Simulating  the  Physical  Effects  of  External  Forces  on  Cloth  Juan  de  Joya  (cs184-­‐ax)  

CS184  Fall  2011  Final  Project    

 Figure  1:  The  simulated  cloth  piece  colliding  with  the  sphere:  the  first  three  images  show  different  patterned  cloth  materials,  and  the  fourth  image  shows  the  demo’s  wireframe  mode.    1    Introduction  Robust  physics  remains  a  popular  topic  in  the  field  of  computer  graphics.  The  real-­‐time  and  offline  demands  of  film,  gaming,  and  research  necessitate  more  realistic  representation  of  computer-­‐generated  simulations,  and  as  such  further  studies  in  the  field  are  no  less  relevant  than  they  were  in  the  last  few  decades.    Building  on  previous  research  and  development,  it  is  now  possible  to  create  isolated  realistic  physical  simulation  without  investing  in  a  full-­‐fledged  physics  engine.  In  my  report  I  present  the  techniques  behind  the  implementation  of  my  cloth  simulator,  the  goal  of  which  is  to  showcase  the  physics  of  wind  and  gravity  on  the  simulated  cloth  as  well  as  collision  detection  when  said  cloth  interacts  with  the  sphere.      2    Background  Soft  body  dynamics  is  the  study  and  development  of  techniques  to  simulate  visually  realistic  physical  motion  and  properties  of  deformable  objects  (or  soft  bodies).  Unlike  rigid  bodies,  soft  bodies  have  a  malleable  shape  when  subject  to  physical  effects.  While  this  means  that  the  relative  distance  of  any  two  points  that  comprise  a  given  soft  body  are  not  fixed,  the  body  is  expected  to  relatively  maintain  its  shape.      Cloth  simulation  in  the  context  of  computer  graphics  is  the  simulation  of  two-­‐dimensional  soft  bodies  that  serve  as  a  representation  of  cloth  in  the  3D  space.  A  common  method  of  simulating  cloth  is  to  represent  the  material  as  a  grid  of  particles  connected  to  each  other  by  springs.  The  mass-­‐spring  model  determines  the  spring  forces  acting  on  the  particles  at  each  time  step,  in  combination  with  gravity  and  other  effective  forces.    Collision  detection  is  an  important  prerequisite  in  creating  a  realistic  cloth  simulation;  the  detection  algorithm  between  cloth  and  most  objects  is  relatively  straightforward  given  an  object  with  a  well-­‐defined  shape.    3    Implementation  In  the  mass-­‐spring  model,  a  continuous  cloth  surface  is  discretized  into  a  finite  number  of  particles  for  drawing;  the  particles  are  then  connected  in  an  orderly  fashion  with  springs.    Each  particle  in  the  grid  contains  the  data  necessary  to  calculate  its  motion.  We  can  then  find  the  new  positions  and  velocities  of  cloth  particles  using  Newton’s  Second  Law  and  a  numerical  integration  method.    

 Newton’s  Second  Law  states  that  the  acceleration  𝑎  of  an  object  is  directly  proportional  to  the  net  force  𝐹  and  indirectly  proportional  to  the  mass  𝑚,  𝑎 = !

!.  Using  this  we  can  solve  the  

differential  equations  of  𝑎  to  find  velocity  𝑣  and  position  𝑝  of  a  particle.  Since  𝑎 = !!!"  and  

𝑣 = !!!",  we  can  integrate  the  acceleration  to  find  the  new  velocity  of  a  given  particle,  which  

in  turn  we  can  integrate  to  find  its  [the  particle’s]  new  position.    There  are  several  numerical  integration  methods  that  can  be  used;  for  future  implementation  (see  Limitations  and  Conclusion  and  Future  Work,  below),  Verlet  Integration  offers  greater  stability  and  accuracy  with  no  computational  cost  in  comparison  to  the  Euler  method.  For  the  purposes  of  this  cloth  simulation  however,  we  will  use  a  modified  Euler  Integration.      

 Figure  2:  Cloth  simulation  using  basic  Euler  Integration  results  in  severe  calculation  errors  over  time.  Within  seconds  of  running,  earlier  builds  of  my  cloth  simulation  exhibited  gross  inaccuracies  leading  to  the  final  picture  shown  above.    Euler  Integration  on  its  own  leads  to  very  inaccurate  results;  there  will  always  be  an  integration  error  that  accumulates  over  time.    In  response,  there  are  two  modifications  in  my  code  that  help  mitigate  error  accumulation.  First,  oversampling  to  decrease  the  time  step  to  𝑑𝑡 = !

