# Homework 1: 3D FEM (Explicit, Jacobi and conjugate gradient method)

## Introduction

An implementation of 3D linear FEM based on Neohookean elasticity model (No damping force).

## Simulation Results

Using Explicit, Jacobi and CG method respectively (Rendered by Blender).

## Analysis

For Explicit method, there is no constraints between nodes when updating, so the system explodes if time step or Young’s modulus is large.

For Jacobi method, the convergence is determined by the spectral radius \rho of the matrix -D^{-1}(L+U) , Jacobi method converges if and only if \rho is less than 1.

For CG method, the system always converges since the coefficient matrix A is positive definite for (solid) FEM problems.

### Same Scene

Explicit and Jacobi method have to update many times using a small time step at each frame.

\rho = 0.0036

### Large Time Step

\rho = 1.09

### Large Young’s Modulus

\rho = 1.13

### Problems of Conjugate Gradient Method

1. Object rotates slowly when time step is large.

This problem is caused by the lack of damping force, add damping force to object can avoid this problem.

1. CG doesn’t converge when both time step and Young’s modulus are large.

I still don’t know why CG doesn’t converge in this condition, I guess this is caused by numerical accuracy.

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hi, @tianlajiangjun ，dP/dF的推导在我提到的参考文献 The classical FEM method and discretization methodology 这篇文章中有介绍。

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