Yao-XunChang, Zen-ChungShih

1 Introduction

The synthesis of rustin seawater

Yao-Xun Chang,Zen-Chung Shih

Department of Computer and Information Science,National Chiao Tung University, Hsinchu, 30050,Taiwan, R.O.C.E-mail: [email protected],

[email protected]

Published online: 28 January 2003c© Springer-Verlag 2003

Computer graphics researchers have recentlyrevealed that the presence of imperfections isan essential part of the illusion of photoreal-ism. An improved model to simulate the rustdevelopment of ferrous metals in seawateris proposed. The tide and the current, underthe gravity of the sun and the moon, affectthe stability of the environment at the bot-tom of the ocean. A simple current model isemployed to elucidate the periodic transitionof currents in the ocean. L-systems, coupledwith the current model, modulate the diffu-sion of rust according to tendencies based onthe object geometry and environmental fac-tors. Changing the underlying distributionsof tendency allows images of a variety ofmetallic rusts to be generated.

Key words: Texture synthesis – Metalliccorrosion – L-systems

Although rendering techniques such as ray tracingand radiosity algorithms – which simulate light andits interaction with the environment – can enhancea virtual scene using effects such as soft shadows andreflections, the illusion of photorealism also dependscrucially on the quality of the underlying materialmodels. Over the past decade, researchers have at-tempted to model the changes in the appearance ofa surface due to aging phenomena, such as dirt ac-cumulation, rain washing, sedimentation of deposits,and metal corrosion (Becket and Badler 1990; Blinn1982; Chang and Shih 2000; Dorsey el al. 1999;Dorsey and Hanrahan 1996; Dorsey et al. 1996; Go-bron and Chiba 1999; Hsu and Wong 1995; Merillouet al. 2001; Miller 1994; Wong et al. 1997). Suchinvestigations have sought to enhance the authen-ticity of computer-generated images, because digi-tally created objects in early computer images lookedtoo clean to be realistic. Accordingly, methods forgenerating various imperfections on a surface havebeen elucidated, showing that the presence of im-perfections is critical in causing a viewer to acceptwillingly the rendered images as real. The appear-ance of a real surface can change over time whenexposed to the surrounding environment. Real-worldobjects are normally covered with a variety of im-perfections that underlie a person perceiving the ob-ject as real. Therefore, this study concentrates onelucidating the basic physical models of aging phe-nomena and modeling the corresponding evolutionof a surface.Imperfections may occur for a number of rea-sons that range from chemical reactions, such asmetallic rusting, to biological growth, such as thecovering of a stone with moss. Many pieces ofresearch have proposed to the simulation of re-lated aging and weathering effects. Becket andBadler (1990) generated a broad class of surfaceblemishes, such as scratches and splotches. Miller(1994) related surface accessibility to the distri-bution of patinas on tarnished surfaces. Hsu andWong (1995) simulated the amount of dust accu-mulation. Furthermore, Wong et al. (1997) proposeda geometry-dependent method to represent blem-ishes according to modeled tendency distribution.Dorsey and Hanrahan (1996) examined a metal-lic surface as a series of layers and then simulatedthe patina with a collection of procedural opera-tors. Dorsey et al. (1999) also modeled the pat-terns of washes and weathered stone. Gobron andChiba (1999) described a 3D-surface cellular au-

The Visual Computer (2003) 19:50–66Digital Object Identifier (DOI) 10.1007/s00371-002-0172-0

Y.-X. Chang, Z.-C. Shih: The synthesis of rust in seawater 51

tomata method, simulating the propagation of pati-nas over surfaces. More recently, Merillou et al.(2001) provided an approach based on experimentaldata to simulate and render the corrosion (rusting) ofmetals.In light of the above developments, our previouswork presented a physically based model to sim-ulate the appearance of patinas on antique objectsthat have for a long time been underground, includ-ing ancient Chinese bronzes (Chang and Shih 2000).The framework presented simulates the develop-ment of layered patinas guided by environmentaltendencies. The distributions of tendency (Wonget al. 1997) representing the potential occurrenceof patinas were defined to model the influence ofenvironmental factors on patination. The structureof patinas was represented as a set of layers. Thisapproach, as well as using a simple rule-based al-gorithm to simulate the complicated diffusion ofdifferent patinas, has the advantage of incorporat-ing the influence of the environment on patinas.The primary concern in simulating patinas on an-cient Chinese bronzes is the bronzes’ burial en-vironment. The underground environment is wellknown to be almost static. Any changes to the an-tiques are small. Accordingly, invariable distribu-tions are assumed throughout the simulation in ourmodel. The model used these distributions to de-termine the propagation of various patinas with-out updating any tendency values. However, notall real environments remain static; for instance,the atmosphere and the ocean provide dynamic en-vironments. The formation of corrosion productsinfluences subsequent corrosion processes, mean-ing that the formation changes the corrosive poten-tial of the immediate environment of the products.Thus, this paper develops an enhanced environmentmodule to describe dynamic environments such asseawater. Feedback to the environment of newlyformed corrosion products is also included in thenew model presented here, to imitate precisely theinteraction between the material and the surroundingenvironment.This study extends the authors’ previous patinationmodel to generate the rusty appearance of ferrousmetals in seawater, as shown in Fig. 1. The proposedmodel consists of two modules – a seawater moduleand a rusting module. The seawater module adoptstendency distributions to reflect the potential of rust-ing in seawater. The rusting module simulates thedevelopment of the diffusion of the rust. A simple

Fig. 1. Samples of real metallic rust in seawater

current model is employed to illustrate the periodictransition of currents in the ocean. Coupling the cur-rent model with the order of iteration, the rustingmodule models the diffusion of rust on surfaces ac-cording to the change of a dynamic environment.Following each iteration of the L-system, the envi-ronment module updates tendency values of verticesat which rusting occurs. Dormancy is accounted forin the rusting module. Rust development may beheld in abeyance, or remain torpid under ill-suitedconditions, and begin propagating again as circum-stances become favorable owing to the dynamic na-ture of tendency distributions. Finally, the symbolstring generated by L-systems is graphically inter-preted by a ray tracer.The rest of this paper is organized as follows. Sec-tion 2 briefly describes the underlying characteris-tics of seawater during rusting and the formationof rust in such an environment. Section 3 specifiesthe overall descriptive framework for interactionsbetween rust and environment. Section 4 presentsthe distribution of tendencies that maps the influ-ence of various factors on the development of rust.Section 5 depicts the L-system, simulating the ef-fect of the environment on the growth of rust. Sec-tion 6 discusses a visualization method that rendersthe blemished model. Section 7 gives experimen-tal results that demonstrate the proposed framework,while conclusions and areas for further research arefinally provided in Sect. 8.

