Computer-Aided Design 41 (2009) 306–314 Contents lists available at ScienceDirect Computer-Aided Design journal homepage: www.elsevier.com/locate/cad Particle-based viscoplastic fluid/solid simulation ✩ Afonso Paiva, Fabiano Petronetto ∗ , Thomas Lewiner, Geovan Tavares Matmidia Lab., Department of Mathematics, PUC, Rio de Janeiro, Brazil article info Article history: Received 16 April 2008 Accepted 3 October 2008 Keywords: Viscoplastic fluid Solid deformation Smoothed particle hydrodynamics Radial basis function Non-Newtonian fluid Physically-based animation abstract Simulations of viscoplastic materials are traditionally governed by continuum mechanics. The viscous behavior is typically modeled as an internal force, defined by diverse quantities. This work introduces a fluid model to simulate the viscoplastic effect of solid materials, such as plastic, wax, clay and polymer. Our method consists in modeling a solid object through a non-Newtonian fluid with high viscosity. This fluid simulation uses the Smoothed Particle Hydrodynamics method and the viscosity is formulated by using the General Newtonian Fluid model. This model concentrates the viscoplasticity in a single parameter. Our results show clear effects of creep, melting, hardening and flowing. © 2008 Elsevier Ltd. All rights reserved. 1. Introduction Physical simulations entered the Computer Graphics commu- nity to produce visually realistic animations. In particular, fluid objects are frequently used for sensational effects, such as water, fire or gas evolutions, large continuous deformations, suspense- ful object ruptures among many others. However, the delicate point for animation purposes remains formulating the underlying physics simple enough to produce a sequence efficiently, and rich enough for the simulation to remain realistic. Moreover, an ani- mation artist should be able to control material behavior through a reduced set of intuitive parameters. This involves finding an adequate formulation for the physical laws, which are usually formulated through differential equations, and a stable numerical approximation scheme for their discretiza- tion. Furthermore, the main parameters can be freely set while al- ways generating a visually realistic simulation. This paper proposes a simplified formulation for viscoplastic objects simulation (see the example of Fig. 1) by keeping the physical intuition behind a sin- gle parameter and avoiding the artist having to cope with technical parts. Viscoplastic materials. Viscoplastic objects are non-Newtonian fluids, i.e. their flow behavior is viscous when weak forces are applied. However, ✩ Expanded version of ‘‘Particle-based non-Newtonian fluid animation for melting objects’’ presented at SIBGRAPI 2006 (XIX Brazilian Symposium of Computer Graphics and Image Processing). ∗ Corresponding author. Tel.: +55 21 3527 1751. E-mail addresses: apneto@mat.puc-rio.br (A. Paiva), fbipetro@mat.puc-rio.br (F. Petronetto), tomlew@mat.puc-rio.br (T. Lewiner), tavares@mat.puc-rio.br (G. Tavares). when under significant force, the material starts to flow like a liquid (toothpaste effect): it changes phase from a non-Newtonian viscous but almost solid behavior to a Newtonian liquid. Unlike viscoelastic objects, viscoplastic materials have no memory: a rupture does not induce a ‘‘spring effect’’ of any part. This work is based on the recent advances of Mendes et al. [1] in formulating a viscosity function for viscoplastic objects that encompasses both viscous and liquid phases. The conciseness and generality of this formulation provides an efficient control of the viscosity, since it mainly depends on one physical parameter, named the jump number, which replaces multiple parameters, such as stiffness, compressibility, plasticity, viscosity, cohesion. . . This rheological model suits better for plastic deformation and it has been recently introduced in melting simulations [2]. Previous techniques for animating viscoplastic objects rely on continuous mechanics formulation, although those materials are essentially fluids. The approach of this work directly follows their physical models, which allows a realistic behavior for almost any material parameter. The jump number can be set to an arbitrary value, generating creep and work hardening effects. It can also be linked to other variable parameters such as temperature. This allows straightforward simulations of freezing and melting behaviors, when parts of the objects alternate continuously between solid and liquid phases, or lava flow simulations. This occurs discontinuously, depending on the terrain’s topography. Computational approximation. Simulating the fluid behavior of a viscoplastic object in its liquid phase requires a computational fluid dynamics (CFD) framework. In the Computer Graphics literature, the most common CFD model relies on Eulerian formulation where physical quantities are sampled on a regular grid. This suits well for classical Newtonian fluids such as water for instance. However, in order to control 0010-4485/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.cad.2008.10.004