pyFormex example scripts

Warning

This document still needs some cleanup!

Sometimes you learn quicker from studying an example than from reading a tutorial or user guide. To help you we have created this collection of annotated examples. Beware that the script texts presented in this document may differ slightly from the corresponding example coming with the pyFormex distribution.

WireStent

To get acquainted with the modus operandi of pyFormex, the WireStent.py script is studied step by step. The lines are numbered for easy referencing, but are not part of the script itself.

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# $Id: examples.html,v 1.4 2016/09/28 20:57:01 bverheg Exp $   *** pyformex ***
##
##  This file is part of pyFormex 1.0.2  (Thu Jun 18 15:35:31 CEST 2015)
##  pyFormex is a tool for generating, manipulating and transforming 3D
##  geometrical models by sequences of mathematical operations.
##  Home page: http://pyformex.org
##  Project page:  http://savannah.nongnu.org/projects/pyformex/
##  Copyright 2004-2015 (C) Benedict Verhegghe (benedict.verhegghe@feops.com)
##  Distributed under the GNU General Public License version 3 or later.
##
##  This program is free software: you can redistribute it and/or modify
##  it under the terms of the GNU General Public License as published by
##  the Free Software Foundation, either version 3 of the License, or
##  (at your option) any later version.
##
##  This program is distributed in the hope that it will be useful,
##  but WITHOUT ANY WARRANTY; without even the implied warranty of
##  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
##  GNU General Public License for more details.
##
##  You should have received a copy of the GNU General Public License
##  along with this program.  If not, see http://www.gnu.org/licenses/.
##
"""Wirestent.py

A pyFormex script to generate a geometrical model of a wire stent.

This version is for inclusion in the pyFormex documentation.
"""

from formex import *

class DoubleHelixStent:
    """Constructs a double helix wire stent.

    A stent is a tubular shape such as used for opening obstructed
    blood vessels. This stent is made frome sets of wires spiraling
    in two directions.
    The geometry is defined by the following parameters:
      L  : approximate length of the stent
      De : external diameter of the stent 
      D  : average stent diameter
      d  : wire diameter
      be : pitch angle (degrees)
      p  : pitch  
      nx : number of wires in one spiral set
      ny : number of modules in axial direction
      ds : extra distance between the wires (default is 0.0 for
           touching wires)
      dz : maximal distance of wire center to average cilinder
      nb : number of elements in a strut (a part of a wire between two
           crossings), default 4
    The stent is created around the z-axis. 
    By default, there will be connectors between the wires at each
    crossing. They can be switched off in the constructor.
    The returned formex has one set of wires with property 1, the
    other with property 3. The connectors have property 2. The wire
    set with property 1 is winding positively around the z-axis.
    """
    def __init__(self,De,L,d,nx,be,ds=0.0,nb=4,connectors=True):
        """Create the Wire Stent."""
        D = De - 2*d - ds
        r = 0.5*D
        dz = 0.5*(ds+d)
        p = math.pi*D*tand(be)
        nx = int(nx)
        ny = int(round(nx*L/p))  # The actual length may differ a bit from L
        # a single bumped strut, oriented along the x-axis
        bump_z=lambda x: 1.-(x/nb)**2
        base = Formex(pattern('1')).replic(nb,1.0).bump1(2,[0.,0.,dz],bump_z,0)
        # scale back to size 1.
        base = base.scale([1./nb,1./nb,1.])
        # NE and SE directed struts
        NE = base.shear(1,0,1.)
        SE = base.reflect(2).shear(1,0,-1.)
        NE.setProp(1)
        SE.setProp(3)
        # a unit cell of crossing struts
        cell1 = (NE+SE).rosette(2,180)
        # add a connector between first points of NE and SE
        if connectors:
            cell1 += Formex([[NE[0][0],SE[0][0]]],2)
        # and create its mirror
        cell2 = cell1.reflect(2)
        # and move both to appropriate place
        self.cell1 = cell1.translate([1.,1.,0.])
        self.cell2 = cell2.translate([-1.,-1.,0.])
        # the base pattern cell1+cell2 now has size [-2,-2]..[2,2]
        # Create the full pattern by replication
        dx = 4.
        dy = 4.
        F = (self.cell1+self.cell2).replic2(nx,ny,dx,dy)
        # fold it into a cylinder
        self.F = F.translate([0.,0.,r]).cylindrical(
            dir=[2,0,1],scale=[1.,360./(nx*dx),p/nx/dy])
        self.ny = ny

    def all(self):
        """Return the Formex with all bar elements."""
        return self.F


if __name__ == "draw":

