Comput Mech (2014) 54:171–191DOI 10.1007/s00466-014-1012-6

ORIGINAL PAPER

Additive particle deposition and selective laser processing-acomputational manufacturing framework

T. I. Zohdi

Received: 25 March 2014 / Accepted: 26 March 2014 / Published online: 12 April 2014© Springer-Verlag Berlin Heidelberg 2014

Abstract Many additive manufacturing technologiesinvolve the deposition of particles onto a surface followed byselective, targeted, laser heating. This paper develops a mod-ular computational framework which combines the varioussteps within this overall process. Specifically, the frameworksynthesizes the following:

• particle dynamics, which primarily entails: (a) the move-ment of the particles induced by contact with the surface,(b) particle-to-particle contact forces and (c) near-fieldinteraction and external electromagnetic fields.

• laser-input, which primarily entails: (a) absorption oflaser energy input and (b) beam interference (attenua-tion) from particles and

• particle thermodynamics, which primarily entails: (a)heat transfer between particles in contact by conductionand (b) subsequent thermal softening of the particles.

Numerical examples are provided and extensions are alsoaddressed for two advanced processing scenarios involvingsolid-liquid-gas phase transformations.

Keywords Deposition · Selective laser processing ·Manufacturing · Particle · Laser

1 Introduction to additive processes

1.1 Motivation

Additive manufacturing (AM), rapid-prototyping (RP) and3-D printing (3DP) have received a great deal of attention for

T. I. Zohdi (B)Department of Mechanical Engineering, University of California,Berkeley, CA 94720-1740, USAe-mail: zohdi@me.berkeley.edu

a number of years. According to the American Society forTesting and Materials (ASTM), AM is defined as the processof joining materials to make objects from 3D model data,usually layer upon layer, as opposed to subtractive manu-facturing methodologies, which remove material. For exam-ple, an emerging and critical area is print-based technologiesemploying deposition of particulate materials. In particular,printed electronics on flexible foundational substrates arebecoming popular, and have a wide-range of applications,such as flexible solar cells and smart electronics.1 Print-basedmethods involving particles are, in theory, ideal for large-scale applications, and provide a framework for assemblingelectronic circuits by mounting printed electronic devices oncompliant substrates.

Over 50 % of the raw materials handled in industry appearin powdered form during the various stages of process-ing. Consistent, high-quality, particles for manufacturingprocesses can be produced in a variety of ways, such as:(a) sublimation from a raw solid to a gas, which condensesinto particles that are recaptured (harvested), (b) atomizationof liquid streams into droplets by breaking jets of metal, (c)reduction of metal oxides and (d) comminution/pulverizingof bulk material. The particles are usually passed througha series of sieves to separate various sizes. Recently, sev-eral additive manufacturing applications have arisen thatinvolve the deposition of particles onto a surface, followed byselective (targeted) laser-assisted processing (Fig. 1). Suchprocesses involve harnessing optical energy which, due tothe monochromatic and collimated nature of lasers, provide

1 For an early history of the printed electronics field, see Gamota et al.[36]. For reviews of optical coatings and photonics, see Nakanishi etal. [65] and Maier and Atwater [58], for biosensors see Alivisatos [3],for catalysts, see Haruta [41] and for MEMS applications, see Full etal. [35] and Ho et al. [44].

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172 Comput Mech (2014) 54:171–191

Fig. 1 The overall process tobe modelled: (1) particledeposition and (2) slective laserprocessing

PROCESSING

PARTICLE

CONDITION DEPOSITIONINITIAL LASER

an accurate way to post-process (soften, anneal, bond, drill,cut, etc.) powdered materials. Simulation of these processesrequires algorithms which account for optical energy prop-agating through the material microstructure, its conversioninto heat, thermal softening, and subsequent phase trans-formations, as well as the dynamic response of particulatesystems in the presence of strong optically-induced ther-mal fields. In many cases, there is significant multifield cou-pling, which requires methods that can capture the uniqueand essential physics of these systems. In this article, themodeling and simulation of the dynamic particle depositionand subsequent laser processing is undertaken.

The key for next generation manufacturers to succeedis to draw upon rigorous theory and high-fidelity compu-tation to guide and simultaneously develop design rules forscaling up to industrial-level manufacturing. Because of theextremely tight profit margins and short turn-around timesin manufacturing of new materials, there is an industrialneed for numerical simulation of these types of processes.However, continuum-based simulation methods, such as thefinite element methods are ill-suited to simulate systems com-prised of discrete units (particles). A relatively new modelingand simulation paradigm for such advanced manufacturingsystems are discrete element/particle-based mechanics andmethods. Particle-based mechanics and numerical methodshave become wide-spread in the natural sciences, industrialapplications, engineering, biology, applied mathematics andmany other areas. The term “particle mechanics/methods”has now come to imply several different areas of researchin the twenty-first century, for example: (1) Particles as aphysical unit in granular media, particulate flows, plasmas,swarms, etc., (2) Particles representing material phases incontinua at the meso-, micro- and nano-scale and (3) Parti-cles as a discretization unit in continua and discontinua innumerical methods. The application areas of particle-basedmethods are quite wide-ranging, for example: (1) Particulateand granular flow problems, motivated by high-tech indus-trial processes such as those stemming from spray, depositionand printing processes, (2) Fluid-structure interaction prob-lems accounting for free surface flow effects on civil andmarine engineering (water jets, wave loads, ship hydrody-

namics and sea keeping situations, debris flows, etc.), (3)Coupled multiphysical phenomena involving solid, fluid,thermal, electromagnetic and optical systems, (4) Mater-ial design/functionalization using particles to modify basematerials, (5) Manufacturing processes involving forming,cutting, compaction, material processing, (6) Biomedicalengineering, involving cell mechanics, molecular dynamicsand scale-bridging, (7) Multifracture and fragmentation ofmaterials and structures under impact and blast loads and(8) Excavation and drilling problems in the oil/gas indus-try and tunneling processes. Particle or Discrete Element-based computation has emerged in multiple fields, and isideal for simulation of additive manufacturing processes,since the physical systems are inherently discontinuous. Theyare advantageous in dealing with domains that break apartor come together, as compared to traditional continuumbased finite difference and finite element methods whichhave severe limitations when dealing with discontinua. Forreviews see, for example, Duran [27], Pöschel and Schwager[75], Onate et al. [67,68], Rojek et al. [80], Carbonell et al.[9], Labra and Onate [52] and Zohdi [114].2

As mentioned, one class of additive processes is 3D print-ing, which was first developed by Hull [46] of the 3D-Systems Corporation in 1984. 3D printing was a 2.2 bil-lion dollar industry in 2012, with applications ranging frommilitary to medical to the arts (30 % motor vehicles, 15 %consumer products, medical 9 % and business 11 %). Typ-ically such a process takes CAD drawings and slices theminto layers, printing layer by layer. The usual resolution ofthis process is 250 and 1600 DPI at the high end. The typesof deposition range from:

• Extrusive deposition-fused deposition modeling usingclays,

2 There has been considerable research activity in processing of pow-ders, in particular by compaction, for example, see Akisanya et al. [2],Anand and Gu [5], Brown and Abou-Chedid [8], Domas [20], Fleck[33], Gethin et al. [37], Gu et al. [39], Lewis et al. [55], Ransing et al.[76], Tatzel [87] and Zohdi [101,102]. The study of “granular” or “par-ticulate” media is wide ranging. Classical examples include the studyof natural materials, such as sand and gravel, associated with coastalerosion, landslides and avalanches.

