finite element method(s-fem)

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1 APCOM & ISCM th December, 013, Sigapore Simulatio of passive myocardium of rabbit vetricles, usig selective smoothed fiite elemet method(s-fem) C. Jiag 1, Zhi-Qia Zhag, *X. Ha 1, G.R. Liu 3 1 State Key Laboratory of Advaced Techology of Desig ad Maufacturig for Vehicle Body, Hua Uiversity, P.R. Chia, Istitute of High Performace Computig, A*STAR, Sigapore, Departmet of Aerospace Egieerig ad Egieerig Mechaics, Uiversity of Ciciati, 851 Woodside Dr, Ciciati, OH 451, USA *Correspodig author: haxu@hu.edu.c Abstract A selective smoothed fiite elemet method (Selective S-FEM) is eloped for dyamic 3D aalysis of extremely large deformatio of icompressible bio-tissues, usig the simplest 4-ode tetrahedro elemets. I the preset selective S-FEM, face-based Smoothed FEM (FS-FEM) ad ode-based Smoothed FEM (NS-FEM) are, respectively, used for the iatoric ad volumetric parts of the deformatio of icompressible bio-tissues. Therefore the overly-soft feature of the NS-FEM is made use of for mitigatig the volumetric lockig that may occur i bio soft tissue. Because the FS-FEM ca provide close-to-exact stiffess, our selective S-FEM ca provide a accurate solutio. I the curret formulatio, the soft tissue is modeled usig the hyperelastic costitutive law. Numerical examples are preseted to simulate a passive fillig process of aatomical rabbit vetricles. It is demostrated that the Selective S-FEM possess good potetial for accurately simulatig the behavior of bio-tissues for reliable solutio.. Keywords: Fiite Elemet Method, Smoothed Fiite Elemet Method, Gradiet Smoothig, Icompressibility, Aisotropy, Myocardium, Tetrahedral, Large Deformatio, Explicit time itegratio Itroductio The explicit dyamic Fiite Elemet Method (FEM) has bee successfully applied to solve trasiet oliear resposes of various material ad structural systems with large deformatio ad strai, impact-cotact ad metal formig problems i automotive, aerospace, ad bioegieerig (Goudreau ad Hallquist 198; Belytschko, Liu, ad Mora 000; Miller et al. 006). I explicit dyamics aalyses usig FEM, four-ode quadrilateral elemet (Q4) ad eight-ode hexahedro elemet (H8) are the most frequetly used with sigle quadrature poit for efficiecy, which requires a hourglass cotrol to remedy hourglass istability (spurious zero-eergy modes) (Belytschko, Liu, ad Mora 000). I additio, the oe-poit quadrature techique ca also mitigate the volumetric lockig i early icompressible solids. Whe quadrilateral ad hexahedral elemets are used, it is laborious ad time-cosumig i preprocessig ad the remeshig for distorted elemets, because of the difficulties i automatically meshig with these types of elemets It is much easier to automatically create ad refie, whe usig Triagular ad Tetrahedral mesh (T-mesh) are used i complicated geometry. I fact, there are ow may commercial ad ope-source softwares, have bee eloped for automotive geeratig T-meshes. However, i the FEM based o the stadard weak formulatio, the performace of T-mesh is usually very poor for its overly stiff behavior. This is i particular true for icompressible solids where volumetric lockig may occur. 1

