cwpearson/astaroth
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

Clarified the syntax for real number literals. 1.0 is the same precision as AcReal, 1.0f is an explicit float and 1.0d is an explicit double.

Browse Source
This commit is contained in:
jpekkila
2019-10-07 18:24:32 +03:00
parent aa6c2b23d9
commit ff12332f06
4 changed files with 184 additions and 171 deletions
Download Patch File Download Diff File Expand all files Collapse all files

181
acc/mhd_solver/stencil_kernel.ac
View File

@@ -9,7 +9,7 @@
#define LUPWD (1)
#define LSINK (0)
#define AC_THERMAL_CONDUCTIVITY (AcReal(0.001)) // TODO: make an actual config parameter
#define AC_THERMAL_CONDUCTIVITY (0.001) // TODO: make an actual config parameter
#define H_CONST (0) // TODO: make an actual config parameter
#define C_CONST (0) // TODO: make an actual config parameter
@@ -140,11 +140,11 @@ der6x_upwd(in ScalarField vertex)
{
Scalar inv_ds = AC_inv_dsx;
return (Scalar){Scalar(1.0 / 60.0) * inv_ds *
(-Scalar(20.0) * vertex[vertexIdx.x, vertexIdx.y, vertexIdx.z] +
Scalar(15.0) * (vertex[vertexIdx.x + 1, vertexIdx.y, vertexIdx.z] +
return (Scalar){(1.0 / 60.0) * inv_ds *
(-20.0 * vertex[vertexIdx.x, vertexIdx.y, vertexIdx.z] +
15.0 * (vertex[vertexIdx.x + 1, vertexIdx.y, vertexIdx.z] +
vertex[vertexIdx.x - 1, vertexIdx.y, vertexIdx.z]) -
Scalar(6.0) * (vertex[vertexIdx.x + 2, vertexIdx.y, vertexIdx.z] +
6.0 * (vertex[vertexIdx.x + 2, vertexIdx.y, vertexIdx.z] +
vertex[vertexIdx.x - 2, vertexIdx.y, vertexIdx.z]) +
vertex[vertexIdx.x + 3, vertexIdx.y, vertexIdx.z] +
vertex[vertexIdx.x - 3, vertexIdx.y, vertexIdx.z])};
@@ -155,11 +155,11 @@ der6y_upwd(in ScalarField vertex)
{
Scalar inv_ds = AC_inv_dsy;
return (Scalar){Scalar(1.0 / 60.0) * inv_ds *
(-Scalar(20.0) * vertex[vertexIdx.x, vertexIdx.y, vertexIdx.z] +
Scalar(15.0) * (vertex[vertexIdx.x, vertexIdx.y + 1, vertexIdx.z] +
return (Scalar){(1.0 / 60.0) * inv_ds *
(-20.0 * vertex[vertexIdx.x, vertexIdx.y, vertexIdx.z] +
15.0 * (vertex[vertexIdx.x, vertexIdx.y + 1, vertexIdx.z] +
vertex[vertexIdx.x, vertexIdx.y - 1, vertexIdx.z]) -
Scalar(6.0) * (vertex[vertexIdx.x, vertexIdx.y + 2, vertexIdx.z] +
6.0 * (vertex[vertexIdx.x, vertexIdx.y + 2, vertexIdx.z] +
vertex[vertexIdx.x, vertexIdx.y - 2, vertexIdx.z]) +
vertex[vertexIdx.x, vertexIdx.y + 3, vertexIdx.z] +
vertex[vertexIdx.x, vertexIdx.y - 3, vertexIdx.z])};
@@ -170,11 +170,11 @@ der6z_upwd(in ScalarField vertex)
{
Scalar inv_ds = AC_inv_dsz;
return (Scalar){Scalar(1.0 / 60.0) * inv_ds *
(-Scalar(20.0) * vertex[vertexIdx.x, vertexIdx.y, vertexIdx.z] +
Scalar(15.0) * (vertex[vertexIdx.x, vertexIdx.y, vertexIdx.z + 1] +
return (Scalar){(1.0 / 60.0) * inv_ds *
(-20.0 * vertex[vertexIdx.x, vertexIdx.y, vertexIdx.z] +
15.0 * (vertex[vertexIdx.x, vertexIdx.y, vertexIdx.z + 1] +
vertex[vertexIdx.x, vertexIdx.y, vertexIdx.z - 1]) -
Scalar(6.0) * (vertex[vertexIdx.x, vertexIdx.y, vertexIdx.z + 2] +
6.0 * (vertex[vertexIdx.x, vertexIdx.y, vertexIdx.z + 2] +
vertex[vertexIdx.x, vertexIdx.y, vertexIdx.z - 2]) +
vertex[vertexIdx.x, vertexIdx.y, vertexIdx.z + 3] +
vertex[vertexIdx.x, vertexIdx.y, vertexIdx.z - 3])};
@@ -182,7 +182,6 @@ der6z_upwd(in ScalarField vertex)
#endif
#if LUPWD
Device Scalar
upwd_der6(in VectorField uu, in ScalarField lnrho)
@@ -202,7 +201,8 @@ gradients(in VectorField uu)