    # show an example

    wireframe()
    reset()

    D = 10.
    L = 80.
    d = 0.2
    n = 12
    b = 30.
    res = askItems([['Diameter',D],
                    ['Length',L],
                    ['WireDiam',d],
                    ['NWires',n],
                    ['Pitch',b]])

    if not res:
        exit()
        
    D = float(res['Diameter'])
    L = float(res['Length'])
    d = float(res['WireDiam'])
    n = int(res['NWires'])
    if (n % 2) != 0:
        warning('Number of wires must be even!')
        exit()
    b = float(res['Pitch'])

    H = DoubleHelixStent(D,L,d,n,b).all()
    clear()
    draw(H,view='iso')
    
    # and save it in a lot of graphics formats
    if ack("Do you want to save this image (in lots of formats) ?"):
        for ext in [ 'bmp', 'jpg', 'pbm', 'png', 'ppm', 'xbm', 'xpm',
                     'eps', 'ps', 'pdf', 'tex' ]: 
            image.save('WireStent.'+ext)

# End

As all pyFormex scripts, it starts with a comments line holding the word pyformex (line 1). This is followed more comments lines specifying the copyright and license notices. If you intend to distribute your scripts, you should give these certainly special consideration.

Next is a documentation string explaining the purpose of the script (lines 25-30). The script then starts by importing all definitions from other modules required to run the WireStent.py script (line 32).

Subsequently, the class DoubleHelixStent is defined which allows the simple use of the geometrical model in other scripts for e.g. parametric, optimization and finite element analyses of braided wire stents. Consequently, the latter scripts do not have to contain the wire stent geometry building and can be condensed and conveniently arranged. The definition of the class starts with a documentation string, explaining its aim and functioning (lines 34-60).

The constructor __init__ of the DoubleHelixStent class requires 8 arguments (line 61):

  • stent external diameter De (mm).
  • stent length L (mm).
  • wire diameter d (mm).
  • Number of wires in one spiral set, i.e. wires with the same orientation, nx (-).
  • Pitch angle \beta (\deg).
  • Extra radial distance between the crossing wires ds (mm). By default, ds is [0.0]mm for crossing wires, corresponding with a centre line distance between two crossing wires of exactly d.
  • Number of elements in a strut, i.e. part of a wire between two crossings, nb (-). As every base element is a straight line, multiple elements are required to approximate the curvature of the stent wires. The default value of 4 elements in a strut is a good assumption.
  • If connectors=True, extra elements are created at the positions where there is physical contact between the crossing wires. These elements are required to enable contact between these wires in finite element analyses.

The virtual construction of the wire stent structure is defined by the following sequence of four operations: (i) Creation of a nearly planar base module of two crossing wires; (ii) Extending the base module with a mirrored and translated copy; (iii) Replicating the extended base module in both directions of the base plane; and (iv) Rolling the nearly planar grid into the cylindrical stent structure, which is easily parametric adaptable.

Creating the base module

(lines 63-71)

Depending on the specified arguments in the constructor, the mean stent diameter D, the average stent radius r, the bump or curvature of the wires dz, the pitch p and the number of base modules in the axial direction ny are calculated with the following script. As the wire stent structure is obtained by braiding, the wires have an undulating course and the bump dz corresponds to the amplitude of the wave. If no extra distance ds is specified, there will be exactly one wire diameter between the centre lines of the crossing wires. The number of modules in the axial direction ny is an integer, therefore, the actual length of the stent model might differ slightly from the specified, desired length L. However, this difference has a negligible impact on the numerical results.

