Example 5532: Passive inflation/isovolumic contraction/ejection of a trilinear prolate spheroid

A single prolate. Geometry is interpolated isochorically (Focus^3.Cosh[lambda].Sinh^2[lambda]). Hydrostatic pressure is interpolated quadratically in Xi3 (element based) and the pole-zero law defines the incompressible material behaviour. The prolate is inflated then contracted isovolumically then ejection is simulated.

The comfile run by this example is as follows:

#Example_5532  Inflation and contraction of an 
#             orthotropic prolate spheroid with fibres and sheets
     
fem
fem define coordinate;r;profib;example #Defines 3 GLOBAL PROLATE SPHEROIDAL
#                                      !  coordinates.
#                                      !  Transmural interpolation is in
#                                      !  Focus^3.Cosh(Lamda).Sinh^2(Lamda).
fem define node;r;;example             #Focus = 37.5. Reads 4 NODES, 3 COORDS,
#                                      !  1 VERSION per node per nj, 0
#                                      !  DERIVATIVES for all coords. Nodal
#                                      !  coordinates, Xj, are (lamda,mu,theta):
#                                      !  1=(0.43,0,0);2=(0.43,120,0);
#                                      !  3=(0.72,0,0); 4=(0.72,120,0).
fem define base;r;;example             #Defines 3 basis functions:
#                                      !  1) Lagrange/Hermite with 3 Xi coords,
#                                      !     Linear Lagrange interpolant and 1
#                                      !     Gauss point in Xi1 dirctn (theta-
#                                      !     circumferential) but 3 in Xi2 (mu-
#                                      !     longitudinal) and Xi3 (lamda-
#                                      !     transmural);
#                                      !  2) An auxiliary basis with 5 auxiliary
#                                      !     element parameters, pressure basis
#                                      !     same number of Gauss points as for
#                                      !     basis function 1, and polynomial
#                                      !     degrees of: 0 for Xi1 and Xi2 for
#                                      !     all 5 parameters; 0 in Xi3 for
#                                      !     parameter 1; 1 for parameter 2; 2
#                                      !     for parameter 3; -1 for parameter
#                                      !     4 (Xi3=0 face pressure bc); & -2
#                                      !     for parameter 5 (Xi3=1 face
#                                      !     pressure bc).
#                                      !  3) 2 Xi coords with linear Lagrange
#                                      !     interpolant, and 3 Gauss points
#                                      !     in Xi1 and Xi2 directions.
fem define element;r;;example          #Defines 1 prolate spheroidal element.
fem define fibre;r;;example            #Defines fibre angle orientation wrt Xi1
#                                      !  direction (theta) in degrees. Uses
#                                      !  basis 1 for fibre angle interpolation
#                                      !  in all elements. No version prompts
#                                      !  or derivs. Nodes 1,2 (inner nodes)
#                                      !  are oriented at 65 degrees while the
#                                      !  outer nodes 3,4 are -55 degrees from
#                                      !  the axis. Defines sheet angles of -45 degrees
#                                      !  at inner nodes 1,2 and +45
#                                      !  degrees at outer nodes 3,4.
fem define element;r;;example fibre    #Defines fibre elements.
fem define window                      #Defines window dimensions in X,Y,Z
#                                      !  directions as (min,max):(-5,5),(-5,5),
#                                      !  (-5,5).
fem draw lines                         #Makes line segments visible on window.
fem define equation;r;;example         #Static 3D finite elasticity problem with
