a review of peridynamics (pd) theory of diffusion based

Review ArticleA Review of Peridynamics (PD) Theory of DiffusionBased Problems

Migbar Assefa Zeleke 12 and Mesfin Belayneh Ageze 3

1Department of Mechanical Engineering Institute of Technology Hawassa University Hawassa Ethiopia2Department of Mechanical Engineering University of Botswana Gaborone 0061 Botswana3Center of Renewable Energy Addis Ababa Institute of Technology Addis Ababa University Addis Ababa Ethiopia

Correspondence should be addressed to Migbar Assefa Zeleke migbarassefagmailcom

Received 27 July 2021 Accepted 1 October 2021 Published 19 October 2021

Academic Editor Roberto Brighenti

Copyright copy 2021 Migbar Assefa Zeleke and Mesfin Belayneh Ageze is is an open access article distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided theoriginal work is properly cited

e study of heat conduction phenomena using peridynamic (PD) theory has a paramount significance on the development ofcomputational heat transfer is is because PD theory has got an interesting feature to deal with the inherent nonlocal nature ofheat transfer processes Since the revolutionary work on PD theory by Silling (2000) extensive investigations have been devoted toPD theoryis paper provides a survey on the recent developments of PD theory mainly focusing on diffusion based peridynamic(PD) formulation Both the bond-based and state-based PD formulations are revisited and numerical examples of two-di-mensional problems are presented

1 Introduction

Modeling and simulation of diffusion constitute an essentialprocess we usually come across in our everyday life and in anumber of industrial processes e basic and fundamentalknowledge of diffusion process is vital in the efficient uti-lization of energy in different cooling and heating processese governing mathematical equations of many physicalphenomena in engineering and science are described bylocal diffusion equations In solving these equations re-searchers employed several numerical techniques in the pastseveral years including finite element method (FEM)boundary element method (BEM) [1] finite differencemethod (FDM) and meshless methods At the continuumlevel diffusion processes are typically defined by localmodels via the famous Fourierrsquos law of heat conduction andFickrsquos law of mass transport e main drawback of the localmodels is that they ignore the nonlocal effect In fact theeffect of nonlocality cannot be ignored due to the fact thatnonlocality plays a significant role in diffusion processes likeheat conduction moisture flow electrical conduction pit-ting corrosion and hydraulic fracturing Hence it is quite

critical to develop a nonlocal model that accommodates boththe nonlocality effect and emerging discontinuities at thesame time

In this regard peridynamics (PD) is a promising andsuitable approach for dealing with the aforementionedchallenges encountered by the local models PD theory wasproposed by Silling [2] It was mainly intended to deal withsolid mechanics problems but later it was effectively ex-tended and realized to solve diffusion related problems Aninteresting breakthrough that proves that PD is beyond solidmechanics problems was presented by Gerstle et al [3] Inthis article the authors showed the applicability of PD insolving multiphysics problems like electromigration ebeauty of PD theory is that it takes into account bothnonlocal interactions and constitutive laws simultaneouslyLater Bobaru and Duangpanya [4] developed a 1D heatconduction PD formulation and then they extended it to 2Dheat conduction problems with discontinuities [5] On theother hand [6] compared and analyzed various dis-cretization schemes of nonlocal diffusion (ND) and linearPD equations by demonstrating illustrative approachesdeveloped for solving nonlocal models with both

HindawiJournal of EngineeringVolume 2021 Article ID 7782326 20 pageshttpsdoiorg10115520217782326

nonintegrable and integrable kernels In this article theauthors paid attention to conforming finite element andquadrature-based finite difference techniques in the dis-cretization of PD and ND models Following the pioneeringworks of Bobaru and Duangpanya [4 5] Agwai [7] andOterkus et al [8] pushed further the bond-based PD for-mulation developed by [4 5] to the state-based PD for-mulation e formulation developed by [7 8] is so generaland can be reduced to the bond-based formulations of [4 5]e only difference between the bond-based formulations in[4 5 7 8] is that they use different response functions(kernel functions) as briefly described by Chen and Bobaru[9]

PD formulation for saturated steady-state pressuredriven porous flow has been demonstrated by Katiyar et al[10] Later Jabakhanji and Mohtar [11] extended the work of[10] to transient moisture flow in unsaturated anisotropicand heterogeneous soils in PD framework A PD diffusionmodel by employing Greenrsquos function has been developed by[12] In this study the steady-state and transient PD Greenrsquosfunctions were derived and implemented to 2D infinite platewhich is heated by a Gauss source PD formulation of heatconduction in functionally graded materials (FGMs) hasbeen developed by Liao et al [13] e authors in this articleemployed a state-based peridynamic (PD) approach tosimulate heat conduction phenomena with insulated cracksin FGMs In the same year [14] addressed the issue oftransient heat transfer by combining the classical and PDformulations simultaneously In this study the inte-grodifferential equations have been solved by employing thespectral technique based Galerkin scheme Another con-tribution in relation to PD diffusion problems by employinga general-purpose finite element analysis software ANSYSwas from [15]e technique implemented in this article wasinteresting in that it reduced the computational time sig-nificantly due to the fact that the authors used implicit timeintegration scheme

e transient advection diffusion PD model has beenpresented by Zhao et al [16] by extending the bond-basedPD approach developed by Bobaru and Duangpanya [5]Xue et al [17] on the other hand addressed thermal contactproblems by utilizing the state-based version of peridy-namic formulation In this article the authors employedthe domain decomposition technique to solve heat transferproblems by considering the thermal flux and temperatureas the primary variables Recently [18] applied PD dif-ferential operator to study a refined bond-based PD heatconduction model by reviewing the present state-based andbond-based PD heat conduction models By extending thework of [13] Tan et al [19] performed their research onheat conduction in FGMs with discontinuities using PD bytaking into consideration the effect of surface correctionnear the crack and domain boundaries Later [20] inves-tigated the effects of interfacial transition zone propertiesand diffusivity on chloride diffusion concrete of arbitrarilydistributed aggregates using multiscale PD model Toimprove the computational precision and diminish the costof computation the authors employed multiscale dis-cretization scheme

PD diffusion model with a capability of handlingunbounded domains using accurate absorbing boundaryconditions (ABCs) has been developed by [21] In this studythe authors employed mesh-free discretization scheme incase of PD diffusion models whereas FEM has beenemployed in the case of local diffusion models Wang et al[22] very recently developed a dual horizon PD formulationto study thermal diffusion problem e Lagrangian for-mulation has been implemented in this study to develop thegoverning equations One of the benefits of this formulationis that it permits the implementation of flexible dis-cretization in the domain of interest which in turn con-tributes a lot to the reduction of computational time Yanet al [23] further implemented the BB-PD formulation tomodel a coupled chemical transport and water flow inunsaturated discontinuous and heterogeneous media In thisstudy processes like dispersion diffusion and advectionhave been considered in partially saturated porous media

e main objective of this article is to revisit recentdevelopments in PD applications that are pertinent to dif-fusion-type problems erefore it is structured as followse first section introduces and reviews PD theory ingeneral e second section is devoted to PD formulation ofdiffusion-type problems it is segmented into PD heatconduction PD electrical conduction and PD moistureflow Finally several illustrative examples of different casesare presented and the performance of PD is investigated

2 Review of Peridynamic Theory

Diffusion related PD formulation cannot be understoodwithout understanding its uniqueness as compared withclassical continuum theory In classical continuum me-chanics material particles interact with the nearby imme-diate material particles as shown in Figure 1(a) In contrastto the classical continuum theory PD material points areallowed to interact with material points within its family (Hi)at a finite distance called horizon as shown in Figure 1(b)

PD theory is a nonlocal integrodifferential mesh-freemethod without spatial derivatives [2 24] It is just thenonlocal version of classical continuum theory its formu-lation mainly depends on an integrodifferential equationunlike the classical counterpart which is based on spatialderivatives [24 25] e term peridynamics was originallycoined by Silling [2] at Sandia National Laboratories in thelate nineties is paper reformulated the continuum basedequation of motion (EOM) to integrodifferential equation todeal with spontaneous emergence and propagation of dis-continuity in solids For the PD formulation of elasticityconsider a body having a region Ω as shown in Figure 2With reference to Figure 1 the EOM for particle i at time t asproposed in [2] is given as (1) e original formulation inthis paper [2] was the bond-based PD theory where internalforces in a body are modeled as a network of pairwise in-teractions e material points interact in a pairwise mannerand are restricted to a specified neighborhood through abond e force of interaction between a pair of materialpoints is dependent on the deformation of the two pointsonly

2 Journal of Engineering

ρi eurou (i t) 1113946Hx

f uj minus ui rij1113872 1113873dVj + b[i t] (1)

where ρ is mass density u is displacement vector field f ispairwise force function (the force vector per unit square ofvolume that the material point j exerts on material point i)b is a body force density Hi is a neighborhood of particle iwith radius δ as shown in Figure 2 rij le δ is the positionvector of the nearest particle pointing to i from j and dVi isthe differential volume of j inside the horizon of i

Since its introduction PD theory has been used to solve avariety of problems Accordingly literature concerningperidynamic theory is quite exhaustive and abundant In thepast two decades several peridynamic studies pertaining toelastic deformation and fracture solids [2 26ndash28] brittlefracture [29ndash33] fatigue failure [34ndash49] and PD applicationof damage in composites [33 43 50ndash65] have been reportedStudies related to crack initiation and propagation usingperidynamics can be found in [30 66ndash69] As far as studiesrelated to plasticity viscoelasticity and viscoplasticity areconcerned the reader is advised to refer to [70ndash81] Figure 3shows the general trend in the number of articles publishedin the referenced journals Figure 4 on the other hand showsthe citation history of the first pioneering article [2] in thearea of PD theory since its first publication It is evident fromthese two figures (Figures 3 and 4) that year after year re-search outputs pertinent to PD theory increase incredibly

erefore the aim of this section is to provide anoverview on the computational aspects of PD theory withemphasis on the modeling of diffusion problems In this

regard the citation trends of the most influential and pio-neering articles in the areas of PD based diffusion models arepresented as shown in Figure 5 [4 8] Figure 5 comparedcitation trends of the bond-based PD thermal diffusiondeveloped by [4 5] and state-based PD thermal diffusionformulation of [8] Similar observations may be realizedfrom Figure 5 too

erefore this section is outlined as follows the first partprovides a review of the state-based PD formulation of heatconduction followed by electrical conduction and moistureflow e discussion in this paper mainly revisits the state-based peridynamic theory and then later we will reduce theSB equations to bond-based (BB) equations as the particularcase of state-based peridynamic theory Although thenonlocal model we propose in this review offers a betterpredictive capability of high strain gradient and fracturemechanics the scope of this review is limited to diffusionbased problems only

21 State-Based PD Formulation for Heat Conduction Ingeneral heat transfer is directly related to temperaturewhether it is radiation convection or conduction Heatconduction is the process by which temperature gradientexists within a body hence its primary objective is to de-termine temperature distribution and exchange of energywithin a body e development of numerical solutions forheat conduction problems is growing as an effective tool inthermal engineering e applicability of continuum basedheat transfer equations to bodies with discontinuities ismathematically awkward due to the fact that these equations

(a)

irij

Hi

j

δ

(b)

Figure 1 Representations of local and nonlocal interaction (a) local theory (b) nonlocal theory

Hi

rij

i

j

δ

Ω

(a)

i-321 i-2 i-1 i+1 i+2 i+3 N-1 N

Hx

Xi

Xi-3 Xi-2 Xi-1 Xi+1 Xi+2 Xi+3

δ

(b)

Figure 2 Peridynamics concept representation (a) peridynamic medium representation (b) 1D peridynamic medium representation

Journal of Engineering 3

contain temperature gradients To deal with the afore-mentioned challenges and limitations of the continuumbased heat transfer equations PD theory gives ideal solutionis section covers PD formulation of heat conductionequations e PD heat conduction equations used in thisarticle are simple and interesting in that thermal conduc-tivity coefficient of a particle is defined within its neigh-borhood at a certain finite distance called horizon [9] unlikecontinuum based Fourier theory where a particle interactsonly with its immediate vicinity Hence PD is a valuable tool

that takes into account both the nonlocality effect anddiscontinuities concurrently across the temperature andother potential fields like electric potential hydraulic po-tential and chemical potential

e present section addresses studies that are pertinentto heat transfer Gerstle et al [3] were the first to propose theanalytical and computational simulation of electromigrationthat accounts for heat transfer in a one-dimensionalproblem Later Bobaru and Duangpanya [4] introduced thebond-based PD formulation for thermal problems with

0

50

100

150

200

250

300

2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 20210 0 2 2 5 4

20 1525 32

4554

49

8299

137

167

208

272

295

187

Cita

tions

per

yea

r

Year of Publication

Figure 4 Publication history of [2] from Scopus database

0

100

200

300

400

500

600

700

2000 2003 2006 2009 2012 2015 2018 2021

1 1 2 4 7 19 20 27 33 49 66 69

119153

199

250

350388

515

599N

umbe

r of A

rtic

les

Year of Publication

Figure 3 Number of publications per year from Scopus database


evolving discontinuities e authors used a constructiveapproach to obtain the PD equations for heat transferFollowing their previous contribution [4] Bobaru andDuangpanya [5] proposed a multidimensional PD formu-lation to solve two-dimensional heat conduction problemswith discontinuities Recently the generalized state-basedPD heat transfer problem using Lagrangian approach wasdemonstrated by Oterkus et al [8] In this work the authorsdetermined the PD material parameter the micro con-ductivity by simplifying the state-based PD heat transferequation to its bond-based PD heat transfer equation eauthors also confirmed that the governing equation repre-sented the conservation of thermal energy Later Chen andBobaru [9] analyzed the behavior of PD solutions fortransient heat diffusion model and studied the convergenceproperties of the one point Gauss quadrature scheme

In PD heat conduction the interaction among materialpoints is due to the exchange of heat Hence we present thegoverning PD equations based on the one proposed byOterkus et al [8] and Agwai [7] for the sake of completenessby considering temperature as primary variable

Based on the generalized state-based PD formulationthe temperature state τ that contains the temperature dif-ference associated with each interaction of a particularmaterial point is given by

τ ilangrijrang Tj minus Ti (2)

where T is the temperature Note that states are representedby variables with underscores and the angular bracketsindicate the bond being operated on

According to [7 8] the SBPD heat flow state is written as

h h τ( 1113857 (3)

erefore the transient form of heat conduction in theframework of SBPD is expressed as [7 8]

ρCv( 1113857i_Ti 1113946

Hi

hilangrijrang minus hjlangrjirang1113872 1113873dVj + si (4)

where h qTKrijwijerefore (4) may be modified as follows

ρCv( 1113857i_Ti 1113946

Hi

qTj Kj + qT

i Ki1113872 1113873rijwijdVj + Si (5)

where si is the heat sink or source q is the classical heat fluxand Cv is the specific heat capacity

