SQUASHING BRAINS Arbitrary Lagrangian-Eulerian Method for

Modelling of Large Distortions of Brain Phantom for Robotic Surgery

Genée du Plessis

B. Engineering Science

This thesis is presented for the degree of

Master of Professional Engineering

Intelligent Systems for Medicine Laboratory

School of Mechanical and Chemical Engineering

The University of Western Australia

2016

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1 ABSTRACT

Biomechanical models have been proposed in the literature for predicting and analysing

the mechanical responses (including deformation and stress fields) of the human body

organs due to surgery and impact loads that result from traffic accidents. Such responses

are difficult (and impossible) to determine using the experimental studies and/or

accident data analysis due to technical limitations and ethical constraints.

Biomechanical models for predicting the body organ responses must employ methods of

computational mechanics that ensure stable and robust solution that occur under large

deformations/strains induced by surgery and impacts. This study uses non-linear finite

element analysis for such prediction.

Two classical formulations existing in computational continuum mechanics include the

Langrangian description (common in solid mechanics) in which each node of the finite

element mesh follows the associated material particle as it moves and the Eulerian

description (common in fluid mechanics) in which the mesh is fixed while the

continuum moves with respect to the grid. The former approach lacks the ability to

ensure stable and robust solution at large distortions without costly, frequent remeshing

operations while the latter may handle large distortions but cannot precisely track a

moving boundary. Since 1964 (Noh 1964), some research has endeavoured to combine

the best of both aforementioned approaches to establish a new formulation named

Arbitrary Lagrangian-Eulerian or ALE. The ability of this approach to accurately

predict the behaviour of soft continua is evaluated in this study. This is done through

modelling of cylindrical soft tissue phantom. The models were implemented using

ABAQUS/Explicit finite element solver due to its advanced ALE capabilities.

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Tissue and brain phantoms were used to eliminate biohazard risk, ethical constraints and

uncertainty associated with the effects of post-mortem delay on soft tissue properties

and variation of soft tissue properties between specimens and subjects. Sylgard® 527

silicone gel was employed to manufacture the phantoms. The gel samples (soft tissue

phantoms) were subjected to 30% compression at rates of 10, 50 and 360 mm/min in

order to determine the strain rate dependency of the material behaviour. An Ogden-

based hyperviscoelastic constitutive model as proposed by Miller and Chinzei (2002)

was selected and the finite element model was calibrated to experimental results in

order to obtain the constitutive equation parameters.

Explicit integration in the time domain was selected as it has been used for applications

involving large deformations/distortion in neurosurgical simulation (Taylor et al. 2011)

and injury biomechanics (Miller 2011). This approach works best with under integrated

elements that tend to exhibit non-physical deformation modes (hourglass modes), which

requires application of countermeasures to prevent ‘hourglassing’ such as hourglass

control.

Compression of a cylindrical tissue phantom was simulated using the ALE method

(henceforth referred to as Case 1). Thereafter, finite element analysis (FEA) with

activated distortion and enhanced hourglass control was applied (henceforth referred to

as Case 2) as this approach is recommended in ABAQUS for hyperelastic continua

undergoing large deformations/distortions. Finite element analysis was selected as it is a

method of choice in computational biomechanics.

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As meshless methods of computational mechanics, that use “clouds of points” as

computational grids, are known (provide references) to provide stable/accurate solutions

‘beyond FEA’, in situations where FEA accuracy/stability deteriorates due to mesh

distortion, it was decided to that a meshless method using an in-house meshless code

would be used and compared to the results obtained using ALE and FEA for

verification.

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2 ACKNOWLEDGEMENTS

I would like to extend a warm thank you to my thesis supervisor, Professor Adam

Wittek- a man of blunt temperament, sharp wit and sheer brilliance.

I would also like to acknowledge the Dr Ashley Horton and Dr Xia Jin (former PhD

students at the UWA Intelligent Systems for Medicine Laboratory) as well as Dr

Guiyong Zhang who contributed to the development of the ISML meshless code and

Grand Joldes for his contributions. I would also like to acknowledge the funding

support from ARC DP1092893 and ARC DP120100402 grants.

Appreciation is also owed to Agnes Kang for sharing my struggles in teasing apart the

complex Matlab® meshless code. A special thanks to Scott List for the generation of

the tissue and brain phantoms as well as for ongoing support and encouragement.

The study also demands the acknowledgement of Maimuna Majimbi (the literal ‘brain

squasher’ herself) who fed junk food to my body and laughter to my soul during many

late night study sessions.

Finally, I’d like to thank my parents, Karen Theunissen and Daniel Theunissen, for

tolerating my shirking of responsibilities in the pursuit of my passions and for

unwavering love and support. I’d also like to acknowledge my dad, Eugene du Plessis

for inspiring lateral thought and curiosity in the Genée, the child, who still continues to

drive Genée,the adult.

Last- but not least- thank you, God. Thank you for it all.

