Mathematical library, including random number generation and tensor representations. More...

Data Types
interface	math_eye

Functions/Subroutines
subroutine	math_init
	initialization of random seed generator and internal checks More...

recursive subroutine	math_sort (a, istart, iend, sortDim)
	Quicksort algorithm for two-dimensional integer arrays. More...

pure real(preal) function, dimension(sum(how))	math_expand (what, how)
	vector expansion More...

pure integer function, dimension(n)	math_range (N)
	range of integers starting at one More...

pure real(preal) function, dimension(d, d)	math_identity2nd (d)
	second rank identity tensor of specified dimension More...

pure real(preal) function, dimension(d, d, d, d)	math_identity4th (d)
	symmetric fourth rank identity tensor of specified dimension More...

real(preal) pure function	math_levicivita (i, j, k)
	permutation tensor e_ijk More...

real(preal) pure function	math_delta (i, j)
	kronecker delta function d_ij More...

pure real(preal) function, dimension(3)	math_cross (A, B)
	cross product a x b More...

pure real(preal) function, dimension(size(a, 1), size(b, 1))	math_outer (A, B)
	outer product of arbitrary sized vectors (A ⊗ B / i,j) More...

real(preal) pure function	math_inner (A, B)
	inner product of arbitrary sized vectors (A · B / i,i) More...

real(preal) pure function	math_tensordot (A, B)
	double contraction of 3x3 matrices (A : B / ij,ij) More...

pure real(preal) function, dimension(3, 3)	math_mul3333xx33 (A, B)
	matrix double contraction 3333x33 = 33 (ijkl,kl) More...

pure real(preal) function, dimension(3, 3, 3, 3)	math_mul3333xx3333 (A, B)
	matrix multiplication 3333x3333 = 3333 (ijkl,klmn) More...

pure real(preal) function, dimension(3, 3)	math_exp33 (A, n)
	3x3 matrix exponential up to series approximation order n (default 5) More...

pure real(preal) function, dimension(3, 3)	math_inv33 (A)
	Cramer inversion of 3x3 matrix (function) More...

pure subroutine	math_invert33 (InvA, DetA, error, A)
	Cramer inversion of 3x3 matrix (subroutine) More...

real(preal) function, dimension(3, 3, 3, 3)	math_invsym3333 (A)
	Inversion of symmetriced 3x3x3x3 matrix. More...

subroutine	math_invert (InvA, error, A)
	invert quadratic matrix of arbitrary dimension More...

pure real(preal) function, dimension(3, 3)	math_symmetric33 (m)
	symmetrize a 3x3 matrix More...

pure real(preal) function, dimension(6, 6)	math_symmetric66 (m)
	symmetrize a 6x6 matrix More...

pure real(preal) function, dimension(3, 3)	math_skew33 (m)
	skew part of a 3x3 matrix More...

pure real(preal) function, dimension(3, 3)	math_spherical33 (m)
	hydrostatic part of a 3x3 matrix More...

pure real(preal) function, dimension(3, 3)	math_deviatoric33 (m)
	deviatoric part of a 3x3 matrix More...

real(preal) pure function	math_trace33 (m)
	trace of a 3x3 matrix More...

real(preal) pure function	math_det33 (m)
	determinant of a 3x3 matrix More...

real(preal) pure function	math_detsym33 (m)
	determinant of a symmetric 3x3 matrix More...

pure real(preal) function, dimension(9)	math_33to9 (m33)
	convert 3x3 matrix into vector 9 More...

pure real(preal) function, dimension(3, 3)	math_9to33 (v9)
	convert 9 vector into 3x3 matrix More...

pure real(preal) function, dimension(6)	math_sym33to6 (m33, weighted)
	convert symmetric 3x3 matrix into 6 vector More...

pure real(preal) function, dimension(3, 3)	math_6tosym33 (v6, weighted)
	convert 6 vector into symmetric 3x3 matrix More...

pure real(preal) function, dimension(9, 9)	math_3333to99 (m3333)
	convert 3x3x3x3 matrix into 9x9 matrix More...

pure real(preal) function, dimension(3, 3, 3, 3)	math_99to3333 (m99)
	convert 9x9 matrix into 3x3x3x3 matrix More...

pure real(preal) function, dimension(6, 6)	math_sym3333to66 (m3333, weighted)
	convert symmetric 3x3x3x3 matrix into 6x6 matrix More...

pure real(preal) function, dimension(3, 3, 3, 3)	math_66tosym3333 (m66, weighted)
	convert 66 matrix into symmetric 3x3x3x3 matrix More...

