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||||
File diff suppressed because it is too large.
Load diff
+254
@@ -0,0 +1,254 @@
|
||||
{
|
||||
"cells": [
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Description\n",
|
||||
"\n",
|
||||
"This notebook shows a way to solve an inverse problem parameterized with a single quantity.\n",
|
||||
"\n",
|
||||
"# Load libraries"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 2,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"import numpy as np\n",
|
||||
"import igl\n",
|
||||
"import meshplot as mp\n",
|
||||
"import time\n",
|
||||
"\n",
|
||||
"import sys as _sys\n",
|
||||
"_sys.path.append(\"../src\")\n",
|
||||
"from elasticsolid import *\n",
|
||||
"from elasticenergy import *\n",
|
||||
"from stretch_utils import *\n",
|
||||
"from matplotlib import gridspec\n",
|
||||
"import matplotlib.pyplot as plt\n",
|
||||
"\n",
|
||||
"shadingOptions = {\n",
|
||||
" \"flat\":True,\n",
|
||||
" \"wireframe\":False, \n",
|
||||
"}\n",
|
||||
"\n",
|
||||
"rot = np.array(\n",
|
||||
" [[1, 0, 0 ],\n",
|
||||
" [0, 0, 1],\n",
|
||||
" [0, -1, 0 ]]\n",
|
||||
")"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Load mesh\n",
|
||||
"\n",
|
||||
"We will use the ball for the current experiment."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 3,
|
||||
"metadata": {
|
||||
"scrolled": false
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"v, _, _, t, _, _ = igl.read_obj(\"../data/ball.obj\")\n",
|
||||
"\n",
|
||||
"be = igl.edges(igl.boundary_facets(t))\n",
|
||||
"bv = np.unique(igl.boundary_facets(t))\n",
|
||||
"e = igl.edges(t)\n",
|
||||
"\n",
|
||||
"aabb = np.max(v, axis=0) - np.min(v, axis=0)\n",
|
||||
"length_scale = np.mean(aabb)\n",
|
||||
"\n",
|
||||
"# p = mp.plot(v @ rot.T, t, shading=shadingOptions)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Linear/Non-Linear Elastic Solid\n",
|
||||
"\n",
|
||||
"## Instantiation\n",
|
||||
"\n",
|
||||
"We first specify the elasticity model to use for the elastic solid, as well as pinned vertices, and volumetric forces."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 4,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"rho = 131 # [kg.m-3]\n",
|
||||
"young = 7e5 # [Pa] \n",
|
||||
"poisson = 0.2\n",
|
||||
"force_mass = np.zeros(shape = (3,))\n",
|
||||
"force_mass[2] = - rho * 9.81\n",
|
||||
"\n",
|
||||
"# Determines the pinned indices and the lowest pinned indext\n",
|
||||
"maxZ = np.max(v[:, 2])\n",
|
||||
"pin_idx = np.arange(v.shape[0])[v[:, 2] > maxZ - 0.1*aabb[2]]\n",
|
||||
"lowest_pin_idx = np.argsort(v[:, 2])[-pin_idx.shape[0]-1]\n",
|
||||
"\n",
|
||||
"ee = NeoHookeanElasticEnergy(young, poisson)\n",
|
||||
"solid = ElasticSolid(v, t, ee, rho=rho, pin_idx=pin_idx, f_mass=force_mass)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Manually search for a solution"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 5,
|
||||
"metadata": {
|
||||
"scrolled": true
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"n_search = 50\n",
|
||||
"n_steps = 20\n",
|
||||
"thresh = 1.\n",
|
||||
"\n",
|
||||
"v_rest_init = v.copy()\n",
|
||||
"v_rest_tmp = v.copy()\n",
|
||||
"v_target = v.copy()[bv]\n",
|
||||
"\n",
|
||||
"stretches = np.linspace(0.55, 1.1, n_search)\n",
|
||||
"\n",
|
||||
"list_v_rest, list_v_eq, target_closeness = report_stretches(solid, v_rest_init, bv, v_target, stretches, lowest_pin_idx)\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 6,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light"
|
||||
},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"plt.plot(stretches, target_closeness)\n",
|
||||
"plt.title(\"Closeness to target\", fontsize=14)\n",
|
||||
"plt.xlabel(\"Stretch along $z$\", fontsize=12)\n",
|
||||
"plt.ylabel(\"Objective\", fontsize=12)\n",
|
||||
"plt.grid()\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Plot best solution"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 7,
|
||||
"metadata": {
|
||||
"scrolled": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"application/vnd.jupyter.widget-view+json": {
|
||||
"model_id": "665a8bf8e742437bb4b51c4b3d384c4c",
|
||||
"version_major": 2,
|
||||
"version_minor": 0
|
||||
},
|
||||
"text/plain": [
|
||||
"Renderer(camera=PerspectiveCamera(children=(DirectionalLight(color='white', intensity=0.6, position=(0.0020764…"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"3"
