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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng (2016)Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.5226

A Green’s discrete transformation meshfree method for simulatingtransient diffusion problems

Weijie Mai1, Soheil Soghrati1,2,3,*,† and Rudolph G. Buchheit2,4

1Department of Materials Science and Engineering, The Ohio State University, Columbus, OH, USA2Department of Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH, USA

3Simulation Innovation and Modeling Center, Columbus, OH, USA4Fontana Corrosion Center, Columbus, OH, USA

SUMMARY

This manuscript presents the formulation and application of the Green’s discrete transformation method(GDTM) for the meshfree simulation of transient diffusion problems, including those with moving bound-aries. The GDTM implements a linear combination of time-dependent Green’s basis functions defined ona set of source points to approximate the field in the form of a solution series. A discrete transformationis implemented to evaluate unknown coefficients of this series, which eliminates the need to use time inte-gration schemes. We will study the optimal number and location of the GDTM source points that yield thehighest level of accuracy, while maintaining a manageable condition number for the resulting linear sys-tem of equations. The optimal values of these parameters, which are inherently independent of the domaingeometry, are determined such that the basis functions have appropriate features for approximating the field.A comprehensive convergence study is presented to show the precision and convergence rate of the GDTMfor modeling various diffusion problems. We also demonstrate the application of this method for simulatingthree diffusion problems with complex and evolving morphologies: heat transfer in a turbine blade, thermalresponse of a porous material, and localized (pitting) corrosion in stainless steel. Copyright © 2016 JohnWiley & Sons, Ltd.

Received 6 July 2015; Revised 14 December 2015; Accepted 25 January 2016

KEY WORDS: meshfree method; Green’s function; transient diffusion; moving boundary; heat transfer;pitting corrosion

NOMENCLATURE

˛ scaling factor to create an enlarged domain to place source pointsˇ parameter to control the shape of basis function! ratio between the number of source points and field pointsx spatial coordinate of field pointt temporal coordinate of field point! spatial coordinate of source point" temporal location of source pointdm problem dimensiond maximum distance between adjacent field pointsG.!; "/ Green’s function defined at source point .!; "/ci constant coefficients for the solution series

*Correspondence to: Soheil Soghrati, Assistant professor, Mechanical and Aerospace Engineering, Materials Scienceand Engineering, The Ohio State University, 201 W. 19th Avenue, Columbus, OH 43210, USA.

†E-mail: [email protected]

Copyright © 2016 John Wiley & Sons, Ltd.

TRANSIENT GREEN’S DISCRETE TRANSFORMATION MESHFREE METHOD

D mass/thermal diffusivity# thermal conductivityu field variable to be evaluatedue equilibrium concentration of dissolved metal ionsusolid atom concentration of the solid metalT temperatureAdiss dissolution affinityEcorr corrosion potential´ average charge number$ overpotential

1. INTRODUCTION

Several engineering problems and physical phenomena, such as the conductive heat and mass trans-fer, are governed by the diffusion law, which necessitates the accurate numerical simulation of thisprocess for a broad range of applications. The standard finite element method (FEM) and its varyingstabilized formulations are among the most reliable and widely used techniques for the approxi-mation of transient diffusion problems [1]. However, the requirement of creating appropriate finiteelement (FE) meshes that conform to the problem geometry can cause difficulties in implementingthis method for modeling problems with intricate morphologies. This limitation becomes particu-larly challenging for simulating moving boundary problems (e.g., localized corrosion [2] and phasetransition [3]), where the evolving geometry of the domain may require the adaptive refinement [4,5] or regeneration of the FE mesh throughout the solution process.

Advanced mesh-independent FEMs such as the extended/generalized FEM [6, 7] and the hier-archical interface-enriched FEM (HIFEM) [8, 9] alleviate this limitation by adding appropriateenrichments to field/gradient discontinuous regions of the solution, which allows the use ofFE meshes that are independent of the problem morphology. The combination of the general-ized/extended FEM and the level set method has successfully been implemented for simulatingmoving boundary problems [10, 11]. It must be noted that the FEM-based approximation of transientdiffusion problems also requires using techniques such as the finite difference [12, 13] or Laplacetransformation [14] for the time integration.

An alternative approach that can considerably simplify the mesh generation process is the bound-ary element method (BEM) [15–17], which enables substituting the FE volume mesh with a lowerdimensional surface mesh for linear problems. In this method, an integral equation is solved to sat-isfy boundary conditions of a problem with existing Green’s basis functions [18]. However, thekey advantage of the BEM, that is, implementing a lower-dimensional mesh for discretizing thedomain, is lost for modeling transient and nonlinear/inhomogeneous problems. In the dual reci-procity boundary element method [19], the homogeneous solution is represented using the BEM,while the particular solution is approximated by collocating on a few internal points to avoid usingvolume integrals for modeling transient/nonlinear problems [20–22].

