Home My Page Projects Code Snippets Project Openings SML/NJ
 Summary Activity Forums Tracker Lists Tasks Docs Surveys News SCM Files

# SCM Repository

[smlnj] View of /sml/trunk/benchmarks/programs/simple/simple.sml
 [smlnj] / sml / trunk / benchmarks / programs / simple / simple.sml

# View of /sml/trunk/benchmarks/programs/simple/simple.sml

Fri Nov 20 17:43:59 1998 UTC (22 years, 5 months ago) by monnier
File size: 30538 byte(s)
```Initial revision
```
```(* Simple
* error: grid_max < 5
*)
functor Simple(val grid_max: int val step_count: int) : BMARK =
struct

fun fold f [] = (fn b => b)
| fold f (a::r) = (fn b => let fun f2(e,[]) = f(e,b)
| f2(e,a::r) = f(e,f2(a,r))
in f2(a,r)
end)

fun min (x:real,y:real) = if x<y then x else y
fun max (x:real,y:real) = if x<y then y else x
exception MaxList
exception MinList
exception SumList
fun max_list [] = raise MaxList | max_list l = fold max l (hd l)
fun min_list [] = raise MinList | min_list l = fold min l (hd l)
fun sum_list [] = raise SumList
| sum_list (l:real list) = fold (op +) l 0.0

fun for {from=start:int,step=delta:int, to=endd:int} body =
if delta>0 andalso endd>=start then
let fun f x = if x > endd then () else (body x; f(x+delta))
in f start
end
else if endd<=start then
let fun f x = if x < endd then () else (body x; f(x+delta))
in f start
end
else ()
fun from(n,m) = if n>m then [] else n::from(n+1,m)
fun flatten [] = []
| flatten (x::xs) = x @ flatten xs
fun pow(x:real,y:int) = if y = 0 then 1.0 else x * pow(x,y-1)
fun array2(bounds as ((l1,u1),(l2,u2)),v) =
(Array2.array((u1-l1+1, u2-l2+1),v), bounds)
fun sub2((A,((lb1:int,ub1:int),(lb2:int,ub2:int))),(k,l)) =
Array2.sub(A, (k-lb1, l-lb2))
fun update2((A,((lb1,_),(lb2,_))),(k,l), v) = Array2.update(A,(k-lb1,l-lb2),v)
fun bounds2(_,b) = b
fun printarray2 (A as (M:real Array2.array2,((l1,u1),(l2,u2)))) =
for {from=l1,step=1,to=u1} (fn i =>
(print "[";
for {from=l2,step=1,to=u2-1} (fn j =>
print (Real.toString (sub2(A,(i,j))) ^ ", "));
print (Real.toString (sub2(A,(i,u2))) ^ "]\n")))
fun array1((l,u),v) = (Array.array(u-l+1,v),(l,u))
fun sub1((A,(l:int,u:int)),i:int) = Array.sub(A,i-l)
fun update1((A,(l,_)),i,v) = Array.update(A,i-l,v)
fun bounds1(_,b) = b

(*
* Specification of the state variable computation
*)
val grid_size = ((2,grid_max), (2,grid_max))

fun north (k,l) = (k-1,l)
fun south (k,l) = (k+1,l)

fun east (k,l) = (k,l+1)
fun west (k,l) = (k,l-1)

val northeast = north o east
val southeast = south o east
val northwest = north o west
val southwest = south o west

fun farnorth x = (north o north ) x
fun farsouth x = (south o south) x
fun fareast x = (east o east) x
fun farwest x = (west o west) x

fun zone_A(k,l) = (k,l)
fun zone_B(k,l) = (k+1,l)

fun zone_C(k,l) = (k+1,l+1)
fun zone_D(k,l) = (k,l+1)

val  zone_corner_northeast = north
val  zone_corner_northwest = northwest
fun  zone_corner_southeast zone = zone
val  zone_corner_southwest = west

val ((kmin,kmax),(lmin,lmax)) 	= grid_size
val dimension_all_nodes   	= ((kmin-1,kmax+1),(lmin-1,lmax+1))
fun for_all_nodes f =
for {from=kmin-1, step=1, to=kmax+1} (fn k =>
for {from=lmin-1, step=1, to=lmax+1} (fn l => f k l))

val dimension_interior_nodes  	= ((kmin,kmax),(lmin,lmax))
fun for_interior_nodes f =
for {from=kmin, step=1, to=kmax} (fn k =>
for {from=lmin, step=1, to=lmax} (fn l => f k l))

val dimension_all_zones  	= ((kmin,kmax+1),(lmin,lmax+1))
fun for_all_zones f =
for {from=kmin, step=1, to=kmax+1} (fn k =>
for {from=lmin, step=1, to=lmax+1} (fn l => f (k,l)))

val dimension_interior_zones  	= ((kmin+1,kmax),(lmin+1,lmax))
