Sample Problems > Usage > Integrals > 3d

{ 3D_INTEGRALS.PDE

This problem demonstrates the specification of various integrals in 3D.

( This is a modification of problem 3D_BRICKS.PDE)

}

title '3D Integrals'

coordinates

cartesian3

variables

definitions

long = 1

wide = 1

K { thermal conductivity -- values supplied later }

Q = 10*max(1-x^2-y^2-z^2,0) { Thermal source }

{ These definitions create a selector that supresses evaluation

of Tp except in region 2 of layer 2 }

flag22=0

check22 = if flag22>0 then Tp else 0

{ These definitions create a selector that supresses evaluation

of Tp except in region 2 of all layers }

flag20=0

check20 = if flag20>0 then Tp else 0

initial values

Tp = 0.

equations

Tp: div(k*grad(Tp)) + Q = 0 { the heat equation }

extrusion

surface "bottom" z = -long

layer 'bottom'

surface "middle" z=0

layer 'top'

surface 'top' z= long { divide Z into two layers }

boundaries

surface 1 value(Tp)=0 { fix bottom surface temp }

surface 3 value(Tp)=0 { fix top surface temp }

Region 1 { define full domain boundary in base plane }

layer 1 k=1 { bottom right brick }

layer 2 k=0.1 { top right brick }

start "outside" (-wide,-wide)

value(Tp) = 0 { fix all side temps }

line to (wide,-wide) { walk outer boundary in base plane }

to (wide,wide)

to (-wide,wide)

to close

Region 2 "Left" { overlay a second region in left half }

flag20=1

layer 1 k=0.2 { bottom left brick }

layer 2 k=0.4 flag22=1 { top left brick }

start(-wide,-wide)

line to (0,-wide) { walk left half boundary in base plane }

to (0,wide)

to (-wide,wide)

to close

monitors

contour(Tp) on surface z=0 as "XY Temp"

contour(Tp) on surface x=0 as "YZ Temp"

contour(Tp) on surface y=0 as "ZX Temp"

elevation(Tp) from (-wide,0,0) to (wide,0,0) as "X-Axis Temp"

elevation(Tp) from (0,-wide,0) to (0,wide,0) as "Y-Axis Temp"

elevation(Tp) from (0,0,-long) to (0,0,long) as "Z-Axis Temp"

plots

contour(Tp) on z=0 as "XY Temp"

contour(Tp) on x=0 as "YZ Temp"

contour(Tp) on y=0 as "ZX Temp"

contour(k*dz(Tp)) on z=-0.001 as "Low Middle Z-Flux"

contour(k*dz(Tp)) on z=0.001 as "High Middle Z-Flux"

summary

report("Compare various forms for integrating over region 2 of layer 2")

report(integral(Tp,2,2))

report(integral(Tp,"Left","Top"))

report(integral(check22))

report '-----'

report("Compare various forms for integrating over region 2 in all layers")

report(integral(Tp,2,0))

report(integral(check20))

report '-----'

report("Compare various forms for integrating over total volume")

report(integral(Tp,"ALL","ALL"))

report(integral(Tp))

report '-----'

report("Compare various forms for integrating over surface 'middle'")

report(sintegral(normal(-k*grad(Tp)),2))

report(sintegral(normal(-k*grad(Tp)),'Middle'))

report(sintegral(-k*dz(Tp),2))

report '-----'

report("Compare various forms for integrating over surfaces")

report(sintegral(normal(-k*grad(Tp)),1)) as "Bottom Flux"

report(sintegral(normal(-k*grad(Tp)),3)) as "Top Flux"

report(sintegral(normal(-k*grad(Tp)),"outside")) as "Side Flux"

report(sintegral(normal(-k*grad(Tp)),1)+sintegral(normal(-k*grad(Tp)),3)

+sintegral(normal(-k*grad(Tp)),"outside")) as "Bottom+Top+Side Flux"

report(sintegral(normal(-k*grad(Tp)))) { surface integral on total outer surface }

report(integral(Q) ) as "Source Integral"

report(sintegral(normal(-k*grad(Tp)),"outside")

+sintegral(normal(-k*grad(Tp)),1)+sintegral(normal(-k*grad(Tp)),3)

-integral(Q) ) as "Energy Error"

report(integral(div(-k*grad(Tp))) ) as "Divergence Integral"

report '-----'

report("Compare surface flux on region 2 of layer 2 to internal divergence integral")

3d_integrals

3d_integrals