applied sciences Article
A Modified Polynomial Expansion Algorithm for Solving the Steady-State Allen-Cahn Equation for Heat Transfer in Thin Films Chih-Wen Chang 1, * 1 2
*
ID
, Chein-Hung Liu 1 and Cheng-Chi Wang 2, *
Department of Mechanical Engineering, National Chung Hsing University, Taichung 40227, Taiwan;
[email protected] Graduate Institute of Precision Manufacturing, National Chin-Yi University of Technology, Taichung 41170, Taiwan Correspondence:
[email protected] (C.-W.C.);
[email protected] (C.-C.W.); Tel.: +886-4-2284-0433 (C.-W.C.); +886-4-2392-4505 (ext. 5163) (C.-C.W.)
Received: 20 April 2018; Accepted: 13 June 2018; Published: 15 June 2018
Featured Application: The specific application of the work is for heat and mass transfer. Abstract: Meshfree algorithms offer a convenient way of solving nonlinear steady-state problems in arbitrary plane areas surrounded by complicated boundary shapes. The simplest of these is the polynomial expansion approach. However, it is rarely utilized as a primary tool for this purpose because of its rather ill-conditioned behavior. A well behaved polynomial expansion algorithm is presented in this paper which can be more effectively used to solve the steady-state Allen-Cahn (AC) equation for heat transfer in thin films. In this method, modified polynomial expansion was used to cope with each iteration of the steady-state Allen-Cahn equation to produce nonlinear algebraic equations where multiple scales are automatically determined by the collocation points. These scales can largely decrease the condition number of the coefficient matrix in each nonlinear system, so that the iteration process converges very quickly. The numerical solutions were found to be accurate and stable against moderate noise to better than 7.5%. Computational results verified the method and showed the steady-state Allen-Cahn equation for heat transfer in thin films could easily be resolved for several arbitrary plane domains. Keywords: steady-state Allen-Cahn equation; meshless approach; modified polynomial expansion; boundary value problems; heat transfer in thin films
1. Introduction Much interest has been shown in the solution of the Allen-Cahn (AC) equation, the primary purpose of which has been depiction of the motion of antiphase boundaries in crystalline solids and phase separation in binary alloys [1]. This equation is now utilized in many moving boundary issues involving thermodynamic driving forces in microstructure evolution, domain evolution in thin films, image processing, fluid dynamics, and materials science through a phase-field approach [2–7]. The Allen-Cahn equation has been employed to model diverse phenomena in nature, for example, Beneš et al. [8] proposed a method of pattern recovery (image segmentation) based on solution of the AC equation. The method is often realized as a regularization of the level-set motion by mean curvature where a particular forcing term is added that allows the initial level set to nearly encompass the pattern in question. This demonstrates convergence of the numerical approach and display function of the method for several artificial and real instances.
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Feng and Prohl [9] constructed some useful a priori error estimates for proposed numerical schemes. In addition, the optimal order and quasi-optimal order error bounds were displayed for semi-discrete and fully discrete schemes under different constraints of mesh and time step size and with different regularity assumptions about the initial datum function. Later, Wheeler et al. [10] identified three stages of temporal evolution for AC equations: the first corresponds to interfacial genesis, which happens very fast; the second is interfacial motion controlled by diffusion and the local energy difference across the interface; the time scale of the last stage was longer and here curvature effects were pivotal. After this, Sabir et al. [11] employed the mathematical formulation of tumor hypoxia-targeting by presenting the decay parameter of oxygen in a model [12,13]. For numerical calculation the conforming Q1 finite element scheme for space discretization, as well as the second-order diagonally implicit fractional step θ-method for temporal discretization, were used. Note that the distribution of nutrients in tissues has substantial influence on tumor structure and growth rate. Later, Zahra [14] developed a numerical solution based on the nonpolynomial B-spline (trigonometric B-spline) collocation method for solving the AC equation. This algorithm combined the trigonometric B-spline interpolant and the θ-weighted scheme for space and time discretization. Von Neumann stability analysis showed the proposed technique to be unconditionally stable. Bulent et al. [15] investigated the numerical solution of the AC equation with constant and degenerate mobility, as well as with polynomial and