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3 CONTENTS

1 Abstract ..................................................................................................................... 1

2 Acknowledgements ................................................................................................... 4

4 List of figures ............................................................................................................ 7

5 Introduction ............................................................................................................... 9

5.1 Overview ............................................................................................................ 9

5.2 Computer integrated surgery (CIS) .................................................................. 10

5.3 Finite element method ...................................................................................... 11

5.3.1 Finite element formulation: ALE .............................................................. 11

5.3.2 Time-integration scheme ........................................................................... 12

5.4 Issues/Limitations ............................................................................................. 13

5.4.1 Distortion control for crushable materials in ABAQUS/Explicit ............. 14

5.4.2 Enhanced hourglass control in ABAQUS/Explicit ................................... 14

5.5 Sylgard® tissue samples and brain phantom .................................................... 15

6 Experiments ............................................................................................................. 16

6.1 Method .............................................................................................................. 16

7 Model ...................................................................................................................... 18

7.1 Major assumptions and simplifications ............................................................ 18

7.2 Verification and validation ............................................................................... 18

7.3 Boundary conditions and loading ..................................................................... 19

7.4 Constitutive model and its parameters ............................................................. 21

7.4.1 Neo-Hookean constitutive model .............................................................. 21

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7.4.2 Ogden’s constitutive model ....................................................................... 22

7.5 Element type and mesh generation ................................................................... 23

7.6 Solution procedure ............................................................................................ 24

8 Results ..................................................................................................................... 25

8.1 Experiment ....................................................................................................... 25

8.1.1 Force-displacement results ........................................................................ 25

8.1.2 Strain rate dependence .............................................................................. 26

8.2 Finite element model ........................................................................................ 28

8.2.1 Finding Ogden material model parameters ............................................... 28

8.2.2 Simulation results ...................................................................................... 31

9 Verifying Algorithm ................................................................................................ 34

9.1 Boundary conditions and loading ..................................................................... 35

9.2 Constitutive model and its parameters ............................................................. 35

9.3 Element type and mesh generation ................................................................... 35

9.4 Results .............................................................................................................. 37

10 Discussion ............................................................................................................... 40

10.1 Constitutive model and parameters ............................................................... 40

10.2 Comparing Case 1 and Case 2 ...................................................................... 41

11 Conclusions and Future work .................................................................................. 42

12 References ............................................................................................................... 44

13 Appendices .............................................................................................................. 48

13.1 Appendix A: Matlab® code for extracting force-displacement data ............ 48

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4 LIST OF FIGURES

Figure 1: Semi-compression uniaxial compression test setup ........................................ 16

Figure 2: Experimental test bench setup (image from Agrawal et al 2014) ................... 17

Figure 3: Compression of cylindrical sample for strains exceeding 10% (Morriss et al.

2007). .............................................................................................................................. 19

Figure 4: 3-4-5 Polynomial loading amplitude (Wittek 2016) ........................................ 20

Figure 5: 3-4-5 Polynomial velocity amplitude (Wittek 2016) ....................................... 21

Figure 6: 3-4-5 Polynomial acceleration amplitude (Wittek 2016) ................................ 21

Figure 7: Meshed cylindrical sample shows rigidly constrained nodes at the top and

bottom with the top nodes free to move in the vertical (z) direction .............................. 24

Figure 8: Force-displacement response for simulation of compression tests on Sylgard®

layers A-F ........................................................................................................................ 26

Figure 9: Colour-coded element sets defined for each layer of the Finite element model

of the brain phantom generated in HyperMesh (List 2016) ............................................ 26

Figure 10: Force-displacement response for Samples A1-A3 at three different loading

rates of 0.17 mm/s, 0.83 mm/s and 6 mm/s. ................................................................... 27

Figure 11: Young’s modulus of layers A-F with varying strain rate showing a slight

increase in stiffness as well as an inconsistency in the results for compression of a

sample from Layer B at a strain rate of 0.83 mm/s ......................................................... 28

Figure 12: Slope of linear region of stress-strain plot for Layer A to determine Young's

modulus ........................................................................................................................... 29

Figure 13: Calibrating the simulation response to the experimental results ................... 30

Figure 14: Compressed cylindrical tissue sample simulation result using ALE ............. 31

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Figure 15: Force-time response of compression simulation using ALE. The simulation

failed at approximately 0.75s. ......................................................................................... 32

Figure 16: Compressed cylindrical tissue sample simulation result using distortion and

enhanced hourglass control ............................................................................................. 33

Figure 17: Force-displacement response of the explicit time integrated hyperelastic

model using distortion and enhanced hourglass control ................................................. 33

Figure 18: Undeformed meshless model used in verification process to compare

simulation results with those obtained using FEM with model generated using in-house

meshless code as in Zhang et al. (2013). ......................................................................... 36

Figure 19: Undeformed ABAQUS model used in verification process to compare

simulation results with those obtained using the meshless method ................................ 36

Figure 20: Tissue sample model compressed by 30% using the meshless method using

in-house meshless code as in Zhang et al. (2013). .......................................................... 37

Figure 21: Tissue sample model compressed by 30% using FEM using explicit time

integration as well as distortion and enhanced hourglass control (Case 2) ..................... 38

Figure 22: Force-displacement response of deformation of up to 30% strain of non-

linear incompressible cylindrical sample using meshless techniques ............................. 39

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5 INTRODUCTION

5.1 Overview

Modern medicine has come a long way. By strength of the backs of many medical

professionals and researchers the lives of millions have been saved and quality of lives

restored. However, humanity’s endeavours have always been limited by our fleshy

bodily form and our efforts to transcend our current status are ever thwarted by mortal

encumbrances including fatigue, anxiety or lapse in judgement. We are after all, as the

saying goes, only human.

As we have strived to supersede these limitations so our ingenuity has birthed areas of

study such as robotics and biomechanics which carry the promise of surgical procedures

that may, one day, be entirely controlled by numerically controlled mechanical devices

unhindered by our shortcomings. However, given the complexity and uniqueness of the

human body, the constant motion of organs within the body and non-linear mechanical

behaviour of biological tissue the quest remains a rocky one with much ground yet to be

covered.

