### Citations

2285 |
Iterative Methods for Sparse Linear Systems,
- Saad
- 1996
(Show Context)
Citation Context ...rict the discussion to the most important iterative method used, the method of conjugate gradients. A more profound discussion can be found in the standard texts of Trefthen[25], Greenbaum[11] or Saad=-=[23]-=-. It uses only matrix-vector multiplications to solve a symmetric system of the form Ax = b, (51)s14 SIGGRAPH 2003 Course #29: Clothing Simulation and Animation with A ∈ IR N×N ,x,b ∈ IR N . Thus the ... |

883 | Elastically deformable models.
- Terzopoulos, Platt, et al.
- 1987
(Show Context)
Citation Context ...ous cases, leading to an implicit ODE Mx ′ 2 = f(x)(3) with a non-diagonal and even singular mass matrix M. A common technique is to use mass lumping, to make M diagonal and reduce computational costs=-=[24, 8]-=-. For the following discussion we assume that the ODE is given in the general explicit form x ′ = f(x)or x ′′ = f(x, x ′ )(4) 2 Methods for Numerical Integration As we have seen in the previous sectio... |

632 |
Solving Ordinary Differential Equations I. Nonstiff Problems,
- Hairer, NZrsett, et al.
- 1987
(Show Context)
Citation Context ...he numerical approximation Y1 produced by an explicit Euler step y(t1) − Y1 = O(h 2 ). (12) If we continue the method using the numerical solution Y1 as a starting value for the next time step we lose=-=[13]-=- apowerofh for the global error y(tn) − Yn = O(h). (13) This means that the explicit Euler method converges linearly or has order 1. We will analyze the stability and efficiency of the method later. A... |

579 | Large steps in cloth simulation,"
- Baraff, Witkin
- 1998
(Show Context)
Citation Context ...7 0.8 0.9 1 Figure 1: Solutions of example 1 for λ = 2 (dashed) and λ = −15 (solid) Throughout the following discussion we will use the following examples: 1. y ′ = λy, y(0)= 1 with λ =2, −15 for t ∈ =-=[0, 1]-=- (figure 1). 2. The overdamped wave equation y ′′ = λ/2y + λy ′ with λ = −5 fort∈ [0.10] and starting values y(0)= 0, y ′ (0)= 1. It has the analytical solution y(t) = 1/15 √ 15e 1/2 (−5+√15)t − 1/15 ... |

449 | discrete differentialgeometry operators for triangulated 2-manifolds,” in
- Meyer, Desbrun, et al.
- 2002
(Show Context)
Citation Context ...difference formulations. This replacement is easily accomplished in 1-space or on structured grids in any dimension. It becomes harder to define finite difference approximations on unstructured meshes=-=[18]-=-, often finite element techniques are used for deriving appropriate schemes[19]. An important trait of finite difference schemes is, that the terms on the right hand side of the ODE are given for a po... |

447 |
Iterative Methods for Solving Linear Systems,
- Greenbaum
- 1997
(Show Context)
Citation Context ...tem. We restrict the discussion to the most important iterative method used, the method of conjugate gradients. A more profound discussion can be found in the standard texts of Trefthen[25], Greenbaum=-=[11]-=- or Saad[23]. It uses only matrix-vector multiplications to solve a symmetric system of the form Ax = b, (51)s14 SIGGRAPH 2003 Course #29: Clothing Simulation and Animation with A ∈ IR N×N ,x,b ∈ IR N... |

446 |
Numerical Linear Algebra.
- Trefethen, Bau
- 1997
(Show Context)
Citation Context ...g nonlinear system. We restrict the discussion to the most important iterative method used, the method of conjugate gradients. A more profound discussion can be found in the standard texts of Trefthen=-=[25]-=-, Greenbaum[11] or Saad[23]. It uses only matrix-vector multiplications to solve a symmetric system of the form Ax = b, (51)s14 SIGGRAPH 2003 Course #29: Clothing Simulation and Animation with A ∈ IR ... |

255 | Deformation constraints in a mass-spring model to describe rigid cloth behaviour.
- Provot
- 1995
(Show Context)
Citation Context ...cal properties are specified by directly defining forces between these mass points. Typical representatives of this approach, that is very popular in cloth simulations, are mass-spring-damper systems =-=[21, 26]-=- and particle systems with forces defined directly by measured curves[6] or low order polynomial fits of this data[2]. 1.2 Finite Differences Another physical modelling concept is to specify physical ... |

