Abstract
The peridynamic nonlocal continuum model for solid mechanics is an integro-differential equation that does not involve spatial derivatives of the displacement field. Several numerical methods such as finite element method and collocation method have been developed and analyzed in many articles. However, there is no theory to give a finite difference method because the model does not involve spatial derivatives of the displacement field. Here, we consider a finite difference scheme to solve a continuous static bond-based peridynamics model of mechanics based on its equivalent partial integro-differential equations. Furthermore, we present a fast solution technique to accelerate Toeplitz matrix-vector multiplications arising from finite difference discretization respectively. This fast solution technique is based on a fast Fourier transform and depends on the special structure of coefficient matrices, and it helps to reduce the computational work from \(O(N^{3})\) required by traditional methods to O(Nlog\(^{2}N)\) and the memory requirement from \(O(N^{2})\) to O(N) without using any lossy compression, where N is the number of unknowns. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.
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References
Bobaru, F., Yang, M., Alves, L.F., Silling, S.A., Askari, E., Xu, J.: Convergence, adaptive refinement, and scaling in 1D peridynamics. Int. J. Numer. Methods Eng. 77, 852–877 (2009)
Böttcher, A., Silbermann, B.: Introduction to Large Truncated Toeplitz Matrices. Springer Science & Business Media, Berlin (2012)
Casey, J., Krishnaswamy, S.: A characterization of internally constrained thermoelastic materials. Math. Mech. Solids 3, 71–89 (1998)
Chen, X., Gunzburger, M.: Continuous and discontinuous finite element methods for a peridynamics model of mechanics. Comput. Methods Appl. Mech. Eng. 200, 1237–1250 (2011)
Dan, G., Rand, O.: Dynamic thermoelastic coupling effects in a rod. Aiaa J. 33, 776–778 (2015)
Ding, H., Li, C., Chen, Y.: High-order algorithms for Riesz derivative and their applications (ii). J. Comput. Phys. 293, 218–237 (2015)
Du, Q., Zhou, K.: Mathematical analysis for the peridynamic nonlocal continuum theory. ESAIM Math. Model. Numer. Anal. 45, 217–234 (2011)
Gray, R.M.: Toeplitz and Circulant Matrices: A Review. Foundations and Trends in Communications and Information Theory 2(3), 155–239 (2006)
Hosseini-Tehrani, P., Eslami, M.R.: Bem analysis of thermal and mechanical shock in a two-dimensional finite domain considering coupled thermoelasticity. Eng. Anal. Bound. Elem. 24, 249–257 (2000)
Li, C., Zeng, F.: Numerical Methods for Fractional Calculus, vol. 24. CRC Press, Boca Raton (2015)
Lin, Y., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys 225, 1533–1552 (2007)
Liu, W., Long, X.: A new nonconforming finite element with a conforming finite element approximation for a coupled continuum pipe-flow/darcy model in karst aquifers. Numer. Methods Partial Differ. Equ. 32, 778–798 (2016)
Liu, W., Kang, Z., Rui, H.: Finite volume element approximation of the coupled continuum pipe-flow/darcy model for flows in karst aquifers. Numer. Methods Partial Differ. Equ. 30, 376–392 (2013)
Liu, Z., Cheng, A., Wang, H.: An hp-Galerkin method with fast solution for linear peridynamic models in one dimension. Comput. Math. Appl. 73, 1546–1565 (2017)
Madenci, E., Oterkus, E.: Peridynamic Theory and Its Applications. Springer, New York (2014)
Nickell, R.E., Sackman, J.L.: Approximate solutions in linear, coupled thermoelasticity. J. Appl. Mech. 35, 255–266 (1968)
Seleson, P., Littlewood, D.J.: Convergence studies in meshfree peridynamic simulations. Comput. Math. Appl. 71, 2432–2448 (2016)
Silling, S.A.: Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48, 175–209 (2000)
Silling, S.A.: Linearized theory of peridynamic states. J. Elast. 99, 85–111 (2010)
Silling, S.A., Askari, E.: A meshfree method based on the peridynamic model of solid mechanics. Comput. Struct. 83, 1526–1535 (2005)
Silling, S.A., Zimmermann, M., Abeyaratne, R.: Deformation of a peridynamic bar. J. Elast. 73, 173–190 (2003)
Taylor, M.J.: Numerical Simulation of Thermo-elasticity, Inelasticity and Rupture in Membrane Theory. Dissertations and Theses-Gradworks (2008)
Tian, X., Du, Q.: Analysis and comparison of different approximations to nonlocal diffusion and linear peridynamic equations. SIAM J. Numer. Anal. 51, 3458–3482 (2013)
Wang, H., Tian, H.: A fast galerkin method with efficient matrix assembly and storage for a peridynamic model. J. Comput. Phys. 231, 7730–7738 (2012)
Wang, H., Tian, H.: A fast and faithful collocation method with efficient matrix assembly for a two-dimensional nonlocal diffusion model. Comput. Methods Appl. Mech. Eng. 273, 19–36 (2014)
Zhou, K., Du, Q.: Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions. SIAM J. Numer. Anal. 48, 1759–1780 (2010)
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Liu, Z., Li, X. A Fast Finite Difference Method for a Continuous Static Linear Bond-Based Peridynamics Model of Mechanics. J Sci Comput 74, 728–742 (2018). https://doi.org/10.1007/s10915-017-0456-1
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DOI: https://doi.org/10.1007/s10915-017-0456-1