Geometric Computing

Mark Pauly 教授跟 Geometric Computing 應該算是我想來 EPFL 的原因之一。

我總覺得 Geometric Computing 是電腦圖學的反義詞；圖學用電腦創造虛擬的世界，Geometric Computing 則在模擬真正的現實。

3.4.2 Gradient Descent with Backtracking line-search

• Dinasour

  • Parameter
    w = 10, obj_tol = 1E-10, theta = 1.0, beta = 0.5, c = 0.01
  • Result
    hit tolerance @ iter #17110 with energy 1.36E-4
    
  • Objective function trend chart
    

• Dance

  • Parameter
    w = 7, obj_tol = 1E-10, theta = 1.0, beta = 0.5, c = 0.01
  • Result
    hit tolerance @ iter #29392 with energy 9.95E-5
    
  • Objective function trend chart
    

• Gym

  • Parameter
    w = 10, obj_tol = 1E-10, theta = 1.0, beta = 0.5, c = 0.02
  • Result
    hit tolerance @ iter #35012 with energy 3.11E-5
    
  • Objective function trend chart
    

3.4.3 BFGS

• Dinasour

  • Parameter
    w = 10, obj_tol = 1E-10, theta = 1.0, beta = 0.5, c = 0.02
  • Result
    hit tolerance @ iter #337 with energy 1.31E-4
    
  • Objective function trend chart
    

• Dance

  • Parameter
    w = 13, obj_tol = 1E-10, theta = 1.0, beta = 0.5, c = 0.02
  • Result
    hit tolerance @ iter #494 with energy 2.01E-5
    
  • Objective function trend chart
    

• Gym

  • Parameter
    w = 10, obj_tol = 1E-10, theta = 1.0, beta = 0.5, c = 0.02
  • Result
    hit tolerance @ iter #460 with energy 2.75E-5
    
  • Objective function trend chart
    

3.4.4 My Shape

• Cat

  • Parameter
    w = 44, obj_tol = 1E-10, theta = 0.1, beta = 0.5, c = 0.02
  • Result
    hit tolerance @ iter #1433 with energy 1E-3
    
  • Objective function trend chart
    

• Wu-Jing (Turnip-Head)

  • Parameter
    w = 9, obj_tol = 1e-10, theta = 0.1, beta = 0.5, c = 0.01
  • Result
    hit tolerance @ iter #1625 with energy 2.8e-6
    
  • Objective function trend chart
    

• Helicopter

  • Parameter
    w = 150, obj_tol = 1e-9, theta = 0.5, beta = 0.5, c = 0.01
  • Result
    hit tolerance @ iter #1367 with energy 4.21e-4
    
  • Objective function trend chart
    

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Laser Cutting

I choosed Wu-Jing (Turnip-Head) for cutting. However, the initial support line is too short for inserting a foot, so I made another one which is stronger.

3.5.4 Equilibrium

• Neo-Hookean elasticity model is used

Beam

• $\rho$ = $131$
• young = $3 \times 10^8$
• poisson = $0.2$
• iteration time = $1000$
• initial step size =
  • $10^{-7}$ (Gradient Descent)
  • $10^{-8}$ (BFGS)

Linear Elasticity model

Gradient Descnet	BFGS	NGC

Residuals	Total Energy

Neo-Hookean Elasticity model

Gradient Descnet	BFGS	NGC

Residuals	Total Energy

Dinosaur

• $\rho$ = $131$
• young = $3 \times 10^8$
• poisson = $0.2$
• iteration time = $1000$
• initial step size =
  • $10^{-8}$ (Gradient Descent)
  • $5 \times 10^{-9}$ (BFGS)

Linear Elasticity model

Gradient Descnet	BFGS	NGC

Residuals	Total Energy

Neo-Hookean Elasticity model

Gradient Descnet	BFGS	NGC

Residuals	Total Energy

Ball

• $\rho$ = $131$
• young = $10^6$
• poisson = $0.2$
• iteration time = $1000$
• initial step size =
  • $10^{-7}$ (Gradient Descent)
  • $10^{-8}$ (BFGS)

Linear Elasticity model

Gradient Descnet	BFGS	NGC

Residuals	Total Energy

Neo-Hookean Elasticity model

Gradient Descnet	BFGS	NGC

Residuals	Total Energy

Beam

Without point load

• Mesh
  
• Objective
  

With point load

• Mesh
  

Dino

Without point load

• Mesh
  
• Objective
  

With point load

• Mesh
  

Mean Curvature Flow

Two Rings

$(\text{Max iter}, \epsilon_1, \epsilon_2) = (1000, 5 \times 10^{-2}, 10^{-3})$

Average Mean Curvature	Final Mini

Half Cube

$(\text{Max iter}, \epsilon_1, \epsilon_2) = (1000, 5 \times 10^{-2}, 10^{-3})$

Average Mean Curvature	Final Mini

Cube

$(\text{Max iter}, \epsilon_1, \epsilon_2) = (2, 1, 1)$

Average Mean Curvature	Final Mini

My Shape

$(\text{Max iter}, \epsilon_1, \epsilon_2) = (300, 0.5, 1e-2)$

Average Mean Curvature	Final Mini

Fitting Output

Bob

Gaussian curvature	Mean curvature

Principal curvature directions	Asymptotic directions

Bob

Gaussian curvature	Mean curvature

Principal curvature directions	Asymptotic directions