Kratos::GeometryUtils Class Reference

This function provides basic routines for working with simplicial meshes. More...

#include <geometry_utilities.h>

Collaboration diagram for Kratos::GeometryUtils:

Public Types
Type Definitions
using	SizeType = std::size_t
	The size type definition. More...
using	IndexType = std::size_t
	The index type definition. More...
using	GeometryType = Geometry< Node >
	Definition of the geometry. More...

Static Public Member Functions
template<>
void	EvaluateHistoricalVariableValueAtGaussPoint (double &rOutput, const GeometryType &rGeometry, const Variable< double > &rVariable, const Vector &rGaussPointShapeFunctionValues, const int Step)
Operations
static std::string	GetGeometryName (const GeometryData::KratosGeometryType TypeOfGeometry)
	This function returns a string equivalent for the geometry type. More...
template<class TGeometryType >
static TGeometryType::CoordinatesArrayType &	PointLocalCoordinatesPlanarFaceTetrahedra (const TGeometryType &rGeometry, typename TGeometryType::CoordinatesArrayType &rResult, const typename TGeometryType::CoordinatesArrayType &rPoint)
	Returns the local coordinates of a given arbitrary point for a given linear tetrahedra. More...
static void	CalculateGeometryData (const GeometryType &rGeometry, BoundedMatrix< double, 4, 3 > &rDN_DX, array_1d< double, 4 > &rN, double &rVolume)
	This function is designed to compute the shape function derivatives, shape functions and volume in 3D. More...
static double	CalculateVolume3D (const GeometryType &rGeometry)
	This function computes the element's volume (with sign) More...
static void	CalculateGeometryData (const GeometryType &rGeometry, BoundedMatrix< double, 3, 2 > &DN_DX, array_1d< double, 3 > &N, double &rArea)
	This function is designed to compute the shape function derivatives, shape functions and area of a triangle. More...
static double	CalculateVolume2D (const GeometryType &rGeometry)
	This function computes the element's volume (with sign) More...
static void	SideLenghts2D (const GeometryType &rGeometry, double &hmin, double &hmax)
	This function compute the maximum and minimum edge lenghts. More...
static void	CalculateGeometryData (const GeometryType &rGeometry, BoundedMatrix< double, 2, 1 > &rDN_DX, array_1d< double, 2 > &rN, double &rLength)
	This function is designed to compute the shape function derivatives, shape functions and length. More...
template<std::size_t TSize>
static void	CalculateTetrahedraDistances (const GeometryType &rGeometry, array_1d< double, TSize > &rDistances)
	Calculate the exact distances to the interface TRIANGLE defined by a set of initial distances. More...
template<std::size_t TSize>
static void	CalculateTriangleDistances (const GeometryType &rGeometry, array_1d< double, TSize > &rDistances)
	Calculate the exact distances to the interface SEGMENT defined by a set of initial distances. More...
template<std::size_t TSize>
static void	CalculateExactDistancesToPlane (const GeometryType &rThisGeometry, array_1d< double, TSize > &rDistances)
	Calculate the exact distances to the plane interface defined by a set of initial distances. More...
template<std::size_t TSize1, std::size_t TSize2>
static int	CalculateTetrahedraIntersectionPoints (const GeometryType &rGeometry, array_1d< double, TSize1 > &rDistances, array_1d< Point, TSize2 > &rIntersectionPoints)
	This function calculates the coordinates of the intersecion points between edges of tetrahedra and a isosurface given by the distances in its corners. More...
static double	PointDistanceToLineSegment3D (const Point &rLinePoint1, const Point &rLinePoint2, const Point &rToPoint)
	This function calculates the distance of a 3D point to a 3D line segment. More...
static double	PointDistanceToTriangle3D (const Point &rTrianglePoint1, const Point &rTrianglePoint2, const Point &rTrianglePoint3, const Point &rPoint)
	This function calculates the distance of a 3D point to a 3D triangle. More...
