The algebraic test for the material conservation I developed is very powerful in that it can be applied to other numerical methods. Nowadays, in the polymer theory community, the use of pseudo-spectral method for the solution of SCFT in Cartesian coordinates is very common [3, 24], and it is worth investigating its material conservation property. Because the pseudo-spectral method can be presented in terms of matrix multiplications as shown below, the algebraic test developed in this thesis is applicable. Let me begin with the material conservation of a two dimensional Cartesian coordinate system (x, y) with periodic boundary conditions in both
6.4 Pseudo-spectral Method directions. For the start of the pseudo-spectral method, Strang splitting is applied to the modified diffusion equation,
q(r, s+ ∆s)'exp
−∆s w(r) 2
exp
a2N∆s∇2 6
exp
−∆s w(r) 2
q(r, s). (6.57) The first step is to calculate (−∆s w(r)/2)q(r, s) in the real space. Using the bra-ket notation in the discrete world, it can be written asX |qni. The next step is to perform the two dimensional discrete Fourier transform (DFT), which is represented by the following matrix,
F ≡ 1
√IJ
1 1 1 · · · 1 1 1 1 · · · 1 · · ·
1 λ λ2 · · · λJ−1 1 λ λ2 · · · λJ−1 · · ·
1 λ2 λ4 · · · λ2(J−1) 1 λ2 λ4 · · · λ2(J−1) · · ·
. .. . .. . ..
1 λJ−1 λ2(J−1) · · · λ(J−1)(J−1) 1 λJ−1 λ2(J−1) · · · λ(J−1)(J−1) · · ·
1 1 1 · · · 1 ω ω ω · · · ω · · ·
1 λ λ2 · · · λJ−1 ω ωλ1 ωλ2 · · · ωλJ−1 · · ·
1 λ2 λ4 · · · λ2(J−1) ω ωλ2 ωλ4 · · · ωλ2(J−1) · · ·
. .. . .. . ..
1 λJ−1 λ2(J−1) · · · λ(J−1)(J−1) ω ωλJ−1 ωλ2(J−1) · · · ωλ(J−1)(J−1) · · ·
. .. . .. . ..
,
(6.58) where ω = exp (2πi/I) and λ = exp (2πi/J) are the principal I-th and J-th roots of unity, respectively. In this notation, the inverse DFT is simply written as F−1 and it is easy to confirmF−1 =F†. The next task is to apply exp a2N∆s∇2/6
operator in the Fourier space, which is a multiplication of a diagonal real matrix with the following shape,
C=
c1,1
c1,2 0
c1,3 . ..
0 cI,J−1
cI,J
. (6.59)
The last step is to perform inverse DFT and to multiply X, and the final evolution matrix becomes U =X F−1CFX. The analysis in the previous sections only considered real matrices and the final task was to check the symmetry of the matrixVU. Now, repeating processes from equation (6.1) to equation (6.13) with the standard complex version of bra-ket notation, it is straightforward to show that the actual condition for the material conservation is that VU is Hermitian. The boundaries are periodic, and thus it is natural to take all the cell volumes to be uniform, V = ∆x∆yI. Applying the complex version of appendix B, the following matrix must be Hermitian for the material conservation,
∆x∆yF−1CF. (6.60)
Since F−1CF†
=F†C F−1†
= F−1CF, the matrix VU is Hermitian and the material con- servation is confirmed. Three dimensional generalization of this proof is straightforward.
For other boundary conditions, discrete cosine and sine transforms are often used in the pseudo-spectral method, and half size cells may be used at the boundaries. Even though the above proof is not directly applicable in those cases, such a problem is mathematically equivalent to a periodic boundary problem with twice the original domain size, and the above proof guarantees the material conservation of those systems.
