zbMATH — the first resource for mathematics

Improved one-point quadrature algorithms for two-dimensional peridynamic models based on analytical calculations. (English) Zbl 1423.74143
Summary: The peridynamics theory is a reformulation of the classical theory of continuum mechanics, based on long-range interactions, suitable for the description of material failure and damage. Integration plays a central role in this theory. We study one-point quadrature algorithms for the discretization of two-dimensional integrals in peridynamics. These algorithms are closely related to meshfree methods; they assume an underlying reference lattice, where each lattice point within a body is assigned a cell. A main challenge in such algorithms is the accurate estimation of the area of the intersecting region between the neighborhood of a point and a neighbor cell. We provide a classification of the different types of intersecting regions in square lattices and present analytical derivations for the exact calculation of their areas, leading to improved integration accuracy. To address convergence issues, geometric centers of intersecting regions are taken as quadrature points, replacing commonly used cell centers. We present analytical derivations for the exact calculation of those geometric centers, based on a decomposition of intersecting regions into subdomains of simple geometry. By using exact values for the areas and geometric centers of intersecting regions, we achieve an asymptotically monotonic convergence in numerical integration; in contrast, other algorithms from the literature exhibit a highly-oscillatory behavior. Numerical results compare the proposed algorithms with others from the literature, for different quantities of interest, and demonstrate their improved accuracy and convergence. Error estimates for the quadrature algorithms are derived, and extended and hybrid algorithms are discussed. Additional numerical studies, using influence functions with a finite support and a controlled regularity, suggest an alternative means to improve the convergence of discretization algorithms.

74B99 Elastic materials
65D30 Numerical integration
Full Text: DOI
[1] Silling, S. A., Reformulation of elasticity theory for discontinuities and long-range forces, J. Mech. Phys. Solids, 48, 175-209, (2000) · Zbl 0970.74030
[2] Silling, S. A.; Epton, M.; Weckner, O.; Xu, J.; Askari, E., Peridynamic states and constitutive modeling, J. Elast., 88, 151-184, (2007) · Zbl 1120.74003
[3] Hu, W.; Wang, Y.; Yu, J.; Yen, C.-F.; Bobaru, F., Impact damage on a thin Glass plate with a thin polycarbonate backing, Int. J. Impact Eng., 62, 152-165, (2013)
[4] Silling, S. A.; Askari, E., A meshfree method based on the peridynamic model of solid mechanics, Comput. Struct., 83, 1526-1535, (2005)
[5] Kilic, B.; Agwai, A.; Madenci, E., Peridynamic theory for progressive damage prediction in center-cracked composite laminates, Compos. Struct., 90, 141-151, (2009)
[6] Oterkus, E.; Madenci, E.; Weckner, O.; Silling, S.; Bogert, P.; Tessler, A., Combined finite element and peridynamic analyses for predicting failure in a stiffened composite curved panel with a central slot, Compos. Struct., 94, 839-850, (2012)
[7] Xu, J.; Askari, A.; Weckner, O.; Silling, S., Peridynamic analysis of impact damage in composite laminates, J. Aerosp. Eng., SPECIAL ISSUE: Impact Mech. Compos. Mater. Aerosp. Appl., 21, 187-194, (2008)
[8] Silling, S. A.; Weckner, O.; Askari, E.; Bobaru, F., Crack nucleation in a peridynamic solid, Int. J. Fract., 162, 219-227, (2010) · Zbl 1425.74045
[9] Bobaru, F., Influence of van der Waals forces on increasing the strength and toughness in dynamic fracture of nanofibre networks: a peridynamic approach, Modelling Simul. Mater. Sci. Eng., 15, 397-417, (2007)
[10] Silling, S. A.; Bobaru, F., Peridynamic modeling of membranes and fibers, Int. J. Non-Linear Mech., 40, 395-409, (2005) · Zbl 1349.74231
[11] Ha, Y. D.; Bobaru, F., Studies of dynamic crack propagation and crack branching with peridynamics, Int J. Fract., 162, 229-244, (2010) · Zbl 1425.74416
[12] Gerstle, W.; Sau, N.; Silling, S., Peridynamic modeling of concrete structures, Nucl. Eng. Des., 237, 1250-1258, (2007)
[13] Eringen, A. C., Nonlocal continuum field theories, (2002), Springer New York · Zbl 1023.74003
[14] Kröner, E., Elasticity theory of materials with long range cohesive forces, Int. J. Solids Struct., 3, 5, 731-742, (1967) · Zbl 0163.19402
[15] Kunin, I. A., Elastic media with microstructure I: one-dimensional models, (Springer Series in Solid State Sciences, vol. 26, (1982), Springer-Verlag Berlin) · Zbl 0527.73002
[16] Kunin, I. A., Elastic media with microstructure II: three-dimensional models, (Springer Series in Solid State Sciences, vol. 44, (1983), Springer-Verlag Berlin) · Zbl 0536.73003