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Comput Mech (2014) 54:171–191 173

• Wire/linear deposition-electron beam freeform for met-als,

• Granular deposition-direct metal laser sintering for met-als and alloys and

• Powder bed deposition-plaster based materials,

to name a few. In particular, for granular and powder basedmaterials, selective thermal processing of materials usinglasers (or variants based on electron beams) is quite attractive.The upper bound for the power of a typical industrial laser isapproximately 6,000 Watts. Typically, the initial beam pro-duced is in the form of collimated (parallel) beams which arethen focused with a lens onto a small focal point (approxi-mately 50 mm away) of no more than about 0.000025 m indiameter. Selective laser sintering, was pioneered by House-holder [43] in 1979 and Deckard and Beamen [16] in the mid-1980s.3 An overall technological goal is to develop compu-tational tools to accelerate the manufacturing of printed elec-tronics. Lasers can play a central role in precisely processingthese systems.

1.2 Specific objectives

The objective in this paper is to develop a direct particle-based computational framework which captures the follow-ing main physical events:

• particle dynamics, which primarily entails: (a) the move-ment of the particles induced by contact with the surface,(b) particle-to-particle contact forces and (c) near-fieldinteraction and external electromagnetic fields.

• laser-input, which primarily entails: (a) absorption oflaser energy input and (b) beam interference (attenua-tion) from particles and

• particle thermodynamics, which primarily entails: (a)heat transfer between particles in contact by conductionand (b) subsequent thermal softening of the particles.

We remark that the inclusion of electromagnetic effects stemsfrom the fact that in many emerging processes, the depositedparticles are endowed with charges and guided to the sur-face with an electromagnetic field, in order to obtain supe-rior deposition control, relative to a charge-free system. Thecharges are achieved through a variety of possible meth-ods, such as: (1) Post-atomization charging-whereby theparticles come into contact with an electrostatic field (pro-duced by electrostatic induction or by electrodes) down-stream of the outlet nozzle, (2) Direct charging-whereby anelectrode is immersed in the coating supply and (3) Tribolog-ical charging-whereby the friction in the nozzle induces an

3 A closely related method, Electron Beam Melting, fully melts thematerial and produces dense solids that are void free.

electrostatic charge on the particles as they rub the surface.There are a variety of industrial deposition techniques, andwe refer the reader to the surveys of the state of the art foundin Martin [60,62], as well as the extensive works of Choi etal. [10–13] and Demko et al. [17].

The overall multiphysical system is strongly-coupled,since the dynamics controls which particles are in mechan-ical contact and also the induced thermal fields, spurred onby selective laser-input, thus softening and binding the mate-rial. The approach taken in the present work is to constructa sub-model for each primary physical process. These sub-models are coupled to one another. In order to resolve thecoupling, a recursive multiphysical staggering scheme is con-structed. The general methodology is as follows (at a giventime increment): (1) each field equation is solved individ-ually, “freezing” the other (coupled) fields in the system,allowing only the primary field to be active and (2) after thesolution of each field equation, the primary field variable isupdated, and the next field equation is treated in a similarmanner. As the physics changes, the field that is most sen-sitive (exhibiting the largest amount of relative nondimen-sional change) dictates the time-step size. This approach canbe classified as an implicit, staggered, time-stepping scheme,in conjunction with an iterative solution method that auto-matically adapts the time-step sizes to control the rates ofconvergence within a time-step. If the process does not con-verge (below an error tolerance) within a preset number ofiterations, the time-step is adapted (reduced) by utilizing anestimate of the spectral radius of the coupled system. Themodular approach allows for easy replacement of submod-els, if needed. For more details, the reader is referred toZohdi [116]. Towards the end of the work, extensions are alsoaddressed for two advanced processing scenarios involvingphase transformations and subsequent multiphase dynamicswhereby: (1) material on the system surface is melted andpenetrates the subsurface and (2) material on the surface ismelted, vaporized and vacuumed away.

2 Direct particle representation/calculations

We consider a group of non-intersecting particles (i =1, 2, . . . , Np). The objects in the system are assumed tobe small enough to be considered (idealized) as particles,spherical in shape, and that the effects of their rotation withrespect to their mass center is unimportant to their overallmotion, although, we will make further remarks on theseeffects shortly. The equation of motion for the i th particle insystem is

mi ri = � toti (r1, r2, . . . , rNp ) = �con

i +�walli

+�bondi +�

dampi +�e+m

i , (2.1)

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174 Comput Mech (2014) 54:171–191

where ri is the position vector of the i th particle and where� tot

i represents all forces acting on particle i , which is decom-posed into the sum of forces due to:

• Inter-particle forces (�coni ) generated by contact with

other particles,• Wall forces (�wall

i ) generated by contact with constrain-ing surfaces,

• Adhesive bonding forces (�bondi ) with other particles and

walls,• Damping forces arising from the surrounding intersti-

tial environment (�dampi ) occurring from potentially vis-

cous, surrounding, interstitial fluids, surfactants and• External electromagnetic forces (�e+m

i ) which can playa key role in small charged or magnetized particles.

In the next sections, we examine of each of the types of forcesin the system in detail.

2.1 Comments on rolling

The introduction of rolling and spin is questionable for asmall object, idealized by a particle, in particular becauseof rolling resistance. In addition to the balance of linearmomentum, mi vi = � tot

i , where the vi is the velocityof the center of mass, the equations of angular momen-

tum read Hi,cm = d(Ii · ωi )

dt= Mtot

i,cm . For spheres, we have

Hi,cm = I i,sωi = 25 mi R2

i ωi and for the time discretization

ωi (t + �t) = ωi (t) + �t

I i,s

(φMtot

i,cm(t + �t)

+(1 − φ)Mtoti,cm(t)

), (2.2)

where Mtoti,cm are the total moments generated by interaction

forces, such as contact forces, rolling resistance, etc. For theapplications at hand, the effects of rolling is generally negli-gible, in particular because the particles are small. However,nonetheless, we formulate the system with rotations where ri

is the position of the center of mass, vi is the velocity of thecenter of mass and ωi is the angular velocity. An importantquantity of interest is the velocity on the surface of the “par-ticles”, which is a potential contact point with other particles,denoted vc

i

vci = vi + ωi × ri→c, (2.3)

where ri→c is the relative position vector from the center tothe possible point of contact. This is discussed further later.

2.2 Particle-to-particle contact forces

Following Zohdi [116], we employ a simple particle overlapmodel to determine the normal contact force contributions

Ψ

con,n

con,ncon,n

con,n

con,f

con,fcon,f

con,fΨ

Ψ

Ψ

Ψ

Ψ

Ψ

Ψ

4

3

i

1

2

3

1

2

4

Fig. 2 Normal contact and friction forces induced by neighboring par-ticles in contact (after Zohdi [116])

from the surrounding particles (Nci ) in contact, �con,ni =

∑Ncij=1 ψ

con,ni j , based on separation distance between particles

in contact (Fig. 2). Generally,

�con,ni j = F(||ri − r j ||, Ri , R j , material parameters).

(2.4)

There is no shortage of contact models, of varying complex-ity, to generate a contact interaction force. Throughout thiswork, we will utilize a particularly simple relation wherebycontact force is proportional to the relative normalized prox-imity of particles i and j in contact, detected by the distancebetween centers being less that the sum of the radii

If ||ri − r j || ≤ Ri + R j ⇒ activate contact, (2.5)

where we define the overlap as

δi jdef= |||ri − r j || − (Ri + R j )|. (2.6)

Accordingly, we consider the following

�con,ci j ∝ −K pi j |Ei j |pp ni j Ac

i j , (2.7)

where 0 < K pi j < ∞ is a particle-to-particle contact com-pliance constant, pp is a material parameter, Ei j is normal-ized/nondimensional (strain-like) deformation metric

Ei j =∣∣∣∣||ri − r j || − (Ri + R j )

(Ri + R j )

∣∣∣∣ = δi j

(Ri + R j )(2.8)

and

ni j = − ri − r j

||ri − r j || = r j − ri

||ri − r j || , (2.9)

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Comput Mech (2014) 54:171–191 175

PARTICLE

WALL

Fig. 3 An example of overlap contact between a wall and a particle.The amount of overlap of the particle with the wall position dictatingthe force

where the Ri and R j are the radii of particles i and j respec-tively. The term Ac

i j is a contact area parameter, which is dis-cussed in the appendix. The appendix also provides a briefreview of alternative models, such as the classical Hertziancontact model.

