Evaluation of contact definitions in a Finite Element model of the

human cervical musculature

Victor S. Alvarez

Degree project in

Solid Mechanics

Second level, 30.0 HEC

Supervisors:

Peter Halldin

Svein Kleiven

1

Evaluation of contact definitions in a Finite Element model of the human cervical

musculature

Victor S. Alvarez

Degree project in Solid Mechanics Second level, 30.0 HEC

Stockholm, Sweden 2012

Abstract The human neck is especially vulnerable to severe injuries why it is of interest to gain a better understanding of the injury mechanisms involved. A 3D Finite Element (FE) model including the geometry of the individual neck muscles has been developed at Kungliga Tekniska Högskolan (KTH) and gives the possibility to study the strains inside the muscles. However, in this model the muscles are only hindered to interpenetrate and can glide relative each other with a roughly estimated coefficient of friction. The focus of this thesis has been to modify the FE‐model in order to model the connectivity on a physiological basis using methods available in the used FE‐software. The main reason for this was to get a more realistic muscle displacement without separations that are present in the current model.

The connective tissue surrounding and connecting the muscles was modeled with two principal approaches. The first was based on a combination of FE‐contacts where the material properties were coupled to the stiffness of the FE‐contacts. The other approach was based on a new element type called Smoothed Particle Hydrodynamics (SPH) elements in combination with FE‐contacts. Both approaches showed that the structural stiffness did not increase significantly, but the strain levels where somewhat elevated. Stability issues arouse with deleted elements and negative element volumes causing the FE‐contact based approach to fail prematurely. The SPH‐based approach had fewest deleted elements and completed the simulation but increased the calculation time with approximately 50 %.

It was concluded that the implementation of a connection between the muscles had a relatively low influence on the strains and kinematics of the neck and could be used to avoid muscle separations. The influence on stability of the model was however more evident and the most stable result increased the calculation time significantly.

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Utvärdering av kontaktdefinitioner i en Finita Element model av den mänskliga

nackmuskulaturen

Victor S. Alvarez

Examensarbete i Hållfasthetslära Avancerad nivå, 30 hp

Stockholm, Sverige 2012

Sammanfattning

Den mänskliga nacken är en väldigt sårbar del av kroppen och det är därför av stort intresse att lära sig mer om de olika skademekanismerna inblandade. På Kungliga Tekniska Högskolan (KTH) har en 3D Finita Element (FE) modell utvecklats som inkluderar musklernas geometri och ger möjlighet att studera töjningarna inuti musklerna. Men i denna modell hindras musklerna endast från penetration och kan glida relativt varandra med en grovt uppskattad friktionskoefficient. I detta examensarbete har fokus legat på att modifiera FE‐modellen för att modellera sammanbindningen mellan musklerna på en fysiologisk basis med hjälp av tillgängliga metoder i den använda FE‐mjukvaran. Huvudmålet med detta var att uppnå mer realistiska muskelrörelser utan separationer som uppstår i den befintliga modellen.

Den bindvävnad som omger och binder samman musklerna modellerades med två grundstrategier. Den första var baserad på en kombination av FE‐kontakter där materialegenskaperna kopplades till styvheten i FE‐kontakterna. Den andra strategin baserades på en ny elementtyp kallad Smoothed Particle Hydrodynamics (SPH) i kombination med FE‐kontakter. Båda strategierna visade att den strukturella styvheten inte ökade med några signifikanta nivåer, men töjningsnivåerna ökade något mer. En del stabilitets problem uppstod som gav raderade element och negativa elementvolymer vilket ledde till att den FE‐kontakt baserade strategin kraschade innan simuleringen slutförts. Den SPH‐baserade strategin resulterade i minst antal raderade element och slutförde simuleringen men ökade beräkningstiden med c.a. 50 %.

Det fastställdes att implementeringen av de nya metoderna hade en relativt liten påverkan på töjningar och kinematiken hos nacken och kan användas för att undvika muskelseparationer. Inverkan på stabiliteten hos modellen var dock mer märkbart och det stabilaste resultatet ökade beräkningstiden betydligt.

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Contents

Introduction ............................................................................................................................................. 5

Objective ................................................................................................................................................. 6

Dynamic Finite Element Method ............................................................................................................. 6

FE‐Contact ............................................................................................................................................... 9

Penalty Method ................................................................................................................................ 10

LS‐DYNA Contact Definitions ............................................................................................................. 11

Smoothed Particle Hydrodynamics ....................................................................................................... 12

Cervical Region ...................................................................................................................................... 14

Cervical Musculature ............................................................................................................................. 15

Connective Tissue .................................................................................................................................. 15

Dissection .......................................................................................................................................... 20

Methods ................................................................................................................................................ 22

Tiebreak Contact ............................................................................................................................... 22

SPH‐Elements .................................................................................................................................... 23

Modeling Approaches ....................................................................................................................... 24

Single surface ................................................................................................................................ 24

Tiebreak ........................................................................................................................................ 24

Tiebreak and single surface .......................................................................................................... 25

SPH and single surface .................................................................................................................. 25

SPH ................................................................................................................................................ 25

Material Parameters ......................................................................................................................... 26

Material Validation ........................................................................................................................... 28

Two Muscle Model ............................................................................................................................ 29

Full Neck Model ................................................................................................................................ 30

Results ................................................................................................................................................... 31

Material Validation ........................................................................................................................... 31

Two Muscle Model ............................................................................................................................ 34

Load in z‐direction ........................................................................................................................ 34

Bending ......................................................................................................................................... 38

Full Neck Model ................................................................................................................................ 41

Discussion .............................................................................................................................................. 47

Material Validation ........................................................................................................................... 47

Two Muscle Model ............................................................................................................................ 47

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Full Neck Model ................................................................................................................................ 48

Conclusions ............................................................................................................................................ 50

Material validation ............................................................................................................................ 50

Two Muscle Model ............................................................................................................................ 50

Full Neck Model ................................................................................................................................ 51

Final Conclusions ............................................................................................................................... 51

Future Work .......................................................................................................................................... 52

Bibliography ........................................................................................................................................... 54

Appendix A – MATLAB code for finding nodes to be included in the tiebreak contact ....................... 56

Appendix B – MATLAB code for finding nodes to be defined as SPH‐elements ................................... 63

Appendix C – Maximum displacements for two muscle model ............................................................ 64

Appendix D – Figures from the two muscle model ............................................................................... 66

Appendix E – Relative vertebrea rotations ........................................................................................... 69

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Introduction

The human neck is responsible for the mobility and flexibility of the head and the general kinematics of the head and neck is a product of all components that comprises it, such as muscles, vertebrae and vertebral discs. But this flexible construction also makes it vulnerable to trauma and since the neck connects the brain with the rest of the body this can lead to serious injuries. If it affects the central nervous system this can lead to partial or total paralysis and even death, as well as more diffuse damages to the spine or soft tissues that still can have grave impact on a person’s life. It is therefore important to study and learn more about the injury mechanisms of the head and neck which amongst other thing could lead to better protection systems. Traditionally the main sources of information aimed to understand injury mechanisms and develop protection systems have been experiments with volunteers, cadavers or detailed dummies. The problem with these methods is that they have many deficiencies and are often time consuming and expensive (Hedenstierna, 2008). A powerful supplement to the conventional methods is numerical computer models where loading conditions and system variables are easily controlled. They can also be used in cases where it is impossible to use voluntary experiments or cadaver tests.

The main problem with numerical modeling is the complex nature of human tissues with a non‐homogenous material composition and very non‐linear material behavior. And in the muscles there is not only the passive elasticity to take into account as in conventional engineering materials, but the active part that contracts the muscle depending on signals being sent throughout the body. Several mathematical models of the human body have been constructed with different approaches and levels of detail. Amongst the ones that include the muscles of the neck, the predominant approach has been to use discrete spring elements with non‐linear constitutive relationships, where later, the active and viscoelastic properties have been integrated (de Jager, 2000; Brolin, 2002). With increasing computational power it has become possible to increase the complexity and detail of the models.

One of the latest models of the head and neck, is a 3D Finite Element (FE) model developed at Kungliga Tekniska Högskolan (KTH) with continuum material properties that has been first to include the actual geometry of the cervical musculature with the use of MR images (Hedenstierna, 2008). This approach has many advantages such as improved kinematic compliance and possibility to study tissue injuries. It can also be used in educational purposes where different injury scenarios are simulated so that for example medical students can get a better understanding of the injury mechanisms. There is currently a project being developed at KTH partly for this purpose (STH, department of Neuronics, 2012). The model has had a good level of success in validation with volunteer testing (Hedenstierna, 2008) but some aspects can still be improved. For example, the activation pattern of the muscles still remains to be modeled on a more physiological basis. Another aspect is that the material model currently used is a combination of two existing program defined models available in the used software. A more complex material model might increase the certainty in the material behavior and thus the certainty in the response on a global and local scale.

Another aspect that has not yet been modeled in detail is how the muscle groups are connected to each other and other surrounding tissue. In the current KTH model there is only a simple interaction in the form of FE‐contacts that avoids interpenetrations and implements friction but with a roughly estimated coefficient of friction. This contact leads to the possibility of separations between the

6

muscle parts and is observed in collision simulations where the muscles buckle and separate on large scales. This is not considered physiological and makes it hard to use for educational purposes. An improvement on this aspect has the possibility of producing a more realistic muscle deformation and making the model more usable as a visualization tool. It is also possible that it changes overall passive stiffness of the neck and the strain distribution within the muscles.

Objective

The main focus of this thesis has been to find methods that can model the connectivity of the muscles on a more physiological basis in the 3D FE‐model created by (Hedenstierna, 2008) using the FE‐software LS‐DYNA. The main goal of this is to improve the displacement pattern of the muscles and hence reduce the muscle separation. A sub‐objective has also been to investigate what effects these changes can have on general kinematics and muscle strains. To confine the problem the restrictions have been to use contact definitions, elements and materials available in LS‐DYNA.

Dynamic Finite Element Method

The Dynamic FE‐method, similar to the conventional FE‐method, is a numerical approximate method for solving Partial Differential Equations (PDE). In solid mechanics this is used to solve the stresses, stains and deformation of a body. In the following section this is briefly explained and for a more detailed description the reader is referred to (Kleiven et al., 2001) or other literature on the subject.

The concept of stresses and strains can most easily be explained in one‐dimension, but can be generalized to three‐dimension with little effort.

Figure 1. A one‐dimensional body exerted to the body force q(x) and external forces at boundaries.

The stress σ(x) in a point in a body in figure 1 is defined as the force F(x) in a cross section divided by its area A(X) as

(1)

and the strain ε(x) is defined as

. (2)

where u(x) is the displacement. The stress and strain is then coupled with a constitutive relation as

L

q(x)

x, u(x)

f f

dx

F(x) F(x+dx)

dx

A(x)

dx+du(x)

After deformation

q(x)

7

(3)

where E(x) is the material description. If the relationship between σ and ε is linear then E is called the Young’s modulus.

Combining equations (1)‐(3) with the equilibrium equation leads to a second order differential equation in the displacement u called the strong formulation of the governing equation (Kleiven et al., 2001). In the principal of virtual work this equation is multiplied with an arbitrary scalar function called a weight function v(x) and integrated over the entire domain resulting in the weak formulation and is for the one‐dimension case

0 0 . (4)

This equation could be solved with a displacement assumption, but for most boundary value problems it is not possible to find a displacement assumption that satisfies the governing differential equation and all the boundary conditions. Therefore the domain can instead be divided into subdomains, or finite elements, for which simpler assumptions can be made that also fulfills continuity over subdomain boundaries. For the one‐dimensional example the approximation for the entire domain can be written as

(5)

where ui is a discrete value at point i, or nodal value, of u and Ni is a so called element shape function. Applying this to equation (4) yields

(6)

where

(7)

and

0 0 (8)

Equation (6) can also be written on matrix form as

(9)

and if the shape function is chosen with certain properties the integration can be made over each element instead of the entire structure and the components are summed up in so called stiffness matrix assembly (Kleiven et al., 2001). The resulting equation however is still as equation (9) but with larger matrices. Similar derivation can be made for two and three dimensions.

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A general three‐dimensional body as in Figure 2 can be approximated as an assemblage of discrete finite elements interconnected with nodal points.

Figure 2. General three dimensional body with an 8‐node three‐dimensional element (Kleiven et al., 2001).

The displacement in the local coordinate system x, y, z can then be written as

, , , , (10)

where N(m) is the displacement interpolation matrix for element m and Û is the vector of the three global displacement components at all nodal points. Then the element strains become

x, y, z ∂ , , ∂ , , , , (11)

and the stresses

(12)

where C(m) is the elasticity matrix of element m and σI(m) are the given element initial stresses. Now the principal of virtual of virtual displacement can be applied which states that for a body to be in equilibrium it is required that for any compatible virtual displacement, the total internal virtual work be equal to the total external virtual work (Kleiven et al., 2001). By including the element inertia and by approximating the element velocities and accelerations in the same way as the element displacement in equation (10), the principal of virtual displacement leads to the equilibrium equations as

(13)

where M is the mass matrix given by

(14)

where V(m) is the volume of element m and K is the stiffness matrix given by

(15)

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and F is the load vector as

,…

(16)

where f B(m) is the body forces for element m and f S(m) is the surface forces for element m. The equations in (13) can then be integrated in a step‐by‐step procedure where the time derivatives are approximated by taking the differences of displacements at various instants of time with a method of choice.

