The adaptive solution of the equations of time-harmonic linear elasticity on unstructured meshes

The aim of this tutorial is to demonstrate the adaptive solution of the time-harmonic equations of linear elasticity in cartesian coordinates on unstructured meshes.

The driver code is very similar to the one presented in another tutorial and we only discuss the changes necessary to deal with the generation of the unstructured mesh and the assignment of different material properties to different parts of the domain.

A test problem: Time-harmonic oscillations of an elastic annulus reinforced by a T-rib

The sketch below shows the problem setup: A 2D elastic annulus which is reinforced with two T-ribs is subjected to a time-periodic pressure load of magnitude

\[ {\bf t} = P ( \exp(\alpha(\varphi-\pi/4)^2) + \exp(\alpha(\varphi-3\pi/4)^2) ) \]

(where $ \varphi $ is the polar angle) along its outer edge. The parameter $ \alpha $ controls the "sharpness" of the pressure load. For $ \alpha=0 $ we obtain a uniform, axisymmetric load; the sketch below shows the pressure distribution (red vectors indicating the traction) for $ \alpha = 1. $

Sketch of the problem setup.

The structure is symmetric and we only discretise the right half ( $ x_1 > 0 $) of the domain and apply symmetry conditions (zero horizontal displacement) on the $ x_2-$ axis.

Results

The figure below shows an animation of the structure's time-harmonic oscillation. The blue shaded region shows the shape of the oscillating structure while the pink region shows its initial configuration. The left half of the plot is used to show the (mirror image of the) adaptive unstructured mesh on which the solution was computed:

Animation of the time-harmonic deformation.

This looks very pretty and shows that we can compute in non-trivial geometries but should you believe the results? Here's an attempt to convince you: If we make the rib much softer than the annulus, the rib will not offer much structural resistance and the annular region will deform as if the rib was not present. If we then set $ \alpha = 0 $ we apply an axisymmetric forcing onto the structure and would expect the resulting displacement field (at least in the annular region) to be axisymmetric. For this case it is easy to find an analytical solution to the problem. The next two figures show a comparison between the analytical (green spheres) and computed solutions (shaded) for the real part of the horizontal and vertical displacements, respectively.

Real part of the horizontal displacement, computed (shaded) and exact (spheres).
Real part of the vertical displacement, computed (shaded) and exact (spheres).

Convinced?

The numerical solution

The driver code for this problem is very similar to the one discussed in another tutorial. Running sdiff on the two driver codes

demo_drivers/time_harmonic_linear_elasticity/elastic_annulus/time_harmonic_elastic_annulus.cc

and

demo_drivers/time_harmonic_linear_elasticity/elastic_annulus/unstructured_time_harmonic_elastic_annulus.cc

shows you the differences, the most important of which are:

  • The provision of multiple elasticity tensors and frequency parameters for the two different regions (the rib and the annulus).

  • The provision of a helper function complete_problem_setup() which rebuilds the elements (by passing the problem parameters to the elements) following the unstructured mesh adaptation. (The need/rationale for such a function is discussed in another tutorial.)

  • The mesh generation – the specification of the curvilinear boundaries and the geometry of the rib is somewhat tedious. We refer to another tutorial for a discussion of how to define the internal mesh boundary that separates the two regions (the rib and the annular region) so that we can assign different material properties to them.

    .
    All of this is reasonably straightforward and provides a powerful code that automatically adapts the mesh while respecting the curvilinear boundaries of the domain. Have a look through the driver code and play with it.

Code listing

Here's a listing of the complete driver code:

//LIC// ====================================================================
//LIC// This file forms part of oomph-lib, the object-oriented,
//LIC// multi-physics finite-element library, available
//LIC// at http://www.oomph-lib.org.
//LIC//
//LIC// Copyright (C) 2006-2023 Matthias Heil and Andrew Hazel
//LIC//
//LIC// This library is free software; you can redistribute it and/or
//LIC// modify it under the terms of the GNU Lesser General Public
//LIC// License as published by the Free Software Foundation; either
//LIC// version 2.1 of the License, or (at your option) any later version.
//LIC//
//LIC// This library is distributed in the hope that it will be useful,
//LIC// but WITHOUT ANY WARRANTY; without even the implied warranty of
//LIC// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
//LIC// Lesser General Public License for more details.
//LIC//
//LIC// You should have received a copy of the GNU Lesser General Public
//LIC// License along with this library; if not, write to the Free Software
//LIC// Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA
//LIC// 02110-1301 USA.
//LIC//
//LIC// The authors may be contacted at oomph-lib@maths.man.ac.uk.
//LIC//
//LIC//====================================================================
// Time-harmonic deformation of an T-rib reinforced elastic annulus subject to
// a nonuniform pressure load.
//Oomph-lib includes
#include "generic.h"
#include "time_harmonic_linear_elasticity.h"
//The meshes
#include "meshes/triangle_mesh.h"
using namespace std;
using namespace oomph;
/// ////////////////////////////////////////////////////////////////////
/// ////////////////////////////////////////////////////////////////////
// Straight line as geometric object
/// ////////////////////////////////////////////////////////////////////
/// ////////////////////////////////////////////////////////////////////
//=========================================================================
/// Straight 1D line in 2D space
//=========================================================================
class MyStraightLine : public GeomObject
{
public:
/// Constructor: Pass start and end points
MyStraightLine(const Vector<double>& r_start, const Vector<double>& r_end)
: GeomObject(1,2), R_start(r_start), R_end(r_end)
{ }
/// Broken copy constructor
MyStraightLine(const MyStraightLine& dummy)
{
BrokenCopy::broken_copy("MyStraightLine");
}
/// Broken assignment operator
void operator=(const MyStraightLine&)
{
BrokenCopy::broken_assign("MyStraightLine");
}
/// Destructor: Empty
~MyStraightLine(){}
/// Position Vector at Lagrangian coordinate zeta
void position(const Vector<double>& zeta, Vector<double>& r) const
{
// Position Vector
r[0] = R_start[0]+(R_end[0]-R_start[0])*zeta[0];
r[1] = R_start[1]+(R_end[1]-R_start[1])*zeta[0];
}
private:
/// Start point of line
Vector<double> R_start;
/// End point of line
Vector<double> R_end;
};
/// ///////////////////////////////////////////////////////////////////
/// ///////////////////////////////////////////////////////////////////
/// ///////////////////////////////////////////////////////////////////
//=======start_namespace==========================================
/// Global variables
//================================================================
namespace Global_Parameters
{
/// Poisson's ratio
double Nu=0.3;
/// Square of non-dim frequency
double Omega_sq=10.0;
/// Square of non-dim frequency for the two regions
Vector<double> Omega_sq_region(2,Omega_sq);
/// The elasticity tensors for the two regions
Vector<TimeHarmonicIsotropicElasticityTensor*> E_pt;
/// Thickness of annulus
double H_annulus=0.1;
/// Uniform pressure
double P = 0.1;
/// Peakiness parameter for pressure load
double Alpha=200.0;
/// Constant pressure load (real and imag part)
void pressure_load(const Vector<double> &x,