2.2.1 Particle-wall contact

Contact of a particle-to-wall contact is handled in the iden-tical manner to particle-to-particle, except that the wall dis-placement is considered given (externally controlled), andindependent of the action with the particles. The contactbetween the wall and the particles is handled exactly in thesame manner as the particle to particle contact, with theamount of overlap of the particle with the wall position dic-tating the force (see Fig. 3).

2.2.2 Contact dissipation

Phenomenological particle contact dissipation can be incor-porated by tracking the relative velocity of the particles incontact. A simple model to account for this is

�con,di j = ccd(vnj − vni )Ac

i j (2.10)

2.3 Regularized contact friction models

Frictional stick is modelled via the following regularized fric-tion algorithm: (at the point of contact)

• Check static friction threshold (K f is a tangential contactfriction compliance constant):

K f ||vcj,τ − vc

i,τ ||Aci j�t against μs ||�con,n||, (2.11)

where ||vcj,τ − vc

i,τ ||�t (of dimensions of length) is therelative tangential velocity at a point of contact, �t is the

time-step used later in the numerical discretization, μs isthe static friction coefficient. This step replaces (regular-izes) a more rigorous, and difficult, step of first assumingno slip, generating the no-slip contact forces, by solvingan entire multibody/multisurface contact problem, �ns ,and checking �ns against the threshold μs ||�con,n|| oneach surface.

• If the threshold not met (K f ||vcj,τ − vc

i,τ ||Aci j�t <

μs ||�con,n||), then

�con, f = K f ||vcj,τ − vc

i,τ ||Aci j�tτ c

i j (2.12)

where

τ ci j = − vc

i,τ − vcj,τ

||vcj,τ − vc

i,τ ||= vc

j,τ − vci,τ

||vcj,τ − vc

i,τ ||, (2.13)

where the subscripts indicate the tangential componentsof velocity. The tangential velocity at the contact point isobtained by subtracting away the normal component ofthe velocity

vct = vc − (vc · n)n. (2.14)

• If the threshold is met or exceeded (K f ||vcj,τ−vc

i,τ ||Aci j�t

≥ μs ||�con,n||), then one adopts a slip model of the form

�con, fi j = μd ||�con,n

i j ||τ ci j , (2.15)

where μd is the dynamic friction coefficient.

2.4 Particle-to-particle bonding relation

For the particles to bond, we adopt a criterion based onexceeding a critical interpenetration distance. Explicitly:

• Recall, if ||ri − r j || ≤ (Ri + R j ), then the particles are

in contact and Ei j = δi j(Ri +R j )

.

• If the particles are in contact and |Ei j | ≥ E∗, then an(adhesive/attractive) normal bond is activated betweenthe particles of the form

�bond,ni j = K nb

i j |Ei j |pb ni j Aci j , (2.16)

where 0 ≤ K nbi j is a bonding constant and pb is a material

parameter.• If the particles have an activated normal bond, then the

particles automatically have a rotational bond equivalentin form to stick friction

�bond,ri j = K rb

i j ||vcj,τ − vc

i,τ ||Aci j�tτ c

i j . (2.17)

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176 Comput Mech (2014) 54:171–191

2.5 Electromagnetic forces

The electromagnetic forces are decomposed into three con-tributions, (1) Lorentz forces (for charged particles), (2) mag-netic forces (for magnetic particles) and (3) inter-particlenear-field forces. We will utilize the decomposition of theelectromagnetic forces generated into a (inter-particle) near-field interaction and the external electromagnetic field

�e+mi = �

lor,e+mi +�

magi +

N∑

j =i

ψn fi j

︸ ︷︷ ︸�

n fi

+

= qi (Eext + vi × Bext )︸ ︷︷ ︸

�lor,e+mi

+�magi +�

n fi , (2.18)

where∑N

j =i ψn fi j represents the interaction between particle

i and all other particles j = 1, 2 . . . N ( j = i),�lor,e+mi rep-

resents external Lorentz-induced forces from the surround-ing environment, for example comprised of Eext and Bext ,which are externally-controlled fields that are independentof the response of the system. The terms Eext and Bext

can be considered as static (or extremely slowly-varying),and thus mutually uncoupled and independently controllable.The self-induced magnetic fields developed between parti-cles is insignificant for the velocity ranges of interest here(well below the speed of light). For the Lorentz force, werecall the following important observations in conjunctionwith electromagnetic phenomena (Jackson [47]):

• If a point charge q experiences a force�lor,e, the electricfield, Eext , at the location of the charge is defined by�lor,e = qEext .

• If the charge is moving, another force may arise, �lor,m ,which is proportional to its velocity v. This other(induced) field is denoted as the “magnetic induction”or just the “magnetic field”, Bext , such that �lor,m =qv × Bext .

• If the forces occur concurrently (the charge is movingthrough the region possessing both electric and mag-netic fields), then the electromagnetic force is�lor,e+m =qEext + qv × Bext .

2.5.1 Inter-particle near-field interaction

Following Zohdi [114], a simple form that captures the essen-tial near-field effects is

�n fi =

Np∑

j =i

⎛

⎜⎝α1i j ||ri − r j ||−β1

︸ ︷︷ ︸attraction

−α2i j ||ri − r j ||−β2

︸ ︷︷ ︸repulsion

⎞

⎟⎠ ni j ,

(2.19)

where the α’s and β’s are empirical material parameters. Thevarious representations (decompositions) of the coefficientsthat appear in Eq. 2.19 are with ci = ±1 (a positive/negativeidentifier)

• mass-based (m = mass): αi j = αi j mi m j ci c j ,• surface area-based (a = surface area): αi j = αi j ai a j ci c j ,• volume-based (V = volume): αi j = αi j Vi Vj ci c j and• charge-based: αi j = αi j qi q j ci c j ,

where the αi j are empirical material parameters. There arevast numbers of empirical representations, for example,found in the field of “molecular dynamics” (MD), whichtypically refers to mathematical models of systems of atomsor molecules where each atom (or molecule) is representedby a material point and is treated as a point mass. The over-all motion of such mass-point systems is dictated by New-tonian mechanics. For an extensive survey of MD-type inter-action forces, which includes comparisons of the theoreti-cal and computational properties of a variety of interactionlaws, we refer the reader to Frenklach and Carmer [34]. Inthe usual MD approach (see Haile [40], for example), themotion of individual atoms is described by Newton’s sec-ond law with the forces computed from differentiating a pre-scribed potential energy function, with applications to solids,liquids, and gases, as well as biological systems (Hase [42],Schlick [83] and Rapaport [77]). The interaction functionsusually take the form of the familiar Mie, Lennard-Jones,and Morse potentials (Moelwyn-Hughes [64]), howeverthree-body terms can be introduced directly into the inter-action functions (Stillinger [86]) or, alternatively, “local”modifications can be made to two-body representations(Tersoff [88]).

2.5.2 Magnetic forces

An additional force can be exerted on magnetic parti-cles, independent of the electrodynamically-induced Lorentzforces. A relatively simple model for the charcterization ofthis force is given by

�mag = ∇(γ Bext · Bext ), (2.20)

where γ is a material parameter that is related to the magneti-zation of the particle, and which is dependent on the magneticdipole properties, the magnetic susceptibility, the magneticpermeability and the internal magnetic moment density ofthe material (see Feynman et al. [32], Cullity and Graham[14], Boyer [7] or Jackson [47]. For the specific applicationsin this paper, �mag

i is considered small.

