UAB - ECTC 2014 PROCEEDINGS - Section 5 Page€¦ · Mobile, Alabama 36688 A. -V. Phan†...

SECTION 5

NUMERICAL METHODS

UAB School of Engineering - ECTC 2014 Proceedings - Vol. 13 113

Proceedings of the Fourteenth Annual Early Career Technical ConferenceThe University of Alabama, Birmingham ECTC 2014November 1 – 2, 2014 - Birmingham, Alabama USA

A FINITE ELEMENT MODEL OF THE SYNTHESIS, DEGRADATION, AND SPATIALSPREAD OF cAMP

R. C. SalterDepartment of Mechanical Engineering

University of South AlabamaMobile, Alabama 36688

K. J. WebbCenter for Lung Biology & Department of Pharmacology

College of MedicineUniversity of South Alabama

Mobile, Alabama 36688

A. -V. Phan†

Department of Mechanical EngineeringUniversity of South Alabama


T. C. RichCenter for Lung Biology & Department of Pharmacology

College of MedicineUniversity of South Alabama


ABSTRACTA small number of intracellular signals, including cyclic

adenosine monophosphate (cAMP) signals, control virtually allcellular phenomena. However, it remains unclear how informa-tion is encoded within these signals. In the last decade severalgroups have proposed mathematical descriptions of cAMP sig-nals and signaling pathways to better understand how informa-tion is relayed within cells. Here, we propose a finite elementmodel based upon a linearized version of the equations that gov-ern the synthesis, degradation, and spatial diffusion of cAMP toestimate agonist-induced changes in cAMP within a cell. A sim-plified circular cell was used to numerically test the validity ofthis model. Simulations indicate that cAMP rapidly equilibrateswithin cells with effective diffusion coefficients of 30 µm2/s,that finite element analysis (FEA) approaches will be effectivetools in the study of the spatial distribution of cAMP levels inmore complex cellular geometries, and that these approachesmay allow us to better understand information content of cAMPsignals.

†Corresponding author. Email: [email protected]

INTRODUCTIONThe second messenger cyclic adenosine monophosphate

(cAMP) is produced in cells by an adenylyl cyclase, regulatesa wide variety of cellular processes, and is eventually degradedby phosphodiesterases (PDEs). Pharmacological agents that trig-ger increases in cAMP levels, such as PDE inhibitors and β -adrenergic agonists, are used to treat diseases including pul-monary hypertension, diabetes, cardiac failure, and asthma. Al-ternatively, β -antagonists or β -blockers, which decrease cAMPlevels, are used to treat cardiac arrhythmias, hypertension, mi-graines, and anxiety. While these drugs are used for the treat-ment of many diseases, we do not fully understand the effects ofcAMP signaling.

The concentration of cAMP within a cell can be estimatedusing a nonlinear differential equation. This governing equationpredicts the concentration C = C(t,x,y), where C is the con-centration of cAMP, by considering the synthesis, degradation,and spatial spread of cyclic AMP. Because of its complexity, thisequation cannot be solved analytically, thus requiring a numer-ical approach. Several studies have used mathematical modelsof cAMP signals. For example, in [1], the modeling environ-


ment “Virtual Cell” was used to study the spatial spread of cAMPwithin a pulmonary microvascular endothelial cell. Other stud-ies have employed the use of techniques such as the fourth-orderRunge-Kutta [2] and large scale stochastic simulations [3] to pre-dict cAMP signaling in a variety of cellular environments.

A popular numerical technique used in engineering applica-tions is the finite element method (FEM). This method has beenused to calculate stress, vibration, and a variety of fluid dynamicquantities. Over time, the engineering community has validatedand embraced FEM as a powerful numerical technique for nu-merous engineering applications. However, to the best knowl-edge of the authors, no previous work on FEM models of cAMPsignals is available in the literature.

In the first step of an attempt to develop an effective three-dimensional FEM model of cAMP signals, the current work willpropose a linearized version of the governing equation and itsfinite element discretization in two dimensions. We have veri-fied that the linearization can be justified for low concentrationsof cAMP (C < 0.5 µM). From this, a numerical implementationwith the three-node triangular element will be carried out to eval-uate the time history of the cAMP concentration at any position(x,y) within a 2-D model of a cell. Although the primary ob-jective of this work is to develop and numerically validate theaforementioned FEM model, the biological significance of thenumerical results will also be briefly discussed.

GOVERNING EQUATIONThe equation governing the synthesis, degradation and spa-

tial spread of the signaling molecule within cells is given by

∂C∂ t

= D∇2C− VmaxC

C+Km(1+ IKI)+EAC(t), (1)

where t denotes time, C is the cAMP concentration, D is the ef-fective diffusion coefficient, Vmax is the maximum cAMP hydrol-ysis rate, Km is the Michaelis-Menten constant for cAMP bind-ing to PDE, I is the concentration of a PDE inhibitor, KI is theinhibition constant, and EAC(t) is the cAMP synthesis rate. Ad-ditionally, terms one through three on the right hand side of Eqn.(1) account for the diffusion, hydrolysis, and synthesis of cAMP,respectively [1, 4].

In this work, we consider a linearized version of Eqn. (1)which is written as

∂C∂ t

= D∇2C− Vmax

KmC+EAC(t). (2)

The possible boundary conditions are

1. C is specified;

2. Normal derivative (concentration flux) is prescribed along aboundary,

∂C∂n

= β , (3)

where β is a constant.

FEA DISCRETIZATION AND IMPLEMENTATIONTo use Galerkin’s method, we multiply the governing equa-

tion (2) by the shape functions Ni (i = 1 . . .m, with m beingthe number of nodes of the finite element employed) chosen asweighting functions and then integrate it over the volume V ofthe element as follows:∫

V

(∂C∂ t−D∇

2C+Vmax

KmC−EAC(t)

)Ni dV = 0. (4)

In this equation, the concentration C of cAMP can be in-terpolated over an element from the nodal values {C} using theshape function matrix [N] as given by

C = [N]{C}, (5)

and thus its time derivative is written as

∂C∂ t

= [N]{C}. (6)

Substitution of Eqns. (5) and (6) into Eq. (4) results in

[K1]{C}+[K2]{C}= {R}, (7)

where

[K1] =∫

V[N]T[N]dV, (8)

[K2] =∫

V[B]T[κ][B]dV +

Vmax

Km

∫V[N]T[N]dV, (9)

[B] =

N1,x 0 0 N2,x 0 0 . . . Nm,x 0 0

0 N1,y 0 0 N2,y 0 . . . 0 Nm,y 0

0 0 N1,z 0 0 N2,z . . . 0 0 Nm,z

N1,y N1,x 0 N2,y N2,x 0 . . . Nm,y Nm,x 0

0 N1,z N1,y 0 N2,z N2,y . . . 0 Nm,z Nm,y

N1,z 0 N1,x N2,z 0 N2,x . . . Nm,z 0 Nm,x

, (10)


[κ] = D[

1 00 1

], (11)

{R}= {R1}+{R2}, (12)

{R1}= EAC(t)∫

V[N]T dV, (13)

{R2}= β

∫So

[N]T dS. (14)

For two-dimensional (2-D) analysis using three-node trian-gular elements, we have [5]

[N] = [1−ξ−η ξ η ], (15)

where ξ ,η = 0 . . .1 are the natural coordinates associated withthe Cartesian coordinates x,y.

In this work, EAC(t) is assumed to be a step function of mag-nitude EAC which represents the synthesis rate for cAMP. In otherwords,

EAC(t) = EACH(t), (16)

where H(t) is the Heaviside step function.A numerical implementation using Eqs. (15) and (16) results

in

[K1] =Aτ

12

2 1 11 2 11 1 2

, (17)

[K2] = Aτ

[B]T [κ][B]+ Vmax

12Km

2 1 11 2 11 1 2

, (18)

[B] =1

2A

[y2− y3 y3− y1 y1− y2x3− x2 x1− x3 x2− x1

], (19)

{R1} =AτEAC

6

111

if t = 0

or {R1} =AτEAC

3

111

if t > 0, (20)

where (xi,yi) are the nodal coordinates of the three-node trian-gular element under consideration and A and τ are the area andthickness, respectively, of that element.

By using the time integration method [5], the concentrationstate {Ci+1} at time ti+1 can be found from {Ci} at time ti as

[1∆t

[K1]+β [K2]

]{Ci+1}=

[1∆t

[K1]− (1−β )[K2]

]{Ci}

+(1−β ){Ri}+β{Ri+1}, (21)

which is simply a linear system of algebraic equations.

MODEL VALIDATION AND VERIFICATIONThe primary objective of this work was to develop a finite el-

ement model for estimating the concentration of cAMP, as shownin the previous section. In this section, a simplified circular cellis chosen to test and validate the 2-D model developed. Availablecytosolic surface area is based on work in [1], where the “VirtualCell” framework was used to examine the effects on cell shape.This is not applicable to more complex systems, hence the desirefor the finite element approach.

Problem Setup and FEAFrom [1], many cell types have a three-dimensional geom-

etry reminiscent of a fried egg, with the yolk being the cell nu-cleus. The current work simplified this geometry to that of atwo-dimensional “donut” as shown in Fig. 1. The outer diameterof the simplified model was 20 µm while its inner diameter was7 µm. Note that the inner diameter of this model representedthe cell nucleus. Biologically, cAMP travels from the outer pe-riphery of the cell inward towards the nucleus. Thus, the area ofinterest lies between the cell wall and the nucleus. Because ofthis, an area representing the nucleus was removed, resulting inthe “donut”-like geometry seen in Fig. 1.

For this 2-D cellular model, the following data were em-ployed: EAC = 1 µM/s, Vmax = 5 µM/s, Km = 2 µM and D =30 µm2/s. No concentration flux (β = 0) is prescribed on boththe inner and outer boundaries while the initial condition (IC) isassumed to be 50 nM (see Fig. 1).

The FEA software ANSYS was used to model and mesh thesimplified cell geometry. Meshes were created from three-nodetriangular elements. In all, four meshes were considered for a


No flux

20 µ m

7 µm

= 50 nM (ICs)C(0,x,y)

FIGURE 1. 2-D CELLULAR GEOMETRY

FIGURE 2. MESH 1

mesh convergence test. Each mesh varied in refinement, with“Mesh 1” representing the coarsest mesh and “Mesh 4” repre-senting the most refined mesh. Meshes 1-4 are shown in Figs. 2,3, 4, and 5, respectively. A series of macros were also writtenfor the extraction of pertinent information regarding the ANSYSmodel. This information included the number of nodes, num-ber of elements, nodal coordinates, element connectivities, andboundary conditions.

The computer programming environment MATLAB wasused to perform all required FEA calculations. These calcula-tions were verified through mesh convergence. To test for meshconvergence, the model was constructed in a way such that theconcentration at the same three nodal locations could be com-pared for each mesh. These nodes included 2, 8, and 9. Node 2

FIGURE 3. MESH 2

FIGURE 4. MESH 3

was located on the outside diameter of the model, node 8 was lo-cated on the inner diameter of the model, and node 9 was locatedexactly between nodes 2 and 8. Reference Figs. 2, 3, 4, and 5 forclarification of nodal locations.

Numerical Results, Validation and VerificationAs stated above, the 2-D FEA code was validated through a

series of mesh convergence tests. The results from each of thethree tests are shown in Figs. 6, 7, and 8.

Figure 6 depicts the time history of the concentration ofcAMP at node 2 for meshes 1-4. The calculated concentrationof cAMP at node 2 is nearly identical for meshes 3 and 4. Be-cause of this, the results are said to converge. Similarly, Fig. 7


FIGURE 5. MESH 4

shows the concentration of cAMP at node 9 as a function of time.Meshes 3 and 4, once again, show convergence of results. Fi-nally, Fig. 8 displays cAMP concentration as a function of timeat node 8. Convergence can be seen in meshes 3 and 4 as theresults seem to reach a steady-state solution. Collectively, all ofthe mesh convergence tests showed convergence in meshes 3 and4 at each of the nodal locations. These results validate the con-structed FEA code. They also show the rapid equilibration of theconcentration of cAMP within a cell as previously predicted [6].

