383 lines
10 KiB
Plaintext
383 lines
10 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Description\n",
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"\n",
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"This notebook intends to gather all the functionalities you'll have to implement for assignment 2.1. You will have to generate an elastic solid, deform it, compute the associated Jacobian of the deformation map $\\phi$, and implement pinning constraints. You will also visualize the eigenvectors and eigenvalues of the metric tensor, given a prescribed deformation.\n",
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"\n",
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"# Load libraries"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {},
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"outputs": [],
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"source": [
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"import numpy as np\n",
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"import igl\n",
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"import meshplot as mp\n",
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"\n",
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"import sys as _sys\n",
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"_sys.path.append(\"../src\")\n",
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"from elasticsolid import *\n",
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"from eigendecomposition_metric import *\n",
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"\n",
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"shadingOptions = {\n",
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" \"flat\":True,\n",
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" \"wireframe\":False, \n",
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"}\n",
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"\n",
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"rot = np.array(\n",
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" [[1, 0, 0 ],\n",
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" [0, 0, 1],\n",
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" [0, -1, 0 ]]\n",
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")"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Load mesh\n",
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"\n",
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"Several meshes are available for you to play with under `data/`: `ball.obj`, `dinosaur.obj`, and `beam.obj`. You can also uncomment the few commented lines below to manipulate a simple mesh made out of 2 tetrahedra."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {
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"scrolled": false
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},
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"outputs": [
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{
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"data": {
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"application/vnd.jupyter.widget-view+json": {
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"model_id": "33e8252ccc4d4c9298094b0dc3675ee4",
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"version_major": 2,
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"version_minor": 0
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},
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"text/plain": [
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"Renderer(camera=PerspectiveCamera(children=(DirectionalLight(color='white', intensity=0.6, position=(-1.987469…"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
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"v, _, _, t, _, _ = igl.read_obj(\"../data/dinosaur.obj\")\n",
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"\n",
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"# t = np.array([\n",
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"# [0, 1, 2, 3],\n",
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"# [1, 2, 3, 4]\n",
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"# ])\n",
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"# v = np.array([\n",
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"# [0., 0., 0.],\n",
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"# [1., 0., 0.],\n",
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"# [0., 1., 0.],\n",
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"# [0., 0., 1.],\n",
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"# [2/3, 2/3, 2/3]\n",
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"# ])\n",
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"\n",
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"aabb = np.max(v, axis=0) - np.min(v, axis=0)\n",
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"length_scale = np.mean(aabb)\n",
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"\n",
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"p = mp.plot(v @ rot.T, t, shading=shadingOptions)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Manipulate elastic solids\n",
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"\n",
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"## Instanciation\n",
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"\n",
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"The rest shape matrices $D_m$ and their inverse matrices $B_m$ are computed during instanciation."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"metadata": {},
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"outputs": [],
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"source": [
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"rho = 131 # [kg.m-3]\n",
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"solid = ElasticSolid(v, t, rho=rho)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Deform the mesh\n",
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"\n",
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"This part involves Jacobian computation which relies on deformed shape matrices $D_s$."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"metadata": {
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"scrolled": false
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},
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"outputs": [
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{
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"data": {
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"application/vnd.jupyter.widget-view+json": {
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"model_id": "02c94de8642645d8aab800877f8ddbce",
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"version_major": 2,
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"version_minor": 0
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},
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"text/plain": [
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"Renderer(camera=PerspectiveCamera(children=(DirectionalLight(color='white', intensity=0.6, position=(-1.987469…"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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},
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{
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"data": {
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"text/plain": [
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"<meshplot.Viewer.Viewer at 0x7fa7dc789070>"
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]
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},
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"execution_count": 4,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"v_def = v.copy()\n",
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"v_def[:, 2] *= 2.\n",
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"solid.update_def_shape(v_def)\n",
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"\n",
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"mp.plot(solid.v_def @ rot.T, solid.t, shading=shadingOptions)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Visualize some properties of the metric tensor\n",
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"\n",
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"The metric tensor measures how stretched and sheared directions in the undeformed space are under the deformation $\\phi$. It is defined from the Jacobian of the deformation $\\mathbf{F}$ as follow (see the handout for a derivation):\n",
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"\n",
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"$$\\mathbf{M} = \\mathbf{F}^T \\mathbf{F}$$\n",
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"\n",
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"We intend to plot the eigenvectors coloured by the corresponding eigenvalues in the next cell."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 11,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"application/vnd.jupyter.widget-view+json": {