2.6 Interstitial damping

Finally, we note that damping from interstitial fluid (or evensmaller-scale paricles, solvents) between particles, such as

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Comput Mech (2014) 54:171–191 177

binding enhancers, surfactants and lubricants is possible. Asimple model to account for this is (a very low Reynoldsnumber “Stokesian” model)

�dampi = ce6π Ri (ve − vi ) (2.21)

where ve is the local average velocity of the external intersti-tial medium, which one may assume to be ve ≈ 0, for mostapplications of interest here. The mechanics of the interstitialfluid is unimportant in problems of interest here. However,for other applications, such as high-speed flow, the motionof the fluid can be important, nessitating more sophisticateddrag laws and/or fully coupled (two-way) particle-fluid inter-action models. This is outside the scope of the present work.Generally, this requires the use of solid-fluid staggering-typeschemes (for example, see Zohdi [109] and Avci and Wrig-gers [6]).

3 Time-stepping

Integrating Eq. 2.1 leads to (using a trapezoidal rule withvariable integration metric, 0 ≤ φ ≤ 1)

vi (t + �t) = vi (t) + 1

mi

t+�t∫

t

� toti dt

≈ vi (t) + �t

mi

(φ� tot

i (t + �t)

+(1 − φ)� toti (t)

), (3.1)

where� toti = �con

i +�walli +�bond

i +�dampi +�e+m

i . Theposition can be computed via application of the trapezoidalrule again:

ri (t + �t) ≈ ri (t) + �t (φvi (t + �t) + (1 − φ)vi (t)),

(3.2)

which can be consolidated into

ri (t + �t) = ri (t) + vi (t)�t + φ(�t)2

mi

× (φ� tot

i (t + �t) + (1 − φ)� toti (t)

). (3.3)

This leads to a coupled system of equations, which are solvedusing an adaptive iterative scheme, building on approachesfound in various forms in Zohdi [107–116]. We note that thematerial contact compliance constants in the various forceterms are functions of temperature, K = K (θ) and the tem-perature is, in turn, a function of the laser input. We willdiscuss the thermal effects shortly, but first indicate how thepreceding dynamical particle system can solved.

3.1 Iterative (implicit) solution method

Following the basic framework in Zohdi [107–116], we writeEq. 3.3 in a slightly more streamlined form for particle i

rL+1i = rL

i + vLi �t + φ(�t)2

mi

×(φ(�

tot,L+1i ) + (1 − φ)(� tot,L)

), (3.4)

which leads to a coupled set equations for i = 1, 2 . . . Np par-ticles, where the superscript L is a time interval counter. Theset of equations represented by Eq. 3.4 can be solved recur-sively. Equation 3.4 can be solved recursively by recastingthe relation as

rL+1,Ki = rL

i + vLi �t + (φ�t)2

mi�

tot,L+1,K−1i

+φ(�t)2

mi(1 − φ)�

tot,Li , (3.5)

which is of the form

rL+1,Ki = G(rL+1,K−1

i ) + Ri , (3.6)

where K = 1, 2, 3, . . . is the index of iteration within timestep L + 1 and

• �tot,L+1,K−1i

def= �tot,L+1,K−1i (rL+1,K−1

1 , rL+1,K−12 . . .

rL+1,K−1N ),

• �tot,Li

def= �tot,Li (rL

1 , rL2 . . . rL

N ),

• G(rL+1,K−1i ) = (φ�t)2

mi�

tot,L+1,K−1i and

• Ri = rLi + vL

i �t + φ(�t)2

mi(1 − φ)�

tot,Li .

The term Ri is a remainder term that does not depend onthe solution. The convergence of such a scheme is dependenton the behavior of G. Namely, a sufficient condition for con-vergence is that G is a contraction mapping for all rL+1,K

i ,K = 1, 2, 3 . . .. In order to investigate this further, we definethe iteration error as

�L+1,Ki

def= rL+1,Ki − rL+1

i . (3.7)

A necessary restriction for convergence is iterative self con-sistency, i.e. the “exact” (discretized) solution must be rep-resented by the scheme, rL+1

i = G(rL+1i ) + Ri . Enforcing

this restriction, a sufficient condition for convergence is theexistence of a contraction mapping

|| rL+1,Ki − rL+1

i︸ ︷︷ ︸�

L+1,Ki

|| = ||G(rL+1,K−1i ) − G(rL+1

i )||

≤ ηL+1,K ||rL+1,K−1i − rL+1

i ||, (3.8)

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178 Comput Mech (2014) 54:171–191

where, if 0 ≤ ηL+1,K < 1 for each iteration K , then�

L+1,Ki → 0 for any arbitrary starting value rL+1,K=0

i , asK → ∞, which is a contraction condition that is sufficient,but not necessary, for convergence. The convergence of Eq.

3.5 is scaled by η ∝ (φ�t)2

mi. Therefore, we see that the con-

traction constant of G is:

• directly dependent on the magnitude of the interactionforces (||�||),

• inversely proportional to the masses mi and• directly proportional to (�t)2.

Thus, decreasing the time step size improves the conver-gence. In order to maximize the time-step sizes (to decreaseoverall computing time) and still meet an error tolerance onthe numerical solution’s accuracy, we build on an approachoriginally developed for continuum thermo-chemical mul-tifield problems (Zohdi [107]), where one assumes: (1)ηL+1,K ≈ S(�t)p, (S is a constant) and (2) the error withinan iteration behaves according to (S(�t)p)K � L+1,0 =� L+1,K , K = 1, 2, . . ., where � L+1,0 = rL+1,K=1 − rL

is the initial norm of the iterative (relative) error and S isintrinsic to the system. For example, for second-order prob-lems, due to the quadratic dependency on �t , p ≈ 2. Theobjective is to meet an error tolerance in exactly a preset (theanalyst sets this) number of iterations. To this end, one writes(S(�ttol)

p)Kd � L+1,0 = T O L , where T O L is a toleranceand where Kd is the number of desired iterations. If the errortolerance is not met in the desired number of iterations, thecontraction constant ηL+1,K is too large. Accordingly, onecan solve for a new smaller step size, under the assumptionthat S is constant,

�ttol = �t

⎛

⎜⎜⎝

(T O L

� L+1,0

) 1pKd

(� L+1,K

� L+1,0

) 1pK

⎞

⎟⎟⎠

def= �t�K . (3.9)

The assumption that S is constant is not critical, since the timesteps are to be recursively refined and unrefined throughoutthe simulation. Clearly, the expression in Eq. 3.9 can also beused for time step enlargement, if convergence is met in lessthan Kd iterations (typically chosen to be between five to teniterations).

3.2 Algorithm

The solution steps are, within a time-step:

• (1): Start a global fixed iteration (set i = 1 (particlecounter) and K = 0 (iteration counter))

• (2): If i > Np then go to (4)

• (3): If i ≤ Np then:

(a) Compute the position rL+1,Ki

(b) Go to (2) for the next particle (i = i + 1)

• (4): Measure error (normalized) quantities

(a) �Kdef=

∑Npi=1 ||rL+1,K

i − rL+1,K−1i ||

∑Npi=1 ||rL+1,K

i − rLi ||

(b) ZKdef= �K

T O Lr

(c) �Kdef=

⎛

⎝( T O L

�0)

1pKd

(�K�0

)1

pK

⎞

⎠

• (5): If the tolerance is met: ZK ≤ 1 and K < Kd then

(a) Increment time: t = t + �t(b) Construct the next time step: (�t)new = �K (�t)old ,(c) Select the minimum size: �t = MIN((�t)lim,

(�t)new) and go to (1)

• (6): If the tolerance is not met: ZK > 1 and K < Kd

then

(a) Update the iteration counter: K = K + 1(b) Reset the particle counter: i = 1(c) Go to (2)

