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New simulation algorithm and architecture for Game Physics

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Combine the rigid-body and deformable body dynamics all together. ... Kookea material and Moony-Rivlin material, Green and Adkins, Murnaghan and so on. ... – PowerPoint PPT presentation

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Title: New simulation algorithm and architecture for Game Physics


1
New simulation algorithm and architecture for
Game Physics
Michael Lin President CEO Reality Matrix Inc.
michael_at_reality-matrix.com
2
What we will be going through?
  • Algorithm
  • Combine the rigid-body and deformable body
    dynamics all together.
  • Parallel able and distributable algorithm
  • Industry level computational continuum algorithm
  • Compare to FEM, traditional explicit FEM
  • Simple, Fast, and Accuracy
  • Architecture
  • Vector form of physics architecture, from concept
    to modeling
  • Extensible architecture
  • How we can do that?
  • Using the new algorithm from new concept!

3
Targets on this presentation
  • Rigid body V.S. Deformable body
  • Major applications for Game or 3D VR
  • Cloth Simulation, Deformable Body Simulation…etc
  • Behavior of Collided Bodies, Impacted Effects
  • Major points of considerations
  • Material properties introduce the difference of
    deformation
  • Object colliding behavior influenced by impact
    factors, e.g. high-speed or the shape of
    penetration
  • No real rigid body in the world!
  • Its only the thing that has high stiffness
    material properties!

4
Algorithm and Architecture
  • Concept -gt Mathematical Modeling -gt Algorithm -gt
    Programming architecture
  • Concept
  • Newtonian, Objects forms from particles
  • Deformation caused by rigid body motion
  • Mathematical Modeling
  • Only consider the numerical modeling, no exact
    solution included!
  • Algorithm
  • Explicit integration causes the simple
    computational architecture
  • Programming Architecture
  • Based on simple architecture to the parallable
    and extensible programming architecture

5
Review on traditional methods
  • The particle-spring modeling
  • The simplest, widely used in the game industry
  • The spring or spring-damper modeling that can not
    fully described the material internal forces.
  • FEM, The Finite Element Method
  • It is based on small deformation assumption and
    variational method.
  • It costs time to days and large memory
    requirement by its matrix solving process. High
    programming complexities.
  • It may not convergence and cause errors.
  • BEM, The Boundary Element Method
  • By using the Greens function, its faster then
    FEM.
  • Its difficult to prepare the preprocessing data
    and not easy to apply.
  • DEM, The Discrete Element Method
  • It change the particle-spring model to use the
    rigid-body-spring model.
  • It also can not fully described the internal
    force of material in the deformation process.

6
Key Concepts
  • No nonlinearity at all, we solve the nonlinear of
    material and nonlinear of geometry in a linear
    way!
  • Newtonian, Balancing forces, and Principle of
    Virtual Work
  • How we can convert the deformations to rigid-body
    kinematics modeling?
  • Large deformation Rigid-Body Motion Small
    Deformation
  • Object comes from particles, and deformation of
    body comes from particle movements.
  • Internal forces of the body comes from small
    deformation, not related to the rigid motion.

7
Mathematics Modeling
  • Explicit time integration causes the simple, time
    stepping and real-time architecture.
  • By using balancing (Virtual works, variation)
    method, we can solve the distribution of internal
    forces.
  • Simple equilibrium math, causes the fundamental
    of soluble numerical model.
  • We can solve the small deformation of object by
    the rigid motions.

8
Algorithm
  • Causing the simplicity of mathematical modeling,
    we can describe versatile of systems, including
    rigid-bodies, deformable bodies there're solid
    and fluid simulation.
  • Causing the simplicity of mathematical modeling,
    we can integrate it into other algorithms.
  • The new algorithm is not including traditional
    abilities, also support the advanced simulation
    ability.
  • Parallel algorithm based on the simplicity.
  • The algorithm combines the different stages of
    rigid-body motion, deformation process and
    fracture simulation.