For large deformations, finite strains Fij in multiple dimensions can be described by the deformation gradient as

(17)

where xi is the deformed coordinate and Xj is the reference coordinate. A strain tensor commonly used for solid elements, where large deformations occurs (Hallquist, 2006), is the Green‐St. Venant strain tensor described by

12

(18)

where Iij is the identity matrix.

FE­Contact

When describing the contacts in FE‐Analysis terms frequently used are slave and master. This is a way of distinguishing between two entities in contact and can be in terms of nodes, surfaces or segment (segments can be a general area comprised of a number of nodes). The reason to call it slave and master is because the nodes on the slave side are the ones that are in some way forced to follow the master side. This is however dependent on the formulation used and not always a necessity. The constraints used to force the bodies to interact are applied on the nodes, as all other external forces.

In LS‐DYNA there are three different methods for handling contacts between bodies. These are, the kinematic constraint method, the distributed parameter method and the penalty method (Hallquist, 2006).

The kinematic constraint method imposes constraints on the global equations by a transformation of the slave nodes displacements components along the contact interface. It is a very stiff formulation because the nodes are constrained to move with the surface. No way has been found to control this stiffness why this method is not considered to be of use in this thesis.

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The distributed parameter mass method is also based on constraints, but uses a method of imposing constraints on the accelerations and velocities of the slave nodes to insure their movement along the master surface. This method is disregarded for the same reasons as the kinematic constraint method.

The last method will be the one of interest for the applications in this thesis since it allows for the stiffness to be controlled and is briefly described below.

Penalty Method

The penalty method checks each slave node for penetration through the master surface, see Figure 3.

Figure 3. Schematic figure of a contact surface (Kleiven et al., 2001).

If a slave node is found to penetrate, an interface force is applied between the contact point and the penetrating node. The magnitude of the force fs on the slave node is proportional to the amount of penetration l as

(19)

where ni is the normal vector of the master segment i and ki is the stiffness factor.

Hence this can be considered as placing interface springs between the penetrating nodes and the contact surface. The stiffness factor ki of these springs can in LS‐DYNA be calculated in three different ways. The default setting is to express it in terms of the bulk modulus Ki, the volume Vi and the face area Ai of the element that contains the master segment si that is being penetrated as

(20)

which is for brick elements and fsi is the scale factor for the interface stiffness that can be changed but is normally defaulted to 0.1. For shell elements the stiffness factor is given by

max

. (21)

There are also some options for setting the penalty stiffness value and they are:

‐ Using the minimum of the master segment and slave node stiffness. (default) ‐ Using master segment stiffness. ‐ Using slave node value. ‐ Using slave node value, area or mass weighted.

Master surface

Material A

Slave node Material B

ni

11

‐ Using slave node value, inversely proportional to the shell thickness.

The second approach is called soft constraint penalty formulation and is based on not only calculating slave and master stiffness according to equation (20) or (21), but also an additional stiffness called the stability contact stiffness kcs given by

0.5 · ·1

Δ (22)

where SOFSCL is the scale factor for the soft constraint penalty formulation, m* is a function of the mass of the slave node and the master nodes. ∆tc is set to the initial solution time step but if the time step grows ∆tc is reset to the current time step in order to prevent unstable behavior. This stiffness is then compared with the stiffness calculated with the first method and in general the maximum of the two is chosen as the used stiffness.

The third approach is called segment‐based penalty formulation and uses a stiffness as

0.5 · ·

1Δ

(23)

where SLSFAC is the same scale factor as fsi and SFS and SFM are a scale factors for slave and master segment respectively defaulted to 1.0. The masses m here are segment masses rather than nodal masses and is for solid element segments equal to half the element mass. Here ∆tc is also set to the initial solution time step, but it is only updated if the solution time step grows more than 5%.

LS­DYNA Contact Definitions

There are many contact definitions in LS‐DYNA that uses different methods and approaches. The ones of interest in this thesis are described in this section. All contacts described use the penalty method.

The first contact definition used is denoted “CONTACT_AUTOMATIC_SINGLE_SURFACE” and will be referred to as singe surface for the remainder of the thesis. This contact only uses a slave side which means that it searches for all slave nodes that penetrate through any surface on the same slave side and applies an interface force on it but does not generate any force at separation. It is possible to apply a coefficient of friction and to use any of the penalty stiffness formulations above. The theory for the friction is not mentioned in detail since it is not a central part of this thesis but it is based on a Coulomb formulation and adds an interaction force to the nodes in contact. For more details the reader is referred to (Hallquist, 2006).

The second contact is denoted “CONTACT_TIEBREAK_SURFACE_TO_SURFACE” and will be referred to as tiebreak for the remainder of this thesis. This contact is originally developed to simulate failure which is not necessary for this application but can also be used to tie segments together using the penalty method unlike other tied contacts. The reaction force for this contact is proportional to the relative displacement between master segments and slave nodes and therefore it applies a penalty force both in separation, penetration and relative sliding. The distance between the nodes in contact can also be large and arbitrary. In this contact master and slave sides have to be defined and it is only possible to use the standard penalty stiffness formulation given by equation (20).

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The last contact is denoted “CONTACT _AUTOMATIC_SURFACE_TO_SURFACE” and will be referred to as surface to surface for the remainder of this thesis. It works in a similar way as the single surface contact with the exception that a slave and master segment have to be defined.

Smoothed Particle Hydrodynamics

Smoothed Particle Hydrodynamics (SPH) is a so called mesh‐free computational method and has the advantages over conventional mesh‐based methods in that it provides a simple and accurate solution at large deformations and an easier discretization of complex geometries, amongst others. The idea is that the continuum is described by individual particles that move in space and carry all computed information, and thus the computational frame for solving the partial differential equations describing the continuum (Vesenjak and Ren, 2007). This is achieved by the fact that the exact value of any function f(x) of a three‐dimensional position vector x can theoretically be determined by taking the convolution of the function with the Dirac delta function δ(x‐xi) as

| Ω (24)

where

∞,0, (25)

But since the Dirac function is theoretical and hard to use in practice a so called smoothing function, or Kernel function, W(x,h) is introduced in place of the Dirac function as a first approximation and is non‐zero in a small domain and zero elsewhere and can for example be defined as

,1

(26)

where d is the number of space dimension and h is the smoothing length determining the influence domain of the smoothing function. When h goes to zero the kernel function can be approximated by the Dirac function. A kernel function commonly used is the cubic B‐spline (Hallquist, 2006) which is gained by choosing Θ as

132

| | 114

2 1 | | 2

0 2 | |

(27)

where C is a constant given by normalization and depends on the number of space dimensions. Applying the Kernel function and further approximating the function f(x) to be solved numerically by discretization of the continuous domain Ω into small patches as

∆Ω (28)

where mj is the mass and ρj is the density of particle j. The particle i has the position xi(t) moving in the vector field v. This yields the so called particle approximation of the i‐th particle as

13

, (29)

where N is the number of neighboring particles and the approximation of gradients of a function is

(30)

Equations (29) and (30) are applied to the equations of conservation and give the local continuum properties as the sums of particles contributions. For example the momentum conservation equation is given by

1 (31)

where v is the velocity vector and α, β are the space indices, and the particle approximation becomes

, , (32)

where

1 || ||. (33)

The energy conservation equation is

(34)

where P is the pressure and the particle approximation becomes

(35)

The SPH method has some numerical difficulties like particle inconsistency, inaccuracy at domain boundaries and instabilities at tensile stress state (Vesenjak and Ren, 2007). The stability and accuracy also depend on the particle density within the influence domain and the time step of time integration scheme. Another problem at large deformation is the change of number of particles in the influence domain. If they are compressed the number of particles increase, leading to increased calculation times. And at tension they decrease, leading to lowered accuracy and stability problems. Therefore h varies in time and space in order to maintain approximately the same number of particles in the influence domain (Vesenjak and Ren, 2007). The influence domain is defined as 2h (Lacome, 2001) and the equation that governs the smoothing length is based on keeping the total mass M of n particles within a sphere with volume V constant. The mass in the sphere is given by

323

(36)

14

and too keep this constant over time equation (36) is differentiated in time as

323

323

(37)

where the left hand side is zero because of mass conservation and with some simplifications this leads to

13

(38)

where div(v) is the divergence of the flow. The initial smoothing length h0 is calculated by LS‐DYNA in

the initialization phase and is computed by, for each particle, searching the shortest distance to another particle. The largest values of these distances is then denoted L and is then scaled with a factor defaulted to 1.2, giving the initial smoothing length. The initial smoothing length can also be set to a user defined value. The mesh of SPH‐element should be as regular as possible and not contain large discrepancies of the mass between the particles (Hallquist, 2006).

Cervical Region

The human backbone, or vertebral column, is divided into five regions (Seeley et al., 2005) and consists of bones called cervical vertebrae separated by intervertebral disks. The region closest to the head is called the cervical region and the region below is called the thoracic region. The main functions of the vertebral column are to support the head and body, protect the spinal cord, provide attachment sites for the muscles and permit movement of the head and body.

Figure 4. The upper part of the vertebral column from C1 to T2 viewed from the left side.

The vertebrae in the cervical region have smaller bodies than the rest of the column and the cervical region is also more flexible then the other regions making it more vulnerable to dislocations and fractures. The vertebrae in the cervical region are named C1 to C7 with C1 as the first cervical vertebra closest to the head and in the thoracic region they are named from T1 to T12 as seen in Figure 4.

Cervic

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15

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16

levels and in some sense creates compartments that separate these muscle components. This can be seen in Figure 6 where the cross‐section of a muscle without its muscle proteins is shown.

Figure 6. Image from scanning electron micrographs after removal of skeletal muscle protein. Top left shows the epimysium (EP) and in the bottom the perimysium (P) and the endomysium (E). The top right shows an enlargement of the endomysium surrounding an individual muscle fiber (Kjaer, 2004).

According to a recent study (Guimberteau et al., 2010) the connective tissue is present as a histological continuum throughout the body with no clear separation between the skin, the hypodermis, the vessels and the muscles. Although structures that allow sliding with very little influence on surrounding tissue are present everywhere, as for example seen on the skin when contracting a muscle. They propose that this continuum consists of a network of microvolumes and microfibrils in a chaotic pattern that adapts when exposed to forces to preserve energy and fill space in an optimal manner, see Figure 7. But since the function and composition of the connective tissue can vary extensively throughout the body, so can the allowed sliding. It is widely accepted that the main contributor for the transmission of the force generated in the muscle to the bones at its binding sites is the myotendinous junction. These are sites that connect the muscle with the extracellular connective tissue (Järvinen et al., 1991). The muscle fibers have also been shown to be connected to the endomysium and the perimysium creating pathways that can transmit forces within the muscles (Purslow, 2002) and therefore the muscle fibers cannot be seen as independently working units.

Figure 7. The multimicrovacuolar system of the connective tissue (Guimberteau et al., 2010).

17

It has also been shown that forces can be transmitted to adjacent muscles via the epimysium and the extracellular matrix of a muscle to surrounding non‐muscular elements of so called compartments containing a group of muscles (Maas et al., 2001) and even antagonistic muscles (Huijing, 2009; Yucesoy et al., 2010). A consequence is that length–force characteristics may differ for an isolated muscle in situ and the same muscles in vivo. That is to say it is dependent on not only its own material properties but also on the surrounding muscles. It is also dependent on the properties, configurations and connective tissue structure in its direct environment. The resulting mechanical behavior of the muscles is therefore determined by the properties of the fibers, the extracellular matrix and the interactions between these (Gao et al., 2008b). In order to study the extramuscular force transmissions a FE‐model was created by (Yucesoy et al., 2003). The muscle extensor digitorum longus from the left anterior crural compartment of a rat was modeled with three times six, solid eight node elements. These were connected with 2‐node spring elements with uniaxial and linear stiffness characteristics to a set of fixed points representing the adjacent muscle. The simulations were performed in the same manner as the experiments and the value of the spring stiffness k was determined so that the simulations provided a good agreement with the experimental results, giving a value of k=0.067 unit force/mm.

In another recent study (Gao et al., 2008a) the epimysium is modeled as a material consisting of ground substance containing pairs of wavy collagen fibers. These are distributed in different angles to the muscle fiber direction and allow straightening and reorientation of the collagen fibers. The model predicts experimental tests fairly well as seen in Figure 8.

Figure 8. Comparison of the stress‐strain relationship of the epimysium between the experimental data (Gao et al., 2008a) and the prediction.

The experimental studies were performed on young and old rats (Gao et al., 2008a). Small sections of the epimysium were dissected from tibialis anterior muscles of rats and each end of the samples were attached to plastic holders with glue and stretched at a velocity of 10 mm/s. It was shown that for low strains the main contributors to the tensile strength are the elastin and the ground substance. At high strains the collagen fibers are straightened and become the main contributor to the tensile strength. These experiments show the tensile strength in plane of the connective tissue, see Figure 9, and though it is not specified in what direction it is assume that it can be approximated as transversely isotropic. However the tissues were also analyzed with electron microscope and thee different layers were identified where the outermost (facing the epimysium of the adjacent muscle) consists of bundles of collagen fibers aligned in a dominant direction, see Figure 10.

