fenl_functors.hpp
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fenl_functors.hpp
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	/*
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	// Kokkos v. 2.0
	// Copyright (2014) Sandia Corporation
	//
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	#ifndef KOKKOS_EXAMPLE_FENLFUNCTORS_HPP
	#define KOKKOS_EXAMPLE_FENLFUNCTORS_HPP

	#include <cstdio>

	#include <iostream>
	#include <fstream>
	#include <iomanip>
	#include <cstdlib>
	#include <cmath>
	#include <limits>

	#include <Kokkos_Pair.hpp>
	#include <Kokkos_UnorderedMap.hpp>

	#include <impl/Kokkos_Timer.hpp>

	#include <BoxElemFixture.hpp>
	#include <HexElement.hpp>
	#include <CGSolve.hpp>

	//----------------------------------------------------------------------------
	//----------------------------------------------------------------------------

	namespace Kokkos {
	namespace Example {
	namespace FENL {

	template< class ElemNodeIdView , class CrsGraphType , unsigned ElemNode >
	class NodeNodeGraph {
	public:

	typedef typename ElemNodeIdView::execution_space execution_space ;
	typedef pair<unsigned,unsigned> key_type ;

	typedef Kokkos::UnorderedMap< key_type, void , execution_space > SetType ;
	typedef typename CrsGraphType::row_map_type::non_const_type RowMapType ;
	typedef Kokkos::View< unsigned , execution_space > UnsignedValue ;

	// Static dimensions of 0 generate compiler warnings or errors.
	typedef Kokkos::View< unsigned*[ElemNode][ElemNode] , execution_space >
	ElemGraphType ;

	struct TagFillNodeSet {};
	struct TagScanNodeCount {};
	struct TagFillGraphEntries {};
	struct TagSortGraphEntries {};
	struct TagFillElementGraph {};

	private:

	enum PhaseType { FILL_NODE_SET ,
	SCAN_NODE_COUNT ,
	FILL_GRAPH_ENTRIES ,
	SORT_GRAPH_ENTRIES ,
	FILL_ELEMENT_GRAPH };

	const unsigned node_count ;
	const ElemNodeIdView elem_node_id ;
	UnsignedValue row_total ;
	RowMapType row_count ;
	RowMapType row_map ;
	SetType node_node_set ;
	PhaseType phase ;

	public:

	CrsGraphType graph ;
	ElemGraphType elem_graph ;

	struct Times
	{
	double ratio;
	double fill_node_set;
	double scan_node_count;
	double fill_graph_entries;
	double sort_graph_entries;
	double fill_element_graph;
	};

	NodeNodeGraph( const ElemNodeIdView & arg_elem_node_id ,
	const unsigned arg_node_count,
	Times & results
	)
	: node_count(arg_node_count)
	, elem_node_id( arg_elem_node_id )
	, row_total( "row_total" )
	, row_count(Kokkos::ViewAllocateWithoutInitializing("row_count") , node_count ) // will deep_copy to 0 inside loop
	, row_map( "graph_row_map" , node_count + 1 )
	, node_node_set()
	, phase( FILL_NODE_SET )
	, graph()
	, elem_graph()
	{
	//--------------------------------
	// Guess at capacity required for the map:

	Kokkos::Timer wall_clock ;

	wall_clock.reset();
	phase = FILL_NODE_SET ;

	// upper bound on the capacity
	size_t set_capacity = (28ull * node_count) / 2;
	unsigned failed_insert_count = 0 ;

	do {
	// Zero the row count to restart the fill
	Kokkos::deep_copy( row_count , 0u );

	node_node_set = SetType( ( set_capacity += failed_insert_count ) );

	// May be larger that requested:
	set_capacity = node_node_set.capacity();

	Kokkos::parallel_reduce( Kokkos::RangePolicy<execution_space,TagFillNodeSet>(0,elem_node_id.dimension_0())
	, *this
	, failed_insert_count );

	} while ( failed_insert_count );

	execution_space::fence();
	results.ratio = (double)node_node_set.size() / (double)node_node_set.capacity();
	results.fill_node_set = wall_clock.seconds();
	//--------------------------------

	wall_clock.reset();
	phase = SCAN_NODE_COUNT ;

	// Exclusive scan of row_count into row_map
	// including the final total in the 'node_count + 1' position.
	// Zero the 'row_count' values.
	Kokkos::parallel_scan( node_count , *this );

	// Zero the row count for the fill:
	Kokkos::deep_copy( row_count , 0u );

	unsigned graph_entry_count = 0 ;

	Kokkos::deep_copy( graph_entry_count , row_total );

	// Assign graph's row_map and allocate graph's entries
	graph.row_map = row_map ;
	graph.entries = typename CrsGraphType::entries_type( "graph_entries" , graph_entry_count );

	//--------------------------------
	// Fill graph's entries from the (node,node) set.