logarithmic energy functionals. They discretized the model equation using the symmetric interior penalty Galerkin method in space, and by the average vector field (AVF) method in time. They also showed that the energy stable AVF method as the time integrator for gradient systems, like the AC equation, satisfies the energy decreasing property for a fully discrete scheme. They found that the discrete energy decreased monotonically, phase separation and metastability phenomena could be observed, and the ripening time was detected correctly. After that, Yang et al. [16] addressed the uniform bounds associated with the AC equation and its numerical discretization schemes. These uniform bounds were different from, and weaker than, the conventional energy dissipation and the maximum principle. However, they can be helpful in the analysis of the numerical approach. Moreover, fully discretized schemes on the basis of the Fourier collocation method for spatial discretization and the Strang splitting method for time discretization also preserved the uniform L2 -bound unconditionally. In all the references above discussions were about time-dependent AC equations and regular domains. There was no mention of time-independent AC equations and irregular domains. Liu and Kuo [17] used a single-scale and multiple Pascal triangle formulation to cope with linear elliptic partial differential equations (PDEs) in a simply connected domain that has a complicated boundary shape and obtained accurate results. After that, Liu and Young [18] employed a multiplescale Pascal polynomial to deal with two-dimensional Stokes and inverse Cauchy-Stokes problems and also got good results. Later, Chang [19] addressed steady-state nonlinear heat conduction problems in an arbitrary plane domain enclosed by a complex boundary utilizing a multiple-scale polynomial expansion scheme. In addition, Chang et al. [20] also applied multiple-scale polynomial expansion to tackle the steady-state modified Burgers’ equation in transport problems and got good results. The rest of this paper is organized as follows: Section 2 displays the steady-state Allen-Cahn (AC) equation and a modified polynomial expansion scheme. The multiple-scale idea for the Pascal triangle expansion algorithm, which is totally resolved by the collocation points. The iterative process for solving steady-state AC equations is presented in Section 3. In Section 4, numerical examples for these issues are given. Section 5 offers some concluding remarks. 2. The Steady-State Allen-Cahn Equation and a Modified Polynomial Expansion Method We start with the steady-state Allen-Cahn equation as follows: ∆u( x, y) = u − u3 + H ( x, y), ( x, y) ∈ Ω,
(1)
u( x, y) = G ( x, y), ( x, y) ∈ Γ1 ,
(2)
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un ( x, y) = F ( x, y), ( x, y) ∈ Γ2 ,
(3)
where ∆ is the Laplacian operator, and F, G and H are given functions. Γ is the boundary of issue domain Ω with Γ = Γ1 ∪ Γ2 , and n is a unit outward normal on Γ. Under the presented Dirichlet boundary condition (2) and the Neumann boundary condition (3), we can resolve Equation (1) to acquire a solution of u(x,y). Then, we employ the polynomial expansion as a trial solution of the PDE and formulate the needed algebraic equations after a suitable collocation in the problem area. However, this is rarely utilized as a major numerical tool to resolve the nonlinear PDEs. The main reason being that the resultant nonlinear algebraic equations (NAEs) are often seriously ill-conditioned. The elements in the polynomial matrix are as follows: 1 y y2 ··· y m −1 ym x xy xy2 · · · xym−1 xym 2 x 2 y x 2 y 2 · · · x 2 y m −1 x 2 y m x (4) .. .. .. .. .. . . . ··· . . m m m 2 m m − 1 m m x x y x y ··· x y x y are often utilized to expand the solution of u(x,y). If the elements are restrained from the upper-left triangle, the result is the famous Pascal triangle expansion: 1 x y x2 xy y2 x3 x2 y xy2 y3 4 x x3 y x2 y2 xy3 y4 x5 x4 y x3 y2 x2 y3 xy4 y5 ··· ··· ··· ··· ··· ··· ··· ···
(5)
Consequently, the solution u(x,y) is expanded by m
u( x, y) =
i
∑ ∑ pij xi− j y j−1 ,
(6)
i =1 j −1
where the coefficients pij are to be resolved, where the number of all elements is n = m(m + 1)/2. The highest order of the above polynomial is m − 1. From Equation (6), the following equations can be directly stated: m
u x ( x, y) =
i
∑ ∑ pij (i − j)xi− j−1 y j−1 ,
(7)
i =1 j −1 m
uy ( x, y) =
i
∑ ∑ pij ( j − 1)xi− j y j−2 ,
(8)
i =1 j −1
m un ( x,y)∈Γ = ∑
i
∑ pij
h
i ( i − j ) x i − j −1 y j −1 n x + ( j − 1 ) x i − j y j −2 n y ,
(9)
i =1 j −1
∆u( x, y) =
m
i
∑ ∑ pij [(i − j)(i − j − 1)xi− j−2 y j−1 + ( j − 1)( j − 2)xi− j y j−3 ].
(10)
i =1 j −1
Introducing these equations to Equations (1)–(3), and selecting n1 and n2 collocation points on the boundary and in the area, to satisfy the boundary condition and the field equation, we obtain a system of NAEs to deal with the n coefficients pij .