For PD heat conduction phenomena the discreteform of (5) may be expressed in the form of finite sum asfollows

there4 ρCv( 1113857i_Ti 1113944

j isin Hi

qTj Kj + qT

i Ki1113872 1113873rijwijVj

1113980radicradicradicradicradicradicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradicradicradicradicradicradic1113981minusnablamiddotqi

+si(6)

nabla middot qi⟶ 1113944jisinHi

qTj Kjrji minus qT

i Kirij1113872 1113873wijVj (7)

where Ki is shape tensor related to thermal field and it isgiven as follows

0

5

10

15

20

25

30

35

2009 2011 2013 2015 2017 2019 2021

1

10

14

19

33

23 24

13

9 8

13

9

15

20

2628

24

17

13

97

1413

16

27

23

17

Num

ber o

f Cita

tion

Year of Publication

Oterkus S Madenci E Agwai A_2014Bobaru F Duangpanya M_2010Bobaru F Duangpanya M_2012

Figure 5 Publication history comparison between authors of [4 5 8] from Scopus database


Ki 1113944j isin Hi

rij otimes rijwijVj⎡⎢⎢⎣ ⎤⎥⎥⎦

minus 1

H 1113944jisinHi

Tj minus Ti1113872 1113873rijwijVj⎡⎢⎢⎣ ⎤⎥⎥⎦

there4nablaTi Ki middot H⟶ 1113944j isin Hi


minus 1

middot 1113944jisinHi


(8)

where wij 1 rij le δ0 rij gt δ

1113896 and

nablaTi is the gradient of temperature

211 Correlations between the Classical Heat Flux and PDHeat Flow State e heat flow scalar state h contains theheat flow densities associated with all the interactions [7 8]erefore heat flow density h has units of heat flow rate pervolume square

1113946Hi

hilangrijrang minus hjlangrjirang1113872 1113873dVj (9)

Equation (9) resembles the divergence of heat flux nabla middot qwhich has units of heat flow rate per volume and is given inthe above equation

erefore the PD heat flow state can be correlated to theheat flux q en the expression that relates the heat flux tothe heat flow state has been borrowed from [7 8]

212 Bond-Based Peridynamic (PD) Heat ConductionFormulation In a bond-based peridynamic model materialpoint i can interact with all neighboring material points j inits horizon in a pairwise manner e change in temperatureat the two end points of a bond is assumed to cause the heatto flow along the central axis of the bond only Whenmaterial points interact in a pairwise manner and are re-stricted to a specified neighborhood through a bond eq (6)may be reduced as follows

ρCv( 1113857i_Ti 1113946

Hi

fhdVj + Si (10)

ρCv( 1113857i_Ti 1113946

Hi

kTj minus Ti

rij

1113888 1113889dVj (11)

fh rij t1113872 1113873 kTj minus Ti

rij

(12)

where k 1113954κVHiis micro conductivity of the connected

thermal bonds that joins point i and j as shown in Figure 2VHi

is the horizon volume of material point centered at i and1113954κ is the PD conductivity of thermal bonds between materialpoints i and j

213 Linking Peridynamic Properties with 7ose of theClassical Counterparts In order to create a relationshipbetween the PD properties and the standard materialproperties we borrow directly expressions from [8] usfor one- two- and three-dimensional analysis the PDthermal micro conductivities are expressed correspondinglyas

k 2κ

Aδ2 for(1 minus D)

k 6κπhδ3

for(2 minus D)

k 6κπhδ4

for(3 minus D)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)

where δ A κ and h are horizon cross-sectional areathermal conductivity and thickness respectively

22 State-Based PD Formulation for Electrical Conductione present section deals with the PD formulation ofelectrical conduction Articles that are pertinent to electricalconduction phenomena alone are quite limited but there arefew about the coupled form e application of PD to thefailure of dielectric solids can be found in [82] On the otherhand Prakash and Seidel [83] explained the effectiveness ofPD model in examining the piezoresistive composite ma-terials e same authors further developed an electrome-chanical PD model to predict the deformation and damageof explosive materials [84 85] Zeleke et al [86ndash88] on theother hand developed a PD formulation for thermoelectricphenomena A recent contribution of Diana and Carvell [89]employed micropolar PD (MPPD) model to solve electro-mechanical problems Very recently Zeleke et al [88]employed PD theory to study discontinuities in electric andthermal fields

erefore in this section we first describe PD electricalconduction using the generalized state-based approach ederivation of the generalized PD electrical conductionequation is established and the peridynamic variables areelucidated Afterward simplifications are made so that thebond-based PD formulation for electrical conduction couldbe developed

For electrical conduction phenomena material pointsexchange electrical current with points inside its neigh-borhood defined by the horizon In this section we derivedthe state-based PD electrical conduction equation byemploying a variational technique like Katiyar et al [10] forpressure driven porous flow Based on state-based PDformulation the potential state φ that comprises the elec-trical potential difference linked with each interaction of aparticular material point is given by

φilangrijrang Φj minusΦi (14)

where Φ is the electric potentialerefore the governing equation for electrical con-

duction in the framework of SBPD is obtained as


_ϱi 1113946Hi

Qilangrijrang minus Q

jlangrjirang1113874 1113875dVj + Ji (15)

where Qiis the electrical current flow state _ϱi is the time rate

of charge density and Ji is the charge source By extendingthe SBPD heat flow state we can write the state-basedcurrent flow state as

Q Q φ1113872 1113873 (16)

where ϕiis PD electrical potential scalar state and Q is

current flow state Q jTKrijwij

_ϱi 1113946Hi

jTj Kj + jTi Ki1113872 1113873rijwijdVj + Ji (17)

where j is classical current flux and Ji is charge sourceEquation (18) in its discrete form may be written as

there4 _ϱi 1113944j isin Hi

jTj Kj + jTi Ki1113872 1113873rijwijVj

1113980radicradicradicradicradicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradicradicradicradicradic1113981minusnablamiddotji

+Ji(18)

where (KEL)i is shape tensor related to electrical field and itis given as follows

KEL( 1113857i 1113944j isin Hi


minus 1

(19)

HEL 1113944jisinHi

Φj minusΦi1113872 1113873rijwijVj (20)

there4nablaΦi KEL( 1113857i middot HEL⟶ 1113944j isin Hi


minus 1


Φj minusΦi1113872 1113873rijwijVj⎡⎢⎢⎣ ⎤⎥⎥⎦

(21)

221 Relationship between Electrical Current Density and PDCurrent Flow State e current flow scalar state Q com-prises the current flow densities linked with all the par-ticipating material points having units of current flow perunit volume square and given as

1113946Hi


jlangrjirang1113874 1113875dVj (22)

Equation (22) resembles the divergence of electric fluxnabla middot j which has units of charge flow rate per volume and isgiven as

nabla middot ji⟶ 1113944jisinHi

jTj Kjrji minus jTi Kirij1113872 1113873wijVj (23)

By extending the PD heat flow state formulation to thecurrent flow state we may have the expression that relatesthe current flux to the current flow state as follows

Q jTKrijwij

Q jTj Kj + jTi Ki1113872 1113873rijwij(24)

222 Bond-Based Peridynamic (PD) Electrical ConductionFormulation In a bond-based peridynamic model materialpoint i can interact with all neighboring material points j inits horizon in a pairwise manner e change in electricpotential at the two points of a bond is assumed to cause theelectric current to flow along the axis of the bond only whichresults in pairwise interaction of material points erefore(19) may be reduced as follows

_ϱi 1113946Hi

fIdVj + Ji

_ϱi 1113946Hi

kE

Φj minusΦi

rij

1113888 1113889dVj

φirij Φj minusΦi

fI rij t1113872 1113873 kE

Φj minusΦi

rij

(25)

where kE 1113954κEVHiis micro conductivity of the associated

electrical bonds that connect points i and j as shown inFigure 2 VHi

is the horizon volume of material pointcentered at i and 1113954κE is the PD conductivity of electricalbonds between material points i and j

223 Linking Peridynamic Properties with 7ose of theClassical Counterparts In order to create a connectionbetween the PD properties and the classical materialproperties we directly borrow expressions from [8] andextend them to electric field us for one- two- and three-dimensional analysis the PD electrical micro conductivitiesare expressed as follows

kE 2κE

Aδ2 for(1 minus D)

kE 6κE

πhδ3 for(2 minus D)

kE 6κE


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)

where δ A κE and h are horizon cross-sectional areaelectrical conductivity and thickness respectively

23 State-Based PD Formulation for Chemical and WaterTransport In this section the PD models of chemicaltransport and water flow in their uncoupled state wererevisited In the realm of PD theory a number of scholarsaddressed the issue of diffusion processes in both saturatedand unsaturated porous media Katiyar et al [10] established


a PD model to study the steady-state water flow in saturatedporous media by taking into account the effect of hetero-geneities and discontinuities Later Jabakhanji and Mohtar[11] addressed the transient nature of moisture flow inunsaturated porous media in PD framework Very recently[23] employed PD theory to simulate a coupled chemicaltransport and water flow In this study the authorsimplemented the BB-PD theory to formulate chemicaltransport and water flow by taking into account diffusionadvection and dispersion processes in partially saturatedporous media

231 Chemical Transport in the Realm of PD Similar to thewater flow due to gravity and heat flow due to temperaturegradient diffusion of chemicals moves from high to lowpotential erefore we can describe Fickrsquos law in PDframework for chemical transport in a similar way to whatwe did for PD based Fourierrsquos equation of heat diffusion

zCi

zt _Ci 1113946

Hi

Ririj minus Rjrji1113872 1113873dVj + θm( 1113857i (27)

where

R JTmKrijwij

zCi

zt _Ci 1113946

Hi

Jm( 1113857T

j Kj + Jm( 1113857T

i Ki1113872 1113873rijwijdVj + θm( 1113857i

(28)

e discrete form of (29) may be written as

_Ci 1113944jisinHi

Jm( 1113857T

j Kj + Jm( 1113857T

i Ki1113872 1113873rijwijVj1113980radicradicradicradicradicradicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradicradicradicradicradicradic1113981minusnablamiddot Jm( )i

+ θm( 1113857i(29)

HCon 1113944jisinHi

Cj minus Ci1113872 1113873rijwijVj (30)

there4nablaCi⟶ 1113944j isin Hi


minus 1

1113944jisinHi

Cj minus Ci1113872 1113873rijwijVj⎡⎢⎢⎣ ⎤⎥⎥⎦

(31)

nabla middot Jm( 1113857i⟶ 1113944jisinHi

Jm( 1113857Tj Kjrji minus Jm( 1113857

Ti Kirij1113872 1113873wijVj (32)

zCi

zt minusnabla middot Jm + θm (33)

where Jm minusDnablaC is the flux of solute D is the diffusivity ofsolute in solvent C is the concentration of solute and θm isthe rate of generation of solute per unit volume of thesolvent

232 Bond-Based Peridynamic (PD) Chemical TransportIn a bond-based PD model point i interacts with neigh-boring material points j in its domain in a pairwise mannere change in chemical concentration at the two end pointsof a bond is assumed to cause the chemical to flow along the

axis of the bond only e pairwise interaction of materialpoints is written as follows

zCi

zt _Ci 1113946

Hi

fCdVj + θm( 1113857i

_Ci 1113946Hi

dCj minus Ci

rij

1113888 1113889dVj

(34)

e response function of diffusion is designated by fC

and expressed as

fC rij t1113872 1113873 dCj minus Ci

rij

(35)

where d 1113954DVHiis micro diffusivity of the associated

chemical bonds that connect points i and j as shown inFigure 2 VHi

is the horizon volume of material pointcentered at i and 1113954D is the PD diffusivity of chemical bondsbetween material points i and j

In order to create a linkage between the PD propertiesand the classical material properties we directly borrowexpressions from [8] and coined them as chemical transportphenomena us for one- two- and three-dimensionalanalysis the PD micro diffusivities are expressed respec-tively as follows d is defined in terms of the classical dif-fusivity D as

d 2D

Aδ2 for(1 minus D)

d 6D


d 6D


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(36)

where A h D and δ are cross-sectional area thicknessdiffusivity of solute and horizon respectively

For the examples that follow the PD heat conductionequation has been solved numerically by replacing thenonlocal integral equation (11) with finite sum

ρCv( 1113857i_Ti1113872 1113873

n 1113944

jisinHi

kTj1113872 1113873

nminus Tj1113872 1113873

n

rij

⎛⎝ ⎞⎠dVj (37)

where n signifies the number of time steps i denotes thepoint of interest and j is the point in the horizon of i Vj isthe volume subdomain related to the material point j eforward difference computational scheme has beenemployed by solving the following equation

Tn+1(i) T

n(i) +ΔtρCv( 1113857i

1113944jisinHi

kTj1113872 1113873

nminus Tj1113872 1113873

n

rij

⎛⎝ ⎞⎠dVj (38)

3 Case Studies

In this section we implemented the state-based PD ap-proach to illustrate the versatility of the PD formulation Five


illustrative examples are simulated and presented the firstexample illustrates two-dimensional heat conduction withsymmetric boundary condition e effect of nonsymmet-rical boundary condition is exemplified in the second ex-ample e third and fourth examples demonstrate thebeauty of PD theory in dealing with discontinuities Finallya single pellet of Bi2Te3 thermoelectric material has beenconsidered to compute the temperature and voltage values

Example 1 In this example square plate with its dimensionsthat are shown in Figure 6 and material properties that areshown in Table 1 has been considered e plate is subjectedto the following symmetric boundary and initial conditions

Boundary conditions

TW

2 y t1113874 1113875 0∘C

T minusW

2 y t1113874 1113875 0∘C

(39)

Initial conditions

T(x y 0) 100∘C minusL

2le xle

L

2 minus

W

2leyle

W

2 (40)

e domain in this demonstrating example is discretizedinto 20 by 20 nodal points in the x and the y directions

y

xW

T-bottom

T-top

q=0 q=0

L

Figure 6 Model geometry of square plate

Table 1 Material properties and dimensions

Geometric parameters Material propertiesLength L 2 cm Thermal conductivity κ 16WKmWidth W 2 cm Heat capacityCv(A) 1544 JKkgickness t 001 cm Density ρ(A) 7740kgm3

0102030405060708090

100

-1 -05 0 05 1

Tem

pera

ture

degC

Y-Division (cm)

PD 10 secPD 20 secPD 40 secPD 60 sec

PD 80 secPD 100 secFEM 100 sec

Figure 7 Temperature values for symmetric boundary conditions


1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

80 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

100 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

40 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

60 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

10 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

20 s

Figure 8 Temperature contours for time values of (a) 10 (b) 20 (c) 40 (d) 60 (e) 80 and (f) 100 seconds


respectively with time step of 10minus 2 seconds Figure 7 il-lustrates PD and FEM comparisons of two-dimensionaltemperature variations As presented in Figure 7 the tem-perature decreases with time and reaches its steady-statevalue Further it is found that PD results are in closeagreement with those of FEM counterpart

From Figure 8 we also noticed a decrease in temperatureinside the plate as the time goes by e temperature in theplate was initially 100 oC and 0oC was imposed at thebottom and top boundaries Figure 8 illustrates the tem-perature contour plot for time t 10 s t 20 s 40 s 60 s 80 sand 100 s

