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DynamicsDiagonalQuad4.h
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DynamicsDiagonalQuad4.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_QUAD4_H
#define GOOSEFEM_DYNAMICS_DIAGONAL_QUAD4_H
#include "Macros.h"
#include <cppmat/cppmat.h>
// -------------------------------------------------------------------------------------------------
namespace GooseFEM {
namespace Dynamics {
namespace Diagonal {
// -------------------------------------------------------------------------------------------------
using vec = cppmat::cartesian2d::vector <double>;
using T2 = cppmat::cartesian2d::tensor2 <double>;
using T2s = cppmat::cartesian2d::tensor2s<double>;
using T2d = cppmat::cartesian2d::tensor2d<double>;
// ========================== N.B. most loops are unrolled for efficiency ==========================
template <class QuadraturePoint>
class Quad4
{
public:
// variables
// ---------
// arrays / matrices
cppmat::tiny::matrix2<double,4,2> xe, ue, ve, xi, xi_n, dNdxi, dNdx;
cppmat::tiny::vector <double,4> w, w_n;
cppmat::tiny::matrix2<double,8,8> M;
cppmat::tiny::vector <double,8> fu, fv;
// tensors
cppmat::cartesian2d::tensor2<double> J, Jinv, gradu, gradv;
// scalars
double Jdet, V;
// sizes (nodes per element, dimensions, quadrature points)
size_t nne=4, ndim=2, nk=4;
// quadrature point
std::shared_ptr<QuadraturePoint> quad;
// constructor
// -----------
Quad4(std::shared_ptr<QuadraturePoint> quad);
// functions
// ---------
void computeM (size_t elem); // mass matrix <- quad->density
void computeFu(size_t elem); // displacement dependent forces <- quad->stressStrain
void computeFv(size_t elem); // displacement dependent forces <- quad->stressStrainRate
void post (size_t elem); // post-process <- quad->stressStrain(Rate)
};
// =================================================================================================
template <class QuadraturePoint>
Quad4<QuadraturePoint>::Quad4(std::shared_ptr<QuadraturePoint> _quad)
{
// quadrature point routines
quad = _quad;
// integration point coordinates/weights: normal Gauss integration
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.;
// integration point coordinates/weights: integration at the nodes
xi_n(0,0) = -1.; xi_n(0,1) = -1.; w_n(0) = 1.;
xi_n(1,0) = +1.; xi_n(1,1) = -1.; w_n(1) = 1.;
xi_n(2,0) = +1.; xi_n(2,1) = +1.; w_n(2) = 1.;
xi_n(3,0) = -1.; xi_n(3,1) = +1.; w_n(3) = 1.;
}
// =================================================================================================
template <class QuadraturePoint>
void Quad4<QuadraturePoint>::computeM(size_t elem)
{
// zero-initialize element mass matrix
M.zeros();
// loop over integration points (coincide with the nodes to get a diagonal mass matrix)
for ( size_t k = 0 ; k < nne ; ++k )
{
// - shape function gradients (local coordinates)
dNdxi(0,0) = -.25*(1.-xi_n(k,1)); dNdxi(0,1) = -.25*(1.-xi_n(k,0));
dNdxi(1,0) = +.25*(1.-xi_n(k,1)); dNdxi(1,1) = -.25*(1.+xi_n(k,0));
dNdxi(2,0) = +.25*(1.+xi_n(k,1)); dNdxi(2,1) = +.25*(1.+xi_n(k,0));
dNdxi(3,0) = -.25*(1.+xi_n(k,1)); dNdxi(3,1) = +.25*(1.-xi_n(k,0));
// - Jacobian
// J(i,j) += dNdxi(m,i) * xe(m,j)
J.zeros();
for ( size_t m = 0 ; m < nne ; ++m ) {
J(0,0) += dNdxi(m,0) * xe(m,0);
J(0,1) += dNdxi(m,0) * xe(m,1);
J(1,0) += dNdxi(m,1) * xe(m,0);
J(1,1) += dNdxi(m,1) * xe(m,1);
}
// - determinant and inverse of the Jacobian
Jdet = J.det();
Jinv = J.inv();
