Internal functions used by VALIB. More...

Average corotation and contrarotation helper functions
VA_DEVICE_FUN void	valib_resvortstrain2D (VA_REAL SO, VA_REAL residual_vorticity, VA_REAL residual_strain, VA_REAL principal_axis)

Numerical integrands on a unit sphere
VA_DEVICE_FUN void	valib_integrand_resvort (VA_REAL n, VA_REAL weight, VA_DEVICE_ADDR VA_REAL A, VA_DEVICE_ADDR VA_REAL avgcorot)

VA_DEVICE_FUN void	valib_integrand_resstrain (VA_REAL n, VA_REAL weight, VA_DEVICE_ADDR VA_REAL A, VA_DEVICE_ADDR VA_REAL S_RAVG)

VA_DEVICE_FUN void	valib_integrand_surface (VA_REAL n, VA_REAL weight, VA_DEVICE_ADDR VA_REAL A, VA_DEVICE_ADDR VA_REAL result)

Small helper functions
VA_DEVICE_FUN VA_REAL	valib_sign (VA_REAL a, VA_REAL b)

VA_DEVICE_FUN VA_REAL	valib_cubic_root (VA_REAL x)

VA_DEVICE_FUN VA_REAL	valib_max (VA_REAL a, VA_REAL b)

VA_DEVICE_FUN VA_REAL	valib_min (VA_REAL a, VA_REAL b)

VA_DEVICE_FUN void	valib_swap_values (VA_REAL a, VA_REAL b)

VA_DEVICE_FUN void	valib_solve_cubic_equation (VA_REAL a, VA_REAL b, VA_REAL c, VA_REAL d, VA_REAL complex x1, VA_REAL complex x2, VA_REAL complex x3, int info)

Operations on vectors of length 3
VA_DEVICE_FUN void	valib_vec_zero3 (VA_REAL *v)

VA_DEVICE_FUN VA_REAL	valib_scalar_prod3 (VA_REAL v1, VA_REAL v2)

VA_DEVICE_FUN void	valib_cross_prod3 (VA_REAL v1, VA_REAL v2, VA_REAL *v3)

Operations on 3x3 matrices
VA_DEVICE_FUN void	valib_mat_zero3 (VA_REAL *A)

VA_DEVICE_FUN void	valib_mat_copy3 (VA_REAL A, VA_REAL B)

VA_DEVICE_FUN void	valib_norm_frobenius3 (VA_REAL A, VA_REAL norm)

VA_DEVICE_FUN void	valib_trace3 (VA_REAL A, VA_REAL tr3)

VA_DEVICE_FUN void	valib_second_invariant3 (VA_REAL A, VA_REAL second_invariant)

VA_DEVICE_FUN void	valib_determinant3 (VA_REAL A, VA_REAL det)

VA_DEVICE_FUN void	valib_transpose3 (VA_REAL *A)

VA_DEVICE_FUN void	valib_sym_antisym3 (VA_REAL *A)

VA_DEVICE_FUN void	valib_deviatorise3 (VA_REAL *A)

VA_DEVICE_FUN void	valib_matvec3_prod (int trans_a, VA_REAL A, VA_REAL x, VA_REAL *y)

VA_DEVICE_FUN void	valib_rank1_update3 (VA_REAL alpha, VA_REAL x, VA_REAL y, VA_REAL *A)

VA_DEVICE_FUN void	valib_tatt3 (VA_REAL A, VA_REAL T, VA_REAL *TATT)

VA_DEVICE_FUN void	valib_matmat3_prod (int trans_a, int trans_b, VA_REAL A, VA_REAL B, VA_REAL *C)

VA_DEVICE_FUN void	valib_gram_schmidt3 (VA_REAL *Q)

VA_DEVICE_FUN void	valib_eigenvalues_sym3 (VA_REAL A, VA_REAL eigval)

VA_DEVICE_FUN int	valib_find_pivot3 (VA_REAL *A, int jcol, int start)

VA_DEVICE_FUN void	valib_find_nullspace3 (VA_REAL A, VA_REAL x, int *info)

VA_DEVICE_FUN void	valib_constructQ3 (VA_REAL sina, VA_REAL cosa, VA_REAL sinb, VA_REAL cosb, VA_REAL sing, VA_REAL cosg, VA_REAL *Q)

