A conservative level-set based method for compressible solid/fluid problems on fixed grids. (English) Zbl 1432.74060
Summary: A three-dimensional Eulerian method is presented for simulating dynamic systems comprising multiple compressible solid and fluid components where internal boundaries are tracked using level-set functions. Aside from the interface interaction calculation within mixed cells, each material is treated independently and the governing constitutive laws solved using a conservative finite volume discretisation based upon the solution of Riemann problems to determine the numerical fluxes. The required reconstruction of mixed cell volume fractions and cut cell geometries is presented in detail using the level-set fields. High-order accuracy is achieved by incorporating the weighted-essentially non-oscillatory (WENO) method and Runge – Kutta time integration. A model for elastoplastic solid dynamics is employed formulated using the tensor of elastic deformation gradients permitting the equations to be written in divergence form. The scheme is demonstrated using selected one-dimensional initial value problems for which exact solutions are derived, a two-dimensional void collapse, and a three-dimensional simulation of a confined explosion.
MSC:
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
| 74S10 | Finite volume methods applied to problems in solid mechanics |
| 76M12 | Finite volume methods applied to problems in fluid mechanics |
Keywords:
solid dynamics; fluid mechanics; level-set; Eulerian; WENO; cut-cell; multi-material; Riemann problemSoftware:
HE-E1GODFReferences:
| [1] | Ahn, H. T.; Shashkov, M., Multi-material interface reconstruction on generalized polyhedral meshes, J. Comput. Phys., 226, 2096 (2007) · Zbl 1388.76232 |
| [2] | Balsara, D. S.; Shu, C., Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order accuracy, J. Comput. Phys., 160, 405 (2000) · Zbl 0961.65078 |
| [3] | Barton, P. T.; Drikakis, D.; Romenski, E.; Titarev, V. A., Exact and approximate solutions of Riemann problems in non-linear elasticity, J. Comput. Phys., 228, 7046 (2009) · Zbl 1172.74032 |
| [4] | Barton, P. T.; Drikakis, D.; Romenski, E., An Eulerian finite-volume scheme for large elastoplastic deformations in solids, Int. J. Numer. Methods Eng., 81, 453 (2010) · Zbl 1183.74331 |
| [5] | Barton, P. T.; Drikakis, D., An Eulerian method for multi-component problems in non-linear elasticity with sliding interfaces, J. Comput. Phys., 229, 5518 (2010) · Zbl 1346.74179 |
| [6] | Colella, P.; Graves, D. T.; Keen, B. J.; Modiano, D., A Cartesian grid embedded boundary method for hyperbolic conservation laws, J. Comput. Phys., 211, 347 (2006) · Zbl 1120.65324 |
| [7] | A. Eberle, Characteristic Flux Averaging Approach to the Solution of Euler’s Equations, in: VKI Lecture Series, Computational Fluid Dynamics, 1987.; A. Eberle, Characteristic Flux Averaging Approach to the Solution of Euler’s Equations, in: VKI Lecture Series, Computational Fluid Dynamics, 1987. |
| [8] | A. Eberle, M. Schmatz, N. Bissinger, Generalized fluxvectors for hypersonic shock-capturing, in: AIAA 90-0390, 1990.; A. Eberle, M. Schmatz, N. Bissinger, Generalized fluxvectors for hypersonic shock-capturing, in: AIAA 90-0390, 1990. |
| [9] | Enright, D.; Fedkiw, R.; Ferziger, J.; Mitchell, I., A hybrid particle level set method for improved interface capturing, J. Comput. Phys., 183, 83 (2002) · Zbl 1021.76044 |
| [10] | Favrie, N.; Gavrilyuk, S. L.; Saurel, R., Solid-fluid diffuse interface model in cases of extreme deformations, J. Comput. Phys., 228, 6037 (2009) · Zbl 1280.74013 |
| [11] | Fedkiw, R. P.; Marquina, A.; Merriman, B., An isobaric fix for the overheating problem in multimaterial compressible flows, J. Comput. Phys., 148, 545 (1999) · Zbl 0933.76075 |
| [12] | Fedkiw, R. P.; Aslam, T.; Merriman, B.; Osher, S., A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method), J. Comput. Phys., 152, 457 (1999) · Zbl 0957.76052 |