0 0.2 0.4 0.6 0.8 1

t (s)

0

0.2

0.4

0.6

0.8

1

C (

µM

)

Mesh 1Mesh 2Mesh 3Mesh 4

C(t) at Node 2

FIGURE 6. MESH CONVERGENCE TEST FOR NODE 2

0 0.2 0.4 0.6 0.8 1

t (s)

0

0.1

0.2

0.3

0.4

C (

µM

)


C(t) at Node 9


0 0.2 0.4 0.6 0.8 1

t (s)

0

0.1

0.2

0.3

0.4C

(µ

M)


C(t) at Node 8


DISCUSSIONOne possible interpretation of the numerical results obtained

is that the effective diffusion coefficient is markedly less than theone used here. These results further provoke the question of howcAMP can regulate processes in one region of the cell withoutthe activation of cAMP-mediated processes in other regions ofthe cell. These results also demonstrate that the FEM can be usedto study the spatial distribution of cAMP in cells with more com-plex realistic geometries and distribution of signaling molecules.

It is expected that a finite element model using higher-orderterms in the approximation of the governing equation (1) willimprove the accuracy of the analysis, and this investigation iscurrently being pursued by the authors. More complex cellular


geometries will also be considered in our future work.

AcknowledgmentThis research was supported in part by NIH awards

P01HL066299 and T32HL076125.

REFERENCES[1] Feinstein, W. P., Zhu, B., Leavesley, S. J., Sayner, S. L., and

Rich, T. C., 2012. “Assessment of cellular mechanisms con-tributing to cAMP compartmentalization in pulmonary mi-crovascular endothelial cells”. American Journal of Physiol-ogy - Cell Physiology, 302(6), pp. C839–C852.

[2] Rich, T. C., Xin, W., Mehats, C., Hassell, K. A., Piggott,L. A., Le, X., Karpen, J. W., and Conti, M., 2007. “Cel-lular mechanisms underlying prostaglandin-induced tran-sient cAMP signals near the plasma membrane of HEK-293cells”. American Journal of Physiology - Cell Physiology,292(1), pp. C319–C331.

[3] Oliveira, R. F., Terrin, A., Di Benedetto, G., Cannon, R. C.,Koh, W., Kim, M., Zaccolo, M., and Blackwell, K. T., 2010.“The role of type 4 phosphodiesterases in generating mi-crodomains of cAMP: large scale stochastic simulations”.PLoS ONE, 5(7), p. e11725.

[4] Rich, T. C., Fagan, K. A., Tse, T. E., Schaack, J., Cooper,D. M. F., and Karpen, J. W., 2001. “A uniform extracellu-lar stimulus triggers distinct cAMP signals in different com-partments of a simple cell”. Proceedings of the NationalAcademy of Sciences, 98(23), pp. 13049–13054.

[5] Bathe, K., 1996. Finite Element Procedures. Prentice-HallInternational Series in. Prentice Hall.

[6] Saucerman, J. J., Greenwald, E. C., and Polanowska-Grabowska, R., 2014. “Mechanisms of cyclic AMP compart-mentation revealed by computational models”. The Journalof General Physiology, 143(1), pp. 39–48.


Proceedings of the Fourteenth Annual Early Career Technical Conference The University of Alabama, Birmingham ECTC 2014 November 1 – 2, 2014 - Birmingham, Alabama USA

A PROPOSED RETURN ALGORITHM FOR ELASTOPLASTIC CONSTITUTIVE

RELATIONS

David A. Cooper and David L. Littlefield University of Alabama at Birmingham

Birmingham, AL, USA

ABSTRACT A new integration scheme is presented to determine the

stress during inelastic deformation within finite element structural analysis codes. Motivation for the new scheme comes from the supposition that traditional integration methods evaluate the inelastic strain rate at a point within the time step where the state of stress is not actually calculated. The new scheme, titled “Time Centered Plastic Strain (TCPS)” seeks to align the evaluations of the inelastic strain and stress, effectively shifting the evaluation of the plastic strain to an intermediate point within the time interval.

This paper is organized as follows. A brief introduction to plasticity theory, through description of the von Mises plasticity model and the Levy-von Mises plasticity relations, is presented. Next, the influential works of Krieg and Krieg [1] and Ortiz and Popov [2] are presented and the schemes they discuss are applied to the perfectly plastic von Mises case. Finally, the TCPS method is explained and compared to the generalized mid-point and trapezoid rules of Ortiz and Popov.

BACKGROUND In classical plasticity theory, material behavior is

characterized by three properties: a yield criterion to specify the states of stress corresponding to the onset of plastic flow, a hardening rule to model the observed change in the yield criterion during plastic flow and a flow rule which relates the plastic strain rate to the current stress [3]. This work is limited to perfectly plastic conditions and, as such, hardening rules will not be discussed.

The yield criterion is subject to certain symmetry conditions [3] and can be expressed in general form as

𝑓�𝜎𝑖𝑗 , 𝑞� ≤ 0 (1)

where 𝜎 is the Cauchy stress tensor, and 𝑞 is a set of additional state variables required by the material model. For isotropic materials, the directional aspects of 𝜎 can be disregarded and 𝑓 can be expressed in terms of the three stress invariants, 𝐼𝑖, only. In that case, Equation (1) reduces to 𝑓(𝐼1, 𝐼2, 𝐼3,𝑞) ≤ 0. Furthermore, to a first approximation, yielding in metals has been shown to be independent of the hydrostatic stress and Equation (1) can be simplified further to 𝑓(𝐽2, 𝐽3,𝑞) ≥ 0 where 𝐽𝑖 are the invariants of the deviatoric stress tensor, 𝑆, given by

𝑆𝑖𝑗 = 𝜎𝑖𝑗 −13𝜎𝑘𝑘𝛿𝑖𝑗 (2)

The von Mises criterion, given by Equation (3), posits that

yielding is a function of 𝐽2 only, a good approximation for ductile metals: 𝑓(𝐽2) = 2𝐽2 −

23𝑌2 = 𝑆𝑖𝑗𝑆𝑖𝑗 − 𝑅2 ≤ 0 (3)

Here 𝑌 is the material yield stress in simple tension. It is instructive to represent Equation (3) graphically as a circle of

radius 𝑅 = �23

𝑌 in the Π-plane or equivalently as a cylinder in

the three dimensional principal stress space as shown in Figure 1:

Figure 1. Graphic interpretation of von Mises yield function. © Rswarbrick, “Yield surfaces.svg" CC BY 2.0

A general form usually assumed for the flow rule is given by 𝜖𝑝𝑖𝑗 = �� 𝜕𝑔

𝜕𝜎𝑖𝑗 (4)

where �� is a scalar rate known as the plastic multiplier and 𝑔 is


a scalar function of stress known as the “plastic potential.” When the plastic potential and the yield function coincide (i.e. 𝑔 = 𝑓), the flow rule is called “associated.” Otherwise the flow rule is called “non-associated.” The associated flow rule was first posited by von Mises and is adequate for describing the response of most metals [4].

Combining the von Mises yield criterion and the associated flow rule yields the governing equations for determining the stress for a given deformation state: �� = ��𝑒 + ��𝑝 (5a) ��𝑒 = 1

2𝐺�� (5b)

��𝑖𝑗 = 2𝐺��𝑖𝑗 −2𝐺𝑅2

(𝑆𝑘𝑚��𝑘𝑚)𝑆𝑖𝑗 (5c) where ��𝑖𝑗 = 𝜖��𝑗 −

13𝜖��𝑘𝛿𝑖𝑗 is the deviatoric strain rate, and 𝐺 is

the shear modulus [1]. There are some important aspects of Equation (5) that should be noted. First, the state of stress varies tangentially with respect to the yield surface (i.e. the plastic portion of the strain rate consists solely of that part which would be normal to the yield surface.) Second, there is no dependence on pressure. Therefore, hydrostatic stress results in purely elastic response.

CURRENT METHODS Equations (5) represent a series of ordinary differential

equations and there are many schemes for integrating these relations. Kojic [5] provides a summary of many of them. Among those surveyed are two single step return mapping algorithms, originally evaluated by Krieg and Krieg [1], the secant stiffness and radial return methods. The accuracy of each method was evaluated using an angular difference between the computed final stress and an analytical solution as shown in Figure 2. Iso-error plots showing this angular error plotted against tangential and radial strain increments were presented. A graphical representation is possible and is informative in describing the algorithms.

Figure 2. Krieg and Krieg error plot explanation.

The secant stiffness method is illustrated in Figure 3 and is described algorithmically as 𝑆𝑖𝑗𝑛+1 = 𝑆𝑖𝑗𝑛 + Δ𝑆𝑖𝑗𝑒 −

1𝑅2

(𝑆𝑘𝑚𝐶 Δ𝑆𝑘𝑚𝑒 )𝑆𝑖𝑗𝐶 (6)

where Δ𝑆𝑖𝑗𝑒 = 2𝐺��𝑖𝑗Δ𝑡 is the predicted stress increment if purely elastic, 𝑆𝑖𝑗𝐵 is the deviatoric stress state where plastic strain begins, and 𝑆𝑖𝑗𝐶 is the midpoint between 𝑆𝑖𝑗𝐵 and 𝑆𝑖𝑗𝑛 +Δ𝑆𝑖𝑗𝑒 .

Figure 3. Secant stiffness method. (Top) Graphic interpretation. (Bottom) Angular error plot (degrees).

The radial return is arguably the most popular method in use today and is given simply by

𝑆𝑖𝑗𝑛+1 = 𝑅

�𝑆𝑘𝑚∗ 𝑆𝑘𝑚

∗𝑆𝑖𝑗∗ (7)

where 𝑆𝑖𝑗∗ = 𝑆𝑛 + Δ𝑆𝑖𝑗𝑒


Figure 4. Radial return method. (Top) Graphic interpretation. (Bottom) Angular error plot (degrees).

Krieg and Krieg recommend the simple radial return method for most perfectly plastic von Mises cases. It enjoys good accuracy for most strain increments, is unconditionally stable, and has of a global maximum error of 12.7°.

Ortiz and Popov [2] generalized the return mapping algorithm using the well known midpoint and trapezoid rules. The generalized trapezoid procedure for the case of associated flow is given by 𝜎𝑖𝑗𝑛+1 = 𝐷𝑖𝑗𝑘𝑙�𝜖𝑘𝑙𝑛+1 − (𝜖𝑝)𝑘𝑙𝑛+1� (8a)

(𝜖𝑝)𝑖𝑗𝑛+1 = (𝜖𝑝)𝑖𝑗𝑛 + 𝜆 �(1 − 𝛼) � 𝜕𝑓𝜕𝜎𝑖𝑗

�𝑛

+ 𝛼 � 𝜕𝑓𝜕𝜎𝑖𝑗

�𝑛+1

� (8b)

𝑞𝑖𝑛+1 = 𝑞𝑖𝑛 + 𝜆[(1− 𝛼)ℎ𝑖𝑛 + 𝛼ℎ𝑖𝑛+1] (8c) 𝑓(𝜎) = 0 (8d) where 𝐷𝑖𝑗𝑘𝑙 is the elastic stiffness tensor, 𝛼 is an algorithmic parameter with values between zero and one, 𝑞 is the vector of plastic variables, and ℎ is the vector of plastic moduli. The generalized midpoint rule is obtained by replacing Equation (8b) and (8c) with

(𝜖𝑝)𝑖𝑗𝑛+1 = (𝜖𝑝)𝑖𝑗𝑛 + 𝜆 � 𝜕𝑓𝜕𝜎𝑖𝑗

�𝑛+𝛼

(9a)

𝑞𝑖𝑛+1 = 𝑞𝑖𝑛 + 𝜆ℎ𝑖𝑛+𝛼 (9b)

The algorithmic parameter 𝛼 together with the flow rule determine the specific return path, and 𝜆 is calculated in order to return the state of stress to the updated yield surface. Equations (8) and (9) represent systems of non-linear equations which require simultaneous solution. The generalized midpoint and trapezoid rules are described graphically in Figure 5.

Figure 5. The generalized trapezoid (Top) and midpoint (bottom) procedures for an associated flow rule.

When Equations (8) and (9) are applied to the case of perfect plasticity, they coincide and the resulting system is given by

𝑆𝑖𝑗𝑛+1 = 𝑆𝑖𝑗𝑛 + Δ𝑆𝑖𝑗𝑒 − 𝜆�(1 − 𝛼)𝑆𝑖𝑗𝐵 + 𝛼𝑆𝑖𝑗𝑛+1� (10a) 𝑆𝑖𝑗𝑛+1𝑆𝑖𝑗𝑛+1 = 𝑅2 (10b)

Figure 6 shows the iso-error maps associated with several values of 𝛼. As evidenced by the plots, for 𝛼 = 0.5 and 𝛼 = 1 Equation (10) reduces to the secant stiffness and the radial return methods respectively. Ortiz and Popov also demonstrate that for 𝛼 = 0.5 the generalized methods achieve 2nd order accuracy even though the algorithm produces larger than optimum error for most cases.


