1. Introduction and Background
2. Continuum Mechanics
3. The Finite Element Model
4. Code Implementation (Maya)
5. Plasticity and Fracture 

CONTINUUM MECHANICS

The field of continuum mechanics is a branch of engineering where a material, body or object is considered to have a continuous distribution of constitutive properties. Such a statement appears to be counter-intuitive given that, from an early age, we are taught that all matter is inherently discontinuous, composed of building blocks at the atomic scale such as nuclei and electrons.

Perhaps a better description would be to consider a continuum as the smallest possible quantity of a material (from both a physical and temporal sense) that can be analysed without having to take into account molecular dynamics or quantum mechanics. This assumption is further validated by considering that all interesting physical phenomena worthy of computer animation take place at lengths much greater than atomic spacing’s (10⁻¹⁰m) and over time periods far larger than atomic bond vibrations (10⁻¹⁵sec).

The discipline unites solid and fluid mechanics into one flexible theory that is concerned with the deformation of a material in response to external loading. For the purposes of this project, solid mechanics is of most interest and can be further subdivided into categories of elasticity and plasticity based on whether or not a material returns to its initial configuration when external forces are removed. Given that all materials must conserve mass, momentum and energy - it is the constitutive relationship between loading and deformation (stress and strain) that distinguishes different types of materials and governs their resulting behaviour.

STRESS AND STRAIN

Stress (σ) refers to the intensity of forces induced by a load on a material, often characterised as the force per unit area, while strain is the resulting deformation. Consider the general one-dimensional case of a rod with initial length L that is elongated to length L⁰ (in the x-direction only) due to the application of an external force. The dimensionless ratio of normal strain () can be considered as the change in length per unit original length and is defined mathematically in the equation below. Such an example is synonymous with an elastic spring but for materials of interest to computer animation, stress and strain will not be uniform from point to point.

Instead further refinement is required by introducing the displacement field u(x) = x⁰ − x. Consider the segment [x_i,x_j] shown in the figure below.

In the un-deformed state L_ij = x_j − x_i and in the deformed state l_ij = x⁰_j − x⁰_i. Using our previous equation it is possible to derive the strain on the entire segment. As already stated this is of little practical use but by considering the segment as infinitesimally small (namely as dx → 0) then the normal strain at an arbitrary point can be defined as follows:

In a similar fashion to strain, the stress on a segment (force per unit area) can be used to determine the force density at an arbitrary point which is defined as:

This strain displacement relationship enables the force and deformation at any point in a material to be defined and is easily extensible from one to three-dimensions. Consider the 3D deformable body shown in the figure below where the displacement field u(x) describes the deformed body Ω⁰ in terms of the un-deformed body Ω.

The point x given in material coordinates can be transformed via a function to give the new location described by point x⁰ in world coordinates where u(x) = x⁰ − x. To determine the transformation function consider the vector d = x₂ − x₁ of the un-deformed body (Ω) which is infinitesimal. In the deformed state (Ω⁰), using the principles previously discussed, this corresponds to the following where F = (I +∇ u) is known as the deformation gradient or Jacobian and ∇u is a 3x3 matrix with coefficients in the three directional axes.

Essentially the deformation gradient can be thought of as describing the change in world coordinates with respect to a change in the material coordinates. As discussed in this fantastic reference book, the Jacobian is invariant to translation (magnitudes are unaffected) but not to rotation and the overall deformation of a given point must be described by measuring the change in length squared in all directions where |d0|2 − |d|2 = d0T d0 − dT d) = d^T (F ^T F − I)d. This leads onto the very important definition of the Green-Lagrange Strain Tensor, a non-linear quadratic entity that is formally defined as:

Analysis of this tensor shows that the quadratic term becomes negligible for small displacements and that by removing the non-linear component we arrive at the definition of the linear Cauchy Strain Tensor, where u is the displacement vector, x is the coordinate and the two indices i and j can vary across the three dimensional coordinates x,y,z.

