Namespaces
	Coordinates

	tk

Classes
class	AsciiDataBase

class	AsciiDataBoundary

class	AsciiDataInitial

class	AsciiDataLayered

class	AsciiDataProfile

class	NaturalCoordinate

class	Operator

class	StructuredDataLookup

struct	ThousandSep

class	VectorFunctionFromVelocityFunctionObject

Typedefs
using	DEAL_II_DEPRECATED = StructuredDataLookup< dim >

Functions
template<typename T >
std::vector< T >	possibly_extend_from_1_to_N (const std::vector< T > &values, const unsigned int N, const std::string &id_text)

std::vector< double >	parse_map_to_double_array (const std::string &key_value_map, const std::vector< std::string > &list_of_keys, const bool expects_background_field, const std::string &property_name, const bool allow_multiple_values_per_key=false, const std::shared_ptr< std::vector< unsigned int > > &n_values_per_key=nullptr, const bool allow_missing_keys=false)

template<typename T >
Table< 2, T >	parse_input_table (const std::string &input_string, const unsigned int n_rows, const unsigned int n_columns, const std::string &property_name)

template<int dim>
std::vector< std::string >	expand_dimensional_variable_names (const std::vector< std::string > &var_declarations)

void	split_by_block (const std::vector< types::global_dof_index > &dofs_per_block, const IndexSet &whole_set, std::vector< IndexSet > &partitioned)

template<int dim>
IndexSet	extract_locally_active_dofs_with_component (const DoFHandler< dim > &dof_handler, const ComponentMask &component_mask)

template<int dim>
bool	polygon_contains_point (const std::vector< Point< 2 > > &point_list, const ::Point< 2 > &point)

template<int dim>
double	signed_distance_to_polygon (const std::vector< Point< 2 > > &point_list, const ::Point< 2 > &point)

double	distance_to_line (const std::array<::Point< 2 >, 2 > &point_list, const ::Point< 2 > &point)

template<int dim>
std::array< Tensor< 1, dim >, dim-1 >	orthogonal_vectors (const Tensor< 1, dim > &v)

std::pair< double, double >	real_spherical_harmonic (unsigned int l, unsigned int m, double theta, double phi)

bool	fexists (const std::string &filename)

bool	filename_is_url (const std::string &filename)

std::string	read_and_distribute_file_content (const std::string &filename, const MPI_Comm &comm)

int	mkdirp (std::string pathname, const mode_t mode=0755)

void	create_directory (const std::string &pathname, const MPI_Comm &comm, bool silent)

void	extract_composition_values_at_q_point (const std::vector< std::vector< double > > &composition_values, const unsigned int q, std::vector< double > &composition_values_at_q_point)

std::string	expand_ASPECT_SOURCE_DIR (const std::string &location)

std::string	parenthesize_if_nonempty (const std::string &s)

bool	has_unique_entries (const std::vector< std::string > &strings)

double	weighted_p_norm_average (const std::vector< double > &weights, const std::vector< double > &values, const double p)

template<typename T >
T	derivative_of_weighted_p_norm_average (const double averaged_parameter, const std::vector< double > &weights, const std::vector< double > &values, const std::vector< T > &derivatives, const double p)

template<int dim>
double	compute_spd_factor (const double eta, const SymmetricTensor< 2, dim > &strain_rate, const SymmetricTensor< 2, dim > &dviscosities_dstrain_rate, const double SPD_safety_factor)

template<int dim>
Point< dim >	convert_array_to_point (const std::array< double, dim > &array)

template<int dim>
std::array< double, dim >	convert_point_to_array (const Point< dim > &point)

std::vector< Operator >	create_model_operator_list (const std::vector< std::string > &operator_names)

const std::string	get_model_operator_options ()

template<int dim>
SymmetricTensor< 2, dim >	nth_basis_for_symmetric_tensors (const unsigned int k)

template<int dim, typename VectorType >
void	project_cellwise (const Mapping< dim > &mapping, const DoFHandler< dim > &dof_handler, const unsigned int component_index, const Quadrature< dim > &quadrature, const std::function< void(const typename DoFHandler< dim >::active_cell_iterator &, const std::vector< Point< dim > > &, std::vector< double > &)> &function, VectorType &vec_result)

Detailed Description

A namespace for utility functions that might be used in many different places to prevent code duplication.