Figure 6. Generalized trapezoid/midpoint rule error plots for perfectly plastic von Mises.

A simple extension to the results of Ortiz and Popov can be arrived at by determining the optimal value for α for a given strain increment. This is presented in Figure 7 below. As is evident, for moderate strain increments smaller values for α result in a more accurate solution.

Figure 7. Optimum 𝛼 for a given strain increment.

TCPS For direct comparison with the results of Ortiz and Popov,

TCPS has been defined in the context of the perfectly plastic von Mises model and a similar accuracy analysis procedure carried out. The perfectly plastic stress rate equation is given by Equation (5). Assuming that the total strain rate is constant over the time step (i.e. ��𝑖𝑗 = 𝑐𝑜𝑛𝑠𝑡), and discretizing Equation (5) yields 𝑆𝑖𝑗𝑛+1 − 𝑆𝑖𝑗𝑛 = Δ𝑆𝑖𝑗𝑒 − 2𝐺Δ𝑡��𝑖𝑗

𝑝�𝑛+𝛼 (11a)

��𝑖𝑗𝑝�𝑛+𝛼 ≡ Δ𝜆�𝛼��𝑛+1𝑆𝑖𝑗𝑛+1 + (1− 𝛼)��𝑛𝑆𝑖𝑗𝑛� (11b)

𝑆𝑖𝑗𝑆𝑖𝑗 = 𝑅2 (11c) �� = � 1

𝑅2� (𝑆𝑘𝑚��𝑘𝑚) (11d)

where Δ𝜆 is introduced to allow for the satisfaction of Equation (11c) and 𝛼 is introduced to allow for optimization of the algorithm similarly to the generalized trapezoid and midpoint methods.

Figure 8. TCPS error plots for comparison to Figure 6.

Optimal values for α are given in Figure 9 below. As is evident, for moderate strain increments smaller values for α result in a more accurate solution.


Figure 9. Optimum 𝛼 values for TCPS.

DISCUSSION Functionally, Equation (11) differs from Equation (10)

simply by including the magnitudes of the plastic strain rates at the beginning and end of the interval. This general similarity between the two methods is evident in the resulting error plots as 𝛼 increases. Both methods converge on radial return as 𝛼 approaches 1. However, as 𝛼 falls toward .5, where second order accuracy is achieved by the generalized methods, TCPS gives more accurate results. Since neither a stability nor rate of convergence analysis has yet been conducted for TCPS, it is not clear if the similarity between the two methods will translate to 2nd order accuracy for TCPS.

It is also worth noting that since the plastic strain rate at the beginning of an interval is proportional to the radial strain increment, TCPS converges on radial return as the strain increment approaches purely tangential.

REFERENCES [1] Krieg, R. D., and Krieg, D. B., 1977, “Accuracies of Numerical Solution Methods for the Elastic-Perfectly Plastic Model,” ASME J. Pressure Vessel Technol., 99, pp. 510–515. [2] Ortiz, M. and Popov, E. P., 1985, “Accuracy and Stability of Integration Algorithms for Elastoplastic Constitutive Relations,” Int. J. Numer. Methods Eng., 21, pp. 1561–1576. [3] Hill R., 1950, The Mathematical Theory of Plasticity, University Press: Oxford. [4] Jirásek, M., and Bazant, Z. P., 2002, Inelastic Analysis of Structures, John Wiley & Sons, Ltd. West Sussex, England, Chap. 15. [5] Kojic, M., 2002, “Stress Integration Procedures for Inelastic Material Models Within the Finite Element Method,” Appl. Mech. Rev. 55(4), pp. 389-414.



IMPLEMENTATION AND VALIDATION OF THE RHT CONCRETE MODEL

Gerald Pekmezi and David Littlefield Department of Mechanical Engineering, The University of Alabama at Birmingham,

Birmingham, Alabama, USA

Bradley Martin Munitions Directorate, Air Force Research Laboratory

Eglin AFB, FL, USA

ABSTRACT In many civilian and military applications it is necessary to

have an accurate model of the behavior of concrete under high pressure, high velocity, and highly dynamic events with large deformations. One such model was developed recently by Riedel, Hiermaier and Thoma, the RHT Model. The model uses a three surface approach, third invariant dependency, the Mie-Gruneisen equation of state, and Hermann’s p-α compaction model.

In this work, the RHT Model has been integrated into the EPIC hydrocode. As part of the integration, stress and deformation measures suitable for use in the model were formulated and integrated into EPIC. The model was integrated in conformance with the interface standard so as to ensure a smooth transition to production-level software. Example calculations are shown for benchmark verification problems to illustrate the veracity of the installation.

INTRODUCTION Of all the materials used in engineering, none has the long

history as well as modern relevance of concrete. The behavior of concrete has been studied extensively in a Structural Engineering, static load design context. However, the mechanical behavior of concrete under high pressures and dynamic conditions is also very important to a number of civilian and military applications. For example, when concrete is perforated as a result of an impact, it is necessary to accurately account for strength of failed concrete under compression as well as the tensile limit of cracked and comminuted material. Localized failure and cracking also introduces anisotropic behavior. Strength differentials in tension and compression result in third-invariant dependence typically not observed for many materials when classical inelasticity theories are applied. Distinctions in behavior under high rates of strain are also obviously important. Because of these reasons, as well as others, there has been a renewed

interest in developing realistic material response models for concrete.

Concrete is essentially a coarse (macroscopic) composite material with stiff aggregate embedded in a porous cement matrix. The macroscopic heterogeneity of the material is responsible for the inherent difficulty in accurately describing the mechanical behavior of concrete. Many different approaches for constitutive modeling the dynamic behavior of concrete have been proposed. In this study, the implementation and validation of a model developed by Riedel, Hiermaier and Thoma [1], called the RHT Model, is described.

The model is based on the concept of a three surface approach and makes use of an elastic limit surface with hardening, a failure surface, and a residual strength surface. The model captures tension/compression strength differentials with use of the Lode angle or J3 dependency. Hydrostatic response is described using Hermann’s p-α model [2] with the Mie-Gruneisen equation of state for the pore-free matrix material. Most of the parameters in the model can be calibrated from standard experiments such as uniaxial, biaxial and hydrostatic compression, pure shear, and dynamic description of model.

Equation of State The compression of concrete is a complex, nonlinear

process due to its macroscopic heterogeneity and matrix porosity. To account for this complex compression behavior, Hermann [2] proposed a two-stage equation of state, later modified by Carroll and Holt [3]. One stage models pore collapse, while the other stage models the compression of the matrix material. To this end, a distension parameter α is introduced in Equation (1) to represent the porosity.

𝛼 ≝ 𝜌𝑠𝜌

(1) where ρ is the porous density of the material, while ρs is the density of the matrix material. The evolution of the distension parameter is described by Equation (2).


𝛼 = 1 + (𝛼0 − 1) � 𝑝𝑙𝑜𝑐𝑘−𝑝

𝑝𝑙𝑜𝑐𝑘−𝑝𝑐𝑟𝑢𝑠ℎ�𝑛

(2) where plock is the lockup pressure, pcrush is the pore crush pressure, and α0 represents the initial porosity.

The pressure is described by a Mie-Gruneisen type EOS as in Equation (3).

𝑝 = 𝜌Γ𝑒 + 𝑝𝐻 �1− 12Γ𝜂� (3)

with the polynomial Hugoniot reference curve pH as in Equation (4).

𝑝𝐻 = 𝐴1𝜂 + 𝐴2𝜂2 + 𝐴3𝜂3 (4) where

𝜂 = 𝜌𝜌0

(5)

Note that Equation (3) uses the original Hermann formulation [2] of the p-α compaction model. This is consistent with the RHT formulation as outlined by its authors [1] [4], and the manner it was implemented in AUTODYN [5]. By contrast, the Carroll and Holt formulation [3] of the p-α compaction model has been used in the LS-DYNA implementation [6] of the RHT Model. The difference lies in that, in the latter the Hugoniot of Equation (3) is divided by the distension α.

Strength Model Concrete is a brittle material, and as such it is susceptible

to fractures under even fairly small tensile loads. Conversely, the strength of concrete increases with increasing compressive pressure. A further anomloly lies in the fact that when confined, even crushed concrete possesses significant residual strength in compression. These factors indicate that a rather complex material strength model is required to capture the varied behavior of concrete under different loading conditions.

The RHT Model is such a model, as it provides mathematical descriptions of each of the aforementioned behaviors. The strength model uses three invariants of the stress tensor to define three surfaces: an evolving elastic limit surface, a failure surface, and a residual strength surface.

The failure surface is in the form of Equation (6).

𝑌𝑓𝑎𝑖𝑙(𝑝, 𝜃, 𝜖) = 𝑌𝑡𝑥𝑐(𝑝)𝑅3(𝜃)𝐹𝑟𝑎𝑡𝑒(𝜖) (6) where Ytxc(p) represents the pressure-dependent compression meridian, while Frate(𝜖) is a rate scaling factor not currently implement in the RHT Model in EPIC, and R3(θ) is a Lode angle scaling factor.

The curved meridian of triaxial compression normalized by the unconfined compression strength fc is given by Equation (7) and is illustrated by the green segment of Figure 1.

𝑌𝑡𝑥𝑐∗ (𝑝) = 𝑌𝑡𝑥𝑐(𝑝)𝑓𝑐

= 𝐴 ∙ (𝑝∗ − 𝐻𝑇𝐿`∗) for 𝑝∗ ≥ 13 (7)

HTL’* is a continuity variable to ensure Ytxc*=1 at p*=1/3. Its value is given by Equation (8).

𝐻𝑇𝐿`∗ = 13− (𝐴)−

1𝑛 (8)

On the tensile side 𝑌𝑡𝑥𝑐∗ (𝑝) is limited by Equation (9).

𝑌𝑡𝑥𝑐∗ (𝑝) = 0 for 𝑝∗ < 𝐻𝑇𝐿`∗ (9)

Meanwhile Ytxc* is interpolated from 𝑝∗ = −𝐻𝑇𝐿∗ to 𝑝∗ = 0 as shown in Equation (10) and illustrated by the blue segment of Figure 1.

𝑌𝑡𝑥𝑐∗ (𝑝) = 𝑓𝑠∗ �1 + 𝑝∗

𝐻𝑇𝐿∗� (10)

Interpolation from 𝑝∗ = 0 to 𝑝∗ = 1/3 is as shown in Equation (11) and illustrated by the red segment of Figure 1.

𝑌𝑡𝑥𝑐∗ (𝑝) = 3𝑝∗ + 𝑓𝑠∗(1− 3𝑝∗) (11) Figure 1 illustrates the triaxial compression meridian described by the RHT Model.

Figure 1. Triaxial Compression Meridian in RHT Model

The Lode angle reduction factor R3(θ) may also be regarded as a J3 factor as it depends on the third invariant of the stress tensor. It is given by Equation (12).

𝑅3(𝜃) =2(1 − 𝑄22) cos 𝜃 + (2𝑄2 − 1)[4(1− 𝑄22) cos2 𝜃 + 5𝑄22 − 4𝑄2]

12

4(1− 𝑄22) cos2 𝜃 + (1 − 2𝑄2)2

(12)

where Q2 is given by Equation (13).

𝑄2 = 𝑄2,0 + 𝐵𝑝∗ (13)

where Q2,0 and B are material parameters. Q2 is limited by Equation (14).


0.5 < 𝑄2 ≤ 1 (14) The relation of the Lode angle θ to the third invariant J3 of the stress tensor is given by Equation (15).

cos 3𝜃 = 3√32

𝐽3

𝐽2

12

= 27𝑑𝑒𝑡(𝑆)2𝜎𝑒𝑞3

(15)

Using Equation (6), the failure surface is illustrated in Figure 2.

Figure 2. Failure Surface of RHT Concrete Model

Note that the failure surface depends on the loading type, the surface in Figure 2 is drawn for monotonic uniaxial as well as hydrostatic loading in tension and compression. The failure surface is the same in uniaxial as well as hydrostatic compressive loading. In tensile loading however, the uniaxial loading failure surface lies below the hydrostatic loading failure surface.

