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## Adaptive Nonlinear Finite Elements for Deformable Body Simulation Using Dynamic Progressive Meshes (2001)

Venue: | Computer Graphics Forum |

Citations: | 85 - 3 self |

### Citations

820 |
Biomechanics: Mechanical Properties of Living Tissues,
- Fung
- 1993
(Show Context)
Citation Context ...the strong form, since the equilibrium law is satisfied at every particle in domain Ω. The weak form used in FEM is obtained by satisfying the strong form in an integral manner: � Ωe � δΦdΩ ==-= Ωe δWdΩ (7) w-=-here δ is the variation operator and Φ is the internal deformation energy or strain energy density function (SEDF). W is the external energy generated by the external forces f , i.e., body force, su... |

777 | Applied Numerical Linear Algebra.
- Demmel
- 1997
(Show Context)
Citation Context ...∂x + � � (4) ∂u ∂u ∂v ∂v ∂w ∂w + + ∂x ∂y ∂x ∂y ∂z ∂y The equation of motion is derived through the Principle of Virtual Work equation in Lagrangian form: in domain Ω, wh=-=ere δΦ(C) = δW (5) δW = f T δu (6) -=-In FEM, this equation is called the strong form, since the equilibrium law is satisfied at every particle in domain Ω. The weak form used in FEM is obtained by satisfying the strong form in an integr... |

266 | Smooth view-dependent level-of-detail control and its application to terrain rendering”.
- Hoppe
- 1998
(Show Context)
Citation Context ...all · is a tensor product or component product to guarantee δΦ is a scalar. Now, the variation of Cauchy-Green tensor C is where δC = (δF) T F + F T (δF) = 2F T δF = δ ∂x ∂(X + u) = δ = �=-=� ∂X ∂X ∂u ∂X (9) � �s ∂�-=-�u F (10) ∂X ∂δu = F (11) ∂x F in this equation is called the deformation gradient. It measures the deformation between the current configuration x and initial configuration X. Because we use t... |

217 | ArtDefo: Accurate Real Time Deformable Objects,”
- L, Pai
- 1999
(Show Context)
Citation Context ...or product or component product to guarantee δΦ is a scalar. Now, the variation of Cauchy-Green tensor C is where δC = (δF) T F + F T (δF) = 2F T δF = δ ∂x ∂(X + u) = δ = δ ∂X ∂X ∂u=-= ∂X (9) � �s ∂δu F (10) ∂X �-=-��δu = F (11) ∂x F in this equation is called the deformation gradient. It measures the deformation between the current configuration x and initial configuration X. Because we use the Lagrangian fo... |

208 | Real-time elastic deformations of soft tissues for surgery simulation,"
- Cotin, Delingette, et al.
- 1999
(Show Context)
Citation Context ...nsiders both material nonlinearity and large/nonlinear geometric deformation, represented by quadratic strain, e.g.: εx = ∂u ∂x + 1 � 2 ( ∂u ∂x )2 + ( ∂u ∂y )2 + ( ∂u ∂z )2� γxy =-== ∂u ∂v + ∂y ∂x + � � (4) ∂u ∂u ∂v ∂v ∂w ∂w-=- + + ∂x ∂y ∂x ∂y ∂z ∂y The equation of motion is derived through the Principle of Virtual Work equation in Lagrangian form: in domain Ω, where δΦ(C) = δW (5) δW = f T δu (6) In FEM, ... |

169 | Progressive simplicial complexes
- Popovic, Hoppe
- 1997
(Show Context)
Citation Context ... configuration is the sum of its initial position and its displacement. The subscript s indicates the symmetric part of that matrix. δΦ can be rewritten as δΦ = � 2F ∂Φ ∂C FT � · � ∂=-= ∂X � s δu = TBδu (12) � �-=-s ∂ where B is the symmetric differential operator ∂X . Since the variation of u can be anything, the weak form of (12) issWu, Downes, Goktekin, & Tendick / Adaptive Nonlinear Finite Elements Usin... |

