(SGP 2006)Quadratic Bending Model

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A Quadratic Bending Model for Inextensible Surfaces(SGP 2006)

预备知识：

Laplace-Beltrami operator: 在微分几何中，拉普拉斯算子可以推广为定义在曲面，或更一般地黎曼流形与伪流形上，函数的算子。这个更一般的算子叫做Laplace-Beltrami operator

1. Introduction

​ Our contribution is to consider the class of isometric surface deformations, arriving at an expression for bending energy which is quadratic in positions.

• 此方法适合于stretching stiffness远大于bending stiffness的情况，即：主要应用于布料

1.1 Continuous setting

Consider the bending energy of a deformable surface \(S\):

\[E_b(S)=\frac12\int_SH^2dA \]

where

• \(H\): the mean curvature
• \(dA\): the differential area

then rewrite it as:

\[E_b(S)=\frac12\int_S\left<\Delta x,\Delta x\right>_{R^3}dA \]

where:

• \(\left<,\right>_{R^3}\): the inner product of \(R^3\).
• \(\Delta\): the Laplace-Beltrami
• \(H=\Delta x\).
• \(x\): the embedding of the surface

1.2 Central observation

• For inextensible surfaces, \(E_b(S)\) is quadratic in positions.

• \(E_b(S)\) together with the assumption of isometric deformation is called isometric bending model(IBM).

Our contribution is to present an analogous discrete IBM that is quadratic in positions. Its linear gradient and constant Hessian present an economic model for computing bending forces and their derivatives, enabling fast time-integration of cloth dynamics.

2. Discrete IBM

2.1 数学模型

Donoting the surface's vertex position vector by: \(x=(x_0,x_1,...,x_{n-1})^T\in R^{3n}\), then we write \(E_b(x)\) as:

\[E_b(x)=\frac12x^TQx=\frac12\sum_{i,j}Q_{ij}\left<x_i,x_j\right>_{R^3} \]

\(E_b(x)\) 需要满足的性质：

1. quadratic in \(x\) under isometric deformations(已满足)
2. invariant under rigid motions of the mesh
   • if and only if \(\sum_iQ_{ij}=\sum_jQ_{ij}=0\)
3. invariant under uniform scaling
   • \(Q\) must scale with \(1/s^2\) if the whole mesh is scaled by a global factor \(s\).
   • since \(E_b\) is an energy, \(Q\) must be positive semi-define.
   • we can then write: \(Q=L^TM^{-1}L\),
     • \(L\) is invariant under scaling and \(\sum_jL_{ij}=0\).
     • \(M\) is symmetric positive define and scales with \(s^2\).

A discrete IBM is then any energy of the form:

\[E_b(x)=\frac12x^T(L^TM^{-1}L)x=\frac12x^TQx \]

One way to obtain a suitable M and L is to discretize the smooth Laplacian, ∆, using the finite element (FE) method:

\[L_{ij}=\int_S\left<\nabla\Phi_i,\nabla\Phi_j\right>dA\\ M_{ij}=\int_S\Phi_i\cdot\Phi_jdA \]

where:

• \(\{\Phi_i\}\): some Finite Element basis
• \(L\): the Finite Element stiffness matrix is discrete Laplacian
• \(M^{-1}\): the Finite Element mass matrix inverse simplifies to division by area in a lumped mass matrix approximation
• \(Lx\): the discrete analogue of the smooth mean curvatur vector, \(\Delta x\).

2.2 Implementation

In a one-time precomputation step, the constant Hessian \(Q\), is assembled by considering contributions from each local matrix, \(Q(e_i)\), centered about interior edge \(e_i\) with stencil consisting of:

• the triangles, \(t_0,t_1\),
• their edges \(e_i\),
• their vertices, \(x_0,x_1,x_2,x_3\).

以上图为例：

\[Q(e_0)=\frac3{(A_0+A_1)}K^T_0K_0 \]

where:

• \(A_i\) the area of triangles \(t_i\).

• \(K_0\) is the row vector

• \[K_0=(c_{03}+c_{04},c_{01}+c_{02},-c_{01}-c_{03},-c_{02}-c_{04}) \]

• where \(c_{jk}=cot\angle e_j,e_k=1/tan\angle e_j,e_k\).

The local energy is obtained by:

\[E_b(e_i)=\frac12(x_0,x_1,x_2,x_4)Q_{e_i}(x_0,x_1,x_2,x_4)^T \]

The global (total) energy of the system is obtained by summing over all local contributions corresponding to interior edges.