18

Figure 9. The stress‐strain relationship of the epimysium (Gao et al., 2008a).

Figure 10. Schematic representation of the ultrastructure of the epimysium in rat TBA muscle. Inner is facing the muscle, outer is facing epimysium of adjacent muscle (Gao et al., 2008a).

The thickness of the epimysium was also measured and was almost the same for old and young rats, 29.7 ± 8.6 and 34.6 ± 14.6 µm respectively. From this study it is possible to extract an approximate Young’s modulus of 0.7 Mpa, see Figure 11.

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20

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21

It covers each individual muscle but also groups of muscle and finally the whole neck. Although to say that it covers may be misleading since it is not individual capsules, but rather as if the space between the different tissues is filled with this material. The connective tissue can also be said to be a continuation of a certain tissue that gradually changes composition as suggested by (Guimberteau et al., 2010). For example, in the center of a muscle the material composition is as described for muscles but towards the outer part of it the composition changes, the active fiber components disappears and the relative amount of collagen fibers, elastin fibers and background material changes and the material becomes connective tissue, but no clear line can be found. Continuing towards another type of tissue, say a tendon, the composition continues to change and transforms into the tendon tissue, which basically just has a different relative amount of the components that make up the connective tissue (Seeley et al., 2005). How fast this transition is probably differs between tissues, anatomical position and ultimately physiological function but is quite hard to determine. This appears to be the same for all other tissues as blood veins and fat which fills up some of the space in‐between the muscles making all off the tissues one continuous entity.

The resulting mechanical behavior of the interactions between the muscles is, because of the nature of the connective tissue, quite complicated. But during examination of the roe deer neck one observation made was that the muscles appear to be able to slide in relation to one another with little or no impact on the neighboring muscle when the relative displacement is not to large. This can be seen in Figure 15 where a muscle is in rest in the first picture and deformed in the next.

Figure 15. Figure showing the effect on surrounding tissue when deforming a muscle piece. No interference in A and a muscle compressed in B.

At the same time the muscles are packed relatively tightly and even if it is possible to drag them apart it is hard to determine how strongly they are tied. It is also hard to determine how much of the strength lie in the thin layer of connective tissue separating the pieces and how much is a result of all the tissues and complex intertwining of them. This makes it very hard to determine the stiffness normal to the muscle tissue. As an analogy of the response to interaction, one can imagine a thin plastic bag filled with muscles and some moist which is emptied of air. The pieces inside can be moved relative to each other but cannot be truly separated since there is no air in the system. To be able to see how the muscles are bonded, some of the fabric must be destroyed, as seen in Figure 16.

Compressed muscle tissue

No deformation A B

22

Figure 16. Picture showing the layer between to muscle pieces.

If a small piece of connective tissue is removed from the muscle it reveals that it in fact is very stiff when stretched and also brittle, but the brittleness is probably because it is so thin.

Combining the ideas of these different articles and dissection, it could be assumed that the tensile stiffness in the plane comes from the outermost parts of the connective tissue. But since the amount of fibers decreases towards the middle this is probably what is responsible for the weakness in shearing. The force transmission mentioned by (Maas et al., 2001) is probably varying in different parts of the body but much of it could be assumed to come from the shear stiffness. No information has been found about the stiffness of the connective tissue perpendicular to its surface, though it should probably be lower than the stiffness in the plane of the surface since the shear stiffness is much lower. Nevertheless the resistance against separation, i.e. the apparent stiffness between the muscles is suggestively much higher than the stiffness perpendicular to the plane. The reason is that the resistance would be a result of the entire connective tissue network and not only of a small sheet of tissue between the muscles, since a large portion of the resistance can come from tensile stretching in the plane of the epimysium.

Methods

As for many other soft biological tissue the material properties of the connective tissue is complex, non‐linear, non‐isotropic etc. To implement this material in the current neck model with conventional FE‐methods would be difficult since it requires a very thin sheet of elements in‐between the muscles if not to remodel them completely. To avoid extremely small brick elements this would have to be done using shell elements but would then not correctly model the stiffness of the connective tissue normal to the muscle surfaces since they are two dimensional. But from the information above a few ideas arose concerning how to mimic the connectivity amongst the tissues in the neck with other approaches which are presented below.

Tiebreak Contact

One approach is to use the predefined contact definitions available in LS‐DYNA and manipulate different options in order to get appropriate properties. The stiffness can then be represented through the penalty stiffness and will be a linear stiffness which the connective tissue is evidently not. But since the total stiffness of the neck (with vertebrae, muscles etc.) is several times larger this

23

would probably not be very noticeable. The stiffness of the connective tissue is also fairly linear in a range of strain that the tissue could be assumed be exposed to during these simulations.

The simplest way to implement this idea is to use a tiebreak contact between the surface nodes of the solid elements and manipulate the penalty stiffness. But since there is only one penalty stiffness factor this could lead to interpenetration amongst the muscles if the stiffness is set to low. From the literature studied it is not clear if the connective tissue has the same stiffness in normal and tangential direction, but from the dissection it is assumed that the apparent resulting stiffness is not equal in shearing and in separation.

The first issue to avoid a linear stiffness cannot be countered without making a new contact definition, and ports outside of this thesis. Interpenetrations on the other hand can be countered by adding an extra single surface or surface to surface contact that prohibits the penetration but does not affect the stiffness in separation and sliding. To easily implement the single surface contact the solid elements can be covered with shell elements of very low thickness and a so called null material that has very little influence, something that is already used in the current model to implement the single surface contact. Another way to implement these extra contacts is to create segments on the same nodes as these shell elements but removing the actual shell elements afterwards. With the surface to surface contact a master and slave surface have to be defined.

One practical difficulty in implementing the tiebreak contact is that every individual muscle part needs to be set as slave or master in contact with surrounding muscles. And since several muscles are in contact with multiple muscles this leads not only a tedious work but also to the fact that many muscles will have multiple contact definitions on single nodes. Moreover, since the tiebreak contact ties nodes with large separation, the nodes on the side of the muscle not facing the adjacent muscle can get tied to the contact surface. To counter these problems a small MATLAB program was created that finds the nodes within a certain distance and outputs the master segments for each part and the slave segments within the given distance, see Appendix A.

SPH­Elements

The other approach is to make use of the SPH‐elements and the fact that they describe a continuum material without actually being tied together. This makes it possible to model the very narrow space confined between the muscles as a 3D material. An easy way to achieve this is to define the nodes on the surface of the muscles as SPH‐elements and set the volume of influence so that the neighboring nodes on the same muscle and on the adjacent muscle fall inside this volume. The SPH‐elements can then be given a material property of choice to describe the connective tissue. However, not all materials were found to be compatible with the SPH‐elements but two materials that are suitable for the application and were found to work are an orthotropic linear‐elastic and a simple isotropic linear‐elastic. With this modeling approach there are also some risks of interpenetration if using the isotropic material for low values on the Young’s modulus that can be countered in the same way as with the tiebreak‐based approaches above using additional contact definitions.

One issue with this approach is that it was found that, when using the SPH‐elements, the simulation time increased quite significantly. In an effort to counter this, another MATLAB program was created, see Appendix B, that finds only the nodes on the neighboring muscles that lie within a certain

24

distance, so to reduce to total amount of SPH‐elements used and thereby the simulation time. Another difficulty with these elements is to implement the orthotropic material. This is because no good way was found to define the material axes since, in LS‐DYNA, this is based on the nodes in the corners of brick elements if they are to be local. The only solution to this was to use a global coordinate system roughly aligned with the desired material axes. The particle distribution in the initial SPH‐mesh is a problem hard to counter since the nodes on the surface of the muscles are not uniformly distributed. Finally, the last issue is that no data was found on the relative amount or density of connective tissue surrounding the muscles and therefore it was assumed to be the same as the muscles, and the mass is then set to the density times the average volume between the muscles. Since the added mass will be fairly low and the strength of the connectivity is not affected by this, it is not considered a big issue.

Modeling Approaches

Based on these ideas a few approaches where created to model the connective tissue in order to compare their performance and find advantages and disadvantages.

Single surface

The first approach simply uses what is in the current neck model which is a single surface contact applied on shell elements comprised by the surface nodes of the solids muscle elements, as depicted in Figure 17. A coefficient of friction of 0.3 is used and the approach is denoted “single_surface_original”. This approach is also used without friction and is then denoted “single_surface_nofric”. The idea with these contacts is to have a reference that can show how the new approaches affect the behavior of the overall model. Both using friction and not using friction can show how much the friction in the current model is actually affecting the behavior.

Figur 1:

Figure 17. Schematic figure showing the nodes on the surface of solid elements constituting also shell elements.

Tiebreak

The second approach uses a tiebreak contact implemented on surface nodes of the solid element and is denoted “tiebreak_Ep”, where Ep is substituted with the equivalent value of the Young’s modulus produced by the penalty stiffness. As for all approaches using the tiebreak contact, the nodes used in this contact are found with the MATLAB program previously mentioned.

Solid element

Shell element

Node i

25

Tiebreak and single surface

The third approach is to use the same tiebreak contact on the solid elements but also the single surface contact without friction on the shell elements from the original model. This approach is denoted “tiebreak_singlesurf_Ep” where the Ep is the same as above. The approach is also implemented by replacing the shell elements with segments of the same nodes, for which it is denoted “tiebreak_singlesurf_segments_Ep”. An extra approach was also created using the same shell elements only without the single surface contact applied on them and it is denoted “tiebreak_and_shells_Ep”. This approach does not try to model the connective tissue in any new way, but is rather used in order to determine the effect of using multiple element and contact definitions on the same nodes. The extra contact used in these approaches could also be implemented using the surface to surface contact, but it is basically the same contact as the single surface. Although it could give a slightly different result, because of searching algorithms and other unknown aspects, it would not be possible to determine if it was a better way of modeling the connective tissue. It is also less convenient to implement because a master and slave need to be defined which would be tedious work in the full neck model and is for these reasons omitted.

SPH and single surface

The fifth approach uses the SPH‐elements with the isotropic linear‐elastic material and has the same single surface contact on shell elements as the first approach. It is denoted “SPH_isotropic_singleurf_ Ep” where the Ep is the Young’s modulus used in the material definition for the SPH‐elements. This approach is, as the third approach, also implemented using segments instead shell elements for the single surface contact and is then denoted “SPH_isotropic_singleurf_segments_Ep”. The reason to use an isotropic material was to see if there are any advantages with this approach in comparison to the contact based ones.

SPH

The sixth approach is also based on SPH‐elements but with the orthotropic material and is denoted “SPH_orthotropic_Eab_Ec”, where Eab are the Young’s moduli for the two global material axes roughly in the plane of the muscle surface facing the adjacent muscle surface. Ec is the Young’s modulus in the direction normal to the same surface.

The two last approaches are divided into two groups, one with SPH‐elements on all the nodes covering the whole muscles surface indicated by a “full” at the end of the model approach description but before the Young’s modulus, as “SPH_***_full_ Ep” and one with only the nodes in‐between indicated in the same manner but with “red”. This is mostly to verify a relationship between the calculation time and the number of SPH‐elements.

The reason that the orthotropic SPH approach does not have a version with contact is that the idea is to have the stiffness in normal direction high enough to prohibit interpenetration on its own. This approach is included mostly to show on possible use of the SPH‐elements as it is not likely to function very well being that the orthotropy will not follow the muscle movement.

26

Material Parameters

Basis for the choice of parameters was the shear modulus G minus the standard deviation resulting in G=0.9 kPa found by (Osamura, et al., 2007), the Young’s modulus of E=0.7 MPa extracted from (Gao et al., 2008a) and the spring stiffness of 67 N/m determined by (Yucesoy et al., 2003). To transform the shear modulus to a Young’s modulus, and vice versa, a homogenous and isotropic material is assumed, since the used methods are based on a linear elastic approach, and the relationship is given by

2 1 ν (39)

where ν is the Poisson’s ratio and is set to 0.49 that is the value used for the muscles. This is since no value was found for the connective tissue but is also probable since it is intimately connected to the muscles. From the shear modulus of 0.9 kPa the equivalent Young’s modulus is approximately 2.7 kPa and from the Young’s modulus of 0.7 Mpa the equivalent shear modulus is 0.2 MPa. The values of the Young’s modulus can then be coupled to the penalty stiffness in equation (20) with the following scheme. If the space between two 8‐node cubic solid elements is modeled with springs the total force f between the solids can be described as

4 Δ (40)

where ∆x is the displacement in the direction of the force, the factor 4 is because there are four nodes on the surface and k is the spring stiffness which is equivalent to the penalty stiffness ki. Equation (40) is divided with the distance t between the solids and the area A of the solid element as

4 Δ (41)

where

(42)

and

Δ. (43)

The distance t should be the thickness of the samples used in the material tests performed by (Osamura et al., 2007) and (Gao et al., 2008a). But since both values will be used in a range it is chosen as an educated guess to 0.2 mm which lies closer to the study by (Gao et al., 2008a) since it is for the epimysium of a rat muscle and the other study is from the human hand but not the epimysium.

