Interactive Real Time Cloth Simulation with Adaptive Level of Detail

Transcription

1 Interactive Real Time Cloth Simulation with Adaptive Level of Detail Olof Häggström February 5, 2009 Master s Thesis in Engineering Physics, 30 credits Supervisor at EA-DICE: Joakim Lord and Torbjörn Söderman Examiner: Martin Servin Umeå University Department of Physics SE UMEÅ SWEDEN

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3 Abstract When simulating physics for computer games much effort is put into making the simulation run fast and stable without losing the visual performance. This thesis present a way of simulating many cloth objects on screen with a total cpu-time of 1ms per frame at 30hz. The core simulation is position based, the positions of the particles are constrained, and the mesh has the ability to change its level of detail making the simulation run much faster when many objects are visible. This method can be extended to arbitrary triangle meshes but here only rectangular meshes are used. Despite that little effort has been put into conservation of physical properties such as energy and momentum the output from the simulation is highly convincing. The work was done at EA Digital Illusion CE and the method met their demands for simulating cloth in their inhouse game engine.

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5 Contents 1 Introduction Simulating Physics Physics in computer games Different aspects of cloth Problem Description 5 3 Theory Newton s laws and numerical integration Equation of motion Steppers and integrators Methods for simulating cloth Particle System Springs Constraints Jacobsen s method Verlet stepper Choosing a suitable method Motivation Position based dynamics Derivation, position projection and constraints Implementation Core Simulation Damping Air resistance Level of detail Attaching the cloth iii

6 iv CONTENTS 6 Result Visual Computational Conclusions Limitations and future work Acknowledgements 35 References 37 A Advanced damping 39 B Figures 41

7 List of Figures 3.1 The position in t 2 is wrongly approximated when using the velocity at t 1 ; the particles position differs from the real path and the error i labeled e Particles in a regular order Structure springs connecting particles in horizontal and vertical direction Shear springs connecting particles to their neighbors on the diagonals The bending springs but illustrated as lines Hookes Law Projection of particle on constraintsurface described by a distanceconstraint between the particles a and b Two triangles adjacent to each other, masspoints are numbered and normals face in the oposite direction The same triangles as in fig.(4.7) seen from another angle showing the angle between the triangles normals One triangle with its nodes p 1 p 3, normal n, velocity v and external wind w Small grid with two lod-levels where the black nodes are coarse grid The eight flags that are used in the timing in fig.(6.2) Different number of sheets of cloth at different lod levels plotted against simulation time [ms] B.1 Bars showing how different parts of the simulation is distributed over total simulation time per frame at different LoD levels v

8 vi LIST OF FIGURES

9 List of Tables vii

10 viii LIST OF TABLES

11 Chapter 1 Introduction 1.1 Simulating Physics Even the most complex system can be described by differential equations and by solving these equations, given some initial conditions, the function describing the movement of the system can be used to get the systems positions at every given time. The problem is that far from all differential equations can be solved analytically. One of the most simple systems, the pendulum, can only be solved analytically for initial angles small enough for the small angle approximation sin(x) = x to be valid. And even this system s analytical solution fails to be valid if something alters the movement, like an impulse from something colliding with the pendulum. So this is one of the main reasons why the dynamics of a physical system has to be simulated and cannot be described by the analytical solutions to the system which, as stated before, seldom exists. 1.2 Physics in computer games The computer game Spacewar from 1962 is often described as the first computer game ever. This simple game consists of two player controlled spaceships that moves around a central star. Bullets and the ships are pulled towards the stars so this first game had some simulation running controlling this gravitational field produced by the star. From the age of Spacewar and up to today physics in a computer game has gone through a tremendous change, the game Quake (1996) was the fist game ever to use real 3D-models 1