VA_DEVICE_FUN void	valib_constructQfromN3 (VA_REAL n, VA_REAL Q)

Helper functions for the Q-criterion and its variants.
VA_DEVICE_FUN void	valib_qcriterion_blending (VA_REAL SO, VA_REAL blending_ratio, VA_REAL qcriterion_blended)

Numerical integrators on a sphere
VA_DEVICE_FUN void	valib_integrator_fibonacci (VA_DEVICE_ADDR void integrand(VA_REAL n, VA_REAL weight, VA_DEVICE_ADDR VA_REAL A, VA_DEVICE_ADDR VA_REAL result), VA_DEVICE_ADDR VA_REAL A, VA_DEVICE_ADDR VA_REAL result)

VA_DEVICE_FUN void	valib_integrator_bazant (VA_DEVICE_ADDR void integrand(VA_REAL n, VA_REAL weight, VA_DEVICE_ADDR VA_REAL A, VA_DEVICE_ADDR VA_REAL result), VA_DEVICE_ADDR VA_REAL A, VA_DEVICE_ADDR VA_REAL result)

VA_DEVICE_FUN void	valib_integrator_rectangle (VA_DEVICE_ADDR void integrand(VA_REAL n, VA_REAL weight, VA_DEVICE_ADDR VA_REAL A, VA_DEVICE_ADDR VA_REAL result), VA_DEVICE_ADDR VA_REAL A, VA_DEVICE_ADDR VA_REAL result, int numIntervals)

Triple decomposition helper functions.
VA_DEVICE_FUN void	valib_get_brf (VA_DEVICE_ADDR VA_REAL A, int num_intervals, VA_REAL alpha_brf, VA_REAL beta_brf, VA_REAL *gamma_brf)

Detailed Description

Internal functions used by VALIB.

Function Documentation

◆ valib_resvortstrain2D()

VA_DEVICE_FUN void valib_resvortstrain2D	(	VA_REAL *	SO,
		VA_REAL *	residual_vorticity,
		VA_REAL *	residual_strain,
		VA_REAL *	principal_axis
	)

Function for determining residual vorticity and residual strain in 2D. The 2D case is seen as the leading principal 2x2 block of the 3x3 matrix. The function implements formulas (16)-(27) from [14].

Definition at line 23 of file corotation_common.h.

◆ valib_integrand_resvort()

VA_DEVICE_FUN void valib_integrand_resvort	(	VA_REAL *	n,
		VA_REAL *	weight,
		VA_DEVICE_ADDR VA_REAL *	A,
		VA_DEVICE_ADDR VA_REAL *	avgcorot
	)

Integrand to be used with different integration rules for determining averaged corotation in a given integration point with a weight, compute the local contribution and add it to the average corotation vector.

Definition at line 94 of file corotation_common.h.

◆ valib_integrand_resstrain()

VA_DEVICE_FUN void valib_integrand_resstrain	(	VA_REAL *	n,
		VA_REAL *	weight,
		VA_DEVICE_ADDR VA_REAL *	A,
		VA_DEVICE_ADDR VA_REAL *	S_RAVG
	)

Integrand to be used with different integration rules for determining averaged contrarotation in a given integration point with a weight, compute the local contribution and add it to the average contrarotation.

Definition at line 163 of file corotation_common.h.

◆ valib_integrand_surface()

VA_DEVICE_FUN void valib_integrand_surface	(	VA_REAL *	n,
		VA_REAL *	weight,
		VA_DEVICE_ADDR VA_REAL *	A,
		VA_DEVICE_ADDR VA_REAL *	result
	)

Integrand to be used for debugging purposes. Integrator should return value of the surface of a unit sphere.

Definition at line 269 of file corotation_common.h.

◆ valib_sign()

VA_DEVICE_FUN VA_REAL valib_sign	(	VA_REAL	a,
		VA_REAL	b
	)

Fortran-like SIGN function. Returns absolute value of a with sign of b.

Definition at line 20 of file linalg3.h.

◆ valib_cubic_root()

VA_DEVICE_FUN VA_REAL valib_cubic_root ( VA_REAL x )

Function for finding the real cubic root.

Definition at line 29 of file linalg3.h.

◆ valib_max()

VA_DEVICE_FUN VA_REAL valib_max	(	VA_REAL	a,
		VA_REAL	b
	)

Fortran-like MAX function. Returns the larger of the two numbers.