Of now, all parameters to describe the stent geometry are specified and available to start the construction of the wire stent. Initially a simple Formex is created using the pattern()-function: a straigth line segment of length 1 oriented along the X-axis (East or 1-direction). The replic()-functionality replicates this line segment nb times with step 1 in the X-direction (0-direction). Subsequently, these nb line segments form a new Formex which is given a one-dimensional bump with the bump1()-function. The Formex undergoes a deformation in the Z-direction (2-direction), forced by the point [0,0,dz]. The bump intensity is specified by the quadratic bump_z function and varies along the X-axis (0-axis). The creation of this single bumped strut, oriented along the X-axis is summarized in the next script and depicted in figures A straight line segment, The line segment with replications and A bumped line segment,.

straight line segment

A straight line segment

line segment with replications

The line segment with replications

bumped line segment

A bumped line segment

The single bumped strut (base) is rescaled homothetically in the XY-plane to size one with the scale()-function. Subsequently, the shear()-functionality generates a new NE Formex by skewing the base Formex in the Y-direction (1-direction) with a skew factor of 1 in the YX-plane. As a result, the Y-coordinates of the base Formex are altered according to the following rule: y_2 = y_1 + skew \* x_1. Similarly a SE Formex is generated by a shear() operation on a mirrored copy of the base Formex. The base copy, mirrored in the direction of the XY-plane (perpendicular to the 2-axis), is obtained by the reflect() command. Both Formices are given a different property number by the setProp()-function, visualised by the different color codes in Figure Unit cell of crossing wires and connectors This number can be used as an entry in a database, which holds some sort of property. The Formex and the database are two seperate entities, only linked by the property numbers. The rosette()-function creates a unit cell of crossing struts by 2 rotational replications with an angular step of [180]:math:deg around the Z-axis (the original Formex is the first of the 2 replicas). If specified in the constructor, an additional Formex with property 2 connects the first points of the NE and SE Formices.

(lines 72-83)

rescaled bumped line segment

Rescaled bumped strut

mirrored and skewed bumped strut

Mirrored and skewed bumped strut

unit cell of wires and connectors

Unit cell of crossing wires and connectors

Extending the base module

Subsequently, a mirrored copy of the base cell is generated (Figure Mirrored unit cell). Both Formices are translated to their appropriate side by side position with the translate()-option and form the complete extended base module with 4 by 4 dimensions as depicted in Figure Completed base module. Furthermore, both Formices are defined as an attribute of the DoubleHelixStent class by the self-statement, allowing their use after every DoubleHelixStent initialisation. Such further use is impossible with local variables, such as for example the NE and SE Formices.

(lines 84-89)

mirrored unit cell

Mirrored unit cell

completed base module

Completed base module

Full nearly planar pattern

The fully nearly planar pattern is obtained by copying the base module in two directions and shown in Figure Full planar topology. replic2() generates this pattern with nx and ny replications with steps dx and dy in respectively, the default X- and Y-direction.

(lines 90-93)

Full planar topology

Full planar topology

orthogonal view of the full planar topology

Orthogonal view of the full planar topology

Cylindrical stent structure

Finally the full pattern is translated over the stent radius r in Z-direction and transformed to the cylindrical stent structure by a coordinate transformation with the Z-coordinates as distance r, the X-coordinates as angle \theta and the Y-coordinates as height z. The scale()-operator rescales the stent structure to the correct circumference and length. The resulting stent geometry is depicted in Figure Cylindrical stent. (lines 94-96)

In addition to the stent initialization, the DoubleHelixStent class script contains a function all() representing the complete stent Formex. Consequently, the DoubleHelixStent class has four attributes: the Formices cell1, cell2 and all; and the number ny. (lines 97-100)

cylindrical stent

Cylindrical stent

orthogonal view of thecylindrical stent

Orthogonal view of the cylindrical stent

Parametric stent geometry

An inherent feature of script-based modeling is the possibility of easily generating lots of variations on the original geometry. This is a huge advantage for parametric analyses and illustrated in figures Stent variant with : these wire stents are all created with the same script, but with other values of the parameters De, nx and \beta. As the script for building the wire stent geometry is defined as a the DoubleHelixStent class in the (WireStent.py) script, it can easily be imported for e.g. this purpose.

