 = N[-thickness/2]; L2 = N[thickness/2]; T01 = 120; T02 = 20; T03 = 120;  Length1 = 0.5;   v= 40/60;  time1 = Length1/v; 

For describing the properties change in space, I use the following equation:

slope = 1000; SmoothedStepFunction[fL_, fmax_, fR_, tsL_, tsR_, m_] :=   Function[t, (fL*Exp[tsL*m] + fmax*Exp[m*t])/(Exp[tsL*m] +       Exp[m*t]) - (fR*Exp[tsR*m] + fmax*Exp[m*t])/(Exp[tsR*m] +       Exp[m*t]) + fR];  rhoCp[x_] :=    SmoothedStepFunction[rhoa*Cpa, rhom*Cpm, rhoa*Cpa, L1, L2, slope][x]; k[x_] := SmoothedStepFunction[10^6*ka, 10^6*km, 10^6*ka, L1, L2,      slope][x]; 

The actual heat balance is solved here:

heateq = rhoCp[x]*D[u[x, t], t] ==     Inactive[Div][{{k[x]}}.Inactive[Grad][u[x, t], {x}], {x}];  ic[x_] :=    Piecewise[{{T01, x < L1}, {T02, L1 <= x <= L2}, {T03, x > L2}}];  sol1 = First[    NDSolve[{heateq, u[x, 0] == ic[x]},      u, {x, -2*thickness, 2*thickness}, {t, 0, time1},      Method -> {"MethodOfLines",        "SpatialDiscretization" -> {"FiniteElement",          "MeshOptions" -> {"MaxCellMeasure" -> 0.01}}}]]; 

Despite it works well, I still have a question: next to the Neumann boundary conditions (Inactive[Div][{{k[x]}}.Inactive[Grad][u[x, t]) that are applied over the interfaces to connect the solutions in each domain, are there other boundary conditions applied implicitly?

Heat Metal – Questions about the components

I’m a rookie DM with an even more rookie group of players. I have some questions about Heat Metal (2nd level transmutation).

It says you need a piece of iron and a flame. It also says the target of the spell is a manufactured metal object.

  • The piece of iron and the manufactured metal object are two separate objects, right? The caster needs to be in possession of the piece of iron, and the enemy would optimally be touching the manufactured metal (any metal – not just iron) object. Is that right?
  • Does the flame need to be close enough to the caster to touch (for example, if the caster would need to hold the piece of iron into the flame), or simply in the range of the spell? Intuitively I would have opted for the latter, as I’d imagine the caster would be drawing energy from nearby flames.
  • If there is no environmental fire nearby, would they have to use one action to light a fire with their tinderbox first? The PHB says "Don’t bother to keep track of material components with negligible cost. Assume you have all you need as long as you have your spell component pouch." Does that apply here? Obviously the flame doesn’t have a cost, so can I just assume the caster.. uhm.. has a flame in their pouch? Or can we say lighting a small piece of tinder is part of the spell action?

I guess some/all of these could be answered with "DM-fiat", but in that case I’d be interested in how seasoned DMs rule it 🙂


Heat Metal on Animated Armour – Can something be both an object and a creature? [duplicate]

Heat metal says (Emphasis mine) :

Choose a manufactured metal object, such as a metal weapon or a suit of heavy or medium metal armor, that you can see within range. You cause the object to glow red-hot. Any creature in physical contact with the object takes 2d8 fire damage when you cast the spell. Until the spell ends, you can use a bonus action on each of your subsequent turns to cause this damage again.

The target in this case was Animated armour (which is a construct).

In our session we did rule that it could work just because it fit well with the story. We played it out that the metal was melting and deforming so the joints wouldn’t work. However the issue we had comes down to:

Can something be both an object and a creature?

The armour needs to be a manufactured object for the spell to work (and it specifically mentions a suit of metal armour) but also needs to be a creature to take the damage.

Can the Heat Metal spell target a set of Dragon Scale Mail made from metallic dragon scales?

The heat metal spell can target "a manufactured metal object", including "a suit of heavy or medium metal armor".

A suit of Dragon Scale Mail is clearly a manufactured object, but if it’s made from metallic dragon scales, is it metal, and thus a valid target? Are bronze dragon scales actually made of bronze, or are they just colored like that?

I’m just assuming chromatic dragon scales are inarguably organic.

Solving the heat equation using Laplace Transforms

I am trying to solve the 1-D heat equation using Laplace Transform theory. The equation is as follows. I don’t have the capability to write the symbols so I will write it out.

                     partial u/partial t = 2(partial squared u/ partial x squared) -x    boundary conditions are partial u/partial x(0,t)=1, partial u/partial x(2,t)=beta. 

The problem asks the following: (a). For what value of beta does there exist a steady-state solution? (b). if the initial temperature is uniform such that u(x,0)=5 and beta takes the value suggested by the answer to part (a), derive the equilibrium temperature distribution.

I was able to get an equation that looks like U(x,s)=c e^(s/2)^1/2 -(1/s)((x/s)-u(x,0)). But I am not sure how to go from here to solve for beta using the boundary conditions. I need some assistance from someone.

Inhomogeneous Heat Equation Formula Not Satisfying IVP

From Wikipedia,


However, when I plug the formula of $ u(x,t)$ into Mathematica 12, it doesn’t seem to satisfy the PDE (does not give $ 0$ ):

enter image description here

Here is the code:

G[x_, t_] := 1/Sqrt[4*Pi*k*t]*Exp[-x^2/(4*k*t)]  u[x_, t_] := Integrate[G[x - y, t - s]*f[y, s], {y, -Infinity, Infinity}, {s, 0, t}]  FullSimplify[D[u[x, t], t] - k*D[u[x, t], {x, 2}] - f[x, t], Assumptions -> t > 0] 

(DnD 5e) How large a target can you cast Heat Metal on?

I was running a game and the party was trying to flee from a flying Illithid raiding skiff. It was made of metal and had two Illithid crew members on the deck. The bard in the party cast Heat Metal on the hull of the skiff. I had no idea how to run it so, to keep the game moving, I ran it as close to RAW as possible. It was a manufactured metal object in range, so everyone on the deck had to take damage and try to make the save. If I had more time I’m not sure how I would have ruled it.

As it stand I can think of a few possible alternatives and want some external input.

  1. As RAW the whole metal hull heats up. (As I played it)
  2. The spell targets on plate of the hull, and heats that up, anyone in contact with that plate are effected by the spell. (Each plate is manufactured, then assembled, kinda nitpicky.)
  3. The spell fails as the entirety of the object was not in range. (Very Nitpicky)
  4. the spell heats only a certain mass or volume, perhaps a 10 ft by 10 ft region of the hull. (Assuming that the armor of a huge giant is only large spread around)

Option one has the implications that you could cast it on an Aircraft Carrier and doom them all.

Option two has the implication that only one piece of a full plate suit of armor is tagered by the spell and the heat is conducted to the rest.

Option three has implication that casting a targeting spell while only only having a creatures hand in range would fail.

Option four has the implication that its heat is limited in scope.