DDR_rmplate

Implementation of the DDR scheme for the Reissner-Mindlin plate problem. More...

Classes
struct	HArDCore2D::RMParameters
	Structure to store model data. More...
struct	HArDCore2D::RMNorms
	Structure to store component norms (for rotation, displacement, Kirchoff term and total energy) More...
struct	HArDCore2D::ReissnerMindlin
	Assemble a RM problem. More...

Typedefs
typedef Eigen::SparseMatrix< double >	HArDCore2D::ReissnerMindlin::SystemMatrixType
typedef std::function< double(const RMParameters &, const Eigen::Vector2d &)>	HArDCore2D::ReissnerMindlin::ForcingTermType
typedef std::function< Eigen::Vector2d(const RMParameters &, const Eigen::Vector2d &)>	HArDCore2D::ReissnerMindlin::SolutionRotationType
typedef std::function< Eigen::Matrix2d(const RMParameters &, const Eigen::Vector2d &)>	HArDCore2D::ReissnerMindlin::GradientRotationType
typedef std::function< double(const RMParameters &, const Eigen::Vector2d &)>	HArDCore2D::ReissnerMindlin::SolutionDisplacementType

Functions
	HArDCore2D::RMParameters::RMParameters (const double thickness, const double young_modulus, const double poisson_ratio)
	Constructor.
	HArDCore2D::RMNorms::RMNorms (double norm_rotation, double norm_displacement, double norm_kirchoff)
	Constructor.
	HArDCore2D::ReissnerMindlin::ReissnerMindlin (const DDRCore &ddrcore, const RMParameters &para, const BoundaryConditions &BC_theta, const BoundaryConditions &BC_u, bool use_threads, std::ostream &output=std::cout)
	Constructor.
void	HArDCore2D::ReissnerMindlin::assembleLinearSystem (const ForcingTermType &f, const SolutionRotationType &theta, const GradientRotationType &grad_theta, const SolutionDisplacementType &u)
	Assemble the global system
size_t	HArDCore2D::ReissnerMindlin::dimensionSpace () const
	Returns the dimension of the rotation + displacement space (with BC)
size_t	HArDCore2D::ReissnerMindlin::nb_bdryDOFs () const
	Returns the nb of DOFs for BC.
size_t	HArDCore2D::ReissnerMindlin::sizeSystem () const
	Returns the size of the system without BC.
const std::vector< std::pair< size_t, size_t > > &	HArDCore2D::ReissnerMindlin::locUKN () const
	Returns the location of the unknowns among the DOFs.
std::vector< size_t >	HArDCore2D::ReissnerMindlin::globalDOFIndices (const Cell &T) const
	Create the vector of DOF indices for cell T, which combines the DOFs for the spaces EXcurl and Xgrad.
const RMParameters &	HArDCore2D::ReissnerMindlin::para () const
	Returns the parameters.
const EXCurl &	HArDCore2D::ReissnerMindlin::exCurl () const
	Returns the space EXCurl.
const XGrad &	HArDCore2D::ReissnerMindlin::xGrad () const
	Returns the space XGrad.
const SystemMatrixType &	HArDCore2D::ReissnerMindlin::systemMatrix () const
	Returns the linear system matrix.
SystemMatrixType &	HArDCore2D::ReissnerMindlin::systemMatrix ()
	Returns the linear system matrix.
const Eigen::VectorXd &	HArDCore2D::ReissnerMindlin::systemVector () const
	Returns the linear system right-hand side vector.
Eigen::VectorXd &	HArDCore2D::ReissnerMindlin::systemVector ()
	Returns the linear system right-hand side vector.
const SystemMatrixType &	HArDCore2D::ReissnerMindlin::bdryMatrix () const
	Returns the Matrix for BC.
const Eigen::VectorXd &	HArDCore2D::ReissnerMindlin::bdryValues () const
	Returns the boundary values.
const double &	HArDCore2D::ReissnerMindlin::stabilizationParameter () const
	Returns the stabilization parameter.
double &	HArDCore2D::ReissnerMindlin::stabilizationParameter ()
	Returns the stabilization parameter.
RMNorms	HArDCore2D::ReissnerMindlin::computeNorms (const Eigen::VectorXd &v) const
	Compute the discrete norms: rotation, displacement, Kirchoff term, and complete energy.
template<typename outValue , typename Fct >
std::function< outValue(const Eigen::Vector2d &)>	HArDCore2D::ReissnerMindlin::contractPara (const Fct &F) const
	Takes a function dependent on RMParameter and a position x, and returns a function depending only on x (using the parameters of this class)