The elastic limit surface is given by Equation (16).

𝑌𝑒𝑙𝑎𝑠𝑡𝑖𝑐 = 𝑌𝑓𝑎𝑖𝑙𝐹𝑒𝑙𝑎𝑠𝑡𝑖𝑐𝐹𝑐𝑎𝑝 (16)

where Felastic is an elastic limit scaling function and Fcap is a parabolic scaling function that caps the elastic limit surface at high pressures. Felastic takes the value ft,el/ft at tensile pressures equal to and beyond ft,el/3ft, while it takes the value fc,el/fc at compressive pressures equal to and beyond ft,el/3ft. Between the tensile and compressive pressure bounds Felastic is interpolated using Equation (17).

𝐹𝑒𝑙𝑎𝑠𝑡𝑖𝑐 = 𝑓𝑡,𝑒𝑙𝑓𝑡

+ 𝑝∗+𝑓𝑡,𝑒𝑙 3𝑓𝑐⁄𝑓𝑐,𝑒𝑙 3𝑓𝑐⁄ +𝑓𝑡,𝑒𝑙 3𝑓𝑐⁄

�𝑓𝑐,𝑒𝑙𝑓𝑐

− 𝑓𝑡,𝑒𝑙𝑓𝑡� (17)

The parabolic scaling function Fcap is defined by Equation (18).

𝐹𝑐𝑎𝑝 =

⎩⎪⎨

⎪⎧ 1 𝑓𝑜𝑟 𝑝 ≤ 𝑝𝑢 = 𝑓𝑐

3�

�1 − � 𝑝−𝑝𝑢𝑝0−𝑝𝑢

�2�12�

𝑓𝑜𝑟 𝑝𝑢 < 𝑝 < 𝑝00 𝑓𝑜𝑟 𝑝 ≥ 𝑝0

� (18)

where the upper cap pressure p0 is initially set to the pore crush pressure for the virgin material pel. As the material compacts, p0 is increased and set equal to pc.

The resulting elastic limit surface is shown in Figure 3 for hydrostatic compressive and tensile loading.

Figure 3. Elastic Limit Surface of RHT Model

Once the initial elastic limit surface is reached, permanent

plastic deformation accumulates prior to reaching the failure surface. The hardened elastic surface is described by Equation (19).

𝑌ℎ𝑎𝑟𝑑 = 𝑌𝑒𝑙𝑎𝑠𝑡𝑖𝑐 +𝜖𝑒𝑞𝑝𝑙

𝜖𝑒𝑞𝑝𝑙,ℎ𝑎𝑟𝑑 �𝑌𝑓𝑎𝑖𝑙 − 𝑌𝑒𝑙𝑎𝑠𝑡𝑖𝑐� (19)

where the total available equivalent plastic strain 𝜖𝑒𝑞

𝑝𝑙,ℎ𝑎𝑟𝑑 is defined by Equation (20).

𝜖𝑒𝑞𝑝𝑙,ℎ𝑎𝑟𝑑 =

�𝑌𝑓𝑎𝑖𝑙−𝑌𝑒𝑙𝑎𝑠𝑡𝑖𝑐�

3𝐺� 𝐺𝑒𝑙𝑎𝑠𝑡𝑖𝑐𝐺𝑒𝑙𝑎𝑠𝑡𝑖𝑐−𝐺𝑝𝑙𝑎𝑠𝑡𝑖𝑐

� (20)

Plastic strain 𝜖𝑒𝑞

𝑝𝑙 is accumulated incrementally using Equation (21).

Δ𝜖𝑒𝑞𝑝𝑙 = �3𝐽2−𝑌𝑒𝑙𝑎𝑠𝑡𝑖𝑐

3𝐺 (21)

The last of the three surfaces inherent to the RHT Model

(along with 𝑌𝑓𝑎𝑖𝑙 and 𝑌𝑒𝑙𝑎𝑠𝑡𝑖𝑐) is the friction resistance surface 𝑌𝑓𝑟𝑖𝑐𝑡 defined by Equation (22).

𝑌𝑓𝑟𝑖𝑐𝑡 = 𝐵𝑝𝑚 (22)

where B and m are friction paramters. Physically, the friction resistance surface represents a residual strength of the failed concrete under a compressive pressure. It is worth noting that there has been heretofore no empirical determination of the constants for the residual strength surface due to difficulty in


testing concrete at high hydrostatic pressures post-failure [4]. Presently the way the residual surface of the RHT Model is typically handled by default, is to set as B=A and m=n [1] [4] [7].

When the hardened yield surface reaches the failure surface Yfail, damage accumulates during any additional inelastic loading. The damage variable D is determined in incremental form from the summation in Equation (23).

𝐷 = ∑ Δ𝜖𝑒𝑞𝑝𝑙

𝜖𝑒𝑞𝑝𝑙,𝑓𝑎𝑖𝑙(𝑝)

(23)

where 𝜖𝑒𝑞

𝑝𝑙,𝑓𝑎𝑖𝑙(𝑝) is the pressure-dependent effective strain to failure given by Equation (24).

𝜖𝑒𝑞𝑝𝑙,𝑓𝑎𝑖𝑙(𝑝∗) = 𝐷1(𝑝∗ − 𝐻𝑇𝐿∗)𝐷2 ≥ 𝑒𝑓𝑚𝑖𝑛 (24)

In Equation (24) efmin is a lower limit for the effective strain to failure.

As damage accumulates, the concrete loses deviatoric strength. This is modeled through the collapsed yield/failure surface, which is linearly interpolated between the failure surface Yfail and the residual friction surface Yfric, as shown in Equation (25).

𝑌𝑑𝑎𝑚𝑎𝑔𝑒𝑑 = 𝑌𝑓𝑎𝑖𝑙 + 𝐷�𝑌𝑓𝑟𝑖𝑐 − 𝑌𝑓𝑎𝑖𝑙� (25)

IMPLEMENTATION IN EPIC In this work, the RHT model has been integrated into the

EPIC hydrocode [8]. As part of the integration, stress and deformation measures suitable for use in the model were formulated and integrated into EPIC. This required a recast of stress and deformation measures traditionally used as well as the reverse transformation for results that were passed back. A novel implementation of the p-α model was developed to ensure numerical stability of the formulation. The model was integrated in conformance with the Material Model Module (MMM) interface in EPIC to ensure a smooth transition to production-level software. One example of the model as run in EPIC is presented in Figure 4, where a cylinder of concrete is impacted into a rigid boundary (the Taylor impact problem).

Figure 4. Taylor Impact Test with RHT Concrete

Verification of the veracity of the RHT Model implementation in EPIC was initially carried out separately from EPIC. The RHTconcrete.f90 Fortran90 material subroutine was extracted from the EPIC source code and executed within a specially-written “wrapper” standalone routine, dubbed RHTconcretetest.f90. The purpose of the wrapper is to emulate the way the material subroutine is used within EPIC, but without actually needing to run EPIC or any subroutine other than the RHT subroutine.

In the wrapper, the strain is increased monotonically along several arbitrary loading paths. At each load increment, the equivalent stress, the elastic limit, and the failure surface were tracked. The results are shown in Figure 5 through Figure 14.

Figure 5 shows σeq/fc, Yelastic, and Yfail for hydrostatic tensile loading, i.e. ∆εx=∆εy=∆εz=-∆ε, ∆εxy=∆εxz=∆εyz=0.

Figure 5. Stress vs. Pressure for Hydrostatic Tension


Figure 6 shows σeq/fc, Yelastic, and Yfail for hydrostatic compressive loading, i.e. ∆εx=∆εy=∆εz=∆ε, ∆εxy=∆εxz=∆εyz=0.

Figure 6. Stress vs. Pressure for Hydrostatic Compression

Note that σeq/fc = εeq = 0 for hydrostatic loading so no stress vs. strain plots are possible for Figure 5 and Figure 6.

Figure 7 and Figure 8 show σeq/fc, Yelastic, and Yfail for uniaxial tensile loading, i.e. ∆εx=-∆ε, εy=∆εz=∆εxy=∆εxz=∆εyz=0.

Figure 7. Stress vs. Pressure for Uniaxial Tension

Figure 8. Stress vs. Strain for Uniaxial Tension

Figure 9 and Figure 10 show σeq/fc, Yelastic, and Yfail for uniaxial compressive loading, i.e. ∆εx=∆ε, ∆εy=∆εz=∆εxy=∆εxz=∆εyz=0.

Figure 9. Stress vs. Pressure for Uniaxial Compression

Figure 10. Stress vs. Strain for Uniaxial Compression

Figure 11 and Figure 12 show σeq/fc, Yelastic, and Yfail for

biaxial compressive loading, i.e. ∆εx=∆εy=∆ε, ∆εz=∆εxy=∆εxz=∆εyz=0.

Figure 11. Stress vs Pressure for Biaxial Compression


Figure 12. Stress vs. Strain for Biaxial Compression

Figure 13 and Figure 14 show σeq/fc, Yelastic, and Yfail for a

loading path dominated by shear with a relatively small amount of compression., ∆εx=∆ε, ∆εxy=∆εxz=∆εyz=2∆ε, ∆εz =∆εy=0.

Figure 13. Stress vs. Pressure for Mostly-Shear Loading

Figure 14. Stress vs. Strain for Mostly-Shear Loading

SAMPLE APPLICATIONS The RHT Concrete Model, newly implemented in EPIC, is

used to run two validation problems. These validation problems are typical problems used to validate concrete behavior in hydrocodes.

Buried Detonation The first validation problem performed was the simulation

of an embedded detonation problem. The problem setup is depicted in Figure 15. A concrete target 40 inches square on the face and 10 inches in thickness is reinforced with two layers of a grid of 0.5 inch square steel bars. The grid is arranged in a 5 inch square pattern. The center of the target contains a cylindrical cavity 1 inch in diameter and 8.5 inches in length. The cavity is filled with 5 inches of Composition C-4 explosive with a 3.5 inch unfilled opening above it.

Figure 15. Buried Detonation Model Setup

A mesh consisting of 995272 hexahedral elements was used to simulate the problem. Two planes of symmetry were used in the calculation so that a quarter of the problem was run. Concrete elements were discarded at an inelastic strain of 1.5 so that finite elements with a large accumulated distortion could be removed from the calculation. This strain this is significantly higher than the failure strain of the concrete, however it is important for the failed material to remain in the calculation, particularly for an embedded detonation simulation, since it can still support compression. Results of the calculation are shown in Figure 16 and Figure 17, where materials and inelastic strain are shown at 1.0 ms after detonation. As is evident, the removed concrete exposes the first square of the rebar grid around the detonation and a portion of the second layer of the rebar grid.

Figure 16. Buried Detonation Material Plot at 1ms


Figure 17. Buried Detonation Strain Contours at 1ms

Ballistic Penetration by Steel Projectile As part of the validation, the impact of a projectile onto a

concrete target was also simulated. The problem setup is depicted in Figure 18. The projectile was a steel penetrator and is 13.6 and 2 inches in length and diameter, respectively. The concrete target had a diameter of 72 inches on the face, and a thickness of 10 inches. A mesh consisting of 476130 tetrahedral elements was used in the calculation. Eroding slidelines were enabled to permit the removal of finite elements that accumulated excessive distortion, with the erosion strain set equal 1.5. A single plane of symmetry was used in the calculation so that half the problem was run.

Figure 18. Ballistic Penetration Model Setup

Results from one of these calculations are shown in Figure 19 through Figure 22, where material plots are shown at 500, 1000, 1500 and 2000 µs. Two slightly different material plots with matching strain contours are shown at each aforementioned time step. The material plot on the right shows

the output with all elements other than the elements that have been eroded at a strain greater than or equal to 1.5. The material plot on the left shows only the elements that have not been fully damaged (i.e. D < 1).

Figure 19. Ballistic Penetration Strain Contours at 0.5ms

Figure 20. Ballistic Penetration Strain Contours at 1ms

Figure 21. Ballistic Penetration Strain Contours at 1.5ms

Figure 22. Ballistic Penetration Strain Contours at 2ms

As may be observed, impact of the projectile results in complete perforation of the target leaving a cavity approximately 10 in diameter. The cavity represents the boundary above which the damage accumulated in the target material is at unity.

Figure 23 shows the time history of the velocity magnitude of the steel projectile during the simulation.