7
Material Conservation: Numerical Test
In this section, both two dimensional cylindrical (ρ, ϕ) and spherical (r, θ) molten brush systems are solved using various numerical SCFT methods I have introduced so far. As a simple example, I consider the case with uniform grafting density and segment density,φ(r) = 1, which results in ϕdirectional symmetry for the cylindrical system andθdirectional symmetry for the spherical system. With this extra symmetry, the ADI method reduces to the Crank-Nicolson method, and I do not need to consider each case separately.
In one dimensional cylindrical brush, it is known that self-consistent field becomes w(r) = wc(r) + 2ΛaN1/2δ(ρ−R1), where wc(r) is the continuous part of the pressure field and the δ- function shape of contribution, 2ΛaN1/2δ(ρ−R1), is naturally induced to avoid over-crowdedness near the grafting surface. The analysis of one dimensional spherical brush is essentially identical with the change ofρ tor. Figures 7-1 and 7-2 exhibit the self-consistent field of the cylindrical and spherical brushes, respectively, after taking the volume average of the mean field to be zero. For the quantitative comparison of each numerical method, the global volume error, free energy and w(R1) are compared in tables 7-1 and 7-2. For both FDM cases, the global volume errors of the cylindrical brush are−2.27×10−5, and these errors are uniformly distributed in the whole grid; i. e. φi = 1.0−2.27×10−5. On the other hand, both FVM solutions conserve material within the floating-point precision of the calculation. A similar behavior is observed for the case of the spherical brush.
Considering that the ADI method is now reduced to the Crank-Nicolson method, the above results are consistent with the conclusion of the previous chapter. When Crank-Nicolson method is used, FVM conserves material while FDM fails to do so for the curved coordinate systems.
The operator splitting did not affect the material conservation in this one dimensional case, but its effect is somewhat complicated. As ∆s approaches to zero, the field obtained without the operator splitting converges to the correct field more rapidly, especially at the grafting point ρ=R1 orr =R1.
2 2.5 3 3.5 4
−20
−10 0 10 20 30
ρ(distance fromz axis)
w(ρ)
Figure 7-1: Self-consistent field of a cylindrical brush obtained by FVM and FDM without op- erator splitting. The difference between FVM and FDM is within the thickness of the line.
The parameters are set toR1= 2, R2= 4, I1= 100, I2= 200 and N= 400.
2 2.5 3 3.5 4
−20
−10 0 10 20 30
r(distance from origin)
w(r)
Figure 7-2: Self-consistent field of a spherical brush obtained by FVM and FDM without op- erator splitting. The difference between FVM and FDM is within the thickness of the line.
The parameters are set toR1= 2, R2= 4, I1= 100, I2= 200 andN = 400.
Method FDM FDM with O.S. FVM FVM with O.S.
Mass error −2.27×10−5 −2.27×10−5 −3.33×10−16 −2.22×10−16
Free Energy 5.024763 5.039581 5.025898 5.040606
w(R1) 159.090731 182.048520 159.502168 182.644092
Table 7-1: Comparison of FDM and FVM for a cylindrical brush without ϕ dependence. The parameters are set to R1 = 2, R2 = 4, I1 = 100, I2 = 200 and N = 400. O.S. stands for operator splitting.
Method FDM FDM with O.S. FVM FVM with O.S.
Mass error −4.38×10−5 −4.38×10−5 2.22×10−16 8.88×10−16
Free Energy 6.165728 6.207328 6.167803 6.209063
w(R1) 259.315619 338.430350 260.653523 341.171206
Table 7-2: Comparison of FDM and FVM for a spherical brush without θ dependence. The parameters are set to R1 = 2, R2 = 4, I1 = 100, I2 = 200 and N = 400. O.S. stands for operator splitting.
Until now I have tested the material conservation of one dimensional systems. In order to break the symmetry and create two dimensional problems, I consider artificial compressible situations that the segment concentration satisfiesφ(r) = 1 + 12sin(ϕ) for the cylindrical brush andφ(r) = 1+12sin(2θ) for the spherical brush. Tables 7-3 and 7-4 summarize the self-consistent solutions using FDM and FVM both with and without operator splitting. These self-consistent solutions have stronger ρ and r directional dependences with relatively weak dependences on
ϕ and θ directions. The mass errors obtained by FDMs are always order of 10−5 and they are equivalent to the one dimensional results, indicating that the source of error is almost independent of the φ(r) asymmetry. They are mostly due to the incorrect Oid±’s produced by the FDM.