Variables
double	HArDCore2D::RMParameters::t
double	HArDCore2D::RMParameters::E
double	HArDCore2D::RMParameters::nu
double	HArDCore2D::RMParameters::beta0
double	HArDCore2D::RMParameters::beta1
double	HArDCore2D::RMParameters::kappa
double	HArDCore2D::RMNorms::rotation
	Norm of rotation.
double	HArDCore2D::RMNorms::displacement
	Norm of displacement.
double	HArDCore2D::RMNorms::kirchoff
	Norm of kirchoff term.
double	HArDCore2D::RMNorms::energy
	Total energy.
static const double	HArDCore2D::PI = boost::math::constants::pi<double>()
static ReissnerMindlin::SolutionRotationType	HArDCore2D::constant_theta
static ReissnerMindlin::GradientRotationType	HArDCore2D::constant_grad_theta
static ReissnerMindlin::SolutionDisplacementType	HArDCore2D::constant_u
static ReissnerMindlin::ForcingTermType	HArDCore2D::constant_f
static ReissnerMindlin::SolutionRotationType	HArDCore2D::polynomial_theta
static ReissnerMindlin::GradientRotationType	HArDCore2D::polynomial_grad_theta
static ReissnerMindlin::SolutionDisplacementType	HArDCore2D::polynomial_u
static ReissnerMindlin::ForcingTermType	HArDCore2D::polynomial_f
static std::function< double(const VectorRd &)>	HArDCore2D::an_g = [](const VectorRd &x)->double { return sin(PI*x(0))*sin(PI*x(1)); }
static std::function< VectorRd(const VectorRd &)>	HArDCore2D::an_GRAD_g
static std::function< MatrixRd(const VectorRd &)>	HArDCore2D::an_HESS_g
static std::function< double(const VectorRd &)>	HArDCore2D::an_LAPL_g = [](const VectorRd &x)->double { return -2*pow(PI,2)*sin(PI*x(0))*sin(PI*x(1)); }
static std::function< double(const VectorRd &)>	HArDCore2D::an_V = [](const VectorRd &x)->double { return x(0) * exp(-x(0))*cos(x(1)); }
static std::function< VectorRd(const VectorRd &)>	HArDCore2D::an_GRAD_V
static std::function< MatrixRd(const VectorRd &)>	HArDCore2D::an_HESS_V
static std::function< double(const VectorRd &)>	HArDCore2D::an_LAPL_V = [](const VectorRd &x)->double { return -2.* exp(-x(0))*cos(x(1)); }
static ReissnerMindlin::SolutionDisplacementType	HArDCore2D::an_v
static ReissnerMindlin::SolutionRotationType	HArDCore2D::an_GRAD_v
static ReissnerMindlin::GradientRotationType	HArDCore2D::an_HESS_v
static ReissnerMindlin::SolutionDisplacementType	HArDCore2D::an_LAPL_v
static ReissnerMindlin::SolutionRotationType	HArDCore2D::analytical_theta = an_GRAD_v
static ReissnerMindlin::GradientRotationType	HArDCore2D::analytical_grad_theta = an_HESS_v
static ReissnerMindlin::SolutionDisplacementType	HArDCore2D::analytical_u
static ReissnerMindlin::ForcingTermType	HArDCore2D::analytical_f
static ReissnerMindlin::SolutionRotationType	HArDCore2D::ukn_theta
static ReissnerMindlin::GradientRotationType	HArDCore2D::ukn_grad_theta
static ReissnerMindlin::SolutionDisplacementType	HArDCore2D::ukn_u
static ReissnerMindlin::ForcingTermType	HArDCore2D::ukn_f
static ReissnerMindlin::SolutionRotationType	HArDCore2D::kir_theta
static ReissnerMindlin::GradientRotationType	HArDCore2D::kir_grad_theta
static ReissnerMindlin::SolutionDisplacementType	HArDCore2D::kir_u
static ReissnerMindlin::ForcingTermType	HArDCore2D::kir_f

