Your SlideShare is downloading. ×
Realistic material damage simulation using real-time Finite Element Analysis
Realistic material damage simulation using real-time Finite Element Analysis
Realistic material damage simulation using real-time Finite Element Analysis
Realistic material damage simulation using real-time Finite Element Analysis
Realistic material damage simulation using real-time Finite Element Analysis
Realistic material damage simulation using real-time Finite Element Analysis
Realistic material damage simulation using real-time Finite Element Analysis
Realistic material damage simulation using real-time Finite Element Analysis
Realistic material damage simulation using real-time Finite Element Analysis
Upcoming SlideShare
Loading in...5
×

Thanks for flagging this SlideShare!

Oops! An error has occurred.

×
Saving this for later? Get the SlideShare app to save on your phone or tablet. Read anywhere, anytime – even offline.
Text the download link to your phone
Standard text messaging rates apply

Realistic material damage simulation using real-time Finite Element Analysis

2,852

Published on

Simulating, rather than animating, materials in real-time allows simulation designers and developers to deploy …

Simulating, rather than animating, materials in real-time allows simulation designers and developers to deploy
kinetically realistic simulations while reducing development time and cost.

1 Comment
0 Likes
Statistics
Notes
  • Be the first to like this

No Downloads
Views
Total Views
2,852
On Slideshare
0
From Embeds
0
Number of Embeds
2
Actions
Shares
0
Downloads
87
Comments
1
Likes
0
Embeds 0
No embeds

Report content
Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
No notes for slide