The algebraic analysis in the previous chapter suggests that not only FDM but also FVM fails to conserve material when ADI algorithm is chosen for the two dimensional system without symmetry. Indeed, the mass errors of the FVM turn out to be above the errors of the floating- point calculation, and my predictions are confirmed. However, they are relatively small, of order 10−8 ∼ 10−9 for the parameter values I used. In a harsher situation, with an abrupt jump of φ(r), the mass error increases to a value of order 10−7, but it is still much lower than the FDM case. The only algorithm with accurate material conservation is the FVM with Crank-Nicolson method. However, in the Crank-Nicolson method, one needs to solve a matrix equation involving the giant matrixU for theqi propagation. In one test run, I performed band matrix calculations such as LU decomposition and multiplication using OpenBLAS library [31], and 45 minutes of computation was required for the full solution of the spherical brush systems I consider with a single core of Xeon E5-2690 v3 CPU. On the other hand, a properly implemented ADI method can find the solutions in less than 5 minutes, which is about a factor of 10 performance enhancement.
Algorithm ADI Crank-Nicolson
Method FDM FVM FVM
Operator Splitting Yes No Yes No Yes No
Mass error −2.27×10−5 −2.27×10−5 2.31×10−9 6.87×10−9 6.66×10−16 4.44×10−16 Free Energy 5.103990 5.089161 5.105015 5.090296 5.105015 5.090296
Table 7-3: Comparison of FDM and FVM for a cylindrical brush withϕdependence. The param- eters are set toR1= 2, R2= 4, I1= 100, I2= 200, J = 101 andN = 400.
Algorithm ADI Crank-Nicolson
Method FDM FVM FVM
Operator Splitting Yes No Yes No Yes No
Mass error −4.38×10−5 −4.37×10−5 9.97×10−9 3.33×10−8 2.22×10−16 4.44×10−16 Free Energy 6.277818 6.236196 6.279552 6.238269 6.279552 6.238269
Table 7-4: Comparison of FDM and FVM for a spherical brush withθdependence. The parameters are set toR1= 2, R2= 4, I1= 100, I2= 200, J= 100 andN = 400.
Discussion and Conclusion
In this thesis, various numerical methods for solving modified diffusion equation of SCFT are derived in the two dimensional Cartesian, cylindrical and spherical coordinate systems. I de- velop a finite volume method (FVM) with half grid size at the boundary which is a natural extension of the three-point finite difference method (FDM) in the finite volume world. The FVM formulations are summarized in Tables 4-1 and 4-3. I take cylindrical and spherical brush systems, as model systems to test the developed numerical methods.
It has been known that SCFT discretized by FDM fails to conserve the amount of material in the system in general. The FVM is designed to and known to perform better, but the material conservation of the modified diffusion equation is not equivalent to the material conservation of normal diffusion equation, because the segment volume of SCFT is obtained by integrating products of two partial partition functions over the space. Therefore, it is possible, and it is known that SCFT discretized by FVM fails to conserve material from time to time.
In this research, in order to prove the utility of FVM in the SCFT, I successfully developed an algebraic test using bra-ket notation which verifies the material conservation of each numerical method. The material conservation condition was equivalent to the condition that the product of volume matrix and the evolution matrix is symmetric,VU = (VU)T. For the Crank-Nicolson method, this requirement is equivalent that the Laplacian is discretized by FVM, Oid+∆Vi = Oi+ˆd−e
d∆Vi+ˆed; thus, in any coordinate system, material is conserved when the FVM is applied.
