# Multi-GPU Acceleration for Finite Element Analysis in Structural Mechanics
## 6 Conclusions  
We have presented our experiences using modern high-level libraries to exploit the computational resources of multi-core and many-core architectures to accelerate finite element analysis in structural mechanics. The numerical results show that using a distributed conjugate gradient solver preconditioned with aggressive AMG is especially rewarding using GPU computing. These massively parallel computer architectures provide a high bandwidth that we can use to accelerate the resolution of structural mechanics system of equations using the proper techniques. The numerical experiments evaluate the weak scalability using distributed multi-core and many-core systems to address structural mechanics problems. Such problems include models with structured and unstructured meshes using 2D and 3D finite elements, which we solve with different relaxing operators. It also presents a quantitative comparison of the use of both approaches with different computational resources and techniques. The results show the limitations of multi-GPU systems due to the device memory requirements and the advantages in computational capabilities.  
The comparison between multi-core and many-core systems shows higher speedups using GPU computing concerning the CPU implementation counterpart. This conclusion is especially true when using multiple graphics devices, which provide a memory bandwidth and latency that can hardly obtain a multi-core system. Another advantage of using many-core architectures is the use of a fewer number of subdomains. A high number of subdomains generates significant problems partitioning large-scale models and increases the use of communications with the global subdomain implementation used in this work. We also have to remark that we perform the numerical experiments on a relatively low-cost workstation, which can address problems of several millions of unknowns in a reasonable time. We adopt, implement, and evaluate techniques that can distribute the computing burden in GPU clusters, which reduces the GPU computing limitations exposed in this work.  
As future developments, we can mention the extension of experiments to other GPU architectures to evaluate the applicability and adaptability of the aggregation AMG method across different hardware platforms. A performance comparison between GMG, classical AMG, and aggregation AMG methods using distributed GPU implementations is also an interesting future work. We also plan to evaluate the GPU implementation using mixed precision operations because GPU hardware specifications show an increasing teraflop rating as reducing the precision of the operations. Finally, we can also propose the integration of the aggregation AMG preconditioner with cloud computing environments using synchronous and asynchronous iterative parallel methods for solving structural mechanic problems.  
Conceptualization, D.H.-P. and H.M.-B.; software, D.H.-P. and H.M.-B.; investigation, D.H.-P. and H.M.-B.; supervision, H.M.-B.; project administration, H.M.-B.; writing--review and editing, D.H.-P. All authors have read and agreed to the published version of the manuscript.  
This research was funded by AEI/FEDER and UE grant number DPI2016-77538-R.  
Figure 9: Wall-clock time of setup and solving stages using GPU computing.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.  
**Conflicts of Interest:** The authors declare no conflicts of interest.