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ElementQuad4.hpp
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/* =================================================================================================
(c - GPLv3) T.W.J. de Geus (Tom) | tom@geus.me | www.geus.me | github.com/tdegeus/GooseFEM
================================================================================================= */
#ifndef GOOSEFEM_ELEMENTQUAD4_CPP
#define GOOSEFEM_ELEMENTQUAD4_CPP
// -------------------------------------------------------------------------------------------------
#include "ElementQuad4.h"
// =================================== GooseFEM::Element::Quad4 ====================================
namespace GooseFEM {
namespace Element {
namespace Quad4 {
// ======================================== tensor algebra =========================================
inline double inv(const T2 &A, T2 &Ainv)
{
// compute determinant
double det = A[0] * A[3] - A[1] * A[2];
// compute inverse
Ainv[0] = A[3] / det;
Ainv[1] = -1. * A[1] / det;
Ainv[2] = -1. * A[2] / det;
Ainv[3] = A[0] / det;
return det;
}
// ================================ GooseFEM::Element::Quad4::Gauss ================================
namespace Gauss {
// --------------------------------- number of integration points ----------------------------------
inline size_t nip()
{
return 4;
}
// ----------------------- integration point coordinates (local coordinates) -----------------------
inline xt::xtensor<double,2> xi()
{
static const size_t nip = 4;
static const size_t ndim = 2;
xt::xtensor<double,2> xi = xt::empty<double>({nip,ndim});
xi(0,0) = -1./std::sqrt(3.); xi(0,1) = -1./std::sqrt(3.);
xi(1,0) = +1./std::sqrt(3.); xi(1,1) = -1./std::sqrt(3.);
xi(2,0) = +1./std::sqrt(3.); xi(2,1) = +1./std::sqrt(3.);
xi(3,0) = -1./std::sqrt(3.); xi(3,1) = +1./std::sqrt(3.);
return xi;
}
// ----------------------------------- integration point weights -----------------------------------
inline xt::xtensor<double,1> w()
{
static const size_t nip = 4;
xt::xtensor<double,1> w = xt::empty<double>({nip});
w(0) = 1.;
w(1) = 1.;
w(2) = 1.;
w(3) = 1.;
return w;
}
// -------------------------------------------------------------------------------------------------
}
// ================================ GooseFEM::Element::Quad4::Nodal ================================
namespace Nodal {
// --------------------------------- number of integration points ----------------------------------
inline size_t nip()
{
return 4;
}
// ----------------------- integration point coordinates (local coordinates) -----------------------
inline xt::xtensor<double,2> xi()
{
static const size_t nip = 4;
static const size_t ndim = 2;
xt::xtensor<double,2> xi = xt::empty<double>({nip,ndim});
xi(0,0) = -1.; xi(0,1) = -1.;
xi(1,0) = +1.; xi(1,1) = -1.;
xi(2,0) = +1.; xi(2,1) = +1.;
xi(3,0) = -1.; xi(3,1) = +1.;
return xi;
}
// ----------------------------------- integration point weights -----------------------------------
inline xt::xtensor<double,1> w()
{
static const size_t nip = 4;
xt::xtensor<double,1> w = xt::empty<double>({nip});
w(0) = 1.;
w(1) = 1.;
w(2) = 1.;
w(3) = 1.;
return w;
}
// -------------------------------------------------------------------------------------------------
