Numerical Modeling and Simulation of Flame Spread Over Charring Materials

Matthew T. McGurn December 14, 2012

A dissertation submitted to the Faculty of the Graduate School of the University at Buﬀalo, State University of New York in partial fulﬁllment of the requirements for the degree of

Doctor of Philosophy

Department of Mechanical & Aerospace Engineering UMI Number: 3554478

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ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, MI 48106 - 1346 Acknowledgments

I would ﬁrst like to express my deepest gratitude to my advisor, Dr. Paul E. DesJardin, for his continuous support, encouragement, and guidance throughout my graduate studies. The research and life experience I gained from my association with him will stay with me for many years to come. I am grateful to Dr. Gary F. Dargush, Dr. James D. Felske, and Dr. Amjad J. Aref for their discussion and suggestions on my research and for serving on my dissertation commit- tee. I would especially like to thank all the current and past members of UB Computational Energy Transport (CET) Laboratory for their continued help, support, and friendship. My appreciation goes to all others who have helped on the research work, but the space is too limited to mention everyone. However, I would like to especially thank Dr. Amanda Dodd from Sandia National Laboratories, Dr. Jim Lua from Global Engineering and Materials and Dr. Brian Lattimer from Virginia Tech for their support on this work. I owe a great deal of gratitude to my entire family, especially my parents, Debra and Thomas McGurn, without whose unwavering love and support I would not be the individual I am today.

Support for this work has been provided by the National Science Foundation (NSF) under Grant CBET-1033328.

Matthew T. McGurn The State University of New York at Buﬀalo December 2012

ii Contents

Acknowledgments ...... ii Abstract ...... x

1 Introduction & Background 1 1.1 Background ...... 1 1.2 Objective ...... 5

2 Porous Media Modeling of Charring Materials 7 2.1 Formulation ...... 8 2.2 Numerical Implementation ...... 10 2.2.1 Weak Formulation ...... 10 2.2.2 Numerical Integration in Time ...... 13 2.2.3 Numerical Integration in Space ...... 14 2.3 Carbon Epoxy Modeling ...... 14 2.3.1 Pyrolysis Rate Modeling ...... 15 2.3.2 Matrix Thermal Properties ...... 16 2.3.3 Composite Swelling ...... 17 2.3.4 Gas Properties and Exothermic Chemical Reactions ...... 20 2.3.5 Results ...... 21 2.3.6 Conclusions ...... 27

3 Fluid-Solid Coupling 36 3.1 Formulation ...... 38 3.1.1 Eulerian Model ...... 38 3.2 Numerical Formulation ...... 42 3.2.1 Eﬃcient Level Set Initialization and Update ...... 43

iii CONTENTS iv

3.2.2 Ghost-Fluid Implementation ...... 47 3.2.3 Coupling Time Interval ...... 52 3.2.4 CFD Numerics ...... 53 3.3 Results ...... 54 3.3.1 Conservation Error Checks ...... 54 3.3.2 Flow Over a Cylinder ...... 55 3.3.3 Isothermal Plate Heat Transfer ...... 55 3.3.4 Surface Blowing ...... 56 3.3.5 Sublimation with Constant Densities ...... 57 3.4 Conclusions ...... 59

4 Flame Spread Simulations 70 4.1 Small Scale Simulations ...... 70 4.2 Large Scale Simulations ...... 72

5 Conclusion 84 5.1 Proposed Future Research ...... 86

A Balsa Wood Modeling 87 A.0.1 Thermal and Transport Properties ...... 88 A.0.2 Water Transport ...... 89 A.0.3 Balsa Results ...... 90

B Finite Element Domain Decomposition 97 B.1 Mathematical and Numerical Formulation ...... 99 B.1.1 Finite Element Methodology ...... 99 B.1.2 Domain Decomposition Methodology ...... 101 B.1.3 Numerical Integration in Time ...... 106 B.1.4 Reduced Interface Problem ...... 107 B.1.5 Solution Algorithm ...... 109 B.2 Coding Implementation ...... 110 B.2.1 Existing Framework ...... 110 B.2.2 Existing Multi-Threading ...... 111 B.2.3 Message Passing in Java ...... 112 B.2.4 Object Layout ...... 113 CONTENTS v

B.3 Results ...... 115 B.3.1 Veriﬁcation ...... 115 B.3.2 Performance and Scaling ...... 116 B.4 Conclusions ...... 123

Bibliography 124 List of Figures

1.1 Sketch of upward ﬂame spread at an inclination angle, θ, showing the ﬂame

height, Sf , and pyrolysis length, Sp...... 2

2.1 Illustration of element expansion process in 1D for (a) temperature and (b) reaction progress variable at t = 0 s and 180 s...... 29 2.2 Coupon scale simulations showing (a) instantaneous snapshots of temperature (solid lines) and α (dashed lines) at t = 775, 1775 and 2025 s for a 10 oC/min heating rate and (b) comparison of predictions of solid mass fraction (solid lines) to TGA data from Quintiere et al. (symbols) and their curve fit using a first-order Arrhenius rate model...... 30 2.3 Time-to-ignition cases with simulated sealed and open right boundaries compared against time-to-ignition data provided by QWC...... 31 2.4 One-sided heating cases with open right boundary showing heat release rate (HRR) comparisons for incident heat fluxes of (a) 25 kW/m2, (b) 50 kW/m2, (c) 75 kW/m2 and (d) 100 kW/m2...... 32 2.5 One-sided heating cases with sealed right boundary showing heat release rate (HRR) comparisons for incident heat fluxes of (a) 25 kW/m2, (b) 50 kW/m2, (c) 75 kW/m2 and (d) 100 kW/m2...... 33

2.6 Instantaneous snapshots of (a) T & α and (b) pg & K at t = 25, 50 and 75 s. Symbols denote position of FE nodes...... 34

2.7 Comparisons of predictions of (a) V/Vo, Vg/Ve & Yrc and (b) Yrcf to data with increasing incident heat ﬂux...... 35

3.1 Classiﬁcation of CFD nodes as either being deﬁned as either inside (xp1, xp3,

xp4) or outside (xp2) the projected volumes associated with the surface mesh. 60

vi LIST OF FIGURES vii

3.2 Identiﬁcation of nearest node (xc) and attached surface elements for the calculation of the of level set function...... 60 3.3 Illustrations of patch level growth starting from the original surface mesh. . 61 3.4 Level set function initialization with patch levels of (a) 0 (original mesh), (b) -1, (c) -2 and (d) -3 for every node n the Eulerian mesh. Patches are indicated as diﬀerent colored lines on the surface mesh (a rectangle). . . . 61

3.5 Ghost-fluid mirroring showing the ghost-fluid node of interest, xG and its evaluation using linear extrapolation across the interface from the gas-phase at location, x...... 62 3.6 Flow over a moving 2D cylinder and a 3D sphere showing (a) vorticity contours around a cylinder, and (b) vorticity contours around a sphere with an isosurface at a vorticity of 5000 and contours of density...... 63 3.7 Mass conservation errors for moving cylinder and sphere cases showing (a) mass conservation errors for high (dashed red) and low (black) resolution cylinder cases, and (b) mass conservation errors for high (dashed red) and low (black) resolution sphere cases...... 64 3.8 Flow shedding over a stationary cylinder showing (a) Von Karman vortex streak, (b) cross-stream velocity along the centerline at 0.5 diameters downstream and (c) pressure coefficient and percent error along the upper surface (0o corresponds to the upstream stagnation point)...... 65 3.9 Conjugate heat transfer for an isothermal plate showing (a) problem sketch,

(b) instantaneous snapshot of temperature contours, (c) NuL vs. RaL for inclination angles of 30o and 60o. Green symbols are cases for which the mesh is aligned with the surface and the gravity vector changed...... 66 3.10 One-dimensional diffusion test problem results showing (a) steady state mass fraction distribution and (b) L2 error for the “standard” ghost-fluid method, area modified ghost-fluid method, and exact interface treatment using an Eulerian method...... 67 3.11 A sketch of the sublimation validation problem of section 3.3.5...... 68 3.12 One-dimensional sublimation test problem results showing (a) the predicted interface location with grid refined compared to the analytical solution and (b) L2 relative error norm for the non-dimensional interface location, velocity, and heat flux...... 69 LIST OF FIGURES viii

4.1 Fully coupled simulation showing the CFD and FE domains along with igniter. 76 4.2 The L2 relative error norm in average surface temperature and total heat to the surface for the fully coupled simulations in one and two dimensions. . . 77 4.3 Fully coupled simulations of the carbon-epoxy composite showing mass frac-

tion of CH1.3O0.2, gas and composite temperatures after (a) 2.5 s (b) 3.5 s and (c) 4 s...... 79 4.4 Fully coupled simulation showing the CFD and FE domains along with inert backer-board and igniter flame region...... 80 4.5 Fully coupled simulations of the carbon-epoxy composite with an uniform applied heat flux of 14.2 kW/m2 after (a) 15 s (b) 30 s and (c) 45 s. . . . . 82 4.6 Time history of integrated heat flux over the composite surface for applied heat fluxes of 0.0, 10.0, 14.2, and 20.0 kW/m2...... 83

A.1 Sketch of a balsa slab response to heating from a fire...... 92 A.2 One-dimensional mesh utilized for the numerical modeling of balsa wood. . 93 A.3 Distributions of volume fraction (a), temperature and pressure (b) and mass fraction and mass source/sink terms (c) and thermal properties (d) after three minutes of heating with low heat flux and 0.012 initial volume fraction of bound water...... 95 A.4 Front location of the bound water, char and temperature in balsa wood for far field temperatures of 1290 K and 1175 K for initial volume fractions of water of 0.012(a) and 0.024(b)...... 96

B.1 A schematic representation of the linking of two FE subdomains with La- grange multipliers...... 105 B.2 A schematic representation of the implementation of threads (b) compared to the original implementation (a) within the UB CET FE framework. . . . 112 B.3 A schematic representation of the FE Domain implementation linking multiple regions...... 114 B.4 The temperature proﬁle of a representative conduction material with an applied heat ﬂux from the left and convection boundary conditions elsewhere. (top) The results were computed using two subdomains (red/blue meshes) to represent the structure. (bottom) Results are shown using a single domain to model the structure...... 117 LIST OF FIGURES ix

B.5 The temperature profile of a representative conduction material with an applied heat flux from the left are shown at a y location of 0.0223 m...... 117 B.6 Computational cost of the solution of linear system using the BiCG with MTJ.118 B.7 Scaling and efficiency of a one meter cubed domain consisting of 64,000 elements on UB’s CCR U2 cluster...... 119 B.8 Pictorial representation of the relative cost of the global and local BiCG solvers.120 B.9 The required allocated memory for a single processor in 3D as a function of number of elements in one dimension...... 121 B.10 The mesh for the 1.56 Million finite element case distributed across 64 subdomains...... 121 B.11 The communication (left) and loading (right) for the scaling problem on 64 processors...... 122 Abstract

The overall objective of this dissertation is the development of a modeling and simulation approach for upward flame spread. This objective is broken into two primary tasks: development of a porous media charring model for carbon-epoxy composites and an algorithm to couple flow and structural solvers. The charring model incorporates pyrolysis decomposition, heat and mass transport, individual species tracking and volumetric swelling using a novel finite element algorithm. Favorable comparisons to experimental data of the heat release rate (HRR) and time-to-ignition as well as the final products (mass fractions, volume percentages, porosity, etc.) are shown. The charring model and flow solvers are coupled using a newly developed conjugate heat and mass transfer algorithm designed for complex geometries in fire environments. Highlights of the coupling algorithm include: a level set description of complex moving geometry, perfect conservation of energy and mass transfer across the interface, a no-slip and no-penetration ghost-fluid interface description, and a patch level set update system that balances accuracy and computational efficiency by reducing the resolution of the Lagrangian model away from the interface. A systematic study of grid convergence order and comparison to analytical benchmark problems is conducted to show the soundness of the approach. The interface methodology is combined with the carbon-epoxy charring model and is used to study burning composites. Comparison of simulations to experimental data show good agreement of composite material response and flame spread (critical heat flux).

x Nomenclature

2 an interface normal acceleration [m/s ] Greek C proportionality constant α reaction progress variable 2 CP specific heat [J/kg − K] αg thermal diffusivity [m /s] d distance vector [m] γ level set function [m] 2 2 Di,m effective binary diffusion coefficient [m /s] Γ surface area [m ] e internal energy [J/kg] ε emissivity E total energy [J/kg] ζ mapped coordinate f body force term [m/s2] η non-dimensional coordinate h enthalpy [J/kg] θ non-dimensional temperature h convection coefficient [W/m2 − K] µ molecular viscosity [P a − s] 3 hpry heat of pyrolysis [J/kg] ρ density [kg/m ] H total enthalpy [J/kg] σ Cauchy stress tensor [P a] ∼ K curvature σ Stefan Boltzmann constant [W/m2 − K4] k thermal conductivity [W/m2 − K] ∼ τ non-dimensional time K permeability [m2] τ viscous stress tensor [P a] ∼ ∼ l patch level φ volume fraction

Ls heat of sublimation [J/kg] ψ basis function

Le Lewis number ωr relaxation factor m00 mass ﬂux, [kg/m2 − s] subscripts/superscripts M mass [kg] app applied m˙ 000 mass source/sink term, [kg/m3 − s] crit critical nk surface normal diff diﬀusion

NuL Nusselt number e element p pressure, [P a] ener energy P r Prandtl number exp expansion q00, q heat ﬂux, [kW/m2] f, c, r ﬁber, char, resin R radius of curvature g, s gas, solid constituent Re Reynolds Number G ghost node

RaL Rayleigh Number i constituent, species Sc Schmidt number int interface St Strouhal number j element node Ste Stefan number k interface constituent t time [s] LES large eddy simulation t tangential unit vector mass mass T temperature, [K] mom momentum u velocity [m/s] n normal direction U momentum [kg − m/s] o, e initial, end state V volume [m3] rad radiation V I interface velocity [m/s] s sublimation W weighting factor t tangent direction x location vector [m] vol volumetric Y species mass fraction Chapter 1

Introduction & Background

The destructive damage from large-scale ﬁre spread ranging from building ﬁres1 to forest

fires has become a growing concern in recent times.2 In total, the cost of fire damage to developed nations is estimated to be 1% of their gross domestic product resulting in billions of dollars.2–4 Characteristic to all fires is their exponential growth which comes from the continued participation of surrounding materials as fuel in the fire event.5 Minimizing fire growth requires a detailed understanding of conjugate heat and mass transfer processes that define fire spread and tools which can yield quantitative predictions of fire growth.

1.1 Background

The understanding and modeling of flame spread has been of interest to the fire science community since the 1960’s. Flame spread is often classified as either counter-current

(opposed) or concurrent. For counter-current flame spread, the direction of flame spread is opposite to that of oxygen diffusion resulting in a steady flame spread rate that has been well described by extensive theoretical,6–10 numerical11 and experimental11, 12 studies.

Concurrent ﬂame spread shown in Fig. 1.1 for vertically oriented surfaces is however much less understood. The fundamental challenge with upward ﬂame spread is an adequate

1 CHAPTER 1. INTRODUCTION & BACKGROUND 2

Figure 1.1: Sketch of upward ﬂame spread at an inclination angle, θ, showing the ﬂame height, Sf , and pyrolysis length, Sp.

description of the time dependent surface heat ﬂux as the ﬂame grows.

In 1976 Williams presented the following “fundamental equation of ﬂame spread” which

describes the governing physics of the problem,13

ρV ∆h =q ˙00 (1.1)

where ρ is the density, V is the ﬂame spread rate, ∆h is the enthalpy change associated with

pyrolysis, andq ˙00 is the heat ﬂux to the surface. Although deceptively simple, this steady

state equation illustrates the fundamental concept behind the phenomena where the ﬂame

spread rate is determined by a balance of heat ﬂux to the surface and energy consumed by

the fuel. Much of the research of the past few decades can be thought of as working to

determine the eﬀective ∆h andq ˙00 for increasing complex ﬂame spread scenarios.

The earliest attempts to complete the fundamental equation of ﬂame spread were ana-

lytical solutions. Due to the complexity of the problem these methods required additional CHAPTER 1. INTRODUCTION & BACKGROUND 3

assumptions in order to close the system. For instance, most of these methods assumed a

laminar ﬂow, non-charring, and either 1D thermally thin or thick heat transfer within the

solid.14, 15

Sibulkin and Kim16 examined turbulent ﬂows by estimating heat ﬂux at the pyrolysis

front with extensions of simple heat transfer experimental correlations. A “leap-frog” de-

scription of turbulent ﬂame spread was modeled assuming a power law relation between

n the ﬂame height (Sf ) and pyrolysis length (Sp), i.e., Sf (t) = βSp (t), to couple the ﬂuid and solid phases.17–20 The β parameters was assumed a function of solid material and en-

vironment, resulting in each parameter being speciﬁc to each study. The time-to-ignition

for each stage of the leap-frogging time was determined assuming constant heat ﬂux during

each stage. As was done by Markstien and de Ris21 and Orloﬀ et al.18 experimental data

was often utilized to determine the parameter β. In the work of Annamalai and Sibulkin15

and Fernandez-Pello and Mao,22 heat ﬂux distributions were allowed to vary with the py-

rolysis length as prescribed by a power law. Delichatsios et al. further improved these

theoretical estimates by developing similarity solutions for turbulent upward ﬂame spread

assuming that the time-to-ignition is a linear function of time.23 While excellent agree-

ment was reported between the theory and detailed numerical predictions of Sp (assuming constant heat ﬂux), less satisfactory agreement was shown between the theory and experi-

mental measurements. As concluded by Drysdale in his review of ﬂame spread over solids

in Ref. [5]: “Accurate predictions of ﬂame spread rates cannot be obtained by analytical

solutions to the governing equations...”. More recent work has focused upon the utilization

of numerical techniques for modeling ﬂame spread.

Much of the work of the last two decades has focused upon modeling ﬂame spread using

numerical techniques. Many of these early numerical solutions were extensions of analytical

expressions in which the underling mathematical system would be impossible to directly

solve. For instance, the analytical Karlsson model24 for ﬂame spread was extended using

numerical techniques by Kokkala and Baroudi25 and later by Grant and Drysdale.26 The CHAPTER 1. INTRODUCTION & BACKGROUND 4

Grant and Drysdale numerical model was capable of incorporating “burn-out” of the solid

fuel using the raw empirical heat release data and an expression for the “burn-out” front

resulting in a highly non linear function of ﬂame height. This was originally neglected

under the Karlsson model because a closed form analytical solution could not be obtained.

Despite the increase in complexity, many of these models still relied upon such empirical

relations for heat ﬂux and therefore limited in applicability and scope to conditions similar

to those where the empirical relations were determined.

In order to relax the need for empirical correlations (e.g. empirical heat release data),

progress was made in coupling a gas phase solution with that of the solid fuel models.

Many of the early numerical simulations of concurrent ﬂame spread relied heavily upon

assumptions. Carrier et al.27, 28 coupled the gas and solid phase conservation equations by

assuming a uniform velocity proﬁle.

With the advent of high performance computing, simulations of the governing equations

of ﬂow and solids are possible for simple conﬁgurations. Numerical studies of Kumar et al.29

and Consalvi et al.30 performed Reynolds-Averaged Navier-Stokes (RANS) calculations and

conﬁrmed the importance of temporally varying heat ﬂux in detailed numerical simulations

of ﬂame spread along Poly(methyl methacrylate) PMMA. In the latter study, the time

dependent evolution of both the solid and gas phases are solved simultaneously for ﬂame

spread over PMMA. Despite the use of these numerical techniques most of the computational

models assume either constant or simple geometries and therefore neglect the resulting aﬀect

that local geometry has on heat ﬂux.

The dependence of heat ﬂux on geometry has also been observed experimentally. Quin-

tiere31 and Drysdale32 examined upward and downward ﬂame spread over (PMMA) at

several angles of inclination, shown as θ in Fig. 1.1. In both of these studies as much as

a factor of two decrease in ﬂame spread rates is observed for both conﬁgurations when the

material is oriented from a vertical position to ±60o. A ﬂame spread theory was developed

by Quintiere for thermally thin materials using a modiﬁed Grashof number to account for CHAPTER 1. INTRODUCTION & BACKGROUND 5

the inclination angle along with experimentally developed heat transfer correlations from

Ahmad and Faeth33 and Roper.34 The resulting simpliﬁed heat ﬂux description to the

preheating region of the material was given in terms of the pyrolysis length, cosine of incli-

nation angle, etc., but the spatial distribution of the heat ﬂux within the preheating zone

is assumed constant. While the resulting predictions were shown to be only in qualitative

agreement with experiments, the model was able to show that the heat ﬂux to the surface

is sensitive to geometry. This can be especially important if the solid fuel surface is moving

(expansion with char/regression with fuel consumption) where the ﬂame spread model must

be capable of accounting for this geometry change.

One of most commonly cited numerical tools for studying ﬂame spread and related phe-

nomena has been the ﬁre dynamics simulator (FDS). FDS was originally released in the

year 2000 by National Institute of Standards and Technologies (NIST).35 In their review,

Kwon and colleagues reviewed the use of FDS for modeling upward ﬂame spread based

upon original research and available literature. It was concluded that the inconsistencies

for the predictions of ﬂame spread are serious enough to dissuade usage.36 Some of these

inconsistencies can be attributed to assumptions incorporated into the FDS formulations.

These include the rectangular treatment of boundaries (thereby limiting the representation

of complex geometry) and limited treatment of the solid fuel. This limited solid fuel model

assumes one-dimensional heat transfer, instantaneous release of volatiles across phases and

neglects porosity eﬀects. The solid fuel model and coupling algorithms assume a station-

ary solid/gas interface and therefore are incapable of accounting for the geometry change

associated with fuel regression or char expansion.35

1.2 Objective

The main objective of this study is to develop the numerical tools needed to simulate and better understand ﬂame spread. In order to achieves these objects, the fundamental physics CHAPTER 1. INTRODUCTION & BACKGROUND 6

behind ﬂame spread must be understood and expressed in mathematical terms. In order

to accurately model ﬂame spread and improve upon the available numerical models, the

following goals are central to this study: 1) develop a solid model capable of capturing

the eﬀects of charring and individual gaseous species transport. 2) develop the algorithms

necessary to couple this charring solid model with a ﬂow solver in a self-consistent manner.

Due to the nature of the charring material, the complete algorithm must be capable of

capturing the coupling of arbitrary moving geometries while accurately resolving the heat

ﬂux. As a method to reach these objectives, the charring model and coupling algorithms

are implemented within an existing computational framework.

The organization of this dissertation is as follows: Chapter 2 discusses the porous media

modeling of charring materials where the formulation and mathematical model are dis-

cussed. Emphasis is placed upon the application of the charring model to carbon-epoxy

composites. Chapter 3 provides the details of the coupling algorithm where attention is

paid to the explicit conservation across interface and capturing of the arbitrary geometry.

Chapter 4 applies these models to the study of ﬂame spread where the results are compared

against experimental data. Lastly, conclusions are drawn in chapter 5. Chapter 2

Porous Media Modeling of Charring Materials

Composite materials are being used at an increasing rate in applications including aerospace vessels and other transport vehicles. The advantages of these structures are their high strength to weight ratio, corrosion resistance, and ease of fabrication. One of the safety challenges to application of composite materials is their susceptibility to ﬁre. When exposed to ﬁre, composites degrade, releasing volatile gases, and producing char, thereby reducing structural integrity. The reliance on composite materials for primary structural components only serves to increase the need for improved modeling techniques.37–43 Carbon-epoxy composite is the material of interest discussed throughout the chapter, while additional information on the application of the charring model to balsa wood can be found in appendix

The modeling of composite materials in fire environments requires knowledge of the temperature field. One of the earliest models for the thermal response of composites is the work of Henderson et al.,44–46 where the composite material was modeled as composed of either virgin or char material. This modeling approach has been expanded and modified, most notably by Sullivan and Salamon,47–49 Springer and colleagues,50–52 Dimitrienko,53–56

7 CHAPTER 2. POROUS MEDIA MODELING OF CHARRING MATERIALS 8

Gibson et al.,57 Mouritz et al.58–60 and DiBlasi et al.61, 62 Some of these models have been

incorporated into several publicly available codes that include Gpyro by Lautenberger,63, 64

the solid phase model in the NIST Fire Dynamics Simulator by McGrattanet al.,65 and

ThermaKin by Stoliarov and Lyon.66, 67

2.1 Formulation

The material modeling for this eﬀort is based on homogenization theories developed for both the thermal and mechanical ﬁelds of polymer composite systems by Luo and DesJardin.68

In this approach, the local governing equations within each constituent (e.g., fiber, resin, gas, char, etc.) are first defined. The equations, representing the local mass, thermal and mechanical response of that material, are assumed to be locally valid within a given constituent. These equations are then averaged over a localized volume. The volume is chosen to be sufficiently large relative to the mesoscopic features of the laminate, (e.g., a unit cell associated with the weave), but small relative to the system, i.e., the entire laminate structure. After averaging the transport equations, additional surface integral terms appear in the equations representing the interphase processes. These terms originate from commuting the averaging operator with differentiation and, in general, are unknown and problem specific.