Figure 23. Ballistic Projectile Velocity Magnitude

The projectile exit velocity magnitude is about 55% of the entry velocity magnitude. Taking into account the difference in entry velocity magnitudes and slab thicknesses, this compares favorably with the slowdown found experimentally by Hansson and Skoglund [7].

CONCLUSIONS The RHT Concrete Model was implemented in the EPIC

hydrocode. All the necessary state, stress, and deformation measures were integrated into the existing EPIC interface where possible and included in the new material implementation where necessary. A number of loading tests were conducted on the material subroutine to verify that programmed behavior matched that described by the mathematical model.

Two fairly complex benchmark models were analyzed using the new concrete model. The results from these examples were plotted and compared with experimental results, where available. Further studies into the RHT Model’s prediction of the behavior of concrete in high energy simulations will be carried out in follow-up studies, with a particular interest in finding an empirical basis for the residual surface formulation.

REFERENCES

[1] W. Riedel, K. Thoma, S. Hiermaier and E. Schmolinske, "Penetration of reinforced concrete by BETA-B-500 numerical analysis using a new macroscopic concrete model for hydrocodes.," in Proceedings of the 9th International Symposium on the Effects of Munitions with Structures, Berlin, 1999.

[2] W. Hermann, "Constitutive equation for the dynamic compaction of ductile porous materials," Journal of Applied Physics, vol. 40, no. 6, pp. 2490-2499, 1969.

[3] M. M. Carroll and A. C. Holt, "Static and Dynamic Pore Collapse Relations for Ductile Porous Materia," Journal of Applied Physics, vol. 43, no. 4, pp. 1626-1636, 1972.

[4] W. Riedel, N. Kawai and K.-i. Kondo, "Numerical Assessment for Impact Strength Measurements in Concrete Materials," International Journal of Impact Engineering, vol. 36, no. 2, pp. 283-293, 2009.

[5] AUTODYN, TM, "Theory Manual Revision 4.3," Century Dynamics, Concord, CA, 2003.

[6] J. Hallquist, "LS-Dyna Material Manual," Livermore Software and Technology Corporation, Livermore, 2014.

[7] H. Hansson and P. Skoglund, "Simulation of Concrete Penetration in 2D and 3D with the RHT Material Model," Swedish Defense Research Agency, Tumba, 2002.

[8] G. Johnson, "A Computer Program for Elastic-Plastic Impact Computations in 2 Dimensions Plus Spin," Honeywell Inc., Hopkins, MN, June (1978).



COMPUTATIONAL FLUID DYNAMICS ANSLYSIS OF TYPICAL LOWER LUNG BIFURCATIONS UNDER MECHANICAL VENTILATION AND NORMAL BREATHING

Raghu Arambakam, Ramana M. Pidaparti College of Engineering, University of Georgia Athens, GA, U.S.A.

Angela M. Reynolds Dept. of Mathematics and Applied Mathematics,

Virginia Commonwealth University Richmond, VA, USA

Rebecca L. Heise Dept. of Biomedical Engineering,

Virginia Commonwealth University Richmond, VA, USA

ABSTRACT This study computationally investigates airflow induced by

mechanical ventilation and normal breathing in a three-dimensional model of lung bifurcation generation 2-3. Air pressure and wall shear stresses, among other flow parameters, under different breathing conditions were computed using ANSYS Fluent Solver. The mechanical ventilation waveform, which mimics inhalation with a constant flow rate and exhalation with an exponentially decreasing flow rate, was used as the inlet boundary condition. The normal breathing was modelled by a modified sinusoidal wave. A C++ subroutine was used to enhance the ANSYS solver capabilities for specifying the inlet boundary condition.

Most importantly aging of lung was modeled by shrinking the lung bifurcation uniformly or at predetermined random locations. The values of pressure and wall shear stress obtained from the simulation results at predetermined locations throughout the lung bifurcations were used to determine the relationships between flow waveform and the pressure/wall shear stress. The levels of pressure and wall shear stress obtained throughout this study indicate that the lung walls at the lower lung bifurcations experience forces close to 0.15 Pa which are significant enough to cause considerable tissue deformation. The results also indicate that the airway pressures obtained in the between normal and aged lung are different wherein the pressure drop across the aged lung bifurcation is larger compared to a normal lung.

This is study is an important prologue for future studies which could incorporate the effects of Fluid-Structure Interaction (FSI) to study the complex and intriguing physical phenomenon of ventilator induced lung injury.

INTRODUCTION It has been a traditional approach to model trachea-

bronchial region of the lung to gain an insight into the breathing function of human lung under mechanical ventilation and during normal breathing conditions [1]. Respiratory flow patterns in lung bifurcations were studied by numerous authors in the past [2-5]. CFD models were also used for studying drug delivery and aerosol deposition in lung airways [6-10]. It is well known in literature that aging causes human rib cage to shrink thereby causing compression of the lung structures. Due to this effect it has been noted that the airways shrink in size thereby adversely affecting breathing [11]. It was also reported previously that the cause of certain lung diseases is due to the contraction of smooth muscle in airway walls and inflammation, which leads to narrowing of the airway lumen [12]. However the amount of shrinkage of lung, to the authors’ knowledge, has not been quantified yet. In this study, computational investigation of airflow induced by mechanical ventilation and normal breathing in a three-dimensional model of typical lower lung bifurcation was conducted. Most importantly aging of the lung geometry was computationally modeled by shrinking the geometry of a reference typical lower lung bifurcation, see Fig. 1a, uniformly by 20% of its original volume, see Fig. 1b. Alternatively the typical lower lung bifurcation was also shrunk at random locations along its length by a predetermined volume percentage of 10% or 20%, see Figs. 1c, 1d.

Air pressure and wall shear stresses, among other flow parameters, under mechanical ventilation conditions were computed using ANSYS Fluent Solver. This study is an important prologue for future studies that could incorporate the effects of Fluid-Structure Interaction (FSI) to study the complex


and intriguing physical phenomenon of ventilator induced lung injury.

HUMAN RESPIRATION CFD analysis of air flow through lungs facilitates an

economical and better understanding of the physics of breathing which in turn provides us with a better insight of lung health. Compared to other common experimental techniques like particle image velocimetry measurements, CFD study of human respiration provides us a framework for detailed investigation of breathing under controlled conditions [13]. CFD studies are especially important in order for us to be able to predict respiratory diseases, say for instance, due to aging, chronic obstructive pulmonary disease, asthma and bronchitis to name a few [14]. A thorough understanding of the air flow patterns would assist early detection and cure of certain diseases. However in order to conduct a CFD study of air flow through the whole respiratory system is computationally expensive due to the different geometrical scales involved in the study and also due to the computational resources available to us.

In order to accurately and economically solve the above described problem, three cases of air way models involving the 2-3 generation was developed. In the first case, see Fig. 1a, an idealized geometry of the 2-3 generation was developed based on the dimensions of lung obtained from Weibel [15]. In order to simulate the effects of aging, as discussed above, the dimensions of the typical lower lung bifurcation were shrunk by 20% uniformly in the second case, see Fig. 1b, due to the well-known fact that the lung bifurcations shrink with age and disease. Finally in the third case, see Fig. 1c, the dimensions of the left and right airways of the typical lower lung bifurcation were shrunk by 20% on the left and 10% on the right airways respectively. It must be noted here that all the shrinkage was assumed to occur within the airways and not within the lumen. All the three cases considered in this study were meshed using ANSYS CFX software with tetrahedron elements. Careful attention was paid to ensure that the mesh density did not affect the accuracy of the CFD calculations. The mass flow rate of air entering and leaving the bifurcation at each time step was monitored and sufficient iterations were allowed so that the net mass flow rate of air passing through the bifurcation at each time step is zero. Each bifurcation was meshed with one million tetrahedron elements.

MODELING OF NORMAL BREATHING AND MECHANICAL VENTILATION

Flow field through the lung bifurcations was predicted using ANSYS Fluent software by solving the Navier-Stokes equations. Flow field through the lung bifurcations is estimated using the mass and momentum conservation equations i.e. the Navier-Stokes equations given as:

( u ) 0iPt

ρ∂+∇ ⋅ =

∂ (1)

2i i i

jj i j

u u uu P

t x x xρ ∂ ∂ ∂

+ = −∂ + ∂ ∂ ∂ ∂ (2)

Here u is the velocity and P is the pressure differential and the equation is expressed in Einstein notation. Equations 1 and 2 were solved subjected to a velocity boundary condition at the inlet and pressure boundary condition at the four outlets assuming flow exits the bifurcation unobstructed. No-slip boundary conditions are imposed at the walls. For the pressure and velocity conditions occurring in the lung geometry, the flow is assumed to be incompressible.

A schematic representation of the boundary conditions imposed and can be found in Fig. 2. The area of inlet is 0.0001328 m2, areas of outlets 1, 2, 3 and 4 are 5.45e-5 m2, 5.45e-5 m2, 6.27e-5 m2 and 6.27e-5 m2 respectively. Two different cases of inhalation, normal breathing and mechanical ventilation were studied.

The inspiration to expiration ratio was set at 1:4 which corresponds to a 0.4 s inhalation and 1.6 s exhalation times. Inhalation waveform with a constant flow rate and exhalation with an exponentially decreasing flow rate was used as the inlet boundary condition in the latter case, see Fig. 3. The time dependent inlet velocity profiles shown in Fig. 3 are used as the inlet boundary conditions. The Reynolds number at the inlet for

a)

Figure 1. Schematic of reference typical lower lung bifurcation (a) shrunk uniformly by 20% of its original volume (b) and shrunk at random locations along its

length by a predetermined volume percentage of 10% or 20% (c)

20% shrinkage

b)

10% shrinkage

20% shrinkage

c)


all the cases was calculated to be 2190 indicating that the flow is laminar. A C++ subroutine was used to enhance the ANSYS solver capabilities for specifying the inlet boundary conditions discussed above. The C++ routine was used as the time dependent velocity profile cannot be specified in the ANSYS Fluent code directly. The pressure outlet boundary condition was imposed at the outlet and a no-slip boundary condition at the wall.

Several transient finite difference simulations with a time step size of 0.01 seconds were carried out using this computational fluid dynamics analysis outlined above. The values of pressure and wall shear stress obtained from the simulation results at predetermined locations throughout the lung bifurcations were used to determine the relationships between flow waveform and the pressure/wall shear stress.

RESULTS AND DISCUSSION Using the methodology described in the previous sections,

a prediction of velocity, pressure, mass flow rate and wall shear stress distributions were obtained. It must be noted here that the total time taken for one complete cycle of normal inhalation to occur is 4 seconds whereas during the mechanical ventilation conditions it is 2 seconds, as it the case during clinical mechanical ventilation. In order to compare different flow and physical properties, a normalized time is defined.

* c

total

ttt

= (3)

Here tc is the instantaneous breathing time and ttotal is the total breathing time for one cycle.

The velocity field of the flow in the bifurcation was obtained from the simulations. A representative figure showing the velocity vectors through a lung bifurcation during normal

inhalation at a t*=1 can be seen in Fig. 4. Further investigation of velocity field at different locations under normal breathing conditions can be seen in the three different cases of lung bifurcation discussed in Fig. 1 can be seen in Fig. 5. The figures 5a-c reveal the velocity magnitudes at the inlet and outlets in the different cases under investigation. This is due to the fact that there is no change in the area of cross section of the inlet. However due to shrinkage of the left and right airways in cases two and three a difference in velocity magnitudes is observed when compared to the first case. 15% change in velocity can be observed indicating obvious effect of outlet cross sectional areas on the velocity magnitude. It should also be noted here that the velocity magnitude follows a piecewise sinusoidal pattern as it is directly influenced by the breathing profile imposed as a boundary condition at the inlet. Similarly velocity at different locations during mechanical ventilation is presented in Fig. 6.

Volumetric flow rate at different cross-sections in the lung bifurcations for normal and mechanical ventilation waveform can be seen in Fig. 7 and Fig. 8 respectively. Careful examination of the results reveal that the mass flow rate of air through different lung airways in the three different cases studied here depend on the airways’ cross sectional area. A negative mass flow rate here indicates that an exhalation event is taking place in the lung and the flow direction is reversed. It can also be seen that the mass flow rate follow the same profiles as the normal and mechanical ventilation wave forms.