Transcript

  • 1. Interservice/Industry Training, Simulation, and Education Conference (I/ITSEC) 2009 Realistic material damage simulation using real-time Finite Element Analysis Steven L. Griffith Objective Interface Systems, Inc Herndon, Virginia steve.griffith@ois.com ABSTRACT The realistic modeling of material damage is a key component in the development of high fidelity virtual simulations. Properly simulated battle damage provides invaluable feedback to the simulation user and produces emergent scenarios and behaviors that more precisely reflect the real world. However, producing simulations that depict objects that realistically deform and break as if they were made from real-world materials is labor-intensive and expensive. Simulation developers have traditionally relied heavily on art swaps, or real-time substitutions of art assets, to model the deformation and fracture of simulation objects; often with disappointing results. Even when combined with rigid body physics systems, art swapping lacks the level of detail required to capture the complex interaction of battle damage and the effect on the battlespace and the warfighter. This paper will describe the use of an advanced, physics-based method to model and simulate material damage. This simulation accounts for the material properties of an object (density, toughness, plasticity, dampening, etc.) and the forces acting on the object. These variables are processed in real-time using advance finite element analysis (FEA) and the object is rendered in a visually realistic deformed or fractured state. This method can be employed to model virtually any solid material including concrete, glass, rubber, terrain, and vegetation. Furthermore, changing a material’s behavior (e.g. replacing standard glass with bullet-resistant glass) is accomplished by simply modifying the objects material properties rather than creating new simulation assets. Simulating, rather than animating, materials in real-time allows simulation designers and developers to deploy kinetically realistic simulations while reducing development time and cost. ABOUT THE AUTHOR Steve Griffith is the director of physical modeling and simulation at Objective Interface Systems. He has more than 20 years of engineering, business development, and management experience in the software industry. 2009 Paper No. 9409 Page 1 of 9
  • 2. Interservice/Industry Training, Simulation, and Education Conference (I/ITSEC) 2009 Realistic material damage simulation using real-time Finite Element Analysis Steven L. Griffith Objective Interface Systems, Inc Herndon, Virginia steve.griffith@ois.com INTRODUCTION damage. This simulation accounts for the material properties of an object (density, toughness, plasticity, The realistic modeling of material damage is a key dampening, etc.) and the forces acting on the object. component in the development of high fidelity virtual These variables are processed in real-time using simulations. However, many simulation developers advanced finite element analysis (FEA) and the object have been reluctant to incorporate this level of detail is rendered in a visually realistic deformed or fractured into their development lifecycle. This reluctance is not state. This method can be employed to model virtually due to a lack of enthusiasm for realistic kinematics; any solid material including concrete, glass, rubber, rather it is a reflection of the cost and complexity of terrain, and vegetation. Furthermore, changing a producing simulations that depict objects that material’s behavior (e.g. replacing standard glass with realistically deform and break as if they were made bullet-resistant glass) is accomplished by simply from real-world materials. modifying the object’s material properties rather than creating new simulation assets. Finally, the paper will Despite the challenges, it is imperative that simulation discuss Digital Molecular Matter (DMM), a COTS developers strive to provide properly simulated battle software implementation of real-time FEA developed damage. This level of detail provides invaluable by Pixelux Entertainment and subsequently adapted for feedback to the simulation user and produces emergent military simulation application and released as scenarios and behaviors that more precisely reflect the DMMfx. Figure 1 shows a tank breaking through some real world. In real-world warfare, the environment is wooden fences in a simulation using DMMfx. constantly changing—terrain craters, buildings crumble, obstacles are eliminated and new ones are created. If warfighters are to “train like they fight and fight like they train,” the physical dynamics of the battlefield need to be simulated with as much fidelity as possible. Furthermore, today’s military demands that warfighters are trained not only to overtake the enemy, but to be aware of the political, economic, social, and infrastructure implications of their actions. Realistic training simulations depicting accurate battlefield damage can help achieve the goal of building and reinforcing this awareness. Simulation developers have traditionally relied heavily on art swaps, or real-time substitutions of art assets, to model the deformation and fracture of simulation objects; often with disappointing results. Even when Figure 1. Simulated Tank Breaking Through an combined with rigid body physics systems, art FEA Simulated Fence. swapping requires the use of pre-defined geometry that lacks the level of detail required to capture the complex WHY REALISM MATTERS interaction of battle damage and the effects on the A growing number of researchers are finding a battlespace and the warfighter. substantial synergy between interactive storytelling and training. Rather than simply reciting facts, figures, or This paper will describe the use of an advanced, procedures; storytelling builds context around critical physics-based method to model and simulate material information and allows the student to more quickly 2009 Paper No. 9409 Page 2 of 9