}
// =================================================================================================
// ------------------------------------------ constructor ------------------------------------------
inline Quadrature::Quadrature(const xt::xtensor<double,3> &x) : m_x(x)
{
assert( m_x.shape()[1] == m_nne );
assert( m_x.shape()[2] == m_ndim );
// set integration scheme
m_xi = Gauss::xi();
m_w = Gauss::w ();
// extract number of elements
m_nelem = m_x.shape()[0];
m_nip = m_w.size();
// allocate arrays
// - shape functions
m_N = xt::empty<double>({m_nip,m_nne});
// - shape function gradients in local coordinates
m_dNxi = xt::empty<double>({m_nip,m_nne,m_ndim});
// - shape function gradients in global coordinates
m_dNx = xt::empty<double>({m_nelem,m_nip,m_nne,m_ndim});
// - integration point volume
m_vol = xt::empty<double>({m_nelem,m_nip});
// shape functions
for ( size_t k = 0 ; k < m_nip ; ++k )
{
m_N(k,0) = .25 * (1.-m_xi(k,0)) * (1.-m_xi(k,1));
m_N(k,1) = .25 * (1.+m_xi(k,0)) * (1.-m_xi(k,1));
m_N(k,2) = .25 * (1.+m_xi(k,0)) * (1.+m_xi(k,1));
m_N(k,3) = .25 * (1.-m_xi(k,0)) * (1.+m_xi(k,1));
}
// shape function gradients in local coordinates
for ( size_t k = 0 ; k < m_nip ; ++k )
{
// - dN / dxi_0
m_dNxi(k,0,0) = -.25*(1.-m_xi(k,1));
m_dNxi(k,1,0) = +.25*(1.-m_xi(k,1));
m_dNxi(k,2,0) = +.25*(1.+m_xi(k,1));
m_dNxi(k,3,0) = -.25*(1.+m_xi(k,1));
// - dN / dxi_1
m_dNxi(k,0,1) = -.25*(1.-m_xi(k,0));
m_dNxi(k,1,1) = -.25*(1.+m_xi(k,0));
m_dNxi(k,2,1) = +.25*(1.+m_xi(k,0));
m_dNxi(k,3,1) = +.25*(1.-m_xi(k,0));
}
// compute the shape function gradients, based on "x"
compute_dN();
}
// ------------------------------------------ constructor ------------------------------------------
inline Quadrature::Quadrature(const xt::xtensor<double,3> &x, const xt::xtensor<double,2> &xi,
const xt::xtensor<double,1> &w) : m_x(x), m_w(w), m_xi(xi)
{
assert( m_x.shape()[1] == m_nne );
assert( m_x.shape()[2] == m_ndim );
// extract shape
m_nelem = m_x.shape()[0];
m_nip = m_w.size();
assert( m_xi.shape()[0] == m_nip );
assert( m_xi.shape()[1] == m_ndim );
assert( m_w .size() == m_nip );
// allocate arrays
// - shape functions
m_N = xt::empty<double>({m_nip,m_nne});
// - shape function gradients in local coordinates
m_dNxi = xt::empty<double>({m_nip,m_nne,m_ndim});
// - shape function gradients in global coordinates
m_dNx = xt::empty<double>({m_nelem,m_nip,m_nne,m_ndim});
// - integration point volume
m_vol = xt::empty<double>({m_nelem,m_nip});
// shape functions
for ( size_t k = 0 ; k < m_nip ; ++k )
{
m_N(k,0) = .25 * (1.-m_xi(k,0)) * (1.-m_xi(k,1));
m_N(k,1) = .25 * (1.+m_xi(k,0)) * (1.-m_xi(k,1));
m_N(k,2) = .25 * (1.+m_xi(k,0)) * (1.+m_xi(k,1));
m_N(k,3) = .25 * (1.-m_xi(k,0)) * (1.+m_xi(k,1));
}
// shape function gradients in local coordinates
for ( size_t k = 0 ; k < m_nip ; ++k )
{
// - dN / dxi_0
m_dNxi(k,0,0) = -.25*(1.-m_xi(k,1));
m_dNxi(k,1,0) = +.25*(1.-m_xi(k,1));
m_dNxi(k,2,0) = +.25*(1.+m_xi(k,1));
m_dNxi(k,3,0) = -.25*(1.+m_xi(k,1));
// - dN / dxi_1
m_dNxi(k,0,1) = -.25*(1.-m_xi(k,0));
m_dNxi(k,1,1) = -.25*(1.+m_xi(k,0));
m_dNxi(k,2,1) = +.25*(1.+m_xi(k,0));
m_dNxi(k,3,1) = +.25*(1.-m_xi(k,0));
}
// compute the shape function gradients, based on "x"
