Example solution of an Advection Diffusion equation on a quadrilateral or triangular mesh using the CVFEM. More...

#include "Intrepid_FunctionSpaceTools.hpp"
#include "Intrepid_CellTools.hpp"
#include "Intrepid_ArrayTools.hpp"
#include "Intrepid_Basis.hpp"
#include "Intrepid_HGRAD_QUAD_C1_FEM.hpp"
#include "Intrepid_HCURL_QUAD_I1_FEM.hpp"
#include "Intrepid_HGRAD_TRI_C1_FEM.hpp"
#include "Intrepid_HCURL_TRI_I1_FEM.hpp"
#include "Intrepid_RealSpaceTools.hpp"
#include "Intrepid_DefaultCubatureFactory.hpp"
#include "Intrepid_Cubature.hpp"
#include "Intrepid_CubatureControlVolume.hpp"
#include "Intrepid_CubatureControlVolumeSide.hpp"
#include "Intrepid_Utils.hpp"
#include "Epetra_Time.h"
#include "Epetra_Map.h"
#include "Epetra_SerialComm.h"
#include "Epetra_FECrsMatrix.h"
#include "Epetra_FEVector.h"
#include "Epetra_Import.h"
#include "Teuchos_oblackholestream.hpp"
#include "Teuchos_RCP.hpp"
#include "Teuchos_BLAS.hpp"
#include "Teuchos_GlobalMPISession.hpp"
#include "Teuchos_XMLParameterListHelpers.hpp"
#include "Shards_CellTopology.hpp"
#include "EpetraExt_RowMatrixOut.h"
#include "EpetraExt_MultiVectorOut.h"
#include "AztecOO.h"
#include "ml_MultiLevelPreconditioner.h"
#include "ml_epetra_utils.h"
#include "Sacado.hpp"

Include dependency graph for example_CVFEM.cpp:

Functions
template<typename Scalar >
const Scalar	exactSolution (const Scalar &x, const Scalar &y)
	User-defined exact solution. More...

template<typename Scalar >
void	advectionVelocity (Scalar advVel[2], const Scalar &x, const Scalar &y, const std::string problem)
	User-defined advection velocity. More...

template<typename Scalar1 , typename Scalar2 >
const Scalar1	diffusivity (const Scalar1 &x, const Scalar1 &y, const Scalar2 &epsilon, const bool variableEpsilon)
	User-defined diffusivity. More...

template<typename Scalar >
void	exactSolutionGrad (Scalar gradExact[2], const Scalar &x, const Scalar &y)
	Computes gradient of the exact solution. Requires user-defined exact solution. More...

template<typename Scalar1 , typename Scalar2 >
const Scalar1	sourceTerm (Scalar1 &x, Scalar1 &y, Scalar2 &epsilon, const bool variableEpsilon, const std::string problem)
	Computes source term: f = -div(J_n). Requires user-defined exact solution and material parameters. More...

template<class ArrayOut , class ArrayIn >
void	evaluateExactSolution (ArrayOut &exactSolutionValues, const ArrayIn &evaluationPoints)
	Computation of the exact solution at array of points in physical space. More...

template<class ArrayOut , class ArrayIn >
void	evaluateExactSolutionGrad (ArrayOut &exactSolutionGradValues, const ArrayIn &evaluationPoints)
	Computation of the gradient of the exact solution at array of points in physical space. More...

template<class ArrayOut1 , class ArrayOut2 , class Scalar >
void	createMesh (ArrayOut1 &elemToNode, ArrayOut2 &nodeCoords, ArrayOut1 &bcLeftId, ArrayOut1 &bcRightId, ArrayOut1 &bcTopId, ArrayOut1 &bcBotId, const std::string &meshType, const Scalar &meshSize)
	Create quadrilateral or triangle mesh on the domain [0,1]x[0,1]. More...

int	main (int argc, char *argv[])

Detailed Description

Example solution of an Advection Diffusion equation on a quadrilateral or triangular mesh using the CVFEM.

 Advection diffusion system:

      - div (epsilon grad phi - u phi) = f in Omega
                                   phi = g on Gamma

   where
         u is the advection velocity
         epsilon is the diffusion coefficient
         f is a given source term


 Corresponding discrete linear system for nodal coefficients(x):

             Kx = b

        K - HGrad stiffness matrix
        b - right hand side vector

Author: K. Peterson

Remarks: Usage:
./TrilinosCouplings_Example_CVFEM.exe <input.xml>; Example requires an xml input file with mesh and solver settings.

