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DynamicsDiagonalSmallStrainQuad4.h
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/* =================================================================================================
(c - GPLv3) T.W.J. de Geus (Tom) | tom@geus.me | www.geus.me | github.com/tdegeus/GooseFEM
================================================================================================= */
#ifndef GOOSEFEM_DYNAMICS_DIAGONAL_SMALLSTRAIN_QUAD4_H
#define GOOSEFEM_DYNAMICS_DIAGONAL_SMALLSTRAIN_QUAD4_H
// -------------------------------------------------------------------------------------------------
#include "Macros.h"
#include <cppmat/cppmat.h>
// -------------------------------------------------------------------------------------------------
namespace GooseFEM {
namespace Element {
namespace Diagonal {
namespace SmallStrain {
// ========================== N.B. most loops are unrolled for efficiency ==========================
template<class Material>
class Quad4
{
public:
// class variables
// ---------------
// constitutive response per integration point, per element
std::unique_ptr<Material> mat;
// matrices to store the element data
cppmat::matrix<double> x, u, v, f, M, D, dNx, dNxi, w, V, dNxi_n, w_n, V_n;
// dimensions
size_t nelem, nip=4, nne=4, ndim=2;
// signals for solvers
bool changed_f=false, changed_M=true, changed_D=true;
// class functions
// ---------------
// constructor
Quad4(std::unique_ptr<Material> mat, size_t nelem);
// recompute relevant quantities after "x", "u", or "v" have been externally updated
void updated_x();
void updated_u();
void updated_v();
};
// =================================================================================================
template<class Material>
inline Quad4<Material>::Quad4(std::unique_ptr<Material> _mat, size_t _nelem)
{
// copy from input
nelem = _nelem;
mat = std::move(_mat);
// allocate matrices
// -
x .resize({nelem,nne,ndim});
u .resize({nelem,nne,ndim});
v .resize({nelem,nne,ndim});
f .resize({nelem,nne,ndim});
// -
M .resize({nelem,nne*ndim,nne*ndim});
D .resize({nelem,nne*ndim,nne*ndim});
// -
dNx .resize({nelem,nip,nne,ndim});
// -
V .resize({nelem,nip});
V_n .resize({nelem,nne});
// -
dNxi .resize({nip,nne,ndim});
dNxi_n.resize({nne,nne,ndim});
// -
w .resize({nip});
w_n .resize({nne});
// zero initialize matrices (only diagonal is written, no new zero-initialization necessary)
M.setZero();
D.setZero();
// shape function gradient at all Gauss points, in local coordinates
// - k == 0
dNxi(0,0,0) = -.25*(1.+1./std::sqrt(3.)); dNxi(0,0,1) = -.25*(1.+1./std::sqrt(3.));
dNxi(0,1,0) = +.25*(1.+1./std::sqrt(3.)); dNxi(0,1,1) = -.25*(1.-1./std::sqrt(3.));
dNxi(0,2,0) = +.25*(1.-1./std::sqrt(3.)); dNxi(0,2,1) = +.25*(1.-1./std::sqrt(3.));
dNxi(0,3,0) = -.25*(1.-1./std::sqrt(3.)); dNxi(0,3,1) = +.25*(1.+1./std::sqrt(3.));
// - k == 1
dNxi(1,0,0) = -.25*(1.+1./std::sqrt(3.)); dNxi(1,0,1) = -.25*(1.-1./std::sqrt(3.));