The gas density and individual gas species are tracked through the phase averaged species mass conservation equations. The total number of gaseous species depends upon the complexity of the pyrolysis model. Phase-averaged species conservation equations are

th solved for the bulk density, ρg, and the mass fraction of the k gas constituent, Yk,g,

∂(φgρg) ∂ ∂pg 000 = ρgK/µg +m ˙ (2.1a) ∂t ∂x ∼ ∂x int CHAPTER 2. POROUS MEDIA MODELING OF CHARRING MATERIALS 9

∂(φgρgYk,g) ∂ ∂pg = ρgYk,g K/µg (2.1b) ∂t ∂x ∼ ∂x ∂ ∂Y + ρ φ D k,g +m ˙ 000 φ +m ˙ 000 ∂x g g k,m ∂x k,vol g k,int

where a Darcy’s Law is employed to approximate the bulk gas transport, i.e., φgug =

−(K/µg)∂pg/∂x requiring the specification of the permeability, K. Fick’s Law of diffusion ∼ accounts for the effects of differential diffusion, i.e., uk,diff = −Dk,m(∂Yk,g/∂x)/Yk,g where

th Dk,m is the effective binary diffusion coefficient for the k species in the mixture. The term,

000 m˙ k,vol, on the RHS accounts for the production or consumption of species from volumetric reactions within the gas phase (e.g., oxidation of pyrolysis gases within the material). The

000 P 000 last terms,m ˙ int = m˙ k,int, account for the production of pyrolysis gases from endothermic decomposition and evaporation processes which occur at phase interfaces. For the solid phases, the species conservation equation simpliﬁes once the density of a given phase is

000 assumed to be constant, i.e., ρk,s∂φk,s/∂t =m ˙ k,int. Thermal equilibrium among the solid, liquid and gas phases is assumed resulting in a single transport equation to describe energy transport,

" K # ∂T ∂T ∼ ∂pg ρCP = ρgCP,g (2.2) ∂t ∂x µg ∂x Ng ∂ ∂T X ∂T ∂Yk,g + k + CP,k ρgφgDk,m ∂x ∼ ∂x ∂x ∂x k=1 N ∂(φgpg) ∂ ∂pg X 000 000 + − pgK/µg − hk(φgm˙ +m ˙ ) ∂t ∂x ∼ ∂x k,vol k,int k=1 P P where ρ(= ρkφk) and CP (= φkρk/ρCP,k) are the bulk density and specific heat, respectively. The terms on the RHS of Eq. (2.2) represent bulk advection (via the Darcy Law approximation), conduction, differential diffusion, pressure work, and oxidation/decomposition rate processes. CHAPTER 2. POROUS MEDIA MODELING OF CHARRING MATERIALS 10

2.2 Numerical Implementation

The Finite Element (FE) numerical method is utilized to solve the density, species mass fraction and energy equations presented in Eqs. (2.1a), (2.1b), and (2.2), respectively. The solution procedure utilized standard Galerkin formulation consisting of linear basis functions. The details of domain decomposition implementation that allows for the solution across multiple processors is provided in Appendix B.

2.2.1 Weak Formulation

The application of the ﬁnite element solution procedure requires a weak form of the governing equations that is obtained by multiplying by a weight function w and integrating over the domain (Ω) resulting in

Z Z ∂ρg pg ∂w ∂ρg w φg dΩ + K dΩ (2.3a) Ω ∂t Ω µg ∂x ∼ ∂x Z 2 Z ρg ∂w ∂RgT 00 = − K dΩ − w m˙ g n dΓ Ω µg ∂x ∼ ∂x Γ Z Z 000 + w m˙ int dΩ − w ρg∂φg/∂t dΩ Ω Ω

Z Z ∂Yk,g ∂Yk,g wφgρg dΩ + wρgφgug dΩ (2.3b) Ω ∂t Ω ∂x Z Z ∂Yk,g ∂w 00 + ρgφgDk,m dΩ = − w m˙ k,gn dΓ Ω ∂x ∂x Γ Z 000 000 000 + w m˙ k,volφg +m ˙ k,int − Yk,gm˙ int dΩ Ω CHAPTER 2. POROUS MEDIA MODELING OF CHARRING MATERIALS 11

Z ∂T w (ρCP − φgρgRg) dΩ (2.3c) Ω ∂t Z ∂T + w (ρgφgCP,gug − ugφgρgRg) dΩ Ω ∂x Z ∂T ∂w + k dΩ Ω ∼ ∂x ∂x Z ∂(u φ ρ R ) ∂(φ ρ R) = − w T g g g g + g g dΩ Ω ∂x ∂t Z N X 000 000 − w hk(φgm˙ k,vol +m ˙ k,int) dΩ Ω k=1 Z − w qn dΓ Γ

where Γ and n deﬁne the boundary and outward normal of the domain. Equation (2.3c)

assumes energy transport due to molecular diﬀusion is negligible which is a reasonable

assumption since conduction through the matrix is the dominant method of heat transfer.

The second (φgρgRg) and seventh (∂(φgρgR)/∂t) terms in Eq. (2.3c) are a result of the substitution of the ideal gas equation of state into the temporal pressure derivative in

Eq. (2.2). Lagrange interpolating polynomial are chosen for the spacial approximation of

69 ρg, Yk,g and T . For a given element, defined by N nodes, the field variable χ(= ρg,Yk,g,T )is assumed to fit the following Lagrange interpolating polynomial,

N e X e e χ(x, t) ≈ χ (x, t) = χj(t)ψj (x) . (2.4) j=1

In order to close the system of equations produced with the substitution of Eq. (2.4) into the weak form of the governing equations, Eq. (2.3), N expressions for the weighting function must be chosen. The required weighting functions are chosen to be equal to the basis functions used to describe the ﬁeld varaiable such that wi = ψ1, ψ2, ...ψN . Substituting the approximations given in Eq. (2.4) and basis function for w the weak form of the governing CHAPTER 2. POROUS MEDIA MODELING OF CHARRING MATERIALS 12

equations can be written in matrix form as,

[M]{ρ˙g} + [K]{ρg} = {Q} + {qb} (2.5a)

Z e e Mij = ψi ψj φg dΩ Ωe Z e e pg ∂ψi ∂ψj Kij = K dΩ ∼ Ωe µg ∂x ∂x Z e 000 Qi = ψi m˙ int − ρg∂φg/∂t dΩ Ωe Z ρ 2 ∂ψe ∂R T − g i K g dΩ ∼ Ωe µg ∂x ∂x Z e 00 qb,i = − ψi m˙ g n dΓ Γe

[M]{Y˙k,g} + [K]{Yk,g} = {Q} + {qb} (2.5b)

Z e e Mij = ψi ψj φgρg dΩ Ωe Z e e ∂ψi ∂ψj Kij = ρgφgDk,m dΩ Ωe ∂x ∂x Z e e ∂ψj + ψi ρgφgug dΩ Ωe ∂x Z e 000 000 000 Qi = ψi m˙ k,volφg +m ˙ k,int − Yk,gm˙ int dΩ Ωe Z e 00 qb,i = − ψi m˙ k,gn dΓ Γe

[M]{T˙ } + [K]{T } = {Q} + {qb} (2.5c) CHAPTER 2. POROUS MEDIA MODELING OF CHARRING MATERIALS 13

Z e e Mij = (ρCP − φgρgRg) ψi ψj dΩ Ωe Z e e ∂ψi ∂ψj Kij = k ∼ Ωe ∂x ∂x ∂ψe +ψe (ρ φ C u − u φ ρ R ) j i g g P,g g g g g g ∂x ∂ ∂(φ ρ R ) − (u φ ρ R ) + g g g ψeψe dΩ ∂x g g g g ∂t i j Z N X 000 000 e Qi = hi(φgm˙ k,vol +m ˙ k,int)ψi dΩ Ωe i=1 Z e qb,i = − ψi qndΓ Γe

2.2.2 Numerical Integration in Time

The time integration of Eq. (2.5), is performed utilizing the alpha family integration scheme.69

The ﬁeld variables, ρg, Yk,g and T , are advanced in time individually, where all other ﬁeld variables are assumed constant over each time step. However, due to the nonlinear relationship between Eqs. (2.5a), (2.5b), and (2.5c),this assumption can lead to considerable errors and numerical instability for highly transient processes. This is remedied with the introduction of a Point Successive Over-Relaxation (PSOR) method where within a single time step the solution is sub cycled.

000 000 The explicit source termsm ˙ k,int,m ˙ k,vol and ∂φg/∂t involving the conversion from one phase/species to another are solved using a fractional step method to mitigate the numerical stiﬀness associated with integrating Arrhenius based reaction rates. In this approach, the source terms in Eq. (2.5) are integrated over the time step, ∆t, using a separate ODE solver assuming the rest of the transport processes are “frozen”. Source terms are then constructed from the ODE result and substituted into the ﬁnite element solver. CHAPTER 2. POROUS MEDIA MODELING OF CHARRING MATERIALS 14

2.2.3 Numerical Integration in Space

The calculation of the elemental matrix presented in Eq. (2.5) requires the integration over element domain Ωe. This procedure is accomplished by mapping to a master element and approximating the integration using Gauss quadrature integration,69 where the integration

is approximated as a sum of the weights multiplied by the integrand evaluated at the

quadrature locations. Standard implementation of this approximation, however, results in

non-physical pressure oscillations in the solution. The root of the problem is the linearization

of the pressure gradient term (as a function of ρg and T ) which may result in non-zero 2 ρgRgT K ρgK values for a uniform pressure. To rectify this problem, the coeﬃcients ∼ and ∼ in µg µg Eq. (2.1a) are assumed uniform throughout the element and are determined by interpolating

to the element center. This implementation satisﬁes the limitings case of uniform pressure,

prescribed by a increasing temperature in space and corresponding decrease in density, as

dictated by the ideal gas law.

2.3 Carbon Epoxy Modeling

The eﬀects of volumetric swelling on thermal response has been examined by Dimitrienko,53, 54

Springer et al.51, 52 and Luo & DesJardin.68 In these approaches swelling is modeled in the

context of a micro-mechanics model using an eﬀective thermal expansion coeﬃcient that

is deduced from experimental measurements. This approach is reasonable when the tem-

peratures are below that associated with pyrolysis where elastic theories can be applied.

At higher temperatures, however, the physical mechanisms of laminate swelling from char

growth is quite complicated - involving fracture of the lamella, crack growth and propa-

gation from gas expansion, and ﬁber fraying. In this study, a simpler phenomenological

description is explored in which expansion is linearly correlated with a pyrolysis reaction

progress variable - similar to the model of Staggs.70 CHAPTER 2. POROUS MEDIA MODELING OF CHARRING MATERIALS 15

The thermal model is based on the extensive property data given by the study of Quin- tiere, Walters and Crowley (QWC).71 Quintiere and colleagues developed a complete set of properties carbon-ﬁber composite including but not limited to thermal properties, kinetics of degradation, and heat of decomposition. The data from QWC is used to construct phenomenological models of: 1) pyrolysis decomposition, 2) composite swelling and 3) thermal transport properties. Validation of these models are compared to the well documented coupon level and one-sided heating experimental results from QWC.

2.3.1 Pyrolysis Rate Modeling

For the purposes of constructing phenomenological Arrhenius based decomposition rates, a reaction progress variable, α, is often introduced deﬁned in terms of the solid mass, m(= mr + mf + mc), as, m − m α = o . (2.6) me − mo

The subscript o and e represent the initial and ﬁnal states of the solid, respectively. As- suming the ﬁber does not participate in the pyrolysis then the rates of decomposition of the resin (r), char (c) and gas (g) can be expressed directly in terms of changes in α,

1 m˙ 000 = − [ρ − (V /V )ρ ]α ˙ r,int 1 − τ o e o e τ m˙ 000 = [ρ − (V /V )ρ ]α ˙ (2.7) c,int 1 − τ o e o e 000 m˙ g,int = [ρo − (Ve/Vo)ρe]α ˙

1−α whereα ˙ is modeled using an Arrhenius rate law suggested by QWC as:α ˙ = 1−µ k(T ) with

µ = (mf +mc)e/mo being deﬁned as the char fraction and k given as k = ap exp(−Ea/RT ).

Values for the activation energy (Ea) and pre-exponential constant (ap) were determined by QWC using TGA data resulting in kinetic parameters of Ea = 182 kJ/mol and ap =

10 −1 9.67 × 10 s . The quantity Ve/Vo in Eq. (2.7) is the overall volumetric expansion ratio assuming complete charring and is set equal to a value of 2.2 to match the measurements CHAPTER 2. POROUS MEDIA MODELING OF CHARRING MATERIALS 16

of QWC. The quantity τ is the mass of char per unit mass of resin and is deﬁned as:

000 m˙ c,int Ve (φcρc)e τ ≡ 000 = . (2.8) m˙ r,int Vo (φrρr)o which can be directly related to the residue fraction as,

ρ µ − [ρ φ ] τ = o f f o (2.9) [φrρr]o

3 3 where ρo, ρf and φf,o are given by QWC as 1530 kg/m , 1230 kg/m and 0.6, respectively.

Neglecting the mass of the initial gas then the initial ﬁber volume fraction is φr,o = 0.398 and gas volume fraction is φg,o = 1 − φr,o − φf,o = 0.007. Substituting the initial volume fractions and densities into Eq. (2.8) results in τ = 0.17 which is consistent with the reported value given by QWC of 0.20 ± 0.05. It is important to note that for modeling purposes, the values of µ and τ should be self-consistent. The heat of decomposition, required to calculate the energy consumed through pyrolysis, is given by QWC as 2.5 × 106J/kg of the original material.

2.3.2 Matrix Thermal Properties

One of the major modeling challenges is accurate models of the transport coeﬃcients K ∼ and k. In the current study, the conductivity of the composite is assumed isotropic and ∼ modeled using a curve ﬁt to the data of QWC given as: k = 0.023T 0.46I W/m2 −o C, where ∼ ∼ T is measured in oC and I is the identify matrix. ∼

The gas is modeled as ﬂow through a bank of circular cylinders with diameter, df for which the permeability is be estimated using the following correlation,72

3 2 φgdf K = 2 (2.10) C(1 − φg)

where C = 144 if the tubes are cylindrical in shape. df is set equal to 0.1 mm based on CHAPTER 2. POROUS MEDIA MODELING OF CHARRING MATERIALS 17

an estimate of the ﬁber toe diameter determined by dividing the thickness of the laminate

(3.2 mm) by twice the total number of plies (16). The rationale is that each layer is composed of overlapping toes in the weave and the resin rich region between layers is small relative to the weave thickness. The resulting initial and ﬁnal permeability using this approach are 2.42 × 10−17 m2 and 2.83 × 10−10 m2, respectively, however, the upper bound of K is

clipped to a value of 1×10−13 to avoid unnecessarily small time steps to maintain numerical stability. For the heat ﬂux ranges used in this study, the results are insensitive to this factor as long as it is chosen to be greater than 1 × 10−14.

The bulk speciﬁc heat of the matrix is modeled using a reaction progress variable description, CP (T, α) = CP,o(T ) + α(CP,e(T ) − CP,o(T )), using temperature dependent virgin

(CP,o(T )) and charred properties (CP,e(T )) from QWC given as,

CP,o = 0.75 + 0.0041T J/kg − K (2.11a)

CP,e = 0.84 + 0.0035T J/kg − K (2.11b) where T is in oC.

2.3.3 Composite Swelling

As discussed by QWC, an overall volumetric expansion ratio of Ve/Vo = 2.2 is observed in the experiments. As will be shown in the results, the effects of volumetric expansion has a pronounced effect on burn-out times because of the changes in gas volume fraction and the overall growth of the composite thickness. To account for these effects, an evolution equation for the kth solid volume fraction can, in principle, be derived,

000 Dφk m˙ k,int = + V˙k,exp (2.12) Dt ρk CHAPTER 2. POROUS MEDIA MODELING OF CHARRING MATERIALS 18

where the subscript k(= f, r, c) represents either ﬁber, resin or char and the corresponding

000 source/sink term,m ˙ k,int, is given in Eq. (2.7). The second term on the RHS of Eq. (2.12) accounts for the increasing solid volume fraction from swelling (expansion) processes and

th can be directly related to the divergence of the k solid material velocity, i.e., V˙k,exp =

∂uk −φk ∂x . To rigorously account for the effects of swelling requires a detailed analysis of the mechanical response of the structure to determine uk that, in turn, will depend on the thermal field (via, thermal expansion, fiber fraying, etc.). To model this coupled thermomechanical system, a presumed mico-mechanics description is often defined in the context of unit cell homogenization approaches.68 The exact nature of the swelling process is, however, quite complicated and potentially difficult to validate experimentally therefore a simpler phenomenological approach is pursued. Similar to the work of Staggs,70 in which the density is calculated as linear interpolation of a progress variable, the solid volume fraction is directly expressed as a linear function of the reaction progress variable,

φk = φk,o + (φk,f − φk,o)α (2.13)

where the initial (φk,o) and ﬁnal (φk,f ) volume fractions of each solid constituent that are summarized in Table 2.1 below.

Table 2.1: Summary of the initial and ﬁnal volume fractions of solid constituents. k φk,o φk,f f 0.600 0.273 r 0.393 0.000 g 0.007 0.706 c 0.000 0.021

This simple model accounts for the leading order eﬀects from composite swelling which is to decrease the solid volume fractions and increase the gas volume fraction that will be shown to be important in the results. In addition, the composite geometry change from swelling is also accounted for using a newly developed element expansion algorithm. In this CHAPTER 2. POROUS MEDIA MODELING OF CHARRING MATERIALS 19

approach, each element node located at position xo is ﬁrst assigned a neighboring reference node, xo,ref . The distance between these nodes at the start of the simulation is deﬁned as: do = xo − xo,ref , as shown in Fig. 2.1. As the expansion process proceeds the distance

between the two nodes is directly related to the expected volumetric expansion, V/Vo, via,

d = doV/Vo, where V/Vo can be determined by re-arranging Eq. (2.6),

V (ρ V /V − ρ )α + ρ = e e o o o (2.14) Vo ρ

P resulting in a non-linear dependence of V/Vo on α. In Eq. (2.14), ρ = φkρk is the total matrix density which is computed at each node. The neighboring reference node

associated with each moving node is selected by a closest search in a direction that is opposite

to that of a user prescribed expansion direction, nexp = −do/|do|. The user speciﬁed direction could, in general, be selected to point anywhere but for this study is chosen to point towards the boundary that is heated (i.e., the left boundary in Fig. 2.1). Element expansion is implemented after each integration time step by moving the element nodes as: x(t) = xref (t) + d(t). The reference node shown in Fig. 2.1, is in turn, a function of its own reference node, and so on for the rest of the nodes. At the boundary farthest from the heating source, the reference node and node of interest are the same, therefore do = 0 and so the nodes farthest from the heated boundary do not move (i.e., the right boundary in Fig.

2.1). Ideally, the nodes farthest away from the heating source should be updated ﬁrst and the nodes closest to the heating source updated last. However, because of the unstructured nature of the data in the FE implementation, it is not straight forward to implement such an update strategy. Alternatively, a time linearization procedure is implemented such that, x(t + ∆t) = xref (t) + d(t + ∆t), where d(t + ∆t) is determined as d(t + ∆t) = doV (t)/Vo with V/Vo evaluated using Eq. (2.14). The resulting algorithm yields a smooth movement of nodes so that V/Vo → Ve/Vo as α → 1. Figure 2.1 illustrates the element expansion algorithm for one-sided heating of a 3.2 mm CHAPTER 2. POROUS MEDIA MODELING OF CHARRING MATERIALS 20

thick carbon-epoxy laminate. The computations are one-dimensional using 10 elements across the through thickness. The right boundary is assumed adiabatic. A constant

60 kW/m2 heat ﬂux is imposed on the left boundary. The initial (a) temperature and

(b) α are shown and after 3 min of heating. As expected, the temperature on the left face of the laminates increases for which decomposition of the laminate occurs as shown by the increase in α. The expansion direction (nexp) denoted in the ﬁgures is directed to the left as that is direction of the applied heat. For illustration purposes, the second node in from

the left boundary is identiﬁed as the node of interest and its self-identiﬁed reference node

is the neighboring node to the right. As shown, the distance between them (d) grows with

time in accordance with Eq. (2.14) such that |d|/|do| and approaches a ﬁnal value of 2.2 corresponding to Ve/Vo.

2.3.4 Gas Properties and Exothermic Chemical Reactions

The mixture of pyrolysis gases and air are treated as ideal gases using CHEMKIN polynomial ﬁts for thermodynamic properties.73 The decomposition the epoxy resign is a complicated process in which a large number of species are released. The hydrocarbon chains in epoxy can either be broken through chain scission or side chain/group elimination.74 How- ever in the decomposition process new chemical bounds may be formed through additional cross-linking and cyclization.75 In epoxy this process can result in the release of phenol,

4-isopropylphenol, bisphenol A, 4-t-butyl-o-cresol, and additional products not yet identi-

ﬁed.74As an approximation proportions of carbon, hydrogen and oxygen in the pyrolysis

76 gas are assumed to be the same as those given by Tewarson as CH1.3O0.2. For lack of data, the sensible enthalpy and speciﬁc heats of the pyrolysis product gas are assumed to be that of methane (CH4) since the molecular weights are similar (i.e., 16.5 vs. 16). Once the pyrolysis gas leaves the composite, it is assumed to burn according to the following CHAPTER 2. POROUS MEDIA MODELING OF CHARRING MATERIALS 21

one-step molar reaction.

CH1.3O0.2 + 1.225(O2 + 3.76N2) → (2.15)

CO2 + 0.65H2O + 1.225(3.76)N2

76 Using a suggested heat of combustion of ∆hC = 28.8 kJ/g by Tewarson, a calculated heat of formation for the pyrolysis gas using Eq. (2.15) is ho = −4.5785 kJ/g resulting in CH1.3O0.2 an adiabatic ﬂame temperature of Tad = 2300 K.

2.3.5 Results

Three sets of data are used from QWC consisting of coupon scale TGA data, time-to- ignition, and one-sided heating, calorimeter tests. In the one-sided heating tests, 15 ×

15 cm samples are heated by a radiant heater. The experiments are carried out according to Title 14 Code of Federal Regulations (CFR) 25.853 a-1 which is the Federal Aviation

Administration (FAA) ﬂammability test method for large surface area materials for aircraft cabin interiors. Reported values of heat release rate (HRR), ﬁnal volume expansion ratio

(V/Vo), thickness expansion ratio, ﬁnal porosity (Vg/Ve), char + resin mass to initial resin mass ratio (Yrc = (Mc + Mr)e/Mr,o) and fraction of remaining mass (Yrcf ) are given by QWC.

Coupon Scale Validation

The heating rates selected for the simulations are 1, 3, 10 and 30 oC/min to match those of QWC and compare the sample mass fraction to the TGA data. The bulk solid mass fraction, Yrcf = Ms/Ms,o = (Mr + Mc + Mf )/Ms,o, is computed by integrating the masses of each constituent element-wise (Ωe) over the solution domain (Ω) using the following CHAPTER 2. POROUS MEDIA MODELING OF CHARRING MATERIALS 22

relation,

Z Z X ∗ Mk = φkρkdV ' J(ζ) φk(ζ)ρk(ζ)dV Ω e Ωe X X = Wj(ζj) J(ζj) φk(ζj)ρk(ζj) e j

where J is the time varying Jacobian associated with the mapping of the physical space (x)

to the reference computational space (ζ) as the elements stretch due to swelling expansion

processes. The quantities Wj and ζj are the quadrature weights and locations in mapped space on the element for the numerical integration.

One-dimensional simulations are conducted using prescribed time dependent tempera-

ture boundaries. The temperature of the boundaries are increased in time to be consistent

with the prescribed heating rate. A total of 10 elements are used to discretize the 3.2 mm

thickness sample which is deemed adequate based on a grid sensitivity check (not shown).

Figure 2.2(a) shows instantaneous snapshots of temperature (solid lines) and α (dashed

lines) for the 10 oC/min heating case at t = 775, 1775 and 2025 s. The temperature is

nearly uniform indicating that a thermally lumped analysis for developing the kinetic rates

is valid. Figure 2.2(b) shows the solid mass fraction compared to TGA results for heating

rates of 3 oC/min, 10 oC/min and 30 oC/min. As shown, the overall agreement is reason-

able - demonstrating that the Arrhenius kinetics are are properly incorporated into the

framework.

Time-to-Ignition Simulations

To further validate the models transient behavior against data, one-dimensional simulations

are conducted using 30 elements for grid independent results. The time-to-ignition is es-

timated based upon a critical mass ﬂow calibrated from experimental data of QWC. It is

assumed that this critical mass flow is sufficient is providing fuel to produce a flame off the

surface. The critical mass ﬂux was calibrated from ignition times of the rough side of the CHAPTER 2. POROUS MEDIA MODELING OF CHARRING MATERIALS 23

composite (with pilot ﬂame) provided by QWC, summarized in Fig. 2.3.

The thermal boundary conditions for the left (L) and right (R) boundaries are treated by defining the net heat flux in terms of incident (i), convective and surface blowing fluxes.