fun for_interior_zones f =
for {from=kmin+1, step=1, to=kmax} (fn k =>
for {from=lmin+1, step=1, to=lmax} (fn l => f (k,l)))

fun map_interior_nodes f =
flatten(map (fn k => (map (fn l => f (k,l))
(from(lmin,lmax))))
(from(kmin,kmax)))
fun map_interior_zones f =
flatten(map (fn k => (map (fn l => f (k,l))
(from(lmin+1,lmax))))
(from(kmin+1,kmax)))

fun for_north_ward_interior_zones f =
for {from=kmax, step= ~1, to=kmin+1} (fn k =>
for {from=lmin+1, step=1, to=lmax} (fn l => f (k,l)))
fun for_west_ward_interior_zones f =
for {from=kmin+1, step=1, to=kmax} (fn k =>
for {from=lmax, step= ~1, to=lmin+1} (fn l => f (k,l)))

fun for_north_zones f = for {from=lmin, step=1, to=lmax+1} (fn l => f (kmin,l))
fun for_south_zones f = for {from=lmin+1, step=1, to=lmax} (fn l => f (kmax+1,l))
fun for_east_zones f = for {from=kmin+1, step=1, to=kmax+1}(fn k => f (k,lmax+1))
fun for_west_zones f = for {from=kmin+1, step=1, to=kmax+1}(fn k => f (k,lmin))

fun reflect dir node A = sub2(A, dir node)
val reflect_north = fn x => reflect north x
val reflect_south = fn x => reflect south x
val reflect_east = fn x => reflect east x
val reflect_west = fn x => reflect west x

fun for_north_nodes f =
for {from=lmin, step=1, to=lmax-1} (fn l => f (kmin-1,l))
fun for_south_nodes f =
for {from=lmin, step=1, to=lmax-1} (fn l => f (kmax+1,l))
fun for_east_nodes f  =
for {from=kmin, step=1, to=kmax-1} (fn k => f (k,lmax+1))
fun for_west_nodes f =
for {from=kmin, step=1, to=kmax-1} (fn k => f (k,lmin-1))

val north_east_corner = (kmin-1,lmax+1)
val north_west_corner = (kmin-1,lmin-1)
val south_east_corner = (kmax+1,lmax+1)
val south_west_corner = (kmax+1,lmin-1)

val west_of_north_east = (kmin-1, lmax)
val west_of_south_east = (kmax+1, lmax)
val north_of_south_east = (kmax, lmax+1)
val north_of_south_west = (kmax, lmin-1)

(*
* Initialization of parameters
*)
val  constant_heat_source = 0.0
val  deltat_maximum = 0.01
val  specific_heat = 0.1
val  p_coeffs = let val M = array2(((0,2),(0,2)), 0.0)
in update2(M, (1,1), 0.06698); M
end
val e_coeffs = let val M = array2(((0,2),(0,2)), 0.0)
in update2(M, (0,1), 0.1); M
end
val p_poly   = array2(((1,4),(1,5)),p_coeffs)

val e_poly   = array2(((1,4),(1,5)), e_coeffs)

val rho_table = let val V = array1((1,3), 0.0)
in  update1(V,2,1.0);
update1(V,3,100.0);
V
end
val theta_table = let val V = array1((1,4), 0.0)
in update1(V,2,3.0);
update1(V,3,300.0);
update1(V,4,3000.0);
V
end

val extract_energy_tables_from_constants  = (e_poly,2,rho_table,theta_table)
val extract_pressure_tables_from_constants = (p_poly,2,rho_table,theta_table)

val nbc = let val M = array2(dimension_all_zones, 1)
in for {from=lmin+1,step=1,to=lmax} (fn j => update2(M,(kmax+1, j),2));
update2(M,(kmin,lmin),4);
update2(M,(kmin,lmax+1),4);
update2(M,(kmax+1,lmin),4);
update2(M,(kmax+1,lmax+1),4);
M
end
val pbb = let val A = array1((1,4), 0.0)
in update1(A,2,6.0); A
end
val pb  = let val A = array1((1,4), 1.0)
in update1(A,2,0.0); update1(A,3,0.0); A
end
val qb  =     pb

val all_zero_nodes = array2(dimension_all_nodes, 0.0)

val all_zero_zones = array2(dimension_all_zones, 0.0)

(*
* Positional Coordinates. (page 9-10)
*)

fun make_position_matrix interior_function =
let val r' = array2(dimension_all_nodes, 0.0)
val z' = array2(dimension_all_nodes, 0.0)
fun boundary_position (rx,zx,ry,zy,ra,za) =
let val (rax, zax)  =  (ra - rx, za - zx)
val (ryx, zyx)  =  (ry - rx, zy - zx)
val omega       =  2.0*(rax*ryx + zax*zyx)/(ryx*ryx + zyx*zyx)
val rb 	    	=  rx - rax + omega*ryx