compute_dN();
}
// --------------------------- integration volume (per tensor-component) ---------------------------
inline void Quadrature::dV(xt::xtensor<double,2> &qscalar) const
{
assert( qscalar.shape()[0] == m_nelem );
assert( qscalar.shape()[1] == m_nip );
#pragma omp parallel for
for ( size_t e = 0 ; e < m_nelem ; ++e )
for ( size_t k = 0 ; k < m_nip ; ++k )
qscalar(e,k) = m_vol(e,k);
}
// -------------------------------------------------------------------------------------------------
inline void Quadrature::dV(xt::xtensor<double,4> &qtensor) const
{
assert( qtensor.shape()[0] == m_nelem );
assert( qtensor.shape()[1] == m_nne );
assert( qtensor.shape()[2] >= m_ndim );
assert( qtensor.shape()[3] >= m_ndim );
#pragma omp parallel for
for ( size_t e = 0 ; e < m_nelem ; ++e )
for ( size_t k = 0 ; k < m_nip ; ++k )
for ( size_t i = 0 ; i < qtensor.shape()[2] ; ++i )
for ( size_t j = 0 ; j < qtensor.shape()[3] ; ++j )
qtensor(e,k,i,j) = m_vol(e,k);
}
// -------------------------------------------------------------------------------------------------
inline xt::xtensor<double,2> Quadrature::dV() const
{
xt::xtensor<double,2> out = xt::empty<double>({m_nelem, m_nip});
this->dV(out);
return out;
}
// -------------------------------------- number of elements ---------------------------------------
inline size_t Quadrature::nelem() const
{
return m_nelem;
}
// ---------------------------------- number of nodes per element ----------------------------------
inline size_t Quadrature::nne() const
{
return m_nne;
}
// ------------------------------------- number of dimensions --------------------------------------
inline size_t Quadrature::ndim() const
{
return m_ndim;
}
// --------------------------------- number of integration points ----------------------------------
inline size_t Quadrature::nip() const
{
return m_nip;
}
// --------------------------------------- update positions ----------------------------------------
inline void Quadrature::update_x(const xt::xtensor<double,3> &x)
{
assert( x.shape()[0] == m_nelem );
assert( x.shape()[1] == m_nne );
assert( x.shape()[2] == m_ndim );
assert( x.size() == m_x.size() );
// update positions
m_x = x;
// update the shape function gradients for the new "x"
compute_dN();
}
// ------------------------ shape function gradients in global coordinates -------------------------
inline void Quadrature::compute_dN()
{
// loop over all elements (in parallel)
#pragma omp parallel
{
// - allocate
T2 J;
T2 Jinv;
#pragma omp for
for ( size_t e = 0 ; e < m_nelem ; ++e )
{
// alias nodal positions
auto x = xt::view(m_x, e, xt::all(), xt::all());
// loop over integration points
for ( size_t k = 0 ; k < m_nip ; ++k )
{
// - alias
auto dNxi = xt::view(m_dNxi, k, xt::all(), xt::all());
auto dNx = xt::view(m_dNx , e, k, xt::all(), xt::all());
// - Jacobian (loops unrolled for efficiency)
// J(i,j) += dNxi(m,i) * x(m,j);
J(0,0) = dNxi(0,0)*x(0,0) + dNxi(1,0)*x(1,0) + dNxi(2,0)*x(2,0) + dNxi(3,0)*x(3,0);
J(0,1) = dNxi(0,0)*x(0,1) + dNxi(1,0)*x(1,1) + dNxi(2,0)*x(2,1) + dNxi(3,0)*x(3,1);
J(1,0) = dNxi(0,1)*x(0,0) + dNxi(1,1)*x(1,0) + dNxi(2,1)*x(2,0) + dNxi(3,1)*x(3,0);
J(1,1) = dNxi(0,1)*x(0,1) + dNxi(1,1)*x(1,1) + dNxi(2,1)*x(2,1) + dNxi(3,1)*x(3,1);