Taylor and Stoianovici (2003) divided surgical robots into two categories namely

surgical computer-aided design/manufacturing (CAD/CAM) and surgical assistants.

Surgical assistants are either operated directly by surgeons in order to extend human

capabilities in performance of surgical tasks or work by the surgeon’s side providing

support such as holding surgical tools. Surgical CAD/CAM systems assist in planning

and intraoperative navigation via the means of reconstructing preoperative images and

forming three-dimensional (3D) models, registering this data to the patient during

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surgery and using robots and displays of image overlays to aid in the execution of the

planned procedures (Moustris et al 2011).

Current technologies such as Magnetic Resonance Imaging (MRI) are able to produce

the aforementioned pre-operative images. However, as the organs are translated and

deformed, the accuracy of the images is disturbed and updating these images requires

time that is often of the essence in a surgical setting. Moreover, precisely quantified

images are required for robotic surgeons, especially in neurosurgical applications.

Physical models may be used to calculate the deformation field within an organ after

which the images may be updated accordingly. The accuracy of these models is

currently crippled by uncertainties concerning the calculation of this deformation due to

geometrical irregularities as well as the complex, incompressible constitutive response

of brain tissue. Moreover, the ability of typically used Finite Element procedures

relying on Lagrange formulation to accurately predict the forces to which human body

organs are subjected to during surgery is compromised due to the loss of accuracy at

large (30%+) strains. This study intends to assess the performance of Finite Element

procedures using Arbitrary Lagrange-Euler formulation in predicting the responses of

soft continua with brain tissue-like constitutive properties subjected to large

compressive strains in the context of predicting soft tissue deformations for surgical

robot control.

5.2 Computer integrated surgery (CIS)

Given the highly interactive nature of the surgical process, requiring constant

intraoperative decisions to be made, the aim is not to replace the surgeon but to provide

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the surgeon with an assisting robot with a set of versatile tools at its disposal which

ultimately extends the abilities of the surgeon. These surgical robot assistants may be

regarded as smart surgical tools that provide advantages such as (Taylor 2006):

• Significantly improving the surgeon’s technical capability by increasing the

speed and accuracy of existing procedures while decreasing the invasiveness,

and

• Increasing surgical safety via improved performance of complex procedures,

providing online monitoring support and providing active assists such as ‘no fly

zones’ which prevent robots from bringing tools too closely to delicate

anatomical areas.

During the preoperative planning phase, 2D and 3D images coupled with patient-

specific information may be generated to produce a computer model of the patient. As

the robotic systems evolve the core challenge today is to develop computationally

stable, accurate and efficient methods for generating these models for use in surgical

procedures (Taylor 2006).

5.3 Finite element method

5.3.1 Finite element formulation: ALE

Two of the most commonly used formulations in commercial finite element programs

include the Total Lagrangian and Updated Lagrangian formulation. For both

formulations there is no material motion relative to the convected mesh. The mesh is

fixed to and distorts along with the mesh, resulting in degradation of the mesh.

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Conversely, in an Eulerian displacement model there is motion of the material with

respect to the mesh and the particles may translate across element boundaries and at

each stage of the material deformation the particle associated with a particular node may

change (Haber, 1984). The disadvantage of the Lagrangian approach is that frequent,

costly remeshing is required if the computational domain is subjected to large

deformations. Eulerian methods can handle large deformations, however resolution of

flow details and interface definition are somewhat compromised. The advantages of

both approaches have since been combined to yield what is referred to as the Arbitrary

Lagrangian-Eulerian or ALE description. ALE permits the mesh nodes to move with the

continuum or be held fixed as required. Via a fusion of the two methods, the continuum

can handle greater distortions (overcoming Lagrangian shortcomings) with better

resolution than is yielded by the Eulerian approach (Donea et al 2004).

5.3.2 Time-integration scheme

Equilibrium equations may be integrated in the time domain using implicit or explicit

methods. The implicit integration methods most commonly applied are unconditionally

stable, but at each time step a set of non-linear algebraic equations must be solved and

iterations must be performed at every implicit integration time step to prevent

divergence and control error.

In explicit time integration, such as the central difference method, computations are

done at the element level and there is no need to compute the stiffness matrix for the

whole model. Therefore, the internal memory requirements and computational cost of

explicit methods is much less than that of implicit methods, making it more suited to

real-time computation of soft tissue deformation. Although, explicit methods are only

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conditionally stable and satisfactory results are only obtained if the step time is largely

restricted. The maximum step time allowed is approximately proportional to the square

root of the material density divided by Young’s modulus. Since the stiffness of

Sylgard® 527 gel is very low (similar to that of brain tissue), this allows a relatively

long time step.

Therefore, ALE will be combined with explicit time integration to compute the large

deformation of hyperelastic soft material (Sylgard® 527 gel).

5.4 Issues/Limitations

According to ABAQUS manual Version 6.14, ALE (or adaptive meshing as it is

referred to in ABAQUS) is not recommended for use in domains modelled with

hyperelastic or hyperfoam materials as elements with distortion control cannot be

included in an adaptive mesh domain. ABAQUS activates distortion control by default

for elements modelled with hyperelastic materials. Better results are predicted to be

obtained using elements with distortion control and using the enhanced hourglass

method (ABAQUS 2014) in a non-adaptive mesh domain.

Should hyperelastic materials be used in an adaptive mesh domain, section controls

must be specified and distortion control deactivated. Regardless of this limitation, the

capabilities of ALE will be explored in this study and compared to the alternate

approach, Case 2: Distortion and enhanced hourglass control.

