funcy Namespace Reference

Main namespace of the funcy library. More...

Namespaces
	linalg
	Functionality from linear algebra such as (modified) principal and mixed matrix invariants.
	mathop
	Mathematical operations and corresponding differentation rules.

Classes
struct	ACos
	Arc cosine function (based on acos(double) in <cmath>). More...
struct	ASin
	Arc sine function (based on asin(double) in <cmath>). More...
struct	Constant
	Wrap a constant. More...
struct	Cos
	Cosine function (based on cos(double) in <cmath>). More...
struct	CumulativeNormalDistribution
	Cumulative standard normal distribution. More...
struct	Decay
	Identity, i.e. Decay<F>::type == F More...
struct	Decay< F, std::void_t< static_check::test::nested_type::PlainObject< F > > >
	Underlying type for expression templates of the Eigen library. More...
struct	Erf
	Error function. More...
struct	Exp
	Exponential function. More...
struct	Exp2
	Function \(2^x\). More...
struct	Identity
	Identity mapping \( f(x)=x \). More...
struct	IndexedType
struct	is_arithmetic
	Specialize this template class to register arithmetic types that are not built-in. More...
struct	LN
	Natural logarithm. More...
struct	Log10
	Common (base 10) logarithm. More...
struct	Log2
	Base 2 logarithm. More...
class	NonSymmetricMatrixException
	Exception for non-symmetric matrices if symmetric matrices are required. More...
class	OutOfDomainException
	Exception for scalar function arguments that are outside the domain of the function. More...
struct	Pow
	Power function with rational exponent \( k = \frac{dividend}{divisor} \) including first three derivatives. More...
struct	Sin
	Sine function (based on sin(double) in <cmath>). More...
struct	Tan
	Tangent function. More...
struct	Variable
	Independent variable. Can be uniquely identified by its id. More...

Typedefs
using	Sqrt = Pow< 1, 2 >
	Square root (based on sqrt(double) in <cmath>).
using	Cbrt = Pow< 1, 3 >
	Third root (based on sqrt(double) in <cmath>).
using	Cbrt2 = Pow< 2, 3 >
	Third root squared (based on sqrt(double) in <cmath>).
template<class F >
using	decay_t = typename Decay< std::decay_t< F > >::type
	Access underlying type (if it is hidden by expression templates).
template<class F >
using	remove_reference_t = typename Decay< std::remove_reference_t< F > >::type
	Access underlying type (if it is hidden by expression templates).
template<class F , int id>
using	Variable_t = typename detail::VariableT< F, id >::type
	Get underlying type of variable with index id.