Combining equations (41), (42) and (43) with Hooke’s law given by

(44)

and some minor manipulation the relation between the penalty stiffness and the Young’s modulus Ep becomes

27

4

. (45)

An easy way to achieve this penalty stiffness is to change the bulk modulus K used in LS‐DYNA, and then the bulk modulus for a certain Young’s modulus is given by combining equations (20) and (45) as

4

(46)

Another option is to change the penalty stiffness scale factor fsi and keeping K constant in which case

the two variable switch place in equation (46). If the Bulk modulus K is not changed and all scale factors affecting the penalty stiffness is set to 1, LS‐DYNA uses the bulk modulus of the material involved in the contact.

With equation (45) a Young’s modulus can be determined from the experimentally determined spring stiffness (Yucesoy et al., 2003) resulting in approximately E=2.1 kPa. It should however be noted that in the simulations performed by (Yucesoy et al., 2003) the muscle was displaced giving a shearing in the elastic springs, but the force in these springs is assumed to only be proportional to the elongation of the springs. So if the two muscles are sheared in respect to each other as illustrated in Figure 18 the force f in the springs would be given by

. (47)

where L is the distance between the muscles and L2 is the length of the spring after displacement ∆x.

Figure 18. Schematic figure of the spring setup performed by (Yucesoy et al., 2003).

In the case of using tiebreak contact with penalty stiffness, the force is proportional to any relative displacement of the master and slave segments in any direction. Therefore, the reaction force for the displacement in Figure 18 would be given by

Δ (48)

where ki is the penalty stiffness. In order for these forces to be equal they could be coupled with some simple geometry and the reaction force would be given by

Δ . (49)

No information about the distance between the muscles used in the simulations done by (Yucesoy et al., 2003) was found, but with the value of t=0.2 mm used above the effective stiffness did not decrease significantly why the value of E=2.1 kPa is kept.

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31

Results

Material Validation

The results from the simulations are presented below in the order of tensile test, shear test and finally compression test. In a first set of simulations presented in Table 1 the bulk modulus in the contact algorithms are kept unchanged in order to extract the effective Young’s modulus when the bulk modulus is defined by LS‐DYNA.

Table 1: Resulting Young’s modulus using LS‐DYNA defined bulk modulus.

Modeling approach Scale factor fsi Bulk modulus K /Mpa

Resulting Ep /MPa

tiebreak_Ep 1 1.32 0.21 tiebreak_singlesurf_Ep 1 5.89 0.94 tiebreak_and_shells_ Ep 1 5.89 0.94 tiebreak_singlesurf_segments_Ep 1 1.32 0.21

From Table 1 it becomes evident that the LS‐DYNA used bulk modulus changes when using the single surface contact together with the tiebreak contact, and consequently the measured Young’s modulus. The “tiebreak_and_shells_Ep” approach, that only uses the tiebreak contact on the solids, gives the same used bulk modulus and resulting Young’s modulus as the “tiebreak_singlesurf_Ep” approach. When using two contact definitions on one single node in the “tiebreak_singlesurf_segments_Ep” approach and no shell elements are present, the used bulk modulus and resulting Young’s modulus are the same as for the “tiebreak_Ep” approach that uses only one contact definition.

The measured bulk modulus’s from Table 1 was used to calculate the penalty scale factors needed to get the desired young’s modulus.

Table 2: Resulting Young modulus for the ”tiebreak_Ep” approach when changing fsi or K.

Modeling approach Scale factor fsi Bulk modulus K /kPa

Resulting Ep /kPa

Desired Ep /kPa

tiebreak_Ep 1 5.625 0.90 0.9 tiebreak_Ep 2.82283e‐4 ‐ 0.90 0.9

In Table 2 the resulting Young’s moduli for the “tiebreak_Ep” approach are listed when setting either the bulk modulus or scale factor according to equation (46) to get the desired Ep. It shows that both methods work equally good for setting the Young’s modulus.

Table 3. Resulting Young modulus for the “tiebreak_singlesurf_segments_Ep” approach when changing fsi or K

Modeling approach Scale factor fsi Bulk modulus K /kPa

Resulting Ep /kPa

Desired Ep /kPa

tiebreak_singlesurf_segments_Ep 1 5.625 0.90 0.9 tiebreak_singlesurf_segments_Ep 2.82283e‐4 ‐ 0.90 0.9

32

The results for the “tiebreak_singlesurf_segments_Ep” approach are presented in Table 3 and are identical to the ones in Table 2, which is not surprising since it uses the same bulk modulus.

It should be noted that the results when using the scale factor and the user defined bulk modulus are not identical as they may appear in Table 2 and Table 3. The values are however rounded to two significant digits, hence the difference is not large and probably due to truncation when calculating the parameters. The scale factors are presented with ten symbols since this is the largest amount possible to input in the LS‐DYNA keyword file. The number of digits when calculating the scale factors needed is larger and therefore the truncation error is probable.

Table 4: Resulting Young modulus for the “tiebreak_singlesurf_Ep” approach when changing fsi or K.

Modeling approach Scale factor fsi Specified bulk modulus K /kPa

Resulting Ep /kPa

Desired Ep /kPa

tiebreak_singlesurf_Ep 1 5.625 942 0.9 tiebreak_singlesurf_Ep 9.54774e‐4 ‐ 0.90 0.9

In Table 4 the same result as in Table 2 are listed but for the “tiebreak_singlesurf_Ep” approach. These show that when using the shell elements and a single surface contact it no longer becomes possible to use the user defined bulk modulus in order to change the penalty stiffness as the resulting Young’s modulus is remained unchanged. The penalty stiffness scale factor however is still possible to use for this purpose.

Table 5. Resulting shear modulus for the ”tiebreak_Ep” approach.

Modeling approach

Scale factor fsi Bulk modulus K /kPa

Resulting shear modulus /kPa

Desired Ep /kPa

tiebreak_Ep 1 5.625 0.90 0.9

For the shear simulations only one result from the “tiebreak_Ep” approach is presented in Table 5. This is because all of the simulations gave very similar results for their tensile test counterparts. This is as expected and justifies the use of the shear modulus of 0.9 kPa as the lower bound for the Young’s modulus.

Figure 23. Stress‐strain curve from compression using the ”tiebreak_and_singlesurf_0.9 kPa” approach and default penalty scale factor on the single surface contact.

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0-6000

-5000

-4000

-3000

-2000

-1000

0

E = 0.89517 [kPa]

ε [ ]

σ [N

/m2 ]

33

In Figure 23 the stress‐strain curve is plotted for the compression simulation using the ”tiebreak_and_singlesurf_Ep” approach. The penalty stiffness scale factor for the tiebreak contact is set to the value that gives an effective Young’s modulus of 0.9 kPa from Table 4. This is also seen in the right part of the plot where the slope coincides with this value. At the strain of approximately ‐1 the slope becomes steeper and consequently the Young’s modulus increases. This increase should not occur at this strain and the reason is as follows. LS‐DYNA places the contact surface at a distance from the middle surface equal to half the shell thickness. The shell thickness is set to 0.1 mm and since there are two surfaces the distance between the contact surfaces is also 0.1 mm and as the distance between the solids is 0.2 mm the contact surfaces should come in contact at a strain of ‐0.5.

Figure 24. . Stress‐strain curve from compression using the ”tiebreak_0.9kPa” approach.

Figure 24 shows the result from the compression simulation using the “tiebreak_0.9kPa” approach. This clearly shows that there is a very low increase in the stiffness caused by the single surface contact used in the ”tiebreak_and_singlesurf_Ep” approach since it is almost identical to Figure 23.

Figure 25. Stress‐strain curve from compression using the ”tiebreak_and_singlesurf_0.9 kPa” approach with an 1000 times augmented penalty scale factor on the single surface contact.

If the penalty stiffness scale factor for the single surface contact in the “tiebreak_and_singlesurf_0.9 kPa” approach is augmented 1000 times the presence of the shell elements contact surface becomes evident and can be seen in Figure 25. In this plot the slope of the curve increases at a strain of ‐0.5 which is when the contact surfaces collide. Like for the plot in Figure 23 the slope in the right part coincides with the Young’s modulus from Table 4. It can also be seen in Figure 25 that the interpenetration of the solids is reduced.

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0-6000

-5000

-4000

-3000

-2000

-1000

0

E = 0.90042 [kPa]

ε [ ]

σ [N

/m2 ]

-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0-8

-7

-6

-5

-4

-3

-2

-1

0x 104

E = 0.8952 [kPa]

ε [ ]

σ [N

/m2 ]

Figure 26.

Figure “tiebreathat resuis equivaof the sstiffness

It shouldthe forcsimulatioprescrib

For the instabilitremain i

Two M

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Figure 27.B: tiebrea

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Muscle Mo

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odel

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at timestep 0.2iebreak_singles

-1.4-7

-6

-5

-4

-3

-2

-1

0x 104

σ [N

/m2 ]

mpression using

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2 s from loadingsurf_segments

-1.2 -1

34

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35

When looking at the deformations of the muscles in this load case the first thing that becomes clear is that only in the original approach using single surface contact, with or without friction, the two muscles separates upon loading. This is shown in Figure 27 where some representative approaches are presented at the same time step. It can also be seen that the deformation mode of the SPH‐based approach is slightly different compared to the tiebreak based models approaches.

Figure 28. Z‐displacement of the loading point as function of time for the models presented in the legend. The load used is 1.6N.

In Figure 28 the displacement in z‐direction of the loading point on the muscles is plotted against the time step, for the original approach and the softest new modeling approaches. A complete table of the maximum z‐displacement from this loading is presented in Appendix C. The plot in Figure 28 shows what can also be seen in Figure 27, that there is a large difference in the overall structural stiffness between the original approach and the other approaches that ties the nodes together in different ways. If not taking into account the different deformation modes, this could indicate a large increase in structural stiffness caused by the stiffness in the contacts. But since only the original approach, with and without friction, gives separation between the muscles and hence a very different deformation mode from the other approaches, they cannot be compared in this manner. The displacements for the original approaches with and without a coefficient of friction are identical and therefore only the original with friction is presented in this plot for better clarity. This is of course expected as the two muscles separate at the beginning of the simulation and the contact algorithms has virtually no influence.

Figure 29. Z‐displacement of the loading point as function of time for the new modeling approaches using the lowest effective Young’s modulus. Load used is 2.6N.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-0.2

-0.18

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

time [s]

z-di

spla

cem

ent [

m]

single_surface_originalnew models with E=0.9 Kpa

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-0.18

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

time [s]

z-di

spla

cem

ent [

m]

SPH_isotropic_singlesurf_segments_red_0.9 kPatiebreak_singlesurf_0.9 kPatiebreak_singlesurf_segments_0.9 kPatiebreak_0.9 kPa

36

Figure 28 also shows a difference in the displacement between the new approaches even though the material properties are set to be the same. This better viewed in Figure 29 where the load is increased to 2.6N and only the new modeling approaches with the lowest effective Young’s modulus are plotted. It can be seen that the tiebreak‐based approaches do not show a very large difference in the structural stiffness. The “tiebreak_singlesurf_0.9 kPa” approach gives the softest result and has the largest difference from the other tiebreak based approaches in the beginning of the simulations. The “tiebreak_0.9 kPa” and “tiebreak_singlesurf_segments_0.9 kPa” approaches have a very similar response for more than half of the simulation time where after the “tiebreak_0.9 kPa” approach becomes stiffer. The SPH‐based approach displays the largest deviation from the other approaches, but the deviation starts at the second half of the simulation, at 0.1 s, from where it becomes stiffer than all the other approaches.

The “tiebreak_Ep” is the only modeling approach in Figure 29 that does not use an extra contact definition to avoid interpenetrations. It is also the only approach that shows a fairly large interpenetration for the Young’s modulus 0.9 kPa of up to 0.003 m (which is slightly more than the average element thickness) and about 0.015 m when using a Young’s modulus of 2.1 kPa. When using the stiffest Young’s modulus of 21.6 kPa there is only such a slight interpenetration that it becomes hard to measure, see figures in Appendix D for details.

Figure 30. Z‐displacement of the loading point for the “tiebreak_singlesurf_segments_Ep” and “tiebreak_Ep” approaches using three effective Young’s moduli and a load of 2.6 N.

In Figure 30 the z‐displacement for both the “tiebreak_singlesurf_segments_Ep” and “tiebreak_Ep” approaches are plotted using all three Young’s moduli. It shows that as the stiffness is increased the difference between the two modeling approaches diminishes and is almost negligible for the effective Young’s modulus 21.6 kPa.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-0.18

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

time [s]

z-di

spla

cem

ent [

m]

tiebreak_singlesurf_segments_0.9 kPatiebreak_singlesurf_segments_2.1 kPatiebreak_singlesurf_segments_21.6 kPatiebreak_0.9 kPatiebreak_2.1 kPatiebreak_21.6 kPa

37

Figure 31. Z‐displacement of the loading point using 2.6N, for the “tiebreak_singlesurf_segments_Ep” approach using three Young’s moduli.

Figure 31 reflects how the effective Young’s modulus Ep influences the structural stiffness for the “tiebreak_singlesurf_segments_Ep” approach. The plot shows that the increased stiffness does not give a very large increase in structural stiffness. Even when the stiffness is increased more than a factor of ten (2.1‐21.6 kPa) the maximum decrease in z‐displacement is approximately 23%.