NOTE: This problem does not currently work on more than one core (csief.nosp@m.er@s.nosp@m.andia.nosp@m..gov 3/17/16)

Function Documentation

template<typename Scalar >

void advectionVelocity	(	Scalar	advVel[2],
		const Scalar &	x,
		const Scalar &	y,
		const std::string	problem
	)

User-defined advection velocity.

Parameters

advVel	[out] advection velocity evaluated at (x,y)
x	[in] x-coordinate of the evaluation point
y	[in] y-coordinate of the evaluation point

Referenced by sourceTerm().

template<class ArrayOut1 , class ArrayOut2 , class Scalar >

void createMesh	(	ArrayOut1 &	elemToNode,
		ArrayOut2 &	nodeCoords,
		ArrayOut1 &	bcLeftId,
		ArrayOut1 &	bcRightId,
		ArrayOut1 &	bcTopId,
		ArrayOut1 &	bcBotId,
		const std::string &	meshType,
		const Scalar &	meshSize
	)

Create quadrilateral or triangle mesh on the domain [0,1]x[0,1].

Parameters

elemToNode	[out] Array with element to node mapping
nodeCoords	[out] Array with nodal coordinates
bcLeftId	[out] Array with ids of left boundary nodes,
bcRightId	[out] Array with ids of right boundary nodes
bcTopId	[out] Array with ids of top boundary nodes
bcBotId	[out] Array with ids of bottom boundary nodes
meshType	[in] Mesh type (quad or tri)
meshSize	[in] Number of elements in each direction

template<typename Scalar1 , typename Scalar2 >

const Scalar1 diffusivity	(	const Scalar1 &	x,
		const Scalar1 &	y,
		const Scalar2 &	epsilon,
		const bool	variableEpsilon
	)

User-defined diffusivity.

Parameters

x	[in] x-coordinate of the evaluation point
y	[in] y-coordinate of the evaluation point
epsilon	[in] diffusion coefficient from xml imput file

Returns: Value of the diffusivity at (x,y)

Referenced by sourceTerm().

template<class ArrayOut , class ArrayIn >

void evaluateExactSolution	(	ArrayOut &	exactSolutionValues,
		const ArrayIn &	evaluationPoints
	)

Computation of the exact solution at array of points in physical space.

Parameters

exactSolutionValues	[out] Rank-2 (C,P) array with the values of the exact solution
evaluationPoints	[in] Rank-3 (C,P,D) array with the evaluation points in physical frame

template<class ArrayOut , class ArrayIn >

void evaluateExactSolutionGrad	(	ArrayOut &	exactSolutionGradValues,
		const ArrayIn &	evaluationPoints
	)

Computation of the gradient of the exact solution at array of points in physical space.

Parameters

exactSolutionGradValues	[out] Rank-3 (C,P,D) array with the values of the gradient of the exact solution
evaluationPoints	[in] Rank-3 (C,P,D) array with the evaluation points in physical frame

template<typename Scalar >

const Scalar exactSolution	(	const Scalar &	x,
		const Scalar &	y
	)

User-defined exact solution.

Parameters

x	[in] x-coordinate of the evaluation point
y	[in] y-coordinate of the evaluation point

Returns: Value of the exact solution at (x,y)

Referenced by exactSolutionGrad(), and sourceTerm().

template<typename Scalar >

void exactSolutionGrad	(	Scalar	gradExact[2],
		const Scalar &	x,
		const Scalar &	y
	)

Computes gradient of the exact solution. Requires user-defined exact solution.

Parameters

gradExact	[out] gradient of the exact solution evaluated at (x,y)
x	[in] x-coordinate of the evaluation point
y	[in] y-coordinate of the evaluation point

References exactSolution().

Referenced by sourceTerm().

template<typename Scalar1 , typename Scalar2 >

const Scalar1 sourceTerm	(	Scalar1 &	x,
		Scalar1 &	y,
		Scalar2 &	epsilon,
		const bool	variableEpsilon,
		const std::string	problem
	)

Computes source term: f = -div(J_n). Requires user-defined exact solution and material parameters.

Parameters

x	[in] x-coordinate of the evaluation point
y	[in] y-coordinate of the evaluation point
epsilon	[in] diffusion coefficient

Returns: Source term corresponding to the user-defined exact solution evaluated at (x,y)

References advectionVelocity(), diffusivity(), exactSolution(), and exactSolutionGrad().

Functions

Detailed Description

Function Documentation