Functions
template<Function F>
auto	acos (const F &f)
	Generate \( \arccos\circ f \). More...
template<Function F>
auto	asin (const F &f)
	Generate \( \arcsin\circ f \). More...
auto	cnd (double x)
template<Function F>
auto	cnd (const F &f)
	Generate \( \mathrm{cnd}\circ f \). More...
template<Function F>
auto	cos (const F &f)
	Generate \( \cos\circ f \). More...
template<Function F>
auto	erf (const F &f)
	Generate \( \mathrm{erf}\circ f \). More...
template<Function F>
auto	exp (const F &f)
	Generate \( \exp(f) \). More...
template<Function F>
auto	exp2 (const F &f)
	Generate \(2^f\). More...
auto	ln (double x)
template<Function F>
auto	ln (const F &f)
	Generate \( \mathrm{ln}\circ f \). More...
template<Function F>
auto	log10 (const F &f)
	Generate \( \mathrm{log}_{10}\circ f \). More...
template<Function F>
auto	log2 (const F &f)
	Generate \( \mathrm{log}_{2}\circ f \). More...
template<Function F>
auto	sqrt (const F &f)
	Generate \( \sqrt{f} \). More...
template<Function F>
auto	cbrt (const F &f)
	Generate \( \sqrt[3]{f} \). More...
template<Function F>
auto	cbrt2 (const F &f)
	Generate \( \sqrt[3]{f^2}\). More...
template<int k, int l, Function F>
auto	pow (const F &f)
	Generate \( f^{k/l}\). More...
template<int k, Function F>
auto	pow (const F &f)
	Generate \( f^k,\ k \in \mathbb{N}\). More...
template<Function F>
auto	sin (const F &f)
	Generate \( \sin\circ f \). More...
template<Function F>
auto	tan (const F &f)
	Generate \( \tan\circ f \). More...
template<class Arg >
auto	const_ref (const Arg &x) noexcept(std::is_nothrow_copy_constructible_v< Arg >)
	Generate a constant function that stores its argument as constant reference. More...
template<class Arg >
constexpr auto	constant (Arg &&x) noexcept((std::is_rvalue_reference_v< Arg > &&std::is_nothrow_move_constructible_v< Arg >)||(std::is_lvalue_reference_v< Arg > &&std::is_nothrow_copy_constructible_v< Arg >))
	Wrap a constant. More...
template<linalg::Matrix Mat, int offset = linalg::dim< Mat >()>
auto	incompressible_adipose_tissue_sommer_holzapfel (double cCells, double k1, double k2, double kappa, const Mat &A, const Mat &F)
	Model for adipose tissue of [Sommer2013]. More...
template<linalg::Matrix Mat, int offset = linalg::dim< Mat >()>
auto	incompressible_adipose_tissue_sommer_holzapfel (const Mat &A, const Mat &F)
	Model for adipose tissue of [Sommer2013]. More...
template<class Inflation , class Compression , linalg::Matrix Mat, int offset = linalg::dim< Mat >()>
auto	compressible_adipose_tissue_sommer_holzapfel (double cCells, double k1, double k2, double kappa, double d0, double d1, const Mat &M, const Mat &F)
	Compressible version of the model for adipose tissue of [Sommer2013]. More...
template<class Inflation , class Compression , linalg::Matrix Mat, int offset = linalg::dim< Mat >()>
auto	compressible_adipose_tissue_sommer_holzapfel (double d0, double d1, const Mat &M, const Mat &F)
	Compressible version of the model for adipose tissue of [Sommer2013]. Material parameters are taken from the same publication, Table 2, i.e. \(c_\mathrm{Cells}=0.15 (\,\mathrm{kPa})\), \(k_1=0.8 (\,\mathrm{kPa})\), \(k_2=47.3\) and \(\kappa=0.09\). More...
template<linalg::Matrix Mat, int offset = linalg::dim< Mat >()>
auto	incompressible_muscle_tissue_martins (double c, double b, double d, double e, const Mat &A, const Mat &F)
	Incompressible version of the model for muscle tissue of [Martins1998]. More...
template<linalg::Matrix Mat, int offset = linalg::dim< Mat >()>
auto	incompressible_muscle_tissue_martins (const Mat &A, const Mat &F)
	Incompressible version of the model for muscle tissue of [Martins1998]. More...
template<class Inflation , class Compression , linalg::Matrix Mat, int offset = linalg::dim< Mat >()>
auto	compressible_muscle_tissue_martins (double c, double b, double A, double a, double d0, double d1, const Mat &M, const Mat &F)
	Compressible version of the model for muscle tissue of [Martins1998]. More...
template<class Inflation , class Compression , linalg::Matrix Mat, int offset = linalg::dim< Mat >()>
auto	compressible_muscle_tissue_martins (double d0, double d1, const Mat &M, const Mat &F)
	Compressible version of the model for muscle tissue of [Martins1998]. More...
template<linalg::Matrix Mat, int n = linalg::dim< Mat >()>
auto	incompressible_skin_hendriks (double c0, double c1, const Mat &F)
	Model for skin tissue of [Hendriks2005]. More...
template<linalg::Matrix Mat, int n = linalg::dim< Mat >()>
auto	incompressible_skin_hendriks (const Mat &F)
	Model for skin tissue of [Hendriks2005]. More...
template<class InflationPenalty , class CompressionPenalty , linalg::Matrix Mat, int n = linalg::dim< Mat >()>
auto	compressible_skin_hendriks (double c0, double c1, double d0, double d1, const Mat &F)
	Compressible version of the model for skin tissue of [Hendriks2005]. More...
template<class InflationPenalty , class CompressionPenalty , linalg::Matrix M, int n = linalg::dim< M >()>
auto	compressible_skin_hendriks (double d0, double d1, const M &F)
	Compressible version of the model for skin tissue of [Hendriks2005]. More...
template<class Scalar , linalg::Vector Vec, idx = 0>
auto	heat_model (double c, double d, Scalar u, const Vec &du)
	Weak model for nonlinear heat transfer \( (c+du^2)\nabla u \). More...
template<linalg::Matrix Mat, int n = linalg::dim< Mat >()>
auto	incompressible_mooney_rivlin (double c0, double c1, const Mat &F)