In meshfree methods (MMs), the problem domain is discretized using only a set of nodes withoutthe need to create elements and element connectivity. Thus, such methods are particularly attractivefor simulating problems with complex and/or evolving geometries by allowing the implementationof a simple algorithm for discretizing the domain. A wide range of MMs have been introduced thatvary in the discretization approach (boundary versus domain), basis functions, and the method usedfor constructing the approximate field. In the method of fundamental solutions (MFS) [23–25], a setof fundamental bases is employed to approximate the solution using the point collocation method.Similarly, the exponential basis functions (EBFs) meshfree method [26, 27] approximates the fieldbased on a linear combination of exponential bases, but implements a discrete transformationtechnique to compute the unknown coefficients of the solution series.

Recently, the Green’s discrete transformation method (GDTM) has been introduced for the mesh-free approximation of Poisson problems, which implements a discrete transformation similar to thatused in the EBFs method, together with the Green’s basis functions for evaluating the solution series.

Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2016)DOI: 10.1002/nme

W. MAI, S. SOGHRATI AND R. G. BUCHHEIT

Similar to the BEM, the GDTM also requires the existence of Green’s functions for the govern-ing differential equations, which, although limits its range of applications, enables the treatmentof problems with complex morphologies [28]. To handle nonlinearity and inhomogeneous sourceterms, similar techniques to those implemented in the dual reciprocity boundary element methodcan be used, where the particular solution is approximated separately and an equivalent homoge-neous problem with modified boundary conditions is solved to evaluate the combined solution [27,29, 30]. Among other MMs, we can enumerate the smoothed particle hydrodynamics (SPH) [31,32], element-free Galerkin method [33, 34], meshless local Petrov–Galerkin method [35], and thelocal radial basis function collocation method [36], although several other successful methods havealso been introduced. A detailed review of MMs and their applications is provided in [37, 38].

While different techniques such as the Laplace transformation can be implemented in the mesh-free approximation of transient problems to eliminate time-dependent terms [27, 29, 30, 39], suchtechniques often increase the computational cost and introduce additional errors. Alternatively,in the MFS and BEM, it is possible to evaluate time-dependent basis functions that can directlybe used for constructing the solution series. This technique was first implemented for simulatinghomogeneous diffusion problems [40] and later expanded to inhomogeneous heat transfer [41] andunsteady Stokes equation [42]. A similar approach is also used in the EBFs method by implementingtime-dependent bases for approximating the field [43].

The current manuscript aims at expanding the GDTM to enable simulating transient diffusionproblems, including those with moving boundaries (i.e., the Stefan problem [44]). This requiresimplementing a set of source points to evaluate the time-dependent Green’s functions, which mustbe created at appropriate locations in time and space to yield the optimal accuracy. A discrete trans-formation similar to that used in [27, 28] is implemented to evaluate the unknown coefficients of thesolution series, without the need to use conventional time marching algorithms such as the Crank-Nicolson [45]. We introduce a systematic approach for creating GDTM source points that providebasis functions with optimal features for approximating the field. Several numerical studies are pro-vided to study the effect of the source points on the accuracy of the GDTM. Similar to the SPH, aLagrangian approach is employed to determine the location of boundary points for the GDTM sim-ulation of problems with evolving morphologies. It must be noted that although this work is focusedon 2D diffusion problems, similar to the BEM, the proposed method has no intrinsic limitation forexpansion to 3D and other types of governing equations with existing Green’s functions.

The outline of the remainder of this article is as follows. Section 2 provides the transient diffusiongoverning equations and the GDTM formulation for approximating this phenomenon. Required con-siderations for simulating moving boundary problems are also discussed in that section. In Section 3,we present a detailed numerical study to determine the optimal location and number of GDTMsource points that yield the highest level of accuracy. A convergence study is provided in Section 4,followed in Section 5 by demonstrating the application of the GDTM for simulating three engineer-ing problems: thermal response of a turbine blade, conductive heat transfer in a porous material, andthe pitting corrosion in stainless steel.

2. PROBLEM FORMULATION AND GREEN’S DISCRETE TRANSFORMATIONMETHOD ALGORITHM

2.1. Problem formulation

Consider a 2D domain%with closure N% and three mutually exclusive boundaries @% D &D[&N [&R corresponding to Dirichlet, Neumann, and Robin boundary conditions, respectively. The strongform of the transient diffusion governing equations for this domain can be written as follows: Findu.x; t / such that

@u

@t.x; t / D Dr2u.x; t / in N%u.x; 0/ D u0.x/ in N%u.x; t / D Nu on &D

!#ru.x; t / " n D Nq on &N!#ru.x; t / " nC hu.x; t / D hu1 on &R;

(1)



where D is the diffusivity, u0.x/ is the initial field distribution, n is the unit vector normal to theboundary, Nu is the fixed field value on &D , Nq is the applied flux, and #, h, and u1 are material-dependent and ambient-dependent constant parameters. In the case of conductive heat transfer, thelast three parameters refer to the thermal conductivity, heat transfer coefficient, and the ambienttemperature, respectively. In moving boundary diffusion problems, the so-called Stefan conditionalong the moving boundary is given by

!Dru.xs; t / " n D Lvn.xs; t /; (2)

where xs is the coordinates of the points located on the moving boundary, vn is the normalcomponent of the velocity at each point, and L is a constant material parameter.