A total of nine strain measures exist as shown in the following equation. Notationally, E_ij represents the normal strain previously explored and γ_ij represents the shear strain. It can be thought of as the change in angle of an angle previously at a right angle and represents the total amount of deformation perpendicular to a given axis. Due to the symmetry of this matrix only six independent strain components are needed meaning that the strain can conveniently be rewritten as follows:

It is important to note that the Green strain tensor is rotation-invariant. This can be shown using polar decomposition where any deformation gradient (F ⁰) can be factored into separate rotation (R) and deformation components (F). Conversely, the Cauchy strain tensor is not rotation-invariant and results in aesthetically displeasing visual artefacts (discussed in a later article).

Stress describes the internal traction forces (T) within a solid - that is the tensile or compressive forces experienced by a continua block - and can be visualised by considering the following figure. E

Externally, the body will experience a force per unit area at a given location on the objects surface. This traction is a bound vector that means it can only be fully described by both the force and surface upon which it acts. Internally the process is very similar and by making an arbitrary slice across the object to expose a surface (dA) with a corresponding normal (n) and force (f):

Typically an infinitesimal block within the object, as shown in (b) above, is decomposed into three mutually orthogonal components resulting in the Cauchy Stress Tensor σ. It can be thought of as a set of internal datum planes that are aligned to the x,y,z coordinate system, enabling the stress at any point to be evaluated. The normal stress component (σ) is perpendicular to the surface and changes the volume of a material while the shear stress component (τ) is tangential to the surface and causes deformation. Similar to the strain tensor, there are only six independent values and the matrix can be simplified to the following tensor. 

CONSTITUTIVE RELATIONS

Stress and strain for a particular material form constitutive equations, a notion that was first suggested by British physicist Robert Hooke and arguably one of the most familiar mathematical expressions where F = −kx. His theory was then generalised by Augustin Cauchy to three dimensional bodies based on the observation that the 6 components of stress are linearly related to the 6 components of strain. This suggests that there are 36 matrix components, true in the general sense but given the conservative nature of most materials and the resulting symmetry of the matrix, this can be reduced to 21 independent components.

The type of material also has a bearing on the complexity of the matrix, as shown in the table above where Triclinic (General Anisotropic) materials have properties dependant on orientation such as the grain on wood. Isotropic materials whose mechanical properties remain the same in all directions are much simpler to consider and the requirements of any computational system are dramatically reduced since only two independent properties are required to define the material - the Young’s Modulus E which embodies the relationship between stress and strain (determining the stiffness of the material in question) and the Poisson Ratio v which is responsible for volume preservation. This can be thought of as materials strained in one direction also experiencing strain in a mutually perpendicular direction despite there being no external loading. Large values of E imply that the material is hard to stretch such as 200GPa for steel or 0.05GPa for more flexible material like rubber.

The Material Property Matrix D can be formed to relate stress to strain:

Further examination of matrix D highlights that, from a computational perspective, only three variables need to be stored giving the following which will greatly reduce storage requirements by eliminating the sparseness of the matrix:

EQUILIBRIUM CONDITIONS

The overall governing equations of an object can be derived by now considering the equilibrium (and boundary) conditions of the body in conjunction with the principles of stress and strain previously discussed. Any object under external loading will reach a state of equilibrium when the net force acting on the body is zero. It is important to note that this balance of forces, or mechanical equilibrium, only occurs when the internal and external forces sum to zero at every point.

As a simple example, consider the figure above where an axial force f_s is applied to a bar in the x-direction only. From early on in our stress/strain equations it is straightforward to see that:

EAu_x(x) + f^b(x) = 0       x ∈]0,1[

The governing Partial Differential Equation (PDE) is completely specified by considering the force constraint as f^s = EAu_x(1) and the displacement constraint (or boundary condition) as u(0) = 0. Mathematically this is considered a Boundary Value Problem (BVP), a framework in which forces, displacements, stresses and strains are connected together before computation for an unknown quantity. The formulation of a BVP, taking into account equilibrium, kinematics and material law, comes in three distinct forms: Strong, Weak and Variational.