dNxi(1,1,0) = +.25*(1.+1./std::sqrt(3.)); dNxi(1,1,1) = -.25*(1.+1./std::sqrt(3.));
dNxi(1,2,0) = +.25*(1.-1./std::sqrt(3.)); dNxi(1,2,1) = +.25*(1.+1./std::sqrt(3.));
dNxi(1,3,0) = -.25*(1.-1./std::sqrt(3.)); dNxi(1,3,1) = +.25*(1.-1./std::sqrt(3.));
// - k == 2
dNxi(2,0,0) = -.25*(1.-1./std::sqrt(3.)); dNxi(2,0,1) = -.25*(1.-1./std::sqrt(3.));
dNxi(2,1,0) = +.25*(1.-1./std::sqrt(3.)); dNxi(2,1,1) = -.25*(1.+1./std::sqrt(3.));
dNxi(2,2,0) = +.25*(1.+1./std::sqrt(3.)); dNxi(2,2,1) = +.25*(1.+1./std::sqrt(3.));
dNxi(2,3,0) = -.25*(1.+1./std::sqrt(3.)); dNxi(2,3,1) = +.25*(1.-1./std::sqrt(3.));
// - k == 3
dNxi(3,0,0) = -.25*(1.-1./std::sqrt(3.)); dNxi(3,0,1) = -.25*(1.+1./std::sqrt(3.));
dNxi(3,1,0) = +.25*(1.-1./std::sqrt(3.)); dNxi(3,1,1) = -.25*(1.-1./std::sqrt(3.));
dNxi(3,2,0) = +.25*(1.+1./std::sqrt(3.)); dNxi(3,2,1) = +.25*(1.-1./std::sqrt(3.));
dNxi(3,3,0) = -.25*(1.+1./std::sqrt(3.)); dNxi(3,3,1) = +.25*(1.+1./std::sqrt(3.));
// integration point weight at all Gauss points
w(0) = 1.;
w(1) = 1.;
w(2) = 1.;
w(3) = 1.;
// shape function gradient at all nodes, in local coordinates
// - k == 0
dNxi_n(0,0,0) = -0.5; dNxi_n(0,0,1) = -0.5;
dNxi_n(0,1,0) = +0.5; dNxi_n(0,1,1) = 0.0;
dNxi_n(0,2,0) = 0.0; dNxi_n(0,2,1) = 0.0;
dNxi_n(0,3,0) = 0.0; dNxi_n(0,3,1) = +0.5;
// - k == 1
dNxi_n(1,0,0) = -0.5; dNxi_n(1,0,1) = 0.0;
dNxi_n(1,1,0) = +0.5; dNxi_n(1,1,1) = -0.5;
dNxi_n(1,2,0) = 0.0; dNxi_n(1,2,1) = +0.5;
dNxi_n(1,3,0) = 0.0; dNxi_n(1,3,1) = 0.0;
// - k == 2
dNxi_n(2,0,0) = 0.0; dNxi_n(2,0,1) = 0.0;
dNxi_n(2,1,0) = 0.0; dNxi_n(2,1,1) = -0.5;
dNxi_n(2,2,0) = +0.5; dNxi_n(2,2,1) = +0.5;
dNxi_n(2,3,0) = -0.5; dNxi_n(2,3,1) = 0.0;
// - k == 3
dNxi_n(3,0,0) = 0.0; dNxi_n(3,0,1) = -0.5;
dNxi_n(3,1,0) = 0.0; dNxi_n(3,1,1) = 0.0;
dNxi_n(3,2,0) = +0.5; dNxi_n(3,2,1) = 0.0;
dNxi_n(3,3,0) = -0.5; dNxi_n(3,3,1) = +0.5;
// integration point weight at all nodes
w_n(0) = 1.;
w_n(1) = 1.;
w_n(2) = 1.;
w_n(3) = 1.;
// Note, the above is a specialization of the following:
//
// - Shape function gradients
//
// dNxi(0,0) = -.25*(1.-xi(k,1)); dNxi(0,1) = -.25*(1.-xi(k,0));
// dNxi(1,0) = +.25*(1.-xi(k,1)); dNxi(1,1) = -.25*(1.+xi(k,0));
// dNxi(2,0) = +.25*(1.+xi(k,1)); dNxi(2,1) = +.25*(1.+xi(k,0));
// dNxi(3,0) = -.25*(1.+xi(k,1)); dNxi(3,1) = +.25*(1.-xi(k,0));
//
// - Gauss point coordinates and weights
//
// xi(0,0) = -1./std::sqrt(3.); xi(0,1) = -1./std::sqrt(3.); w(0) = 1.;
// xi(1,0) = +1./std::sqrt(3.); xi(1,1) = -1./std::sqrt(3.); w(1) = 1.;
// xi(2,0) = +1./std::sqrt(3.); xi(2,1) = +1./std::sqrt(3.); w(2) = 1.;
// xi(3,0) = -1./std::sqrt(3.); xi(3,1) = +1./std::sqrt(3.); w(3) = 1.;
//
// - Nodal coordinates and weights
//
// xi(0,0) = -1.; xi(0,1) = -1.; w(0) = 1.;
// xi(1,0) = +1.; xi(1,1) = -1.; w(1) = 1.;
// xi(2,0) = +1.; xi(2,1) = +1.; w(2) = 1.;
// xi(3,0) = -1.; xi(3,1) = +1.; w(3) = 1.;
}
// =================================================================================================
template<class Material>
inline void Quad4<Material>::updated_x()
{