	execution_space::fence();
	results.scan_node_count = wall_clock.seconds();

	wall_clock.reset();
	phase = FILL_GRAPH_ENTRIES ;
	Kokkos::parallel_for( node_node_set.capacity() , *this );

	execution_space::fence();
	results.fill_graph_entries = wall_clock.seconds();

	//--------------------------------
	// Done with the temporary sets and arrays
	wall_clock.reset();
	phase = SORT_GRAPH_ENTRIES ;

	row_total = UnsignedValue();
	row_count = RowMapType();
	row_map = RowMapType();
	node_node_set.clear();

	//--------------------------------

	Kokkos::parallel_for( node_count , *this );

	execution_space::fence();
	results.sort_graph_entries = wall_clock.seconds();

	//--------------------------------
	// Element-to-graph mapping:
	wall_clock.reset();
	phase = FILL_ELEMENT_GRAPH ;
	elem_graph = ElemGraphType("elem_graph", elem_node_id.dimension_0() );
	Kokkos::parallel_for( elem_node_id.dimension_0() , *this );

	execution_space::fence();
	results.fill_element_graph = wall_clock.seconds();
	}

	//------------------------------------
	// parallel_for: create map and count row length

	KOKKOS_INLINE_FUNCTION
	void operator()( const TagFillNodeSet & , unsigned ielem , unsigned & count ) const
	{
	// Loop over element's (row_local_node,col_local_node) pairs:
	for ( unsigned row_local_node = 0 ; row_local_node < elem_node_id.dimension_1() ; ++row_local_node ) {

	const unsigned row_node = elem_node_id( ielem , row_local_node );

	for ( unsigned col_local_node = row_local_node ; col_local_node < elem_node_id.dimension_1() ; ++col_local_node ) {

	const unsigned col_node = elem_node_id( ielem , col_local_node );

	// If either node is locally owned then insert the pair into the unordered map:

	if ( row_node < row_count.dimension_0() \|\| col_node < row_count.dimension_0() ) {

	const key_type key = (row_node < col_node) ? make_pair( row_node, col_node ) : make_pair( col_node, row_node ) ;

	const typename SetType::insert_result result = node_node_set.insert( key );

	// A successfull insert: the first time this pair was added
	if ( result.success() ) {

	// If row node is owned then increment count
	if ( row_node < row_count.dimension_0() ) { atomic_fetch_add( & row_count( row_node ) , 1 ); }

	// If column node is owned and not equal to row node then increment count
	if ( col_node < row_count.dimension_0() && col_node != row_node ) { atomic_fetch_add( & row_count( col_node ) , 1 ); }
	}
	else if ( result.failed() ) {
	++count ;
	}
	}
	}
	}
	}

	KOKKOS_INLINE_FUNCTION
	void fill_graph_entries( const unsigned iset ) const
	{
	if ( node_node_set.valid_at(iset) ) {
	// Add each entry to the graph entries.

	const key_type key = node_node_set.key_at(iset) ;
	const unsigned row_node = key.first ;
	const unsigned col_node = key.second ;

	if ( row_node < row_count.dimension_0() ) {
	const unsigned offset = graph.row_map( row_node ) + atomic_fetch_add( & row_count( row_node ) , 1 );
	graph.entries( offset ) = col_node ;
	}

	if ( col_node < row_count.dimension_0() && col_node != row_node ) {
	const unsigned offset = graph.row_map( col_node ) + atomic_fetch_add( & row_count( col_node ) , 1 );
	graph.entries( offset ) = row_node ;
	}
	}
	}

	KOKKOS_INLINE_FUNCTION
	void sort_graph_entries( const unsigned irow ) const
	{
	const unsigned row_beg = graph.row_map( irow );
	const unsigned row_end = graph.row_map( irow + 1 );
	for ( unsigned i = row_beg + 1 ; i < row_end ; ++i ) {
	const unsigned col = graph.entries(i);
	unsigned j = i ;
	for ( ; row_beg < j && col < graph.entries(j-1) ; --j ) {
	graph.entries(j) = graph.entries(j-1);
	}
	graph.entries(j) = col ;
	}
	}

	KOKKOS_INLINE_FUNCTION
	void fill_elem_graph_map( const unsigned ielem ) const
	{
	for ( unsigned row_local_node = 0 ; row_local_node < elem_node_id.dimension_1() ; ++row_local_node ) {

	const unsigned row_node = elem_node_id( ielem , row_local_node );

	for ( unsigned col_local_node = 0 ; col_local_node < elem_node_id.dimension_1() ; ++col_local_node ) {

	const unsigned col_node = elem_node_id( ielem , col_local_node );

	unsigned entry = ~0u ;

	if ( row_node + 1 < graph.row_map.dimension_0() ) {

	const unsigned entry_end = graph.row_map( row_node + 1 );

	entry = graph.row_map( row_node );

	for ( ; entry < entry_end && graph.entries(entry) != col_node ; ++entry );

	if ( entry == entry_end ) entry = ~0u ;
	}

	elem_graph( ielem , row_local_node , col_local_node ) = entry ;
	}
	}
	}

	KOKKOS_INLINE_FUNCTION
	void operator()( const unsigned iwork ) const
	{
	/*
	if ( phase == FILL_NODE_SET ) {
	operator()( TagFillNodeSet() , iwork );
	}
	else */
	if ( phase == FILL_GRAPH_ENTRIES ) {
	fill_graph_entries( iwork );
	}
	else if ( phase == SORT_GRAPH_ENTRIES ) {
	sort_graph_entries( iwork );
	}
	else if ( phase == FILL_ELEMENT_GRAPH ) {
	fill_elem_graph_map( iwork );
	}
	}