2.2. Green’s discrete transformation method approximation

The GDTM implements Green’s basis functions for approximating the field [28]. For (1), theGreen’s function is evaluated by solving the following equation:

@u

@t.x; t / !Dr2u.x; t / D ı.x; t I !; "/; (3)

where ı.x; t I !; "/ is the Dirac delta function defined at the source point .!; "/. The resulting Green’sfunction is given by

G.x; t I !; "/ D Œ4'D.t ! "/(!dm2 exp!! .x ! !/24D.t ! "/

"; (4)

where dm is the problem dimension. Note that the basis function given in (4) is singular at " D t ,which indicates that source points must be placed at a different time level than that of the problembeing solved, while there is no restriction on their spatial distribution. Details on how to determinethe optimal location of source points are provided in the next section.

The GDTM approximation of u.x; t / is written as a linear combination of the Green’s basisfunctions,

Ou.x; t / DNspXiD1

ciG.x; t I !i ; "i /; (5)

where ci are unknown constant coefficients and Nsp is the number of Green’s functions or equiva-lently the number of source points used to approximate the field. Because the Green’s functions usedin this solution series are time-dependent, no time integration scheme is needed for simulating thetransient field. The unknown coefficients ci are evaluated using a discrete transformation technique[27], such that the solution series satisfies the initial and boundary conditions of the problem at eachtime step. In this method, we first discretize the domain boundaries usingMB DMDCMN CMR

points corresponding to the number of points discretizing &D , &N , and &R boundaries, respec-tively. To enforce initial conditions, we also create MF field points to discretize the open domain%. Unknown coefficients ci associated with the i th Green’s function (or alternatively the i th sourcepoint) are then evaluated as [27]

ci D VTi R NU; (6)

where R is the projection matrix (to be evaluated later) and NU is a vector holding initial/boundaryvalues corresponding to each field/boundary point,

NU D Œ¹ Nuº.1!MD/; ¹ Nqº.1!MN /; ¹hu1º.1!MR/; ¹u0.x/º.1!MF /(T : (7)

Also, the j th component of vector Vi is computed by applying the left-hand-side operator in (1)corresponding to the initial/boundary condition stored at the j th component of NU on the Green’s



function and evaluating that at the i th source point .!i ; "i /. For example, if the j th component of NUrefers to point xj with the Neumann boundary condition !#ru.xj ; t / "nj D Nqj , the j th componentof Vi is evaluated as !#rG.xj ; tj I !i ; "i / " nj . Thus, Vi can be written as

Vi D Œ¹G.x; t I !i ; "i /º.1!MD/; ¹!#rG.x; t I !i ; "i / " nº.1!MN /;¹!#rG.x; t I !i ; "i / " nC hG.x; t I !i ; "i /º.1!MR/;¹G.x; t I !i ; "i /º.1!MF /(T :

(8)

To evaluate the unknown projection matrix R, we replace (6) in (5) and re-evaluate theinitial/boundary values of the problem, which yields

NU DNspXiD1

ViVTi R NU: (9)

Matrix R can then be computed from (9) as

R D

0@NspXiD1

ViVTi

1AC

D GC; (10)

where .:/C is the pseudo inverse operator [46]. After evaluating R, the coefficients ci needed forconstructing the solution series can be evaluated from (6). Note that this approach leads to the con-struction of a dense matrix G, for which computing the pseudo inverse can be computationallyexpensive. However, this disadvantage is compensated by the fact that an accurate GDTM simu-lation requires relatively few degrees of freedom for discretizing the domain, as the interpolationfunctions used in this method satisfy the governing equations. Furthermore, because the GDTM doesnot require a time integration scheme, according to (6), the solution at each time step is obtainedvia a matrix multiplication rather than solving a linear system of equations, which can considerablyreduce the computational cost.

For moving boundary problems, (2) indicates that the velocity of each boundary point is propor-tional to the normal derivative of the field at that point. After evaluating coefficients ci , the normalcomponent of the velocity at the current time step can be computed as

vn.xs; t / D!DL

NspXiD1

cirG.xs; t I !i ; "i / " ns: (11)

Because of the meshfree nature of the GDTM, the locations of boundary points at the next timestep can be easily determined by moving them by a distance of vn)t in the directions of theirnormals. This feature is similar to that used in the SPH, which eliminates the need to implementalternative numerical techniques (e.g., the level set method) to implicitly track the interface loca-tion. Moreover, because no mass/density is assigned to the GDTM field/boundary points, one caneasily add (remove) them as the domain expands (shrinks) throughout the simulation of movingboundary problems.

