Large Steps in Cloth Simulation - SIGGRAPH 98 박 강 수박 강 수

Cloth Simulation

Issues in Cloth Simulation Large time steps - stability Damping forces - oscillation Constraints - contact or fix conditions Solving a large sparse linear system - conjugate gradient iteration

Cloth and Mass-Spring Model Discrete cloth model

Differential Equation of Spring x : geometric state vector(position) M : mass distribution matrix of cloth E : scalar function of x (internal energy) F : other forces (air-drag, damping, contact)

Simulation Overview Notation and Geometry Position of world space Forces Planer coordinate

Simulation Overview Energy and Forces Internal forces - Stretch, Shear, Bending Damping force Combining all forces

Simulation Overview Sparse Matrices Very sparse system - n particles : n x n matrix - nonzero entry : dense 3x3 matrices of scalar Modified conjugate gradient iterative method

Implicit Integration Explicit forward Euler method

Implicit Integration(cont.) Implicit backward Euler method Nonlinear, need iteration By Taylor series expansion to f, first order approximation

Implicit Integration(cont.) Implicit backward Euler method Rewrite this approximated equation,

Implicit Integration(cont.) Taking the bottom row of below equation and substituting top row yields,

Implicit Integration(cont.) Letting I denote the identity matrix, and regrouping,

Forces The force f arising from energy E Impractical approach - Expressing E as a single monolithic function - Taking derivatives Batter approach - Decompose E into a sum of sparse energy functions

Forces But decomposing method is not enough. - Sensible damping function problem Instead, we define vector condition C(x) which is, - Formulating internal behavior - To be zero Define associated energy k is a stiffness constant

Forces (Forces & Force Derivatives) Block form of f Sparse matrix Derivative matrix K Sparse, Symmetric Matrix

Forces (Stretch Forces) Stretch force UV coordinates

Forces (Stretch Forces) Stretch force can be measured by Unstretched condition

Forces (Stretch Forces) i jk Approximate w(u,v) as a linear function over each triangle,

Forces (Stretch Forces) Stretch energy Usually, we set

Forces(Shear & Bend) Shear forceBending force Idea : Inner product Idea : angle of adjacent triangles

Forces(Damping) Strong stretch force ⇔ Strong damping force ☞ Prevent oscillation Damping force Eq: Damping directionDamping strength

Forces(Damping) Differentiate the damping eq. Asymmetric, Sparse Matrix breaks symmetry, so we omitted this term.

Constraints Unsuitable approaches - Reduced Coordinates - Penalty Methods - Lagrange Multipliers

Constraints(Mass Modification) xy-plane constraint : Generalization

Constraints(Mass Modification) Rewrite previous eq. z is change in velocity along the constrained direction

Constraints(Implementation) Multiply M, Symmetric(Positive definite) These two systems have a same solution Δv.

Constraints(Implementation) W is singular Two conditions : residual will be zero, Linear System Ax=b

Modified Conjugate Gradient Method

Collisions(Initiation) For collision detect, Coherency-based bounding-box approach is used. Penalty force(moving positions) “Jumpy” behavior in local regions.

Collisions(Position Alteration) Particle’s position in next step If collisions occur, Considering collisions,

Results

Thank you Questions || Comments ?

Conjugate gradient Method? Isn’t there more simple ways to implement mass-spring systems?