#pragma omp parallel
{
// intermediate quantities
cppmat::cartesian2d::tensor2<double> J_, Jinv_;
double Jdet_;
// local views of the global arrays (speeds up indexing, and increases readability)
cppmat::tiny::matrix2<double,8,8> M_, D_;
cppmat::tiny::matrix2<double,4,2> dNxi_, dNx_, x_;
cppmat::tiny::vector <double,4> w_, V_;
// loop over all elements (in parallel)
#pragma omp for
for ( size_t e = 0 ; e < nelem ; ++e )
{
// pointer to element positions
x_.map(&x(e));
// nodal quadrature
// ----------------
// pointer to element mass/damping matrix, nodal volume, and integration weight
M_.map(&M (e));
D_.map(&D (e));
V_.map(&V_n(e));
w_.map(&w_n(0));
// loop over nodes, i.e. the integration points
for ( size_t k = 0 ; k < nne ; ++k )
{
// - pointer to the shape function gradients (local coordinates)
dNxi_.map(&dNxi_n(k));
// - Jacobian
// J(i,j) += dNxi(m,i) * xe(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 of the Jacobian
Jdet_ = J_.det();
// - integration point volume
V_(k) = w_(k) * Jdet_;
// - assemble element mass matrix
// M(m+i,n+i) += N(m) * rho * V * N(n);
M_(k*2 ,k*2 ) = mat->rho(e,k) * V_(k);
M_(k*2+1,k*2+1) = mat->rho(e,k) * V_(k);
// - assemble element non-Galilean damping matrix
// D(m+i,n+i) += N(m) * alpha * V * N(n);
D_(k*2 ,k*2 ) = mat->alpha(e,k) * V_(k);
D_(k*2+1,k*2+1) = mat->alpha(e,k) * V_(k);
}
// Gaussian quadrature
// -------------------
// pointer to element integration volume and weight
V_.map(&V(e));
w_.map(&w(0));
// loop over Gauss points
for ( size_t k = 0 ; k < nip ; ++k )
{
// - pointer to the shape function gradients (local/global coordinates)
dNxi_.map(&dNxi( k));
dNx_ .map(&dNx (e,k));
// - Jacobian
// J(i,j) += dNxi(m,i) * xe(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
Jdet_ = J_.det();
Jinv_ = J_.inv();
// - integration point volume
V_(k) = w_(k) * Jdet_;
// - shape function gradients (global coordinates)
// dNx(m,i) += Jinv(i,j) * dNxi(m,j)
for ( size_t m = 0 ; 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);
}
}
}
// set signals
changed_f = false;
changed_M = true;
changed_D = true;
} // #pragma omp parallel
}
// =================================================================================================
template<class Material>
inline void Quad4<Material>::updated_u()
{
#pragma omp parallel
{
// intermediate quantities
cppmat::cartesian2d::tensor2 <double> gradu_;
cppmat::cartesian2d::tensor2s<double> eps_, sig_;
// local views of the global arrays (speeds up indexing, and increases readability)
cppmat::tiny::matrix2<double,4,2> dNx_, u_, f_;
cppmat::tiny::vector <double,4> V_;
// loop over all elements (in parallel)
#pragma omp for
for ( size_t e = 0 ; e < nelem ; ++e )
{
// pointer to element forces, displacements, and integration volume
f_.map(&f(e));
u_.map(&u(e));
V_.map(&V(e));
// zero initialize forces
f_.setZero();
// loop over all integration points in element "e"
for ( size_t k = 0 ; k < nip ; ++k )
{
// - pointer to the shape function gradients, strain and stress tensor (stored symmetric)