12 2 Chapter 1. Introduction and from there the evolution has boomed into where we are today. A computer games visuals and gameplay is today much dependent on the physics simulation; the dynamics of characters, vehicles, smoke, water, etc. is today most commonly controlled by a physics engine 1. One of the main reasons is that the experience of the computer game is enhanced if the game feels and looks as if it is a real world. Take for instance a flag, one can animate a flag based on the pattern from an off line simulation and get perfectly realistic movement but the movement has to, at some point, repeat itself and therefor a repetitive pattern can be spotted. Interaction with the environment is also difficult to achieve with an animation, if the flag is attached to a vehicle is hard to animate especially if the car is controlled by the player making the movement impossible to predict. I will not state that it is impossible to animate movement of cloth only that it can be a hard and timeconsuming work. By simulating, in this case the flag in real time, the user can get accurate feedback directly from her or his actions and this enhances the overall experience of the game. The same goes for everything else in a game so the game companies tries to simulate as much as possible, making the game look and feel more realistic. Of course there is a limitation in CPU and memory so what and what s not simulated has to be carefully chosen. 1.3 Different aspects of cloth Cloth is as hard to simulate as it is easy to know what is expected of the simulations visual output. Through the history of mathematically describing the dynamics of cloth a quite important question has been asked; Is cloth a mechanical material with mechanical interaction between threads or a continuous material, as metal, but with low resistance to bending? [2] Everyone has come in contact with cloth; we wear it all the time without thinking about these things. The thing is that the more one studies the dynamics of cloth the more one understands the complexity. A lot of people has, since the dawn of computer animation and physics simulation, tried to described the dynamics of cloth and for an extensive overview i suggest the book 1 A physics engine is a computer program that simulates Newtonian physics models, using variables such as mass, velocity, friction and wind resistance. It can simulate and predict effects under different conditions that would approximate what happens in real life or in a fantasy world. Its main uses are in scientific simulation and in video games.-wikipedia, the free encyclopedia-

13 1.3. Different aspects of cloth 3 Cloth Modeling and Animation [2], where the authors has collected many references to articles about cloth written up until the year 2000.

14 4 Chapter 1. Introduction

15 Chapter 2 Problem Description Simulating cloth, likely to occur in a Battlefield game is the core statement given by DICE prior to the start of this project. Speed is important and the CPU budget is 1ms per frame making it important to have dynamic level of detail (lod) enabling the number of simulation nodes to decrease with distance from the camera. DICE does not currently use simulated cloth in their Battlefield games so the purpose of this thesis project is to make a cloth simulation that is fast enough to make it into the Frostbite game engine. The visual result has to be convincing for the player without breaking the CPU-budget and this will certainly make the physics less accurate but again as long as the visual result is convincing, the accuracy on the actual physics is less important. The implementation will be done in the Frostbite Engine 1. 1 Inhouse game engine written from ground up at EA Digital Illusion CE (DICE). 5

16 6 Chapter 2. Problem Description

17 Chapter 3 Theory First some notations that are used throughout this work, x is the positions of the particles p is the estimates for the new positions v is the velocities C is the set of constraints for the system t timestep F ext external force m mass w = 1 m k d some damping for the velocities 3.1 Newton s laws and numerical integration To be able to understand even the most basic theory in cloth simulation one has to remember some basic laws of physics and a little about of numerical integration. It s assumed that the reader has basic skills in calculus and linear algebra, still the explanations and arguments is made as simple as possible. 7

18 8 Chapter 3. Theory Equation of motion Newton s second law states that the acceleration a on an mass point is proportional to the sum of all external forces acting on it. Since the acceleration is the second timederivative of the position Newton s second law can be written as, F tot = m d2 dt 2 x (3.1) Where m is the mass of the object and x is the position. From here on the time derivative on x will be denoted ẋ so the relation, eq.(3.1) can be written as F tot = mẋ and, if all the external forces are known, it can be used to calculate the change in position Steppers and integrators What follows is a brief description of, and short comment on, some of the numerical integrators considered for this project. The derivations and discussions are quite short so for a thorough analysis around the stability for different steppers i strongly suggest Claude Lacoursières Phd. thesis Ghosts and machines: regularized variational methods for interactive simulations of multibodies with dry frictional contacts [4]. Explicit Eq.(3.1) is a second order differential equation and can be split up into a system of first order equation, v = F m (3.2) v = ẋ (3.3) By approximating these derivatives with v = v n+1 v n t + O( t 2 ) (3.4) ẋ = x n+1 x n t + O( t 2 ) (3.5)