3. GREEN’S DISCRETE TRANSFORMATION METHOD FIELD AND SOURCE POINTS

As noted in the previous section, the GDTM approximation of the transient diffusion problemrequires creating a set of M D MB CMF boundary/field points to discretize the domain and a setof Nsp source points to construct the Green’s basis functions. This process is schematically shownin Figure 1 for approximating the field at the time level tj C )t using the initial and boundaryconditions previously evaluated at tj . To discretize the problem in the spatial domain, we first dis-cretize its boundaries using a set of MB boundary points with a maximum distance of d . As shownin Figure 1(a), each boundary point is then moved inward in the direction of its normal vector by a



(a) (b)

Figure 1. (a) Schematic of the distribution of field points at tj for the Green’s discrete transformation methodapproximation of a transient diffusion problem in an arbitrary-shaped domain; (b) distribution of sourcepoints in an enlarged spatial domain of the problem and boundary points created at tjC1 to approximate the

field at the next time level.

distance of 0:5d to create a layer of field points in the vicinity of the boundary. Although this stepis not necessary, it ensures that an adequate number of field points exists in a controlled distancefrom the boundary, which can provide a more accurate approximation of boundary conditions. Theremainder of field points are simply created on a structured grid with spacing d inside the domain.As depicted in Figure 1(b), in addition to discretizing the domain at tj , a set of boundary points isalso created at tj C )t to enforce the boundary conditions at the next time level, where the fieldneeds to be approximated.

According to (4), there are an infinite number of choices for creating the GDTM source points, asthe only restriction imposed on their locations in the time-space domain is "i ¤ tj to avoid the con-struction of singular basis functions. The simplest approach for creating the source points is to usethe same spatial coordinates as those of field points but at a lower time level ("i D " < tj ), whichhas also been implemented in the context of the MFS [40, 42]. Alternatively, Chantasiriwan [47]showed that placing some of the source points on an imaginary domain created by enlarging thedomain boundaries can improve the accuracy of the MFS solution. Tsai et al. [42] empirically con-cluded that the optimal value of " is a function of the grid size used for discretizing the domain.While similar approaches can be implemented in the GDTM, the discrete transformation (6) allowsselecting the number of source points independent of those of the boundary/field points, which canalso affect their optimal time-space locations. In this work, we introduce a simple algorithm forselecting the number and locations of GDTM source points, which is based on providing appro-priate features for the resulting Green’s basis functions to accurately approximate the field. Thus,the proposed approach for creating the source points is independent of the domain geometry andinitial/boundary conditions of the problem.

As schematically shown in Figure 1(b), in the proposed approach the GDTM source points areplaced on an enlarged virtual domain created by moving the boundaries of the physical domain inthe direction of their normal vectors by a distance of ˛d (˛ is the scaling factor). We also create atotal number of Nsp D !M source points on a structured grid inside this virtual domain, where !is a constant coefficient. The number and locations of the GDTM source points can be then fullycharacterized by three parameters: ˛, which determines the size of the virtual domain accommo-dating the source points, ! , which determines the ratio of the number of source points to that offield points, and " , which determines the location of source points in time. Next, we study the effectof each parameter on the characteristics of resulting Green’s basis functions used for approximat-ing the field for domains with various geometries and initial/boundary conditions. For briefness, we



Figure 2. Domain description, initial/boundary conditions, and distribution of field points in the benchmarkproblem used to study the effect of Green’s discrete transformation method parameters.

first present a detailed study on the impact of these parameters on the GDTM approximation of atransient diffusion problem defined on a 10 $ 10 m square domain with D D 0:5 m2/s and ini-tial/boundary conditions illustrated in Figure 2. The analytical solution for the transient field in thisdomain is given by

uexact.x; y; t/ D exp.Dt C 10 ! y/ ! 1: (12)

For the GDTM approximation of the field in this benchmark problem, the space-time domain isdiscretized using M D 225 and )t = 0.1 s, which ensures an adequate level of refinement in boththe space and time domains for discretizing the problem. In the following studies, we monitor thevariation of the relative L2-norm of error versus " , ˛, and ! at the last time step of the solution,which is evaluated as

ke.t/kL2 DsR

. Ou ! u/2 d%Ru2 d%

(13)

3.1. Optimal value of "

As noted previously, " (temporal coordinates of source points) determines the shape of the Green’sbasis functions at each time step, which can have an important effect on the GDTM approximationof the field. To better elucidate this effect, Figure 3 shows the normalized basis functions associatedwith tj ! " D 1; 5; 10; 100 s created at source point ! D .2:5; 2:5/ and with D D 0:01 m2/s.As depicted in that figure, while for small values of tj ! " the Green’s function rapidly decays toG.x; tj I !; "/ % 0, increasing tj ! " flattens the basis function (decays at a slower rate). There-fore, the fundamental solutions corresponding to very small values of tj ! " are highly localizedfunctions that are linearly independent of one another, which results in the construction of a well-conditioned matrix G. However, such spike-shaped functions are not appropriate for simulating acontinuum field, as the resulting solution will be highly oscillatory due to the inappropriate shapes ofbasis functions.

While the basis functions associated with larger values of tj ! " can obviate this deficiency,an excessively large value of tj ! " results in bases that have nearly identical values at severalboundary/field points. Such bases are not only inappropriate for approximating the field but alsolead to the construction of a highly ill-conditioned matrix G, which may not be treated using thepseudo inverse algorithm, and thus can considerably deteriorate the GDTM accuracy. This effect is



(a)

(c)

(b)

(d)

Figure 3. Effect of " on the shapes of normalized Green’s basis functions. The source point is located at! D .2:5; 2:5/ and the diffusivity of the problem is D D 0:01 m2/s.