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5.4.1 Distortion control for crushable materials in ABAQUS/Explicit

Hyperelastic materials tend to stiffen when subjected to large deformations. This

stiffening behaviour may be enough to prevent excessive distortion such as negative

element volumes from occurring in a finer mesh, however for a coarse mesh the

analysis may fail prematurely. Distortion control is available in ABAQUS/Explicit and

prevents the solid elements from inverting and/or distorting excessively.

ABAQUS/Explicit practices a constraint method that prevents nodes from punching

inward toward the element centre, causing the element to become non-convex

(ABAQUS 2014). Constraints are enforced by using a penalty approach and the

parameter ‘distortion length ratio’ may be input.

When the node moves a specific small distance away from the plane of constraint, the

constraint penalty forces are applied. This limits the reduction of time increment due to

shortening of the element characteristic length and improves the robustness of the

method. The distortion length ratio times the initial element characteristic length is

equal to this offset distance. The default value for the distortion length ratio is 0.1

(ABAQUS 2014).

5.4.2 Enhanced hourglass control in ABAQUS/Explicit

The enhanced hourglass control approach is the default for hyperelastic materials in

ABAQUS /Explicit and it offers a refinement of the pure stiffness method. The stiffness

coefficients are based on the enhanced assumed strain method. This method provides

increased resistance to hourglassing for non-linear materials. Furthermore, in

ABAQUS/Explicit enhanced hourglass control will more accurately predict the return to

the original configuration upon removal of the load. Enhanced hourglass control cannot

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be used for hyperelastic materials when adaptive meshing is applied to the domain

(ABAQUS 2014).

5.5 Sylgard® tissue samples and brain phantom

Tissue samples as well as a brain (human) phantom were manufactured by Scott List

for his final year project (List 2016) using Sylgard® 527 silicone gel which is

recognised by biomechanics community as closely representing brain tissue constitutive

behaviour (Brands et al. 2000).

Sylgard® 527 is a two-part silicone gel with a batch and curing agent mixed in a 1:1

ratio and cured for 24 hours (List 2015). Six batches were mixed in total and 3 samples

per batch were generated. To represent the internal structure of the human skull, a

translucent human skull anatomical cast by 3B Scientific was used in which to construct

the brain phantom. The phantom was constructed layer-by-layer in order to minimise

the amount of air bubbles trapped in the material as it cured as well as to allow the

embedding of rare earth magnets between the layers. The magnets were used as markers

in the phantom to track the deformation field. Due to slight variations in the mixing

ratio of the Sylgard® agents, the constitutive properties varied slightly for each layer.

Therefore, 3 cylindrical samples per layer were generated in order to extract material

model parameters for each layer used in the phantom (Ma et al. 2010, List 2015). There

were six layers in total and will be referred to as layers A-F in the order in which the

batches were generated.

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In this study, the samples were used in order to extract Ogden material model

parameters for Sylgard® 527 gel in order to define the constitutive behaviour of the

samples in the commercial finite element solver, ABAQUS. Once a relatively robust

finite element procedure is obtained for the cylindrical sample model, the procedure

may be repeated for the more complex geometry of the brain phantom. This task was

not completed in this study but is recommended for future exploration in Section 11:

Conclusions and Future work.

6 EXPERIMENTS

6.1 Method

Semi-confined uniaxial compression tests were performed on Sylgard® 527 tissue

phantom samples. The experimental procedure may be seen in Figure 1.

Figure 1: Semi-compression uniaxial compression test setup

Three tests were performed on each sample at three different loading rates including

0.17 mm/s, 0.83 mm/s and 6 mm/s (upper limit of the machine’s loading rate capacity).

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Each sample was compressed to 30% of its initial height (7.2 mm of 24 mm). Three

samples per layer were tested for 6 layers with 3 tests per sample which equates to 54

loading cycles in total. The diameter of the samples was 38 mm and the height was 24

mm.

The tests were performed on an in-house tension-compression rig as shown in Figure 2.

Figure 2: Experimental test bench setup (image from Agrawal et al 2014)

The device has two sensors: one to measure displacement of the platen and one to

measure the force applied to the platen. The force is measured by a Burster 8523 load

cell with a capacity of 20 N; however it is only run on half the loading capacity as the

dynamic performance of the load cell is specified as 50% of its capacity. The lower

platform is fixed while the upper platen may move up and down along the vertical axis.

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Sandpaper was glued to the top and bottom platens so as to ensure a fixed boundary

condition at the top and bottom faces of the cylindrical sample. Given the sticky nature

of the material, it would stick to the sandpaper, consistent with the fixed boundary

condition definition.

7 MODEL

7.1 Major assumptions and simplifications

It is assumed that Sylgard® as well as brain material may be assumed to be isotropic

and homogenous. Furthermore, as previously mentioned compression of Sylgard® 527

gel is assumed to be strain rate independent (for Ogden’s material model, the

viscoelastic term may be ignored). The Poisson’s ratio for an incompressible material is

taken to be approximately 0.5 and a no-sliding assumption between the sample and

compression platens is made.

7.2 Verification and validation

Miller (2005) proposed an analytical solution for the state of deformation within the

sample; however it assumes that the planes perpendicular to the direction of the applied

force remain plane. However, as shown in Figure 3, the cylindrical part of the sample

comes into contact with the platens at approximately 15% compression resulting in an

increase of contact surface area and violating the perpendicular plane assumption.

