Session: 12-03-01: Minisymposium on Peridynamic Modeling of Materials’ Behavior

Paper Number: 77354

Start Time: Monday, 12:05 PM

77354 - The Fast Convolution-Based Method for Nonlocal Models 

Computational cost of peridynamic (PD) simulations (and nonlocal models in general) is relatively higher compared to that of corresponding local models, due to the higher “connectivity” between nodes. Existing discretization methods for PD, such as the meshfree discretization or the finite element method, have the computational complexity of O(N²), with N being the total number of degrees of freedom. These discretizations usually require neighbor search during initialization and storing neighbor information. This leads to a memory allocation that scales as O(N²) as well. As a result, large scale computation and simulations with very fine resolution of engineering problems at practically relevant scales may be beyond the reach of even the largest supercomputers.

In this study, we introduce a fast convolution-based method (FCBM) for peridynamic problems on bounded domains with arbitrary shapes and arbitrary boundary conditions. In the FCBM discretization of a given PD problem, one first expresses PD integrals in terms of convolutions. Then, the convolving functions are approximated with truncated Fourier series. Consequently, a quadrature approximation of the convolutions can be computed via fast Fourier transform (FFT) and inverse FFT operations at the cost of O(Nlog₂N). Having the quadrature computed in the Fourier space, neighbor identification and storage is not needed, which leads to significant savings in memory allocation, which now scales as O(N). Since, Fourier series approximation requires periodicity of the domain, we introduce an Embedded Constraint method to incorporate any desired volume constraint (nonlocal boundary conditions) into the FCBM framework. The main requirement for the applicability of the FCBM is finding a convolutional structure for a given PD integral formulation. In many cases this structure exists and is easy to write, as in the case of the linearized bond-based and state-based elastic material models, or for linear diffusion problems. For nonlinear models, finding the convolutional structure is carried out on a case-by-case basis. In certain cases, ingenious modifications of the PD model need to be devised in order to reach a convolutional structure amenable to the FCBM. For example, we introduced a new pointwise energy-based damage model to replace the critical bond-strain damage model in PD formulations of brittle fracture. With the new damage model, a convolutional structure can be found.

To date, the FCBM has been developed and implemented for a variety of PD problems including transient diffusion (linear, or nonlinear with certain types of nonlinearities), dynamic deformation and fracture of elastic and brittle materials (certain bond-based and state-based elastic materials with brittle fracture, PD correspondence models), and PD corrosion damage models. Here we show examples of obtaining the convolutional forms and numerical results for illustrative examples in 2D and 3D to demonstrate the remarkable efficiency gains in running time and memory allocation provided by the FCBM. Our results show that computations that potentially needs years to run (on a single processor), are now possible within a few days if not hours. Moreover, savings in terms of memory allocation requirements is several orders of magnitude. With the FCBM, fast and accurate simulations of complex damage problems like brittle fracture with branching and corrosion damage are now possible.

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., Mousavi, F., Larios, A., & Bobaru, F. (2021). A general and fast convolution-based method for peridynamics: applications to elasticity and brittle fracture (in review) arXiv preprints: 2105.06055.

[2] Jafarzadeh, S., Wang, L., Larios, A., & Bobaru, F. (2021). Computer Methods in Applied Mechanics and Engineering 375, 113633.

[3] Jafarzadeh, S., Larios, A., & Bobaru, F. (2020).  Journal of Peridynamics and Nonlocal Modeling 2, 185-110.

Presenting Author: Siavash Jafarzadeh University of Nebraska-Lincoln

Authors:

Siavash Jafarzadeh University of Nebraska-Lincoln
Farzaneh Mousavi University of Nebraska-Lincoln
Longzhen Wang University of Nebraska-Lincoln
Adam Larios University of Nebraska-Lincoln
Florin Bobaru University of Nebraska-Lincoln