(a) (b) (c)

Figure 4. Variations of the relative L2-norm of the error versus (a) ˇ, (b) ˛, and (c) ! , while keeping theother two parameters constant in each plot.

similar to that reported in the so-called localized radial basis functions method [36, 48], where thearea of influence of radial functions are confined to improve the condition number of the discretizedproblem. A similar trend can also be observed in the SPH method, where the smoothing length scaleof the Kernel function has a crucial impact on the accuracy of the method [49–51].

According to the discussion previously, the optimal value of " must yield basis functions thathave the ability to properly reconstruct the solution field and simultaneously avoid the formation ofan excessively ill-conditioned matrix G. We propose the following equation for evaluating " as afunction of the grid spacing d used for creating the boundary/field points

" D tj !.ˇd/2

8D ln.10/; (14)

where ˇ is a constant coefficient. This relationship is obtained by enforcing the Green’s basis func-tion to decay to 1% of its maximum value (which occurs at the same spatial coordinate as that ofthe source point) within a radius of jrj D ˇd , that is,

G.! C r; tj I !; "/ D 10"2G.!; tj I !; "/; (15)

The variation of the relative L2-norm of the error versus different values of ˇ for the benchmarkproblem (Figure 2) is illustrated in Figure 4(a). As shown there, ˇ D 2 does not yield a satisfactory



accuracy for approximating the field, which can be attributed to the small overlap between thebasis functions due to their inappropriate shapes. Also, using ˇ > 6 does not yield a satisfactoryprecision, which is due to the excessively high condition number of G; leading to large errors duringthe evaluation of the projection matrix in (10). According to this study, ˇ D 4 yields the optimalaccuracy by providing proper basis functions for approximating the field. It must be noted that thisoptimal value is independent of the values of ˛ and ! (as presented in Figure 4(a)), and also themorphology of the domain (to be studied in Section 3.4).

3.2. Optimal value of ˛

To determine the optimal value of ˛, which governs the size of the enlarged virtual spatial domainfor creating the source points, a similar study as that presented in the Section 3.1 is conductedfor different values of ˛. The variation of the relative L2-norm of the error versus ˛ is depictedin Figure 4(b). While the effect of ˛ varies with different ˇ and ! combinations, using ˛ D 5yields high precision. This observation is repeatable for problems with other domain geometries.The main reason for the improvement in the GDTM accuracy while increasing ˛ is the contributionof the source points located within the enlarged portion of the domain in reconstructing the fieldin the vicinity of boundaries. Note that by using ˇ D 4 for creating the Green’s functions, theresulting bases have a 99% decay over a distance of 4d . Thus, if the source points are far fromthe boundaries in the spatial domain, they decay to zero on the domain boundaries and thereforetheir contributions to approximating the field will be negligible. This explains why using ˛ > 5is generally not effective in improving the GDTM precision. Also, note that while implementing˛ > 5 does not deteriorate the accuracy, it is increasing the total number of source points and thushas a negative impact on the computational cost.

3.3. Optimal value of !

Implementing the discrete transformation (6) in GDTM allows for using a larger number of sourcepoints (Nsp D !M ) than the number of field points (M ). While using more source points canpotentially improve the accuracy, increasing Nsp also increases the computational cost. Therefore,determining the optimal value of ! has an important impact on both the accuracy and efficiency ofthe GDTM. Figure 4(c) illustrates the variations of theL2-norm of the error versus ! for three differ-ent combinations of ˛ and ˇ values (a similar trend is observed for other combinations). As shownin that figure, increasing ! from 1 to 2 significantly improves the accuracy of the GDTM. Moni-toring the condition number of G shows that it is also considerably smaller for ! D 2 comparedwith ! D 1. However, while using ! > 2 increases the computational cost, it has no meaningfulimpact on the accuracy. Thus, ! D 2 is clearly the optimal value for this parameter. This obser-vation is also consistent with that previously reported for the GDTM approximation of Poissonproblems in [28].

3.4. Effect of the domain geometry

As noted previously, the optimal values of ˛, ˇ, and ! for approximating the benchmark prob-lem (Figure 2) also yield the highest accuracy for simulating problems with other geometries andinitial/boundary conditions. The optimal values of these parameters are determined such that theresulting basis functions have appropriate shapes (governed by ˇ) and provide adequate supportinside the domain (governed by !) and in the vicinity of its boundaries (governed by ˛) for approxi-mating the field. Thus, these optimal values are inherently independent of the problem morphology.To verify this statement, the relative L2-norm of the error associated with the last time step of theGDTM simulation of four conductive heat transfer problems with different domain geometries andinitial/boundary conditions are presented in Table I. The values of the error are computed usingstandard FEM reference solutions with highly refined meshes (more than 5$ 104 four-node quadri-lateral elements) and )t D 10"4 s. The distribution of the absolute error is also shown in this tablefor each domain, which better demonstrates the ability of the GDTM for the accurate approximationof the field.



Table I. Error distribution and relative L2-norm of the error at the last time step of the GDTM approxi-mation of the thermal responses of four problems with different domain geometries and boundary

conditions (d D 0:015 m, ) t = 5 $ 10!3 s) using the optimized GDTM parameters(˛ D 5, ˇ D 4, ! D 2).

Note that the contours of the absolute error distribution are warped by the value of the error for more clarity.