00 00 00 q˙L =q ˙i + h(T∞,L − TL) − m˙ g,nhg|L (2.16a)

00 00 q˙R = h(T∞,R − TR) − m˙ g,nhg|R (2.16b)

00 00 00 whereq ˙ = −qn andm ˙ g,n =m ˙ g n with n being the outward normal from the solid surface (n = −1, 1). The convective coeﬃcients for the left and right boundaries are assumed to

be constant and equal to h = 10 W/m2 − K. The far-ﬁeld temperatures on the surfaces are

assumed constant and equal to 300 K. For the gas transport, the gas density is determined

on the left boundary assuming a constant pressure, via an equation of state and calculated

surface temperature. For the mass fractions, Yk, given in Eq. (2.1b), the following outﬂow convective boundary condition is imposed.

if ug,n > 0 then DYk/Dt = 0 else Yk(t + ∆t) = Yk(t) (2.17)

where ug,n = ugn. The specification of the boundary conditions for gas transport on the right boundary is less certain. While aluminum foil was used to seal the back surfaces in the experiments, it is not clear that it provided an effective seal since tests with and without the foil showed little difference in the resulting flame observed jetting out the front face.71

To explore the sensitivity of the calculations to this uncertainty, cases are conducted using both open (i.e., constant pressure and the convective boundary of Eq. (2.17)) and closed

(sealed) right boundaries. For the closed cases the ﬂux for the gases on the right boundary are set equal to zero.

In lieu of a coupled ﬂuid combustion model a critical mass ﬂux criteria is utilized to CHAPTER 2. POROUS MEDIA MODELING OF CHARRING MATERIALS 24

estimate the time-to-ignition. This criterion requires that the critical mass flux be calibrated for each simulation, sealed and open right boundaries. The applied heat flux at the calibration point was chosen from the available experimental data as the minimum applied heat flux that could support piloted ignition. This point, shown in Fig. 2.3, corresponds to the time-to-ignition from QWC data for an applied heat flux of 19.5 kW/m2. For this heat load the time-to-ignition is 330 seconds. At 330 seconds the predicted mass fluxes are 3.35 g/m2 − s (sealed) and 2.40 g/m2 − s (open), respectively. The predicted critical mass fluxes are used to estimate time-to-ignition for the rest of the applied heat loads of

10, 17.5, 25, 31.6, 50, 75, and 100 kW/m2.

Figure 2.3 shows comparisons of the simulation results to measurements. The differences between the predictions and experimental time-to-ignition are 7.4, 5.9, 21.4, and 38.2% for applied heat fluxes of 25, 50, 75, and 100 kW/m2, respectively, which is within experimental uncertainty for the lower heating rates. The time-to-ignition predictions using the open boundary condition are slightly lower then the sealed cases due to the gas mass loss through the right boundary. For values less then the critical heat flux, indicated Fig. 2.3, the surface mass flux is insufficient to produce a flame.

Intermediate Scale Validation using One-sided Heating Experiments

One-sided heating simulations are conducted using a flame heat flux model and compared against the available data. Measured bulk quantities of the final products are directly computed in the FE model by integrating over the entire solution domain similar to that for the coupon level samples. To estimate the heat release rate, it is assumed that the mass

ﬂux blowing oﬀ of the heated surface instantly burns with the surrounding air. The heat

00 release rate per unit surface area may then be estimated as: HRR =m ˙ g,n|L ∆hC , where 00 m˙ g,n|L is the mass ﬂux from the heated left surface of the FE model. In this approximation all mass that is released is assumed to contribute to the HRR after the critical mass ﬂux is reached. The experimental measurements of HRR, however, consist of primarily the CHAPTER 2. POROUS MEDIA MODELING OF CHARRING MATERIALS 25

convective contributions of the HRR and therefore it is expected that the model will over- predict the peak HRR.

The thermal boundary conditions are deﬁned as they are in Eq. (2.16a) and Eq. (2.16b), however, the far-ﬁeld temperature on the left surface is assumed equal to 300K until ignition

00 00 00 2 2 whenm ˙ g,n|L > m˙ crit. The value ofm ˙ crit is either 3.35 g/m − s (sealed) or 2.40 g/m − s

(open). After ignition the far-field temperature is set to T∞,L = 0.5(Tad + TL) to account for the additional heat flux from the near-wall flame with Tad = 2300 K determined from

00 00 Eq. (2.15). After sufficient burning, them ˙ g,n|L again falls belowm ˙ crit and the flame is assumed to extinguish. When this occurs the left far-field temperature is reset to T∞,L = 300 K. The gas transport boundary conditions are treated identically to the time-to-ignition cases where both the open and sealed right boundaries are examined.

Figure 2.4 shows comparisons of the heat release rate for four diﬀerent heating rates

00 2 of (a)q ˙i = 25, (b) 50, (c) 75 and (d) 100 kW/m . For all cases, the right boundary is assumed open. For each heat ﬂux case, simulations are conducted both with (solid lines) and without (dashed lines) the expansion model activated and are compared to data

(symbols) consisting of 4 to 5 separate runs. Overall the agreement of the model to the data is reasonable considering the simplifications in estimating the heat release rate and the variability in the experimental data. In all cases, the model under-predicts ignition times with errors consistent with the ignition time predictions of Fig. 2.3. Comparing simulation cases without (w/o) and with (w/ ) the expansion model shows that accounting for volumetric expansion processes extends the burnout time by at least 50% for all cases resulting in much better agreement to the data. The reason for the premature burnout times w/o expansion is due to the effective reduced mass of the sample by using the final volume fractions summarized in Table 2.1 without increasing the overall size of the sample.

While the peak heat ﬂux and burn out times predicted are reasonable in Fig. 2.4, the R duration and overall heat release, ∆Q = t HRR(t) dt, are signiﬁcantly smaller than the experiments. This may be expected since the gas escaping out the right boundary was not CHAPTER 2. POROUS MEDIA MODELING OF CHARRING MATERIALS 26

accounted for in the overall heat release. To explore this possibility, additional simulations are conducted sealing the right boundary. Figure 2.5 are the same cases as those of Fig. 2.4 but with the right boundary sealed. It is apparent that the sealing of the right boundary has a pronounced eﬀect on ∆Q as well as the time history of HRR. A secondary plateau in heat release is observed that is more consistent with the double peak HRR history observed in the experiments. Since the simulations with the expansion model with open (Fig. 2.4) and sealed (Fig. 2.5) appear to bound the HRR data, it is reasonable to assume that the actual boundary from the experiments lies between these limits. This observation shows the importance of properly characterizing the unheated boundary for model validation purposes

- the easiest approach may be to simply leave the unheated surface completely open or insulated with a very porous thermal blanket (e.g., superwool).

QWC offers an explanation for the twin peak HRR in Figs. 2.4 & 2.5. They attribute the appearance of the first peak to the composite seeking to achieve a steady-state thermal distribution for the resin binder as an insulation layer is built up and the second peak to thermal heating wave reaching the back of the composite and being inhibited thereafter from the insulation backer board. A more quantitative complementary explanation can be explored by examining the instantaneous distributions of through thickness temperature and pressure. Figure 2.6 shows snapshots of (a) T & α and (b) pg & K at t = 25, 50 and 150 s. The temperature smoothly rises with increasing time as the decomposition front penetrates deeper into the laminate (denoted by α). The gas pressure, however, shows a very different behavior with a local peak pressure at the decomposition front where the permeability is low. The pressure continues to grow with time to a peak value of 1.5 atm until the decomposition front reaches the back face of the laminate at t = 75 s. Between 75 s and 150 s, the residual resin decomposes resulting in a capacitance of pressure that slowly vents off creating the plateau region of Fig. 2.5(d). While the current model qualitatively captures the double peak history, the predicted second peak in HRR is lower than measured in the experiments. The reasons for this are two-fold. The first is the simplified estimate of CHAPTER 2. POROUS MEDIA MODELING OF CHARRING MATERIALS 27

HRR that doesn’t account for the eﬀects of of turbulent mixing already discussed. Future fully coupled simulations will explore relaxing this assumption. The second reason is the limitations of the current permeability model. The exact time history of the venting process out the front face will be very sensitive to how the permeability is modeled. As shown in

Fig. 2.6(b), the model for permeability results in a fairly monotonic permeability proﬁle through the decomposition front (even though it is a strong non-linear function of porosity, see Eq. (2.10)). If, however, the permeability decreases abruptly in the charred region due

ﬁber clumping and/or collapse then the gas could potentially be limited enough to cause the more pronounced dip in HRR observed in the data. To account for these eﬀects would require a much more sophisticated permeability model that depends on mechanical response.

Nonetheless, the ﬁnal product volume and mass fractions are well predicted as shown in Fig.

2.7 showing comparisons of (a) V/Vo, Vg/Ve & Yrc and (b) Yrcf to data from QWC using the

simulation results with the sealed right boundary. The overall agreement for V/Vo, Vg/Ve

& Yrc is quite good with errors less than 15% over the entire range of incident heat ﬂuxes considered. The ﬁnal mass predictions show in Fig. 2.7(b) appear to under-predict the

data at lower heat ﬂuxes, however, considering the repeatability uncertainty in the data it

is difficult to draw definitive conclusions about deficiencies in the modeling.

2.3.6 Conclusions

A thermal model for a carbon-epoxy laminate is developed based on the data of Quintiere et

al.71 The model includes pyrolysis decomposition, heat and mass transport, and volumetric

swelling using a novel ﬁnite element algorithm. Model validation runs are conducted using

TGA and one-sided heating experiments. Overall good agreement is observed between

the model and data for the overall heat release rate and time-to-ignition. Neglecting the

effects of the composite swelling resulted in significant under-predictions of flame burnout -

highlighting the importance of accounting for laminate swelling for the current epoxy-carbon

material. Remaining discrepancies in HRR predictions are attributed to three factors. CHAPTER 2. POROUS MEDIA MODELING OF CHARRING MATERIALS 28

The ﬁrst is the uncertainty as to the speciﬁcation of the unheated back boundary for the gas transport. Simulation results bound the HRR data using either an open or perfectly closed boundary indicting that the experimental conditions lie someplace in-between. This emphasizes the importance of being able to accurately characterize this boundary for future validation level experiments.

The second factor is the simpliﬁed estimate of HRR using the decoupled calculations that do not account for important gas-phase turbulent combustion processes. Future fully- coupled simulations will attempt to relax this assumption. The third factor is the permeability model which is monotonic through the decomposition front therefore does not account for potentially important matrix collapse processes that may change the time history of the

HRR. However, the overall heat release predicted by the model seems to be in qualitative agreement with the data. Future efforts will try to quantify these differences by directly computing the overall heat released. Lastly, comparisons of the final volumetric expansion ratio, porosity and final char mass to the data are quite good, indicating that the overall thermal modeling approach is sound. CHAPTER 2. POROUS MEDIA MODELING OF CHARRING MATERIALS 29

(a)

(b)

Figure 2.1: Illustration of element expansion process in 1D for (a) temperature and (b) reaction progress variable at t = 0 s and 180 s. CHAPTER 2. POROUS MEDIA MODELING OF CHARRING MATERIALS 30

(a)

1 Simulation Fit (QWC) Data (QWC) 0.95 3 oC/min

0.9 30 oC/min 10 oC/min

0.85 Mass Fraction

0.8

0.75

100 200 300 400 500 600 700 Temperature (oC)

(b)

Figure 2.2: Coupon scale simulations showing (a) instantaneous snapshots of temperature (solid lines) and α (dashed lines) at t = 775, 1775 and 2025 s for a 10 oC/min heating rate and (b) comparison of predictions of solid mass fraction (solid lines) to TGA data from Quintiere et al. (symbols) and their curve ﬁt using a ﬁrst-order Arrhenius rate model. CHAPTER 2. POROUS MEDIA MODELING OF CHARRING MATERIALS 31

350 Calibration Point sim (sealed) 300 sim (open) 2 250 data - Pilot Smooth data - Pilot Rough 200 data - Auto Rough

150 Time (s)

100

50 No Piloted Ignition@ 17.5 kW/m 0 10 30 50 70 90 110 Incident heat Flux (kW/m2)

Figure 2.3: Time-to-ignition cases with simulated sealed and open right boundaries compared against time-to-ignition data provided by QWC. CHAPTER 2. POROUS MEDIA MODELING OF CHARRING MATERIALS 32

500 500 sim (w/o exp) sim (w/o exp) 450 sim (w/ exp) 450 sim (w/ exp) data - set 1 data - set 1 400 data - set 2 400 data - set 2 350 data - set 3 350 data - set 3 ) )

2 data - set 4 2 data - set 4 300 300

250 250

200 200 HRR (kW/m HRR (kW/m 150 150

100 100

50 50

0 0 0 100 200 300 400 500 0 100 200 300 400 500 time (s) time (s) (a) (b)

500 500 sim (w/o exp) sim (w/o exp) 450 sim (w/ exp) 450 sim (w/ exp) data - set 1 data - set 1 400 data - set 2 400 data - set 2 350 data - set 3 350 data - set 3 ) )

2 data - set 4 2 data - set 4 300 data - set 5 300 250 250

200 200 HRR (kW/m HRR (kW/m 150 150

100 100

50 50

0 0 0 100 200 300 400 500 0 100 200 300 400 500 time (s) time (s) (c) (d)

Figure 2.4: One-sided heating cases with open right boundary showing heat release rate (HRR) comparisons for incident heat ﬂuxes of (a) 25 kW/m2, (b) 50 kW/m2, (c) 75 kW/m2 and (d) 100 kW/m2. CHAPTER 2. POROUS MEDIA MODELING OF CHARRING MATERIALS 33

500 sim (w/o exp) 500 sim (w/o exp) 450 sim (w/ exp) 450 sim (w/ exp) data - set 1 data - set 1 400 data - set 2 400 data - set 2 350 data - set 3 350 data - set 3 ) ) 2

data - set 4 2 data - set 4 300 300

250 250

200 200 HRR (kW/m HRR (kW/m 150 150

100 100

50 50

0 0 0 100 200 300 400 500 0 100 200 300 400 500 time (s) time (s) (a) (b)

500 sim (w/o exp) 500 sim (w/o exp) 450 sim (w/ exp) 450 sim (w/ exp) data - set 1 data - set 1 400 data - set 2 400 data - set 2 350 data - set 3 350 data - set 3 ) )

2 data - set 4 2 data - set 4 300 data - set 5 300

250 250

200 200 HRR (kW/m HRR (kW/m 150 150

100 100

50 50

0 0 0 100 200 300 400 500 0 100 200 300 400 500 time (s) time (s) (c) (d)

Figure 2.5: One-sided heating cases with sealed right boundary showing heat release rate (HRR) comparisons for incident heat ﬂuxes of (a) 25 kW/m2, (b) 50 kW/m2, (c) 75 kW/m2 and (d) 100 kW/m2. CHAPTER 2. POROUS MEDIA MODELING OF CHARRING MATERIALS 34

1800

1 1600

1400 0.8

1200 0.6

1000 _ T(K)

800 0.4 T(t=25s) 600 T(t=50s) T(t=150s) _ (t=25 s) 0.2 400 _ (t=50 s) _ (t=150 s) 200 0 -0.003 -0.002 -0.001 0 0.001 0.002 0.003 X(m) (a)

150 10-13

140 10-14 P(t=25s) P(t=50s) 130 P(t=150) ) K(t=25s) 2 -15 K(t=50s) 10 m K(t=150s) ( K P(kPa) 120

10-16 110

100 10-17 -0.003 -0.002 -0.001 0 0.001 0.002 0.003 X(m) (b)

Figure 2.6: Instantaneous snapshots of (a) T & α and (b) pg & K at t = 25, 50 and 75 s. Symbols denote position of FE nodes. CHAPTER 2. POROUS MEDIA MODELING OF CHARRING MATERIALS 35

1.5 Simulation V/Vo (data) Porosity (data) Yrcf (data) 1 Fraction

0.5

0 0 20 40 60 80 100 Incident Heat Flux (kW/m2) (a)

0.95 sim data

0.9

0.85

0.8

0.75 rcinofFraction Mass Remaining

0.7 20 40 60 80 100 Incident Heat Flux (kW/m2) (b)

Figure 2.7: Comparisons of predictions of (a) V/Vo, Vg/Ve & Yrc and (b) Yrcf to data with increasing incident heat ﬂux. Chapter 3

Fluid-Solid Coupling

In order to simulate flame spread, a coupling algorithm between the porous media charring model and flow solver is developed. One of the outstanding issues in fire science is a better understanding of flame spread which is controlled by the conjugate heat and mass transfer processes at the fluid-solid interface. The heating of a combustible structural material often results in pyrolysis of the material that generates volatile gases which, in turn, burn in the surrounding air providing additional heat for further pyrolysis. While there is a rich history of modeling flame spread, the models are often limited to only providing qualitative estimates of flame spread velocity. More quantitative predictions rely on coupled simulations of mass and heat transfer processes.77, 78

The origin of immersed interface methodologies has its roots in the methods developed by Peskin to simulate cardiac mechanics.79 Since then, the use of immersed interfaces for describing fluid-solid interactions has expanded greatly to examine a range of applications for incompressible80, 81 and compressible82–85 flows. Recents reviews on the topic can be found by Osher and Fedkiw86 as well as Mittal and Iaccarino.87 The focus of the current effort is to develop a ghost-fluid based immersed boundary methodology for describing the conjugate heat and mass transfer processes associated with burning solids. The flow for these simulations is solved using Eulerian discretization approaches and the solid is solved

36 CHAPTER 3. FLUID-SOLID COUPLING 37

using a Lagrangian finite element method (FE). To the author’s knowledge, this study presents the first application of ghost-fluid methodology to charring solids.

The main challenge for coupling these two different numerical methods is the disparately different mesh topologies and maintaining numerical stability during the coupling. The components for a coupling scheme include: 1) an efficient and accurate update of moving interfaces, 2) the treatment of no-slip and no-penetration boundary conditions at the solid interface as viewed by the flow solver and 3) exchange of energy and mass transfer across interface. Two commonly used approaches are often applied to impose interface constraints associated with the last two items. The first approach is to impose the constraints as point source forcing functions. These approaches, based on the early methodologies developed by

Peskin,79 are often used in incompressible flow formulations80 because of their relative ease of implementation when solving a Poisson equation for pressure. The second approach is a ghost-fluid approach where the interface constraints are imposed via specification of ghost-

fluid properties by reflection and interpolation.86 Ghost-fluid approaches are often used for compressible flow formulations for which local solutions (i.e., a Riemann solution) can be used to specify the state of the ghost-cells and can be readily adapted with use in explicit time marching schemes. For the current problem, a combination of these two approaches is found useful for developing a robust coupling algorithm for describing conjugate heat and mass transfer processes.

The rest of this chapter begins with a description of the governing equations and a discussion of the partitioning of various terms for solution using the level-set based immersed interface. The numerical formulation follows with details on the re-initialization of the level- set function using an adaptive surface mesh representation of moving solid interfaces, the ghost-ﬂuid implementation, and treatment of source terms associated with the generation of pyrolysis gases due to thermal decomposition of the solid. In the results, several benchmark validation problems are ﬁrst presented showing increasing levels of complexity with regard to the level of coupling. CHAPTER 3. FLUID-SOLID COUPLING 38

3.1 Formulation

3.1.1 Eulerian Model

The ﬂow-ﬁeld is described by the Navier-Stokes equations for a fully compressible, reactive gas.

∂ρ + ∇ · (ρu) = 0 (3.1a) ∂t ∂(ρY ) i + ∇ · (ρY u) = −∇ · m00diff +m ˙ 000 (3.1b) ∂t i i i ∂(ρu) + ∇ · (ρuu + p) = ∇ · τ + ρf (3.1c) ∂t ∼ ∂(ρE) + ∇ · (ρuH) = ∇ · (u · τ ) − ∇ · q + u · ρf (3.1d) ∂t ∼

where E(= e + u · u/2) is the total energy, H(= h + u · u/2) is the total enthalpy, and

000 m˙ i are source or sink terms associated with chemical reactions. The viscous stress tensor (τ ), species diffusion (m00diff ), and heat fluxes (q), are modeled using Newton’s, Fick’s and ∼ i Fourier’s laws, respectively, with simple models of diffusion,

2 τ = µ(T )(− ∇ · u + (∇u + (∇u)T )) (3.2a) ∼ 3

00diff mi = −µ(T )∇Yi/Sc (3.2b)

q = −µ(T )(Cp(1 − 1/Le)∇T/P r + ∇h/Sc) . (3.2c)

The molecular Schmidt (Sc), Prandtl (P r), and Lewis (Le) numbers are set to 1.0, 0.7, and

1.43, respectively. The molecular viscosity is assumed to depend on a Sutherland’s Law,

3/2 −5 µ = µo(To + S)/(T + S)(T/To) , with constants µo, To and S set equal to 1.716 × 10 , 273 and 111, respectively, corresponding to air.88 The molecular diﬀusivities of all species CHAPTER 3. FLUID-SOLID COUPLING 39

are assumed equal but may vary as a function of temperature through the dependence on the prescribed Sc number. The quantity f in Eqs. (3.1c) and (3.1d) is the body force term.

To develop a ghost-ﬂuid algorithm, a set of general interface jump conditions are considered for a moving and blowing surface,

I 00 [[ρk(uk − V k) · nk]] = [[m ˙ k]] = 0 (3.3a)

00 00diff [[Yi,km˙ k +m ˙ k,i ]] = 0 (3.3b)

00 [[m ˙ kuk + σ · nk]] = 0 (3.3c) ∼k

X 00 00 [[ (m ˙ k,iEk,i) + (σ · uk) · nk − q˙k ]] = 0 (3.3d) ∼k i P where the [[( )]] notation denotes k( ) where the subscript k = g, s represents a property I of either the gas or solid side of the interface. The quantity nk is the surface normal, V is

00 the interface velocity, andq ˙k (= kk∇Tk · nk) is the normal heat conduction. The quantity σ is the Cauchy stress tensor for the kth phase. The treatment of pyrolysis is described ∼k using two diﬀerent modeling approaches depending on if a char layer is formed. For charring materials such as wood or thermosetting polymers, pyrolysis can be considered a volumetric phenomena and a porous media ﬂow modeling approach is often used.58, 68 The solid phase of the interface in this case is actually a mixture of gas and solid char that is generated from the products of pyrolysis. For non-charring materials such as PMMA, pyrolysis may be treated as surface phenomena. In both cases, the Lagrangian model computes the extent of pyrolysis and associated amount of mass, momentum and energy transferred to the Eulerian

ﬂow (to be discussed in section 3.2.2).

The general jump relations given in Eq. (3.3) are further simpliﬁed for the low Mach number case of interest. Taking (3.1c)·uk results in,

2 |ut,k| (∇ · σ ) · nk = ρk an,k − − f k (3.4) ∼k Rk CHAPTER 3. FLUID-SOLID COUPLING 40

where an,k = D(uk · nk)/Dt is the interface normal acceleration, Rk = 1/(∇ · nk) is the radius of curvature and ut,k is the tangential velocity to the interface. Taking ∇·(3.3c) and substituting in Eq. (3.4) for the solid phase, results in the following expression for gas phase normal pressure gradient,

2 ∂pg |ut,k| ps − pg D = ρs − an + f · ng − − (σ − τ ): ∇ng + (∇ · τ ) · ng (3.5) ∂ng Rg Rg ∼s ∼g ∼g

where an = D(us · ng)/Dt is the solid acceleration defined using the gas unit normal direction and σD is the devatoric component of Cauchy stress tensor for the solid phase. ∼s Equation (3.5) is used to populate the gas pressure in the ghost-fluid cells using an interpolation procedure (to be discussed in section 3.2.2). Including only the first term on the

RHS of Eq. (3.5) results in the same balance relation as that used in Ref. [89] who showed limited success with this type of boundary conditions for strong shock interactions. For low Mach number flows, pg ≈ ps, across the interface and the second term on the RHS of Eq. (3.5) is identically equal to zero. For high Mach flows, however, this is not the case and may help explain some of the discrepancies discussed in Ref. [89] when a simplified version of Eq. (3.5) is compared to the use of a Riemann solver for populating the gas pressure in the ghost-fluid cells. For the current study, the flow is at a low Mach number and therefore the second term on the RHS of Eq. (3.5) is very small and is ignored. The third and fourth terms on the RHS of Eq. (3.5) account for the effects of viscous shear and the response of the solid to those shear forces. These terms ensure the pressure gradient is adjusted such that a no-slip condition is imposed for the tangential component of the velocity. Rather than including these terms, no-slip and no-penetration boundary conditions are explicitly enforced for populating the gas velocity in the ghost-fluid,

[[uk · nk]] = 0 (3.6a)

[[uk · tk]] = 0 (3.6b) CHAPTER 3. FLUID-SOLID COUPLING 41

where tk is the tangential unit vector. Since velocity continuity across the interface is assumed (a low Mach number assumption) the energy jump condition of Eq. (3.3d) can be readily simpliﬁed by subtracting uk·(Eq. (3.3c))/2 from it to remove the mechanical energy

(it doesn’t matter if uk is that of the gas or solid since they are assumed to be the same at the interface) resulting in,

X 00 00 X 00 00 [[ (ek,im˙ k,i) − q˙k ]] = [[ek,im˙ k.i]] − [[q ˙k ]] (3.7) i i

00 00 00diff th th wherem ˙ k,i(= Ykm˙ k +m ˙ i ) is the mass ﬂux of the i species in the k material across the interface. For charring materials, a detailed model of the movement of pyrolysis gases within

the solid is used and the interface is treated as a blowing surface. Under these circumstances,

00 00 [[m ˙ k,i]] = [[m ˙ kYg,i − µ∇Yi/Sc]] = 0, which could be used to deﬁne a reﬂection condition for

Yg,i by either knowing the mass ﬂux at the interface or the species mass fractions in the solid porous material. However, it was found to be diﬃcult to ensure mass conservation across

the interface at low mesh resolutions and the resulting simulation often became unstable.