const Vector<double> &n,
Vector<std::complex<double> >&traction)
{
double phi=atan2(x[1],x[0]);
double magnitude=exp(-Alpha*pow(phi-0.25*MathematicalConstants::Pi,2));
unsigned dim = traction.size();
for(unsigned i=0;i<dim;i++)
{
traction[i] = complex<double>(-magnitude*P*n[i],magnitude*P*n[i]);
}
} // end_of_pressure_load
/// Output directory
string Directory="RESLT";
} //end namespace
//=============begin_problem============================================
/// Annular disk
//======================================================================
template<class ELASTICITY_ELEMENT>
class RingWithTRibProblem : public Problem
{
public:
/// Constructor:
RingWithTRibProblem();
/// Update function (empty)
void actions_after_newton_solve() {}
/// Update function (empty)
void actions_before_newton_solve() {}
/// Actions before adapt: Wipe the mesh of traction elements
void actions_before_adapt();
/// Actions after adapt: Rebuild the mesh of traction elements
void actions_after_adapt();
/// Doc the solution
void doc_solution();
private:
/// Create traction elements
void create_traction_elements();
/// Delete traction elements
void delete_traction_elements();
/// Helper function to complete problem setup
void complete_problem_setup();
#ifdef ADAPTIVE
/// Pointer to refineable solid mesh
RefineableTriangleMesh<ELASTICITY_ELEMENT>* Solid_mesh_pt;
#else
/// Pointer to solid mesh
TriangleMesh<ELASTICITY_ELEMENT>* Solid_mesh_pt;
#endif
/// Pointer to mesh of traction elements
Mesh* Traction_mesh_pt;
/// DocInfo object for output
DocInfo Doc_info;
/// Boundary ID of upper symmetry boundary
unsigned Upper_symmetry_boundary_id;
/// Boundary ID of lower symmetry boundary
unsigned Lower_symmetry_boundary_id;
/// Boundary ID of outer boundary
unsigned Outer_boundary_id;
};
//===========start_of_constructor=======================================
/// Constructor:
//======================================================================
template<class ELASTICITY_ELEMENT>
RingWithTRibProblem<ELASTICITY_ELEMENT>::RingWithTRibProblem()
{
// Solid mesh
//-----------
// Start and end coordinates
Vector<double> r_start(2);
Vector<double> r_end(2);
// Outer radius of hull
double r_outer = 1.0;
// Inner radius of hull
double r_inner = r_outer-Global_Parameters::H_annulus;
// Thickness of rib
double rib_thick=0.05;
// Depth of rib
double rib_depth=0.2;
// Total width of T
double t_width=0.2;
// Thickness of T
double t_thick=0.05;
// Half-opening angle of rib
double half_phi_rib=asin(0.5*rib_thick/r_inner);
// Pointer to the closed curve that defines the outer boundary
TriangleMeshClosedCurve* closed_curve_pt=0;
// Provide storage for pointers to the parts of the curvilinear boundary
Vector<TriangleMeshCurveSection*> curvilinear_boundary_pt;
// Outer boundary
//---------------
Ellipse* outer_boundary_circle_pt = new Ellipse(r_outer,r_outer);
double zeta_start=-0.5*MathematicalConstants::Pi;
double zeta_end=0.5*MathematicalConstants::Pi;
unsigned nsegment=50;
unsigned boundary_id=curvilinear_boundary_pt.size();
curvilinear_boundary_pt.push_back(
new TriangleMeshCurviLine(
outer_boundary_circle_pt,zeta_start,zeta_end,nsegment,boundary_id));
// Remember it
Outer_boundary_id=boundary_id;
// Upper straight line segment on symmetry axis
//---------------------------------------------
r_start[0]=0.0;
r_start[1]=r_outer;
r_end[0]=0.0;
r_end[1]=r_inner;
MyStraightLine* upper_sym_pt = new MyStraightLine(r_start,r_end);
zeta_start=0.0;
zeta_end=1.0;
nsegment=1;
boundary_id=curvilinear_boundary_pt.size();
curvilinear_boundary_pt.push_back(
new TriangleMeshCurviLine(
upper_sym_pt,zeta_start,zeta_end,nsegment,boundary_id));
// Remember it
Upper_symmetry_boundary_id=boundary_id;
// Upper part of inner boundary
//-----------------------------
Ellipse* upper_inner_boundary_pt =