pure real(preal) function, dimension(3, 3, 3, 3)	math_voigt66to3333 (m66)
	convert 66 Voigt matrix into symmetric 3x3x3x3 matrix More...

real(preal) function	math_samplegaussvar (meanvalue, stddev, width)
	draw a random sample from Gauss variable More...

subroutine	math_eigh (m, w, v, error)
	eigenvalues and eigenvectors of symmetric matrix More...

subroutine	math_eigh33 (m, w, v)
	eigenvalues and eigenvectors of symmetric 3x3 matrix using an analytical expression and the general LAPACK powered version for arbritrary sized matrices as fallback More...

real(preal) function, dimension(3, 3)	math_rotationalpart (m)
	rotational part from polar decomposition of 3x3 tensor More...

real(preal) function, dimension(size(m, 1))	math_eigvalsh (m)
	Eigenvalues of symmetric matrix. More...

real(preal) function, dimension(3)	math_eigvalsh33 (m)
	eigenvalues of symmetric 3x3 matrix using an analytical expression More...

pure real(preal) function, dimension(3)	math_invariantssym33 (m)
	invariants of symmetrix 3x3 matrix More...

integer pure function	math_factorial (n)
	factorial More...

integer pure function	math_binomial (n, k)
	binomial coefficient More...

integer pure function	math_multinomial (alpha)
	multinomial coefficient More...

real(preal) pure function	math_voltetrahedron (v1, v2, v3, v4)
	volume of tetrahedron given by four vertices More...

real(preal) pure function	math_areatriangle (v1, v2, v3)
	area of triangle given by three vertices More...

real(preal) pure elemental function	math_clip (a, left, right)
	limits a scalar value to a certain range (either one or two sided) More...

subroutine, private	unittest
	check correctness of some math functions More...

Variables
real(preal), parameter	pi = acos(-1.0_pReal)
	ratio of a circle's circumference to its diameter More...

real(preal), parameter	indeg = 180.0_pReal/PI
	conversion from radian into degree More...

real(preal), parameter	inrad = PI/180.0_pReal
	conversion from degree into radian More...

complex(preal), parameter	twopiimg = cmplx(0.0_pReal, 2.0_pReal*PI)
	Re(0.0), Im(2xPi) More...

real(preal), dimension(3, 3), parameter	math_i3 = reshape([ 1.0_pReal,0.0_pReal,0.0_pReal, 0.0_pReal,1.0_pReal,0.0_pReal, 0.0_pReal,0.0_pReal,1.0_pReal ], [3,3])
	3x3 Identity More...

real(preal), dimension(6), parameter, private	nrmmandel = [ 1.0_pReal, 1.0_pReal, 1.0_pReal, sqrt(2.0_pReal), sqrt(2.0_pReal), sqrt(2.0_pReal) ]
	forward weighting for Mandel notation More...

real(preal), dimension(6), parameter, private	invnrmmandel = 1.0_pReal/NRMMANDEL
	backward weighting for Mandel notation More...

integer, dimension(2, 6), parameter, private	mapnye = reshape([ 1,1, 2,2, 3,3, 1,2, 2,3, 1,3 ], [2,6])
	arrangement in Nye notation. More...

integer, dimension(2, 6), parameter, private	mapvoigt = reshape([ 1,1, 2,2, 3,3, 2,3, 1,3, 1,2 ], [2,6])
	arrangement in Voigt notation More...

integer, dimension(2, 9), parameter, private	mapplain = reshape([ 1,1, 1,2, 1,3, 2,1, 2,2, 2,3, 3,1, 3,2, 3,3 ], [2,9])
	arrangement in Plain notation More...

Detailed Description

Mathematical library, including random number generation and tensor representations.

Author: Franz Roters, Max-Planck-Institut für Eisenforschung GmbH; Philip Eisenlohr, Max-Planck-Institut für Eisenforschung GmbH; Christoph Kords, Max-Planck-Institut für Eisenforschung GmbH; Martin Diehl, Max-Planck-Institut für Eisenforschung GmbH

Function/Subroutine Documentation

◆ math_3333to99()

pure real(preal) function, dimension(9,9) math::math_3333to99 ( real(preal), dimension(3,3,3,3), intent(in) m3333 )

convert 3x3x3x3 matrix into 9x9 matrix

Definition at line 741 of file math.f90.

References mapplain.

Referenced by crystallite::crystallite_stresstangent(), crystallite::integratestress(), unittest(), and spectral_utilities::utilities_maskedcompliance().

