## 3 Performance Evaluation
### Triaxial Test  
The micro characteristics of the granular material are often calibrated using the triaxial element test by comparing the macro response of the samples. Simulation of triaxial tests was performed to evaluate the performance of the GPU parallel strategy. A cylindrical element of a standard sand specimen, with a height \(H\) of 8 cm and a diameter \(D\) of 4 cm, was modelled and enclosed by smooth boundary walls, as shown in Figure 4a. The Fujian standard sand--a marine sand--with the particle size distribution shown in Figure 4b, was considered. In order to reduce the computing time, particle diameters in the simulations were scaled up to 7.5 times the real sand particles, which were represented as \(d_{10}=2.4\) mm, \(d_{50}=3.2\) mm and \(d_{90}=5\) mm. The uniformity coefficient was \(C_{\text{u}}=d_{60}/d_{10}=1.57\) (well sorted). The initial void ratio was 0.592. Therefore, a total of 3920 spheres were generated and randomly distributed, corresponding to the characteristic ratio \(D/d_{50}\) as 12.5. Given the real particle sizes are used, more than 1,650,000 particles would be required. The density of the particles was 2643 kg/m\({}^{3}\). The self-gravity of the particles was not considered. The normal and tangential stiffness (\(K^{\text{n}}\), \(K^{\text{s}}\)) were taken as 300 MPa and 200 MPa, respectively, and the friction coefficient was 0.5. A local damping ratio of 0.7 was used to accelerate the equilibrium process. A confining pressure of 300 kPa was imposed prior to the loading procedure through a servo system on the boundary walls. The under-compaction method was adopted to prepare a homogenous specimen [61], then the specimen was loaded with the top and bottom plates moving inward at a velocity of 1 cm/s (i.e., a strain rate of 12.5%), while the confining pressure in the lateral direction was maintained. The stress-strain curves from the CPU and GPU computations are shown in Figure 5a. Considering the uncertainty due to the relatively small particle numbers, the predictions with different tools agree well with each other. The derivative stress converges to a critical value of 320 kPa. Figure 6 presents the transient velocity, the accumulative displacement, and the force chains on the particles at an axial strain of 6.2%. Another sequential CPU analysis using the widely used software package Particle Flow Code in 3 Dimensions (PFC3D) was also performed, and the results agree well with the CPU and GPU curves at the plastic stage of deformation, which verifies the accuracy of the tools developed in this study. An average runtime of 618 s was consumed in the CPU sequential calculations for 10,000 incremental steps, in which around 78% of the efforts were taken over the procedure \"Calculate contact forces\", and 22% over the procedure \"Update kinematic state of particles\". As 10 threads were invoked in the CPU parallel computation, a speedup of 4.3 was obtained over the CPU sequential calculation (Table 1). In comparison, the GPU parallelisation accelerates the computations 33 times over the CPU calculations using a single thread. In particular, the speedups of the GPU parallelisation for the procedures \"Calculate contact forces\" and \"Update kinematic state of particles\" are 26 and 272 (Table 2), respectively. Then, the ratio of the specimen diameter to the mean particle diameter, \(D/d_{50}\), was increased from 25 to 31.5, 40, 50, 72, 100, 126, 158, 200 and 316 by enlarging the model sizes to increase the computational scale, while the aspect ratio \(H/D\) of the specimen remained as 2; the corresponding particle numbers increased from 3920 to 7840, 15,680, 31,360, 100,880, 250,880, 501,760, 2,007,040 and 4,997,600. Theoretically, the space of the global memory (12 GB) of the GPU used in this study is capable of accommodating particles up to 27,000,000, for which the total runtime to finish an integrated case would be unacceptably long. Therefore, the maximum number of particles was kept below 5,000,000 as the concern of many geotechnical problems.