// - integration point volume (== volume associated with the node, in this element)
V = w_n(k) * Jdet;
// - assemble to element mass matrix (use the delta properties of the shape functions)
// M(m+i,n+i) = N(m) * rho * V * N(n);
M(k*2 ,k*2 ) = quad->density(elem,k,V) * V;
M(k*2+1,k*2+1) = quad->density(elem,k,V) * V;
}
}
// =================================================================================================
template <class QuadraturePoint>
void Quad4<QuadraturePoint>::computeFu(size_t elem)
{
// zero-initialize element force
fu.zeros();
// loop over integration points
for ( size_t k = 0 ; k < nk ; ++k )
{
// - shape function gradients (local coordinates)
dNdxi(0,0) = -.25*(1.-xi(k,1)); dNdxi(0,1) = -.25*(1.-xi(k,0));
dNdxi(1,0) = +.25*(1.-xi(k,1)); dNdxi(1,1) = -.25*(1.+xi(k,0));
dNdxi(2,0) = +.25*(1.+xi(k,1)); dNdxi(2,1) = +.25*(1.+xi(k,0));
dNdxi(3,0) = -.25*(1.+xi(k,1)); dNdxi(3,1) = +.25*(1.-xi(k,0));
// - Jacobian
// J(i,j) += dNdxi(m,i) * xe(m,j)
J.zeros();
for ( size_t m = 0 ; m < nne ; ++m ) {
J(0,0) += dNdxi(m,0) * xe(m,0);
J(0,1) += dNdxi(m,0) * xe(m,1);
J(1,0) += dNdxi(m,1) * xe(m,0);
J(1,1) += dNdxi(m,1) * xe(m,1);
}
// - determinant and inverse of the Jacobian
Jdet = J.det();
Jinv = J.inv();
// - integration point volume
V = w(k) * Jdet;
// - shape function gradients (global coordinates)
// dNdx(m,i) += Jinv(i,j) * dNdxi(m,j)
for ( size_t m = 0 ; m < nne ; ++m ) {
dNdx(m,0) = Jinv(0,0) * dNdxi(m,0) + Jinv(0,1) * dNdxi(m,1);
dNdx(m,1) = Jinv(1,0) * dNdxi(m,0) + Jinv(1,1) * dNdxi(m,1);
}
// - displacement gradient
// gradu(i,j) += dNdx(m,i) * ue(m,j)
gradu.zeros();
for ( size_t m = 0 ; m < nne ; ++m ) {
gradu(0,0) += dNdx(m,0) * ue(m,0);
gradu(0,1) += dNdx(m,0) * ue(m,1);
gradu(1,0) += dNdx(m,1) * ue(m,0);
gradu(1,1) += dNdx(m,1) * ue(m,1);
}
// - strain tensor (symmetric part of "gradu")
// quad->eps(i,j) = .5 * ( gradu(i,j) + gradu(j,i) )
quad->eps(0,0) = gradu(0,0);
quad->eps(0,1) = .5 * ( gradu(0,1) + gradu(1,0) );
quad->eps(1,0) = quad->eps(0,1);
quad->eps(1,1) = gradu(1,1);
// - constitutive response
quad->stressStrain(elem,k,V);
// - assemble to element force
// fu(m*ndim+j) += dNdx(m,i) * quad->sig(i,j) * V;
for ( size_t m = 0 ; m < nne ; ++m ) {
fu(m*ndim+0) += dNdx(m,0) * quad->sig(0,0) * V + dNdx(m,1) * quad->sig(1,0) * V;
fu(m*ndim+1) += dNdx(m,0) * quad->sig(0,1) * V + dNdx(m,1) * quad->sig(1,1) * V;
}
}
}
// =================================================================================================
template <class QuadraturePoint>
void Quad4<QuadraturePoint>::computeFv(size_t elem)
{
// zero-initialize element force
fv.zeros();
// loop over integration points
for ( size_t k = 0 ; k < nk ; ++k )
{
// - shape function gradients (local coordinates)
dNdxi(0,0) = -.25*(1.-xi(k,1)); dNdxi(0,1) = -.25*(1.-xi(k,0));
dNdxi(1,0) = +.25*(1.-xi(k,1)); dNdxi(1,1) = -.25*(1.+xi(k,0));
dNdxi(2,0) = +.25*(1.+xi(k,1)); dNdxi(2,1) = +.25*(1.+xi(k,0));
dNdxi(3,0) = -.25*(1.+xi(k,1)); dNdxi(3,1) = +.25*(1.-xi(k,0));
// - Jacobian
// J(i,j) += dNdxi(m,i) * xe(m,j)
J.zeros();
for ( size_t m = 0 ; m < nne ; ++m ) {
J(0,0) += dNdxi(m,0) * xe(m,0);
J(0,1) += dNdxi(m,0) * xe(m,1);
J(1,0) += dNdxi(m,1) * xe(m,0);
J(1,1) += dNdxi(m,1) * xe(m,1);
}
// - determinant and inverse of the Jacobian
Jdet = J.det();
Jinv = J.inv();
// - integration point volume
V = w(k) * Jdet;
// - shape function gradients (global coordinates)
// dNdx(m,i) += Jinv(i,j) * dNdxi(m,j)
for ( size_t m = 0 ; m < nne ; ++m ) {