9
Our knowledge base
  • Rigid-Body Kinematics and Dynamics
  • To compute the particle movement
  • The Finite Element Method
  • To compute the piece of bodys internal forces
    (cohesion forces)
  • Numerical Integration Method
  • To compute the stepping of displacement, to form
    the trend of body movements.
  • We can use implicit or explicit integration
    method. But the explicit method causes the simple
    programming architecture.
  • Explicit algorithm causes the vector form of
    programming architecture.

10
Our key technology Zigma
  • Modeling of algorithm and mechanical math
    modeling.
  • The estimate method of the rigid-body movement.
  • Estimate the internal forces in the body more
    accuracy.
  • We use the incorrect explicit integration method
    to become the self-equilibrium system.
  • The vector architecture forms the foundation of
    parallelism.

11
A Review on Game Physics Approaches
  • A rigid body type description
  • Matrix form, describe the whole object
  • Using 3x4 float points
  • A mesh type description
  • Vertex type description, for each vertex
  • For deformable body

12
Rigid Body
  • A rigid body by Newton
  • A continuum can be defined by a group of
    particles, they connected to each other.
  • the rigid-body type description, cannot to
    describe the deformable body movement.

13
Deformable body
  • why do we need a deformable body?
  • In the latest game, the existence of deformable
    body is everywhere. Just like the real world we
    lived, thats only the deformable body. The
    rigid-body actually does not really exist in the
    world.
  • All about the cloth system, wave and water
    effects, theyre everywhere. Even we can use the
    vertex shader technology, it is an operation
    about the mesh level or using the particle
    method. Thats different from rigid body
    descriptions.

14
The finite element method
  • In fact, in the engineering field, let go back
    1970, people develop the finite element method to
    solve the deformable body simulations, in
    actually words we say, structure analysis.
  • From now, we have saw many faces and keywords,
    continuity, non-continuity, rigid-body,
    deformable body, they seems troublesome.
    Actually, we will put all together in our
    algorithm in below.

15
Whats the difficulty about the FEM in game
industry?
  • A problem about the consuming of the
    computational power
  • A problem about the memory consuming
  • Finite element method is based on small
    deformation assumption.
  • In the small deformation assumptions, the finite
    element method has its difficulties on analysis
    the large deformation about bodies, including
    cloth or fracture phenomenon.

16
Deformation
  • Deformation, in mechanics analysis, we usually
    separate it into large deformation and small
    deformation analysis.
  • In the small deformation assumptions, the finite
    element method has its difficulties on analysis
    the large deformation about bodies, including
    cloth or fracture phenomenon.
  • Because the clothing problem, itself a large
    deformation process. In mechanics, its a
    nonlinear problem. The nonlinearity will be
    included geometry and material nonlinearity.

17
Material Modeling
  • Hooks law
  • (dF) (k)(dX)
  • it is the basic concept of the constitutive law
    in material mechanics. That describes the
    stress-strain relationships.
  • The material will cause the permanent deformation
    and fluid type it will need the complex
    mathematics and physics modeling. They are usual
    described of partial differential equations.

18
An introduction to a game physics algorithm
  • The modeling is similar to the classical
    approaches used in structures and mechanical
    vibrations. The physical concept is easy to
    incorporate with other physical phenomena.
  • The computer codes are small and simple.
  • The algorithm is straightforward. Special
    numerical techniques are not required.

19
Why we dont use traditional FEM?
  • The computing power of finite element is too
    huge, its unable to use in the real time gaming
    purpose.
  • The finite element is using the matrix solving
    techniques to expand its domain it can not be
    applied on limited memory storage console
    platforms.
  • The basic is that the finite element is based on
    SMALL-DEFORMATION. So it naturally can not be
    applied to large deformation calculations. Well
    describe that below.
  • The finite element method is can not to handle
    the fracture problem in nature, because it needs
    the continuity about an element.
  • Of course, most of the finite element codes have
    been adopted into parallel forms on super
    computers. But naturally it is a matrix form
    solving problem, if we can use the vector type
    algorithm, then it will take the most advantage
    in modern vector accelerated chips.

20
Explicit finite element method
  • The explicit finite element method its since
    1970, from the national laboratory in Livermore
    and Argon. It is widely used on nuclear
    simulation and reactor safety simulations. It is
    most widely used on explosion and penetration
    topics. It is so-called transient finite element
    analysis or explicit finite element analysis.
  • We adopted it into the game industry for several
    years research.
  • Something was wrong in traditional explicit
    finite element method, we will prove that in
    below.