The pressure field across the different bifurcations was

obtained from the simulations. A representative contour plot of the pressure for case 1 of the lung bifurcation is shown in Fig. 9 for a t*=1. Similarly the pressure gradient across each location indicated in Fig. 2 is presented in Fig. 10 for normal breathing condition and in Fig. 11 for mechanical ventilation condition. From the figures it can be seen that, for similar cases, there is a higher value of pressure gradient in the location where a higher mass flow rate is observed in Figs. 7 and 8. Also approximately 10-20% of increase in pressure drop is seen in places with volume shrinkage (cases 2 and 3) when compared to case 1.

Figure 3. Inhalation waveforms

0 1 2 3 4-4

-3

-2

-1

0

1

2

3

4

time (seconds)

velo

city

(m/s

)

mechanical ventilationnormal breathing

Figure 2. Schematic of the lung bifurcation indicating the boundary conditions and locations


Finally in Figs. 12 and 13 the area weighted average values

of the wall shear stress over the entire bifurcation at different time steps for the three different cases are presented. It can be seen that in both breathing conditions, 15-20% higher values of wall shear stresses are obtained in cases 2 and 3 compared to case 1. This is due to the fact that lowering the area of cross section of the bifurcations will result in larger mass flow rate given that the tidal volume of the lung remains constant. Also by comparing the wall shear stress vales of normal breathing and mechanical breathing conditions, it can be seen that the vales of wall shear stress are larger by about 43% in the case of mechanically ventilated lung. This indicates that strain induced inflammation is higher by 27% in the case of lung bifurcations which have shrunk by 20% in volume and more importantly during mechanical ventilation.

CONCLUSION The effects of lung shrinkage and aging were quantitatively

predicted using the current technique. Effect of bifurcation geometry on velocity, pressure, mass flow rate and wall shear stress distributions during two different breathing conditions was studied. From the studies, the levels of pressure and wall shear stress obtained indicate that the lung walls at the typical lower lung bifurcation generation experience forces close to 0.2 Pa that are significant enough to cause considerable tissue deformation. The results also indicate that the airway pressures obtained in normal bifurcation are lower by 15% compared to the aged lung bifurcation. Obviously, this is due to the fact that the shrinkage in the lung bifurcation due to aging causes a restricted air passage. Due to the shrinkage of the lung causing increased pressure drop; the amount wall shear stress experienced by the lung tissue also increases.

AKNOWLEDGEMENT This work was supported by the National Institute of Health grant # R01AG041823. The authors gratefully acknowledge their support.

Figure 5. Variation of velocity at different normalized time (t*=tc/ttotal) under normal breathing at different locations. (a) normal lung bifurcation, (b) values for case 1 lung

bifurcation, (c) values for case 2 lung bifurcation

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

4

time

velo

city

(m/s)

inletoutlet 1outlet 2outlet 3outlet 4

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

4

time

velo

city

(m/s)


0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

4

time

velo

city

(m/s)


a)

c)

b)

Figure 4. Velocity vectors at t*=1 in a normal lung


Figure 7. Variation of mass flow rate at different normalized time (t*=tc/ttotal) under normal breathing at

different locations. (a) normal lung bifurcation, (b) values for case 1 lung bifurcation, (c) values for case

2 lung bifurcation

0.2 0.4 0.6 0.8 1

-6

-4

-2

0

2

4

6

x 10-4

time

q'' (

m3 /s

)

plane 1plane 2plane 3plane 4plane 5plane 6

0.2 0.4 0.6 0.8 1

-6

-4

-2

0

2

4

6

x 10-4

time

q'' (

m3 /s

)


0.2 0.4 0.6 0.8 1

-6

-4

-2

0

2

4

6

x 10-4

time

q'' (

m3 /s

)


a)

b)

c)

Figure 6. Variation of velocity at different normalized time (t*=tc/ttotal) under mechanical ventilation at


2 lung bifurcation

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

4

time

velo

city

(m/s

)


0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

4

time

velo

city

(m/s

)


0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

4

time

velo

city

(m/s

)


a)

b)

c)


Figure 10. Variation of pressure (Pa) at different normalized time (t*=tc/ttotal) under normal breathing at


2 lung bifurcation

0.2 0.4 0.6 0.8 110-6

10-5

10-4

10-3

10-2

10-1

100

101

time

pres

sure

(Pa)


0.2 0.4 0.6 0.8 110-6

10-5

10-4

10-3

10-2

10-1

100

101

time

pres

sure

(Pa)


0.2 0.4 0.6 0.8 110-6

10-5

10-4

10-3

10-2

10-1

100

101

time

pres

sure

(Pa)


a)

b)

c)

Figure 9. Contours of pressure in Pa at time 1 in a normal lung bifurcation

Figure 8. Variation of mass flow rate at different normalized time (t*=tc/ttotal) under mechanical

ventilation at different locations. (a) normal lung bifurcation, (b) values for case 1 lung bifurcation, (c)

values for case 2 lung bifurcation

0.2 0.4 0.6 0.8 1

-6

-4

-2

0

2

4

6

x 10-4

time

q'' (

m3 /s

)


0.2 0.4 0.6 0.8 1

-6

-4

-2

0

2

4

6

x 10-4

time

q'' (

m3 /s

)


0.2 0.4 0.6 0.8 1

-6

-4

-2

0

2

4

6

x 10-4

time

q'' (

m3 /s

)


a)

b)

c)


Figure 12. Variation of wall shear stress (Pa) at different normalized time (t*=tc/ttotal) under normal breathing at different locations. (a) normal lung

bifurcation, (b) values for case 1 lung bifurcation, (c) values for case 2 lung bifurcation

0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

time

σ (P

a)

0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

time

σ (P

a)

0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

time

σ (P

a)

a)

b)

c)

Figure 11. Variation of pressure (Pa) at different normalized time (t*=tc/ttotal) under mechanical

ventilation at different locations. (a) normal lung bifurcation, (b) values for case 1 lung bifurcation, (c)

values for case 2 lung bifurcation

0.2 0.4 0.6 0.8 110-6

10-5

10-4

10-3

10-2

10-1

100

101

time

pres

sure

(Pa)


0.2 0.4 0.6 0.8 110-6

10-5

10-4

10-3

10-2

10-1

100

101

time

pres

sure

(Pa)


0.2 0.4 0.6 0.8 110-6

10-5

10-4

10-3

10-2

10-1

100

101

time

pres

sure

(Pa)


a)

b)

c)


REFERENCES [1] Comerford A., Forster C. and Wall W. A., Structured Tree Impedance Outflow Boundary Conditions for 3D Lung Simulations, J. Biomech. 132, 081002-1 – 081002-10 (2010). [2] Calay, R.K., Kurujareon, J., Holdo, A.E., 2002. Numerical simulation of respiratory flow patterns within the human lung. Respiratory Physiology & Neurobiology 130, 201-221. [3] Lee, D.Y., Lee, J.W., 2002. Dispersion of aerosol bolus during one respiration cycle in a model of pulmonary airways. J Aerosol Sci 33, 1219-1234. [4] Liu, Y., So, R.M.C., Zhang, C.H., 2003. Modeling the bifurcating flow in an asymmetric human pulmonary airway. J. Biomech. 36, 951-959. [5] Van Ertbruggen, C., Hirsch, C, Paiva, M., 2005. Anatomically based three-dimensional model of airways to simulate flow and particle transport using computational fluid dynamics. J. Appl. Physiol. 98(3), 970-980. [6] H.Y.Luo, Y.Liu, X.L.Yang, Particle deposition in obstructed airways, J.Biomech.40 (2007)3096–3104. [7] C.Kleinstreuer, H.Shi, Z.Zhang, Computational analyses of a pressurized metered dose inhaler and a new drug-aerosol targeting methodology, J. Aerosol.Med. 20(2007) 294–309. [8] Snyder, B., Dantzker, D.R., Jaeger, M.J., 1981. Flow partitioning in symmetric cascades of branches. . J. Appl. Physiol. 51(3), 598-506. [9] Zhao, Y., Lieber, B.B., 1994. Steady inspiratory flow in a model symmetric bifurcation. J Biomech Eng-T Asme 116, 488-496. [10] Liu, Y., So, R.M.C., Zhang, C.H., 2003. Modeling the bifurcating flow in an asymmetric human pulmonary airway. J. Biomech. 36, 951-959. [11] Gibson G. J. and Pride N. B., Pulmonary mechanics in fibrosing alveolitis: the effects of lung shrinkage, The American Review of Respiratory Disease, 1977, 116(4):637-647 [12] J.B. West, Pulmonary Pathophysiology: The Essentials, Wolters Kluwer Health/ Lippincott Williams & Wilkins, Philadelphia, 2008. [13] Koombua K. and Pidaparti R. M., 2008, “Inhalation induced Stresses and Flow charecteristics in Human airways through Fluid-structure interaction Analysis”, Modelling and Simulation in Engineering, 358748 (8). [14] J.Vestbo, S.S.Hurd, A.G.Agusti, P.W.Jones, C.Vogelmeier, A.Anzueto, P.J.Barnes, L.M.Fabbri, F.J.Martinez, M.Nishimura, R.A.Stockley, D.D.Sin, R. Rodriguez-Roisin, Global Strategy for the diagnosis ,management, and prevention of chronic obstructive pulmonary disease:GOLD executive sum-mary, Am.J.Respir.Crit.CareMed.187(2013)347–365. [15] Weibel, E.R., 1963. Morphometry of the Human Lung, Springer Verlag, Berlin.

Figure 13. Variation of wall shear stress (Pa) at different normalized time (t*=tc/ttotal) under

mechanical ventilation at different locations. (a) normal lung bifurcation, (b) values for case 1 lung bifurcation, (c) values for case 2 lung bifurcation

0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

time

σ (P

a)

0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

time

σ (P

a)

0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

time

σ (P

a)a)

b)

c)



PREDICTION OF NATURAL MODE SHAPES AND FREQUENCIES OF LUNG AIRWAYS USING FINITE ELEMENT ANALYSIS

Dawn A. Dagen, Raghu Arambakam, Ramana M. Pidaparti College of Engineering, University of Georgia

Athens, GA USA

ABSTRACT This study focuses on the modal analysis of lung airway

bifurcations to estimate their natural mode shapes and frequencies. The maximum principle elastic strain under deformation that develops due to the first five modes in two different lung bifurcations was studied. The bifurcation was assumed to be structurally fixed at three different locations. The natural mode shapes and frequencies of the lung airway bifurcation under these different cases were also studied. This study is especially important in cases where a person is subjected to mechanical ventilation, as the forcing of air into the respiratory system by external means causes the lungs to resonate at certain frequencies. It is important to estimate the natural mode shapes and frequencies of the lungs in order to minimize injury due to vibration and deformation caused by artificial respiration. From the results it was observed that the third mode of vibration caused the maximum deformation in the lung airways. Also the maximum shear strain occurs during the third vibrational mode. It was also found that material properties of the lung significantly affected deformation and shear strain values.

INTRODUCTION Structural characteristics of human airways are very

important for studying the phenomenon of gas exchange and drug delivery. Understanding the stresses and strains which occur in the lung airways during respiration process is important in predicting and diagnosing various respiratory diseases. It is not uncommon to subject diseased patients to assisted breathing techniques such as mechanical ventilation. Mechanical ventilation requires air to be pumped into the lungs with external force, which causes the lungs to vibrate at a different frequency than during natural breathing. The different ventilator modes used for mechanical ventilation can cause varying deformation of lung tissues [1-4]. In order to estimate deformations and vibrations occurring in the lungs during breathing, computational and experimental studies have been conducted in the past [4-6]. It was also discovered that lung tissue hemorrhage occurs due to use of ultra sound based imaging techniques [6].

In this study, the natural mode shapes and frequencies of lung bifurcations were predicted using modal analysis in ANSYS. The software was able to estimate the shear stresses caused by the natural resonance of lung bifurcations due to vibrations induced by mechanical ventilation. Maximum principle elastic shear stress was measured in three different cases of structurally fixed lung bifurcation models. The four different cases where the lung bifurcations were assumed to be made of a tissue having different Young’s moduli and densities were also studied. In one case Young’s modulus was reduced by 20%, and reduced by 40% in a second case with respect to the normal tissue Young’s Modulus. In these two cases the density of the tissue material was assumed to be constant. In another case, the tissue density was reduced by 10% keeping the Young’s Modulus constant.

LUNG AIRWAY DEFORMATION Due to cyclic loading of the lung tissue in natural breathing

and, in some cases mechanical ventilation, inflammation of the lung tissue occurs [7]. Studies have been conducted to measure/predict strain induced in airway tissues due to deformation of the same airway tissue [8]. It is also well known from literature that structural changes occur in lung airways due to patient aging or disease [9-12].