dNx_.map(&dNx (e,k));
eps_.map(&mat->eps(e,k));
sig_.map(&mat->sig(e,k));
// - displacement gradient
// gradu_(i,j) += dNx(m,i) * ue(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);
// - strain (stored symmetric)
// eps(i,j) = .5 * ( gradu_(i,j) + gradu_(j,i) )
eps_(0,0) = gradu_(0,0);
eps_(0,1) = .5 * ( gradu_(0,1) + gradu_(1,0) );
eps_(1,1) = gradu_(1,1);
// - constitutive response
mat->updated_eps(e,k);
// - assemble to element force
// f(m,j) += dNx(m,i) * sig(i,j) * V;
for ( size_t m = 0 ; m < nne ; ++m )
{
f_(m,0) += dNx_(m,0) * sig_(0,0) * V_(k) + dNx_(m,1) * sig_(1,0) * V_(k);
f_(m,1) += dNx_(m,0) * sig_(0,1) * V_(k) + dNx_(m,1) * sig_(1,1) * V_(k);
}
}
}
// set signals
changed_f = true;
changed_M = false;
changed_D = false;
}
}
// =================================================================================================
template<class Material>
inline void Quad4<Material>::updated_v()
{
#pragma omp parallel
{
// intermediate quantities
cppmat::cartesian2d::tensor2 <double> gradv_;
cppmat::cartesian2d::tensor2s<double> epsdot_, sig_;
// local views of the global arrays (speeds up indexing, and increases readability)
cppmat::tiny::matrix2<double,4,2> dNx_, v_, f_;
cppmat::tiny::vector <double,4> V_;
// loop over all elements (in parallel)
#pragma omp for
for ( size_t e = 0 ; e < nelem ; ++e )
{
// pointer to element forces, displacements, and integration volume
f_.map(&f(e));
v_.map(&v(e));
V_.map(&V(e));
// zero initialize forces
f_.setZero();
// loop over all integration points in element "e"
for ( size_t k = 0 ; k < nip ; ++k )
{
// - pointer to the shape function gradients, strain-rate and stress tensor (stored symmetric)
dNx_ .map(&dNx (e,k));
epsdot_.map(&mat->epsdot(e,k));
sig_ .map(&mat->sig (e,k));
// - displacement gradient
// gradv_(i,j) += dNx(m,i) * ue(m,j)
gradv_(0,0) = dNx_(0,0)*v_(0,0) + dNx_(1,0)*v_(1,0) + dNx_(2,0)*v_(2,0) + dNx_(3,0)*v_(3,0);
gradv_(0,1) = dNx_(0,0)*v_(0,1) + dNx_(1,0)*v_(1,1) + dNx_(2,0)*v_(2,1) + dNx_(3,0)*v_(3,1);
gradv_(1,0) = dNx_(0,1)*v_(0,0) + dNx_(1,1)*v_(1,0) + dNx_(2,1)*v_(2,0) + dNx_(3,1)*v_(3,0);
gradv_(1,1) = dNx_(0,1)*v_(0,1) + dNx_(1,1)*v_(1,1) + dNx_(2,1)*v_(2,1) + dNx_(3,1)*v_(3,1);
// - strain (stored symmetric)
// epsdot(i,j) = .5 * ( gradv_(i,j) + gradv_(j,i) )
epsdot_(0,0) = gradv_(0,0);
epsdot_(0,1) = .5 * ( gradv_(0,1) + gradv_(1,0) );
epsdot_(1,1) = gradv_(1,1);
// - constitutive response
mat->updated_epsdot(e,k);
// - assemble to element force
// f(m,j) += dNdx(m,i) * sig(i,j) * V;
for ( size_t m = 0 ; m < nne ; ++m )
{
f_(m,0) += dNx_(m,0) * sig_(0,0) * V_(k) + dNx_(m,1) * sig_(1,0) * V_(k);
f_(m,1) += dNx_(m,0) * sig_(0,1) * V_(k) + dNx_(m,1) * sig_(1,1) * V_(k);
}
}
}
// set signals
changed_f = true;
changed_M = false;
changed_D = false;
}
}
// =================================================================================================
}}}} // namespace ...
#endif
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