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Unlike geometric techniques, computationally intensive physics-based techniques can yield real material behavior of a multiple-material/non-homogeneous virtual model [Knopf, 2005]. Physics-based deformation models give the designer, more opportunities to try different types of materials during the interactive design phase and validate product models in real-time. Important physics-based techniques are discussed in the following sub-sections.

2.7.2.1 Mass spring method

Mass spring systems are simple physical model with well understood dynamics. In physically-based techniques on mass-spring-damper models, an elastic object is constructed by applying a mass at each point of a mesh and using springs to link the points as edges and diagonals [Cotin, 2000; Lin, 2002; Nedel, 1998]. Elastic forces and damping forces act on mass points as internal forces, and gravity and the user induced forces act on them as external forces. These are easy to construct and computational cost is moderate. It is possible to achieve interactive real-time interaction with the virtual model using mass spring system, on the ubiquitous desktop computers. Mass spring systems are also suitable for parallel processing and hence can benefit from multi-core

processors being used in desktop computers. The pioneering work of Terzopoulos et al.

[Terzopoulos, 1987], Waters [Waters, 1992], and Platt and Barr [Platt, 1988] has shown the advantages of physically based models over geometric-based computer animation techniques.

However, mass-spring-damper models have drawbacks. The model is a significant approximation of the true physics that occurs in a continuous body. The lattice is tuned through its spring constant, and proper values for these constants are not always easy to derive from measured material properties. The physical accuracy of modeling is often not sufficient and cannot realize the global deformation. The above models are linear, and to simulate nonlinear force responses, it is necessary to use a precise integration mechanism such as the finite element method (FEM). However, such a method generally cannot provide update rates that are sufficient for haptic interactions [Luo, 2007]. In addition, certain constraints are not naturally expressed in model such as incompressible

volumetric objects, or thin surfaces resistant to bending. Mass spring systems also

sometimes exhibit stiffness problem which can occur when very large spring constants

are used. Stiff systems are very problematic because they have poor stability even with longer time steps.

Terzopoulos et al. [Terzopoulos, 1991] described a mass spring model for

deformable bodies that experience state transition from solid to liquid. The deformable model features non-rigid dynamics governed by Lagrangian equations of motion and conductive heat transfer governed by the heat equation for non-homogeneous, non- isotropic media. In its solid state, the discretized model is an assembly of hexahedral finite elements in which thermo-elastic units interconnect particles situated in a lattice. A discretized form of the heat equation is used to compute the diffusion of heat through the material. At melting point the stiffness reaches zero thereby severing the bond. The

molten state of the model involves a molecular dynamics simulation in which fluid

particles that have broken free from the lattice interact through long-range attraction forces and short-range repulsion forces. Tu and Terzopoulos [Tu, 1994] used mass spring

system to generate artificial fish with internal contractile muscles that are activated to

produce the desired motions. The fish model comprised of 23 mass points and 91 springs. The spring-mass system is simulated using implicit Euler method which maintains the

stability of the simulation over the large dynamic range of forces. Christensen et al.

[Christensen, 1997] embedded objects in a cubic eight nodes lattice connected by 28 damped linear springs and used dynamic simulation and FFD's to animate the embedded objects. The simple Coulomb friction model is used, though friction is permitted to be stronger in one preferred direction if the animator so specifies. The resulting equations of motion are solved numerically by a variable time-step, fifth-order Runge-Kutta integration procedure. Actuation of the mass-spring lattices is achieved by varying the rest lengths of the springs.

Mass-spring system has been extensively used for facial and modeling and animation. Terzopoulos and Waters [Terzopoulos, 1990] incorporated physically based approximation to facial tissue and muscle actuators. Different spring constants were used

to model different layers based on tissue properties. Lee et al. [Lee, 1995] presented a

nonlinear deformation properties, a muscle layer knit together under the skin, and an impenetrable skull structure beneath the muscle layer. For improved realism, the model used a constraint which prevented muscles and facial nodes form penetrating the scull.