	//------------------------------------
	// parallel_scan: row offsets

	typedef unsigned value_type ;

	KOKKOS_INLINE_FUNCTION
	void operator()( const unsigned irow , unsigned & update , const bool final ) const
	{
	// exclusive scan
	if ( final ) { row_map( irow ) = update ; }

	update += row_count( irow );

	if ( final ) {
	if ( irow + 1 == row_count.dimension_0() ) {
	row_map( irow + 1 ) = update ;
	row_total() = update ;
	}
	}
	}

	// For the reduce phase:
	KOKKOS_INLINE_FUNCTION
	void init( const TagFillNodeSet & , unsigned & update ) const { update = 0 ; }

	KOKKOS_INLINE_FUNCTION
	void join( const TagFillNodeSet &
	, volatile unsigned & update
	, volatile const unsigned & input ) const { update += input ; }

	// For the scan phase::
	KOKKOS_INLINE_FUNCTION
	void init( unsigned & update ) const { update = 0 ; }

	KOKKOS_INLINE_FUNCTION
	void join( volatile unsigned & update
	, volatile const unsigned & input ) const { update += input ; }

	//------------------------------------
	};

	} /* namespace FENL */
	} /* namespace Example */
	} /* namespace Kokkos */

	//----------------------------------------------------------------------------

	namespace Kokkos {
	namespace Example {
	namespace FENL {

	template< class ElemCompType >
	class NodeElemGatherFill {
	public:

	typedef typename ElemCompType::execution_space execution_space ;
	typedef typename ElemCompType::vector_type vector_type ;
	typedef typename ElemCompType::sparse_matrix_type sparse_matrix_type ;
	typedef typename ElemCompType::elem_node_type elem_node_type ;
	typedef typename ElemCompType::elem_vectors_type elem_vectors_type ;
	typedef typename ElemCompType::elem_matrices_type elem_matrices_type ;
	typedef typename ElemCompType::elem_graph_type elem_graph_type ;

	static const unsigned ElemNodeCount = ElemCompType::ElemNodeCount ;

	//------------------------------------

	private:

	typedef Kokkos::StaticCrsGraph< unsigned[2] , execution_space > CrsGraphType ;
	typedef typename CrsGraphType::row_map_type::non_const_type RowMapType ;
	typedef Kokkos::View< unsigned , execution_space > UnsignedValue ;

	enum PhaseType { FILL_NODE_COUNT ,
	SCAN_NODE_COUNT ,
	FILL_GRAPH_ENTRIES ,
	SORT_GRAPH_ENTRIES ,
	GATHER_FILL };

	const elem_node_type elem_node_id ;
	const elem_graph_type elem_graph ;
	UnsignedValue row_total ;
	RowMapType row_count ;
	RowMapType row_map ;
	CrsGraphType graph ;
	vector_type residual ;
	sparse_matrix_type jacobian ;
	elem_vectors_type elem_residual ;
	elem_matrices_type elem_jacobian ;
	PhaseType phase ;

	public:

	NodeElemGatherFill()
	: elem_node_id()
	, elem_graph()
	, row_total()
	, row_count()
	, row_map()
	, graph()
	, residual()
	, jacobian()
	, elem_residual()
	, elem_jacobian()
	, phase( FILL_NODE_COUNT )
	{}

	NodeElemGatherFill( const NodeElemGatherFill & rhs )
	: elem_node_id( rhs.elem_node_id )
	, elem_graph( rhs.elem_graph )
	, row_total( rhs.row_total )
	, row_count( rhs.row_count )
	, row_map( rhs.row_map )
	, graph( rhs.graph )
	, residual( rhs.residual )
	, jacobian( rhs.jacobian )
	, elem_residual( rhs.elem_residual )
	, elem_jacobian( rhs.elem_jacobian )
	, phase( rhs.phase )
	{}

	NodeElemGatherFill( const elem_node_type & arg_elem_node_id ,
	const elem_graph_type & arg_elem_graph ,
	const vector_type & arg_residual ,
	const sparse_matrix_type & arg_jacobian ,
	const elem_vectors_type & arg_elem_residual ,
	const elem_matrices_type & arg_elem_jacobian )
	: elem_node_id( arg_elem_node_id )
	, elem_graph( arg_elem_graph )
	, row_total( "row_total" )
	, row_count( "row_count" , arg_residual.dimension_0() )
	, row_map( "graph_row_map" , arg_residual.dimension_0() + 1 )
	, graph()
	, residual( arg_residual )
	, jacobian( arg_jacobian )
	, elem_residual( arg_elem_residual )
	, elem_jacobian( arg_elem_jacobian )
	, phase( FILL_NODE_COUNT )
	{
	//--------------------------------
	// Count node->element relations

	phase = FILL_NODE_COUNT ;