  • 3. Interservice/Industry Training, Simulation, and Education Conference (I/ITSEC) 2009 internalize knowledge (Mantovani, 2001). As develop the simulation. A breaking pane of glass, for interactive media has become ubiquitous, visual effects example, will break the same way if struck by a bullet have taken on a big role in today’s digital storytelling. or struck by a rock. Should a simulation require a Especially for the younger generations, computer change of material, such as the addition of bullet- games, interactivity, immersion in synthetic scenarios, resistant glass, new art assets need to be created and are as normal and accessible as other media like scripted to depict the new behavior. The time required internet, television, radio or books. (Ponder, et al, to produce art swaps to depict kinetic effects drives up 2003). It follows; therefore, the efficacy of a simulation the cost of simulation development and can make the is influenced greatly by how immersive and realistic it cost of updating an existing simulation prohibitive. is. Many simulation developers combine RBD systems Today’s 3D graphics technology is capable of with art swapping in an effort to improve kinetic rendering visually stunning scenes. Off the shelf, fidelity and generate emergent behaviors. This however, the technology does little to provide greater approach has several disadvantages. kinetic realism. Kinetic realism should be considered just as important, if not more important, than visual Unconstrained emergent behaviors tend to produce realism – especially in military simulations. unintended consequences and side effects, especially as the number of interactions between objects increases. Human perception is highly tuned to movement, and so Developers need to be able to constrain emergent kinetic fidelity is a major visual cue in providing behavior depending on the training objective. immersive simulations. Because visual fidelity has Simulations for manual skills training that require a seen so much advancement over the past 10 years, it great deal of practice may require little or no emergent has served to exasperate the problem of a lack of behavior that might interfere with the repetitive nature kinetic fidelity. In the field of animation, it is well of the procedure being learned. Simulations that build understood how important it is for the visual fidelity to psychological skills, such as decision making, can be less convincing than the kinetic fidelity in order to benefit from having a large range of emergent provide a convincing element of animation. Pixar, in behaviors that render the simulation game play less fact, has kept their cartoonish style chiefly because predictable (Ponder, et al, 2003). their lighting models are so good that bumping the visual fidelity up to its true potential would cause them Additionally, RBD combined with art swapping does a problem of having to increase the kinetic fidelity of not consider the consequences of secondary effects in their animations up to a level not possible with manual complex kinematic scenarios. When a bomb explodes animation. near a vehicle, pieces of the vehicle may then become projectiles that may, in turn, damage other nearby The advent of Rigid Body Dynamics (RBD) has materials. Simulating this type of complex interaction improved the situation, but physics engines do not quickly becomes impractical with rigid body systems provide an accurate portrayal of materials reacting to because they do not allow for the deformation and high-energy forces such as munitions (Mann, et al, fracture of materials. 2008). Finally, RBD is a very limited way of representing the The Problem with Art Swapping physical properties of an object. Simulation developers using RBD have only 10 variables at their disposal to Simulation developers have traditionally relied heavily describe very complex materials: 3 rotations, 3 on art swaps, or real-time substitutions of art assets, to translations, mass, inertia, dampening, and coefficient show the deformation and fracture of simulation of restitution (bounciness). objects. When a projectile strikes a brick wall, artwork of wall fragments are swapped in and keyframe Training simulation developers are facing an animated to show the wall crumbling. The result is a increasingly sophisticated audience that is demanding wall that always breaks the same way regardless of the more immersive and realistic synthetic environments direction and force of the projectile. To create this that capture their attention and engage them with effect, artists have to draw hundreds of individual visually impressive digital storytelling. To do this, we frames to show the slightest bit of motion or movement. need a new approach that provides greater control and freedom to define and render the kinetic behavior of This approach limits an object’s behaviors while simulation objects. These behaviors can no longer be greatly increasing the effort and time required to scripted and animated, they need to be simulated. 2009 Paper No. 9409 Page 3 of 9
  • 4. Interservice/Industry Training, Simulation, and Education Conference (I/ITSEC) 2009 technology. In 1964, a review of NASA's structural The logical evolution of virtual simulation is the dynamics research determined that the various research accurate modeling of the kinetic properties of physical centers were duplicating there efforts to develop materials. One of the most promising approaches to structural analysis software. The review recommended kinetic fidelity is to utilize FEA in real time. FEA has that a single generic software program should be used been a proven method to analyze the effects of force on instead. A cooperative project was started to develop solid materials in less than real-time simulations for this software and created a specification that outlined over fifty years. Using today’s CPUs and GPUs, it is the capabilities for the software (MacNeal, 1972). possible to implement a real-time FEA physics engine to create a material physics simulator that renders A contract was awarded to Computer Sciences objects in a virtual world that behave as if they were Corporation (CSC) to develop the software. The name made from real-world materials. of the program is an acronym formed from NAsa STRuctural ANalysis. The NASTRAN system was A BRIEF HISTORY OF FINITE ELEMENT released to NASA in 1968. ANALYSIS By the early 1970s, FEA was being applied to solve a Finite Element Analysis as discussed here (also referred wide variety of engineering problems in aerospace, to as the Finite Element Method) was first developed in automotive, and civil engineering (Strange, et al, 1973). 