Typedef Documentation

§ DEAL_II_DEPRECATED

using aspect::Utilities::DEAL_II_DEPRECATED = typedef StructuredDataLookup<dim>

Definition at line 667 of file structured_data.h.

Function Documentation

§ possibly_extend_from_1_to_N()

template<typename T >

std::vector< T > aspect::Utilities::possibly_extend_from_1_to_N	(	const std::vector< T > &	values,
		const unsigned int	N,
		const std::string &	id_text
	)

inline

Given an array values, consider three cases:

If it has size N, return the original array.
If it has size one, return an array of size N where all elements are equal to the one element of value.
If it has any other size, throw an exception that uses id_text as an identifying string.

This function is typically used for parameter lists that can either contain different values for each of a set of objects (e.g., for each compositional field), or contain a single value that is then used for each object.

Definition at line 523 of file utilities.h.

References AssertThrow, compute_spd_factor(), convert_array_to_point(), convert_point_to_array(), derivative_of_weighted_p_norm_average(), ExcMessage(), expand_ASPECT_SOURCE_DIR(), has_unique_entries(), parenthesize_if_nonempty(), point(), aspect::MaterialModel::NonlinearDependence::strain_rate, Utilities::to_string(), values, weighted_p_norm_average(), and weights.

§ parse_map_to_double_array()

std::vector<double> aspect::Utilities::parse_map_to_double_array	(	const std::string &	key_value_map,
		const std::vector< std::string > &	list_of_keys,
		const bool	expects_background_field,
		const std::string &	property_name,
		const bool	allow_multiple_values_per_key = `false`,
		const std::shared_ptr< std::vector< unsigned int > > &	n_values_per_key = `nullptr`,
		const bool	allow_missing_keys = `false`
	)

This function takes a string argument that is interpreted as a map of the form "key1 : value1, key2 : value2, etc", and then parses it to return a vector of these values where the values are ordered in the same order as a given set of keys. If allow_multiple_values_per_key is set to 'true' it also allows entries of the form "key1: value1|value2|value3, etc", in which case the returned vector will have more entries than the provided list_of_keys.

This function also considers a number of special cases:

If the input string consists of only a comma separated set of values "value1, value2, value3, ..." (i.e., without the "keyx :" part), then the input string is interpreted as if it had had the form "key1 : value1, key2 : value2, ..." where "key1", "key2", ... are exactly the keys provided by the list_of_keys in the same order as provided. In this situation, if a background field is required, the background value is assigned to the first element of the output vector. This form only allows a single value per key.
Whether or not a background field is required depends on expects_background_field. Requiring a background field inserts "background" into the list_of_keys as the first entry in the list. Some calling functions (like some material models) require values for background fields, while others may not need background values.
Two special keys are recognized: all –> Assign the associated value to all fields. Only one key is allowed in this case. background –> Assign associated value to the background.

Parameters

[in]	key_value_map	The string representation of the map to be parsed.
[in]	list_of_keys	A list of N valid key names that are allowed to appear in the map. The order of these keys determines the order of values that are returned by this function.
[in]	expects_background_field	If true, expect N+1 values and allow setting of the background using the key "background".
[in]	property_name	A name that identifies the type of property that is being parsed by this function and that is used in generating error messages if the map does not conform to the expected format.
[in]	allow_multiple_values_per_key	If true allow having multiple values for each key. If false only allow a single value per key. In either case each key is only allowed to appear once.
[in,out]	n_values_per_key	A pointer to a vector of unsigned integers. If no pointer is handed over nothing happens. If a pointer to an empty vector is handed over, the vector is resized with as many components as keys (+1 if there is a background field). Each value then stores how many values were found for this key. The sum over all entries is the length of the return value of this function. If a pointer to an existing vector with one or more entries is handed over (e.g. a n_values_per_key vector created by a previous call to this function) then this vector is used as expected structure of the input parameter, and it is checked that `key_value_map` fulfills this structure. This can be used to assert that several input parameters prescribe the same number of values to each key.
[in]	allow_missing_keys	Whether to allow that some keys are not set to any values, i.e. they do not appear at all in the `key_value_map`. This also allows a completely empty map.

Returns: A vector of values that are parsed from the map, provided in the order in which the keys appear in the list_of_keys argument. If multiple values per key are allowed, the vector contains first all values for key 1, then all values for key 2 and so forth. Using the n_values_per_key vector allows the caller to associate entries in the returned vector with specific keys.