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◆ math_33to9()

pure real(preal) function, dimension(9) math::math_33to9 ( real(preal), dimension(3,3), intent(in) m33 )

convert 3x3 matrix into vector 9

Definition at line 650 of file math.f90.

References mapplain.

Referenced by crystallite::integratestress(), and unittest().

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◆ math_66tosym3333()

pure real(preal) function, dimension(3,3,3,3) math::math_66tosym3333	(	real(preal), dimension(6,6), intent(in)	m66,
		logical, intent(in), optional	weighted
	)

convert 66 matrix into symmetric 3x3x3x3 matrix

Weighted conversion (default) rearranges according to Nye and weights shear

Parameters

[in]	m66	6x6 matrix
[in]	weighted	weight according to Mandel (.true. by default)

Definition at line 806 of file math.f90.

References invnrmmandel, and mapnye.

Referenced by constitutive::constitutive_hooke_sanditstangents(), math_invsym3333(), rotations::rottensor4sym(), and unittest().

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◆ math_6tosym33()

pure real(preal) function, dimension(3,3) math::math_6tosym33	(	real(preal), dimension(6), intent(in)	v6,
		logical, intent(in), optional	weighted
	)

convert 6 vector into symmetric 3x3 matrix

Weighted conversion (default) rearranges according to Nye and weights shear

Parameters

[in]	v6	6 vector
[in]	weighted	weight according to Mandel (.true. by default)

Definition at line 715 of file math.f90.

References invnrmmandel, and mapnye.

Referenced by unittest().

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◆ math_99to3333()

pure real(preal) function, dimension(3,3,3,3) math::math_99to3333 ( real(preal), dimension(9,9), intent(in) m99 )

convert 9x9 matrix into 3x3x3x3 matrix

Definition at line 758 of file math.f90.

References mapplain.

Referenced by crystallite::crystallite_stresstangent(), unittest(), and spectral_utilities::utilities_maskedcompliance().

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◆ math_9to33()

pure real(preal) function, dimension(3,3) math::math_9to33 ( real(preal), dimension(9), intent(in) v9 )

convert 9 vector into 3x3 matrix

Definition at line 667 of file math.f90.

References mapplain.

Referenced by damask_spectral(), crystallite::integratestress(), and unittest().

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◆ math_areatriangle()

real(preal) pure function math::math_areatriangle	(	real(preal), dimension (3), intent(in)	v1,
		real(preal), dimension (3), intent(in)	v2,
		real(preal), dimension (3), intent(in)	v3
	)

area of triangle given by three vertices

Definition at line 1184 of file math.f90.

References math_cross().

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◆ math_binomial()

integer pure function math::math_binomial	(	integer, intent(in)	n,
		integer, intent(in)	k
	)

binomial coefficient

Definition at line 1132 of file math.f90.

Referenced by math_multinomial(), and unittest().

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◆ math_clip()

real(preal) pure elemental function math::math_clip	(	real(preal), intent(in)	a,
		real(preal), intent(in), optional	left,
		real(preal), intent(in), optional	right
	)

limits a scalar value to a certain range (either one or two sided)

Definition at line 1197 of file math.f90.

Referenced by eigenvectorbasis(), lambert::lambert_balltocube(), math_eigvalsh33(), rotations::om2ax(), rotations::qu2ax(), rotations::qu2ho(), rotations::qu2ro(), and unittest().

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◆ math_cross()

pure real(preal) function, dimension(3) math::math_cross	(	real(preal), dimension(3), intent(in)	A,
		real(preal), dimension(3), intent(in)	B
	)

cross product a x b

Definition at line 312 of file math.f90.

Referenced by lattice::buildcoordinatesystem(), lattice::buildtransformationsystem(), lattice::lattice_nonschmidmatrix(), math_areatriangle(), math_eigh33(), and unittest().

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◆ math_delta()

real(preal) pure function math::math_delta	(	integer, intent(in)	i,
		integer, intent(in)	j
	)

kronecker delta function d_ij

Definition at line 300 of file math.f90.

◆ math_det33()

real(preal) pure function math::math_det33 ( real(preal), dimension(3,3), intent(in) m )

determinant of a 3x3 matrix

Definition at line 623 of file math.f90.

Referenced by constitutive::constitutive_lianditstangents(), crystallite::crystallite_init(), rotations::frommatrix(), crystallite::integratestress(), math_voltetrahedron(), unittest(), spectral_utilities::utilities_fouriergammaconvolution(), and spectral_utilities::utilities_updategamma().