// - determinant and inverse of the Jacobian
double Jdet = inv(J, Jinv);
// - shape function gradients wrt global coordinates (loops partly unrolled for efficiency)
// dNx(m,i) += Jinv(i,j) * dNxi(m,j);
for ( size_t m = 0 ; m < m_nne ; ++m )
{
dNx(m,0) = Jinv(0,0) * dNxi(m,0) + Jinv(0,1) * dNxi(m,1);
dNx(m,1) = Jinv(1,0) * dNxi(m,0) + Jinv(1,1) * dNxi(m,1);
}
// - copy to matrix: integration point volume
m_vol(e,k) = m_w(k) * Jdet;
}
}
}
}
// ------------------- dyadic product "qtensor(i,j) = dNdx(m,i) * elemvec(m,j)" --------------------
inline void Quadrature::gradN_vector(
const xt::xtensor<double,3> &elemvec, xt::xtensor<double,4> &qtensor) const
{
assert( elemvec.shape()[0] == m_nelem );
assert( elemvec.shape()[1] == m_nne );
assert( elemvec.shape()[2] == m_ndim );
assert( qtensor.shape()[0] == m_nelem );
assert( qtensor.shape()[1] == m_nne );
assert( qtensor.shape()[2] >= m_ndim );
assert( qtensor.shape()[3] >= m_ndim );
// zero-initialize output: matrix of tensors
qtensor *= 0.0;
// loop over all elements (in parallel)
#pragma omp parallel for
for ( size_t e = 0 ; e < m_nelem ; ++e )
{
// alias element vector (e.g. nodal displacements)
auto u = xt::view(elemvec, e, xt::all(), xt::all());
// loop over all integration points in element "e"
for ( size_t k = 0 ; k < m_nip ; ++k )
{
// - alias
auto dNx = xt::view(m_dNx , e, k, xt::all() , xt::all() );
auto gradu = xt::view(qtensor, e, k, xt::range(0,m_ndim), xt::range(0,m_ndim));
// - evaluate dyadic product (loops unrolled for efficiency)
// gradu(i,j) += dNx(m,i) * u(m,j)
gradu(0,0) = dNx(0,0)*u(0,0) + dNx(1,0)*u(1,0) + dNx(2,0)*u(2,0) + dNx(3,0)*u(3,0);
gradu(0,1) = dNx(0,0)*u(0,1) + dNx(1,0)*u(1,1) + dNx(2,0)*u(2,1) + dNx(3,0)*u(3,1);
gradu(1,0) = dNx(0,1)*u(0,0) + dNx(1,1)*u(1,0) + dNx(2,1)*u(2,0) + dNx(3,1)*u(3,0);
gradu(1,1) = dNx(0,1)*u(0,1) + dNx(1,1)*u(1,1) + dNx(2,1)*u(2,1) + dNx(3,1)*u(3,1);
}
}
}
// -------------------------------------------------------------------------------------------------
inline xt::xtensor<double,4> Quadrature::gradN_vector(const xt::xtensor<double,3> &elemvec) const
{
xt::xtensor<double,4> qtensor = xt::empty<double>({m_nelem, m_nip, m_ndim, m_ndim});
this->gradN_vector(elemvec, qtensor);
return qtensor;
}
// ---------------------------------- transpose of "gradN_vector" ----------------------------------
inline void Quadrature::gradN_vector_T(
const xt::xtensor<double,3> &elemvec, xt::xtensor<double,4> &qtensor) const
{
assert( elemvec.shape()[0] == m_nelem );
assert( elemvec.shape()[1] == m_nne );
assert( elemvec.shape()[2] == m_ndim );
assert( qtensor.shape()[0] == m_nelem );
assert( qtensor.shape()[1] == m_nne );
assert( qtensor.shape()[2] >= m_ndim );
assert( qtensor.shape()[3] >= m_ndim );
// zero-initialize output: matrix of tensors
qtensor *= 0.0;
// loop over all elements (in parallel)
#pragma omp parallel for
for ( size_t e = 0 ; e < m_nelem ; ++e )
{
// alias element vector (e.g. nodal displacements)
auto u = xt::view(elemvec, e, xt::all(), xt::all());
// loop over all integration points in element "e"
for ( size_t k = 0 ; k < m_nip ; ++k )
{
// - alias
auto dNx = xt::view(m_dNx , e, k, xt::all() , xt::all() );
auto gradu = xt::view(qtensor, e, k, xt::range(0,m_ndim), xt::range(0,m_ndim));