val zb 	    	=  zx - zax + omega*zyx
in (rb, zb)
end

fun reflect_node (x_dir, y_dir, a_dir, node) =
let val rx = reflect x_dir  node  r'
val zx = reflect x_dir  node  z'
val ry = reflect y_dir  node  r'
val zy = reflect y_dir  node  z'
val ra = reflect a_dir  node  r'
val za = reflect a_dir  node  z'
in boundary_position (rx, zx, ry, zy, ra, za)
end
fun u2 (rv,zv) n = (update2(r',n,rv); update2(z',n,zv))
in
for_interior_nodes (fn k => fn l => u2 (interior_function (k,l)) (k,l));
for_north_nodes(fn n => u2 (reflect_node(south,southeast,farsouth,n)) n);
for_south_nodes (fn n => u2(reflect_node(north,northeast,farnorth,n)) n);
for_east_nodes (fn n => u2(reflect_node(west, southwest, farwest, n)) n);
for_west_nodes (fn n => u2(reflect_node(east, southeast, fareast, n)) n);
u2 (reflect_node(south, southwest, farsouth, west_of_north_east))
west_of_north_east;
u2 (reflect_node(north, northwest, farnorth, west_of_south_east))
west_of_south_east;
u2 (reflect_node(west, northwest, farwest, north_of_south_east))
north_of_south_east;
u2 (reflect_node(east, northeast, fareast, north_of_south_west))
north_of_south_west;
u2 (reflect_node(southwest, west, farwest, north_east_corner))
north_east_corner;
u2 (reflect_node(northwest, west, farwest, south_east_corner))
south_east_corner;
u2 (reflect_node(southeast, south, farsouth, north_west_corner))
north_west_corner;
u2 (reflect_node(northeast, east, fareast, south_west_corner))
south_west_corner;
(r',z')
end

(*
* Physical Properties of a Zone (page 10)
*)
fun zone_area_vol ((r,z), zone) =
let val (r1,z1)=(sub2(r,zone_corner_southwest zone),
sub2(z,zone_corner_southwest zone))
val (r2,z2)=(sub2(r,zone_corner_southeast zone),
sub2(z,zone_corner_southeast zone))
val (r3,z3)=(sub2(r,zone_corner_northeast zone),
sub2(z,zone_corner_northeast zone))
val (r4,z4)=(sub2(r,zone_corner_northwest zone),
sub2(z,zone_corner_northwest zone))
val area1  =  (r2-r1)*(z3-z1) - (r3-r2)*(z3-z2)
val radius1  =  0.3333  *(r1+r2+r3)
val volume1  =  area1 * radius1
val area2  =  (r3-r1)*(z4-z3) - (r4-r3)*(z3-z1)
val radius2  =  0.3333 *(r1+r3+r4)
val volume2  =  area2 * radius2
in  (area1+area2, volume1+volume2)
end

(*
* Velocity (page 8)
*)
fun make_velocity((u,w),(r,z),p,q,alpha,rho,delta_t) =
let fun line_integral (p,z,node) : real =
sub2(p,zone_A node)*(sub2(z,west node) - sub2(z,north node)) +
sub2(p,zone_B node)*(sub2(z,south node) - sub2(z,west node)) +
sub2(p,zone_C node)*(sub2(z,east node) - sub2(z,south node)) +
sub2(p,zone_D node)*(sub2(z,north node) - sub2(z,east node))
fun regional_mass node =
0.5 * (sub2(rho, zone_A node)*sub2(alpha,zone_A node) +
sub2(rho, zone_B node)*sub2(alpha,zone_B node) +
sub2(rho, zone_C node)*sub2(alpha,zone_C node) +
sub2(rho, zone_D node)*sub2(alpha,zone_D node))
fun velocity node =
let val d = regional_mass node
val n1 = ~(line_integral(p,z,node)) - line_integral(q,z,node)
val n2 = line_integral(p,r,node) + line_integral(q,r,node)
val u_dot = n1/d
val w_dot = n2/d
in (sub2(u,node)+delta_t*u_dot, sub2(w,node)+delta_t*w_dot)
end
val U = array2(dimension_interior_nodes,0.0)
val W = array2(dimension_interior_nodes,0.0)
in for_interior_nodes (fn k => fn l => let val (uv,wv) = velocity (k,l)
in update2(U,(k,l),uv);
update2(W,(k,l),wv)
end);
(U,W)
end

fun make_position ((r,z),delta_t,(u',w')) =
let fun interior_position node =
(sub2(r,node) + delta_t*sub2(u',node),
sub2(z,node) + delta_t*sub2(w',node))
in make_position_matrix interior_position
end

fun make_area_density_volume(rho, s, x') =
let val alpha' = array2(dimension_all_zones, 0.0)
