A simple and systematic scheme implemented in explicit FEM solver for surface tension effects in powder injection moulding process

International Journal of Material Forming - Tập 12 - Trang 123-134 - 2018
Jianjun Shi1, Thierry Barriere2, Baosheng Liu3, Zhiqiang Cheng3
1Shock and Vibration of Engineering Materials and Structures Key Laboratory of Sichuan Province, Southwest University of Science and Technology, Mianyang, China
2Department of Applied Mechanics, Femto-ST Institute, University Bourgogne-Franche-Comté, COMUE UBFC, Besançon, France
3Department of Mechanics and Engineering, Southwest Jiaotong University, Chengdu, China

Tóm tắt

In order to simulate more accurately the powder injection moulding process, the explicit finite element method solver is extended with the surface tension effect. The evaluation of surface tension takes the notion of pressure boundary method, while a simple and systematic scheme is proposed to fit the finite element method solver for the Laplacian operator. Because of the difference in dimension for filling function and velocity function, the integration of filling function in second derivative is not suitable to be transformed into the boundary integration and the integration of function in lower order derivative. To evaluate conveniently the curvature of filling front, hence the force of surface tension, a simple and systematic scheme is suggested and implemented into the finite element method solver. This specific scheme includes only the vectorial operations in low cost, and is completely systematic without piecemeal operations. Fitness of the proposed method is proved by the numerical examples of filling flow in a small-scaled channel. It shows the considerable effect of surface tension for the problems in micro-scale of sub-millimeter sizes, in which the boundary conditions at front surface are not negligible in powder injection moulding process. The surface tension effect becomes the dominating role for governing the trace and shape of filling front, which can no longer be neglected.