	Generate an "incompressible" Mooney-Rivlin material law \( W(F)=c_0\iota_1(F^T F) + c_1\iota_2(F^T F) \), where \(\iota_1\) is the first and \(\iota_2\) the second principal matrix invariant.
template<class InflationPenalty , class CompressionPenalty , linalg::Matrix Mat, int n = linalg::dim< Mat >()>
auto	compressible_mooney_rivlin (double c0, double c1, double d0, double d1, const Mat &F)
	Generate a compressible Mooney-Rivlin material law \( W(F)=c_0\iota_1(F^T F) + c_1\iota_2(F^T F) + d_0\Gamma_\mathrm{In}(\det(F))+d_1\Gamma_\mathrm{Co}(\det(F)) \), where \(\iota_1\) is the first and \(\iota_2\) the second principal matrix invariant.
template<linalg::Matrix Mat, class InflationPenalty , class CompressionPenalty >
auto	create_mooney_rivlin_from_lame_constants (double lambda, double mu)
	Generate a compressible Mooney-Rivlin material law \( W(F)=c_0\iota_1(F^T F) + c_1\iota_2(F^T F) + d_0\Gamma_\mathrm{In}(\det(F))+d_1\Gamma_\mathrm{Co}(\det(F)) \), where \(\iota_1\) is the first and \(\iota_2\) the second principal matrix invariant. The parameters \(c_0,c_1,d_0,d_1\) are chosen such that for \(F\rightarrow I\) the model asymptotically yields Hooke's model of linearized elasticity with Lam\'e constants \(\lambda,\mu\). Here \(I\) denotes the unit matrix. More...
template<linalg::Matrix Mat, class InflationPenalty , class CompressionPenalty >
auto	create_mooney_rivlin_from_material_constants (double E, double nu)
	Generate a compressible Mooney-Rivlin material law \( W(F)=c_0\iota_1(F^T F) + c_1\iota_2(F^T F) + d_0\Gamma_\mathrm{In}(\det(F))+d_1\Gamma_\mathrm{Co}(\det(F)) \), where \(\iota_1\) is the first and \(\iota_2\) the second principal matrix invariant. The parameters \(c_0,c_1,d_0,d_1\) are chosen such that for \(F\rightarrow I\) the model asymptotically yields Hooke's model of linearized elasticity with Young's modulus \(E\) and Poisson ratio \(\nu\). Here \(I\) denotes the unit matrix. More...
template<linalg::Matrix M, int n = linalg::dim< M >()>
auto	incompressible_neo_hooke (double c, const M &F)
	Generate an "incompressible" neo-Hookean material law \( W(F)=c\iota_1(F^T F) \), where \(\iota_1\) is the first principal matrix invariant .
template<linalg::Matrix M, int n = linalg::dim< M >()>
auto	modified_incompressible_neo_hooke (double c, const M &F)
	Generate an "incompressible" neo-Hookean material law \( W(F)=c\bar\iota_1(F^T F) \), where \(\bar\iota_1\) is the modified first principal matrix invariant.
template<class InflationPenalty , class CompressionPenalty , linalg::Matrix M, int n = linalg::dim< M >()>
auto	compressible_neo_hooke (double c, double d0, double d1, const M &F)
	Generate a compressible neo-Hookean material law \( W(F)=c\iota_1(F^T F)+d_0\Gamma_\mathrm{In}(\det(F))+d_1\Gamma_\mathrm{Co}(\det(F)) \), where \(\iota_1\) is the first principal matrix invariant.
template<class InflationPenalty , class CompressionPenalty , linalg::Matrix M, int n = linalg::dim< M >()>
auto	modified_compressible_neo_hooke (double c, double d0, double d1, const M &F)
	Generate a compressible neo-Hookean material law \( W(F)=c\bar\iota_1(F^T F)+d_0\Gamma_\mathrm{In}(\det(F))+d_1\Gamma_\mathrm{Co}(\det(F)) \), where \(\bar\iota_1\) is the modified first principal matrix invariant.
template<class Inflation , class Compression , linalg::Matrix M>
auto	volumetric_penalty (double d0, double d1, const M &A)
template<linalg::Matrix M>
auto	volumetric_quad_and_log (double d0, double d1, const M &A)
	Create the volumetric penalty function \( d_0 j^2 + d_1 \log(j),\ j=\det(A) \).
template<linalg::Matrix M>
auto	volumetric_hartmann_neff (double d0, double d1, const M &A)
	Create the volumetric penalty function \( d_0 j^5 + d_1 j^{-5},\ j=\det(A) \).
template<linalg::Matrix M>
auto	yield_surface (double beta, double offset, M sigma=linalg::unit_matrix< M >())
template<Function F>
auto	finalize (F &&f) requires(static_check
	Finish function definition. More...
template<Function F, Function G>
auto	operator+ (F &&f, G &&g)
	overload of "+"-operator for the generation of functions. More...
template<Function F, Function G>
auto	operator* (F &&f, G &&g)
	overload of "*"-operator for the generation of functions. More...
template<Function F, Function G>
auto	dot (F &&f, G &&g)
	overload of "dot"-function for the generation of functions. More...
template<Function F>
auto	squared (F &&f)
	Generate squared function. More...
template<Function F, Function G>
auto	operator- (F &&f, G &&g)
	overload of "-"-operator for the generation of functions. More...
template<class F , class G >
auto	operator/ (F &&f, G &&g) requires(requires(G g)
	overload of "/"-operator for the generation of functions. More...
template<class F , Arithmetic G>
auto	operator/ (F &&f, G g)
	overload of "/"-operator for the generation of functions. More...
template<class Arg >
auto	identity (Arg &&x)
	Construct Identity<Arg>(x).
template<Function F, Function G>
decltype(auto)	max (F &&f, G &&g)
template<Function F, Function G>
decltype(auto)	min (F &&f, G &&g)
template<Function F, class Arg >
void	update_if_present (F &&, Arg &&)
template<int id, Function F, class Arg >
void	update_if_present (F &&, Arg &&)
template<Function F, class... IndexedArgs>
void	bulk_update_if_present (F &&, IndexedArgs &&...)
template<Function F, class IndexedArg , class... IndexedArgs>
void	bulk_update_if_present (F &&f, IndexedArg &&x0, IndexedArgs &&... x)
template<int id, class T >
Variable< T, id >	variable (const T &t)
	Generate variable from input type.

