Investigation of Neutron-Irradiated Microstructure of Fe-Cr System: A - PDF document

019 Investigation of Neutron-Irradiated Microstructure of Fe-Cr System: A GPU Accelerated Phase-field method Jeonghwan Lee, Kunok Chang * a Nuclear Engineering, Kyung Hee Univ., Republic of Korea * Corresponding author: kunok.chang@khu.ac.kr The molar free energy F(r, t) in Eq.(1) is given by 1. Introduction Eq. (2), The gradient coefficient κ is given by Ferritic martensitic steels are promising due to their low swelling rate under the fast neutron irradiation. 1 κ = 2 𝑀 𝐺𝑓𝐷𝑠 (4) 6 𝑠 0 However, in the case of Fe-Cr steels, precipitation of Cr rich phase (α΄ phase) near 475 ° C is still pointed out as a where 𝑠 0 is the lattice parameter and 𝑀 𝐺𝑓𝐷𝑠 is the weak point in the view point of structural integrity. [1][2] regular solution interaction parameter. therefore, understanding the spinodal decomposition The mobility M in the Cahn-Hilliard-Cook equation is behavior of the Fe-Cr system under the fast neutron assumed to be independent of the concentration field irradiation is important topic in studying the integrity of Therefore, rearranged Eq. (1) as structural materials. Herein, we analyze the spinodal decomposition 𝜀𝑔(𝐷) 𝜖𝑑(𝑠,𝑢) = ∇ 2 [( 𝜀c ) − 𝜆∇ 2 c(r, t)] (5) behavior under the fast neutron irradiation of the Fe-Cr 𝜖𝑢 system using a graphics processing unit(GPU)- 𝜀𝐺(𝑠,𝑢) 𝜖𝑑̃(𝑙,𝑢) = −𝑙 2 ( 𝑙 − 𝜆𝑙 4 𝑑̃(k, t) (6) ) accelerated phase-field method. Since high energy 𝜖𝑢 𝜀𝑑 particles produce a point defect, we quantify the where k = (𝑙 1 , 𝑙 2 ) is the reciprocal vector in the microstructure evolution behavior of the Fe-Cr system 2 and 𝑑̃(𝑙, 𝑢) under the fast neutron irradiation. 2 +𝑙 2 Fourier space of magnitude k = √𝑙 1 In addition, we consider the different computational 𝜀𝐺(𝑠,𝑢) and ( 𝜀𝑑 ) 𝑙 are the Fourier transforms of c(r, t) and technique. Since the quantitative prediction of the real 𝜀𝑔(𝐷) material system is highly computationally expensive, we ( 𝜀c ) respectively. Then, we applied an explicit Euler have implement a parallel computing scheme based on Fourier spectral treatment to this equation, yielding the compute unified device architecture (CUDA) to improve computational efficiency [3], comparing it with 𝑜 𝜖𝑑̃ 𝑜+1 (𝑙,𝑢)−𝑑̃ 𝑜 (𝑙,𝑢) = −𝑙 2 ( 𝜀𝐺(𝑠,𝑢) − 𝜆𝑙 4 𝑑̃(k, t) (7) 𝜀𝑑 ) parallelized code using CUDA when solving the Cahn- ∆𝑢 𝑙 Hilliard diffusion equation [4] using a semi-implicit so spectral method [5]. Although CUDA has previously been applied to the 𝑜 𝑑̃ 𝑜 (𝑙,𝑢)−∆t𝑙 2 ( 𝜀𝐺(𝑠,𝑢) ) 𝜀𝑑 phase-field method, it was used to create an explicit 𝑑̃ 𝑜+1 (𝑙, 𝑢) = 𝑙 (8) 1+∆t𝜆𝑙 4 solver [6], [7]. Herein, we instead use it to implement a semi-implicit spectral method and compare the 2.2 Modified CALPHAD-type free energy performance of OpenMP- and CUDA-accelerated code. Results will help to guide any researchers aiming to solve The molar chemical free energy f(c) in Eq. (2) is given the Cahn-Hilliard equation using fast Fourier transform. by[ 주석 ] 2. CALPHAD-based phase-field methods 0 + c𝐻 𝐷𝑠 0 + 𝑀 𝐺𝑓𝐷𝑠 𝑑(1 − 𝑑) f(c) = (1 − 𝑑)𝐻 𝐺𝑓 + 𝑆𝑈[𝑑𝑚𝑜𝑑 + (1 − 𝑑) ln(1 − 𝑑)] 2.1 Semi-implicit Fourier spectral method + 𝐻 𝑛 (J/mol) (9) We simulate the evolution of the Cr concentration 0 and 𝐻 𝐷𝑠 0 are the molar Gibbs free energies where 𝐻 𝐺𝑓 field by solving the following Cahn-Hilliard equation [8]: for pure elemental Fe and Cr, respectively, 𝑀 𝐺𝑓𝐷𝑠 is the interaction parameter between Fe and Cr, R (= 8.314J/ 𝜖𝑑(𝑠,𝑢) 𝜀𝐺(𝑠,𝑢) 2 ∇ ∙ [𝑁(𝑠, 𝑢) ∙ ∇ ( = 𝑊 𝜀𝑑 )] (1) mol ∙ K) is the gas constant, T is the system’s ab solute 𝑛 𝜖𝑢 temperature, which is 563 K herein, and 𝐻 𝑛 is the molar 1 1 2 𝜆(∇𝑑) 2 ]}𝑒𝑊 F(r, t) = ∫ { 𝑛 [𝑔(𝑑) + (2) 𝑊 𝑊 Gibbs free energy of the magnetic ordering effect. These were calculated as follows: where c is the Cr concentration, κ is the gradient 0 = +1225.7 + 124.134 × T − 23.5143 × T × lnT 𝐻 𝐺𝑝 energy coefficient, F(r, t) and f(c) are the system’s − 0.00439752 × 𝑈 2 − 5.89269 molar free energy and molar chemical free energy, × 10 −8 × 𝑈 3 + 77358.5 × 𝑈 −1 respectively. f(c) is discussed in the following section.