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◆ math_detsym33()

real(preal) pure function math::math_detsym33 ( real(preal), dimension(3,3), intent(in) m )

determinant of a symmetric 3x3 matrix

Definition at line 637 of file math.f90.

Referenced by math_invariantssym33(), and unittest().

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◆ math_deviatoric33()

pure real(preal) function, dimension(3,3) math::math_deviatoric33 ( real(preal), dimension(3,3), intent(in) m )

deviatoric part of a 3x3 matrix

Definition at line 598 of file math.f90.

References math_spherical33().

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◆ math_eigh()

subroutine math::math_eigh	(	real(preal), dimension(:,:), intent(in)	m,
		real(preal), dimension(size(m,1)), intent(out)	w,
		real(preal), dimension(size(m,1),size(m,1)), intent(out)	v,
		logical, intent(out)	error
	)

eigenvalues and eigenvectors of symmetric matrix

Parameters

[in]	m	quadratic matrix to compute eigenvectors and values of
[out]	w	eigenvalues
[out]	v	eigenvectors

Definition at line 888 of file math.f90.

Referenced by math_eigh33().

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◆ math_eigh33()

subroutine math::math_eigh33	(	real(preal), dimension(3,3), intent(in)	m,
		real(preal), dimension(3), intent(out)	w,
		real(preal), dimension(3,3), intent(out)	v
	)

eigenvalues and eigenvectors of symmetric 3x3 matrix using an analytical expression and the general LAPACK powered version for arbritrary sized matrices as fallback

Author: Joachim Kopp, Max-Planck-Institut für Kernphysik, Heidelberg (Copyright (C) 2006); Martin Diehl, Max-Planck-Institut für Eisenforschung GmbH

See http://arxiv.org/abs/physics/0610206 (DSYEVH3)

Parameters

[in]	m	3x3 matrix to compute eigenvectors and values of
[out]	w	eigenvalues
[out]	v	eigenvectors

Definition at line 915 of file math.f90.

References math_cross(), math_eigh(), and math_eigvalsh33().

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◆ math_eigvalsh()

real(preal) function, dimension(size(m,1)) math::math_eigvalsh ( real(preal), dimension(:,:), intent(in) m )

Eigenvalues of symmetric matrix.

Parameters

[in] m symmetric matrix to compute eigenvalues of

Definition at line 1046 of file math.f90.

Referenced by math_eigvalsh33().

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◆ math_eigvalsh33()

real(preal) function, dimension(3) math::math_eigvalsh33 ( real(preal), dimension(3,3), intent(in) m )

eigenvalues of symmetric 3x3 matrix using an analytical expression

Author: Martin Diehl, Max-Planck-Institut für Eisenforschung GmbH

similar to http://arxiv.org/abs/physics/0610206 (DSYEVC3) but apparently more stable solution and has general LAPACK powered version for arbritrary sized matrices as fallback

Parameters

[in] m 3x3 symmetric matrix to compute eigenvalues of

Definition at line 1071 of file math.f90.

References math_clip(), math_eigvalsh(), math_invariantssym33(), and pi.

Referenced by math_eigh33().

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◆ math_exp33()

pure real(preal) function, dimension(3,3) math::math_exp33	(	real(preal), dimension(3,3), intent(in)	A,
		integer, intent(in), optional	n
	)

3x3 matrix exponential up to series approximation order n (default 5)

Definition at line 402 of file math.f90.

References math_i3.

Referenced by unittest().

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◆ math_expand()

pure real(preal) function, dimension(sum(how)) math::math_expand	(	real(preal), dimension(:), intent(in)	what,
		integer, dimension(:), intent(in)	how
	)

vector expansion

takes a set of numbers (a,b,c,...) and corresponding multiples (x,y,z,...) to return a vector of x times a, y times b, z times c, ...

Definition at line 207 of file math.f90.

Referenced by kinematics_cleavage_opening::kinematics_cleavage_opening_init(), kinematics_slipplane_opening::kinematics_slipplane_opening_init(), source_damage_anisobrittle::source_damage_anisobrittle_init(), source_damage_anisoductile::source_damage_anisoductile_init(), and unittest().

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◆ math_factorial()

integer pure function math::math_factorial ( integer, intent(in) n )

factorial

Definition at line 1120 of file math.f90.

References math_range().

Referenced by unittest().

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◆ math_identity2nd()

pure real(preal) function, dimension(d,d) math::math_identity2nd ( integer, intent(in) d )

second rank identity tensor of specified dimension

Parameters

[in] d tensor dimension

Definition at line 240 of file math.f90.