2 May efforts have bee made to ehace the capability of T-mesh i hadlig the icompressibility, icludig the F-bar method (De Souza Neto et al. 1996), the method usig hydrostatic pressure as a additioal idepedet variable (Ziekiewicz et al. 1998), odal pressure average treatmet (Boet ad Burto 1998), Hu-Washizu three fields variatioal theorem (Taylor 000), ad odal averagig treatmet of deformatio gradiet tesor (Boet, Marriott, ad Hassa 001). The odebased strai smoothig has also bee successfully applied to meshfree methods for stabilizatio (Che, Yoo, ad Wu 00). Models usig smoothed strais ca have a theoretical foudatio o G space theory (Liu ad Zhag 009; Liu 009; Liu ad Zhag 013). Because the G space theory allows the use of discotiuous fuctios, such a formulatio is also termed as weakeed weak (or W) forms. Typical W formulatio method is the Smoothed Poit Iterpolatio Methods (S-PIM) (Liu ad Zhag 013). Whe fuctios i a FEM space are used i a W formulatio, the so-called the smoothed fiite elemet method or S-FEM. I terms of the way to costruct the smoothig domais, S-FEM ca have a family of methods of uique properties, which ca be classified as cell-based smoothed FEM (CS-FEM) (Liu, Dai, ad Nguye 006), ode-based smoothed FEM (NS-FEM) (Nguye-Thoi et al. 010), edge-based S-FEM (ES-FEM) (Liu, Nguye-Thoi, ad Lam 009) for D problems, ad face-based S-FEM (FS-FEM) for 3D problems (Nguye-Thoi et al. 009). I these S-FEM methods, the NS-FEM has a uique property of volumetric lockig free, due to its strog softeig effects. However, spurious o-zero-eergy eige-modes ad temporal istabilities arise for NS-FEM, despite the fact that it is prove spatially stable (Liu 009; Liu, Dai, ad Nguye 006). Differig from NS-FEM, the ES-FEM or FS-FEM that uses edge-based or face-based smoothig domais are foud both spatial ad temporal stable, because of usig more smooth domais. However, ES-FEM or FS-FEM usually produces slightly overestimated stiffess, ad hece it ca also suffer from the volumetric lockig. A method usig selective gradiet-smoothig techique, so called Selective S- FEM, has bee proposed i order to elimiate volumetric lockig, at the same time, improve the performace of the simplex TRI3 ad TET4 elemets. The employmet of the combied edgebased ad ode-based smoothig operatios eables effective hadlig of elemet distortio i extremely large deformatio with low computatio cost. Biological tissues, like ski, myocardium, arterial layer, ca ofte treated as hyperelastic material. This is because they ca udergo very large deformatio, ad after uloadig, they ca recover to origial state. These strai eergy fuctios ca be divided ito two groups by their variables. The first group is the strai-based strai eergy fuctios, such as the Fug form for arterial layer (Chuog ad Fug 1986) ad the McCulloch expoetial form for myocardium (Vetter ad McCulloch 000), which regard strais as idepedet variables. The aother group is the ivariat-based strai eergy fuctios, like the Hozapfel-Gasser-Odge form for arterial layer (Gasser, Ogde, ad Holzapfel 006), the Li form (Li ad Yi 1998) for rabbit heart, ad the Ogde form (Ogde 197) for extremely large deformatio. I this paper, we first itroduce the basic formulatios of explicit dyamic oliear Selective S- FEM i Sectio 1, ad the briefly review the strai eergy fuctios of bio-tissues i Sectio. Followig umerical examples of trasversely isotropic hyperelastic plate ad passive bi-vetricles i Sectio 3, coclusios are draw i Sectio 4. Explicit Dyamic oliear Selective S-FEM I dyamic aalysis, iertia ad dampig effects must be cosidered, ad the discrete form of the goverig equatio ca be give i geeral as follow

3 where u is odal displacemets vector, ad T C N cn d ( c is dampig coefficiet vector), Mu Cu F F (1) it ext T M N ρn d ( ρ is mass desity vector), F N b N t. T T ext d d t t I geeral S-FEM, odal iteral forces should be computed from smoothed strai matrix ad smoothed stress, T Fit B σ d () Figure 1. Cofiguratio ad motio of a cotiuum body, ad smoothig domais I this paper, total Lagrage formulatio is employed to solve geometrical oliearity. The iitial referece cofiguratio of material poit i a body is deoted by X, ad displacemet at time t is deoted as u the the curret deformed cofiguratio is expressed as x X u (3) There is the deformatio gradiet tesor of X which is the measure of deformatio from referece cofiguratio, as i Figure 1. It is give by x F (4) X I our S-FEM, the deformatio gradiet tesor eeds also be writte i smoothed versio, as below s F ij F 0 e, 0 s ij L d u 0 0 s i s s jd ij ij L ij A X L L L A X (5) L where 0 A is the area of iitial smooth domais. s L For hyperelastic icompressible biology material, the correspodig strai eergy fuctio is ofte split ito volumetric part ad iatoric part. The smoothed secod Piola-Kirchoff (PK) stress tesor ca be calculated by give strai eergy fuctio ad smoothed deformatio gradiet as follows vol S ij NS-FEM FS-FEM E ij C ij C ij C (6) ij where E ij is the Gree strai tesor, the smoothed right Cauchy-Gree strai tesor C ij F kif kj. After gettig the PK stress tesor, substitute it ito Eq.() to get iteral odal force vector. I the Selective S-FEM, its strai smoothig is performed usig a combiatio of edge-based smoothig i D problem (face-based smoothig i 3D) ad ode-based smoothig. These two types of smooth domais i Selective S-FEM used here are plotted i Figure. I Selective S-FEM, volumetric part of PK stress is computed by the over-softly NS-FEM, ad the iatoric part of 3 Figure. Illustratios of smoothig domais of (left) FS-FEM-TET4 (right) NS-FEM- TET4