Detailed Description

Implementation of the DDR scheme for the Reissner-Mindlin plate problem.

Typedef Documentation

ForcingTermType

typedef std::function<double(const RMParameters &, const Eigen::Vector2d &)> HArDCore2D::ReissnerMindlin::ForcingTermType

GradientRotationType

typedef std::function<Eigen::Matrix2d(const RMParameters &, const Eigen::Vector2d &)> HArDCore2D::ReissnerMindlin::GradientRotationType

SolutionDisplacementType

typedef std::function<double(const RMParameters &, const Eigen::Vector2d &)> HArDCore2D::ReissnerMindlin::SolutionDisplacementType

SolutionRotationType

typedef std::function<Eigen::Vector2d(const RMParameters &, const Eigen::Vector2d &)> HArDCore2D::ReissnerMindlin::SolutionRotationType

SystemMatrixType

typedef Eigen::SparseMatrix<double> HArDCore2D::ReissnerMindlin::SystemMatrixType

Function Documentation

assembleLinearSystem()

void ReissnerMindlin::assembleLinearSystem	(	const ForcingTermType &	f,
		const SolutionRotationType &	theta,
		const GradientRotationType &	grad_theta,
		const SolutionDisplacementType &	u
	)

Assemble the global system

Parameters

f	Forcing term
theta	Boundary value
grad_theta	Boundary value
u	Boundary value

bdryMatrix()

const SystemMatrixType & HArDCore2D::ReissnerMindlin::bdryMatrix ( ) const

inline

Returns the Matrix for BC.

bdryValues()

const Eigen::VectorXd & HArDCore2D::ReissnerMindlin::bdryValues ( ) const

inline

Returns the boundary values.

computeNorms()

RMNorms ReissnerMindlin::computeNorms

(

const Eigen::VectorXd &

v

)

const

Compute the discrete norms: rotation, displacement, Kirchoff term, and complete energy.

Parameters

v	The vector

contractPara()

template<typename outValue , typename Fct >

std::function< outValue(const Eigen::Vector2d &)> HArDCore2D::ReissnerMindlin::contractPara ( const Fct &  F ) const

inline

Takes a function dependent on RMParameter and a position x, and returns a function depending only on x (using the parameters of this class)

dimensionSpace()

size_t HArDCore2D::ReissnerMindlin::dimensionSpace ( ) const

inline

Returns the dimension of the rotation + displacement space (with BC)

exCurl()

const EXCurl & HArDCore2D::ReissnerMindlin::exCurl ( ) const

inline

Returns the space EXCurl.

globalDOFIndices()

std::vector< size_t > HArDCore2D::ReissnerMindlin::globalDOFIndices ( const Cell &  T ) const

inline

Create the vector of DOF indices for cell T, which combines the DOFs for the spaces EXcurl and Xgrad.

locUKN()

const std::vector< std::pair< size_t, size_t > > & HArDCore2D::ReissnerMindlin::locUKN ( ) const

inline

Returns the location of the unknowns among the DOFs.

nb_bdryDOFs()

size_t HArDCore2D::ReissnerMindlin::nb_bdryDOFs ( ) const

inline

Returns the nb of DOFs for BC.

para()

const RMParameters & HArDCore2D::ReissnerMindlin::para ( ) const

inline

Returns the parameters.