val s' = array2(dimension_all_zones, 0.0)
val rho' = array2(dimension_all_zones, 0.0)
fun interior_area zone =
let val (area, vol) = zone_area_vol (x', zone)
val density =  sub2(rho,zone)*sub2(s,zone) / vol
in (area,vol,density)
end
fun reflect_area_vol_density reflect_function =
(reflect_function alpha',reflect_function s',reflect_function rho')
fun update_asr (zone,(a,s,r)) = (update2(alpha',zone,a);
update2(s',zone,s);
update2(rho',zone,r))
fun r_area_vol_den (reflect_dir,zone) =
let val asr =  reflect_area_vol_density (reflect_dir zone)
in update_asr(zone, asr)
end
in
for_interior_zones (fn zone => update_asr(zone, interior_area zone));
for_south_zones (fn zone => r_area_vol_den(reflect_north, zone));
for_east_zones (fn zone => r_area_vol_den(reflect_west, zone));
for_west_zones (fn zone => r_area_vol_den(reflect_east, zone));
for_north_zones (fn zone => r_area_vol_den(reflect_south, zone));
(alpha', rho', s')
end

(*
* Artifical Viscosity (page 11)
*)
fun make_viscosity(p,(u',w'),(r',z'), alpha',rho') =
let	fun interior_viscosity zone =
let fun upper_del f =
0.5 * ((sub2(f,zone_corner_southeast zone) -
sub2(f,zone_corner_northeast zone)) +
(sub2(f,zone_corner_southwest zone) -
sub2(f,zone_corner_northwest zone)))
fun lower_del f =
0.5 * ((sub2(f,zone_corner_southeast zone) -
sub2(f,zone_corner_southwest zone)) +
(sub2(f,zone_corner_northeast zone) -
sub2(f,zone_corner_northwest zone)))
val xi 	= pow(upper_del   r',2) + pow(upper_del   z',2)
val eta = pow(lower_del   r',2) + pow(lower_del   z',2)
val upper_disc =  (upper_del r')*(lower_del w') -
(upper_del z')*(lower_del u')
val lower_disc = (upper_del u')*(lower_del z') -
(upper_del w') * (lower_del r')
val upper_ubar = if upper_disc<0.0   then upper_disc/xi    else 0.0
val lower_ubar = if lower_disc<0.0   then lower_disc/eta    else 0.0
val gamma    = 1.6
val speed_of_sound  =  gamma*sub2(p,zone)/sub2(rho',zone)
val ubar  =  pow(upper_ubar,2) + pow(lower_ubar,2)
val viscosity   =
sub2(rho',zone)*(1.5*ubar + 0.5*speed_of_sound*(Math.sqrt ubar))
val length   = Math.sqrt(pow(upper_del r',2) + pow(lower_del r',2))
val courant_delta = 0.5* sub2(alpha',zone)/(speed_of_sound*length)
in (viscosity, courant_delta)
end
val q' = array2(dimension_all_zones, 0.0)
val d  = array2(dimension_all_zones, 0.0)
fun reflect_viscosity_cdelta (direction, zone) =
sub2(q',direction zone) * sub1(qb, sub2(nbc,zone))
fun do_zones (dir,zone) =
update2(q',zone,reflect_viscosity_cdelta (dir,zone))
in
for_interior_zones (fn zone => let val (qv,dv) = interior_viscosity zone
in update2(q',zone,qv);
update2(d,zone,dv)
end);
for_south_zones (fn zone => do_zones(north,zone));
for_east_zones (fn zone => do_zones(west,zone));
for_west_zones (fn zone => do_zones(east,zone));
for_north_zones (fn zone => do_zones(south,zone));
(q', d)
end

(*
* Pressure and Energy Polynomial (page 12)
*)

fun polynomial(G,degree,rho_table,theta_table,rho_value,theta_value) =
let fun table_search (table, value) =
let val (low, high) = bounds1  table
fun search_down  i =  if  value > sub1(table,i-1)  then i
else search_down (i-1)
in
if  value>sub1(table,high)  then  high+1
else if  value <= sub1(table,low)  then low
else search_down   high
end
val rho_index = table_search(rho_table, rho_value)
val theta_index = table_search(theta_table, theta_value)
val A =  sub2(G, (rho_index, theta_index))
fun from(n,m) = if n>m then [] else n::from(n+1,m)
fun f(i,j) = sub2(A,(i,j))*pow(rho_value,i)*pow(theta_value,j)
in
sum_list (map (fn i => sum_list(map (fn j => f (i,j)) (from(0,degree))))