4 PK stress is hadled by more precise ES-FEM usig smoothed deformatio gradiet tesor from Eq.(5). For the time itegratio, the well-established explicit cetral differetial method is used. Although explicit scheme is coditioal stable, it is much easier i programmig ad o eed to form the global taget stiffess matrix. The time icremets are defied as 1/ 1, 1/ t t t t t t /, 1/ t t t 1/. (7) The displacemet ad velocity ca be update by 1 1/ u u v t 1/, (8) 1/ 1/ v v t a, (9) The acceleratio a u is obtaied by solvig the followig equatio Ma f ext u, t f it u, t Cv. (10) If a lumped mass matrix is employed i Eq.(10), M becomes a diagoal matrix, ad a ca be computed via trivial operatios without ivokig a liear algebra equatio solver. Hyperelastic strai eergy fuctios for bio-tissue Hyperelastic costitutive laws are ofte employed to model the mechaical respose of bio-tissues. Based o experimetal data, variety of isotropic ad aisotropic strai eergy fuctios have bee proposed i recet years. I all these forms of strai eergy fuctio, they ca be grouped ito two categories, i terms of their idepedet variables. The first group is the strai-based forms group, which is expressed directly i terms of the compoets of suitable strai tesor (ofte is Gree strai tesor E ).The geeral decoupled form of this group is give below vol ( E ). (11) I the begiig of the elopmet of biomechaics, almost all strai eergy fuctios are formed usig Gree strai tesor, like Fug s form (Chuog ad Fug 1986), Pole-zero form (Nash ad Huter 000), ad McCulloch form (Vetter ad McCulloch 000). Although these forms are the aturally trasited from elastic costitutive law, they are borig i fidig the material parameters ad costructig the local material coordiate systems for their eeds of a orthogoal coordiate system. Aother group is the ivariat-based forms group, strai eergy fuctios are expressed i terms of ivariats of right Cauchy-Gree strai tesor I 1, I ad fiber directios I i (i=4,5, ), ad the geeral form is give as below vol I1, I, J, I4,... I1, I, I4,... J, (1) where J is the volume ratio ad J I3. Note that i S-FEM, all ivariats should use the smoothed versio. Material parameters of ivariat-based form are less tha strai-based form ad ofte just eed the fiber orietatio. With these advatages, this type is more ad more popular i biomechaics, like the recetly proposed Holzapfel-Gasser-Odge (H.G.O) form (Gasser, Ogde, ad Holzapfel 006) for advetitial layers. There is also a group of strai eergy fuctio i terms of pricipal stretches, such as the Ogde form (Ogde 197). Although they should be categorized i aother group for its variables, the pricipal stretches ca be derived form ivariats of right Cauchy-Gree strai tesor, so we regard they are just special versio of the ivariat-based forms. 4

5 Numerical examples 1. Trasversely isotropic hyperelastic plate uder uiform pressure Figure 3. (a) Problems settigs of example 3, regular mesh (b) ad irregular mesh (c) at iitial cofiguratio The model cofiguratio of oe family of fiber-reiforced hyperelastic plate is showed i Figure 3, where L = 1m, W = 1m, H = 0.1m, ad a is the uit vector i fiber orietatio. The regular mesh ad oe kid of irregular mesh are plotted i Figure 3(b)(c). Nearly-icompressible Mooey-Rivli model is employed as the isotropic part of strai eergy with the parameters: C10 1kPa, 3 C01 0.5kPa, 1000mPa ad desity kg/m. Weiss polyomial form (Weiss, Maker, ad Govidjee 1996) is employed as iatoric part of strai eergy fuctio with parameter: A1 5kPa. This strai eergy fuctio ca be writte i followig form. vol I1, I, J, I4,... iso I1, I ai I1, I, I4,... J, A1 (13) iso I1, I C01( I1 3) C10 I 3, ai I1, I, I4, I5 I4 1. The boudary coditio ad iitial coditio are give as follows. Iitial coditio: v 0 0 ad 0 u 0. Boudary coditio: o the four lateral surface u x u y uz 0. Loadig: o the upper surface, a give uiform pressure is applied. This example is aimed to illustrate the ability S-FEM to solve large aisotropic deformatio caused by trasversely isotropic costitutive law. Figure 4. Deformed cofiguratios for regular (above) ad irregular meshed (below) example 1 with differet fiber directios Explicit aalyses with covetioal dyamic relaxatio techique are oly performed usig Selective S-FEM with 7058 TET4 elemets (5088 odes) to achieve the quasi-static solutios. Five differet fiber orietatios cases are calculated here, ad the correspodig orietatio vectors 5