Definition at line 39 of file linalg3.h.

◆ valib_min()

VA_DEVICE_FUN VA_REAL valib_min	(	VA_REAL	a,
		VA_REAL	b
	)

Fortran-like MIN function. Returns the smaller of the two numbers.

Definition at line 49 of file linalg3.h.

◆ valib_swap_values()

VA_DEVICE_FUN void valib_swap_values	(	VA_REAL *	a,
		VA_REAL *	b
	)

Simple function for swapping two values.

Definition at line 58 of file linalg3.h.

◆ valib_solve_cubic_equation()

VA_DEVICE_FUN void valib_solve_cubic_equation	(	VA_REAL	a,
		VA_REAL	b,
		VA_REAL	c,
		VA_REAL	d,
		VA_REAL complex *	x1,
		VA_REAL complex *	x2,
		VA_REAL complex *	x3,
		int *	info
	)

Function for solving cubic equation using Cardano formulas based on K. Rektorys: Prehled uzite matematiky, p. 39 a*x^3 + b*x^2 + c*x + d = 0 If determinant is positive, it cannot solve the problem, info = -1. If determinant is negative, it finds the three roots, info = 0.

Definition at line 74 of file linalg3.h.

◆ valib_vec_zero3()

VA_DEVICE_FUN void valib_vec_zero3 ( VA_REAL * v )

Zero a vector of length 3.

Definition at line 184 of file linalg3.h.

◆ valib_scalar_prod3()

VA_DEVICE_FUN VA_REAL valib_scalar_prod3	(	VA_REAL *	v1,
		VA_REAL *	v2
	)

Function for scalar product of two vectors

\[ result = v_1 \cdot v_2 \]

.

Definition at line 196 of file linalg3.h.

◆ valib_cross_prod3()

VA_DEVICE_FUN void valib_cross_prod3	(	VA_REAL *	v1,
		VA_REAL *	v2,
		VA_REAL *	v3
	)

Function for cross product of two vectors

\[ v_3 = v_1 \times v_2 \]

.

Definition at line 211 of file linalg3.h.

◆ valib_mat_zero3()

VA_DEVICE_FUN void valib_mat_zero3 ( VA_REAL * A )

Zero a 3x3 matrix.

Definition at line 229 of file linalg3.h.

◆ valib_mat_copy3()

VA_DEVICE_FUN void valib_mat_copy3	(	VA_REAL *	A,
		VA_REAL *	B
	)

Copy a 3x3 matrix A to a 3x3 matrix B.

Definition at line 243 of file linalg3.h.

◆ valib_norm_frobenius3()

VA_DEVICE_FUN void valib_norm_frobenius3	(	VA_REAL *	A,
		VA_REAL *	norm
	)

Compute Frobenius norm of a 3x3 matrix.

Definition at line 257 of file linalg3.h.

◆ valib_trace3()

VA_DEVICE_FUN void valib_trace3	(	VA_REAL *	A,
		VA_REAL *	tr3
	)

Compute the trace of a 3x3 matrix (1st tensor invariant).

Definition at line 273 of file linalg3.h.

◆ valib_second_invariant3()

VA_DEVICE_FUN void valib_second_invariant3	(	VA_REAL *	A,
		VA_REAL *	second_invariant
	)

Compute the second invariant of a 3x3 matrix (2nd tensor invariant).

Definition at line 282 of file linalg3.h.

◆ valib_determinant3()

VA_DEVICE_FUN void valib_determinant3	(	VA_REAL *	A,
		VA_REAL *	det
	)

Compute the determinant of a 3x3 matrix (3rd tensor invariant).

Definition at line 293 of file linalg3.h.

◆ valib_transpose3()

VA_DEVICE_FUN void valib_transpose3 ( VA_REAL * A )

Transpose A inplace.

Definition at line 307 of file linalg3.h.

◆ valib_sym_antisym3()

VA_DEVICE_FUN void valib_sym_antisym3 ( VA_REAL * A )

Find the symmetric and the antisymmetric parts of a 3x3 matrix. Store the symmetric part in the upper triangle, and the antisymmetric part in the lower triangle.

Definition at line 320 of file linalg3.h.

◆ valib_deviatorise3()

VA_DEVICE_FUN void valib_deviatorise3 ( VA_REAL * A )

Find the deviatoric part of a 3x3 matrix, i.e. shift the diagonal such that the matrix has zero trace,

\[ A_D = A - \frac{1}{3}\mbox{trace}(A)\ I. \]

Definition at line 344 of file linalg3.h.

