You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
1415 lines
57 KiB
1415 lines
57 KiB
// David Eberly, Geometric Tools, Redmond WA 98052
|
|
// Copyright (c) 1998-2021
|
|
// Distributed under the Boost Software License, Version 1.0.
|
|
// https://www.boost.org/LICENSE_1_0.txt
|
|
// https://www.geometrictools.com/License/Boost/LICENSE_1_0.txt
|
|
// Version: 4.0.2019.08.13
|
|
|
|
#pragma once
|
|
|
|
#include <Mathematics/Logger.h>
|
|
#include <Mathematics/Math.h>
|
|
#include <Mathematics/Array3.h>
|
|
#include <algorithm>
|
|
#include <array>
|
|
#include <cstring>
|
|
|
|
// The interpolator is for uniformly spaced(x,y z)-values. The input samples
|
|
// must be stored in lexicographical order to represent f(x,y,z); that is,
|
|
// F[c + xBound*(r + yBound*s)] corresponds to f(x,y,z), where c is the index
|
|
// corresponding to x, r is the index corresponding to y, and s is the index
|
|
// corresponding to z.
|
|
|
|
namespace gte
|
|
{
|
|
template <typename Real>
|
|
class IntpAkimaUniform3
|
|
{
|
|
public:
|
|
// Construction and destruction.
|
|
IntpAkimaUniform3(int xBound, int yBound, int zBound, Real xMin,
|
|
Real xSpacing, Real yMin, Real ySpacing, Real zMin, Real zSpacing,
|
|
Real const* F)
|
|
:
|
|
mXBound(xBound),
|
|
mYBound(yBound),
|
|
mZBound(zBound),
|
|
mQuantity(xBound* yBound* zBound),
|
|
mXMin(xMin),
|
|
mXSpacing(xSpacing),
|
|
mYMin(yMin),
|
|
mYSpacing(ySpacing),
|
|
mZMin(zMin),
|
|
mZSpacing(zSpacing),
|
|
mF(F),
|
|
mPoly(xBound - 1, yBound - 1, zBound - 1)
|
|
{
|
|
// At least a 3x3x3 block of data points is needed to construct
|
|
// the estimates of the boundary derivatives.
|
|
LogAssert(mXBound >= 3 && mYBound >= 3 && mZBound >= 3 && mF != nullptr, "Invalid input.");
|
|
LogAssert(mXSpacing > (Real)0 && mYSpacing > (Real)0 && mZSpacing > (Real)0, "Invalid input.");
|
|
|
|
mXMax = mXMin + mXSpacing * static_cast<Real>(mXBound - 1);
|
|
mYMax = mYMin + mYSpacing * static_cast<Real>(mYBound - 1);
|
|
mZMax = mZMin + mZSpacing * static_cast<Real>(mZBound - 1);
|
|
|
|
// Create a 3D wrapper for the 1D samples.
|
|
Array3<Real> Fmap(mXBound, mYBound, mZBound, const_cast<Real*>(mF));
|
|
|
|
// Construct first-order derivatives.
|
|
Array3<Real> FX(mXBound, mYBound, mZBound);
|
|
Array3<Real> FY(mXBound, mYBound, mZBound);
|
|
Array3<Real> FZ(mXBound, mYBound, mZBound);
|
|
GetFX(Fmap, FX);
|
|
GetFX(Fmap, FY);
|
|
GetFX(Fmap, FZ);
|
|
|
|
// Construct second-order derivatives.
|
|
Array3<Real> FXY(mXBound, mYBound, mZBound);
|
|
Array3<Real> FXZ(mXBound, mYBound, mZBound);
|
|
Array3<Real> FYZ(mXBound, mYBound, mZBound);
|
|
GetFX(Fmap, FXY);
|
|
GetFX(Fmap, FXZ);
|
|
GetFX(Fmap, FYZ);
|
|
|
|
// Construct third-order derivatives.
|
|
Array3<Real> FXYZ(mXBound, mYBound, mZBound);
|
|
GetFXYZ(Fmap, FXYZ);
|
|
|
|
// Construct polynomials.
|
|
GetPolynomials(Fmap, FX, FY, FZ, FXY, FXZ, FYZ, FXYZ);
|
|
}
|
|
|
|
~IntpAkimaUniform3() = default;
|
|
|
|
// Member access.
|
|
inline int GetXBound() const
|
|
{
|
|
return mXBound;
|
|
}
|
|
|
|
inline int GetYBound() const
|
|
{
|
|
return mYBound;
|
|
}
|
|
|
|
inline int GetZBound() const
|
|
{
|
|
return mZBound;
|
|
}
|
|
|
|
inline int GetQuantity() const
|
|
{
|
|
return mQuantity;
|
|
}
|
|
|
|
inline Real const* GetF() const
|
|
{
|
|
return mF;
|
|
}
|
|
|
|
inline Real GetXMin() const
|
|
{
|
|
return mXMin;
|
|
}
|
|
|
|
inline Real GetXMax() const
|
|
{
|
|
return mXMax;
|
|
}
|
|
|
|
inline Real GetXSpacing() const
|
|
{
|
|
return mXSpacing;
|
|
}
|
|
|
|
inline Real GetYMin() const
|
|
{
|
|
return mYMin;
|
|
}
|
|
|
|
inline Real GetYMax() const
|
|
{
|
|
return mYMax;
|
|
}
|
|
|
|
inline Real GetYSpacing() const
|
|
{
|
|
return mYSpacing;
|
|
}
|
|
|
|
inline Real GetZMin() const
|
|
{
|
|
return mZMin;
|
|
}
|
|
|
|
inline Real GetZMax() const
|
|
{
|
|
return mZMax;
|
|
}
|
|
|
|
inline Real GetZSpacing() const
|
|
{
|
|
return mZSpacing;
|
|
}
|
|
|
|
// Evaluate the function and its derivatives. The functions clamp the
|
|
// inputs to xmin <= x <= xmax, ymin <= y <= ymax and
|
|
// zmin <= z <= zmax. The first operator is for function evaluation.
|
|
// The second operator is for function or derivative evaluations. The
|
|
// xOrder argument is the order of the x-derivative, the yOrder
|
|
// argument is the order of the y-derivative, and the zOrder argument
|
|
// is the order of the z-derivative. All orders are zero to get the
|
|
// function value itself.
|
|
Real operator()(Real x, Real y, Real z) const
|
|
{
|
|
x = std::min(std::max(x, mXMin), mXMax);
|
|
y = std::min(std::max(y, mYMin), mYMax);
|
|
z = std::min(std::max(z, mZMin), mZMax);
|
|
int ix, iy, iz;
|
|
Real dx, dy, dz;
|
|
XLookup(x, ix, dx);
|
|
YLookup(y, iy, dy);
|
|
ZLookup(z, iz, dz);
|
|
return mPoly[iz][iy][ix](dx, dy, dz);
|
|
}
|
|
|
|
Real operator()(int xOrder, int yOrder, int zOrder, Real x, Real y, Real z) const
|
|
{
|
|
x = std::min(std::max(x, mXMin), mXMax);
|
|
y = std::min(std::max(y, mYMin), mYMax);
|
|
z = std::min(std::max(z, mZMin), mZMax);
|
|
int ix, iy, iz;
|
|
Real dx, dy, dz;
|
|
XLookup(x, ix, dx);
|
|
YLookup(y, iy, dy);
|
|
ZLookup(z, iz, dz);
|
|
return mPoly[iz][iy][ix](xOrder, yOrder, zOrder, dx, dy, dz);
|
|
}
|
|
|
|
private:
|
|
class Polynomial
|
|
{
|
|
public:
|
|
Polynomial()
|
|
{
|
|
for (size_t ix = 0; ix < 4; ++ix)
|
|
{
|
|
for (size_t iy = 0; iy < 4; ++iy)
|
|
{
|
|
mCoeff[ix][iy].fill((Real)0);
|
|
}
|
|
}
|
|
}
|
|
|
|
// P(x,y,z) = sum_{i=0}^3 sum_{j=0}^3 sum_{k=0}^3 a_{ijk} x^i y^j z^k.
|
|
// The tensor term A[ix][iy][iz] corresponds to the polynomial term
|
|
// x^{ix} y^{iy} z^{iz}.