Referenced by crystallite::crystallite_stresstangent(), crystallite::integratestress(), math_identity4th(), unittest(), and spectral_utilities::utilities_maskedcompliance().

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◆ math_identity4th()

pure real(preal) function, dimension(d,d,d,d) math::math_identity4th ( integer, intent(in) d )

symmetric fourth rank identity tensor of specified dimension

Parameters

[in] d tensor dimension

Definition at line 258 of file math.f90.

References math_identity2nd().

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◆ math_init()

subroutine math::math_init

initialization of random seed generator and internal checks

Definition at line 95 of file math.f90.

References numerics::randomseed, and io::unittest().

Referenced by cpfem2::cpfem_initall().

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◆ math_inner()

real(preal) pure function math::math_inner	(	real(preal), dimension(:), intent(in)	A,
		real(preal), dimension(size(a,1)), intent(in)	B
	)

inner product of arbitrary sized vectors (A · B / i,i)

Definition at line 343 of file math.f90.

Referenced by lattice::slipprojection_direction(), and lattice::slipprojection_transverse().

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◆ math_inv33()

pure real(preal) function, dimension(3,3) math::math_inv33 ( real(preal), dimension(3,3), intent(in) A )

Cramer inversion of 3x3 matrix (function)

Direct Cramer inversion of matrix A. Returns all zeroes if not possible, i.e.

Definition at line 435 of file math.f90.

References math_invert33().

Referenced by constitutive::constitutive_lianditstangents(), crystallite::crystallite_init(), crystallite::crystallite_push33toref(), crystallite::crystallite_stress(), crystallite::crystallite_stresstangent(), crystallite::integratestress(), math_rotationalpart(), and unittest().

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◆ math_invariantssym33()

pure real(preal) function, dimension(3) math::math_invariantssym33 ( real(preal), dimension(3,3), intent(in) m )

invariants of symmetrix 3x3 matrix

Definition at line 1104 of file math.f90.

References math_detsym33(), and math_trace33().

Referenced by eigenvectorbasis(), and math_eigvalsh33().

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◆ math_invert()

subroutine math::math_invert	(	real(preal), dimension(size(a,1),size(a,1)), intent(out)	InvA,
		logical, intent(out)	error,
		real(preal), dimension(:,:), intent(in)	A
	)

invert quadratic matrix of arbitrary dimension

Definition at line 521 of file math.f90.

Referenced by crystallite::crystallite_stresstangent(), lattice::equivalent_mu(), lattice::equivalent_nu(), unittest(), spectral_utilities::utilities_fouriergammaconvolution(), spectral_utilities::utilities_maskedcompliance(), and spectral_utilities::utilities_updategamma().

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◆ math_invert33()

pure subroutine math::math_invert33	(	real(preal), dimension(3,3), intent(out)	InvA,
		real(preal), intent(out)	DetA,
		logical, intent(out)	error,
		real(preal), dimension(3,3), intent(in)	A
	)

Cramer inversion of 3x3 matrix (subroutine)

Direct Cramer inversion of matrix A. Also returns determinant

Definition at line 454 of file math.f90.

Referenced by crystallite::integratestress(), math_inv33(), and unittest().

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◆ math_invsym3333()

real(preal) function, dimension(3,3,3,3) math::math_invsym3333 ( real(preal), dimension(3,3,3,3), intent(in) A )

Inversion of symmetriced 3x3x3x3 matrix.

Definition at line 489 of file math.f90.

References math_66tosym3333(), and math_sym3333to66().

Referenced by grid_mech_spectral_polarisation::formresidual(), grid_mech_spectral_polarisation::grid_mech_spectral_polarisation_init(), and grid_mech_spectral_polarisation::grid_mech_spectral_polarisation_solution().

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◆ math_levicivita()

real(preal) pure function math::math_levicivita	(	integer, intent(in)	i,
		integer, intent(in)	j,
		integer, intent(in)	k
	)

permutation tensor e_ijk

Definition at line 280 of file math.f90.

Referenced by unittest().

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◆ math_mul3333xx33()

pure real(preal) function, dimension(3,3) math::math_mul3333xx33	(	real(preal), dimension(3,3,3,3), intent(in)	A,
		real(preal), dimension(3,3), intent(in)	B
	)

matrix double contraction 3333x33 = 33 (ijkl,kl)

Definition at line 368 of file math.f90.