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The multi-scale Pascal triangle is then introduced as follows. Because x and y in the issue area Ω may be arbitrarily large, the above expansion would lead to a divergence of the powers xm and ym . To acquire an accurate solution of an AC equation employing the modified Pascal triangle polynomial expansion scheme, we have to develop a more accurate and effective method to cope with these NAEs by reducing the condition numbers. A new multiple-scale Pascal triangle is obtained by expansion of u(x,y) by m
u( x, y) =
i
∑ ∑ pij sij xi− j y j−1 ,
(11)
i =1 j −1
in which the scales sij are decided as below. The coefficients pij utilized in the expansion (6) can be demonstrated as an n-dimensional vector p with components pk , k = 1, . . . , n by k=0 Do i = 1, m Do j = 1, i (12) k=k+1 pk = pij Enddo. After that, for a generic point (x, y)∈ Ω, the term u(x, y) can be displayed as an inner product of a vector a with p, i.e., p1 p 2 T p u( x, y) = [ 1 x y x2 xy y2 x3 x2 y xy2 y3 . . . ] (13) 3 = a p. . . . pn Likewise, for a generic point (x, y)∈ Ω, the term ∆u(x, y) can be displayed as an inner product of a vector q with p, in which the components qk are of the type qk = (i − j)(i − j − 1) xi− j−2 y j−1 + ( j − 1)( j − 2) xi− j y j−3 . Then, while we select n1 points (xi , yi ), i = 1, . . . , n1 on the boundary Γ to satisfy the boundary condition, and n2 points (xi , yi ), i = 1, . . . , n2 on the area Ω to satisfy the field equation. For example, for Laplace’s equation we have T a1 G ( x1 , y1 ) . .. .. . aT G(x , y ) n1 n1 n1 A = T , b = (14) . q1 0 .. .. . . T 0 qn2 We can cope with a normal linear system in place of Ap = b:
where
Kp = b1
(15)
b1 = AT b, K = AT A > 0.
(16)
The conjugate gradient method (CGM) can be employed to resolve Equation (15). If we investigate the norm of each column of the coefficient matrix of A is equal, the multiple-scale sij is equal to kp1 k/kpk k, where s11 = 1 and pk denotes the kth column of A in Equation (16). Such that in the new system
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Ip = b,
(17)
the n column norms of the new coefficient matrix B are equal. Let Mk = sij , and we can present a post conditioning matrix: Q := diag( M1 , . . . , Mn ),
(18)
such that the above equilibrated multiple-scale skill is equivalent to acquiring the new coefficient I by I = AQ.
(19)
3. The Iterative Process for the Steady-State Allen-Cahn Equation At the start, we utilize pij = p0ij and (u, ux , uy ) are calculated by m
u( x, y) =
i
∑ ∑ pij sij xi− j y j−1 ,
(20)
i =1 j −1 m
u x ( x, y) =
i
∑ ∑ pij (i − j)sij xi− j−1 y j−1 ,
(21)
i =1 j −1 m
uy ( x, y) =
i
∑ ∑ pij ( j − 1)sij xi− j y j−2 ,
(22)
i =1 j −1
After that, upon collocating nk points to satisfy the boundary conditions (2) and (3) and the field equation Ap = b, we can formulate a coefficient matrix A from the left-hand side of the field equation with the aid of Equations (7)–(10), and then by employing the multiple-scale sij , therefore P via Equation (18). Meanwhile a tentative right-hand side b0 is acquired from the right-hand side of Equation (1) by introducing Equations (20)–(22). Hence, we have a linear system with a different right-hand side: AQp = bk , (23) whose normal form is handled by the CGM to generate a new coefficient p1ij . Iteration is continued until convergence. The numerical processes this algorithm uses to resolve the steady-state AC equation are outlined in the following steps. (i) (ii)
Given n1 , n2 (nk = n1 × n2 ), and m [n = m(m + 1)/2]. Given p0ij .
(iii) Given collocation points (xi , yi ), i = 1, . . . , nk . (iv) One-time only: Generate sij , sij = kp1 k/kpk k, Generate Q from Mk = sij and Equation (18), I = AQ. (v)
(24)
For k = 0, 1, 2 , . . . , reiteration as follows: Compute u, ux , uy from Equations (20)–(22) Generate bk from the right-hand side of Equation (1) Resolve the normal form of Ip = bk to acquire pijk+1
(25)
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r If pijk converges in accordance with a given stopping criterion
2
∑im=1 ∑ij=1 ( pijk+1 − pijk ) < ε 2 ,
which is a named relative distance coefficient, then stop; otherwise, go to step (iv). 4. Numerical Examples 4.1. Example 1 We first consider the following AC equation in heat transfer of thin films in an arbitrary domain (peanut shape, Cassini shape, gear shape, and amoeba-like irregular shape) as shown in Figure 1. ∆u = u − u3 + H ( x, y), r ρ(θ ) = 0.3
cos(2θ ) +
ρ(θ ) = cos 4θ +
q
q
(26)
1.1 − sin2 (2θ ), 2
18/5 − sin 4θ
(27)
1/3 ,
x = 0.5[2 + 0.5 sin(7θ )] cos[θ + 0.5 sin(7θ )], y = 0.5[2 + 0.5 sin(7θ )] sin[θ + 0.5 sin(7θ )], ρ(θ ) = exp[sin(θ )] sin2 (2θ ) + exp[cos(θ )] cos2 (2θ ),
(28) (29) (30)