Alternatively, the integrated mass ﬂux across the interface is used to deposit the exchange

of mass, momentum and energy from surface blowing into the Eulerian ﬁeld using a source

deposition procedure (to be discussed in section 3.2.2) which ensures conservation principles

are strictly enforced resulting in a robust coupling algorithm. For this case, the role of the

ghost-ﬂuid mirroring is to enforce a zero penetration of the species mass ﬂux,

∇Yg,i · ng = 0 (3.8)

since the actual mass transfer is already accounted for using the source term deposition pro-

cedure. In the numerical implementation, however, this condition is not precisely satisﬁed.

The reason is the discrepancy between the interpolants used to populate the ghost-ﬂuid and

that used to calculate species flux near the fluid-solid interface. A simple fix for eliminating CHAPTER 3. FLUID-SOLID COUPLING 42

the mass leakage is given in section 3.2.2.

For non-charring materials, the pyrolysis process is treated as a surface phenomena and the eﬀects of surface blowing are treated in a similar fashion with the Lagrangian model providing the total amount of mass, momentum and energy over a given coupling interval. In this case, Eq. (3.7) is used to determine the surface pyrolysis rate per unit area,

00 00 00 m˙ pyr = (eg + ∆hpyr)/(q ˙g +q ˙s ) where ∆hpyr is the heat of pyrolysis. Since a separate source term deposition approach is used for both the charring and non- charring materials then the eﬀects of surface blowing may be ignored in Eq. (3.7) resulting in the following jump conditions that are used to specify the gas-phase temperature in the ghost-ﬂuid,

00 [[q ˙k ]] = [[kk∂T/∂nk]] = 0. (3.9)

00 For adiabatic walls,q ˙s = 0, therefore the condition for the ghost-ﬂuid speciﬁcation is

∂Tg/∂ng = 0. For non-adiabatic walls, the temperature at the interface is speciﬁed by the Lagrangian model and the gas-phase temperature is extrapolated into the ghost-ﬂuid.

3.2 Numerical Formulation

The overall computational approach is based on the coupling of Computational Fluid Dy- namics (CFD) with FE representations of solids. The flow field modeling is based on the use of Large Eddy Simulation (LES) developed for examining buoyancy driven plumes90 and pool fires.91, 92 A summary of the numerics used in the finite volume formulation of the LES and FE formulation of the structure are given in section 3.2.4 and chapter 2, respectively.

In the following subsections, the treatment of the fluid-solid interface is described using a level set description that makes use of an efficient re-initialization procedure to update the level set for moving interfaces. Next, the ghost-fluid update is described in section 3.2.2 using a reflection and interpolation procedure with details of the surface blowing in section

3.2.2. CHAPTER 3. FLUID-SOLID COUPLING 43

3.2.1 Eﬃcient Level Set Initialization and Update

The ﬂuid-solid interface is deﬁned as the zeroth level of a level set function, γk, a signed

th distance function deﬁned as, γk(x, t) < 0 when xk lies in the k material region and for

2 which the surface normal and curvature are deﬁned as, nk = ∇γk and Kk = −∇ γk, respectively. One of the advantages of using a level set function to describe the interface is

00 the ease of computing wall normal heat ﬂux to the solid from the gas,q ˙g = −kg∇Tg · ng =

−kg∇Tg · ∇γg where simple central diﬀerencing may be used to approximate ∇T and ∇γg

since both T and γg are continuous across the interface. For moving interfaces, additional computational cost is introduced for the re-initialization

of the level set function on the Eulerian mesh at each coupling interval. Eﬃcient numerical

algorithms to re-initialize the level set include: the fast marching methods,93 pseudo-time

relaxation procedures94 and method of characteristics.95 All of these methods have an pre-

deﬁned error tolerance that is set to result in accurate estimates of the level set function

near the ﬂuid-solid interface and coarser estimates for locations far from the interface.

In this study, a new methodology is developed that results in the exact value of the level

set function near the interface without iteration and is suitable for execution on parallel

computers. A triangulated surface mesh is ﬁrst extracted from the surface of the FE model

and sent to each region of the CFD over all processors. Since the surface mesh typically

has a low memory footprint, repeated storage of it over all processors doesn’t present a

problem. The surface mesh serves as the basis for deﬁning the interface for each CFD block

region. The accuracy of the re-initialization is therefore as accurate as the triangulated

surface mesh representation of the geometry. A normalized level set measure is introduced

to define regions of the flow from near to far from the fluid-solid interface to allow a balance

∗ of accuracy and computational cost, γ (= γg/γg,max), with γg,max being a characteristic

∗ size of the computational domain. The location γo ≡ C∆xmax/γg,max defines the value of ∗ γ that separates the near-field from the far-field where ∆xmax is the largest grid cell and C is a proportionality constant that is set to the maximum interpolant stencil size. CHAPTER 3. FLUID-SOLID COUPLING 44

∗ ∗ For values of γ ≤ γo , the CFD node of interest, xp, is in the near-ﬁeld and the level set

(γg) is initialized using the triangulated surface mesh. Two categories of nodes are deﬁned for this operation and are classiﬁed as either “inside” or “outside” the projected volume of

a surface element shown in Fig. 3.1 for a 2D surface. The projected volume is deﬁned as

the volume created by sweeping the element surface area of the triangle along its normal

direction. Using this classiﬁcation, points 1, 3, and 4 are deemed as inside nodes while node

2 is an outside node. Note, this classiﬁcation has no relation to if the nodes lie in the gas

or solid phases - as shown in Fig. 3.1 nodes 1 and 3 are in one phase and node 4 in the

other, yet they are all classiﬁed as inside nodes. For these nodes, the shortest distance is

the surface normal distance (d~p1, d~p3 and d~p4). Node 2 in Fig. 3.1, however, does not lie within the projected volume of any of the elements and therefore is classiﬁed as an outside node. For this node, the shortest distance (dp,2) is the closest node on the surface mesh.

The algorithm for sorting CFD nodes and determining γg begins with a nodal search

to determine the closest surface node, xc, to the the CFD node, xp, as shown in Fig. 3.2. The purpose of this search is to quickly reduce the number surface elements considered for

determining the shortest distance to the surface. A collection of candidate elements are

then deﬁned which are the elements that share xc in common, as identiﬁed with dashed lines in Fig. 3.2 (4 in total). A connectivity list relating nodes to candidate elements is pre-

computed at the beginning of the simulation and stored and used thereafter, therefore the

search for candidate elements only occurs once. Next, an estimate of the shortest distance,

|de|, from xp to the surface is computed for each element. The evaluation of |de| for each

candidate element diﬀers depending on whether xp is either inside or outside the projected area of element e. The classiﬁcation is established based on the element surface normal

(ne) and the surface normals (n1, n2, n3), shown in Fig. 3.2, and are deﬁned as:

ne = x1→2 × x1→3 (3.10a) CHAPTER 3. FLUID-SOLID COUPLING 45

n1 = x2→3 × ne (3.10b)

n2 = x3→1 × ne (3.10c)

n3 = x1→2 × ne (3.10d)

where xi→j ≡ xj −xi. If the result of projecting all vectors from xp to the element vertexes onto the surface normal of the corresponding side are negative,

[(x1→p · n3 ≤ 0), (x2→p · n1 ≤ 0), (x3→p · n2 ≤ 0)], then the point is deemed inside the pro-

jected element volume and the magnitude of the normal distance vector between xp and

the the element plane is deﬁned as: |de| = ne · xc. If the point is not inside then it must lie outside of the projected volume. In this case, the shortest distance is deﬁned as the

minimum distance from xp to each of the nodes comprising the element. The ﬁnal value of

the shortest distance from the surface to node xp (in magnitude) is taken as the minimum

N value over all candidate elements, i.e., value of |d| = mine=1(|de|). The direction of d is

For xp nodes that are classiﬁed as outside, more than one element will always be selected since xc is shared by more than one element. To uniquely identify the candidate element that is closer, the sides of each triangular element are shifted slightly inward along the normal by a tiny constant (1 × 10−10m) thereby separating each element and producing unique results.

∗ ∗ If γ > γo then xp is in the far-field and only a crude, inexpensive, estimate of the signed distance is required. In the far-field, the signed distance is not used directly in the ghost-fluid method, rather it only serves as an indicator for when more refined estimates are needed. To reduce computational cost, a sequence of “patches” are constructed from the original surface mesh as illustrated in Fig. 3.3 and associated (negative) patch levels are CHAPTER 3. FLUID-SOLID COUPLING 46

deﬁned as:

∗ ∗ ∗ l = int[lmin(γ − γo )/(1 − γo )] (3.11)

∗ which satisﬁes the limits that as γg → γmax then γ → 1 & l → lmin < 0. The appeal of the patches is to reduce the number of distance calculations required to estimate the

distance function by using the patch topology rather than the elements. Starting from a

patch level of zero (the baseline surface mesh) decreasing levels are constructed using a

grouping algorithm of the triangles ﬁrst and then of the patches at lower patch levels. A

starting node is ﬁrst identiﬁed to construct patches at level -1 and is (arbitrarily) chosen

to be the first node in the data array defining the unstructured list of nodes defining the

surface mesh. This starting node deﬁnes the origin of the ﬁrst patch at level −1. This

ﬁrst patch is the collection of surface nodes connected to the starting node through surface

elements as shown in red in Fig. 3.3. The origin of the next patch at level -1 is the next

node in the surface mesh data array that has not been used in patch 1 and an associated

patch is deﬁned for it (patch 2). This process is repeated until all of the surface nodes are

grouped into patches at level −1. Patch connectivity at level −1 is established if two or

more patches share the same surface mesh node. The connectivity list is used to group

patches at level −1 to form super patches at level −2, as shown in Fig. 3.3. The origin of

patch 1 at level −2 is set equal to that of the ﬁrst patch at level −1. The origin of the next

super patch at level −2 (patch 2) is deﬁned using the next available patch at level −1 from

the list of all patches at level −1 that has not been already used in patch 1. The grouping

of patches continues and associated (negative) levels are deﬁned until a single super patch

is created that contains all of the surface nodes. Figure 3.3 illustrates this progression up

to a patch level of -4.

Since the patches do not retain surface normal information, the sign of γg cannot be

readily deduced - only its magnitude. To determine the sign of γg, the near-ﬁeld algorithm is CHAPTER 3. FLUID-SOLID COUPLING 47

used only once at the beginning of the simulation and stored for all nodes. Subsequent far-

field evaluations using patches then update the magnitude of γg until the interface becomes sufficiently close where the near-field algorithm is employed. The evaluation of the local level

∗ using the linear interpolation given by Eq. (3.11) uses γg from the previous coupling time

step. This provides a continuous degradation in the estimate of γg but with an associated savings in computational cost. While more sophisticated far-ﬁeld level indicators could be

devised, the linear degradation model is found to be satisfactory for all of the problems

considered in this study.

To illustrate the use of patches, Fig. 3.4 shows the use of patches for a simple 2D problem

for which the surface mesh is represented by a collection of lines. To the left of each diagram

is the solid represented by a surface mesh. To the right is an Eulerian mesh used by the

CFD. The original surface mesh resolution is shown in Fig. 3.4(a) corresponding to patch

0. A coarsening of the surface mesh using patches are shown in Figs. 3.4(b), (c) and (d)

for patch levels of -1, -2 and -3, respectively. Each patch is represented by a diﬀerent

colored line, e.g., 5 patches are shown at level −2 in Fig. 3.4(c). Plotted on the right of each

surface mesh are contours of contours of the level set function γg using the original surface mesh (Fig. 3.4(a)) and then using progressive coarsening of it using patches. It should be

emphasized that the initialization of γg near the surface is intentionally poor to illustrate

∗ ∗ the use of patches. In practice, near the surface γ < γo and therefore the use of patches is not appropriate and the near-ﬁeld algorithm would be used instead to provide the exact

value of γg.

3.2.2 Ghost-Fluid Implementation

The update of the Eulerian solver near the immersed interface is accomplished using a ghost-

fluid methodology. In this approach, a relatively narrow range of ghost-fluid cells are defined

near the boundary of the ﬂuid-solid interface. The means of populating the properties of

the ghost-ﬂuid cells and interface description distinguishes one embedded methodology from CHAPTER 3. FLUID-SOLID COUPLING 48

another. Several variants of ghost-fluid cell updates have been tailored for the use of level set interface descriptions for both incompressible86 and compressible83, 96–99 flows and are problem dependent. In the original ghost-fluid method for Eulerian-Eulerian multi-material descriptions, continuous variables (e.g., pressure and velocity) from one material populate the ghost-cells of a neighboring material while discontinuous variables (e.g., entropy) are extrapolated.98 For the Eulerian-Lagrangian extension of the ghost-fluid method, the ghost- cell pressure is determined from the Eulerian field through extrapolation (i.e.,[p] = 0) and velocity extrapolated from the Lagrangian field to the ghost-cells (i.e.,[un] = 0). Recently Sambasivan and UdayKumar have summarized the more popular methods to update the ghost-cells using 1) a simple reflection, 2) a local Riemann solver, and 3) method of characteristics.89 In their study, they show the issues of over/under heating errors from shock impact on solids using a simple reflection methodology for which several fixes have been proposed in the literature (e.g., Fedkiw’s isobaric fix86). For the current low Mach study, these errors do not arise even though a fully compressible flow formulation is employed therefore only a simple reflection strategy is required.

The linear mirroring extrapolation of Ye et al.100 is employed that has been used in ghost-fluid studies of both incompressible80 and compressible89, 99 flows. In this approach, linear extrapolation to the ghost region is achieved by first computing the location of the mirror image of a given ghost node, x, that is defined along the line normal to the interface at a location that is equidistant (γg) from the interface in the (solved) Eulerian gas field, as shown in Fig 3.5,

x = xG + 2γg(xG)ng (3.12)

where xG is the location of the ghost node of interest. Once x is identiﬁed, then a local stencil can be deﬁned in the vicinity of that location for a Newton interpolating polynomial.

Using the interpolating polynomial, the interface constraints summarized in Eqs. (3.5), (3.6),

(3.8) and (3.9) are imposed at the intersection between the line connecting points xG and CHAPTER 3. FLUID-SOLID COUPLING 49

x and the ﬂuid-solid interface. The result is a local 4 × 4 matrix in 2D or a 8 × 8 matrix in

3D that must be inverted to solve for the unknown scalar value at xG. Additional details on this approach can be found in Refs.89, 100 For a ﬁxed mesh, the matrix inversion is pre-computed a priori for each ghost node and associated interpolation weights are stored for either a Neumann type (e.g., pressure in Eq. (3.5)) or Dirichlet type (e.g., velocity in

Eq. (3.6)) boundary to avoid unnecessary repetitive matrix inversions. The weights are then recomputed every time the Lagrangian mesh is moved.

Mass Transfer Implementation

As discussed in section 3.1, the effects of surface blowing from off-gassing are superimposed through an explicit source term deposition into the Eulerian gas field. To avoid double counting, the implicit mass flux through the fluid-solid interface must therefore be identically equal to zero. While the errors using a standard ghost-fluid method are sufficiently low for capturing solid boundaries, it is not sufficiently accurate for problems involving ablation or surface blowing since the numerical errors are large relative to the flux of mass, momentum and energy. The error is especially pronounced for flame spread problems. A relatively low flux error in mass flux will create an imbalance of chemical energy to the gas phase that will falsify the flame spread rate. One way to reduce the mass leakage is to either use higher-order approximations to the linear mirroring or use a finer mesh in the vicinity of the fluid-solid interface. Both of of these solutions result in additional computational expense. A simpler solution is used in this study where the mass flux through cell faces containing the fluid-solid interface are simply set equal to zero by setting the associated area of that surface to zero in the context of the finite volume solution. The advantage of this mass transfer approach is to ensure that overall mass (∆Mi & ∆M), momentum (∆U) and energy (∆E) exchanged across the interface is exactly preserved. In this approach, the mass, momentum and energy to be transferred to the gas field is integrated over the surface of the solid Lagragian model, Γs, and over a defined coupling time, ∆tc, (to be defined in CHAPTER 3. FLUID-SOLID COUPLING 50

section 3.2.3),

Z t+∆tc Z e 00 e ∆Mi = m˙ i dΓ dt (3.13a) t Γe

e X e ∆M = Mi (3.13b) i

Z t+∆tc Z e 00 e ∆U = m˙ i usdΓ dt (3.13c) t Γe

Z t+∆tc Z e 00 1 e ∆E = m˙ i (eg + ||us||)dΓ dt (3.13d) t Γe 2

e where superscript e represents an element with surface Γ that lies on ∂Ωs so that ∂Ω = P e e e Γ . The source term for each element is distributed in the gas ﬁeld centered about x e P Γ Γ Γ which is deﬁned as the element face centroid, x = ( i xi )/N , where xi is the position of the element face vertices and N Γ is the total number of vertices on the element face. The

e e e e e element face source term, S = {∆Mi , ∆M , ∆U , ∆E }, is distributed in the ﬂow ﬁeld e about x evenly amongst deposition nodes xj. The deposition nodes are selected adjacent to the element face covering the approximate area of the element. This is accomplished by

tagging the ﬁrst node along the outward surface normal that is in the gas phase (γg < 0). A sphere is then deﬁned, centered about xe, with a radius that result in an area equal to Γe

when a slice is taken though the center of the sphere. In three dimensions these assumptions

1 results in a radius of re ≈ (Γe/π) 2 . If there are no tagged nodes within the sphere the radius is repeatedly increased until at least one node if identiﬁed. The source term weights are pre-computed and stored to minimize computational cost of multiple source term use. The weights are only re-initialized when the Lagrangian mesh moves.

Heat Flux Implementation

The heat ﬂux (q ˙00) from the CFD is computed using a DNS approach where the heat ﬂux is computed in terms of the normal temperature gradient which is expressed in terms of CHAPTER 3. FLUID-SOLID COUPLING 51

gradients in the temperature and level set function,

00 q˙ = −kg ∂T/∂ng = −kg∇T · ∇φg (3.14)

where ng is outward surface normal direction from the gas phase. The gradients are computed using high-order Lagrangian interpolating polynomials which are constructed from

CFD nodes in the vicinity of the point of interest where the heat ﬂux is required. In this approach, the scalar variables are represented using either 2D or 3D basis functions (ψ).

For example, the temperature in 2D may be expressed as,

X X T (x, t) = ψi,j(ζ(x))Ti,j(t) (3.15) i j

where x is the point of interest to evaluate the heat ﬂux and ζ is the associated curvi-

linear coordinate in the body ﬁtted coordinate system for the CFD mesh. The subscript

(i, j) corresponds to the location in a 2 × 2 matrix of nodes deﬁning the extent of the

interpolant. The basis functions are expressed as the product of 1D basis functions in each

of the respective curvi-linear directions,

1D 1D ψi,j = ψi (ζ1)ψj (ζ2) (3.16)

1D 1D where ψi (ζ1) and ψj (ζ2) are the associated basis function for the ﬁrst and second curvilinear directions that are expressed in terms of generalized Lagrange interpolating polynomials

of order N,

1D N+1 ζ1 − ζp ψi (ζ1) = Πp=0,p6=i (3.17a) ζi − ζp

1D N+1 ζ2 − ζp ψj (ζ2) = Πp=0,p6=j (3.17b) ζj − ζp CHAPTER 3. FLUID-SOLID COUPLING 52

Using Eq. (3.15), the gradient of temperature can be determined analytically and expressed in terms of derivatives of the basis functions from the curvilinear coordinate system,

k ∇T = g ∂T/∂ζk 1D 1D X X ∂ψ X X ∂ψj = g1 T (t) i ψ1D + g2 T (t) ψ1D (3.18) i,j ∂ζ j i,j ∂ζ i i j 1 i j 2

k 101 where g (≡ eˆj∂ζk/∂xj) is the contra-variant vector that is evaluated locally using the

1D same set of basis function as that used for the temperature. The derivatives of ψi and 1D ψj in Eq. (3.18) can be determined analytically resulting in the following result.

1D N+1 ∂ψi X 1 N+1 ζ1 − ζq = Πq=0,q6=i (3.19a) ∂ζ1 ζi − ζp ζi − ζq p=0,p6=i 1D N+1 ∂ψj X 1 N+1 ζ2 − ζq = Πq=0,q6=j . (3.19b) ∂ζ2 ζj − ζp ζj − ζq p=0,p6=j

For the present study, second-order polynomials are used (i.e., N = 2) resulting in 3 × 3 node stencil in 2D or 3 × 3 × 3 node stencil in 3D for the construction of the heat ﬂux.

The stencil nodes are selected such that all of the nodes are located within the ﬂuid. Once the nodes are selected then the gradients may be determined using Eqs. (3.18) and (3.19).

1D 1D In practice, the computed weights (e.g., ψj ∂ψi /∂ζ1) that multiply Ti,j are precomputed and stored for later use each time the interface moves to minimize computational cost of repetitive calculations when computing gradient quantities at the interface.

3.2.3 Coupling Time Interval

If the source terms from surface blowing, discussed in section 3.2.2, are directly deposited over a single CFD time step (∆tCFD) then an artiﬁcially small time step is required to maintain numerical stability. To mitigate this potential problem, a dynamic time stepping algorithm is developed for coupling along with a robust mechanism to integrate the source CHAPTER 3. FLUID-SOLID COUPLING 53

n+1 terms into the gas phase solution. In this approach, a future coupling time step, ∆tc n is estimated using a current estimate of the coupling time step, ∆tc , and the previous n−1 n+1 n−1 n coupling time step, ∆tc , using a time relaxation relation, ∆tc = ∆tc + ωr(∆tc − n−1 n ∆c ), where ωr is a relaxation factor and set equal to a value of 0.1. The value of ∆tc is

the minimum coupling time scales for mass (∆tc,mass), momentum (∆tc,mom) and energy

(∆tc,ener) exchange over all Eulerian nodes that are aﬀected by the source term deposition, and are computed as follows.

e ∆tc,mass = Fmass (ρg Vg)/∆Mj (3.20a)

e ∆tc,mom = Fmom MIN (ρg ukVg)/∆Uj,k (3.20b)

e ∆tc,ener = Fener (ρg Eg Vg)/∆Ej (3.20c)

where Fmass, Fmom and Fener are safety factors that are all set equal to 0.1. The source terms that are passed from the Lagrangian model to the Eulerian ﬁeld are integrated into

n+1 the gas-ﬁeld solution over a time, ∆tc , by distributing the total source terms over the n+1 time remaining until the next coupling time, tc = t + ∆tc . In this approach, the total source of mass, momentum and energy from surface blowing for a given Eulerian node

is ﬁrst determined by summing over all elements at the start of a coupling interval, i.e.,

P e Si,j,k = e Si,j,k. During the coupling time interval the total source is decremented a factor

∆Si,j,k = Si,j,k(tc − t)/∆tCFD and ∆Si,j,k is treated as a constant source during that CFD time step. The process is repeated over each CFD time step until the residual source is

driven to zero at time tc.

3.2.4 CFD Numerics

The flow solver used is based on a finite volume formulation using the AUSM family of flux

splitting schemes102, 103 using a low Mach number preconditioning for eﬃcient integration CHAPTER 3. FLUID-SOLID COUPLING 54

at low speeds.90 Fluxes at the cell faces are interpolated using upwind biased stencils104 for momentum and essentially non-oscillatory polynomials (ENO)105 for density, energy and reacting species. The equations are integrated in time using a second order Runge-Kutta with standard weights. A block-structured domain decomposition is employed to partition the solution integration over multiple processors. A region communicates with neighboring blocks through a ghost cell (not to be confused with ghost-ﬂuid cells) strategy using a native

Java version of the message passing interface (MPI).106

3.3 Results

In the following subsections several benchmark problems are first explored to assess the accuracy of the algorithm. The first three benchmark problems explore the errors associated with moving interfaces, flow over a cylinder and heat transfer from an isothermal plate.

3.3.1 Conservation Error Checks

It is well known that ghost-ﬂuid methods are not conservative.82, 98 In order to determine the conservation errors associated with the immersed boundary method used in this study, a sinusoidally translating cylinder (2D) and sphere (3D) with isothermal solid wall boundary conditions at the domain edges is examined. The total mass should stay constant throughout the problem. For all cases the diameter of the sphere and cylinder is 1.6 cm, and the domain is 10 cm × 3 cm in 2D and 10 cm × 3 cm × 3 cm in 3D. For all cases the object is moved periodically from x = 0 to 8 cm with a period of 5 ms, which represents an average velocity of 32 m/s. For both objects low and high resolution cases are examined, corresponding to a mesh spacing of 1 mm and 0.25 mm respectively. For all cases the object boundary condition is adiabatic with no-slip and the domain is initialized with air at standard atmospheric conditions.

Figures 3.6(a,b) show the vorticity at a time of 2.55s. The resulting complex ﬂow ﬁeld CHAPTER 3. FLUID-SOLID COUPLING 55

shows that the ghost-fluid immersed boundary method is able to accurately capture the moving boundary condition in both 2D and 3D. Figures 3.7(a,b) show the percent mass conservation errors as a function of time for the cylinder and sphere cases, respectively. The conservation error oscillates rapidly, but is shown to be less than ±0.06% for all times. The errors for the high resolution cases are seen to oscillate less and have lower conservation errors on average. These results demonstrate that the version of the ghost-fluid method for this study accurately captures the effects of moving solid boundaries with minimal errors in conservation of mass. The error of ±0.06% is definitely a worst case scenario since the surface expansion velocities associated charring materials are much lower.