	Kokkos::parallel_for( elem_node_id.dimension_0() , *this );

	//--------------------------------

	phase = SCAN_NODE_COUNT ;

	// Exclusive scan of row_count into row_map
	// including the final total in the 'node_count + 1' position.
	// Zero the 'row_count' values.
	Kokkos::parallel_scan( residual.dimension_0() , *this );

	// Zero the row count for the fill:
	Kokkos::deep_copy( row_count , typename RowMapType::value_type(0) );

	unsigned graph_entry_count = 0 ;

	Kokkos::deep_copy( graph_entry_count , row_total );

	// Assign graph's row_map and allocate graph's entries
	graph.row_map = row_map ;

	typedef typename CrsGraphType::entries_type graph_entries_type ;

	graph.entries = graph_entries_type( "graph_entries" , graph_entry_count );

	//--------------------------------
	// Fill graph's entries from the (node,node) set.

	phase = FILL_GRAPH_ENTRIES ;

	Kokkos::deep_copy( row_count , 0u );
	Kokkos::parallel_for( elem_node_id.dimension_0() , *this );

	execution_space::fence();

	//--------------------------------
	// Done with the temporary sets and arrays

	row_total = UnsignedValue();
	row_count = RowMapType();
	row_map = RowMapType();

	//--------------------------------

	phase = SORT_GRAPH_ENTRIES ;
	Kokkos::parallel_for( residual.dimension_0() , *this );

	execution_space::fence();

	phase = GATHER_FILL ;
	}

	void apply() const
	{
	Kokkos::parallel_for( residual.dimension_0() , *this );
	}

	//------------------------------------
	//------------------------------------
	// parallel_for: Count node->element pairs

	KOKKOS_INLINE_FUNCTION
	void fill_node_count( const unsigned ielem ) const
	{
	for ( unsigned row_local_node = 0 ; row_local_node < elem_node_id.dimension_1() ; ++row_local_node ) {

	const unsigned row_node = elem_node_id( ielem , row_local_node );

	if ( row_node < row_count.dimension_0() ) {
	atomic_fetch_add( & row_count( row_node ) , 1 );
	}
	}
	}

	KOKKOS_INLINE_FUNCTION
	void fill_graph_entries( const unsigned ielem ) const
	{
	for ( unsigned row_local_node = 0 ; row_local_node < elem_node_id.dimension_1() ; ++row_local_node ) {

	const unsigned row_node = elem_node_id( ielem , row_local_node );

	if ( row_node < row_count.dimension_0() ) {

	const unsigned offset = graph.row_map( row_node ) + atomic_fetch_add( & row_count( row_node ) , 1 );

	graph.entries( offset , 0 ) = ielem ;
	graph.entries( offset , 1 ) = row_local_node ;
	}
	}
	}

	KOKKOS_INLINE_FUNCTION
	void sort_graph_entries( const unsigned irow ) const
	{
	const unsigned row_beg = graph.row_map( irow );
	const unsigned row_end = graph.row_map( irow + 1 );
	for ( unsigned i = row_beg + 1 ; i < row_end ; ++i ) {
	const unsigned elem = graph.entries(i,0);
	const unsigned local = graph.entries(i,1);
	unsigned j = i ;
	for ( ; row_beg < j && elem < graph.entries(j-1,0) ; --j ) {
	graph.entries(j,0) = graph.entries(j-1,0);
	graph.entries(j,1) = graph.entries(j-1,1);
	}
	graph.entries(j,0) = elem ;
	graph.entries(j,1) = local ;
	}
	}

	//------------------------------------

	KOKKOS_INLINE_FUNCTION
	void gather_fill( const unsigned irow ) const
	{
	const unsigned node_elem_begin = graph.row_map(irow);
	const unsigned node_elem_end = graph.row_map(irow+1);

	// for each element that a node belongs to

	for ( unsigned i = node_elem_begin ; i < node_elem_end ; i++ ) {

	const unsigned elem_id = graph.entries( i, 0);
	const unsigned row_index = graph.entries( i, 1);

	residual(irow) += elem_residual(elem_id, row_index);

	// for each node in a particular related element
	// gather the contents of the element stiffness
	// matrix that belong in irow

	for ( unsigned j = 0 ; j < ElemNodeCount ; ++j ) {
	const unsigned A_index = elem_graph( elem_id , row_index , j );

	jacobian.coeff( A_index ) += elem_jacobian( elem_id, row_index, j );
	}
	}
	}

	//------------------------------------

	KOKKOS_INLINE_FUNCTION
	void operator()( const unsigned iwork ) const
	{
	if ( phase == FILL_NODE_COUNT ) {
	fill_node_count( iwork );
	}
	else if ( phase == FILL_GRAPH_ENTRIES ) {
	fill_graph_entries( iwork );
	}
	else if ( phase == SORT_GRAPH_ENTRIES ) {
	sort_graph_entries( iwork );
	}
	else if ( phase == GATHER_FILL ) {
	gather_fill( iwork );
	}
	}