Tài liệu tham khảo

Sha B, Dimov S, Griffiths C, Packianather MS (2007) Investigation of micro-injection moulding: factors affecting the replication quality. J Mater Process Technol 183(2-3):284–296 Foudzi FM, Muhamad N, Sulong AB, Zakaria H (2013) Yttria stabilized zirconia formed by micro ceramic injection molding: rheological properties and debinding effects on the sintered part. Ceram Int 39(3):2665–2674 Cheng ZQ, Barriere T, Liu BS, Gelin JC (2009) A new explicit simulation for injection molding and its validation. Polym Eng Sci 49(6):1243–1252 Cao W, Hassanger O, Wang Y (2008) Surface tension effect on micro-injection molding. Proceedings of the polymer processing society, 24th annual meeting, PPS 24, June 15–19. Salerno (Italy) Kim DS, Lee K, Kwon TH, Lee SS (2002) Micro-channel filling flow considering surface tension effect. J Micromech Microeng 12:3 Liovic P, Lakehal D (2012) Subgrid-scale modelling of surface tension within interface tracking-based large Eddy and interface simulation of 3D interfacial flows. Comput Fluids 63:27–46 Raessi M, Mostaghimi J, Bussmann M (2010) A volume-of-fluid interfacial flow solver with advected normals. Comput Fluids 39(8):1401–1410 Eyiyurekli M, Breen D (2010) Interactive free-form level-set surface-editing operators. Comput Graph 34(5):621–638 Moelans N, Blanpain B, Wollants P (2008) An introduction to phase-field modeling of microstructure evolution. Calphad 32(2):268–294 Pianet G, Vincent S, Leboi J, Caltagirone JP, Anderhuber M (2010) Simulating compressible gas bubbles with a smooth volume tracking 1-fluid method. Int J Multiphase Flow 36(4):273–283 Bourlioux AA (1995) A coupled level set volume of fluid algorithm for tracking material interfaces. In: Dwyer HA (ed) Proceedings of the sixth international symposium on computational fluid dynamics, Lake Tahoe, p 15–22 Sussman M, Puckett EG (2000) A coupled level set and volume-of-fluid method for computing 3D and axisymmetric incompressible two-phase flows. J Comput Phys 162(2):301–337 Son G, Hur N (2002) A couple level set and volume-of-fluid method for the buoyancy-driven motion of fluid particles. Numer Heat Transf B Fund 42(6):523–542 Ménard T, Tanguy S, Berlemont A (2007) Coupling level set/VOF/ghost fluid methods: validation and application to 3D simulation of the primary break-up of a liquid jet. Int J Multiphase Flow 33(5):510–524 Abadie T, Aubin J, Legendre D (2015) On the combined effects of surface tension force calculation and interface advection on spurious currents within Volume of Fluid and Level Set frameworks. J Comput Phys 297:611–636 Li Q, Ou YJ, Yang B, Li X (2012) Numerical simulation of gas-assisted injection molding using CLSVOF method. Appl Math Model 36(5):2262–2274 Sun DL, Tao WQ (2010) A coupled volume-of-fluid and level set (VOSET) method for computing incompressible two-phase flows. Int J Heat Mass Transf 53(4):645–655 Shu CW, Osher S (1989) Efficient implementation of essentially non-oscillatory shock-capturing schemes, II. J Comput Phys 83(1):32–78 Lv X, Zou Q, Zhao Y, Reeve D (2010) A novel coupled level set and volume of fluid method for sharp interface capturing on 3D tetrahedral grids. J Comput Phys 229(7):2573–2604 Hysing S (2012) Mixed element FEM level set method for numerical simulation of immiscible fluids. J Comput Phys 231(6):2449–2465 Dréau K, Chevaugeon N, Moës N (2010) Studied X-FEM enrichment to handle material interfaces with higher order finite element. Comput Method Appl Mech Eng 199(29-32):1922–1936 Yang X, James AJ, Lowengrub J, Zheng X, Cristini V (2006) An adaptive coupled level-set/volume-of-fluid interface capturing method for unstructured triangular grids. J Comput Phys 217(2):364–394 Macklin P, Lowengrub J (2006) An improved geometry-aware curvature discretization for level set methods: application to tumor growth. J Comput Phys 215(2):392–401 Ramšak M, Škerget L (2007) 3D multidomain BEM for solving the Laplace equation. Eng Anal Bound Elem 31(6):528–538 Trontin P, Vincent S, Estivalezes JL, Caltagirone JP (2012) A subgrid computation of the curvature by a particle/level-set method. Application to a front-tracking/ghost-fluid method for incompressible flows. J Comput Phys 231(20):6990–7010 Rangogni R (1986) Numerical solution of the generalized Laplace equation by coupling the boundary element method and the perturbation method. Appl Math Model 10(4):266–270 Meier M, Yadigaroglu G, Smith BL (2002) A novel technique for including surface tension in PLIC-VOF methods. Eur J Mech B Fluids 21(1):61–73 Raessi M, Mostaghimi J, Bussmann M (2007) Advecting normal vectors: a new method for calculating interface normals and curvatures when modeling two-phase flows. J Comput Phys 226(1):774–797 Tong AY, Wang Z (2007) A numerical method for capillarity-dominant free surface flows. J Comput Phys 221(2):506–523 Brackbill JU, Kothe DB, Zemach C (1992) A continuum method for modeling surface tension. J Comput Phys 100(2):335–354 Osher S, Sethian JA (1988) Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J Comput Phys 79(1):12–49 Cheng ZQ, Barriere T, Liu B, Gelin JC (2012) A vectorial algorithm with finite element method for prediction of powder segregation in metal injection molding. Int J Numer Methods Fluids 70(10):1290–1304 Denner F, Evrad F, van Wechem BGM (2017) Artificial viscosity model to mitigate numerical artefacts at fluid interfaces with surface tension. Comput Fluids 143:59–72 Dağ İ, Canıvar A, Şahin A (2011) Taylor–Galerkin and Taylor-collocation methods for the numerical solutions of Burgers’ equation using B-splines. Commun Nonlinear Sci Numer Simulat 16(7):2696–2708 Li CT, Lai FC (2010) Visualization of the surface tension and gravitational effects on flow injection in center-gated disks. Int Commun Heat Mass Transfer 37(3):230–233