|
|
Real& A(int ix, int iy, int iz)
|
|
{
|
|
return mCoeff[ix][iy][iz];
|
|
}
|
|
|
|
Real operator()(Real x, Real y, Real z) const
|
|
{
|
|
std::array<Real, 4> xPow = { (Real)1, x, x * x, x * x * x };
|
|
std::array<Real, 4> yPow = { (Real)1, y, y * y, y * y * y };
|
|
std::array<Real, 4> zPow = { (Real)1, z, z * z, z * z * z };
|
|
|
|
Real p = (Real)0;
|
|
for (size_t iz = 0; iz <= 3; ++iz)
|
|
{
|
|
for (size_t iy = 0; iy <= 3; ++iy)
|
|
{
|
|
for (size_t ix = 0; ix <= 3; ++ix)
|
|
{
|
|
p += mCoeff[ix][iy][iz] * xPow[ix] * yPow[iy] * zPow[iz];
|
|
}
|
|
}
|
|
}
|
|
|
|
return p;
|
|
}
|
|
|
|
Real operator()(int xOrder, int yOrder, int zOrder, Real x, Real y, Real z) const
|
|
{
|
|
std::array<Real, 4> xPow;
|
|
switch (xOrder)
|
|
{
|
|
case 0:
|
|
xPow[0] = (Real)1;
|
|
xPow[1] = x;
|
|
xPow[2] = x * x;
|
|
xPow[3] = x * x * x;
|
|
break;
|
|
case 1:
|
|
xPow[0] = (Real)0;
|
|
xPow[1] = (Real)1;
|
|
xPow[2] = (Real)2 * x;
|
|
xPow[3] = (Real)3 * x * x;
|
|
break;
|
|
case 2:
|
|
xPow[0] = (Real)0;
|
|
xPow[1] = (Real)0;
|
|
xPow[2] = (Real)2;
|
|
xPow[3] = (Real)6 * x;
|
|
break;
|
|
case 3:
|
|
xPow[0] = (Real)0;
|
|
xPow[1] = (Real)0;
|
|
xPow[2] = (Real)0;
|
|
xPow[3] = (Real)6;
|
|
break;
|
|
default:
|
|
return (Real)0;
|
|
}
|
|
|
|
std::array<Real, 4> yPow;
|
|
switch (yOrder)
|
|
{
|
|
case 0:
|
|
yPow[0] = (Real)1;
|
|
yPow[1] = y;
|
|
yPow[2] = y * y;
|
|
yPow[3] = y * y * y;
|
|
break;
|
|
case 1:
|
|
yPow[0] = (Real)0;
|
|
yPow[1] = (Real)1;
|
|
yPow[2] = (Real)2 * y;
|
|
yPow[3] = (Real)3 * y * y;
|
|
break;
|
|
case 2:
|
|
yPow[0] = (Real)0;
|
|
yPow[1] = (Real)0;
|
|
yPow[2] = (Real)2;
|
|
yPow[3] = (Real)6 * y;
|
|
break;
|
|
case 3:
|
|
yPow[0] = (Real)0;
|
|
yPow[1] = (Real)0;
|
|
yPow[2] = (Real)0;
|
|
yPow[3] = (Real)6;
|
|
break;
|
|
default:
|
|
return (Real)0;
|
|
}
|
|
|
|
std::array<Real, 4> zPow;
|
|
switch (zOrder)
|
|
{
|
|
case 0:
|
|
zPow[0] = (Real)1;
|
|
zPow[1] = z;
|
|
zPow[2] = z * z;
|
|
zPow[3] = z * z * z;
|
|
break;
|
|
case 1:
|
|
zPow[0] = (Real)0;
|
|
zPow[1] = (Real)1;
|
|
zPow[2] = (Real)2 * z;
|
|
zPow[3] = (Real)3 * z * z;
|
|
break;
|
|
case 2:
|
|
zPow[0] = (Real)0;
|
|
zPow[1] = (Real)0;
|
|
zPow[2] = (Real)2;
|
|
zPow[3] = (Real)6 * z;
|
|
break;
|
|
case 3:
|
|
zPow[0] = (Real)0;
|
|
zPow[1] = (Real)0;
|
|
zPow[2] = (Real)0;
|
|
zPow[3] = (Real)6;
|
|
break;
|
|
default:
|
|
return (Real)0;
|
|
}
|
|
|
|
Real p = (Real)0;
|
|
|
|
for (size_t iz = 0; iz <= 3; ++iz)
|
|
{
|
|
for (size_t iy = 0; iy <= 3; ++iy)
|
|
{
|
|
for (size_t ix = 0; ix <= 3; ++ix)
|
|
{
|
|
p += mCoeff[ix][iy][iz] * xPow[ix] * yPow[iy] * zPow[iz];
|
|
}
|
|
}
|
|
}
|
|
|
|
return p;
|
|
}
|
|
|
|
private:
|
|
std::array<std::array<std::array<Real, 4>, 4>, 4> mCoeff;
|
|
};
|
|
|
|
// Support for construction.
|
|
void GetFX(Array3<Real> const& F, Array3<Real>& FX)
|
|
{
|
|
Array3<Real> slope(mXBound + 3, mYBound, mZBound);
|
|
Real invDX = (Real)1 / mXSpacing;
|
|
int ix, iy, iz;
|
|
for (iz = 0; iz < mZBound; ++iz)
|
|
{
|
|
for (iy = 0; iy < mYBound; ++iy)
|
|
{
|
|
for (ix = 0; ix < mXBound - 1; ++ix)
|
|
{
|
|
slope[iz][iy][ix + 2] = (F[iz][iy][ix + 1] - F[iz][iy][ix]) * invDX;
|
|
}
|
|
|
|
slope[iz][iy][1] = (Real)2 * slope[iz][iy][2] - slope[iz][iy][3];
|
|
slope[iz][iy][0] = (Real)2 * slope[iz][iy][1] - slope[iz][iy][2];
|
|
slope[iz][iy][mXBound + 1] = (Real)2 * slope[iz][iy][mXBound] - slope[iz][iy][mXBound - 1];
|
|
slope[iz][iy][mXBound + 2] = (Real)2 * slope[iz][iy][mXBound + 1] - slope[iz][iy][mXBound];
|
|
}
|
|
}
|
|
|
|
for (iz = 0; iz < mZBound; ++iz)
|
|
{
|
|
for (iy = 0; iy < mYBound; ++iy)
|
|
{
|
|
for (ix = 0; ix < mXBound; ++ix)
|
|
{
|
|
FX[iz][iy][ix] = ComputeDerivative(slope[iz][iy] + ix);
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
void GetFY(Array3<Real> const& F, Array3<Real>& FY)
|
|
{
|
|
Array3<Real> slope(mYBound + 3, mXBound, mZBound);
|
|
Real invDY = (Real)1 / mYSpacing;
|
|
int ix, iy, iz;
|
|
for (iz = 0; iz < mZBound; ++iz)
|
|
{
|
|
for (ix = 0; ix < mXBound; ++ix)
|
|
{
|
|
for (iy = 0; iy < mYBound - 1; ++iy)
|
|
{
|
|
slope[iz][ix][iy + 2] = (F[iz][iy + 1][ix] - F[iz][iy][ix]) * invDY;
|
|
}
|
|
|
|
slope[iz][ix][1] = (Real)2 * slope[iz][ix][2] - slope[iz][ix][3];
|
|
slope[iz][ix][0] = (Real)2 * slope[iz][ix][1] - slope[iz][ix][2];
|
|
slope[iz][ix][mYBound + 1] = (Real)2 * slope[iz][ix][mYBound] - slope[iz][ix][mYBound - 1];
|
|
slope[iz][ix][mYBound + 2] = (Real)2 * slope[iz][ix][mYBound + 1] - slope[iz][ix][mYBound];
|
|
}
|
|
}
|
|
|
|
for (iz = 0; iz < mZBound; ++iz)
|
|
{
|
|
for (ix = 0; ix < mXBound; ++ix)
|
|
{
|
|
for (iy = 0; iy < mYBound; ++iy)
|
|
{
|
|
FY[iz][iy][ix] = ComputeDerivative(slope[iz][ix] + iy);
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
void GetFZ(Array3<Real> const& F, Array3<Real>& FZ)
|
|
{
|
|
Array3<Real> slope(mZBound + 3, mXBound, mYBound);
|
|
Real invDZ = (Real)1 / mZSpacing;
|
|
int ix, iy, iz;
|
|
for (iy = 0; iy < mYBound; ++iy)
|
|
{
|
|
for (ix = 0; ix < mXBound; ++ix)
|
|
{
|
|
for (iz = 0; iz < mZBound - 1; ++iz)
|
|
{
|
|
slope[iy][ix][iz + 2] = (F[iz + 1][iy][ix] - F[iz][iy][ix]) * invDZ;
|
|
}
|
|
|
|
slope[iy][ix][1] = (Real)2 * slope[iy][ix][2] - slope[iy][ix][3];
|
|
slope[iy][ix][0] = (Real)2 * slope[iy][ix][1] - slope[iy][ix][2];
|
|