#                                      !  geometric coords as the dependent
#                                      !  variables. Solution is by Galerkin
#                                      !  finite elements in a nonlinear
#                                      !  analysis of a nonlinear equation.
#                                      !  Incompressible material with basis 1
#                                      !  in all elements for dep vars 1,2,3
#                                      !  and basis 2 in all elements for dep
#                                      !  var 4 (hydrostatic pressure).
fem define material;r;;example         #Stresses in constitutive law are
#                                      !  referred to body coords.  Defines a
#                                      !  hyperelastic constitutive law for
#                                      !  which the strain energy function, W,
#                                      !  is a function of fibre/transverse
#                                      !  strains that is pole-zero in strains
#                                      !  with 16 constant constitutive law
#                                      !  parameters. The Lagrangian strain
#                                      !  components are referred to the
#                                      !  undeformed fibre coordinate system
#                                      !  where the 1-direction is aligned with
#                                      !  fibre axis, 2-direction is the sheet
#                                      !  axis (in the plane of the sheet,
#                                      !  perpendicular to the fibres), and the
#                                      !  3-direction is the sheet-normal coordinate.
fem define initial;r;;example          #Boundary pressure increments are entered
#                                      !  and hydrostatic pressure is matched
#                                      !  across boundaries with adjacent Xi3
#                                      !  faces. Initial displacements are all
#                                      !  zero. For dependent variable 2
#                                      !  (mu-dirn) fix all nodes (1..4);
#                                      !  Dependent variable 3 (theta-dirn),
#                                      !  fix node 2. No force bcs.
#                                      !  Dependent variable 4 (hyd press) fix
#                                      !  element 1, parameter 4 (Xi3=0 face
#                                      !  inside prolate) with increment
#                                      !  magnitude 2, and parameter 5 (Xi3=1
#                                      !  face outside prolate) with increment
#                                      !  magnitude 0. No force bcs.
fem export nodes;heart1 field as heart #export the undeformed nodes
fem export elements;heart1 field as heart #export the undeformed elements
fem define solve;r;profib;example     #  Read in solution information.
fem solve increment 0.2 iter 3         #Solve the problem.
#                                      !Incrementing of the ventricular
#                                      !  pressure is neccessary to avoid
#                                      !  unstable solutions. This depends on
#                                      !  the constitutive law and pressure
#                                      !  the constitutive law and pressure
#                                      !  magnitude, as the wall is gradually
#                                      !  stiffened.
fem export nodes;heart2 field as heart #export the deformed nodes
fem export elements;heart2 field as heart #export the deformed elements
#
fem solve increment 0.3 iter 3 
fem export nodes;heart3 field as heart 
fem export elements;heart3 field as heart 
#                              
fem solve increment 0.5        
fem export nodes;heart4 field as heart
fem export elements;heart4 field as heart
#                              
#                              
#                              
fem draw lines def dotted      #Draw deformed mesh on window.
fem list strain at 1 ref       #List strain information wrt reference
#                              !  coordinates.
fem list stress at 1 ref       #List stress information wrt reference
#                              !  coordinates.!Example_5521  Contraction of an inflated orthotropic prolate element