Figure 32. Z‐displacement of the loading point as function of time, comparing the SPH‐based approaches using full covered muscles and reduced. The single surface contact in the SPH‐based approaches uses the segment‐based penalty formulation. Load used is 2.6N and the “tiebreak_E_0.9kPa” is also plotted as a reference.

Figure 32 show that the two SPH approaches of reduced and full coverage of the muscles do not give the same results. In the simulations plotted the radius of influence for the SPH‐elements is set automatically by LS‐DYNA and becomes so big that the particles on opposite side of the muscles can influence the nodes in‐between. This could be changed manually by setting the initial smoothing length to a suitable value, but no effort has been put in this since the calculation time for the full SPH model is many times longer than any other model as seen in Table 6 and is not considered reasonable.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

time [s]

z-di

spla

cem

ent [

m]

tiebreak_singlesurf_segments_0.9 kPatiebreak_singlesurf_segments_2.1 kPatiebreak_singlesurf_segments_21.6 kPa

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-0.18

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

time [s]

z-di

spla

cem

ent [

m]

SPH_isotropic_singlesurf_segments_full_0.9 kPaSPH_isotropic_singlesurf_segments_red_0.9 kPatiebreak_0.9 kPa

38

Figure 33. Z‐displacement of the loading point for the “SPH_isotropic_singlesurf_segments_Ep” and “tiebreak_Ep” approaches using three effective Young’s moduli and a load of 2.6 N. Plot to the left uses Standard penalty formulation for the single surface contact and plot to the right uses the Segment‐based penalty formulation.

Figure 33 compares the difference when using segment‐based penalty formulation and standard penalty formulation on the single surface contact in the SPH‐based approach. The choice affects the result to a quite large extent. What can be observed when using the standard penalty formulation is that the structural stiffness for the lowest Young’s modulus of 0.9 kPa on the SPH‐elements is significantly lower than for the lowest tiebreak based approach. When raising this value to 2.1 kPa the structural stiffness of the SPH‐based approach is raised to the highest of all curves in the plot. And when increasing to 21.6 kPa, which should have the largest effect, there is actually a small decrease.

When using the standard penalty formulation in the SPH‐based approach the model becomes stiffer for all values on the Young’s modulus but the increase in structural stiffness is more in proportion to the increase in the Young’s modulus, as it is for the “tiebreak_Ep” approach.

Bending

Figure 34. Y‐displacement of the loading point as function of time for the models presented in the legend.

In Figure 34 the displacement of the cylinder that is pressed against the muscles is plotted in the direction of the load for the new modeling approaches with the lowest effective Young’s modulus and the original single surface based approach. In this plot the differences between the “single_surface_original”, “tiebreak_0.9kPa” and “SPH_isotropic_singlesurf_segments_red_0.9kPa”

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-0.25

-0.2

-0.15

-0.1

-0.05

0Standard penalty formulation

time [s]

z-di

spla

cem

ent [

m]

SPH_isotropic_singlesurf_segments_red_0.9 kPaSPH_isotropic_singlesurf_segments_red_2.1 kPaSPH_isotropic_singlesurf_segments_red_21.6 kPatiebreak_0.9 kPatiebreak_2.1 kPatiebreak_21.6 kPa

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-0.18

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0Segment-based penalty formulation

time [s]

z-di

spla

cem

ent [

m]

SPH_isotropic_singlesurf_segments_red_0.9 kPaSPH_isotropic_singlesurf_segments_red_2.1 kPaSPH_isotropic_singlesurf_segments_red_21.6 kPatiebreak_0.9 kPatiebreak_2.1 kPatiebreak_21.6 kPa

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-0.09

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

time [s]

y-di

spla

cem

ent [

m]

single_surface_originalSPH_isotropic_singlesurf_segments_red_0.9 kPatiebreak_singlesurf_0.9 kPatiebreak_singlesurf_segments_0.9 kPatiebreak_0.9 kPasingle_surface_nofric

39

approaches are very small, although the difference between the curves are not constant throughout the simulations, and hence shows upon some deviating moving pattern. The “tiebreak_singlesurf_segments_0.9kPa” approach shows the largest deviation but is in percentage roughly the same as the maximum deviation in the z‐loading case although in this case the deviation starts earlier in the simulation. As in the z‐loading loading case the “tiebreak_singlesurf_0.9kPa” approach shows a softer response than the segment based counterpart, but is in this case stiffer and not softer than the “tiebreak_0.9kPa” approach. The plot also shows that the approach only using a single surface contact and no friction has the softest response. As for the longitudinal loading case, using the “tiebreak_0.9kPa” approach, the two muscles interpenetrate to a fairly large extent (0.002 m at some locations) and also decreases in the same way as the penalty stiffness is increased, see Appendix D.

Figure 35. Y‐displacement of the loading point as function of time for the “tiebreak_singlesurf_segments_Ep” and “tiebreak_Ep” approaches using three effective Young’s moduli.

Figure 35 shows that as the effective Young’s modulus increase, the difference between the two approaches decreases in the same manner as for the z‐loading case and is also almost equal for the highest effective Young’s modulus of 21.6 kPa. And also the increase in structural stiffness is not increased very much by an increased Young’s modulus.

Figure 36. Y‐displacement of the loading point as function of time for the “tiebreak_singlesurf_segments_21.6 kPa” and “tiebreak_21. 6kPa” approaches using segment‐based penalty formulation for the single surface contact.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

time [s]

y-di

spla

cem

ent [

m]

tiebreak_singlesurf_segments_0.9 kPatiebreak_singlesurf_segments_2.1 kPatiebreak_singlesurf_segments_21.6 kPatiebreak_0.9 kPatiebreak_2.1 kPatiebreak_21.6 kPa

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0Segment-based Penalty Formulation

time [s]

y-di

spla

cem

ent [

m]

tiebreak_singlesurf_segments_21.6 kPatiebreak_21.6 kPa

40

In Figure 36 the single surface contact uses the segment‐based penalty formulation rather than the standard penalty formulation that is used for the simulations in Figure 35. Only the “tiebreak_singlesurf_segments_21.6 kPa” approach is plotted with the “tiebreak_21. kPa” approach as a reference. In this plot the curves are not equal as they are in Figure 35 rather the “tiebreak_singlesurf_segments_21.6k Pa” curve is slightly above the “tiebreak_21.6 kPa” curve. This is only shown for the bending load case since it has the same tendency for the other load case.

Figure 37. Y‐displacement of the loading point for the “SPH_isotropic_singlesurf_segments_red_Ep” and “tiebreak_Ep” approaches using three effective Young’s moduli. Plot to the left uses Standard penalty formulation for the single surface contact and plot to the right uses the segment‐based penalty formulation.

Figure 37 compares the difference when using segment‐based penalty formulation and standard penalty formulation on the single surface contact in the SPH‐based approach. The difference is not as significant as for the longitudinal loading case. The largest difference is for the highest Young’s modulus which becomes stiffer than the tiebreak based approach when using segments‐based penalty stiffness, and softer when using the standard penalty stiffness. For both plots there is a quite small increase in structural stiffness as the Young’s modulus is increased from 0.9 to 2.1 kPa.

Table 6. Calculation times for used approaches in the two muscle model in bending.

Modeling approach

single_surface_original tiebreak_0.9kPa tiebreak_singlesurf_segments_0.9kPa

Total CPU time

4 min 27 sec

4 min 38 sec

5 min 9 sec

Modeling approach

SPH_isotropic_singleurf_segments_red_0.9kPa

SPH_isotropic_singleurf_segments_full_0.9kPa

Total CPU time

17 min 21 sec

35 min 44 sec

In Table 6 the calculation times for some chosen modeling approaches are listed. When comparing the “single_surface_original” and the “tiebreak_0.9 kPa” approach the calculation time is only slightly increased. When adding an additional contact in the “tiebreak_singlesurf_segments_0.9kPa” the calculation time increases with around 15% from the original model. When instead using the SPH‐based models the calculation time increases more significantly and is for the reduced version around 4 times longer and for the full version around 8 times longer.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

time [s]

y-di

spla

cem

ent [

m]

Standard penalty formulation

SPH_isotropic_singlesurf_segments_red_0.9 kPaSPH_isotropic_singlesurf_segments_red_2.1 kPaSPH_isotropic_singlesurf_segments_red_21.6 kPatiebreak_0.9 kPatiebreak_2.1 kPatiebreak_21.6 kPa

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

time [s]y-

disp

lace

men

t [m

]

Segment-based penalty formulation

SPH_isotropic_singlesurf_segments_red_0.9 kPaSPH_isotropic_singlesurf_segments_red_2.1 kPaSPH_isotropic_singlesurf_segments_red_21.6 kPatiebreak_0.9 kPatiebreak_2.1 kPatiebreak_21.6 kPa

41

The “SPH_orthotropic_Eab_Ec” model did, as expected, not give the desired properties since the orthotropy was not local and only resulted in incorrect results where the muscles passed through each other as they were bent away from the stiffer direction, see Appendix D. A complete table of the maximum y‐displacement from this loading is presented in Appendix C

Full Neck Model

The modeling approaches used in this simulation was, the “single_surface_original”, the “tiebreak_singlesurf_segments_0.9kPa” and the “SPH_singlesurf_segments_red_0.9kPa”. When starting to implement these approaches several problems emerged. When trying to implement the “tiebreak_singlesurf_segments_0.9kPa” approach, several elements on the muscles got negative volumes and the simulation failed. In an effort to counter this problem the maximum distance between nodes to be tied was decreased in order to reduce the amount of nodes in contact and some segments in the tiebreak contact that were in the area were elements failed were removed manually. An option in LS‐DYNA to remove elements with negative energy and not stop the simulation was also chosen. After these changes the simulation still crashed after 85 % of the simulation time and several elements in the vertebrae disks failed and were deleted for unknown reasons. The single surface contact in this case used the standard penalty based formulation.

Using the “SPH_singlesurf_segments_red_0.9kPa” approach, and also the standard penalty based formulation on the single surface contact, the simulation cleared but even more elements in the vertebral discs were deleted.

When the single surface contact is changed to the segment‐based penalty formulation the situation is worsened. With the “tiebreak_singlesurf_segments_0.9kPa” approach the simulation is not able to start and with the “SPH_singlesurf_segments_red_0.9kPa” approach the simulation crashes after 80% of the simulation time. However for the SPH‐based approach the stiffness scale factor for the single surface contact was reduced from 1 to 0.01 where after the simulation cleared and fewer elements in the vertebral discs were deleted than for the SPH‐approach using standard penalty formulation.

Figure 38. Plot of the relative y‐rotation in degrees between the head and the T1 vertebra. The penalty formulation indicated in the legend refers to the single surface contact in the modeling approaches.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-20

0

20

40

60

80

100Rotation of head realtive to vertebra T1

time [s]

Rel

ativ

e y-

rota

tion

[Deg

]

single_surface_originalSPH_isotropic_singlesurf_segments_red_0.9 kPaStandard penalty formulationSPH_isotropic_singlesurf_segments_red_0.9 kPaSegment-based penalty formulationtiebreak_singlesurf_segments_0.9 kPaStandard penalty formulation

42

Despite these problems it was possible to compare the approaches with the data gained. In Figure 38 the overall neck stiffness is reflected as it plots the relative y‐rotation between the head and the T1 vertebra on which the acceleration is applied and it can be seen that all curves lie relatively near each other. The SPH‐based approaches exhibit a slightly stiffer spring‐back behavior as can be seen in the end of the curves.

Figure 39. Plot to the right is the relative x‐displacement between the head and the T1 vertebra and plot to the right is the relative z‐displacement.

The same equality in structural stiffness can be seen in Figure 39 and especially in the plot for the relative x‐displacement where the curves almost overlap for most of the simulation time. In the plot for the relative z‐displacement the same spring‐back discrepancy can be observed between the original approach and the SPH‐based approaches as in Figure 38.

Figure 40. Plots of the relative y‐rotation, relative x‐displacement and relative z‐displacement between head and T1 for the approaches shown in the legend.

The “single_surface_original” approach was changed to use segments rather than shell elements to implement the contact. The result of this is shown in Figure 40 and the resulting curves are in this

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.05

0.1

0.15

0.2

0.25X-displacement of head realtive to vertebra T1

time [s]

Rel

ativ

e x-

disp

lace

men

t [m

]

single_surface_originalSPH_isotropic_singlesurf_segments_red_0.9 kPaStandard penalty formulationSPH_isotropic_singlesurf_segments_red_0.9 kPaSegment-based penalty formulationtiebreak_singlesurf_segments_0.9 kPaStandard penalty formulation

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-0.05

0

0.05

0.1

0.15

0.2Z-displacement of head realtive to vertebra T1

time [s]R

elat

ive

z-di

spla

cem

ent [

m]

single_surface_originalSPH_isotropic_singlesurf_segments_red_0.9 kPaStandard penalty formulationSPH_isotropic_singlesurf_segments_red_0.9 kPaSegment-based penalty formulationtiebreak_singlesurf_segments_0.9 kPaStandard penalty formulation

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]

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0.25X-displacement of head realtive to vertebra T1

time [s]

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ativ

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disp

lace

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t [m

]

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-0.05

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time [s]

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disp

lace

men

t [m

]

single_surface_originalSPH_isotropic_singlesurf_segments_red_0.9 kPaStandard penalty formulationsingle_surface_segments

43

case closer the SPH‐based approach. This is especially evident in the relative y‐rotation and relative z‐displacement where the last parts of the curves have the same spring back behavior.