6 are (cos 0,si 0,0), (cos30,si 30,0), (cos 45,si 45,0), (cos 60,si 60,0), ad (cos90,si 90,0). The deformatio cofiguratios of differet cases are plotted i Figure 4. It is obvious that differet embedded fiber directio will lead to a quite differet deformatio. Moreover, accordig to defiitio of trasversely isotropic, deformatio cofiguratios of cases with fiber directio (cos 0,si 0,0) ad (cos90,si 90,0) should be idetical, so as to (cos30,si 30,0) ad (cos 60,si 60,0). For clear demostratio of this pheomeo, displacemets of Z-axis i the two upper face diagoal lies are plotted i Figure 5. I Figure 5, odes with the same distace to the ceter of plate will have the same deflectios i every differet fiber directio cases. Also i this figure, results of regular ad irregular mesh are matched very well. This example demostrates that the preset selective S-FEM usig TET4 is capable to hadle the trasversely isotropic hyperelastic material i large deformatio. Figure 5. The deflectios o the diagoal lie (0, 0, 0.1) (1, 1, 0.1) (left) ad (1, 0, 0.1) (0, 1, 0.1) (right) versus distaces to poit (0, 0) of regular ad irregular meshed cases with differet fiber orietatios.. Passive fillig of rabbit vetricles Heart, as the most importat orga i ay species of aimals, pumps blood to other orgas ad muscles by repeated, rhythmic cotractios ad expasios. Two phases ca be divided i the cardiac cycle; oe is the cotractio period, referred as vetricular systole, aother is the vetricular diastole which the heart is relaxed ad refilled for the ext cycle. This heart pump depeds maily o the active mechaical property of left vetricle myocardium. Here for the sake of simplicity, the passive fillig process is 6 Figure 6. Stress-strai curves of fiber, cross fiber ad fitted Ogde forms. calculated to approximate the diastole phase, because of its easier passive mechaical property ad loadig. Usually, myocardium is fiber-reiforced, ad its fibers are complex distributed. Accordig to referece (Li ad Yi 1998) ad referece (Vetter ad McCulloch 000), they provide a ivariat-based trasversely isotropic strai eergy fuctio for rabbit myocardium ad correspodig equibiaxial stress-strai curves. Luckily, because the correlatio of the fiber ad crossfiber strai-stress curves is , aisotropy ca be igored. Igorig the aisotropy of fiber architecture really ca reduce may works o recostructig the fiber orietatios. We fit these two curves ito Ogde form strai eergy fuctio (Ogde 197) as i Figure 6. I Figure 6, the

7 d ad 4th order Ogde forms ca well approximate the curves i the fiber directio ad cross fiber directio. For the sake of less parameters, the d Ogde form here is employed as the costitutive law, the decoupled form is N i 1 i i i ( J 1) (14) i1 i 1/3 where modified pricipal stretches a J a ( a 1,,3), N= for d Ogde form, i (Pa) ad are the material coefficiets, is the bulk modulus. Here, for the rabbit myocardium, i Pa, , 5.96Pa, 5.0, ad Pa. To validate our Selective S-FEM, stadard FEM with TET10 elemets is employed here to do a compariso. The rabbit vetricle model is discretized by 3914 elemets, 9069 odes for Selective S-FEM, 639 odes for FEM, see it i Figure 7 (left). Boudary coditio: LV ad RV edocardial base is costraied the logitude (X-directio) displacemets, epicardial base are fixed costraied. Loadig: The 5mmHg pressure is smoothly applied o the LV surface to simulate the passive fillig process, ad the RV is uloaded here. A slice of rabbit vetricles i X-Y plae is drew i Figure 7 (Right) as a simple validatio of TET4 Selective S-FEM ad TET10 FEM. I Figure 7 (Right), the blue dash curve is the udeformed outlie of TET10 FEM slice, the red dot curve is the deformed outlie of TET10 FEM slice, but the cotour of displacemet magitude belogs to TET4 FS/NS-FEM. The two deformed outlie match with each other very well, except some mismatches i the septum. This example shows the ear quadratic tetrahedro elemet accuracy of Selective S-FEM. Figure 7. Mesh of rabbit vetricles (left) ad a slice i X-Y plae (right, outlie belogs to FEM TET10, cotour belogs to FS/NS-FEM). Coclusios I this paper, Selective FS/NS-FEM is applied ito isotropic ad aisotropic icompressible biotissues. Several coclusios ca be derived from this paper. (1) Selective S-FEM usig TET4 elemet ca easily mesh complex geometry, such as vetricles. () FS-FEM is ideal for the iatoric part of deformatio of icompressible isotropic or aisotropic solids, so as to greatly improve the performace of the liear elemets. 7

