4.4 Discussion and Conclusion
BCP films but also on the lamella-forming BCP films. In the case of lamella-forming BCPs, a more accurate renormalization is possible using the Morse Calibration, and the simulation is easier to perform because its morphology is simpler than that of the cylindrical phase. Using the (χN)ODT equation of symmetric BCPs with respect to ¯N proposed by Morse and Matsen, [27, 28,100] it is possible to accurately compare the results with the theoretical prediction in an experimentally relevant regime, without using the TIM.
In chapter 3, the effect of the wetting condition on the ODT behavior of thin films was investigated using the SCFT adopting finite-ranged interaction, but in chapter 4, the effect of fluctuation on the ODT of thin film was investigated using the discrete chain model without the finite-ranged interaction. In order to simulate a polymeric system with a more realistic polymer model, it is necessary to incorporate the finite-ranged interaction with FTS. In chapter 2, the field transform using an arbitrary pair interaction potential u(r) has already been intro- duced, and it seems possible to develop an FTS adopting finite-ranged interaction. Since the renormalization methods known so far have been developed on the assumption of a continuous Gaussian chain model, it is also necessary to modify the renormalization method. One possible suggestion is to perform the RPA theory adopting finite-ranged interaction and try an RPA renormalization.
Unlike the SCFT which is a mean field theory, the FTS can account for the fluctuation effect, but it is practically difficult to perform FTS in a large polymeric system because the simulation time increases by a factor of thousands to tens of thousands of times compared to the SCFT calculation. In this study, the simulation was performed efficiently by parallelizing the calculation using GPUs, but further improvements are required in order to perform L- FTS of various systems which had been mostly studied by SCFT. One possible suggestion is to accelerate L-FTS using the machine learning method, which is rapidly evolving in recent years. In L-FTS, it is necessary to find the partial saddle point at every Langevin step. Since at least tens of thousands Langevin steps are performed in one L-FTS execution, data on the corresponding pressure field in a given exchange field can be collected as the training data.
Using this data and online learning, L-FTS could be accelerated tens of times by generating the pressure field directly through machine learning method during each simulation. This method is very versatile and could be applied to a variety of systems without prior training of a neutral network model.
Appendix A
Gaussian Integral
The one-dimensional Gaussian integral of a quadratic equation is evaluated as Z ∞
−∞
dxexp −ax2/2 +Jx
= 2π
a 1/2
exp
J2/(2a)
, (A.1)
where a constanta >0 andJ are real. This Gaussian integral can be expanded to multivariable integral ofx= (x1, x2, . . . , xN)T
Z
dxexp −xTAx/2 +JTx
= (2π)N/2
(detA)1/2 exp JTA−1J/2
, (A.2)
whereAis a symmetric and positive definiteN×N matrix, and a vector J= (J1, J2, . . . , JN)T is a real-valued vector. The functional integral version of Gaussian integral is
R Dfexp
−(1/2)R dxR
dx0f(x)A(x, x0)f(x0) +R
dxJ(x)f(x) R Dfexp
−(1/2)R dxR
dx0f(x)A(x, x0)f(x0)
= exp 1
2 Z
dx Z
dx.J(x)A−1(x, x0)J(x0)
. (A.3)
The function A(x, x0) is also symmetric and positive definite, and A−1(x, x0) is a functional inverse which satisfies
Z
dx00A(x, x00)A−1(x00, x0) =δ(x−x0), (A.4) whereδ(x) is the Dirac delta distribution. [61]
Appendix B
GPU Optimization
Currently, the main trend of general-purpose computing on graphics processing units (GPGPU) is to use GPU hardware manufactured by NVIDIA and to develop programs on its platform, compute unified device architecture (CUDA). In our L-FTS study, GPU parallelization was implemented using CUDA, and this appendix is devoted to describe the optimization of GPU parallelization using CUDA.
The GPU hardware has tens of streaming multiprocessors (SMs), and each SM consists of many CUDA cores. For example, the GeForce RTX 3080 we use has 68 SMs and each SM has 128 CUDA cores; there are 8704 CUDA cores. Each CUDA core has relatively limited computing power compared to a single CPU core, but it shows high parallel performance using the large number of GPU cores. In order to reduce possible bottlenecks, the GPU has many types of memories and the most important ones are global and shared memories. The main task of GPU optimization is to reduce the bottlenecks caused by data transfer between memories and to maximize usage of GPU cores.
Delaney and Fredrickson first published a paper about polymeric field-based simulations on GPUs. [98] The computation employing pseudo-spectral method was parallelized using CUDA.