slope[iy][ix][mZBound + 1] = (Real)2 * slope[iy][ix][mZBound] - slope[iy][ix][mZBound - 1];
|
|
slope[iy][ix][mZBound + 2] = (Real)2 * slope[iy][ix][mZBound + 1] - slope[iy][ix][mZBound];
|
|
}
|
|
}
|
|
|
|
for (iy = 0; iy < mYBound; ++iy)
|
|
{
|
|
for (ix = 0; ix < mXBound; ++ix)
|
|
{
|
|
for (iz = 0; iz < mZBound; ++iz)
|
|
{
|
|
FZ[iz][iy][ix] = ComputeDerivative(slope[iy][ix] + iz);
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
void GetFXY(Array3<Real> const& F, Array3<Real>& FXY)
|
|
{
|
|
int xBoundM1 = mXBound - 1;
|
|
int yBoundM1 = mYBound - 1;
|
|
int ix0 = xBoundM1, ix1 = ix0 - 1, ix2 = ix1 - 1;
|
|
int iy0 = yBoundM1, iy1 = iy0 - 1, iy2 = iy1 - 1;
|
|
int ix, iy, iz;
|
|
|
|
Real invDXDY = (Real)1 / (mXSpacing * mYSpacing);
|
|
for (iz = 0; iz < mZBound; ++iz)
|
|
{
|
|
// corners of z-slice
|
|
FXY[iz][0][0] = (Real)0.25 * invDXDY * (
|
|
(Real)9 * F[iz][0][0]
|
|
- (Real)12 * F[iz][0][1]
|
|
+ (Real)3 * F[iz][0][2]
|
|
- (Real)12 * F[iz][1][0]
|
|
+ (Real)16 * F[iz][1][1]
|
|
- (Real)4 * F[iz][1][2]
|
|
+ (Real)3 * F[iz][2][0]
|
|
- (Real)4 * F[iz][2][1]
|
|
+ F[iz][2][2]);
|
|
|
|
FXY[iz][0][xBoundM1] = (Real)0.25 * invDXDY * (
|
|
(Real)9 * F[iz][0][ix0]
|
|
- (Real)12 * F[iz][0][ix1]
|
|
+ (Real)3 * F[iz][0][ix2]
|
|
- (Real)12 * F[iz][1][ix0]
|
|
+ (Real)16 * F[iz][1][ix1]
|
|
- (Real)4 * F[iz][1][ix2]
|
|
+ (Real)3 * F[iz][2][ix0]
|
|
- (Real)4 * F[iz][2][ix1]
|
|
+ F[iz][2][ix2]);
|
|
|
|
FXY[iz][yBoundM1][0] = (Real)0.25 * invDXDY * (
|
|
(Real)9 * F[iz][iy0][0]
|
|
- (Real)12 * F[iz][iy0][1]
|
|
+ (Real)3 * F[iz][iy0][2]
|
|
- (Real)12 * F[iz][iy1][0]
|
|
+ (Real)16 * F[iz][iy1][1]
|
|
- (Real)4 * F[iz][iy1][2]
|
|
+ (Real)3 * F[iz][iy2][0]
|
|
- (Real)4 * F[iz][iy2][1]
|
|
+ F[iz][iy2][2]);
|
|
|
|
FXY[iz][yBoundM1][xBoundM1] = (Real)0.25 * invDXDY * (
|
|
(Real)9 * F[iz][iy0][ix0]
|
|
- (Real)12 * F[iz][iy0][ix1]
|
|
+ (Real)3 * F[iz][iy0][ix2]
|
|
- (Real)12 * F[iz][iy1][ix0]
|
|
+ (Real)16 * F[iz][iy1][ix1]
|
|
- (Real)4 * F[iz][iy1][ix2]
|
|
+ (Real)3 * F[iz][iy2][ix0]
|
|
- (Real)4 * F[iz][iy2][ix1]
|
|
+ F[iz][iy2][ix2]);
|
|
|
|
// x-edges of z-slice
|
|
for (ix = 1; ix < xBoundM1; ++ix)
|
|
{
|
|
FXY[iz][0][ix] = (Real)0.25 * invDXDY * (
|
|
(Real)3 * (F[iz][0][ix - 1] - F[iz][0][ix + 1]) -
|
|
(Real)4 * (F[iz][1][ix - 1] - F[iz][1][ix + 1]) +
|
|
(F[iz][2][ix - 1] - F[iz][2][ix + 1]));
|
|
|
|
FXY[iz][yBoundM1][ix] = (Real)0.25 * invDXDY * (
|
|
(Real)3 * (F[iz][iy0][ix - 1] - F[iz][iy0][ix + 1])
|
|
- (Real)4 * (F[iz][iy1][ix - 1] - F[iz][iy1][ix + 1]) +
|
|
(F[iz][iy2][ix - 1] - F[iz][iy2][ix + 1]));
|
|
}
|
|
|
|
// y-edges of z-slice
|
|
for (iy = 1; iy < yBoundM1; ++iy)
|
|
{
|
|
FXY[iz][iy][0] = (Real)0.25 * invDXDY * (
|
|
(Real)3 * (F[iz][iy - 1][0] - F[iz][iy + 1][0]) -
|
|
(Real)4 * (F[iz][iy - 1][1] - F[iz][iy + 1][1]) +
|
|
(F[iz][iy - 1][2] - F[iz][iy + 1][2]));
|
|
|
|
FXY[iz][iy][xBoundM1] = (Real)0.25 * invDXDY * (
|
|
(Real)3 * (F[iz][iy - 1][ix0] - F[iz][iy + 1][ix0])
|
|
- (Real)4 * (F[iz][iy - 1][ix1] - F[iz][iy + 1][ix1]) +
|
|
(F[iz][iy - 1][ix2] - F[iz][iy + 1][ix2]));
|
|
}
|
|
|
|
// interior of z-slice
|
|
for (iy = 1; iy < yBoundM1; ++iy)
|
|
{
|
|
for (ix = 1; ix < xBoundM1; ++ix)
|
|
{
|
|
FXY[iz][iy][ix] = (Real)0.25 * invDXDY * (
|
|
F[iz][iy - 1][ix - 1] - F[iz][iy - 1][ix + 1] -
|
|
F[iz][iy + 1][ix - 1] + F[iz][iy + 1][ix + 1]);
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
void GetFXZ(Array3<Real> const& F, Array3<Real> & FXZ)
|
|
{
|
|
int xBoundM1 = mXBound - 1;
|
|
int zBoundM1 = mZBound - 1;
|
|
int ix0 = xBoundM1, ix1 = ix0 - 1, ix2 = ix1 - 1;
|
|
int iz0 = zBoundM1, iz1 = iz0 - 1, iz2 = iz1 - 1;
|
|
int ix, iy, iz;
|
|
|
|
Real invDXDZ = (Real)1 / (mXSpacing * mZSpacing);
|
|
for (iy = 0; iy < mYBound; ++iy)
|
|
{
|
|
// corners of z-slice
|
|
FXZ[0][iy][0] = (Real)0.25 * invDXDZ * (
|
|
(Real)9 * F[0][iy][0]
|
|
- (Real)12 * F[0][iy][1]
|
|
+ (Real)3 * F[0][iy][2]
|
|
- (Real)12 * F[1][iy][0]
|
|
+ (Real)16 * F[1][iy][1]
|
|
- (Real)4 * F[1][iy][2]
|
|
+ (Real)3 * F[2][iy][0]
|
|
- (Real)4 * F[2][iy][1]
|
|
+ F[2][iy][2]);
|
|
|
|
FXZ[0][iy][xBoundM1] = (Real)0.25 * invDXDZ * (
|
|
(Real)9 * F[0][iy][ix0]
|
|
- (Real)12 * F[0][iy][ix1]
|
|
+ (Real)3 * F[0][iy][ix2]
|
|
- (Real)12 * F[1][iy][ix0]
|
|
+ (Real)16 * F[1][iy][ix1]
|
|
- (Real)4 * F[1][iy][ix2]
|
|
+ (Real)3 * F[2][iy][ix0]
|
|
- (Real)4 * F[2][iy][ix1]
|
|
+ F[2][iy][ix2]);
|
|
|
|
FXZ[zBoundM1][iy][0] = (Real)0.25 * invDXDZ * (
|
|
(Real)9 * F[iz0][iy][0]
|
|
- (Real)12 * F[iz0][iy][1]
|
|
+ (Real)3 * F[iz0][iy][2]
|
|
- (Real)12 * F[iz1][iy][0]
|
|
+ (Real)16 * F[iz1][iy][1]
|
|
- (Real)4 * F[iz1][iy][2]
|
|
+ (Real)3 * F[iz2][iy][0]
|
|
- (Real)4 * F[iz2][iy][1]
|
|
+ F[iz2][iy][2]);
|
|
|
|
FXZ[zBoundM1][iy][xBoundM1] = (Real)0.25 * invDXDZ * (
|
|
(Real)9 * F[iz0][iy][ix0]
|
|
- (Real)12 * F[iz0][iy][ix1]
|
|
+ (Real)3 * F[iz0][iy][ix2]
|
|
- (Real)12 * F[iz1][iy][ix0]
|
|
+ (Real)16 * F[iz1][iy][ix1]
|
|
- (Real)4 * F[iz1][iy][ix2]
|
|
+ (Real)3 * F[iz2][iy][ix0]
|
|
- (Real)4 * F[iz2][iy][ix1]
|
|
+ F[iz2][iy][ix2]);
|
|
|
|
// x-edges of y-slice
|
|
for (ix = 1; ix < xBoundM1; ++ix)
|
|
{
|
|
FXZ[0][iy][ix] = (Real)0.25 * invDXDZ * (
|
|
(Real)3 * (F[0][iy][ix - 1] - F[0][iy][ix + 1]) -
|
|
(Real)4 * (F[1][iy][ix - 1] - F[1][iy][ix + 1]) +
|
|