#if LSINK
Device Vector
sink_gravity(int3 globalVertexIdx){
sink_gravity(int3 globalVertexIdx)
{
int accretion_switch = int(AC_switch_accretion);
if (accretion_switch == 1) {
Vector force_gravity;
@@ -210,33 +210,34 @@ sink_gravity(int3 globalVertexIdx){
(globalVertexIdx.y - AC_ny_min) * AC_dsy,
(globalVertexIdx.z - AC_nz_min) * AC_dsz};
const Scalar sink_mass = AC_M_sink;
const Vector sink_pos = (Vector){AC_sink_pos_x,
AC_sink_pos_y,
AC_sink_pos_z};
const Vector sink_pos = (Vector){AC_sink_pos_x, AC_sink_pos_y, AC_sink_pos_z};
const Scalar distance = length(grid_pos - sink_pos);
const Scalar soft = AC_soft;
//MV: The commit 083ff59 had AC_G_const defined wrong here in DSL making it exxessively strong.
//MV: Scalar gravity_magnitude = ... below is correct!
const Scalar gravity_magnitude = (AC_G_const * sink_mass) / pow(((distance * distance) + soft*soft), 1.5);
// MV: The commit 083ff59 had AC_G_const defined wrong here in DSL making it exxessively
// strong. MV: Scalar gravity_magnitude = ... below is correct!
const Scalar gravity_magnitude = (AC_G_const * sink_mass) /
pow(((distance * distance) + soft * soft), 1.5);
const Vector direction = (Vector){(sink_pos.x - grid_pos.x) / distance,
(sink_pos.y - grid_pos.y) / distance,
(sink_pos.z - grid_pos.z) / distance};
force_gravity = gravity_magnitude * direction;
return force_gravity;
} else {
}
else {
return (Vector){0.0, 0.0, 0.0};
}
}
#endif
#if LSINK
// Give Truelove density
Device Scalar
truelove_density(in ScalarField lnrho){
truelove_density(in ScalarField lnrho)
{
const Scalar rho = exp(value(lnrho));
const Scalar Jeans_length_squared = (M_PI * AC_cs2_sound) / (AC_G_const * rho);
const Scalar TJ_rho = ((M_PI) * ((AC_dsx * AC_dsx) / Jeans_length_squared) * AC_cs2_sound) / (AC_G_const * AC_dsx * AC_dsx);
const Scalar TJ_rho = ((M_PI) * ((AC_dsx * AC_dsx) / Jeans_length_squared) * AC_cs2_sound) /
(AC_G_const * AC_dsx * AC_dsx);
// TODO: AC_dsx will cancel out, deal with it later for optimization.
Scalar accretion_rho = TJ_rho;
@@ -246,13 +247,12 @@ truelove_density(in ScalarField lnrho){
// This controls accretion of density/mass to the sink particle.
Device Scalar
sink_accretion(int3 globalVertexIdx, in ScalarField lnrho, Scalar dt){
sink_accretion(int3 globalVertexIdx, in ScalarField lnrho, Scalar dt)
{
const Vector grid_pos = (Vector){(globalVertexIdx.x - AC_nx_min) * AC_dsx,
(globalVertexIdx.y - AC_ny_min) * AC_dsy,
(globalVertexIdx.z - AC_nz_min) * AC_dsz};
const Vector sink_pos = (Vector){AC_sink_pos_x,
AC_sink_pos_y,
AC_sink_pos_z};
const Vector sink_pos = (Vector){AC_sink_pos_x, AC_sink_pos_y, AC_sink_pos_z};
const Scalar profile_range = AC_accretion_range;
const Scalar accretion_distance = length(grid_pos - sink_pos);
int accretion_switch = AC_switch_accretion;
@@ -261,12 +261,13 @@ sink_accretion(int3 globalVertexIdx, in ScalarField lnrho, Scalar dt){
if (accretion_switch == 1) {
if ((accretion_distance) <= profile_range) {
//weight = Scalar(1.0);
// weight = 1.0;
// Hann window function
Scalar window_ratio = accretion_distance / profile_range;
weight = Scalar(0.5)*(Scalar(1.0) - cos(Scalar(2.0)*M_PI*window_ratio));
} else {
weight = Scalar(0.0);
weight = 0.5 * (1.0 - cos(2.0 * M_PI * window_ratio));
}
else {
weight = 0.0;
}
// Truelove criterion is used as a kind of arbitrary density floor.