52 Y.-X. Chang, Z.-C. Shih: The synthesis of rust in seawater

2 The rusting in seawater

Research in the field of corrosion (Craig 1989;Fontana 1986; Jones 1992; Newman and Sieradzki1994; Ross 1977; Shreir 1976) has revealed thatthe chemical properties of a given metal dependon the precise prevailing environmental conditions.Corrosion can be basically considered to be a het-erogeneous chemical reaction that occurs at a metal–environment interface. The rusting of ferrous met-als – one kind of corrosion – is an electrochemicalprocess that can be broken down into anodic andcathodic reactions (Chandler 1985; Fontana 1986;Jones 1992; Rogers 1968; Ross 1977; Shreir 1976).The products of anodic and cathodic reactions, Fe2+and OH− respectively, migrate in the electrolyte andreact to form ferrous hydroxide that is eventually ox-idized to ferric salt: Fe2+ + 2OH− → Fe(OH)2 →Fe2O3 · H2O. Where the supply of oxygen is re-stricted, black Fe3O4 is formed. This description issimplified but serves to illustrate the electrochemicalprocess of rusting and the essential role of moistureand oxygen supplied by the environment. Exactlyhow seawater influences rusting must be investigatedto support a lifelike simulation of rusting in the sea.

2.1 Seawater

Seawater is a common environment in which metal-lic products are exposed. The chloride content of sea-water supports an electric conductivity much higherthan that of air, possibly leading to higher corrosiv-ity. In fact, the corrosive nature of seawater is alsoaffected by the concentration and availability of dis-solved oxygen, the salinity, the concentration of mi-nor ions, and the concentration and nature of pol-lutants. Other factors, including temperature, depthand ocean currents, are often equally or more impor-tant. However, not all factors that govern corrosioncan be easily isolated and measured.

• Dissolved gases. Oxygen and carbon dioxide arethe most important gases dissolved in seawater.At ordinary temperatures, dissolved oxygen isa prerequisite for any appreciable corrosion ofiron or steel. Moreover, differences in dissolvedoxygen transport readily generate differential aer-ation cells, resulting in localized corrosion on theiron surface. Carbon dioxide significantly affectsthe pH of seawater and influences the formation

of protective carbonate scales. In seawater, the bi-carbonate ions form insoluble calcareous scales,resulting from cathodic protection. These coher-ent and adherent scales limit access by dissolvedoxygen, markedly reducing the rate of corrosion.

• pH. In the intermediate pH4 through pH10 range,a loose, porous ferrous-oxide deposit shields themetal surface. The rate of corrosion is nearly con-stant. In solutions more acidic than pH4, oxideis soluble and corrosion increases owing to theavailability of H+ ions. At pH values exceed-ing 10, corrosion is slow due to the formation ofa passive ferric oxide film in dissolved oxygen.

• Temperature. Changes in temperature alter therate at which chemical and electrochemical reac-tions proceed. For example, a rise in temperaturewould be expected to lead to an increase in corro-sion. However, such a rise also decreases the solu-bility of the dissolved oxygen, thereby promotingthe formation of a protective magnetite film onsurfaces. The temperature of seawater is subjectto seasonal variations depending on the prevailingwinds, ocean currents, tidal streams and elevationof the sun.

• Depth. Oxygen varies with an increasing depth,tending to drop from 1000 m to 2000 m and thenincrease again. The temperature declines withdepth. The rate of corrosion generally decreaseswith depth. Apart from the general corrosion rate,factors such as pressure and mechanical effectsvarying with the depth may be important in deter-mining the types of corrosion that occur.

• Velocity. The velocity of seawater can markedlyinfluence corrosion by controlling the supply ofoxygen to the rusting surface and removing prod-ucts that would otherwise stifle further rusting.The motion of water may cause the formation ofdifferentially aerated cells. Several specific formsof corrosion, such as cavitation, can manifest athigh seawater velocities. Corrosion in seawaterincreases with velocity, because of the high con-centration of chloride ions.

2.2 Forms of corrosion

Corrosion can affect metals in a variety of ways,depending on its type and the precise prevailing en-vironmental conditions. Corrosion can range fromfairly uniform wastage, resulting in general thin-ning, to highly localized attack, resulting in pitting


a

b

c

d

e

Fig. 2a–e. Different types of corrosion: a uniform cor-rosion; b pitting; c crevice corrosion; d stress corro-sion crack; e hydrogen blister (left column is the cross-section, right column is exposed surfaces)

and cracking. As such, various forms of corrosionhave been categorised according to the appearance ofthe corroded metal (Chandler 1985; Fontana 1986;Jones 1992; Rogers 1971; Trethewey and Chamber-lain 1988). Figure 2 schematically depicts severaltypical forms of corrosion.• Uniform corrosion. A uniform, regular removal

of metal from a surface is the most common formof corrosion and is typified by steel rusting. “Uni-form” implies that a chemical or electrochemicalreaction proceeds over the entire exposed surfaceat approximately the same rate, but the loss ofmetal is rarely completely uniform.

• Pitting corrosion. Pitting is a form of extremelylocalized attack that penetrates small, discrete ar-eas. Commonly, the penetration through the sur-face is deeper than that caused by uniform corro-sion. All aqueous environments can support this

kind of attack and no metal is immune from it.Although the likelihood and course of pitting isalways unpredictable, some conditions encour-age pitting, such as (1) the presence of chlorides,(2) stagnant conditions, and (3) discontinuities inprotective coatings.