| [13] | W. Fickett, L.M. Scherr, Numerical calculation of the cylinder test, in: LA-5906, 1975.; W. Fickett, L.M. Scherr, Numerical calculation of the cylinder test, in: LA-5906, 1975. |
| [14] | Godunov, S. K.; Romenskii, E. I., Elements of Continuum Mechanics and Conservation Laws (2003), Kluwer Academic/Plenum Publishers · Zbl 1031.74004 |
| [15] | Goldman, R. N., Area of planar polygons and volume of polyhedra, (Arvo, J., Graphics Gems II, vol. 2 (1994), Morgan Kaufmann) |
| [16] | Hartmann, D.; Meinke, M.; Schröder, W., The constrained reinitialization equation for level set methods, J. Comput. Phys., 229, 1514 (2010) · Zbl 1329.76259 |
| [17] | Hu, X. Y.; Khoo, B. C.; Adams, N. A.; Huang, F. L., A conservative interface method for compressible flows, J. Comput. Phys., 219, 553 (2006) · Zbl 1102.76038 |
| [18] | Ji, H.; Lien, F.-S.; Yee, E., Numerical simulation of detonation using an adaptive Cartesian cut-cell method combined with a cell-merging technique, Comput. Fluids, 39, 1041 (2010) · Zbl 1242.76168 |
| [19] | Jiang, G. S.; Shu, C. W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126, 202 (1996) · Zbl 0877.65065 |
| [20] | Liu, T. G.; Khoo, B. C.; Yeo, K. S., Ghost fluid method for strong shock impacting on material interface, J. Comput. Phys., 190, 651 (2003) · Zbl 1076.76592 |
| [21] | Lorensen, W. E.; Cline, H. E., Marching cubes: A high resolution 3D surface construction algorithm, Comput. Graphics, 21, 163 (1987) |
| [22] | Merzhievsky, L. A.; Tyagel’sky, A. V., Relation of dislocation kinetics with dynamic characteristics in modelling mechanical behaviour of materials, J. de Physique C3, 49, 457-466 (1988) |
| [23] | Miller, G. H.; Colella, P., A high-order Eulerian Godunov method for elastic-plastic flow in solids, J. Comput. Phys., 167, 131 (2001) · Zbl 0997.74078 |
| [24] | Miller, G. H.; Colella, P., A conservative three-dimensional Eulerian method for coupled solid-fluid shock capturing, J. Comput. Phys., 183, 26 (2002) · Zbl 1057.76558 |
| [25] | Nourgaliev, R. R.; Theofanous, T. G., High-fidelity interface tracking in compressible flows: unlimited anchored adaptive level set, J. Comput. Phys., 224, 836 (2007) · Zbl 1124.76043 |
| [26] | Nourgaliev, R. R.; Liou, M.-S.; Theofanous, T. G., Numerical prediction of interfacial instabilities: sharp interface method (SIM), J. Comput. Phys., 227, 3940 (2008) · Zbl 1275.76164 |
| [27] | Pember, R. B.; Bell, J. B.; Collela, P.; Crutchfield, W. Y.; Welcome, M. L., An adaptive Cartesian grid method for unsteady compressible flow in irregular regions, J. Comput. Phys., 120, 278 (1995) · Zbl 0842.76056 |
| [28] | Peng, D.; Merriman, B.; Osher, S.; Zhao, H.; Kang, M., A PDE-based fast local level set method, J. Comput. Phys., 155, 410 (1999) · Zbl 0964.76069 |
| [29] | Romenski, E. I., Thermodynamics and hyperbolic systems of balance laws in continuum mechanics, (Toro, E. F., Godunov Methods: Theory and Applications (2001), Kluwer Academic/Plenum Publishers) · Zbl 1017.74004 |
| [30] | Titarev, V. A.; Romenski, E.; Toro, E. F., MUSTA-type upwind fluxes for non-linear elasticity, Int. J. Numer. Methods Eng., 73, 897 (2008) · Zbl 1159.74046 |
| [31] | Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction (1999), Springer · Zbl 0923.76004 |
| [32] | Tran, L. B.; Udaykumar, H. S., A particle-level set-based sharp interface cartesian grid method for impact, penetration, and void collapse, J. Comput. Phys., 193, 469 (2004) · Zbl 1109.74376 |
| [33] | Yang, G.; Causon, D. M.; Ingram, D. M., Calculation of compressible flows about complex moving geometries using a three-dimensional Cartesian cut cell method, Int. J. Numer. Methods Eng., 33, 1121 (2000) · Zbl 0980.76052 |
| [34] | Zhao, H.-K.; Osher, S.; Merriman, B.; Kang, M., Implicit and nonparametric shape reconstruction from unorganized data using a variational level set method, Comput. Vis. Image Und., 80, 295 (2000) · Zbl 1011.68538 |
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.