(from (0,degree)))
end
fun zonal_pressure  (rho_value:real,  theta_value:real)  =
let val (G,degree,rho_table,theta_table) =
extract_pressure_tables_from_constants
in polynomial(G, degree, rho_table, theta_table, rho_value, theta_value)
end

fun zonal_energy (rho_value, theta_value) =
let val (G, degree, rho_table, theta_table) =
extract_energy_tables_from_constants
in polynomial(G, degree, rho_table, theta_table, rho_value, theta_value)
end
val dx =   0.000001
val tiny = 0.000001

fun newton_raphson (f,x) =
let	fun iter (x,fx) =
if fx > tiny then
let val fxdx = f(x+dx)
val denom = fxdx - fx
in if denom < tiny then iter(x,tiny)
else iter(x-fx*dx/denom, fxdx)
end
else x
in iter(x, f x)
end

(*
* Temperature (page 13-14)
*)

fun make_temperature(p,epsilon,rho,theta,rho_prime,q_prime) =
let fun interior_temperature   zone =
let val qkl = sub2(q_prime,zone)
val rho_kl = sub2(rho,zone)
val rho_prime_kl = sub2(rho_prime,zone)
val tau_kl = (1.0 /rho_prime_kl - 1.0/rho_kl)
fun energy_equation epsilon_kl   theta_kl =
epsilon_kl -  zonal_energy(rho_kl,theta_kl)
val epsilon_0 = sub2(epsilon,zone)
fun revised_energy pkl =  epsilon_0 - (pkl + qkl) * tau_kl
fun revised_temperature  epsilon_kl   theta_kl =
newton_raphson ((energy_equation epsilon_kl), theta_kl)
fun revised_pressure  theta_kl = zonal_pressure(rho_kl, theta_kl)
val p_0 = sub2(p,zone)
val theta_0 = sub2(theta,zone)
val epsilon_1 =  revised_energy    p_0
val theta_1 =  revised_temperature    epsilon_1    theta_0
val p_1 =  revised_pressure    theta_1
val epsilon_2 =  revised_energy    p_1
val theta_2 =  revised_temperature    epsilon_2    theta_1
in  theta_2
end
val M = array2(dimension_all_zones, constant_heat_source)
in
for_interior_zones
(fn zone => update2(M, zone, interior_temperature zone));
M
end

(*
* Heat conduction
*)

fun make_cc(alpha_prime, theta_hat) =
let fun interior_cc zone =
(0.0001 * pow(sub2(theta_hat,zone),2) *
(Math.sqrt (abs(sub2(theta_hat,zone)))) / sub2(alpha_prime,zone))
handle Sqrt => (print (Real.toString (sub2(theta_hat, zone)));
print ("\nzone =(" ^ Int.toString (#1 zone) ^ "," ^
Int.toString (#2 zone) ^ ")\n");
printarray2 theta_hat;
raise Sqrt)
val cc = array2(dimension_all_zones, 0.0)
in
for_interior_zones(fn zone => update2(cc,zone, interior_cc zone));
for_south_zones(fn zone => update2(cc,zone, reflect_north zone cc));
for_west_zones(fn zone => update2(cc,zone,reflect_east zone cc));
for_east_zones(fn zone => update2(cc,zone,reflect_west zone cc));
for_north_zones(fn zone => update2(cc,zone, reflect_south zone cc));
cc
end

fun make_sigma(deltat, rho_prime, alpha_prime) =
let fun interior_sigma   zone =
sub2(rho_prime,zone)*sub2(alpha_prime,zone)*specific_heat/ deltat
val M = array2(dimension_interior_zones, 0.0)
fun ohandle zone =
(print (Real.toString (sub2(rho_prime, zone)) ^ " ");
print (Real.toString (sub2(alpha_prime, zone)) ^ " ");
print (Real.toString specific_heat ^ " ");
print (Real.toString deltat ^ "\n");
raise Overflow)

in  if !Control.trace
then print ("\t\tmake_sigma:deltat = " ^ Real.toString deltat ^ "\n")
else ();
(***	for_interior_zones(fn zone => update2(M,zone, interior_sigma zone)) **)
for_interior_zones(fn zone => (update2(M,zone, interior_sigma zone)
handle Overflow => ohandle zone));
M
end

fun make_gamma  ((r_prime,z_prime), cc, succeeding, adjacent) =
let fun interior_gamma   zone =
let val r1 = sub2(r_prime, zone_corner_southeast   zone)
val z1 = sub2(z_prime, zone_corner_southeast   zone)
val r2 = sub2(r_prime, zone_corner_southeast (adjacent   zone))