21
Our approach Intrinsic Vector form FEM
  • Compute the motion and deformation due to the
    application of non-equilibrium external forces.
    Mathematically, it can solve displacement, force
    or mixed boundary value problems. The essential
    boundary conditions do not have to be prescribed.
  • Handle multiple continuous bodies and their
    interactions.
  • Handle body fragmentation or merger based on
    prescribed failure criteria.
  • Compute crack initiation, and crack propagation
    in a continuous body.
  • Compute very large deformation.
  • Handle complicated, inelastic, or discontinuous
    material properties.
  • Compute transient response. With a slowly applied
    force or a dynamic relaxation process, obtain
    quasi-static response.

22
Matrix form and Vector form
  • Matrix form is highly coupled. That means it is a
    sequence about the row and column operations.
    Instead of the matrix architecture, purely vector
    form is very easy to use the hardware and
    software acceleration features to improve the
    solving speed.
  • The matrix method widely knows is related to the
    level of square. In the common language in
    physics, that is dependent on the degree of
    freedom.
  • The vector form solving architecture, it just
    grows in linear with the degree of freedom. This
    is benefit for memory consuming issue.

23
Zigma Technology
  • The Zigma technology, or by its neutrality, we
    call it vector form intrinsic finite element
    analysis.
  • Our demands
  • First, it deals with a lot of continuous body
    combine with rigid body movements and they can
    interact with each other.
  • Second, it deals with the non-linearity and
    discontinuous material effectively.
  • Third, it deals with the geometry deformation
    about the continuum.

24
Target
  • This approach can be stable to handling the basic
    problem from one continuum into multiple
    continua.
  • Handling the large rigid-body motion and
    deformation at the same time
  • Handling the unbalance forces work corresponding
    to the rigid body motion
  • Handling the complex material properties modeling
  • Handling single continua into multiple bodies
  • Handling the single continuum interact to other
    bodies
  • Handling the unbalance forces
  • Calculating the deformation precisely, that means
    handle the deformation in large rotation
  • Avoiding the iterations

25
Nonlinear continuum mechanics
  • The real object has obviously geometric
    deformation while destroy and cracking.
  • Nonlinear continuum mechanics once have deeper
    discussion to the large deformation theory such
    as Malvern.
  • To apply the nonlinear continuum mechanics to the
    large deformation process, it brings a lot of
    restriction from the initial shape definition.

26
Current form and Init form
  • Because in practices problems, the real time
    shape and the initial shape usually has great
    difference. At the same time, some mathematics
    requirements may not be satisfied, something like
    the positive defines about the matrix.
  • We noticed that in some literatures about the
    large deformation analysis, like the elastica
    problems and rubber elasticity problems, they
    most add the moderately large deformation
    assumptions.

27
Large displacement deformation
  • In finite element analysis, large displacement
    and large deformation is a subject paid
    attention.
  • When the practical problems involving the
    complicated object geometry and changes about the
    material properties, the algorithms mostly use
    the iterative method to solve such problems.

28
Our idea
  • To set up a practically algorithm that can
    predict the object is destroyed and cracking, it
    should not be limited with the amount of the
    deformation and displacement.
  • Meanwhile, it should be stable and precisely to
    obtain the results. On such basis, we adopted
    some explicit algorithms to a new simulation
    algorithm and concepts.
  • Use the current form as the basic frame to define
    the incremental stress and incremental strain. We
    also use the incremental constitutive law.

29
accuracy
  • For an elastic body is passing through the
    moderate or similar to the initial form
    deformation process, it might not be so complete
    and accuracy as the total analysis.
  • Because the error occurs in each incremental
    calculation, and its accumulated.

30
Our approach
  • the vector form of equation of motion
  • explicit time integration
  • the co-rotational coordinate method to separate
    the rigid body motion and deformation
  • the moving convected coordinate method to
    handling the large deformation and large
    displacement parts.
  • We dont have the variational form in our
    approach and also dont use the partial
    differential equation by the expression about the
    stress.

