stdderiv.h

Go to the documentation of this file.

#pragma once
#ifndef STENCIL_ORDER
#define STENCIL_ORDER (6)
#endif
 
uniform Scalar AC_dsx     = 0.04908738521;
uniform Scalar AC_dsy     = 0.04908738521;
uniform Scalar AC_dsz     = 0.04908738521;
uniform Scalar AC_inv_dsx = 1.0 / AC_dsx;
uniform Scalar AC_inv_dsy = 1.0 / AC_dsy;
uniform Scalar AC_inv_dsz = 1.0 / AC_dsz;
 
Scalar
first_derivative(Scalar pencil[], Scalar inv_ds)
{
#if STENCIL_ORDER == 2
    Scalar coefficients[] = {0, 1.0 / 2.0};
#elif STENCIL_ORDER == 4
    Scalar coefficients[] = {0, 2.0 / 3.0, -1.0 / 12.0};
#elif STENCIL_ORDER == 6
    Scalar coefficients[] = {0, 3.0 / 4.0, -3.0 / 20.0, 1.0 / 60.0};
#elif STENCIL_ORDER == 8
    Scalar coefficients[] = {0, 4.0 / 5.0, -1.0 / 5.0, 4.0 / 105.0, -1.0 / 280.0};
#endif
 
#define MID (STENCIL_ORDER / 2)
 
    Scalar res = 0;
 
    for (int i = 1; i <= MID; ++i) {
        res += coefficients[i] * (pencil[MID + i] - pencil[MID - i]);
    }
 
    return res * inv_ds;
}
 
Scalar
second_derivative(Scalar pencil[], Scalar inv_ds)
{
#if STENCIL_ORDER == 2
    Scalar coefficients[] = {-2.0, 1.0};
#elif STENCIL_ORDER == 4
    Scalar coefficients[] = {-5.0 / 2.0, 4.0 / 3.0, -1.0 / 12.0};
#elif STENCIL_ORDER == 6
    Scalar coefficients[] = {-49.0 / 18.0, 3.0 / 2.0, -3.0 / 20.0, 1.0 / 90.0};
#elif STENCIL_ORDER == 8
    Scalar coefficients[] = {-205.0 / 72.0, 8.0 / 5.0, -1.0 / 5.0, 8.0 / 315.0, -1.0 / 560.0};
#endif
 
#define MID (STENCIL_ORDER / 2)
    Scalar res = coefficients[0] * pencil[MID];
 
    for (int i = 1; i <= MID; ++i) {
        res += coefficients[i] * (pencil[MID + i] + pencil[MID - i]);
    }
 
    return res * inv_ds * inv_ds;
}
 
Scalar
cross_derivative(Scalar pencil_a[], Scalar pencil_b[], Scalar inv_ds_a, Scalar inv_ds_b)
{
#if STENCIL_ORDER == 2
    Scalar coefficients[] = {0, 1.0 / 4.0};
#elif STENCIL_ORDER == 4
    Scalar coefficients[] = {0, 1.0 / 32.0,
                             1.0 / 64.0}; // TODO correct coefficients, these are just placeholders
#elif STENCIL_ORDER == 6
    Scalar fac            = 1.0 / 720.0;
    Scalar coefficients[] = {0.0 * fac, 270.0 * fac, -27.0 * fac, 2.0 * fac};
#elif STENCIL_ORDER == 8
    Scalar fac            = (1.0 / 20160.0);
    Scalar coefficients[] = {0.0 * fac, 8064.0 * fac, -1008.0 * fac, 128.0 * fac, -9.0 * fac};
#endif
 
#define MID (STENCIL_ORDER / 2)
    Scalar res = 0.0;
 
    for (int i = 1; i <= MID; ++i) {
        res += coefficients[i] *
               (pencil_a[MID + i] + pencil_a[MID - i] - pencil_b[MID + i] - pencil_b[MID - i]);
    }
    return res * inv_ds_a * inv_ds_b;
}
 