• Crevice corrosion. This type of corrosion usuallyoccurs where a crevice is formed between twosurfaces, such as in lapped joints, under wash-ers and in loose bolts. Crevice corrosion resultssimply from differences between the metal ion oroxygen concentration in the crevice and that in itssurroundings. A large cathodic current acting onthe small anodic area in the crevice facilitates anintense local attack.

• Stress corrosion crack. Stress corrosion crackingoccurs under a static tensile stress within specificcorrosive environmental conditions. For exam-ple, stainless steel is susceptible in hot chloridesand carbon steel in nitrates. Such corrosion canlead to rapid failure of an alloy. The major factorsthat influence such corrosion include the environ-mental conditions, the stress conditions and thealloy under consideration. However, the mecha-nisms involved are not well understood.

• Hydrogen blister. A hydrogen blister is formedwhen atomic hydrogen migrates from the surfaceto internal defects and voids, in which molecularhydrogen can nucleate, generating sufficient in-ternal pressure locally to deform and rupture themetal. This process normally occurs in storagetanks and in refining processes.

This work represents metallic rust as patinas – thatis, as a series of layers (Dorsey and Hanrahan 1996)with zero thickness and with each layer containing atleast one corrosion form, as illustrated in Fig. 3. Thelowest layer on the base metal is the oxide layer, andis composed of two chemical constituents – reddishhematite and black magnetite. The next layer is thehydrogen blister that simulates the bubbles formedbetween the oxide layer and the base metal. Thislayer deforms surface geometry, but does not changethe constituents of the rust. A deposition layer im-mediately next to the blister layer is formed whendissociated ferrous ions in seawater react with saltssuch as carbonates and sulfates. Pitting and crackingare feathering in the top layer. The upper three lay-ers need not coexist; for example, pits or cracks maybe observed directly on the oxide layer. However, thesequence of these layers cannot change.


Fig. 3. A layer structure of rust

3 Framework for rusting

The transport of metallic ions and electrons playsa critical role in the formation of rust, since corrosioncan be defined as an electrochemical reaction. Forexample, in anodic areas, iron is dissolved in seawa-ter as ferrous ions. These ions migrate through sea-water to the cathodic areas and react with some com-pound in seawater such as a carbonate or sulfide, toyield precipitates on metal surfaces. The opportuni-ties to form rust increase with the number of ferrousions absorbed. The system undergoes transition to itslowest energy state without the application of exter-nal forces if the change in free energy associated withthe transition is negative. Corrosion reactions behavein exactly the same manner. The reactions are highlylikely to take place to reduce the free energy of themetal (Trethewey and Chamberlain 1988). In the an-odic areas, corrosion becomes stifled over a periodand new anodic areas adjacent to the original onesbecome active. Although the propagation of corro-sion is not point-to-point, it follows a kind of branch-ing structure, as evidenced by the skeleton of corro-sion pattern. Accordingly, L-systems are introducedto simulate the diffusion of rusting on metal surfacesin seawater.Early L-systems (Hanan 1992; Lindenmayer 1968;Prusinkiewicz and Lindenmayer 1990) have beenwidely adopted as a general framework for plantmodeling. Those systems simulated the geometry ofplants under the control of endogenous or exoge-nous mechanisms. Although the results are impres-sive, the interactions between plants and their sur-rounding environment are neglected in most models.Prusinkiewicz et al. (1994) first introduced an en-

vironmentally sensitive L-system in 1994. The au-thors simulated the influence of the local environ-ment on plant structures and behaviors, by utiliz-ing the simple function of a turtle’s position to limitthe growth of plants. Based on this model, Mechand Prusinkiewicz (1996) then proposed an openL-system that incorporated a bi-directional commu-nication module between two concurrent processes– the plant and the environment. The growth ofplants is impacted by the environment, which is it-self altered by the growth of the plants, throughthe communication module. Noser and Thalmann(1999) extended the concept to simulate living or-ganisms in a systemic sense. They presented a be-havioral animation system for humanoids. Parish andMüller (2001) also proposed a procedural approachto generate the layout of a city, depending on theuser’s needs. Following the new trend in open L-systems, our work presents a novel framework tosimulate rusting in seawater, as guided by environ-mental conditions.Figure 4 depicts the framework for rusting, in whicha communication interface acts as a bridge betweentwo major parts of the framework – the rusting mod-ule and the seawater module. The interface includesa query element and an update element, togethergoverning the communication between these twomodules. The seawater module includes a databasethat records present tendency distributions on themetal surface. The following section describes thetendency to rust. Environmental functions of the sea-water module are stimulated by the rusting module,to evaluate or update forthcoming tendency distribu-tions. The rusting module, itself an L-system com-posed of an alphabet, an initial string “axiom” anda set of production rules, simulates the evolution andthe propagation of rust. At the beginning of each iter-ative step, the rusting module sends a message to theseawater module, stating the position of nodes (ver-tices of the triangular mesh), the type of rust, and thelifetime of nodes. The seawater module then deter-mines the subsequent state of the propagation, usingenvironmental functions limited by some criteria; themodule keeps a record of the propagation. Similarly,the results of queries are returned to the rusting mod-ule. The derivation step then establishes the propa-gation of rust, according to the production rules ofL-systems, after all feedback from the environmenthas been received. The rusting module then sendsanother message that causes the seawater module toupdate all tendency values of new rusty vertices and


Fig. 4. A rusting framework

their neighbors before the following iteration. Theproposed framework can model the complete evolu-tion of rust in seawater by repeating this process.