static double	PointDistanceToTriangle3D (const Point &rTrianglePoint1, const Point &rTrianglePoint2, const Point &rTrianglePoint3, const Point &rTrianglePoint4, const Point &rTrianglePoint5, const Point &rTrianglePoint6, const Point &rPoint)
	This function calculates the distance of a 3D point to a 3D quadratic triangle. More...
static double	PointDistanceToQuadrilateral3D (const Point &rQuadrilateralPoint1, const Point &rQuadrilateralPoint2, const Point &rQuadrilateralPoint3, const Point &rQuadrilateralPoint4, const Point &rPoint)
	This function calculates the distance of a 3D point to a 3D quadrilateral. More...
template<class TMatrix1 , class TMatrix2 , class TMatrix3 >
static void	ShapeFunctionsGradients (TMatrix1 const &rDN_De, TMatrix2 const &rInvJ, TMatrix3 &rDN_DX)
	Calculate the gradients of shape functions. More...
template<class TMatrix1 , class TMatrix2 , class TMatrix3 >
static void	DeformationGradient (TMatrix1 const &rJ, TMatrix2 const &rInvJ0, TMatrix3 &rF)
	Calculate the deformation gradient. More...
static void	JacobianOnInitialConfiguration (GeometryType const &rGeom, GeometryType::CoordinatesArrayType const &rCoords, Matrix &rJ0)
	Calculate the Jacobian on the initial configuration. More...
template<class TMatrix >
static void	DirectJacobianOnCurrentConfiguration (GeometryType const &rGeometry, GeometryType::CoordinatesArrayType const &rCoords, TMatrix &rJ)
	Calculate the Jacobian on the initial configuration. More...
template<class TMatrix >
static void	DirectJacobianOnInitialConfiguration (GeometryType const &rGeometry, TMatrix &rJ0, const IndexType PointNumber, const GeometryType::IntegrationMethod &rIntegrationMethod)
	Calculate the Jacobian on the initial configuration. More...
template<class TDataType >
static void	EvaluateHistoricalVariableValueAtGaussPoint (TDataType &rOutput, const GeometryType &rGeometry, const Variable< TDataType > &rVariable, const Vector &rGaussPointShapeFunctionValues, const int Step=0)
	Evaluates variable value at gauss point. More...
static void	EvaluateHistoricalVariableGradientAtGaussPoint (array_1d< double, 3 > &rOutput, const GeometryType &rGeometry, const Variable< double > &rVariable, const Matrix &rGaussPointShapeFunctionDerivativeValues, const int Step=0)
	Evaluates gradient of scalar at gauss point. More...
static void	EvaluateHistoricalVariableGradientAtGaussPoint (BoundedMatrix< double, 3, 3 > &rOutput, const GeometryType &rGeometry, const Variable< array_1d< double, 3 >> &rVariable, const Matrix &rGaussPointShapeFunctionDerivativeValues, const int Step=0)
	Evaluates gradient of a 3d vector at gauss point. More...
static void	ShapeFunctionsSecondDerivativesTransformOnAllIntegrationPoints (DenseVector< DenseVector< Matrix >> &rResult, const GeometryType &rGeometry, const GeometryType::IntegrationMethod &rIntegrationMethod)
	This method gives the transform of shape functions second derivatives from isoparametric to natural coordinates evaluated in all integration points. More...
static void	ShapeFunctionsSecondDerivativesTransformOnIntegrationPoint (const Matrix &DN_DX, const GeometryType &rGeometry, const GeometryType::CoordinatesArrayType &rLocalIntegrationPointCoordinates, DenseVector< Matrix > &rResult)
	This method gives the transform of shape functions second derivatives from isoparametric to natural coordinates evaluated on an integration point. More...
static bool	ProjectedIsInside (const GeometryType &rGeometry, const GeometryType::CoordinatesArrayType &rPointGlobalCoordinates, GeometryType::CoordinatesArrayType &rResult, const double Tolerance=std::numeric_limits< double >::epsilon())
	Checks if given point in global space coordinates is inside the geometry boundaries. More...
template<class TGeometryType >
static double	CalculateDistanceFrom3DGeometry (const TGeometryType &rGeometry, const typename TGeometryType::CoordinatesArrayType &rPointGlobalCoordinates, const double Tolerance=std::numeric_limits< double >::epsilon())
	Computes the distance between an point in global coordinates and the closest point of this geometry. More...

