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chemPointISAT.H: Corrected documentation

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Henry Weller
2020-07-03 15:45:31 +01:00
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src/thermophysicalModels/chemistryModel/chemistryModel/TDACChemistryModel/tabulation/ISAT/chemPointISAT/chemPointISAT.H
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@ -2,7 +2,7 @@
========= |
\\ / F ield | OpenFOAM: The Open Source CFD Toolbox
\\ / O peration | Website: https://openfoam.org
\\ / A nd | Copyright (C) 2016-2019 OpenFOAM Foundation
\\ / A nd | Copyright (C) 2016-2020 OpenFOAM Foundation
\\/ M anipulation |
-------------------------------------------------------------------------------
License
@ -30,73 +30,76 @@ Description
composition Rphi, the mapping gradient matrix A and the matrix describing
the Ellipsoid Of Accuracy (EOA).
1)When the chemPoint is created the region of accuracy is approximated by
an ellipsoid E centered in 'phi' (obtained with the constant):
E = {x| ||L^T.(x-phi)|| <= 1},
with x a point in the composition space and L^T the transpose of an upper
triangular matrix describing the EOA (see below: "Computation of L" ).
1) When the chemPoint is created the region of accuracy is approximated by
an ellipsoid E centered in 'phi' (obtained with the constant):
E = {x| ||L^T.(x-phi)|| <= 1},
with x a point in the composition space and L^T the transpose of an
upper triangular matrix describing the EOA (see below: "Computation of
L" ).
2)To RETRIEVE the mapping from the chemPoint phi, the query point phiq has to
be in the EOA of phi. It follows that, dphi=phiq-phi and to test if phiq
is in the ellipsoid there are two methods. First, compare r=||dphi|| with
rmin and rmax. If r < rmin, phiq is in the EOA. If r > rmax, phiq is out of
the EOA. This operations is O(completeSpaceSize) and is performed first.
If rmin < r < rmax, then the second method is used:
||L^T.dphi|| <= 1 to be in the EOA.
2) To RETRIEVE the mapping from the chemPoint phi, the query point phiq has
to be in the EOA of phi. It follows that, dphi=phiq-phi and to test if
phiq is in the ellipsoid there are two methods. First, compare
r=||dphi|| with rmin and rmax. If r < rmin, phiq is in the EOA. If r >
rmax, phiq is out of the EOA. This operations is O(completeSpaceSize)
and is performed first. If rmin < r < rmax, then the second method is
used:
||L^T.dphi|| <= 1 to be in the EOA.
If phiq is in the EOA, Rphiq is obtained by linear interpolation:
Rphiq= Rphi + A.dphi.
If phiq is in the EOA, Rphiq is obtained by linear interpolation:
Rphiq= Rphi + A.dphi.
3) If phiq is not in the EOA, then the mapping is computed. But as the EOA
is a conservative approximation of the region of accuracy surrounding
the point phi, we could expand it by comparing the computed results with
the one obtained by linear interpolation. The error epsGrow is
calculated:
epsGrow = ||B.(dR - dRl)||,
with dR = Rphiq - Rphi, dRl = A.dphi and B the diagonal scale factor
matrix. If epsGrow <= tolerance, the EOA is too conservative and a GROW
is performed otherwise, the newly computed mapping is associated to the
initial composition and added to the tree.
3)If phiq is not in the EOA, then the mapping is computed. But as the EOA
is a conservative approximation of the region of accuracy surrounding the
point phi, we could expand it by comparing the computed results with the
one obtained by linear interpolation. The error epsGrow is calculated:
epsGrow = ||B.(dR - dRl)||,
with dR = Rphiq - Rphi, dRl = A.dphi and B the diagonal scale factor
matrix.
If epsGrow <= tolerance, the EOA is too conservative and a GROW is performed
otherwise, the newly computed mapping is associated to the initial
composition and added to the tree.