val z2 = sub2(z_prime, zone_corner_southeast (adjacent   zone))
val cross_section = 0.5*(r1+r2)*(pow(r1 - r2,2)+pow(z1 - z2,2))
val (c1,c2) = (sub2(cc, zone), sub2(cc, succeeding zone))
val specific_conductivity =  2.0 * c1 * c2 / (c1 + c2)
in cross_section  *  specific_conductivity
end
val M = array2(dimension_all_zones, 0.0)
in
for_interior_zones(fn zone => update2(M,zone,interior_gamma zone));
M
end

fun make_ab(theta, sigma, Gamma, preceding) =
let val a = array2(dimension_all_zones, 0.0)
val b = array2(dimension_all_zones, 0.0)
fun interior_ab   zone =
let val denom = sub2(sigma, zone) + sub2(Gamma, zone) +
sub2(Gamma, preceding zone) *
(1.0 - sub2(a, preceding zone))
val nume1 = sub2(Gamma,zone)
val nume2 = sub2(Gamma,preceding zone)*sub2(b,preceding zone) +
sub2(sigma,zone) * sub2(theta,zone)
in  (nume1/denom,  nume2 / denom)
end
val f  = fn zone => update2(b,zone,sub2(theta,zone))
in
for_north_zones f;
for_south_zones f;
for_west_zones f;
for_east_zones f;
for_interior_zones(fn zone => let val ab = interior_ab zone
in update2(a,zone,#1 ab);
update2(b,zone,#2 ab)
end);
(a,b)
end

fun make_theta (a, b, succeeding, int_zones) =
let val theta = array2(dimension_all_zones, constant_heat_source)
fun interior_theta zone =
sub2(a,zone) * sub2(theta,succeeding zone)+ sub2(b,zone)
in
int_zones (fn (k,l) => update2(theta, (k,l), interior_theta (k,l)));
theta
end

fun compute_heat_conduction(theta_hat, deltat, x', alpha', rho') =
let val sigma 	= make_sigma(deltat,  rho',  alpha')
val _ = if !Control.trace then print "\tdone make_sigma\n" else ()

val cc 		= make_cc(alpha',  theta_hat)
val _ = if !Control.trace then print "\tdone make_cc\n" else ()

val Gamma_k  	= make_gamma(  x', cc, north, east)
val _ = if !Control.trace then print "\tdone make_gamma\n" else ()

val (a_k,b_k)  	= make_ab(theta_hat, sigma, Gamma_k, north)
val _ = if !Control.trace then print "\tdone make_ab\n" else ()

val theta_k  	= make_theta(a_k,b_k,south,for_north_ward_interior_zones)
val _ = if !Control.trace then print "\tdone make_theta\n" else ()

val Gamma_l  	= make_gamma(x', cc, west, south)
val _ = if !Control.trace then print "\tdone make_gamma\n" else ()

val (a_l,b_l) 	= make_ab(theta_k, sigma, Gamma_l, west)
val _ = if !Control.trace then print "\tdone make_ab\n" else ()

val theta_l  	= make_theta(a_l,b_l,east,for_west_ward_interior_zones)
val _ = if !Control.trace then print "\tdone make_theta\n" else ()
in  (theta_l, Gamma_k, Gamma_l)
end

(*
* Final Pressure and Energy calculation
*)
fun make_pressure(rho', theta')  =
let val p = array2(dimension_all_zones, 0.0)
fun boundary_p(direction, zone) =
sub1(pbb, sub2(nbc, zone)) +
sub1(pb,sub2(nbc,zone)) * sub2(p, direction zone)
in
for_interior_zones
(fn zone => update2(p,zone,zonal_pressure(sub2(rho',zone),
sub2(theta',zone))));
for_south_zones(fn zone => update2(p,zone,boundary_p(north,zone)));
for_east_zones(fn zone => update2(p,zone,boundary_p(west,zone)));
for_west_zones(fn zone => update2(p,zone,boundary_p(east,zone)));
for_north_zones(fn zone => update2(p,zone,boundary_p(south,zone)));
p
end

fun make_energy(rho', theta')  =
let val epsilon' = array2(dimension_all_zones, 0.0)
in
for_interior_zones
(fn zone => update2(epsilon', zone, zonal_energy(sub2(rho',zone),
sub2(theta',zone))));
for_south_zones
(fn zone => update2(epsilon',zone, reflect_north zone epsilon'));
for_west_zones
(fn zone => update2(epsilon',zone, reflect_east zone epsilon'));
for_east_zones
(fn zone => update2(epsilon',zone, reflect_west  zone epsilon'));
for_north_zones