new Ellipse(r_inner,r_inner);
zeta_start=0.5*MathematicalConstants::Pi;
zeta_end=half_phi_rib;
nsegment=20;
boundary_id=curvilinear_boundary_pt.size();
curvilinear_boundary_pt.push_back(
new TriangleMeshCurviLine(
upper_inner_boundary_pt,
zeta_start,zeta_end,nsegment,boundary_id));
// Upper half of inward rib
//-------------------------
r_start[0]=r_inner*cos(half_phi_rib);
r_start[1]=r_inner*sin(half_phi_rib);
r_end[0]=r_start[0]-rib_depth;
r_end[1]=r_start[1];
MyStraightLine* upper_inward_rib_pt = new MyStraightLine(r_start,r_end);
zeta_start=0.0;
zeta_end=1.0;
nsegment=1;
boundary_id=curvilinear_boundary_pt.size();
TriangleMeshCurviLine* upper_inward_rib_curviline_pt=
new TriangleMeshCurviLine(
upper_inward_rib_pt,zeta_start,zeta_end,nsegment,boundary_id);
curvilinear_boundary_pt.push_back(upper_inward_rib_curviline_pt);
// Vertical upper bit of T
//------------------------
r_start[0]=r_end[0];
r_start[1]=r_end[1];
r_end[0]=r_start[0];
r_end[1]=r_start[1]+0.5*(t_width-rib_thick);
MyStraightLine* vertical_upper_t_rib_pt = new MyStraightLine(r_start,r_end);
zeta_start=0.0;
zeta_end=1.0;
nsegment=1;
boundary_id=curvilinear_boundary_pt.size();
curvilinear_boundary_pt.push_back(
new TriangleMeshCurviLine(
vertical_upper_t_rib_pt,zeta_start,zeta_end,nsegment,boundary_id));
// Horizontal upper bit of T
//-----------==-------------
r_start[0]=r_end[0];
r_start[1]=r_end[1];
r_end[0]=r_start[0]-t_thick;
r_end[1]=r_start[1];
MyStraightLine* horizontal_upper_t_rib_pt = new MyStraightLine(r_start,r_end);
zeta_start=0.0;
zeta_end=1.0;
nsegment=1;
boundary_id=curvilinear_boundary_pt.size();
curvilinear_boundary_pt.push_back(
new TriangleMeshCurviLine(
horizontal_upper_t_rib_pt,zeta_start,zeta_end,nsegment,boundary_id));
// Vertical end of rib end
//------------------------
r_start[0]=r_end[0];
r_start[1]=r_end[1];
r_end[0]=r_start[0];
r_end[1]=-r_start[1];
MyStraightLine* inner_vertical_rib_pt = new MyStraightLine(r_start,r_end);
zeta_start=0.0;
zeta_end=1.0;
nsegment=1;
boundary_id=curvilinear_boundary_pt.size();
curvilinear_boundary_pt.push_back(
new TriangleMeshCurviLine(
inner_vertical_rib_pt,zeta_start,zeta_end,nsegment,boundary_id));
// Horizontal lower bit of T
//-----------==-------------
r_start[0]=r_end[0];
r_start[1]=r_end[1];
r_end[0]=r_start[0]+t_thick;
r_end[1]=r_start[1];
MyStraightLine* horizontal_lower_t_rib_pt = new MyStraightLine(r_start,r_end);
zeta_start=0.0;
zeta_end=1.0;
nsegment=1;
boundary_id=curvilinear_boundary_pt.size();
curvilinear_boundary_pt.push_back(
new TriangleMeshCurviLine(
horizontal_lower_t_rib_pt,zeta_start,zeta_end,nsegment,boundary_id));
// Vertical lower bit of T
//------------------------
r_start[0]=r_end[0];
r_start[1]=r_end[1];
r_end[0]=r_start[0];
r_end[1]=r_start[1]+0.5*(t_width-rib_thick);
MyStraightLine* vertical_lower_t_rib_pt = new MyStraightLine(r_start,r_end);
zeta_start=0.0;
zeta_end=1.0;
nsegment=1;
boundary_id=curvilinear_boundary_pt.size();
curvilinear_boundary_pt.push_back(
new TriangleMeshCurviLine(
vertical_lower_t_rib_pt,zeta_start,zeta_end,nsegment,boundary_id));
// Lower half of inward rib
//-------------------------
r_end[0]=r_inner*cos(half_phi_rib);
r_end[1]=-r_inner*sin(half_phi_rib);
r_start[0]=r_end[0]-rib_depth;
r_start[1]=r_end[1];
MyStraightLine* lower_inward_rib_pt = new MyStraightLine(r_start,r_end);
zeta_start=0.0;
zeta_end=1.0;
nsegment=1;
boundary_id=curvilinear_boundary_pt.size();
TriangleMeshCurviLine* lower_inward_rib_curviline_pt=
new TriangleMeshCurviLine(
lower_inward_rib_pt,zeta_start,zeta_end,nsegment,boundary_id);
curvilinear_boundary_pt.push_back(lower_inward_rib_curviline_pt);
// Lower part of inner boundary
//-----------------------------
Ellipse* lower_inner_boundary_circle_pt = new Ellipse(r_inner,r_inner);
zeta_start=-half_phi_rib;
zeta_end=-0.5*MathematicalConstants::Pi;