(a) (b) (c)

Figure 5. First example problem: (a) domain geometry and boundary conditions; (b) Green’s discretetransformation method (GDTM) and finite element method (FEM) approximations of variations of the tem-perature versus time at points A(0.25, 0.5), B(0.25, 0.25), C(0.13, 0.13). The variation of the relative errorversus time at point C is also presented; (c) variations of the GDTM relative L2-norm of the error and

H1-seminorm of the error versus the number of field points M .

4. CONVERGENCE STUDY

In this section, we further study the accuracy and convergence rate of the GDTM for simulatingthree transient diffusion problems using the optimal parameters evaluated in the previous section forcreating source points: ˛ D 5, ˇ D 4, and ! D 2.

4.1. Heat transfer: square domain

In this example problem, we implement the GDTM to simulate the conductive heat transfer in a0:5 $ 0:5 m square domain with D D 1:13 $ 10"2 m2/s, # D 31 W/(m K), and initial/boundary



conditions depicted in Figure 5(a). The temperature along the bottom edge of the domain is fixed atNu D 0 ıC, side edges are insulated, and the top edge has the convective condition with u1 D 300 ıCand h D 20W/(m2 K). The initial temperature is given by u.x; y; 0/ D 31:2Œexp.y/! 1( (the originof the coordinate system is at the bottom left corner of the domain).

Figure 5(b) presents a comparison between the GDTM and the standard FEM simulations ofthe thermal responses at points A(0.25, 0.5), B(0.25, 0.25), and C(0.13, 0.13), which are labeledsimilarly in Figure 5(a). The GDTM results are obtained usingM D 169 field points and)t D 0:05s, while 4$104 four-node quadrilateral elements with)t D 10"3 s are used to evaluate the standardFEM reference solution. As shown in Figure 5(b), the GDTM results are in good agreement withthose of the FEM. This figure also presents the variation of the relative error versus time at pointC. After an initial small increase in the error, the relative error constantly decreases and approacheszero in the subsequent time steps. The initial increase in the error is due to the sharp gradient ofthe field at the first few time steps in the vicinity of the convective boundary, while using a coarsegrid for discretizing the domain in the GDTM approximation. Note that unlike the FEM, the GDTMdoes not introduce any error associated with the implementation of time marching schemes, whichcan explain why the errors do not keep increasing as the GDTM simulation proceeds. It must benoted that a similar trend is observed for the variations of the relative error versus time at points Aand B, although the maximum error is smaller at these points.

Variations of the relative L2-norm and H1-seminorm of the error versus the number of fieldpoints M at the last time step (t D 40 s) are presented in Figure 5(c). As shown there, increasingM constantly improves the GDTM accuracy, with an expected lower convergence rate for the H1-seminorm than the L2-norm of the error. It must be noted that the H1-seminorm of the error iscalculated similar to the L2-norm of the error, but with respect to the distributional derivatives ofthe approximate field.

4.2. Heat transfer: domain with a curved boundary

In this example, we study the performance of the GDTM for simulating the conductive heat transferin the domain shown in Figure 6(a). The initial temperature of the domain is u.x; y; 0/ D 20 ıC,its left edge has a convective boundary condition with u1 D 100 ıC and h D 500W/(m K), andother edges are insulated. Figure 6(b) shows the GDTM (M D 226,)t D 4$10"3 s) and the FEMapproximations of variations of the temperature versus time at points A(0.3, 0) and B(0.46, 0.36).Similar to the previous example problem, the GDTM yields nearly identical results to those of theFEM reference solution. The variation of the relative error versus time at point A is also plotted inFigure 6(b), which shows a similar trend to the results presented in Figure 5(b). Note that point Ais located at the corner of the domain, which is expected to exhibit the largest error in this problem.The variation of the GDTM relative L2-norm of the error versus different numbers of field points att D 2 s is also depicted in Figure 6(c).

(a) (b) (c)

Figure 6. Second example problem: (a) geometry and boundary conditions; (b) Green’s discrete transfor-mation method (GDTM) and finite element method (FEM) approximations of variations of the temperatureversus time at points A(0.3, 0) and B(0.46, 0.36). The variation of the relative error versus time at point A isalso presented; (c) variation of the GDTM relative L2-norm of the error versus the number of field points at

t D 2 s.



It must be noted that for simulating this problem, the computational cost associated with theGDTM is considerably less than that of the FEM approximation with similar number of degrees offreedom. At each time step of the GDTM solution, the unknown coefficients of the solution series arecomputed using (6), which only requires performing a matrix multiplication. Thus, after computingthe projection matrix R at the first time step, evaluating the GDTM approximation at next timesteps is a computationally inexpensive task. For example, while using 190 field points in the GDTMyields a relative L2-norm of the error of 2 $ 10"4 at t D 2 s, the error associated with the FEMsimulation conducted using the same number of degrees of freedom and temporal discretizationlevel is 4.38 times larger. In addition to yielding a lower accuracy, the FEM approximation requirestime integration, which leads to solving a linear system of equations at each time step and thereforea simulation time of approximately six times larger than that of the GDTM. To measure the CPUtimes needed for this comparison, the ABAQUS software and a MATLAB code are used to performthe FEM and the GDTM simulations, respectively.