!""𝑡ℎ  of  a  second  each  Euler  integration  will  lead  to  results  much  closer  to  the  

ideal  solution.    More  importantly,  damping,  which  is  the  proportional  velocity  lost,  is  factored  into  the  calculation  of  velocity  at  each  time  step.    In  this  simulation,  damping  is  introduced  to  tweak  the  look  of  the  cloth  material  and  make  the  simulation  stable.  A  high  damping  factor  makes  the  cloth  material  more  rigid,  while  a  low  damping  factor  results  in  a  lightweight  cloth  material.      

       Default    

Damping:  0.6  Time:  16.0  seconds  

Slightly  Lighter  Damping:  0.55  

Time:  12.0  seconds  

Slightly  Rigid  Damping:  0.7  

Time:  22  seconds  

More  Rigid  Damping:  0.9  

Time:  34  Seconds  Table  1:  The  following  cloth  materials  available  in  the  cloth  simulator,  listed  with  their  respective  damping  factors  and  the  time  it  took  for  them  to  reach  the  rest  position  shown  above.    

Because  the  mass-­‐spring  system  of  my  cloth  simulator  is  implemented  as  a  particle  system,  each  particle  is  connected  to  24  neighboring  particles;  these  interconnections  represent  the  “stretch”,  “bend”,  and  “shear”  springs.  The  “stretch”  and  “bend”  springs  prevent  the  cloth  from  stretching  too  much  and  folding  in  on  itself  respectively,  while  the  “shear”  springs  allow  the  cloth  to  resist  shearing  movement  by  maintaining  the  diagonal  shape  of  the  cloth.  For  my  cloth  body,  I  have  generated  a  7x7  rectangular  cloth  piece  made  out  of  50x50  particles,  resulting  in  60000  spring  connections.  Two  triangles  are  produced  for  every  four  particles  forming  a  grid  square.    

 Figure  3:  The  cloth  simulator’s  wireframe  mode.  When  rendering  this  surface,  the  masses  and  springs  are  not  typically  drawn;  instead  they  are  used  to  generate  triangle  vertices.      The  nature  of  cloth  simulation  is  essentially  a  constraint-­‐solving  problem.  It  involves  solving  for  the  positions  of  the  particles  in  the  grid  at  each  frame  of  the  simulation,  where  the  positions  are  affected  by  the  springs  keeping  the  particles  together  as  well  as  external  forces  acting  on  the  particles.      In  my  cloth  simulator,  I  implemented  my  cloth  patch  to  react  to  the  external  forces  of  gravity,  wind,  and  collision  with  the  sphere.  When  releasing  the  cloth  from  its  inert  state,  gravity  naturally  pulls  downward  on  the  particles.  Wind  forces  can  be  applied  to  push  the  cloth  towards  any  direction  in  the  3D  space;  given  enough  wind  pushing  upwards  (in  the  +y  direction  in  3D  space),  the  cloth  patch  will  float  and  defy  the  initial  gravity  forces  pulling  it  downward.    Both  wind  and  gravity  are  altered  and  added  to  the  net  force  (accumulatedForce  in  the  source  code)  affecting  the  particle,  which  will  then  be  used  to  offset  the  new  positions  and  velocities.      Implementing  collision  detection  between  the  cloth  patch  and  the  sphere  is  easy  considering  that  the  method  is  not  dissimilar  from  the  circle-­‐ray  intersection  methods  for  ray  tracing.  Our  sphere  is  defined  by  a  center  and  radius;  as  such,  we  can  detect  collision  by  asking  whether  a  particle  is  inside  the  radius  of  the  sphere.  If  it  is,  the  simulation  resolves  the  collision  by  moving  the  particle  outside  of  the  sphere  along  the  vector  from  the  center  to  the  particle;  the  resulting  position  of  the  particle  will  be  the  distance  equal  to  the  radius  of  the  sphere.    To  provide  user  interactivity  with  the  simulation,  I  implemented  a  basic  user  interface  that  informs  the  user  of  the  features  that  allow  them  to  manipulate  the  external  forces  affecting  the  cloth,  and  the  type  of  cloth  being  used  in  the  simulation.  Several  other  features  include  switching  between  shaded  and  wireframe  mode,  switching  between  window  sizes,  rotation  and  zooming,  and  the  ability  to  pause  or  un-­‐pause  the  animation.  