19 3.1. Newton s laws and numerical integration 9 The system can be written as v n+1 = v n + t F (x n, v n ) m (3.6) x n+1 = x n + tv n (3.7) And this is the explicit Euler stepper, given a force (F ), time step ( t) and initial conditions at time n the new velocities and positions at time n + 1 can be calculated. This stepper is the easiest there is and of course this comes with some drawbacks. The main issue with explicit steppers are that they are not stable, using the velocity and position at one timestep to approximate the state in the next step accumulate the error term and finally the system will explode. p e real path time t 1 t 2 Figure 3.1: The position in t 2 is wrongly approximated when using the velocity at t 1 ; the particles position differs from the real path and the error i labeled e. Simply by using the velocity at time n + 1 when updating positions we get a stepper known as symplectic Euler or Leapfrog. v n+1 = v n + t F (x n, v n ) m (3.8)

20 10 Chapter 3. Theory x n+1 = x n + tv n+1 (3.9) This method is as easy to use as the explicit Euler but conserves energy much better. Implicit Until now the previous steppers has predicted the next step using only information in the current step. To get more stable simulations we can use an implicit stepper v n+1 = v n + t F (x n+1) m (3.10) x n+1 = x n + tv n+1 (3.11) This stepper uses information about the state in the next time step, and since we don t have that information yet we have to form a system of algebraic equation. Note that the velocities in the function describing the force has been removed since this implicit method will introduce a lot of numerical damping and also for simplicity. The unknowns in this system is what we want namely the velocities and positions in the coming time step. From here we can assemble and solve matrix system for all the equations.

21 Chapter 4 Methods for simulating cloth 4.1 Particle System Figure 4.1: Particles in a regular order This, fig.(4.1), is the basic setup for simulating a rectangular piece of cloth. Masspoints are evenly distributed over a rectangular area and are then kept in place by either constraints or springs. Two different methods are discusses and one of them where chosen due to stability and other properties discussed later Springs The most straightforward way of simulating a piece of cloth is to build a grid consisting of mass points fig.(4.1) and springs. The masses are connected with mass less springs in 11

22 12 Chapter 4. Methods for simulating cloth different ways and what follows is a description of these different springs. The different springs have different primary functions but they all contribute to govern how much the cloth will stretch. Structural springs These springs fig.(4.2) primary task is to keep the structure of the cloth, they connect particles with neighboring particles in the horizontal and vertical direction. Figure 4.2: Structure springs connecting particles in horizontal and vertical direction Shear springs The shear springs fig.(4.3) are there to keep the cloth from, as the name implies, shear. These springs connect a pointmass to the neighbors on the diagonal. Figure 4.3: Shear springs connecting particles to their neighbors on the diagonals

23 4.1. Particle System 13 Bending springs The final set of springs is called bending springs fig.(4.4) and these springs connect each point mass with masses in the horizontal and vertical direction skipping the closest one. The bending springs keeps the cloth from bending so without these springs the cloth would be able to fold, without resistance, over the vertical and horizontal direction. Figure 4.4: The bending springs but illustrated as lines Hooke s law Hooke s law, eq.(4.1), states that the force from a spring is proportional to the spring s deviation from its rest length. F spring = k(x 0 x) (4.1) The value k is the spring constant which governs how large the force will be relative the elongation of the spring. This model can be extended to account for damping. The damping is relative the derivative of the elongation of the spring hence the velocity of the point mass. Force from a spring can, in one dimension, be rewritten as F spring = k s (x 0 x) k d v, (4.2) where the term k d v is the damping proportional to the velocity. This is only valid in scalar or one dimensional form. So by introducing vector-notation the equation describing the force from a spring can be written as

24 14 Chapter 4. Methods for simulating cloth k x 0 x Figure 4.5: Hookes Law [ ] v ab x ab xab F ab = k s ( x ab x 0 ) + k d x ab x ab (4.3) Here the subscript denotes the force, distance and velocity relative two particles, a and b, connected by a spring. Note that the force from a to b is the same, in magnitude, as the force from b to a but its direction is inverted. This property comes from Newton s third law that states that from every action there is an equal and opposite reaction. Force By using Hooke s law the force from each spring on each particle can be calculated, the calculations are simplified by using Newtons third law. When all the forces from the springs are calculated the component from the gravity, contacts and air resistance are applied. (More about air resistance later) If all forces are known the system can be stepped forward in time using a stepper/integrator of choice. Note that the explicit stepper will not work here since the dynamics from the springs will give oscillating movement between the particles, fig.(3.1).

