Pullback, pushforward, volume

$f_i=f(x({\zeta }_i,{\eta }_i),y({\zeta }_i,{\eta }_i))$, $\sqrt{g}$ More...

Collaboration diagram for Pullback, pushforward, volume:

Functions
template<class Functor , class Geometry >
Geometry::host_vector	dg::pullback (const Functor &f, const Geometry &g)
	$f_i=f(x({\zeta }_i,{\eta }_i),y({\zeta }_i,{\eta }_i))$
template<class Functor1 , class Functor2 , class container , class Geometry >
void	dg::pushForwardPerp (const Functor1 &vR, const Functor2 &vZ, container &vx, container &vy, const Geometry &g)
	$\overline{v}=Jv$
template<class Functor1 , class Functor2 , class Functor3 , class container , class Geometry >
void	dg::pushForward (const Functor1 &vR, const Functor2 &vZ, const Functor3 &vPhi, container &vx, container &vy, container &vz, const Geometry &g)
	$\overline{v}=Jv$
template<class FunctorRR , class FunctorRZ , class FunctorZZ , class container , class Geometry >
void	dg::pushForwardPerp (const FunctorRR &chiRR, const FunctorRZ &chiRZ, const FunctorZZ &chiZZ, SparseTensor< container > &chi, const Geometry &g)
	$\overline{\chi }=J\chi J^T$
template<class Geometry >
Geometry::host_vector	dg::create::volume (const Geometry &g)
	Create the volume element on the grid (including weights!!)
template<class Geometry >
Geometry::host_vector	dg::create::inv_volume (const Geometry &g)
	Create the inverse volume element on the grid (including weights!!)

Detailed Description

$f_i=f(x({\zeta }_i,{\eta }_i),y({\zeta }_i,{\eta }_i))$, $\sqrt{g}$

Function Documentation

inv_volume()

template<class Geometry >

Geometry::host_vector dg::create::inv_volume

(

const Geometry &

g

)

Create the inverse volume element on the grid (including weights!!)

This is the same as the inv_weights divided by the volume form $\sqrt{g}$

Template Parameters

Geometry

A type that is or derives from one of the abstract geometry base classes ( aGeometry2d, aGeometry3d, aMPIGeometry2d, ...).

Parameters

g	Geometry object

Returns

The inverse volume form

pullback()

template<class Functor , class Geometry >

Geometry::host_vector dg::pullback	(	const Functor &	f,
		const Geometry &	g )

$f_i=f(x({\zeta }_i,{\eta }_i),y({\zeta }_i,{\eta }_i))$

Pull back a function defined in physical coordinates to the curvilinear (computational) coordinate system. The pullback is equivalent to the following:

1. generate the list of physical space coordinates (e.g. in 2d $x_i=x({\zeta }_i,{\eta }_i),y_i=y({\zeta }_i,{\eta }_i)$ for all i) using the map member of the grid e.g. aRealGeometry2d::map()
2. evaluate the given function or functor at these coordinates and store the result in the output vector (e.g. in 2d $v_i=f(x_i,y_i)$ for all i)

Note

the grid defines what its physical coordinates are, i.e. it could be either Cartesian or Cylindrical coordinates

Template Parameters

Functor

The binary (for 2d grids) or ternary (for 3d grids) function or functor with signature: real_type ( real_type x, real_type y) ; real_type ( real_type x, real_type y, real_type z)

Parameters

f	The function defined in physical coordinates
g	a two- or three dimensional Geometry (g.map() is used to evaluate f)

Note

Template deduction for the Functor will fail if you overload functions with different dimensionality (e.g. real_type sine( real_type x) and real_type sine(real_type x, real_type y) ) You will want to rename those uniquely

Returns

The output vector v as a host vector

See also

If the function is defined in computational space coordinates, then use dg::evaluate

pushForward()

template<class Functor1 , class Functor2 , class Functor3 , class container , class Geometry >

void dg::pushForward	(	const Functor1 &	vR,
		const Functor2 &	vZ,
		const Functor3 &	vPhi,
		container &	vx,
		container &	vy,
		container &	vz,
		const Geometry &	g )

$\overline{v}=Jv$

Push forward a vector from cylindrical or Cartesian to a new coordinate system. Applies the Jacobian matrix $\overline{v}=Jv$. With $v^R=v^R(R(x,y,z),Z(x,y,z),\phi (x,y,z))$ and analogous $v^Z$, $v^{\phi }$, and the elements of $J=J(x,y,z)$:

$$\begin{array}{r}{v}^x(x,y,z)=x_R{v}^R+x_Z{v}^Z+x_{\phi }{v}^{\phi }\\ v^y(x,y,z)=y_R{v}^R+y_Z{v}^Z+y_{\phi }{v}^{\phi }\\ v^z(x,y,z)=z_R{v}^R+z_Z{v}^Z+z_{\phi }{v}^{\phi }\end{array}$$

where $x_R=\frac{\partial x}{\partial R}$, ...