31
Our approach
  • It deal with a lot of continua and combine with
    rigid body and they interact with each other.
  • It deal with the nonlinearity and discontinuous
    material effectively.
  • It deal with the geometry deformation of the
    continuum.

32
Displacement and Deformation
  • Since the continuum is deformable, the procedure
    needs to handle the rigid body movement and
    deformation at the same time.
  • The rigid body displacement can be far greater
    than deformation.

33
The moving convected coordinate architecture
  • definition about the strain, stress and virtual
    works
  • It derives an incremental process to calculate
    the large displace and large deformation
    movement.

34
Our idea
  • Adopt the incremental calculation to setup an
    explicit deformation procedure. In this
    procedure, avoid to use the iterations.
  • Use the current form as the basic frame to define
    the incremental stress and incremental strain. We
    will also use the incremental constitutive law.

35
Co-rotational coordinate approach
  • We assumed that theres fixed global frame and a
    co-rotational frame fowling the element rotated
    and translated.
  • We can describe any displacement of a point
    within the element as two parts

36
Large displacement
  • However, we can prove that, when the large
    displacement causes the object with large
    rotations, the traditional co-rotation method is
    lake of the accuracy. So if it does not add the
    improvement, it is only suitable for limited
    large displacement motions.
  • Second, the co-rotational method seems unsuitable
    for two or three dimensional problems.
  • This is because the large displacement and large
    deformation usually happens simultaneously in a
    continuum. The simplify modeling is not suitable
    for small deformation adding on large rigid body
    movements.

37
The basic assumptions and discretization
  • We consider a body It is composed by multiple
    rigid bodies and deformable continua.
  • When the body is applied by external forces, each
    body will change their direction and position,
    and might change their geometry shape.
  • Some of them will collide each other or assemble
    to a new continuum. For any one of them, it might
    be separated into more individuals too.

38
Equations of motion
  • Equations of motion
  • is the reaction force vector that is the external
    force.
  • is the resistance force vector comes from the a
    continuum media around a particle, that is
    global internal force vector.

39
  • The commonly used method of discretization is
    that we divide a continuum in to several proper
    sub-regions that is also elements.
  • However, the rule of division is arbitrary in
    theoretically. So the particle and the node
    should not be in the same place.

40
Element analysis
  • For each element, they dont have mass property
    so itself satisfy the static equilibrium.
  • are the internal forces acting on elements or
    nodes that are the element internal nodal forces.
  • In traditional finite element analysis, we can
    define the nodal forces by the virtual work
    principle. That is

41
  • The summation about the global nodal internal
    forces acting on each node is sum of element
    nodal internal forces.
  • In the same way, we can process the definition
    about the element external forces in the same way
    as follows
  • is the external force vector on alpha node of k
    element.

42
FEM Shape function
  • When we calculate the virtual work d, we can use
    the traditional finite element method.
  • The function N satisfy the basic continuity
    requirement and on the boundary of element.

43
the virtual external work
  • According to d'Alembert theory, we consider the
    virtual external work for each element by the
    inertia forces,
  • is the nodal mass for alpha element.

44
Summarize
  • On the node, the motion of the particle satisfy
  • Within the element, the internal nodal forces
    satisfy
  • And the displacement vector satisfy the
    Cn-continuity.
  • On the boundary surfaces, force and displacement
    satisfy the boundary conditions.
  • On the contact surfaces, force and displacement
    satisfy the collision conditions or the
    continuity condition.

45
Some conclusions
  • This algorithm about the bodies basically
    simulates a limited number of particles. This is
    different from traditional finite element method
    that needs the system equilibrium.
  • The absolute displacement of the objects nodes
    comes from the equations of motion. The
    displacement on each objects displacement is the
    same so the continuity condition is satisfied
    between the elements.
  • This approach we purposed introduces the
    co-rotational method to separate the rigid body
    displacement by deformation displacement.

46
Time integration Explicit or Implicit methods
  • We use explicit time integration, that is
    convenient for handling the non-elastic and
    non-continues material properties.
  • It is avoid the iterations in solving the
    equations of motion.
  • However, the basic theory on finite element is
    not limit to explicit time integration methods.
    Other time integration methods, like Newmark-beta
    implicit time integration method, is suitable to
    solve the equations of motion.