Furthermore, generally such a problem would be solved by using Hooke’s law:

( 1 )

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( 2 )

𝜎 = 𝐸𝜀

Where 𝜎 is normal stress 𝜀 is normal strain and E is Young’s modulus. However, this

equation assumes that the cross-sectional area remains constant as well as that

deformations are small. These assumptions are also violated.

Therefore, the analytical solution fails and, considering that there is no existing

benchmark to which the solution can be compared, the model cannot be verified via a

comparison with an analytical solution. It may be verified, however, by comparison

with other existing methods such as the meshless method as will be discussed in Section

9: Verifying Algorithm.

Figure 3: Compression of cylindrical sample for strains exceeding 10% (Morriss et al. 2007).

7.3 Boundary conditions and loading

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The nodes on the cylinder’s bottom face are fixed while the nodes on the top face are

free to move in the negative z-direction (vertical direction). The sample model was

loaded via prescribed nodal displacements of the top surface nodes to 7.2 mm in the

negative z-direction. The loading amplitude used was tabular and defined using the 3-4-

5 Polynomial with a smoothing factor of 0.5 to ensure smooth loading with zero

acceleration and velocity at the start and end of the load application.

The 3-4-5 Polynomial is given by (Wittek 2016):

( 3 )

𝐹 𝑡 = 6𝜃!

𝛽! − 15𝜃!

𝛽! − 10𝜃!

𝛽!

Figure 4 shows the 3-4-5 Polynomial loading amplitude while Figure 5 and Figure 6

show the first and second derivatives (velocity and acceleration curves) of the 3-4-5

Polynomial.

Figure 4: 3-4-5 Polynomial loading amplitude (Wittek 2016)

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Figure 5: 3-4-5 Polynomial velocity amplitude (Wittek 2016)

Figure 6: 3-4-5 Polynomial acceleration amplitude (Wittek 2016)

7.4 Constitutive model and its parameters

7.4.1 Neo-Hookean constitutive model

The Neo-Hookean model is generally used due to its simplicity. The strain energy

density function for an incompressible Neo-Hookean material is as follows:

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( 4 )

𝑊 = 𝐶!(𝐼! − 3)

where 𝐶! is a material constant, and 𝐼! is the first invariant of the left Cauchy-Green

deformation tensor. The Ogden model will reduce to the Neo-Hookean model if 𝑁 =

1 and 𝛼 = 2.

In order to verify the final model a version of MTLED algorithm implemented in

MATLAB® was used. The Neo-Hookean material model was implemented in this

version, but the Ogden model could have been implemented via an alteration of the

code. However, since the focus of the study is not on constitutive modelling but rather

on the analysis technique itself, as well as due to time limitations, the neo-Hookean

model was used. Though the forces obtained using the Neo-Hookean material model

should be lower than when Ogden’s model is used for the finite element model, if the

Neo-Hookean model is consistently used in the FEM method and the meshless method

during the verification process, the results should agree if the results obtained from both

the methods are accurate. If the results agree, the algorithms are verified.

7.4.2 Ogden’s constitutive model

The Ogden form of strain energy potential is given by (ABAQUS™ manual, Version

6.14):

( 5 )

𝑈 = 2𝜇!𝛼!!

!

!!!

𝜆!!! + 𝜆!

!! + 𝜆!!! − 3 +

1𝐷!(𝐽!" − 1)!!

!

!!!

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Where µ is the initial shear modulus, U represents the potential function, 𝐷! is the

compressibility factor and 𝜆! are deviatoric principal stretches, 𝜆! are the principal

stretches and alpha may assume any real value without restriction (Miller & Chinzei

2002). The compressibility factor is approximately equal to the inverse of the bulk

modulus and according to ABAQUS manual, Version 6.14 it is determined to be zero

for incompressible materials.

The initial shear modulus can be calculated by:

( 6 )

𝜇 = 𝐸

2(𝜐 + 1)

Where E is Young’s modulus and 𝜐 is Poisson’s ratio. An incompressible material is

assumed to have a Poisson’s ratio of 0.5. Finally, the model was solved as a geometric

non-linear problem which involves the activation of the NLGEOM function in

ABAQUS.

7.5 Element type and mesh generation

As in the experiments, the dimensions of the tissue sample model are a diameter of 38

mm and a height of 24 mm. The mesh was generated using 8-node hexahedral

continuum (solid) elements with reduced integration and hourglass control.

Incompressible materials tend to produce an overly stiff response and volumetric

locking may occur. Volumetric locking occurs when there are too many

incompressibility constraints imposed on the discretised solution, relative to the number

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of degrees of freedom in the solution. Fully Integrated Elements are frequently prone to

volumetric locking. This may be remedied by using reduced integration elements.

However, reduced integration may result in ‘hourglassing’. Hourglassing is a spurious

deformation mode of a Finite Element Mesh and is nonphysical. These zero energy

modes produce zero strain and stress. Therefore hourglass control is implemented in

order to counteract this phenomenon. The generated mesh as well as constrained nodes

may be seen in Figure 7.

Figure 7: Meshed cylindrical sample shows rigidly constrained nodes at the top and bottom with the top nodes free to move in the vertical (z) direction

7.6 Solution procedure

The results obtained using explicit time integrated, hyperelastic, geometrically non-

linear analysis of the model without distortion and enhanced hourglass control but using

ALE (Case 1) was compared to those obtained from the model with distortion and

Fixed boundary condition

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enhanced hourglass control in a non-adaptive mesh domain (Case 2) in order to assess

their respective performances. Distortion control and enhanced hourglass control is

implemented by ABAQUS/Explicit when the distortion control option is toggled on/off

(and length ratio is specified) and the hourglass control is set to ‘enhanced’ when

specifying the element controls of the element type. For distortion control the defaults

length ratio value of 0.1 was used. Further, for both cases density of 1000 kg/m3 was

used and mass scaling was disabled throughout the analysis.