@@ -274,25 +275,26 @@ sink_accretion(int3 globalVertexIdx, in ScalarField lnrho, Scalar dt){
Scalar rate;
if (value(lnrho) > lnrho_min) {
rate = (exp(value(lnrho)) - exp(lnrho_min)) / dt;
} else {
rate = Scalar(0.0);
}
else {
rate = 0.0;
}
accretion_density = weight * rate;
} else {
accretion_density = Scalar(0.0);
}
else {
accretion_density = 0.0;
}
return accretion_density;
}
// This controls accretion of velocity to the sink particle.
Device Vector
sink_accretion_velocity(int3 globalVertexIdx, in VectorField uu, Scalar dt) {
sink_accretion_velocity(int3 globalVertexIdx, in VectorField uu, Scalar dt)
{
const Vector grid_pos = (Vector){(globalVertexIdx.x - AC_nx_min) * AC_dsx,
(globalVertexIdx.y - AC_ny_min) * AC_dsy,
(globalVertexIdx.z - AC_nz_min) * AC_dsz};
const Vector sink_pos = (Vector){AC_sink_pos_x,
AC_sink_pos_y,
AC_sink_pos_z};
const Vector sink_pos = (Vector){AC_sink_pos_x, AC_sink_pos_y, AC_sink_pos_z};
const Scalar profile_range = AC_accretion_range;
const Scalar accretion_distance = length(grid_pos - sink_pos);
int accretion_switch = AC_switch_accretion;
@@ -304,32 +306,33 @@ sink_accretion_velocity(int3 globalVertexIdx, in VectorField uu, Scalar dt) {
// Arch of a cosine function?
// Cubic spline x^3 - x in range [-0.5 , 0.5]
if ((accretion_distance) <= profile_range) {
//weight = Scalar(1.0);
// weight = 1.0;
// Hann window function
Scalar window_ratio = accretion_distance / profile_range;
weight = Scalar(0.5)*(Scalar(1.0) - cos(Scalar(2.0)*M_PI*window_ratio));
} else {
weight = Scalar(0.0);
weight = 0.5 * (1.0 - cos(2.0 * M_PI * window_ratio));
}
else {
weight = 0.0;
}
Vector rate;
// MV: Could we use divergence here ephasize velocitie which are compressive and
// MV: not absorbins stuff that would not be accreted anyway?
if (length(value(uu)) > Scalar(0.0)) {
rate = (Scalar(1.0)/dt) * value(uu);
} else {
if (length(value(uu)) > 0.0) {
rate = (1.0 / dt) * value(uu);
}
else {
rate = (Vector){0.0, 0.0, 0.0};
}
accretion_velocity = weight * rate;
} else {
}
else {
accretion_velocity = (Vector){0.0, 0.0, 0.0};
}
return accretion_velocity;
}
#endif
Device Scalar
continuity(int3 globalVertexIdx, in VectorField uu, in ScalarField lnrho, Scalar dt)
{
@@ -344,35 +347,34 @@ continuity(int3 globalVertexIdx, in VectorField uu, in ScalarField lnrho, Scalar
- divergence(uu);
}
#if LENTROPY
Device Vector
momentum(int3 globalVertexIdx, in VectorField uu, in ScalarField lnrho, in ScalarField ss, in VectorField aa, Scalar dt)
momentum(int3 globalVertexIdx, in VectorField uu, in ScalarField lnrho, in ScalarField ss,
in VectorField aa, Scalar dt)
{
const Matrix S = stress_tensor(uu);
const Scalar cs2 = AC_cs2_sound * exp(AC_gamma * value(ss) / AC_cp_sound +
(AC_gamma - 1) * (value(lnrho) - AC_lnrho0));
const Vector j = (Scalar(1.0) / AC_mu0) *
const Vector j = (1.0 / AC_mu0) *
(gradient_of_divergence(aa) - laplace_vec(aa)); // Current density
const Vector B = curl(aa);
// TODO: DOES INTHERMAL VERSTION INCLUDE THE MAGNETIC FIELD?