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"model_id": "6ff42798692c4a29ad4a37f40ca77a84",
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"version_major": 2,
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"version_minor": 0
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},
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"text/plain": [
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"Renderer(camera=PerspectiveCamera(children=(DirectionalLight(color='white', intensity=0.6, position=(-1.987469…"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
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"# We limit ourselves to stretching the mesh in the z direction\n",
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"# Feel free to experiment with other kinds of deformations!\n",
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"\n",
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"v_def = v.copy()\n",
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"v_def[:, 2] *= 2.0\n",
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"solid.update_def_shape(v_def)\n",
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"\n",
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"squared_eigvals, eigvecs = compute_eigendecomposition_metric(solid.F)\n",
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"plot_eigendecomposition_metric(solid, squared_eigvals, eigvecs, rot, scale=0.05)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Pin vertices of the mesh\n",
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"\n",
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"Pass a `pin_idx` to the constructor, compute the mask for deformations."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"metadata": {},
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"outputs": [],
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"source": [
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"maxZ = np.max(solid.v_rest[:, 2])\n",
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"pin_idx = np.arange(solid.v_rest.shape[0])[solid.v_rest[:, 2] > maxZ - 0.1 * aabb[2]]\n",
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"\n",
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"v_def = v.copy()\n",
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"v_def[:, 2] -= 0.1 * aabb[2]\n",
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"solid.update_def_shape(v_def)\n",
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"\n",
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"solid_pinned = ElasticSolid(v, t, rho=rho, pin_idx=pin_idx)\n",
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"solid_pinned.update_def_shape(v_def)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"application/vnd.jupyter.widget-view+json": {
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"model_id": "13f958a33ec945ca979aca15d7381e01",
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"version_major": 2,
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"version_minor": 0
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},
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"text/plain": [
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"Renderer(camera=PerspectiveCamera(children=(DirectionalLight(color='white', intensity=0.6, position=(-1.987469…"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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},
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{
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"data": {
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"text/plain": [
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"1"
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]
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},
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"execution_count": 7,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"p = mp.plot(solid_pinned.v_def @ rot.T, t, shading=shadingOptions)\n",
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"p.add_points(solid_pinned.v_def[pin_idx, :] @ rot.T, shading={\"point_color\":\"black\", \"point_size\": 0.1 * length_scale})"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 1.2.4 Derivation"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Volume of tetrahedron ($X_1, X_2, X_3, X_4$) is given by $\\frac{1}{6}|det(D_m)|$\n",
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"\n",
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"We can decompose the provided formula:\n",
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"\n",
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"$Vol = \\frac{1}{6}|det(D_m)| = \\frac{1}{6}|D_{m1}^T \\cdot (D_{m2} \\times D_{m3})|$\n",
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"\n",
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"\n",
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"Next, we build 3 vectors:\n",
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"\n",
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"$V_0 = X_1 - X_4 = \\begin{pmatrix}X_{x1}-X_{x4} \\\\ Y_{x1}-Y_{x4} \\\\ Z_{x1}- Z_{x4} \\end{pmatrix}$\n",
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"\n",
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"$V_1 = X_2 - X_4 = \\begin{pmatrix}X_{x2}-X_{x4} \\\\ Y_{x2}-Y_{x4} \\\\ Z_{x2}- Z_{x4} \\end{pmatrix}$\n",
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"\n",
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"$V_2 = X_3 - X_4 = \\begin{pmatrix}X_{x3}-X_{x4} \\\\ Y_{x3}-Y_{x4} \\\\ Z_{x3}- Z_{x4} \\end{pmatrix}$\n",
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"\n",
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"We then compute the base as:\n",
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"\n",
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"$B = V_1 \\times V_2$\n",
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"\n",
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"And also the height as:\n",
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"\n",
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"$H = V_0$\n",
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"\n",
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"From this we can compute the volume:\n",
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"\n",
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"$Vol = \\frac{1}{6} \\cdot B \\cdot H = \\frac{1}{6} V_0 \\cdot (V_1 \\times V_2)$\n",
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"\n",
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"We can note the similarity between both fomulas. We can note the equality between both formulas: \n",
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"\n",
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"$V_0 = D_{m1}, V_1 = D_{m2}, V_2 = D_{m3}$\n",
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"\n",
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"Note that the absolute value is not directly taken into account since its absolute volume and not signed volume.\n",
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"\n",
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"\n",
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"\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 1.2.7 Derivation"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Show that: $Vol(x_1, x_2, x_3, x_4)/Vol(X_1, X_2, X_3, X_4) = |det(F)|$\n",
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"\n",
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"Since we know that $F = D_s D_m^{-1}$, and that: $V = \\frac{1}{6}|det(D_m)|$\n",
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"\n",
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"We can drag the det and abs into it:\n",
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"\n",
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"$|det(F)| = \\frac{|det(D_s)|}{|det(D_m)|} = \\frac{\\frac{1}{6}|det(D_s)|}{\\frac{1}{6}|det(D_m)|} = \\frac{Vol(x_1, x_2, x_3, x_4)}{Vol(X_1, X_2, X_3, X_4)}$\n",
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"\n",
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"\n",
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"\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": []
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}
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],
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"display_name": "Python 3 (ipykernel)",
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"pygments_lexer": "ipython3",
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"version": "3.9.7"
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},
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"toc": {
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