4 Seawater module

Corrosion in seawater deteriorates metals by thechemical, mechanical, and biological actions of theseawater environment. A seawater module is intro-duced to represent the environment in which met-als are immersed and corroded over a period. Theprimary functions of the seawater module are dy-namically maintaining a record of present tendencydistributions, and evaluating the propagation andevolution of rust. Whenever a query is received fromthe rusting module, the seawater module begins bygathering corresponding tendencies from neighbor-ing vertices of the vertex of interest, except those atwhich rusting has already occurred. Rust is producedat a neighboring vertex if its tendency value exceedsboth the average value of the corresponding tendencydistribution and the tendency of the queried vertex.As well as determining the propagation direction ofpresent rust, the seawater module also determineswhether another type of rust is triggered. While gath-ering tendencies from the neighborhood, the seawa-ter module also collects other tendencies to rust fromupper layers at the queried vertex. The module trig-gers another type of rust that is most likely to developat the same vertex, when the difference between thelocal tendency and the average of the tendency dis-tribution exceeds the standard variation of the same

distribution. The result is then attached to the mes-sage returned to the rusting module.

4.1 Tendency distribution

Recent investigations (Chandler 1985; Mercer 1990;Shreir 1976) have demonstrated that the corrosionpotential of metals in seawater is a function of wa-ter salinity, temperature, oxygen content, velocity,and the surface condition of the metal. Varying onefactor may change the conditions of the other fac-tors and, in some cases, may differentially promoteor suppress corrosion. Such complex properties andthe composition of seawater may explain the diffi-culty of simulating the real mechanism of metalliccorrosion in seawater. The detail of the corrosion ofmetals is highly complex and scientists’ understand-ing of the process is far from complete. However,the underlying systematic distribution can be more orless observed. Thus, a tendency T , representing thepotential occurrence of rust at a surface point of in-terest, is defined as follows (Wong et al. 1997):

T = α∗ M +β ∗ G +γ ∗ A + δ∗ O +λ∗ F +ω∗ S,

where α, β, γ , δ, λ, and ω are weighting constants.The letters M, G, A, O, F, and S represent the maxi-mum principle curvature, Gaussian curvature, acces-sibility, orientation, water current and distribution ofimportant salts in seawater respectively. These fac-tors can be separated into two categories: surfacecurvature and accessibility are local geometric prop-erties; and orientation, currents and salt distributionsare global environmental properties.


Concave areas shield a metal from current attacks,but convex areas provide more opportunities forcontact with dissolved oxygen. Any sort of geo-metric configuration that makes the distribution ofdissolved oxygen unequal may initiate corrosion.The surface curvature is frequently considered to de-scribe surface geometric properties such as bumps,hollows and saddles. Two approaches, proposed byTurk (1992) and Calladine (1986), are adopted toestimate the principle curvature and Gaussian curva-ture of metal surfaces respectively, and a polygonalrepresentation is employed to model smooth ob-jects. Principle curvature specifies whether a pointis concave or convex, and Gaussian curvature dis-tinguishes saddle points. Surface accessibility mea-sures how easily a surface would be touched andaffects the dimensionality of the contact between anobject’s surface and dissolved oxygen. Miller (1994)presented a tangent-sphere method that employsa maximum sphere tangent at the vertex, without in-tersecting any triangles of objects, to approximatethe accessibility of that vertex. Tests have demon-strated that inclined specimens corroded more thanvertical specimens, and undersides corroded morethan topsides (Chandler 1985). A dot product of thesurface normal and the direction of gravity indicatesthe influence of orientation on corrosion, since anobject’s orientation affects the impact of currents onits surfaces. Seawater is a complex, heterogeneousenvironment. The distribution of significant salts inseawater that may form scales on surfaces, can noteasily be measured. These scales can locally changea corrosive environment, affecting corrosion in someway. Thus, a noise solid texture is employed to de-scribe in general the distribution of these salts inseawater. This solid texture also specifies the localvariation in acidity on metallic surfaces.The most significant feature of seawater is its move-ment. Turbulence, tides and seasonal currents maybe present. Currents are very important in corrosionreactions, and they change surface conditions whenthey flow onto objects’ surfaces, just as when thesun shines on objects. However, currents and sun-light differ in that, when striking an object’s surface,water, unlike light, is neither absorbed nor reflectedbut instead exhibits a slightly altered flow direction,as it passes along the object’s surface, as schemati-cally depicted in Fig. 5a. For simplicity, an improvedmatte shading method with a distant area light sourceseems to simulate appropriately the interaction be-tween currents and surfaces. An area light source is

5a 5b

6

Fig. 5. The schematic diagram: a current; b area light sourceFig. 6. Tendency distributions (from left to right and top to bot-tom: Fe3O4, Fe2O3, blister, deposit, pit, crack)

introduced to overcome the difference between lightand water in this regard. Consider Fig. 5b. The um-bra on the surfaces represents a region in which thecurrent is nearly quiet. The velocity of currents canbe represented by the intensity of light in the shadingmodel. Then the current tendency F of a vertex is de-fined as F = 1

n

∑ni=1 I f ∗ (Di · N )∗ S(Di , P), where

n is the number of sample points over the light sourceand N is the surface normal of a vertex P. Di is a unitvector from P to the ith sample point. The light in-tensity, representing current velocity, is I f . As fornormal shading, S(Di, P) represents a shadow testand a returned value of zero indicates that vertex P isin a shadow that corresponds to the ith sample pointof the area light source. But, when the vertex facesdirectly away from the light, the returned value isset to 1. A negative value of Di · N represents sta-tionary water. Figure 6 displays experimental resultsof tendency distributions, where the intense yellow


Fig. 7. A dynamic current model

represents a high tendency value and a high prob-ability of rusting. Different tendency distributionsrepresent preferable conditions for different types ofrusting.