	//------------------------------------
	// parallel_scan: row offsets

	typedef unsigned value_type ;

	KOKKOS_INLINE_FUNCTION
	void operator()( const unsigned irow , unsigned & update , const bool final ) const
	{
	// exclusive scan
	if ( final ) { row_map( irow ) = update ; }

	update += row_count( irow );

	if ( final ) {
	if ( irow + 1 == row_count.dimension_0() ) {
	row_map( irow + 1 ) = update ;
	row_total() = update ;
	}
	}
	}

	KOKKOS_INLINE_FUNCTION
	void init( unsigned & update ) const { update = 0 ; }

	KOKKOS_INLINE_FUNCTION
	void join( volatile unsigned & update , const volatile unsigned & input ) const { update += input ; }
	};

	} /* namespace FENL */
	} /* namespace Example */
	} /* namespace Kokkos */

	//----------------------------------------------------------------------------

	namespace Kokkos {
	namespace Example {
	namespace FENL {

	template< class FiniteElementMeshType , class SparseMatrixType >
	class ElementComputation ;


	template< class ExecSpace , BoxElemPart::ElemOrder Order , class CoordinateMap , typename ScalarType >
	class ElementComputation<
	Kokkos::Example::BoxElemFixture< ExecSpace , Order , CoordinateMap > ,
	Kokkos::Example::CrsMatrix< ScalarType , ExecSpace > >
	{
	public:

	typedef Kokkos::Example::BoxElemFixture< ExecSpace, Order, CoordinateMap > mesh_type ;
	typedef Kokkos::Example::HexElement_Data< mesh_type::ElemNode > element_data_type ;

	typedef Kokkos::Example::CrsMatrix< ScalarType , ExecSpace > sparse_matrix_type ;
	typedef typename sparse_matrix_type::StaticCrsGraphType sparse_graph_type ;

	typedef ExecSpace execution_space ;
	typedef ScalarType scalar_type ;

	static const unsigned SpatialDim = element_data_type::spatial_dimension ;
	static const unsigned TensorDim = SpatialDim * SpatialDim ;
	static const unsigned ElemNodeCount = element_data_type::element_node_count ;
	static const unsigned FunctionCount = element_data_type::function_count ;
	static const unsigned IntegrationCount = element_data_type::integration_count ;

	//------------------------------------

	typedef typename mesh_type::node_coord_type node_coord_type ;
	typedef typename mesh_type::elem_node_type elem_node_type ;
	typedef Kokkos::View< scalar_type*[FunctionCount][FunctionCount] , execution_space > elem_matrices_type ;
	typedef Kokkos::View< scalar_type*[FunctionCount] , execution_space > elem_vectors_type ;
	typedef Kokkos::View< scalar_type* , execution_space > vector_type ;

	typedef typename NodeNodeGraph< elem_node_type , sparse_graph_type , ElemNodeCount >::ElemGraphType elem_graph_type ;

	//------------------------------------


	//------------------------------------
	// Computational data:

	const element_data_type elem_data ;
	const elem_node_type elem_node_ids ;
	const node_coord_type node_coords ;
	const elem_graph_type elem_graph ;
	const elem_matrices_type elem_jacobians ;
	const elem_vectors_type elem_residuals ;
	const vector_type solution ;
	const vector_type residual ;
	const sparse_matrix_type jacobian ;
	const scalar_type coeff_K ;

	ElementComputation( const ElementComputation & rhs )
	: elem_data()
	, elem_node_ids( rhs.elem_node_ids )
	, node_coords( rhs.node_coords )
	, elem_graph( rhs.elem_graph )
	, elem_jacobians( rhs.elem_jacobians )
	, elem_residuals( rhs.elem_residuals )
	, solution( rhs.solution )
	, residual( rhs.residual )
	, jacobian( rhs.jacobian )
	, coeff_K( rhs.coeff_K )
	{}

	// If the element->sparse_matrix graph is provided then perform atomic updates
	// Otherwise fill per-element contributions for subequent gather-add into a residual and jacobian.
	ElementComputation( const mesh_type & arg_mesh ,
	const scalar_type arg_coeff_K ,
	const vector_type & arg_solution ,
	const elem_graph_type & arg_elem_graph ,
	const sparse_matrix_type & arg_jacobian ,
	const vector_type & arg_residual )
	: elem_data()
	, elem_node_ids( arg_mesh.elem_node() )
	, node_coords( arg_mesh.node_coord() )
	, elem_graph( arg_elem_graph )
	, elem_jacobians()
	, elem_residuals()
	, solution( arg_solution )
	, residual( arg_residual )
	, jacobian( arg_jacobian )
	, coeff_K( arg_coeff_K )
	{}