where ρ is the radius function of Γ, H(x,y) = u3 − u + 6exp(xy3 + x3 y) − 6x2 y(x2 + 2y2 ) and the closed-form solution is u = x3 y3 (exp − x ). (31) Under the Dirichlet boundary condition, for the peanut domain, with n1 = 27, n2 = 4 (nk = 108), m = 7, and n = 28, we employed the multiple-scale expansion method by using the CGM under the convergence criterion ε1 = 10− 8 . At ε2 = 10− 8 , the iterative process reaches convergence after 7 iterations and the results are shown in Figure 2a,b. The numerical error is shown in Figure 3, the maximum error being 1.88×10−10 . For the Cassini-shaped domain, with n1 = 26, n2 = 2 (nk = 52), m = 8, n = 36, and a convergence criterion of ε1 = 10− 12 . At ε2 = 10− 12 , the iterative process reaches convergence after 11 iterations and the results are shown in Figure 4a,b. The numerical error is plotted in Figure 5, the maximum error being 1.40 × 10−13 . For the gear irregular domain, under n1 = 25, n2 = 2 (nk = 50), the same m, n, and convergence criterion ε1 = 10−3 . At ε2 = 10−3 , convergence is reached after 8 iterations and the results are shown in Figure 6a,b. The numerical error is plotted in Figure 7, the maximum error being 3.09. Comparison with the maximum absolute value of 124.88 of u(x, y), shows this error to be acceptable. For the amoeba-like irregular domain, with n1 = 50, n2 = 2 (nk = 100), the same m, n, and convergence criterion, convergence is reached after 11 iterations and the results are shown in Figure 8a,b. The numerical error is plotted in Figure 9 and the maximum error is 9.04 × 10−9 . Note that the multiple-scale method is highly efficient, and it can provide a very accurate solution. Furthermore, to the authors’ best knowledge, there are no existing published reports of numerical solutions to this problem that provide more accurate results than these. To address concern about the stability of this scheme when boundary data were perturbed by random noise, an investigation was carried out with random noise added to the boundary data. For the amoeba-like irregular domain, with n1 = 60, n2 = 2 (nk = 120), m = 8, n = 36, and convergence criterion ε1 = 10−4 . At ε2 = 10−4 , convergence was reached after 100 iterations and the results are shown in Figure 10a,b. The numerical error plot is shown in Figure 11 and the maximum error was 0.36. Note that the multiple-scale method still gives an accurate result with imposed noise equal to 0.5.
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Figure 1. Cont.
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Figure 1. Four geometric configurations of the Allen-Cahn (AC) equation in heat transfer of thin films
Figure 1. Four geometric configurations of the Allen-Cahn (AC) equation in heat transfer of thin films are shown in (a) the peanut shape, in (b) the shape,(AC) in (c) the gearinirregular shape, (d) Figure 1. Four geometric theCassini Allen-Cahn transfer ofand thinin films are shown in (a) the peanutconfigurations shape, in (b) of the Cassini shape, in equation (c) the gearheat irregular shape, and in (d) the amoeba-like irregular shape. are shown in (a) the peanut shape, in (b)ofthe shape, in (c) the gearinirregular shape, (d) Figure 1. Four geometric configurations theCassini Allen-Cahn (AC) equation heat transfer of and thin in films the amoeba-like irregular shape. the shown amoeba-like shape. are in (a) irregular the peanut shape, in (b) the Cassini shape, in (c) the gear irregular shape, and in (d) the amoeba-like irregular shape.
Figure 2. The exact solutions for the steady-state AC equation with peanut-shaped domain are shown in (a), and in (b) thesolutions proposedfor approach solutionAC without random Figure 2. The exact the steady-state equation withnoise. peanut-shaped domain are shown
Figure 2. The exact solutions for the steady-state AC equation with peanut-shaped domain are shown in (a), and in (b) thesolutions proposed solutionAC without random Figure 2. The exact forapproach the steady-state equation with noise. peanut-shaped domain are shown in (a), and in (b) the proposed approach solution without random noise. in (a), and in (b) the proposed approach solution without random noise.
Figure 3. The numerical errors of the proposed solution method for the steady-state AC equation with peanut-shaped domain. errors of the proposed solution method for the steady-state AC equation with Figure 3. The numerical peanut-shaped domain.errors of the proposed solution method for the steady-state AC equation with Figure 3. The numerical
Figure 3. The numerical errors of the proposed solution method for the steady-state AC equation with peanut-shaped domain. peanut-shaped domain.
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Figure 4. The are shown Figure Theexact exactsolutions solutionsfor forthe thesteady-state steady-stateAC ACequation equationwith withCassini-shaped Cassini-shapeddomain domain Figure 4. 4. The exact solutions for the steady-state AC equation with Cassini-shaped domain are are shown shown in (a), and in (b) the proposed approach solution without random noise. Figure 4. The exact solutions forapproach the steady-state AC equation with Cassini-shaped domain are shown in (a), and in (b) the proposed solution without random noise. in (a), and in (b) the proposed approach solution without random noise. in (a), and in (b) the proposed approach solution without random noise.