§ parse_input_table()

template<typename T >

Table<2,T> aspect::Utilities::parse_input_table	(	const std::string &	input_string,
		const unsigned int	n_rows,
		const unsigned int	n_columns,
		const std::string &	property_name
	)

This function takes a string argument that is assumed to represent an input table, in which each row is separated by semicolons, and each column separated by commas. The function returns the parsed entries as a table. In addition this function utilizes the possibly_extend_from_1_to_N() function to accept inputs with only a single column/row, which will be extended to n_rows or n_columns respectively, to allow abbreviating the input string (e.g. you can provide a single value instead of n_rows by n_columns identical values). This function can for example be used by material models to read densities for different compositions and different phases for each composition.

§ expand_dimensional_variable_names()

template<int dim>

std::vector<std::string> aspect::Utilities::expand_dimensional_variable_names ( const std::vector< std::string > & var_declarations )

Given a vector var_declarations expand any entries of the form vector(str) or tensor(str) to sublists with component names of the form str_x, str_y, str_z or str_xx, str_xy... for the correct dimension value.

This function is to be used for expanding lists of variable names where one or more such variable is actually intended to be a list of components.

Returns the generated list of variable names

§ split_by_block()

void aspect::Utilities::split_by_block	(	const std::vector< types::global_dof_index > &	dofs_per_block,
		const IndexSet &	whole_set,
		std::vector< IndexSet > &	partitioned
	)

Split the set of DoFs (typically locally owned or relevant) in whole_set into blocks given by the dofs_per_block structure.

The numbers of dofs per block need to add up to the size of the index space described by whole_set.

§ extract_locally_active_dofs_with_component()

template<int dim>

IndexSet aspect::Utilities::extract_locally_active_dofs_with_component	(	const DoFHandler< dim > &	dof_handler,
		const ComponentMask &	component_mask
	)

Returns an IndexSet that contains all locally active DoFs that belong to the given component_mask.

This function should be moved into deal.II at some point.

§ polygon_contains_point()

template<int dim>

bool aspect::Utilities::polygon_contains_point	(	const std::vector< Point< 2 > > &	point_list,
		const ::Point< 2 > &	point
	)

Given a 2d point and a list of points which form a polygon, computes if the point falls within the polygon.

§ signed_distance_to_polygon()

template<int dim>

double aspect::Utilities::signed_distance_to_polygon	(	const std::vector< Point< 2 > > &	point_list,
		const ::Point< 2 > &	point
	)

Given a 2d point and a list of points which form a polygon, compute the smallest distance of the point to the polygon. The sign is negative for points outside of the polygon and positive for points inside the polygon.

§ distance_to_line()

double aspect::Utilities::distance_to_line	(	const std::array<::Point< 2 >, 2 > &	point_list,
		const ::Point< 2 > &	point
	)

Given a 2d point and a list of two points that define a line, compute the smallest distance of the point to the line segment. When the point's perpendicular base does not lie on the line segment, the smallest distance to the segment's end points is calculated.

§ orthogonal_vectors()

template<int dim>

std::array<Tensor<1,dim>,dim-1> aspect::Utilities::orthogonal_vectors ( const Tensor< 1, dim > & v )

Given a vector v in dim dimensional space, return a set of (dim-1) vectors that are orthogonal to v and to each other. The length of each of these vectors equals that of the original vector v to ensure that the resulting set of vectors represents a well-conditioned basis.

§ real_spherical_harmonic()

std::pair<double,double> aspect::Utilities::real_spherical_harmonic	(	unsigned int	l,
		unsigned int	m,
		double	theta,
		double	phi
	)

A function for evaluating real spherical harmonics. It takes the degree (l) and the order (m) of the spherical harmonic, where $l \geq 0$ and $0 \leq m \leq l$. It also takes the colatitude (theta) and longitude (phi), which are in radians.

There are an unfortunate number of normalization conventions in existence for spherical harmonics. Here we use fully normalized spherical harmonics including the Condon-Shortley phase. This corresponds to the definitions given in equations B.72 and B.99-B.102 in Dahlen and Tromp (1998, ISBN: 9780691001241). The functional form of the real spherical harmonic is given by

\[ Y_{lm}(\theta, \phi) = \sqrt{2} X_{l \left| m \right| }(\theta) \cos m \phi \qquad \mathrm{if} \qquad -l \le m < 0 \]

\[ Y_{lm}(\theta, \phi) = X_{l 0 }(\theta) \qquad \mathrm{if} \qquad m = 0 \]

\[ Y_{lm}(\theta, \phi) = \sqrt{2} X_{lm}(\theta) \sin m \phi \qquad \mathrm{if} \qquad 0< m \le m \]

where $X_{lm}( \theta )$ is an associated Legendre function.