3.3.2 Flow Over a Cylinder

Figure 3.8(a) shows vortex shedding behind a 1 mm diameter cylinder. A uniform mesh of

400 × 100 is used with open boundary conditions and a uniform inlet velocity of 18.9m/s resulting in a Re = 1000. Figures 3.8(b) shows the cross-stream velocity behind the cylinder along the centerline at 0.5 diameters downstream. A Strouhal number of St = 0.23 is computed and agrees well with measured values of shedding frequency of 0.22.80, 107 Fig- ure 3.8(c) is a comparison of the computed coeﬃcient of pressure with the results of Ref.80 where direct numerical simulation (DNS) was used to compute the exact solution. As shown, the percent error is within 3% for all data points.

3.3.3 Isothermal Plate Heat Transfer

The next validation problem considered is the heating of gas from an inclined isothermal plate as shown in Fig. 3.9(a). The temperature of the plate is varied to achieve a range of Rayleigh numbers and the far-ﬁeld gas temperature is initialized to T∞ = 300 K. Fig- ure 3.9(b) is an instantaneous temperature contour showing the transitionally turbulent na-

00 ture of the ﬂow. Figure 3.9(c) is a comparison of Nusselt number, NuL(= q¯˙ L/[(Ts−T∞)kg]) 3 vs. Rayleigh number, Ra (= g β(Ts−T∞)L cosθ ) from the simulations (lines with symbols) L νgαg CHAPTER 3. FLUID-SOLID COUPLING 56

and a standard heat transfer correlation for the lower surface of inclined plates given as,108, 109

" 1/6 #2 0.387RaL NuL = 0.825 + (3.21) 1 + (0.492/P r)9/168/27 where θ is the inclination angle. The experimental data used for the Nusselt number correlation is also shown (black squares). The overall agreement is very good with a maximum diﬀerence of 8%. Additional simulations are also conducted with the mesh aligned with the

ﬂuid-solid interface and the direction of gravity modiﬁed to account for the inclination angle

(green circles). The diﬀerence in NuL predictions and those using the immersed interface are less than 2% and is attributed to statistical errors in computing the time averaged heat

ﬂux, q¯˙00.

3.3.4 Surface Blowing

Errors in mass, momentum and energy exchange across the Eulerian-Lagrangian interface are important to minimize for modeling surface blowing phenomena such as ﬂame spread.

To explore the error associated with surface blowing a simple one dimensional diﬀusion problem is examined. The problem considered is the diﬀusion of one species (species A) into another species (species B) that is stagnant. The analytical expression for YA(x) in

terms of the mass fraction of species A on the interface (YA,s) is

Sc Y (x) = 1 − (1 − Y ) exp m˙ 00 x 110. (3.22) A A,s µ(T ) A

In the present problem, species A and B are both assumed to be nitrogen (N2). A mass ﬂux

00 −3 2 ofm ˙ A = 1.8107 × 10 kg/m − s and Sc/µ(T ) = 38280.25 m − s/kg is selected to produce a surface mass fraction YA,s = 0.5 at x = 0. CHAPTER 3. FLUID-SOLID COUPLING 57

Three interface methodologies are examined; “standard” ghost-fluid method, area modified ghost-fluid method, and an exact interface treatment using an Eulerian method. Fig- ure 3.10(a) compares the steady state results of the three cases against the analytical solution. Due to errors in the standard ghost-fluid method the mass fraction predicted on the interface is 0.543. This error is shown to approach 20% in Fig. 3.10(b) where the L2 error norm for the three cases is provided as a function of nodes solely in the gas phase. These errors are attributed to the mass leakage across the interface discussed in detail in section

3.2.2. Shown in the same figure, the area modified ghost-fluid and exact Eulerian methods converge first order to the analytical solution.

3.3.5 Sublimation with Constant Densities

The sublimation validation problem is chosen to test all components of the coupling algorithm which includes the moving interface and mass/energy exchange. This validation problem not only tests each of the coupling algorithms but how they interact.

The problem setup is shown in Fig. 3.11. The initial temperature of the domain is set equal to the solid sublimation temperature, Ts. The left boundary is then raised to To resulting in the generation of gas and movement of the gas/solid interface to the right. The dimensionless form of the governing equation for the gas temperature ﬁeld is,111

∂2θ ∂θ = (3.23) ∂η2 ∂τ

T (x, t) − Ts 2 θ = , η = x/b, τ = αgt/b To − Ts

where b is a characteristic length (chosen to be the domain size) and αg = k/(ρCP ) is the thermal diﬀusivity. The constraints at the normalized interface location, δ, are θ = 0 and

∂θ − ∂η = (1/Ste)∂δ/∂τ, where Ste is the Stefan number deﬁned as: Ste = CP (To − Ts)/Ls

where Ls is the heat of sublimation. The gas and solid densities are assumed identical and constant therefore there is no advection within the gas. The analytical solution for the CHAPTER 3. FLUID-SOLID COUPLING 58

temperature ﬁeld and interface position are given by,111

h i √η erf 2 τ θ(η, τ) = 1 − (3.24) erf(λ) √ δ(τ) = 2λ τ (3.25)

√ where λ can be solved for numericaly from the constraint equation πλeλ2 erf(λ) = Ste.

The solid portion of the sublimation model consists of a Lagrangian mesh where the left boundary is initially at η = 0. This solid model assumes that all energy passed from the Eulerian solver (Sec. 3.2.2) serves to sublimate the solid thereby releasing gas and moving the interface in the positive η direction. The interface velocity is calculated as:

I 00 V =q ˙ /(ρLs). The temperature of the solid model is held at the sublimation temperature. An additional Lagrangian mesh is used to enforce the isothermal boundary condition on the left boundary of the Eulerian domain.

The sublimation problem examined is deﬁned in terms of the parameters outlined in

Table 3.1. The calculated interface position is compared against the analytical solution in

Fig. 3.12(a). As the Eulerian grid is reﬁned from a spacing of ∆η = 0.003 to 0.114, the predicted interface location approaches the analytical solution. The rate of convergence for the interface location is shown to be nominally ﬁrst order in Fig. 3.12(b) where the L2 relative error norm drops from 24% to less than 1.5%. The convergence shown in Fig.

3.12(b) indicates that the coupling proposed algorithms not only work separately but when combined together.

Table 3.1: The parameters used to deﬁne sublimation validation problem.

Parameter Value Ste 0.0674 −6 2 αg 6.88 × 10 m /s To − Ts 5 K CHAPTER 3. FLUID-SOLID COUPLING 59

3.4 Conclusions

An immersed boundary methodology for the modeling of conjugate heat and mass transfer is presented with specific application to fire environments. The model addresses the need for an efficient and accurate update of the moving interfaces, a no-slip and no-penetration solid interface and exchange of energy and mass transfer across the interfaces. The need for an efficient and accurate interface update is accomplished with the introduction of a patch system which systematically reduces the resolution of the Lagrangian model as a function of distance from the interface. A combination of ghost-fluid approaches and point source forcing functions at the interface are used to impose the last two constraints.

This methodology is shown to satisfy conservation principles with minimal error. Further validation is performed with modeling of ﬂow over a cylinder with excellent results. The heat transfer aspect of the methodology is veriﬁed with an isothermal plate in buoyancy driven

flow. The surface blowing case verified the need for area modification in the ghost-fluid method in order to accurately capture low speed flows. CHAPTER 3. FLUID-SOLID COUPLING 60

! x (inside) x (outside) p1 p2

d p1 d p2

d p4

x (inside) x (inside) d p3 p4 p3

Figure 3.1: Classiﬁcation of CFD nodes as either being deﬁned as either inside (xp1, xp3, xp4) or outside (xp2) the projected volumes associated with the surface mesh.

n2 x3, xc x1

n3 n1 x p

Figure 3.2: Identiﬁcation of nearest node (xc) and attached surface elements for the calculation of the of level set function. CHAPTER 3. FLUID-SOLID COUPLING 61

&8< &8; &8: &89 &4

=#".>10& ?57(&

,-./0/1"*&2(+%3&

!"#$%&'()(*+& ,!"#$%&4&+%561&/1&.(73&

Figure 3.3: Illustrations of patch level growth starting from the original surface mesh.

4 4 4 4

2 2 2 2 γ

•0.1 •0.2 •0.3 •0.4

y 0 y 0 y 0 y 0 •0.5 •0.6 •0.7 •0.8 •2 •2 •2 •2 •0.9 •1 •1.1

•4 •4 •4 •4 Lagrangian Surface Mesh Eulerian Mesh Lagrangian Surface Mesh Eulerian Mesh Lagrangian Surface Mesh Eulerian Mesh Lagrangian Surface Mesh Eulerian Mesh

•1 0 1 •1 0 1 •1 0 1 •1 0 1 x x x x (a) (b) (c) (d)

Figure 3.4: Level set function initialization with patch levels of (a) 0 (original mesh), (b) -1, (c) -2 and (d) -3 for every node n the Eulerian mesh. Patches are indicated as diﬀerent colored lines on the surface mesh (a rectangle). CHAPTER 3. FLUID-SOLID COUPLING 62

Figure 3.5: Ghost-ﬂuid mirroring showing the ghost-ﬂuid node of interest, xG and its evaluation using linear extrapolation across the interface from the gas-phase at location, x. CHAPTER 3. FLUID-SOLID COUPLING 63

(a)

(b)

Figure 3.6: Flow over a moving 2D cylinder and a 3D sphere showing (a) vorticity contours around a cylinder, and (b) vorticity contours around a sphere with an isosurface at a vorticity of 5000 and contours of density. CHAPTER 3. FLUID-SOLID COUPLING 64

(a)

(b)

Figure 3.7: Mass conservation errors for moving cylinder and sphere cases showing (a) mass conservation errors for high (dashed red) and low (black) resolution cylinder cases, and (b) mass conservation errors for high (dashed red) and low (black) resolution sphere cases. CHAPTER 3. FLUID-SOLID COUPLING 65

(a)

(b)

(c)

Figure 3.8: Flow shedding over a stationary cylinder showing (a) Von Karman vortex streak, (b) cross-stream velocity along the centerline at 0.5 diameters downstream and (c) pressure coeﬃcient and percent error along the upper surface (0o corresponds to the upstream stagnation point). CHAPTER 3. FLUID-SOLID COUPLING 66

(a) (b)

(c)

Figure 3.9: Conjugate heat transfer for an isothermal plate showing (a) problem sketch, (b) instantaneous snapshot of temperature contours, (c) NuL vs. RaL for inclination angles of 30o and 60o. Green symbols are cases for which the mesh is aligned with the surface and the gravity vector changed. CHAPTER 3. FLUID-SOLID COUPLING 67

0.6 Standard Ghost•Fluid 0.5 Modified Ghost•Fluid Exact Eulerian 0.4 Analytical

A 0.3 Y

0.2

0.1

0 0 0.002 0.004 0.006 0.008 0.01 x (a)

0.25 0.2 0.15 0.1

0.05 Error Norm 2

L Standard Ghost•Fluid Modified Ghost•Fluid Exact Eulerian

50 100 150 200 250 300 Nodes (b)

Figure 3.10: One-dimensional diffusion test problem results showing (a) steady state mass fraction distribution and (b) L2 error for the “standard” ghost-fluid method, area modified ghost-fluid method, and exact interface treatment using an Eulerian method. CHAPTER 3. FLUID-SOLID COUPLING 68

! ! ! ! ! ! ! !

!! gas solid

!=1!

interface

!=0!

!(τ)! !!

Figure 3.11: A sketch of the sublimation validation problem of section 3.3.5. CHAPTER 3. FLUID-SOLID COUPLING 69

1 )

0.8

0.6

Analytical 0.4 0.114 0.057 0.028 0.014 0.2 Interface Location ( 0.007 0.003

0 0 2 4 6 8 Time () (a)

10-1

10-2 Interface Location eaieErrNorm Error Relative

2 Interface Velocity L Heat Flux 10-3 0.1 0.08 0.06 0.04 0.02 Non-Dimensional Spacing (b)

Figure 3.12: One-dimensional sublimation test problem results showing (a) the predicted interface location with grid reﬁned compared to the analytical solution and (b) L2 relative error norm for the non-dimensional interface location, velocity, and heat ﬂux. Chapter 4

Flame Spread Simulations

The porous media charring model and coupling algorithms outlined in chapters 2 and 3, respectively, are utilized to examine fully coupled simulations. Two types of simulations are performed; a small scale simulation utilized primarily for a convergence analysis and a large scale simulation matching an experimental setup. The carbon-epoxy material, given in detail in section 2.3, is used throughout. The small scale simulations are performed with few assumptions, i.e., direct numerical calculation of heat flux as the gradient of the temperature field and utilization of a single step chemical kinetics model. Due to the significant computational cost of this detailed analysis additional models are implemented for the large scale simulations. The details of both simulations are given in the remainder of this chapter.

4.1 Small Scale Simulations

The small scale simulations consist of a mesh reﬁnement study of the fully coupled problem, outlined in Fig. 4.4, performed in both one and two dimensions. The two dimensional problem setup consists a 2.5 cm × 5.0 m ﬂuid domain. The inset in Fig. 4.4 provides the detail of the 3.2 mm × 2 cm carbon epoxy sample composed of 8 × 50 elements. A circular

70 CHAPTER 4. FLAME SPREAD SIMULATIONS 71

ignition source, placed at 1.0 cm × 0.75 cm with a surface temperature of 900K, serves to initially ignite the gas released from the composite. The one dimensional cases are equivalent to a slice taken at y = 1.5 cm from the two dimensional problem. Consistent with the analysis described in Ref. [112], two additional heat fluxes are applied to the left hand side of the sample. The first is a uniform background heat flux of 25 kW/m2 (from a radiant heating panel) and the second is an ignition heat flux of 60 kW/m2 over the lower quarter of the sample as indicated in Fig. 4.4 (to account for burner induced ignition).

The ﬂow solver outlined in chapter 3.1 is used with a single step global reaction combustion model to describe the gas phase combustion processes for the small scale simulations.

The rate law governing reaction (4.1a) is speciﬁed assuming an Arrhenius relation given in

Eq. (4.1b).

CH1.3O0.2 + 1.225(O2 + 3.76N2) → CO2 + 0.65H2O + 1.225(3.76)N2 (4.1a)

d [CH1.3O0.2] −Ea 0.2 1.3 = −A exp [CH1.3O0.2] [O2] (4.1b) dt RuT

The resulting system of ordinary diﬀerential equations is simultaneously solved using a customized version of the Semi-Implicit Euler numeric integrator found in Ref. [113]. As an approximation proportions of carbon, hydrogen and oxygen in the pyrolysis gas are assumed

76 to be the same as those given by Tewarson as CH1.3O0.2. The reaction in Eq. (4.1a) is chosen to be similar to that for methane where the Arrhenius parameters are obtained from

Ref. [114]. Further details on the treatment of the epoxy gas is available in Ref. [112].

The convergence analysis is performed by varying the number of Eulerian nodes from a

spacing of 0.5 mm to 0.016 mm for the 1D cases and 1.0 mm to 0.125 mm for the 2D cases.

Since an exact solution isn’t available, the most reﬁned case is chosen as the basis for the

L2 relative error norm. Two parameters are utilized to study convergence, the total heating ˙ ˙ R rate (Q) and average surface temperature. The total heat is calculated as Q = Γ (q ˙) dΓ. The area of interest (Γ) for both parameters is limited to the left hand side of the panel. CHAPTER 4. FLAME SPREAD SIMULATIONS 72

The rate of convergence for both parameters is shown to be nominally first order in Fig. 4.2 where there is clear overlap between the 1D and 2D L2 relative error norms. The error associated with average surface temperature is approximately two orders of magnitude less than that of total heat error. As described in section 3.2.2, the heat flux is calculated based upon the gradients of temperature thereby introducing additional error to that of the temperature field. The convergence shown in Fig. 4.2 indicates that the coupling proposed algorithms not only work separately but when combined together.

Figure 4.3 illustrates the typical progression of heating, composite pyrolysis, and combustion. As shown in Fig. 4.3(a), the lower region of the composite ﬁrst decomposes from the applied heat load thereby releasing volatile gases indicated with the mass fraction of

CH1.3O0.2. When the volatile gases and available oxygen reach the ignition temperature they ignite. The flame shortly after ignition is shown in Fig. 4.3(b). This flame increases the overall heat returned to the surface causing additional pyrolysis and volatile gas release shown in Fig. 4.3(c). The pyrolysis of the composite material is accompanied by expansion also shown in this figure. As the pyrolysis region grows from the additional heat from the

ﬂame the primary mechanism of ﬂame spread is modeled.

4.2 Large Scale Simulations

Due to the detailed numerics of the small scale simulations outlined in the previous section additional assumptions were introduced to speed computation. The two primary computational constraints of the simulation are the solution of flow chemical kinetics and required explicit time step associated with the Eulerian mesh spacing needed to directly capture the heat flux. These constraints were relaxed with the implementation of a flamelet and heat

ﬂux models.

A ﬂamelet based presumed PDF combustion model is used to describe the gas phase combustion processes. Standard LES subgrid scale (SGS) models are employed consisting CHAPTER 4. FLAME SPREAD SIMULATIONS 73

of a Smagorinsky model for momentum and gradient diﬀusion models for mixture fraction and energy transport. Further details on the LES and implementation of SGS models can be found in Refs.90, 91 The heat ﬂux to the surface of the composite is modeled in the LES as,

00 00 00 00 q˙ =q ˙LES +q ˙rad +q ˙app (4.2a)

00 q˙LES = MAX(hF , hN )(T∞ − Ts) (4.2b)

00 4 4 q˙rad = εσ T∞ − Ts (4.2c)

where hN is the natural convection coeﬃcient, hF is the forced convection coeﬃcient and

00 q˙app is the applied heat ﬂux including the igniter ﬂame. The solid surface temperature,

Ts, is determined from the FE model. The natural convection coeﬃcient is modeled using

1/3 2 1/3 115 the relation, hN = C(T∞ − Ts) where C = 1.5 W/m − K . The forced convection coeﬃcient is computed using a Nusselt (Nu) based number correlation, i.e., h = Nu kg/L, where the characteristic length, L, is set equal to an estimate of the local grid cell size and

Nu is modeled as: Nu = 0.037Re0.8P r1/3 where the Prandtl (P r) number is set equal to

0.7 and the Reynolds number, Re = |ug,∞| L/νg,∞, is computed using “far-field” properties (denoted with subscript ∞)116 which are interpolated from the fluid normal to the surface at a distance of ∆x, where ∆x is the spacing associated with the closest cut-cell. The far-field temperature, velocity and kinematic viscosity (needed for Re) are interpolated at a location normal to the fluid-solid interface located at a distance equal to L. The contribution of thermal radiation from the near-wall flame are modeled assuming blackbody radiation from the flame surface, where ε is the emissivity (assumed unity) and σ is the Stefan-Boltzmann constant.

Figure 4.4 shows a subsection of the large scale simulation setup with a 0.9 m × 0.9 m

ﬂuid domain consisting of a 114×114 uniform Eulerian mesh. The inset in Fig. 4.4 provides the detail of the 3.2 mm×25 cm carbon epoxy panel composed of 30×30 elements. An inert CHAPTER 4. FLAME SPREAD SIMULATIONS 74

backer-board section is added to the back of the panel. The thickness of the backer-board is selected to be 4.68cm to provide a minimum of 6 nodes to accommodate the ghost-ﬂuid methodology. This setup was designed to match the experimental conditions of Quintiere,

Walters and Crowley (QWC).71 Predictions from the decoupled study of carbon-epoxy composites including heat release rate, time-to-ignition and ﬁnal products (mass fractions, volume percentages, porosity, etc.) are compared to this experimental data in section 2.3, where agreement is shown to be quite good.

For most charring materials self-sustained burning will not occur unless a separately imposed heat flux is applied. This heat flux is in addition to that from the flame. The mini-

00 mum heat flux to sustain burning is the critical heat flux,q ˙crit. In the study of QWC, several 00 2 flame spread experiments were conducted and a critical heat flux ofq ˙crit = 14.2 kW/m was determined. The experiments are conducted by heating a 3.2 mm thick by 6 cm × 25 cm carbon-epoxy sample with radiant heaters and a piloted flame near the bottom of the sample. The igniter flame produces an estimated net heat flux of 60 kW/m2 over approximately

6cm of the bottom of the sample. The purpose of the full scale simulations is to see if the current modeling capabilities can reproduce the critical heat ﬂux determined by QWC.

Figure 4.5 illustrates the typical progression of heating, composite pyrolysis, and combustion. As shown in Fig. 4.5(a), the lower region of the composite ﬁrst decomposes from the applied heat load, as indicated by the contour plot of the reaction progress variable. The decomposition serves to both release volatile gases and induce swelling within the composite.

The released gases burn, increasing the ﬂame surface area and the associated preheat zone.

This coupled processes expands the pyrolysis region shown in Fig. 4.5(b). If the applied heat flux is greater than 14.2 kW/m2 the combined heating from the flame will continue resulting in the decomposition of the exposed laminate surfaces after sufficient time, shown in Fig. 4.5c.

A parametric study is performed to study the burn sensitivity of the critical heat ﬂux.

The heat ﬂuxes examined, applied to the entire front face of the panel, include 0, 10, 14.2 CHAPTER 4. FLAME SPREAD SIMULATIONS 75

and 20 kW/m2. For the first twenty seconds of simulation the heat flux is composed from the igniter flame, applied, and heat flux from the flame, after which the igniter flame is removed. The total heating rate (Q˙ ) delivered to the panel from the flame is presented ˙ R 00 00 in Fig. 4.6. The total heat is calculated as Q = Γ (q ˙LES +q ˙rad) dΓ, where Γ is the total exposed area of the laminate. After the removal of the igniter flame, for cases with an

applied heat ﬂux less then 14.2 kW/m2 the ﬂame extinguishes resulting in panel cooling

(negative heating rate). Consistent with the results of QWC, applied heat ﬂuxes equal to

or greater then the critical heat ﬂux results in ﬂame spread. CHAPTER 4. FLAME SPREAD SIMULATIONS 76

0.0325 T 0.05 T 780 760 1900 0.0275 740 1700 720 0.04 700 1500 680 y(m) 1300 660 1100 0.0225 640 900 620 0.03 600 700 580 500 560 300 0.0175 y(m) 0.02

0.0125 0.0135 0.0185 0.01 x(m) Igniter

0 0 0.01 0.02 x(m)

Figure 4.1: Fully coupled simulation showing the CFD and FE domains along with igniter. CHAPTER 4. FLAME SPREAD SIMULATIONS 77

100 Total Heat (1D) Total Heat (2D) Avg. Surf. T (1D) Avg. Surf. T (2D)

10-1

10-2 eaieErrNorm Error Relative 2 L

10-3

1 0.8 0.6 0.4 0.2 Spacing (m)

Figure 4.2: The L2 relative error norm in average surface temperature and total heat to the surface for the fully coupled simulations in one and two dimensions. CHAPTER 4. FLAME SPREAD SIMULATIONS 78

(a)

(b) CHAPTER 4. FLAME SPREAD SIMULATIONS 79

(c)

Figure 4.3: Fully coupled simulations of the carbon-epoxy composite showing mass fraction of CH1.3O0.2, gas and composite temperatures after (a) 2.5 s (b) 3.5 s and (c) 4 s. CHAPTER 4. FLAME SPREAD SIMULATIONS 80

0.8

T 0.35 T

1400 1100 1300 1050 0.3 1200 1000 950 1100 900 0.5 1000 0.25 850 900 800 750 800 y 700

y 700 0.2 650 600 600 Igniter 500 Flame 0.2 400 0.15 300

0.1 45 55 0.14 1 0.15 1 0. x 0. Inert Backer•Board •0.1 •0.35 •0.05 0.25 0.55 x

Figure 4.4: Fully coupled simulation showing the CFD and FE domains along with inert backer-board and igniter ﬂame region. CHAPTER 4. FLAME SPREAD SIMULATIONS 81

0.8 0.35 T (K) α 0.95 1400 0.9 1300 0.3 0.85 1200 0.8 1100 0.75 0.5 1000 0.7 900 0.25 0.65 800 0.6 700 y y 0.55 600 0.5 500 0.2 0.45 400 0.2 0.4 300 0.35 0.3 0.15 0.25 0.2 0.15 0.1 •0.1 0.1 0.05 •0.35 •0.05 0.25 0.55 x Time: 15 s 0.15 Heat Flux: 14.2 kW/m2 0.145 x 0.155 (a)

Figure 4.5: Fully coupled simulations of the carbon-epoxy composite with an uniform applied heat ﬂux of 14.2 kW/m2 after (a) 15 s (b) 30 s and (c) 45 s. CHAPTER 4. FLAME SPREAD SIMULATIONS 83

14 Igniter Flame Removal 0.0 kW/m2 12 10.0 kW/m2 10 14.2 kW/m2 20.0 kW/m2 8

-2 Time Average Total Heat (kW) -4

-6 0 10 20 30 40 50 Time (s)

Figure 4.6: Time history of integrated heat ﬂux over the composite surface for applied heat ﬂuxes of 0.0, 10.0, 14.2, and 20.0 kW/m2. Chapter 5

Conclusion

In his 1968 review article “A survey of knowledge about idealized fire spread over surfaces”, Friedman identified several characteristics needed for any flame spread model to be complete. These characteristics of completeness include flame aerodynamics, chemical phenomena, fuel geometry and heat transfer to the fuel.117 In this study numerical techniques were introduced to better model complex fuel geometry and associated heat transfer in

ﬂame spread simulations. This required two primary tasks 1) extension to model additional charring materials and 2) introduction of a coupling algorithms between the charring and

ﬂow solvers. The charring model includes pyrolysis decomposition, heat and mass transport, individual species tracking, and volumetric swelling using a novel ﬁnite element algorithm.