8 (3) NS-FEM is ideal for the volumetric part of deformatio of icompressible solids, ad ca effectively mitigate the volumetric lockig i icompressible materials, eve if liear elemets are used. (4) Selective S-FEM is capable to hadle the large rotatio of fibers i aisotropic bio-tissues with regular or irregular mesh. (5) Selective S-FEM ca ehace the accuracy of liear TET4 elemet to be closer to quadratic TET10 elemet. Ackowledgmets This work is partially supported by NSF with Grat No awarded to the seior author, ad the A*Star, Sigapore to the d author. It is also partially supported by the Ope Research Fud Program of the State Key Laboratory of Advaced Techology of Desig ad Maufacturig for Vehicle Body, Hua Uiversity, P.R.Chia uder the grat umber Refereces Belytschko, Ted, W.K Liu, ad B. Mora. (000). Noliear Fiite Elemets for Cotiua ad Structures. Wiley. Boet, J., ad A. J. Burto. (1998). A Simple Average Nodal Pressure Tetrahedral Elemet for Icompressible ad Nearly Icompressible Dyamic Explicit Applicatios. Commuicatios i Numerical Methods i Egieerig. 14. (5).pp: Boet, J., H. Marriott, ad O. Hassa. (001). A Averaged Nodal Deformatio Gradiet Liear Tetrahedral Elemet for Large Strai Explicit Dyamic Applicatios. Commuicatios i Numerical Methods i Egieerig. 17. (8).pp: Che, Jiu-Shya, Sagpil Yoo, ad Cheg-Tag Wu. (00). No-liear Versio of Stabilized Coformig Nodal Itegratio for Galerki Mesh-free Methods. Iteratioal Joural for Numerical Methods i Egieerig. 53. (1).pp: Chuog, C. J., ad Y. C. Fug. (1986). O Residual Stresses i Arteries. Joural of Biomechaical Egieerig ().pp: 189. Gasser, T Christia, Ray W Ogde, ad Gerhard a Holzapfel. (006). Hyperelastic Modellig of Arterial Layers with Distributed Collage Fibre Orietatios. Joural of the Royal Society, Iterface / the Royal Society. 3. (6).pp: Goudreau, G.L., ad J.O. Hallquist. (198). Recet Developmets i Large-scale Fiite Elemet Lagragia Hydrocode Techology. Computer Methods i Applied Mechaics ad Egieerig. 33. (1-3).pp: Li, D. H. S., ad F. C. P. Yi. (1998). A Multiaxial Costitutive Law for Mammalia Left Vetricular Myocardium i Steady-State Barium Cotracture or Tetaus. Joural of Biomechaical Egieerig. 10. (4).pp: 504. Liu, G. R. (009). A G Space Theory ad a Weakeed Weak (W ) Form for a Uified Formulatio of Compatible ad Icompatible Methods: Part II Applicatios to Solid Mechaics Problems. Iteratioal Joural for Numerical Methods i Egieerig.pp: /a /a. Liu, G. R., K. Y. Dai, ad T. T. Nguye. (006). A Smoothed Fiite Elemet Method for Mechaics Problems. Computatioal Mechaics. 39. (6).pp: Liu, G. R., ad G. Y. Zhag. (009). A Normed G Space ad Weakeed Formulatio of a Cellbased Smoothed Poit Iterpolatio Method. Iteratioal Joural of Computatioal Methods. 06. (01).pp: Liu, G.R., T. Nguye-Thoi, ad K.Y. Lam. (009). A Edge-based Smoothed Fiite Elemet Method (ES-FEM) for Static, Free ad Forced Vibratio Aalyses of Solids. Joural of Soud ad Vibratio. 30. (4-5).pp:

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