> Dear All,
>
> I recently thought about using the following structural elements:
>
> * 1D: Euler-Bernoulli or Timoshenko beam
> * 2D: axially symmetric continuum elements
> * 1D: axially symmetric flat plate (like an axially symmetric beam)
>
> Before I start working on any of these elements. Maybe there is somebody who
> already implemented any of these. (Like the truss element in a recent email.)
> Or somebody has a few hints for me e.g. how to implement axial symmetry in
> general.
> Christian Fischer
Hello,
Continuing the thread about implementation of axisymmetric problems.
I have prepared quick and dirty example of calculation of system matrix
for heat conductivity problem in a cylinder (in attachment).
In this case element conductivity matrix is
S = 2 * pi \int_(Le) k * (dH/dr)^T * dH/dr * r dr
where Le - element length
H - matrix of shape functions
(I use notation from J.E. Akin Finite Element Analysis with Error Estimators,
-- there was just a calculated example I could refer to).
(the trick there is to multiply differential equation by r to remove
1/r from the left hand side ).
The example illustrates the use of mesh_fem_global_function.
In this case the global function f(r) = r.
It is not needed in this example but for complete implementation one could
also implement calculation of gradient and Hessian of this function (to be
honest I haven't figured this yet :).
I hope that the code will be of any help.
Regards
Roman
--
Roman Putanowicz, PhD < [email protected] >
Institute for Computational Civil Engng (L-5)
Dept. of Civil Engng, Cracow Univ. of Technology
www.l5.pk.edu.pl, tel. +48 12 628 2569, fax 2034
#include <getfem/getfem_mesh.h>
#include <getfem/getfem_mesh_im.h>
#include <getfem/bgeot_mesh_structure.h>
#include <getfem/getfem_export.h>
#include <getfem/getfem_regular_meshes.h>
#include <getfem/getfem_assembling.h>
#include <getfem/getfem_mesh_region.h>
#include <getfem/getfem_mesh_fem_global_function.h>
#include <iostream>
#include <vector>
/* This program calculates global conduction matrix for heat conduction
* in a cylinder using axisymmetrical setup
*
*
* Compare J.E. Akin "Finite Element Analysis with Error Estimators"
* Chapter 8, "Cylindrical analysis problems", page 218.
*/
int main(int argc, char *argv[]) {
try {
/* Create mesh by building it from scratch
*
* Inner radious r=1
* Outer radious r=2
* 4 linear elements of equal length
*/
getfem::mesh mesh;
int nnodes = 5;
std::vector<bgeot::size_type> ind(nnodes);
double dr = 0.25;
double r0 = 1.0;
for (int i = 0; i<nnodes; i++) {
ind[i] = mesh.add_point(bgeot::base_node(r0 + i*dr, 0.0));
}
for (int i = 0; i<nnodes-1; i++) {
mesh.add_segment(ind[i], ind[i+1]);
}
/* Create FEM description of the problem */
getfem::mesh_fem femT(mesh);
getfem::pfem pf = getfem::fem_descriptor("FEM_PK(1,1)");
femT.set_finite_element(pf);
getfem::mesh_im mim(mesh);
mim.set_integration_method(
getfem::int_method_descriptor("IM_GAUSS1D(2)"));
getfem::size_type Ndof = femT.nb_dof();
/* Create matrix. Since the problem is small
* dense matrices are used here which also helps in printing them.
*
* The equation: M*T = P
*/
gmm::dense_matrix<double> M(Ndof, Ndof); /* characteristic matrix */
struct global_function_r : public getfem::global_function {
getfem::scalar_type
val(const getfem::fem_interpolation_context& c) const {
getfem::base_node x = c.xreal();
return gmm::vect_norm2(x);
}
/* one should also provide methods for gradient and hesian
* calculations if they are used. Here we can skip this
*/
};
global_function_r r_gf;
getfem::pglobal_function pgf_r = new global_function_r;
getfem::mesh_fem_global_function r_fem(mesh);
r_fem.set_functions(pgf_r);
getfem::generic_assembly assem;
assem.push_mi(mim);
assem.push_mf(femT);
assem.push_mf(r_fem);
assem.push_mat(M);
assem.set(
"t=2*comp(Grad(#1).Grad(#1).Base(#2))(:,k,:,k,1);"
"M$1(#1,#1)+=t;"
);
/* CAUTION : the calculated matrix should be scaled by pi * K
* where K is the conductivity constant to make a true conductivity
* matrix
*/
assem.assembly();
std::cout << M << "\n";
} GMM_STANDARD_CATCH_ERROR;
}
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