×

zbMATH — the first resource for mathematics

Development of general finite differences for complex geometries using a sharp interface formulation. (English) Zbl 1456.65142
Summary: We apply sharp interface boundary conditions to collocated meshfree approximations. Using the sharp interface formulation with the general finite difference (GFD) shape functions, we solve several test cases on meshfree grids with uniform and variable resolutions. Numerical results for the 2D Poisson equation indicate that the sharp interface formulation performs as well as the traditional collocated boundary approach with respect to both the convergence rate and accuracy. The analytic lid driven cavity results indicate that our discretization of a semi-implicit approximate projection scheme is indeed consistent and spatially converges at a second order rate under diffusive time stepping and at a first order rate under convective time stepping due to the first order in time splitting error. To resolve sharp gradients, variable resolution meshfree grids are considered for the classic lid driven cavity problem and for the uniform flow over a cylinder problem. Comparison to appropriate data sets indicate that our approach accurately estimates the various flow parameters using small cloud sizes of 13 neighbors while using high clustering ratios.
MSC:
65N06 Finite difference methods for boundary value problems involving PDEs
65N08 Finite volume methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
Software:
DistMesh; Eigen; ROOT
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Belytschko, T.; Lu, Y. Y.; Gu, L., Element-free Galerkin methods, Int J Numer Methods Eng, 37, 2, 229-256, (1994) · Zbl 0796.73077
[2] Atluri, S. N.; Zhu, T., A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Comput Mech, 22, 2, 117-127, (1998) · Zbl 0932.76067
[3] Liszka, T.; Orkisz, J., The finite difference method at arbitrary irregular grids and its application in applied mechanics, Comput Struct, 11, 83-95, (1979) · Zbl 0427.73077
[4] Liu, M.; Liu, G., Smoothed particle hydrodynamics (SPH): an overview and recent developments, Arch Comput Methods Eng, 17, 1, 25-76, (2010) · Zbl 1348.76117
[5] Tseng, Y.-H.; Ferziger, J. H., A ghost-cell immersed boundary method for flow in complex geometry, J Comput Phys, 192, 2, 593-623, (2003) · Zbl 1047.76575
[6] Mittal, R.; Iaccarino, G., Immersed boundary methods, Annu Rev Fluid Mech, 37, 239-261, (2005) · Zbl 1117.76049
[7] Belytschko, T.; Krongauz, Y.; Organ, D.; Fleming, M.; Krysl, P., Meshless methods: an overview and recent developments, Comput Methods Appl Mech Eng, 139, 1-4, 3-47, (1996) · Zbl 0891.73075
[8] Onate, E.; Idelsohn, S.; Zienkiewicz, O. C.; Taylor, R. L., A finite point method in computational mechanics. applications to convective transport and fluid flow, Int J Numer Methods Eng, 39, 3839-3866, (1996) · Zbl 0884.76068
[9] Ye, T.; Mittal, R.; Udaykumar, H.; Shyy, W., An accurate cartesian grid method for viscous incompressible flows with complex immersed boundaries, J Comput Phys, 156, 2, 209-240, (1999) · Zbl 0957.76043
[10] Perrone, N.; Kao, R., A general finite difference method for arbitrary meshes, Comput Struct, 5, 1, 45-57, (1975)
[11] Lancaster, P.; Salkauskas, K., Surfaces generated by moving least squares methods, Math Comp, 37, 155, 141-158, (1981) · Zbl 0469.41005
[12] Brun, R.; Rademakers, F., Root-an object oriented data analysis framework, Nucl Instrum Methods Phys Res Sect A, 389, 1-2, 81-86, (1997)
[13] Guennebaud G., Jacob B., et al. Eigen v3. http://eigen.tuxfamily.org; 2010.
[14] Schechter, H.; Bridson, R., Ghost SPH for animating water, ACM Trans Graph (ProcSIGGRAPH 2012), 31, 4, (2012)
[15] Jin, X.; Li, G.; Aluru, N., Positivity conditions in meshless collocation methods, Comput Methods Appl Mech Eng, 193, 12, 1171-1202, (2004) · Zbl 1060.74667
[16] Li, Z., A fast iterative algorithm for elliptic interface problems, SIAM J Numer Anal, 35, 1, 230-254, (1998) · Zbl 0915.65121
[17] Liu, X.-D.; Fedkiw, R. P.; Kang, M., A boundary condition capturing method for Poisson’s equation on irregular domains, J Comput Phys, 160, 1, 151-178, (2000) · Zbl 0958.65105
[18] Gibou, F.; Fedkiw, R. P.; Cheng, L.-T.; Kang, M., A second-order-accurate symmetric discretization of the Poisson equation on irregular domains, J Comput Phys, 176, 1, 205-227, (2002) · Zbl 0996.65108
[19] Chow, E.; Patel, A., Fine-grained parallel incomplete lu factorization, SIAM J Scient Comput, 37, 2, C169-C193, (2015) · Zbl 1320.65048
[20] Almgren, A. S.; Bell, J. B.; Crutchfield, W. Y., Approximate projection methods: part i. inviscid analysis, SIAM J Scient Comput, 22, 4, 1139-1159, (2000) · Zbl 0995.76059
[21] Cummins, S. J.; Rudman, M., An SPH projection method, J Comput Phys, 152, 2, 584-607, (1999) · Zbl 0954.76074
[22] Sousa, F. S.; Oishi, C. M.; Buscaglia, G. C., Spurious transients of projection methods in microflow simulations, Comput Methods Appl Mech Eng, 285, 659-693, (2015) · Zbl 1423.76103
[23] Oberkampf, W. L.; Trucano, T. G., Verification and validation in computational fluid dynamics, Prog Aerosp Sci, 38, 3, 209-272, (2002)
[24] Shih, T.; Tan, C.; Hwang, B., Effects of grid staggering on numerical schemes, Int J Numer Methods Fluids, 9, 2, 193-212, (1989) · Zbl 0661.76024
[25] Ghia, U.; Ghia, K. N.; Shin, C., High-re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, J Comput Phys, 48, 3, 387-411, (1982) · Zbl 0511.76031
[26] Persson, P.-O.; Strang, G., A simple mesh generator in matlab, SIAM Rev, 46, 2, 329-345, (2004) · Zbl 1061.65134
[27] Dennis, S.; Chang, G.-Z., Numerical solutions for steady flow past a circular cylinder at Reynolds numbers up to 100, J Fluid Mech, 42, 3, 471-489, (1970) · Zbl 0193.26202
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.