Figure 5. The numerical errors of the proposed method solution for the steady-state AC equation with Figure 5. 5. The numerical errors the proposed method solution for the steady-state AC equation with Figure Thenumerical numerical errorsofof ofthe theproposed proposedmethod methodsolution solutionfor for thesteady-state steady-stateAC AC equationwith with Cassini-shaped domain.errors Figure 5. The the equation Cassini-shaped domain. domain. Cassini-shaped Cassini-shaped domain.
Figure 6. The exact solutions for the steady-state AC equation with gear irregular domain are shown Figure 6. The exact solutions for the steady-state AC equation with gear irregular domain are shown in (a), and in (b) thesolutions proposedfor approach solution AC without random noise. Figure 6.6.The exact the equation with gear Figure The exact thesteady-state steady-state equation withnoise. gearirregular irregulardomain domainare areshown shown in (a), and in (b) thesolutions proposedfor approach solution AC without random inin(a), (a),and andinin(b) (b)the theproposed proposedapproach approachsolution solutionwithout withoutrandom randomnoise. noise.
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Figure The numerical errors the proposed method solution for the steady-state AC equation with Figure7.7. 7.The Thenumerical numericalerrors errorsofof ofthe theproposed proposedmethod methodsolution solutionfor forthe thesteady-state steady-stateAC ACequation equationwith with Figure Figure 7. The numerical errors of the proposed method solution for the steady-state AC equation with gear irregular domain. gear irregular domain. gear irregular domain. gear irregular domain.
Figure 8. The exact solutions for the steady-state AC equation with amoeba-like irregular domain are Figure 8. The Figure Theexact exactsolutions solutionsfor forthe thesteady-state steady-stateAC ACequation equationwith withamoeba-like amoeba-likeirregular irregulardomain domainare are Figure 8. 8. The exact for the steady-state AC equation with amoeba-like shown in (a), and insolutions (b) the proposed approach solution without random noise. irregular domain are shown inin(a), and in (b) the proposed approach solution without random noise. shown (a), and in (b) the proposed approach solution without random noise. shown in (a), and in (b) the proposed approach solution without random noise.
Figure 9. The numerical errors of the proposed method solution for the steady-state AC equation with Figure 9. The numerical errors of the proposed method solution for the steady-state AC equation with Figure 9. Theirregular numerical errors of the proposed method solution for the steady-state AC equation with amoeba-like domain. Figure 9. The numerical errors of the proposed method solution for the steady-state AC equation with amoeba-like irregular domain. amoeba-like irregular domain. amoeba-like irregular domain.
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Figure 10. Thenumerical numerical solutions for the steady-state AC equation with amoeba-like irregular Figure 10. for for the the steady-state AC equation with amoeba-like irregularirregular domain Figure 10. The The numericalsolutions solutions steady-state AC equation with amoeba-like domain are shown in (a), and in (b) the proposed approach solution with random noise. are shown in (a), and in (b) the proposed approach solution with random noise. domain are shown in (a), and in (b) the proposed approach solution with random noise.
Figure 11. The numerical errors with random noise. Figure Figure 11. 11. The The numerical numerical errors errors with with random random noise. noise.
4.2. Example 2 4.2. 4.2. Example Example 22 A steady-state AC equation was derived for these four shape domains, where H(x,y) = u33 − A AC equation equation was was derived derived for these four shape domains, where H(x,y) = u3 − A steady-state steady-state 3exp·sin(x)sin(y) andAC the closed-form solution is:for these four shape domains, where H(x,y) = u − 3exp·sin(x)sin(y) and the closed-form solution 3exp·sin(x)sin(y) and the closed-form solution is: is: u exp sin( x)sin( y ). (32) u exp sin( x)sin( y ). (32) u = exp · sin( x ) sin(y). (32) Under the Dirichlet boundary condition, for the peanut domain, with n1 = 28, n2 = 4 (nk = 112), m Under the Dirichlet boundary condition, for the peanut domain, with n1 = 28, n2 = 4 (nk = 112), m = 7, and n = 28, a multiple-scale expansion approach was utilized by applying the CGM with a Under boundaryexpansion condition,approach for the peanut domain,by with n1 = 28,the n2 =CGM 4 (nk with = 112), = 7, and n =the 28,Dirichlet a multiple-scale was utilized applying a −11. At ε2 = 10−11, convergence was reached after 6 iterations and the convergence criterion of ε1 = 10−11 m = 7, and n criterion = 28, a multiple-scale expansion was was utilized by applying the CGMand with convergence of ε1 = 10 . At ε2 = 10−11,approach convergence reached after 6 iterations thea results are shown in Figure 12a,b. The numerical error is shown in Figure 13, the maximum error convergence criterion of ε1 = 12a,b. 10−11 . The At ε2numerical = 10−11 , convergence wasinreached 6 iterations and the results are shown in Figure error is shown Figure after 13, the maximum error −9 being 5.82 × 10−9 . For the Cassini-shaped domain, with n1 = 50, n2 = 10 (nk = 500), m = 12, n = 78, and results5.82 are ×shown in Figure 12a,b. The numerical errorn1is=shown 13, the being 10 . For the Cassini-shaped domain, with 50, n2 =in10Figure (nk = 