Strong-form approach to elasticity: hybrid finite difference-meshless collocation method (FDMCM). (English) Zbl 1480.65353
Summary: We propose a numerical method that combines the finite difference (FD) and strong form (collocation) meshless method (MM) for solving linear elasticity equations. We call this new method FDMCM. The FDMCM scheme uses a uniform Cartesian grid embedded in complex geometries and applies both methods to calculate spatial derivatives. The spatial domain is represented by a set of nodes categorized as (i) boundary and near boundary nodes, and (ii) interior nodes. For boundary and near boundary nodes, where the finite difference stencil cannot be defined, the Discretization Corrected Particle Strength Exchange (DC PSE) scheme is used for derivative evaluation, while for interior nodes standard second order finite differences are used. FDMCM method combines the advantages of both FD and DC PSE methods. It supports a fast and simple generation of grids and provides convergence rates comparable to weak formulations. We demonstrate the appropriateness and robustness of the proposed scheme through various benchmark problems in 2D and 3D. Numerical results show good accuracy and \(h\)-convergence properties. The ease of computational grid generation makes the method particularly suited for problems where geometries are very complicated and known only imperfectly from images, frequently occurring in e.g. geomechanics and patient-specific biomechanics, where the proposed FDMCM method, after its extension to non-linear regime, appears to be a promising alternative to the traditional weak form-based numerical schemes used in the field.
MSC:
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
| 74B05 | Classical linear elasticity |
Keywords:
meshless method; strong form; Cartesian grid embedded; finite difference method; discretization correction particle strength exchange; elastostaticsReferences:
| [1] | Schrader, B.; Reboux, S.; Sbalzarini, I. F., Discretization correction of general integral PSE operators for particle methods, J. Comput. Phys., 229, 4159-4182 (2010) · Zbl 1334.65196 |
| [2] | Bourantas, G. C.; Cheesman, B. L.; Ramaswamy, R.; Sbalzarini, I. F., Using DC PSE operator discretization in Eulerian meshless collocation methods improves their robustness in complex geometries, Comput. Fluids, 136, 285-300 (2016) · Zbl 1390.76645 |
| [3] | LeVeque, R. J.; Li, Z., The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal., 31, 1019 (1994) · Zbl 0811.65083 |
| [4] | Peskin, C. S.; Printz, B. F., Improved volume conservation in the computation of flows with immersed elastic boundaries, J. Comput. Phys., 105, 33 (1993) · Zbl 0762.92011 |
| [5] | Li, Z., A fast iterative method for elliptic interface problems, SIAM J. Numer. Anal., 35, 230 (1998) · Zbl 0915.65121 |
| [6] | Berger, M. J.; Colella, P., Local adaptive mesh refinement for shock hydrodynamics, J. Comput. Phys., 82, 64 (1989) · Zbl 0665.76070 |
| [7] | Berger, M. J.; Oliger, J., Adaptive mesh refinement for hyperbolic partial differential equations, J. Comput. Phys., 53, 484 (1984) · Zbl 0536.65071 |
| [8] | Bramble, J. H.; Ewing, R. E.; Pasciak, J. E.; Shen, J., The analysis of multigrid algorithms for cell-centered finite difference methods, Adv. Comput. Math., 5, 15 (1996) · Zbl 0848.65082 |
| [9] | Sanmiguel-Rojas, E.; Ortega-Casanova, J.; del Pino, C.; Fernandez-Feria, R., A Cartesian grid finite-difference method for 2D incompressible viscous flows in irregular geometries, J. Comput. Phys., 204, 302-318 (2005) · Zbl 1143.76486 |
| [10] | Li, S.; Liu, W. K., Meshfree Particle Methods (2004), Springer-Verlag: Springer-Verlag Berlin · Zbl 1073.65002 |
| [11] | Liu, G. R., Mesh Free Methods: Moving Beyond the Finite Element Method (2003), CRC Press: CRC Press Boca Raton · Zbl 1031.74001 |
| [12] | Liu, G. R.; Gu, Y. T., An Introduction to Meshfree Methods and Their Programming (2005), Springer: Springer The Netherlands |
| [13] | Atluri, S. N.; Shen, S. P., The Meshless Local Petrov-Galerkin (MLPG) Method (2002), Tech Science Press: Tech Science Press Encino, USA · Zbl 1012.65116 |