Figure 41. The relative y‐rotation in degrees between a vertebra and underlying vertebra, ranging from C3 to T1 for the “SPH_singlesurf_segments_red_0.9kPa” approach and the “single_surface_original” approach.

In Figure 41 the relative rotation between a vertebra and its closest underlying vertebra is plotted for the simulation using the “SPH_singlesurf_segments_red_0.9kPa” approach. The plots show that there is some deviation when changing the modeling approach but the general shape is similar between the two. However the difference in the curves is most evident for the C7‐T1 plot where the relative rotation using the SPH‐based approach is very low.

Figure 42.The relative y‐rotation in degrees between a vertebra and underlying vertebra, ranging from C3 to T1 for the “SPH_singlesurf_segments_red_0.9kPa” approach and the “single_surface_original” approach.

In Figure 42 the same plots are shown as in Figure 41 but the single surface contact in the SPH‐based approach uses the standard penalty formulation. In most plots there are some kinks in the curves for the SPH‐based approach indicating abrupt movements in the vertebrae which is also present for the “tiebreak_singlesurf_segments_0.9kPa” approach, see Appendix E for details.

0 0.1 0.2 0.3 0.4-20

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]

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time [s]

single_surface_originalSPH_isotropic_singlesurf_segments_red_0.9 kPaSegment-based penalty formulation

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time [s]

single_surface_originalSPH_isotropic_singlesurf_segments_red_0.9 kPaStandard penalty formulation

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46

shows that there is a slightly larger separation for the tiebreak‐based approach than the SPH‐based approach for the muscles at the lower part of the model.

Table 7. Calculation times for used approaches in the full neck model.

Modeling Approach

single_surface_original

SPH_singlesurf_segments_red_0.9kPa Standard penalty formulation

SPH_singlesurf_segments_red_0.9kPa Segment‐based penalty formulation

tiebreak_singlesurf_segments_0.9kPa

Total CPU time

36 hrs 55 min 32 sec

48 hrs 33 min 31 sec 56 hrs 17 min 41 sec ‐‐‐

estimated total CPU time

31 hrs 54 min 40 hrs 46 min 53 hrs 18 min 27 hrs 25 min

In Table 7 the calculation times for the three approaches are presented. Unfortunately since the “tiebreak_singlesurf_segments_0.9kPa” approach did not finish at normal termination time the actual calculation time is unknown. Therefore the estimated total CPU time also listed, this is an estimation made by LS‐DYNA in the beginning of the simulation. As can be seen there is a certain level of uncertainty in this value and in the two cases that the simulation clears it is a slight underestimation. For the SPH‐based approach there is however an increase in calculation time but in this case only approximately 30%, or 1.3 times longer using the standard penalty formulation and approximately 50% using the segment‐based penalty formulation.

Figure 47. Plots of energies in three approaches named in the legend.

The energy from the external work done on the model not converted to kinetic energy is taken up by the deformations and stresses of the elements, as well as the contacts. In Figure 47 this is plotted for the original model, the SPH‐based approach and the tiebreak based approach. The first plot of the internal energy includes the deformations and stresses in all the elements and the sliding energy plot shows the energy in all the contacts. These show that there is some difference between the original and SPH‐based approach but not to any larger extent. It can also be seen that the energy from the contact is not changed significantly between these and that this energy is several times smaller than

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time [s]Ene

rgy

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time [s]

single_surface_originalSPH_singlesurf_segments_red_0.9kPatiebreak_singlesurf_segments_0.9kPa

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0.2

0.4

0.6

0.8Internal Energy SPH-elements

time [s]

47

the internal energy. The last plot shows the internal energy in the SPH‐elements and is several times lower than the sliding energy. However for the tiebreak based approach there is an abrupt increase in internal energy and for the sliding energy there is an abrupt increase in negative energy. Moreover the level of the sliding energy for the tiebreak based approach is in the same order of magnitude as the internal energy.

Discussion

Material Validation

The bulk moduli calculated in the results from the material validation are based on equation (20) which is for solid elements why the calculated bulk modulus for the models using shell elements is not valid. This is however not of interest since the equation can still be used to calculate the penalty stiffness scale factor giving the desired Young’s modulus. Moreover it is not known how LS‐DYNA calculates the penalty stiffness when using both element types, but is not investigated since this is not needed to change the penalty stiffness.

In the results from the material validation tests in compression it was possible to draw some conclusions. But exactly what happens when the solids have collided is not fully understood since the stiffness is augmented slightly after the solids penetrate each other (Figure 23 and Figure 24), independent on the presence or absence of an additional contact. This is however not of any mayor interest since interpenetration should be avoided and the stiffness could be modified empirically to achieve this if necessary.

As noted in the results for the material validation, the compression tests were modified so that only the most distal nodes in respect to the adjacent solid were given a prescribed displacement. A consequence of this is that the measured Young’s modulus will have a contribution coming from the deformation of the solid element. Hence the values from the measured Young’s modulus before collision in Figure 23 ‐ Figure 26 differ from the tensile tests in Table 4. The objective of these simulations is however not to measure the Young’s modulus and hence not an issue.

Two Muscle Model

The reason why the “tiebreak_singlesurf_Ep” approach is always softer than the other approaches in the z‐loading case of the two muscle model could have to do with the fact that the single surface contact becomes relevant when the distance between the muscle surfaces is equal to the shell thickness. Hence the single surface contact becomes active earlier than when not using shell elements and prohibits the relative movement of the nodes. However, this issue is not investigated further since this modeling approach is dependent on both the material parameters and thickness of the shell elements. If any of these aspects need to be changed, the bulk modulus used by LS‐DYNA is changed and need to be re‐measured in order to make it possible to calculate the new penalty stiffness scale factor. One option could also be to figure out the mathematical relation between these parameters and the used bulk modulus, but is not investigated since it is not considered to have any gain.

48

For the SPH‐based approach the reason for the differences is probably more complex. However one important aspect is that the deformation mode differs and probably is one of the largest contributors to the stiffening, for as can be seen in Figure 27, it is a more structurally stable shape. What supports this argument is the fact that this structural stiffening is not seen when the two muscles are bent instead and the deformation mode is much more equal for the different approaches. In this case the SPH‐based approach is softer than the others but present virtually no interpenetration due to the high stiffness in the single surface contact. Any good explanation for this difference in deformation has however not been found other than that the SPH‐based approach actually has the properties of an isotropic material. This could lead to weaker stiffness in shearing because of equation (39) where the shear modulus will become lower than the Young’s modulus. This is not the case for the tiebreak based approaches since the stiffness is always proportional to the relative displacement of the nodes tied and, effectively, the shear modulus is equal to the Young’s modulus.

Based on the two muscle model, the SPH‐based approaches seam to exhibit a more erratic behavior than the contact‐based approaches which is probably due to the instability in the SPH‐elements. It is also not clear from the simulations which penalty formulation is most favorable to use for the single surface contact in the approach. However the result from the z‐loading showed a very unrealistic behavior for the segment‐based penalty stiffness when increasing the Young’s modulus.

In the two muscle model bending simulations it was found from Figure 36 that the penalty formulation used in the single surface contact affected the structural stiffness of the model. When using the standard penalty formulation, very slight interpenetrations were also observed for the lowest penalty stiffness. When instead using the segment‐based penalty formulation there were no interpenetrations, but the structural stiffness was slightly higher and the “tiebreak_singlesurf_segments_Ep” and “tiebreak_Ep” did not become almost equal for the highest penalty stiffness as it did when using standard penalty formulation. The reason for this could be the difference in their formulations or the searching algorithms but it has not been investigated further. But since for the highest penalty stiffness no interpenetration was seen in either approach this could indicate that the second contact was in some way affecting the strength of the bonds, and therefore be an argument to use the standard penalty formulation in this approach.

Full Neck Model

In the full neck model an issue was that elements in the vertebrae disks were deleted because of some sort of element failure. This also seemed to cause the model to stop prematurely. It was observed that when using segment‐based penalty formulation on the single surface contact in the “tiebreak_singlesurf_segments_0.9kPa” approach, fewer elements were deleted. But this is probably only because the simulation was also stopped earlier, at 85%, and hence there is less time for the elements to be deleted.

When using the “tiebreak_singlesurf_segments_0.9kPa” approach there was some interpenetration in some of the muscles. This indicates some sort of error in the simulation since the contacts would not allow this if the interface forces are not very high. This could of course be the case, but the strain levels at this area are not so many times higher than in the other approaches and therefor indicate some other failure in the contact algorithm. The fact that the muscle deformation and interpenetration is non‐symmetric also indicates some calculation problem since the model is

49

otherwise symmetric. Moreover the fact that the negative sliding energy almost mirrors the internal energy in Figure 47 would indicate that the problem is very local (Dynasupport.com, 2012) and might be resolved if the contact segments are further modified manually.

It could be said that the SPH‐based approach using the segment‐based penalty stiffness on the single surface contact gives better results than when using the standard penalty stiffness. Fewer elements in the vertebral disk are deleted, but it also gives a smoother movement seen in the relative vertebral rotations, which may also be related. Further the strain levels in the cross‐section in Figure 44 don not present as elevated levels in small areas as for the standard penalty formulation. However the obvious drawback is the increase in calculation time with around 15% from the other SPH‐approach. The calculation times for the full neck model however also show that the increase in calculation time is not as large as for the two muscle model. This is probably due to the fact that there are a lower proportion of SPH‐elements in this model since it in this model are many other elements that do not have SPH‐elements on them. To compare the calculation time for the “tiebreak_singlesurf_segments_0.9kPa” approach is not considered to have much meaning since the actual calculation time could be somewhere around the estimated time. If only looking at the estimated times it would also indicate that it actually decreases when adding the extra contact which is not considered very likely. An assumption that could be made is that if the simulation would function better, the time would increase but perhaps not to any significant extent based on the results from the two muscle model.

Throughout this thesis, some versions of modeling approaches initially proposed have been left unused in some simulations. This is because they have been considered to have properties that are unwanted or unpractical. Before the approaches that were considered to have possibilities of giving good result were implemented in the full neck model, it was thought that tiebreak‐based approach would be the most stable and simple alternative. However, problems that were not foreseen emerged and some efforts were made to solve these. It is believed that one aspect that could be causing the elements to get negative volumes and crash the simulations is that nodes become over constrained. The MATLAB program that finds the nodes to be tied was modified so that any single node could not be tied to nodes belonging to different muscles. It was also modified so that two nodes belonging to the same element were not tied to different muscles. This improved the result to some extent, but some muscles were still tied to multiple other muscles and the problem seemed to be concentrated in the areas were the muscles branches out like fingers. Only after some manual modification of the contact segments based on these assumptions the simulations was able to reach 85% of the time. It is however not known if this is the only or even one problem with this approach. Another aspect can be that the penalty stiffness might vary to a quite large extent since it is calculated with the area and volume of the segment involved. And since some segments have varying areas and volumes this might cause problems. It might be possible to find out why this approach does not work very well and see if this is something that can be avoided, but since no good way of doing this was found it would have to be made by trial and error. And since there are many aspects that could be changed one at the time and the time for one simulation is about 28 hours this was not possible to make within this thesis. It could also be questioned if the method is even worth investigating further if such much effort needs to be made in order for it to function properly, not knowing if it will even be possible.

50

Conclusions

Material validation

Based on the results in the material validation section it becomes clear that the change in measured Young’s modulus and Bulk modulus is due to the usage of shell elements in the model and not multiple contact definitions. This is shown by the fact that the change also occurs when layer of shell element are added but no additional contact is implemented (see Table 1). Moreover the results are identical when one or two contact definitions are used but no shell elements are present. It can also be concluded that it is the usage of shell elements that inhibits the possibility to use the user defined bulk modulus to change the penalty stiffness. This can be said since this problem is only present when the shell elements are included in the model.

The results from the compression tests showed that the single surface contact has almost no influence on the stiffness in the contact. This could be explained by the fact that the contact stiffness of the single surface contact is several times smaller than the tiebreak contact and therefore is not noticed. These tests also show that the additional contact definition used does not affect the interface stiffness until collision since the stiffness is increased only when the contact interfaces have collided.

Two Muscle Model

When analyzing the results from the two muscle model exerted to a load in the longitudinal, or z‐direction, it could be concluded that the difference between the “tiebreak_Ep” and “tiebreak_singlesurf_segments_Ep” approaches seems to be directly related to the interpenetration seen in the “tiebreak_Ep” approach, see Appendix D, being that the difference almost completely disappears when the penalty stiffness is increased, see Figure 30, causing the interpenetrations disappear. It is also assumed that the interpenetration has a part in causing the structural stiffness to decrease since there is more movement allowed giving a softer response.