Koch et al. [Koch, 1996] used a mass spring model to predict the postoperative

appearance of patients whose underlying bone structure for surgical planning and prediction of human facial shape after craniofacial and maxillofacial surgery for patients with facial deformities.

Mass-spring models have also been extensively used in surgical simulations due to their simplicity of implementation and their relatively low computational complexity [Baumann, 1996; Kuehnapfel, 1993]. Kuehnapfel and Neisius [Kuehnapfel, 1993] presented a simulation of endoscopic surgery based on a surface spring-mass model.

Cotin et al. [Cotin, 2000] presented a hybrid model consisting of mass spring system and

tensor-mass model based on continuum mechanics and linear elasticity theory. The tensor-mass model is used for the simulation of tearing and cutting.

Mass-spring system has also been used for concept design validation. Igwe et al.

[Igwe, 2008b] proposed to generate hexahedral mesh of mass spring system by using

volumetric self-organizing feature map (VSOFM). The VSOFM exploits the adaptive and

self-organizing ability of Kohonen’s original algorithm [Kohonen, 2001] to develop a 3D mesh where the position the exterior nodes represent surface points of the underlying object. Material removal and tearing are achieved by eliminating selected mass points

and spring coefficients in the evolving mesh. Pungotra et al. [Pungotra, 2009a] used the

mass spring system based on VSOFM for validating the concept design.

The underlying geometry of mass spring models can easily be modified to represent topology changes. However, spring-mass models are discrete representations of a continuum, and the update of stiffness and mass values is hard to handle. This becomes a problem when complex models are to be deformed in real time. To avoid this problem, iterative method was used to solve the deformation at any localized region.

2.7.2.2 Finite element method

In the finite element method, the model's solution is subject to the constraints at the node points and the element boundaries so as to achieve continuity between the elements.

Unfortunately, finite element calculations are notoriously slow, making them not very appealing for real-time applications. Finite element methods are often considered to be less efficient than spring-mass models. In FEM, the applied forces must be converted to their equivalent force vectors. This requires numerically integrating distributed forces over the volume at each time step. This can lead to a significant pre-processing time for finite element methods. If the topology of the object changes during the simulation, mass and stiffness matrices must be re-evaluated during the simulation. Traditional FEM is more accurate in modeling materials such as metals, where the amount of deformation is limited. As the model deforms, the volume over which equivalent force, mass, and stiffness matrix integrations are performed will change. Real-time finite element modeling requires high computational power to achieve visual realism [Berkley, 2004]. FEM has been used for fairly simple physical systems to simulate tissue deformation [Cotin, 1996; Sagar, 1994].

However, the finite element method (FEM) is a very common and accurate way to solve continuum-mechanical boundary-value problems [Bathe, 1996; Zienkiewicz, 2005]. Finite element methods provide a more physically realistic simulation than mass-spring system with fewer nodes. There is a growing trend in using finite element soft tissue

models for real-time computation, as shown for instance by Székely et al. [Székely,

2000] who simulated the deformation of a nonlinearly elastic material using a parallel processing architecture.

2.7.2.3 Continuum method

Instead of considering the model as a discrete object model, it can be considered as a continuum that is the solid bodies with mass and energies distributed throughout. Models can be discrete or continuous but the method used to solve in computer simulation is always discrete. The numerical integration techniques used to solve the model approximate the system at discrete time steps. However, unlike the mass-spring model, continuum models are derived from equations of continuum mechanics. The continuum model of a deformable object considers the equilibrium of a general body acted upon by external forces. The object deformation system is a function of these acting forces and

object's material properties. The object reaches equilibrium when its potential energy is at a minimum.

Several authors have based their soft tissue models on continuum mechanics theory, and the use of elastic solids is widely described in the literature [Bainville, 1995; Speeter,

1992]. Bayville et al. [Bainville, 1995] define the evolution of a set of rigid and

deformable solids under the influence of various forces. In this case, the deformation law is represented by a hyper-elastic, quasi-static model, associated with a finite element method for the numerical resolution. Unfortunately, the computation time makes this approach impractical for real-time simulations.