| [14] | Belytschko, T.; Lu, Y. Y.; Gu, L., Element free Galerkin method, Int. J. Numer. Methods Eng., 37, 229-256 (1994) · Zbl 0796.73077 |
| [15] | Lu, Y. Y.; Belytschko, T.; Gu, L., A new implementation of the element free Galerkin method, Comput. Methods Appl. Mech. Eng., 113, 397-414 (1994) · Zbl 0847.73064 |
| [16] | Metsis, P.; Lantzounis, N.; Papadrakakis, M., A new hierarchical partition of unity formulation of EFG meshless methods, Comput. Methods Appl. Mech. Eng., 283, 782-805 (2015) · Zbl 1423.74906 |
| [17] | Metsis, P.; Papadrakakis, M., Overlapping and non-overlapping domain decomposition methods for large-scale meshless EFG simulations, Comput. Methods Appl. Mech. Eng., 229-232, 128-141 (2012) · Zbl 1253.74110 |
| [18] | Zhang, T.; Dong, L.; Alotaibi, A.; Atluri, S. N., Application of the MLPG mixed collocation method for solving inverse problems of linear isotropic/anisotropic elasticity with simply/multiply-connected domains, CMES, 94, 1, 1-28 (2013) · Zbl 1356.74079 |
| [19] | Sellountos, E. J.; Polyzos, D.; Atluri, S. N., A new and simple meshless LBIE-RBF numerical scheme in linear elasticity, CMES, 89, 6, 513-551 (2012) · Zbl 1356.74256 |
| [20] | Atluri, S. N.; Sladek, J.; Sladek, V.; Zhu, T., The local boundary integral equation (LBIE) and it’s meshless implementation for linear elasticity, Comput. Mech., 25, 180-198 (2000) · Zbl 1020.74048 |
| [21] | Sladek, J.; Sladeck, V.; Van Keer, R., Meshless local boundary integral equation for 2D elastodynamic problems, Int. J. Numer. Methods Eng., 235-249 (2003) · Zbl 1062.74643 |
| [22] | Arroyo, M.; Ortiz, M., Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods Int, J. Numer. Meth. Engng, 65, 2167-2202 (2006) · Zbl 1146.74048 |
| [23] | Lancaster, P.; Salkauskas, K., Surfaces generated by moving least squares method, Math. Comput., 37, 141-158 (1981) · Zbl 0469.41005 |
| [24] | Joldes, G. R.; Chowdhuty, H. A.; Wittek, A.; Doyle, B.; Miller, K., Modified moving least squares with polynomial bases for scattered data approximation, Appl. Math. Comput., 266, 893-902 (2015) · Zbl 1410.65019 |
| [25] | Netuzhylov, H.; Zilian, A., Space – time meshfree collocation method: methodology and application to initial-boundary value problems, Int. J. Numer. Methods Eng., 80, 3, 355-380 (2009) · Zbl 1176.65114 |
| [26] | Mai-Duy, N.; Tran-Cong, T., Approximation of function and its derivatives using radial basis function networks, Appl. Math. Model., 27, 197-220 (2003) · Zbl 1024.65012 |
| [27] | Sukumar, N., Construction of polygonal interpolants: a maximum entropy approach, Int. J. Numer. Meth. Engng, 61, 2159-2181 (2004) · Zbl 1073.65505 |
| [28] | Fasshauer, G. E., Meshfree Approximation Methods With MATLAB (2007), World Scientific · Zbl 1123.65001 |
| [29] | Tolstykh, A., On using RBF-based differencing formulas for unstructured and mixed structured – unstructured grid calculations, (Proceedings of the Sixteenth IMACS World Congress. Proceedings of the Sixteenth IMACS World Congress, Lausanne (2000)), 1-6 |
| [30] | Shu, C.; Ding, H.; Yeo, K. S., Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 192, 7-8, 941-954 (2003) · Zbl 1025.76036 |
| [31] | Tolstykh, A. I.; Shirobokov, D. A., On using radial basis functions in a finite difference mode” with applications to elasticity problems, Computational Mechanics, 33, 68-79 (2003), Springer, December · Zbl 1063.74104 |
| [32] | Cecil, T.; Qian, J.; Osher, S., Numerical methods for high dimensional Hamilton-Jacobi equations using radial basis functions, J. Comput. Phys., 196, 327-347 (2004) · Zbl 1053.65086 |
| [33] | Perrone, N.; Kao, R., A general finite difference method for arbitrary meshes, Comput. Struct., 5, 45-58 (1975) |
| [34] | Jensen, P. S., Finite difference techniques for variable grids, Comput. Struct., 2, 17-29 (1972) |
| [35] | Liszka, T.; Orkisz, J., The finite difference method at arbitrary irregular grids and its application in applied mechanics, Comput. Struct., 11, 83-95 (1980) · Zbl 0427.73077 |
| [36] | Liszka, T., An interpolation method for an irregular net of nodes, Int. J. Numer. Meth. Eng., 20, 1599-1612 (1984) · Zbl 0544.65006 |
| [37] | Wen, P. H.; Aliabadi, M. H., A hybrid finite difference and moving least square method for elasticity problems, Eng. Anal. Bound. Elem., 36, 4, 600-605 (2012) · Zbl 1351.74161 |