Detailed Description

This function provides basic routines for working with simplicial meshes.

It is faster than using Geometry as it is more specialized

Author

Riccardo Rossi

Member Typedef Documentation

GeometryType

using Kratos::GeometryUtils::GeometryType = Geometry<Node>

Definition of the geometry.

IndexType

using Kratos::GeometryUtils::IndexType = std::size_t

The index type definition.

SizeType

using Kratos::GeometryUtils::SizeType = std::size_t

The size type definition.

Member Function Documentation

CalculateDistanceFrom3DGeometry()

template<class TGeometryType >

static double Kratos::GeometryUtils::CalculateDistanceFrom3DGeometry ( const TGeometryType &  rGeometry, const typename TGeometryType::CoordinatesArrayType &  rPointGlobalCoordinates, const double  Tolerance = std::numeric_limits<double>::epsilon()  )

inlinestatic

Computes the distance between an point in global coordinates and the closest point of this geometry.

Parameters

rGeometry	the geometry to compute the distance to.
rPointGlobalCoordinates	the point to which the closest point has to be found.
Tolerance	accepted orthogonal error.

Returns

Distance to geometry. positive -> outside of to the geometry (for 2D and solids) 0 -> on/ in the geometry.

CalculateExactDistancesToPlane()

template<std::size_t TSize>

static void Kratos::GeometryUtils::CalculateExactDistancesToPlane ( const GeometryType &  rThisGeometry, array_1d< double, TSize > &  rDistances  )

inlinestatic

Calculate the exact distances to the plane interface defined by a set of initial distances.

Same argument is used to give the calculated exact distances back

Parameters

rThisGeometry	Geometry can be either a triangle or a tetrahedra
rDistances	The distances which define the isosurface as input.

CalculateGeometryData() [1/3]

static void Kratos::GeometryUtils::CalculateGeometryData ( const GeometryType &  rGeometry, BoundedMatrix< double, 2, 1 > &  rDN_DX, array_1d< double, 2 > &  rN, double &  rLength  )

inlinestatic

This function is designed to compute the shape function derivatives, shape functions and length.

Parameters

rGeometry	it is the array of nodes. It is expected to be a line
rDN_DX	a stack matrix of size 3*2 to store the shape function's derivatives
rN	an array_1d to store the shape functions at the barycenter
rLength	the volume of the element

CalculateGeometryData() [2/3]

static void Kratos::GeometryUtils::CalculateGeometryData ( const GeometryType &  rGeometry, BoundedMatrix< double, 3, 2 > &  DN_DX, array_1d< double, 3 > &  N, double &  rArea  )

inlinestatic

This function is designed to compute the shape function derivatives, shape functions and area of a triangle.

Parameters

rGeometry	it is the array of nodes. It is expected to be a triangle
DN_DX	a stack matrix of size 3*2 to store the shape function's derivatives
N	an array_1d to store the shape functions at the barycenter
Area	the volume of the element

CalculateGeometryData() [3/3]

static void Kratos::GeometryUtils::CalculateGeometryData ( const GeometryType &  rGeometry, BoundedMatrix< double, 4, 3 > &  rDN_DX, array_1d< double, 4 > &  rN, double &  rVolume  )

inlinestatic

This function is designed to compute the shape function derivatives, shape functions and volume in 3D.

Parameters

rGeometry	it is the array of nodes. It is expected to be a tetrahedra
rDN_DX	a stack matrix of size 4*3 to store the shape function's derivatives
rN	an array_1d to store the shape functions at the barycenter
rVolume	the volume of the element

CalculateTetrahedraDistances()

template<std::size_t TSize>

static void Kratos::GeometryUtils::CalculateTetrahedraDistances ( const GeometryType &  rGeometry, array_1d< double, TSize > &  rDistances  )

inlinestatic

Calculate the exact distances to the interface TRIANGLE defined by a set of initial distances.

Parameters

rGeometry	The tetrahedra itself. Note: If the geometry is not a tetrahedra the result is undefined and may cause memory error.
rDistances	The distances which define the isosurface as input and the same argument is used to give the calculated exact distance

CalculateTetrahedraIntersectionPoints()

template<std::size_t TSize1, std::size_t TSize2>

static int Kratos::GeometryUtils::CalculateTetrahedraIntersectionPoints ( const GeometryType &  rGeometry, array_1d< double, TSize1 > &  rDistances, array_1d< Point, TSize2 > &  rIntersectionPoints  )

inlinestatic

This function calculates the coordinates of the intersecion points between edges of tetrahedra and a isosurface given by the distances in its corners.

Parameters

rGeometry	The tetrahedra itself. Note: If the geometry is not a tetrahedra the result is undefined and may cause memory error.
rDistances	The distances of the 4 nodes of the tetrahedra to the iso surface.
rIntersectionPoints	The result intersection points

Returns

Number of intersection points.

CalculateTriangleDistances()

template<std::size_t TSize>

static void Kratos::GeometryUtils::CalculateTriangleDistances ( const GeometryType &  rGeometry, array_1d< double, TSize > &  rDistances  )

inlinestatic

Calculate the exact distances to the interface SEGMENT defined by a set of initial distances.