dNdx(m,0) = Jinv(0,0) * dNdxi(m,0) + Jinv(0,1) * dNdxi(m,1);
dNdx(m,1) = Jinv(1,0) * dNdxi(m,0) + Jinv(1,1) * dNdxi(m,1);
}
// - velocity gradient
// gradv(i,j) += dNdx(m,i) * ve(m,j)
gradv.zeros();
for ( size_t m = 0 ; m < nne ; ++m ) {
gradv(0,0) += dNdx(m,0) * ve(m,0);
gradv(0,1) += dNdx(m,0) * ve(m,1);
gradv(1,0) += dNdx(m,1) * ve(m,0);
gradv(1,1) += dNdx(m,1) * ve(m,1);
}
// - strain-rate tensor (symmetric part of "gradv")
// quad->epsdot(i,j) = .5 * ( gradv(i,j) + gradv(j,i) )
quad->epsdot(0,0) = gradv(0,0);
quad->epsdot(0,1) = .5 * ( gradv(0,1) + gradv(1,0) );
quad->epsdot(1,0) = quad->epsdot(0,1);
quad->epsdot(1,1) = gradv(1,1);
// - constitutive response
quad->stressStrainRate(elem,k,V);
// - assemble to element force
// fv(m*ndim+j) += dNdx(m,i) * quad->sig(i,j) * V;
for ( size_t m = 0 ; m < nne ; ++m ) {
fv(m*2+0) += dNdx(m,0) * quad->sig(0,0) * V + dNdx(m,1) * quad->sig(1,0) * V;
fv(m*2+1) += dNdx(m,0) * quad->sig(0,1) * V + dNdx(m,1) * quad->sig(1,1) * V;
}
}
}
// =================================================================================================
template <class QuadraturePoint>
void Quad4<QuadraturePoint>::post(size_t elem)
{
// loop over integration points
for ( size_t k = 0 ; k < nk ; ++k )
{
// - shape function gradients (local coordinates)
dNdxi(0,0) = -.25*(1.-xi(k,1)); dNdxi(0,1) = -.25*(1.-xi(k,0));
dNdxi(1,0) = +.25*(1.-xi(k,1)); dNdxi(1,1) = -.25*(1.+xi(k,0));
dNdxi(2,0) = +.25*(1.+xi(k,1)); dNdxi(2,1) = +.25*(1.+xi(k,0));
dNdxi(3,0) = -.25*(1.+xi(k,1)); dNdxi(3,1) = +.25*(1.-xi(k,0));
// - Jacobian
// J(i,j) += dNdxi(m,i) * xe(m,j)
J.zeros();
for ( size_t m = 0 ; m < nne ; ++m ) {
J(0,0) += dNdxi(m,0) * xe(m,0);
J(0,1) += dNdxi(m,0) * xe(m,1);
J(1,0) += dNdxi(m,1) * xe(m,0);
J(1,1) += dNdxi(m,1) * xe(m,1);
}
// - determinant and inverse of the Jacobian
Jdet = J.det();
Jinv = J.inv();
// - integration point volume
V = w(k) * Jdet;
// - shape function gradients (global coordinates)
// dNdx(m,i) += Jinv(i,j) * dNdxi(m,j)
for ( size_t m = 0 ; m < nne ; ++m ) {
dNdx(m,0) = Jinv(0,0) * dNdxi(m,0) + Jinv(0,1) * dNdxi(m,1);
dNdx(m,1) = Jinv(1,0) * dNdxi(m,0) + Jinv(1,1) * dNdxi(m,1);
}
// - displacement gradient
// gradu(i,j) += dNdx(m,i) * ue(m,j)
gradu.zeros();
for ( size_t m = 0 ; m < nne ; ++m ) {
gradu(0,0) += dNdx(m,0) * ue(m,0);
gradu(0,1) += dNdx(m,0) * ue(m,1);
gradu(1,0) += dNdx(m,1) * ue(m,0);
gradu(1,1) += dNdx(m,1) * ue(m,1);
}
// - strain tensor (symmetric part of "gradu")
// quad->eps(i,j) = .5 * ( gradu(i,j) + gradu(j,i) )
quad->eps(0,0) = gradu(0,0);
quad->eps(0,1) = .5 * ( gradu(0,1) + gradu(1,0) );
quad->eps(1,0) = quad->eps(0,1);
quad->eps(1,1) = gradu(1,1);
// - velocity gradient
// gradv(i,j) += dNdx(m,i) * ve(m,j)
gradv.zeros();
for ( size_t m = 0 ; m < nne ; ++m ) {
gradv(0,0) += dNdx(m,0) * ve(m,0);
gradv(0,1) += dNdx(m,0) * ve(m,1);
gradv(1,0) += dNdx(m,1) * ve(m,0);
gradv(1,1) += dNdx(m,1) * ve(m,1);
}
// - strain-rate tensor (symmetric part of "gradv")
// quad->epsdot(i,j) = .5 * ( gradv(i,j) + gradv(j,i) )
quad->epsdot(0,0) = gradv(0,0);
quad->epsdot(0,1) = .5 * ( gradv(0,1) + gradv(1,0) );
quad->epsdot(1,0) = quad->epsdot(0,1);
quad->epsdot(1,1) = gradv(1,1);
// - constitutive response
quad->stressStrainPost (elem,k,V);
quad->stressStrainRatePost(elem,k,V);
}
}
// =================================================================================================
}}} // namespace GooseFEM::Dynamics::Diagonal
#endif

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