1943 by Richard Courant. While analyzing problems However, FEA required tremendous computing power involving vibration, Courant proposed breaking a and was limited to the most high priority projects. continuous material into triangular regions to simplify the approximation of the properties of a material During the 1980s and 1990s the application of FEA (Courant 1943). In the mid 1950s a group of engineers expanded into the areas of electromagnetics, fluid from academia and the Boeing Airplane Company dynamics, and thermal dynamics (Strang, 1973). As published an article in the Journal of Aeronautical the number of problems addressed by FEA increased, Sciences analyzing the stiffness of wing design by so did the demand on computing power. dividing the wing structure into triangular segments. It is about this time that the term finite element method By the year 2000 the groundwork was laid for FEA in was coined (Turner, et al, 1956). the simulation and gaming environment with Dr. James O’Brien’s seminal work on graphically modeling and Offline FEA simulations have been used in the animating the realistic behavior of materials that manufacturing industry for many years. FEA fracture and deformation under stress (O’Brien, et al, simulations are used to test and refine designs before 1999). At the time of O’Brien’s original writings, the the prototype phase of production – reducing the time required to render the shattering of his example number of prototypes required, improving time-to- subject, a teapot, was almost an entire day. In just a few market and reducing costs (Figure 2). years, technological advances would reduce that time from hours to seconds. FINITE ELEMENT ANALYSIS CORE CONCEPTS We will recall from mathematics that a differential equation states how a rate of change in a single independent variable is related to other variables and that partial differential equations are a type of differential equation involving multiple independent variables. Partial differential equations are the most common mathematical description of physical systems. They are used to solve problems such as those involving mechanics, thermodynamics, fluid dynamics, Figure 2. Visual Representation of FEA Simulation and elasticity. of an Automobile Crash. Advances in FEA continued throughout the second half Finite Element Analysis is a mathematical technique for of the 20th century, paralleling advances in computer approximating solutions of partial differential equations. The approach is to render the partial 2009 Paper No. 9409 Page 4 of 9
  • 5. Interservice/Industry Training, Simulation, and Education Conference (I/ITSEC) 2009 differential equation into an approximation of ordinary differential equations that are more easily solved. The result, and the key feature of FEA in simulations, is the discretization of a continuous object into a mesh of finite triangular constituent elements. FEA is a good choice for solving partial differential equations involving complex objects, such as vehicles or buildings, which undergo change (such as collisions with obstacles or projectiles). It is also useful when a variable level of precision is desired. For instance, when ordinance detonates in a simulated street near a Figure 4. Tetrahedral mesh for a simple object. In building, it is possible to increase the accuracy of the (a), only the external faces of the tetrahedra are simulation in more critical areas (such as a storefront) drawn; in (b) the internal structure is shown and reduce the precision in areas facing away from the (O’Brien 1999) street. This approach offers the opportunity to tune the performance of the simulation to achieve an optimal Calculations are then applied to these elements to result. create a visualization where objects bend and twist and reveal the distribution of stresses and displacements. The degree to which the forces are distributed through FEA provides thousands of degrees of freedom the material are determined by the material properties Creating simulation objects in a FEA-capable assigned to the object at design time or at run time. environment starts with a detailed, artist-created surface Utilizing a real-time FEA solver allows for vastly more mesh. This mesh is then used as the basis for the realistic representations of a simulated material. creation of a tetrahedral cage (tet cage) or shell, of points around the surface mesh (Figure 3). Armed with FEA in real-time, simulation developers have thousands of degrees of freedom in describing how each discrete element can move and interact with the simulation environment. Moreover, the properties of these elements can be set to accurately behave like real-life materials. Wood doesn't simply break apart along a predetermined seam every time – instead it splinters into countless pieces from the exact point of impact, also taking into account the amount of sheer force exerted. Likewise, concrete crumbles; metal bends, deforms, and tears; and glass shatters realistically. The result is kinetic fidelity never before seen in real-time simulations. Using FEA, stresses applied to an object as a whole are interpreted as stresses to the individual elements. The result is a more Figure 3. Detailed Surface Mesh (left) and a Lower granular and realistic view of how an object reacts to Resolution Tetrahedral Cage (right). stress. The tet cage encapsulates the visible surface mesh and What is more, art objects developed with an FEA mesh is usually less detailed. The tet cage is in turn used to are created once, and their fracture and deformation create a “tet mesh”—a tetrahedral tessellation of the behavior is determined by their material properties and volume bounded by the tet cage (Figure 4). If the rendered in real time – eliminating the need for art object is breakable, the tet mesh has to be clipped swapping. against the surface mesh and have internal faces added to tetrahedral boundaries, which will be visible when Objects in real time FEA simulations can realistically the object breaks. The tet mesh represents the pre- react to forces according to their physical properties calculated fracture points of the solid object. whether big, small, dense, thin, floppy or rigid – FEA causes it to react appropriately. For example, an aerial refueling hose-and-drogue can react to air turbulence 2009 Paper No. 9409 Page 5 of 9
