The solving capability of GPU was verified by simulation in Sect. 3, the performance of GPU-based testing system will be evaluated by shaking table RTHS testing in this section.
4.1 Schemes of testing
Figure 6 shows the principle of RTHS based on shaking table. The frame structure was installed on the shaking table as physical substructure. It interacted with numerical substructure in real time to test the dynamic performance of integral structure. The schemes of RTHS testing in this work are shown in Fig. 7. GPU was used to solve numerical model, and the solving time of numerical substructure was controlled by the loop structure of time control in LABVIEW. After solving numerical substructure based on GPU, the interface response yN was sent to physical substructure by signal transmission part. The physical substructure was loaded by shaking table after the loading system receiving the response. The dynamic response of the physical substructure f was measured by sensors and transmitted to numerical substructure. The dynamic characteristics of the shaking table were compensated by an out-loop controller, which was added between the signal transmission part and the shaking table. The command signal \(y_{N}^{'}\) was generated by the controller. The out-loop controller program was written in SIMULINK and then downloaded into dSPACE to run in real time.
4.2 Testing implementation
The parameters of experiment modal were same as Fig. 3 in Sect. 3.1. In this shaking table RTHS testing, the loading facility is a 0.5 m×0.5 m electromagnetic shaking table, solving part of numerical substructure is same as the GPU server configuration in Table 1, and the other parts are shown in Table 4.
Table 4
Testing system configuration parameters
Parts
|
configuration parameters
|
Signal transmission
|
NICompactDAQ-9174、NI-9201、NI-9263
|
Real-time simulation
|
dSPACE MicroLabBOX-1202
|
The layout of the physical substructure and the sensors were shown in Fig. 8.During the testing process, the sensors were connected to the control system of the shaking table. The acceleration sensors were installed on the top of the physical substructure and the shaking table. And the displacement of the physical substructure was measured by a laser displacement meter. The displacement of the shaking table was obtained from the control system of the shaking table. The physical substructure is an aluminum single-layer steel frame, and the bottom was fixed on the shaking table by bolts. The top mass of the physical substructure is 7 kg, and the mass of the four struts is 0.48 kg. As the struts mass only accounts for a small part of the total mass, the sum of the 1/2 struts mass and the top mass were added as the mass of the physical substructure, that is m = 7.24 kg. White noise displacement command signal with frequency range of 0.2–15 Hz and amplitude of 2 mm was inputted to the shaking table, and the stiffness and damping of the physical substructure are identified as k = 753.25 N/m and c = 0.44 N/(m/s). The Kobe and El-Centro waves with PGA = 0.5 g were input to shaking table, and the displacement of the physical substructure was compared with integral simulation through central difference method. Figure 9(i) shows the displacement time history results. In Fig. 9(ii), X-axis is the displacement time history in simulation, and Y-axis is the displacement time history of physical substructure in RTHS testing. The inevitable error existed when the frame was treated as a linear time-invariant system. Therefore, the measured and simulated response in Fig. 9 was not consistent perfectly. However, this work focuses on the feasibility of GPU for RTHS testing. The model of this frame is still accurate enough for evaluating the testing capacity of GPU-based RTHS system.
Furthermore, a white noise displacement signal with frequency range of 0.2–15 Hz and amplitude of 2 mm was input to the shaking table, the input signal and displacement of the shaking table were recorded. A fourth-order transfer function was used to identify characteristics of the shaking table, and the transfer function is shown in Eq. (3).
$${G_{st}}=\frac{{1.023e9}}{{{s^4}+427.5{s^3}+1.481e05{s^2}+3.041e07s+1.017e09}}$$
3
Figure 10 shows the transfer function of the shaking table and identified results. The amplitude and phase errors of the shaking table signals are very small when the frequency between 0-3 Hz, hence the shaking table command can be accurately loaded in this frequency range. When the frequency exceeds 3 Hz, the differences of amplitude and phase increase with the rise of frequency, so it is necessary to add an out-loop controller to improve the accuracy of loading.
Kobe wave with PGA of 0.5g as input command of the shaking table. Figure 12 shows the expected time histories and the achieved response with or without FSCS controller. The time histories with FSCS controller are much closer to the expectation than those without FSCS. The results show that the FSCS controlled shaking table is suitable for the RTHS testing.
In this testing framework, LABVIEW and MATLAB used in the numerical substructure were based on Windows operating system (OS). Windows is not a real-time OS, the real-time performance is unstable[36], the maximum of Windows clock frequency is 1kHz. In order to ensure that more CPU and memory resources are allocated to the numerical calculation, the priority of LABVIEW and MATLAB can be set higher in Windows OS task manager. There is a delay in data communication when LABVIEW invoked MATLAB, which leaded to a delay of nearly 3 ms in data transmission between MATLAB and LABVIEW. That is to say, after the dynamic analysis of numerical substructure is completed, it takes 3 ms to send the data to the signal transmission part. The MATLAB script stops calculating within this 3 ms. Therefore, the minimum time step in the RTHS testing in this work is shown as Eq. (4).
$$\Delta t={t_{solve}}+3$$
4
\(\Delta t\) in Eq. (4) is the actual time step of numerical integration, and \({t_{solve}}\) is the actual time cost. For example, the time step would be 4 ms when the time cost of solution is 1 ms. The accuracy of numerical integration is greatly affected by the time step[37], hence the maximum time step of RTHS testing is 20 ms in this work, because of the 3 ms in data communication, the actual time cost of solution is 17 ms.