nsegment=20;
boundary_id=curvilinear_boundary_pt.size();
curvilinear_boundary_pt.push_back(
new TriangleMeshCurviLine(
lower_inner_boundary_circle_pt,zeta_start,zeta_end,nsegment,boundary_id));
// Lower straight line segment on symmetry axis
//---------------------------------------------
r_start[0]=0.0;
r_start[1]=-r_inner;
r_end[0]=0.0;
r_end[1]=-r_outer;
MyStraightLine* lower_sym_pt = new MyStraightLine(r_start,r_end);
zeta_start=0.0;
zeta_end=1.0;
nsegment=1;
boundary_id=curvilinear_boundary_pt.size();
curvilinear_boundary_pt.push_back(
new TriangleMeshCurviLine(
lower_sym_pt,zeta_start,zeta_end,nsegment,boundary_id));
// Remember it
Lower_symmetry_boundary_id=boundary_id;
// Combine to curvilinear boundary
//--------------------------------
closed_curve_pt=
new TriangleMeshClosedCurve(curvilinear_boundary_pt);
// Vertical dividing line across base of T-rib
//--------------------------------------------
Vector<TriangleMeshCurveSection*> internal_polyline_pt(1);
r_start[0]=r_inner*cos(half_phi_rib);
r_start[1]=r_inner*sin(half_phi_rib);
r_end[0]=r_inner*cos(half_phi_rib);
r_end[1]=-r_inner*sin(half_phi_rib);
Vector<Vector<double> > boundary_vertices(2);
boundary_vertices[0]=r_start;
boundary_vertices[1]=r_end;
boundary_id=100;
TriangleMeshPolyLine* rib_divider_pt=
new TriangleMeshPolyLine(boundary_vertices,boundary_id);
internal_polyline_pt[0]=rib_divider_pt;
// Make connection
double s_connect=0.0;
internal_polyline_pt[0]->connect_initial_vertex_to_curviline(
upper_inward_rib_curviline_pt,s_connect);
// Make connection
s_connect=1.0;
internal_polyline_pt[0]->connect_final_vertex_to_curviline(
lower_inward_rib_curviline_pt,s_connect);
// Create open curve that defines internal bondary
Vector<TriangleMeshOpenCurve*> inner_boundary_pt;
inner_boundary_pt.push_back(new TriangleMeshOpenCurve(internal_polyline_pt));
// Define coordinates of a point inside the rib
Vector<double> rib_center(2);
rib_center[0]=r_inner-rib_depth;
rib_center[1]=0.0;
// Now build the mesh
//===================
// Use the TriangleMeshParameters object for helping on the manage of the
// TriangleMesh parameters. The only parameter that needs to take is the
// outer boundary.
TriangleMeshParameters triangle_mesh_parameters(closed_curve_pt);
// Target area
triangle_mesh_parameters.element_area()=0.2;
// Specify the internal open boundary
triangle_mesh_parameters.internal_open_curves_pt()=inner_boundary_pt;
// Define the region
triangle_mesh_parameters.add_region_coordinates(1,rib_center);
#ifdef ADAPTIVE
// Build the mesh
Solid_mesh_pt=new
RefineableTriangleMesh<ELASTICITY_ELEMENT>(triangle_mesh_parameters);
// Set error estimator
Solid_mesh_pt->spatial_error_estimator_pt()=new Z2ErrorEstimator;
#else
// Build the mesh
Solid_mesh_pt=new
TriangleMesh<ELASTICITY_ELEMENT>(triangle_mesh_parameters);
#endif
// Let's have a look where the boundaries are
Solid_mesh_pt->output("solid_mesh.dat");
Solid_mesh_pt->output_boundaries("solid_mesh_boundary.dat");
// Construct the traction element mesh
Traction_mesh_pt=new Mesh;
create_traction_elements();
// Solid mesh is first sub-mesh
add_sub_mesh(Solid_mesh_pt);
// Add traction sub-mesh
add_sub_mesh(Traction_mesh_pt);
// Build combined "global" mesh
build_global_mesh();
// Create elasticity tensors
Global_Parameters::E_pt.resize(2);
Global_Parameters::E_pt[0]=new TimeHarmonicIsotropicElasticityTensor(
Global_Parameters::Nu);
Global_Parameters::E_pt[1]=new TimeHarmonicIsotropicElasticityTensor(
Global_Parameters::Nu);
// Complete problem setup
complete_problem_setup();
//Assign equation numbers
cout << assign_eqn_numbers() << std::endl;
// Set output directory
Doc_info.set_directory(Global_Parameters::Directory);
} //end_of_constructor
//=====================start_of_complete_problem_setup====================
/// Complete problem setup