25 4.2. Jacobsen s method Constraints The particles in fig.(4.1) can be kept in place by constraints in the same way as with the springs. A constraint is basically a condition that must be satisfied, here the constraints are distances between particles of angles between adjacent triangles in a mesh. 4.2 Jacobsen s method Tomas Jakobsen presented a totally constraint based method for simulating cloth [3], rigid body movement and also plants and he states that a simulation probably cannot be done faster than this. His method is straightforward and relatively simple and requires, according to himself, only one pass over every constraint in the system per time step. This method was used in the game Hitman: Codename 47 by IO-Interactive back in The method was used for simulation all the physics in the game even plants so the method performs well when it comes to speed. On drawback is that it is heavily damped but this should not be a big problem when simulating cloth because cloth is quite damped and if carefully done numerical damping can be used to simulate physical damping, hence exclude a lot of the physical damping from the simulation. The method presented is, as stated, simple and straight forward. Starting from the rectangular mesh fig.(4.1) a series of constraints is setup between the points. The constraints are the distances between the nodes in the horizontal and vertical direction furthermore constraints that are the distances between neighbors on the diagonal is also set up. The only forces that come into this method are external forces, gravity for example. Choose a stepper, more about the stepper later, and step a time step and in the end of the time step make sure that the constraint is fulfilled one by one, simply by moving particles together. This method, moving particles consequentially, is called relaxation. Relaxation is simply moving particles, to fulfill local constraints, and by iterating over all constraints a number of times each time step all the constraints will, hopefully, be fulfilled in the end of the time step Verlet stepper In his article [3], Jacobsen uses the Verlet-scheme and this stepper is much more stable than the simple explicit Euler, and as stable as symplectic Euler, but it still as easy to

26 16 Chapter 4. Methods for simulating cloth implement. What follows is a derivation of the stepper by a Taylor-expansion of the position in forward and backward direction. x t+dt = x t + x t dt + x t 2 dt x t 3 dt3 + O(dt 4 ) x t+dt = x t x t dt + x t 2 dt 2... x t 3 dt3 + O(dt 4 ) (4.4) Adding the two equations gives x t dt + x t+dt = 2 x t + x t dt 2 + O(dt 4 ) (4.5) Moving the position at t+dt to the RHS, using Newtons second law on the acceleration (second time derivative of the position) and dropping the error-term yields the Verlet scheme. x t+dt = 2 x t x t dt + F m dt2 (4.6) Notice that this stepped does not use the velocity explicitly to calculate the position in the coming timestep. This stepper is of 4th order with respect to the timestep and compared to the explicit Euler which is of 2nd order. There is a way of calculating velocities in every timestep using v n = x t dt x t dt + O(dt 2 ) (4.7) This introduces an 2nd order errorterm so this approximation of the velocity is not very accurate but later, after implementation, there will be clear if this approximation needs to be improved. 4.3 Choosing a suitable method There are many methods available for cloth simulation and there are advantages and disadvantages with every method. There are many methods that have been considered for this work, some where rejected early in the work due to the apprehension that it would be too computationally expensive. For a more or less complete survey i suggest reading Physically based deformable models in computer graphics [1].