The newly introduced swelling model accounts for the leading order effects from composite swelling which is to decrease the solid volume fractions and increase the gas volume fraction. Overall good agreement is observed between the model and data for the overall heat release rate and time-to-ignition. Neglecting the effects of the composite swelling resulted in significant under-predictions of flame burnout - highlighting the importance of accounting for laminate swelling in carbon-epoxy material.

An immersed boundary coupling methodology for the modeling of conjugate heat and

84 CHAPTER 5. CONCLUSION 85

mass transfer is presented with specific application to fire environments. The model addresses the need for an efficient and accurate update of the moving complex interfaces, a no-slip and no-penetration solid interface and exchange of energy and mass transfer across the interfaces. An efficient and accurate interface update algorithm was accomplished with the introduction of a patch system which systematically reduces the resolution of the La- grangian model as a function of distance from the interface. A new combination of ghost-

fluid approaches and point source forcing functions at the interface are used to impose the last two constraints. This methodology is shown to satisfy conservation principles with minimal error. Further validation is performed with modeling of flow over a cylinder with excellent results. The heat transfer aspect of the methodology is verified with an isothermal plate in buoyancy driven flow. The surface blowing case verified the need for area modification in the ghost-fluid method in order to accurately capture low speed flows.

The newly introduced interface methodology is combined with the carbon-epoxy charring model and is utilized to study the burning of composites. The combined algorithms used for the fully coupled simulations are shown to be numerically consistent and mesh independent. Lastly, the newly introduced methodologies are utilized to study the full scale burning of carbon-epoxy composites with predictions closely matching that of experimental results. A parametric study is performed to study the burn sensitivity of the critical heat

flux, where the critical heat flux is defined as the minimum externally applied heat flux required to sustain flame spread. When this heat flux, often applied from an radiant heater, falls below that of the critical heat flux the flame extinguishes. This behavior was predicted numerically where a critical heat flux 14.2 kW/m2 was determined, which is consistent with experimental measurements. CHAPTER 5. CONCLUSION 86

5.1 Proposed Future Research

This research focused upon the development of numerical methods to accurately model

ﬂame spread. However, very large scale simulations are still computationally prohibitive.

This cost can be further reduced with the introduction sub grid scale models such as the embedded ﬂame spread model (EFSM).78 The basic idea of the methodology is to express the near-wall ﬂame gas composition (i.e., mass fractions (Yi) and temperature (T )) and the

00 surface normal heat flux (q ˙n) in terms of a reduced number of reaction progress variables and their wall-normal gradients. This methodology is far from complete and has not been tested for complex geometries, (charring) materials and turbulent flows. The advantage of this modeling approach is the ability to obtain quantitatively accurate estimates of heat flux to solid surfaces without excessive computational costs so that it may be used in practical

ﬁre safety analysis. The numerical methods developed can be utilized to construct the

EFSM models for sample sized simulations which then can be used as a SGS model in LES to explore ﬂame spread over construction components such as wall and ceiling assemblies.

The focus of this research as been the study of ﬂame spread over charring materials.

However the numerical methodologies developed can be applied to model a variety of fire scenarios where the explicit conservation of mass, momentum, and energy is important across complex geometry. Recent work as focused upon one such application. The charring model is based upon homogenization theories where the local field equations are averaged over a localized volume chosen to be sufficiently large relative to the mesoscopic features of the solid but small relative to the system. This procedure allows material proprieties such as conductivity and permeability to be specified for the homogenized material, where the properties are often determined using experimentally derived relations. The introduced methodologies will allow simulations of mesoscopic unit cell where the material response and interface can be explicitly tracked. This simulations would allow the construction of properties that can be applied to the homogenized material from first principles. Appendix A

Balsa Wood Modeling

The combustion of wood can be modeled as a series of zones that are depicted in Fig. A.1.

Initially the entire content can be considered in zone one where the wood is present in its original state with signiﬁcant moisture content. Upon heating, the available moisture evap- orates into vapor where it migrates towards the surface or further into the wood where it can condense. Zone three consists of the portion of wood behind the evaporation front and before the pyrolysis zone. Pyrolysis is the degradation of solid fuels due to temperature.118

The constituents of wood are considered to degrade in diﬀerent temperature ranges: hemi- cellulose (498 − 598K), cellulose (598 − 648K) and lignin (523 − 773K).62 The pyrolysis of lignocellulosic fuels results in products that are often classiﬁed into groups of tar, char and volatile vapors. As the pyrolysis zone progresses the char zone is produced thereby changing the physical and thermal properties of the wood. The volatile vapors that are produced escape through surface of the solid fuel which burn and thereby contribute to the heating.118

Consistent with the combustion of all lignocellulosic fuels, moisture is present in both liquid and vapor forms. The introduction of liquid requires additional transport equations to account for the evaporation and diﬀusion of water. Pyrolysis of lignocellulosic fuel results in the production of species that can be classiﬁed as tar, volatile vapors, and char. This,

87 APPENDIX A. BALSA WOOD MODELING 88

Table A.1: Property values utilized for the numerical model for balsa wood.

Property Phase Value Source Permeability gas 5.432 × 1013(m2) 119 gas 299.8+5.4037T −1.60×10−3T 2(J/kg−◦C) ◦ Speciﬁc Heat Cw,v = 0.68T + 1420(J/kg − C) 119 wood ◦ Cw,d = 1.33T + 3194(J/kg − C) Thermal k = 0.06 + 9.21 × 10−8T 2.503 (W/m2 −◦ wood w,v 119 Conductivity C) −6 1.89 2 ◦ kw,d = 0.008+2.223×10 T (W/m − C) Intrinsic wood 334.5kg/m3 120 Density char 102.8kgm3 Heat of De- wood −500kJ/kg 119 composition Heat of Va- bound −2, 440kJ/kg 119 porization water

in addition to the water vapor present in all wood, requires the tracking of multiple species with homogenous chemical reactions throughout the gas phase.

A.0.1 Thermal and Transport Properties

Temperature dependent speciﬁc heats and conductivity for both the virgin and decomposed balsa are based on those developed by Lattimer et al.119 and summarized in table A.1 for a direction parallel to the grain. The bulk properties are determined using a reaction progress variable model, e.g., C = FCw,v + (1 − F )Cw,d where, F = (ρ − ρd)/(ρv − ρd), is the reaction progress variable, subscripts v and d denote virgin and fully decomposed

materials, respectively. Both the bulk and differential diffusion effects are considered for

the gas flow processes. The bulk flow and differential diffusion are modeled using Darcy’s

Law and Fick’s law, respectively. APPENDIX A. BALSA WOOD MODELING 89

Table A.2: The Arrhenius kinetic parameters used to model balsa wood, reproduced from Ref. [120].

Reaction Order Activation Energy Pre-Exp. Factor Dehydration 1.205 47.4(kJ/mol) 2.55 × 106 Decomposition 3.94 187.3(kJ/mol) 73.7

A.0.2 Water Transport

The majority of pyrolysis models neglect the eﬀect of liquid moisture content transport.

This assumption cannot be made when modeling the combustion of moist woods because of its effect on delaying the pyrolysis process. Moisture can exist in three forms within lignocellulosic fuels: bound water, free or capillary liquid water, and water vapor. All liquid water within wood can be considered bound until the percentage by weight exceeds that of the fiber saturation point (FSP). Above the FSP the woods cells are considered saturated and additional water exists in free form, where it located in capillaries and voids in the wood. The fiber saturation point for most lingocelluosic fuels is approximately 30% by weight.121

The implementation of water transport is speciﬁc for the material to be studied, balsa wood. The balsa wood within the sandwich panel has water content 7% by weight and therefore can be consider bound water.122 The transport of bound water within wood

can be modeled using Ficks law and a modified diffusion coefficient.123 The evaporation

rate from bound water to water vapor and is described using an experientially determined

Arrhenius reaction rate given as:

dρ ρ − ρf E = −(ρi − ρf ) A exp (A.1) dt ρi − ρf RT where the Arrhenius kinetic parameters were obtained from Ref. [120] and given in Table

A.2. APPENDIX A. BALSA WOOD MODELING 90

Table A.3: The initial and ﬁnal volume fractions of balsa wood constituents determined experimentally.120

Constituents Initial Final Gas 0.38 0.55 Virgin Wood 0.61 0 Char 0 0.45 Bound Water 0.012 0

A.0.3 Balsa Results

The heat and mass transfer processes in a direction parallel to the wood grain are considered.

The system is solved using a high-order accurate spectral ﬁnite element method. A total of

50 elements are employed with each element consisting of a fourth order accurate Lagrange polynomial resulting in a 201 nodes, as shown in Fig. A.2. The use of higher order elements is required in order to capture the effects of Darcys and Ficks laws represented in the source terms of the finite element equations. The second derivative is calculated based upon the element basis functions and therefore require greater than first order basis functions to produce a physical representation. The nodes are clustered towards the applied heat flux according to the transformation:

(β + 1) − (β − 1) [β + 1)/(β − 1)]1−η x = H (A.2) ([(β + 1)/(β − 1)]1−η + 1) where H is the thickness of sample and β = 1.05 is a mesh compression factor. The applied boundary conditions consist of atmospheric pressure and applied heat ﬂux at x equal to zero and adiabatic and zero mass ﬂux at x equal to 0.05 meters. The baseline initial and

ﬁnal composition of the gas, wood, char and bound water are summarized below in Table

A.3.

A variable heat ﬂux is applied at the boundary of the wood using a grey gas assumption, where the far-ﬁeld is a black body radiator and the surface is assumed black, i.e., APPENDIX A. BALSA WOOD MODELING 91

00 4 4 q = σ(T∞ − Ts ). Two far-ﬁeld temperatures of T∞ equal to 1175K or 1290K are chosen corresponding to low and high heating cases.

Figure A.3 presents proﬁles of (a) constituent volume fractions, (b) temperature and gas pressure, (c) mass loss rates and (d) thermal properties for the baseline case of T = 1175K.

Consistent with the sketch of decomposition in Fig. A.1, two distinct decomposition regions can be identified. The first is associated with dehydration processes where the bound water is liberated and evaporation occurs. This region can identified in Fig. A.3(c) where the

000 evaporation rate,m ˙ boundwater , is non-zero. In this region, ϕboundwater goes to zero and a slight increase in ϕgas is observed in Fig. A.3(a). The release of bound water and evaporation serves to increase the density of the gas within the balsa structure resulting in increased pressure (Fig. A.3(b)) and mass fraction of water vapor (Fig. A.3(c)).

000 The second region is that associated with wood charring wherem ˙ balsa is non-zero in Fig. A.3(c). In between dehydration and charring regions, the temperature undergoes a rapid rise (Fig. A.3(b)) associated with the rapid heating of the dry wood. The charring of the wood converts the virgin dry wood to char and gas resulting in a rapid increase of ϕgas and

ϕchar to 0.6 and 0.4, respectively. The release of gaseous wood products further increases the density of the air and therefore the pore gas pressure to a peak value of 114kP a, shown in Fig. A.3(b). At the ﬁnal stage of pyrolysis the bulk speciﬁc heat is a factor of two larger than the virgin material and the conductivity increases by a factor of 8.

Three separate fronts can be identified in Fig. A.3 associated with an initial temperature rise (XT ), water vaporization (Xwater) and char formation (Xchar). To examine the sensitivity to initial moisture content and heat flux, each front is tracked and compared. For these cases, the temperature front is defined as the spatial location when the temperature

first reaches 305K. The water and char fronts are determined by finding the volume fraction location corresponding to the average of the initial and final values. Figure A.4 presents the temperature, water and char fronts for far field black body radiation temperatures of

1175K and 1290K, and for initial volume fractions of (a) 0.012 and (b) 0.024, respectively. APPENDIX A. BALSA WOOD MODELING 92

Front progression is shown to be highly sensitive to initial volume fraction of water and applied heat flux (Fig. A.4). As expected, the high heat flux produces temperature, water, and char progression rates higher than those for the low heat flux. For all fronts, the increase in initial volume fraction of water serves to delay front progression. This can be attributed to both an increase in thermal mass and an increase in advection energy transport. Primary energy transport through conduction serves to heat the material while gas flow toward the heated surface serves to transport energy away from the pyrolysis zone. At the higher heat

fluxes, the rate of dehydration and charring is larger resulting in a corresponding larger pore pressure near those regions than for the lower heat flux case. The larger pressure gradient drives gas flow away from the pyrolysis front resulting in a net reduction in the front speed.

The increase in water content from 0.012 to 0.024 volume fraction slowed temperature front

progression by 12% and 9% for the high and low heat ﬂux cases, respectively. It should

be noted, however, that these trends are sensitive to the value of the permeability and its

dependence on the decomposition process.

1. Wet wood 2. Evap. Zone 3. Dry wood 4. Pyrolysis zone 222 5. Char zone 6. Flame

1 2 3 4 5 6 222 Figure A.1: Sketch of a balsa slab response to heating from a ﬁre. APPENDIX A. BALSA WOOD MODELING 93

Figure A.2: One-dimensional mesh utilized for the numerical modeling of balsa wood. APPENDIX A. BALSA WOOD MODELING 94

(a)

(b) APPENDIX A. BALSA WOOD MODELING 95

(c)

(d) Figure A.3: Distributions of volume fraction (a), temperature and pressure (b) and mass fraction and mass source/sink terms (c) and thermal properties (d) after three minutes of heating with low heat ﬂux and 0.012 initial volume fraction of bound water. APPENDIX A. BALSA WOOD MODELING 96

(a)

(b) Figure A.4: Front location of the bound water, char and temperature in balsa wood for far ﬁeld temperatures of 1290 K and 1175 K for initial volume fractions of water of 0.012(a) and 0.024(b). Appendix B

Finite Element Domain Decomposition

The introduction of domain decomposition techniques into FE models requires the division of the entire domain Ω into multiple subdomains Ωs for ease of computation across mul-

tiple processors/computers. The subdivided domain must then be reconnected or “glued”

in order to solve the global system. Three of the most popular methods for “gluing”

subdomains back together include Lagrange multipliers, penalty-based, and mortar based

techniques. Lagrange multipliers arise through the variational derivation of the FE gov-

erning equations but can also represent physical quantities between subdomains, such as

heat ﬂux. This technique allows meshes of equal discretization to maintain conservation

principles and continuity across the interface.124 The mortar and dual mortar methods for

coupling ﬁnite element subdomains124, 125 relax the equal mesh discretization requirements,

thereby allowing for diﬀerent mesh resolutions in each subdomain. Other domain decompo-

sition methods include the penalty-based interfaces,124, 126 where the number of unknowns

is reduced with the elimination of the Lagrange Multipliers. This is done through the in-

troduction of a penalty parameter. These methods only approximately maintain continuity

between subdomains (as a function of the penalty parameter) whereas Lagrange multiplier

97 APPENDIX B. FINITE ELEMENT DOMAIN DECOMPOSITION 98

based methods enforce the constraints exactly. The use of penalty techniques may also introduce additional numerical problems into the solution (locking or instability) and often require change to the existing numerical implementation.124

The focus of this chapter is the implementation of Lagrange multipliers for the coupling of FE subdomains. As with much of the early finite element research, domain decomposition with Lagrange multipliers research focused on the solution to linear elastic equations as applied to structures.127, 128 Many of these early methods had the possibility of introducing “floating” subdomains. These subdomains, named because of the early focus upon structures, were subdomains not connected to a Dirichlet boundary condition. The traditional solution to floating subdomains required the inverse to a singular matrix and therefore prevented arbitrary mesh decomposition.127 The problem of “floating” subdo-

mains was addressed by Farhat and Roux129 with the introduction of the Finite Element

Tearing and Interconnecting (FETI) methodology. This method, also known as the dual

Schur technique,130 has been widely implemented in both research and commercial frame-

works.131 Since its inception, the FETI method has been applied to a variety of problems

beside the initial static linear elastic equations. Applications include transient response,132 advection-diﬀusion equation,133 simple transport equations134 and eletro-magnetics.135 One of the reasons for the success of the FETI method is the scalability of the algorithm. The

H 3 H condition number of the global problem is approximately given by 1 + log h , where h is the ratio of subdomain to element. This implies that if the number of processors utilized to solve the problem scales with the domain size, the total number of iterations to solve the global system remains ﬁxed, thereby producing a scalable algorithm.136

The organization of the rest of this chapter is as follows: in Section B.1 the mathematical and numerical formulation are discussed. The coding implementation with a detailed discussion of message passing in Java is summarized in Section B.2. The veriﬁcation and performance results are provided in the Section B.3. Section B.4 summarizes the ﬁndings from this study. APPENDIX B. FINITE ELEMENT DOMAIN DECOMPOSITION 99

B.1 Mathematical and Numerical Formulation

B.1.1 Finite Element Methodology

Finite elements methods provide a powerful mathematical tool for analyzing a variety of problems. This project will focus upon analyzing transient heat conduction in a continuous media assuming constant properties given below108

∂T ρc = ∇ · (k∇T ) +q ˙ . (B.1) p ∂t

A “weak” or “variational” form is obtained by multiplying equation Eq. (B.1) by a weight

function w and integrating over the domain (Ω)69 resulting in

Z ∂T Z Z ρcp wdΩ = ∇ · (k∇T ) wdΩ + qwd˙ Ω . (B.2) Ω ∂t Ω Ω

The second term on the right hand side of Eq. (B.2) can be rewritten using the integration by parts theorem,

w∇ · (k∇T ) = ∇ · (wk∇T ) − ∇w · k∇T. (B.3)

The ﬁrst volume integral on the right hand side of Eq. (B.3) can be expressed as a surface integration by applying the Gauss divergence theorem,

w∇ · (k∇T ) = ∇ · (wk∇T ) − ∇w · k∇T (B.4) wheren ˆ is the outward normal and Γ is the surface boundary. Using Fourier’s Law, k∇T ·nˆ can be redeﬁned as the normal heat ﬂux leaving the surface or qn resulting ,

Z Z Z ∇ · (wk∇T ) dΩ = wk∇T · ndˆ Γ = wqndΓ . (B.5) Ω Γ Γ APPENDIX B. FINITE ELEMENT DOMAIN DECOMPOSITION 100

Substituting these changes into Eq. (B.2) produces the weak form of the heat transfer equation Z ∂T Z Z Z ρcp wdΩ + ∇w · k∇T dΩ = wqndΓ + qwd˙ Ω . (B.6) Ω ∂t Ω Γ Ω

In Eq. (B.6) the highest order derivative has been reduced to ﬁrst order in space for both

temperature and the weight function w.

Until this point, no assumptions about the solution to Eq. (B.1) have been employed.

However, in order to develop numerical solutions to the weak form of the heat transfer

equation assumptions about the temperature ﬁeld must be made. The Rayleigh-Ritz ap-

proximation is chosen in development of the ﬁnite element method.69 For a given element,

deﬁned by N nodes, the temperature distribution is assumed to ﬁt a Lagrange interpolating

polynomial, N e X e e T (~x,t) ≈ T (~x,t) = Tj (t)ψj (~x) . (B.7) j=1

It is important to note that temperature, a function of space and time, is represented by

nodal temperatures and basis functions which are only functions of time or space, respec-

tively.

Substituting, the approximation for temperature into the weak form of the heat transfer

equation, Eq. (B.6), results in the following for a single element,

N e Z N Z Z Z X ∂Tj X ρc ψewdΩ + ∇w · k∇ψedΩT e = wq dΓ . + qwd˙ Ω . (B.8) ∂t p j j j n j=1 Ωe j=1 Ωe Γe Ω

The temperature at each node is assumed unknown in Eq. (B.8) resulting in N unknowns and requiring N independent algebraic equations to solve the system. Consistent with the

Rayleigh-Ritz method, N expressions for the weight function w are chosen to be equal to

th the basis functions used to describe temperature, wi = ψ1, ψ1, ...ψ1. For the i system APPENDIX B. FINITE ELEMENT DOMAIN DECOMPOSITION 101

equation, corresponding to the weighting function equal to ψi, Eq. (B.8) simpliﬁes to

N e Z N Z Z Z X ∂Tj X ρc ψeψedΩ + k∇ψe · ∇ψedΩT e = ψeq dΓ + qψ˙ edΩ . (B.9) ∂t p j i i j j i n i j=1 Ωe j=1 Ωe Γe Ωe

The entire system of equations, including contributions from each element, can be compactly written in matrix form as,

[M]{T˙ } + [K]{T } = {Q} + {qb} (B.10)

where [M], [K], {Q}, {qb} are the mass, “stiﬀness”, source and surface terms, respectively, and are deﬁned as follows,

Z e e Mi,j = ρcpψj ψi dΩ (B.11) Ωe Z e e Ki,j = k∇ψi · ∇ψj dΩ (B.12) Ωe Z e Qi = qψ˙ i dΩ (B.13) Ωe Z e qb(i) = ψi qndΓ . (B.14) Γe

B.1.2 Domain Decomposition Methodology

The proposed domain decomposition method will link each subdomain using Lagrange multipliers (λ) and continuous temperature interface conditions. In order to introduce the Lagrange multipliers into ﬁnite element formulation Eq. (B.8) must be expressed in variational form as a functional. The functional Π is a scalar function of temperature Π(T ).

The equivalency of the variational approach and Eq. (B.10) can be shown by taking the APPENDIX B. FINITE ELEMENT DOMAIN DECOMPOSITION 102

ﬁrst variation with respect to δT of Π,124 where the variation in the functional is

1 δΠ = {δT }T [M]{T˙ } + {δT }T [K]{T } (B.15) 2 1 + {T }T [K]{δT } − {δT }T {Q} − {δT }T {q } = 0 . 2 b

If the stiﬀness matrix is symmetric, as it is for the heat conduction equation, terms in Eq.

(B.15) can be combined as follows,

1 1 {δT }T [K]{T } + {T }T [K]{δT } = {δT }T [K]{T } 2 2 where Eq. (B.15) simpliﬁes to,

T T T T δΠ = {δT } [M]{T˙ } + {δT } [K]{T } − {δT } {Q} − {δT } {qb} = 0 . (B.16)

The variation in T , δT , can be factored from Eq. (B.17) resulting in,

T δΠ = {δT } [M]{T˙ } + [K]{T } − {Q} − {qb} = 0 (B.17) therefore indicating that,

[M]{T˙ } + [K]{T } − {Q} − {qb} = 0 (B.18) which is equivalent to Eq. (B.10). Thus far Dirichlet type boundary conditions have not been considered. The enforcement of Dirichlet boundary conditions simply state that the temperature T at a given point on the surface must be equal to a prescribed temperature T¯.

These additional constraints can be introduced directly into the functional form by deﬁning a new functional Π¯ as,

Z Π(¯ T, λ) = Π(T ) + λ T − T¯ dΓ . (B.19) Γλ APPENDIX B. FINITE ELEMENT DOMAIN DECOMPOSITION 103

where Γλ is the surface area in which the Dirichlet constraint applies. Taking the variation of Eq. (B.19) results in,

Z Z δΠ(¯ T, λ) = δΠ(T ) + δT λdΓ + δλ T − T¯ dΓ = 0 . (B.20) Γλ Γλ

In order to solve for λ, a numerical method for approximating λ based upon nodal values must established. For simplicity and consistency the form for λ is assumed to be λ ≈

Pn th k=1 ψk(~x)λk, where λk is the value for the Lagrange multiplier at the k node on the surface. Utilizing this approximation for λ and a Lagrange interpolating polynomial for the

δT , consistent with the Rayleigh-Ritz method, R δT λdΓ can be written in vector form as, Γλ

{δT }T [q]{λ} (B.21) where the coupling matrix [q] is given by the following,

Z q(i,k) = ψiψkdΓ . (B.22) Γλ

Substituting Eq. (B.22) into Eq. (B.20) results in the following simpliﬁed system,

T δΠ(¯ T, λ) = {δT } [M]{T˙ } + [K]{T } − {Q} − {qb} + [q]{λ} (B.23) Z + δλ T − T¯ dΓ = 0 . Γλ

With the requirement that Eq. (B.23) must remain valid independent of δT implies that,

[M]{T˙ } + [K]{T } − {Q} − {qb} + [q]{λ} = 0 . (B.24)

This statement also implies that the following must be independently true,

Z δλ T − T¯ dΓ = 0 (B.25) Γλ APPENDIX B. FINITE ELEMENT DOMAIN DECOMPOSITION 104

or approximately by employing Eq. (B.7)

  Z N N X X ¯ ψk  Tjψj(~x) − Tjψj(~x) dΓ = 0 (B.26) Γλ j=1 j=1

Equation (B.26) can be expressed in matrix form as,

[qD] {T } − [qD] {T¯} = 0 (B.27)

where [qD], the continuous temperature boundary condition coupling matrix is deﬁned as,

Z qD(i,k) = ψjψkdΓ . (B.28) Γλ

For the heat transfer equation being studied the [qD] coupling matrix is just the matrix transpose of the coupling matrix [q]. For a single region with Dirichlet boundary conditions applied using Lagrange multipliers the FE system can be summarized as,

[M]{T˙ } + [K]{T } − {Q} − {q} + [q]{λ} = 0 (B.29)

[q]T {T } − [q]T {T¯} = 0 . (B.30)

Domain decomposition can be introduced into the FE model by subdividing the domain

Ω into subdomains s = 1...Ns, where Ns is the total number of subdomains. For simplicity the rest of this derivation will examine only two subdomains (Ns = 2), as shown in Fig. B.1. The surface of each subdomain is divided into two portions, interface and original boundary represented as ΓI and Γb, respectively. After splitting the two domains, they can be “glued” together by applying Dirchelt boundary conditions across the ΓI interface using the variational principles introduced in Eq. (B.19). The FE system equations can be APPENDIX B. FINITE ELEMENT DOMAIN DECOMPOSITION 105

2 ! "b

2 $ "I ! " !

nˆ1

! " nˆ !" #! 2 1 ! "I ! 1 "b !