(F[2][iy][ix - 1] - F[2][iy][ix + 1]));
|
|
|
|
FXZ[zBoundM1][iy][ix] = (Real)0.25 * invDXDZ * (
|
|
(Real)3 * (F[iz0][iy][ix - 1] - F[iz0][iy][ix + 1])
|
|
- (Real)4 * (F[iz1][iy][ix - 1] - F[iz1][iy][ix + 1]) +
|
|
(F[iz2][iy][ix - 1] - F[iz2][iy][ix + 1]));
|
|
}
|
|
|
|
// z-edges of y-slice
|
|
for (iz = 1; iz < zBoundM1; ++iz)
|
|
{
|
|
FXZ[iz][iy][0] = (Real)0.25 * invDXDZ * (
|
|
(Real)3 * (F[iz - 1][iy][0] - F[iz + 1][iy][0]) -
|
|
(Real)4 * (F[iz - 1][iy][1] - F[iz + 1][iy][1]) +
|
|
(F[iz - 1][iy][2] - F[iz + 1][iy][2]));
|
|
|
|
FXZ[iz][iy][xBoundM1] = (Real)0.25 * invDXDZ * (
|
|
(Real)3 * (F[iz - 1][iy][ix0] - F[iz + 1][iy][ix0])
|
|
- (Real)4 * (F[iz - 1][iy][ix1] - F[iz + 1][iy][ix1]) +
|
|
(F[iz - 1][iy][ix2] - F[iz + 1][iy][ix2]));
|
|
}
|
|
|
|
// interior of y-slice
|
|
for (iz = 1; iz < zBoundM1; ++iz)
|
|
{
|
|
for (ix = 1; ix < xBoundM1; ++ix)
|
|
{
|
|
FXZ[iz][iy][ix] = ((Real)0.25) * invDXDZ * (
|
|
F[iz - 1][iy][ix - 1] - F[iz - 1][iy][ix + 1] -
|
|
F[iz + 1][iy][ix - 1] + F[iz + 1][iy][ix + 1]);
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
void GetFYZ(Array3<Real> const& F, Array3<Real> & FYZ)
|
|
{
|
|
int yBoundM1 = mYBound - 1;
|
|
int zBoundM1 = mZBound - 1;
|
|
int iy0 = yBoundM1, iy1 = iy0 - 1, iy2 = iy1 - 1;
|
|
int iz0 = zBoundM1, iz1 = iz0 - 1, iz2 = iz1 - 1;
|
|
int ix, iy, iz;
|
|
|
|
Real invDYDZ = (Real)1 / (mYSpacing * mZSpacing);
|
|
for (ix = 0; ix < mXBound; ++ix)
|
|
{
|
|
// corners of x-slice
|
|
FYZ[0][0][ix] = (Real)0.25 * invDYDZ * (
|
|
(Real)9 * F[0][0][ix]
|
|
- (Real)12 * F[0][1][ix]
|
|
+ (Real)3 * F[0][2][ix]
|
|
- (Real)12 * F[1][0][ix]
|
|
+ (Real)16 * F[1][1][ix]
|
|
- (Real)4 * F[1][2][ix]
|
|
+ (Real)3 * F[2][0][ix]
|
|
- (Real)4 * F[2][1][ix]
|
|
+ F[2][2][ix]);
|
|
|
|
FYZ[0][yBoundM1][ix] = (Real)0.25 * invDYDZ * (
|
|
(Real)9 * F[0][iy0][ix]
|
|
- (Real)12 * F[0][iy1][ix]
|
|
+ (Real)3 * F[0][iy2][ix]
|
|
- (Real)12 * F[1][iy0][ix]
|
|
+ (Real)16 * F[1][iy1][ix]
|
|
- (Real)4 * F[1][iy2][ix]
|
|
+ (Real)3 * F[2][iy0][ix]
|
|
- (Real)4 * F[2][iy1][ix]
|
|
+ F[2][iy2][ix]);
|
|
|
|
FYZ[zBoundM1][0][ix] = (Real)0.25 * invDYDZ * (
|
|
(Real)9 * F[iz0][0][ix]
|
|
- (Real)12 * F[iz0][1][ix]
|
|
+ (Real)3 * F[iz0][2][ix]
|
|
- (Real)12 * F[iz1][0][ix]
|
|
+ (Real)16 * F[iz1][1][ix]
|
|
- (Real)4 * F[iz1][2][ix]
|
|
+ (Real)3 * F[iz2][0][ix]
|
|
- (Real)4 * F[iz2][1][ix]
|
|
+ F[iz2][2][ix]);
|
|
|
|
FYZ[zBoundM1][yBoundM1][ix] = (Real)0.25 * invDYDZ * (
|
|
(Real)9 * F[iz0][iy0][ix]
|
|
- (Real)12 * F[iz0][iy1][ix]
|
|
+ (Real)3 * F[iz0][iy2][ix]
|
|
- (Real)12 * F[iz1][iy0][ix]
|
|
+ (Real)16 * F[iz1][iy1][ix]
|
|
- (Real)4 * F[iz1][iy2][ix]
|
|
+ (Real)3 * F[iz2][iy0][ix]
|
|
- (Real)4 * F[iz2][iy1][ix]
|
|
+ F[iz2][iy2][ix]);
|
|
|
|
// y-edges of x-slice
|
|
for (iy = 1; iy < yBoundM1; ++iy)
|
|
{
|
|
FYZ[0][iy][ix] = (Real)0.25 * invDYDZ * (
|
|
(Real)3 * (F[0][iy - 1][ix] - F[0][iy + 1][ix]) -
|
|
(Real)4 * (F[1][iy - 1][ix] - F[1][iy + 1][ix]) +
|
|
(F[2][iy - 1][ix] - F[2][iy + 1][ix]));
|
|
|
|
FYZ[zBoundM1][iy][ix] = (Real)0.25 * invDYDZ * (
|
|
(Real)3 * (F[iz0][iy - 1][ix] - F[iz0][iy + 1][ix])
|
|
- (Real)4 * (F[iz1][iy - 1][ix] - F[iz1][iy + 1][ix]) +
|
|
(F[iz2][iy - 1][ix] - F[iz2][iy + 1][ix]));
|
|
}
|
|
|
|
// z-edges of x-slice
|
|
for (iz = 1; iz < zBoundM1; ++iz)
|
|
{
|
|
FYZ[iz][0][ix] = (Real)0.25 * invDYDZ * (
|
|
(Real)3 * (F[iz - 1][0][ix] - F[iz + 1][0][ix]) -
|
|
(Real)4 * (F[iz - 1][1][ix] - F[iz + 1][1][ix]) +
|
|
(F[iz - 1][2][ix] - F[iz + 1][2][ix]));
|
|
|
|
FYZ[iz][yBoundM1][ix] = (Real)0.25 * invDYDZ * (
|
|
(Real)3 * (F[iz - 1][iy0][ix] - F[iz + 1][iy0][ix])
|
|
- (Real)4 * (F[iz - 1][iy1][ix] - F[iz + 1][iy1][ix]) +
|
|
(F[iz - 1][iy2][ix] - F[iz + 1][iy2][ix]));
|
|
}
|
|
|
|
// interior of x-slice
|
|
for (iz = 1; iz < zBoundM1; ++iz)
|
|
{
|
|
for (iy = 1; iy < yBoundM1; ++iy)
|
|
{
|
|
FYZ[iz][iy][ix] = (Real)0.25 * invDYDZ * (
|
|
F[iz - 1][iy - 1][ix] - F[iz - 1][iy + 1][ix] -
|
|
F[iz + 1][iy - 1][ix] + F[iz + 1][iy + 1][ix]);
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
void GetFXYZ(Array3<Real> const& F, Array3<Real> & FXYZ)
|
|
{
|
|
int xBoundM1 = mXBound - 1;
|
|
int yBoundM1 = mYBound - 1;
|
|
int zBoundM1 = mZBound - 1;
|
|
int ix, iy, iz, ix0, iy0, iz0;
|
|
|
|
Real invDXDYDZ = ((Real)1) / (mXSpacing * mYSpacing * mZSpacing);
|
|
|
|
// convolution masks
|
|
// centered difference, O(h^2)
|
|
Real CDer[3] = { -(Real)0.5, (Real)0, (Real)0.5 };
|
|
// one-sided difference, O(h^2)
|
|
Real ODer[3] = { -(Real)1.5, (Real)2, -(Real)0.5 };
|
|
Real mask;
|
|
|
|
// corners
|
|
FXYZ[0][0][0] = (Real)0;
|
|
FXYZ[0][0][xBoundM1] = (Real)0;
|
|
FXYZ[0][yBoundM1][0] = (Real)0;
|
|
FXYZ[0][yBoundM1][xBoundM1] = (Real)0;
|
|
FXYZ[zBoundM1][0][0] = (Real)0;
|
|
FXYZ[zBoundM1][0][xBoundM1] = (Real)0;
|
|
FXYZ[zBoundM1][yBoundM1][0] = (Real)0;
|
|
FXYZ[zBoundM1][yBoundM1][xBoundM1] = (Real)0;
|
|
for (iz = 0; iz <= 2; ++iz)
|
|
{
|
|
for (iy = 0; iy <= 2; ++iy)
|
|
{
|
|
for (ix = 0; ix <= 2; ++ix)
|
|
{
|
|
mask = invDXDYDZ * ODer[ix] * ODer[iy] * ODer[iz];
|
|
FXYZ[0][0][0] += mask * F[iz][iy][ix];
|
|
FXYZ[0][0][xBoundM1] += mask * F[iz][iy][xBoundM1 - ix];
|
|
FXYZ[0][yBoundM1][0] += mask * F[iz][yBoundM1 - iy][ix];
|
|
FXYZ[0][yBoundM1][xBoundM1] += mask * F[iz][yBoundM1 - iy][xBoundM1 - ix];