• (7): If the tolerance is not met (ZK > 1) and K = Kd

then

(a) Construct a new time step: (�t)new = �K (�t)old

(b) Restart at time t and go to (1)

Time-step size adaptivity is critical, since the system’sdynamics and configuration can dramatically change overthe course of time, possibly requiring quite different timestep sizes to control the iterative error. However, to main-tain the accuracy of the time-stepping scheme, one mustrespect an upper bound dictated by the discretization error,i.e., �t ≤ �t lim . Note that in step (5), �K may enlarge thetime-step if the error is lower than the preset tolerance. Thealgorithm will be modified shortly to account for coupledthermal effects. As mentioned, the parameters, such as K pi j

can be thermally dependent since the particles can thermallysoften. For example, the compliance constant for the particlesin the contact law can be written as:

K pi = MAX

(

K pio

(

e−ai

θiθ∗i

−1)

, K limpi

)

, (3.10)

and for particle j

K pj = MAX

(

K pjo

(

e−a j

θ jθ∗

j−1

)

, K limpj

)

, (3.11)

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Comput Mech (2014) 54:171–191 179

and take the average at the interface and the value in thecontact law:

K pi j = 1

2

(K pi + K pj

). (3.12)

There is a multitude of possible representations, and it isrelatively easy to select one or the other, and to embed in thestaggering framework developed.

4 Thermal fields

4.1 Heat-transfer model

We assume that (infrared) radiative, convective and strain-rate effects are negligible for the particle’s thermodynam-ics. Only laser-input heating and conduction are consideredimportant. Thus, for each particle i = 1, 2, . . . Np,

mi Ci θi = Qi + Hi , (4.1)

where Qi represents the conductive contribution from sur-rounding particles in contact (including walls), and Hi rep-resents the external heating term. It is assumed that the tem-perature fields are uniform within the (small) particles. Thisassumption is justified, i.e. a lumped thermal model, ignoringtemperature gradients and assuming a uniform temperaturewithin a particle, when the Biot number is small. The Biotnumber for spheres scales with the ratio of the particle volume

(V ) to the particle surface area (As), VAs

= 43 π R3

4π R2 = R3 , which

indicates that a uniform temperature distribution is appropri-ate, since the particles, by definition, are small. Assumingthat the fields are uniform in each particle allows for the fol-lowing (for particle i)

Q = −∫

∂ω

Q · n d A ≈Npc∑

j=1

IKi jθ j − θi

||ri − r j || Aci j , (4.2)

where the summation extends over all particles j =1, 2, 3 . . . Npc that are in contact with particle i (Fig. 4).4

This yields

mi Cidθi

dt=

Npc∑

j=1

IKi jθ j − θi

||ri − r j || Aci j

︸ ︷︷ ︸Qi

+Hi , (4.3)

4 IKi j can be approximated by an average interfacial value of the i −j pair, IKi j ≈ IKi +IK j

2 . If the materials are the same, this collapsesto simply IK . As for the mechanical contact, Ac

i j is the contact areaassociated with the particle pair (i j).

Q

Q

Q

Q4

3

i

1

2

Fig. 4 Heat flux exchange for a particle induced by neighboring par-ticles in contact

where the specific form of laser-induced heating is

Hi ≈ ai Ii Vi , (4.4)

where aIi is the absorbed laser-input for particle i and 0 ≤ai ≤ 1 is an absorption constant. More remarks on laser-inputwill be given next.

4.2 Lasers-various levels of description

There are varying degrees of sophistication for modeling theheat input from a laser, for example:

• Method 1-A very rough overall energy balance wherebythe total energy input W tot (given by integrating thepower output from the laser) and performing an energybalance

MC�θ = W tot ⇒ �θ = W tot

MC(4.5)

where the total mass is M = ∑Ni=1 mi and C is the heat

capacity. This provides no information about conductivelosses, the distribution of the heat, etc. It simply gives arough estimate of the overall effects of laser power input.

• Method 2-Ultilization of the Beer-Lambert law, wherebyone performs an overall power balance (Fig. 5):

(I + �I ) − I + L�z = 0 ⇒ d I

dz=≈ −aI, (4.6)

where aI is the absorbed irradiance (per unit area) fromthe laser and z is the penetration depth, which is solvedto yield

I (z) = Ioe−az . (4.7)

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180 Comput Mech (2014) 54:171–191

Fig. 5 Representations of laser penetration based on modification (right) of the classical Beer–Lambert relation (left)

One can modify this relation to account for a parti-cle at depth z from particle interference from abovethe particle in question by determining the intersectionswith particles above particle at depth z, for example,z∗ = ∑N

i=1 �zi and modifying the Beer-Lambert rela-tion to read

I (z) = Ioe−az∗. (4.8)

• Method 3-Discretization of the irradiant beam into raysand performing a full-blown ray-tracking scheme (seeZohdi [103–106]).

• Method 4-Discretization of the beam into it’s electro-magnetic field components via Maxwell’s equations (seeZohdi [112,113]).

Method 1 is rather simplistic, and can be used more as arough semianalytical guide to laser input selection. Meth-ods 3 and 4 provide extremely detailed field information atthe smallest scales. They are extraordinarily computationallyexpensive, and for the applications in this work, unwarranted.This leaves Method 2, which we will employ for the remain-der of the work. In such an approach, one can easily introducenonuniform beam profiles, for example

I (d) = I (d = 0)e−bd , (4.9)

where d is the distance from the center of the incident beamline. In the case of b = 0 we recapture a flat beam, I (d) =I (d = 0).

4.3 Numerical integration

Integrating the energy equation yields, for each particle (i =1, 2, . . . , Np) yields

θi (t + �t) = θi (t) + 1

mi Ci

⎛

⎝t+�t∫

t

Qi dt +t+�t∫

t

Hi dt

⎞

⎠

(4.10)

≈ θi (t) + �t

mi Ci(φ(Qi (t + �t) + Hi (t + �t))

+(1 − φ)(Qi (t) + Hi (t)))

We note that Eq. 4.10 represents a coupled system of the gen-eral form (similar to the equation that arises for the particledynamics)

θi (t + �t) = Gi (θi (t + �t)) + Ri , (4.11)

where for the “remainder” term, Ri = Ri (θi (t + �t)), andwhere Gi ’s behavior is controlled by the magnitude of �t .Clearly, the temperature is coupled to the mechanical behav-ior of the system. Shortly, we develop a multiphysical stag-gering scheme to solve the overall system.

5 Total system coupling: multiphysical staggeringscheme

We now extend the iterative solution process introduced ear-lier for the particle dynamics to a multifield setting by (at agiven time increment): (1) solving each field equation indi-vidually, “freezing” the other (coupled) fields in the system,allowing only the primary field to be active and (2) updat-ing the primary field variable after the solution of each fieldequation. The next field equation is treated in a similar man-ner where, as the physics changes, the field that is mostsensitive (exhibits the largest amount of relative nondimen-sional change) dictates the time-step size. This is an implicit,staggered, adaptive time-stepping scheme. Such approacheshave a long history in the computational mechanics commu-nity. For example, Zienkiewicz [99], Zienkiewicz et al. [100],Lewis et al. [57], Lewis and Schrefler [56], Park and Felippa

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Comput Mech (2014) 54:171–191 181

[71], Farhat et al. [29], Farhat and Lesoinne [30], Farhat etal. [31], Piperno [72], Piperno et al. [73], Piperno and Farhat[74] and Michopoulos et al. [61], Lesoinne and Farhat [53]and Le Tallec and Muoro [54].