	ElementComputation( const mesh_type & arg_mesh ,
	const scalar_type arg_coeff_K ,
	const vector_type & arg_solution )
	: elem_data()
	, elem_node_ids( arg_mesh.elem_node() )
	, node_coords( arg_mesh.node_coord() )
	, elem_graph()
	, elem_jacobians( "elem_jacobians" , arg_mesh.elem_count() )
	, elem_residuals( "elem_residuals" , arg_mesh.elem_count() )
	, solution( arg_solution )
	, residual()
	, jacobian()
	, coeff_K( arg_coeff_K )
	{}

	//------------------------------------

	void apply() const
	{
	parallel_for( elem_node_ids.dimension_0() , *this );
	}

	//------------------------------------

	static const unsigned FLOPS_transform_gradients =
	/* Jacobian / FunctionCount TensorDim * 2 +
	/* Inverse jacobian / TensorDim 6 + 6 +
	/* Gradient transform / FunctionCount 15 ;

	KOKKOS_INLINE_FUNCTION
	float transform_gradients(
	const float grad[][ FunctionCount ] , // Gradient of bases master element
	const double x[] ,
	const double y[] ,
	const double z[] ,
	float dpsidx[] ,
	float dpsidy[] ,
	float dpsidz[] ) const
	{
	enum { j11 = 0 , j12 = 1 , j13 = 2 ,
	j21 = 3 , j22 = 4 , j23 = 5 ,
	j31 = 6 , j32 = 7 , j33 = 8 };

	// Jacobian accumulation:

	double J[ TensorDim ] = { 0, 0, 0, 0, 0, 0, 0, 0, 0 };

	for( unsigned i = 0; i < FunctionCount ; ++i ) {
	const double x1 = x[i] ;
	const double x2 = y[i] ;
	const double x3 = z[i] ;

	const float g1 = grad[0][i] ;
	const float g2 = grad[1][i] ;
	const float g3 = grad[2][i] ;

	J[j11] += g1 * x1 ;
	J[j12] += g1 * x2 ;
	J[j13] += g1 * x3 ;

	J[j21] += g2 * x1 ;
	J[j22] += g2 * x2 ;
	J[j23] += g2 * x3 ;

	J[j31] += g3 * x1 ;
	J[j32] += g3 * x2 ;
	J[j33] += g3 * x3 ;
	}

	// Inverse jacobian:

	float invJ[ TensorDim ] = {
	static_cast<float>( J[j22] * J[j33] - J[j23] * J[j32] ) ,
	static_cast<float>( J[j13] * J[j32] - J[j12] * J[j33] ) ,
	static_cast<float>( J[j12] * J[j23] - J[j13] * J[j22] ) ,

	static_cast<float>( J[j23] * J[j31] - J[j21] * J[j33] ) ,
	static_cast<float>( J[j11] * J[j33] - J[j13] * J[j31] ) ,
	static_cast<float>( J[j13] * J[j21] - J[j11] * J[j23] ) ,

	static_cast<float>( J[j21] * J[j32] - J[j22] * J[j31] ) ,
	static_cast<float>( J[j12] * J[j31] - J[j11] * J[j32] ) ,
	static_cast<float>( J[j11] * J[j22] - J[j12] * J[j21] ) };

	const float detJ = J[j11] * invJ[j11] +
	J[j21] * invJ[j12] +
	J[j31] * invJ[j13] ;

	const float detJinv = 1.0 / detJ ;

	for ( unsigned i = 0 ; i < TensorDim ; ++i ) { invJ[i] *= detJinv ; }

	// Transform gradients:

	for( unsigned i = 0; i < FunctionCount ; ++i ) {
	const float g0 = grad[0][i];
	const float g1 = grad[1][i];
	const float g2 = grad[2][i];

	dpsidx[i] = g0 * invJ[j11] + g1 * invJ[j12] + g2 * invJ[j13];
	dpsidy[i] = g0 * invJ[j21] + g1 * invJ[j22] + g2 * invJ[j23];
	dpsidz[i] = g0 * invJ[j31] + g1 * invJ[j32] + g2 * invJ[j33];
	}

	return detJ ;
	}

	KOKKOS_INLINE_FUNCTION
	void contributeResidualJacobian(
	const float coeff_k ,
	const double dof_values[] ,
	const float dpsidx[] ,
	const float dpsidy[] ,
	const float dpsidz[] ,
	const float detJ ,
	const float integ_weight ,
	const float bases_vals[] ,
	double elem_res[] ,
	double elem_mat[][ FunctionCount ] ) const
	{
	double value_at_pt = 0 ;
	double gradx_at_pt = 0 ;
	double grady_at_pt = 0 ;
	double gradz_at_pt = 0 ;

	for ( unsigned m = 0 ; m < FunctionCount ; m++ ) {
	value_at_pt += dof_values[m] * bases_vals[m] ;
	gradx_at_pt += dof_values[m] * dpsidx[m] ;
	grady_at_pt += dof_values[m] * dpsidy[m] ;
	gradz_at_pt += dof_values[m] * dpsidz[m] ;
	}

	const scalar_type k_detJ_weight = coeff_k * detJ * integ_weight ;
	const double res_val = value_at_pt * value_at_pt * detJ * integ_weight ;
	const double mat_val = 2.0 * value_at_pt * detJ * integ_weight ;