//========================================================================
template<class ELASTICITY_ELEMENT>
void RingWithTRibProblem<ELASTICITY_ELEMENT>::complete_problem_setup()
{
#ifdef ADAPTIVE
// Min element size allowed during adaptation
if (!CommandLineArgs::command_line_flag_has_been_set("--validation"))
{
Solid_mesh_pt->min_element_size()=1.0e-5;
}
#endif
//Assign the physical properties to the elements
//----------------------------------------------
unsigned nreg=Solid_mesh_pt->nregion();
for (unsigned r=0;r<nreg;r++)
{
unsigned nel=Solid_mesh_pt->nregion_element(r);
for (unsigned e=0;e<nel;e++)
{
//Cast to a solid element
ELASTICITY_ELEMENT *el_pt =
dynamic_cast<ELASTICITY_ELEMENT*>(Solid_mesh_pt->region_element_pt(r,e));
// Set the constitutive law
el_pt->elasticity_tensor_pt() = Global_Parameters::E_pt[r];
// Square of non-dim frequency
el_pt->omega_sq_pt()= &Global_Parameters::Omega_sq_region[r];
}
}
// Solid boundary conditions:
//---------------------------
// Pin real and imag part of horizontal displacement components
//-------------------------------------------------------------
// on vertical boundaries
//-----------------------
{
//Loop over the nodes to pin and assign boundary displacements on
//solid boundary
unsigned n_node = Solid_mesh_pt->nboundary_node(Upper_symmetry_boundary_id);
for(unsigned i=0;i<n_node;i++)
{
Node* nod_pt=Solid_mesh_pt->boundary_node_pt(Upper_symmetry_boundary_id,i);
// Real part of x-displacement
nod_pt->pin(0);
nod_pt->set_value(0,0.0);
// Imag part of x-displacement
nod_pt->pin(2);
nod_pt->set_value(2,0.0);
}
}
{
//Loop over the nodes to pin and assign boundary displacements on
//solid boundary
unsigned n_node = Solid_mesh_pt->nboundary_node(Lower_symmetry_boundary_id);
for(unsigned i=0;i<n_node;i++)
{
Node* nod_pt=Solid_mesh_pt->boundary_node_pt(Lower_symmetry_boundary_id,i);
// Real part of x-displacement
nod_pt->pin(0);
nod_pt->set_value(0,0.0);
// Imag part of x-displacement
nod_pt->pin(2);
nod_pt->set_value(2,0.0);
}
}
}
//=====================start_of_actions_before_adapt======================
/// Actions before adapt: Wipe the mesh of traction elements
//========================================================================
template<class ELASTICITY_ELEMENT>
void RingWithTRibProblem<ELASTICITY_ELEMENT>::actions_before_adapt()
{
// Kill the traction elements and wipe surface mesh
delete_traction_elements();
// Rebuild the Problem's global mesh from its various sub-meshes
rebuild_global_mesh();
}// end of actions_before_adapt
//=====================start_of_actions_after_adapt=======================
/// Actions after adapt: Rebuild the mesh of traction elements
//========================================================================
template<class ELASTICITY_ELEMENT>
void RingWithTRibProblem<ELASTICITY_ELEMENT>::actions_after_adapt()
{
// Create traction elements from all elements that are
// adjacent to FSI boundaries and add them to surface meshes
create_traction_elements();
// Rebuild the Problem's global mesh from its various sub-meshes
rebuild_global_mesh();
// Complete problem setup
complete_problem_setup();
}// end of actions_after_adapt
//============start_of_create_traction_elements==============================
/// Create traction elements
//=======================================================================
template<class ELASTICITY_ELEMENT>
void RingWithTRibProblem<ELASTICITY_ELEMENT>::create_traction_elements()
{
unsigned b=Outer_boundary_id;
{
// How many bulk elements are adjacent to boundary b?
unsigned n_element = Solid_mesh_pt->nboundary_element(b);
// Loop over the bulk elements adjacent to boundary b
for(unsigned e=0;e<n_element;e++)
{
// Get pointer to the bulk element that is adjacent to boundary b
ELASTICITY_ELEMENT* bulk_elem_pt = dynamic_cast<ELASTICITY_ELEMENT*>(
