Skip to content
Permalink
Fetching contributors…
Cannot retrieve contributors at this time
140 lines (98 sloc) 4.85 KB
// This code gives the static deflection and mechanical vibration eigenfrequencies and eigenmodes
// around that static deflection for a 3D bilayer disk (800um diameter, 10um thick) whose top layer is prestressed,
// whose bottom layer is clamped at its external boundary and on top of which the atmospheric pressure is pushing.
//
// Small-strain GEOMETRIC NONLINEARITY is used to simulate this prestressed membrane.
//
// With minimal extra effort the buckling of the membrane can be simulated with the Newmark time
// resolution available in sparselizard.
#include "sparselizardbase.h"
using namespace mathop;
void sparselizard(void)
{
// The domain regions as defined in 'disk.geo':
int botlayer = 1, toplayer = 2, sur = 3, top = 4;
// The mesh can be curved!
mesh mymesh("disk.msh");
// Define the whole domain for convenience:
int vol = regionunion({botlayer, toplayer});
// Nodal shape functions 'h1' with 3 components.
// Field u is the membrane deflection.
field u("h1xyz");
// Use interpolation order 2 on the whole domain:
u.setorder(vol, 2);
// Clamp on surface 'sur' (i.e. 0 valued-Dirichlet conditions):
u.setconstraint(sur);
// E is Young's modulus [Pa]. nu is Poisson's ratio []. rho is the density [kg/m^3].
// The material is a polymer.
parameter E, nu, rho;
E|botlayer = 10e9; nu|botlayer = 0.4; rho|botlayer = 1500;
E|toplayer = 15e9; nu|toplayer = 0.4; rho|toplayer = 2000;
formulation elasticity;
// The linear elasticity formulation with geometric nonlinearity for small strains is predefined:
elasticity += integral(botlayer, predefinedelasticity(dof(u), tf(u), u, E, nu, 0.0));
// The top layer is prestressed with 10 MPa in the x and y direction (sigma xx and yy):
expression prestress(6,1,{10e6,10e6,0,0,0,0});
elasticity += integral(toplayer, predefinedelasticity(dof(u), tf(u), u, E, nu, prestress));
// Add the atmospheric pressure force at the top face (perpendicular to it).
// With u provided as second argument all calculations are performed on the deformed
// mesh for this term (thus the normal is updated with the deflection).
double pressure = 1e5;
elasticity += integral(top, u, -pressure*normal(top)*tf(u));
///// STEP 1: GET THE STATIC DEFLECTION WITH A NONLINEAR LOOP:
// Start with an all-zero deflection:
vec solu(elasticity);
double relres = 1;
while (relres > 1e-5)
{
// Generate and get the algebraic matrix A and vector b of the static Ax = b problem:
elasticity.generate();
mat A = elasticity.A();
vec b = elasticity.b();
// Compute the relative residual at the previous iteration:
relres = (b-A*solu).norm() / b.norm();
// Solve for x in Ax = b:
solu = solve(A, b);
// Make solution available in field u:
u.setdata(vol, solu);
// Write the deflection on the top surface of the membrane with an order 2 interpolation:
u.write(top, "ustatic.pos", 2);
std::cout << "Relative residual is: " << relres << std::endl;
}
std::cout << std::endl << "Peak static deflection: " << norm(u).max(vol,4)[0]*1e6 << " um" << std::endl;
///// STEP 2: COMPUTE THE EIGENFREQUENCIES AND EIGENMODES AROUND THE STATIC DEFLECTION:
// Add the inertia terms to the formulation (required for eigenvalue computation):
elasticity += integral(vol, -rho*dtdt(dof(u))*tf(u));
elasticity.generate();
// Get the stiffness and mass matrix:
mat K = elasticity.K();
mat M = elasticity.M();
// Remove the rows and columns corresponding to the 0 constraints:
K.removeconstraints();
M.removeconstraints();
// Create the object to solve the generalised eigenvalue problem K*x = lambda*M*x :
eigenvalue eig(K, M);
// Compute the 5 eigenvalues closest to the target magnitude 0.0 (i.e. the 5 first ones):
eig.compute(5, 0.0);
// Print the eigenfrequencies:
eig.printeigenfrequencies();
// The eigenvectors are real thus we only need the real part:
std::vector<vec> myeigenvectors = eig.geteigenvectorrealpart();
// Loop on all eigenvectors found:
for (int i = 0; i < myeigenvectors.size(); i++)
{
// Transfer the data from the ith eigenvector to field u:
u.setdata(top, myeigenvectors[i]);
// Write the deflection on the top surface of the membrane with an order 2 interpolation:
u.write(top, "ueigenmode"+std::to_string(i)+".pos", 2);
}
// Code validation line. Can be removed.
std::cout << (eig.geteigenvaluerealpart()[0] < 1.1453e+12 && eig.geteigenvaluerealpart()[0] > 1.1451e+12);
}
int main(void)
{
SlepcInitialize(0,{},0,0);
sparselizard();
SlepcFinalize();
return 0;
}
You can’t perform that action at this time.