|
|
FXYZ[zBoundM1][0][0] += mask * F[zBoundM1 - iz][iy][ix];
|
|
FXYZ[zBoundM1][0][xBoundM1] += mask * F[zBoundM1 - iz][iy][xBoundM1 - ix];
|
|
FXYZ[zBoundM1][yBoundM1][0] += mask * F[zBoundM1 - iz][yBoundM1 - iy][ix];
|
|
FXYZ[zBoundM1][yBoundM1][xBoundM1] += mask * F[zBoundM1 - iz][yBoundM1 - iy][xBoundM1 - ix];
|
|
}
|
|
}
|
|
}
|
|
|
|
// x-edges
|
|
for (ix0 = 1; ix0 < xBoundM1; ++ix0)
|
|
{
|
|
FXYZ[0][0][ix0] = (Real)0;
|
|
FXYZ[0][yBoundM1][ix0] = (Real)0;
|
|
FXYZ[zBoundM1][0][ix0] = (Real)0;
|
|
FXYZ[zBoundM1][yBoundM1][ix0] = (Real)0;
|
|
for (iz = 0; iz <= 2; ++iz)
|
|
{
|
|
for (iy = 0; iy <= 2; ++iy)
|
|
{
|
|
for (ix = 0; ix <= 2; ++ix)
|
|
{
|
|
mask = invDXDYDZ * CDer[ix] * ODer[iy] * ODer[iz];
|
|
FXYZ[0][0][ix0] += mask * F[iz][iy][ix0 + ix - 1];
|
|
FXYZ[0][yBoundM1][ix0] += mask * F[iz][yBoundM1 - iy][ix0 + ix - 1];
|
|
FXYZ[zBoundM1][0][ix0] += mask * F[zBoundM1 - iz][iy][ix0 + ix - 1];
|
|
FXYZ[zBoundM1][yBoundM1][ix0] += mask * F[zBoundM1 - iz][yBoundM1 - iy][ix0 + ix - 1];
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
// y-edges
|
|
for (iy0 = 1; iy0 < yBoundM1; ++iy0)
|
|
{
|
|
FXYZ[0][iy0][0] = (Real)0;
|
|
FXYZ[0][iy0][xBoundM1] = (Real)0;
|
|
FXYZ[zBoundM1][iy0][0] = (Real)0;
|
|
FXYZ[zBoundM1][iy0][xBoundM1] = (Real)0;
|
|
for (iz = 0; iz <= 2; ++iz)
|
|
{
|
|
for (iy = 0; iy <= 2; ++iy)
|
|
{
|
|
for (ix = 0; ix <= 2; ++ix)
|
|
{
|
|
mask = invDXDYDZ * ODer[ix] * CDer[iy] * ODer[iz];
|
|
FXYZ[0][iy0][0] += mask * F[iz][iy0 + iy - 1][ix];
|
|
FXYZ[0][iy0][xBoundM1] += mask * F[iz][iy0 + iy - 1][xBoundM1 - ix];
|
|
FXYZ[zBoundM1][iy0][0] += mask * F[zBoundM1 - iz][iy0 + iy - 1][ix];
|
|
FXYZ[zBoundM1][iy0][xBoundM1] += mask * F[zBoundM1 - iz][iy0 + iy - 1][xBoundM1 - ix];
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
// z-edges
|
|
for (iz0 = 1; iz0 < zBoundM1; ++iz0)
|
|
{
|
|
FXYZ[iz0][0][0] = (Real)0;
|
|
FXYZ[iz0][0][xBoundM1] = (Real)0;
|
|
FXYZ[iz0][yBoundM1][0] = (Real)0;
|
|
FXYZ[iz0][yBoundM1][xBoundM1] = (Real)0;
|
|
for (iz = 0; iz <= 2; ++iz)
|
|
{
|
|
for (iy = 0; iy <= 2; ++iy)
|
|
{
|
|
for (ix = 0; ix <= 2; ++ix)
|
|
{
|
|
mask = invDXDYDZ * ODer[ix] * ODer[iy] * CDer[iz];
|
|
FXYZ[iz0][0][0] += mask * F[iz0 + iz - 1][iy][ix];
|
|
FXYZ[iz0][0][xBoundM1] += mask * F[iz0 + iz - 1][iy][xBoundM1 - ix];
|
|
FXYZ[iz0][yBoundM1][0] += mask * F[iz0 + iz - 1][yBoundM1 - iy][ix];
|
|
FXYZ[iz0][yBoundM1][xBoundM1] += mask * F[iz0 + iz - 1][yBoundM1 - iy][xBoundM1 - ix];
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
// xy-faces
|
|
for (iy0 = 1; iy0 < yBoundM1; ++iy0)
|
|
{
|
|
for (ix0 = 1; ix0 < xBoundM1; ++ix0)
|
|
{
|
|
FXYZ[0][iy0][ix0] = (Real)0;
|
|
FXYZ[zBoundM1][iy0][ix0] = (Real)0;
|
|
for (iz = 0; iz <= 2; ++iz)
|
|
{
|
|
for (iy = 0; iy <= 2; ++iy)
|
|
{
|
|
for (ix = 0; ix <= 2; ++ix)
|
|
{
|
|
mask = invDXDYDZ * CDer[ix] * CDer[iy] * ODer[iz];
|
|
FXYZ[0][iy0][ix0] += mask * F[iz][iy0 + iy - 1][ix0 + ix - 1];
|
|
FXYZ[zBoundM1][iy0][ix0] += mask * F[zBoundM1 - iz][iy0 + iy - 1][ix0 + ix - 1];
|
|
}
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
// xz-faces
|
|
for (iz0 = 1; iz0 < zBoundM1; ++iz0)
|
|
{
|
|
for (ix0 = 1; ix0 < xBoundM1; ++ix0)
|
|
{
|
|
FXYZ[iz0][0][ix0] = (Real)0;
|
|
FXYZ[iz0][yBoundM1][ix0] = (Real)0;
|
|
for (iz = 0; iz <= 2; ++iz)
|
|
{
|
|
for (iy = 0; iy <= 2; ++iy)
|
|
{
|
|
for (ix = 0; ix <= 2; ++ix)
|
|
{
|
|
mask = invDXDYDZ * CDer[ix] * ODer[iy] * CDer[iz];
|
|
FXYZ[iz0][0][ix0] += mask * F[iz0 + iz - 1][iy][ix0 + ix - 1];
|
|
FXYZ[iz0][yBoundM1][ix0] += mask * F[iz0 + iz - 1][yBoundM1 - iy][ix0 + ix - 1];
|
|
}
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
// yz-faces
|
|
for (iz0 = 1; iz0 < zBoundM1; ++iz0)
|
|
{
|
|
for (iy0 = 1; iy0 < yBoundM1; ++iy0)
|
|
{
|
|
FXYZ[iz0][iy0][0] = (Real)0;
|
|
FXYZ[iz0][iy0][xBoundM1] = (Real)0;
|
|
for (iz = 0; iz <= 2; ++iz)
|
|
{
|
|
for (iy = 0; iy <= 2; ++iy)
|
|
{
|
|
for (ix = 0; ix <= 2; ++ix)
|
|
{
|
|
mask = invDXDYDZ * ODer[ix] * CDer[iy] * CDer[iz];
|
|
FXYZ[iz0][iy0][0] += mask * F[iz0 + iz - 1][iy0 + iy - 1][ix];
|
|
FXYZ[iz0][iy0][xBoundM1] += mask * F[iz0 + iz - 1][iy0 + iy - 1][xBoundM1 - ix];
|
|
}
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
// interiors
|
|
for (iz0 = 1; iz0 < zBoundM1; ++iz0)
|
|
{
|
|
for (iy0 = 1; iy0 < yBoundM1; ++iy0)
|
|
{
|
|
for (ix0 = 1; ix0 < xBoundM1; ++ix0)
|
|
{
|
|
FXYZ[iz0][iy0][ix0] = (Real)0;
|
|
|
|
for (iz = 0; iz <= 2; ++iz)
|
|
{
|
|
for (iy = 0; iy <= 2; ++iy)
|
|
{
|
|
for (ix = 0; ix <= 2; ++ix)
|
|
{
|
|
mask = invDXDYDZ * CDer[ix] * CDer[iy] * CDer[iz];
|
|
FXYZ[iz0][iy0][ix0] += mask * F[iz0 + iz - 1][iy0 + iy - 1][ix0 + ix - 1];
|
|
}
|
|
}
|
|
}
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
void GetPolynomials(Array3<Real> const& F, Array3<Real> const& FX,
|
|
Array3<Real> const& FY, Array3<Real> const& FZ, Array3<Real> const& FXY,
|
|
Array3<Real> const& FXZ, Array3<Real> const& FYZ, Array3<Real> const& FXYZ)
|
|
{
|
|
int xBoundM1 = mXBound - 1;
|
|
int yBoundM1 = mYBound - 1;
|
|
int zBoundM1 = mZBound - 1;
|
|
for (int iz = 0; iz < zBoundM1; ++iz)
|
|
{
|
|
for (int iy = 0; iy < yBoundM1; ++iy)
|
|
{
|
|
for (int ix = 0; ix < xBoundM1; ++ix)
|
|
{
|
|
// Note the 'transposing' of the 2x2x2 blocks (to match
|
|
// notation used in the polynomial definition).