Referenced by constitutive::constitutive_hooke_sanditstangents(), grid_mech_spectral_basic::formresidual(), grid_mech_fem::formresidual(), grid_mech_spectral_polarisation::formresidual(), and grid_mech_spectral_polarisation::grid_mech_spectral_polarisation_forward().

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◆ math_mul3333xx3333()

pure real(preal) function, dimension(3,3,3,3) math::math_mul3333xx3333	(	real(preal), dimension(3,3,3,3), intent(in)	A,
		real(preal), dimension(3,3,3,3), intent(in)	B
	)

matrix multiplication 3333x3333 = 3333 (ijkl,klmn)

Definition at line 385 of file math.f90.

Referenced by crystallite::crystallite_stresstangent(), and crystallite::integratestress().

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◆ math_multinomial()

integer pure function math::math_multinomial ( integer, dimension(:), intent(in) alpha )

multinomial coefficient

Definition at line 1151 of file math.f90.

References math_binomial().

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◆ math_outer()

pure real(preal) function, dimension(size(a,1),size(b,1)) math::math_outer	(	real(preal), dimension(:), intent(in)	A,
		real(preal), dimension(:), intent(in)	B
	)

outer product of arbitrary sized vectors (A ⊗ B / i,j)

Definition at line 327 of file math.f90.

Referenced by lattice::buildtransformationsystem(), kinematics_slipplane_opening::kinematics_slipplane_opening_init(), lattice::lattice_nonschmidmatrix(), lattice::lattice_schmidmatrix_cleavage(), lattice::lattice_schmidmatrix_slip(), and lattice::lattice_schmidmatrix_twin().

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◆ math_range()

pure integer function, dimension(n) math::math_range ( integer, intent(in) N )

range of integers starting at one

Parameters

[in] n length of range

Definition at line 226 of file math.f90.

Referenced by math_factorial(), and unittest().

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◆ math_rotationalpart()

real(preal) function, dimension(3,3) math::math_rotationalpart ( real(preal), dimension(3,3), intent(in) m )

rotational part from polar decomposition of 3x3 tensor

Definition at line 962 of file math.f90.

References eigenvectorbasis(), math_i3, and math_inv33().

Referenced by crystallite::crystallite_orientations(), and unittest().

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◆ math_samplegaussvar()

real(preal) function math::math_samplegaussvar	(	real(preal), intent(in)	meanvalue,
		real(preal), intent(in)	stddev,
		real(preal), intent(in), optional	width
	)

draw a random sample from Gauss variable

Parameters

[in]	stddev	standard deviation of gauss distribution
[in]	width	width of considered values as multiples of standard deviation

Parameters

meanvalue meanvalue of gauss distribution

Definition at line 853 of file math.f90.

◆ math_skew33()

pure real(preal) function, dimension(3,3) math::math_skew33 ( real(preal), dimension(3,3), intent(in) m )

skew part of a 3x3 matrix

Definition at line 572 of file math.f90.

References math_symmetric33().

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◆ math_sort()

recursive subroutine math::math_sort	(	integer, dimension(:,:), intent(inout)	a,
		integer, intent(in), optional	istart,
		integer, intent(in), optional	iend,
		integer, intent(in), optional	sortDim
	)

Quicksort algorithm for two-dimensional integer arrays.

Definition at line 132 of file math.f90.

References qsort_partition().

Referenced by unittest().

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◆ math_spherical33()

pure real(preal) function, dimension(3,3) math::math_spherical33 ( real(preal), dimension(3,3), intent(in) m )

hydrostatic part of a 3x3 matrix

Definition at line 585 of file math.f90.

References math_i3, and math_trace33().

Referenced by math_deviatoric33().

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◆ math_sym3333to66()

pure real(preal) function, dimension(6,6) math::math_sym3333to66	(	real(preal), dimension(3,3,3,3), intent(in)	m3333,
		logical, intent(in), optional	weighted
	)

convert symmetric 3x3x3x3 matrix into 6x6 matrix

Weighted conversion (default) rearranges according to Nye and weights shear

Parameters

[in]	m3333	symmetric 3x3x3x3 matrix (no internal check)
[in]	weighted	weight according to Mandel (.true. by default)

Definition at line 778 of file math.f90.

References mapnye, and nrmmandel.

Referenced by lattice::lattice_init(), math_invsym3333(), rotations::rottensor4sym(), and unittest().