8 RESULTS

8.1 Experiment

8.1.1 Force-displacement results

The force-displacement responses of Layers A-F are shown in Figure 8. As is evident

from the figure, the responses are not significantly different for each layer.

The layers of Sylgard® 527 gel mentioned in Section 5.5: Sylgard® tissue samples and

brain phantom are shown in Figure 9. The figure shows the element sets generated in

Hypermesh as in List (2015), a finite element pre-processor commonly used to generate

meshes for models of highly complex geometries. The element sets are colour coded

and assist in the visualisation of the aforementioned generated Sylgard® 527 gel layers.

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Figure 8: Force-displacement response for simulation of compression tests on Sylgard® layers A-F

Figure 9: Colour-coded element sets defined for each layer of the Finite element model of the brain phantom generated in HyperMesh (List 2016)

8.1.2 Strain rate dependence

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The strain rate dependency of the Sylgard® gel was tested by performing the

compression tests at rates of 0.17 mm/s, 0.83 mm/s and 6 mm/s. As may be seen in

Figure 10, in agreement with previous studies (Ma et al 2010; Ma 2006), the

experimental results indicate negligibly minor strain rate dependency for strain rate

increase from 0.17 mm/s to 6 mm/s (approximately 35 times faster). Therefore, the

viscoelastic term in Ogden’s material model may be ignored.

Figure 10: Force-displacement response for Samples A1-A3 at three different loading rates of 0.17 mm/s, 0.83 mm/s and 6 mm/s.

The change in Young’s modulus as strain rate increases may be seen in Figure 11. For

compression at a strain rate of 6 mm/s the stiffness of the material is approximately

double that of the material compressed at a strain rate of 0.17 mm/s (35 times less). The

general trend of the graph is an increase with stiffness for an increase in strain rate,

except for layer B at 0.83 mm/s. This discrepancy may be due to errors in data

processing.

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Figure 11: Young’s modulus of layers A-F with varying strain rate showing a slight increase in stiffness as well as an inconsistency in the results for compression of a sample from Layer B at a strain rate of 0.83 mm/s

8.2 Finite element model

8.2.1 Finding Ogden material model parameters

Young’s modulus may be obtained by calculating the slope of the tangent line to the

linear region of the curve which passes through the point at zero strain. The slope of the

curve was calculated using the mathematical solver package, Matlab®, via numerical

differentiation (see Figure 12). For sample A1 the slope was estimated to be

approximately 3000 Pa. Using Equation ( 6 ) this value for Young’s modulus was used

to calculate the initial shear modulus of approximately 1000 Pa. This was adopted as a

first-guess value for 𝜇 along with an α-value of 4 as gauged from fitted values from Ma

(2006).

0

5,000

10,000

15,000

20,000

25,000

30,000

35,000

0.17 0.83 6

Youn

g's M

odul

us (P

a)

Strain rate (mm/s)

Young's Modulus for layers A-F

A B C D E F

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Figure 12: Slope of linear region of stress-strain plot for Layer A to determine Young's modulus

The model response was fitted to the experimental response and the parameters were

extracted for the Ogden model. Figure 13 below shows the simulation response (Mu

1875 Pa_alpha 4_9) fitted to the experimental results (Layer A experimental data) as

well as the response of the model with initial-guess Ogden material parameters (Mu

1000 Pa_alpha 4).

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Figure 13: Calibrating the simulation response to the experimental results

The initial shear modulus and alpha parameters obtained for layer A may be seen in

Table 1. These values are very similar to those in Ma (2006).

Table 1: Ogden material parameters obtained by calibrating the finite element model to experimental semi-confined uniaxial compression tests

Layer Initial Shear Modulus, µ (Pa) Alpha, α

A 1875 4.9

B 1875 6.5

C 2165 6

D 3000 6

E 3000 5.7

F 3100 5.5

Forc

e (N

)

Time (s)

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8.2.2 Simulation results

The following section will compare results obtained from the simulation of compression

of the model using the arbitrary Lagrangian-Eulerian method (Case 1) and those

obtained from the simulation of compression of the model using distortion and

enhanced hourglass control (Case 2).

As expected, the computational cost of Case 1 is much greater than that of Case 2, due

to the costly mesh and advection sweeps. The frequency of mesh sweeps was set to 35

with the number of mesh sweeps set to 5. A total of 1 advection sweep was performed

during the analysis as was determined automatically by ABAQUS.

The simulation aborted at a compression of approximately 23% of the initial height after

2 hours and 45 minutes of computational time. The force-displacement output may be

seen in Figure 15 below while the distorted model may be viewed in Figure 14.

Figure 14: Compressed cylindrical tissue sample simulation result using ALE

Curling edge

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The plane of nodes below the top plane seem to curl up over the top plane of nodes and

result in the raised edge along the cylinder circumference as seen in Figure 14.

The instability and costly computational time of the solution procedure make it unfit for

surgical simulation applications.

Figure 15: Force-time response of compression simulation using ALE indicating failure at approximately 0.75s.

In the force-time plot in Figure 15, a sudden drop indicates where the analysis aborts.