47 | Multirate simulation for high fidelity haptic interaction with deformable objects in virtual environments.
- Cavusoglu, Tendick
- 2000
(Show Context)
Citation Context ...in in the velocity field is viscosity, which appears when there is creep during deformation. We do not yet consider its effect in this paper. Thus the resulting system equation is Mü + D˙u + R(u) = =-=0 (3)-=- which is similar to the linear approach in equation 2 except that the stiffness term is no longer linear in the displacement field. Solving the full system is extremely timeand storage-consuming. Sze... |

42 |
An Introduction to the Finite Element Method 2nd ed.,
- Reddy
- 1993
(Show Context)
Citation Context ...nce the variation of u can be anything, the weak form of (12) issWu, Downes, Goktekin, & Tendick / Adaptive Nonlinear Finite Elements Using Dynamic Progressive Meshes � Ω e � T δΦdΩ = δu Ωe =-=(TB) T dΩ (13) The external v-=-irtual work of each element comes from its inertia force ρeü T , damping force βe ˙u T , body force ρedΩ e g[0 0 − 1], and surface traction force f e : δW = δWI + δWD + δWB + δWT (14) �... |

22 | Anisotropic elasticity and force extrapolation to improve realism of surgery simulation.
- Picinbono, Lombardo, et al.
- 2000
(Show Context)
Citation Context ...mponent product to guarantee δΦ is a scalar. Now, the variation of Cauchy-Green tensor C is where δC = (δF) T F + F T (δF) = 2F T δF = δ ∂x ∂(X + u) = δ = δ ∂X ∂X ∂u ∂X (9) � ��=-=�s ∂δu F (10) ∂X ∂δu = F (11) �-=-��x F in this equation is called the deformation gradient. It measures the deformation between the current configuration x and initial configuration X. Because we use the Lagrangian formulation, the c... |

18 |
Fast finite elements for surgery simulation
- Bro-Nielsen
- 1997
(Show Context)
Citation Context ...x + ∂u ∂z where ε’s and γ’s are components of strain and u, v, and w are displacements in x, y, and z. Using the linear formulation, the resulting system equations have the form (1) Mü + D�=-=�u + Ku = f (2)-=- where the mass matrix M, the damping matrix D and the stiffness matrix K are constant. These matrices are generally sparse and have size N 2 . u is an N-dimensional vector of the discrete displacemen... |

8 |
Measuring
- Brouwer, Ustin, et al.
(Show Context)
Citation Context ...umbers and store onto nodes of the hierarchy tree of DPM offline. For instance, in Figure 3, before edge collapse, J0[0] is stored in an array for the corresponding face. After Vs −Vt is collapsed, =-=J0[1]-=- is pushed into the array. During online simulation, we use the prestored J0[1] if there is not local refinement. When Vpar is split, we extract J0[0] from that array without recomputing it. During th... |

1 |
c○ The Eurographics Association and Blackwell Publishers 2001. Adaptive simulation of soft bodies in real-time
- Debunne, Desbrun, et al.
- 2000
(Show Context)
Citation Context ...xy = ∂u ∂v + ∂y ∂x + � � (4) ∂u ∂u ∂v ∂v ∂w ∂w + + ∂x ∂y ∂x ∂y ∂z ∂y The equation of motion is derived through the Principle of Virtual Work equation in Lagrangian =-=form: in domain Ω, where δΦ(C) = δW (5) δ-=-W = f T δu (6) In FEM, this equation is called the strong form, since the equilibrium law is satisfied at every particle in domain Ω. The weak form used in FEM is obtained by satisfying the strong f... |

1 |
Mesh generation for domains with
- Shewchuk
(Show Context)
Citation Context ...Ω (13) The external virtual work of each element comes from its inertia force ρeü T , damping force βe ˙u T , body force ρedΩ e g[0 0 − 1], and surface traction force f e : δW = δWI + δWD =-=+ δWB + δWT (14) � = ρeü -=-T + βe ˙u T + ρedΩ e � T g[0 0 − 1] + f δu We currently model a linear viscous resistance as our friction force, but a nonlinear friction force could be added with ease. The body force is gra... |