• Strong Form

Satisfies each point in the problem domain with strong continuity of the dependant field variables (namely the displacements u,v, and w), with the obvious example being Newton’s second law (F = ma). The PDE discussed above can be considered as being represented in the strong form but, despite offering an exact solution, is difficult to realise practically. Typically the Finite Difference Method (FDM) can be used to discretise the problem so to obtain a finite set of algebraic equations for an approximate solution. This works well for one-dimensional systems (often seen in audio applications) but becomes complicated for non-regular and complex three dimensional domains. From a visual perspective this results in coarse aesthetically displeasing boundary edges, known as the staircase effect, that can suffer from large discretisation errors. This is simply unacceptable for high-quality VFX applications.

• Weak Form

Established using either the integration by parts on a Weighted Residual equation or directly by the Principle of Virtual Work. In a dynamic system the principle of virtual work is replaced by d’Alembert’s Principle which includes an additional inertia term. In both instances only average (integrated) satisfaction of equilibrium is achieved. Such statements were personally quite daunting when first encountered but are simple to conceptualise by referring back to the bar shown in figure 2.4. Consider a test function ¯u that approximates the strong form equation by integrating over the domain: 

It then follows that integration by parts of the above equation and the imposition of boundary conditions (f^s = EAu_x(1),¯u(0) = 0) yields the complete governing PDE in weak form, also known as the Virtual Work Equation:

Given that any force F acting through a given distance u performs work such that W = Fu then we can interpret the test function ¯u as a virtual displacement (∂u). It follows that the virtual displacements across the domain perform no work if and only if the system is in (static) equilibrium. In general, the principle of virtual work states that the stresses, body forces (f^b) and tractions (f^s) are in equilibrium if the internal work equals the external work for every virtual displacement field. The concept is explained in a general sense here and can be applied to solids or fluids.

• Variational Form

Derived from the Principle of Minimum Potential Energy for static systems or using Hamilton’s Principle for the inclusion of dynamics, which again includes an additional inertia term. It is a form only possible for conservative systems, namely those where work done within the system can be recovered. This form is unattainable when non-conservative forces such as friction are encountered. Owing to the complexity of full variational calculus and that the principle of minimum potential energy is actually derived from the principles of virtual work, the variational form of a BVP will not be explored in this project but interested readers are directed to [15] for further information. Another way of looking at the situation is to consider mechanics, in general, as either Vectorial (based on Newtonian equations) or Analytical (based on energy principles). With this in mind both the weak and variational forms are preferential to the strong form because they use these energy principles to establish fundamental physical insights and functionally express all properties of the model. They are invariant to coordinate transformations since functions are scalar and lead naturally to conservation laws. Both forms are discretised by the Finite Element Method

The following diagram summarises the formulation and subsequent discretisation of the continuum based equations explored thus far:

As with stress and strain, it is relatively straightforward to extend these principles from one to three dimensions by considering the infinitely small block (dx,dy,dz → 0). In the diagram below we assemble the components highlighted in green (which represent the traction forces acting in the x-direction) and the subsequent equation represents the static three-dimensional equilibrium where f_x represents an external body force.

A similar process is used to derive the y and z components to give:

As with the one-dimensional case, the balance of internal and external forces formulates the governing PDE in strong form:∇σ + f^b = 0                                                              

It is then possible to formulate the weak form by integrating the strong form, multiplying by the virtual displacement vector (∂u) and integrating by parts:

REFERENCES

Graham Charles Archer. Object-Oriented Finite Element Analysis. PhD thesis, University of California, Berkeley, 1996.

Bernhard Thomaszewski. Continuum Mechanics and Finite Elements for Deformable Solid Simulation. Computer Graphics Laboratory ETH Zurich, 2011.

Demetri Terzopoulos. Elastically Deformable Models. SIGGRAPH Computer Graphics, 21(4):205–214, 1987.

This was an observation that I made long ago in 2011 when finite element techniques, a branch of traditional structural engineering that simulates incredibly realistic deformation and fracture, was in its real time computational infancy - namely it took a long time to get any meaningful results!