Parameters

rGeometry	The Triangle itself. Note: If the geometry is not a triangle the result is undefined and may cause memory error.
rDistances	The distances which define the isosurface as input and the same argument is used to give the calculated exact distance

CalculateVolume2D()

static double Kratos::GeometryUtils::CalculateVolume2D ( const GeometryType &  rGeometry )

inlinestatic

This function computes the element's volume (with sign)

Parameters

rGeometry

it is the array of nodes. It expects a triangle

Deprecated:

This method can be replaced by geometry function without loosing performance

CalculateVolume3D()

static double Kratos::GeometryUtils::CalculateVolume3D ( const GeometryType &  rGeometry )

inlinestatic

This function computes the element's volume (with sign)

Parameters

rGeometry

it is the array of nodes. It expects a tetrahedra

Deprecated:

This method can be replaced by geometry function without loosing performance

DeformationGradient()

template<class TMatrix1 , class TMatrix2 , class TMatrix3 >

static void Kratos::GeometryUtils::DeformationGradient ( TMatrix1 const &  rJ, TMatrix2 const &  rInvJ0, TMatrix3 &  rF  )

inlinestatic

Calculate the deformation gradient.

See, e.g., P. Wriggers, Nonlinear Finite Element Methods, Springer, 2008.

Parameters

rJ	element Jacobian.
rInvJ0	inverse of the element Jacobian of the initial configuration.
rF	deformation gradient.

DirectJacobianOnCurrentConfiguration()

template<class TMatrix >

static void Kratos::GeometryUtils::DirectJacobianOnCurrentConfiguration ( GeometryType const &  rGeometry, GeometryType::CoordinatesArrayType const &  rCoords, TMatrix &  rJ  )

inlinestatic

Calculate the Jacobian on the initial configuration.

Parameters

rGeom	element geometry.
rCoords	local coordinates of the current integration point.
rJ0	Jacobian on the initial configuration.

DirectJacobianOnInitialConfiguration()

template<class TMatrix >

static void Kratos::GeometryUtils::DirectJacobianOnInitialConfiguration ( GeometryType const &  rGeometry, TMatrix &  rJ0, const IndexType  PointNumber, const GeometryType::IntegrationMethod &  rIntegrationMethod  )

inlinestatic

Calculate the Jacobian on the initial configuration.

Parameters

rGeom	element geometry
rJ0	Jacobian on the initial configuration
PointNumber	The integration point considered
rIntegrationMethod	The integration method considered

EvaluateHistoricalVariableGradientAtGaussPoint() [1/2]

void Kratos::GeometryUtils::EvaluateHistoricalVariableGradientAtGaussPoint ( array_1d< double, 3 > &  rOutput, const GeometryType &  rGeometry, const Variable< double > &  rVariable, const Matrix &  rGaussPointShapeFunctionDerivativeValues, const int  Step = 0  )

static

Evaluates gradient of scalar at gauss point.

This method evaluates gradient of a scalar variable for given shape function derivative values. For 2D, it returns 3rd component as zero.

Parameters

rOutput	Output 3d variable containing gradients at gauss point
rGeometry	Geometry from which gauss point values are interpolated
rVariable	Variable for value interpolation
rGaussPointShapeFunctionDerivativeValues	Shape function derivatives evaluated at gauss point
Step	Step to be used in historical variable value interpolation

EvaluateHistoricalVariableGradientAtGaussPoint() [2/2]

void Kratos::GeometryUtils::EvaluateHistoricalVariableGradientAtGaussPoint ( BoundedMatrix< double, 3, 3 > &  rOutput, const GeometryType &  rGeometry, const Variable< array_1d< double, 3 >> &  rVariable, const Matrix &  rGaussPointShapeFunctionDerivativeValues, const int  Step = 0  )

static

Evaluates gradient of a 3d vector at gauss point.

This method evaluates gradient of a vector variable for given shape function derivative values. [ rOutput(i, j) = \frac{\partial u_i}{\partial x_j} ] Where $u_i$ is the component of 3D rVariable, and $x_j$ is the cartesian coordinate component. In 2D, it returns 3rd row and 3rd column with zeros.