176 | H.-S.: Stable but responsive cloth.
- CHOI, KO
- 2002
(Show Context)
Citation Context ...or solving the linear equations. Again this may lead to artificial slowdowns. Recently more advanced methods gain importance. Hauth et. al.[15] used BDF and the implicit midpoint rule, and Choi and Ko=-=[3]-=- alsousedBDF,bothcombining it with an iterative cg solver. In Hauth et. al. [16] there the complete BDF-2 algorithm, including variable step sizes, is presented in pseudocode, derived in the presented... |

158 | Interactive animation of structured deformable objects,".
- Desbrun, Schröder, et al.
- 1991
(Show Context)
Citation Context ...ow downs as observed by Volino[28] and Eberhardt[7]. Provot[21] proposed a simple model only incorporating linear springs, combining it with an explicit method. This model was used by Desbrun et. al. =-=[5]-=- who also use only a linearized implicit method. But instead of linearizing the whole system they split it in a linear and nonlinear part and use a precomputed inverse of A for solving the linear part... |

144 | Predicting the Drape of Woven Cloth Using Interacting Particles.
- Breen, House, et al.
- 1994
(Show Context)
Citation Context ...ch, that is very popular in cloth simulations, are mass-spring-damper systems [21, 26] and particle systems with forces defined directly by measured curves[6] or low order polynomial fits of this data=-=[2]-=-. 1.2 Finite Differences Another physical modelling concept is to specify physical behavior by minimizing some energy functionals defined on a continuous solid. The arising October 29, 2003 1 equation... |

118 |
Solving Ordinary Differential Equations II,
- Hairer, Wanner
- 2010
(Show Context)
Citation Context ...hf t + h/2, Y1 + Y0 2 � , (32) using a simplified notation for advancing one step, i.e. writing Y0 and Y1 instead of Yn and Yn+1. Alternatively the midpoint rule can be derived as a collocation method=-=[12]-=- withs = 1 internal nodes, i.e. by constructing a polynomial interpolating the particle trajectories at a given, fixed set of s nodes[12]. This idea allows for the construction of implicit Runge-Kutta... |

108 |
Flexible, Particle-System Model for Cloth Draping,”
- Eberhardt, Weber, et al.
- 1996
(Show Context)
Citation Context ...oints. Typical representatives of this approach, that is very popular in cloth simulations, are mass-spring-damper systems [21, 26] and particle systems with forces defined directly by measured curves=-=[6]-=- or low order polynomial fits of this data[2]. 1.2 Finite Differences Another physical modelling concept is to specify physical behavior by minimizing some energy functionals defined on a continuous s... |

70 | Implementing fast cloth simulation with collision response,”
- Volino, Magnenat-Thalmann
- 2000
(Show Context)
Citation Context ...is introduces a lot of non-zero elements into the factors. Although reordering techniques alleviate the effects, this approach may be too expensive for the present application. The majority of authors=-=[1, 27, 15, 3]-=- use iterative methods to solve the linear system. We will also use the conjugate gradient method here to solve the linear systems in each Newton step. Unfortunately this changes the convergence behav... |

58 |
Methods for Solving Systems of Nonlinear Equations.
- Rheinboldt
- 1998
(Show Context)
Citation Context ... gradient method here to solve the linear systems in each Newton step. Unfortunately this changes the convergence behaviour of the outer Newton method, which is referred to as an inexact Newton method=-=[22]-=-, given by algorithm 2. Algorithm 2: Inexact Newton Method explicit Euler for evaluation of correctness. Does the solution suddenly diverge for h>h ? crit Yes No System is supposed to be non-stiff. Is... |

49 | A high performance solver for the animation of deformable objects using advanced numerical methods,”
- Hauth, Etzmuss
- 2001
(Show Context)
Citation Context ...ore they just do a single iteration of a Jacobi-like scheme for solving the linear equations. Again this may lead to artificial slowdowns. Recently more advanced methods gain importance. Hauth et. al.=-=[15]-=- used BDF and the implicit midpoint rule, and Choi and Ko[3] alsousedBDF,bothcombining it with an iterative cg solver. In Hauth et. al. [16] there the complete BDF-2 algorithm, including variable step... |

39 | Implicit-explicit schemes for fast animation with particle systems
- Eberhardt, Etzmu, et al.
- 2000
(Show Context)
Citation Context ...ion of a nonlinear system with only one Newton iteration. Because the nonlinear part is not integrated, with high stiffness one may encounter similar slow downs as observed by Volino[28] and Eberhardt=-=[7]-=-. Provot[21] proposed a simple model only incorporating linear springs, combining it with an explicit method. This model was used by Desbrun et. al. [5] who also use only a linearized implicit method.... |