4.3. Precipitate growth

In this example, we implement the GDTM to simulate the diffusion-controlled growth of a sim-ple precipitate to verify the accuracy of this method for modeling moving boundary problems. Theprecipitate growth is the spontaneous aggregation of atoms from the supersaturated phase to theequilibrium phase, that is, the precipitate. If the growth rate is determined by the transportationof atoms to the phase boundary, the surface concentration is fixed at the equilibrium concentra-tion ueq [52]. The geometry and boundary conditions of the precipitate growth problem studiedin this example are depicted in Figure 7(a). The initial concentration of atoms in the precipitate

(a) (b)

Figure 7. Third example problem: (a) initial geometry and initial/boundary conditions of the precipitategrowth problem. The gray region shows the dissolved matrix after reaching the saturation concentration.(b) Variation of the Green’s discrete transformation method (GDTM) approximation of the precipitate area

versus time. The dashed line shows the analytically computed area of the fully grown precipitate.

(a) (b) (c)

Figure 8. Third example problem: Green’s discrete transformation method simulation of the concentrationfield and the evolving geometry of the precipitate at three different times.



is u.x; y; 0/ D ueq

#x2

a2C y2

b2

$, where a and b are diameters of the ellipse and ueq D 5 mol/L.

According to (2), the velocity of the moving boundary is given by

vn DDru " nuM ! ueq

; (16)

where D D 10"7 m2/s and uM D 10 mol/L is the concentration of atoms in the surroundingmatrix. As the matrix dissolves into the precipitate, the average concentration within the precipitateincreases and eventually reaches the equilibrium concentration ueq. At this moment, ru D 0 andtherefore the precipitate growth stops. Although no analytical solution exists for this problem, thearea of the fully grown precipitate A (Figure 7(a)) can be computed using the mass conservationlaw, that is,

m0 C .A ! A0/uM D Aueq; (17)

where m0 is the initial mass and A0 is the initial area of the precipitate.The GDTM simulations of the concentration field and evolving morphology of the precipitate at

three different time steps using M D 277 field points and )t D 10"3 s are depicted in Figure 8.As the precipitate geometry evolves, new field points are automatically added on a structured gridinside the domain to ensure the distance between field points does not exceed a maximum value.Figure 7(b) illustrates the variation of the precipitate area versus time, showing its decreasing rateof growth, which eventually stops when it reaches the analytically computed area of the fully grownprecipitate with a relative error of 0:051%.

Figure 9. First application problem: (a) geometry and boundary conditions of the cross section of a turbineblade, together with distribution of the Green’s discrete transformation method field points; (b–d) Green’s

discrete transformation method approximations of the temperature fields at different time steps.



5. APPLICATION PROBLEMS

In this section, we demonstrate the application of the GDTM for solving three diffusion problemswith complex and evolving geometries: heat transfer in a turbine blade, thermal response of a porousmaterial, and the pitting corrosion phenomenon in the stainless steel.

5.1. Heat transfer in a turbine blade

Figure 9(a) shows the geometry and boundary conditions of the cross section of an aircraft turbineblade studied in this example problem. This figure also illustrates the distribution of field pointsused for discretizing the domain. The blade has a thermal conductivity of # D 237 W/(m K), aninitial temperature of u.x; y; 0/ D 20 ıC, and convective boundary conditions with h D 100 W/(mK) and u1 D 100 ıC. The GDTM approximations of the temperature field at three time lev-els are depicted in Figures 9(b) to 9(d), which show the ability of this method for simulating theequilibrium condition.

5.2. Heat transfer in porous material

This example demonstrates the application of the GDTM for approximating the thermal response ofa porous material with the geometry and boundary conditions shown in Figure 10(a). The domainhas a thermal conductivity of # = 31 W/(m K) and an initial temperature of u.x; y; 0/ D 20 ıC.Convective and Dirichlet boundary conditions with h D 20 W/(m K), u1 D 300 ıC, and Nu D 0 ıCare assigned along the top and bottom edges, respectively, while other edges are insulated. Thedistribution of field points (M = 2874) used for discretizing the domain is depicted in Figure 10(b).

(a) (b)

Figure 10. Second application problem: (a) geometry and boundary conditions of a porous material; (b)distribution of field points created for discretizing the domain.

Figure 11. Second application problem: GDTM approximations of the temperature field in a porous materialat three different time levels.



The algorithm used for creating these points was explained in Section 3. Figure 11 illustrates theGDTM approximation of the thermal responses of this porous material at three different times.

5.3. Pitting corrosion

In this final application problem, we implement the GDTM to simulate the pitting corrosion phe-nomenon in the stainless steel. Pitting is a common form of corrosion in high-strength alloys, whichoccurs due to the localized attack of aggressive anions (e.g., chloride) on the metal surface afterthe breakdown of its protective film [53, 54]. The propagating pit induces stress concentrations inthe corroding material, which can lead to crack nucleation and accelerated mechanical failure [55].Therefore, the ability to accurately simulate the growth rate and the morphology of corrosion pits iscrucial to the reliable design and health monitoring of structures that are prone to this phenomenon(e.g., aircrafts, ships, and bridges). While the complex evolving geometry of the propagating pit hin-ders the straightforward application of the standard FEM for modeling this problem, the meshfreeapproach of the GDTM can considerably facilitate its simulation.