Template Parameters

Functor1	Binary or Ternary functor
Functor2	Binary or Ternary functor
Functor3	Binary or Ternary functor

Template Parameters

Container

A data container class for which the blas1 functionality is overloaded and to which the return type of blas1::subroutine() can be converted using dg::construct. We assume that Container is copyable/assignable and has a swap member function. In connection with Geometry this is one of dg::HVec, dg::DVec when Geometry is a shared memory geometry dg::MHVec or dg::MDVec when Geometry is one of the MPI geometries

Geometry

A type that is or derives from one of the abstract geometry base classes ( aGeometry2d, aGeometry3d, aMPIGeometry2d, ...). Geometry determines which Container type can be used.

Parameters

vR	input R-component in cartesian or cylindrical coordinates
vZ	input Z-component in cartesian or cylindrical coordinates
vPhi	input Z-component in cartesian or cylindrical coordinates
vx	x-component of vector (gets properly resized)
vy	y-component of vector (gets properly resized)
vz	z-component of vector (gets properly resized)
g	The geometry object

pushForwardPerp() [1/2]

template<class Functor1 , class Functor2 , class container , class Geometry >

void dg::pushForwardPerp	(	const Functor1 &	vR,
		const Functor2 &	vZ,
		container &	vx,
		container &	vy,
		const Geometry &	g )

$\overline{v}=Jv$

Push forward a vector from cylindrical or Cartesian to a new coordinate system. Applies the Jacobian matrix $\overline{v}=Jv$:

$$\begin{array}{r}{v}^x(x,y)=x_R(x,y)v^R(R(x,y),Z(x,y))+x_Z{v}^Z(R(x,y),Z(x,y))\\ v^y(x,y)=y_R(x,y)v^R(R(x,y),Z(x,y))+y_Z{v}^Z(R(x,y),Z(x,y))\end{array}$$

where $x_R=\frac{\partial x}{\partial R}$, ...

Template Parameters

Functor1	Binary or Ternary functor
Functor2	Binary or Ternary functor

Template Parameters

Container

A data container class for which the blas1 functionality is overloaded and to which the return type of blas1::subroutine() can be converted using dg::construct. We assume that Container is copyable/assignable and has a swap member function. In connection with Geometry this is one of dg::HVec, dg::DVec when Geometry is a shared memory geometry dg::MHVec or dg::MDVec when Geometry is one of the MPI geometries

Geometry

A type that is or derives from one of the abstract geometry base classes ( aGeometry2d, aGeometry3d, aMPIGeometry2d, ...). Geometry determines which Container type can be used.

Parameters

vR	input R-component in cylindrical coordinates
vZ	input Z-component in cylindrical coordinates
vx	x-component of vector (gets properly resized)
vy	y-component of vector (gets properly resized)
g	The geometry object

pushForwardPerp() [2/2]

template<class FunctorRR , class FunctorRZ , class FunctorZZ , class container , class Geometry >

void dg::pushForwardPerp	(	const FunctorRR &	chiRR,
		const FunctorRZ &	chiRZ,
		const FunctorZZ &	chiZZ,
		SparseTensor< container > &	chi,
		const Geometry &	g )

$\overline{\chi }=J\chi J^T$

Push forward a symmetric 2d tensor from cylindrical or Cartesian to a new coordinate system. Applies the Jacobian matrix $\overline{\chi }=J\chi J^T$:

$$\begin{array}{r}{\chi }^{xx}(x,y)=x_R^2{\chi }^{RR}+2x_R{x}_Z{\chi }^{RZ}+x_Z^2{\chi }^{ZZ}\\ {\chi }^{xy}(x,y)=x_R{y}_R{\chi }^{RR}+(x_R{y}_Z+y_R{x}_Z){\chi }^{RZ}+x_Z{y}_Z{\chi }^{ZZ}\\ {\chi }^{yy}(x,y)=y_R^2{\chi }^{RR}+2y_R{y}_Z{\chi }^{RZ}+y_Z^2{\chi }^{ZZ}\end{array}$$

where $x_R=\frac{\partial x}{\partial R}$, ...

Template Parameters

FunctorRR	Binary or Ternary functor
FunctorRZ	Binary or Ternary functor
FunctorZZ	Binary or Ternary functor

Template Parameters

Container

A data container class for which the blas1 functionality is overloaded and to which the return type of blas1::subroutine() can be converted using dg::construct. We assume that Container is copyable/assignable and has a swap member function. In connection with Geometry this is one of dg::HVec, dg::DVec when Geometry is a shared memory geometry dg::MHVec or dg::MDVec when Geometry is one of the MPI geometries

Geometry

A type that is or derives from one of the abstract geometry base classes ( aGeometry2d, aGeometry3d, aMPIGeometry2d, ...). Geometry determines which Container type can be used.

Parameters

chiRR	input RR-component in cylindrical coordinates
chiRZ	input RZ-component in cylindrical coordinates
chiZZ	input ZZ-component in cylindrical coordinates
chi	tensor (gets properly resized)
g	The geometry object

volume()

template<class Geometry >

Geometry::host_vector dg::create::volume

(

const Geometry &

g

)

Create the volume element on the grid (including weights!!)

This is the same as the weights multiplied by the volume form $\sqrt{g}$

Template Parameters

Geometry

A type that is or derives from one of the abstract geometry base classes ( aGeometry2d, aGeometry3d, aMPIGeometry2d, ...).

Parameters

g	Geometry object

Returns

The volume form