Fortunately for me (and my bank account) Framestore (the guys that went on to make Gravity - its awesome!) were interested in exploring the notion of a non-traditional physics engine and lone behold I was given six months to prove my worth. The outcome of my endeavours was the development of a prototype Maya plug-in that demonstrated the elastic and plastic deformation of linear tetrahedron before fracture occurs in a visually plausible manner...

Anyone understood that? It's a pretty messy area of engineering - crazy mathematics, algorithmic principles, optimisation techniques...but fear not because this blog series intends to explain the entire process end to end - heck I will even throw in some lovely C++ code - so that you can start developing your very own physics engine 

1. Introduction and Background
2. Continuum Mechanics
3. The Finite Element Model
4. Code Implementation (Maya)
5. Plasticity and Fracture 

INTRODUCTION AND BACKGROUND

Physical realism is becoming an increasingly mandatory requirement for computer graphics. People have an instinctive ability to detect when an image is unnatural or implausible, destroying the suspension of disbelief and removing their sense of immersion. This pursuit of enhanced realism has led to a range of significant innovations towards graphics that are visually indistinguishable from the real world. However, as the overall look or visual fidelity to computer generated sequences improves through increasingly accurate lighting algorithms and more powerful rendering programs, it is inaccuracies in the behaviour or kinetic fidelity of a simulation that become more glaringly obvious to the viewer.

Typically kinematics has been used to achieve pre-configured animation, an approach that works well for animating articulated figures that possess joints or hinges. Artists use a technique called key-framing to explicitly define an objects start and end positions, interpolating in-between to get the full motion (think back to school and flicking through a notebook of a stick man):  

However, realism can only be judged subjectively and its application to other simulations such as destruction can be a long and tedious process. Therefore, a compelling argument to enhance the overall believability is to incorporate the laws of physics into models for more realistic procedural movement. Recent efforts into this highly-interdisciplinary area of interest, encompassing advanced topics of engineering and mathematics, have culminated in a field of study called Physics-Based Animation.

In the film industry, traditional forms of practical special effects involve the use of live pyrotechnics, complicated stunt work or intricate miniature models. The first widely acknowledged special effect can be seen in 'The Execution of Mary Queen of Scots (1895)' while you will undoubtable be aware of the iconic model used in Star Wars (1977) as well as the incredible live action shots from 'Saving Private Ryan (1998)'. Such techniques have proven to be effective in telling a story and prevailed as the de-facto choice of directors for many years, ultimately due to a lack of viable alternative. However, in certain situations, artistic vision stipulates effects that are simply too dangerous and/or expensive to be realised practically, such as the perilous trip by Tom Hanks and crews to the moon in 'Apollo 13 (1995)'. Beyond safety and cost, logistics can cause problems if locations required for re-shoots are unavailable and need to be recreated digitally.

These fundamental problems have been alleviated by the digital revolution of the last 30 years, beginning with the unprecedented efforts to create the ground breaking visual effects (VFX) seen in 'Tron (1981)' - a watershed moment that heralded in a new era of visual entertainment and spawning companies such as Pixar, Weta Digital and Framestore. Impossible, previously infeasible effects such as the large scale destruction seem in the film '2010' and 'Transformers Dark of the Moon'  cannot be realised without the assistance of visually realistic computer simulations that  behave in a manner identical to an actual building collapsing or vehicle exploding. Thus, the new and exciting discipline of physics-based animation has (general) usurped traditional approaches - although a hybrid of the two is often seen. Modern cinema, saturated with a range if ground-breaking visual effects showcases many algorithms for physics based animation. This wide gamut of computer vision techniques such as de-warping, feature tracking and image segmentation have been adapted for incredible VFX applications.

Here is a summary of the main types of forward dynamic simulations:  

• Particle Systems

These systems can be considered the earliest and simplest form of dynamic simulation that can be used to replicate a variety of simple phenomena such as explosions, smoke, snow and rain. Typically, particles have a position and velocity but no orientation and are generally implemented in three-dimensional space as point masses. Application of Newtonian law govern the potential forces action on particles that, when couple together, model so-called fuzzy objects.