◆ valib_matvec3_prod()

VA_DEVICE_FUN void valib_matvec3_prod	(	int	trans_a,
		VA_REAL *	A,
		VA_REAL *	x,
		VA_REAL *	y
	)

Multiplication of a 3x3 matrix with a vector.

\[ y = A x \]

.

Definition at line 361 of file linalg3.h.

◆ valib_rank1_update3()

VA_DEVICE_FUN void valib_rank1_update3	(	VA_REAL	alpha,
		VA_REAL *	x,
		VA_REAL *	y,
		VA_REAL *	A
	)

Rank-1 update of a matrix A of dimensions 3x3, computes

\[ A = A + \alpha x y^T \]

.

Definition at line 383 of file linalg3.h.

◆ valib_tatt3()

VA_DEVICE_FUN void valib_tatt3	(	VA_REAL *	A,
		VA_REAL *	T,
		VA_REAL *	TATT
	)

Transformation of a square 3x3 matrix A

\[ TATT = T A T^T \]

.

Definition at line 406 of file linalg3.h.

◆ valib_matmat3_prod()

VA_DEVICE_FUN void valib_matmat3_prod	(	int	trans_a,
		int	trans_b,
		VA_REAL *	A,
		VA_REAL *	B,
		VA_REAL *	C
	)

Multiplication of two 3x3 matrices.

\[ C = A B + C \]

.

Definition at line 429 of file linalg3.h.

◆ valib_gram_schmidt3()

VA_DEVICE_FUN void valib_gram_schmidt3 ( VA_REAL * Q )

Gram-Schmidt orthogonalization of a 3x3 matrix.

Definition at line 457 of file linalg3.h.

◆ valib_eigenvalues_sym3()

VA_DEVICE_FUN void valib_eigenvalues_sym3	(	VA_REAL *	A,
		VA_REAL *	eigval
	)

Computing eigenvalues. Uses the method given in [17].

Definition at line 486 of file linalg3.h.

◆ valib_find_pivot3()

VA_DEVICE_FUN int valib_find_pivot3	(	VA_REAL *	A,
		int	jcol,
		int	start
	)

Find a the index of the pivot in the column of a matrix. It searches the jcol-th column of matrix A from index start to 3, i.e. index of the maximum of absolute value of A(start:,jcol).

Definition at line 542 of file linalg3.h.

◆ valib_find_nullspace3()

VA_DEVICE_FUN void valib_find_nullspace3	(	VA_REAL *	A,
		VA_REAL *	x,
		int *	info
	)

Find a nontrivial solution to the equation Ax = 0. This function is used for searching the eigenvector corresponding to real eigenvalue in case of simple eigenvalues. It expects one-dimensional nullspace, otherwise returns an error.

Definition at line 569 of file linalg3.h.

◆ valib_constructQ3()

VA_DEVICE_FUN void valib_constructQ3	(	VA_REAL	sina,
		VA_REAL	cosa,
		VA_REAL	sinb,
		VA_REAL	cosb,
		VA_REAL	sing,
		VA_REAL	cosg,
		VA_REAL *	Q
	)

Get orthogonal matrix Q column-wise based on sines and cosines of 3 angles of rotations of coordinate system along coordinate axes by z, y', and z'', see [14].

Definition at line 743 of file linalg3.h.

◆ valib_constructQfromN3()

VA_DEVICE_FUN void valib_constructQfromN3	(	VA_REAL *	n,
		VA_REAL *	Q
	)

Get orthogonal matrix Q column-wise based on components of a normal vector n which corresponds to the last row of Q. Vector n need not be normalized to \( \|n\|_2 = 1\).

Definition at line 765 of file linalg3.h.

◆ valib_qcriterion_blending()

VA_DEVICE_FUN void valib_qcriterion_blending	(	VA_REAL *	SO,
		VA_REAL	blending_ratio,
		VA_REAL *	qcriterion_blended
	)

inline

Function common for \(Q\), \(Q_D\), \(Q_W\), and \(Q_M\) criterions with prescribed blending factor \(\beta\) of the norms \( \| \Omega \|_F^2 \) and \( \| S_X \|_F^2\). Evaluates

\[ Q = \frac{1}{2} ( \| \Omega \|_F^2 - \beta \| S_X \|_F^2 ) > 0, \]

where \( \Omega = \frac{1}{2} ( \nabla u - ( \nabla u )^T ) \) is the antisymmetric part and \( S = \frac{1}{2} ( \nabla u + ( \nabla u )^T ) \) is the symmetric part of the velocity gradient \( \nabla u \).