Parameters

rOutput	Output matrix containing gradients at gauss point
rGeometry	Geometry from which gauss point values are interpolated
rVariable	Variable for value interpolation
rGaussPointShapeFunctionDerivativeValues	Shape function derivatives evaluated at gauss point
Step	Step to be used in historical variable value interpolation

EvaluateHistoricalVariableValueAtGaussPoint() [1/2]

template<>

void Kratos::GeometryUtils::EvaluateHistoricalVariableValueAtGaussPoint ( double &  rOutput, const GeometryType &  rGeometry, const Variable< double > &  rVariable, const Vector &  rGaussPointShapeFunctionValues, const int  Step  )

static

EvaluateHistoricalVariableValueAtGaussPoint() [2/2]

template<class TDataType >

void Kratos::GeometryUtils::EvaluateHistoricalVariableValueAtGaussPoint ( TDataType &  rOutput, const GeometryType &  rGeometry, const Variable< TDataType > &  rVariable, const Vector &  rGaussPointShapeFunctionValues, const int  Step = 0  )

static

Evaluates variable value at gauss point.

This method evaluates variable value at gauss point given by gauss point shape function values.

Template Parameters

TDataType

Data type

Parameters

rOutput	Output which holds evaluated value
rGeometry	Geometry from which gauss point values are interpolated
rVariable	Variable for value interpolation
rGaussPointShapeFunctionValues	Shape function values evaluated at gauss point
Step	Step to be used in historical variable value interpolation

GetGeometryName()

std::string Kratos::GeometryUtils::GetGeometryName ( const GeometryData::KratosGeometryType  TypeOfGeometry )

static

This function returns a string equivalent for the geometry type.

Parameters

TypeOfGeometry

The geometry type

JacobianOnInitialConfiguration()

static void Kratos::GeometryUtils::JacobianOnInitialConfiguration ( GeometryType const &  rGeom, GeometryType::CoordinatesArrayType const &  rCoords, Matrix &  rJ0  )

inlinestatic

Calculate the Jacobian on the initial configuration.

Parameters

rGeom	element geometry.
rCoords	local coordinates of the current integration point.
rJ0	Jacobian on the initial configuration.

PointDistanceToLineSegment3D()

double Kratos::GeometryUtils::PointDistanceToLineSegment3D ( const Point &  rLinePoint1, const Point &  rLinePoint2, const Point &  rToPoint  )

static

This function calculates the distance of a 3D point to a 3D line segment.

Parameters

rLinePoint1	First point of the line segment
rLinePoint2	End point of the line segment
rToPoint	The point which distance is required

Returns

The distance between the point and the line

PointDistanceToQuadrilateral3D()

double Kratos::GeometryUtils::PointDistanceToQuadrilateral3D ( const Point &  rQuadrilateralPoint1, const Point &  rQuadrilateralPoint2, const Point &  rQuadrilateralPoint3, const Point &  rQuadrilateralPoint4, const Point &  rPoint  )

static

This function calculates the distance of a 3D point to a 3D quadrilateral.

The implementation is done by decomposing the quadrilateral into 2 triangles and calling PointDistanceToTriangle3D

Parameters

rQuadrilateralPoint1	First point of quadrilateral
rQuadrilateralPoint2	Second point of quadrilateral
rQuadrilateralPoint3	Third point of quadrilateral
rQuadrilateralPoint4	Third point of quadrilateral
rPoint	The point which distance is required

Returns

The distance between the point and the quadrilateral

PointDistanceToTriangle3D() [1/2]

double Kratos::GeometryUtils::PointDistanceToTriangle3D ( const Point &  rTrianglePoint1, const Point &  rTrianglePoint2, const Point &  rTrianglePoint3, const Point &  rPoint  )

static

This function calculates the distance of a 3D point to a 3D triangle.

The implementation is done using following reference: http://www.geometrictools.com/Documentation/DistancePoint3Triangle3.pdf

Parameters

rTrianglePoint1	First point of triangle
rTrianglePoint2	Second point of triangle
rTrianglePoint3	Third point of triangle
rPoint	The point which distance is required

Returns

The distance between the point and the triangle

PointDistanceToTriangle3D() [2/2]

double Kratos::GeometryUtils::PointDistanceToTriangle3D ( const Point &  rTrianglePoint1, const Point &  rTrianglePoint2, const Point &  rTrianglePoint3, const Point &  rTrianglePoint4, const Point &  rTrianglePoint5, const Point &  rTrianglePoint6, const Point &  rPoint  )

static

This function calculates the distance of a 3D point to a 3D quadratic triangle.