• Rigid Body Simulation

Rigid Body Dynamics (RBD) employ non-penetrating multi-body systems composed of unyielding materials that are assigned physical characteristics along with the imposition of constraints. They differ from particle systems in that they occupy a physical space and are geometrically more complex. Various physics engines such as Havok, PhysX and PhysBAM have been employed by facilities to create such simulation. Most of these are closed source engines that cost money to incorporate into a pipeline. Instead open source engines are more inviting as they allow developers to customise the code to suit specific needs. Previously, ODE has been a popular choice but this has recently been displaced by Bullet as the de-facto engine of choice in many studios.  

• Deformable Body Simulation

Deformable objects, as the name suggests, are not rigid and include a range of different objects such as cloth, fur, hair and rope. The complexities of self intersections and collision response make them a challenging problem. The academy award winning efforts of Terzopoulos in 1987 has been followed up by further pioneering work in inelastic deformation including plasticity and fracture. Generally, mass-spring systems are sued where particles are connected together based on the principles of Hooke's Law, but are unable to capture to volumetric effects or the precision stipulated by more complex systems. Alternatively, Finite Differences can be used to simulates deformable models based on continuum models.

• Computational Fluid Dynamics

The simulation of fluid phenomena including complex fire, smoke and water environments (that particle systems cannot handle) is accurately described by the Navier-Stokes equation. In its original form dating back to 1821 the equation simply describes the physics of fluids but has since been extended to account for turbulence, shallowness and high velocities. The challenges faced by computes graphics has been tackled by many including Digital Domain for 'The Day After Tomorrow' and DreamWork's for the pioneering work seen in 'Antz'. Current state of the art software for fluid simulation is Naiad by Exotic Matter while proprietary software developed by VFX facility Scanline earned an Oscar nomination of the tsunami sequence seen in the film 'Hereafter'.

• Finite Element Methods

Based on theories taken from continuum mechanics, the finite element method can be used to discretise an object based on stress-strain relations and more accurately model deformable objects, as well as fracture. Originally, this has been used from a medical perspective and then the work of O'Brien at the turn of the centure looked at non-real time non-linear animation of brittle and ductile materials. The general consensus in the community is that such methods are the future of physics based animation for visual effects.

A caveat to suing physical laws in the production of visual effects for film is that it is difficult to obey the laws of nature as well as the Principles of Animation, a set of suggested rules used by animators to exaggerate motion and engage the audience. For example, the squash and stretch principle is often seen in cartoons or animations to provide surrealistic movements favoured by younger audiences, enabling unnatural character poses and impossible physical scenarios. Further to this, directors may actively seek out unrealistic effects that cannot occur in the real world such as dream sequences, alien worlds or fantasy environments. Thus, it is imperative to retain perspective when developing new tools for VFX - an inherently artistic endeavour that regularly looks to push the boundaries o acceptability and manipulate algorithms in ways that were not necessarily intended for.

Further to this, typical requirements encountered in traditional engineering applications - where emphasis is very much on accuracy - are not appropriate to VFX. Instead it is speed and predictability that are of paramount importance so that artists are able to convey interesting object motion with complete control. To keep pace with ever-changing production demands and the evolution of technology, tools must be extensible while maintaining stability and efficient.  

Fortunately, in the past 12 months I have travelled to Munich, Seattle and Atlanta for some fantastic training events; gained exposure across the entire Microsoft technology portfolio and made the decision to specialise in SharePoint - arguably the most successful, innovative and inspiring platform offered by the company. As part of the team I am now in, we maintain a SharePoint specific blog series (http://blogs.msdn.com/b/uksharepoint/) that focusses on the ideas, concepts and principles we encounter during our work from an infrastructure, architectural and development perspective.

But given my wider passion for technology and engineering background, where despite its amazing capabilities there are other interesting technologies beyond SharePoint, I wanted to kick off my second year with Microsoft with my own blog series on other activities and interests that I have to promote the other platforms, solutions and now even devices that we offer.

More to follow shortly...