47
Vector form FEM
  • According the Newtons basic assumption, we
    define a continuum as a group of particle mass
    assemblage. So the finite element calculation is
    to formulate a set of vector equation.
  • Adopting the explicit time integration method to
    solve the particle motion.
  • Adopting the co-rotational frame architecture to
    resolute the rigid body displacement and
    deformable displacement.
  • We purpose a moving convected material reference
    frame approach to formulate the large deformation
    and large displacement approach.

48
Traditional Explicit Finite Element
  • X is the global position vector on time 0 of a
    particle within the continuum
  • when time t, its global position is x-head
  • the deformation gradient is Fd

49
Traditional Co-rotational coordinate
  • If we set a co-rotation coordinate fixed on a
    particle and assume that the Fd has no rotational
    deformation, then dX and dX-head can be treated
    as the relative position by the co-rotational
    coordinate.
  • Let the particle doing the rigid body rotation R,
    then
  • If then

50
Lagrangian strain Cauchy Stress
  • Lagrangian strain
  • The Cauchy stress from X to x

51
  • If we define the Cauchy stress is sigma-head from
    X to x-head, then sigma-head is also the Cauchy
    stress within the co-rotational coordinate.
  • We can compare the two equations listed above,
  • So the Piola stress function S and the strain
    function E, does not change with the rotation.

52
  • To simplify the fem analysis and make the shape
    function to satisfy the boundary condition of
    continuity, we set another co-rotation frame.
  • So we change the position vector X , x and x
    prime to transform as X-head, Q is the
    transformation matrix

53
  • Although the constitutive equation has to satisfy
    the principle of objectivity, or the principle of
    material frame indifference, Malvern.
  • According to this principle , the form of S has
    to satisfy
  • So S and S-head comes from the same type. This
    conclusion is the same with traditional
    co-rotational coordinate.

54
Some conclusion
  • In our approach, the co-rotational coordinate
    definition seems different form traditional. We
    not limited to small deformation and principle of
    superposition.
  • The transformation matrix and rotational matrix
    can be different, so the co-rotational coordinate
    can be set in another way.
  • But the definition of strain, E-prime, or the
    constitutive equation S, and the form of
    calculation about the virtual internal work is
    the same with traditional co-rotational method.

55
  • the Piola stress seems the same in our approach
    by global coordinate and co-rotational
    coordinate, the Cauchy stress is not the same.
  • If the rotational matrix and transformation
    matrix comes the same, then

56
Whats going wrong?
  • When large deformation come with large rotation,
    the deformation and rigid body motion can not be
    defined separately.
  • When particle mass was moving by external force,
    the transformation matrix is including
    deformation and rotation the superposition is no
    longer exist!

57
  • we use the vector form finite element method to
    handling the large displacement and large
    deformation.
  • the Piola stress seems the same in our approach
    by global coordinate and co-rotational
    coordinate, the Cauchy stress is not the same.

58
  • When the current form of object is obviously
    different from the initial form, some mathematics
    condition may not be satisfied.
  • For example the deformation gradient, we often
    assume it as the positive definite, when the
    large strain comes, Jacobian may be positive.
  • So in planning the computing process, it seems to
    be suitable by using the incremental of
    deformation.

59
Nonlinear Material Modeling
  • To handle the large deformation material
    properties, in the past we usually use the
    total-stress-strain relationship. For example for
    the rubber elasticity proposed neo-Kookea
    material and Moony-Rivlin material, Green and
    Adkins, Murnaghan and so on.
  • We thought that they are because the traditional
    mechanics of large deformation approach to
    formulate the simple geometry model, to make a
    assumption to be corresponding the equilibrium
    equations and analytic solutions.

60
Some benefits
  • Fist, if we reduce the increment of force, the
    total stress-strain relationship, according to
    the differential and mean-value theory, we can
    express them using the linear incremental
    relationship. We compare the total stress model
    and plastic flow theory about the elastic-plastic
    materials we might see the advantage of the
    incremental model.
  • Second, when an object is going through the large
    deformation, it can not maintain the homogenous
    deformation state. That means, the history of
    large deformation is no longer a proportional
    loading process. It should be use the incremental
    form of constitutive equation.So using the
    incremental constitutive law is reasonable and
    simple.