| [38] | Degond, P.; Mas-Gallic, S., The weighted particle method for convection-di_usion equations. Part 2: The anisotropic case, Math. Comput., 53, 188, 509-525 (1989) · Zbl 0676.65122 |
| [39] | Eldredge, J. D.; Leonard, A.; Colonius, T., A general deterministic treatment of derivatives in particle methods, J. Comput. Phys., 180, 2, 686-709 (2002) · Zbl 1143.76550 |
| [40] | Schrader, B., Discretization-corrected PSE operators for adaptive multiresolution particle methods (Ph.D. thesis) (2011), Diss., Eidgenossische Technische Hochschule: Diss., Eidgenossische Technische Hochschule ETH Zurich |
| [41] | Schrader, B.; Reboux, S.; Sbalzarini, I. F., Choosing the best kernel: performance models for diffusion operators in 561 particle methods, SIAM J. Sci. Comput., 34, 3, A1607-A1634 (2012) · Zbl 1246.15010 |
| [42] | Reboux, S.; Schrader, B.; Sbalzarini, I. F., A self-organizing Lagrangian particle method for adaptive-resolution advection – diffusion simulations, J. Comput. Phys., 231, 9, 3623-3646 (2012) · Zbl 1402.65130 |
| [43] | Viccione, G.; Bovolin, V.; Carratelli, E. P., Defining and optimizing algorithms for neighbouring particle identification in SPH fluid simulations, Int. J. Numer. Methods Fluids, 58, 625-638 (2008) · Zbl 1283.76060 |
| [44] | Barnes, J.; Hut, P., A hierarchical \(O(N\) log \(N)\) force-calculation algorithm, Nature, 324, 446-449 (1986) |
| [45] | Greengard, L.; Gropp, W. D., A parallel version of the fast multipole method, Comput. Math. Appl., 20, 63-71 (1990) · Zbl 0715.65015 |
| [46] | Hockney, R. W.; Eastwood, J. W., Computer Simulation using Particles (1988), Institute of Physics Publishing · Zbl 0662.76002 |
| [47] | Verlet, L., Computer experiments on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules, Phys. Rev., 159, 1, 98-103 (1967) |
| [48] | Zienkiewicz, O. C.; Zhu, J. Z., A simple error estimator and adaptive procedure for practical engineering analysis, Int. J. Numer. Methods Eng., 24, 337-357 (1987) · Zbl 0602.73063 |
| [49] | Stevens, D.; Power, H.; Meng, C. Y.; Howard, D.; Cliffe, K. A., An alternative local collocation strategy for high-convergence meshless PDE solutions, using radial basis functions, J. Comput. Phys., 254, 1, 52-75 (2013) · Zbl 1349.65657 |
| [50] | Bishop, J. E., A displacement-based finite element formulation for general polyhedra using harmonic shape functions, Int. J. Numer. Meth. Engng, 97, 1-31 (2014) · Zbl 1352.74326 |
| [51] | Barber, J., Elasticity (2010), Springer: Springer New York · Zbl 1068.74001 |
| [52] | Seibold, B., Performance of algebraic multigrid methods for non-symmetric matrices arising in particle methods, Numer. Linear Algebra Appl., 17, 433-451 (2010) · Zbl 1240.76047 |
| [53] | Miller, K.; Joldes, G.; Lance, D.; Wittek, A., TLED total Lagrangian explicit dynamics finite element algorithm for computing soft tissue deformation, Commun. Numer. Methods Eng., 23, 2, 121-134 (2007) · Zbl 1154.74045 |
| [54] | Horton, A.; Wittek, A.; Joldes, G. R.; Miller, K., A meshless total Lagrangian explicit dynamics algorithm for surgical simulation, Int. J. Numer. Methods Biomed. Eng., 26, 8, 977-998 (2010) · Zbl 1193.92056 |
| [55] | Zhang, G.; Wittek, A.; Joldes, G.; Jin, X.; Miller, K., A three-dimensional nonlinear meshfree algorithm for simulating mechanical responses of soft tissue, Eng. Anal. Bound. Elem., 42, 60-66 (2014) · Zbl 1297.74076 |
| [56] | Miller, K.; Horton, A.; Joldes, G. R.; Wittek, A., Beyond finite elements: a comprehensive, patient-specific neurosurgical simulation utilizing a meshless method, J. Biomech., 45, 15, 2698-2701 (2012) |
| [57] | Li, M.; Miller, K.; Joldes, G. R.; Kikinis, R.; Wittek, A., Biomechanical model for computing deformations for whole-body image registration: a meshless approach, Int. J. Numer. Methods biomed. Eng. (2016) |
| [58] | Wittek, A.; Grosland, N. M.; Joldes, G. R.; Magnotta, V.; Miller, K., From finite element meshes to clouds of points: a review of methods for generation of computational biomechanics models for patient-specific applications, Annal. Biomed. Eng., 44, 1, 3-15 (2016) |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