|
|
Real G[2][2][2] =
|
|
{
|
|
{
|
|
{
|
|
F[iz][iy][ix],
|
|
F[iz + 1][iy][ix]
|
|
},
|
|
{
|
|
F[iz][iy + 1][ix],
|
|
F[iz + 1][iy + 1][ix]
|
|
}
|
|
},
|
|
{
|
|
{
|
|
F[iz][iy][ix + 1],
|
|
F[iz + 1][iy][ix + 1]
|
|
},
|
|
{
|
|
F[iz][iy + 1][ix + 1],
|
|
F[iz + 1][iy + 1][ix + 1]
|
|
}
|
|
}
|
|
};
|
|
|
|
Real GX[2][2][2] =
|
|
{
|
|
{
|
|
{
|
|
FX[iz][iy][ix],
|
|
FX[iz + 1][iy][ix]
|
|
},
|
|
{
|
|
FX[iz][iy + 1][ix],
|
|
FX[iz + 1][iy + 1][ix]
|
|
}
|
|
},
|
|
{
|
|
{
|
|
FX[iz][iy][ix + 1],
|
|
FX[iz + 1][iy][ix + 1]
|
|
},
|
|
{
|
|
FX[iz][iy + 1][ix + 1],
|
|
FX[iz + 1][iy + 1][ix + 1]
|
|
}
|
|
}
|
|
};
|
|
|
|
Real GY[2][2][2] =
|
|
{
|
|
{
|
|
{
|
|
FY[iz][iy][ix],
|
|
FY[iz + 1][iy][ix]
|
|
},
|
|
{
|
|
FY[iz][iy + 1][ix],
|
|
FY[iz + 1][iy + 1][ix]
|
|
}
|
|
},
|
|
{
|
|
{
|
|
FY[iz][iy][ix + 1],
|
|
FY[iz + 1][iy][ix + 1]
|
|
},
|
|
{
|
|
FY[iz][iy + 1][ix + 1],
|
|
FY[iz + 1][iy + 1][ix + 1]
|
|
}
|
|
}
|
|
};
|
|
|
|
Real GZ[2][2][2] =
|
|
{
|
|
{
|
|
{
|
|
FZ[iz][iy][ix],
|
|
FZ[iz + 1][iy][ix]
|
|
},
|
|
{
|
|
FZ[iz][iy + 1][ix],
|
|
FZ[iz + 1][iy + 1][ix]
|
|
}
|
|
},
|
|
{
|
|
{
|
|
FZ[iz][iy][ix + 1],
|
|
FZ[iz + 1][iy][ix + 1]
|
|
},
|
|
{
|
|
FZ[iz][iy + 1][ix + 1],
|
|
FZ[iz + 1][iy + 1][ix + 1]
|
|
}
|
|
}
|
|
};
|
|
|
|
Real GXY[2][2][2] =
|
|
{
|
|
{
|
|
{
|
|
FXY[iz][iy][ix],
|
|
FXY[iz + 1][iy][ix]
|
|
},
|
|
{
|
|
FXY[iz][iy + 1][ix],
|
|
FXY[iz + 1][iy + 1][ix]
|
|
}
|
|
},
|
|
{
|
|
{
|
|
FXY[iz][iy][ix + 1],
|
|
FXY[iz + 1][iy][ix + 1]
|
|
},
|
|
{
|
|
FXY[iz][iy + 1][ix + 1],
|
|
FXY[iz + 1][iy + 1][ix + 1]
|
|
}
|
|
}
|
|
};
|
|
|
|
Real GXZ[2][2][2] =
|
|
{
|
|
{
|
|
{
|
|
FXZ[iz][iy][ix],
|
|
FXZ[iz + 1][iy][ix]
|
|
},
|
|
{
|
|
FXZ[iz][iy + 1][ix],
|
|
FXZ[iz + 1][iy + 1][ix]
|
|
}
|
|
},
|
|
{
|
|
{
|
|
FXZ[iz][iy][ix + 1],
|
|
FXZ[iz + 1][iy][ix + 1]
|
|
},
|
|
{
|
|
FXZ[iz][iy + 1][ix + 1],
|
|
FXZ[iz + 1][iy + 1][ix + 1]
|
|
}
|
|
}
|
|
};
|
|
|
|
Real GYZ[2][2][2] =
|
|
{
|
|
{
|
|
{
|
|
FYZ[iz][iy][ix],
|
|
FYZ[iz + 1][iy][ix]
|
|
},
|
|
{
|
|
FYZ[iz][iy + 1][ix],
|
|
FYZ[iz + 1][iy + 1][ix]
|
|
}
|
|
},
|
|
{
|
|
{
|
|
FYZ[iz][iy][ix + 1],
|
|
FYZ[iz + 1][iy][ix + 1]
|
|
},
|
|
{
|
|
FYZ[iz][iy + 1][ix + 1],
|
|
FYZ[iz + 1][iy + 1][ix + 1]
|
|
}
|
|
}
|
|
};
|
|
|
|
Real GXYZ[2][2][2] =
|
|
{
|
|
{
|
|
{
|
|
FXYZ[iz][iy][ix],
|
|
FXYZ[iz + 1][iy][ix]
|
|
},
|
|
{
|
|
FXYZ[iz][iy + 1][ix],
|
|
FXYZ[iz + 1][iy + 1][ix]
|
|
}
|
|
},
|
|
{
|
|
{
|
|
FXYZ[iz][iy][ix + 1],
|
|
FXYZ[iz + 1][iy][ix + 1]
|
|
},
|
|
{
|
|
FXYZ[iz][iy + 1][ix + 1],
|
|
FXYZ[iz + 1][iy + 1][ix + 1]
|
|
}
|
|
}
|
|
};
|
|
|
|
Construct(mPoly[iz][iy][ix], G, GX, GY, GZ, GXY, GXZ, GYZ, GXYZ);
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
Real ComputeDerivative(Real const* slope) const
|
|
{
|
|
if (slope[1] != slope[2])
|
|
{
|
|
if (slope[0] != slope[1])
|
|
{
|
|
if (slope[2] != slope[3])
|
|
{
|
|
Real ad0 = std::fabs(slope[3] - slope[2]);
|
|
Real ad1 = std::fabs(slope[0] - slope[1]);
|
|
return (ad0 * slope[1] + ad1 * slope[2]) / (ad0 + ad1);
|
|
}
|
|
else
|
|
{
|
|
return slope[2];
|
|
}
|
|
}
|
|
else
|
|
{
|
|
if (slope[2] != slope[3])
|
|
{
|
|
return slope[1];
|
|
}
|
|
else
|
|
{
|
|
return (Real)0.5 * (slope[1] + slope[2]);
|
|
}
|
|
}
|
|
}
|
|
else
|
|
{
|
|
return slope[1];
|
|
}
|
|
}
|
|
|
|
void Construct(Polynomial& poly,
|
|
Real const F[2][2][2], Real const FX[2][2][2], Real const FY[2][2][2],
|
|
Real const FZ[2][2][2], Real const FXY[2][2][2], Real const FXZ[2][2][2],
|
|
Real const FYZ[2][2][2], Real const FXYZ[2][2][2])
|
|
{
|
|
Real dx = mXSpacing, dy = mYSpacing, dz = mZSpacing;
|
|
Real invDX = (Real)1 / dx, invDX2 = invDX * invDX;
|
|
Real invDY = (Real)1 / dy, invDY2 = invDY * invDY;
|
|
Real invDZ = (Real)1 / dz, invDZ2 = invDZ * invDZ;
|
|
Real b0, b1, b2, b3, b4, b5, b6, b7;
|
|
|
|
poly.A(0, 0, 0) = F[0][0][0];
|
|
poly.A(1, 0, 0) = FX[0][0][0];
|
|
poly.A(0, 1, 0) = FY[0][0][0];
|
|
poly.A(0, 0, 1) = FZ[0][0][0];
|
|
poly.A(1, 1, 0) = FXY[0][0][0];
|
|
poly.A(1, 0, 1) = FXZ[0][0][0];
|
|
poly.A(0, 1, 1) = FYZ[0][0][0];
|
|
poly.A(1, 1, 1) = FXYZ[0][0][0];
|
|
|
|
// solve for Aij0
|
|
b0 = (F[1][0][0] - poly(0, 0, 0, dx, (Real)0, (Real)0)) * invDX2;
|
|
b1 = (FX[1][0][0] - poly(1, 0, 0, dx, (Real)0, (Real)0)) * invDX;
|
|
poly.A(2, 0, 0) = (Real)3 * b0 - b1;
|
|
poly.A(3, 0, 0) = ((Real)-2 * b0 + b1) * invDX;
|
|
|
|
b0 = (F[0][1][0] - poly(0, 0, 0, (Real)0, dy, (Real)0)) * invDY2;
|
|
b1 = (FY[0][1][0] - poly(0, 1, 0, (Real)0, dy, (Real)0)) * invDY;
|
|
poly.A(0, 2, 0) = (Real)3 * b0 - b1;
|
|
poly.A(0, 3, 0) = ((Real)-2 * b0 + b1) * invDY;
|
|
|
|
b0 = (FY[1][0][0] - poly(0, 1, 0, dx, (Real)0, (Real)0)) * invDX2;
