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// tankex.fe
// Equilibrium shape of liquid in flat-ended cylindrical tank.
// Tank has axis along y-axis and flat bottom in x-z plane. This
// is so gravity acting vertically draws liquid toward wall.
// Straight edges cannot conform exactly to curved walls.
// We need to give them an area so that area cannot shrink by straght edges
// pulling away from the walls. The gaps are also accounted for
// in volume and gravitational energy.
SYMMETRIC_CONTENT // for volume calculations
// Contact angles, initially for 45 degrees.
PARAMETER ENDT = 0.707 /* surface tension of uncovered base */
PARAMETER WALLT = 0.707 /* surface tension of uncovered side wall */
// Gravity components
PARAMETER GY = 0
PARAMETER GZ = -1
SPRING_CONSTANT 1 // for most accurate gap areas for constraint 2
#define TR 1.00 /* tank radius */
#define RHO 1.00 /* fuel density */
constraint 1 // flat base
function: y = 0
energy: // for contact energy line integral
e1: -ENDT*z
e2: 0
e3: 0
#define wstuff (WALLT*TR*y/(x^2 + z^2)) // common wall area term
#define vstuff (TR^2/3*y/(x^2 + z^2)) // common wall volume term
#define gstuff (GY*TR^2*y^2/4/(x^2 + z^2) + GZ*TR^3*y*z/3/(x^2+z^2)**1.5)
// common gap gravity term
constraint 2 CONVEX // cylindrical wall
function: x^2 + z^2 = TR^2
energy: // for contact energy and gravity
e1: -wstuff*z + RHO*GY*y^2*z/4 + RHO*GZ*y*z^2/3 - RHO*gstuff*z
e2: 0
e3: wstuff*x - RHO*GY*y^2*x/4 - RHO*GZ*y*z*x/3 + RHO*gstuff*x
content: // so volumes calculated correctly
c1: vstuff*z - z*y/6 + vstuff*z/2
c2: 0
c3: -vstuff*x + x*y/6 - vstuff*x/2
// named quantity for arbitrary direction gravity on facets
quantity arb_grav energy method facet_vector_integral global
vector_integrand:
q1: 0
q2: -RHO*GY*y^2/2 - RHO*GZ*y*z
q3: 0
// Now the specification of the initial shape
vertices
1 0.5 0.0 0.5 constraint 1
2 0.5 0.0 -0.5 constraint 1
3 -0.5 0.0 -0.5 constraint 1
4 -0.5 0.0 0.5 constraint 1
5 1.0 0.5 0.0 constraint 2
6 0.0 0.5 -1.0 constraint 2
7 -1.0 0.5 0.0 constraint 2
8 0.0 0.5 1.0 constraint 2
edges
1 2 1 constraint 1
2 1 4 constraint 1
3 4 3 constraint 1
4 3 2 constraint 1
5 5 6 constraint 2
6 6 7 constraint 2
7 7 8 constraint 2
8 8 5 constraint 2
9 1 8
10 1 5
11 2 5
12 2 6
13 3 6
14 3 7
15 4 7
16 4 8
faces
1 13 6 -14
2 3 14 -15
3 15 7 -16
4 2 16 -9
5 9 8 -10
6 1 10 -11
7 11 5 -12
8 4 12 -13
bodies
1 1 2 3 4 5 6 7 8 volume 0.6 density 0 /* no default gravity */
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