(fn zone => update2(epsilon',zone, reflect_south zone epsilon'));
epsilon'
end

(*
* Energy Error Calculation (page 20)
*)

fun compute_energy_error  ((u',w'),(r',z'),p',q',epsilon',theta',rho',alpha',
Gamma_k,Gamma_l,deltat)  =
let fun mass zone =  sub2(rho',zone) * sub2(alpha',zone):real
val internal_energy =
sum_list (map_interior_zones (fn z => sub2(epsilon',z)*(mass z)))
fun kinetic   node =
let val average_mass =  0.25*((mass (zone_A  node)) +
(mass (zone_B  node)) +
(mass (zone_C  node)) +
(mass (zone_D  node)))
val v_square = pow(sub2(u',node),2) + pow(sub2(w',node),2)
in 0.5 * average_mass * v_square
end
val kinetic_energy =  sum_list (map_interior_nodes kinetic)
fun work_done (node1, node2)  =
let val (r1, r2) = (sub2(r',node1), sub2(r',node2))
val (z1, z2) = (sub2(z',node1), sub2(z',node2))
val (u1, u2) = (sub2(p',node1), sub2(p',node2))
val (w1, w2) = (sub2(z',node1), sub2(z',node2))
val (p1, p2) = (sub2(p',node1), sub2(p',node2))
val (q1, q2) = (sub2(q',node1), sub2(q',node2))
val force =  0.5*(p1+p2+q1+q2)
val radius = 0.5* (r1+r2)
val area =   0.5* ((r1-r2)*(u1-u2) - (z1-z2)*(w1-w2))
in  force * radius * area * deltat
end

fun from(n,m) = if n > m then [] else n::from(n+1,m)
val north_line =
map (fn l => (west(kmin,l),(kmin,l))) (from(lmin+1,lmax))
val south_line =
map (fn l => (west(kmax,l),(kmax,l))) (from(lmin+1,lmax))
val east_line  =
map (fn k => (south(k,lmax),(k,lmax))) (from(kmin+1,kmax))
val west_line  =
map (fn k => (south(k,lmin+1),(k,lmin+1))) (from(kmin+1,kmax))

val w1 = sum_list (map work_done north_line)
val w2  = sum_list (map work_done south_line)
val w3  = sum_list (map work_done east_line)
val w4  = sum_list (map work_done west_line)
val boundary_work =  w1 + w2 + w3 + w4

fun heat_flow  Gamma (zone1,zone2)  =
deltat * sub2(Gamma, zone1) * (sub2(theta',zone1) - sub2(theta',zone2))

val north_flow =
let val k = kmin+1
in map (fn l => (north(k,l),(k,l))) (from(lmin+1,lmax))
end
val south_flow =
let val k = kmax
in map (fn l => (south(k,l),(k,l))) (from(lmin+2,lmax-1))
end
val east_flow  =
let val l = lmax
in map (fn k => (east(k,l),(k,l))) (from(kmin+2,kmax))
end
val west_flow  =
let val l = lmin+1
in map (fn k => (west(k,l),(k,l))) (from(kmin+2,kmax))
end

val h1  = sum_list    (map (heat_flow  Gamma_k)   north_flow)
val h2  = sum_list    (map (heat_flow  Gamma_k)   south_flow)
val h3  = sum_list    (map (heat_flow  Gamma_l)   east_flow)
val h4  = sum_list    (map (heat_flow  Gamma_l)   west_flow)
val boundary_heat =  h1 + h2 + h3 + h4
in
internal_energy  +  kinetic_energy  -  boundary_heat  -  boundary_work
end

fun compute_time_step(d, theta_hat,  theta') =
let val deltat_courant =
min_list (map_interior_zones (fn zone => sub2(d,zone)))
val deltat_conduct =
max_list (map_interior_zones
(fn z => (abs(sub2(theta_hat,z) - sub2(theta', z))/
sub2(theta_hat,z))))
val deltat_minimum = min (deltat_courant, deltat_conduct)
in min   (deltat_maximum,  deltat_minimum)
end

fun compute_initial_state () =
let
val v  = (all_zero_nodes, all_zero_nodes)
val x  = let fun interior_position  (k,l)  =
let val pi = 3.1415926535898
val rp = real (lmax - lmin)
val z1 = real(10 + k - kmin)
val zz = (~0.5 + real(l - lmin) / rp) * pi
in (z1 * Math.cos zz,  z1 * Math.sin zz)
end
in  make_position_matrix interior_position
end
val (alpha,s) =
let val (alpha_prime,s_prime) =
let val A = array2(dimension_all_zones, 0.0)
val S = array2(dimension_all_zones, 0.0)
fun reflect_area_vol f = (f A, f S)

fun u2 (f,z) =
let val (a,s) = reflect_area_vol(f z)
in update2(A,z,a);
update2(S,z,s)
end
in