4.2 Dynamic tendency distribution

Several current directions are defined to describeseasonal variations, and a swing-like order sim-ply specifies the sequence of these currents. Fig-ure 7 depicts three current directions. The sea-water module, which includes a dynamic currentmodel, is informed to update the tendency distribu-tions when the current changes direction. Schemesthat apply the current sequence to the iterationsteps cause the tendency distributions to change inthe simulation, because the rusting module inter-prets the iterative steps as time steps (Prusinkiewiczand Lindenmayer 1990). As presented in Table 1,the water current can change direction at each

Table 1. Current scheme ( f1, f2, and f3 represent three currents with different directions)

iteration 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

frequency = 1 f1 f2 f3 f2 f1 f2 f3 f2 f1 f2 f3 f2 f1 f2 f3frequency = 1/2 f1 f1 f2 f2 f3 f3 f2 f2 f1 f1 f2 f2 f3 f3 f2frequency = 1/3 f1 f1 f1 f2 f2 f2 f3 f3 f3 f2 f2 f2 f1 f1 f1

consecutive or second iteration, or even less fre-quently. The same current can be maintained tomodel steady flow. Figure 8 shows the period ofcurrents: a higher intensity of blue color repre-sents water striking the surface more directly anda higher intensity of red color represents a quieterflow.While the environment is essential in corrosion, theeffect of the metal–environment interaction uponthe environment is also important. The metallicions formed by corrosion can affect the corrosiv-ity of the environment. Metallic ions and elec-trons in a rusty area are consumed when a rustyvertex is generated, and new ions and electronsmove into this area from the immediately surround-ing area to achieve a new local equilibrium. Thetendency values are therefore updated after eachderivation step. In the seawater module, a thresh-old value is defined to reduce the tendency valueof the new rusty vertex. The tendencies of verticesin the surrounding area also decrease according tothe distance to the new rusty vertex. Figure 9 illus-trates the paradigm; the red point is the new rustyvertex.

5 Rusting module

Although areas with high corrosive potential willrust severely, rusting may still occur in areas withlow potential. Hence, a set of seed vertices is ran-domly selected from vertices at which the tendencyvalue of the first layer exceeds a predefined thresh-old. A uniform random distribution is used. Thesevertices constitute the axiom of L-systems. Whenthe results of queries are returned, rust developmentfrom seed vertices simultaneously proceeds alongthe edges connected to these vertices, according tothe production rules of L-systems. The essential fea-tures of the rusting module are described by the fol-lowing production rules.


Fig. 8. A period of currents (from left to right: f1, f2, f3, f2)

#de fine L1 80 /* initial lifespan of apices */#de fine P 10 /* the increment of parameter t */

initial :p1 :Q(M, n, c)< A(p, l, t):c == 1 & t == 0 �−→ Q(M, 0, 0)A(p, l, c)p2 :Q(M, n, c)< A(p, l, t):c == 2 & t == 0 �−→ Q(M, 0, 0)A(p, l, c)

termination :p3 :Q(M, n, c)< A(p, l, t):l ≤ 0 ‖ n < 0 �−→ R(p, l, t)p4 :Q(M, n, c)< A(p, l, t):n == 0 & t ≥ m P �−→ R(p, l, t)

p5 :Q(M, n, c)< A(p, l, t):n == 0 & t < m P �−→ B(p, l −1, t, T1)

propagation :p6 :Q(M, n, c)< A(p, l, t):c == 0 & t%P == 6 �−→ R(p, l, t)Q(M, 0, c+1)

A(M[0], l −1, t)Q(M, 0, c+1)A(M[1], l −1, t)p7 :Q(M, n, c)< A(p, l, t):c > 0 & t%P == 6 �−→ R(p, l, t)Q(M, 0, c+1)A(M[0], l −1, t)p8 :Q(M, n, c)< A(p, l, t):c == 0 & t%P == 3 �−→ R(p, l, t)Q(M, 0, 0)A(M[0], l −1, t)Q(M, 0, 0)A(M[1], l −1, t) · · ·

Q(M, 0, 0)A(M[n −1], l −1, t)p9 :Q(M, n, c)< A(p, l, t):c == 0 �−→ R(p, l, t)Q(M, 0, 0)A(M[0], l −1, t)Q(M, 0, 0)A(M[1], l −2, t) · · ·

Q(M, 0, 0)A(M[n −1], l −n, t)

triggering :p10:Q(M, n, c)< A(p, l, t):c > 4 & t%P < 5 �−→ R(p, l, t)Q(M, 0, 0)A(M[0], l −1, t)Q(M, 0, 0)A(M[1], l −2, t) · · ·

Q(M, 0, 0)A(M[n −1], l −n, t)B(p, l −1, c, T2)p11:Q(M, n, c)< A(p, l, t):c == 4 & t%P < 4 �−→ R(p, l, t)Q(M, 0, 0)A(p, l −1, c)Q(M, 0, 0)A(M[0], l −1, t)Q(M, 0, 0)

A(M[1], l −2, t) · · · Q(M, 0, 0)A(M[n −1], l −n, t)p12:Q(M, n, c)< A(p, l, t):c == 3 & t%P < 3 �−→ R(p, l, t)B(p, l −1, c, T2)Q(M, 0, 0)A(M[0], l −1, t)Q(M, 0, 0)

A(M[1], l −2, t) · · · Q(M, 0, 0)A(M[n −1], l −n, t)

p13: Q(M, n, c) : ∗ �−→ ε

tor pidity&bud :p14:B(p, l, t, d):l ≤ 0 �−→ εp15:B(p, l, t, d):d < −1 �−→ B(p, l −1, t, d +1)p16:B(p, l, t, d):d == −1 �−→ Q(M, 0, 0)A(p, l −1, t + P)p17:B(p, l, t, d):d > 1 �−→ B(p, l −1, t, d −1)p18:B(p, l, t, d):d == 1 & t == 3 �−→ Q(M, 0, 0)A(p, T3, t)p19:B(p, l, t, d):d == 1 & t == 5 �−→ R(p, l −1, t)p20:B(p, l, t, d):d == 1 & t == 6 �−→ Q(M, 0, 0)A(p, l −1, t)

blistering :p21: R(p, l, t) :t%P == 3 & l%10 > 0 �−→ R(p, l +9, t)

axiom : Q(M, 0, 0)A(V1, L1, 0)Q(M, 0, 0)A(V2 , L1, 0) · · · Q(M, 0)A(Vk , L1, 0)