500), m =maximum 12, n = 78,error and −10. At ε2 = 10−10, convergence was reached after 17 iterations and the convergence criterion of ε1 = 10−10 −10, convergence being 5.82 × criterion 10−9 . Forofthe with n1 was = 50,reached n2 = 10 after (nk =17 500), m = 12,and n =the 78, convergence ε1 =Cassini-shaped 10 . At ε2 = 10domain, iterations results are shown in Figure 14a,b. The numerical error is shown in Figure 15, and the maximum error − 10 − 10 and convergence ε1 = 10 . At ε2 = 10error , convergence was reached iterationserror and results are showncriterion in Figureof14a,b. The numerical is shown in Figure 15, andafter the 17 maximum −9. For the irregular gear domain, with n1 = 65, n2 = 3 (nk = 195), m = 5, n = 15, and was 2.45 × 10−9 the results in Figure 14a,b. numerical 15, m and then maximum was 2.45 × are 10 shown . For the irregular gearThe domain, witherror n1 = is 65,shown n2 = 3in(nFigure k = 195), = 5, = 15, and −2. At ε2 = 10−2, convergence was reached after 15 iterations and the convergence criterion of ε1 = 10−2 error was 2.45 × 10−9 .ofFor with was n1 = reached 65, n2 = after 3 (nk 15 = 195), m = 5,and n =the 15, convergence criterion ε1 =the 10irregular . At ε2 =gear 10−2,domain, convergence iterations results are shown in Figure 16a,b. The numerical error is shown in Figure 17, and the maximum error − 2 − 2 and convergence ε1 = 10 . At ε2 = 10error , convergence reached after iterationserror and results are shown criterion in Figureof16a,b. The numerical is shown in was Figure 17, and the15 maximum was 0.61. Comparison with the maximum absolute value of 2.71 of u(x, y) shows the above error to the results are shown in Figure The numerical errorofis 2.71 shown in Figure 17, and the maximum was 0.61. Comparison with the 16a,b. maximum absolute value of u(x, y) shows the above error to be acceptable. For the amoeba-like irregular domain, with n1 = 50, n2 = 2 (nk = 100), m = 11, n = 66, and error was 0.61.For Comparison with the maximum absolute value u(x, shows be acceptable. the amoeba-like irregular domain, with n1 = of 50,2.71 n2 =of 2 (n k =y) 100), m =the 11, above n = 66,error and −7. At ε2 = 10−7, convergence was reached after more than 100 iterations convergence criterion of ε1 = 10−7 −7 to be acceptable. For the domain, with = 50, n2after = 2 (n = 11, n = 66, convergence criterion of εamoeba-like 1 = 10 . At ε2irregular = 10 , convergence wasn1reached more thanm 100 iterations k = 100), and the results are shown in Figure 18a,b. The numerical error is shown in Figure 19, and the and the results are shown in Figure 18a,b. The numerical error is shown in Figure 19, and the maximum error was 1.70 × 10−3. Note that the multiple-scale scheme is highly efficient, and it can offer maximum error was 1.70 × 10−3. Note that the multiple-scale scheme is highly efficient, and it can offer
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and convergence criterion of ε1 = 10−7 . At ε2 = 10−7 , convergence was reached after more than 100 iterations and the results are shown in Figure 18a,b. The numerical error is shown in Figure 19, and the Appl. Sci. 2018, 8, x 12 of 16 − 3 Appl. Sci. 2018, 8, x 12 of 16 maximum error was 1.70 × 10 . Note that the multiple-scale scheme is highly efficient, and it can accurate solution. Toauthors’ the authors’ knowledge, there no existing published reports aoffer verya very accurate solution. To the bestbest knowledge, there are are no existing published reports of a very accurate solution. To the authors’ best knowledge, there are no existing published reports of of numerical solutions to this problem that provide more accurate results than these. numerical solutions to this problem that provide more accurate results than these. numerical solutions to this problem that provide more accurate results than these. Concern about about the the stability stability of of this this scheme, scheme, where where the the boundary boundary data data were were perturbed perturbed by by random random Concern Concern about the stability of this scheme, where the boundary data were perturbed by random noise, was investigated by adding random noise to the boundary data. For the amoeba-like irregular noise, was investigated by adding random noise to the boundary data. For the amoeba-like irregular noise, was investigated by adding random noise to the boundary data. For the amoeba-like −2 . At εirregular −2 domain, with domain, with nn11 == 35, 35, nn22 == 22 (n (nkk = = 70), 70), m= m= 7, 7, nn == 28, 28, and andconvergence convergencecriterion criterionofofε1ε=1 =1010−2 . At ε22==10 10−2,, domain, with n1 = 35, n2 = 2 (nk = 70), m= 7, n = 28, and convergence criterion of ε1 = 10−2. At ε2 = 10−2, convergence was was reached reached after after more more than than 100 100 iterations iterations and and the the results results are are shown shown in in Figure Figure 20a,b. 20a,b. convergence convergence was reached after more than 100 iterations and the results are shown in Figure 20a,b. The numerical error is shown Figure 21, and the maximum error was 0.23. Note that the multiple-scale The numerical error is shown Figure 21, and the maximum error was 0.23. Note that the multipleThe numerical error is shown Figure 21, and the maximum error was 0.23. Note that the multiplemethod can acquire accurate resultsresults with imposed noise noise as large 7.5%. scale method can acquire accurate with imposed as as large as 7.5%. scale method can acquire accurate results with imposed noise as large as 7.5%.