for_interior_zones
(fn z => let val (a,s) = zone_area_vol(x, z)
in update2(A,z,a);
update2(S,z,s)
end);
for_south_zones (fn z => u2 (reflect_north, z));
for_east_zones (fn z => u2 (reflect_west, z));
for_west_zones (fn z => u2 (reflect_east, z));
for_north_zones (fn z => u2 (reflect_south, z));
(A,S)
end
in  (alpha_prime,s_prime)
end
val rho  = let val R = array2(dimension_all_zones, 0.0)
in for_all_zones (fn z => update2(R,z,1.4)); R
end
val theta =
let val T = array2(dimension_all_zones, constant_heat_source)
in for_interior_zones(fn z => update2(T,z,0.0001));
T
end
val p       = make_pressure(rho, theta)
val q       = all_zero_zones
val epsilon = make_energy(rho, theta)
val  deltat  = 0.01
val  c       = 0.0
in
(v,x,alpha,s,rho,p,q,epsilon,theta,deltat,c)
end

fun compute_next_state state =
let
val (v,x,alpha,s,rho,p,q,epsilon,theta,deltat,c) = state
val v'  = make_velocity (v, x, p, q, alpha, rho, deltat)
val _ = if !Control.trace then print "done make_velocity\n" else ()

val x'  = make_position(x,deltat,v')
handle Overflow =>(printarray2 (#1 v');
printarray2 (#2 v');
raise Overflow)
val _ = if !Control.trace then print "done make_position\n" else ()

val (alpha',rho',s')  = make_area_density_volume (rho,  s , x')
val _ = if !Control.trace then print "done make_area_density_volume\n"
else ()

val (q',d)  = make_viscosity (p,  v',  x',  alpha',  rho')
val _ = if !Control.trace then print "done make_viscosity\n" else ()

val theta_hat  = make_temperature (p, epsilon, rho, theta, rho', q')
val _ = if !Control.trace then print "done make_temperature\n" else ()

val (theta',Gamma_k,Gamma_l) =
compute_heat_conduction (theta_hat, deltat, x', alpha', rho')
val _ = if !Control.trace then print "done compute_heat_conduction\n"
else ()

val p'  = make_pressure(rho', theta')
val _ = if !Control.trace then print "done make_pressure\n" else ()

val epsilon'  = make_energy (rho', theta')
val _ = if !Control.trace then print "done make_energy\n" else ()

val c'  = compute_energy_error (v', x', p', q', epsilon', theta', rho',
alpha', Gamma_k, Gamma_l,  deltat)
val _ = if !Control.trace then print "done compute_energy_error\n"
else ()

val deltat'  = compute_time_step (d, theta_hat, theta')
val _ = if !Control.trace then print "done compute_time_step\n\n" else ()
in
(v',x',alpha',s',rho',p',q',  epsilon',theta',deltat',c')
end

fun runit () =
let fun iter (i,state) = if i = 0 then state
else (print ".";
iter(i-1, compute_next_state state))
in iter(step_count, compute_initial_state())
end

fun print_state ((v1,v2),(r,z),alpha,s,rho,p,q,epsilon,theta,deltat,c) = (
print "Velocity matrices = \n";
printarray2 v1; print "\n\n";
printarray2 v2;

print "\n\nPosition matrices = \n";
printarray2 r; print "\n\n";
printarray2 z;

print "\n\nalpha = \n";
printarray2 alpha;

print "\n\ns = \n";
printarray2 s;

print "\n\nrho = \n";
printarray2 rho;

print "\n\nPressure = \n";
printarray2 p;

print "\n\nq = \n";
printarray2 q;

print "\n\nepsilon = \n";
printarray2 epsilon;

print "\n\ntheta = \n";
printarray2 theta;

print ("delatat = " ^ Real.toString (deltat : real)^ "\n");
print ("c = " ^ Real.toString (c : real) ^ "\n"))

fun testit outstrm = print_state (runit())

fun doit () = let
val (_, _, _, _, _, _, _, _, _, delta', c') = runit()
val delta = Real.trunc delta'
val c = Real.trunc (c' * 10000.0)
in
if (c = 6787 andalso delta = ~33093)
then ()
else TextIO.output (TextIO.stdErr, "*** ERROR ***\n")
end

end; (* functor Simple *)
```

 root@smlnj-gforge.cs.uchicago.edu ViewVC Help Powered by ViewVC 1.0.0