5.1 A general iterative framework

The staggering scheme introduced earlier utilizes approachesfound in Zohdi [107–116] and proceeds by considering anabstract setting, whereby one solves for the particle positions,assuming the thermal fields fixed,

A1(rL+1,K , θ L+1,K−1) = F1(rL+1,K−1, θ L+1,K−1), (5.1)

then one solves for the thermal fields, assuming the particlepositions fixed,

A2(rL+1,K , θ L+1,K ) = F2(rL+1,K , θ L+1,K−1), (5.2)

where only the underlined variable is “active”, L indicatesthe time step and K indicates the iteration counter. Withinthe staggering scheme, implicit time-stepping methods, withtime step size adaptivity, will be used throughout the upcom-ing analysis. We define the normalized errors within eachtime step, for the two fields,

�r Kdef= ||rL+1,K − rL+1,K−1||

||rL+1,K − rL || and

�θ Kdef= ||θ L+1,K − θ L+1,K−1||

||θ L+1,K − θ L || . (5.3)

We define maximum “violation ratio”, i.e. as the larger ofthe ratios of each field variable’s error to its corresponding

tolerance, by ZKdef= MAX(zr K , zθ K ), where

zr Kdef= �r K

T O Lrand zθ K

def= �θ K

T O Lθ

, (5.4)

with the minimum scaling factor defined as�Kdef= MIN(�r K ,

�θ K ), where

�r Kdef=

⎛

⎝(

T O Lr�r0

) 1pKd

(�r K�r0

) 1pK

⎞

⎠ , �θ Kdef=

⎛

⎝(

T O Lθ�θ0

) 1pKd

(�θ K�θ0

) 1pK

⎞

⎠

(5.5)

5.2 Overall solution algorithm

The algorithm is as follows:

(1) GLOBAL FIXED − POINT ITERATION :(SET i = 1 AND K = 0) :

(2) IF i > Np THEN GO TO (4)

(3) IF i ≤ Np THEN : (FOR PARTICLE i)(a) COMPUTE POSITION :rL+1,K

i(b) COMPUTE TEMPERATURE :θ L+1,K

i(c) GO TO (2) AND NEXT PARTICLE (i = i + 1)

(4) ERROR MEASURES(normalized) :(a)�r K

def=∑Np

i=1 ||rL+1,Ki − rL+1,K−1

i ||∑Np

i=1 ||rL+1,Ki − rL

i ||�θ K

def=∑Np

i=1 ||θ L+1,Ki − θ

L+1,K−1i ||

∑Npi=1 ||θ L+1,K

i − θ Li ||

(b)Z Kdef= MAX(zr K , zθ K ) where

zr Kdef= �r K

T O Lr, zθ K

def= �θ K

T O Lθ

(c)�Kdef= MIN(�r K ,�θ K ) where

�r Kdef=

⎛

⎝( T O Lr

�r0)

1pKd

(�r K�r0

)1

pK

⎞

⎠,

�θ Kdef=

⎛

⎝( T O Lθ

�θ0)

1pKd

(�θ K�θ0

)1

pK

⎞

⎠

(5)IF TOL. NOT MET (Z K > 1) AND K < Kd REPEATITERATION (K = K + 1)

(6) IF TOL. MET (Z K ≤ 1) AND K < Kd THEN :(a) INCREMENT TIME : t = t + �t(b) CONSTRUCT NEW TIME STEP : �t = �K �t,(c) SELECT MINIMUM :�t = MIN(�t lim ,�t)(d) UPDATE LASER FIELD Ii (FOR ALL PARTICLES

ITERATIVELY)

(e) AND GO TO (1)

(7) IF TOL. NOT MET (Z K > 1) AND K = Kd THEN :(a) CONSTRUCT NEW TIME STEP :�t = �K �t(b) UPDATE LASER FIELD Ii (FOR ALL PARTICLES

ITERATIVELY)

(c) RESTART AT TIME = t AND GO TO (1)

(5.6)

The overall goal is to deliver solutions where staggering(incomplete coupling) error is controlled and the temporaldiscretization accuracy dictates the upper limits on the timestep size (�t lim).

5.3 Interaction lists

There are a variety of options for simulation speed up involv-ing particle calculations. For example, one can construct a so-called interaction list of neighboring particles within a radiusof influence with which a specific particles interacts for a fewtime steps. As the overall system configuration changes, onecan periodically update the lists (see Pöschel and Schwager[75]). For example, this dramatically reduces the computa-tion time used in contact search, which is an N 2 operation.In this paper, we adopt the following process:

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182 Comput Mech (2014) 54:171–191

REFINE TIME−STEP

UPDATE MECHANICAL FIELD

UPDATE THERMAL FIELD

CHECK FOR CONVERGENCE

YES: GO TO NEXT TIME STEP

(ASSUME PARTICLE POSITIONS ARE FIXED)

THERMAL FIELD: SOLVE FOR THE PARTICLE TEMPERATURES

UPDATE ALL STATE VARIABLES AND COMPUTE ERROR METRICS

MECHANICAL FIELD: SOLVE PARTICLE DYNAMICS AND CONTACT

(ASSUME THERMAL STATES ARE FIXED)

NO ITERATION LIMIT MET?

NO

YESITERATION

AND RESTART

Fig. 6 A flow chart for the modular, staggered, computation

• An initial full-blown contact search is initiated at thebeginning of the simulation to generate an interactionlist of nearest neighbors for each particle,

• The interaction lists are used for all inter-particle calcula-tions for a select subinterval of the total simulation time,0 ≤ �T ≤ T and

• After a select subinterval of the total simulation time(�T ) the interaction lists are updated for each particle.

Interaction lists were used throughout the upcoming simu-lations, with the results being tested against direct computa-tion (no interaction truncation) in order to determine appro-priate parameter settings for the truncation radius and thefrequency of list updating. The upcoming simulation resultswere insensitive to the truncations and are on the order of 50–100 times faster than direct computation. Finally, we remarkthat one can also employ Domain Decomposition techniqueswhereby the domain is partitioned into subdomains, the par-ticles within each subdomain are sent to a processor andstepped forward in time, but with the positions of the par-ticles outside of the subdomain fixed (relative to the parti-cles in that subdomain). This is done for all of the subdo-mains separately, then the position of all of the particles areupdated and this information is shared between processors,with the procedure being repeated as needed. It is impor-tant to note that a significant acceleration in the computationcan be achieved via sorting and binning methods, which pro-ceed by partitioning the whole domain into bins. The parti-cles are sorted by the bins in which they reside. The particleinteraction proceeds, bin by bin, where the particles withina bin potentially only interact with particles in other near-est neighbor bins. Essentially, for a given particle in a bin,

contact searches are conducted with particles in the neigh-boring bins only. The approach is relatively straightforwardto implement and can speed up the computation significantly(see Zohdi [111,116]). There are a variety of related tech-niques to further accelerate computations. For example, onecan assume that particles stay in the bins for a few time steps,and that one does not need to re-sort immediately. This canbe also used in conjunction with interaction lists, althoughthis was not done in this work (Fig. 6).

6 Numerical examples

We consider a model problem of a droplet of equally-sizedparticles (Fig. 7) to illustrate how to assemble the system. Theabsolute dimensions are unimportant for the model problem,and have been normalized so that the initial droplet diame-ter was on the order of unity.5 In order to generate an initialparticle configuration, we randomly dispersed Np = 2, 000non-overlapping particles within a spherical domain, thendropped onto a large surface. The configuration of the sam-ple, before it was dropped, was generated using a classicalrandom sequential addition (RSA) algorithm (Widom [96]),which places non-overlapping particles randomly into thedomain of interest. One could start with a denser startingconfiguration by using equilibrium-driven Metropolis algo-rithm or alternative methods based on simultaneous particleflow and growth (see Kansaal et. al [49], Donev et. al [21–23]and Torquato [89]), although this was not necessary for this

5 All system parameters can be scaled to describe any specific systemof interest. They were selected simply for illustration purposes.