[17] Rogula, D., Introduction to nonlocal theory of material media, (Rogula, D., Nonlocal Theory of Material Media, (1982), Springer-Verlag Berlin), 125-222
[18] Bobaru, F.; Duangpanya, M., The peridynamic formulation for transient heat conduction, Int. J. Heat Mass Transfer, 53, 4047-4059, (2010) · Zbl 1194.80010
[19] Bobaru, F.; Duangpanya, M., A peridynamic formulation for transient heat conduction in bodies with evolving discontinuities, J. Comput. Phys., 231, 2764-2785, (2012) · Zbl 1253.80002
[20] Burch, N.; Lehoucq, R., Classical, nonlocal, and fractional diffusion equations on bounded domains, Int. J. Multiscale Comput. Eng., 9, 661-674, (2011)
[21] Du, Q.; Gunzburger, M.; Lehoucq, R. B.; Zhou, K., Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev., 54, 4, 667-696, (2012) · Zbl 1422.76168
[22] Seleson, P.; Gunzburger, M.; Parks, M. L., Interface problems in nonlocal diffusion and sharp transitions between local and nonlocal domains, Comput. Methods Appl. Mech. Engrg., 266, 185-204, (2013) · Zbl 1286.74010
[23] Parks, M. L.; Seleson, P.; Plimpton, S. J.; Lehoucq, R. B.; Silling, S. A., Peridynamics with LAMMPS: a user guide, tech. rep. SAND2010-5549, sandia national laboratories, (2010)
[24] Parks, M. L.; Lehoucq, R. B.; Plimpton, S. J.; Silling, S. A., Implementing peridynamics within a molecular dynamics code, Comput. Phys. Commun., 179, 11, 777-783, (2008) · Zbl 1197.82014
[25] Seleson, P.; Parks, M. L.; Gunzburger, M.; Lehoucq, R. B., Peridynamics as an upscaling of molecular dynamics, Multiscale Model. Simul., 8, 1, 204-227, (2009) · Zbl 1375.82073
[26] Seleson, P.; Parks, M. L.; Gunzburger, M., Peridynamic state-based models and the embedded-atom model, Commun. Comput. Phys., 15, 1, 179-205, (2014) · Zbl 1373.82029
[27] Kilic, B.; Madenci, E., Structural stability and failure analysis using peridynamic theory, Int. J. Non-Linear Mech., 44, 845-854, (2009) · Zbl 1203.74045
[28] Yu, K.; Xin, X. J.; Lease, K. B., A new adaptive integration method for the peridynamic theory, Modelling Simul. Mater. Sci. Eng., 19, 045003, (2011)
[29] Henke, S. F.; Shanbhag, S., Mesh sensitivity in peridynamic simulations, Comput. Phys. Commun., 185, 181-193, (2014) · Zbl 1344.65115
[30] Du, Q.; Ju, L.; Tian, L.; Zhou, K., A posteriori error analysis of finite element method for linear nonlocal diffusion and peridynamic models, Math. Comp., 82, 284, 1889-1922, (2013) · Zbl 1327.65101
[31] Chen, X.; Gunzburger, M., Continuous and discontinuous finite element methods for a peridynamics model of mechanics, Comput. Methods Appl. Mech. Engrg., 200, 1237-1250, (2011) · Zbl 1225.74082
[32] S. Bond, Quadrature for nonlocal mechanics and peridynamics, Presentation at the 12th U.S. National Congress on Computational Mechanics, July 22-25, 2013, Raleigh, NC, USA.
[33] Hu, W.; Ha, Y. D.; Bobaru, F., Numerical integration in peridynamics, tech. rep., university of nebraska-lincoln, department of mechanical & materials engineering, (September 2010)
[34] Emmrich, E.; Weckner, O., The peridynamic equation and its spatial discretisation, Math. Model. Anal., 12, 1, 17-27, (2007) · Zbl 1121.65073
[35] Silling, S. A., Linearized theory of peridynamic states, J. Elast., 99, 85-111, (2010) · Zbl 1188.74008
[36] Bobaru, F.; Yang, M.; Alves, L. F.; Silling, S. A.; Askari, E.; Xu, J., Convergence, adaptive refinement, and scaling in 1D peridynamics, Internat. J. Numer. Methods Engrg., 77, 852-877, (2009) · Zbl 1156.74399
[37] Seleson, P.; Parks, M. L., On the role of the influence function in the peridynamic theory, Int. J. Multiscale Comput. Eng., 9, 6, 689-706, (2011)
[38] Seleson, P., Peridynamic multiscale models for the mechanics of materials: constitutive relations, upscaling from atomistic systems, and interface problems, (2010), Florida State University, (Ph.D. thesis)
[39] Foster, J. T.; Silling, S. A.; Chen, W. W., Viscoplasticity using peridynamics, Internat. J. Numer. Methods Engrg., 81, 1242-1258, (2010) · Zbl 1183.74035
[40] Foster, J. T., Dynamic crack initiation toughness: experiments and peridynamic modeling, tech. rep. SAND2009-7217, sandia national laboratories, (2009)
[41] Tupek, M. R.; Rimoli, J. J.; Radovitzky, R., An approach for incorporating classical continuum damage models in state-based peridynamics, Comput. Methods Appl. Mech. Engrg., 263, 20-26, (2013) · Zbl 1286.74022
[42] Kilic, B.; Madenci, E., Prediction of crack paths in a quenched Glass plate by using peridynamic theory, Int. J. Fract., 156, 165-177, (2009) · Zbl 1273.74455
[43] Kilic, B.; Madenci, E., Coupling of peridynamic theory and the finite element method, J. Mech. Mater. Struct., 5, 5, 707-733, (2010)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.