Solid_mesh_pt->boundary_element_pt(b,e));
//Find the index of the face of element e along boundary b
int face_index = Solid_mesh_pt->face_index_at_boundary(b,e);
// Create element
TimeHarmonicLinearElasticityTractionElement<ELASTICITY_ELEMENT>* el_pt=
new TimeHarmonicLinearElasticityTractionElement<ELASTICITY_ELEMENT>
(bulk_elem_pt,face_index);
// Add to mesh
Traction_mesh_pt->add_element_pt(el_pt);
// Associate element with bulk boundary (to allow it to access
// the boundary coordinates in the bulk mesh)
el_pt->set_boundary_number_in_bulk_mesh(b);
//Set the traction function
el_pt->traction_fct_pt() = Global_Parameters::pressure_load;
}
}
} // end_of_create_traction_elements
//============start_of_delete_traction_elements==============================
/// Delete traction elements and wipe the traction meshes
//=======================================================================
template<class ELASTICITY_ELEMENT>
void RingWithTRibProblem<ELASTICITY_ELEMENT>::delete_traction_elements()
{
// How many surface elements are in the surface mesh
unsigned n_element = Traction_mesh_pt->nelement();
// Loop over the surface elements
for(unsigned e=0;e<n_element;e++)
{
// Kill surface element
delete Traction_mesh_pt->element_pt(e);
}
// Wipe the mesh
Traction_mesh_pt->flush_element_and_node_storage();
} // end of delete_traction_elements
//==============start_doc===========================================
/// Doc the solution
//==================================================================
template<class ELASTICITY_ELEMENT>
void RingWithTRibProblem<ELASTICITY_ELEMENT>::doc_solution()
{
ofstream some_file;
char filename[100];
// Number of plot points
unsigned n_plot=5;
// Output displacement field
//--------------------------
sprintf(filename,"%s/elast_soln%i.dat",Doc_info.directory().c_str(),
Doc_info.number());
some_file.open(filename);
Solid_mesh_pt->output(some_file,n_plot);
some_file.close();
// Output traction elements
//-------------------------
sprintf(filename,"%s/traction_soln%i.dat",Doc_info.directory().c_str(),
Doc_info.number());
some_file.open(filename);
Traction_mesh_pt->output(some_file,n_plot);
some_file.close();
// Output regions
//---------------
unsigned nreg=Solid_mesh_pt->nregion();
for (unsigned r=0;r<nreg;r++)
{
sprintf(filename,"%s/region%i_%i.dat",Doc_info.directory().c_str(),
r,Doc_info.number());
some_file.open(filename);
unsigned nel=Solid_mesh_pt->nregion_element(r);
for (unsigned e=0;e<nel;e++)
{
FiniteElement* el_pt=Solid_mesh_pt->region_element_pt(r,e);
el_pt->output(some_file,n_plot);
}
some_file.close();
}
// Output norm of solution (to allow validation of solution even
// if triangle generates a slightly different mesh)
sprintf(filename,"%s/norm%i.dat",Doc_info.directory().c_str(),
Doc_info.number());
some_file.open(filename);
double norm=0.0;
unsigned nel=Solid_mesh_pt->nelement();
for (unsigned e=0;e<nel;e++)
{
double el_norm=0.0;
Solid_mesh_pt->compute_norm(el_norm);
norm+=el_norm;
}
some_file << norm << std::endl;
// Increment label for output files
Doc_info.number()++;
} //end doc
//=======start_of_main==================================================
/// Driver for annular disk loaded by pressure
//======================================================================
int main(int argc, char **argv)
{
// Store command line arguments
CommandLineArgs::setup(argc,argv);
// Define possible command line arguments and parse the ones that
// were actually specified
#ifdef ADAPTIVE
// Max. number of adaptations
unsigned max_adapt=3;
CommandLineArgs::specify_command_line_flag("--max_adapt",&max_adapt);
#endif
// Peakiness parameter for loading
CommandLineArgs::specify_command_line_flag("--alpha",
&Global_Parameters::Alpha);
// Validation run?
CommandLineArgs::specify_command_line_flag("--validation");