Example 2 Heat conduction with nonsymmetric boundaryIn this example a temperature of 0degC was enforced at the

bottom of the plate and 300degC at the top and the temper-ature on the rest of the plate was subjected to initial value of100degC as shown below

Initial conditions


2lexle

L

2 minus

W

2leyle

W

2(41)

Boundary conditions

TW

2 y t1113874 1113875 300∘C (Top)

T minusW

2 y t1113874 1113875 0∘C(Bottom)

(42)

e temperature field across the plate is illustrated inFigure 9 e temperature distribution inside the plate be-comes closer to linear distribution as simulation time in-creases In this example we also compared temperaturevalues from PD and FEM at t 100 s As can be seen fromFigure 9 temperature distributions tend to be interestinglycloser Further Figure 10 depicts the temperature contour ofthe plate up to 100-second simulation time erefore fromthe above two examples we may draw the conclusion thatPD theory is an interesting theory that can deal with dif-fusion problems correctly

Example 3 Adiabatic crack with constant heat flux per-pendicular to crack surface

To validate the proposed PD method in handling dis-continuities we considered here an inclined crack withadiabatic inclined crack as shown in Figure 11 In this ex-ample the dimensions are the same as the previous examplesand β 45deg Equal and opposite magnitude of temperaturehas been imposed on the top and bottom edge of the plate inorder to keep the heat flux constant In themeantime the leftand right edges of the plate are heat-insulated e inclinedcrack is modeled as an adiabatic crack in order to capture thediscontinuity in temperature profile using PD theory asshown in Figure 12

Figure 12 depicts the comparison between PD resultsand FEM in the case of adiabatic inclined crack Figure 12proves that our PD solution agrees well with the solutionfrom FEM Hence our proposed peridynamic model is able

-1 -08 -06 -04 -02 0 02 04 06 08 10

50

100

150

200

250

300

Tem

pera

ture

degCY-Division (cm)

PD t=25 secPD t=5 secPD t=10 sec

PD t=20 secFEM t=100secPD t=100sec

Figure 9 Temperature values for nonsymmetric boundary conditions


1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

10 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

20 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

25 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

5 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

40 s

1

05

0

-05

-1

Y

-1 -05 0X

100 s

05 1

Temperature3002852702552402252101951801651501351201059075604530150



to capture the transient heat conduction of a plate with aninsulated inclined crack

Further Figure 13 illustrates the PD and FEM tem-perature variations along the Y-axis for t 100 s As we cansee from Figure 13 PD and FEM results are in closeagreement

Example 4 Isothermal crack with specified temperature atthe crack surface

is example considers an inclined crack with isother-mal crack condition as shown in Figure 14 Once again thedimensions are the same as the previous examples andβ 45deg Here the temperature values of equal magnitude arespecified at the four edges of the plate and an essentialboundary condition is defined on the crack surface For thisset of boundaries we choose T2 gtT1

Similar to Example 3 we compared our PD results withthose of FEM It is also observed from Figures 15 and 16 thatour solution is consistent with solution from FEM

Example 5 Single pellet of bismuth telluride (Bi2Te3)e main target of this example is to show the effec-

tiveness of PD theory in dealing with couple fields Acomparison between PD solution and results from literature[90] has been considered by taking into account constantmaterial properties and temperature dependent materialproperties of bismuth telluride (Bi2Te3)

Case 1 Constant material propertiesMaterial properties and model geometry are given in

Table 2 and Figure 17 [90] respectively e problem

y

xW

T-top

q=0

q=0q=0

q=0

T-bottomL

β=45deg2a

Figure 11 Model geometry and boundary conditions for a square plate with adiabatic inclined crack

1

05

0

-05

-1

Y

-1 -05 0X

05 1

100833333666667503333331666672E-08-166667-333333-50-666667-833333-100

temperature

(a)

+1000e+02NT11

+8333e+01+6667e+01+5000e+01+3333e+01+1667e+01-1144e-05-1667e+01-3333e+01-5000e+01-6667e+01-8333e+01-1000e+02

(b)

Figure 12 A comparison of temperature contour in case of adiabatic inclined crack when t 100 s and at X 0 or (L2) (a) PD (b) FEM


considered in this example is enforced as one-dimensionallinear problem [90]

Temperature and voltage boundary conditions areenforced as

T(0 t) 273∘K

T(L t) 298∘K

V(L) 0v

(43)

Considering constant material properties temperatureresults from PD simulation have been obtained and com-pared with results from [90] as depicted in Figure 18 FromFigure 18 it is observed that PD results quite agree withthose from [90]

A further study on electric potential values using PD hasbeen conducted Simulation results from PD and thoseof from [90] have been compared in Figure 19 It is clear

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 0005 001 0015 002

Tem

pera

ture

degC

Width of Plate (cm)

PD t=100 secFEM t=100 sec

Figure 13 Comparison of PD and FEM solutions at t 100 s

y

xW

T2

T2

T2T2

T1

T1

L

β=45deg2a

Figure 14 Square plate domain with an inclined crack isothermal crack T2gtT1


from Figure 19 that our PD results agreed well with thosefrom [90]

Case 2 Temperature dependence of material propertiesTable 3 depicts the dimensions of model geometry and

material properties [90] Similar to Case 1 temperature andelectric potential values have been computed using PDBoundary conditions and dimensions are similar to Case 1

Here we also made a comparison between PDtemperature values and those from [90] As can be seenfrom Figure 20 PD results smoothly agree with thosefrom [90]

Temperature dependent electric potential values usingPD have been computed and compared with results from[90] It is quite evident from Figure 21 that PD results agreevery well with those from [90]

1

05

0

-05

-1

Y

-1 -05 0X

05 1

0-208333-416667-625-833333-104167-125-145833-166667-1875-208333-229167-250

temperature

(a)

+0000e+00

-2083e+01

-4167e+01

-6250e+01

-8333e+01

-1042e+02

-1250e-02

-1458e+02

-1667e+02

-1875e+02

-2083e+02

-2292e+02

-2500e+02

NT11

(b)

Figure 15 Temperature contour in case of isothermal inclined crack at t 100 s (a) PD solution (b) FEM solution

Plate width (cm)-250

-200

-150

-100

-50

0-1 -08 -06 -04 -02 0 02 04 06 08 1

Tem

pera

ture

degC

FEM t=100 secPD t=100 sec

Figure 16 Temperature in case of isothermal inclined crack at t 100 s along X 0 or Y 0

Table 2 Geometric dimensions and material properties [90]

Geometric parameters Material propertiesLength L 1524mm α 1849 times 10minus 4 vKWidth W 14mm κ 1701Wkm


xL

W

y

T0

V0

TL

VL

jx

qx

Figure 17 Model geometry and boundary conditions [90]

270

275

280

285

290

295

300

0 02 04 06 08 1 12 14 16Horizontal distance (mm)

Tem

pera

ture

(degK)

PDRef

Figure 18 Comparison of temperature values from PD and [90]

00E+0

50E-4

10E-3

15E-3

20E-3

25E-3

30E-3

35E-3

40E-3

45E-3

50E-3

Elec

tric

pot

entia

l (V

)


PDRef

Figure 19 Comparison of potential values from PD and [90]


4 Conclusion

is article revisited the PD computational scheme that ispertinent to diffusion based problems like heat diffusionelectrical conduction and chemical transport In this reviewPD functional integrals play a vital role in replacing

gradients of hydraulic potentials solute concentrationtemperature and electrical conduction is is becausefunctional integrals are valid anywhere in the domain de-spite the presence of discontinuities Here the state-basedPD diffusion equation developed by [7 8] has been bor-rowed to write electrical conduction and chemical transport

270

275

280

285

290

295

300

Tem

pera

ture

(degK)


PDRef

Figure 20 Comparison of temperature values in case of temperature dependent material properties from PD and [90]

00E+0

50E-4

10E-3

15E-3

20E-3

25E-3

30E-3

35E-3

40E-3

45E-3

50E-3

Elec

tric

pot

entia

l (V

)


PDRef

Figure 21 Comparison of potential values in case of temperature dependent material properties from PD and [90]

Table 3 Dimensions and material properties (temperature dependent)

Geometric parameters Material propertiesLength L 1524mm α 1804 times 10minus 4 + 3598 times 10minus 7(T minus 273)

Width W 14mm κ 1754 minus 4260 times 10minus 3(T minus 273)

T is in Kelvin


equations Five examples have been demonstrated to showthe versatility of the PD theory and the results werecompared with results from [90] and FEM Temperaturevalues inside a square plate have been determined in the firstexample by considering symmetric temperature boundaryIn the second example nonsymmetric temperatureboundary has been considered and results were presentedIn both examples the PD results have been compared withFEM results and close agreement has been obtained In thethird and fourth demonstrations we proved the competenceof PD in handling discontinuities In these examples resultsfrom PD and FEM have been also compared and found to beinteresting Finally we solved one-dimensional thermo-electric phenomenon by comparing PD results with thosefrom [90] and proved the capability of PD to take care ofcoupled fields Hence we may conclude that PD theory isway beyond solving fracture and solid mechanics problemsIt is also versatile in dealing with diffusion based models andtheir coupled fields

Conflicts of Interest

e authors wish to confirm that there are no knownconflicts of interest associated with this publication

References

[1] M I Azis ldquoStandard-BEM solutions to two types of aniso-tropic-diffusion convection reaction equations with variablecoefficientsrdquo Engineering Analysis with Boundary Elementsvol 105 pp 87ndash93 2019

[2] S A Silling ldquoReformulation of elasticity theory for discon-tinuities and long-range forcesrdquo Journal of the Mechanics andPhysics of Solids vol 48 no 1 pp 175ndash209 2000

[3] W Gerstle and A Silling D Read V Tewary and R LehoucqldquoPeridynamic simulation of electromigrationrdquo ComputersMaterials amp Continua vol 8 no 2 pp 75ndash92 2008

[4] F Bobaru and M Duangpanya ldquoe peridynamic formula-tion for transient heat conductionrdquo International Journal ofHeat and Mass Transfer vol 53 no 19-20 pp 4047ndash40592010

[5] F Bobaru and M Duangpanya ldquoA peridynamic formulationfor transient heat conduction in bodies with evolving dis-continuitiesrdquo Journal of Computational Physics vol 231no 7 pp 2764ndash2785 2012

[6] X Tian and Q Du ldquoAnalysis and comparison of differentapproximations to nonlocal diffusion and linear peridynamicequationsrdquo SIAM Journal on Numerical Analysis vol 51no 6 pp 3458ndash3482 2013

[7] A Agwai Peridynamic Approach for Coupled Fields Uni-versity of Arizona Tucson AZ USA 2011

[8] S Oterkus E Madenci and A Agwai ldquoPeridynamic thermaldiffusionrdquo Journal of Computational Physics vol 265pp 71ndash96 2014

[9] Z Chen and F Bobaru ldquoSelecting the kernel in a peridynamicformulationA study for transient heat diffusionrdquo ComputerPhysics Communications 2015 In Press

[10] A Katiyar J T Foster H Ouchi and M M Sharma ldquoAperidynamic formulation of pressure driven convective fluidtransport in porous mediardquo Journal of Computational Physicsvol 261 pp 209ndash229 2014

[11] R Jabakhanji and R H Mohtar ldquoA peridynamic model offlow in porous mediardquo Advances in Water Resources vol 78pp 22ndash35 2015

[12] L J Wang J F Xu and J XWang ldquoeGreenrsquos functions forperidynamic non-local diffusionrdquo Proceedings of the RoyalSociety A Mathematical Physical amp Engineering Sciencesvol 472 no 2193 Article ID 20160185 2016

[13] Y Liao L Liu Q Liu X Lai M Assefa and J Liu ldquoPeri-dynamic simulation of transient heat conduction problems infunctionally gradient materials with cracksrdquo Journal of7ermal Stresses vol 40 no 12 pp 1484ndash1501 2017

[14] A Jafari R Bahaaddini and H Jahanbakhsh ldquoNumericalanalysis of peridynamic and classical models in transient heattransfer employing Galerkin approachrdquo Heat Transfer-AsianResearch vol 47 no 3 pp 531ndash555 2017

[15] C Diyaroglu S Oterkus E Oterkus and E MadencildquoPeridynamic modeling of diffusion by using finite-elementanalysisrdquo IEEE Transactions on Components Packaging andManufacturing Technology vol 7 no 11 pp 1823ndash1831 2017

[16] J Zhao Z Chen J Mehrmashhadi and F Bobaru ldquoCon-struction of a peridynamic model for transient advection-diffusion problemsrdquo International Journal of Heat and MassTransfer vol 126 pp 1253ndash1266 2018

[17] T Xue X Zhang and K K Tamma ldquoA two-field state-basedperidynamic theory for thermal contact problemsrdquo Journal ofComputational Physics vol 374 pp 1180ndash1195 2018

[18] X Gu Q Zhang and E Madenci ldquoRefined bond-basedperidynamics for thermal diffusionrdquo Engineering Computa-tions vol 36 no 8 pp 2557ndash2587 2019

[19] Y Tan Q Liu L Zhang L Liu and X Lai ldquoPeridynamicsmodel with surface correction near insulated cracks fortransient heat conduction in functionally graded materialsrdquoMaterials vol 13 no 6 p 1340 2020

[20] L Guo X Zhang W Li and X Zhou ldquoMulti-scale peridy-namic formulations for chloride diffusion in concreterdquo En-gineering Analysis with Boundary Elements vol 120pp 107ndash117 2020

[21] A Shojaei A Hermann P Seleson and C J CyronldquoDirichlet absorbing boundary conditions for classical andperidynamic diffusion-type modelsrdquo Computational Me-chanics vol 66 no 4 pp 773ndash793 2020

[22] B Wang S Oterkus and E Oterkus ldquoermal diffusionanalysis by using dual horizon peridynamicsrdquo Journal of7ermal Stresses vol 44 no 1 pp 51ndash74 2021

[23] H Yan M Sedighi and A P Jivkov ldquoPeridynamics mod-elling of coupled water flow and chemical transport in un-saturated porous mediardquo Journal of Hydrology vol 591Article ID 125648 2020

[24] S A Silling M Zimmermann and R Abeyaratne ldquoDefor-mation of a peridynamic barrdquo Journal of Elasticity vol 73no 1-3 pp 173ndash190 2003

[25] I A Kunin Elastic Media with Microstructure I One Di-mensional Models Springer-Verlag Berlin Germany 1982

[26] T L Warren ldquoA non-ordinary state-based peridynamicmethod to model solid material deformation and fracturerdquoInternational Journal of Solids and Structures vol 46pp 1186ndash1195 2009

[27] S A Silling M Epton O Weckner J Xu and E AskarildquoPeridynamic states and constitutive modellingrdquo Journal ofElasticity vol 88 no 2 pp 151ndash184 2007

[28] J OrsquoGrady and J Foster ldquoPeridynamic beams a non-ordi-nary state-based modelrdquo International Journal of Solids andStructures vol 51 no 18 pp 3177ndash3183 2014