37 |
Fast cloth animation on walking avatars
- Vassilev, Spanlang, et al.
- 2001
(Show Context)
Citation Context ...cal properties are specified by directly defining forces between these mass points. Typical representatives of this approach, that is very popular in cloth simulations, are mass-spring-damper systems =-=[21, 26]-=- and particle systems with forces defined directly by measured curves[6] or low order polynomial fits of this data[2]. 1.2 Finite Differences Another physical modelling concept is to specify physical ... |

32 |
Analysis of numerical methods for the simulation of deformable models
- Hauth, Etzmuß, et al.
- 2003
(Show Context)
Citation Context ...ecently more advanced methods gain importance. Hauth et. al.[15] used BDF and the implicit midpoint rule, and Choi and Ko[3] alsousedBDF,bothcombining it with an iterative cg solver. In Hauth et. al. =-=[16]-=- there the complete BDF-2 algorithm, including variable step sizes, is presented in pseudocode, derived in the presented framework. 2.7 Selectingan efficient method Which method is best for a certain ... |

29 | Computational aspects of discrete minimal surfaces, Global Theory of Minimal Surfaces
- Polthier
- 2005
(Show Context)
Citation Context ...on structured grids in any dimension. It becomes harder to define finite difference approximations on unstructured meshes[18], often finite element techniques are used for deriving appropriate schemes=-=[19]-=-. An important trait of finite difference schemes is, that the terms on the right hand side of the ODE are given for a point and its neighbors, i.e. are specified on the edges of the discretization. T... |

28 |
MAGNENAT-THALMANN N.: Comparing efficiency of integration methods for cloth simulation
- VOLINO
(Show Context)
Citation Context ...ponds to the solution of a nonlinear system with only one Newton iteration. Because the nonlinear part is not integrated, with high stiffness one may encounter similar slow downs as observed by Volino=-=[28]-=- and Eberhardt[7]. Provot[21] proposed a simple model only incorporating linear springs, combining it with an explicit method. This model was used by Desbrun et. al. [5] who also use only a linearized... |

26 |
Stiff Differential Equations Solved by Radau Methods,”
- Hairer, Wanner
- 1999
(Show Context)
Citation Context ...achieves a faster convergence. On the other hand, the number of Newton iterations necessary is a criteria for the behaviour of the integrator and has already been used for an order selection algorithm=-=[14]-=- of the Radau integrator. In our implementation the number of Newton iterations decides whether to increase or decrease the time step h. 4 Linear Systems In a last section we briefly present the last ... |

23 |
Deriving a particle system from continuum mechanics for the animation of deformable objects.
- Etzmuss, Gross, et al.
- 2003
(Show Context)
Citation Context ...iscretization. The resulting equations are structured very similar to these from particle systems, indeed finite difference techniques can be used to derive a particle system from continuous equations=-=[9]-=-. 1.3 Finite Elements Finite elements, in their beginning, were designed to overcome the difficulties of finite differences with unstructured meshes. They also start with a continuous model, usually g... |

22 | On the use of two QMR algorithms for solving singular systems and applications in Markov chain modeling, RIACS - Freund, Hochbruck - 1991 |

17 | C.: Real-time animation technique for flexible and thin objects
- KANG, CHOI, et al.
(Show Context)
Citation Context ...Instead the angular momentum is corrected to account for the nonlinear part. With this algorithm one can neither change the stepsize h nor deal with an A depending on t. Based on this work Kang et al.=-=[17]-=-didsomefurther simplification to avoid solving the linear system. In order to update the solution vector in one step they divide by diagonal entry of the matrix of the linear system. Therfore they jus... |

15 |
Nadia Magnenat Thalmann. Versatile and efficient techniques for simulating cloth and other deformable objects
- Courshesnes, Volino
- 1995
(Show Context)
Citation Context ...the arising ODE. Later publications focused on explicit integration methods, e.g. Eberhardt et al.[6] preferred RK4 and the Burlisch-Stoer extrapolation method as suggested in Press et al.[20]. Volino=-=[4]-=- used an explicit midpoint rule. Implicit methods again became popular with the work of Baraff and Witkin. They used a linearized implicit Euler method and achieved simulations about an order of magni... |

1 |
Finite-element modeling and SIGGRAPH 2003 Course #29: Clothing Simulation and Animation control of flexible fabric parts
- Eischen, Deng, et al.
- 1996
(Show Context)
Citation Context ...ous cases, leading to an implicit ODE Mx ′ 2 = f(x)(3) with a non-diagonal and even singular mass matrix M. A common technique is to use mass lumping, to make M diagonal and reduce computational costs=-=[24, 8]-=-. For the following discussion we assume that the ODE is given in the general explicit form x ′ = f(x)or x ′′ = f(x, x ′ )(4) 2 Methods for Numerical Integration As we have seen in the previous sectio... |