We introduce an ultrafast Fourier spectral method for peridynamic (PD) volume-constrained/boundary-value problems. The efficiency of our method stems from the conversion of the PD integral operators into multiplications when using the Fast Fourier Transform (FFT), which reduces the complexity of PD solvers from O(*N*^2^) to O(*N*log*N*). *N* being the total number of nodes in the model. However, unlike the conventional Fourier spectral methods which are limited to problems on periodic domains, our boundary adapted spectral method (BASM) is applicable to general problems with arbitrary shaped domains with boundary conditions, by using a novel “embedded constraint” (EC) approach. To this aim we replace the boundary value problem on the bounded domain, with a constructed equivalent problem on extended periodic domain with volume constraints being embedded inside. We first discuss the BASM-EC formulation and show verification and convergence studies using exact nonlocal solutions. Then we present solutions for several PD problems in 2D and 3D: linear and nonlinear diffusion, quasi static elastic deformation, and dynamic brittle fracture. Our examples show that simulations that would require more than a century using the conventional meshfree method often used for discretizing PD models, are now possible within hours using the new BASM. Extra efficiency is easily achieved by utilizing MATLAB’s built-in FFT solvers that perform multi-threaded and GPU computation. Note that the efficiency of this method is only in the case of PD integral operators that possess some form of convolutional structure. While all linear PD operators have convolutional structure, some nonlinear PD operators cannot be expressed in terms of convolutions. We discuss how one can develop/modify PD operators with convolutional structures to serves the same purpose as the ones without this property.   

Acknowledgements: This work has been supported by NSF grant No. 1953346, and by a Nebraska System Science award from the Nebraska Research Initiative. This work was completed utilizing the Holland Computing Center of the University of Nebraska, which receives support from the Nebraska Research Initiative.

References

\[1\] Jafarzadeh, S., Larios, A., & Bobaru, F. (2020). Efficient solutions for nonlocal diffusion problems via boundary-adapted spectral methods. *Journal of Peridynamics and Nonlocal Modeling*, 1-26.

\[2\] Jafarzadeh, S., Wang, L., Larios, A., & Bobaru, F. (2020). A fast Fourier spectral method for peridynamic transient diffusion in arbitrary domains. *(in review)*.

\[3\] D'Elia, M., Du, Q., Glusa, C., Gunzburger, M., Tian, X., & Zhou, Z. (2020). Numerical methods for nonlocal and fractional models. *arXiv preprint arXiv:2002.01401*.


A Boundary Adapted Spectral Method for Peridynamic Problems on Arbitrary Domains

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Paper Proceedings: https://asmedigitalcollection.asme.org/IMECE/IMECE2020/volume/84478

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