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◆ math_sym33to6()

pure real(preal) function, dimension(6) math::math_sym33to6	(	real(preal), dimension(3,3), intent(in)	m33,
		logical, intent(in), optional	weighted
	)

convert symmetric 3x3 matrix into 6 vector

Weighted conversion (default) rearranges according to Nye and weights shear

Parameters

[in]	m33	symmetric 3x3 matrix (no internal check)
[in]	weighted	weight according to Mandel (.true. by default)

Definition at line 687 of file math.f90.

References mapnye, and nrmmandel.

Referenced by unittest().

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◆ math_symmetric33()

pure real(preal) function, dimension(3,3) math::math_symmetric33 ( real(preal), dimension(3,3), intent(in) m )

symmetrize a 3x3 matrix

Definition at line 546 of file math.f90.

Referenced by math_skew33(), and unittest().

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◆ math_symmetric66()

pure real(preal) function, dimension(6,6) math::math_symmetric66 ( real(preal), dimension(6,6), intent(in) m )

symmetrize a 6x6 matrix

Definition at line 559 of file math.f90.

◆ math_tensordot()

real(preal) pure function math::math_tensordot	(	real(preal), dimension(3,3), intent(in)	A,
		real(preal), dimension(3,3), intent(in)	B
	)

double contraction of 3x3 matrices (A : B / ij,ij)

Definition at line 356 of file math.f90.

◆ math_trace33()

real(preal) pure function math::math_trace33 ( real(preal), dimension(3,3), intent(in) m )

trace of a 3x3 matrix

Definition at line 611 of file math.f90.

Referenced by lattice::lattice_schmidmatrix_slip(), lattice::lattice_schmidmatrix_twin(), math_invariantssym33(), math_spherical33(), and rotations::om2ax().

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◆ math_voigt66to3333()

pure real(preal) function, dimension(3,3,3,3) math::math_voigt66to3333 ( real(preal), dimension(6,6), intent(in) m66 )

convert 66 Voigt matrix into symmetric 3x3x3x3 matrix

Parameters

[in] m66 6x6 matrix

Definition at line 834 of file math.f90.

References mapvoigt.

Referenced by lattice::lattice_init().

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◆ math_voltetrahedron()

real(preal) pure function math::math_voltetrahedron	(	real(preal), dimension (3), intent(in)	v1,
		real(preal), dimension (3), intent(in)	v2,
		real(preal), dimension (3), intent(in)	v3,
		real(preal), dimension (3), intent(in)	v4
	)

volume of tetrahedron given by four vertices

Definition at line 1167 of file math.f90.

References math_det33().

Referenced by unittest().

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◆ unittest()

subroutine, private math::unittest

private

check correctness of some math functions

Definition at line 1214 of file math.f90.

References math_3333to99(), math_33to9(), math_66tosym3333(), math_6tosym33(), math_99to3333(), math_9to33(), math_binomial(), math_clip(), math_cross(), math_det33(), math_detsym33(), math_exp33(), math_expand(), math_factorial(), math_i3, math_identity2nd(), math_inv33(), math_invert(), math_invert33(), math_levicivita(), math_range(), math_rotationalpart(), math_sort(), math_sym3333to66(), math_sym33to6(), math_symmetric33(), and math_voltetrahedron().

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Variable Documentation

◆ indeg

real(preal), parameter math::indeg = 180.0_pReal/PI

conversion from radian into degree

Definition at line 28 of file math.f90.

◆ inrad

real(preal), parameter math::inrad = PI/180.0_pReal

conversion from degree into radian

Definition at line 29 of file math.f90.

Referenced by rotations::fromaxisangle(), and rotations::fromeulers().

◆ invnrmmandel

real(preal), dimension(6), parameter, private math::invnrmmandel = 1.0_pReal/NRMMANDEL

private

backward weighting for Mandel notation

Definition at line 44 of file math.f90.

Referenced by math_66tosym3333(), and math_6tosym33().

◆ mapnye

integer, dimension (2,6), parameter, private math::mapnye = reshape([ 1,1, 2,2, 3,3, 1,2, 2,3, 1,3 ], [2,6])

private

arrangement in Nye notation.

Definition at line 47 of file math.f90.

Referenced by math_66tosym3333(), math_6tosym33(), math_sym3333to66(), and math_sym33to6().

◆ mapplain

integer, dimension (2,9), parameter, private math::mapplain = reshape([ 1,1, 1,2, 1,3, 2,1, 2,2, 2,3, 3,1, 3,2, 3,3 ], [2,9])

private

arrangement in Plain notation

Definition at line 67 of file math.f90.

Referenced by math_3333to99(), math_33to9(), math_99to3333(), and math_9to33().