27 4.4. Position based dynamics Motivation The aim for this work was to get a fast, stable simulation with accepted visual results. The methods require solving a linear system, FEM and implicit was discharged because of the apprehension that it would take to much time considering the 1ms per frame budget. So the choice stood between position based, constraint method and a system with springs. There is quite easy to motivate that the position based method is the method of choice, solving the problem using springs requires that, at every time step, one pass over the springs is required to keep the springs at an accepted length since a spring that is to long gives a force to large for the system to handle [7]. This means that one pass over every spring per timestep can be seen as one pass over every constraint so the pass over the springs just to keep them at an accepted length is the actual pass for keeping the system together in the position based method. There was also a thought that the lod would be easier to solve if having a method that worked directly on the positions. To attach a piece of cloth to a moving object felt easier as well since, when working on positions directly, the attached particles can simply be set to follow some object. The method used is, because of this, the position based method proposed by Müller et al. since this method fits best into the demands from DICE prior to the start of the project. It has great performance and gives complete control over the positions of the nodes in the mesh. 4.4 Position based dynamics Müller et al. [5] presents a method where they generalizes Jakobsen s method taking any constraint and applying projections based on the gradient of the constraint function. This method is not as easy to implement as Jacobsen s but uses the Explicit Euler scheme to calculate the new positions at every timestep. They also presents a constraint that keeps the angle between adjacent triangles so that one can specify bending stiffness without having to adjust the stiffness of the stretching constraints. In this method, working on the positions directly, one solves for the equilibrium and project the position. This gives a system of second order equations and the system is solved by iterating over the set of constraint and solving projection positions for one constraint at a time. This is exactly what Jakobsen proposed in his article but Jakobsen only handled distance constraints. Müller et al. presents a way of applying position

28 18 Chapter 4. Methods for simulating cloth a b Figure 4.6: Projection of particle on constraintsurface described by a distanceconstraint between the particles a and b. projection for any constraint. This work handles a bending constraint and a distance constraint, but the model applies to any so first the method is explained in a general way followed by a derivation of the two types of constraints handled here Derivation, position projection and constraints Let C( p) be some constraint function to a given position p, then we want a p such that C( p + p) = 0. By expanding this in a 1st order Taylor serie we get C( p + p) C( p) + p C( p) p = 0 (4.8) p must be in the direction of p C( p) so we want to find λ so that p = λ p C( p) (4.9) by inserting eq.(4.9) into eq.(4.8), solving for λ and putting λ back in eq.(4.9) we get an expression for valid p.

29 4.4. Position based dynamics 19 Now a correction for a single point can be written as, p = pc( p) C( p) (4.10) p C( p) 2 p i = s pi C( p 1,..., p n ) (4.11) with, s = C( p 1,..., p n ) j p j C( p 1,..., p n ) 2 (4.12) The scaling factor eq.(4.12) is constant for all points involved in the constraint. If the point have different masses the projection has to be scaled with respect to the masses, because a light particle won t affect the system as much as a heavy on, so the scaling factor becomes, s = C( p 1,..., p n ) j w i pj C( p 1,..., p n ) 2 (4.13) where w = 1/m, m being the mass of a particle. Finally the correction can be written as, p i = sw i pi C( p 1,..., p n ). (4.14) In this thesis two special cases has been implemented, both are taken from the article by Müller et al [5]. The two cases are distance and bending constraint and the reason for the separation is that it seems like a good thing to be able to control bending separately from the stretching of the cloth. The constraints are solved one by one at every timestep making the system converge towards the correct solution however the system will not converge exactly if the number of iteration over the constraints are kept low. This method is called relaxation. Distance constraint These constraints purpose is to keep the shape of the cloth and there are two different ones, shear and structure. The constraint function for a distance constraint can be written as,

30 20 Chapter 4. Methods for simulating cloth C( p 1, p 2 ) = p 1 p 2 d = 0 (4.15) with derivatives, p1 C( p 1, p 2 ) = p2 C( p 1, p 2 ) = p 1 p 2 p 1 p 2. (4.16) Using eq.(4.14), eq.(4.15) and eq.(4.16) the final corrections for the projection of a distance constraint becomes p 1 = w 1 w 1 + w 2 ( p 1 p 2 d) p 1 p 2 p 1 p 2 (4.17) and p 2 = + w 2 w 1 + w 2 ( p 1 p 2 d) p 1 p 2 p 1 p 2. (4.18) Bending constraint The other constraint is the constraint that keeps the normals of adjacent triangles at a certain angle. The constraint for bending is C( p 1, p 2, p 3, p 4 ) = arccos( n 1, n 2 ) ϕ 0 (4.19) Where ϕ 0 is the desired angle between the two normals. Computing the projection for the points involved in this constraint is not as straight forward as with the distance constraints because it is required to compute the gradient of the normalized cross product. Given a normalized cross product between the vectors v 1 and v 2, the derivatives with respect to the vectors are n = v 1 v 2 v 1 v 2, (4.20) n 1 1 = v 1 v 1 v 2 ( v 2 + n( n v 2 ) T ) (4.21)