[29] C Xin ldquoA non-ordinary state based peridynamic modeling offractures in quasi-brittle materialsrdquo International Journal ofImpact Engineering vol 111 pp 130ndash146 2018

[30] Y Ha and F Bobaru ldquoCharacteristics of dynamic brittlefracture captured with peridynamicsrdquo Engineering FractureMechanics vol 78 pp 1156ndash1168 2011

[31] F Li J Pan and C Sinka ldquoModelling brittle impact failure ofdisc particles using material point methodrdquo InternationalJournal of Impact Engineering vol 38 pp 653ndash660 2011

[32] W Liu and J W Hong ldquoDiscretized peridynamics for brittleand ductile solidsrdquo International Journal for NumericalMethods in Engineering vol 89 pp 1028ndash1046 2012

[33] E Postek T Sadowski and M Boniecki ldquoImpact of brittlecomposites peridynamics modellingrdquo Materials TodayProceedings vol 45 pp 4268ndash4274 2021

[34] G Zhang and F Bobaru ldquoModeling the evolution of fatiguefailure with peridynamicsrdquo 7e Romanian Journal of Tech-nical Sciences and Applied Mechanics vol 61 no 1 pp 22ndash402016

[35] G Zhang Q Le A Loghin A Subramaniyan and F BobaruldquoValidation of a peridynamic model for fatigue crackingrdquoEngineering Fracture Mechanics vol 162 pp 76ndash94 2016

[36] Y L Hu and E Madenci ldquoPeridynamics for fatigue life andresidual strength prediction of composite laminatesrdquo Com-posite Structures vol 160 pp 169ndash184 2017

[37] J Jung and J Seok ldquoMixed-mode fatigue crack growthanalysis using peridynamic approachrdquo International Journalof Fatigue vol 103 pp 591ndash603 2017

[38] F Wang Y E Ma Y Guo andW Huang ldquoStudies on quasi-static and fatigue crack propagation behaviours in friction stirwelded joints using peridynamic theoryrdquo Advances in Ma-terials Science and Engineering vol 2019 Article ID 510561216 pages 2019

[39] S Bazazzadeh M Zaccariotto and U Galvanetto ldquoFatiguedegradation strategies to simulate crack propagation usingperidynamic based computational methodsrdquo Latin AmericanJournal of Solids and Structures vol 16 no 2 2019

[40] L I U Binchao B A O Rui and S U I Fucheng ldquoA fatiguedamage-cumulative model in peridynamicsrdquo Chinese Journalof Aeronautics vol 34 no 2 pp 329ndash342 2021

[41] E Oterkus I Guven and E Madenci ldquoFatigue failure modelwith peridynamic theoryrdquo in Proceedings of the 2010 12thIEEE Intersociety Conference on 7ermal and 7ermo-mechanical Phenomena in Electronic Systems pp 1ndash6 LasVegas NV USA June 2010

[42] S A Silling and A Askari ldquoPeridynamic model for fatiguecrackingrdquo University of Nebraska Lincoln Nebraska 2014

[43] E Madenci ldquoCombined peridynamics and kinetic theory offracture for fatigue failure of composites under constant andvariable amplitude loadingrdquo7eoretical and Applied FractureMechanics vol 112 2021

[44] T Nguyen S Oterkus and E Oterkus ldquoAn energy-basedperidynamic model for fatigue crackingrdquo Engineering Frac-ture Mechanics vol 241 Article ID 107373 2021

[45] J Han and W Chen ldquoAn ordinary state-based peridynamicmodel for fatigue cracking of ferrite and pearlite wheel ma-terialrdquo Applied Sciences vol 10 no 12 p 4325 2020

[46] N Zhu C Kochan E Oterkus and S Oterkus ldquoFatigueanalysis of polycrystalline materials using Peridynamic e-ory with a novel crack tip detection algorithmrdquo Ocean En-gineering vol 222 Article ID 108572 2021

[47] X Ma ldquoA 2D peridynamic model for fatigue crack initiationof railheadsrdquo International Journal of Fatigue vol 135 ArticleID 105536 2020

[48] Y Liu L Deng W Zhong J Xu and W Xiong ldquoA newfatigue reliability analysis method for steel bridges based onperidynamic theoryrdquo Engineering Fracture Mechanicsvol 236 Article ID 107214 2020

[49] F Baber and I Guven ldquoSolder joint fatigue life predictionusing peridynamic approachrdquo Microelectronics Reliabilityvol 79 pp 20ndash31 2017

[50] Y Yu and H Wang ldquoPeridynamic analytical method forprogressive damage in notched composite laminatesrdquo Com-posite Structures vol 108 pp 801ndash810 2014

[51] Y Zhang and P Qiao ldquoA fully-discrete peridynamic modelingapproach for tensile fracture of fiber-reinforced cementitiouscompositesrdquo Engineering Fracture Mechanics vol 242 ArticleID 107454 2021

[52] A Jenabidehkordi R Abadi and T Rabczuk ldquoComputationalmodeling of meso-scale fracture in polymer matrix com-posites employing peridynamicsrdquo Composite Structuresvol 253 Article ID 112740 2020

[53] B M Baykan U Yolum E Ozaslan M A Guler andB Yıldırım ldquoFailure prediction of composite open hole tensiletest specimens using bond based peridynamic theoryrdquo Pro-cedia Structural Integrity vol 28 pp 2055ndash2064 2020

[54] Y L Hu Y Yu and E Madenci ldquoPeridynamic modeling ofcomposite laminates with material coupling and transverseshear deformationrdquo Composite Structures vol 253 Article ID112760 2020

[55] C Mitts S Naboulsi C Przybyla and E Madenci ldquoAxi-symmetric peridynamic analysis of crack deflection in a singlestrand ceramic matrix compositerdquo Engineering FractureMechanics vol 235 Article ID 107074 2020

[56] E Gok U Yolum and M A Guler ldquoMode II and mixedmode delamination growth in composite materials usingperidynamic theoryrdquo Procedia Structural Integrity vol 28pp 2043ndash2054 2020

[57] W Zhou D Liu and N Liu ldquoAnalyzing dynamic fractureprocess in fiber-reinforced composite materials with a peri-dynamic modelrdquo Engineering Fracture Mechanics vol 178pp 60ndash76 2017

[58] W Hu Y D Ha and F Bobaru ldquoModeling dynamic fractureand damage in fiber-reinforced composites with peridy-namicsrdquo International Journal for Multiscale ComputationalEngineering vol 9 pp 707ndash726 2011

[59] M Radel C Willberg and D Krause ldquoPeridynamic analysisof fibre-matrix debond and matrix failure mechanisms incomposites under transverse tensile load by an energy-baseddamage criterionrdquo Composites Part B Engineering vol 158pp 18ndash27 2019

[60] Y L Hu and E Madenci ldquoBond-based peridynamic modelingof composite laminates with arbitrary fiber orientation andstacking sequencerdquo Composite Structures vol 153 pp 139ndash175 2016

[61] C Sun and Z Huang ldquoPeridynamic simulation to impactingdamage in composite laminaterdquo Composite Structuresvol 138 pp 335ndash341 2016

[62] B Ren C T Wu P Seleson D Zeng and D Lyu ldquoAperidynamic failure analysis of fiber-reinforced compositelaminates using finite element discontinuous Galerkin ap-proximationsrdquo International Journal of Fracture vol 214no 1 pp 49ndash68 2018

[63] E Askari ldquoPeridynamics for multiscale materials modelingrdquoJournal of Physics vol 125 pp 1ndash11 2008

[64] B Kilic and E Madenci ldquoPrediction of crack paths in aquenched glass plate by using peridynamic theoryrdquo Inter-national Journal of Fracture vol 156 pp 165ndash177 2009


[65] E Oterkus and E Madenci ldquoPeridynamic theory for damageinitiation and growth in composite laminaterdquo Key Engi-neering Materials vol 488 pp 355ndash358 2012

[66] S A Silling O Weckner E Askari and F Bobaru ldquoCracknucleation in a peridynamic solidrdquo International Journal ofFracture vol 162 no 1-2 pp 219ndash227 2010

[67] Y Ha and F Bobaru ldquoStudies of dynamic crack propagationand crack branching with peridynamicsrdquo InternationalJournal of Fracture vol 162 no 1-2 pp 229ndash244 2010

[68] A Agwai I Guven and E Madenci ldquoPredicting crackpropagation with peridynamics a comparative studyrdquo In-ternational Journal of Fracture vol 171 no 1 pp 65ndash78 2011

[69] Z Cheng G Zhang Y Wang and F Bobaru ldquoA peridynamicmodel for dynamic fracture in functionally graded materialsrdquoComposite Structures vol 133 pp 529ndash546 2015

[70] J T Foster S A Silling and W W Chen ldquoViscoplasticityusing peridynamicsrdquo International Journal for NumericalMethods in Engineering vol 81 pp 1242ndash1258 2010

[71] A Lakshmanan J Luo I Javaheri and V Sundararaghavanldquoree-dimensional crystal plasticity simulations using per-idynamics theory and experimental comparisonrdquo Interna-tional Journal of Plasticity vol 142 Article ID 102991 2021

[72] A Pathrikar M M Rahaman and D Roy ldquoA thermody-namically consistent peridynamics model for visco-plasticityand damagerdquo Computer Methods in Applied Mechanics andEngineering vol 348 pp 29ndash63 2019

[73] Y D Ha ldquoDynamic fracture analysis of high-speed impact ongranite with peridynamic plasticityrdquo Journal of the Compu-tational Structural Engineering Institute of Korea vol 32no 1 pp 37ndash44 2019

[74] O Weckner and N A N Mohamed ldquoViscoelastic materialmodels in peridynamicsrdquo Applied Mathematics and Com-putation vol 219 no 11 pp 6039ndash6043 2013

[75] R Delorme I Tabiai L L Lebel and M Levesque ldquoGen-eralization of the ordinary state-based peridynamic model forisotropic linear viscoelasticityrdquo Mechanics of Time-dependentMaterials vol 21 no 4 pp 549ndash575 2017

[76] E Madenci and S Oterkus ldquoOrdinary state-based peridy-namics for thermoviscoelastic deformationrdquo EngineeringFracture Mechanics vol 175 pp 31ndash45 2017

[77] L Wu D Huang and F Bobaru ldquoA reformulated rate-de-pendent visco-elastic model for dynamic deformation andfracture of PMMA with peridynamicsrdquo International Journalof Impact Engineering vol 149 Article ID 103791 2021

[78] D Behera P Roy and E Madenci ldquoPeridynamic modeling ofbonded-lap joints with viscoelastic adhesives in the presenceof finite deformationrdquo Computer Methods in Applied Me-chanics and Engineering vol 374 Article ID 113584 2021

[79] J A Mitchell ldquoA nonlocal ordinary state-based plasticitymodel for peridynamicsrdquo Sandia National LaboratoriesAlbuquerque NM USA pp SAND2011ndashSAND3166 2011

[80] J A Mitchell A Non-local Ordinary-State-Based Viscoelas-ticity Model for Peridynamics pp SAND2011ndashSAND8064Sandia National Laboratories Albuquerque NM USA 2011

[81] S Sun and V Sundararaghavan ldquoA peridynamic imple-mentation of crystal plasticityrdquo International Journal of Solidsand Structures vol 51 pp 3350ndash3360 2014

[82] A Raymond S Wildman and A G George ldquoA Dynamicelectro-thermo-mechanical model of dielectric breakdown insolids using PERIDYNAMICSrdquo Journal of Mechanics ofMaterials and Structures vol 10 no 5 pp 613ndash630 2015

[83] N Prakash and G D Seidel ldquoElectromechanical peridy-namics modeling of piezoresistive response of carbon

nanotube nanocompositesrdquo Computational Materials Sciencevol 113 pp 154ndash170 2016

[84] N Prakash and G D Seidel ldquoComputational electrome-chanical peridynamics modeling of strain and damage sensingin nanocomposite bonded explosive materials (NCBX)rdquoEngineering Fracture Mechanics vol 177 pp 180ndash202 2017

[85] N Prakash and G D Seidel ldquoEffects of microscale damageevolution on piezoresistive sensing in nanocomposite bondedexplosives under dynamic loading via electromechanicalperidynamicsrdquoModelling and Simulation in Materials Scienceand Engineering vol 26 no 1 Article ID 015003 2017

[86] A Migbar L Xin L Lisheng and L Yang ldquoPeridynamicformulation for coupled thermoelectric phenomenardquo Ad-vances in Materials Science and Engineering vol 2017 ArticleID 9836741 10 pages 2017

[87] A Migbar L Xin and L Liu ldquoBond based peridynamicformulation for thermoelectric materialsrdquo Materials ScienceForum vol 883 pp 51ndash59 2016

[88] M A Zeleke X Lai and L Liu ldquoA peridynamic computa-tional scheme for thermoelectric fieldsrdquo Materials vol 13no 11 p 2546 2020

[89] V Diana and V Carvell ldquoAn electromechanical micropolarperidynamic modelrdquo Computer Methods in Applied Me-chanics and Engineering vol 365 Article ID 112998 2020

[90] B L Wang ldquoA finite element computational scheme fortransient and nonlinear coupling thermoelectric fields and theassociated thermal stresses in thermoelectric materialsrdquoApplied 7ermal Engineering vol 110 no 5 pp 136ndash1432017


nonintegrable and integrable kernels In this article theauthors paid attention to conforming finite element andquadrature-based finite difference techniques in the dis-cretization of PD and ND models Following the pioneeringworks of Bobaru and Duangpanya [4 5] Agwai [7] andOterkus et al [8] pushed further the bond-based PD for-mulation developed by [4 5] to the state-based PD for-mulation e formulation developed by [7 8] is so generaland can be reduced to the bond-based formulations of [4 5]e only difference between the bond-based formulations in[4 5 7 8] is that they use different response functions(kernel functions) as briefly described by Chen and Bobaru[9]

PD formulation for saturated steady-state pressuredriven porous flow has been demonstrated by Katiyar et al[10] Later Jabakhanji and Mohtar [11] extended the work of[10] to transient moisture flow in unsaturated anisotropicand heterogeneous soils in PD framework A PD diffusionmodel by employing Greenrsquos function has been developed by[12] In this study the steady-state and transient PD Greenrsquosfunctions were derived and implemented to 2D infinite platewhich is heated by a Gauss source PD formulation of heatconduction in functionally graded materials (FGMs) hasbeen developed by Liao et al [13] e authors in this articleemployed a state-based peridynamic (PD) approach tosimulate heat conduction phenomena with insulated cracksin FGMs In the same year [14] addressed the issue oftransient heat transfer by combining the classical and PDformulations simultaneously In this study the inte-grodifferential equations have been solved by employing thespectral technique based Galerkin scheme Another con-tribution in relation to PD diffusion problems by employinga general-purpose finite element analysis software ANSYSwas from [15]e technique implemented in this article wasinteresting in that it reduced the computational time sig-nificantly due to the fact that the authors used implicit timeintegration scheme

e transient advection diffusion PD model has beenpresented by Zhao et al [16] by extending the bond-basedPD approach developed by Bobaru and Duangpanya [5]Xue et al [17] on the other hand addressed thermal contactproblems by utilizing the state-based version of peridy-namic formulation In this article the authors employedthe domain decomposition technique to solve heat transferproblems by considering the thermal flux and temperatureas the primary variables Recently [18] applied PD dif-ferential operator to study a refined bond-based PD heatconduction model by reviewing the present state-based andbond-based PD heat conduction models By extending thework of [13] Tan et al [19] performed their research onheat conduction in FGMs with discontinuities using PD bytaking into consideration the effect of surface correctionnear the crack and domain boundaries Later [20] inves-tigated the effects of interfacial transition zone propertiesand diffusivity on chloride diffusion concrete of arbitrarilydistributed aggregates using multiscale PD model Toimprove the computational precision and diminish the costof computation the authors employed multiscale dis-cretization scheme