To approximate the distribution of dissolved ions in the electrolyte solution formed inside thecorrosion pit and its moving boundary velocity, we adopt the activation-diffusion-controlled modeldescribed in [56]. This model assumes that after the concentration of dissolved ions reaches thesaturation concentration usat, a salt film is formed on the pit surface (Figure 12), which avoids furtherincrease of the concentration along this boundary. The velocity of the moving pit boundary is thenevaluated using the Rankine–Hugoniot condition [57], which is given by

ŒDruC .usat ! usolid/v( " n D 0 8x 2 & W u.x; t / D usat; (18)

where usolid is the concentration of atoms in the metallic material. Because the saturated concen-tration of the salt film is constant, the Dirichlet condition with Nu D usat is prescribed along the pitboundary at this stage.

Before the formation of the salt film, an activation-controlled model is implemented to evaluatethe velocity of the evolving pit boundary. In this model, an electrochemical kinetics of Butler–Volmer type describes the metal dissolution and the Faraday’s second law [54] is employed to relatethe electric current to the dissolution rate [57]. The normal component of the moving boundaryvelocity for u < usat is evaluated as

vn D v " n D Adiss

usolidexp

%´F.Ecorr C ˛$/

RT

&8x 2 & W u.x; t / < usat; (19)

where Ecorr is the corrosion potential of the metallic material, $ is the over-potential, Adiss is thedissolution affinity, F D 96485 A/mol is the Faraday’s constant, R D 8:314 J/(mol K) is the Gasconstant, and ´ is the average charge number. Unlike the diffusion-controlled case ( Nu D usat), thevelocity of the moving boundary can be computed in advance from (19). By replacing vn in (18),one can assign the following Robin condition along unsaturated portions of the pit boundary

Figure 12. Third application problem: pitting corrosion problem description, illustrating the pit morphology,the electrolyte solution, and the salt film formed on the pit boundary.



Table II. Values of different parameters used for defining the pittingcorrosion example problem.

Parameter Value Unit Parameter Value Unit

D 8:5 $ 10"6 cm2/s Ecorr !0.24 Vusat 5.1 mol/L $ 0.13 Vusolid 143 mol/L T 298.15 KAdiss 4 mol/(cm2 s) ´ 2.19 !

Figure 13. Third application problem: Green’s discrete transformation method simulation of the evolvingmorphologies and dissolved ions concentrations in three corrosion pits at different time levels. Insets of the

figures show the distribution of field points in the vicinity of the moving boundary.

Dru.x; t / " nC u.x; t /vn D usolidvn 8x 2 & W u.x; t / < usat: (20)

The initial morphology and boundary conditions of one of the corrosion pits simulated in thisexample are depicted in Figure 12. A constant Dirichlet boundary condition of Nu D 0 mol/L isassigned along the pit mouth, while the initial concentration of ions in the electrolyte is zero every-where. Values of other parameters needed for defining this problem are given in Table II. The pittingcorrosion begins as an activation-controlled process until portions of the pit surface reach the satu-ration concentration usat, which leads to the formation of the salt film. At this stage, we switch tothe diffusion-controlled model to evaluate the velocity of the moving pit boundary.

The GDTM simulation of evolving morphologies and concentrations of dissolved ions in theelectrolyte solution for three corrosion pits at different time levels are depicted in Figure 13. Foreach problem, the inset of the top figure shows the distribution of boundary/field points used fordiscretizing the domain. Throughout the simulation, new field points are automatically generatedon a structured grid and added to the discretized model to maintain a maximum distance of 5*mbetween the field points in the expanding domain. As shown in Figure 13, the GDTM simulatesthe smoothening of the growing pit morphology, which occurs due to similar processes to theelectropolishing phenomenon.

6. CONCLUSION

A new meshfree method (GDTM) was introduced that relies on Green’s basis functions, togetherwith a discrete transformation technique to simulate transient diffusion problems, including those



with moving boundaries. The solution field is approximated as a linear combination of Green’s func-tions, which automatically satisfies the governing equation and thus eliminates the need to employtime integration schemes. We proposed a straightforward algorithm for constructing the sourcepoints defining the Green’s functions such that the resulting basis functions provide optimal charac-teristics for approximating the field. A convergence study was presented to investigate the accuracyand convergence rate of the GDTM for simulating varying diffusion problems, showing that thismeshfree method yields a similar level of accuracy as that of the FEM using less number of degreesof freedom for discretizing the domain. Finally, the GDTM was employed to simulate three diffu-sion problems with complex and evolving morphologies: conductive heat transfer in a gas turbineblade, thermal response of porous material, and pitting corrosion in stainless steel.

ACKNOWLEDGEMENT

This work has been supported by funding from the Department of Materials Science and Engi-neering at The Ohio State University and in part by an allocation of computing time from the OhioSupercomputer Center and the Simulation Innovation and Modeling Center (SIMCenter).

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