  • 6. Interservice/Industry Training, Simulation, and Education Conference (I/ITSEC) 2009 just as easily as a stone wall can be made to crumble. cost. Simulations no longer need be scripted scenarios, Any solid substance imaginable can be simulated. and time-to-deployment is exponentially faster. The user experience is enhanced because objects can Table 1. Common Material Properties Used in FEA now react in entirely new ways each time the user engages in the simulation. So when a tank fires a Material Property Description projectile at a building at different angles, the building will crumble differently each time. These emergent behaviors can reinforce decision training and deliver a Young’s modulus Denotes the elasticity (flexibility) realistic user experience that will keep trainees of a material. It is the ratio of engaged. stress (the force on a material) over strain (deformation of the Material Adjustment material) While the FEA equation solver does the heavy lifting, Young’s The material’s capacity to the most valuable asset within a material physics Dampening dissipate the energy simulation engine are the material properties variables. The material properties of an object are assigned at design time, but are not hard coded into the object so Young’s Creep Change in Young’s Modulus as a that it is possible to change the properties of an object material deforms without having to re-create the object itself. For example, you may decide to up-armor a vehicle which Poisson’s Ratio Specifies the amount of volume would include adding ballistic-resistant glass. In fact, preservation a material has when you can adjust material properties at run-time to reflect subjected to stress changes in the environment. For example, the properties of a steel beam can be altered to simulate Poisson’s Affects the velocity at which softening and deformation due to heat from fire. Damping something changes shape Likewise, a rubber refueling hose can become more rigid and even shatter as the ambient temperature drops Density Specifies how much a material in a simulation scene. These properties can also be weighs per unit volume manipulated to allowing the user to create effects visualize “what if” scenarios within the simulation Denotes the strength of a material Toughness itself. (how breakable something is) Material adjustments can also be used to fine-tune an Toughness Creep Change in Toughness as a object. Watching a brick wall slightly bend before it material deforms crumbles provides a familiar visual cue that can enhance decision training. Changes in the material Plastic Yield Determines how much something properties and the deformation of simulation objects has to deform before it will no can be used as feedback to the simulator’s sound longer return to it’s original shape system so that the creaking sound of a wooden door can be heard before it cracks open. Maximum Yield Limits how much a material may The material properties of objects in a real-time FEA deform. If you strain a material simulation are exactly the same as what you might more than this than the material expect to find in a materials science textbook (Table 1). will not deform any more Material properties may be determined by standardized test methods. Many such test methods have been Plastic Creep Determines how quickly documented by their respective user communities and deformation occurs published by ASTM International. Using real-time FEA technology, simulation developers Friction Controls how slippery a material can vastly improve the visual and kinetic fidelity of is their simulations while reducing asset creation time and 2009 Paper No. 9409 Page 6 of 9
  • 7. Interservice/Industry Training, Simulation, and Education Conference (I/ITSEC) 2009 IMPLEMENTATION OF FINITE ELEMENT ANALYSIS IN SIMULATIONS Real-time Finite Element Analysis has been deployed in a new technology called Digital Molecular Matter (DMM). DMM technology is implemented as a real- time engine subsystem that runs independently of the primary simulation engine; it also includes the tools required to convert ordinary meshes created by artists into finite element volumetric meshes. A key advantage of DMM is the ability to add FEA effects to both new objects as they are created or to existing objects for enhanced capability. With minimal effort, simulation developers can leverage their existing investments by adding DMM capability to legacy simulations. Based on James O’Brien’s original work, DMM was developed and brought to market by Pixelux Entertainment for the gaming industry. DMM attracted the attention of LucasArts, who wanted to deliver state- of-the art gameplay technology and take their video games to the next level of realism and reduce production costs. In late 2005, Pixelux began working in partnership with LucasArts to develop and refine Figure 5. A Shaped Charge and a Steel Plate DMM into an artist-friendly technology that could Modeled as DMM objects before (top) and After deliver the promise of real-time finite element physics. Detonation. DMM is used extensively in their newly released video game “Star Wars: The Force Unleashed.” Creating DMM Assets Pixelux subsequently partnered with Objective Creating breakable/deformable assets for use with Interface Systems (OIS) to adapt DMM to the military DMM starts by creating a watertight, non-self