 Figure  4:  This  image  shows  the  basic  user  interface  available  with  the  program.    4    Results  The  keyboard  functions  listed  in  the  simulation’s  user  interface  can  be  directly  used  to  generate  the  physical  effects  of  wind  and  gravity.  Available  at  the  end  of  this  report  are  pictures  showcasing  my  results.  The  source  code,  additional  photos,  and  a  supplementary  video  are  included  in  the  attached  CD.    5    Limitations  Although  my  project  showcases  the  appropriate  realistic  dynamic  behavior,  I  would  like  to  use  a  different  numerical  integration  method  when  I  do  future  cloth  simulations.  While  the  modified  Euler  Integration  used  in  the  project  does  its  job,  it  would  be  better  to  use  more  accurate  methods  for  more  complicated  simulations.  The  damping  factor  works  well  in  differentiating  the  type  of  cloth  and  mitigating  errors,  but  a  damping  factor  less  than  0.5  leads  to  marginally  diminishing  accuracy;  at  a  damping  factor  of  0.2,  for  example,  the  particles  near  the  edge  of  the  grid  continue  to  jump  around  even  when  the  cloth  is  at  rest.  That  being  said,  the  method  I  initially  used  for  this  project  was  Verlet  Integration;  unfortunately,  I  discovered  bugs  in  my  implementation  that  would  require  adjusting  the  entire  source  code.  Given  the  time  constraints  of  working  alone,  I  opted  to  use  a  modified  Euler  Integration  that,  while  not  ideal  for  complicated  simulations,  works  appropriately  for  the  scope  of  this  project.    6    Conclusion  and  Future  Work  I  have  written  above  the  implementation  techniques  I  used  to  create  a  cloth  simulation  with  a  focus  on  reproducing  realistic  physical  effects  generated  by  wind,  gravity,  and  collision.  The  results  of  the  simulation  produce  striking  imagery  and  animations  showcasing  the  effects  of  these  external  forces  when  enacted  on  the  cloth  patch.      Working  on  the  project  on  my  own,  while  challenging,  was  immeasurably  gratifying.  Having  received  an  underprivileged  education,  I  have  always  felt  intimidated  by  the  level  of  competency  of  the  rest  of  my  peers  in  the  class.  Being  able  to  do  this  project  on  my  own  and  successfully  opened  my  eyes  to  how  far  I’ve  gotten  since  at  the  beginning  of  this  course.  When  I  saw  my  cloth  patch  realistically  react  to  the  wind  and  gravity  I  realized  how  much  I  

enjoyed  what  I  have  been  doing  in  this  class,  and  I  want  to  continue  pursuing  other  projects  that  stimulate  me  on  the  same  level  as  this  experience  has.      Looking  into  the  future,  I’m  looking  forward  to  further  improving  my  cloth  simulation  by  solving  existing  technical  limitations.  I  am  interested  in  implementing  a  more  accurate  integration  method  as  well  as  more  complicated  physical  effects  such  as  cloth  tearing,  multiple  collisions  with  other  types  of  objects,  and  self-­‐intersection  collision  detection.    References    Baraff,  D.,  and  Witkin,  A.  1998.  Large  Steps  in  Cloth  Simulation.  Computer  Graphics  Proceedings,  Annual  Conference  Series,  SIGGRAPH  98,  43-­‐54.    Lander,  J.  1999.  Devil  in  the  Blue  Faceted  Dress:  Real-­‐time  Cloth  Animation.  Game  Developer  Magazine,  17-­‐22.  http://www.darwin3d.com/gamedev/articles/col0599.pdf    Lee,  S.  2008.  Real-­‐Time  Cloth  Simulation.  CG:  Skeelogy,  Skeel’s  CG,  Games  &  Tech  Portfolio.  http://cg.skeelogy.com/real-­‐time-­‐cloth-­‐simulation/    Mosegaard,  J.  2009.  Mosegaards  Cloth  Simulation  Coding  Tutorial.  Computer  Graphics  Lab  at  the  Alexandra  Institute.  http://cg.alexandra.dk/2009/06/02/mosegaards-­‐cloth-­‐simulation-­‐coding-­‐tutorial/    Rotenberg,  S.  2005.  Cloth  Simulation.  CSE  169:  Computer  Animation,  UCSD  Winter  2005.  graphics.ucsd.edu/courses/cse169_w05/CSE169_16.ppt      

     

 Figure  5-­‐7:  More  results  from  the  final  project.  Please  note  the  realistic  cloth  folds  and  

wrinkles  exhibit  as  a  result  of  wind  and  collision  effects.