31 4.4. Position based dynamics 21 2 n n 1 1 Figure 4.7: Two triangles adjacent to each other, masspoints are numbered and normals face in the oposite direction. n 2 1 = v 1 v 1 v 2 ( v 1 + n( n v 1 ) T ) (4.22) where the matrix v has the property v x = v x Setting p 1 = 0 and replacing the normals in eq.(4.19) with n 1 = p 2 p 3 p 2 p 3 (4.23) and n 2 = p 2 p 4 p 2 p 4 (4.24) eq.(4.19) becomes C( p 1, p 2, p 3, p 4 ) = arccos(d) ϕ 0, (4.25) with d = n T 1 n 2. d Since, dx arccos(d) = 1 1 d, the gradients of eq.(4.19) becomes 2

32 22 Chapter 4. Methods for simulating cloth n 2 1,2 4 3 n 1 Figure 4.8: The same triangles as in fig.(4.7) seen from another angle showing the angle between the triangles normals. 1 p3 C = (( n 1 ) T n 2 ) (4.26) 1 d 2 p 3 1 p4 C = (( n 2 ) T n 1 ) (4.27) 1 d 2 p 4 1 p2 C = (( n 1 ) T n 2 + ( n 2 ) T n 1 ) (4.28) 1 d 2 p 2 p 2 p1 C = p2 C p3 C p4 C. (4.29) By using the relations in eg.(4.21) and 4.22 the derivatives in eq.(4.26) - eq.(4.29) can be computed as, q 3 = p 2 n 2 + ( n 1 p 2 )d p 2 p 3 (4.30)

33 4.4. Position based dynamics 23 q 4 = p 2 n 1 + ( n 2 p 2 )d p 2 p 3 (4.31) q 3 = p 3 n 2 + ( n 1 p 3 )d p 2 p 3 p 4 n 1 + ( n 2 p 4 )d p 2 p 4 (4.32) q 1 = q 2 q 3 q 4. (4.33) Finally the correction for a bending constraint can be written as, p i = 4w i 1 d2 (arccos(d) ϕ 0 ) j w j j q j 2 q j (4.34) The denominators in eq.( ) can become invalid if one of the vectors are close to zero or if the triangle is highly elongated so in the implementation one has to check for theses special cases.

34 24 Chapter 4. Methods for simulating cloth

35 Chapter 5 Implementation 5.1 Core Simulation The core simulation is straight forward making is easy to implement and to understand. The simulation of the system is done as follows; 1.1 Setup the initial positions, velocities and masses. 1.2 loop as long as system is active assemble external force (wind, gravity) integrate and damp velocities, v = v + twf ext k d integrate positions to get new estimates p = x + tv generate collision constraints solve constraints update velocities v = (p x) t update position x = p calculate triangle normals Note that the LOD-level i chosen before step and is determined by the distance to the camera. The constraint projection i.e solving the constraints, works on the estimates and after that the velocities are set to values that would have taken the initial positions to the position after the projection, making the velocities valid for the next turn in the loop. 25

36 26 Chapter 5. Implementation Damping Müller et al. presents a way to damp the velocities that doesn t affect the global movement of the particle system. This method works good but the computational cost is to high because it involves computing the inertia tensor and taking its inverse. The total cost for using this more advanced damping model is approximately 25% of the total computation time per frame. In this work damping is simply performed by multiplying the velocities by a scalar between 0 and 1, this gives a nonrealistic behavior if the value is not close to 1 but computational speed is greatly improved compared to the more advanced approach. The more advanced damping model is explained in the appendix Air resistance Force from both wind and the drag due to the cloths own velocity is modeled for each triangle in the mesh and the distributed equally to every node in the triangle. n p 2 v p 1 p 3 w Figure 5.1: One triangle with its nodes p 1 p 3, normal n, velocity v and external wind w. The drag due to the triangles velocity is modeled as