In the two muscle model the fact that the SPH models with full surface coverage is stiffer than the reduced version could be explained with that, as mentioned, the radius of influence is so large that it is almost as putting an additional muscle in the entire domain of the two muscles but with material properties of the connective tissue. Therefore it does not simulate a thin layer on the surface of the muscle, rather a superposed muscle with the material properties of the connective tissue.

A conclusion drawn from Figure 34 is that in the bending simulations the resistance to shear of the bonds is better reflected. One argument is that the model using only single surface contact and no friction is softer than the one using friction. These approaches are otherwise identical and therefor reflect, as expected, that an inhibition for relative sliding stiffens the structure. This also implies that the SPH‐based model is actually softer in shearing than the other contact based models, but since the deformation is different, this is not reflected in the z‐loading case.

The calculation times for the two muscle model are only presented for the bending simulation since all approaches are exerted to the same load in contrast to the z‐load case. From these result it could be said that though the calculation times in Table 6 increases slightly when the contact is changed

51

from single surface to tiebreak, this could be accounted for by chance, because the time can be slightly different each time a simulations is done. But also the algorithm could be slightly more time consuming, and a fact is that the forces in the tiebreak contact will be calculated in every time step, whereas the forces in the single surface will only be calculated when the searching algorithm finds a node to penetrate a contact surface. The increased calculation time when adding an additional contact can be explained with that LS‐DYNA in this case has a little more to calculate. But since the difference is still small it is apparently not the most time consuming aspect of the simulation. As expected the calculation time is increased significantly when using SPH‐elements and is apparently proportional to the amount elements used.

Full Neck Model

From the full neck model simulations it can be concluded that the general stiffness of the neck is only changed slightly when using the effective Young’s modulus of 0.9 kPa in the new approaches. In other words the added connections between the muscles do not, on a significant level, change the general kinematics of the head and neck. For the SPH‐based approach this can be explained by that most of the energy from the head movement is taken up by the tension and stresses in the elements in the model (Figure 47). For the tiebreak based approach it is harder to make the same conclusion. On a more local scale there are larger differences, as seen in the relative vertebrae rotations, for which the new approaches generally shows a stiffer behavior.

When changing the “single_surface_original” approach to apply the contact algorithm on segments rather than shell elements, the results seem to indicate that it is not the SPH‐elements that are the main cause for the difference in the head kinematics. This is a very interesting result that would imply both that the SPH‐elements does not influence the general kinematics as much as it could appear initially, but still keep the muscles more tied together, and that the choice of how to implement even the single surface contact does influence the kinematics.

It can finally be concluded that it is possible to use the proposed tiebreak‐ and SPH‐based approaches in order to some extent reduce the separation of the muscles. The reduction is however not very large and the stability of the model is affected negatively, especially for the tiebreak‐based model. The strains in the muscles are also affected since nodes that display large relative sliding are being hindered by the new connections between them.

Final Conclusions

It could be said that the SPH‐based approach gives the most stable results even though it was thought to be much to unstable before implementing it in the full neck model. A big drawback is of course the increased calculation time but there is also a large uncertainty in their ability to give correct results because of the rather unconventional manner in which they are implemented. Examples of this are that the initial geometry should be as symmetric as possible and that the density of the SPH‐particles should be high in proportion to the problem. The particles lie at a fairly random distance from each other and can be very different depending on the direction. The density could also not be considered very high, since in the direction transversely to the thickness of the connective tissue, meaning the gap between the muscles, there are only two particles. It is therefore hard to say if the approach in a correct manner simulates the force transmission that would occur

52

because of the connection between the muscles. The SPH‐approach give lower shearing strength than the tiebreak‐based approach, but it is hard to determine if this is because of inaccuracy in using the SPH‐elements. However, if the main interest would be only to avoid muscle separation the approach is at least able to deliver this and not change the properties of the present model significantly.

Though the two new approaches finally used are able to produce some results for the simulation with the full neck model it might be argued that gain in reduced muscle separation is not as large as the reduction in stability. Furthermore it is hard two determine if the change in muscle strains is physiologically correct. One possibility is however that the constraints put on the surface nodes of the muscles are too severe and causes unnaturally high local strains.

Future Work

The full extent of the problems associated with the tiebreak‐based approach is not known and further investigation on this aspect could perhaps make it more usable, or completely discard it.

A more advanced method of implementing the same idea could be to construct a custom contact algorithm within LS‐DYNA that better mimics the material behavior of the connective tissue with anisotropy and non‐linearity. The same principal idea could also be applied to the SPH‐based approach by constructing a custom material that is applicable to the SPH‐elements. It would also be necessary to find a new way defining the local material axes that is practically implementable.

Another aspect that could also use some additional work is the material description and its parameters. The present models are relatively simplified and not many investigations on this subject have been found in the literature. A more specialized description might be able to better reflect how the muscles are tied together and the forces are transmitted between them.

One aspect that might improve the SPH‐based model is to find a way to fill the space between the muscles so that they are distributed more uniformly and also making it possible to have a higher density of SPH‐elements. Though it would be possible to increase the total amount of SPH‐elements, this might not result in a higher calculation time because of the even distribution. The SPH‐elements could then be connected to the muscle mesh with regular contact algorithms.

A more drastic idea that might better reflect the forces and strain distribution in all part of the neck is an almost complete remodeling. This is based on the fact that in reality there is no empty space in the body as it is modeled today. When looking at a cross‐section of, for example the neck, all points in the neck are occupied by some sort of material and this material merely changes its composition but has no clear separations. Hence another way of constructing the model would be to not base it on separate parts of muscle and bone etc. but to create a single continuous mesh where the material properties are dependent on the position in the model. An issue that immediately arises with this idea is that it would inevitably lead to an increased amount of element needed and the element size is already relatively large in the current model in order to keep the calculation times to a reasonable level. However with this method the elements could be given higher quality in element shape and a more regular element size. Furthermore there would be no need for contact algorithms. Another problem is that it could be difficult to control the transition of material properties in areas where the

53

transition is very abrupt within a small distance, as for example between two muscle separated by a small distance. Though there are elements with the capability of having transitional material properties within themselves, this might not be enough and element size might become very small. But perhaps one of the biggest issues is that the combination of large deformations and a continuous mesh would cause the element distortion to increase, which is already a source of problems. Many of these problems might however be countered if the ordinary FE‐mesh were replaced with a SPH‐mesh. Firstly the initial distribution of the SPH‐elements could be kept very even because only the outer surface would be irregular. This could probably keep down the increase in calculation time caused by the usage of SPH‐elements. Further, the transition of material could in this case be made easier by controlling the material parameter and the smoothing length of the elements. However the largest gain lays in the fact that element distortion would no longer be a problem and it is even possible that this could keep the calculation times within the same levels as in the current model and make it more stable.

There is a need for further investigation on this subject, both in the aspect of calculation time in practical implementation but also theoretical possibilities and functionality. First after this it would be possible to understand if there is a way to implement these ideas. It could on the other hand be questioned how large the impact would be on a final model, and what gains could actually be made from this, something that would have to be considered first.

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Maas, H., Baan, G.C., and Huijing, P. a (2001). Intermuscular interaction via myofascial force transmission: effects of tibialis anterior and extensor hallucis longus length on force transmission from rat extensor digitorum longus muscle. Journal of Biomechanics 34, 927–940.

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Purslow, P.P. (2002). The structure and functional significance of variations in the connective tissue within muscle. Comparative Biochemistry and Physiology. Part A, Molecular & Integrative Physiology 133, 947–966.

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Appendix A – MATLAB code for finding nodes to be included in the tiebreak contact

clc clear close all elements = importdata('surface_elements_full.k'); nodes = importdata('nodes_full.k'); s = unique(round(nodes(:,1)/10000)); s = [s;s(end)+1]; %part number n = s*10000; maxd = 0.003; %max distance between nodes to be set in contact % must be 10 symbols sfm = ' 1'; sfs = ' 1'; sldstf = ' 5.625e3'; a=1; b=1; for y=1:length(n)‐1 %A finds the positions in vector "nodes" within one part. A=find( nodes(:,1)>=n(y) & nodes(:,1)<n(y+1) ); B=find( nodes(:,1)>n(y+1)); %Finds the nodes in one part within specified distance to the other %nodes for t=1:length(A); for i=1:length(B), d = sqrt((nodes(A(t),2)‐nodes(B(i),2))^2+(nodes(A(t),3)... ‐nodes(B(i),3))^2+(nodes(A(t),4)‐nodes(B(i),4))^2); if d<maxd nearnodesA(a,y) = nodes(A(t)); nearnodesB(b,y) = nodes(B(i)); a=a+1; b=b+1; end end end end contactsegmentATOT=[]; contactsegmentBTOT=[]; w=1; q=1; f=1; for x=1:size(nearnodesA,2) rowsA =[]; nearnodesAx=nearnodesA(:,x); NearnodesAx=nearnodesAx(nearnodesAx>0); if NearnodesAx~=0

57

contactrowsA=[]; for i=1:length(NearnodesAx) [rowsA column] = find(elements==NearnodesAx(i)); if length(rowsA)<7 rowsA(7)=0; end contactrowsA(:,i) = rowsA; end contactrowsA=(unique(contactrowsA)); contactrowsA=contactrowsA(contactrowsA>0); contactsegmentA = elements(unique(contactrowsA),3:10); a=contactsegmentA/10000; a=(round(a(:,1))); part = num2str(s(x)); part = ['part_' part(1:3)]; rowsB =[]; nearnodesBx=nearnodesB(:,x); NearnodesBx=nearnodesBx(nearnodesBx>0); contactrowsB=[]; for i=1:length(NearnodesBx) [rowsB column] = find(elements==NearnodesBx(i)); if length(rowsB)<6 rowsB(6)=0; end contactrowsB(:,i) = rowsB; end contactrowsB=(unique(contactrowsB)); contactrowsB=contactrowsB(contactrowsB>0); contactsegmentB = elements(unique(contactrowsB),3:10); [rowsBB colum] = find(ismember(contactsegmentB(:,1:4),contactsegmentBTOT)); [rowsBA colum]= find(ismember(contactsegmentB(:,1:4),contactsegmentATOT)); duplicatesBB = unique(rowsBB); duplicatesBA = unique(rowsBA); contactsegmentB(duplicatesBB,:) = []; contactsegmentB(duplicatesBA,:) = []; contactsegmentBTOT(q:size(contactsegmentB,1)+q‐1,1:4)=contactsegmentB(:,1:4); q = q+size(contactsegmentB,1); [rowsAA colum] = find(ismember(contactsegmentA(:,1:4),contactsegmentATOT));%onödig [rowsAB colum] = find(ismember(contactsegmentA(:,1:4),contactsegmentBTOT)); duplicatesAA = unique(rowsAA); duplicatesAB = unique(rowsAB); contactsegmentA(duplicatesAA,:) = []; contactsegmentA(duplicatesAB,:) = []; contactsegmentATOT(w:size(contactsegmentA,1)+w‐1,1:4)=contactsegmentA(:,1:4); w = w+size(contactsegmentA,1); a=contactsegmentB/10000; a=(round(a(:,1)));

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tiebreak = ['*CONTACT_TIEBREAK_SURFACE_TO_SURFACE_ID\n'... '$# cid title\n'... ' ',part(6:8),'\n'... '$# ssid msid sstyp mstyp sboxid mboxid spr mpr\n'... ' 1',part(7:8),' ',part(6:8),' 0 0 0 0 1 1\n'... '$# fs fd dc vc vdc penchk bt dt\n'... ' 0.000 0.000 0.000 0.000 20.00 0 0.0001.0000E+20\n'... '$# sfs sfm sst mst sfst sfmt fsf vsf\n',... sfs,sfm,' 0.000 0.000 1.000000 1.000000 1.000000 1.000000\n'... '$# nfls sfls tblcid thkoff\n'... ' 1e20 1e20 0 0\n'... '$# soft sofscl lcidab maxpar sbopt depth bsort frcfrq\n'... ' 0 0.1 0 1.025000 2.000000 2 0 1\n'... '$# penmax thkopt shlthk snlog isym i2d3d sldthk sldstf\n'... ' 0.000 0 0 0 0 0 0',sldstf,'\n'... '$# igap ignore dprfac dtstiff unused unused flangl cid_rcf\n'... ' 1 0 0.000 0 0 0 0.000 0\n']; headingA = ['*SET_SEGMENT\n',... '$# sid da1 da2 da3 da4 solver\n',... ' ',part(6:8),' 0.000 0.000 0.000 0.000MECH\n',... '$# n1 n2 n3 n4 a1 a2 a3 a4\n']; headingB = ['*SET_SEGMENT\n',... '$# sid da1 da2 da3 da4 solver\n',... ' 1',part(7:8),' 0.000 0.000 0.000 0.000MECH\n',... '$# n1 n2 n3 n4 a1 a2 a3 a4\n']; a0=[num2str(contactsegmentA)]; w=1; for i =1:size(a0,1) a1 = a0(i,:); stringadA(w:w+82‐1) = [' ',a1(1:9),' ',a1(10:18),' ',a1(19:27),' ',a1(28:36),... ' ',a1(37:45),' ',a1(46:54),' ',a1(55:63),' ',a1(64:end),'\n']; w=w+82; end b0=[num2str(contactsegmentB)]; w=1; for i =1:size(b0,1) b1 = b0(i,:); stringadB(w:w+82‐1) = [' ',b1(1:9),' ',b1(10:18),' ',b1(19:27),' ',b1(28:36),... ' ',b1(37:45),' ',b1(46:54),' ',b1(55:63),' ',b1(64:end),'\n']; w=w+82; end asemb = [tiebreak headingA,stringadA,headingB,stringadB]; Asemb(f:length(asemb)+f‐1) = asemb; f=f+length(asemb);