Figure B.1: A schematic representation of the linking of two FE subdomains with Lagrange multipliers. ! written for domains one and two as,

1 ˙ 1 1 1 1 1 1 M {T } + [K ]{T } − {Q } − {qb } + [qI ]{λ} = 0 (B.31) 2 ˙ 2 2 2 2 2 2 M {T } + [K ]{T } − {Q } − {qb } + [qI ]{λ} = 0 (B.32)

respectively, where [qI ] has been redeﬁned as the coupling matrix to the interface and the

2 sign of [qI ] has been reversed resulting in,

Z 1 1 qI(i,k) = ψi ψkdΓ (B.33) ΓI Z 2 2 qI(i,k) = − ψi ψkdΓ . (B.34) ΓI

The Dirchlet boundary condition enforces continuity not between a surface temperature

and a known value but the interface temperatures of each subdomain. This change in Eq. APPENDIX B. FINITE ELEMENT DOMAIN DECOMPOSITION 106

(B.26) is reﬂected in the following equation,

  Z N N X 1 e X 2 e ψk  Tj (t)ψj (~x) − Tj (t)ψj (~x) dΓ = 0 . (B.35) ΓI j=1 j=1

Equation (B.35) can be expressed in matrix form as,

1T 1 2T 2 qI {T } + qI {T } = 0 (B.36)

1 2 where qI and qI are the continuous temperature interface condition coupling matrixes are deﬁned in Eq. (B.33) and Eq. (B.34), respectively. For this simple two subdomain decomposition the resulting system of equations can be summarized as,

          [M 1] 0 0 {T˙ 1} [K1] 0 [q1] {T 1} {Q1} + {q1}      I     b             0 [M 2] 0 {T˙ 2} +  0 [K2][q2] {T 2} = {Q2} + {q2} . (B.37)    I  b             1 T 2 T     0 0 0  {λ}  [qI ] [qI ] 0  {λ}   0 

For clarity, the formulation was performed assuming two subdomains, however it can

be readily extended to an arbitrary number of subdomains as shown,

s ˙ s s s s s s s [M ]{T } + [K ]{T } + [qI ]{λ} = {Q } + {qb } = {F } (B.38) where the superscript “s” represents the sth subdomain. The continuous temperature interface condition for all subdomains is written as,

Ns X s T s [qI ] {T } = 0 . (B.39) s=1

B.1.3 Numerical Integration in Time

Utilizing the alpha family integration scheme, provided in Ref.,69 Eq. (B.38) can be integrated in time thereby reducing the number of unknowns to {T s}n+1 and {λ}. The reduced APPENDIX B. FINITE ELEMENT DOMAIN DECOMPOSITION 107

system is presented below for each individual subdomain s, where n is the time step number and ∆t is the change in time.

[Kˆ s]n+1{T s}n+1 + [ˆqs]n+1{λ}n+1 = [K¯ s]n{T s}n + [¯qs]n{λ}n + {F¯s}n,n+1 (B.40) where,

[Kˆ s]n+1 = α∆t[Ks]n+1 + [M s] (B.41)

[ˆqs]n+1 = α∆t[qs]n+1 (B.42)

[K¯ s]n = −(1 − α)∆t[Ks]n + [M s] (B.43)

[¯qs]n = −(1 − α)∆t[qs]n (B.44)

{F¯s}n,n+1 = ∆t α{F s}n + (1 − α){F s}n+1 . (B.45)

The α parameter is the numerical integrating factor that determines the integration scheme, details provided in Table B.1. This result can be further simpliﬁed by deﬁning all known quantities as {F s} such that,

{F s} = [K¯ s]n{T s}n + [¯qs]n{λ}n + {F¯s}n,n+1 (B.46) thereby simplifying the governing equation in each subdomain s for each iteration to

[Kˆ s]{T s} + [ˆqs]{λ} = {F s} . (B.47)

B.1.4 Reduced Interface Problem

If a system of equations was solved as presented in Eq. (B.47) there would be little practical beneﬁt over the single processor problem. However, rearranging the system of equations can produce a reduced interface problem. Assuming that the inverse of [Kˆ s] is readily available APPENDIX B. FINITE ELEMENT DOMAIN DECOMPOSITION 108

Table B.1: Table of numerical integration schemes corresponding to the α parameter, reproduced from Ref.69 α Numerical Integration Schemes 0 forward diﬀerence scheme 1/2 the Crank-Nicolson scheme 2/3 the Galerkin method 1 backward diﬀerence scheme

and non-singular, temperature in each subdomain can be explicitly solved for from Eq.

(B.47) as, −1 {T s} = [Kˆ s] ({F s} − [ˆqs]{λ}) . (B.48)

Substituting {T s} in terms of λ into the continuous temperature interface condition Eq.

(B.39) produces, Ns X −1 [qs]T [Kˆ s] ({F s} − [ˆqs]{λ}) = 0 (B.49) s=1 which further simpliﬁes to,

Ns Ns X −1 X −1 [qs]T [Kˆ s] {F s} = [qs]T [Kˆ s] [ˆqs]{λ} . (B.50) s=1 s=1

Equation (B.50) can be compacted by deﬁning matrix FI and vector d as

Ns X s s s T ˆ s −1 s FI = [FI ] where, [FI ] = [q ] [K ] [ˆq ] (B.51) s=1

Ns X −1 d = {ds} where, {ds} = [qs]T [Kˆ s] {F s} (B.52) s=1 thereby simplifying the reduced interface problem to

FI λ = d . (B.53) APPENDIX B. FINITE ELEMENT DOMAIN DECOMPOSITION 109

B.1.5 Solution Algorithm

The solution algorithm describes the techniques for the solution of Eq. (B.53), the global reduced interface problem. Two distinct BiConjugate Gradient matrix (BiCG) solvers are utilized throughout the solution process; global system and local system solvers. The global

BiCG solver was custom written to solve a linear system across multiple processors. The implementation does not require the explicit construction of the global matrix and therefore reduces the overall memory footprint on any given node. The local BiCG solver can be used when the entire linear system is contained on a single processor, as in Eq. (B.48). In order to speed development a freely available numerical package, Matrix Toolkits Java (MTJ),137 was utilized for the local BiCG. For additional details concerning the BiCG solver methodology see Ref.138 A general outline of the solution algorithm is provided below. It assumes all initialization procedures have been performed.

1. Calculate minimum time step across all domains.

2. Update all physical boundary conditions through sidesets.

3. Build local [Kˆ s] and {F s}.

4. Solve Eq. (B.53) using global BiCG

(a) Generate local residual vectors. (b) Generate a global residual vector that is shared with all domains. (c) Calculate scalar correction factors on each domain based upon global residual vectors. (d) Update vectors and repeat until convergence. Upon convergence the solution for λ will be known.

5. Solve for the ﬁeld variable with Eq. (B.48) and λ using the local BiGC solver.

6. Update material properties from ﬁeld variable.

7. Advance time and repeat.

s The generation of the local residual vectors in Step 4a requires the calculation of [FI ] s s s ˆ s and {d }. The direct calculation of [FI ] and {d } requires the inverse of [K ], a computation APPENDIX B. FINITE ELEMENT DOMAIN DECOMPOSITION 110

that is too computationally expensive for practical use. This calculation can be bypassed s s ˆ s −1 s for {d } by deﬁning a temporary vector {Vd } equal to [K ] {F } such that,

s ˆ s −1 s ˆ s s s {Vd } = [K ] {F } → [K ]{Vd } = {F }

s thereby allowing {Vd } to be solved using an iterative solver such as the local BiCG. Once s s s s T s {Vd } is determined, the local vector {d } can be readily calculated as {d } = [q ] {Vd }.A s similar procedure can be applied to the calculation of [FI ]{λ}, but requires a “guess” for λ. s The {λ} used to calculate [FI ]{λ} is provided as a function of the iterative matrix solver. s ˆ s −1 s For a given {λ}, a temporary vector {VF } equal to [K ] [ˆq ]{λ} is deﬁned such that,

s ˆ s −1 s ˆ s s s {VF } = [K ] [ˆq ]{λ} → [K ]{VF } = [ˆq ]{λ}

s thereby allowing {VF } to be solved using an iterative solver as a function λ. The calculation s s T s of [FI ]{λ} is then given by [q ] {VF }.

B.2 Coding Implementation

B.2.1 Existing Framework

In order to understand the multi-processor implementation method used for the UB CET framework a general understanding of the pre-existing code is required. The object-oriented nature of Java allows for development of a framework where each object serves a specific function. The UB CET FE framework utilizes a hierarchical system of object definitions where each subsequent object contributes methods that are unique to the specific application. For instance, the Carbon Epoxy Material class inherits the behavior shared between all polymer composites from Polymer Composite Material, but also contributes properties unique to carbon epoxy materials. The existing UB CET framework is divided into the following primary categories; material, mesh, region, sidesets, and additional support objects. APPENDIX B. FINITE ELEMENT DOMAIN DECOMPOSITION 111

Within each category multiple classes contribute to the behavior of a single object.

• Material: Material classes are used to deﬁne and store the properties associated with

said material. The Simple Conduction Material class contains all the required ma-

terial properties such as thermal conductivity, density and speciﬁc heats as well as

holding primitive variables (Temperature).

• Mesh: The FEM Mesh object, speciﬁc to the FE framework, holds all nodal locations,

array of elements and required support methods.

• Region: The region is the portion of the code that is speciﬁc to the problem being

solved and holds all of the decode and update methods.

• Process: The process classes are responsible for building the global stiﬀness, mass and

forcing matrix/vectors for the speciﬁc equation being solved. The matrix is solved

using a pre packaged matrix solver (Matrix Toolkit for Java).139

• SideSets: Within the framework the sideset classes serve to impose the boundary

conditions upon the system. A sideset must be instantiated for each of the boundary

conditions, speciﬁc to the equation being solved.

B.2.2 Existing Multi-Threading

Previous work to speed up computational time focused upon Java multi-threading on a per process basis. For the solution of complex physical systems different equations representing different physical processes were solved independently. For instance the energy transport, gas diffusion and mechanics were linearized such that the independent solutions were based solely on the previous time step. Generally, in FE implementations the most time intensive step of the computation is embedded in the assembly and solution of the global system matrixes. These facts naturally lead to the desire to build/solve these linearized and therefore independent equations simultaneously. APPENDIX B. FINITE ELEMENT DOMAIN DECOMPOSITION 112

In Java, threads can be implemented as an instantiation of a single object such that each object represents a single thread.140 This methodology was used to generate an array of processes that run simultaneously. This is diagramed in Fig. B.2 where Assemble T, ρ, and

Mech represent the assembly and solution to the FE equations for energy conservation, mass conservation and mechanics, respectively. Although increasing the speed of computation by a speedup factor less than that of number equations, there were obvious limitations. As with most shared memory implementations Java multi-threading is limited to a single machine and therefore limited in scalability. This implementation was further limited to where the maximum speed-up was less than the number of equations solved. Therefore there was no eﬀective speedup for problems such as heat conduction with only a single process.

Assemble T

Region Region "%$! Assemble !

Assemble Mech

! Assemble T

"#$! Region Region Assemble !

Assemble Mech

&'()!

Figure B.2: A schematic representation of the implementation of threads (b) compared to the original implementation (a) within the UB CET FE framework.

B.2.3 Message Passing in Java

The Message Passing Interface (MPI) is a speciﬁcation for a standard application programing interface that allows an individual program access to the required libraries and functions in order communicate between computers.141 The MPI standard, maintained by the MPI

Forum, provides official implementations for Fortran, C, and recently C++. However, there are no official bindings for Java and therefore an unofficial binding must be used. Due to the APPENDIX B. FINITE ELEMENT DOMAIN DECOMPOSITION 113

heterogeneous programing environments within the CET research group (Microsoft Win- dows, Linux, Mac OS X) a highly portable MPI implementation had to be chosen. These requirements led to the selection of MPJ Express developed by Shaﬁ.139

MPJ Express consists of two main portions; the MPJ Express Java Libraries and MPJ

Express runtime infrastructure. The MPJ Express runtime infrastructure consists of the daemons and mpjrun module, both speciﬁc to the machine architecture and run as an

OS service. However the libraries are written in pure Java and are therefore portable to any machine capable of running Java. The MPJ Express implementation of MPI can be implemented with multiple network interfaces including Gigabit Ethernet and Myrinet.142

However, despite best eﬀorts, at the current time Myrinet communication cannot be utilized on CCR’s U2 Cluster. Continued work includes the implementation of the Myrinet eXpress libraries as applied to MPJ Express. For additional information on MPJ Express refer to the oﬃcial website at .

B.2.4 Object Layout

The introduction of a message passing interface, unlike shared memory implementations, often requires restructuring much of the existing framework. One of the primary goals was to minimize such restructuring thereby allowing other users of the FE framework uninter- rupted use. This design technique allows users to continue to run FE problems on a single machine or to simply extend their existing regions to utilize MPJ. This was accomplished by abstracting the existing framework to a higher level by introducing a domain class. The

FE domain class contains most of the code responsible for controlling communication and solution to the global system, Eq. (B.53). Although a separate domain is instantiated on each node, the domain class is viewed as a super class containing all such regions, Fig. B.3.

This can be perceived because all of the communication between nodes is controlled by the FE domain object. The domain utilizes the majority of functions within pre-existing regions with only minor modiﬁcation. This technique allows new users to the framework to APPENDIX B. FINITE ELEMENT DOMAIN DECOMPOSITION 114

develop a FE region, speciﬁc to a set of physics, without knowledge of MPJ. When the need to apply this region to a larger scale is encountered, the user may simply add the required function calls to utilize the MPJ solver.

FEM Domain

Region Region

mesh mat

Asmble Asmble Asmble obj. 1 obj. i obj. sp

! i F i d i BC Processor Thread Computer Node n

Figure B.3: A schematic representation of the FE Domain implementation linking multiple regions.

A similar technique was utilized for the modiﬁcation of the assembly processes and sidesets. The construction of the system matrixes Eq. (B.10) for both multi-process and single process systems is identical. However, the assembly procedure is unique to the physical problem and the material type being studied. Both needs were met by placing all of the assembly machinery within a general abstract assemble FEM class. This class takes a standard FE equation and performs the required modiﬁcations for either a single or multi- processor solution. For a multi-processor solution, the results from the assemble classes are APPENDIX B. FINITE ELEMENT DOMAIN DECOMPOSITION 115

passed through the region to the domain where the required communication is performed.

All of the MPJ express function calls are passed through a Global comm communication wrapper. This is implemented such that all classes written in the CET framework will pass communication through the abstract object Global comm which would be an instantiation of the MPI global comm. The abstraction of the MPJ express function calls serve two primary purposes, the ﬁrst of which is to future proof the code from changing MPI in Java standards.

The future of messaging within Java is a dynamic one and therefore a new or more complete binding may need to be implemented. The Global comm wrapper allows for an installation of a new massage passing system with the creation of a single new class with parallel functionality to the MPI global comm. All user function calls to the Global comm will remain functional with either MPI implementation, allowing either to be used. The second reason for the abstraction of MPJ express is to allow the deconstruction and communication of complex objects. MPJ Express is capable of communicating entire objects from one node to another. However, this process requires the serialization of the object and therefore is slower than the transmission of primitive data types. The Global comm wrapper allows the communicator to take apart an object and send the basic components as arrays of primitives thereby increasing the speed of communication. Once the primitives are received by the other subdomain, the Global comm reassembles them and returns a copy of the original object.

B.3 Results

B.3.1 Veriﬁcation

With an implementation of any new code into an existing framework it is important to verify results against accepted standards. A simple verification was performed utilizing two subdomains and a simple conduction material. A specified constant heat flux was applied to the left boundary with convection sidesets elsewhere. As indicated in Fig. B.4 the results APPENDIX B. FINITE ELEMENT DOMAIN DECOMPOSITION 116

using one domain were visually identical to those using two domains after 60 seconds. This is further conﬁrmed in Fig. B.5 by comparing a single line extracted from the domain.

It should be noted that the results of a single domain and multiple domains may not be exactly equal to an arbitrary number of significant figures. Unlike many explicit codes, where the solution technique is independent of the parallelization technique, e.g. ghost cells, the solution of implicit codes depends upon every other node. This inherent linkage makes the solution a function of the accuracy of the matrix solver. Because of the differences in matrix solvers and number of instances in which the solver is called upon one can expect different numerical errors between a single block and multi-block implementation of implicit codes.

B.3.2 Performance and Scaling

The eﬃciency of all implicit ﬁnite element solvers must be examined in reference to the solution to the linear systems. The two primary solution techniques for linear systems, direct and iterative, do not scale linearly with the size of a system. For instance, the

2 3 popular direct solver “Gaussian Elimination” scales as 3 m , where m is the rank of the matrix. In other words, if the size of the matrix doubled the solution would take about eight times as long to solve. This non linear growth, to a lesser extent, is also exhibited by the iterative solvers such as BiCG. In ideal cases iterative solvers scale with m, however improvements in more practical applications range from m to m3.138 The local BiCG solver

implemented in MTJ137 is shown to scale as approximately m2.5, Fig. B.6.

Scaling and performance tests were run on a one meter cubed domain with all adiabatic boundary conditions except the applied boundary which was held at 1000 K. A total of 50 time steps were taken with the required number of sub-iterations for convergence, error less than 1 × 10−6. The scaling was performed on a constant domain consisting of 64,000 linear elements. Division between domains was performed in order to ensure that the total number of elements remained constant. The scaling for this domain conducted on UB’s Center for APPENDIX B. FINITE ELEMENT DOMAIN DECOMPOSITION 117

Figure B.4: The temperature proﬁle of a representative conduction material with an applied heat ﬂux from the left and convection boundary conditions elsewhere. (top) The results were computed using two subdomains (red/blue meshes) to represent the structure. (bottom) Results are shown using a single domain to model the structure.

Figure B.5: The temperature proﬁle of a representative conduction material with an applied heat ﬂux from the left are shown at a y location of 0.0223 m. APPENDIX B. FINITE ELEMENT DOMAIN DECOMPOSITION 118

3%/#'4+589/)$+5"&:)+6;',<7+ )&''''!

)'''''!

$&''''!

$'''''!

(&''''!

('''''! !"!#$%&!! &''''!

!"#$%&'()*+,"$-./%0"1%&+2'$)+ '! '! ('''! $'''! )'''! *'''! &'''! +'''! 3%/#'4+5'()+6$7+

Figure B.6: Computational cost of the solution of linear system using the BiCG with MTJ.

Computational Research (CCR) U2 cluster is provided in Fig. B.7, in increments of powers of two.

As indicated in Fig. B.7, the scaling behavior for implicit algorithms can drastically vary from that of explicit algorithms. The scaling for processors up to 32 produced super linear scaling with a maximum speed-up of 61.5 for 32 processors. Continued increase in processors served to decrease the speed-up below that expected from a linear behavior.

However, in these cases the speed-up did exceed unity and therefore produced results faster than that for a single processor. Similar behavior was exhibited by the parallel eﬃciency.

The maximum parallel efficiency (defined out of unity) reached 3.5 and occurred with 16 processors. Further increase in the number of processors decreased the parallel efficiency to less than unity.

This behavior can be explained by the nature of linear system solvers. As discussed, the cost of linear system solvers grows approximately as m2.5, where the growth depends upon the required convergence criteria and system characteristics. The scaling behavior APPENDIX B. FINITE ELEMENT DOMAIN DECOMPOSITION 119

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)!" '#$" ,-../01-" '" $!" 2"3",145" &#$" (!" &" '!" )*%%+,-*( %#$" !"#$%&#'( &!" %"

%!" !#$"

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Figure B.7: Scaling and eﬃciency of a one meter cubed domain consisting of 64,000 elements on UB’s CCR U2 cluster. can be interpreted in terms of a time limiting portion of the solution procedure, either local solver limited or global solver limited. A representative schematic of this behavior is provided in Fig. B.8. For the case where there is only one subdomain, the number of processors is one, the primary cost of the solution is within the local BiCG solver. As this region is subdivided into multiple subdomains, the relative cost associated with the local BiCG solver decreases super linearly. However, as the region is further subdivided the global system increases in size thereby increasing the cost of the global BiCG solver and associated communication. The total cost of the local and global BiCG solvers represent the bulk of the computational cost and is the controlling factor in scaling. When this total cost is minimized the maximum speed-up is achieved. For the case studied this occurred with 32 processors, which corresponds to the location in which the cost of local and global

BiCG intersect. Further increasing the number of processors only serves to increase the computational cost associated with global BiCG resulting in a decrease in eﬃciency.

As previously discussed, the initial implementation of CET’s FE solver was memory APPENDIX B. FINITE ELEMENT DOMAIN DECOMPOSITION 120

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Figure B.8: Pictorial representation of the relative cost of the global and local BiCG solvers.

limited. As indicated in Fig. B.9, the total memory allocated for a single region grows with the number elements in a single direction cubed. For a single computer with two gigabytes of memory the practical limit is 250,000 elements or 633 elements in one direction within a domain. However, through the implementation of MPJ and a global domain solver this restriction was removed. Figure B.10 shows the mesh for a 1.56 million element case run across 64 processors. Utilizing the JobVis program available of CCR U2 cluster, an anecdotal analysis of cpu loading was performed. As indicated in Fig. B.11 for the 1.56 million element case, the distribution of work across nodes is approximately equal. This is consistent with the fact that all nodes simultaneously solve the global system. The communication behavior is similar because all nodes must communicate to all other nodes. APPENDIX B. FINITE ELEMENT DOMAIN DECOMPOSITION 121

Figure B.9: The required allocated memory for a single processor in 3D as a function of number of elements in one dimension.

Figure B.10: The mesh for the 1.56 Million ﬁnite element case distributed across 64 subdomains. APPENDIX B. FINITE ELEMENT DOMAIN DECOMPOSITION 122

Figure B.11: The communication (left) and loading (right) for the scaling problem on 64 processors. APPENDIX B. FINITE ELEMENT DOMAIN DECOMPOSITION 123

B.4 Conclusions

The implementation of a MPI finite element solver into an existing Java framework was explored. The implementation was successful against the first goal of speed-up with super- linear scaling up to 16 nodes on an 64,000 element case. The second goal of memory distribution was accomplished and verified with the distribution of a 1.56 million element case across 64 processors. Two primary limitations encountered throughout the development were message passing in Java and the inherent limitations to a solution to an implicit system. Since the development of Java is relatively new, many of the packages expected for numerical analysis do not existent or are not mature. Some of these examples include

MPI, Linear-Algebra Packages, and Input/Output Libraries (e.g. Tecplot IO, HDF support). However, with recent advances along with internal code development this limitation was overcome to produce an MPI based FE solver. The limitation inherent in all implicit system solvers cannot be as easily overcome and therefore limits the scalability of implicit codes. Future work includes the minimization of such limitations by improving communication with Myrinet, reducing the amount of required communication with global matrix pre-conditioners, and additional code reﬁnement. Bibliography

[1] G. Corely, World trade center study: Data collection, preliminary observations, and recommendations, Tech. rep., Federal Emergency Management Agency (2002).

[2] D. Gantenbein, Burning questions, Scientiﬁc American (2002) 82–89.

[3] J. Snell, Towards engineered ﬁre safety: A global research strategy for the 21st century (2000).

[4] G. Cox, Fire research in the 21st century, Fire Safety Journal 32 (1999) 203–219.

[5] D. Drysdale, An Introduction to Fire Dynamics, John Wiley and Sons, New York, NY, 1998.

[6] J. N. deRis, Spread of a laminar diﬀusion ﬂame, in: Proceedings of 12th Symp. (Int.) on Combustion, The Combustion Institute, 1969, pp. 241–252.

[7] I. S. Wichman, F. A. Williams, I. Glassman, Theoretical aspects of flame spread in an opposed flow over flat surfaces of solid fuels (1982).

[8] I. S. Wichman, F. A. Williams, A simplified model of flame spread in an opposed flow along a flat surface of a semi-infinite solid, Combust. Sci. and Tech. 32 (1983) 91–123.

[9] I. S. Wichman, Material ﬂammability, combustion, toxicity and ﬁre hazard in transportation, Prog. Energy Combust. Sci. 29 (2003) 247–299.