37 5.1. Core Simulation 27 F drag = ρ( v n) 2 A v v, (5.1) where ρ 1 is some drag coefficient A is the area of the triangle. Force from an explicit wind is modeled in a similar fashion, F wind = ρ( w n)a n n. (5.2) The force due to drag eq.(5.1) cannot change the direction of the triangles velocity so if this happens simply set it to Level of detail Level of detail (lod) is handled by multiple meshes that represent the same piece of cloth and which mesh is used is determined by the distance to the viewer. At initialization of the system every lod, despite for the level with the highest resolution are given indices to it s neighbors in the level below so that the position and velocity for a node introduced when going from coarser to finer mesh can be interpolated as the average from the assigned nodes. The different lod-meshes have different masses for each node since mass is calculated from one third of each triangle it borders and every triangle gets a mass during initialization calculated from a user defined density and the triangles area. In fig.(5.2) node 5 knows about nodes 1, 3, 7, and 9 so when going from a coarser to a finer grid node 5 gets it s velocity and position from these nodes. Going from a finer to a coarser grid is much more simple sine the nodes left in the coarser grid keeps its position and velocity. The level of detail is the most important thing when it comes to performance; at distance one can hardly tell the difference between a coarse and a fine grid. The overhead for the level of detail is mainly in memory, not keeping the same positions etc. for nodes shared by lod-levels takes up more memory. There are not so much computation done when going between different levels and furthermore the code becomes more cache 2 efficient if data is more closely packed at different levels. 1 in a more accorate model ρ would consist of two part; one describing the viscosity of the fluid and the other a material property. But here the two are combined to reduced the variables required. 2 A cache is a temporary storage area where frequently accessed data can be stored for rapid access.

38 28 Chapter 5. Implementation Figure 5.2: Small grid with two lod-levels where the black nodes are coarse grid Attaching the cloth By using the Frostbite Engine it was interesting to attach cloth to vehicles; this is done by simply choosing one side of the cloth, give the particles on that side infinite mass and finally let the particles follow the vehicles transformation. A particle with infinite mass has zero inverse mass so there are no reason to keep the actual mass.

39 Chapter 6 Result I have based the result on the initial request from DICE and how well this work had met these demands, very little focus has been put into the actual physical correctness of the simulation. The results are all based on computational and visual performance but as stated by many others, if it looks convincing it is probably close to a physically correct simulation [2] p.327. Game companies normally use visual results to ensure quality so visual and computational performance is the measurements that have been made to ensure quality. 6.1 Visual The visual result are good, a clothlike behavior that meets the demands from DICE. Especially drapes and flags. The parasol looks ok and there is really no need to simulate more than the outer parts since the actual parasol doesn t move much in the wind. It is not easy to setup the stiffness so that the lodding is made easy. There are some issues with the cloth appearing stiffer at distance because of the fact that the solver solves the system better with fewer constraints. This problem can be solved if the stiffness for the constraints are chosen so that the solver returns a less accurate solution and/or if some constraints are simply not included into more sparser meshes. i.e, remove shear and bend constraints. 29

40 30 Chapter 6. Result Figure 6.1: The eight flags that are used in the timing in fig.(6.2). 6.2 Computational The simulation had a budget, 1ms per frame and at 30 Hz the simulation meets this budget. It is not possible to have too many flags onscreen at the highest lod-level but if the game is setup with this in mind I see no reason why this method won t fit into the engine. The most expensive step in the simulation is the bending constraints and the damping. The bending constraints requires 4 cross products per constraint and the damping runs slow due to the fact that an inverse of a matrix has to be calculated in every timestep and this is the reason why the more advanced was implemented but not activated in the final product. If only visual performance is the goal, tests have shown that the damping can be replaced by a much more simple model and thus solving this issue. The bending constraints can be replaced by using distance constraints in the same way as the bending springs in fig.(4.4). The ability to change the meshes resolution greatly increase performance; fig.(6.2) shows that the time needed to simulate a sheet of cloth decreases almost at a factor 1 x2 with respect to the lod level. Furthermore fig.(b.1) shows how the total time needed in every frame is divided between different parts of the simulation at different lod-levels. As can be seen in fig.(6.2) a scene with 8 flags at highest lod-level takes approximately 1.6ms and the same scene at the lowest level takes up only 0.2ms so there are a lot of gain in computational time with an efficient lod-system.