The natural frequencies and mode shapes are important parameters in the analysis of deformation in the lung airways under dynamic loading conditions such as mechanical ventilation. Since it is not possible to economically perform experiments to investigate the mode shapes of the lung airways under mechanical loading conditions, Finite Element Analysis (FEA) of such problems will facilitate a better understanding of mechanical ventilation induced lung injuries.

In order to accurately and economically solve the above described problem, four cases of air way models involving the 4-5 generation was developed (see Fig. 1). In the first case (see Fig. 1) geometry of the 4-5 generation was developed based on lung dimensions obtained from Weibel [13]. In order to simulate the effect of lung stiffness on natural mode shapes and shear stress, Young’s modulus and density of the lung airway generation was varied. Firstly the 4-5 lung generations were subjected to the first three natural modes with respectively


using Young’s modulus and lung density obtained from the literature [14, 15]. The lung tissue airways were assumed to have homogenous material properties. All cases considered in this study were meshed using ANSYS software with tetrahedron elements. Careful attention was paid to ensure that the mesh density did not affect accuracy of the numerical calculations. Each bifurcation was meshed with one million tetrahedron elements.

MODELING OF LUNG AIRWAYS’ NATURAL MODE SHAPES AND FREQUENCIES

The natural mode shapes and frequencies of the lung airways were computed using the dynamic modal analysis in ANSYS Mechanical software assuming free vibraations. The lung bifurcations shown in Fig. 1 were discretized using half a million tetrahedron elements.

Figure 1. Schematic of the 4-5 lung bifurcation used in the

current study airway is shown as shadow. The natural mode shapes due to the first three modes were studied by solving the force balance equation: [ ][ ] [F]v k = (1)

where v is the velocity matrix, k is the stiffness matrix and F is the resultant force vector. The unit quantities are all in SI. The stiffness matrix remains constant throughout the simulations with a velocity matrix that varies based on imposed mode. This equation is solved using the ANSYS Mechanical software to obtain mode shape and frequencies for the 3D lung bifurcation model.

The inlet and outlet walls of the lung airways were assumed to be fixed; therefore the fixed support boundary condition was imposed in the ANSYS Mechanical software. In the first case the lung model was assumed to have a Young’s modulus of 102.48 kPa from the work of Prakash and Hyatt [15]. A corresponding density value of 1356.6 kg/m3 was used along with a Poisson’s ratio of 0.45 taken from the works of Sera et al. and Prakash and Hyatt respectively [14, 15]. In the second and third cases, values of Poisson’s ratio and density remained unchanged. In the second case the value of Young’s modulus was reduced by 20% to 81.98 kPa. Similarly in the third case the Young’s modulus was reduced by 40% to 61.488 kPa. In the fourth case the density of the airway tissue was reduced by 10% with a density of 1220.94 kg/m3.

Using the above material properties and boundary conditions the lung airway bifurcations were subjected to their respective first three natural modes. The corresponding mode shapes and maximum shear elastic strains were monitored.

RESULTS AND DISCUSSION Maximum elastic shear strain occurring at different mode

shapes in lung airway bifurcations with different Young’s moduli and densities was obtained using FEA analysis outlined in the previous sections. Shapes of the lung airways during the first three natural modes were obtained.

The mode shapes of the lung bifurcation during the first three natural modes can be seen in Figs. 2-4 using the material properties of case 1. For the sake of brevity only the maximum elastic shear strain contours are shown at un-deformed and maximum deformed positions. From Figs. 2-4 it can be observed that the maximum deformation occurs in the lung airways during the third mode occurring at a frequency of 2.7987 Hz (see Fig. 4c). It can also be noted from this figure that the maximum shear strain occurs during this maximum deformed mode shape.

Similarly, the mode shapes of the lung bifurcation during the first three natural modes can be seen in Figs. 5-7 using the material properties of case 2. The maximum elastic shear strain contours are shown at undeformed and maximum deformed positions. From Figs. 5-7 it can be observed that the maximum deformation occurs in the lung airways during the third mode occurring at a frequency of 2.5033 Hz (see Fig. 7c). It can also be noted from this figure that the maximum shear strain occurs during this maximum deformed mode shape.

In addition, the mode shapes of the lung bifurcation during the first three natural modes can be seen in Figs. 8-10 using the material properties of case 3. The maximum elastic shear strain

outlet 1outlet 2

outlet 3outlet 4

inlet


contours are shown at undeformed and maximum deformed positions. From Figs. 5-7 it can be observed that the maximum deformation occurs in the lung airways during the third mode occurring at a frequency of 2.7987 Hz (see Fig. 10c). It can also be noted from this figure that the maximum shear strain occurs during this maximum deformed mode shape.

By studying these first three cases it is evident that the maximum deformed mode shape occurs during the third vibrational mode in bifurcation.

Also the effect of lung airway tissue density was studied in the case of a third vibrational mode with the material properties of case 4 (see Fig. 11). It was verified that varying the density, while keeping the Poisson’s ratio and Young’s modulus constant, will have a natural modal frequency of 2.7987 Hz. It was also verified that the third natural mode yielded a maximum deformation in shape and hence the maximum shear strain. However, results of the first two modes with the material properties of case 4 were not presented here.

CONCLUSION Effect of lung airway bifurcation geometry and material

properties on the mode shapes, natural frequencies and shear strains were studied using Finite element analysis. From the studies, modal shape (i.e. deformation) and shear strains experienced by the 4-5 generation of the lung airway bifurcation show that the maximum shear and deformation occurs during the third mode. The results also indicate that the shear strain values vary significantly with change in material properties. This is due to the fact that deformation caused by the modal vibrations is a function of material properties.

AKNOWLEDGEMENT This work was supported by the Unites States National

Science Foundation grant CMMI-1430379. The authors gratefully acknowledge their support.

Figure 2. Contours of Maximum shear elastic strain occurring in a lung bifurcation with material

properties of case 1 subjected to the first vibrational mode during maximum modal

deformations (a) and (c) and with no modal deformation (c). The original position of the

lung.

a)

b)

c)


a)

b)

c)

Figure 4. Contours of Maximum shear elastic strain occurring in a lung bifurcation with

material properties of case 1 subjected to the third vibrational mode during maximum modal

deformations (a) and (c) and with no modal deformation (c). The original position of the lung

airway is shown as a shadow.

a)

b)

c)


material properties of case 1 subjected to the second vibrational mode during maximum modal


air way is shown as a shadow.


a)

b)

c)





a)

b)

c)


material properties of case 2 subjected to the first vibrational mode during maximum modal deformations (a) and (c) and with no modal

deformation (c). The original position of the lung airway is shown as a shadow.


a)

b)

c)


material properties of case 3 subjected to the first vibrational mode during maximum modal deformations (a) and (c) and with no modal

deformation (c). The original position of the lung airway is shown as a shadow.

a)

b)

c)






a)

b)

c)





a)

b)

c)






REFERENCES [1] Gluck E., Sarrigianidis A., Dellinger R.P., 2001, “Mechanical Ventilation. In Critical Care Medicine: Principles of Diagnosis and Management in the Adult”, 2nd edition. Parrillo J.E., Dellinger R.P., St. Louis: Mosby, pp.137-161. [2] Esteban A, Hanzueto A., Alia I., et al., 2000,: “How is mechanical ventilation employed in the intensive care unit? An international utilization review” Am J Respir Crit Care Med, pp. 161; 1450-1458. [3] Baker A.B., Colliss J.E., Cowie R.W.,1977, “Effect of varying inspiratory flow waveform and time in intermittent positive pressure ventilation”, Various physiological variables, Br J. Anaesth, 49, pp.1221-1234. [4] Koombua K. and Pidaparti R. M., 2008, “Inhalation induced Stresses and Flow charecteristics in Human airways through Fluid-structure interaction Analysis”, Modelling and Simulation in Engineering, 358748 (8). [5] R. Phillip Dellinger, Smith Jean, Ismail Cinel, Christina Tay, Susmita Rajanala, Yael A. Glickman and Joseph E. Parrillo, 2007, “Regional distribution of acoustic-based lung vibration as a function of mechanical ventilation mode”, Critical care, 11 (1). [6] D. John Jabaraj and Mohamad Suhaimi Jaafar, 2013, Vibration Analysis of Circular Membrane Model of Alveolar Wall in examining Ultrasound-induced Lung Hemorrhage, Journal of Medical Ultrasound (21), pp. 81-91. [7] Pidaparti R.M. and Koombus K., Tissue straininduced in airways due to mechanical ventilation, MCB 180, 1-20 (2010). [8] M. A. Swartz, D. J. Tschumperlin, R. D. Kamm, and J. M. Drazen, 2001, “Mechanical stress is communicated between different cell types to elicit matrix remodeling”, PNAS, pp.98, 6180-6185. [9] Chetta A., Foresi A., del Donno M., Bertorelli G., Pesci A. & Olivieri, D.,1997, “Airways Remodeling Is a Distinctive Feature of Asthma and Is Related to Severity of Disease, Chest,” pp. 111, 852–857. [10] Laitinen, A. & Laitinen, L. A., 1994, Am. J. Respir. Crit. Care Med.,pp. 150, S14–S17. [11] Kuwano, K., Bosken, C. H., Pare, P. D., Bai, T. R., Wiggs, B. R. & Hogg, J. C., Airway morphology: epithelium/basement membrane, 1993, Am. Rev. Respir. Dis. 148, 1220–1225. [12] Jeffery, P. K., 1992, “Histological features of the airways in asthma and COPD”, Respiration (59), Suppl. 1, pp. 13–16. [13] Weibel, E.R., 1963, “Morphometry of the Human Lung”, Springer Verlag, Berlin. [14] T. Sera, S. Satoh, H. Horinouchi, K. Kobayashi, and K. Tanishita, 2003, “Respiratory flow in a realistic tracheostenosis model,” Journal of Biomechanical Engineering, vol. 125, no. 4, pp. 461–471. [15] U. B. S. Prakash and R. E. Hyatt, 1978, “Static mechanical properties of bronchi in normal excised human lungs,” Journal of Applied Physiology, vol. 45, no. 1, pp. 45–50

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GRAVITATIONAL STABILIZATION METHODS FOR HYDROCODES

Gerald Pekmezi and David Littlefield Department of Mechanical Engineering, The University of Alabama at Birmingham,

Birmingham, AL USA

Stephen Akers and Kent Danielson US Army Corps of Engineers, Engineer Research and Development Center

Vicksburg, MS USA

ABSTRACT Highly dynamic events in many civilian and military

applications are modeled using a specialized type of simulation code, called a hydrocode. Because of the large deformations typical to these types of events, explicit time-stepping is used to advance the solution. A common drawback of this approach is the inability to model long term quasi-static loading, such as that enacted by gravity and atmospheric pressure.

In this paper, two damping approaches for stabilizing an explicit Finite Element model with gravity are implemented and evaluated. The latter approach, dynamic relaxation, is implemented in the hydrocode package Epic as well as in the package Pronto3D. Some examples are provided of the applicability and effect of these methods.

INTRODUCTION In the last three decades, with the advent of popular

computing, numerical methods and their computational implementations have become essential to engineering applications. The Finite Element Method and the Finite Volume Method have become particularly popular in modeling solids and fluids, respectively.

One area where these simulation tools have seen increasing use, is in the modeling of high velocity, high energy, and highly dynamic events with large deformations. However, the Finite Element formulation typically used to model solid materials has a difficult time dealing with such extreme events, which necessitates a different approach. This approach is one that treats solids more like liquids, by using Conservation of Mass

𝜕𝜌𝜕𝑡

+ ∆ ∙ (𝜌𝒖) = 0 (1)

Conservation of Energy:

𝜕𝐸𝜕𝑡

+ ∆ ∙ (𝒖(𝐸 + 𝜌)) = 0 (2)

and Conservation of Momentum

𝜕(𝜌𝒖)𝜕𝑡

+ ∆ ∙ (𝒖 × (𝜌𝒖) = 0 (3) An Equation of State describes how pressure affects density and internal energy.

𝑝 = 𝑝(𝜌, 𝐼) (4) Finally, a Constitutive Model is of course required to describe the stress in the material due to the strain, strain rate effects, internal energy, and damage.