4.3 Testing results
Table 5 shows the maximum DOFs of the numerical substructure model solved by GPU and CPU, with different time steps using double and single precision. Figure 13 shows time histories of the physical substructure solved by GPU and CPU, with Δt =4 ms and double precision data. The results of RTHS testing were consistent with the results of integral simulation, which indicated that the testing framework can meet the accuracy requirements of RTHS testing. Figure 14 shows displacement time histories of physical substructure obtained from GPU-based RTHS testing and integral simulation under working conditions NO.2-9 in Table 5. Groups (i) and (ii) in Figure 14 are 4 ms and 20 ms respectively. The displacement time histories of simulation are taken as X-axis, and the displacement time histories of the GPU-based RTHS testing are taken as Y-axis. All plots of GPU-based RTHS and simulation are basically linear lines. In some cases, the simulated response cannot match perfectly with RTHS testing results because of the modelling errors of physical substructure shown in Figure 9 but not the calculation errors resulted from GPU or CPU. Therefore, we are still sure that, with the same numerical substructure, the precision of RTHS testing based on GPU is same as CPU.
Table 5 Performance comparisons of numerical solution by GPU and CPU
NO.
|
Δt (ms)
|
Solving hardware
|
Precision
|
DOFs
|
1
|
4
|
CPU
|
double
|
1500
|
2
|
4
|
GPU
|
double
|
1500
|
3
|
4
|
GPU
|
single
|
3168
|
4
|
4
|
CPU
|
double
|
1080
|
5
|
4
|
CPU
|
single
|
1500
|
6
|
20
|
GPU
|
double
|
18876
|
7
|
20
|
GPU
|
single
|
27000
|
8
|
20
|
CPU
|
double
|
2904
|
9
|
20
|
CPU
|
single
|
3888
|
10
|
5
|
GPU
|
single
|
3888
|
11
|
20
|
GPU
|
single
|
3888
|
In order to discuss the influence of integration step in GPU-based RTHS testing, a numerical substructure model with 3888 DOFs was solved by GPU and CPU. NO.8 and 9 in Table 5 show that the minimum Δt are 5 ms and 20 ms respectively. Figure 15 shows displacement and acceleration time histories of physical substructure in the RTHS testing and integral simulation. The peak error of displacement between the RTHS testing and the simulation is 6.74% when Δt =5 ms, and 13.45% when Δt =20 ms. The peak error of acceleration between the RTHS testing and the simulation is 6.76% when Δt =5 ms, and 16.91% when Δt =20 ms. Therefore, GPU solution can carry out the RTHS testing with smaller time step than CPU solution and improve the accuracy of solution.
4.4 Discussion of testing results
NO. 2-5 in Table 5 shows that, with Δt =4 ms and double precision, the maximum DOFs is 1500 solved by GPU and 1080 solved by CPU, the advantage of GPU solving is not obvious. When the precision is single, the maximum DOFs is 3168 solved by GPU, and 1500 solved by CPU. The advance of solution by GPU with single precision is higher than double precision.
NO. 6-9 in Table 5 shows that, with Δt =20 ms and double precision, the maximum DOFs is 18,876 solved by GPU, and 2904 solved by CPU is. When the precision is single, the maximum DOFs is 27,000 solved by GPU, and 3888 solved by CPU. It can be seen that when Δt =20 ms, the advantage of DOFs solved by GPU is obvious, the maximum DOFs solved by GPU far exceed those solved by CPU. The RTHS testing with large-scale numerical substructure can be realized by GPU solution, which cannot be solved by CPU with the same time step.
In RTHS testing, the size of integration step needs to meet the requirement of real time. Using the GPU server and PCT, the RTHS testing with 27,000 DOFs can be carried out with Δt =20 ms, it needs 926 ms to solve by CPU at least. Solution by CPU is far from meeting the real time requirement of numerical solution. When the numerical substructure is 3888 DOFs, the time step is reduced from 20 ms(CPU-based) to 5 ms(GPU-based). According to results of the different time steps, the solution accuracy can be improved with smaller time step. The time step can be reduced, and the testing accuracy can be improved through the GPU-based solution in RTHS testing.
Due to the instability of clock frequency in Windows OS and the time delay caused by communication, Duo to the minimum execution time of task is 1 ms on Windows OS and the time delay caused by communication between LABVIEW and MATLAB, when Δt is 4 ms, the actual time cost of solution is 1 ms. The performance of GPU-based RTHS testing is limited. The maximum DOFs is 27,000 in the GPU-based RTHS testing because of the limitation of MATLAB in this work. Hence, the method of GPU solution, GPU parallel calculation, resource allocation and data communication still can be optimized, which may further improve the scale and efficiency of the solutions.