PD diffusion model with a capability of handlingunbounded domains using accurate absorbing boundaryconditions (ABCs) has been developed by [21] In this studythe authors employed mesh-free discretization scheme incase of PD diffusion models whereas FEM has beenemployed in the case of local diffusion models Wang et al[22] very recently developed a dual horizon PD formulationto study thermal diffusion problem e Lagrangian for-mulation has been implemented in this study to develop thegoverning equations One of the benefits of this formulationis that it permits the implementation of flexible dis-cretization in the domain of interest which in turn con-tributes a lot to the reduction of computational time Yanet al [23] further implemented the BB-PD formulation tomodel a coupled chemical transport and water flow inunsaturated discontinuous and heterogeneous media In thisstudy processes like dispersion diffusion and advectionhave been considered in partially saturated porous media

e main objective of this article is to revisit recentdevelopments in PD applications that are pertinent to dif-fusion-type problems erefore it is structured as followse first section introduces and reviews PD theory ingeneral e second section is devoted to PD formulation ofdiffusion-type problems it is segmented into PD heatconduction PD electrical conduction and PD moistureflow Finally several illustrative examples of different casesare presented and the performance of PD is investigated

2 Review of Peridynamic Theory

Diffusion related PD formulation cannot be understoodwithout understanding its uniqueness as compared withclassical continuum theory In classical continuum me-chanics material particles interact with the nearby imme-diate material particles as shown in Figure 1(a) In contrastto the classical continuum theory PD material points areallowed to interact with material points within its family (Hi)at a finite distance called horizon as shown in Figure 1(b)

PD theory is a nonlocal integrodifferential mesh-freemethod without spatial derivatives [2 24] It is just thenonlocal version of classical continuum theory its formu-lation mainly depends on an integrodifferential equationunlike the classical counterpart which is based on spatialderivatives [24 25] e term peridynamics was originallycoined by Silling [2] at Sandia National Laboratories in thelate nineties is paper reformulated the continuum basedequation of motion (EOM) to integrodifferential equation todeal with spontaneous emergence and propagation of dis-continuity in solids For the PD formulation of elasticityconsider a body having a region Ω as shown in Figure 2With reference to Figure 1 the EOM for particle i at time t asproposed in [2] is given as (1) e original formulation inthis paper [2] was the bond-based PD theory where internalforces in a body are modeled as a network of pairwise in-teractions e material points interact in a pairwise mannerand are restricted to a specified neighborhood through abond e force of interaction between a pair of materialpoints is dependent on the deformation of the two pointsonly










(a)

irij

Hi

j

δ

(b)


Hi

rij

i

j

δ

Ω

(a)

i-321 i-2 i-1 i+1 i+2 i+3 N-1 N

Hx

Xi


δ

(b)






0

50

100

150

200

250

300

2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 20210 0 2 2 5 4

20 1525 32

4554

49

8299

137

167

208

272

295

187

Cita

tions

per

yea

r

Year of Publication


0

100

200

300

400

500

600

700

2000 2003 2006 2009 2012 2015 2018 2021

1 1 2 4 7 19 20 27 33 49 66 69

119153

199

250

350388

515

599N

umbe

r of A

rtic

les

Year of Publication









h h τ( 1113857 (3)


ρCv( 1113857i_Ti 1113946

Hi



ρCv( 1113857i_Ti 1113946

Hi

qTj Kj + qT

i Ki1113872 1113873rijwijdVj + Si (5)



there4 ρCv( 1113857i_Ti 1113944

j isin Hi

qTj Kj + qT



+si(6)


qTj Kjrji minus qT

i Kirij1113872 1113873wijVj (7)


0

5

10

15

20

25

30

35

2009 2011 2013 2015 2017 2019 2021

1

10

14

19

33

23 24

13

9 8

13

9

15

20

2628

24

17

13

97

1413

16

27

23

17

Num

ber o

f Cita

tion

Year of Publication




Ki 1113944j isin Hi


minus 1

H 1113944jisinHi




minus 1



(8)


1113896 and



1113946Hi





ρCv( 1113857i_Ti 1113946

Hi

fhdVj + Si (10)

ρCv( 1113857i_Ti 1113946

Hi

kTj minus Ti

rij

1113888 1113889dVj (11)


rij

(12)





k 2κ

Aδ2 for(1 minus D)

k 6κπhδ3

for(2 minus D)

k 6κπhδ4

for(3 minus D)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)









_ϱi 1113946Hi





Q Q φ1113872 1113873 (16)



_ϱi 1113946Hi






+Ji(18)


KEL( 1113857i 1113944j isin Hi


minus 1

(19)

HEL 1113944jisinHi




minus 1



(21)


1113946Hi







Q jTKrijwij



_ϱi 1113946Hi

fIdVj + Ji

_ϱi 1113946Hi

kE

Φj minusΦi

rij

1113888 1113889dVj

φirij Φj minusΦi

fI rij t1113872 1113873 kE

Φj minusΦi

rij

(25)





kE 2κE

Aδ2 for(1 minus D)

kE 6κE


kE 6κE


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)






zCi

zt _Ci 1113946

Hi


where

R JTmKrijwij

zCi

zt _Ci 1113946

Hi

Jm( 1113857T

j Kj + Jm( 1113857T


(28)


_Ci 1113944jisinHi

Jm( 1113857T

j Kj + Jm( 1113857T


+ θm( 1113857i(29)

HCon 1113944jisinHi




minus 1

1113944jisinHi


(31)



Ti Kirij1113872 1113873wijVj (32)

zCi





zCi

zt _Ci 1113946

Hi


_Ci 1113946Hi

dCj minus Ci

rij

1113888 1113889dVj

(34)


and expressed as


rij

(35)





d 2D

Aδ2 for(1 minus D)

d 6D


d 6D


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(36)



ρCv( 1113857i_Ti1113872 1113873

n 1113944

jisinHi

kTj1113872 1113873

nminus Tj1113872 1113873

n

rij

⎛⎝ ⎞⎠dVj (37)


Tn+1(i) T

n(i) +ΔtρCv( 1113857i

1113944jisinHi

kTj1113872 1113873

nminus Tj1113872 1113873

n

rij

⎛⎝ ⎞⎠dVj (38)

3 Case Studies





Boundary conditions

TW

2 y t1113874 1113875 0∘C

T minusW

2 y t1113874 1113875 0∘C

(39)

Initial conditions


2le xle

L

2 minus

W

2leyle

W

2 (40)


y

xW

T-bottom

T-top

q=0 q=0

L




0102030405060708090

100

-1 -05 0 05 1

Tem

pera

ture

degC

Y-Division (cm)





1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

80 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

100 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

40 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

60 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

10 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

20 s







Initial conditions


2lexle

L

2 minus

W

2leyle

W

2(41)

Boundary conditions

TW

2 y t1113874 1113875 300∘C (Top)

T minusW

2 y t1113874 1113875 0∘C(Bottom)

(42)





-1 -08 -06 -04 -02 0 02 04 06 08 10

50

100

150

200

250

300

Tem

pera

ture

degCY-Division (cm)





1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

10 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

20 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

25 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

5 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

40 s

1

05

0

-05

-1

Y

-1 -05 0X

100 s

05 1

Temperature3002852702552402252101951801651501351201059075604530150












y

xW

T-top

q=0

q=0q=0

q=0

T-bottomL

β=45deg2a


1

05

0

-05

-1

Y

-1 -05 0X

05 1

100833333666667503333331666672E-08-166667-333333-50-666667-833333-100

temperature

(a)

+1000e+02NT11

+8333e+01+6667e+01+5000e+01+3333e+01+1667e+01-1144e-05-1667e+01-3333e+01-5000e+01-6667e+01-8333e+01-1000e+02

(b)





T(0 t) 273∘K

T(L t) 298∘K

V(L) 0v

(43)



-100

-80

-60

-40

-20

0

20

40

60

80

100

0 0005 001 0015 002

Tem

pera

ture

degC

Width of Plate (cm)



y

xW

T2

T2

T2T2

T1

T1

L

β=45deg2a








1

05

0

-05

-1

Y

-1 -05 0X

05 1

0-208333-416667-625-833333-104167-125-145833-166667-1875-208333-229167-250

temperature

(a)

+0000e+00

-2083e+01

-4167e+01

-6250e+01

-8333e+01

-1042e+02

-1250e-02

-1458e+02

-1667e+02

-1875e+02

-2083e+02

-2292e+02

-2500e+02

NT11

(b)



-200

-150

-100

-50

0-1 -08 -06 -04 -02 0 02 04 06 08 1

Tem

pera

ture

degC






xL

W

y

T0

V0

TL

VL

jx

qx


270

275

280

285

290

295

300


Tem

pera

ture

(degK)

PDRef


00E+0

50E-4

10E-3

15E-3

20E-3

25E-3

30E-3

35E-3

40E-3

45E-3

50E-3

Elec

tric

pot

entia

l (V

)


PDRef



4 Conclusion



270

275

280

285

290

295

300

Tem

pera

ture

(degK)


PDRef


00E+0

50E-4

10E-3

15E-3

20E-3

25E-3

30E-3

35E-3

40E-3

45E-3

50E-3

Elec

tric

pot

entia

l (V

)


PDRef





T is in Kelvin





References







































































































(a)

irij

Hi

j

δ

(b)


Hi

rij

i

j

δ

Ω

(a)

i-321 i-2 i-1 i+1 i+2 i+3 N-1 N

Hx

Xi


δ

(b)






0

50

100

150

200

250

300

2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 20210 0 2 2 5 4

20 1525 32

4554

49

8299

137

167

208

272

295

187

Cita

tions

per

yea

r

Year of Publication


0

100

200

300

400

500

600

700

2000 2003 2006 2009 2012 2015 2018 2021

1 1 2 4 7 19 20 27 33 49 66 69

119153

199

250

350388

515

599N

umbe

r of A

rtic

les

Year of Publication









h h τ( 1113857 (3)


ρCv( 1113857i_Ti 1113946

Hi



ρCv( 1113857i_Ti 1113946

Hi

qTj Kj + qT

i Ki1113872 1113873rijwijdVj + Si (5)



there4 ρCv( 1113857i_Ti 1113944

j isin Hi

qTj Kj + qT



+si(6)


qTj Kjrji minus qT

i Kirij1113872 1113873wijVj (7)


0

5

10

15

20

25

30

35

2009 2011 2013 2015 2017 2019 2021

1

10

14

19

33

23 24

13

9 8

13

9

15

20

2628

24

17

13

97

1413

16

27

23

17

Num

ber o

f Cita

tion

Year of Publication




Ki 1113944j isin Hi


minus 1

H 1113944jisinHi




minus 1



(8)


1113896 and



1113946Hi





ρCv( 1113857i_Ti 1113946

Hi

fhdVj + Si (10)

ρCv( 1113857i_Ti 1113946

Hi

kTj minus Ti

rij

1113888 1113889dVj (11)


rij

(12)





k 2κ

Aδ2 for(1 minus D)

k 6κπhδ3

for(2 minus D)

k 6κπhδ4

for(3 minus D)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)









_ϱi 1113946Hi





Q Q φ1113872 1113873 (16)



_ϱi 1113946Hi






+Ji(18)


KEL( 1113857i 1113944j isin Hi


minus 1

(19)

HEL 1113944jisinHi




minus 1



(21)


1113946Hi







Q jTKrijwij



_ϱi 1113946Hi

fIdVj + Ji

_ϱi 1113946Hi

kE

Φj minusΦi

rij

1113888 1113889dVj

φirij Φj minusΦi

fI rij t1113872 1113873 kE

Φj minusΦi

rij

(25)





kE 2κE

Aδ2 for(1 minus D)

kE 6κE


kE 6κE


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)






zCi

zt _Ci 1113946

Hi


where

R JTmKrijwij

zCi

zt _Ci 1113946

Hi

Jm( 1113857T

j Kj + Jm( 1113857T


(28)


_Ci 1113944jisinHi

Jm( 1113857T

j Kj + Jm( 1113857T


+ θm( 1113857i(29)

HCon 1113944jisinHi




minus 1

1113944jisinHi


(31)



Ti Kirij1113872 1113873wijVj (32)

zCi





zCi

zt _Ci 1113946

Hi


_Ci 1113946Hi

dCj minus Ci

rij

1113888 1113889dVj

(34)


and expressed as


rij

(35)





d 2D

Aδ2 for(1 minus D)

d 6D


d 6D


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(36)



ρCv( 1113857i_Ti1113872 1113873

n 1113944

jisinHi

kTj1113872 1113873

nminus Tj1113872 1113873

n

rij

⎛⎝ ⎞⎠dVj (37)


Tn+1(i) T

n(i) +ΔtρCv( 1113857i

1113944jisinHi

kTj1113872 1113873

nminus Tj1113872 1113873

n

rij

⎛⎝ ⎞⎠dVj (38)

3 Case Studies





Boundary conditions

TW

2 y t1113874 1113875 0∘C

T minusW

2 y t1113874 1113875 0∘C

(39)

Initial conditions


2le xle

L

2 minus

W

2leyle

W

2 (40)


y

xW

T-bottom

T-top

q=0 q=0

L




0102030405060708090

100

-1 -05 0 05 1

Tem

pera

ture

degC

Y-Division (cm)





1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

80 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

100 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

40 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

60 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

10 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

20 s







Initial conditions


2lexle

L

2 minus

W

2leyle

W

2(41)

Boundary conditions

TW

2 y t1113874 1113875 300∘C (Top)

T minusW

2 y t1113874 1113875 0∘C(Bottom)

(42)





-1 -08 -06 -04 -02 0 02 04 06 08 10

50

100

150

200

250

300

Tem

pera

ture

degCY-Division (cm)





1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

10 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

20 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

25 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

5 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

40 s

1

05

0

-05

-1

Y

-1 -05 0X

100 s

05 1

Temperature3002852702552402252101951801651501351201059075604530150












y

xW

T-top

q=0

q=0q=0

q=0

T-bottomL

β=45deg2a


1

05

0

-05

-1

Y

-1 -05 0X

05 1

100833333666667503333331666672E-08-166667-333333-50-666667-833333-100

temperature

(a)