Variables
template<class T >
concept	Function
	Function concept. More...
template<class T >
concept	Arithmetic = is_arithmetic< T >::value
	Extensible concept for arithmetic types. More...
template<class F >
constexpr bool	is_arithmetic_v = is_arithmetic< F >::value

Detailed Description

Main namespace of the funcy library.

See also

Mathematical Operations

Common Mathematical Functions

Linear Algebra

Function Documentation

const_ref()

template<class Arg >

auto funcy::const_ref ( const Arg &  x )

noexcept

Generate a constant function that stores its argument as constant reference.

This admits to use variable constant arguments, i.e. parameters that we want to study.

constant()

template<class Arg >

constexpr auto funcy::constant ( Arg &&  x )

noexcept

Wrap a constant.

Returns

Constant<Arg>(x)

dot()

template<Function F, Function G>

auto funcy::dot	(	F &&	f,
		G &&	g
	)

overload of "dot"-function for the generation of functions.

If the resulting type represents a polynomial of order smaller than two, than you need to wrap it into Finalize to generate missing derivatives.

finalize()

template<Function F>

auto funcy::finalize

(

F &&

f

)

Finish function definition.

Adds the definition of possibly undefined vanishing higher order derivatives. If the template class Variable is not used, then no ids must be provided for the update-function and derivatives.

heat_model()

template<class Scalar , linalg::Vector Vec, idx = 0>

auto funcy::heat_model	(	double	c,
		double	d,
		Scalar	u,
		const Vec &	du
	)

Weak model for nonlinear heat transfer \( (c+du^2)\nabla u \).

Parameters

c	weighing of linearity
d	weighing of nonlinearity
u	heat
du	heat gradient

operator*()

template<Function F, Function G>

auto funcy::operator*	(	F &&	f,
		G &&	g
	)

overload of "*"-operator for the generation of functions.

If the resulting type represents a polynomial of order smaller than two, than you need to wrap it into Finalize to generate missing derivatives.

operator+()

template<Function F, Function G>

auto funcy::operator+	(	F &&	f,
		G &&	g
	)

overload of "+"-operator for the generation of functions.

If the resulting type represents a polynomial of order smaller than two, than you need to wrap it into Finalize to generate missing derivatives.

operator-()

template<Function F, Function G>

auto funcy::operator-	(	F &&	f,
		G &&	g
	)

overload of "-"-operator for the generation of functions.

If the resulting type represents a polynomial of order smaller than two, than you need to wrap it into Finalize to generate missing derivatives.

operator/() [1/2]

template<class F , class G >

auto funcy::operator/	(	F &&	f,
		G &&	g
	)

overload of "/"-operator for the generation of functions.

Only for functions that return types that are convertible to double If the resulting type represents a polynomial of order smaller than two, than you need to wrap it into Finalize to generate missing derivatives.

operator/() [2/2]

template<class F , Arithmetic G>

auto funcy::operator/	(	F &&	f,
		G	g
	)

overload of "/"-operator for the generation of functions.

Only for functions that return types that are convertible to double If the resulting type represents a polynomial of order smaller than two, than you need to wrap it into Finalize to generate missing derivatives.

squared()

template<Function F>

auto funcy::squared

(

F &&

f

)

Generate squared function.

If the resulting type represents a polynomial of order smaller than two, than you need to wrap it into Finalize to generate missing derivatives.

volumetric_penalty()

template<class Inflation , class Compression , linalg::Matrix M>

auto funcy::volumetric_penalty	(	double	d0,
		double	d1,
		const M &	A
	)

Create volumetric penalty function composed of a penalty for inflation and one for compression.

yield_surface()

template<linalg::Matrix M>

auto funcy::yield_surface	(	double	beta,
		double	offset,
		M	sigma = linalg::unit_matrix< M >()
	)

Yield surface \( \frac{\beta}{3}\iota_1(\sigma) + J_2(\sigma)-offset \), where \(\iota_1\) is the first principal and \(J_2\) is the second deviatoric invariant.