Figure 12. The The exactsolutions solutions steady-state AC equation peanut-shaped domain are Figure forfor thethe steady-state AC equation with with peanut-shaped domain are shown Figure 12. 12. Theexact exact solutions for the steady-state AC equation with peanut-shaped domain are shown in (a), andthe in proposed (b) the proposed approach solution without random noise. in (a), and in (b) approach solution without random noise. shown in (a), and in (b) the proposed approach solution without random noise.
Figure 13. The numerical errors of proposed method solution for the steady-state AC equation with Figure 13. The numerical errors of proposed method solution for the steady-state AC equation with 13. The numerical aFigure peanut-shaped domain. errors of proposed method solution for the steady-state AC equation with a a peanut-shaped domain. peanut-shaped domain.
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Figure 14. The The exactsolutions solutionsforfor steady-state equation Cassini-shaped domain are Figure 14. thethe steady-state AC AC equation with with Cassini-shaped domain are shown Figure 14. Theexact exact solutions for the steady-state AC equation with Cassini-shaped domain are shown in (a), and in (b) the proposed noise. Figure 14. The exact solutions for theapproach steady-state AC without equation with Cassini-shaped domain are in (a), and in (b) the proposed approach solutionsolution without randomrandom noise. shown in (a), and in (b) the proposed approach solution without random noise. shown in (a), and in (b) the proposed approach solution without random noise.
Figure 15. The numerical errors of proposed approach solution for the steady-state AC equation with Figure 15. The numerical errors of of proposed proposedapproach approachsolution solutionfor forthe thesteady-state steady-state ACequation equation witha 15.The Thenumerical numerical errors aFigure Cassini-shaped domain.errors Figure 15. of proposed approach solution for the steady-state AC AC equationwith with aCassini-shaped Cassini-shapeddomain. domain. a Cassini-shaped domain.
Figure 16. The exact solutions for the steady-state AC equation with irregular gear shaped domain Figure 16. The exact solutions for the steady-state AC equation with irregular gear shaped domain are displayed (a), and in (b) for thethe proposed approach solution without random noise. Figure 16. exact solutions the steady-state ACequation equationwith withirregular irregular gear shaped domain Figure 16.The Thein solutions steady-state AC gear shaped domain are are displayed inexact (a), and in (b)for the proposed approach solution without random noise. are displayed in (a), and in (b) the proposed approach solution without random noise. displayed in (a), and in (b) the proposed approach solution without random noise.
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Figure 17. The numerical errors the proposed proposed method method solution solution for for the the steady-state steady-state AC AC equation equation Figure 17. The numerical errors of of the Figure 17. The numerical errors of the proposed method solution for the steady-state AC equation with the gear shaped irregular domain. Figure Theshaped numerical errors of the proposed method solution for the steady-state AC equation with the17. gear irregular domain. with the gear shaped irregular domain. with the gear shaped irregular domain.
Figure 18. The exact solutions for the steady-state AC equation with amoeba-like irregular domain Figure 18. The exact exactsolutions solutionsfor forthe the steady-stateAC AC equation with amoeba-like irregular domain Figure 18. equation with amoeba-like domain are are shown in (a), and solutions in (b) the for proposed scheme solution without random noise.irregular Figure 18. The The exact thesteady-state steady-state AC equation with amoeba-like irregular domain are shown in (a), and in the (b) proposed the proposed scheme solution without random noise. shown in (a), and in (b) scheme solution without random noise. are shown in (a), and in (b) the proposed scheme solution without random noise.