// Parse command line
CommandLineArgs::parse_and_assign();
// Doc what has actually been specified on the command line
CommandLineArgs::doc_specified_flags();
#ifdef ADAPTIVE
//Set up the problem
RingWithTRibProblem<ProjectableTimeHarmonicLinearElasticityElement
<TTimeHarmonicLinearElasticityElement<2,3> > >
problem;
#else
//Set up the problem
RingWithTRibProblem<TTimeHarmonicLinearElasticityElement<2,3> > problem;
#endif
// Initial values for parameter values
Global_Parameters::P=0.1;
//Parameter incrementation
unsigned nstep=2;
for(unsigned i=0;i<nstep;i++)
{
#ifdef ADAPTIVE
// Solve the problem with Newton's method, allowing
// up to max_adapt mesh adaptations after every solve.
problem.newton_solve(max_adapt);
#else
// Solve the problem using Newton's method
problem.newton_solve();
#endif
// Doc solution
problem.doc_solution();
// Now reduce the stiffness of the rib and set its inertia to
// zero, then re-solve. Resulting displacement field should
// be approximately axisymmetric in the annular region.
Global_Parameters::E_pt[1]->update_constitutive_parameters(
Global_Parameters::Nu,1.0e-10);
Global_Parameters::Omega_sq_region[1]=0.0;
}
} //end of main
MyStraightLine
////////////////////////////////////////////////////////////////////
Definition: unstructured_time_harmonic_elastic_annulus.cc:51
RingWithTRibProblem::actions_before_adapt
void actions_before_adapt()
Actions before adapt: Wipe the mesh of traction elements.
Definition: unstructured_time_harmonic_elastic_annulus.cc:648
RingWithTRibProblem::complete_problem_setup
void complete_problem_setup()
Helper function to complete problem setup.
Definition: unstructured_time_harmonic_elastic_annulus.cc:566
RingWithTRibProblem::actions_after_adapt
void actions_after_adapt()
Actions after adapt: Rebuild the mesh of traction elements.
Definition: unstructured_time_harmonic_elastic_annulus.cc:664
RingWithTRibProblem::delete_traction_elements
void delete_traction_elements()
Delete traction elements.
Definition: unstructured_time_harmonic_elastic_annulus.cc:728
RingWithTRibProblem::create_traction_elements
void create_traction_elements()
Create traction elements.
Definition: unstructured_time_harmonic_elastic_annulus.cc:683
Global_Parameters
/////////////////////////////////////////////////////////////////// /////////////////////////////////...
Definition: time_harmonic_elastic_annulus.cc:50
Global_Parameters::H_annulus
double H_annulus
Thickness of annulus.
Definition: time_harmonic_elastic_annulus.cc:62
Global_Parameters::pressure_load
void pressure_load(const Vector< double > &x, const Vector< double > &n, Vector< std::complex< double > > &traction)
Constant pressure load (real and imag part)
Definition: unstructured_time_harmonic_elastic_annulus.cc:130
Global_Parameters::Nu
double Nu
Poisson's ratio.
Definition: time_harmonic_elastic_annulus.cc:53
Global_Parameters::Directory
string Directory
Output directory.
Definition: time_harmonic_elastic_annulus.cc:97
Global_Parameters::P
double P
Uniform pressure.
Definition: time_harmonic_elastic_annulus.cc:81
Global_Parameters::Omega_sq_region
Vector< double > Omega_sq_region(2, Omega_sq)
Square of non-dim frequency for the two regions.
Global_Parameters::E_pt
Vector< TimeHarmonicIsotropicElasticityTensor * > E_pt
The elasticity tensors for the two regions.
Definition: unstructured_time_harmonic_elastic_annulus.cc:118
Global_Parameters::Omega_sq
double Omega_sq
Square of non-dim frequency.
Definition: time_harmonic_elastic_annulus.cc:56
Global_Parameters::Alpha
double Alpha
Peakiness parameter for pressure load.
Definition: unstructured_time_harmonic_elastic_annulus.cc:127
main
int main(int argc, char **argv)
Driver for annular disk loaded by pressure.
Definition: time_harmonic_elastic_annulus.cc:646

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