In practice it is often convenient to compute the sine ( $-l \le m < 0$) and cosine ( $0 < m \le l$) variants of the real spherical harmonic at the same time. That is the approach taken here, where we return a pair of numbers, the first corresponding the cosine part and the second corresponding to the sine part. Given this, it is no longer necessary to distinguish between positive and negative $m$, so this function only accepts $ m \ge 0$. For $m = 0$, there is only one part, which is stored in the first entry of the pair.

Note: This function uses the Boost spherical harmonics implementation internally, which is not designed for very high order (> 100) spherical harmonics computation. If you use spherical harmonics of a high order be sure to confirm the accuracy first. For more information, see: http://www.boost.org/doc/libs/1_49_0/libs/math/doc/sf_and_dist/html/math_toolkit/special/sf_poly/sph_harm.html

§ fexists()

bool aspect::Utilities::fexists ( const std::string & filename )

Checks whether a file named filename exists and is readable.

Parameters

filename File to check existence

Referenced by aspect::Utilities::ThousandSep::do_grouping().

§ filename_is_url()

bool aspect::Utilities::filename_is_url ( const std::string & filename )

Checks to see if the user is trying to use data from a url.

Parameters

filename File to check

Referenced by aspect::Utilities::ThousandSep::do_grouping().

§ read_and_distribute_file_content()

std::string aspect::Utilities::read_and_distribute_file_content	(	const std::string &	filename,
		const MPI_Comm &	comm
	)

Reads the content of the ascii file filename on process 0 and distributes the content by MPI_Bcast to all processes. The function returns the content of the file on all processes.

Parameters

[in]	filename	The name of the ascii file to load.
[in]	comm	The MPI communicator in which the content is distributed.

Returns: A string which contains the data in filename.

Referenced by aspect::Utilities::ThousandSep::do_grouping().

§ mkdirp()

int aspect::Utilities::mkdirp	(	std::string	pathname,
		const mode_t	mode = `0755`
	)

Creates a path as if created by the shell command "mkdir -p", therefore generating directories from the highest to the lowest level if they are not already existing.

Parameters

pathname	String that contains the path to create. '/' is used as directory separator.
mode	Permissions (mode bits) of the created directories. See the documentation of the chmod() command for more information.

Returns: The function returns the error value of the last mkdir call inside. It returns zero on success. See the man page of mkdir() for more information.

Referenced by aspect::Utilities::ThousandSep::do_grouping().

§ create_directory()

void aspect::Utilities::create_directory	(	const std::string &	pathname,
		const MPI_Comm &	comm,
		bool	silent
	)

Create directory pathname, optionally printing a message.

Parameters

pathname	String that contains path to create. '/' is used as directory separator.
comm	MPI communicator, used to limit creation of directory to processor 0.
silent	Print a nicely formatted message on processor 0 if set to true.

Referenced by aspect::Utilities::ThousandSep::do_grouping().

§ extract_composition_values_at_q_point()

void aspect::Utilities::extract_composition_values_at_q_point	(	const std::vector< std::vector< double > > &	composition_values,
		const unsigned int	q,
		std::vector< double > &	composition_values_at_q_point
	)

inline

Extract the compositional values at a single quadrature point with index q from composition_values, which is indexed by compositional index and quadrature point, and write them into composition_values_at_q_point. In other words, this extracts composition_values[i][q] for all i.

Definition at line 504 of file utilities.h.

References Assert, and ExcInternalError().

§ expand_ASPECT_SOURCE_DIR()

std::string aspect::Utilities::expand_ASPECT_SOURCE_DIR ( const std::string & location )

Replace the string $ASPECT_SOURCE_DIR in location by the current source directory of ASPECT and return the resulting string.

Referenced by possibly_extend_from_1_to_N().

§ parenthesize_if_nonempty()

std::string aspect::Utilities::parenthesize_if_nonempty ( const std::string & s )

Given a string s, return it in the form ' ("s")' if nonempty. Otherwise just return the empty string itself.

Referenced by possibly_extend_from_1_to_N().

§ has_unique_entries()

bool aspect::Utilities::has_unique_entries ( const std::vector< std::string > & strings )

Returns if a vector of strings strings only contains unique entries.