Figure 19. The numerical errors of the proposed method solution for the steady-state AC equation Figure 19. The numerical errors of the proposed method solution for the steady-state AC equation with the19. amoeba-like irregular domain. Figure 19. The errors of proposed method method solution the steady-state steady-state AC equation Figure The numerical numerical errors of the the proposed solution for for the AC equation with the amoeba-like irregular domain. with the the amoeba-like amoeba-like irregular irregular domain. domain. with
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Figure 20. The The numericalsolutions solutions steady-state AC equation with amoeba-like Figure 20. forfor the the steady-state AC equation with amoeba-like irregularirregular domain Figure 20. Thenumerical numerical solutions for the steady-state AC equation with amoeba-like irregular domain are shown in (a), and in (b) the proposed approach solution with random noise. are shown in (a), and in (b) the proposed approach solution with random noise. domain are shown in (a), and in (b) the proposed approach solution with random noise.
Figure 21. Numerical errors with random noise. Figure noise. Figure 21. 21. Numerical Numerical errors errors with with random random noise.
5. Conclusions 5. Conclusions 5. Conclusions In this study a new meshless algorithm was developed to solve steady-state Allen-Cahn In this study study a new new meshless meshless algorithm algorithm was was developed developed to to solve solve steady-state steady-state Allen-Cahn Allen-Cahn In this equations for heat atransfer in thin films with several arbitrary plane domains. In the modified equations for heat transfer in thin films with several arbitrary plane domains. In the modified equations for heat transfer in thin films with several plane domains. In the modified polynomial expansion algorithm proposed, we were ablearbitrary to make decisions about better values of sij polynomial expansion algorithm we were able to make decisions about better values of sij polynomial algorithmproposed, proposed, were to make decisions about better values based on theexpansion idea of an equilibrated matrix to we acquire a able multiple-scale sij in a closed-form which was based on the idea of an equilibrated matrix to acquire a multiple-scale s ij in a closed-form which was of s based on the idea of an equilibrated matrix to acquire a multiple-scale s in a closed-form ij ij completely determined by the collocation points. We found that the presented approach is applicable completely determineddetermined by the collocation We found that the presented approach is applicable which was completely by thepoints. collocation points. found presented approach to two-dimensional steady-state Allen-Cahn equations andWe based onthat thethenumerical examples, to two-dimensional steady-state Allen-Cahn equations and based on the numerical examples, is applicable to two-dimensional Allen-Cahn equations and on in the computationally very efficient, evensteady-state with large amounts of random noise (upbased to 7.5%) thenumerical amoebacomputationally very efficient, even with large amounts of amounts random noise (up to noise 7.5%) (up in the amoeba−1) with examples, computationally very efficient, even with large of random to in like irregular domain. The maximum numerical errors of our scheme are in the order of O(107.5%) −1) with like irregular domain. The maximum numerical errors of our scheme are in the order of O(10 the amoeba-like numerical errors of our scheme arewith in the of the amoeba-like irregular irregular domain. domain. The Themaximum current algorithm can be extended to deal theorder threethe amoeba-like domain. Thedomain. current algorithm can be extended deal with the − 1 ) with the irregular O(10 amoeba-like irregular The current algorithm can betosteady-state extended to PDEs dealthreewith dimensional and complex steady-state nonlinear PDEs, fourth-order nonlinear and dimensional and complex nonlinear nonlinear PDEs, fourth-order nonlinear steady-state PDEs and the three-dimensional andsteady-state complex steady-state PDEs, fourth-order nonlinear steady-state many other practical engineering issues. many and other practical issues. PDEs many otherengineering practical engineering issues. Author Contributions: C.-W.C. designed and performed the numerical experiments and analyzed the data. Author Contributions: C.-W.C. designed and numerical experiments andand analyzed the the data.data. C.-W.C. designed andperformed performedthe the numerical experiments analyzed Author Contributions: C.-W.C., C.-H.L. and C.-C.W. wrote the paper. C.-W.C., C.-W.C., C.-H.L. C.-H.L. and and C.-C.W. C.-C.W.wrote wrotethe thepaper. paper. Funding: This research was funded by Ministry of Science and Technology of the Republic of China under grants Funding: This research research was was funded funded by by Ministry Ministry of of Science Science and and Technology Technology of the Republic Republic of of China China under under grants grants Funding: This of the MOST 107-2634-F-005-001,MOST MOST 107-2823-8-005-002, MOST-106-3114-8-005-003, MOST-106-2221-E-167-012MOST MOST-106-3114-8-005-003, MOST-106-2221-E-167-012-MY2 MOST 107-2634-F-005-001, 107-2634-F-005-001, MOST107-2823-8-005-002, 107-2823-8-005-002, MOST-106-3114-8-005-003, MOST-106-2221-E-167-012MY2 and MOST-106-2622-E-167-010-CC3. and MOST-106-2622-E-167-010-CC3. MY2 and MOST-106-2622-E-167-010-CC3. Acknowledgments: The authors would like to express their deepest appreciations to three anonymous Acknowledgments: The authors would like to express their deepest appreciations to three anonymous reviewers, who provided constructive comments on the paper. reviewers, who provided constructive comments on the paper.
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Acknowledgments: The authors would like to express their deepest appreciations to three anonymous reviewers, who provided constructive comments on the paper. Conflicts of Interest: The authors declare no conflict of interest.
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