61
Incremental stress and strain
  • The incremental stress, incremental strain and
    the virtual work can be expressed as follows
  • Where,

62
Incremental Constitutive Law
  • If large deformation process can be expressed by
    incremental stress and incremental displacement
    as a basis, then the material model can be
    expressed be incremental method.
  • When an object is going through the large
    deformation, it can not maintain the homogenous
    deformation state.
  • That means, the history of large deformation is
    no longer a proportional loading process. It
    should be use the incremental form of
    constitutive equation.

63
Some considerations about the fracture algorithm
  • If the material model is non continuous,
    something like traditional elastic-plastic
    material model or viscous elastic material type,
    it should be added in the iteration process in
    formulating the equation sets.
  • After the object is cracked, the geometry form of
    the body will have great changes. For a small
    individual, it will face to large displacement
    and rigid body motion on it.
  • Our approach mainly focus on a new vector form
    algorithm about continuum and naturally support
    fracture simulation, nonlinear material models,
    large deformation and the core support the
    parallel and distributed computing from
    mathematically to code architecture.

64
The real-time fragmentation method
  • The judgment of nodal failure criterion
  • The failure process
  • This is a general algorithm can be adapted to any
    kind of mesh types.

65
  • Basically, we just use the maximum principle
    stress direction to judge the direction about the
    failure. When the internal tensile strength in
    exceed the material tensile strength, of course,
    thats why the object will be break.

66
  • First, we should set up the unit vector for each
    element and nodes. In co-rotational coordinate,
    this unit vector will rotate by time and geometry
    changes. So it should be updated on each time
    step will be the correct boundary position of the
    element.
  • When the system is receiving the permit about the
    failure along the element boundary, we start the
    failure procedures. In element-based method, we
    are happy to use the nodal connect table because
    it is simple.

67
Fragmentation
68
Issues about the synchronization
  • In parallel computing for large computational
    project, the synchronization is a very important
    issue.
  • Too much or under considerations will cause the
    system performance down or loss control of each
    CPU. That will cause the whole system crash and
    give us the wrong answer.
  • Thats we should notice about parallel and
    nonparallel part in the algorithm and codes.

69
Generalization
  • Our approach has introduced a general description
    of particle motion. The handling of finite
    element rigid body motion with deformation is
    different from the classical continuum mechanics.

70
Progress
  • We may assume that mass points are inside the
    element, and the total number varies according to
    the stress or deformation intensity. Their
    positions may also be adjusted, and thus give a
    more accurate stress distribution.
  • To satisfy continuity requirements of a
    continuum, we choose to use the standard element
    shape functions to calculate internal forces
    acting on the mass particle. Our approach is not
    limited to that. For specific physical problems,
    we may choose to use a mixture of different
    models for the calculation. They include finite
    difference procedures with differential
    equations, formulas deduced from test data and
    mathematical analyses.
  • Since our approach assumes that the body is a
    collection of particles, different models can be
    calculated independently and then assembled for
    each particle.

71
Outlook
  • Failure analysis
  • Program based on our approach can be developed to
    provide quantitative predictions of the
    initiation of failure, subsequent failure
    sequence and a total collapse or disintegration
    of the structure.
  • Penetration mechanics
  • Our approach is able to model moving projectiles,
    multiple continuous media, and fragmentation.
    Thus, it is possible to study the time history of
    entire penetration process as a function of
    projectile geometry, traveling speed, impact
    angle and material properties.

72
  • Impact and collision
  • Explicit finite elements have been effectively
    implemented to conduct safety analyses of reactor
    structures, to evaluate crash-worthiness of
    trucks and automobiles, and to model explosion
    and demolition.

73
  • Thank you
  • Were happy to cooperate with anyone who loves
    game physics and interest about more advanced
    algorithm.
  • Reality Matrix Inc.
  • Zigma Technology.
  • michael_at_reality-matrix.com
  • I am welcome for your email
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