ReissnerMindlin()

ReissnerMindlin::ReissnerMindlin	(	const DDRCore &	ddrcore,
		const RMParameters &	para,
		const BoundaryConditions &	BC_theta,
		const BoundaryConditions &	BC_u,
		bool	use_threads,
		std::ostream &	output = std::cout
	)

Constructor.

Parameters

ddrcore	Core for the DDR space sequence
para	Physical parameters
BC_theta	Boundary conditions for rotation
BC_u	Boundary conditions for displacement
use_threads	True for parallel execution, false for sequential execution
output	Output stream to print status messages

RMNorms()

HArDCore2D::RMNorms::RMNorms ( double  norm_rotation, double  norm_displacement, double  norm_kirchoff  )

inline

Constructor.

RMParameters()

HArDCore2D::RMParameters::RMParameters ( const double  thickness, const double  young_modulus, const double  poisson_ratio  )

inline

Constructor.

sizeSystem()

size_t HArDCore2D::ReissnerMindlin::sizeSystem ( ) const

inline

Returns the size of the system without BC.

stabilizationParameter() [1/2]

double & HArDCore2D::ReissnerMindlin::stabilizationParameter ( )

inline

Returns the stabilization parameter.

stabilizationParameter() [2/2]

const double & HArDCore2D::ReissnerMindlin::stabilizationParameter ( ) const

inline

Returns the stabilization parameter.

systemMatrix() [1/2]

SystemMatrixType & HArDCore2D::ReissnerMindlin::systemMatrix ( )

inline

Returns the linear system matrix.

systemMatrix() [2/2]

const SystemMatrixType & HArDCore2D::ReissnerMindlin::systemMatrix ( ) const

inline

Returns the linear system matrix.

systemVector() [1/2]

Eigen::VectorXd & HArDCore2D::ReissnerMindlin::systemVector ( )

inline

Returns the linear system right-hand side vector.

systemVector() [2/2]

const Eigen::VectorXd & HArDCore2D::ReissnerMindlin::systemVector ( ) const

inline

Returns the linear system right-hand side vector.

xGrad()

const XGrad & HArDCore2D::ReissnerMindlin::xGrad ( ) const

inline

Returns the space XGrad.

Variable Documentation

an_g

std::function<double(const VectorRd &)> HArDCore2D::an_g = [](const VectorRd &x)->double { return sin(PI*x(0))*sin(PI*x(1)); }

static

an_GRAD_g

std::function<VectorRd(const VectorRd &)> HArDCore2D::an_GRAD_g

static

Initial value:

= [](const VectorRd & x) -> VectorRd {
                 return VectorRd( PI*cos(PI*x(0))*sin(PI*x(1)), PI*sin(PI*x(0))*cos(PI*x(1)) ); 
               }

an_GRAD_V

std::function<VectorRd(const VectorRd &)> HArDCore2D::an_GRAD_V

static

Initial value:

= [](const VectorRd & x) -> VectorRd {
                 return VectorRd((1.-x(0))*exp(-x(0))*cos(x(1)), -x(0)*exp(-x(0))*sin(x(1))); 
               }

an_GRAD_v

ReissnerMindlin::SolutionRotationType HArDCore2D::an_GRAD_v

static

Initial value:

= [](const RMParameters & para, const VectorRd & x) -> VectorRd {
                 return pow(para.t, 2) * an_GRAD_V(x/para.t) + an_GRAD_g(x);
               }

an_HESS_g

std::function<MatrixRd(const VectorRd &)> HArDCore2D::an_HESS_g

static

Initial value:

= [](const VectorRd & x) -> MatrixRd {
                MatrixRd H = MatrixRd::Zero();
                H.row(0) << -PI*PI*sin(PI*x(0))*sin(PI*x(1)), PI*PI*cos(PI*x(0))*cos(PI*x(1));
                H.row(1) << PI*PI*cos(PI*x(0))*cos(PI*x(1)), -PI*PI*sin(PI*x(0))*sin(PI*x(1));
                return H;
              }

an_HESS_V

std::function<MatrixRd(const VectorRd &)> HArDCore2D::an_HESS_V

static

Initial value:

= [](const VectorRd & x) -> MatrixRd {
                MatrixRd H = MatrixRd::Zero();
                H.row(0) << (x(0)-2.)*exp(-x(0))*cos(x(1)), -(1.-x(0))*exp(-x(0))*sin(x(1));
                H.row(1) << -(1.-x(0))*exp(-x(0))*sin(x(1)), -x(0)*exp(-x(0))*cos(x(1));
                return H;
              }

an_HESS_v

ReissnerMindlin::GradientRotationType HArDCore2D::an_HESS_v

static

Initial value:

= [](const RMParameters & para, const VectorRd & x) -> MatrixRd {
                  return para.t * an_HESS_V(x/para.t) + an_HESS_g(x);
              }

an_LAPL_g

std::function<double(const VectorRd &)> HArDCore2D::an_LAPL_g = [](const VectorRd &x)->double { return -2*pow(PI,2)*sin(PI*x(0))*sin(PI*x(1)); }

static

an_LAPL_V

std::function<double(const VectorRd &)> HArDCore2D::an_LAPL_V = [](const VectorRd &x)->double { return -2.* exp(-x(0))*cos(x(1)); }

static

an_LAPL_v

ReissnerMindlin::SolutionDisplacementType HArDCore2D::an_LAPL_v

static

Initial value:

= [](const RMParameters & para, const VectorRd & x) -> double {
                     return para.t * (an_LAPL_V)(x/para.t) + an_LAPL_g(x);
                   }

an_V

std::function<double(const VectorRd &)> HArDCore2D::an_V = [](const VectorRd &x)->double { return x(0) * exp(-x(0))*cos(x(1)); }

static

an_v

ReissnerMindlin::SolutionDisplacementType HArDCore2D::an_v

static

Initial value:

= [](const RMParameters & para, const VectorRd & x) -> double {
                     return pow(para.t,3) * an_V(x/para.t) + an_g(x);
                   }

analytical_f

ReissnerMindlin::ForcingTermType HArDCore2D::analytical_f

static

Initial value:

= [](const RMParameters & para, const VectorRd & x) -> double {
                 return (para.beta0+para.beta1) * 4 * pow(PI,4)*sin(PI*x(0))*sin(PI*x(1));
               }

analytical_grad_theta

ReissnerMindlin::GradientRotationType HArDCore2D::analytical_grad_theta = an_HESS_v

static

analytical_theta

ReissnerMindlin::SolutionRotationType HArDCore2D::analytical_theta = an_GRAD_v

static

analytical_u

ReissnerMindlin::SolutionDisplacementType HArDCore2D::analytical_u

static

Initial value:

= [](const RMParameters & para, const VectorRd & x) -> double {
                     return an_v(para, x) - pow(para.t,2) * ( (para.beta0+para.beta1)/para.kappa ) * an_LAPL_v(para, x);
                   }

beta0

double HArDCore2D::RMParameters::beta0

beta1

double HArDCore2D::RMParameters::beta1

constant_f

ReissnerMindlin::ForcingTermType HArDCore2D::constant_f

static

Initial value:

= [](const RMParameters & para, const VectorRd & x) -> double {
                 return 0.;
               }

constant_grad_theta

ReissnerMindlin::GradientRotationType HArDCore2D::constant_grad_theta

static

Initial value:

= [](const RMParameters & para, const VectorRd & x) -> Eigen::Matrix2d {
                 return Eigen::Matrix2d::Zero();
               }

constant_theta

ReissnerMindlin::SolutionRotationType HArDCore2D::constant_theta

static

Initial value:

= [](const RMParameters & para, const VectorRd & x) -> VectorRd {
                 return VectorRd(1., 2.);
               }

constant_u

ReissnerMindlin::SolutionDisplacementType HArDCore2D::constant_u

static

Initial value:

= [](const RMParameters & para, const VectorRd & x) -> double {
                     return 1.;
                   }

displacement

double HArDCore2D::RMNorms::displacement

Norm of displacement.