Fig. 9. Tendency reduction process

The communication symbol Q(M, n, c) details a re-quest to the seawater module and the received re-sult of the request for deriving L-systems, where Mrepresents an array which stores candidates for theforthcoming propagation of rust that are returned bythe seawater module, and n is the number of candi-dates. The third parameter, c, indicates the formationof different types of rust in the upper layers. An apexA with a lifespan l denotes a cell that can continuallymove on the surfaces, and the parameter t denotes thetype of rust. The lifespan of an apex declines gradu-ally until it reaches zero, at which point the apex diesnaturally. When rust t is formed at vertex p, a sym-bol R(p, l, t) is introduced to outline the formation ofrust.These production rules can be divided into severalparts. The initial part specifies the initialization ofrust in the first layer and the termination rules governthe termination of the growth of the rust. Differenttypes of rust develop according to different rules, aslisted in the propagation set. Under the appropriateconditions, the triggering rules breed another typeof rust in the upper layer. The “torpidity & bud”part describes the initial manifestation of rust or anabeyance apex. Detailed production rules follow.Development of rust begins with a set of seed ver-tices V1, V2, . . . Vk, chosen randomly, and their cor-responding communication Q symbols. These seedvertices on the base metal initially transform intoferrous or ferric oxides in the first layer of rust, ac-cording to the query results returned by the seawater

module. These transformations are depicted in theproductions p1 and p2. Each derivation step beginsafter all query processes have been answered.First, the rusting module checks whether the life ofthe apex passed, according to the context-sensitiveproduction p3. The module ends the growth of theapex and replaces both the apex A and the commu-nication symbol Q by a rust node R if the lifespanequals or is less than zero. The apex is fixed at n ≤ 0when the seawater module determines that no spaceis available for new rust in the same layer. An apexshould not die immediately when no proper neigh-bors are available to grow the same type of rust, sincethe seawater environment continuously varies. A pe-riod of torpidity may be entered, with a period of |T1|derivation steps due to production p5. The constantT1 pertains to the number of currents and the fre-quency of current variation. The purpose of torpidityis to halt the growth of an apex until the immedi-ate environment changes. The symbol B(p, l, t, d)locates an apex of rust t which is torpid at vertex pover |d| derivation steps. Production p4 terminatesthe growth in a manner similar to production p3,when an apex has become torpid torpidity over mtimes.The propagation of a crack is described by produc-tions p6 and p7. A crack is generated in the stressspectrum direction with the highest intensity. Al-though many crack types exist, a crack may be as-sumed to begin and propagate in two directions. Anisotropic propagation rule, p8, explains why blistersare commonly round. The seawater module checkswhether the environment at each new rusty vertexsupports the breeding of another type of rust in theupper layer. Production p9 describes the primarygrowth of rust that does not stimulate another type ofrusting. An apex A(p, l, t) with a query Q(M, n, c)is replaced by a rust node R(p, l, t) and many newapices A’s led by query nodes Q’s in each iteration.A new apex with a higher tendency has a longer lifes-pan than those with lower tendencies.The rusting module follows different productionrules to specify each new kind of rust triggered inupper layers. Pits and cracks are triggered by pro-duction p10. Productions p11 and p12 represent thetriggering of deposits and blisters respectively.When an apex enters the torpid state, it rests for a pe-riod governed by production p15 and is then stimu-lated (by production p16) to propagate again over thesurfaces. In the triggering rules, the torpidity sym-bol B is used to sketch a bud of rust. Each bud has


a b c

d e f

Fig. 10a–f. An example of rust development: a iteration 1; b iteration 2; c iteration 3; d iteration 3+|T1|; e iteration 3+ T2;f iteration 6+ T2

a period of T2 derivation steps to prepare for germi-nation (production p17), because the upper rust doesnot necessarily appear simultaneously with the cur-rent rust. At germination, buds are replaced by nodesof different types of rust, accounted for by produc-tions p18, p19, and p20. Torpified apices and buds dienaturally (production p14).Production p21 specifies the difference between blis-ters and other types of rust. A stimulated blister grad-ually swells the surface to a height that is related tothe time since it began to develop. Finally, produc-tion p13 removes communication symbols after theyhave performed their tasks.An example is used herein to describe the relation-ship between these sets of rules. A seed vertex isinitially transformed into ferric oxide, according toproduction p2, in the first layer and then diffuses out-ward in the same layer (production p9), as shown inFig. 10a,b. The bud of a crack in the upper layer is

bred on the oxide, under appropriate environmentalconditions (production p10). The bud is present fora period (production p17) and germinates accordingto production p20, as shown in Fig. 10e. The crackingproceeds and is guided by production p6, as depictedin Fig. 10f. An apex of ferric oxide may enter a torpidstate (production p5) during propagation, remainingdormant for a while (production p15) before reprop-agating (production p16), as illustrated in Fig. 10c,d.The growth of an apex is terminated by productionp3 when its lifespan is passed.

6 Rendering

The Whitted (1980) illumination model with a sim-ple ray tracer, simulating the reflection and transmis-sion of light through rust layers, is used to renderthe appearance of metal corroded in seawater. The


color of a point P on an object’s surface is simplyapproximated by mixing the colors of various lay-ers at that point. Each layer exhibits its own mate-rial structure and optical properties. A more compactlayer allows less light, reflected from lower layers,to be transmitted, thereby affecting the final colorof the surface. More rust may accumulate at an areawith higher tendency values; the tendency distribu-tion reasonably describes the density of rust. Thecoefficient of rust transmittance is adjusted accord-ing to the relative tendency values, to simulate thetransmission of light through layers of varying densi-ties. The rust constituent of a point P, at which a rayis incident, is determined on the vertices of the tri-angle that contains that point. Different methods ofassessment are applied for different combinations ofrust constituents at three triangle vertices. For exam-ple, the boundary of deposits generally crosses themiddle of a triangle and pits are commonly aroundthe vertices. Perlin’s noise function and fractal sub-divisions with different thresholds are adopted to de-termine the rust at surface points. Accordingly, therust boundary appears to be irregular. The bottom ofthe pits and cracks usually contain black magnetite,while the surrounding surfaces show the characteris-tic red rust color of hematite. The triangle is there-fore subdivided into bottom, edge and original partsto depict the features of the pits and cracks. A sim-ple sinusoidal function is applied at the boundariesof the rust, as a bump function, to achieve the illu-sion of thickness, since the rust layers themselveshave zero thickness. The variations of the sinusoidalfunction can bump up surfaces surrounding the pitsand cracks where the roughness differs slightly fromthat of other rust. Scales, such as oxides and ferrousdeposits, on surfaces generally appear rugged. Thenormal vectors of interior points of such rust are agi-tated by noise to generate uneven surfaces. Hydrogen