// - evaluate transpose of dyadic product (loops unrolled for efficiency)
// gradu(j,i) += dNx(m,i) * u(m,j)
gradu(0,0) = dNx(0,0)*u(0,0) + dNx(1,0)*u(1,0) + dNx(2,0)*u(2,0) + dNx(3,0)*u(3,0);
gradu(1,0) = dNx(0,0)*u(0,1) + dNx(1,0)*u(1,1) + dNx(2,0)*u(2,1) + dNx(3,0)*u(3,1);
gradu(0,1) = dNx(0,1)*u(0,0) + dNx(1,1)*u(1,0) + dNx(2,1)*u(2,0) + dNx(3,1)*u(3,0);
gradu(1,1) = dNx(0,1)*u(0,1) + dNx(1,1)*u(1,1) + dNx(2,1)*u(2,1) + dNx(3,1)*u(3,1);
}
}
}
// -------------------------------------------------------------------------------------------------
inline xt::xtensor<double,4> Quadrature::gradN_vector_T(const xt::xtensor<double,3> &elemvec) const
{
xt::xtensor<double,4> qtensor = xt::empty<double>({m_nelem, m_nip, m_ndim, m_ndim});
this->gradN_vector_T(elemvec, qtensor);
return qtensor;
}
// ------------------------------- symmetric part of "gradN_vector" --------------------------------
inline void Quadrature::symGradN_vector(
const xt::xtensor<double,3> &elemvec, xt::xtensor<double,4> &qtensor) const
{
assert( elemvec.shape()[0] == m_nelem );
assert( elemvec.shape()[1] == m_nne );
assert( elemvec.shape()[2] == m_ndim );
assert( qtensor.shape()[0] == m_nelem );
assert( qtensor.shape()[1] == m_nne );
assert( qtensor.shape()[2] >= m_ndim );
assert( qtensor.shape()[3] >= m_ndim );
// zero-initialize output: matrix of tensors
qtensor *= 0.0;
// loop over all elements (in parallel)
#pragma omp parallel for
for ( size_t e = 0 ; e < m_nelem ; ++e )
{
// alias element vector (e.g. nodal displacements)
auto u = xt::view(elemvec, e, xt::all(), xt::all());
// loop over all integration points in element "e"
for ( size_t k = 0 ; k < m_nip ; ++k )
{
// - alias shape function gradients (local coordinates)
auto dNx = xt::view(m_dNx , e, k, xt::all() , xt::all() );
auto eps = xt::view(qtensor, e, k, xt::range(0,m_ndim), xt::range(0,m_ndim));
// - evaluate symmetrized dyadic product (loops unrolled for efficiency)
// grad(i,j) += dNx(m,i) * u(m,j)
// eps (j,i) = 0.5 * ( grad(i,j) + grad(j,i) )
eps(0,0) = dNx(0,0)*u(0,0) + dNx(1,0)*u(1,0) + dNx(2,0)*u(2,0) + dNx(3,0)*u(3,0);
eps(1,1) = dNx(0,1)*u(0,1) + dNx(1,1)*u(1,1) + dNx(2,1)*u(2,1) + dNx(3,1)*u(3,1);
eps(0,1) = ( dNx(0,1)*u(0,0) + dNx(1,1)*u(1,0) + dNx(2,1)*u(2,0) + dNx(3,1)*u(3,0) +
dNx(0,0)*u(0,1) + dNx(1,0)*u(1,1) + dNx(2,0)*u(2,1) + dNx(3,0)*u(3,1) ) / 2.;
eps(1,0) = eps(0,1);
}
}
}
// -------------------------------------------------------------------------------------------------
inline xt::xtensor<double,4> Quadrature::symGradN_vector(const xt::xtensor<double,3> &elemvec) const
{
xt::xtensor<double,4> qtensor = xt::empty<double>({m_nelem, m_nip, m_ndim, m_ndim});
this->symGradN_vector(elemvec, qtensor);
return qtensor;
}
// ------- scalar product "elemmat(m*ndim+i,n*ndim+i) = N(m) * qscalar * N(n)"; for all "i" --------
inline void Quadrature::int_N_scalar_NT_dV(
const xt::xtensor<double,2> &qscalar, xt::xtensor<double,3> &elemmat) const
{
assert( qscalar.shape()[0] == m_nelem );
assert( qscalar.shape()[1] == m_nip );
assert( elemmat.shape()[0] == m_nelem );
assert( elemmat.shape()[1] == m_nne*m_ndim );
assert( elemmat.shape()[2] == m_nne*m_ndim );
// zero-initialize: matrix of matrices