8.2.2.1 Applying distortion and hourglass control (Case 2)

The cylinder was subjected 30% compression in ABAQUS/CAE. In order to prevent

excessive distortion of the elements, distortion control was toggled on and enhanced

hourglass control was activated. The analysis was run for 1 second with a stable time

increment of 2.18E-05. The analysis completed in a CPU time of approximately 23

minutes. The deformed cylinder may be seen in Figure 16.

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Figure 16: Compressed cylindrical tissue sample simulation result using distortion and enhanced hourglass control

The force-time response results, as seen in Figure 17, agree with the experimental

results, showing a total force of approximately 3.5 N for 30% compression.

Figure 17: Force-displacement response of the explicit time integrated hyperelastic model using distortion and enhanced hourglass control

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9 VERIFYING ALGORITHM

In order to verify the accuracy of the algorithm used to solve the model, the simulation

force-displacement results were compared to those of a colleague, Agnes Kang. Agnes

used the Meshless Total Lagrangian Explicit Dynamics (MTLED) algorithm with a

Neo-Hookean constitutive model implemented using Matlab® in order to analyse the

deformation of incompressible, hyperelastic materials.

With some exceptions (e.g. LS-DYNA) commercial FAE/CAE software packages

(including ABAQUS) have no meshless capabilities and if they have (such as LS-

DYNA) they are very limited. Therefore, an in-house meshless code created at UWA

Intelligent Systems for Medicine Laboratory (ISML) was used. The meshless method

was applied using explicit integration in time domain and Total Lagrange formulation

(for accuracy and efficiency).

PhD students Ashely Horton and Xia Jin as well as Dr Guiyong Zhang contributed to

the ISML meshless code development (Zhang et al. 2013). In addition Grand Joldes

contributed an implementation which incorporates subroutines in NVIDIA’s CUDA to

facilitate parallel computations on GPU or Graphics Processing Unit when conducting

explicit integration in order to reduce computation time. The code was created with the

funding support from ARC DP1092893 and ARC DP120100402 grants.

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9.1 Boundary conditions and loading

Again, in ABAQUS the loading (displacement) was applied using a 3-5-5 Polynomial

that ensures zero velocity and acceleration at the start and end of loading. The face of

the cylinder at z = 0 is rigidly constrained while the opposite face at z = 0.1 is displaced.

Both models were compressed by 0.03 m of their original height (30% strain).

9.2 Constitutive model and its parameters

The code for the meshless method used the Neo-Hookean material model as mentioned

in Section 7.4.1: Neo-Hookean constitutive model, with a Young′s modulus in

undeformed state of 3000 Pa, Poisson′ s ratio of 0.495, and density of 1000 kg/m3. The

same constitutive model and parameters were implemented using FEM in ABAQUS.

The dimensions of the meshless and ABAQUS models are a diameter and height of 0.1

m.

9.3 Element type and mesh generation

In ABAQUS a fine mesh of 8-node linear brick, reduced integration, hourglass control

was used with distortion control (length ratio = 0.1) along with the dynamic explicit

solver. The mesh consisted of 6741 nodes and 6000 elements.

For the meshless simulation, the problem domain was discretized using 6710 nodes.

The difference in node numbers is negligibly small and should not affect the results.

The undeformed meshless model is shown in Figure 18 while the undeformed

ABAQUS model is shown in Figure 19.

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Figure 18: Undeformed meshless model used in verification process to compare simulation results with those obtained using FEM with model generated using in-house meshless code as in Zhang et al. (2013).

Figure 19: Undeformed ABAQUS model used in verification process to compare simulation results with those obtained using the meshless method

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9.4 Results

The meshless code imports the model mesh geometry as an input file of node numbers

and coordinates and compresses the cylinder by 30% using the meshless method. The

compressed model obtained using the meshless model is shown in Figure 20 while the

compressed model using ABAQUS is shown in Figure 21.

Figure 20: Tissue sample model compressed by 30% using the meshless method using in-house meshless code as in Zhang et al. (2013)

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Figure 21: Tissue sample model compressed by 30% using FEM using explicit time integration as well as distortion and enhanced hourglass control (Case 2)

The force-displacement results were extracted for the meshless code using the Matlab®

code as may be seen in Appendix A: Matlab® code for extracting force-displacement

data.

A comparison between the results obtained from the FEM results for Case 2 and those

obtained from MTLED may be seen in Figure 22. The meshless method predicts forces

up to approximately 30 per cent higher than FEM at 30% strain. This divergence in

results increases as the amount of deformation increases due to a loss of accuracy at

higher strains.

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Figure 22: Force-displacement response of deformation of up to 30% strain of non-linear incompressible cylindrical sample using meshless techniques

It is also clear that the meshless method is more sensitive to the fineness of the mesh at

higher strains as may be seen from the deviation between the curves for the mesh of

4663 nodes and for the mesh of 6710 nodes at 30% strain. However, at lower strains,

the meshless method is less sensitive to mesh fineness than FEM with the 4663 and

6710 curves overlapping exactly for up to 20% strain.

In Zhang et al (2013) for a comparison between the meshless method and FEM the

maximum relative difference in both displacement and reaction force was shown to be

smaller than 5% for 20% strain. This suggests that analysis technique of the current

study has some inaccuracy. Zhang et al (2013) used a fine mesh of 20-node quadratic

brick, hybrid, linear pressure elements (C3D20H) and the dynamic implicit solver.