const Scalar inv_rho = Scalar(1.0) / exp(value(lnrho));
const Scalar inv_rho = 1.0 / exp(value(lnrho));
// Regex replace CPU constants with get\(AC_([a-zA-Z_0-9]*)\)
// \1
const Vector mom = -mul(gradients(uu), value(uu)) -
cs2 * ((Scalar(1.0) / AC_cp_sound) * gradient(ss) + gradient(lnrho)) +
cs2 * ((1.0 / AC_cp_sound) * gradient(ss) + gradient(lnrho)) +
inv_rho * cross(j, B) +
AC_nu_visc *
(laplace_vec(uu) + Scalar(1.0 / 3.0) * gradient_of_divergence(uu) +
Scalar(2.0) * mul(S, gradient(lnrho))) +
AC_nu_visc * (laplace_vec(uu) + (1.0 / 3.0) * gradient_of_divergence(uu) +
2.0 * mul(S, gradient(lnrho))) +
AC_zeta * gradient_of_divergence(uu)
#if LSINK
// Gravity term
+ sink_gravity(globalVertexIdx)
// Corresponding loss of momentum
- //(Scalar(1.0) / Scalar( (AC_dsx*AC_dsy*AC_dsz) * exp(value(lnrho)))) * // Correction factor by unit mass
- //(1.0 / ( (AC_dsx*AC_dsy*AC_dsz) * exp(value(lnrho)))) * //
// Correction factor by unit mass
sink_accretion_velocity(globalVertexIdx, uu, dt) // As in Lee et al.(2014)
;
#else
@@ -392,8 +394,8 @@ momentum(int3 globalVertexIdx, in VectorField uu, in ScalarField lnrho, in Scala
(gradient(tt) + value(tt) * gradient(lnrho));
mom = -mul(gradients(uu), value(uu)) - pressure_term +
AC_nu_visc * (laplace_vec(uu) + Scalar(1.0 / 3.0) * gradient_of_divergence(uu) +
Scalar(2.0) * mul(S, gradient(lnrho))) +
AC_nu_visc * (laplace_vec(uu) + (1.0 / 3.0) * gradient_of_divergence(uu) +
2.0 * mul(S, gradient(lnrho))) +
AC_zeta * gradient_of_divergence(uu)
#if LSINK
+ sink_gravity(globalVertexIdx);
@@ -417,13 +419,14 @@ momentum(int3 globalVertexIdx, in VectorField uu, in ScalarField lnrho, Scalar d
// Isothermal: we have constant speed of sound
mom = -mul(gradients(uu), value(uu)) - AC_cs2_sound * gradient(lnrho) +
AC_nu_visc * (laplace_vec(uu) + Scalar(1.0 / 3.0) * gradient_of_divergence(uu) +
Scalar(2.0) * mul(S, gradient(lnrho))) +
AC_nu_visc * (laplace_vec(uu) + (1.0 / 3.0) * gradient_of_divergence(uu) +
2.0 * mul(S, gradient(lnrho))) +
AC_zeta * gradient_of_divergence(uu)
#if LSINK
+ sink_gravity(globalVertexIdx)
// Corresponding loss of momentum
- //(Scalar(1.0) / Scalar( (AC_dsx*AC_dsy*AC_dsz) * exp(value(lnrho)))) * // Correction factor by unit mass
- //(1.0 / ( (AC_dsx*AC_dsy*AC_dsz) * exp(value(lnrho)))) * // Correction
// factor by unit mass
sink_accretion_velocity(globalVertexIdx, uu, dt) // As in Lee et al.(2014)
;
#else
@@ -460,21 +463,21 @@ Device Scalar
lnT(in ScalarField ss, in ScalarField lnrho)
{
return AC_lnT0 + AC_gamma * value(ss) / AC_cp_sound +
(AC_gamma - Scalar(1.0)) * (value(lnrho) - AC_lnrho0);
(AC_gamma - 1.0) * (value(lnrho) - AC_lnrho0);
}
// Nabla dot (K nabla T) / (rho T)
Device Scalar
heat_conduction(in ScalarField ss, in ScalarField lnrho)
{
const Scalar inv_AC_cp_sound = AcReal(1.0) / AC_cp_sound;
const Scalar inv_AC_cp_sound = 1.0 / AC_cp_sound;
const Vector grad_ln_chi = -gradient(lnrho);
const Scalar first_term = AC_gamma * inv_AC_cp_sound * laplace(ss) +