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clear stringadA clear stringadB end end END =['*END']; Asemb(1:length(Asemb)+4) = [Asemb,END]; fid = fopen(['tiebreak_settup.k'], 'w'); fprintf(fid,Asemb); fclose(fid); %% %Left side muscles clc clear close all elements = importdata('surface_elements_full_2.k'); nodes = importdata('nodes_full_2.k'); s = unique(round(nodes(:,1)/10000)); s = [s;s(end)+1]; n = s*10000; maxd = 0.003; %max distance between nodes to be set in contact % must be 10 symbols sfm = ' 1'; sfs = ' 1'; sldstf = ' 5.625e3'; a=1; b=1; for y=1:length(n)‐1 %A finds the positions in vector "nodes" within one part. A=find( nodes(:,1)>=n(y) & nodes(:,1)<n(y+1) ); B=find( nodes(:,1)>n(y+1)); %Finds the nodes in one part within specified distance to the other %nodes for t=1:length(A); for i=1:length(B), d = sqrt((nodes(A(t),2)‐nodes(B(i),2))^2+(nodes(A(t),3)... ‐nodes(B(i),3))^2+(nodes(A(t),4)‐nodes(B(i),4))^2); if d<maxd nearnodesA(a,y) = nodes(A(t)); nearnodesB(b,y) = nodes(B(i)); a=a+1; b=b+1; end end end end

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contactsegmentATOT=[]; contactsegmentBTOT=[]; w=1; q=1; f=1; for x=1:size(nearnodesA,2) rowsA =[]; nearnodesAx=nearnodesA(:,x); NearnodesAx=nearnodesAx(nearnodesAx>0); if NearnodesAx~=0 contactrowsA=[]; for i=1:length(NearnodesAx) [rowsA column] = find(elements==NearnodesAx(i)); if length(rowsA)<7 rowsA(7)=0; end contactrowsA(:,i) = rowsA; end contactrowsA=(unique(contactrowsA)); contactrowsA=contactrowsA(contactrowsA>0); contactsegmentA = elements(unique(contactrowsA),3:10); a=contactsegmentA/10000; a=(round(a(:,1))); part = num2str(s(x)); part = ['part_' part(1:3)]; rowsB =[]; nearnodesBx=nearnodesB(:,x); NearnodesBx=nearnodesBx(nearnodesBx>0); contactrowsB=[]; for i=1:length(NearnodesBx) [rowsB column] = find(elements==NearnodesBx(i)); if length(rowsB)<6 rowsB(6)=0; end contactrowsB(:,i) = rowsB; end contactrowsB=(unique(contactrowsB)); contactrowsB=contactrowsB(contactrowsB>0); contactsegmentB = elements(unique(contactrowsB),3:10); [rowsBB colum] = find(ismember(contactsegmentB(:,1:4),contactsegmentBTOT)); [rowsBA colum]= find(ismember(contactsegmentB(:,1:4),contactsegmentATOT)); duplicatesBB = unique(rowsBB); duplicatesBA = unique(rowsBA); contactsegmentB(duplicatesBB,:) = []; contactsegmentB(duplicatesBA,:) = []; contactsegmentBTOT(q:size(contactsegmentB,1)+q‐1,1:4)=contactsegmentB(:,1:4);

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q = q+size(contactsegmentB,1); [rowsAA colum] = find(ismember(contactsegmentA(:,1:4),contactsegmentATOT));%onödig [rowsAB colum] = find(ismember(contactsegmentA(:,1:4),contactsegmentBTOT)); duplicatesAA = unique(rowsAA); duplicatesAB = unique(rowsAB); contactsegmentA(duplicatesAA,:) = []; contactsegmentA(duplicatesAB,:) = []; contactsegmentATOT(w:size(contactsegmentA,1)+w‐1,1:4)=contactsegmentA(:,1:4); w = w+size(contactsegmentA,1); a=contactsegmentB/10000; a=(round(a(:,1))); tiebreak = ['*CONTACT_TIEBREAK_SURFACE_TO_SURFACE_ID\n'... '$# cid title\n'... ' 2',part(6:8),'\n'... '$# ssid msid sstyp mstyp sboxid mboxid spr mpr\n'... ' 21',part(7:8),' 2',part(6:8),' 0 0 0 0 1 1\n'... '$# fs fd dc vc vdc penchk bt dt\n'... ' 0.000 0.000 0.000 0.000 20.00 0 0.0001.0000E+20\n'... '$# sfs sfm sst mst sfst sfmt fsf vsf\n',... sfs,sfm,' 0.000 0.000 1.000000 1.000000 1.000000 1.000000\n'... '$# nfls sfls tblcid thkoff\n'... ' 1e20 1e20 0 0\n'... '$# soft sofscl lcidab maxpar sbopt depth bsort frcfrq\n'... ' 0 0.1 0 1.025000 2.000000 2 0 1\n'... '$# penmax thkopt shlthk snlog isym i2d3d sldthk sldstf\n'... ' 0.000 0 0 0 0 0 0',sldstf,'\n'... '$# igap ignore dprfac dtstiff unused unused flangl cid_rcf\n'... ' 1 0 0.000 0 0 0 0.000 0\n']; headingA = ['*SET_SEGMENT\n',... '$# sid da1 da2 da3 da4 solver\n',... ' 2',part(6:8),' 0.000 0.000 0.000 0.000MECH\n',... '$# n1 n2 n3 n4 a1 a2 a3 a4\n']; headingB = ['*SET_SEGMENT\n',... '$# sid da1 da2 da3 da4 solver\n',... ' 21',part(7:8),' 0.000 0.000 0.000 0.000MECH\n',... '$# n1 n2 n3 n4 a1 a2 a3 a4\n']; a0=[num2str(contactsegmentA)]; w=1; for i =1:size(a0,1) a1 = a0(i,:); stringadA(w:w+82‐1) = [' ',a1(1:9),' ',a1(10:18),' ',a1(19:27),' ',a1(28:36),... ' ',a1(37:45),' ',a1(46:54),' ',a1(55:63),' ',a1(64:end),'\n']; w=w+82; end

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b0=[num2str(contactsegmentB)]; w=1; for i =1:size(b0,1) b1 = b0(i,:); stringadB(w:w+82‐1) = [' ',b1(1:9),' ',b1(10:18),' ',b1(19:27),' ',b1(28:36),... ' ',b1(37:45),' ',b1(46:54),' ',b1(55:63),' ',b1(64:end),'\n']; w=w+82; end asemb = [tiebreak headingA,stringadA,headingB,stringadB]; Asemb(f:length(asemb)+f‐1) = asemb; f=f+length(asemb); clear stringadA clear stringadB end end END =['*END']; Asemb(1:length(Asemb)+4) = [Asemb,END]; fid = fopen(['tiebreak_settup_2.k'], 'w'); fprintf(fid,Asemb); fclose(fid);

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Appendix B – MATLAB code for finding nodes to be defined as SPH­elements

clc clear close all nodes = importdata('nodes.k'); s = unique(round(nodes(:,1)/10000)); s = [s;s(end)+1]; n = s*10000; dmax = 0.003; %max distance between nodes to be set in contact v=1; for y=1:length(n)‐1 %A & B finds the positions in vector nodes within one part A=find( nodes(:,1)>=n(y) & nodes(:,1)<n(y+1) ); for x=1:length(n)‐1 B=find( nodes(:,1)>=n(x) & nodes(:,1)<n(x+1) ); if A(1)==B(1) break end for t=1:length(A); for i=1:length(B), d = sqrt((nodes(A(t),2)‐nodes(B(i),2))^2+(nodes(A(t),3)... ‐nodes(B(i),3))^2+(nodes(A(t),4)‐nodes(B(i),4))^2); if d<dmax p_a(v) = A(t); p_b(v) = B(i); v=v+1; end end end end end rho = 1000; nnodes = 1122; V=0.005*0.005*0.0005; mass=rho*V/4; mass = 6.0e‐006; red_nodes = unique(nodes([p_a, p_b])); red_nodes(2,:) = 999; red_nodes(3,:) = mass; fid = fopen('red_sph.k', 'w'); fprintf(fid, '%8.0f%8.0f %15.3e\n', red_nodes); fclose(fid);

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Appendix C – Maximum displacements for two muscle model Table 8. Maximum z‐displacement for all modeling approaches used in the two muscle model with 2.6 N z‐loading.

Modeling approach Penalty formulation for single surface contact

Young’s modulus Ep; Eab; Ec /kPa

Z‐displacement (absolute value)/m

tiebreak_Ep

‐ 0.9 0.1640 2.1 0.1393 21.6 0.0989

tiebreak_singlesurf_Ep Segment‐based

0.9 0.1742 2.1 0.1538 21.6 0.1144

tiebreak_singlesurf_segments_Ep Standard

0.9 0.1475 2.1 0.1316 21.6 0.0984

SPH_isotropic_singlesurf_segments_red_Ep Standard

0.9 0.2106 2.1 0.0849 21.6 0.1015

SPH_isotropic_singlesurf_segments_red_Ep Segment‐based

0.9 0.0912 2.1 0.0849 21.6 0.0495

SPH_isotropic_singlesurf_red_Ep Segment‐based 0.9 0.1208 SPH_isotropic_singlesurf_segments_full_Ep Segment‐based 0.9 0.0690 SPH_orthotropic_Eab_Ec_red ‐ ‐ ; 0.9; 21.6 0.1532

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Table 2. Maximum y‐displacement for all modeling approaches used in the two muscle model with 2.8 N bending.

Modeling approach Penalty formulation for single surface contact

Young’s modulus Ep; Eab; Ec /kPa

Y‐displacement (absolute value)/m

single_surface_original Segment‐based ‐ 0.0750 single_surface_nofric Segment‐based ‐ 0.0846 tiebreak_Ep

‐ 0.9 0.0727 2.1 0.0673 21.6 0.0566

tiebreak_singlesurf_Ep Standard

0.9 0.0684 2.1 0.0650 21.6 0.0588

tiebreak_singlesurf_segments_Ep Standard

0.9 0.0649 2.1 0.0626 21.6 0.0565

tiebreak_singlesurf_segments_Ep Segment‐based 0.9 0.0637 2.1 0.0613 21.6 0.0537

SPH_isotropic_singlesurf_segments_red_Ep Standard

0.9 0.0747 2.1 0.0743 21.6 0.0641

SPH_isotropic_singlesurf_segments_red_Ep Segment‐based

0.9 0.0734 2.1 0.0727 21.6 0.0547

SPH_isotropic_singlesurf_red_Ep Standard 0.9 0.0744 SPH_isotropic_singlesurf_segments_full_Ep Standard 0.9 0.0760 SPH_orthotropic_Eab_Ec_red ‐ ‐ ; 0.9; 21.6 0.0770

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Appendix D – Figures from the two muscle model

Figure 48. Cross‐section of the two muscle model showing the interpenetration of the two muscles from the 2.6 N z‐loading simulation using the ”tiebreak_0.9 kPa” approach.

Figure 49. Cross‐section of the two muscle model showing the interpenetration of the two muscles from the 2.6 N z‐loading simulation using the ”tiebreak_2.1 kPa” approach.

Figure 50. Cross‐section of the two muscle model showing the interpenetration of the two muscles from the 2.6 N z‐loading simulation using the ”tiebreak_21.6 kPa” approach.

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Figure 51. Cross‐section of the two muscle model showing the interpenetration of the two muscles from the bending simulation using the ”tiebreak_0.9 kPa” approach.

Figure 52. Cross‐section of the two muscle model showing the interpenetration of the two muscles from the bending simulation using the ”tiebreak_2.1 kPa” approach.

Figure 53. Cross‐section of the two muscle model showing the interpenetration of the two muscles from the bending simulation using the ”tiebreak_21.6 kPa” approach.

Figure 54.

Deformation ffrom loading in

n z‐direction wi

68

ith 2.6N using t

the “SPH_orthootropic_0.9kPaa_21.6kPa” appproach.

69

Appendix E – Relative vertebrea rotations

Figure 55. The relative y‐rotation in degrees between a vertebra and underlying vertebra, ranging from C3 to T1 for the “tiebreak_singlesurf_segments_0.9kPa” approach and the “single_surface_original” approach.

0 0.1 0.2 0.3 0.4-20

-10

0

10C3-C4

Rel

ativ

e y-

rota

tion

[Deg

]

0 0.1 0.2 0.3 0.4-15

-10

-5

0

5C4-C5

0 0.1 0.2 0.3 0.4-20

-10

0

10C5-C6

0 0.1 0.2 0.3 0.4-20

-10

0

10C6-C7

0 0.1 0.2 0.3 0.4-10

-5

0

5C7-T1

time [s]

single_surface_originaltiebreak_singlesurf_segments_0.9 kPaStandard penalty formulation

70