For the \(Q\)-criterion [8] \( S_X = S \) and \(\beta = 1\).

For the \(Q_D\)-criterion [15] \( S_X = S_D \) and \(\beta = 1\).

For the \(Q_M\)-criterion [11] \( S_X = S_D \) and \(\beta = 1.5\).

For the \(Q_W\)-criterion [12] \( S_X = S_D \) and \(\beta = 1.2\).

Here, \( S_D \) is the deviatoric part of \( S \), which is the same as the symmetric part of the deviatoric part of the velocity gradient,

\[ ( \nabla u )_D = \nabla u - \frac{1}{3} \mbox{trace}(\nabla u)\ I. \]

Definition at line 50 of file qcriterion_common.h.

◆ valib_integrator_fibonacci()

VA_DEVICE_FUN void valib_integrator_fibonacci	(	VA_DEVICE_ADDR void	integrandVA_REAL n, VA_REAL weight, VA_DEVICE_ADDR VA_REAL A, VA_DEVICE_ADDR VA_REAL result,
		VA_DEVICE_ADDR VA_REAL *	A,
		VA_DEVICE_ADDR VA_REAL *	result
	)

Integrator on a unit sphere using Fibonacci numbers, [7]. Programmed following Appendix A from [5].

Definition at line 20 of file spherical_integrators.h.

◆ valib_integrator_bazant()

VA_DEVICE_FUN void valib_integrator_bazant	(	VA_DEVICE_ADDR void	integrandVA_REAL n, VA_REAL weight, VA_DEVICE_ADDR VA_REAL A, VA_DEVICE_ADDR VA_REAL result,
		VA_DEVICE_ADDR VA_REAL *	A,
		VA_DEVICE_ADDR VA_REAL *	result
	)

Integrator on a unit sphere following [1].

Definition at line 74 of file spherical_integrators.h.

◆ valib_integrator_rectangle()

VA_DEVICE_FUN void valib_integrator_rectangle	(	VA_DEVICE_ADDR void	integrandVA_REAL n, VA_REAL weight, VA_DEVICE_ADDR VA_REAL A, VA_DEVICE_ADDR VA_REAL result,
		VA_DEVICE_ADDR VA_REAL *	A,
		VA_DEVICE_ADDR VA_REAL *	result,
		int	numIntervals
	)

Integrator on a unit sphere using the simplest rectangular rule. Corresponds to the approach described in [13].

Definition at line 116 of file spherical_integrators.h.

◆ valib_get_brf()

VA_DEVICE_FUN void valib_get_brf	(	VA_DEVICE_ADDR VA_REAL *	A,
		int *	num_intervals,
		VA_REAL *	alpha_brf,
		VA_REAL *	beta_brf,
		VA_REAL *	gamma_brf
	)

Function for determining the Basic Reference Frame (BRF) introduced in [14]. This is the coordinate frame in which the shear is maximized and in which it is later eliminated. The BRF is found as an optimization problem, searching the maxima in formula (10) of [14], i.e.

\[ \max_{\alpha \in [0,\pi], \\ \beta \in [0,\pi], \\ \gamma \in [0,\pi/2]} \left( |S_{12}\Omega_{12}| + |S_{23}\Omega_{23}| + |S_{31}\Omega_{31}| \right), \]

where \(\alpha\), \(\beta\), and \(\gamma\) are the three angles of rotation of the coordinate frame. The optimal value is determined by an exhaustive search over all possible values of the angles. The resolution is determined as \( h = \pi / n_i \), with \(n_i\) the number of intervals (the num_intervals variable). For example, num_intervals = 180 corresponds to 1 degree resolution. This optimization problem can become costly for large number of intervals.

Definition at line 38 of file triple_decomposition.h.

Average corotation and contrarotation helper functions

Numerical integrands on a unit sphere

Small helper functions

Operations on vectors of length 3

Operations on 3x3 matrices

Helper functions for the Q-criterion and its variants.

Numerical integrators on a sphere

Triple decomposition helper functions.