and aerospace simulation market. The resulting intersecting poly mesh and then turning that mesh into a product, DMMfx, was introduced at I/ITSEC 2007 and DMM Object. The result is a tetrahedral mesh assigned represents a way to provide realistic deformation and with default physical materials that will be controlled fracture in real-time within military simulations. Wood by the DMM simulator. DMM provides command line breaks like wood, metal bends and tears like metal, and tools to perform the conversion. These tools are also glass shatters like glass. DMM achieves this capability implemented as plug-ins for Autodesk’s Maya and 3DS by modeling the stress within a scene through finite Max modeling applications. In the examples below, element representations of the art assets in simulation. Autodesk Maya is used. Greatly desired damage features such as buckling, collateral effects, tearing and fracture can now occur in The next step is to define the material properties of the completely expected ways (Figure 5), providing object. When the object is created, it is assigned a simulations with the unpredictability and realism default material, or a material which can be selected necessary to ensure their effectiveness. Virtually any from a library that includes glass, concrete, brick, wood solid object can be modeled and simulated including and many others. Figure 6 shows the dialog box used to architectural elements, terrain, and vegetation. modify material properties at design time. 2009 Paper No. 9409 Page 7 of 9
  • 8. Interservice/Industry Training, Simulation, and Education Conference (I/ITSEC) 2009 These types of objects can be forced to break on pre- defined boundaries by attaching an additional “splinter” cage to the object. A block wall, for example will fracture at the mortar joints (Figure 8). Figure 6. Design time material property Figure 8. Block wall fracturing at mortar joints adjustments. Min Iter, Max Iter, Split Limit and Relative Error are used to control and optimize the Figure 9, below, is a visible representation of how the amount of time consumed by the simulator. DMM object creation and simulation inputs and outputs relate together. At this point, other modifications can be made that will affect the object’s behavior. Forces can be applied, objects can be “glued” using a spring/dampener force, and selected regions of objects can be made “passive” so they are not processed by the simulation. The simulation can now be run and the object will behave according to it’s properties and the forces acting on it— including gravity which is adjustable and turned on by default. Making a surface mesh breakable clips it with the Tet Mesh (Figure 7). The fracture geometry is very angular and straight. This is fine for crystalline materials but not for things made of other materials like wood or bricks. Figure 9. DMM Mesh Preparation and Simulation. Figure 7. Glass cube shattering after fall 2009 Paper No. 9409 Page 8 of 9
  • 9. Interservice/Industry Training, Simulation, and Education Conference (I/ITSEC) 2009 Once the DMM scene is complete the simulation is run Bulletin of the American Mathematical Society 49: 1– in the modeling environment and behaviors are fine- 23. tuned to produce the desired results. The scene is then exported to be run within a game engine. DMM is MacNeal, R., (1972), The NASTRAN Theoretical implemented as static library the is ready to link into Manual. December 1972 your simulation. Mann, J., Lyons, J., del la Cruz, H. (2008) Using Real- CLOSING REMARKS time Physics to Enhance Game Based Effects. Interservice/Industry Training, Simulation, and Although physics-based modeling is not by any means Education Conference (I/ITSEC) 2008 a new field, recent advances in hardware and software now make it possible and cost-effective to deploy Mantovani, F., VR learning: Potential and Challenges virtual simulations that utilized verifiable, real-time for the Use of 3D Environments in Education and FEA physics modeling to improve kinetic fidelity. Training, Chapter 12 in Towards Cyber-Psychology: Mind, Cognitions and Society in the Internet Age, This improved fidelity has important training Amsterdam, IOS Press, © 2001, 2002, 2003. implications, especially in the area of decision training. Furthermore, as younger generations enter the services O’Brien, J., Bargteil, A., Hodgins, J., (2002). Graphical with a history of video game play, virtual training Modeling and Animation of Ductile Fracture, ACM simulations will have to deliver a user experience that SIGGRAPH 2002 rivals that of the gaming world in order to keep them engaged. O’Brien, J., Hodgins, J. (1999). Graphical Modeling and Animation of Brittle Fracture, ACM SIGGRAPH Finally, simulation rather than animating material 1999 damage can save hundreds, even thousands of hours of modeling time. In today’s dynamic, asymmetric warfare Ponder, M., Herbelin, B., Molet, T., Schertenlieb, S., environment high-fidelity simulations, deployed Ulicny, B., Papagiannakis, G., Magnenat-Thalmann, rapidly, will allow our troops to “Train to Fight” and N., Thalmann, D. (2003). Immersice VR Decision “Fight to Win. Training: Telling Interactive Stories Featuring Advanced Virtual Human Simulation Technologies, Proc. of the 9th Eurographics Workshop on Virtual ACKNOWLEDGEMENTS Environments (IPT/EGVE 2003), pp. 97-106, 2003. The author would like to thank Eric Parker, Vik Sohal, Strang, G., Fix, G. (1973). An Analysis of the Finite and Olivier Basille or Pixelux Entertainment for their Element Method. Englewood Cliffs: Prentice-Hall. assistance with this paper. O’Brien, J. (2000). Graphical Modeling and Animation Fracture, Thesis, Georgia Institute of Technology. REFERENCES Turner, M.J. Clough, R., Martin, H., and Topp, L., Bimber, O. (2009) Visual Effects and Beyond. (1956).Stiffness and Deflection Analysis of Complex Computer, Vol. 42, no. 7, July 2009 Structures. Journal of the Aeronautical Sciences 23: 805–82 Courant, R. (1943). Variational Methods for the Solution of Problems of Equilibrium and Vibration. 2009 Paper No. 9409 Page 9 of 9

×