	// $$ R_i = \int_{\Omega} \nabla \phi_i \cdot (k \nabla T) + \phi_i T^2 d \Omega $$
	// $$ J_{i,j} = \frac{\partial R_i}{\partial T_j} = \int_{\Omega} k \nabla \phi_i \cdot \nabla \phi_j + 2 \phi_i \phi_j T d \Omega $$

	for ( unsigned m = 0; m < FunctionCount; ++m) {
	double * const mat = elem_mat[m] ;
	const float bases_val_m = bases_vals[m];
	const float dpsidx_m = dpsidx[m] ;
	const float dpsidy_m = dpsidy[m] ;
	const float dpsidz_m = dpsidz[m] ;

	elem_res[m] += k_detJ_weight * ( dpsidx_m * gradx_at_pt +
	dpsidy_m * grady_at_pt +
	dpsidz_m * gradz_at_pt ) +
	res_val * bases_val_m ;

	for( unsigned n = 0; n < FunctionCount; n++) {

	mat[n] += k_detJ_weight * ( dpsidx_m * dpsidx[n] +
	dpsidy_m * dpsidy[n] +
	dpsidz_m * dpsidz[n] ) +
	mat_val * bases_val_m * bases_vals[n];
	}
	}
	}

	KOKKOS_INLINE_FUNCTION
	void operator()( const unsigned ielem ) const
	{
	// Gather nodal coordinates and solution vector:

	double x[ FunctionCount ] ;
	double y[ FunctionCount ] ;
	double z[ FunctionCount ] ;
	double val[ FunctionCount ] ;
	unsigned node_index[ ElemNodeCount ];

	for ( unsigned i = 0 ; i < ElemNodeCount ; ++i ) {
	const unsigned ni = elem_node_ids( ielem , i );

	node_index[i] = ni ;

	x[i] = node_coords( ni , 0 );
	y[i] = node_coords( ni , 1 );
	z[i] = node_coords( ni , 2 );

	val[i] = solution( ni );
	}


	double elem_vec[ FunctionCount ] ;
	double elem_mat[ FunctionCount ][ FunctionCount ] ;

	for( unsigned i = 0; i < FunctionCount ; i++ ) {
	elem_vec[i] = 0 ;
	for( unsigned j = 0; j < FunctionCount ; j++){
	elem_mat[i][j] = 0 ;
	}
	}


	for ( unsigned i = 0 ; i < IntegrationCount ; ++i ) {
	float dpsidx[ FunctionCount ] ;
	float dpsidy[ FunctionCount ] ;
	float dpsidz[ FunctionCount ] ;

	const float detJ =
	transform_gradients( elem_data.gradients[i] , x , y , z ,
	dpsidx , dpsidy , dpsidz );

	contributeResidualJacobian( coeff_K ,
	val , dpsidx , dpsidy , dpsidz ,
	detJ ,
	elem_data.weights[i] ,
	elem_data.values[i] ,
	elem_vec , elem_mat );
	}

	#if 0

	if ( 1 == ielem ) {
	printf("ElemResidual { %f %f %f %f %f %f %f %f }\n",
	elem_vec[0], elem_vec[1], elem_vec[2], elem_vec[3],
	elem_vec[4], elem_vec[5], elem_vec[6], elem_vec[7]);

	printf("ElemJacobian {\n");

	for ( unsigned j = 0 ; j < FunctionCount ; ++j ) {
	printf(" { %f %f %f %f %f %f %f %f }\n",
	elem_mat[j][0], elem_mat[j][1], elem_mat[j][2], elem_mat[j][3],
	elem_mat[j][4], elem_mat[j][5], elem_mat[j][6], elem_mat[j][7]);
	}
	printf("}\n");
	}

	#endif

	if ( ! residual.dimension_0() ) {
	for( unsigned i = 0; i < FunctionCount ; i++){
	elem_residuals(ielem, i) = elem_vec[i] ;
	for( unsigned j = 0; j < FunctionCount ; j++){
	elem_jacobians(ielem, i, j) = elem_mat[i][j] ;
	}
	}
	}
	else {
	for( unsigned i = 0 ; i < FunctionCount ; i++ ) {
	const unsigned row = node_index[i] ;
	if ( row < residual.dimension_0() ) {
	atomic_fetch_add( & residual( row ) , elem_vec[i] );

	for( unsigned j = 0 ; j < FunctionCount ; j++ ) {
	const unsigned entry = elem_graph( ielem , i , j );
	if ( entry != ~0u ) {
	atomic_fetch_add( & jacobian.coeff( entry ) , elem_mat[i][j] );
	}
	}
	}
	}
	}
	}
	}; /* ElementComputation */