|
|
b1 = (FXY[1][0][0] - poly(1, 1, 0, dx, (Real)0, (Real)0)) * invDX;
|
|
poly.A(2, 1, 0) = (Real)3 * b0 - b1;
|
|
poly.A(3, 1, 0) = ((Real)-2 * b0 + b1) * invDX;
|
|
|
|
b0 = (FX[0][1][0] - poly(1, 0, 0, (Real)0, dy, (Real)0)) * invDY2;
|
|
b1 = (FXY[0][1][0] - poly(1, 1, 0, (Real)0, dy, (Real)0)) * invDY;
|
|
poly.A(1, 2, 0) = (Real)3 * b0 - b1;
|
|
poly.A(1, 3, 0) = ((Real)-2 * b0 + b1) * invDY;
|
|
|
|
b0 = (F[1][1][0] - poly(0, 0, 0, dx, dy, (Real)0)) * invDX2 * invDY2;
|
|
b1 = (FX[1][1][0] - poly(1, 0, 0, dx, dy, (Real)0)) * invDX * invDY2;
|
|
b2 = (FY[1][1][0] - poly(0, 1, 0, dx, dy, (Real)0)) * invDX2 * invDY;
|
|
b3 = (FXY[1][1][0] - poly(1, 1, 0, dx, dy, (Real)0)) * invDX * invDY;
|
|
poly.A(2, 2, 0) = (Real)9 * b0 - (Real)3 * b1 - (Real)3 * b2 + b3;
|
|
poly.A(3, 2, 0) = ((Real)-6 * b0 + (Real)3 * b1 + (Real)2 * b2 - b3) * invDX;
|
|
poly.A(2, 3, 0) = ((Real)-6 * b0 + (Real)2 * b1 + (Real)3 * b2 - b3) * invDY;
|
|
poly.A(3, 3, 0) = ((Real)4 * b0 - (Real)2 * b1 - (Real)2 * b2 + b3) * invDX * invDY;
|
|
|
|
// solve for Ai0k
|
|
b0 = (F[0][0][1] - poly(0, 0, 0, (Real)0, (Real)0, dz)) * invDZ2;
|
|
b1 = (FZ[0][0][1] - poly(0, 0, 1, (Real)0, (Real)0, dz)) * invDZ;
|
|
poly.A(0, 0, 2) = (Real)3 * b0 - b1;
|
|
poly.A(0, 0, 3) = ((Real)-2 * b0 + b1) * invDZ;
|
|
|
|
b0 = (FZ[1][0][0] - poly(0, 0, 1, dx, (Real)0, (Real)0)) * invDX2;
|
|
b1 = (FXZ[1][0][0] - poly(1, 0, 1, dx, (Real)0, (Real)0)) * invDX;
|
|
poly.A(2, 0, 1) = (Real)3 * b0 - b1;
|
|
poly.A(3, 0, 1) = ((Real)-2 * b0 + b1) * invDX;
|
|
|
|
b0 = (FX[0][0][1] - poly(1, 0, 0, (Real)0, (Real)0, dz)) * invDZ2;
|
|
b1 = (FXZ[0][0][1] - poly(1, 0, 1, (Real)0, (Real)0, dz)) * invDZ;
|
|
poly.A(1, 0, 2) = (Real)3 * b0 - b1;
|
|
poly.A(1, 0, 3) = ((Real)-2 * b0 + b1) * invDZ;
|
|
|
|
b0 = (F[1][0][1] - poly(0, 0, 0, dx, (Real)0, dz)) * invDX2 * invDZ2;
|
|
b1 = (FX[1][0][1] - poly(1, 0, 0, dx, (Real)0, dz)) * invDX * invDZ2;
|
|
b2 = (FZ[1][0][1] - poly(0, 0, 1, dx, (Real)0, dz)) * invDX2 * invDZ;
|
|
b3 = (FXZ[1][0][1] - poly(1, 0, 1, dx, (Real)0, dz)) * invDX * invDZ;
|
|
poly.A(2, 0, 2) = (Real)9 * b0 - (Real)3 * b1 - (Real)3 * b2 + b3;
|
|
poly.A(3, 0, 2) = ((Real)-6 * b0 + (Real)3 * b1 + (Real)2 * b2 - b3) * invDX;
|
|
poly.A(2, 0, 3) = ((Real)-6 * b0 + (Real)2 * b1 + (Real)3 * b2 - b3) * invDZ;
|
|
poly.A(3, 0, 3) = ((Real)4 * b0 - (Real)2 * b1 - (Real)2 * b2 + b3) * invDX * invDZ;
|
|
|
|
// solve for A0jk
|
|
b0 = (FZ[0][1][0] - poly(0, 0, 1, (Real)0, dy, (Real)0)) * invDY2;
|
|
b1 = (FYZ[0][1][0] - poly(0, 1, 1, (Real)0, dy, (Real)0)) * invDY;
|
|
poly.A(0, 2, 1) = (Real)3 * b0 - b1;
|
|
poly.A(0, 3, 1) = ((Real)-2 * b0 + b1) * invDY;
|
|
|
|
b0 = (FY[0][0][1] - poly(0, 1, 0, (Real)0, (Real)0, dz)) * invDZ2;
|
|
b1 = (FYZ[0][0][1] - poly(0, 1, 1, (Real)0, (Real)0, dz)) * invDZ;
|
|
poly.A(0, 1, 2) = (Real)3 * b0 - b1;
|
|
poly.A(0, 1, 3) = ((Real)-2 * b0 + b1) * invDZ;
|
|
|
|
b0 = (F[0][1][1] - poly(0, 0, 0, (Real)0, dy, dz)) * invDY2 * invDZ2;
|
|
b1 = (FY[0][1][1] - poly(0, 1, 0, (Real)0, dy, dz)) * invDY * invDZ2;
|
|
b2 = (FZ[0][1][1] - poly(0, 0, 1, (Real)0, dy, dz)) * invDY2 * invDZ;
|
|
b3 = (FYZ[0][1][1] - poly(0, 1, 1, (Real)0, dy, dz)) * invDY * invDZ;
|
|
poly.A(0, 2, 2) = (Real)9 * b0 - (Real)3 * b1 - (Real)3 * b2 + b3;
|
|
poly.A(0, 3, 2) = ((Real)-6 * b0 + (Real)3 * b1 + (Real)2 * b2 - b3) * invDY;
|
|
poly.A(0, 2, 3) = ((Real)-6 * b0 + (Real)2 * b1 + (Real)3 * b2 - b3) * invDZ;
|
|
poly.A(0, 3, 3) = ((Real)4 * b0 - (Real)2 * b1 - (Real)2 * b2 + b3) * invDY * invDZ;
|
|
|
|
// solve for Aij1
|
|
b0 = (FYZ[1][0][0] - poly(0, 1, 1, dx, (Real)0, (Real)0)) * invDX2;
|
|
b1 = (FXYZ[1][0][0] - poly(1, 1, 1, dx, (Real)0, (Real)0)) * invDX;
|
|
poly.A(2, 1, 1) = (Real)3 * b0 - b1;
|
|
poly.A(3, 1, 1) = ((Real)-2 * b0 + b1) * invDX;
|
|
|
|
b0 = (FXZ[0][1][0] - poly(1, 0, 1, (Real)0, dy, (Real)0)) * invDY2;
|
|
b1 = (FXYZ[0][1][0] - poly(1, 1, 1, (Real)0, dy, (Real)0)) * invDY;
|
|
poly.A(1, 2, 1) = (Real)3 * b0 - b1;
|
|
poly.A(1, 3, 1) = ((Real)-2 * b0 + b1) * invDY;
|
|
|
|
b0 = (FZ[1][1][0] - poly(0, 0, 1, dx, dy, (Real)0)) * invDX2 * invDY2;
|
|
b1 = (FXZ[1][1][0] - poly(1, 0, 1, dx, dy, (Real)0)) * invDX * invDY2;
|
|
b2 = (FYZ[1][1][0] - poly(0, 1, 1, dx, dy, (Real)0)) * invDX2 * invDY;
|
|
b3 = (FXYZ[1][1][0] - poly(1, 1, 1, dx, dy, (Real)0)) * invDX * invDY;
|
|
poly.A(2, 2, 1) = (Real)9 * b0 - (Real)3 * b1 - (Real)3 * b2 + b3;
|
|
poly.A(3, 2, 1) = ((Real)-6 * b0 + (Real)3 * b1 + (Real)2 * b2 - b3) * invDX;
|
|
poly.A(2, 3, 1) = ((Real)-6 * b0 + (Real)2 * b1 + (Real)3 * b2 - b3) * invDY;
|
|
poly.A(3, 3, 1) = ((Real)4 * b0 - (Real)2 * b1 - (Real)2 * b2 + b3) * invDX * invDY;