𝜎𝑖𝑗 = 𝑔(𝜖𝑖𝑗 , 𝜖��𝑗 , 𝐼.𝐷) (5) Notwithstanding the solid-like strength model and the strict contact handling used, such a code is essentially a Computational Fluid Dynamics code, hence the name “hydrocode”.

Popular commercial hydrocodes include LS-DYNA, ABAQUS (Explicit), AUTODYN. Other popular hydrocodes were developed by and remain in use for primarily government and military applications. The latter include CTH, EPIC [1], and PRONTO3D [2].

EPIC and PRONTO3D are used extensively to model a variety of situations by the CSM group at UAB Mechanical Engineering, including detonations, high velocity impacts etc. In a number of simulations carried out, the behavior of geomaterials (soil and rock) has been of particular interest.

The stress and deformation in geomaterials resulting from loading by energetic materials or impact from projectiles is a specific problem of interest in a variety of civilian and military applications. One concrete example is the loading of earthen dams and berms from explosive charges. In order to conduct accurate and meaningful simulations of this problem, it is sometimes necessary to include gravitational forces in the


calculation. Late-time evolution of the crater is one feature in particular that is influenced significantly by gravity. While hydrocodes are particularly well-suited for simulation of explosive loading and impact by projectiles, they are not designed to model inherently steady-state phenomena such as stable gravitational stress fields. Using a hydrocode to initialize a gravitational stress field typically results in ringing of the solution around a value at or near its steady-state response.

In the present work, two means for initializing stable gravitational stress fields for use in hydrocodes are discussed. The basic premise of both techniques is that a means for artificial damping must be included in the solution. The damping must be dissipative enough to remove oscillations in the solution but weak enough as to not influence the final values for the stress field.

DAMPING BY ARTIFICIAL VISCOSITY The first technique exploits the bulk artificial viscosity

algorithm incorporated into nearly all hydrocodes. The need for such an algorithm arises from the difficulty in effectively capturing and handling shocks with the finite length cells and elements of discretized-model simulations. The difficulty is twofold. First, high velocity gradients that accompany the shock can cause collapse of an element if the velocity exceeds the speed of the structural response. Second, oscillations in the structural response will cause unceasing “ringing” of the model if left undamped.

The technique typically applied to mitigate these potential pitfalls consists of adding the viscous term of Equation (6) to the volumetric (bulk) response [3].

𝑞 = 𝑏1𝜌𝑐��𝑉

+ 𝜌 �𝑏2𝑙��𝑉�2 (6)

The effect of this term may physically be described as adding a viscous “pressure” which smears the shock through the quadratic term b2 and damps out oscillations through the linear term b1.

By increasing linear and quadratic artificial viscosity coefficients, a means for effectively damping the stress field is obtained. Typically it is the quadratic term that is of great importance, as insufficient resolution of a shock can lead to instability, while excessive smearing of the shock will mask the real structural response. However, the present application requires smoothing of the response to a constant, long-term, and shock-free gravitational force. Obviously, the quadratic term is then less important than the linear one.

An example application of this method is demonstrated in Figures 1 and 2. In this calculation a 1 inch square, 100 inch deep column of soil is subjected to a gravity load, with a 14.7 psi load applied at the top of the column. The calculation was performed using the EPIC code [1]. Progression of the stress at the base of the column for different values of the linear viscosity coefficient is shown in Figure 1.

Figure 1. Normal Stress Oscillations at Base of Soil

The green line represents the undamped response and the

blue line the damped response using a coefficient of 25. As is evident, the solution oscillates about its expected value of 22 psi. Critical damping (i.e. damping with no oscillation or rebound whatsoever) is achieved with a coefficient of 336 and converges to the expected result; this is shown in Figure 2.

Figure 2. Near-critically Damped Normal Stress

There are two issues that arise from practical application of

this technique. First, the value for critical damping would ideally be used

in a calculation to achieve the desired steady-state stress field. However, this value is unknown a priori and in practice can only be found from numerical experiments.

Second, this type of damping affects the stable time increment that can be used to advance a simulation. In fact the relationship between the linear damping term and the stable time increment is described by Equation (7)

∆𝑡

∆𝑡𝑢𝑛𝑑𝑎𝑚𝑝= �1 + 𝑏1

2 − 𝑏1 (7)


The left hand term of equation may be considered a time step reduction factor. The relationship is further illustrated by the plot of Figure 3

Figure 3. Time Step Reduction Factor for Bulk Viscosity

Increasing the value for the linear viscosity coefficient to very large value can severely impair the ability to conduct a simulation in a reasonable length of time. To achieve near-critical damping in the example above, the linear coefficient was increased to a value of 336, which led to a time step 600 times smaller than the undamped stable time step. For smaller models, such as the example above, this is still feasible. However it obviously represents a severe problem for larger, more complex models.

DAMPING BY DYNAMIC RELAXATION The second technique available makes use of another

artificial construct to introduce damping into the solution [4]. The implementation is easily illustrated by considering the momentum equation solved in discretized form given by Equation (8).

𝑀𝑎𝑛 + 𝐶𝑣𝑛 + 𝑄𝑛 = 0 (8)

where M is the mass matrix, a is the acceleration, v is the velocity, Q represents the resultant forces, and the superscript n refers to time step n [5]. Here C represents an artificial damping matrix included in the solution to introduce the required damping. The velocity field is obtained from the difference formula of Equation (9).

𝑎𝑛 = 1

∆𝑡�𝑣𝑛+1/2 − 𝑣𝑛−1/2� (9)

Hence an update for the velocity field at the (n+1/2) time step is given by Equation (10) [6].

𝑣𝑛+12 = 2−𝐶∆𝑡

2+𝐶∆𝑡𝑣𝑛−

12 + 2∆𝑡

2+𝐶∆𝑡𝑀−1𝑄𝑛 (10)

An approximate form which neglects the contribution of damping to the second term is given by Equation (11).

𝑣𝑛+

12 = 𝜇𝑣𝑛−

12 + 𝛼𝑛∆𝑡 (11)

where 𝜇 is a prescribed coefficient.

This method was implemented into both the EPIC as well as PRONTO3D codes and the soil column problem described previously was rerun with 𝜇 = 0.995. Figure 4 illustrates the relaxed and unrelaxed normal stress response at the bottom of the soil column in EPIC.

Figure 4. Relaxed vs. Unrelaxed Response in EPIC

Figure 5 illustrates the relaxed and unrelaxed normal stress response at the bottom of the soil column in PRONTO3D. It should be noted the PRONTO3D plot is run out to a further simulation time in order to further contrast the difference between the relaxed and unrelaxed response.

Figure 5. Relaxed vs. Unrelaxed Response in PRONTO3D

0 100 200 3000.001

0.01

0.1

1

Linear Coefficient

Tim

e St

ep F

acto

r


It is important to take note of both the similarity and contrast between the responses obtained by the two damping methods. An identical steady state solution to that shown in Figure 2 was obtained, but the stable time step remained unaltered. The stable time step with critical damping through artificial viscosity was 600 times smaller, which means the analysis damped through Dynamic Relaxation ran 600 times faster than the one damped through Artificial Viscosity.

One limitation that persists with Dynamic Relaxation, is the inability to have a priori knowledge of the ideal factor for critical damping. However, the default implemented factor of 0.995 is typically good enough to ensure near critical damping [5]. Figure 6 shows the influence of the relaxation factor on the stress evolution of the example model analyzed with EPIC. The stress response history due to a number of different relaxation factors is plotted along with the “No Relaxation” history.

Figure 6. Stress History for Different Relaxation Factors

SAMPLE APPLICATIONS Gravitational stabilization through Dynamic

Relaxation is utilized in two sample applications. These examples will illustrate the need for such a tool.

Buried Charge Benchmark The first application is a ‘buried charge’ benchmark

model. A disk of explosive material is buried just below the surface of a soil material and a steel plate is suspended above the soil. Figure 7 illustrates the geometry of the model and the discretization used to run the simulation.

Figure 7. Buried Charge Benchmark Model

Figure 8 illustrates the settlement displacements in the soil after stabilization under gravity and atmospheric pressure.

Figure 8. Settlement Displacement in Benchmark Model

Figure 9 shows the stress response in the soil after gravity initialization at different relaxation factors.


Figure 9: Stress Response of Benchmark Model

Figure 9 illustrates how subcritical damping of the model can lead to inaccurate results. The underdamped stress response history at the relaxation value of 0.9999 converges to a stress value about 33% lower than the supercritically damped models dictate. The reason for this is that once the stress exceeds the supercritically converged value, excessive permanent (plastic) strains accumulate. Once the stress rebounds these excessive plastic strains lead to lower long-term stress values. Figure 10 and Figure 11 show the undamped and the damped simulations side by side. Figure 10 shows the models 4 ms after detonation. At this point in time, the responses still look nearly identical.

Figure 10. Response 4ms after Detonation

Figure 11 shows the response 11 ms after detonation. By this point the responses have clearly diverged.

Figure 11. Response 11ms after Detonation

Figure 11 illustrates one of the important motivations for the current work, previously mentioned in the introduction. The later-time evolution of the crater of a detonation, is influenced by gravity.

Concrete Dam Model The second application of damping through Dynamic

Relaxation tests the ability of the hydrocode to capture a fairly complex stress field. The model is of a 180 ft. tall concrete dam scaled down by a factor of 30. Figure 12 shows the geometry as well as the mesh of the model.

Figure 12. Concrete Dam Geometry and Mesh

Damped Undamped

Damped Undamped


This model represents a further challenge due to the “stiffness” of concrete. The speed of sound in concrete is about 30 times higher than in the intermediate soil from the first example. Proper stabilization of such a model relies as much on a reasonable relaxation factor as on a gentle “ramping up” of gravity. In fact, a number of runs with different combinations of relaxation factors and gravity ramp durations were tried before a reasonable combination was found. At a relaxation factor of 0.995 and a ramp period of 30 seconds the response of the concrete dam model was sufficiently close to critically damped to be deemed stabilized. Figure 13 shows the stress response history at a number of points in the model.

Figure 13. Stress Histories in the Concrete Dam Model

The vertical (gravity) stress field along a cross-section of the dam is shown in Figure 14 along with the stress field from an implicit analysis in LS-DYNA. The implicit analysis does not require any damping and therefore represents a good comparison for the gravity stress field.

Figure 14. Stress Field in the Dam Cross-section

The stress field found from the relaxed PRONTO3D analysis matches quite well with the stress field from the implicit analysis.

CONCLUSIONS Gravitational stabilization was used in two Lagrangian

hydrocodes: PRONTO3D and EPIC. Two damping methods were compared, Artificial Bulk Viscosity and Dynamic Relaxation. Artificial Bulk Viscosity was already implemented in both codes, but turned out to be extremely computationally expensive for practical applications. In contrast, damping through Dynamic Relaxation had not previously been implemented in the codes. After the codes were modified to use Dynamic Relaxation, it was found this was the most adequate method to use for gravitational stabilization.

Two sample models were analyzed using the Dynamic Relaxation technique. The results from the analyses were as shown to be quite reasonable and promising for future use. Further studies into the effects of gravitational stabilization on geotechnical material behavior in high energy simulations will be carried out in follow-up studies.

REFERENCES [1] G. Johnson, "A Computer Program for Elastic-Plastic

Impact Computations in 2 Dimensions Plus Spin," Honeywell Inc., Hopkins, MN, June (1978).

[2] L. Taylor and D. Flanagan, "PRONTO3D: A Three-Dimensional Transient Solid Dynamics Program," Sandia National Laboratories, Albuquerque, NM, 1989.

[3] M. Wilkins, "Use of Artificial Vicosity in Multidimensional Fluid Dynamic Calculations," Journal of Computational Physics, vol. 36, no. 3, pp. 281-303, 1980.

[4] P. Taylor, A. Stewart, D. Hughes, D. Bamman and M. Chiesa, "A Multi-Level Code for Metallurgical Effects in Metal-Forming Processes," Sandia National Laboratories, Albuquerque, NM, 1997.

[5] J. Hallquist, "LS-Dyna Theory Manual," Livermore Software and Technology Corporation, Livermore, 2006.

[6] P. Underwood, "Dynamic Relaxation in Structural Transient Analysis," in Computational Methods for Transient Analysis (A 84-29160 12-64), Amsterdam, North-Holland, 1983.

[7] M. Papadrakakis, "A Method for the Automatic Evaluation of the Dynamic Relaxation Parameters," Computer Methods in Applied Mechanics and Engineering, vol. 25, no. 1, pp. 35-48, 1981.

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