	//----------------------------------------------------------------------------

	template< class FixtureType , class SparseMatrixType >
	class DirichletComputation ;

	template< class ExecSpace , BoxElemPart::ElemOrder Order , class CoordinateMap , typename ScalarType >
	class DirichletComputation<
	Kokkos::Example::BoxElemFixture< ExecSpace , Order , CoordinateMap > ,
	Kokkos::Example::CrsMatrix< ScalarType , ExecSpace > >
	{
	public:

	typedef Kokkos::Example::BoxElemFixture< ExecSpace, Order, CoordinateMap > mesh_type ;
	typedef typename mesh_type::node_coord_type node_coord_type ;
	typedef typename node_coord_type::value_type scalar_coord_type ;

	typedef Kokkos::Example::CrsMatrix< ScalarType , ExecSpace > sparse_matrix_type ;
	typedef typename sparse_matrix_type::StaticCrsGraphType sparse_graph_type ;

	typedef ExecSpace execution_space ;
	typedef ScalarType scalar_type ;

	//------------------------------------

	typedef Kokkos::View< scalar_type* , execution_space > vector_type ;

	//------------------------------------
	// Computational data:

	const node_coord_type node_coords ;
	const vector_type solution ;
	const sparse_matrix_type jacobian ;
	const vector_type residual ;
	const scalar_type bc_lower_value ;
	const scalar_type bc_upper_value ;
	const scalar_coord_type bc_lower_limit ;
	const scalar_coord_type bc_upper_limit ;
	const unsigned bc_plane ;
	const unsigned node_count ;
	bool init ;


	DirichletComputation( const mesh_type & arg_mesh ,
	const vector_type & arg_solution ,
	const sparse_matrix_type & arg_jacobian ,
	const vector_type & arg_residual ,
	const unsigned arg_bc_plane ,
	const scalar_type arg_bc_lower_value ,
	const scalar_type arg_bc_upper_value )
	: node_coords( arg_mesh.node_coord() )
	, solution( arg_solution )
	, jacobian( arg_jacobian )
	, residual( arg_residual )
	, bc_lower_value( arg_bc_lower_value )
	, bc_upper_value( arg_bc_upper_value )
	, bc_lower_limit( std::numeric_limits<scalar_coord_type>::epsilon() )
	, bc_upper_limit( scalar_coord_type(1) - std::numeric_limits<scalar_coord_type>::epsilon() )
	, bc_plane( arg_bc_plane )
	, node_count( arg_mesh.node_count_owned() )
	, init( false )
	{
	parallel_for( node_count , *this );
	init = true ;
	}

	void apply() const
	{
	parallel_for( node_count , *this );
	}

	//------------------------------------

	KOKKOS_INLINE_FUNCTION
	void operator()( const unsigned inode ) const
	{
	// Apply dirichlet boundary condition on the Solution and Residual vectors.
	// To maintain the symmetry of the original global stiffness matrix,
	// zero out the columns that correspond to boundary conditions, and
	// update the residual vector accordingly

	const unsigned iBeg = jacobian.graph.row_map[inode];
	const unsigned iEnd = jacobian.graph.row_map[inode+1];

	const scalar_coord_type c = node_coords(inode,bc_plane);
	const bool bc_lower = c <= bc_lower_limit ;
	const bool bc_upper = bc_upper_limit <= c ;

	if ( ! init ) {
	solution(inode) = bc_lower ? bc_lower_value : (
	bc_upper ? bc_upper_value : 0 );
	}
	else {
	if ( bc_lower \|\| bc_upper ) {

	residual(inode) = 0 ;

	// zero each value on the row, and leave a one
	// on the diagonal

	for( unsigned i = iBeg ; i < iEnd ; ++i ) {
	jacobian.coeff(i) = int(inode) == int(jacobian.graph.entries(i)) ? 1 : 0 ;
	}
	}
	else {

	// Find any columns that are boundary conditions.
	// Clear them and adjust the residual vector

	for( unsigned i = iBeg ; i < iEnd ; ++i ) {
	const unsigned cnode = jacobian.graph.entries(i) ;
	const scalar_coord_type cc = node_coords(cnode,bc_plane);

	if ( ( cc <= bc_lower_limit ) \|\| ( bc_upper_limit <= cc ) ) {
	jacobian.coeff(i) = 0 ;
	}
	}
	}
	}
	}
	};

	} /* namespace FENL */
	} /* namespace Example */
	} /* namespace Kokkos */

	//----------------------------------------------------------------------------

	/* A Cuda-specific specialization for the element computation functor. */
	#if defined( __CUDACC__ )
	// #include <NonlinearElement_Cuda.hpp>
	#endif

	//----------------------------------------------------------------------------

	#endif /* #ifndef KOKKOS_EXAMPLE_FENLFUNCTORS_HPP */

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