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{ 3D_HELIX_WRAPPED.PDE
This problem shows the use of the function definition facility of FlexPDE to
create a helix of square crosssection in 3D.
The mesh generation facility of FlexPDE extrudes a 2D figure along a straight
path in Z, so that it is not possible to directly define a helical shape.
However, by defining a coordinate transformation, we can build a straight rod
in 3D and interpret the coordinates in a rotating frame.
Define the twisting coordinates by the transformation
xt = x*cos(y/R);
yt = x*sin(y/R);
zt = z
In this transformation, x and y are the coordinates FlexPDE believes it is working
with, and they are the coordinates that move with the twisting.
xt and yt are the "lab coordinates" of the twisted figure.
The chain rule gives
dF/d(xt) = (dx/dxt)*(dF/dx) + (dy/dxt)*(dF/dy) + (dz/dxt)*(dF/dz)
with similar rules for yt and zt.
Some tedious algebra gives
dx/dxt = cos(y/R) dy/dxt = (R/x)*sin(y/R) dz/dxt = 0
dx/dyt = sin(y/R) dy/dyt = (R/x)*cos(y/R) dz/dyt = 0
dx/dzt = dy/dzt = 0 dz/dzt = 1
These relations are defined in the definitions section, and used in the equations
section, perhaps nested as in the heat equation shown here.
Notice that this formulation produces the upward motion by tilting the bar in
the untwisted space and wrapping the resulting figure around a cylinder.
We have added a cylindrical mounting pad at each end of the helix.
See "3d_helix_layered.pde" for a different approach to constructing a helix.
See "Usage/CAD_Import/helix_OBJimport.pde" for how to import a helix from an OBJ file.
}
title '3D Helix  transformation with no shear'
coordinates
cartesian3
select
ngrid=160 { generate enough mesh cells to resolve the twist }
variables
Tp
definitions
zlong = 60
turns = 4
pitch = zlong/turns { z rise per turn }
xwide = 4.5
zhigh = 4.5
Rc = 22  xwide/2 { center radius }
alpha = y/Rc
zstub = 5*zhigh { rod pieces at each end }
sturn = Rc*2*pi { arc length per turn }
yolap = pi*Rc*zhigh/pitch
slong = turns*sturn { arc length of spring }
stot = slong + 2*sturn { add one turn at each end for rod }
xin = Rcxwide/2
xout = Rc+xwide/2
xbore = Rc/2
{ transformations }
rise = pitch/(2*pi) { zrise per radian }
c = cos(alpha)
s = sin(alpha)
xt = x*c
yt = x*s
zt = zzlong/2
{ functional definition of derivatives }
dxt(f) = c*dx(f)  s*(Rc/x)*dy(f)
dyt(f) = s*dx(f) + c*(Rc/x)*dy(f)
dzt(f) = dz(f)
{ Thermal source }
Q = 10*exp((xtRc)^2yt^2zt^2)
z1 = zstub
z2 = max( 0, min(zlong, pitch*y/sturn  zhigh/2))
z3 = max(0, min(zlong, pitch*y/sturn + zhigh/2))
z4 = zlong + zstub
initial values
Tp = 0.
equations
{ the heat equation using transformed derivative operators }
Tp: dxt(dxt(Tp)) + dyt(dyt(Tp)) + dzt(dzt(Tp)) + Q = 0
extrusion z = z1, z2, z3, z4
boundaries
Limited Region 1 { the spring }
layer 2
start(xin,yolap)
line to (xout,yolap)
line to (xout, slongyolap)
line to (xin,slongyolap)
line to close
Limited Region 2 { top rod overlap with coil }
surface 4 value(Tp)=0 {cold at the end of the rod }
layer 2 layer 3
start(xbore,slongyolap)
line to (xout,slongyolap) to (xout,slong+yolap) to (xbore,slong+yolap) to close
Limited Region 3 { top rod free of coil }
surface 4 value(Tp)=0 {cold at the end of the rod }
layer 2 layer 3
start(xbore,slong+yolap)
line to (xout,slong+yolap) to (xout,slong+sturnyolap) to (xbore,slong+sturnyolap)
to close
Limited Region 4 { bottom rod overlap with coil }
surface 1 value(Tp)=0 {cold at the end of the rod }
layer 1 layer 2
start(xbore,yolap)
line to (xout,yolap) to (xout,yolap) to (xbore,yolap) to close
Limited Region 5 { bottom rod free of coil }
surface 1 value(Tp)=0 {cold at the end of the rod }
layer 1 layer 2
start(xbore,sturn+yolap)
line to (xout,sturn+yolap) to (xout,yolap) to (xbore,yolap) to close
monitors
grid(xt,yt,zt) paintregions { the twisted shape }
plots
grid(xt,yt,zt) paintregions { the twisted shape again }
{ In the following, recall that x is really radius, and y is really azimuthal distance.
It is not possible at present to construct a cut in the "lab" coordinates. }
grid(x,z) on y=0
contour(Tp) on y=0 as "ZX Temp"
contour(Tp) on z=0 as "XY Temp"
elevation(Tp) from(Rc,0,0) to (Rc,slong,zlong) { centerline of coil }
end