fem def initial;w;profib_def
fem eval reac


#
#
# Contraction 
#
#


# ############################################ 
# Inflated solution is converged OK! 
# ############################################ 

# Command updated by fixcom.sh on Wed Aug 23 17:49:03 NZT 2000
# Old command: assign variable WALL 1
$WALL = 1


# Command updated by fixcom.sh on Wed Aug 23 17:49:03 NZT 2000
# Old command: assign variable CAVITY 2
$CAVITY = 2

# 
# Set up cavity region geometry 
fem define region;r;coupled;example 
fem define coord;r;cavity;example region $CAVITY
fem define;add base;r;cavity;example 
fem define node;r;cavity;example region $CAVITY
fem define element;r;cavity;example region $CAVITY
# 
# Set up equations, material properties and initial conditions 
# for cavity region 
fem define equation;r;coupled;example region $WALL,$CAVITY lock
fem define material;r;cavity_isovol;example region $CAVITY
fem define initial;r;cavity;example region $CAVITY
# 
# Define cavity-to-wall coupling and solution procedure information 
fem define coupling;r;coupled;example 
fem define solve;r;coupled;example coupled region $WALL,$CAVITY
# 
# Transfer wall deformation from the inflation problem to 
# geometric dependent variables for the cavity region 
fem update solution coupled source_region $WALL
# 
# Set up the reference state for the cavity region to be the deformed 
# configuration from the inflation problem.  To set up the 
# 'inflated cavity', the x-coord ('in 1' represents the 1st RC coord) 
# for the central cavity node (node 6) is set to the average of the 
# x-coords for the adjacent interface nodes (node 2 - this could 
# be a list of nodes) 
fem update solution cavity_reference average 6 in 1 node 2 region $CAVITY
# 
# List undeformed (pre-inflation) and inflated cavity volumes 
fem list element total region $CAVITY
fem list element deformed total region $CAVITY
# ############################################ 
# cavity volume inflated from 40ml to 65ml. 
# ############################################ 
# 
fem draw lines region $WALL,$CAVITY
fem draw lines dot def region $WALL,$CAVITY
# 
# Check that the current coupled solution is converged (as expected!) 
fem solve coupled increment 0.0 
# ############################################ 
# inflated state is a converged solution for the coupled problem! 
# ############################################ 
# 
# ISOVOLUMIC CONTRACTION 
# 
# Increment intracell calcium conc and solve for isovolumic contraction 
fem define material;r;profib_active;example 
fem define active;r;profib_contract;example 
fem solve coupled increment 0.0 
# ############################################ 
# The solution procedure hasn't theoretically converged, but 
# if you follow the progress of the solver and the final 
# constr vs unconstr resids, it's very close to converged. 
# Note that the ratio monotonically decreased, 
# which is also a sign of a stable solution process. 
# ############################################ 
#fem define initial;r;profib_endisovolcontr;example region WALL,CAVITY 
# 
# Save deformed configurations for end ejection 
fem define initial;w;profib_endisovolcontr region $WALL,$CAVITY
# 
fem draw lines deformed dotted rgb=blue region $WALL,$CAVITY
# 
# Check that cavity volume has not changed 
fem list element deformed total region $CAVITY
# ############################################ 
# This contraction phase was indeed isovolumic! 
# (cavity volume still 65ml). 
# ############################################ 
# 
# EJECTION 
# 
# Decrease cavity 'stiffness parameter' and solve for ejection 
fem define material;r;cavity_eject;example region $CAVITY
fem solve coupled increment 0.0 
# ############################################ 
# Ejection problem is SLOW to converge, but note that the ratio 
# monotonically decreases, indicating a stable solution. 
# May need to repeat the last solve command to get better convergence, 
# but will probably not get down to 10^(-10) tolerance. 
# ############################################ 
# 
# Save deformed configurations for end ejection 
fem define initial;w;profib_endeject region $WALL,$CAVITY
# 
fem draw;add lines deformed dotted region $WALL,$CAVITY
# 
# Determine cavity volume at end-ejection. 
# NOTE: to calc cavity volume, the deformed position of the central 
# cavity node must be recalc'ed from the averaged adjacent wall nodes. 
# After doing this the current solution is meaningless, ie. best to save 
# deformed configurations for both regions before this (as above). 
fem update solution cavity_reference average 6 in 1 node 2 region $CAVITY
fem list element deformed total region $CAVITY
# ############################################ 
# Not a huge ejection fraction (!!!), but neverless a decrease 
# in volume from 65ml to 63ml. 
# ############################################

Additional testing commands:

fem list initial region $WALL,$CAVITY
fem write iod

Files used by this example are:

Name                          Modified     Size


example_5532.com              09-Jul-2001   12k
cavity.ipbase                 10-Apr-2000  1.1k
cavity.ipcoor                 10-Apr-2000  707 
cavity.ipelem                 10-Apr-2000  404 
cavity.ipinit                 12-Dec-2002  1.4k
cavity.ipnode                 10-Apr-2000  1.3k
cavity_eject.iod              10-Apr-2000   69k
cavity_eject.ipmate           10-Apr-2000  386 
cavity_isovol.ipmate          10-Apr-2000  390 
coupled.ipcoup                21-Aug-2002  437 
coupled.ipcoup.old            10-Apr-2000  386 
coupled.ipregi                10-Apr-2000   94 
coupled.irequa                02-May-2004  3.8k
coupled.irsolv                16-Aug-2010  3.1k
coupled.irsolv.old            13-Apr-2007  2.9k
heart1.exelem                 10-Apr-2000  5.5k
heart1.exnode                 10-Apr-2000  1.2k
heart2.exelem                 10-Apr-2000  5.5k
heart2.exnode                 13-Dec-2002  1.2k
heart3.exelem                 10-Apr-2000  5.5k
heart3.exnode                 13-Dec-2002  1.2k
heart4.exelem                 10-Apr-2000  5.5k
heart4.exnode                 13-Dec-2002  1.2k
profib.ipacti                 03-Mar-2004  735 
profib.ipbase                 10-Apr-2000  4.2k
profib.ipcoor                 10-Apr-2000  707 
profib.ipelem                 10-Apr-2000  409 
profib.ipelfb                 10-Apr-2000  349 
profib.ipelfd                 10-Apr-2000  276 
profib.ipequa                 02-May-2004  2.0k
profib.ipfibr                 30-Jan-2001  1.3k
profib.ipfiel                 10-Apr-2000  601 
profib.ipfit                  13-Apr-2007  1.5k
profib.ipinit                 12-Dec-2002  1.8k
profib.ipmate                 12-Dec-2002  5.7k
profib.ipnode                 10-Apr-2000  1.2k
profib.ipsolv                 16-Aug-2010  2.3k
profib.ipsolv.old             13-Apr-2007  2.1k
profib.ipwind                 10-Apr-2000  262 
profib.ps                     10-Apr-2000   11k
profib_active.ipmate          12-Dec-2002  6.2k
profib_contract.ipacti        03-Mar-2004  735 
profib_def.ipinit             13-Dec-2002  6.2k
profib_def_test.ipinit        12-Dec-2002  6.2k
profib_endeject.irinit        13-Dec-2002   11k
profib_endisovolcontr.irinit  13-Dec-2002   11k
prol.ippara                   12-Nov-2002  5.9k
test_output.com               10-Apr-2000   52

Download the entire example:

Name                           Modified     Size


examples_5_55_553_5532.tar.gz  17-Aug-2010   30k

Testing status by version:

StatusTestedReal time (s)
i686-linux
cmFailureSun Mar 6 00:10:14 20162
last breakTue Feb 24 03:17:00 20152
last successSun Apr 17 00:51:00 20115
cm-debugFailureSun Mar 6 00:16:39 20165
last breakTue Feb 24 03:23:00 20155
last successFri Apr 8 00:20:00 20118
mips-irix
cmSuccessSun Aug 19 01:47:57 200715
cm-debugSuccessWed Aug 15 01:42:15 200739
cm-debug-clear-mallocSuccessSat Aug 18 01:50:56 200745
cm-debug-clear-malloc7SuccessMon Aug 20 01:47:12 200745
cm64SuccessSun Aug 19 01:48:11 200716
cm64-debugSuccessTue Aug 21 01:41:00 200740
cm64-debug-clear-mallocSuccessFri May 20 10:33:31 200520
rs6000-aix
cmSuccessWed Mar 4 01:09:07 20093
cm-debugSuccessMon Mar 2 01:12:05 200912
cm64SuccessWed Mar 4 01:09:37 20092
cm64-debugSuccessTue Mar 3 01:16:11 200910
x86_64-linux
cmFailureSun Mar 6 00:01:25 20162
last breakTue Aug 12 00:02:00 20141
last successSun Apr 17 00:26:00 20112
cm-debugFailureSun Mar 6 00:01:41 20162
last breakTue Aug 12 00:22:00 20143
last successFri Apr 8 00:06:00 20113

Testing status by file:

Html last generated: Sun Mar 6 05:50:25 2016

Input last modified: Mon Aug 16 11:19:20 2010

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