Generally, an implicit algorithm is more accurate albeit more time intensive. This may

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account for the discrepancies between the findings of the current study and that of

Zhang et al (2013).

10 DISCUSSION

In this study Sylgard® gel brain phantom samples were subjected to uniaxial, semi-

confined compression tests at different strain rates in order to determine the constitutive

parameters of the hyperelastic material model governing the constitutive response of the

gel. A tissue phantom model was generated in the commercial finite element solver

ABAQUS and these parameters were used in the Ogden hyperelastic material model.

The model was then subjected to large (30%+) strains and the response was evaluated.

Finally, the Arbitrary Lagrangian-Eulerian method (Case 1) was applied to the model in

order to increase the accuracy and stability of the solution. Thereafter, a compression

simulation was performed the same model without ALE but with distortion and

enhanced hourglass control (Case 2). The algorithm used in Case 2 was then compared

to the Meshless Total Lagrangian Explicit Dynamic (MTLED) method for verification.

10.1 Constitutive model and parameters

Ogden’s model was successfully used to approximate the constitutive response of the

Sylgard® 527 gel. Upon fitting the uniaxial semi-confined compression simulation

results to the experimental results, the initial shear modulus, µ, and alpha, α, parameters

obtained were in close agreement with those in Ma (2006). The effects of strain rate on

the deemed negligibly small and the viscoelastic term in Ogden’s material model was

ignored.

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10.2 Comparing Case 1 and Case 2

Arbitrary Lagrangian-Eulerian techniques are not recommended for hyperelastic

materials using ABAQUS due to the fact that elements using distortion control may not

be included in an adaptive mesh domain and enhanced hourglass control may not be

used for hyperelastic materials modelled in an adaptive mesh domain. Therefore, if it is

desired to use ALE for hyperelastic materials, a different platform or finite element

solver is recommended. Explicit-time integrated solution of a hyperelastic material

using elements with distortion and enhanced hourglass control is recommended for

modelling of large deformation of hyperelastic materials in ABAQUS finite element

solver.

Therefore, the simulation was performed for two cases. Case 1 involved the simulation

of compression tests of the phantom tissue sample using ALE and in Case 2 the

simulation was repeated for the model but ALE was disabled and instead the element

type was altered to include enhanced hourglass and distortion control.

Case 1: ALE was more robust and resulted in shorter computation time. Deformation of

up to 50% was simulated without termination of the program. This approach is more

robust than that of Case 2. However, the fact that the analysis completed, does not

imply accuracy and the algorithm was to be compared to the MTLED method for

verification.

In Case 2: Distortion and hourglass control the simulation terminated at approximately

24% strain and the computational time was significantly larger than in Case 1: ALE.

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This was expected due to the fact that if distortion control as well as hourglass control

are disabled, the elements are free to distort aggressively due to the excessive

deformation (30%+) as well as suffer from hourglassing which is a result of using

reduced integration. However, reduced integration is necessary to prevent volumetric

locking from occurring due to the incompressibility of the model. Therefore, it may be

concluded that using adaptive meshing (ALE) for hyperelastic materials is not

recommended but that distortion an enhanced hourglass control in a non-adaptive mesh

domain (ALE disabled) should be used.

Finally, the simulation results for MTLED are 30% less than those obtained for the

finite element analysis Case 2. Zhang et al (2013) reports a variation in displacement

and force results of only 5% at approximately 20% strain. The force results of the

current study are up to 30% smaller than the meshless method force results. These

inaccuracies may be due to the higher accuracy of implicit time integrated solutions

when compared to explicit time integrated solutions.

11 CONCLUSIONS AND FUTURE WORK

The way a specific algorithm is implemented in different commercial finite element

solvers differs along with its parameters. The ALE approach in ABAQUS does not

allow the use of distortion and enhanced hourglass controls in adaptive mesh domains

for hyperelastic materials; however this may not be the case in other solvers such as LS-

DYNA. The current study is limited to the application of ALE in ABAQUS. Future

applications could utilise an alternate finite element solver.

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Furthermore, the cylindrical models generated in this study in ABAQUS/CAE are

geometrically uncomplicated. The brain phantom model, however, has varying

curvature and consists of planes with varying constitutive properties along the vertical

axis. In order to test the accuracy and stability of Case 2: Distortion and enhanced

hourglass control, it must be applied to models with more complex geometry such as the

brain phantom model. The parameters obtained in the current study for the constitutive

model for each layer may then be used to define the properties of each layer (or set of

nodes defining each layer) of the model in the finite element solver. The computational

cost, robustness and accuracy of a solution of a more realistic complex geometry will

determine whether a specific analysis technique or algorithm is suitable for application

in a surgical setting.

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13 APPENDICES

13.1 Appendix A: Matlab® code for extracting force-displacement

data

%=============== GetZforce ========================================== % % G du Plessis; May 2016 %==================================================================== dimen = size(forces_time); % Determines amount of time % increments zforces = forces_time(:,3,:); % For all nodes in mesh extracts % forces in z-direction prealloc=zeros(dimen(1),dimen(3)); % Preallocates space prealloc(1:dimen(1),1:dimen(3))=zforces(1:dimen(1),:,1:dimen(3)); % Writes forces in z-direction for each node to preallocated matrix forcedispset = prealloc(node_set_disp(:,1),:); % Extracts force-time % data for displaced % nodes only d = size(forcedispset); % Determines amount of % time increments dispinc = linspace(0,0.03, d(2)); % Creates linearly % spaced vector of % displacement plot(dispinc, sum(forcedispset)) % Plots force- % displacement