The implementation is done by decomposing the quadratic triangle into 3 triangles and calling PointDistanceToTriangle3D

Parameters

rTrianglePoint1	First point of triangle
rTrianglePoint2	Second point of triangle
rTrianglePoint3	Third point of triangle
rTrianglePoint4	Fourth point of triangle
rTrianglePoint5	Fifth point of triangle
rTrianglePoint6	Sixth point of triangle
rPoint	The point which distance is required

Returns

The distance between the point and the triangle

PointLocalCoordinatesPlanarFaceTetrahedra()

template<class TGeometryType >

static TGeometryType::CoordinatesArrayType& Kratos::GeometryUtils::PointLocalCoordinatesPlanarFaceTetrahedra ( const TGeometryType &  rGeometry, typename TGeometryType::CoordinatesArrayType &  rResult, const typename TGeometryType::CoordinatesArrayType &  rPoint  )

inlinestatic

Returns the local coordinates of a given arbitrary point for a given linear tetrahedra.

Based on https://www.colorado.edu/engineering/CAS/courses.d/AFEM.d/AFEM.Ch09.d/AFEM.Ch09.pdf. Section 9.1.6

Parameters

rGeometry	The geometry to be considered
rResult	The vector containing the local coordinates of the point
rPoint	The point in global coordinates

Returns

The vector containing the local coordinates of the point

ProjectedIsInside()

bool Kratos::GeometryUtils::ProjectedIsInside ( const GeometryType &  rGeometry, const GeometryType::CoordinatesArrayType &  rPointGlobalCoordinates, GeometryType::CoordinatesArrayType &  rResult, const double  Tolerance = std::numeric_limits<double>::epsilon()  )

static

Checks if given point in global space coordinates is inside the geometry boundaries.

This function computes the local coordinates and checks then if this point lays within the boundaries after projecting the points.

Parameters

rPointGlobalCoordinates	the global coordinates of the external point.
rResult	the local coordinates of the point.
Tolerance	the tolerance to the boundary.

Returns

true if the point is inside, false otherwise

ShapeFunctionsGradients()

template<class TMatrix1 , class TMatrix2 , class TMatrix3 >

static void Kratos::GeometryUtils::ShapeFunctionsGradients ( TMatrix1 const &  rDN_De, TMatrix2 const &  rInvJ, TMatrix3 &  rDN_DX  )

inlinestatic

Calculate the gradients of shape functions.

Parameters

rDN_De	local gradient of shape functions.
rInvJ	inverse of the element Jacobian.
rDN_DX	gradient of shape functions.

ShapeFunctionsSecondDerivativesTransformOnAllIntegrationPoints()

void Kratos::GeometryUtils::ShapeFunctionsSecondDerivativesTransformOnAllIntegrationPoints ( DenseVector< DenseVector< Matrix >> &  rResult, const GeometryType &  rGeometry, const GeometryType::IntegrationMethod &  rIntegrationMethod  )

static

This method gives the transform of shape functions second derivatives from isoparametric to natural coordinates evaluated in all integration points.

Parameters

rResult

the transform of the second derivative of the shape function on the integration point

ShapeFunctionsSecondDerivativesTransformOnIntegrationPoint()

void Kratos::GeometryUtils::ShapeFunctionsSecondDerivativesTransformOnIntegrationPoint ( const Matrix &  DN_DX, const GeometryType &  rGeometry, const GeometryType::CoordinatesArrayType &  rLocalIntegrationPointCoordinates, DenseVector< Matrix > &  rResult  )

static

This method gives the transform of shape functions second derivatives from isoparametric to natural coordinates evaluated on an integration point.

The method used can be found here. https://scicomp.stackexchange.com/questions/25196/implementing-higher-order-derivatives-for-finite-element

Parameters

rLocalIntegrationPointCoordinates	the local coordinates of the integration point
rResult	the transform of the second derivative of the shape function on the integration point

SideLenghts2D()

static void Kratos::GeometryUtils::SideLenghts2D ( const GeometryType &  rGeometry, double &  hmin, double &  hmax  )

inlinestatic

This function compute the maximum and minimum edge lenghts.

It assumes that it is a triangle

Parameters

rGeometry	it is the array of nodes. It expects a triangle
hmin	The minimum length
hmax	The maximum length

---

The documentation for this class was generated from the following files:

• /home/runner/work/Documentation/Documentation/master/kratos/utilities/geometry_utilities.h
• /home/runner/work/Documentation/Documentation/master/kratos/utilities/geometry_utilities.cpp