Transactions of the Korean Nuclear Society Autumn Meeting Goyang, Korea, October 24-25, 2019 0 = −8856.94 + 157.48 × T − 26.908 × T × lnT 𝐻 𝐷𝑠 due to CUDA is most effective when solving the Cahn- + 0.00189435 × 𝑈 2 − 1.47721 Hilliard equation [9]. A semi-implicit Fourier spectral × 10 −6 × 𝑈 3 + 139250 × 𝑈 −1 method, as described in the previous section, was 𝑀 𝐺𝑓𝐷𝑠 = +20500 − 9.68𝑈 implemented by utilizing cuFFT for the CUDA code. For this benchmark, we conducted 2D spinodal 𝐻 𝑛 = 𝑆𝑈𝑚𝑜(𝛾 + 1) λ (τ), (J/mol) decomposition simulations that describe the microstructure evolution behavior of the Fe-Cr system Where 𝛾 is the atomic magnetic moment, calculated in under the fast neutron irradiation. We measured the time terms the Bohr magneton as taken to calculate 100,000 time steps using the Linux time command, which gives the real elapsed time. 𝛾 = 2.22(1 − 𝑑) − 0.008𝑑 − 0.85𝑑(1 − 𝑑) . The function λ (τ) is expressed as the following polynomial: λ (τ) = −0.90530 τ −1 + 1.0 − 0.153τ 3 − 6.8 × 10 −3 τ 9 − 1.53 × 10 −3 τ 15 (τ > 1) = −0.06417 τ −5 − 2.037 × 10 −3 τ −15 − 4.278 × 10 −4 τ −25 (τ < 1) τ = 𝑈/𝑈 Where is critical magnetic ordering 𝑑 temperature given by 𝑑 = 1043(1 − 𝑑) − 311.5𝑑 + 𝑑(1 − 𝑑)[1650 𝑈 + 550(2𝑑 − 1)](𝑗𝑜 𝐿) Eq. (2) includes a magnetic ordering contribution to the free energy. Some previous studies have neglected Fig 2: Time consumption for the microstructural magnetic ordering effects. However, as shown in Fig. 1, the Fe- Cr system’s free energy at 563 K varies evolution simulation with serial (i9-9900K 3.6 GHz CPU) and CUDA (1060, 2080ti and Tesla V100) substantially depending on whether or not magnetic ordering effects are included. We compared the efficiencies of the CUDA-based code on the same or a comparable computer, and the results obtained are shown in Fig. 2. We conducted these comparisons for seven different numbers of dimensions, namely 128, 256, 512, 1024, and 2048. Here, a dimensionality of 128 (say) means that the system cell size was 128∆x × 128∆y . Fig 1: Free energy curve for the Fe-Cr at 563 K with considering magnetic ordering effects. The equilibrium Cr concentration are 𝐷 𝐷𝑠 = 0.05 and 𝐷 𝐷𝑠 = 0.98 Fig 3: CUDA code efficiency compared to serial code To increase the computational efficiency, we used dimensionless values herein. Specifically, our As shown in Fig. 2 the computational cost of the simulations used the normalized values 𝑠 ∗ = 𝑠/𝑚 , ∇ ∗ = 𝜖/𝜖(𝑠/𝑚) , 𝑢 ∗ = 𝑢𝐸/𝑚 2 , 𝑁 ∗ = 𝑊 CUDA code better than serial code. Also, Fig 3, as the 𝑛 𝑆𝑈 ∗ 𝑁/𝐸 , 𝑔 ∗ (𝑑) = system size increases, the efficiency of the CUDA code 𝑔(𝑑)/(3𝑆𝑈 ∗ ) , and 𝜆 ∗ = 𝜆/𝑆𝑈 ∗ 𝑚 2 with 𝐸 = 10 −24 𝑛 2 /𝑡 , increases up to 103 times. However, CUDA code is 2% 𝑈 ∗ = 900𝐿 , and l = 2.856 Å , where is 𝑏 0 value in Eq. slow when system size is 64∆x × 64∆y . (6). We used 𝜆 ∗ = 2.4901 when considering magnetic ordering effects [9]. 2.4 Simulation results and analysis 2.3 Performance benchmark To improve the computational efficiency, we apply parallelization technique. In this study, CUDA was used