|
|
|
|
// solve for Ai1k
|
|
b0 = (FXY[0][0][1] - poly(1, 1, 0, (Real)0, (Real)0, dz)) * invDZ2;
|
|
b1 = (FXYZ[0][0][1] - poly(1, 1, 1, (Real)0, (Real)0, dz)) * invDZ;
|
|
poly.A(1, 1, 2) = (Real)3 * b0 - b1;
|
|
poly.A(1, 1, 3) = ((Real)-2 * b0 + b1) * invDZ;
|
|
|
|
b0 = (FY[1][0][1] - poly(0, 1, 0, dx, (Real)0, dz)) * invDX2 * invDZ2;
|
|
b1 = (FXY[1][0][1] - poly(1, 1, 0, dx, (Real)0, dz)) * invDX * invDZ2;
|
|
b2 = (FYZ[1][0][1] - poly(0, 1, 1, dx, (Real)0, dz)) * invDX2 * invDZ;
|
|
b3 = (FXYZ[1][0][1] - poly(1, 1, 1, dx, (Real)0, dz)) * invDX * invDZ;
|
|
poly.A(2, 1, 2) = (Real)9 * b0 - (Real)3 * b1 - (Real)3 * b2 + b3;
|
|
poly.A(3, 1, 2) = ((Real)-6 * b0 + (Real)3 * b1 + (Real)2 * b2 - b3) * invDX;
|
|
poly.A(2, 1, 3) = ((Real)-6 * b0 + (Real)2 * b1 + (Real)3 * b2 - b3) * invDZ;
|
|
poly.A(3, 1, 3) = ((Real)4 * b0 - (Real)2 * b1 - (Real)2 * b2 + b3) * invDX * invDZ;
|
|
|
|
// solve for A1jk
|
|
b0 = (FX[0][1][1] - poly(1, 0, 0, (Real)0, dy, dz)) * invDY2 * invDZ2;
|
|
b1 = (FXY[0][1][1] - poly(1, 1, 0, (Real)0, dy, dz)) * invDY * invDZ2;
|
|
b2 = (FXZ[0][1][1] - poly(1, 0, 1, (Real)0, dy, dz)) * invDY2 * invDZ;
|
|
b3 = (FXYZ[0][1][1] - poly(1, 1, 1, (Real)0, dy, dz)) * invDY * invDZ;
|
|
poly.A(1, 2, 2) = (Real)9 * b0 - (Real)3 * b1 - (Real)3 * b2 + b3;
|
|
poly.A(1, 3, 2) = ((Real)-6 * b0 + (Real)3 * b1 + (Real)2 * b2 - b3) * invDY;
|
|
poly.A(1, 2, 3) = ((Real)-6 * b0 + (Real)2 * b1 + (Real)3 * b2 - b3) * invDZ;
|
|
poly.A(1, 3, 3) = ((Real)4 * b0 - (Real)2 * b1 - (Real)2 * b2 + b3) * invDY * invDZ;
|
|
|
|
// solve for remaining Aijk with i >= 2, j >= 2, k >= 2
|
|
b0 = (F[1][1][1] - poly(0, 0, 0, dx, dy, dz)) * invDX2 * invDY2 * invDZ2;
|
|
b1 = (FX[1][1][1] - poly(1, 0, 0, dx, dy, dz)) * invDX * invDY2 * invDZ2;
|
|
b2 = (FY[1][1][1] - poly(0, 1, 0, dx, dy, dz)) * invDX2 * invDY * invDZ2;
|
|
b3 = (FZ[1][1][1] - poly(0, 0, 1, dx, dy, dz)) * invDX2 * invDY2 * invDZ;
|
|
b4 = (FXY[1][1][1] - poly(1, 1, 0, dx, dy, dz)) * invDX * invDY * invDZ2;
|
|
b5 = (FXZ[1][1][1] - poly(1, 0, 1, dx, dy, dz)) * invDX * invDY2 * invDZ;
|
|
b6 = (FYZ[1][1][1] - poly(0, 1, 1, dx, dy, dz)) * invDX2 * invDY * invDZ;
|
|
b7 = (FXYZ[1][1][1] - poly(1, 1, 1, dx, dy, dz)) * invDX * invDY * invDZ;
|
|
poly.A(2, 2, 2) = (Real)27 * b0 - (Real)9 * b1 - (Real)9 * b2 -
|
|
(Real)9 * b3 + (Real)3 * b4 + (Real)3 * b5 + (Real)3 * b6 - b7;
|
|
poly.A(3, 2, 2) = ((Real)-18 * b0 + (Real)9 * b1 + (Real)6 * b2 +
|
|
(Real)6 * b3 - (Real)3 * b4 - (Real)3 * b5 - (Real)2 * b6 + b7) * invDX;
|
|
poly.A(2, 3, 2) = ((Real)-18 * b0 + (Real)6 * b1 + (Real)9 * b2 +
|
|
(Real)6 * b3 - (Real)3 * b4 - (Real)2 * b5 - (Real)3 * b6 + b7) * invDY;
|
|
poly.A(2, 2, 3) = ((Real)-18 * b0 + (Real)6 * b1 + (Real)6 * b2 +
|
|
(Real)9 * b3 - (Real)2 * b4 - (Real)3 * b5 - (Real)3 * b6 + b7) * invDZ;
|
|
poly.A(3, 3, 2) = ((Real)12 * b0 - (Real)6 * b1 - (Real)6 * b2 -
|
|
(Real)4 * b3 + (Real)3 * b4 + (Real)2 * b5 + (Real)2 * b6 - b7) *
|
|
invDX * invDY;
|
|
poly.A(3, 2, 3) = ((Real)12 * b0 - (Real)6 * b1 - (Real)4 * b2 -
|
|
(Real)6 * b3 + (Real)2 * b4 + (Real)3 * b5 + (Real)2 * b6 - b7) *
|
|
invDX * invDZ;
|
|
poly.A(2, 3, 3) = ((Real)12 * b0 - (Real)4 * b1 - (Real)6 * b2 -
|
|
(Real)6 * b3 + (Real)2 * b4 + (Real)2 * b5 + (Real)3 * b6 - b7) *
|
|
invDY * invDZ;
|
|
poly.A(3, 3, 3) = ((Real)-8 * b0 + (Real)4 * b1 + (Real)4 * b2 +
|
|
(Real)4 * b3 - (Real)2 * b4 - (Real)2 * b5 - (Real)2 * b6 + b7) *
|
|
invDX * invDY * invDZ;
|
|
}
|
|
|
|
void XLookup(Real x, int& xIndex, Real& dx) const
|
|
{
|
|
for (xIndex = 0; xIndex + 1 < mXBound; ++xIndex)
|
|
{
|
|
if (x < mXMin + mXSpacing * (xIndex + 1))
|
|
{
|
|
dx = x - (mXMin + mXSpacing * xIndex);
|
|
return;
|
|
}
|
|
}
|
|
|
|
--xIndex;
|
|
dx = x - (mXMin + mXSpacing * xIndex);
|
|
}
|
|
|
|
void YLookup(Real y, int& yIndex, Real & dy) const
|
|
{
|
|
for (yIndex = 0; yIndex + 1 < mYBound; ++yIndex)
|
|
{
|
|
if (y < mYMin + mYSpacing * (yIndex + 1))
|
|
{
|
|
dy = y - (mYMin + mYSpacing * yIndex);
|
|
return;
|
|
}
|
|
}
|
|
|
|
--yIndex;
|
|
dy = y - (mYMin + mYSpacing * yIndex);
|
|
}
|
|
|
|
void ZLookup(Real z, int& zIndex, Real & dz) const
|
|
{
|
|
for (zIndex = 0; zIndex + 1 < mZBound; ++zIndex)
|
|
{
|
|
if (z < mZMin + mZSpacing * (zIndex + 1))
|
|
{
|
|
dz = z - (mZMin + mZSpacing * zIndex);
|
|
return;
|
|
}
|
|
}
|
|
|
|
--zIndex;
|
|
dz = z - (mZMin + mZSpacing * zIndex);
|
|
}
|
|
|
|
int mXBound, mYBound, mZBound, mQuantity;
|
|
Real mXMin, mXMax, mXSpacing;
|
|
Real mYMin, mYMax, mYSpacing;
|
|
Real mZMin, mZMax, mZSpacing;
|
|
Real const* mF;
|
|
Array3<Polynomial> mPoly;
|
|
};
|
|
}
|
|
|