E

double HArDCore2D::RMParameters::E

energy

double HArDCore2D::RMNorms::energy

Total energy.

kappa

double HArDCore2D::RMParameters::kappa

kir_f

ReissnerMindlin::ForcingTermType HArDCore2D::kir_f

static

Initial value:

= [](const RMParameters & para, const VectorRd & x) -> double {
                return sin(PI*x(0))*sin(PI*x(1));
               }

kir_grad_theta

ReissnerMindlin::GradientRotationType HArDCore2D::kir_grad_theta

static

Initial value:

= [](const RMParameters & para, const VectorRd & x) -> Eigen::Matrix2d {
                 double coef = 12.*(1-pow(para.nu,2))/ (para.E * 4.*pow(PI, 4));
                 Eigen::Matrix2d H = Eigen::Matrix2d::Zero();
                 H.row(0) << -PI*PI*sin(PI*x(0))*sin(PI*x(1)), PI*PI*cos(PI*x(0))*cos(PI*x(1));
                 H.row(1) <<  PI*PI*cos(PI*x(0))*cos(PI*x(1)), -PI*PI*sin(PI*x(0))*sin(PI*x(1));
                 return coef * H;
               }

kir_theta

ReissnerMindlin::SolutionRotationType HArDCore2D::kir_theta

static

Initial value:

= [](const RMParameters & para, const VectorRd & x) -> VectorRd {
                double coef = 12.*(1-pow(para.nu,2))/ (para.E * 4.*pow(PI, 4));
                return coef * PI * VectorRd(cos(PI*x(0))*sin(PI*x(1)), sin(PI*x(0))*cos(PI*x(1)));
               }

kir_u

ReissnerMindlin::SolutionDisplacementType HArDCore2D::kir_u

static

Initial value:

= [](const RMParameters & para, const VectorRd & x) -> double {
                  double coef = 12.*(1-pow(para.nu,2))/ (para.E * 4.*pow(PI, 4));
                  return coef  * sin(PI*x(0))*sin(PI*x(1));
               }

kirchoff

double HArDCore2D::RMNorms::kirchoff

Norm of kirchoff term.

nu

double HArDCore2D::RMParameters::nu

PI

const double HArDCore2D::PI = boost::math::constants::pi<double>()

static

polynomial_f

ReissnerMindlin::ForcingTermType HArDCore2D::polynomial_f

static

Initial value:

= [](const RMParameters & para, const VectorRd & x) -> double {
                double val = 0;
                
                val += 12*x(1)*(x(1)-1)*(5*x(0)*x(0)-5*x(0)+1) * 
                        ( 2*x(1)*x(1)*(x(1)-1)*(x(1)-1) + x(0)*(x(0)-1)*(5*x(1)*x(1)-5*x(1)+1) );
                val += 12*x(0)*(x(0)-1)*(5*x(1)*x(1)-5*x(1)+1) * 
                        ( 2*x(0)*x(0)*(x(0)-1)*(x(0)-1) + x(1)*(x(1)-1)*(5*x(0)*x(0)-5*x(0)+1) );
                
                return val * para.E / (12* (1.-para.nu*para.nu) );
               }

polynomial_grad_theta

ReissnerMindlin::GradientRotationType HArDCore2D::polynomial_grad_theta

static

Initial value:

= [](const RMParameters & para, const VectorRd & x) -> Eigen::Matrix2d {
                 Eigen::Matrix2d G = Eigen::Matrix2d::Zero();
                 G(0,0) = 2.*pow(x(0),2)*pow(x(1),3)*pow(x(0)-1.0,2)*pow(x(1)-1.0,3)+2.*x(0)*pow(x(1),3)*(2.*x(0)-1.)*pow(x(0)-1.,2)*pow(x(1)-1.0,3)+pow(x(0),2)*pow(x(1),3)*(2.*x(0)-1.)*(2.*x(0)-2.)*pow(x(1)-1.,3);
                 G(0,1) = 3.*pow(x(0)*x(1),2)*(2.*x(0)-1.)*pow(x(0)-1.,2)*pow(x(1)-1.,3)+3.*pow(x(0),2)*pow(x(1),3)*(2.*x(0)-1.)*pow(x(0)-1.,2)*pow(x(1)-1.,2);
                 G(1,0) = 2.*pow(x(1),2)*pow(x(0),3)*pow(x(1)-1.0,2)*pow(x(0)-1.0,3)+2.*x(1)*pow(x(0),3)*(2.*x(1)-1.)*pow(x(1)-1.,2)*pow(x(0)-1.0,3)+pow(x(1),2)*pow(x(0),3)*(2.*x(1)-1.)*(2.*x(1)-2.)*pow(x(0)-1.,3);
                 G(1,1) = 3.*pow(x(1)*x(0),2)*(2.*x(1)-1.)*pow(x(1)-1.,2)*pow(x(0)-1.,3)+3.*pow(x(1),2)*pow(x(0),3)*(2.*x(1)-1.)*pow(x(1)-1.,2)*pow(x(0)-1.,2);
 
                 return G;
               }

polynomial_theta

ReissnerMindlin::SolutionRotationType HArDCore2D::polynomial_theta

static

Initial value:

= [](const RMParameters & para, const VectorRd & x) -> VectorRd {
                 return VectorRd(
                          pow(x(1),3)*pow(x(1)-1,3)*pow(x(0),2)*pow(x(0)-1,2)*(2*x(0)-1),
                          pow(x(0),3)*pow(x(0)-1,3)*pow(x(1),2)*pow(x(1)-1,2)*(2*x(1)-1)
                          );
               }

polynomial_u

ReissnerMindlin::SolutionDisplacementType HArDCore2D::polynomial_u

static

Initial value:

= [](const RMParameters & para, const VectorRd & x) -> double {
                 double val = 0;
                 
                 val += (1./3.)*pow(x(0),3)*pow(x(0)-1,3)*pow(x(1),3)*pow(x(1)-1,3);
                 
                 val -= ( 2*pow(para.t,2) / (5*(1.-para.nu)) ) *
                          ( pow(x(1),3)*pow(x(1)-1,3)*x(0)*(x(0)-1)*(5*x(0)*x(0)-5*x(0)+1)
                            + pow(x(0),3)*pow(x(0)-1,3)*x(1)*(x(1)-1)*(5*x(1)*x(1)-5*x(1)+1) );
                 
                 return val;
                 }

rotation

double HArDCore2D::RMNorms::rotation

Norm of rotation.

t

double HArDCore2D::RMParameters::t

ukn_f

ReissnerMindlin::ForcingTermType HArDCore2D::ukn_f

static

Initial value:

= [](const RMParameters & para, const VectorRd & x) -> double {
                double val = 1.;
                if (x(0)<.5){
                  val = -1.;
                }
                return val;
               }

ukn_grad_theta

ReissnerMindlin::GradientRotationType HArDCore2D::ukn_grad_theta

static

Initial value:

= [](const RMParameters & para, const VectorRd & x) -> Eigen::Matrix2d {
                 return Eigen::Matrix2d::Zero();
               }

ukn_theta

ReissnerMindlin::SolutionRotationType HArDCore2D::ukn_theta

static

Initial value:

= [](const RMParameters & para, const VectorRd & x) -> VectorRd {
                 return VectorRd(0., 0.);
               }

ukn_u

ReissnerMindlin::SolutionDisplacementType HArDCore2D::ukn_u

static

Initial value:

= [](const RMParameters & para, const VectorRd & x) -> double {
                     return x(0);
                   }