Referenced by possibly_extend_from_1_to_N().

§ weighted_p_norm_average()

double aspect::Utilities::weighted_p_norm_average	(	const std::vector< double > &	weights,
		const std::vector< double > &	values,
		const double	p
	)

This function computes the weighted average $\bar y$ of $y_i$ for a weighted p norm. This leads for a general p to: $\bar y = \left(\frac{\sum_{i=1}^k w_i y_i^p}{\sum_{i=1}^k w_i}\right)^{\frac{1}{p}}$. When p = 0 we take the geometric average: $\bar y = \exp\left(\frac{\sum_{i=1}^k w_i \log\left(y_i\right)}{\sum_{i=1}^k w_i}\right)$, and when $p \le -1000$ or $p \ge 1000$ we take the minimum and maximum norm respectively. This means that the smallest and largest value is respectively taken taken.

This function has been set up to be very tolerant to strange values, such as negative weights. The only things we require in for the general p is that the sum of the weights and the sum of the weights times the values to the power p may not be smaller or equal to zero. Furthermore, when a value is zero, the exponent is smaller then one and the correspondent weight is non-zero, this corresponds to no resistance in a parallel system. This means that this 'path' will be followed, and we return zero.

The implemented special cases (which are minimum (p <= -1000), harmonic average (p = -1), geometric average (p = 0), arithmetic average (p = 1), quadratic average (RMS) (p = 2), cubic average (p = 3) and maximum (p >= 1000) ) is, except for the harmonic and quadratic averages even more tolerant of negative values, because they only require the sum of weights to be non-zero.

Referenced by possibly_extend_from_1_to_N().

§ derivative_of_weighted_p_norm_average()

template<typename T >

T aspect::Utilities::derivative_of_weighted_p_norm_average	(	const double	averaged_parameter,
		const std::vector< double > &	weights,
		const std::vector< double > &	values,
		const std::vector< T > &	derivatives,
		const double	p
	)

This function computes the derivative ( $\frac{\partial\bar y}{\partial x}$) of an average of the values $y_i(x)$ with regard to $x$, using $\frac{\partial y_i(x)}{\partial x}$. This leads for a general p to: $\frac{\partial\bar y}{\partial x} = \frac{1}{p}\left(\frac{\sum_{i=1}^k w_i y_i^p}{\sum_{i=1}^k w_i}\right)^{\frac{1}{p}-1} \frac{\sum_{i=1}^k w_i p y_i^{p-1} y'_i}{\sum_{i=1}^k w_i}$. When p = 0 we take the geometric average as a reference, which results in: $\frac{\partial\bar y}{\partial x} = \exp\left(\frac{\sum_{i=1}^k w_i \log\left(y_i\right)}{\sum_{i=1}^k w_i}\right) \frac{\sum_{i=1}^k\frac{w_i y'_i}{y_i}}{\sum_{i=1}^k w_i}$ and when $p \le -1000$ or $p \ge 1000$ we take the min and max norm respectively. This means that the derivative is taken which has the min/max value.

This function has, like the function weighted_p_norm_average been set up to be very tolerant to strange values, such as negative weights. The only things we require in for the general p is that the sum of the weights and the sum of the weights times the values to the power p may not be smaller or equal to zero. Furthermore, when a value is zero, the exponent is smaller then one and the correspondent weight is non-zero, this corresponds to no resistance in a parallel system. This means that this 'path' will be followed, and we return the corresponding derivative.

The implemented special cases (which are minimum (p <= -1000), harmonic average (p = -1), geometric average (p = 0), arithmetic average (p = 1), and maximum (p >= 1000) ) is, except for the harmonic average even more tolerant of negative values, because they only require the sum of weights to be non-zero.

Referenced by possibly_extend_from_1_to_N().

§ compute_spd_factor()

template<int dim>

double aspect::Utilities::compute_spd_factor	(	const double	eta,
		const SymmetricTensor< 2, dim > &	strain_rate,
		const SymmetricTensor< 2, dim > &	dviscosities_dstrain_rate,
		const double	SPD_safety_factor
	)

This function computes a factor which can be used to make sure that the Jacobian remains positive definite.