41 6.2. Computational 31 n 2 1,2 4 3 n 1 Figure 6.2: Different number of sheets of cloth at different lod levels plotted against simulation time [ms].

42 32 Chapter 6. Result

43 Chapter 7 Conclusions The method presented here where suited for DICE; the simulation handled lod in an efficient way and the visual output is very good. Computational performance met the demand with some limitations in number of high resolution flags close to the camera and total number of flags on screen. The work was integrated with the engine but is currently not in use, still the work is a proof of concept for the selected method. 7.1 Limitations and future work Despite the visual quality from this simulation there are many things that can be improved so the list of limitation can be made long and a lot of them can be removed by continue working on the project. Better tools for setting up the cloth, attachments and more general meshes, so that artists and level designers can make use of the software. Vectorizing the code using the SIMD instruction set would speed up the calculations. Better renderer and shaders. Müller [6] presented in November 2000 improvement of the position based dynamics model for high resolution grids using a multigrid solver. Since this work is based on multiple grids for one patch of cloth it would be easy to integrate this idea making the solver much more accurate. 33

44 34 Chapter 7. Conclusions A more general collision model that handles more primitives and general triangle meshes. Performance can be increased by first looking at fig.(b.1). Implement tearing.

45 Chapter 8 Acknowledgements I would like to thank everyone at DICE who has helped me during this project; Joakim Lord and Torbjörn Söderman who has been my supervisors. Christofer Stegmayr for all the good talks and ideas. Mattias Widmark, who was very helpful and patient when he helped me implement the renderer. Henrik Karlsson who made thing clear about the Entity and Component-system in Frostbite. Peppe and Gidlund for helping me with the physics in Frostbite. Hans Mannby for good talks at the early stage. Elin Fällman for helping me with the project plan. Martin Servin who has been my examinator. My parents for supporting my actions in life up to this point. And finally everyone that I have forgotten... 35

46 36 Chapter 8. Acknowledgements

47 References [1] R. Kaiser E. Boxerman M. Carlson A Nealen, M. Müller. Physically based deformable models in computer graphics. Eurographics 2005 state of the art report, [2] Donald H. House and David E. Breen. Cloth Modeling and Animation [3] Thomas Jakobsen. Advanced Character Physics. Technical report, Copenhagen, Denmark. [4] Claude Lacoursière. Ghosts and machines: regularized variational methods for interactive simulations of multibodies with dry frictional contacts. Umeå: Umeå University, Faculty of Science and Technology, Computing Science, ISBN: [5] M. Hennix J. Ratcliff M. Müller, B. Heidelberger. Position based dynamics. Proceedings of Virtual Reality Interactions and Physical Simulations (VRIPhys), pages 71 80, [6] M. Müller. Hierarchical position based dynamics. Workshop on Virtual Reality Interaction and Physical Simulation VRIPHYS (VRIPhys), [7] Xavier Protov. Deformation constraints in a mass-spring model to describe rigid cloth behavior. Graphics Interface, pages ,

48 38 REFERENCES

49 Appendix A Advanced damping If the system has to be damped without damping the global movement of the system a more advanced model than simply decreasing the velocity at every timestep the following algorithm works great but is to computational expensive to be considered in this work. The model has been implemented but fell out due to the previously mentioned fact. First calculate the center of mass and its velocity x cm = i x im i i m i (A.1) v cm = i v im i. (A.2) i m i Second, calculate the angular momentum of the particle system L = i r i m i m i v i. (A.3) The angular velocity is calculated by multiplying the inverse of the inertia tensor with the angular momentum ω = Ī 1 L (A.4) and the inertia tensor is calculated Ī = i r i r T i m i (A.5) 39

50 40 Chapter A. Advanced damping where r is a matrix with the property r x = r x. Finally for all vertices i do v i = v cm + ω r i v i v i = v i + k d v i (A.6) and the term k d is the damping coefficient [0..1].

51 Appendix B Figures Figure B.1: Bars showing how different parts of the simulation is distributed over total simulation time per frame at different LoD levels. 41