◆ mapvoigt

integer, dimension (2,6), parameter, private math::mapvoigt = reshape([ 1,1, 2,2, 3,3, 2,3, 1,3, 1,2 ], [2,6])

private

arrangement in Voigt notation

Definition at line 57 of file math.f90.

Referenced by math_voigt66to3333().

◆ math_i3

real(preal), dimension(3,3), parameter math::math_i3 = reshape([ 1.0_pReal,0.0_pReal,0.0_pReal, 0.0_pReal,1.0_pReal,0.0_pReal, 0.0_pReal,0.0_pReal,1.0_pReal ], [3,3])

3x3 Identity

Definition at line 32 of file math.f90.

Referenced by lattice::buildtransformationsystem(), constitutive::constitutive_hooke_sanditstangents(), constitutive::constitutive_initialfi(), constitutive::constitutive_lianditstangents(), grid_mech_spectral_polarisation::converged(), crystallite::crystallite_init(), damask_spectral(), eigenvectorbasis(), grid_mech_spectral_polarisation::formresidual(), grid_mech_fem::grid_mech_fem_init(), grid_mech_spectral_basic::grid_mech_spectral_basic_init(), grid_mech_spectral_polarisation::grid_mech_spectral_polarisation_forward(), grid_mech_spectral_polarisation::grid_mech_spectral_polarisation_init(), homogenization::homogenization_init(), crystallite::integratestress(), math_exp33(), math_rotationalpart(), math_spherical33(), unittest(), rotations::unittest(), and lattice::unittest().

◆ nrmmandel

real(preal), dimension(6), parameter, private math::nrmmandel = [ 1.0_pReal, 1.0_pReal, 1.0_pReal, sqrt(2.0_pReal), sqrt(2.0_pReal), sqrt(2.0_pReal) ]

private

forward weighting for Mandel notation

Definition at line 39 of file math.f90.

Referenced by math_sym3333to66(), and math_sym33to6().

◆ pi

real(preal), parameter math::pi = acos(-1.0_pReal)

ratio of a circle's circumference to its diameter

Definition at line 27 of file math.f90.

Referenced by rotations::ax2ro(), eigenvectorbasis(), rotations::eu2ax(), rotations::eu2ro(), rotations::fromaxisangle(), rotations::fromeulers(), lambert::lambert_cubetoball(), lattice::lattice_c66_twin(), math_eigvalsh33(), rotations::om2eu(), rotations::qu2ax(), rotations::qu2eu(), rotations::ro2ax(), rotations::ro2ho(), rotations::unittest(), and spectral_utilities::utilities_getfreqderivative().

◆ twopiimg

complex(preal), parameter math::twopiimg = cmplx(0.0_pReal, 2.0_pReal*PI)

Re(0.0), Im(2xPi)

Definition at line 30 of file math.f90.

Data Types

Functions/Subroutines

Variables

Detailed Description

Function/Subroutine Documentation

◆ math_3333to99()

◆ math_33to9()

◆ math_66tosym3333()

◆ math_6tosym33()

◆ math_99to3333()

◆ math_9to33()

◆ math_areatriangle()

◆ math_binomial()

◆ math_clip()

◆ math_cross()

◆ math_delta()

◆ math_det33()

◆ math_detsym33()

◆ math_deviatoric33()

◆ math_eigh()

◆ math_eigh33()

◆ math_eigvalsh()

◆ math_eigvalsh33()

◆ math_exp33()

◆ math_expand()

◆ math_factorial()

◆ math_identity2nd()

◆ math_identity4th()

◆ math_init()

◆ math_inner()

◆ math_inv33()

◆ math_invariantssym33()

◆ math_invert()

◆ math_invert33()

◆ math_invsym3333()

◆ math_levicivita()

◆ math_mul3333xx33()

◆ math_mul3333xx3333()

◆ math_multinomial()

◆ math_outer()

◆ math_range()

◆ math_rotationalpart()

◆ math_samplegaussvar()

◆ math_skew33()

◆ math_sort()

◆ math_spherical33()

◆ math_sym3333to66()

◆ math_sym33to6()

◆ math_symmetric33()

◆ math_symmetric66()

◆ math_tensordot()

◆ math_trace33()

◆ math_voigt66to3333()

◆ math_voltetrahedron()

◆ unittest()

Variable Documentation

◆ indeg

◆ inrad

◆ invnrmmandel

◆ mapnye

◆ mapplain

◆ mapvoigt

◆ math_i3

◆ nrmmandel

◆ pi

◆ twopiimg