[10] H. R. Baum, A. Atreya, A model of transport of fuel gases in a charring solid and its application to opposed-ﬂow ﬂame spread, Proceedings of the Combustion Institute 31 (2) (2007) 2633–2641.

[11] K. K. Wu, W. F. Fan, C. H. Chen, T. M. Liou, I. J. Pan, Downward ﬂame spread over a thick pmma slab in an opposed ﬂow environment: experiment and modeling, Combust. Flame 132 (2003) 697–707.

[12] A. Ito, Y. Kudo, H. Oyama, Propagation and extinction mechanisms of opposed-flow flame spread over pmma for different sample orientations, Combust. Flame 143 (2005) 428–437.

124 BIBLIOGRAPHY 125

[13] F. Williams, Mechanisms of ﬁre spread (invited review), in: Symposium (Interna- tional) on Combustion, Vol. 16, Combustion Institute, 1977, pp. 1281–1294.

[14] A. Fern´andez-Pello, A theoretical model for the upward laminar spread of ﬂames over vertical fuel surfaces, Combust. and Flame 31 (1978) 135–148.

[15] K. Annamalai, M. Sibulkin, Flame spread over combustible surfaces for laminar flow systems - 2. flame heights and fire spread rates., Combustion Science and Technology 19 (5-6) (1979) 185–193.

[16] M. Sibulkin, J. Kim, Combust. Sci. and Tech. 17 (1977) 39–.

[17] G. H. Markstein, J. de Ris, Upward ﬁre spread over textiles, Symposium (Interna- tional) on Combustion 14 (1) (1973) 1085–1097.

[18] L. Orloﬀ, J. de Ris, G. H. Markstein, Upward turbulent ﬁre spread and burning of fuel surface (1974).

[19] K. Saito, J. G. Quintiere, F. A. Williams, Upward turbulent ﬂame spread (1986).

[20] K. Saito, F. A. Williams, I. Wichman, J. G. Quintiere, Upward turbulent ﬂame spread on wood under external radiation, J. of Heat Transfer 111 (1989) 438–445.

[21] N. B. Mason, D. B. Spalding, notitle (1973).

[22] C. Ferbandez-Pello, C. P. Mao, A uniﬁed analysis of concurrent modes of ﬂame spread, Combustion Science and Technology 26 (3-4) (1981) 147–155.

[23] M. A. Delichatsios, M. Delichatsios, Y. Chen, Y. Hasemi, Similarity solutions and applications to turbulent upward ﬂame spread on noncharring materials, Combust. Flame 102 (1995) 357–370.

[24] B. Karlsson, A mathematical model for calculating heat release rate in the room corner test, Fire Safety Journal 20 (2) (1993) 93–113.

[25] M. Kokkala, D. Baroudi, A thermal model to predict ﬁre growth on a wall, Vol. 418, 1995.

[26] G. Grant, D. Drysdale, Numerical modelling of early ﬂame spread in warehouse ﬁres, Fire Safety Journal 24 (3) (1995) 247–278.

[27] G. F. Carrier, F. E. Fendell, P. S. Feldman, Wind-aided ﬂame spread along a horizontal fuel slab., Combustion science and technology 23 (1-2) (1980) 41–78.

[28] G. Carrier, F. Fendell, S. Fink, Towards wind-aided ﬂame spread along a horizontal charring slab: The steady-ﬂow problem., Tech. rep., TRW, Inc., Redondo Beach, CA. (02 1982). BIBLIOGRAPHY 126

[29] A. Kumar, H. Y. Shih, J. S. T’ien, A comparison of extinction limits and spreading rates in opposed and concurrent spreading ﬂames over thin solids, Combust. Flame 132 (2003) 667–677.

[30] J. L. Consalvi, Y. Pizzo, B. Porterie, Numerical analysis of the heating process in upward ﬂame spread over thick pmma slabs, Fire Safety Journal 43 (2008) 351–362.

[31] J. Q. Quintiere, The eﬀects of angular orientation on ﬂame spread over thin materials, Fire Safety Journal 36 (2001) 291–312.

[32] D. Drysdale, A. Macmillan, Flame spread on inclinded surfaces, Fire Safety Journal 18 (1992) 245–254.

[33] T. Ahmad, G. M. Faeth, An investigation of the laminar overﬁre region along upright surfaces, J. of Heat Transfer 100 (1978) 112–119.

[34] F. G. Roper, C. Smith, A. Cunningham, The prediction of laminar jet diffusion flame sizes: Part ii: experimental verification, Combust. and Flame 71 (1988) 71–189.

[35] K. McGrattan, S. Hostikka, J. Floyd, H. Baum, R. Rehm, W. Mell, R. McDermott, Fire dynamics simulator (version 5)–technical reference guide, volume 1: Mathemat- ical model, NIST Special Publication (2009) 1018–5.

[36] J.-W. Kwon, N. Dembsey, C. Lautenberger, Evaluation of fds v.4: Upward ﬂame spread, Fire Technology 43 (4) (2007) 255–284.

[37] C. A. Ulven, J. Mosbrucker, B. Miller, S. Pillay, S. Vaddi, U. K. Vaidya, Loss of balsa wood core structural integrity during ﬁre exposure of sandwich composites, in: 38th SAMPE Fall Technical Conference: Global Advances in Materials and Process Engineering, International SAMPE Technical Conference, Soc. for the Advancement of Material and Process Engineering, Covina, CA 91724-3748, United States, 2006, p. 8.

[38] A. Mouritz, Z. Mathys, Post-ﬁre mechanical properties of marine polymer composites, Composite Structures 47 (1999) 643–653.

[39] A. Mouritz, Z. Mathys, Post-ﬁre mechanical properties of glass-reinforced polyester composites, Composites Science and Technology 61 (2001) 475–490.

[40] A. Mouritz, E. Gellert, P. Burchill, K. Challis, Review of advanced composite structures for naval ships and submarines, Composite Structures 53 (2001) 21–41.

[41] A. Mouritz, C. Gardiner, Compression properties of ﬁre-damaged polymer sandwich composites, Composites 33 (2002) 609–620.

[42] A. G. Gibson, P. N. H. Wright, Y. S. Wu, A. P. Mouritz, Z. Mathys, C. P. Gardiner, The integrity of polymer composites during and after ﬁre, Journal of Composite Ma- terials 38 (Compendex) (2004) 1283–1307. BIBLIOGRAPHY 127

[43] G. A. Kardomateas, L. Liu, V. Birman, J. W. Holmes, G. J. Simitses, Thermal buck- ling of a heat-exposed, axially restrained composite column, Composites Part A (Ap- plied Science and Manufacturing) 37 (Copyright 2006, The Institution of Engineering and Technology) (2006) 972–80.

[44] J. B. Henderson, J. A. Wiebelt, M. R. Tant, A model for the thermal response of polymer composite materials with experimental veriﬁcation, Journal of Composite Materials 19 (1985) 579–595.

[45] J. B. Henderson, T. E. Wiecek, A mathematical model to predict the thermal response of decomposing expanding polymer composites, Journal of Composite Materials 21 (1987) 373–393.

[46] J. J. Florio, J. B. Henderson, F. L. Test, R. Hariharan, A study of eﬀects of the assummption of local-thermal equilibrium on the overall thermally-induced response of a decomposing glass-ﬁlled polymer composite, J. of Heat and Mass Transfer 34 (1991) 135–146.

[47] R. M. Sullivan, N. J. Salamon, A ﬁnite element method for the thermochemical decomposition of polymeric materials–i. theory, International Journal of Engineering Science 30 (4) (1992) 431–441.

[48] R. M. Sullivan, N. J. Salamon, A ﬁnite element method for the thermochemical decomposition of polymeric materials–ii. carbon phenolic composites, International Journal of Engineering Science 30 (7) (1992) 939–951.

[49] R. M. Sullivan, A coupled solution method for predicting the thermostructural response of decomposing, expanding polymeric composites, Journal of Composite Ma- terials 27 (4) (1993) 408–434.

[50] G. A. Pering, P. V. Farrell, G. S. Springer, Degradation of tensile and shear properties of composites exposed to ﬁre or high temperature, Journal of Composite Materials 14 (Compendex) (1980) 54–68.

[51] H. L. N. McManus, G. S. Springer, High temperature thermomechanical behavior of carbon-phenolic and carbon-carbon composites, i. analysis, Journal of Composite Materials 26 (2) (1992) 206–229.

[52] H. L. N. McManus, G. S. Springer, High temperature thermomechanical behavior of carbon-phenolic and carbon-carbon composites, ii. results, Journal of Composite Materials 26 (2) (1992) 230–255.

[53] Y. I. Dimitrienko, Thermomechanical behavior of composite materials and structures under high temperatures: 1. materials, Composites Part A 28 (1997) 453–461.

[54] Y. I. Dimitrienko, Thermomechanical behavior of composite materials and structures under high temperatures: 2. structures, Composites Part A 28 (1997) 463–471. BIBLIOGRAPHY 128

[55] Y. I. Dimitrienko, Internal heat-mass transfer and stresses in thin-walled structures of ablating materials, Int. J. Heat Mass Transfer 40 (7) (1997) 1701–1711.

[56] Y. I. Dimitrienko, Thermomechanical behavior of composites under local intense heating by irradiation, Composites: Part A 31 (2000) 591–598.

[57] A. G. Gibson, Y. S. Wu, H. W. Chandler, J. A. D. Wilcox, P. Bettess, Model for the thermal performance of thick composite laminates in hydrocarbon ﬁres, Revue de l’Institute Francais du Petrole 50 (Compendex) (1995) 69–74.

[58] A. Mouritz, S. Feih, E. Kandare, Z. Mathys, A. G. Gibson, P. DesJardin, S. W. Case, L. B.Y., Review of ﬁre structural modelling of polymer composites, Composites 40 (2009) 1800–1814.

[59] L. Couchman, A. P. Mouritz, Modeling of Naval Composite Structures in Fire, CRC for Advanced Composite Structures, Templestowe, Victoria, Australia, 2006.

[60] A. P. Mouritz, A. G. Gibson, Fire Properties of Polymer Composite Materials, Solid mechanics and its applications, Springer, Dordrecht, The Netherlands, 2006.

[61] C. DiBlasi, The state of the art of transport models for charring solid degradiation, Polymer International 49 (2000) 1133–1146.

[62] C. Di Blasi, Modeling chemical and physical processes of wood and biomass pyrolysis, Progress in Energy and Combustion Science 34 (1) (2008) 47–90.

[63] C. Lautenberger, C. Fernandez-Pello, Generalized pyrolysis model for combustible solids, Fire Safety Journal 44 (6) (2009) 819–839.

[64] C. Lautenberger, Gpyro – A Generalized Pyrolysis Model for Combustible Solids – Technical Reference Guide, 0th Edition (January 2009).

[65] K. McGrattan, S. Hostikka, J. Floyd, H. Baum, R. Rehm, Fire Dynamics Simulator (Verion 5) Technical Reference Guide, National Institute of Standards and Technology (2007).

[66] S. I. Stoliarov, R. E. Lyon, Thermo-kinetic model of burning, Tech. Rep. DOT/FAA/AR-TN08/17, U.S. Department of Transportation, Federal Aviation Ad- ministration (2008).

[67] S. I. Stoliarov, R. E. Lyon, Thermo-kinetic model of burning for pyrolyzing materials, in: Fire Safety Science 9, 2009, pp. 1141–1152.

[68] C. Luo, P. E. DesJardin, Thermo-mechanical damage modeling for a glass-ﬁber phenolic-resin composite material, Composites Science and Technology 67 (2007) 1475–1488.

[69] J. N. Reddy, Finite Element Method, McGraw Hill, New York, NY, 1993. BIBLIOGRAPHY 129

[70] J. E. J. Staggs, A simple model of polymer pyrolysis including transport of volatiles, Fire Safety Journal 34 (1) (2000) 69–80.

[71] J. G. Quintiere, R. N. Walters, S. Crowley, Flammability properties of aircraft carbon- ﬁber structural composite, Tech. Rep. DOT/FAA/AR-07/57, U.S. Department of Transportation, Federal Aviation Administration (2007).

[72] F. Kuwahara, Y. Kameyama, S. Yamashita, A. Nakayama, Numerical modeling of turbulent ﬂow in porous media using a spatially periodic arrary, J. of Porous Media 1 (1998) 47–55.

[73] R. Kee, F. Rupley, J. Miller, The chemkin thermodynamic data base, Tech. rep., Sandia National Laboratory (1987).

[74] K. Erickson, Thermal decomposition mechanisms common to polyurethane, epoxy, poly(diallyl phthalate), polycarbonate and poly(phenylene sulﬁde), Journal of Ther- mal Analysis and Calorimetry 89 (2) (2007) 427–440–440.

[75] C. L. Beyler, M. M. Hirschler, Thermal decomposition of polymers, in: P. DiNenno (Ed.), The SFPE Handbook of Fire Protection Engineering, National Fire Protection Association, 2002, pp. 1–110 – 1–131.

[76] A. Tewarson, Generation of heat and chemical compounds in ﬁres, in: P. DiNenno (Ed.), The SFPE Handbook of Fire Protection Engineering, National Fire Protection Association, 2002, pp. 3–82 – 3–161.

[77] W. Xie, P. E. DesJardin, A level set embedded interface method for conjugate heat transfer simulations of low speed 2d ﬂows, Computers & Fluids 37 (2008) 1262–1275.

[78] W. Xie, P. DesJardin, An embedded upward ﬂame spread model using 2d direct numerical simulations, Combustion and Flame 156 (2) (2009) 522–530.

[80] Y. Tseng, J. H. Ferziger, A ghost-cell immersed boundary method for ﬂow in complex geometry, J. Comput. Phys. 192 (2003) 593–623.

[81] D. Lacanette, S. Vincent, A. Sarthou, P. Malaurent, J.-P. Caltagirone, An eulerian/lagrangian method for the numerical simulation of incompressible convection ﬂows interacting with complex obstacles: Application to the natural convection in the lascaux cave, International Journal of Heat and Mass Transfer 52 (Compendex) (2009) 2528–2542.

[82] T. G. Liu, B. C. Khoo, Y. S. Yeo, Ghost ﬂuid method for strong shock impacting on material interface, Journal of Computational Physics 190 (Copyright 2004, IEE) (2003) 651–81, double Check Ref. BIBLIOGRAPHY 130

[83] C. Wag, H. Tang, T. Liu, An adaptive fhost ﬂuid ﬁnite volume method for compressible gas-water simulations, J. Comput Phys. 227 (2008) 6385–6409.

[84] C. Farhat, A. Rallu, S. Shankaran, A higher-order generalized ghost ﬂuid method for the poor for the three-dimensional two-phase ﬂow computation of underwater implosions, J. Comput. Phys 227 (2008) 7674–7700.

[85] C. Farhat, A. Rallu, K. Wang, T. Belytschko, Robust and provably second-order explicit–explicit and implicit–explicit staggered time-integrators for highly non-linear compressible ﬂuid–structure interaction problems, International Journal for Numeri- cal Methods in Engineering 84 (1) (2010) 73–107.

[86] S. Osher, R. Fedkiw, Level set methods and dynamic implicit surfaces, Springer, New York, 2003.

[87] R. Mittal, G. Iaccarino, Immersed boundary methods, Ann. Rev. Fluid Mech. 37 (2005) 239–261.

[88] F. M. White, Viscous Fluid Flow, McGraw-Hill, Inc., New York, NY, 1991.

[89] S. K. Sambasivan, H. UdayKumar, Ghost ﬂuid method for strong shock interactions part 2: Immersed solid boundaries, AIAA J. 47 (12) (2009) 2923–2937.

[90] P. E. DesJardin, T. J. O’Hern, S. R. Tieszen, Large eddy simulation and experimental measurements of the near-ﬁeld of a large turbulent helium plume, Phys. Fluids 16 (2004) 1866–1883.

[91] P. E. DesJardin, Modeling of conditional dissipation rate for ﬂamelet models with application to large eddy simulation of ﬁre plumes, Combust. Sci. and Tech. 177 (2005) 1881–1914.

[92] P. DesJardin, H. Shihn, M. D. Carrara, State-of-the-art combustion sgs modeling approaches for large eddy simulation of ﬁres, in: B. Sunden, M. Faghri (Eds.), Transport Phenomena of Fires, WIT Press, UK, Southhampton, UK, 2008, Ch. 9, pp. 327–356.

[93] J. A. Sethian, Level Set Methods and Fast Marching Methods, Cambridge University Press, Cambridge, UK, 1999.

[94] M. Sussman, P. Smereka, S. Osher, A level set approach for computing solutions to incompressible two-phase ﬂow, J. Comput. Phys. 114 (1994) 146–159.

[95] S. Mauch, Eﬃcient algorithms for solving static hamilton-jacobi equations, Ph.D. thesis, California Institute of Technology (2003).

[96] S. Xu, T. Aslam, D. S. Stewart, High resolution numerical simulation of ideal and non-ideal compressible reacting ﬂows with embedded internal boundaries, Combust. Theory Modelling 1 (1997) 113–142. BIBLIOGRAPHY 131

[97] R. K. Fedkiw, T. Aslam, B. Merriman, S. Osher, A non-oscillatory eulerian approach to interfaces in multimaterials ﬂows (the ghost ﬂuid method), J. Comput. Phys. 152 (1999) 457–492.

[98] R. Fedkiw, Coupling an Eulerian ﬂuid calculation to a Lagrangian solid calculation with the ghost ﬂuid method, J. Comput. Phys. 175 (2002) 200–224.

[99] M. Arienti, P. Hung, E. Morano, J. E. Shepherd, A level set approach to Eulerian- Lagrangian coupling, J. Comp. Phys. 185 (2003) 213–251.

[100] T. Ye, R. Mittal, H. Udaykumar, W. Shyy, An accurate cartesian grid method for viscous incompressible ﬂows with complex immersed boundaries, J. Comput Phys. 156 (2) (1999) 209–240.

[101] G. S. James, A brief on tensor analysis, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1994.

[102] M.-S. Liou, A sequel to ausm: Ausm+, J. Comput. Phys. 129 (1996) 364–382.

[103] M.-S. Liou, A sequel to ausm, part ii: Ausm+up for all speeds, J. Comp. Physics 214 (1) (2006) 137 – 70.

[104] Y. Li, Wavenumber-extended high-order upwind-biased ﬁnite-diﬀerence schemes for convective scalar transport, J. Comput. Phys. 133 (1997) 235–255.

[105] C.-W. Shu, S. Osher, Eﬃcient implementation of essentially non-oscillatory shock- capturing schemes, J. Comput. Phys. 77 (1988) 364–382.

[106] M. Baker, B. Carpenter, Shaﬁ, Mpj express: Towards thread safe java hpc, in: IEEE International Conference on Cluster Computing, 2006.

[107] U. Fey, M. Konig, H. Eckelmann, A new Strouhal-Reynolds number relationship for circular cyliner in the range 47 < re < 2 × 105, Phys. Fluids 19 (1998) 1547–1550.

[108] F. P. Incropera, D. P. DeWitt, Fundamentals of heat and mass transfer, 5th Edition, J. Wiley, New York, 2002.

[109] B. R. Rich, An investigation of heat transfer from an inclined ﬂat plate in free convection, Trans. ASME 75 (1953) 489–499.

[110] S. R. Turns, An Introduction to Combustion, McGraw-Hill, Inc., New York, NY, 1996.

[111] M. N. Ozisik, Heat Conduction, 2nd Edition, Wiley-Interscience publication, Wiley, New York, NY, 1993.

[112] M. T. McGurn, P. E. DesJardin, A. B. Dodd, Numerical simulation of expansion and charring of carbon-epoxy laminates in ﬁre environments, International Journal of Heat and Mass Transfer 55 (1–3) (2012) 272–281. BIBLIOGRAPHY 132

[113] W. Press, S. Teukolsky, W. Vetterling, B. Flannery, Numerical Recipes: The Art of Scientiﬁc Computing, Cambridge University Press, 2007.

[114] C. Westbrook, F. Dryer, Simpliﬁed reaction mechanisms for the oxidation of hydrocarbon fuels in ﬂames, Combust. Sci. and Tech. 27 (1981) 31–43.

[115] K. McGrattan, H. Baum, R. G. Rehm, A. Hamins, G. Forney, Fire dynamics simulator - technical reference manual, Tech. rep., National Institute of Standards and Technology (2000).

[116] F. P. Incopera, D. D. P., Fundamentals of Heat and Mass Transfer, John Wiley & Sons, New York, NY, 1990.

[117] Friedman, A survey of knowledge about idealized ﬁre spread over surfaces, Fire Re- search Abstracts and Reviews 10 (1968) 1–8.

[118] L. J. Marc, Modeling of the thermal degradation of structural wood members exposed to ﬁre, Fire and Materials 28 (2-4) (2004) 199–207, 10.1002/fam.848.

[119] B. Lattimer, J. Oullette, J. Trelles, Thermal Response of Composite Material to Elevated Temperatures, Acclaim Printing, 2006, pp. 1–49.

[120] M. McGurn, P. DesJardin, T. Goodrich, B. Lattimer, J. Lua, Thermal modeoling of balsa wood for ﬁre environments, in: Society for the Advancement of Material and Process Engineering (SAMPE) Symposium & Exhibition, 2009.

[121] M. Behdad, The state-of-the-art in pyrolysis modelling of lignocellulosic solid fuels, Fire and Materials 30 (1) (2006) 1–34, 10.1002/fam.891.

[122] P. E. DesJardin, B. Lattimer, Thermal response of balsa wood for ﬁre environments (2009).

[123] C. Di Blasi, Multi-phase moisture transfer in the high-temperature drying of wood particles, Chemical Engineering Science 53 (2) (1998) 353–366.

[124] O. C. Zienkiewicz, R. L. Taylor, The ﬁnite element method, 5th Edition, Butterworth- Heinemann, Oxford ; Boston, 2000.

[125] Y. Maday, C. Mavriplis, A. Patera, Nonconforming mortar element methods: Appli- cation to spectral discretizations, Tech. rep. (1988).

[126] A. Pantano, R. C. Averill, A penalty-based ﬁnite element interface technology, Com- puters and Structures 80 (22) (2002) 1725–48.

[127] F.-X. Roux, P. Destuynder, A parallel solver for the linear elasticity equations on a composite beam (14-16 Jan. 1988).

[128] C. Farhat, A multigrid-like semi-iterative algorithm for the massively parallel solution of large scale ﬁnite element systems (9-14 Apr. 1989). BIBLIOGRAPHY 133

[129] C. Farhat, F.-X. Roux, A method of ﬁnite element tearing and interconnecting and its parallel solution algorithm, International Journal for Numerical Methods in Engi- neering 32 (6) (1991) 1205–1227.

[130] R. K. Denis Vanderstraeten, A parallel solver based on the dual schur decomposition of general ﬁnite element matrices, International Journal for Numerical Methods in Fluids 28 (1) (1998) 23–46.

[131] C. Farhat, P.-S. Chen, F. Risler, F.-X. Roux, Uniﬁed framework for accelerating the convergence of iterative substructuring methods with lagrange multipliers, Interna- tional Journal for Numerical Methods in Engineering 42 (2) (1998) 257–288.

[132] C. Farhat, P.-S. Chen, J. Mandel, A scalable lagrange multiplier based domain decomposition method for time-dependent problems, International Journal for Numerical Methods in Engineering 38 (22) (1995) 3831–3853.

[133] A. Toselli, Feti domain decomposition methods for scalar advection-diﬀusion problems, Computer Methods in Applied Mechanics and Engineering 190 (43-44) (2001) 5759–5776.

[134] B. Maxime, L. Bruno, re, R. Pierre, R. Jean, A domain decomposition method applied to the simpliﬁed transport equations (2008).

[135] L. Du, C. Ru Shan, J. L. Yang, Z. H. Fan, Finite-element time-domain method combined with ﬁnite-element tearing and interconnecting algorithm and its application for electromagnetic band gap structure, in: Microwave Conference, 2008. APMC 2008. Asia-Paciﬁc, 2008, pp. 1–4.

[136] U. D. Navsariwala, S. D. Gedney, An efficient implementation of the finite-element time-domain algorithm on parallel computers using a finite-element tearing and interconnecting algorithm, Microwave and Optical Technology Letters 16 (4) (1997) 204–208.

[137] Matrix-toolkits-java, retrieved 12/01/2009, from http://code.google.com/p/matrix- toolkits-java/ (2009).

[138] L. N. Trefethen, D. Bau, Numerical linear algebra, Society for Industrial and Applied Mathematics, Philadelphia, 1997.

[139] A. Shaﬁ, B. Carpenter, M. Baker, Java for high performance computing, retrieved 12/01/2009, from http://mpj-express.org/ (2008).

[140] Lesson: Concurrency (2009). URL http://java.sun.com/docs/books/tutorial/essential/concurrency

[141] W. Gropp, E. Lusk, A. Skjellum, Using MPI : portable parallel programming with the message-passing interface, 2nd Edition, Scientiﬁc and engineering computation series, MIT Press, Cambridge, Mass., 1999. BIBLIOGRAPHY 134

[142] M. Baker, B. Carpenter, A. Shaﬁ, Mpj express: Towards thread safe java hpc, in: 2006 IEEE International Conference on Cluster Computing, Cluster 2006, September 25, 2006 - September 28, 2006, Proceedings - IEEE International Conference on Cluster Computing, ICCC, Institute of Electrical and Electronics Engineers Inc., Barcelona, Spain, 2006.