Stent variant

Stent variant with De=16, nx=6, \beta=25

Stent variant

Stent variant with De=16, nx=6, \beta=50

Stent variant

Stent variant with De=16, nx=10, \beta=25

Stent variant

Stent variant with De=16, nx=10, \beta=50

Stent variant

Stent variant with De=32, nx=6, \beta=25

Stent variant

Stent variant with De=32, nx=6, \beta=50

Stent variant

Stent variant with De=32, nx=10, \beta=25

Stent variant

Stent variant with De=32, nx=10, \beta=50

# $Id: examples.html,v 1.4 2016/09/28 20:57:01 bverheg Exp $   *** pyformex ***
##
##  This file is part of pyFormex 1.0.2  (Thu Jun 18 15:35:31 CEST 2015)
##  pyFormex is a tool for generating, manipulating and transforming 3D
##  geometrical models by sequences of mathematical operations.
##  Home page: http://pyformex.org
##  Project page:  http://savannah.nongnu.org/projects/pyformex/
##  Copyright 2004-2015 (C) Benedict Verhegghe (benedict.verhegghe@feops.com)
##  Distributed under the GNU General Public License version 3 or later.
##
##  This program is free software: you can redistribute it and/or modify
##  it under the terms of the GNU General Public License as published by
##  the Free Software Foundation, either version 3 of the License, or
##  (at your option) any later version.
##
##  This program is distributed in the hope that it will be useful,
##  but WITHOUT ANY WARRANTY; without even the implied warranty of
##  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
##  GNU General Public License for more details.
##
##  You should have received a copy of the GNU General Public License
##  along with this program.  If not, see http://www.gnu.org/licenses/.
##


from examples.WireStent import DoubleHelixStent

for De in [16.,32.]:
    for nx in [6,10]:
        for beta in [25,50]:
            stent = DoubleHelixStent(De,40.,0.22,nx,beta).all()
            draw(stent,view='iso')
            pause()
            clear()
            

Obviously, generating such parametric wire stent geometries with classical CAD methodologies is feasible, though probably (very) time consuming. However, as provides a multitude of features (such as parametric modeling, finite element pre- and postprocessing, optimization strategies, etcetera) in one single consistent environment, it appears to be the obvious way to go when studying the mechanical behavior of braided wire stents.

Operating on surface meshes

Besides being used for creating geometries, also offers interesting possibilities for executing specialized operations on surface meshes, usually STL type triangulated meshes originating from medical scan (CT) images. Some of the algorithms developed were included in .

Unroll stent

A stent is a medical device used to reopen narrowed arteries. The vast majority of stents are balloon-expandable, which means that the metal structure is deployed by inflating a balloon, located inside the stent. Figure Triangulated mesh of a Cypher® stent shows an example of such a stent prior to expansion (balloon not shown). The 3D surface is obtained by micro CT and consists of triangles.

Cypher stent

Triangulated mesh of a Cypher® stent

The structure of such a device can be quite complex and difficult to analyse. The same functions offers for creating geometries can also be employed to investigate triangulated meshes. A simple unroll operation of the stent gives a much better overview of the complete geometrical structure and allows easier analysis (see figure Result of the unroll operation).

F = F.toCylindrical().scale([1.,2*radius*pi/360,1.])

Cypher stent unrolled

Result of the unroll operation

The unrolled geometry can then be used for further investigations. An important property of such a stent is the circumference of a single stent cell. The clip() method can be used to isolate a single stent cell. In order to obtain a line describing the stent cell, the function intersectionLinesWithPlane() has been used. The result can be seen in figures Part of the intersection with a plane.

Intersection with plane

Part of the intersection with a plane

Finally, one connected circumference of a stent cell is selected (figure Circumference of a stent cell) and the length() function returns its length, which is 9.19 mm.

circumference of stent cell

Circumference of a stent cell