The goal of this function is to find a factor $\alpha$ so that $2\eta(\varepsilon(\mathbf u)) I \otimes I + \alpha\left[a \otimes b + b \otimes a\right]$ remains a positive definite matrix. Here, $a=\varepsilon(\mathbf u)$ is the strain_rate and $b=\frac{\partial\eta(\varepsilon(\mathbf u),p)}{\partial \varepsilon}$ is the derivative of the viscosity with respect to the strain rate and is given by dviscosities_dstrain_rate. Since the viscosity $\eta$ must be positive, there is always a value of $\alpha$ (possibly small) so that the result is a positive definite matrix. In the best case, we want to choose $\alpha=1$ because that corresponds to the full Newton step, and so the function never returns anything larger than one.

The factor is defined by: $\frac{2\eta(\varepsilon(\mathbf u))}{\left[1-\frac{b:a}{\|a\| \|b\|} \right]^2\|a\|\|b\|}$. Alpha is reset to a maximum of one, and if it is smaller then one, a safety_factor scales the alpha to make sure that the 1-alpha won't get to close to zero.

Referenced by possibly_extend_from_1_to_N().

§ convert_array_to_point()

template<int dim>

Point<dim> aspect::Utilities::convert_array_to_point ( const std::array< double, dim > & array )

Converts an array of size dim to a Point of size dim.

Referenced by possibly_extend_from_1_to_N().

§ convert_point_to_array()

template<int dim>

std::array<double,dim> aspect::Utilities::convert_point_to_array ( const Point< dim > & point )

Converts a Point of size dim to an array of size dim.

Referenced by possibly_extend_from_1_to_N().

§ create_model_operator_list()

std::vector<Operator> aspect::Utilities::create_model_operator_list ( const std::vector< std::string > & operator_names )

Create a vector of operator objects out of a list of strings. Each entry in the list must match one of the allowed operations.

§ get_model_operator_options()

const std::string aspect::Utilities::get_model_operator_options ( )

Create a string of model operators for use in declare_parameters

§ nth_basis_for_symmetric_tensors()

template<int dim>

SymmetricTensor<2,dim> aspect::Utilities::nth_basis_for_symmetric_tensors ( const unsigned int k )

A function that returns a SymmetricTensor, whose entries are zero, except for the k'th component, which is set to one. If k is not on the main diagonal the resulting tensor is symmetrized.

§ project_cellwise()

template<int dim, typename VectorType >

void aspect::Utilities::project_cellwise	(	const Mapping< dim > &	mapping,
		const DoFHandler< dim > &	dof_handler,
		const unsigned int	component_index,
		const Quadrature< dim > &	quadrature,
		const std::function< void(const typename DoFHandler< dim >::active_cell_iterator &, const std::vector< Point< dim > > &, std::vector< double > &)> &	function,
		VectorType &	vec_result
	)

Compute the cellwise projection of component component_index of function to the finite element space described by dof_handler.

Parameters

[in]	mapping	The mapping object to use.
[in]	dof_handler	The DoFHandler the describes the finite element space to project into and that corresponds to `vec_result`.
[in]	component_index	The component index of the `dof_handler` for which the projection is being performed. This component should be described by a DG finite element.
[in]	quadrature	The quadrature formula to use to evaluate `function` on each cell.
[in]	function	The function to project into the finite element space. This function should store the value of the function at the points described by the 2nd argument into the 3rd argument as an std::vector<double> of size quadrature.size().
[out]	vec_result	The output vector where the projected function will be stored in.

Namespaces

Classes

Typedefs

Functions

Detailed Description

Typedef Documentation

§ DEAL_II_DEPRECATED

Function Documentation

§ possibly_extend_from_1_to_N()

§ parse_map_to_double_array()

§ parse_input_table()

§ expand_dimensional_variable_names()

§ split_by_block()

§ extract_locally_active_dofs_with_component()

§ polygon_contains_point()

§ signed_distance_to_polygon()

§ distance_to_line()

§ orthogonal_vectors()

§ real_spherical_harmonic()

§ fexists()

§ filename_is_url()

§ read_and_distribute_file_content()

§ mkdirp()

§ create_directory()

§ extract_composition_values_at_q_point()

§ expand_ASPECT_SOURCE_DIR()

§ parenthesize_if_nonempty()

§ has_unique_entries()

§ weighted_p_norm_average()

§ derivative_of_weighted_p_norm_average()

§ compute_spd_factor()

§ convert_array_to_point()

§ convert_point_to_array()

§ create_model_operator_list()

§ get_model_operator_options()

§ nth_basis_for_symmetric_tensors()

§ project_cellwise()