4)To GROW the EOA, we expand it to include the previous EOA and the query
point phiq. The rank-one matrix method is used. The EOA is transformed
to a hypersphere centered at the origin. Then it is expanded to include
the transformed point phiq' on its boundary. Then the inverse transformation
give the modified matrix L' (see below: "Grow the EOA").
4) To GROW the EOA, we expand it to include the previous EOA and the query
point phiq. The rank-one matrix method is used. The EOA is transformed
to a hypersphere centered at the origin. Then it is expanded to include
the transformed point phiq' on its boundary. Then the inverse
transformation give the modified matrix L' (see below: "Grow the EOA").
Computation of L :
In [1], the EOA of the constant approximation is given by
Computation of L :
In [1], the EOA of the constant approximation is given by
E = {x| ||B.A/tolerance.(x-phi)|| <= 1},
with B a scale factor diagonal matrix, A the mapping gradient matrix and
tolerance the absolute tolerance. If we take the QR decomposition of
(B.A)/tolerance= Q.R, with Q an orthogonal matrix and R an upper triangular
matrix such that the EOA is described by
(phiq-phi0)^T.R^T.R.(phiq-phi0) <= 1
L^T = R, both Cholesky decomposition of A^T.B^T.B.A/tolerance^2
This representation of the ellipsoid is used in [2] and in order to avoid
large value of semi-axe length in certain direction, a Singular Value
Decomposition (SVD) is performed on the L matrix:
L = UDV^T,
with the orthogonal matrix U giving the directions of the principal axes
and 1/di the inverse of the element of the diagonal matrix D giving the
length of the principal semi-axes. To avoid very large value of those
length,
di' = max(di, 1/(alphaEOA*sqrt(tolerance))), with alphaEOA = 0.1 (see [2])
di' = max(di, 1/2), see [1]. The latter will be used in this implementation.
And L' = UD'V^T, with D' the diagonal matrix with the modified di'.
Grow the EOA :
More details about the minimum-volume ellipsoid covering an ellipsoid E and
a point p are found in [3]. Here is the main steps to obtain the modified
matrix L' describing the new ellipsoid.
1) calculate the point p' in the transformed space :
p' = L^T.(p-phi)
2) compute the rank-one decomposition:
G = I + gamma.p'.p'^T,
with gamma = (1/|p'|-1)*1/|p'|^2
3) compute L':
L' = L.G.
with B a scale factor diagonal matrix, A the mapping gradient matrix and
tolerance the absolute tolerance. If we take the QR decomposition of
(B.A)/tolerance= Q.R, with Q an orthogonal matrix and R an upper
triangular matrix such that the EOA is described by
(phiq-phi0)^T.R^T.R.(phiq-phi0) <= 1
L^T = R, both Cholesky decomposition of A^T.B^T.B.A/tolerance^2
This representation of the ellipsoid is used in [2] and in order to avoid
large value of semi-axe length in certain direction, a Singular Value
Decomposition (SVD) is performed on the L matrix:
L = UDV^T,
with the orthogonal matrix U giving the directions of the principal axes
and 1/di the inverse of the element of the diagonal matrix D giving the
length of the principal semi-axes. To avoid very large value of those
length, di' = max(di, 1/(alphaEOA*sqrt(tolerance))), with alphaEOA = 0.1
(see [2]) di' = max(di, 1/2), see [1]. The latter will be used in this
implementation. And L' = UD'V^T, with D' the diagonal matrix with the
modified di'.
Grow the EOA :
More details about the minimum-volume ellipsoid covering an ellipsoid E
and a point p are found in [3]. Here is the main steps to obtain the
modified matrix L' describing the new ellipsoid.
1) calculate the point p' in the transformed space :
p' = L^T.(p-phi)
2) compute the rank-one decomposition:
G = I + gamma.p'.p'^T,
with gamma = (1/|p'|-1)*1/|p'|^2
3) compute L':
L' = L.G.
References:
\verbatim