/*===========================================================================*/
const kvs::Real32 QuadraticTetrahedralCell::volume() const
{
const kvs::Vec3 v0( 0, 0, 0 );
const kvs::Vec3 v1( 1, 0, 0 );
const kvs::Vec3 v2( 0, 0, 1 );
const kvs::Vec3 v3( 0, 1, 0 );
const kvs::Vec3 v4( 0.5, 0, 0 );
const kvs::Vec3 v5( 0, 0,0.5 );
const kvs::Vec3 v6( 0,0.5, 0 );
const kvs::Vec3 v7( 0.5, 0,0.5 );
const kvs::Vec3 v8( 0,0.5,0.5 );
const kvs::Vec3 v9( 0.5,0.5, 0 );
const kvs::Vec3 c[8] = {
( v0 + v4 + v5 + v6 ) * 0.25,
( v4 + v1 + v7 + v9 ) * 0.25,
( v5 + v7 + v2 + v8 ) * 0.25,
( v6 + v9 + v8 + v3 ) * 0.25,
( v4 + v7 + v5 + v6 ) * 0.25,
( v4 + v9 + v7 + v6 ) * 0.25,
( v8 + v6 + v5 + v7 ) * 0.25,
( v8 + v7 + v9 + v6 ) * 0.25 };
float sum_metric = 0;
for( size_t i = 0 ; i < 8 ; i++ )
{
BaseClass::setLocalPoint( c[i] );
const kvs::Mat3 J = BaseClass::JacobiMatrix();
const float metric_element = J.determinant();
sum_metric += kvs::Math::Abs<float>( metric_element );
}
return sum_metric / ( 6.0f * 8.0f );
}
/*===========================================================================*/
const kvs::Mat3 CellBase::gradientTensor() const
{
KVS_ASSERT( m_veclen == 3 );
// Calculate a gradient tensor in the local coordinate.
const kvs::UInt32 nnodes = m_nnodes;
const float* dNdp = m_differential_functions;
const float* dNdq = m_differential_functions + nnodes;
const float* dNdr = m_differential_functions + nnodes + nnodes;
const kvs::Real32* Su = m_values;
const kvs::Real32* Sv = Su + nnodes;
const kvs::Real32* Sw = Sv + nnodes;
const float dudp = this->interpolateValue( Su, dNdp, nnodes );
const float dudq = this->interpolateValue( Su, dNdq, nnodes );
const float dudr = this->interpolateValue( Su, dNdr, nnodes );
const float dvdp = this->interpolateValue( Sv, dNdp, nnodes );
const float dvdq = this->interpolateValue( Sv, dNdq, nnodes );
const float dvdr = this->interpolateValue( Sv, dNdr, nnodes );
const float dwdp = this->interpolateValue( Sw, dNdp, nnodes );
const float dwdq = this->interpolateValue( Sw, dNdq, nnodes );
const float dwdr = this->interpolateValue( Sw, dNdr, nnodes );
const kvs::Mat3 t( dudp, dvdp, dwdp, dudq, dvdq, dwdq, dudr, dvdr, dwdr );
// Calculate a gradient tensor in the global coordinate.
const kvs::Mat3 J = this->JacobiMatrix();
float determinant = 0.0f;
const kvs::Mat3 T = 3.0f * J.inverted( &determinant ) * t;
return kvs::Math::IsZero( determinant ) ? kvs::Mat3::Zero() : T;
}
kvs::Mat3 PrismCell::gradient()
{
const kvs::Mat3 t = this->localGradient();
const kvs::Mat3 J = this->JacobiMatrix();
kvs::Real32 det = 0.0f;
const kvs::Mat3 T = 3.0f * J.inverted( &det ) * t;
return kvs::Math::IsZero( det ) ? kvs::Mat3::Zero() : T;
}
/*===========================================================================*/
kvs::Vec3 TetrahedralCell::globalToLocal( const kvs::Vec3& global ) const
{
const kvs::Vec3 v3( BaseClass::coord(3) );
const kvs::Vec3 v03( BaseClass::coord(0) - v3 );
const kvs::Vec3 v13( BaseClass::coord(1) - v3 );
const kvs::Vec3 v23( BaseClass::coord(2) - v3 );
const kvs::Mat3 M(
v03.x(), v13.x(), v23.x(),
v03.y(), v13.y(), v23.y(),
v03.z(), v13.z(), v23.z() );
return M.inverted() * ( global - v3 );
}
kvs::Vec3 PrismCell::globalToLocal( const kvs::Vec3 point )
{
const kvs::Vec3 X( point );
const float TinyValue = static_cast<float>( 1.e-6 );
const size_t MaxLoop = 100;
kvs::Vec3 x0( 0.3f, 0.3f, 0.5f );
for ( size_t i = 0; i < MaxLoop; i++ )
{
this->setLocalPoint( x0 );
const kvs::Vec3 X0( this->localToGlobal( x0 ) );
const kvs::Vec3 dX( X - X0 );
const kvs::Mat3 J( this->JacobiMatrix() );
const kvs::Vec3 dx = J.transposed().inverted() * dX;
if ( dx.length() < TinyValue ) break; // Converged.
x0 += dx;
}
return x0;
}
/*===========================================================================*/
const kvs::Vec3 CellBase::gradientVector() const
{
KVS_ASSERT( m_veclen == 1 );
// Calculate a gradient vector in the local coordinate.
const kvs::UInt32 nnodes = m_nnodes;
const float* dNdp = m_differential_functions;
const float* dNdq = m_differential_functions + nnodes;
const float* dNdr = m_differential_functions + nnodes + nnodes;
const kvs::Real32* S = m_values;
const float dSdp = this->interpolateValue( S, dNdp, nnodes );
const float dSdq = this->interpolateValue( S, dNdq, nnodes );
const float dSdr = this->interpolateValue( S, dNdr, nnodes );
const kvs::Vec3 g( dSdp, dSdq, dSdr );
// Calculate a gradient vector in the global coordinate.
const kvs::Mat3 J = this->JacobiMatrix();
float determinant = 0.0f;
const kvs::Vec3 G = 3.0f * J.inverted( &determinant ) * g;
return kvs::Math::IsZero( determinant ) ? kvs::Vec3::Zero() : G;
}
/*===========================================================================*/
const kvs::Vec3 CellBase::globalToLocal( const kvs::Vec3& global ) const
{
const kvs::Vec3 X( global );
// Calculate the coordinate of 'global' in the local coordinate
// by using Newton-Raphson method.
const float TinyValue = static_cast<float>( 1.e-6 );
const size_t MaxLoop = 100;
kvs::Vec3 x0( 0.25f, 0.25f, 0.25f ); // Initial point in local coordinate.
for ( size_t i = 0; i < MaxLoop; i++ )
{
const kvs::Vec3 X0( this->localToGlobal( x0 ) );
const kvs::Vec3 dX( X - X0 );
const kvs::Mat3 J( this->JacobiMatrix() );
const kvs::Vec3 dx = J.transposed().inverted() * dX;
if ( dx.length() < TinyValue ) break; // Converged.
x0 += dx;
}
return x0;
}