Scalar
derx(int3 vertexIdx, in ScalarField arr)
{
    Scalar pencil[STENCIL_ORDER + 1];
 
    for (int offset = 0; offset < STENCIL_ORDER + 1; ++offset) {
        pencil[offset] = arr[vertexIdx.x + offset - STENCIL_ORDER / 2, vertexIdx.y, vertexIdx.z];
    }
 
    return first_derivative(pencil, AC_inv_dsx);
}
 
Scalar
derxx(int3 vertexIdx, in ScalarField arr)
{
    Scalar pencil[STENCIL_ORDER + 1];
 
    for (int offset = 0; offset < STENCIL_ORDER + 1; ++offset) {
        pencil[offset] = arr[vertexIdx.x + offset - STENCIL_ORDER / 2, vertexIdx.y, vertexIdx.z];
    }
 
    return second_derivative(pencil, AC_inv_dsx);
}
 
Scalar
derxy(int3 vertexIdx, in ScalarField arr)
{
    Scalar pencil_a[STENCIL_ORDER + 1];
 
    for (int offset = 0; offset < STENCIL_ORDER + 1; ++offset) {
        pencil_a[offset] = arr[vertexIdx.x + offset - STENCIL_ORDER / 2,
                               vertexIdx.y + offset - STENCIL_ORDER / 2, vertexIdx.z];
    }
 
    Scalar pencil_b[STENCIL_ORDER + 1];
 
    for (int offset = 0; offset < STENCIL_ORDER + 1; ++offset) {
        pencil_b[offset] = arr[vertexIdx.x + offset - STENCIL_ORDER / 2,
                               vertexIdx.y + STENCIL_ORDER / 2 - offset, vertexIdx.z];
    }
 
    return cross_derivative(pencil_a, pencil_b, AC_inv_dsx, AC_inv_dsy);
}
 
Scalar
derxz(int3 vertexIdx, in ScalarField arr)
{
    Scalar pencil_a[STENCIL_ORDER + 1];
 
    for (int offset = 0; offset < STENCIL_ORDER + 1; ++offset) {
        pencil_a[offset] = arr[vertexIdx.x + offset - STENCIL_ORDER / 2, vertexIdx.y,
                               vertexIdx.z + offset - STENCIL_ORDER / 2];
    }
 
    Scalar pencil_b[STENCIL_ORDER + 1];
    for (int offset = 0; offset < STENCIL_ORDER + 1; ++offset) {
        pencil_b[offset] = arr[vertexIdx.x + offset - STENCIL_ORDER / 2, vertexIdx.y,
                               vertexIdx.z + STENCIL_ORDER / 2 - offset];
    }
 
    return cross_derivative(pencil_a, pencil_b, AC_inv_dsx, AC_inv_dsz);
}
 
Scalar
dery(int3 vertexIdx, in ScalarField arr)
{
    Scalar pencil[STENCIL_ORDER + 1];
 
    for (int offset = 0; offset < STENCIL_ORDER + 1; ++offset) {
        pencil[offset] = arr[vertexIdx.x, vertexIdx.y + offset - STENCIL_ORDER / 2, vertexIdx.z];
    }
 
    return first_derivative(pencil, AC_inv_dsy);
}
 
Scalar
deryy(int3 vertexIdx, in ScalarField arr)
{
    Scalar pencil[STENCIL_ORDER + 1];
 
    for (int offset = 0; offset < STENCIL_ORDER + 1; ++offset) {
        pencil[offset] = arr[vertexIdx.x, vertexIdx.y + offset - STENCIL_ORDER / 2, vertexIdx.z];
    }
 
    return second_derivative(pencil, AC_inv_dsy);
}
 
Scalar
deryz(int3 vertexIdx, in ScalarField arr)
{
    Scalar pencil_a[STENCIL_ORDER + 1];
 
    for (int offset = 0; offset < STENCIL_ORDER + 1; ++offset) {
        pencil_a[offset] = arr[vertexIdx.x, vertexIdx.y + offset - STENCIL_ORDER / 2,
                               vertexIdx.z + offset - STENCIL_ORDER / 2];
    }
 
    Scalar pencil_b[STENCIL_ORDER + 1];
 
    for (int offset = 0; offset < STENCIL_ORDER + 1; ++offset) {
        pencil_b[offset] = arr[vertexIdx.x, vertexIdx.y + offset - STENCIL_ORDER / 2,
                               vertexIdx.z + STENCIL_ORDER / 2 - offset];
    }
 
    return cross_derivative(pencil_a, pencil_b, AC_inv_dsy, AC_inv_dsz);
}
 
Scalar
derz(int3 vertexIdx, in ScalarField arr)
{
    Scalar pencil[STENCIL_ORDER + 1];
 
    for (int offset = 0; offset < STENCIL_ORDER + 1; ++offset) {
        pencil[offset] = arr[vertexIdx.x, vertexIdx.y, vertexIdx.z + offset - STENCIL_ORDER / 2];
    }
 