+1000e+02NT11

+8333e+01+6667e+01+5000e+01+3333e+01+1667e+01-1144e-05-1667e+01-3333e+01-5000e+01-6667e+01-8333e+01-1000e+02

(b)





T(0 t) 273∘K

T(L t) 298∘K

V(L) 0v

(43)



-100

-80

-60

-40

-20

0

20

40

60

80

100

0 0005 001 0015 002

Tem

pera

ture

degC

Width of Plate (cm)



y

xW

T2

T2

T2T2

T1

T1

L

β=45deg2a








1

05

0

-05

-1

Y

-1 -05 0X

05 1

0-208333-416667-625-833333-104167-125-145833-166667-1875-208333-229167-250

temperature

(a)

+0000e+00

-2083e+01

-4167e+01

-6250e+01

-8333e+01

-1042e+02

-1250e-02

-1458e+02

-1667e+02

-1875e+02

-2083e+02

-2292e+02

-2500e+02

NT11

(b)



-200

-150

-100

-50

0-1 -08 -06 -04 -02 0 02 04 06 08 1

Tem

pera

ture

degC






xL

W

y

T0

V0

TL

VL

jx

qx


270

275

280

285

290

295

300


Tem

pera

ture

(degK)

PDRef


00E+0

50E-4

10E-3

15E-3

20E-3

25E-3

30E-3

35E-3

40E-3

45E-3

50E-3

Elec

tric

pot

entia

l (V

)


PDRef



4 Conclusion



270

275

280

285

290

295

300

Tem

pera

ture

(degK)


PDRef


00E+0

50E-4

10E-3

15E-3

20E-3

25E-3

30E-3

35E-3

40E-3

45E-3

50E-3

Elec

tric

pot

entia

l (V

)


PDRef





T is in Kelvin





References


































































































0

50

100

150

200

250

300

2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 20210 0 2 2 5 4

20 1525 32

4554

49

8299

137

167

208

272

295

187

Cita

tions

per

yea

r

Year of Publication


0

100

200

300

400

500

600

700

2000 2003 2006 2009 2012 2015 2018 2021

1 1 2 4 7 19 20 27 33 49 66 69

119153

199

250

350388

515

599N

umbe

r of A

rtic

les

Year of Publication









h h τ( 1113857 (3)


ρCv( 1113857i_Ti 1113946

Hi



ρCv( 1113857i_Ti 1113946

Hi

qTj Kj + qT

i Ki1113872 1113873rijwijdVj + Si (5)



there4 ρCv( 1113857i_Ti 1113944

j isin Hi

qTj Kj + qT



+si(6)


qTj Kjrji minus qT

i Kirij1113872 1113873wijVj (7)


0

5

10

15

20

25

30

35

2009 2011 2013 2015 2017 2019 2021

1

10

14

19

33

23 24

13

9 8

13

9

15

20

2628

24

17

13

97

1413

16

27

23

17

Num

ber o

f Cita

tion

Year of Publication




Ki 1113944j isin Hi


minus 1

H 1113944jisinHi




minus 1



(8)


1113896 and



1113946Hi





ρCv( 1113857i_Ti 1113946

Hi

fhdVj + Si (10)

ρCv( 1113857i_Ti 1113946

Hi

kTj minus Ti

rij

1113888 1113889dVj (11)


rij

(12)





k 2κ

Aδ2 for(1 minus D)

k 6κπhδ3

for(2 minus D)

k 6κπhδ4

for(3 minus D)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)









_ϱi 1113946Hi





Q Q φ1113872 1113873 (16)



_ϱi 1113946Hi






+Ji(18)


KEL( 1113857i 1113944j isin Hi


minus 1

(19)

HEL 1113944jisinHi




minus 1



(21)


1113946Hi







Q jTKrijwij



_ϱi 1113946Hi

fIdVj + Ji

_ϱi 1113946Hi

kE

Φj minusΦi

rij

1113888 1113889dVj

φirij Φj minusΦi

fI rij t1113872 1113873 kE

Φj minusΦi

rij

(25)





kE 2κE

Aδ2 for(1 minus D)

kE 6κE


kE 6κE


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)






zCi

zt _Ci 1113946

Hi


where

R JTmKrijwij

zCi

zt _Ci 1113946

Hi

Jm( 1113857T

j Kj + Jm( 1113857T


(28)


_Ci 1113944jisinHi

Jm( 1113857T

j Kj + Jm( 1113857T


+ θm( 1113857i(29)

HCon 1113944jisinHi




minus 1

1113944jisinHi


(31)



Ti Kirij1113872 1113873wijVj (32)

zCi





zCi

zt _Ci 1113946

Hi


_Ci 1113946Hi

dCj minus Ci

rij

1113888 1113889dVj

(34)


and expressed as


rij

(35)





d 2D

Aδ2 for(1 minus D)

d 6D


d 6D


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(36)



ρCv( 1113857i_Ti1113872 1113873

n 1113944

jisinHi

kTj1113872 1113873

nminus Tj1113872 1113873

n

rij

⎛⎝ ⎞⎠dVj (37)


Tn+1(i) T

n(i) +ΔtρCv( 1113857i

1113944jisinHi

kTj1113872 1113873

nminus Tj1113872 1113873

n

rij

⎛⎝ ⎞⎠dVj (38)

3 Case Studies





Boundary conditions

TW

2 y t1113874 1113875 0∘C

T minusW

2 y t1113874 1113875 0∘C

(39)

Initial conditions


2le xle

L

2 minus

W

2leyle

W

2 (40)


y

xW

T-bottom

T-top

q=0 q=0

L




0102030405060708090

100

-1 -05 0 05 1

Tem

pera

ture

degC

Y-Division (cm)





1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

80 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

100 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

40 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

60 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

10 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

20 s







Initial conditions


2lexle

L

2 minus

W

2leyle

W

2(41)

Boundary conditions

TW

2 y t1113874 1113875 300∘C (Top)

T minusW

2 y t1113874 1113875 0∘C(Bottom)

(42)





-1 -08 -06 -04 -02 0 02 04 06 08 10

50

100

150

200

250

300

Tem

pera

ture

degCY-Division (cm)





1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

10 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

20 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

25 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

5 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

40 s

1

05

0

-05

-1

Y

-1 -05 0X

100 s

05 1

Temperature3002852702552402252101951801651501351201059075604530150












y

xW

T-top

q=0

q=0q=0

q=0

T-bottomL

β=45deg2a


1

05

0

-05

-1

Y

-1 -05 0X

05 1

100833333666667503333331666672E-08-166667-333333-50-666667-833333-100

temperature

(a)

+1000e+02NT11

+8333e+01+6667e+01+5000e+01+3333e+01+1667e+01-1144e-05-1667e+01-3333e+01-5000e+01-6667e+01-8333e+01-1000e+02

(b)





T(0 t) 273∘K

T(L t) 298∘K

V(L) 0v

(43)



-100

-80

-60

-40

-20

0

20

40

60

80

100

0 0005 001 0015 002

Tem

pera

ture

degC

Width of Plate (cm)



y

xW

T2

T2

T2T2

T1

T1

L

β=45deg2a








1

05

0

-05

-1

Y

-1 -05 0X

05 1

0-208333-416667-625-833333-104167-125-145833-166667-1875-208333-229167-250

temperature

(a)

+0000e+00

-2083e+01

-4167e+01

-6250e+01

-8333e+01

-1042e+02

-1250e-02

-1458e+02

-1667e+02

-1875e+02

-2083e+02

-2292e+02

-2500e+02

NT11

(b)



-200

-150

-100

-50

0-1 -08 -06 -04 -02 0 02 04 06 08 1

Tem

pera

ture

degC






xL

W

y

T0

V0

TL

VL

jx

qx


270

275

280

285

290

295

300


Tem

pera

ture

(degK)

PDRef


00E+0

50E-4

10E-3

15E-3

20E-3

25E-3

30E-3

35E-3

40E-3

45E-3

50E-3

Elec

tric

pot

entia

l (V

)


PDRef



4 Conclusion



270

275

280

285

290

295

300

Tem

pera

ture

(degK)


PDRef


00E+0

50E-4

10E-3

15E-3

20E-3

25E-3

30E-3

35E-3

40E-3

45E-3

50E-3

Elec

tric

pot

entia

l (V

)


PDRef





T is in Kelvin





References





































































































h h τ( 1113857 (3)


ρCv( 1113857i_Ti 1113946

Hi



ρCv( 1113857i_Ti 1113946

Hi

qTj Kj + qT

i Ki1113872 1113873rijwijdVj + Si (5)



there4 ρCv( 1113857i_Ti 1113944

j isin Hi

qTj Kj + qT



+si(6)


qTj Kjrji minus qT

i Kirij1113872 1113873wijVj (7)


0

5

10

15

20

25

30

35

2009 2011 2013 2015 2017 2019 2021

1

10

14

19

33

23 24

13

9 8

13

9

15

20

2628

24

17

13

97

1413

16

27

23

17

Num

ber o

f Cita

tion

Year of Publication




Ki 1113944j isin Hi


minus 1

H 1113944jisinHi




minus 1



(8)


1113896 and



1113946Hi





ρCv( 1113857i_Ti 1113946

Hi

fhdVj + Si (10)

ρCv( 1113857i_Ti 1113946

Hi

kTj minus Ti

rij

1113888 1113889dVj (11)


rij

(12)





k 2κ

Aδ2 for(1 minus D)

k 6κπhδ3

for(2 minus D)

k 6κπhδ4

for(3 minus D)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)









_ϱi 1113946Hi





Q Q φ1113872 1113873 (16)



_ϱi 1113946Hi






+Ji(18)


KEL( 1113857i 1113944j isin Hi


minus 1

(19)

HEL 1113944jisinHi




minus 1



(21)


1113946Hi







Q jTKrijwij



_ϱi 1113946Hi

fIdVj + Ji

_ϱi 1113946Hi

kE

Φj minusΦi

rij

1113888 1113889dVj

φirij Φj minusΦi

fI rij t1113872 1113873 kE

Φj minusΦi

rij

(25)





kE 2κE

Aδ2 for(1 minus D)

kE 6κE


kE 6κE


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)






zCi

zt _Ci 1113946

Hi


where

R JTmKrijwij

zCi

zt _Ci 1113946

Hi

Jm( 1113857T

j Kj + Jm( 1113857T


(28)


_Ci 1113944jisinHi

Jm( 1113857T

j Kj + Jm( 1113857T


+ θm( 1113857i(29)

HCon 1113944jisinHi




minus 1

1113944jisinHi


(31)



Ti Kirij1113872 1113873wijVj (32)

zCi





zCi

zt _Ci 1113946

Hi


_Ci 1113946Hi

dCj minus Ci

rij

1113888 1113889dVj

(34)


and expressed as


rij

(35)





d 2D

Aδ2 for(1 minus D)

d 6D


d 6D


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(36)



ρCv( 1113857i_Ti1113872 1113873

n 1113944

jisinHi

kTj1113872 1113873

nminus Tj1113872 1113873

n

rij

⎛⎝ ⎞⎠dVj (37)


Tn+1(i) T

n(i) +ΔtρCv( 1113857i

1113944jisinHi

kTj1113872 1113873

nminus Tj1113872 1113873

n

rij

⎛⎝ ⎞⎠dVj (38)

3 Case Studies





Boundary conditions

TW

2 y t1113874 1113875 0∘C

T minusW

2 y t1113874 1113875 0∘C

(39)

Initial conditions


2le xle

L

2 minus

W

2leyle

W

2 (40)


y

xW

T-bottom

T-top

q=0 q=0

L




0102030405060708090

100

-1 -05 0 05 1

Tem

pera

ture

degC

Y-Division (cm)





1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

80 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

100 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

40 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

60 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

10 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

20 s







Initial conditions


2lexle

L

2 minus

W

2leyle

W

2(41)

Boundary conditions

TW

2 y t1113874 1113875 300∘C (Top)

T minusW

2 y t1113874 1113875 0∘C(Bottom)

(42)





-1 -08 -06 -04 -02 0 02 04 06 08 10

50

100

150

200

250

300

Tem

pera

ture

degCY-Division (cm)





1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

10 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

20 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

25 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

5 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

40 s

1

05

0

-05

-1

Y

-1 -05 0X

100 s

05 1

Temperature3002852702552402252101951801651501351201059075604530150












y

xW

T-top

q=0

q=0q=0

q=0

T-bottomL

β=45deg2a


1

05

0

-05

-1

Y

-1 -05 0X

05 1

100833333666667503333331666672E-08-166667-333333-50-666667-833333-100

temperature

(a)

+1000e+02NT11

+8333e+01+6667e+01+5000e+01+3333e+01+1667e+01-1144e-05-1667e+01-3333e+01-5000e+01-6667e+01-8333e+01-1000e+02

(b)





T(0 t) 273∘K

T(L t) 298∘K

V(L) 0v

(43)



-100

-80

-60

-40

-20

0

20

40

60

80

100

0 0005 001 0015 002

Tem

pera

ture

degC

Width of Plate (cm)



y

xW

T2

T2

T2T2

T1

T1

L

β=45deg2a








1

05

0

-05

-1

Y

-1 -05 0X

05 1

0-208333-416667-625-833333-104167-125-145833-166667-1875-208333-229167-250

temperature

(a)

+0000e+00

-2083e+01

-4167e+01

-6250e+01

-8333e+01

-1042e+02

-1250e-02

-1458e+02

-1667e+02

-1875e+02

-2083e+02

-2292e+02

-2500e+02

NT11

(b)



-200

-150

-100

-50

0-1 -08 -06 -04 -02 0 02 04 06 08 1

Tem

pera

ture

degC






xL

W

y

T0

V0

TL

VL

jx

qx


270

275

280

285

290

295

300


Tem

pera

ture

(degK)

PDRef


00E+0

50E-4

10E-3

15E-3

20E-3

25E-3

30E-3

35E-3

40E-3

45E-3

50E-3

Elec

tric

pot

entia

l (V

)


PDRef



4 Conclusion



270

275

280

285

290

295

300

Tem

pera

ture

(degK)


PDRef


00E+0

50E-4

10E-3

15E-3

20E-3

25E-3

30E-3

35E-3

40E-3

45E-3

50E-3

Elec

tric

pot

entia

l (V

)


PDRef





T is in Kelvin





References































































































Ki 1113944j isin Hi


minus 1

H 1113944jisinHi




minus 1



(8)


1113896 and



1113946Hi





ρCv( 1113857i_Ti 1113946

Hi

fhdVj + Si (10)

ρCv( 1113857i_Ti 1113946

Hi

kTj minus Ti

rij

1113888 1113889dVj (11)


rij

(12)





k 2κ

Aδ2 for(1 minus D)

k 6κπhδ3

for(2 minus D)

k 6κπhδ4

for(3 minus D)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)









_ϱi 1113946Hi





Q Q φ1113872 1113873 (16)



_ϱi 1113946Hi






+Ji(18)


KEL( 1113857i 1113944j isin Hi


minus 1

(19)

HEL 1113944jisinHi




minus 1



(21)