elemmat *= 0.0;
// loop over all elements (in parallel)
#pragma omp parallel for
for ( size_t e = 0 ; e < m_nelem ; ++e )
{
// alias (e.g. mass matrix)
auto M = xt::view(elemmat, e, xt::all(), xt::all());
// loop over all integration points in element "e"
for ( size_t k = 0 ; k < m_nip ; ++k )
{
// - alias shape functions
auto N = xt::view(m_N, k, xt::all());
// - alias
double vol = m_vol (e,k); // integration point volume
double rho = qscalar(e,k); // integration point scalar (e.g. density)
// - evaluate scalar product, for all dimensions, and assemble
// M(m*ndim+i,n*ndim+i) += N(m) * scalar * N(n) * dV
for ( size_t m = 0 ; m < m_nne ; ++m ) {
for ( size_t n = 0 ; n < m_nne ; ++n ) {
M(m*m_ndim+0, n*m_ndim+0) += N(m) * rho * N(n) * vol;
M(m*m_ndim+1, n*m_ndim+1) += N(m) * rho * N(n) * vol;
}
}
}
}
}
// -------------------------------------------------------------------------------------------------
inline xt::xtensor<double,3> Quadrature::int_N_scalar_NT_dV(const xt::xtensor<double,2> &qscalar) const
{
xt::xtensor<double,3> elemmat = xt::empty<double>({m_nelem, m_nne*m_ndim, m_nne*m_ndim});
this->int_N_scalar_NT_dV(qscalar, elemmat);
return elemmat;
}
// ------------ integral of dot product "elemvec(m,j) += dNdx(m,i) * qtensor(i,j) * dV" ------------
inline void Quadrature::int_gradN_dot_tensor2_dV(const xt::xtensor<double,4> &qtensor,
xt::xtensor<double,3> &elemvec) const
{
assert( qtensor.shape()[0] == m_nelem ); // number of elements
assert( qtensor.shape()[1] == m_nip ); // number of integration points
assert( qtensor.shape()[2] >= m_ndim ); // number of dimensions
assert( qtensor.shape()[3] >= m_ndim ); // number of dimensions
assert( elemvec.shape()[0] == m_nelem ); // number of elements
assert( elemvec.shape()[1] == m_nne ); // number of nodes per element
assert( elemvec.shape()[2] == m_ndim ); // number of dimensions
// zero-initialize output: matrix of vectors
elemvec *= 0.0;
// loop over all elements (in parallel)
#pragma omp parallel for
for ( size_t e = 0 ; e < m_nelem ; ++e )
{
// alias (e.g. nodal force)
auto f = xt::view(elemvec, e, xt::all(), xt::all());
// loop over all integration points in element "e"
for ( size_t k = 0 ; k < m_nip ; ++k )
{
// - alias
auto dNx = xt::view(m_dNx , e, k, xt::all(), xt::all());
auto sig = xt::view(qtensor, e, k, xt::range(0,m_ndim), xt::range(0,m_ndim));
double vol = m_vol(e,k);
// - evaluate dot product, and assemble
for ( size_t m = 0 ; m < m_nne ; ++m )
{
f(m,0) += ( dNx(m,0) * sig(0,0) + dNx(m,1) * sig(1,0) ) * vol;
f(m,1) += ( dNx(m,0) * sig(0,1) + dNx(m,1) * sig(1,1) ) * vol;
}
}
}
}
// -------------------------------------------------------------------------------------------------
inline xt::xtensor<double,3> Quadrature::int_gradN_dot_tensor2_dV(const xt::xtensor<double,4> &qtensor) const
{
xt::xtensor<double,3> elemvec = xt::empty<double>({m_nelem, m_nne, m_ndim});
this->int_gradN_dot_tensor2_dV(qtensor, elemvec);
return elemvec;
}
// -------------------------------------------------------------------------------------------------
}}} // namespace ...
// =================================================================================================
#endif

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