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example.cpp
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Sat, Jul 12, 14:36

example.cpp
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#define _USE_MATH_DEFINES // to use "M_PI" from "math.h"
#include <iostream>
#include <math.h>
#include <Eigen/Eigen>
#include <pybind11/pybind11.h>
#include <pybind11/eigen.h>
typedef Eigen::Matrix<double, Eigen::Dynamic, Eigen::Dynamic, Eigen::RowMajor> MatD;
typedef Eigen::Matrix<double, Eigen::Dynamic, 1, Eigen::ColMajor> ColD;
typedef Eigen::Matrix<size_t, Eigen::Dynamic, 1, Eigen::ColMajor> ColS;
// -------------------------------------------------------------------------------------------------
ColD computeM(const ColD &x, double rho)
{
ColD dNdxi(2), dNdx(2), xe(2), M(x.size());
// integration variables
double w = 1; // weight (N.B. such that total volume matches)
double J, V;
// shape function gradients : local coordinates
dNdxi(0) = -0.5;
dNdxi(1) = +0.5;
// extract number of elements : number of nodes - 1
size_t nelem = x.size() - 1;
// zero initialize force
M.setZero();
// loop over elements
for ( size_t e = 0 ; e < nelem ; ++e )
{
// element coordinates
xe(0) = x(e );
xe(1) = x(e+1);
// Jacobian
J = dNdxi(0) * xe(0) + dNdxi(1) * xe(1);
// element size
V = J * w;
// add to force
M(e ) += rho * V;
M(e+1) += rho * V;
}
return M;
}
// -------------------------------------------------------------------------------------------------
ColD computeFu(const ColD &x, const ColD &u, double G)
{
ColD dNdxi(2), dNdx(2), xe(2), ue(2), F(x.size());
// integration variables
double w = 2; // weight (N.B. xi == 0)
double J, V;
// local constitutive response
double sig, eps;
// shape function gradients : local coordinates
dNdxi(0) = -0.5;
dNdxi(1) = +0.5;
// extract number of elements : number of nodes - 1
size_t nelem = x.size() - 1;
// zero initialize force
F.setZero();
// loop over elements
for ( size_t e = 0 ; e < nelem ; ++e )
{
// element coordinates
xe(0) = x(e );
xe(1) = x(e+1);
// element displacements
ue(0) = u(e );
ue(1) = u(e+1);
// Jacobian
J = dNdxi(0) * xe(0) + dNdxi(1) * xe(1);
// element size
V = J * w;
// shape function gradients : global coordinates (constant over the element)
dNdx = ( 1./J ) * dNdxi;
// strain (constant over the element)
eps = dNdx(0) * ue(0) + dNdx(1) * ue(1);
// stress (constant over the element)
sig = G * eps;
// add to force
F(e ) += dNdx(0) * sig * V;
F(e+1) += dNdx(1) * sig * V;
}
return F;
}
// -------------------------------------------------------------------------------------------------
ColD computeFv(const ColD &x, const ColD &v, double eta)
{
ColD dNdxi(2), dNdx(2), xe(2), ve(2), F(x.size());
// integration variables
double w = 2; // weight (N.B. xi == 0)
double J, V;
// local constitutive response
double sig, epsdot;
// shape function gradients : local coordinates
dNdxi(0) = -0.5;
dNdxi(1) = +0.5;
// extract number of elements : number of nodes - 1
size_t nelem = x.size() - 1;
// zero initialize force
F.setZero();
// loop over elements
for ( size_t e = 0 ; e < nelem ; ++e )
{
// element coordinates
xe(0) = x(e );
xe(1) = x(e+1);
// element displacements
ve(0) = v(e );
ve(1) = v(e+1);
// Jacobian
J = dNdxi(0) * xe(0) + dNdxi(1) * xe(1);
// element size
V = J * w;
// shape function gradients : global coordinates (constant over the element)
dNdx = ( 1./J ) * dNdxi;
// strain (constant over the element)
epsdot = dNdx(0) * ve(0) + dNdx(1) * ve(1);
// stress (constant over the element)
sig = eta * epsdot;
// add to force
F(e ) += dNdx(0) * sig * V;
F(e+1) += dNdx(1) * sig * V;
}
return F;
}
// -------------------------------------------------------------------------------------------------
MatD compute(
double rho, double G, double eta, double h, const ColD &Fext,
double dt, size_t ninc, size_t save_every, const ColS &save_nodes
)
{
size_t nel = static_cast<size_t>(Fext.size())-1;
double L = h * static_cast<double>(nel);
double t = 0.;
size_t isave = 0;
size_t jsave = static_cast<size_t>(std::floor(static_cast<double>(ninc)/static_cast<double>(save_every)));
size_t nsave = static_cast<size_t>(save_nodes.size());
MatD out(jsave,nsave);
ColD x (nel+1);
ColD u (nel+1);
ColD v (nel+1);
ColD v_n (nel+1);
ColD a (nel+1);
ColD a_n (nel+1);
ColD M (nel+1);
ColD Minv(nel+1);
ColD Fu (nel+1);
ColD Fv (nel+1);
x = ColD::LinSpaced( nel+1, 0.0, L );
u .setZero();
v .setZero();
a .setZero();
a_n.setZero();
v_n.setZero();
M = computeM(x,rho);
Minv = M.cwiseInverse();
for ( size_t inc = 0 ; inc < ninc ; ++inc )
{
// (1) new nodal positions (displacements)
// - apply update (nodal) : x_{n+1} = x_n + dt * v_n + .5 * dt^2 * a_n"
u += dt * v + ( .5 * std::pow(dt,2.) ) * a;
// - compute forces that are dependent on the displacement, but not on the velocity
Fu = computeFu(x,u,G);
// (2a) estimate nodal velocities
// - update velocities (DOFs)
v.noalias() = v_n + dt * a;
// - compute forces that are dependent on the velocity
Fv = computeFv(x,v,eta);
// - solve for accelerations (DOFs)
a.noalias() = Minv.cwiseProduct( Fext - Fu - Fv );
// - apply boundary conditions
a(0 ) = 0.;
a(nel) = 0.;
// - update velocities (DOFs)
v.noalias() = v_n + ( .5 * dt ) * ( a_n + a );
// - compute forces that are dependent on the velocity
Fv = computeFv(x,v,eta);
// (2b) new nodal velocities
// - solve for accelerations (DOFs)
a.noalias() = Minv.cwiseProduct( Fext - Fu - Fv );
// - apply boundary conditions
a(0 ) = 0.;
a(nel) = 0.;
// - update velocities (DOFs)
v.noalias() = v_n + ( .5 * dt ) * ( a_n + a );
// - compute forces that are dependent on the velocity
Fv = computeFv(x,v,eta);
// (3) new nodal accelerations
// - solve for accelerations (DOFs)
a.noalias() = Minv.cwiseProduct( Fext - Fu - Fv );
// - apply boundary conditions
a(0 ) = 0.;
a(nel) = 0.;
// store history
a_n = a; // accelerations (DOFs)
v_n = v; // nodal velocities
t += dt; // current time instance
// store output
if ( inc % save_every == 0 )
{
for ( size_t i=0; i<nsave; ++i ) out(isave,i) = u(save_nodes(i));
++isave;
}
}
return out;
}
// -------------------------------------------------------------------------------------------------
namespace py = pybind11;
PYBIND11_MODULE(example,m)
{
m.def(
"compute", &compute,
py::arg("rho"), py::arg("G"), py::arg("eta"), py::arg("h"), py::arg("Fext"), py::arg("dt"),
py::arg("ninc"), py::arg("save_every"), py::arg("save_nodes")
);
}

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