1113946Hi







Q jTKrijwij



_ϱi 1113946Hi

fIdVj + Ji

_ϱi 1113946Hi

kE

Φj minusΦi

rij

1113888 1113889dVj

φirij Φj minusΦi

fI rij t1113872 1113873 kE

Φj minusΦi

rij

(25)





kE 2κE

Aδ2 for(1 minus D)

kE 6κE


kE 6κE


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)






zCi

zt _Ci 1113946

Hi


where

R JTmKrijwij

zCi

zt _Ci 1113946

Hi

Jm( 1113857T

j Kj + Jm( 1113857T


(28)


_Ci 1113944jisinHi

Jm( 1113857T

j Kj + Jm( 1113857T


+ θm( 1113857i(29)

HCon 1113944jisinHi




minus 1

1113944jisinHi


(31)



Ti Kirij1113872 1113873wijVj (32)

zCi





zCi

zt _Ci 1113946

Hi


_Ci 1113946Hi

dCj minus Ci

rij

1113888 1113889dVj

(34)


and expressed as


rij

(35)





d 2D

Aδ2 for(1 minus D)

d 6D


d 6D


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(36)



ρCv( 1113857i_Ti1113872 1113873

n 1113944

jisinHi

kTj1113872 1113873

nminus Tj1113872 1113873

n

rij

⎛⎝ ⎞⎠dVj (37)


Tn+1(i) T

n(i) +ΔtρCv( 1113857i

1113944jisinHi

kTj1113872 1113873

nminus Tj1113872 1113873

n

rij

⎛⎝ ⎞⎠dVj (38)

3 Case Studies





Boundary conditions

TW

2 y t1113874 1113875 0∘C

T minusW

2 y t1113874 1113875 0∘C

(39)

Initial conditions


2le xle

L

2 minus

W

2leyle

W

2 (40)


y

xW

T-bottom

T-top

q=0 q=0

L




0102030405060708090

100

-1 -05 0 05 1

Tem

pera

ture

degC

Y-Division (cm)





1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

80 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

100 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

40 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

60 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

10 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

20 s







Initial conditions


2lexle

L

2 minus

W

2leyle

W

2(41)

Boundary conditions

TW

2 y t1113874 1113875 300∘C (Top)

T minusW

2 y t1113874 1113875 0∘C(Bottom)

(42)





-1 -08 -06 -04 -02 0 02 04 06 08 10

50

100

150

200

250

300

Tem

pera

ture

degCY-Division (cm)





1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

10 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

20 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

25 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

5 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

40 s

1

05

0

-05

-1

Y

-1 -05 0X

100 s

05 1

Temperature3002852702552402252101951801651501351201059075604530150












y

xW

T-top

q=0

q=0q=0

q=0

T-bottomL

β=45deg2a


1

05

0

-05

-1

Y

-1 -05 0X

05 1

100833333666667503333331666672E-08-166667-333333-50-666667-833333-100

temperature

(a)

+1000e+02NT11

+8333e+01+6667e+01+5000e+01+3333e+01+1667e+01-1144e-05-1667e+01-3333e+01-5000e+01-6667e+01-8333e+01-1000e+02

(b)





T(0 t) 273∘K

T(L t) 298∘K

V(L) 0v

(43)



-100

-80

-60

-40

-20

0

20

40

60

80

100

0 0005 001 0015 002

Tem

pera

ture

degC

Width of Plate (cm)



y

xW

T2

T2

T2T2

T1

T1

L

β=45deg2a








1

05

0

-05

-1

Y

-1 -05 0X

05 1

0-208333-416667-625-833333-104167-125-145833-166667-1875-208333-229167-250

temperature

(a)

+0000e+00

-2083e+01

-4167e+01

-6250e+01

-8333e+01

-1042e+02

-1250e-02

-1458e+02

-1667e+02

-1875e+02

-2083e+02

-2292e+02

-2500e+02

NT11

(b)



-200

-150

-100

-50

0-1 -08 -06 -04 -02 0 02 04 06 08 1

Tem

pera

ture

degC






xL

W

y

T0

V0

TL

VL

jx

qx


270

275

280

285

290

295

300


Tem

pera

ture

(degK)

PDRef


00E+0

50E-4

10E-3

15E-3

20E-3

25E-3

30E-3

35E-3

40E-3

45E-3

50E-3

Elec

tric

pot

entia

l (V

)


PDRef



4 Conclusion



270

275

280

285

290

295

300

Tem

pera

ture

(degK)


PDRef


00E+0

50E-4

10E-3

15E-3

20E-3

25E-3

30E-3

35E-3

40E-3

45E-3

50E-3

Elec

tric

pot

entia

l (V

)


PDRef





T is in Kelvin





References































































































_ϱi 1113946Hi





Q Q φ1113872 1113873 (16)



_ϱi 1113946Hi






+Ji(18)


KEL( 1113857i 1113944j isin Hi


minus 1

(19)

HEL 1113944jisinHi




minus 1



(21)


1113946Hi







Q jTKrijwij



_ϱi 1113946Hi

fIdVj + Ji

_ϱi 1113946Hi

kE

Φj minusΦi

rij

1113888 1113889dVj

φirij Φj minusΦi

fI rij t1113872 1113873 kE

Φj minusΦi

rij

(25)





kE 2κE

Aδ2 for(1 minus D)

kE 6κE


kE 6κE


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)






zCi

zt _Ci 1113946

Hi


where

R JTmKrijwij

zCi

zt _Ci 1113946

Hi

Jm( 1113857T

j Kj + Jm( 1113857T


(28)


_Ci 1113944jisinHi

Jm( 1113857T

j Kj + Jm( 1113857T


+ θm( 1113857i(29)

HCon 1113944jisinHi




minus 1

1113944jisinHi


(31)



Ti Kirij1113872 1113873wijVj (32)

zCi





zCi

zt _Ci 1113946

Hi


_Ci 1113946Hi

dCj minus Ci

rij

1113888 1113889dVj

(34)


and expressed as


rij

(35)





d 2D

Aδ2 for(1 minus D)

d 6D


d 6D


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(36)



ρCv( 1113857i_Ti1113872 1113873

n 1113944

jisinHi

kTj1113872 1113873

nminus Tj1113872 1113873

n

rij

⎛⎝ ⎞⎠dVj (37)


Tn+1(i) T

n(i) +ΔtρCv( 1113857i

1113944jisinHi

kTj1113872 1113873

nminus Tj1113872 1113873

n

rij

⎛⎝ ⎞⎠dVj (38)

3 Case Studies





Boundary conditions

TW

2 y t1113874 1113875 0∘C

T minusW

2 y t1113874 1113875 0∘C

(39)

Initial conditions


2le xle

L

2 minus

W

2leyle

W

2 (40)


y

xW

T-bottom

T-top

q=0 q=0

L




0102030405060708090

100

-1 -05 0 05 1

Tem

pera

ture

degC

Y-Division (cm)





1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

80 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

100 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

40 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

60 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

10 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

20 s







Initial conditions


2lexle

L

2 minus

W

2leyle

W

2(41)

Boundary conditions

TW

2 y t1113874 1113875 300∘C (Top)

T minusW

2 y t1113874 1113875 0∘C(Bottom)

(42)





-1 -08 -06 -04 -02 0 02 04 06 08 10

50

100

150

200

250

300

Tem

pera

ture

degCY-Division (cm)





1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

10 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

20 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

25 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

5 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

40 s

1

05

0

-05

-1

Y

-1 -05 0X

100 s

05 1

Temperature3002852702552402252101951801651501351201059075604530150












y

xW

T-top

q=0

q=0q=0

q=0

T-bottomL

β=45deg2a


1

05

0

-05

-1

Y

-1 -05 0X

05 1

100833333666667503333331666672E-08-166667-333333-50-666667-833333-100

temperature

(a)

+1000e+02NT11

+8333e+01+6667e+01+5000e+01+3333e+01+1667e+01-1144e-05-1667e+01-3333e+01-5000e+01-6667e+01-8333e+01-1000e+02

(b)





T(0 t) 273∘K

T(L t) 298∘K

V(L) 0v

(43)



-100

-80

-60

-40

-20

0

20

40

60

80

100

0 0005 001 0015 002

Tem

pera

ture

degC

Width of Plate (cm)



y

xW

T2

T2

T2T2

T1

T1

L

β=45deg2a








1

05

0

-05

-1

Y

-1 -05 0X

05 1

0-208333-416667-625-833333-104167-125-145833-166667-1875-208333-229167-250

temperature

(a)

+0000e+00

-2083e+01

-4167e+01

-6250e+01

-8333e+01

-1042e+02

-1250e-02

-1458e+02

-1667e+02

-1875e+02

-2083e+02

-2292e+02

-2500e+02

NT11

(b)



-200

-150

-100

-50

0-1 -08 -06 -04 -02 0 02 04 06 08 1

Tem

pera

ture

degC






xL

W

y

T0

V0

TL

VL

jx

qx


270

275

280

285

290

295

300


Tem

pera

ture

(degK)

PDRef


00E+0

50E-4

10E-3

15E-3

20E-3

25E-3

30E-3

35E-3

40E-3

45E-3

50E-3

Elec

tric

pot

entia

l (V

)


PDRef



4 Conclusion



270

275

280

285

290

295

300

Tem

pera

ture

(degK)


PDRef


00E+0

50E-4

10E-3

15E-3

20E-3

25E-3

30E-3

35E-3

40E-3

45E-3

50E-3

Elec

tric

pot

entia

l (V

)


PDRef





T is in Kelvin





References

































































































zCi

zt _Ci 1113946

Hi


where

R JTmKrijwij

zCi

zt _Ci 1113946

Hi

Jm( 1113857T

j Kj + Jm( 1113857T


(28)


_Ci 1113944jisinHi

Jm( 1113857T

j Kj + Jm( 1113857T


+ θm( 1113857i(29)

HCon 1113944jisinHi




minus 1

1113944jisinHi


(31)



Ti Kirij1113872 1113873wijVj (32)

zCi





zCi

zt _Ci 1113946

Hi


_Ci 1113946Hi

dCj minus Ci

rij

1113888 1113889dVj

(34)


and expressed as


rij

(35)





d 2D

Aδ2 for(1 minus D)

d 6D


d 6D


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(36)



ρCv( 1113857i_Ti1113872 1113873

n 1113944

jisinHi

kTj1113872 1113873

nminus Tj1113872 1113873

n

rij

⎛⎝ ⎞⎠dVj (37)


Tn+1(i) T

n(i) +ΔtρCv( 1113857i

1113944jisinHi

kTj1113872 1113873

nminus Tj1113872 1113873

n

rij

⎛⎝ ⎞⎠dVj (38)

3 Case Studies





Boundary conditions

TW

2 y t1113874 1113875 0∘C

T minusW

2 y t1113874 1113875 0∘C

(39)

Initial conditions


2le xle

L

2 minus

W

2leyle

W

2 (40)


y

xW

T-bottom

T-top

q=0 q=0

L




0102030405060708090

100

-1 -05 0 05 1

Tem

pera

ture

degC

Y-Division (cm)





1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

80 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

100 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

40 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

60 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

10 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

20 s







Initial conditions


2lexle

L

2 minus

W

2leyle

W

2(41)

Boundary conditions

TW

2 y t1113874 1113875 300∘C (Top)

T minusW

2 y t1113874 1113875 0∘C(Bottom)

(42)





-1 -08 -06 -04 -02 0 02 04 06 08 10

50

100

150

200

250

300

Tem

pera

ture

degCY-Division (cm)





1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

10 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

20 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

25 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

5 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

40 s

1

05

0

-05

-1

Y

-1 -05 0X

100 s

05 1

Temperature3002852702552402252101951801651501351201059075604530150












y

xW

T-top

q=0

q=0q=0

q=0

T-bottomL

β=45deg2a


1

05

0

-05

-1

Y

-1 -05 0X

05 1

100833333666667503333331666672E-08-166667-333333-50-666667-833333-100

temperature

(a)

+1000e+02NT11

+8333e+01+6667e+01+5000e+01+3333e+01+1667e+01-1144e-05-1667e+01-3333e+01-5000e+01-6667e+01-8333e+01-1000e+02

(b)





T(0 t) 273∘K

T(L t) 298∘K

V(L) 0v

(43)



-100

-80

-60

-40

-20

0

20

40

60

80

100

0 0005 001 0015 002

Tem

pera

ture

degC

Width of Plate (cm)



y

xW

T2

T2

T2T2

T1

T1

L

β=45deg2a








1

05

0

-05

-1

Y

-1 -05 0X

05 1

0-208333-416667-625-833333-104167-125-145833-166667-1875-208333-229167-250

temperature

(a)

+0000e+00

-2083e+01

-4167e+01

-6250e+01

-8333e+01

-1042e+02

-1250e-02

-1458e+02

-1667e+02

-1875e+02

-2083e+02

-2292e+02

-2500e+02

NT11

(b)



-200

-150

-100

-50

0-1 -08 -06 -04 -02 0 02 04 06 08 1

Tem

pera

ture

degC






xL

W

y

T0

V0

TL

VL

jx

qx


270

275

280

285

290

295

300


Tem

pera

ture

(degK)

PDRef


00E+0

50E-4

10E-3

15E-3

20E-3

25E-3

30E-3

35E-3

40E-3

45E-3

50E-3

Elec

tric

pot

entia

l (V

)


PDRef



4 Conclusion



270

275

280

285

290

295

300

Tem

pera

ture

(degK)


PDRef


00E+0

50E-4

10E-3

15E-3

20E-3

25E-3

30E-3

35E-3

40E-3

45E-3

50E-3

Elec

tric

pot

entia

l (V

)


PDRef





T is in Kelvin





References

































































































Boundary conditions

TW

2 y t1113874 1113875 0∘C

T minusW

2 y t1113874 1113875 0∘C

(39)

Initial conditions


2le xle

L

2 minus

W

2leyle

W

2 (40)


y

xW

T-bottom

T-top

q=0 q=0

L




0102030405060708090

100

-1 -05 0 05 1

Tem

pera

ture

degC

Y-Division (cm)





1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

80 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

100 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

40 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

60 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

10 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

20 s







Initial conditions


2lexle

L

2 minus

W

2leyle

W

2(41)

Boundary conditions

TW

2 y t1113874 1113875 300∘C (Top)

T minusW

2 y t1113874 1113875 0∘C(Bottom)

(42)





-1 -08 -06 -04 -02 0 02 04 06 08 10

50

100

150

200

250

300

Tem

pera

ture

degCY-Division (cm)





1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

10 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

20 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

25 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

5 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

40 s

1

05

0

-05

-1

Y

-1 -05 0X

100 s

05 1

Temperature3002852702552402252101951801651501351201059075604530150












y

xW

T-top

q=0

q=0q=0

q=0

T-bottomL

β=45deg2a


1

05

0

-05

-1

Y

-1 -05 0X

05 1

100833333666667503333331666672E-08-166667-333333-50-666667-833333-100

temperature

(a)

+1000e+02NT11

+8333e+01+6667e+01+5000e+01+3333e+01+1667e+01-1144e-05-1667e+01-3333e+01-5000e+01-6667e+01-8333e+01-1000e+02