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Comput Mech (2014) 54:171–191 183

Fig. 7 Scenario 1: left to right and top to bottom, a sequence of frames for deposition of particles without laser input

example. As an example, the relevant simulation parameterschosen were (in SI-units if not explicitly stated):

• the particle radii, Ri = 0.05 m,• The normal contact parameter was K po = 107 N/m2,

with thermal softening K p = MAX(K po

(e−a θ

θ∗ −1)

,

K limp ), where θ∗ = 500o K elvin, K lim

p = 106 N/m2

and the exponent in the contact law was set to pp = 2,

• The stiffness of the wall in contact law, Kwo =109 N/m2, with thermal softening Kw = MAX(Kwo(

e−a θθ∗

), K lim

w ), where θ∗ = 500o K elvin, K limp =

108 N/m2 and the exponent in the contact law was setto pw = 2,

• The contact damping parameter, ccd = 105,• The friction contact parameter K f = 107,• The coefficient of static friction, μs = 0.4,• The coefficient of dynamic friction, μd = 0.3,

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184 Comput Mech (2014) 54:171–191

Fig. 8 Scenario 2: left to right and top to bottom, a sequence of frames for deposition of particles with laser input

• The normal bond parameter, K nb = 106 N/m2 and theexponent in the binding law was set to pb = 2,

• The rotational bond parameter, K rb = 103,• The nearfield parameters, α1 = 0.5, β1 = 1, α2 = 0.01,

β2 = 2, where the α is per unit mass,• The effective charge of the particles was q = 1 (per unit

mass), the electric field was Eext = (−10, 0, 0), the mag-netic field was Bext = (0, 0, 0) and the magnetizationparameter was γ = 0,

• The interstitial damping coefficient ce = 1 (assumedStokesian-like),

• The density of the particles, ρ = 2000kg/m3,• The heating absorption coefficient, a = 0.5,• The conductivity IK = 100 Watts/m − K elvin,• The initial material temperature, θi (t = 0) = 300o

K elvin,• The wall temperature, θw(t = 0) = 500o K elvin,• The heat capacity, C = 100 J/kg − K elvin,

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Comput Mech (2014) 54:171–191 185

NORMALIZED TIME

TE

MP

ER

AT

UR

E (

KE

LV

IN)

0 0.2 0.4 0.6 0.8300

300.5

301

301.5

NORMALIZED TIME

TE

MP

ER

AT

UR

E (

KE

LV

IN)

0 0.2 0.4 0.6 0.8

400

600

800

1000

1200

Fig. 9 The temperatures for scenario 1 (no laser input) and scenario 2 (with selective laser input)

• The target number of fixed point iterations, Kd = 10,• The trapezoidal time-stepping parameter, φ = 0.5,• The simulation duration, 2 s,• The initial time step size, 0.00025 s,• The time step upper bound, 0.00025 s and• The tolerance for the fixed-point iteration, 5 × 10−4.

To illustrate the results of the computational framework,two examples were considered:

• Scenario 1: The droplet is dropped onto a slightly heatedlower surface (by gravity with an initial uniform down-ward velocity), and allowed to freely spread/deformaccording to its interaction with the surface, as shownin the Fig. 7, the temperature values are depicted by thecolors. The overall temperature is shown in Fig. 9. Thetemperatures rise very slightly, since the contact dura-tion time is a fraction of a second, which does not allowenough time for significant conduction to occur (as wellas because of the small individual contact areas betweenthe particles and the surface).

• Scenario 2: The droplet is dropped onto a slightly heatedlower surface (as in scenario 1, by gravity with an ini-tial uniform downward velocity), and allowed to freelyspread/deform according to its interaction with the sur-face. However, after waiting a period for the droplet tosettle, a laser source is initiated (at one-quarter of thesimulation time), moving back and forth in a linear man-ner, as shown in the Fig. 8, the temperature values aredepicted by the colors. The overall temperature is shownin Fig. 9. The temperatures rise significantly, due to theinput of the laser.

It is important to note that this framework can also handledepositions that are comprised of multiple particle types, aswell as particles with a fluid (solvent) binder (Fig. 10). Thissimply requires a state variable, per particle, which indicates

Fig. 10 A multiphase droplet representation which can be used fordepositions that are comprised of multiple particle types, as well asparticles with a fluid (solvent) binder

what type of particle it is, and the types of interaction ruleswith various other types of particles. Otherwise, the frame-work is identical. We remark that there are several appli-cations where the deposition is particle-functionalized ink (acollection of particles with a solvent binder). This is commonin industrial “additive” processes such as inkjet and electro-hydrodynamic printing, which do not allow precise controlover the structure of the printed lines, and often results in lineswith scalloped edges or non-uniform width, thus offering lim-ited control over the height of the printed features (Ridley etal. [79], Huang et al. [45], Sirringhaus et al. [84], Ahmad etal. [1], Samarasinghe et al. [82] and Wang et al. [90]). Also,recently, imprint lithography has been proposed as a means ofdecreasing the feature size of patterned particles while allow-ing more precise control over the structure of the printed lines(Park et al. [69], Ko et al. [50,51] and Park et al. [70]). In thisfabrication method, the particle-laden inks are patterned bypressing with an elastomer mold and the particles are dried

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186 Comput Mech (2014) 54:171–191

into their final configuration. While the resolution of imprintlithography is improved over inkjet printing, there exists aresidual layer on the substrate that must be etched away afterpatterning. Control over the height of features can be cor-rupted by capillary action between the mold and the dryingink, in particular along the length of longer features. Thus,as a possible alternative to imprint lithography, particle self-assembly methods, based on capillary filling of photoresisttemplates have been proposed (Demko et al. [17]), and appearto be promising. Clearly, selective laser post-processing canplay a role in all of the previously mentioned approaches,in order to add a degree of precision that is otherwisemissing.

7 Closing

This work developed a modular framework for the depositionof particulate materials and selective laser heating, using adirect particle representation in conjunction with a staggeringscheme to couple submodels of each type of physics together.The physics of this system are strongly coupled since thedynamics controls which particles are in contact, which alsodictates the contacts, which in turn controls the conductiveheating and the induced thermal fields, which softens andbinds the material. The strongly coupled system was solvediteratively within each time-step using a staggering schemewhich employs temporal adaptivity to control the error. Thesubmodular approach allows for easy replacement of models,if so desired. Numerical examples were provided. There area number of enhancements that can be made, for example,complete melting and vaporization of the material. However,this requires modifications of the existing framework. Specif-ically, the state of phase-transformed particles (for examplefrom a solid to a liquid) would need to be tracked accord-ing to whether they are currently solid, melted or vaporized,and whether they are in the process of transitioning fromone state to the other, since the latent heats of melting andvaporization affect the heat transfer (see Appendix). Appli-cations where melting is important include densification ofdeposited particulate materials (Fig. 11), whereby the toplayer of the material is melted, allowing the liquid mater-ial to seep into the voids between the particles underneath.Applications where vaporization is important include cut-ting groves, trenches, etc, in the deposited material. Thereare many possibilities for modeling the melted or vaporizedstate of a particle into a liquid. For example, one approachis to break the original particle into smaller particles gov-erned by different interaction rules (Fig. 10).The approachwould be the same for the vaporization into a gas. This iscomputationally expensive, but would offer simulations thatare quite realistic. A simpler, ad hoc approach is to assigna solid particle which has become melted a different set of

LASER

PENETRATION

SURFACE MELT

Fig. 11 Melting of a deposited material top layer and subsequent den-sification of the lower layer by allowing the liquid to seep into the voidsin between the particles below

properties, such as a reduced stiffness, loss of bonds with sur-rounding particles, etc. An alternative approach is to treat thephase transformed material as a continuum, and to then simu-late a hybrid discrete particle surrounding continuum model.These techniques are currently under investigation by theauthor.