(AC_gamma - AcReal(1.0)) * laplace(lnrho);
(AC_gamma - 1.0) * laplace(lnrho);
const Vector second_term = AC_gamma * inv_AC_cp_sound * gradient(ss) +
(AC_gamma - AcReal(1.0)) * gradient(lnrho);
(AC_gamma - 1.0) * gradient(lnrho);
const Vector third_term = AC_gamma * (inv_AC_cp_sound * gradient(ss) + gradient(lnrho)) +
grad_ln_chi;
@@ -492,11 +495,11 @@ Device Scalar
entropy(in ScalarField ss, in VectorField uu, in ScalarField lnrho, in VectorField aa)
{
const Matrix S = stress_tensor(uu);
const Scalar inv_pT = Scalar(1.0) / (exp(value(lnrho)) * exp(lnT(ss, lnrho)));
const Vector j = (Scalar(1.0) / AC_mu0) *
const Scalar inv_pT = 1.0 / (exp(value(lnrho)) * exp(lnT(ss, lnrho)));
const Vector j = (1.0 / AC_mu0) *
(gradient_of_divergence(aa) - laplace_vec(aa)); // Current density
const Scalar RHS = H_CONST - C_CONST + AC_eta * (AC_mu0)*dot(j, j) +
Scalar(2.0) * exp(value(lnrho)) * AC_nu_visc * contract(S) +
2.0 * exp(value(lnrho)) * AC_nu_visc * contract(S) +
AC_zeta * exp(value(lnrho)) * divergence(uu) * divergence(uu);
return -dot(value(uu), gradient(ss)) + inv_pT * RHS + heat_conduction(ss, lnrho);
@@ -511,28 +514,32 @@ heat_transfer(in VectorField uu, in ScalarField lnrho, in ScalarField tt)
const Scalar heat_diffusivity_k = 0.0008; // 8e-4;
return -dot(value(uu), gradient(tt)) + heat_diffusivity_k * laplace(tt) +
heat_diffusivity_k * dot(gradient(lnrho), gradient(tt)) +
AC_nu_visc * contract(S) * (Scalar(1.0) / AC_cv_sound) -
AC_nu_visc * contract(S) * (1.0 / AC_cv_sound) -
(AC_gamma - 1) * value(tt) * divergence(uu);
}
#endif
#if LFORCING
Device Vector
simple_vortex_forcing(Vector a, Vector b, Scalar magnitude){
simple_vortex_forcing(Vector a, Vector b, Scalar magnitude)
{
int accretion_switch = AC_switch_accretion;
if (accretion_switch == 0) {
return magnitude * cross(normalized(b - a), (Vector){0, 0, 1}); // Vortex
} else {
}
else {
return (Vector){0, 0, 0};
}
}
Device Vector
simple_outward_flow_forcing(Vector a, Vector b, Scalar magnitude){
simple_outward_flow_forcing(Vector a, Vector b, Scalar magnitude)
{
int accretion_switch = AC_switch_accretion;
if (accretion_switch == 0) {
return magnitude * (1 / length(b - a)) * normalized(b - a); // Outward flow
} else {
}
else {
return (Vector){0, 0, 0};
}
}
@@ -579,8 +586,7 @@ forcing(int3 globalVertexIdx, Scalar dt)
int accretion_switch = AC_switch_accretion;
if (accretion_switch == 0) {
Vector a = Scalar(0.5) * (Vector){globalGridN.x * AC_dsx,
globalGridN.y * AC_dsy,
Vector a = 0.5 * (Vector){globalGridN.x * AC_dsx, globalGridN.y * AC_dsy,
globalGridN.z * AC_dsz}; // source (origin)
Vector xx = (Vector){(globalVertexIdx.x - AC_nx_min) * AC_dsx,
(globalVertexIdx.y - AC_ny_min) * AC_dsy,
@@ -595,7 +601,6 @@ forcing(int3 globalVertexIdx, Scalar dt)
Vector ff_re = (Vector){AC_ff_hel_rex, AC_ff_hel_rey, AC_ff_hel_rez};
Vector ff_im = (Vector){AC_ff_hel_imx, AC_ff_hel_imy, AC_ff_hel_imz};
// Determine that forcing funtion type at this point.