Table 2. The weighting parameters of tendencies

Max Gaussian Accessibility Orientation Current Saltscurvature curvature

Fe3O4 0.17 0.17 0.17 0.17 0.17 0.15Fe2O3 0.2 0.1 0.3 −0.15 0.15 0.1Blister 0.1 0.1 0.25 0.1 0.2 0.25Deposit −0.05 0.05 −0.2 −0.25 −0.3 0.05

Pit 0.2 0.0 0.15 −0.05 −0.3 0.3Crack 0.15 0.3 0.2 0.0 0.25 0.1

Fig. 11. Rust on ferrous chains

blisters are simulated by pulling up the surface ver-tices some distance along their vertex normal. Theblisters are located by the seawater module, and thepulling distance is proportional to the lifespan of theblister, the ratio of which is predefined. In contrast toother rusts, the blister deforms only the surface ge-ometry and does not change the constituents of therust.

7 Results

This section presents several impressive results ofthe approach. As shown in Fig. 11, ferrous chains arecorroded in seawater under four periodically vary-ing ocean currents with the same speed. The transi-tion of currents is accompanied by the iteration of


12a 12b

13a 13b

14a 14b

Fig. 12. The difference be-tween a dynamic current andb steady current

Fig. 13a,b. The influence ofcurrent velocity on rusting:a fixed velocity; b varied ve-locity

Fig. 14a,b. The results of dif-ferent reduction of tendency:a reduction value of 0.01;b reduction value of 0.1

L-systems. Table 2 states the weighting coefficientsof environmental and geometric factors for varioustypes of rust. Figure 6 displays the correspondingdistributions at the beginning of the simulation.

The introduction of the dynamic current model isa characteristic feature of the proposed framework.Although the variation of currents is more importantthan any other environmental factor, the influence


15a 15b 15c

15d 15e 15f

16a 16b

Fig. 15a–f. A sequence of images corresponding to different iteration steps: a iteration 1; b iteration 5; c iteration 15; d itera-tion 30; e iteration 48; f iteration 69Fig. 16a,b. Different current schemes: a frequency of 1/3; b frequency of 1

of the velocity and the direction of currents shouldbe demonstrated. Pitting is considered to exemplifythe impact of the current on the distribution of rust.Figure 12 presents the difference between the pit dis-tributions for the dynamic current and those for the

steady current. Figure 8 displays the periodic vari-ation of ocean currents. In the case of the steadycurrent, one of those dynamic currents is continuallyapplied to the model, as the seawater can only flowin one direction. The surfaces sheltered from the cur-


Fig. 17. Rusty objects (the bottom row iszooming from the top row)

rents may contain more pits than those facing the cur-rents because pitting tends to be promoted by stag-nant conditions. Hence, few pits are observed on theleft face of the head under a steady current. More pitsare observed in the example of the dynamic current,since the left face is also sheltered from the strikingof current when the direction of current changes.Figure 13 demonstrates the influence of current ve-locity under the same scheme of current directions.The velocity of the currents first increases as the cur-rent swings from the first direction to the last, andthe velocity decreases when the current swings back.The difference is manifest only in the number of pits,and not in their distribution. Another characteristicof the proposed approach is the local variation of ten-dency distributions caused by the corrosion. In theseawater module, a reduction in tendency is definedto redistribute tendencies around the new rust. A re-action that involves a great reduction in free energy ismore likely to happen than one that does not. There-fore, rust with a high reduction value of tendencypropagates faster than that with a lower value, as il-lustrated in Fig. 14.Most L-systems use the number of iterations to por-tray the time passed in the simulation. Figure 15

shows a sequence of images with varying numbersof iterations to illustrate aging phenomenon. Further-more, a current may last for several iterative stepsbefore transitioning to the next direction (Table 1).For a specific current period, corrosion at low fre-quency is a little slower than that at high frequency.Figure 16 indicates that different frequencies yielda slight difference in the rate and distribution of cor-rosion. This method also applies to other models, asshown in Fig. 17.

8 Conclusions

A physically based model for simulating the devel-opment of metallic rust development in seawater wasproposed. Two modules responsible for the growthof rust and changes in the surrounding environmentwere elucidated, based on the framework of openL-systems. The environment determines the distribu-tions of different instances of rust, and the corrosionchanges the immediate environment. The essentialfeature of this model, besides its consideration ofenvironmental influences, is that the seawater envi-ronment can change over time. The seawater module


includes a dynamic current model to characterize theperiodic variation of ocean currents. Moreover, theseawater environment is characterized as a series oftendency distributions that represent the effects ofseawater on rusting.The sea is mysterious and complex, and the seawa-ter module is not yet an exact model. Scientists donot thoroughly understand many aspects of corrosionin seawater, or in other environments. Nevertheless,many interesting research directions are available.For instance, a relevant topic is how to develop morecomprehensive L-systems that can simulate differentforms of corrosion, such as cavitations and crevices.Furthermore, the proposed model can be modifiedfor a broad range of metallic corrosion processes andsurrounding environments.

Acknowledgements. We would like to thank the National Science Coun-cil of the Republic of China for financially supporting this researchunder Contract No. NSC 90-2213-E-009-128.