    return first_derivative(pencil, AC_inv_dsz);
}
 
Scalar
derzz(int3 vertexIdx, in ScalarField arr)
{
    Scalar pencil[STENCIL_ORDER + 1];
 
    for (int offset = 0; offset < STENCIL_ORDER + 1; ++offset) {
        pencil[offset] = arr[vertexIdx.x, vertexIdx.y, vertexIdx.z + offset - STENCIL_ORDER / 2];
    }
 
    return second_derivative(pencil, AC_inv_dsz);
}
 
Preprocessed Scalar
value(in ScalarField vertex)
{
    return vertex[vertexIdx];
}
 
Device Vector
value(in VectorField uu)
{
    return (Vector){value(uu.x), value(uu.y), value(uu.z)};
}
 
Preprocessed Vector
gradient(in ScalarField vertex)
{
    assert(AC_dsx > 0);
    assert(AC_dsy > 0);
    assert(AC_dsz > 0);
 
    assert(AC_inv_dsx > 0);
    assert(AC_inv_dsy > 0);
    assert(AC_inv_dsz > 0);
 
    return (Vector){derx(vertexIdx, vertex), dery(vertexIdx, vertex), derz(vertexIdx, vertex)};
}
 
Preprocessed Matrix
hessian(in ScalarField vertex)
{
    assert(AC_dsx > 0);
    assert(AC_dsy > 0);
    assert(AC_dsz > 0);
 
    assert(AC_inv_dsx > 0);
    assert(AC_inv_dsy > 0);
    assert(AC_inv_dsz > 0);
 
    Matrix mat;
 
    mat.row[0] = (Vector){derxx(vertexIdx, vertex), derxy(vertexIdx, vertex),
                          derxz(vertexIdx, vertex)};
    mat.row[1] = (Vector){mat.row[0].y, deryy(vertexIdx, vertex), deryz(vertexIdx, vertex)};
    mat.row[2] = (Vector){mat.row[0].z, mat.row[1].z, derzz(vertexIdx, vertex)};
 
    return mat;
}
 
 
Device Scalar
laplace(in ScalarField data)
{
    return hessian(data).row[0].x + hessian(data).row[1].y + hessian(data).row[2].z;
}
 
Device Scalar
divergence(in VectorField vec)
{
    return gradient(vec.x).x + gradient(vec.y).y + gradient(vec.z).z;
}
 
Device Vector
laplace_vec(in VectorField vec)
{
    return (Vector){laplace(vec.x), laplace(vec.y), laplace(vec.z)};
}
 
Device Vector
curl(in VectorField vec)
{
    return (Vector){gradient(vec.z).y - gradient(vec.y).z, gradient(vec.x).z - gradient(vec.z).x,
                    gradient(vec.y).x - gradient(vec.x).y};
}
 
Device Vector
gradient_of_divergence(in VectorField vec)
{
    return (Vector){hessian(vec.x).row[0].x + hessian(vec.y).row[0].y + hessian(vec.z).row[0].z,
                    hessian(vec.x).row[1].x + hessian(vec.y).row[1].y + hessian(vec.z).row[1].z,
                    hessian(vec.x).row[2].x + hessian(vec.y).row[2].y + hessian(vec.z).row[2].z};
}
 
// Takes uu gradients and returns S
Device Matrix
stress_tensor(in VectorField vec)
{
    Matrix S;
 
    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 = (2.0 / 3.0) * gradient(vec.y).y -
                 (1.0 / 3.0) * (gradient(vec.x).x + gradient(vec.z).z);
 
    S.row[1].z = (1.0 / 2.0) * (gradient(vec.y).z + gradient(vec.z).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;
    S.row[2].y = S.row[1].z;
 
    return S;
}
 
Device Scalar
contract(const Matrix mat)
{
    Scalar res = 0;
 
    for (int i = 0; i < 3; ++i) {
        res += dot(mat.row[i], mat.row[i]);
    }
 
    return res;
}
 
 
Device Scalar
length(const Vector vec)
{
    return sqrt(vec.x * vec.x + vec.y * vec.y + vec.z * vec.z);
}
 
Device Scalar
reciprocal_len(const Vector vec)
{
    return rsqrt(vec.x * vec.x + vec.y * vec.y + vec.z * vec.z);
}
 
Device Vector
normalized(const Vector vec)
{
    const Scalar inv_len = reciprocal_len(vec);
    return inv_len * vec;
}