The calculations using the pseudo-spectral method are largely element-wise multiplications and FFT operations in between them. In their study, element-wise multiplications were calculated using CUDA kernels, and FFT operations were performed using FFT functions provided by the CUDA library. [87] In order to reduce the time loss in data transfer between the main memory and the global memory, all partition function steps were stored in the global memory, and only the calculated segment densities were transferred to the main memory. In addition, by storing the product of the partial partition function and the complementary partition function instead of storing two partition functions separately, the memory usage could be reduced by 50 percents.
With such techniques, they achieved a speedup of 30 in a simulation of 1283 grids compared to a single CPU calculation using double precision FFTW. [101]
Recently, Cheong et al. provides an SCFT implementation utilizing GPU as an open source.
[99] In their implementation, the reduction operations are also conducted in GPU to efficiently calculate the total partition function and the inner product of the Anderson mixing. The re- duction operation on GPU using shared memory is a very non-trivial task due to issues such as bank conflict. They use the most optimized reduction method suggested by the NVIDIA.
The FFT library provided by CUDA internally determine the number of threads and blocks depending on the simulation grid number. If the simulation grid number is small, it appears that all SMs are not utilized. In Cheong et al., the FFT parallelization efficiency in the case of a small grid number is doubled by mixing calculations of both partial partition functions and by using batching system of FFT operations. Instead of using the FFT batch process, we utilize the concurrent kernel execution. In this approach, two concurrent streams are created and each is responsible for one partial partition function. Compared to the implementation of Cheong et al., the performance is the same, but it is easier to upgrade for multiple GPU implementation.
Figure B1 explains the performance difference between each implementation as a time- line. The FFT parallelization efficiency is doubled by the batch process compared to the basic implementation. With batch process, the calculation time of two FFT operations equals the calculation time of one FFT operation. Similarly, in the case of using double streams, there is a performance gain in the FFT calculations. Note that this illustration is only valid when the simulation grids number is small.
Figure B1: Comparison of GPU implementations. (left) Implementation without batch process and streams. (middle) Two FFT operations performed using the batch process. (right) Each partial partition function is performed simultaneously in each stream.
The performance of GPU parallelization increases as the simulation grid number increases.
Since all partial partition steps are stored in the global memory, the maximum limit of grid number is about 1283due to limit of GPU memory space when using a typical graphics card such as GeForce RTX 3080. Delaney and Fredrickson proposed a method that transfers the partial
partition functions to the main memory as they are computed in order to reduce the GPU memory usage when the simulation grid number is large. This method doubles the execution time in the simulation of 1283 grid number due to the cost of transferring data and loss of parallelization in density calculation.
When using large grid numbers, we implemented another method that saves GPU memory to deal with this case. The GPU supports kernel execution and data transfer overlapping, which simultaneously executes kernel and transfers data between main and global memories. For this, we first create two streams. The first stream is responsible for the kernel execution, and the second stream is responsible for the data transfer. Then, we allocate memory space that can store two steps of each partial partition function in the global memory. For the first half of the memory space, the partial partition function calculated in the previous step is stored, and the other half is used to calculate the partial partition function in the next step, using the result of the previous step. These calculations are performed on the first stream. In the meantime, the second stream transfers the calculated partial partition function to the main memory. The two streams are synchronized at the end of each ∆s step. When calculation of one ∆s step is finished, the roles of the two memory spaces are exchanged and the above process is repeated.
Figure B2:Illustration of overlapping of kernel execution and data transfer. The yellow and green bars denote data transfer and kernel execution times, respectively. The data transfer can be com- pleted during the kernel execution when the simulation grid number is large enough.
By repeating this process, the calculation of the partial partial function q(r) is completed, and the complementary partial partition functionq†(r) calculation begins. In this process, unlike the previous case, the result is not transferred to the main memory, butq(r) stored in the main memory is transferred to the global memory at every contour step by the second stream. At the same time, in the first stream, the calculation ofq†(r) is finished and the density ofsNth segment is computed using the two partial partition functions in the global memory. We synchronize the streams at the end of each ∆s step during the computation of the complementary partial partition function. Figure B2 explains the efficiency of the computation with this overlapping method. If the simulation grid number is large enough, the data transfer can be completed during the kernel execution because the data transfer time is shorter than the kernel execution
time. This method is supposed to be somewhat slower than the previous double stream method when the grid size is small, but for a simulation in 1283 grid, this method is faster because of the reduction of the data transfer time, provided that there are no bottlenecks in the bandwidth between main and global memories.
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