References1. Becket W, Badler NI (1990) Imperfection for realistic im-

age synthesis. J Vis Comput Anim 1(1):26–322. Blinn JF (1982) Light reflection functions for simulation of

clouds and dusty surfaces (SIGGRAPH ’82 Proc.). ComputGraph 16(3):21–29

3. Calladine CR (1986) Gaussian curvature and shell struc-tures. In: Gregory JA (ed) The mathematics of surfaces.Oxford University Press, Oxford, pp 179–196

4. Chandler KA (1985) Marine and offshore corrosion. Butter-worths, London

5. Chang YX, Shih ZC (2000) Physically-based patination forunderground objects. Comput Graph Forum 19(3):109–117

6. Craig BD (ed) (1989) Handbook of corrosion data. ASMInternational

7. Dorsey J, Edelman A, Jensen HW, Legakis J, Pedersen HK(1999) Modeling and rendering of weathered stone (SIG-GRAPH ’99 Proc.). Comput Graph 33:225–234

8. Dorsey J, Hanrahan P (1996) Modeling and rendering ofmetallic patinas (SIGGRAPH ’96 Proc.). Comput Graph30:387–396

9. Dorsey J, Pedersen HK, Hanrahan P (1996) Flow andchanges in appearance (SIGGRAPH ’96 Proc.). ComputGraph 30:411–420

10. Dorsey J, Hanrahan P (2000) Digital materials and virtualweathering. Sci Am 282(2):46–53

11. Fontana MG (1986) Corrosion engineering, 3rd edn.McGraw-Hill, New York

12. Gobron S, Chiba N (1999) 3D surface cellular automata andtheir applications. J Visual Comput Animat 10(3):143–158

13. Haase CS, Meyer GW (1992) Modeling pigmented ma-terials for realistic image synthesis. ACM Trans Graph11(4):305–335

14. Hall R (1989) Illumination and Color in Computer Gener-ated Imagery. Springer, Berlin Heidelberg New York

15. Hanan JS (1992) Parametric L-systems and their appli-cation to the modelling and visualization of plants. PhDthesis, University of Regina

16. Heckbert PS (1986) Survey of texture mapping. IEEE Com-put Graph Appl 6(11):56–67

17. Hsu SC, Wong TT (1995) Simulating dust accumulation.IEEE Comput Graph Appl 15(1):18–22

18. Jones DA (1992) Principles And Prevention of Corrosion.Macmillan, New York

19. Judd DB, Wyszecki G (1975) Color in business, science andindustry, 3rd edn. Wiley, New York

20. Lindenmayer A (1968) Mathematical models for cellularinteractions in development, Parts I and II. J Theoret Biol18:280–315

21. Mech R, Prusinkiewicz P (1996) Visual models of plantsinteracting with their environment (SIGGRAPH ’96 Proc.).Comput Graph 30:397–410

22. Mercer AD (1990) Corrosion in seawater systems. EllisHorwood, New York

23. Merillou S, Dischler JM, Ghazanfarpour D (2001) Corro-sion: simulating and rendering. In: Proceedings of GraphicsInterface, pp 167–174

24. Miller G (1994) Efficient algorithms for local and global ac-cessibility shading (SIGGRAPH ’94 Proc.). Comput Graph28:319–326

25. Newman RC, Sieradzki K (1994) Metallic corrosion. Sci-ence 263(25):1708–1709

26. Noser H, Thalmann D (1999) A rule-based interactive be-havioral animation system for humanoids. IEEE Trans Vi-sual Comput Graph 5(4):281–307

27. Parish YIH, Müller P (2001) Procedural modeling of cities(SIGGRAPH ’01 Proc.). Comput Graph 35:301–308

28. Perlin K (1985) An image synthesizer (SIGGRAPH ’85Proc.). Comput Graph 19(3):287–296

29. Prusinkiewicz P, James M, Mech R (1994) Synthetic topi-ary (SIGGRAPH ’94 Proc.). Comput Graph 28:351–358

30. Prusinkiewicz P, Lindenmayer A (1990) The algorithmicbeauty of plants. Springer, Berlin Heidelberg New York

31. Rogers TH (1971) Marine corrosion.32. Ross TK (1977) Metal corrosion. Oxford University Press,

Oxford33. Shreir LL (ed) (1976) Corrosion: metal/environment reac-

tions. Newnes-Butterworths, London34. Touloukian YS (1967) Thermophysical properties of high

temperature solid materials. Macmillan, New York35. Trethewey KR, Chamberlain J (1988) Corrosion for stu-

dents of science and engineering. Wiley, New York36. Turk G (1992) Re-tiling polygonal surfaces (SIGGRAPH

’92 Proc.). Comput Graph 26(2):55–6437. Warniers R (1998) Dirty pictures. Comput Graph World

21(6):51–6038. Whitted T (1980) An improved illumination model for

shaded display. Commun ACM 23(6):343–34939. Wong TT, Ng WY, Heng PA (1997) A geometry depen-

dent texture generation framework for simulating surfaceimperfections. In: Proceedings of Eurographics Workshopon Rendering, pp 139–150


YAO-XUN CHANG obtainedhis BS degree in applied mathe-matics from National Chung Hs-ing University, Taiwan, in 1993.He is currently a PhD candi-date in the Department of Com-puter and Information Scienceat National Chiao Tung Univer-sity, Taiwan. His research inter-ests are in the area of procedu-ral texture synthesis, virtual re-ality and non-photorealistic ren-dering.

ZEN-CHUNG SHIH wasborn on 10th February 1959,in Taipei, Taiwan, Republic ofChina. He received his BS de-gree in Computer Science fromChung-Yuan Christian Univer-sity in 1980, MS degree in1982 and PhD degree in 1985in computer science from theNational Tsing Hua University.Currently, he is a professor inthe Department of Computer andInformation Science at the Na-tional Chiao Tung University inHsinchu. His current research in-

terests include procedural texture synthesis, non-photorealisticrendering, global illumination, and virtual reality.

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Yao-XunChang, Zen-ChungShih