// Vector force = simple_vortex_forcing(a, xx, magnitude);
// Vector force = simple_outward_flow_forcing(a, xx, magnitude);
@@ -608,9 +613,14 @@ forcing(int3 globalVertexIdx, Scalar dt)
force.y = sqrt(dt) * NN * force.y;
force.z = sqrt(dt) * NN * force.z;
if (is_valid(force)) { return force; }
else { return (Vector){0, 0, 0}; }
} else {
if (is_valid(force)) {
return force;
}
else {
return (Vector){0, 0, 0};
}
}
else {
return (Vector){0, 0, 0};
}
}
@@ -671,7 +681,8 @@ solve()
#endif
#if LSINK
out_accretion = rk3(out_accretion, accretion, sink_accretion(globalVertexIdx, lnrho, dt), dt);// unit now is rho!
out_accretion = rk3(out_accretion, accretion, sink_accretion(globalVertexIdx, lnrho, dt),
dt); // unit now is rho!
if (step_number == 2) {
out_accretion = out_accretion * AC_dsx * AC_dsy * AC_dsz; // unit is now mass!

3
acc/src/acc.l
View File

@@ -39,7 +39,8 @@ L [a-zA-Z_]
"return" { return RETURN; }
{D}+"."{D}+ { return REAL_NUMBER; } /* Literals */
{D}+"."{D}+[fd]+ { return NUMBER; }
{D}+"."{D}+d+ { return DOUBLE_NUMBER; }
{D}+"."{D}+f+ { return NUMBER; }
{D}+[lu]* { return NUMBER; }
{L}({L}|{D})* { return IDENTIFIER; }
\"(.)*\" { return IDENTIFIER; } /* String */

3
acc/src/acc.y
View File

@@ -14,7 +14,7 @@ int yyget_lineno();
%}
%token CONSTANT IN OUT UNIFORM
%token IDENTIFIER NUMBER REAL_NUMBER
%token IDENTIFIER NUMBER REAL_NUMBER DOUBLE_NUMBER
%token RETURN
%token SCALAR VECTOR MATRIX SCALARFIELD SCALARARRAY
%token VOID INT INT3 COMPLEX
@@ -222,6 +222,7 @@ identifier: IDENTIFIER
;
number: REAL_NUMBER { $$ = astnode_create(NODE_REAL_NUMBER, NULL, NULL); astnode_set_buffer(yytext, $$); }
| DOUBLE_NUMBER { $$ = astnode_create(NODE_UNKNOWN, NULL, NULL); astnode_set_buffer(yytext, $$); $$->buffer[strlen($$->buffer) - 1] = '\0'; }
| NUMBER { $$ = astnode_create(NODE_UNKNOWN, NULL, NULL); astnode_set_buffer(yytext, $$); }
;

18
acc/stdlib/stdderiv.h
View File

@@ -302,18 +302,18 @@ stress_tensor(in VectorField vec)
{
Matrix S;
S.row[0].x = Scalar(2.0 / 3.0) * gradient(vec.x).x -
Scalar(1.0 / 3.0) * (gradient(vec.y).y + gradient(vec.z).z);
S.row[0].y = Scalar(1.0 / 2.0) * (gradient(vec.x).y + gradient(vec.y).x);
S.row[0].z = Scalar(1.0 / 2.0) * (gradient(vec.x).z + gradient(vec.z).x);
S.row[0].x = (2.0 / 3.0) * gradient(vec.x).x -
(1.0 / 3.0) * (gradient(vec.y).y + gradient(vec.z).z);
S.row[0].y = (1.0 / 2.0) * (gradient(vec.x).y + gradient(vec.y).x);
S.row[0].z = (1.0 / 2.0) * (gradient(vec.x).z + gradient(vec.z).x);
S.row[1].y = Scalar(2.0 / 3.0) * gradient(vec.y).y -
Scalar(1.0 / 3.0) * (gradient(vec.x).x + gradient(vec.z).z);
S.row[1].y = (2.0 / 3.0) * gradient(vec.y).y -
(1.0 / 3.0) * (gradient(vec.x).x + gradient(vec.z).z);
S.row[1].z = Scalar(1.0 / 2.0) * (gradient(vec.y).z + gradient(vec.z).y);
S.row[1].z = (1.0 / 2.0) * (gradient(vec.y).z + gradient(vec.z).y);
S.row[2].z = Scalar(2.0 / 3.0) * gradient(vec.z).z -
Scalar(1.0 / 3.0) * (gradient(vec.x).x + gradient(vec.y).y);
S.row[2].z = (2.0 / 3.0) * gradient(vec.z).z -
(1.0 / 3.0) * (gradient(vec.x).x + gradient(vec.y).y);
S.row[1].x = S.row[0].y;
S.row[2].x = S.row[0].z;