(b)





T(0 t) 273∘K

T(L t) 298∘K

V(L) 0v

(43)



-100

-80

-60

-40

-20

0

20

40

60

80

100

0 0005 001 0015 002

Tem

pera

ture

degC

Width of Plate (cm)



y

xW

T2

T2

T2T2

T1

T1

L

β=45deg2a








1

05

0

-05

-1

Y

-1 -05 0X

05 1

0-208333-416667-625-833333-104167-125-145833-166667-1875-208333-229167-250

temperature

(a)

+0000e+00

-2083e+01

-4167e+01

-6250e+01

-8333e+01

-1042e+02

-1250e-02

-1458e+02

-1667e+02

-1875e+02

-2083e+02

-2292e+02

-2500e+02

NT11

(b)



-200

-150

-100

-50

0-1 -08 -06 -04 -02 0 02 04 06 08 1

Tem

pera

ture

degC






xL

W

y

T0

V0

TL

VL

jx

qx


270

275

280

285

290

295

300


Tem

pera

ture

(degK)

PDRef


00E+0

50E-4

10E-3

15E-3

20E-3

25E-3

30E-3

35E-3

40E-3

45E-3

50E-3

Elec

tric

pot

entia

l (V

)


PDRef



4 Conclusion



270

275

280

285

290

295

300

Tem

pera

ture

(degK)


PDRef


00E+0

50E-4

10E-3

15E-3

20E-3

25E-3

30E-3

35E-3

40E-3

45E-3

50E-3

Elec

tric

pot

entia

l (V

)


PDRef





T is in Kelvin





References































































































1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

80 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

100 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

40 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

60 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

10 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature10095908580757065605550454035302520151050

20 s







Initial conditions


2lexle

L

2 minus

W

2leyle

W

2(41)

Boundary conditions

TW

2 y t1113874 1113875 300∘C (Top)

T minusW

2 y t1113874 1113875 0∘C(Bottom)

(42)





-1 -08 -06 -04 -02 0 02 04 06 08 10

50

100

150

200

250

300

Tem

pera

ture

degCY-Division (cm)





1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

10 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

20 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

25 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

5 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

40 s

1

05

0

-05

-1

Y

-1 -05 0X

100 s

05 1

Temperature3002852702552402252101951801651501351201059075604530150












y

xW

T-top

q=0

q=0q=0

q=0

T-bottomL

β=45deg2a


1

05

0

-05

-1

Y

-1 -05 0X

05 1

100833333666667503333331666672E-08-166667-333333-50-666667-833333-100

temperature

(a)

+1000e+02NT11

+8333e+01+6667e+01+5000e+01+3333e+01+1667e+01-1144e-05-1667e+01-3333e+01-5000e+01-6667e+01-8333e+01-1000e+02

(b)





T(0 t) 273∘K

T(L t) 298∘K

V(L) 0v

(43)



-100

-80

-60

-40

-20

0

20

40

60

80

100

0 0005 001 0015 002

Tem

pera

ture

degC

Width of Plate (cm)



y

xW

T2

T2

T2T2

T1

T1

L

β=45deg2a








1

05

0

-05

-1

Y

-1 -05 0X

05 1

0-208333-416667-625-833333-104167-125-145833-166667-1875-208333-229167-250

temperature

(a)

+0000e+00

-2083e+01

-4167e+01

-6250e+01

-8333e+01

-1042e+02

-1250e-02

-1458e+02

-1667e+02

-1875e+02

-2083e+02

-2292e+02

-2500e+02

NT11

(b)



-200

-150

-100

-50

0-1 -08 -06 -04 -02 0 02 04 06 08 1

Tem

pera

ture

degC






xL

W

y

T0

V0

TL

VL

jx

qx


270

275

280

285

290

295

300


Tem

pera

ture

(degK)

PDRef


00E+0

50E-4

10E-3

15E-3

20E-3

25E-3

30E-3

35E-3

40E-3

45E-3

50E-3

Elec

tric

pot

entia

l (V

)


PDRef



4 Conclusion



270

275

280

285

290

295

300

Tem

pera

ture

(degK)


PDRef


00E+0

50E-4

10E-3

15E-3

20E-3

25E-3

30E-3

35E-3

40E-3

45E-3

50E-3

Elec

tric

pot

entia

l (V

)


PDRef





T is in Kelvin





References



































































































Initial conditions


2lexle

L

2 minus

W

2leyle

W

2(41)

Boundary conditions

TW

2 y t1113874 1113875 300∘C (Top)

T minusW

2 y t1113874 1113875 0∘C(Bottom)

(42)





-1 -08 -06 -04 -02 0 02 04 06 08 10

50

100

150

200

250

300

Tem

pera

ture

degCY-Division (cm)





1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

10 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

20 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

25 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

5 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

40 s

1

05

0

-05

-1

Y

-1 -05 0X

100 s

05 1

Temperature3002852702552402252101951801651501351201059075604530150












y

xW

T-top

q=0

q=0q=0

q=0

T-bottomL

β=45deg2a


1

05

0

-05

-1

Y

-1 -05 0X

05 1

100833333666667503333331666672E-08-166667-333333-50-666667-833333-100

temperature

(a)

+1000e+02NT11

+8333e+01+6667e+01+5000e+01+3333e+01+1667e+01-1144e-05-1667e+01-3333e+01-5000e+01-6667e+01-8333e+01-1000e+02

(b)





T(0 t) 273∘K

T(L t) 298∘K

V(L) 0v

(43)



-100

-80

-60

-40

-20

0

20

40

60

80

100

0 0005 001 0015 002

Tem

pera

ture

degC

Width of Plate (cm)



y

xW

T2

T2

T2T2

T1

T1

L

β=45deg2a








1

05

0

-05

-1

Y

-1 -05 0X

05 1

0-208333-416667-625-833333-104167-125-145833-166667-1875-208333-229167-250

temperature

(a)

+0000e+00

-2083e+01

-4167e+01

-6250e+01

-8333e+01

-1042e+02

-1250e-02

-1458e+02

-1667e+02

-1875e+02

-2083e+02

-2292e+02

-2500e+02

NT11

(b)



-200

-150

-100

-50

0-1 -08 -06 -04 -02 0 02 04 06 08 1

Tem

pera

ture

degC






xL

W

y

T0

V0

TL

VL

jx

qx


270

275

280

285

290

295

300


Tem

pera

ture

(degK)

PDRef


00E+0

50E-4

10E-3

15E-3

20E-3

25E-3

30E-3

35E-3

40E-3

45E-3

50E-3

Elec

tric

pot

entia

l (V

)


PDRef



4 Conclusion



270

275

280

285

290

295

300

Tem

pera

ture

(degK)


PDRef


00E+0

50E-4

10E-3

15E-3

20E-3

25E-3

30E-3

35E-3

40E-3

45E-3

50E-3

Elec

tric

pot

entia

l (V

)


PDRef





T is in Kelvin





References































































































1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

10 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

20 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

25 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

5 s

1

05

0

-05

-1

Y

-1 -05 0X

05 1

Temperature3002852702552402252101951801651501351201059075604530150

40 s

1

05

0

-05

-1

Y

-1 -05 0X

100 s

05 1

Temperature3002852702552402252101951801651501351201059075604530150












y

xW

T-top

q=0

q=0q=0

q=0

T-bottomL

β=45deg2a


1

05

0

-05

-1

Y

-1 -05 0X

05 1

100833333666667503333331666672E-08-166667-333333-50-666667-833333-100

temperature

(a)

+1000e+02NT11

+8333e+01+6667e+01+5000e+01+3333e+01+1667e+01-1144e-05-1667e+01-3333e+01-5000e+01-6667e+01-8333e+01-1000e+02

(b)





T(0 t) 273∘K

T(L t) 298∘K

V(L) 0v

(43)



-100

-80

-60

-40

-20

0

20

40

60

80

100

0 0005 001 0015 002

Tem

pera

ture

degC

Width of Plate (cm)



y

xW

T2

T2

T2T2

T1

T1

L

β=45deg2a








1

05

0

-05

-1

Y

-1 -05 0X

05 1

0-208333-416667-625-833333-104167-125-145833-166667-1875-208333-229167-250

temperature

(a)

+0000e+00

-2083e+01

-4167e+01

-6250e+01

-8333e+01

-1042e+02

-1250e-02

-1458e+02

-1667e+02

-1875e+02

-2083e+02

-2292e+02

-2500e+02

NT11

(b)



-200

-150

-100

-50

0-1 -08 -06 -04 -02 0 02 04 06 08 1

Tem

pera

ture

degC






xL

W

y

T0

V0

TL

VL

jx

qx


270

275

280

285

290

295

300


Tem

pera

ture

(degK)

PDRef


00E+0

50E-4

10E-3

15E-3

20E-3

25E-3

30E-3

35E-3

40E-3

45E-3

50E-3

Elec

tric

pot

entia

l (V

)


PDRef



4 Conclusion



270

275

280

285

290

295

300

Tem

pera

ture

(degK)


PDRef


00E+0

50E-4

10E-3

15E-3

20E-3

25E-3

30E-3

35E-3

40E-3

45E-3

50E-3

Elec

tric

pot

entia

l (V

)


PDRef





T is in Kelvin





References








































































































y

xW

T-top

q=0

q=0q=0

q=0

T-bottomL

β=45deg2a


1

05

0

-05

-1

Y

-1 -05 0X

05 1

100833333666667503333331666672E-08-166667-333333-50-666667-833333-100

temperature

(a)

+1000e+02NT11

+8333e+01+6667e+01+5000e+01+3333e+01+1667e+01-1144e-05-1667e+01-3333e+01-5000e+01-6667e+01-8333e+01-1000e+02

(b)





T(0 t) 273∘K

T(L t) 298∘K

V(L) 0v

(43)



-100

-80

-60

-40

-20

0

20

40

60

80

100

0 0005 001 0015 002

Tem

pera

ture

degC

Width of Plate (cm)



y

xW

T2

T2

T2T2

T1

T1

L

β=45deg2a








1

05

0

-05

-1

Y

-1 -05 0X

05 1

0-208333-416667-625-833333-104167-125-145833-166667-1875-208333-229167-250

temperature

(a)

+0000e+00

-2083e+01

-4167e+01

-6250e+01

-8333e+01

-1042e+02

-1250e-02

-1458e+02

-1667e+02

-1875e+02

-2083e+02

-2292e+02

-2500e+02

NT11

(b)



-200

-150

-100

-50

0-1 -08 -06 -04 -02 0 02 04 06 08 1

Tem

pera

ture

degC






xL

W

y

T0

V0

TL

VL

jx

qx


270

275

280

285

290

295

300


Tem

pera

ture

(degK)

PDRef


00E+0

50E-4

10E-3

15E-3

20E-3

25E-3

30E-3

35E-3

40E-3

45E-3

50E-3

Elec

tric

pot

entia

l (V

)


PDRef



4 Conclusion



270

275

280

285

290

295

300

Tem

pera

ture

(degK)


PDRef


00E+0

50E-4

10E-3

15E-3

20E-3

25E-3

30E-3

35E-3

40E-3

45E-3

50E-3

Elec

tric

pot

entia

l (V

)


PDRef





T is in Kelvin





References

































































































T(0 t) 273∘K

T(L t) 298∘K

V(L) 0v

(43)



-100

-80

-60

-40

-20

0

20

40

60

80

100

0 0005 001 0015 002

Tem

pera

ture

degC

Width of Plate (cm)



y

xW

T2

T2

T2T2

T1

T1

L

β=45deg2a








1

05

0

-05

-1

Y

-1 -05 0X

05 1

0-208333-416667-625-833333-104167-125-145833-166667-1875-208333-229167-250

temperature

(a)

+0000e+00

-2083e+01

-4167e+01

-6250e+01

-8333e+01

-1042e+02

-1250e-02

-1458e+02

-1667e+02

-1875e+02

-2083e+02

-2292e+02

-2500e+02

NT11

(b)



-200

-150

-100

-50

0-1 -08 -06 -04 -02 0 02 04 06 08 1

Tem

pera

ture

degC






xL

W

y

T0

V0

TL

VL

jx

qx


270

275

280

285

290

295

300


Tem

pera

ture

(degK)

PDRef


00E+0

50E-4

10E-3

15E-3

20E-3

25E-3

30E-3

35E-3

40E-3

45E-3

50E-3

Elec

tric

pot

entia

l (V

)


PDRef



4 Conclusion



270

275

280

285

290

295

300

Tem

pera

ture

(degK)


PDRef


00E+0

50E-4

10E-3

15E-3

20E-3

25E-3

30E-3

35E-3

40E-3

45E-3

50E-3

Elec

tric

pot

entia

l (V

)


PDRef





T is in Kelvin





References




































































































1

05

0

-05

-1

Y

-1 -05 0X

05 1

0-208333-416667-625-833333-104167-125-145833-166667-1875-208333-229167-250

temperature

(a)

+0000e+00

-2083e+01

-4167e+01

-6250e+01

-8333e+01

-1042e+02

-1250e-02

-1458e+02

-1667e+02

-1875e+02

-2083e+02

-2292e+02

-2500e+02

NT11

(b)



-200

-150

-100

-50

0-1 -08 -06 -04 -02 0 02 04 06 08 1

Tem

pera

ture

degC






xL

W

y

T0

V0

TL

VL

jx

qx


270

275

280

285

290

295

300


Tem

pera

ture

(degK)

PDRef


00E+0

50E-4

10E-3

15E-3

20E-3

25E-3

30E-3

35E-3

40E-3

45E-3

50E-3

Elec

tric

pot

entia

l (V

)


PDRef



4 Conclusion



270

275

280

285

290

295

300

Tem

pera

ture

(degK)


PDRef


00E+0

50E-4

10E-3

15E-3

20E-3

25E-3

30E-3

35E-3

40E-3

45E-3

50E-3

Elec

tric

pot

entia

l (V

)


PDRef





T is in Kelvin





References































































































xL

W

y

T0

V0

TL

VL

jx

qx


270

275

280

285

290

295

300


Tem

pera

ture

(degK)

PDRef


00E+0

50E-4

10E-3

15E-3

20E-3

25E-3

30E-3

35E-3

40E-3

45E-3

50E-3

Elec

tric

pot

entia

l (V

)


PDRef



4 Conclusion



270

275

280

285

290

295

300

Tem

pera

ture

(degK)


PDRef


00E+0

50E-4

10E-3

15E-3

20E-3

25E-3

30E-3

35E-3

40E-3

45E-3

50E-3

Elec

tric

pot

entia

l (V

)


PDRef





T is in Kelvin





References































































































4 Conclusion



270

275

280

285

290

295

300

Tem

pera

ture

(degK)


PDRef


00E+0

50E-4

10E-3

15E-3

20E-3

25E-3

30E-3

35E-3

40E-3

45E-3

50E-3

Elec

tric

pot

entia

l (V

)


PDRef





T is in Kelvin





References


































































































References































































































a review of peridynamics (pd) theory of diffusion based

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