Generally, the deposition of particles to build or enhancecomponents is a subclass of additive manufacturing, which inturn is a subfield of advanced manufacturing. Broadly speak-ing, advanced manufacturing of materials is frequently citedas a blend of techniques from various fields. The “green-ness” of an overall process is often considered as a criterionfor the approach to be judged as competitive (Dornfeld andWright [26], Allwood [4], Reich-Weiser et al. [78] and Rosenet al. [81]). The true green-ness of a combination of tech-niques which, individually, may be safe and sustainable, isunclear. Take for example, laser ablation of rare earth materi-als, functionalized inks laden with small-scale particles, etc.,which, if done in mass, can produce a highly toxic envi-ronment in a non-vacuum processing area. Thus, ablationis of particular interest, due to the potential for dangerousejecta for large-scale processes. For example, thermal or pho-tothermal ablation usually refers to laser light conversion tolattice vibration before breaking of bonds liberate atomic-scale material, while photochemical/electronic ablation isa direct electronically-induced vibration. Hydrodynamicalablation refers to micrometer droplets following from themolten phase, whereas exfoliation is an erosive mechanismby which material is removed as flakes. Note that all of themechanisms can occur simultaneously. We refer the reader tothe state-of-the-art in Grigoropoulos [38], which categorizesvarious mechanisms for ablation. In order to determine therealistic contribution of parameters for specific applications,an order-of-magnitude analysis would clearly be useful, andis currently under way by the author.

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Comput Mech (2014) 54:171–191 187

8 Appendix 1: Contact area parameter and alternativemodels

8.1 Contact area parameter

Following Zohdi [116], and referring to Fig. 12, one can solvefor an approximation of the common contact radius ai j (andthe contact area, Ac

i j = πa2i j ) by solving the following three

equations,

a2i j + L2

i = R2i , (8.1)

and

a2i j + L2

j = R2j , (8.2)

and

Li + L j = ||ri − r j ||, (8.3)

where Ri is the radius of particle i , R j is the radius of par-ticle j , Li is the distance from the center of particle i andthe common contact interpenetration line and L j is the dis-tance from the center of particle j and the common contactinterpenetration line, where the extent of interpenetration is

δi j = Ri + R j − ||ri − r j ||. (8.4)

The above equations yield an expression ai j , which yields anexpression for the contact area parameter

Aci j = πa2

i j = π(R2i − L2

i ), (8.5)

PARTICLE j

2a ij

Ri

Rj

PARAMETERCONTACT AREA

j

iL

L

PARTICLE i

Fig. 12 An approximation of the contact area parameter for two par-ticles in contact (Zohdi [116])

where

Li = 1

2

(

||ri − r j || − R2j − R2

i

||ri − r j ||

)

. (8.6)

8.2 Aternative models

One could easily construct more elaborate relations connect-ing the relative proximity of the particles and other metrics tothe contact force,�con,n

i j ∝ F(ri , r j , ni j , Ri , R j , . . .), build-ing on, for example, Hertzian contact models. This poses nodifficulty in the direct numerical method developed. For theremainder of the analysis, we shall use the deformation metricin Eq. 2.8. For detailed treatments, see Wellman et al. [91–95]and Avci and Wriggers [6]. We note that with the appropri-ate definition of parameters, one can recover Hertz, Bradley,Johnson–Kendel–Roberts, Derjaguin–Muller–Toporov con-tact models. For example, Hertzian contact is widely used,with the assumptions being

• frictionless, continuous, surfaces,• each of contacting bodies are elastic half-spaces, whereby

the contact area dimensions are smaller radii of the bodiesand,

• the bodies remain elastic (infinitesimal strains),

results in the following contact force:

�con,ni j = 4

3(R∗)1/2 E∗δ3/2

i j , (8.7)

which has the general form of �con,n = K ∗i jδ

pi j , where

• R∗ =(

1Ri

+ 1R j

)−1and

• E∗ =(

1−ν2i

Ei+ 1−ν2

jE j

)−1

,

where E is the Young’s modulus and ν is the Poisson ratio,The contact area with such a model has already been incor-porated in the relation above, and is equal to Ac

i j = πa2

where a = √R∗δi j . For more details, we refer the reader to

Johnson [48]. Furthermore, we remark that the normal con-tact between a particle and a wall, with a Hertzian model isgiven by

�wall,ni = 4

3(R∗)1/2 E∗δ3/2

iw = K ∗iwδ

piw, (8.8)

where R j = Rw = ∞ (see Eq. 8.7)

• R∗ = Ri and

• E∗ =(

1−ν2i

Ei+ 1−ν2

jE j

)−1

.

123

188 Comput Mech (2014) 54:171–191

It is obvious that for a deeper understanding of the defor-mation within a particle, it must be treated as a deformablecontinuum, which would require a highly-resolved spatialdiscretization, for example using the finite element methodfor the contacting bodies. This requires a large computationaleffort. For the state of the art in finite element methods andContact Mechanics, see the books of Wriggers [97,98]. Forwork specifically focusing on the continuum mechanics ofparticles, see Zohdi and Wriggers [111].

9 Appendix 2: Phase transformations

To include phase transformations, we consider seven cases,which are implemented in a predictor-corrector manner byfirst solving

mi Ci θi = Qi + Hi (9.1)

to obtain predicted temperature, and then checking the fol-lowing:

• Solid → solid-no melting with Ci = CS : If θ(t) < θm

and θ(t +�t) < θm then retain Eq. 9.1 with C(θ) = CS ,• Solid → liquid-melting with Ci = CS : If θ(t) < θm

and θ(t + �t) ≥ θm then re-solve Eq. 9.1 with C(θ) =CS + δP S→L

δθ,

• Liquid → liquid-melted with Ci = CL : If θ(t) ≥ θm andθ(t + �t) ≥ θm then retain Eq. 9.1 with C(θ) = CL ,

• Liquid→ solid-solidification with Ci = CL : If θ(t) ≥ θm

and θ(t + �t) < θm then re-solve Eq. 9.1 with C(θ) =CL + δP L→S

δθ,

• Liquid → vapor-vaporization with Ci = CL : If θ(t) <

θv and θ(t +�t) ≥ θv then re-solve Eq. 9.1 with C(θ) =CL + δP L→V

δθ,

• Vapor → vapor-remains a vapor with Ci = CV : If θ(t) ≥θv and θ(t + �t) ≥ θv then retain Eq. 9.1 with C(θ) =CV ,

• Vapor → liquid-condensation with Ci = Cv: If θ(t) ≥ θv

and θ(t + �t) < θv then re-solve Eq. 9.1 with C(θ) =CV + δPV →L

δθ,

where CS is the heat capacity of the solid, CL is the heatcapacity of the liquid and CV is the heat capacity of thevapor and

• 0 < δP S→L is the latent heat of melting,• 0 < δPL→S is the latent heat of solidification,• 0 < δPL→V is the latent heat of vaporization,• 0 < δPV →L is the latent heat of condensation and

• 0 < δθ is small and can be thought of as a “bandwidth”for a phase transformation. For more details on meltingprocesses, see Davis [15].

We note that latent heats have a tendency to resist the phasetransformations, achieved by adding the positive terms in thedenominator, thus enforcing reduced temperature (during thephase transformation).6 This approach is relatively straight-forward to include within the staggering framework. As aconsequence, the number of particles in the system and theirheat capacities will also change in the algorithm, and Eq.4.10 becomes

θi (t + �t) = θi (t) + 1

mi Ci

⎛

⎝t+�t∫

t

Qi dt +t+�t∫

t

Hi dt

⎞

⎠

≈ θi (t) + �t(φ(QK

i (t + �t) + HKi (t + �t))

)

mi C Ki (t + �t)

+�t (φ(Qi (t) + Hi (t)))

mi Ci (t). (9.2)

The subsequent convergence of the thermal calculation israther quick, since the time steps are extremely small.For more details on convergence on iterative time-steppingschemes, see Zohdi [107–116].

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