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The finite element square reduced (\(FE^{2R}\)) method with GPU acceleration: towards three-dimensional two-scale simulations. (English) Zbl 1352.74352

Summary: The \(FE^{2R}\) method is a renown computational multiscale simulation technique for solid materials with fine-scale microstructure. It allows for the accurate prediction of the mechanical behavior of structures made of heterogeneous materials with nonlinear material behavior. However, the \(FE^{2}\) method leads to excessive CPU time and storage requirements, even for academic two-dimensional problems. In order to allow for realistic three-dimensional two-scale simulations, a significant reduction of the CPU and memory usage is required. For this purpose, the authors have recently proposed a reduced basis homogenization scheme based on a mixed incremental variational principle. The approach exploits the potential structure of generalized standard materials. Thereby, important speed-ups and memory savings can be achieved. Using high-performance GPUs, the reduced-basis method can be further accelerated. In the present contribution, our previous works are combined and extended to form the \(FE^{2}\)-reduced method: the \(FE^{2R}\). The \(FE^{2R}\) can be used to simulate three-dimensional structural problems with consideration of the nonlinearity and microstructure of the underlying material at acceptable computational cost. Thereby, it allows for a new level of complexity in nonlinear multiscale simulations. Numerical examples illustrate the capabilities of the chosen approach.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65Y10 Numerical algorithms for specific classes of architectures
74M25 Micromechanics of solids

Software:

CUDA
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References:

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