Sample Problems > Usage > Constraints > 3d

{ 3D_CONSTRAINT.PDE

This problem demonstrates the specification of region-specific CONSTRAINTS in 3D.

This is a modification of problem 3D_BRICKS.PDE.

We apply a constraint on the integral of temperature in a single region/layer compartment.

For validation, we define a check function that has nonzero value only in the selected

compartment and compare its integral to the region-selection form of the integral

statement.

Value boundary conditions are applied, so the solution is unique, so the constraint

acts as a source or sink to maintain the constrained value, we report

the energy lost to the constraining mechanism.

}

title '3D constraint'

coordinates

cartesian3

variables

definitions

long = 1

wide = 1

Q = 10*exp(-x^2-y^2-z^2) { Thermal source }

flag22=0 { build a test function for region 2, layer 2 }

check22 = if flag22>0 then Tp else 0

initial values

Tp = 0.

equations

Tp: div(k*grad(Tp)) + Q = 0

constraints

{ constrain temperature integral in region 2 of layer 2 }

integral(Tp,2,2) = 1

extrusion

surface "bottom" z = -long

layer 'lower'

surface "middle" z=0

layer 'upper'

surface 'top' z= long

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(-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 }

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"

summary

report("Compare integral forms in region 2 of layer 2 (should be 1.00):")

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

report(integral(check22))

report("-----")

report "Constraint acts as an energy sink:"

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

report(sintegral(normal(-k*grad(Tp)))) as "Surface integral on total outer surface "

report(integral(Q)-sintegral(normal(-k*grad(Tp)))) as "Energy lost to constraint "

end

3d_constraint

3d_constraint