For example, the global volume error of FVM using two dimensional Crank-Nicolson method is practically zero in the numerical test. The global volume error of FDM turns out to beO 10−5
, even though the point-wise error of FDM isO(∆s2) +O(∆x12) +O ∆x22
which is the same as the FVM case; thus, FVM provides a way to achieve near-perfect material conservation without loss of point-wise precision. By applying the same algebraic test to the alternating direction implicit (ADI) method and the Douglas-Gunn method, I show that the FVM in the cylindrical and spherical coordinate systems cannot conserve material in general, but careful numerical tests show that depending on the problem, the use of FVM may still improve material conservation compared to the FDM case.
In the standard FDM and FVM, the field term is combined with the differentiation part to construct the evolution matrix. By separating the field term with operator splitting method, new types of finite difference and finite volume methods are developed, and analytical tests show that this technique has a benefit in material conservation in the Cartesian coordinate system, as was confirmed in a previous report [12]. However, there is also a disadvantage that a sufficiently small contour step ∆sis required to obtain an accurate solution near the grafting surface of the brush. All the issues of material conservation, accuracy and speed of numerical methods are summarized in table 8-1. The table only shows if the material conservation is perfect or not.
An analytical method identifying the degree of material conservation is yet to be found. One possible suggestion is to calculate the norm ofVU −(VU)T and to find its effect to the material conservation.
Because the numerical methods I consider implicitly assume that the q and w functions are continuous, one may question their validity for the brush systems in which q has a δ- function shape initial condition and self-consistent field w requires a δ-function contribution at the grafting surface. Even though the current numerical SCFT seems to produce somewhat reliable results, it is worth considering high resolution FVM which is designed to work efficiently for a system with discontinuous solution in the future.
The material conservation test presented in this thesis is very versatile and its application to the pseudo-spectral method was also possible. Even though I restricted my discussion to two dimensional systems in this thesis, three dimensional version of each numerical method can be tested, and the results seem to be always the same as the two dimensional case, provided the three dimensional generalization is properly made. There are many different versions of FDM, FVM and pseudo-spectral method, and one also has the freedom of choosing the volume integral method. The accuracy of a given combination can be tested individually using the algebraic tool I provided, and I recommend one to wisely select a method with a good material conservation property.
Method FDM FVM
Algorithm Crank-Nicolson ADI Crank-Nicolson ADI
Operator Splitting Yes No Yes No Yes No Yes No
Material Conservation No Symmetry
Cart O O O X O O O X
Cyl & Sph X X X X O O X X
Material Conservation x2Symmetry
Cart O O O O O O O O
Cyl & Sph X X X X O O O O
Delta Function Field Accuracy Low High Low High Low High Low High
Computation Speed Slow Fast Slow Fast
Table 8-1: Summary of material conservation for each numerical method. The row with “x2
symmetry” presents the material conservation when the initial conditions and field are symmetric in that direction, and the row above it is the case without such a symmetry. Material conservation of the Douglas-Gunn method is equivalent to the ADI case.
Symmetry of Inverse Matrix
With an arbitrary invertible matrixA, ifA is symmetric,
I = (AA−1)T = (A−1)TAT = (A−1)TA, (A.1) Since inverse matrix is unique, (A−1)T = A−1, hence its inverse is symmetric. In a similar manner, if A−1 is symmetric, Ais symmetric.
I = (AA−1)T = (A−1)TAT =A−1AT, (A.2) Therefore, Ais symmetric, if and only if A−1 is symmetric.
Appendix B
Removing X Matrices
A new symbol X ≡ exp (−W) is used for a diagonal matrix whose components are Xii = exp(−Wii). With an arbitrary matrixA, for the matrixVX AX to be symmetric, the required equation is
VX AX =X ATX V. (B.1)
Since V andX are diagonal,V and X commutes, and the above equation is equivalent to
X VAX =X ATVX. (B.2)
The matrix X is invertible, because Xii is always nonzero; thus, I obtain
VA=ATV. (B.3)
Now it is clear that the matrixVX AX is symmetric if and only if the matrixVAis symmetric.
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