|
||||
]
|
||||
},
|
||||
"execution_count": 7,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"min_stretch = stretches[np.argmin(target_closeness)]\n",
|
||||
"\n",
|
||||
"stretch_diag = np.array([1., 1., min_stretch])\n",
|
||||
"v_rest_opt = list_v_rest[np.argmin(target_closeness)]\n",
|
||||
"v_eq_opt = list_v_eq[np.argmin(target_closeness)]\n",
|
||||
"\n",
|
||||
"p = mp.plot(v_eq_opt @ rot.T, t, shading=shadingOptions)\n",
|
||||
"p.add_points(v_eq_opt[solid.pin_idx, :] @ rot.T, shading={\"point_color\":\"black\", \"point_size\": 0.1 * length_scale})\n",
|
||||
"p.add_edges(v_rest_opt @ rot.T, be, shading={\"line_color\": \"blue\"})\n",
|
||||
"p.add_edges(v_target @ rot.T, be, shading={\"line_color\": \"red\"})"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": []
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"kernelspec": {
|
||||
"display_name": "Python 3 (ipykernel)",
|
||||
"language": "python",
|
||||
"name": "python3"
|
||||
},
|
||||
"language_info": {
|
||||
"codemirror_mode": {
|
||||
"name": "ipython",
|
||||
"version": 3
|
||||
},
|
||||
"file_extension": ".py",
|
||||
"mimetype": "text/x-python",
|
||||
"name": "python",
|
||||
"nbconvert_exporter": "python",
|
||||
"pygments_lexer": "ipython3",
|
||||
"version": "3.9.7"
|
||||
},
|
||||
"toc": {
|
||||
"base_numbering": 1,
|
||||
"nav_menu": {},
|
||||
"number_sections": true,
|
||||
"sideBar": true,
|
||||
"skip_h1_title": false,
|
||||
"title_cell": "Table of Contents",
|
||||
"title_sidebar": "Contents",
|
||||
"toc_cell": false,
|
||||
"toc_position": {},
|
||||
"toc_section_display": true,
|
||||
"toc_window_display": false
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 4
|
||||
}
|
||||
@@ -0,0 +1,413 @@
|
||||
{
|
||||
"cells": [
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Description\n",
|
||||
"\n",
|
||||
"This notebook intends to gather all the functionalities you'll have to implement for assignment 2.3.\n",
|
||||
"\n",
|
||||
"# Load libraries"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 1,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"import numpy as np\n",
|
||||
"import igl\n",
|
||||
"import meshplot as mp\n",
|
||||
"import time\n",
|
||||
"\n",
|
||||
"import sys as _sys\n",
|
||||
"_sys.path.append(\"../src\")\n",
|
||||
"from elasticsolid import *\n",
|
||||
"from elasticenergy import *\n",
|
||||
"from matplotlib import gridspec\n",
|
||||
"import matplotlib.pyplot as plt\n",
|
||||
"\n",
|
||||
"shadingOptions = {\n",
|
||||
" \"flat\":True,\n",
|
||||
" \"wireframe\":False, \n",
|
||||
"}\n",
|
||||
"\n",
|
||||
"rot = np.array(\n",
|
||||
" [[1, 0, 0 ],\n",
|
||||
" [0, 0, 1],\n",
|
||||
" [0, -1, 0 ]]\n",
|
||||
")"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Load mesh\n",
|
||||
"\n",
|
||||
"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."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 2,
|
||||
"metadata": {
|
||||
"scrolled": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"application/vnd.jupyter.widget-view+json": {
|
||||
"model_id": "cde9c07200964ca49ba30dc0805f86b4",
|
||||
"version_major": 2,
|
||||
"version_minor": 0
|
||||
},
|
||||
"text/plain": [
|
||||
"Renderer(camera=PerspectiveCamera(children=(DirectionalLight(color='white', intensity=0.6, position=(-1.987469…"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"v, _, _, t, _, _ = igl.read_obj(\"../data/dinosaur.obj\")\n",
|
||||
"# v, _, _, t, _, _ = igl.read_obj(\"../data/beam.obj\")\n",
|
||||
"\n",
|
||||
"# t = np.array([\n",
|
||||
"# [0, 1, 2, 3],\n",
|
||||
"# [1, 2, 3, 4]\n",
|
||||
"# ])\n",
|
||||
"# v = np.array([\n",
|
||||
"# [0., 0., 0.],\n",
|
||||
"# [1., 0., 0.],\n",
|
||||
"# [0., 1., 0.],\n",
|
||||
"# [0., 0., 1.],\n",
|
||||
"# [2/3, 2/3, 2/3]\n",
|
||||
"# ])\n",
|
||||
"\n",
|
||||
"be = igl.edges(igl.boundary_facets(t))\n",
|
||||
"e = igl.edges(t)\n",
|
||||
"\n",
|
||||
"aabb = np.max(v, axis=0) - np.min(v, axis=0)\n",
|
||||
"length_scale = np.mean(aabb)\n",
|
||||
"\n",
|
||||
"p = mp.plot(v @ rot.T, t, shading=shadingOptions)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Linear/Non-Linear Elastic Solid\n",
|
||||
"\n",
|
||||
"## Instantiation\n",
|
||||
"\n",
|
||||
"We first specify the elasticity model to use for the elastic solid, as well as pinned vertices, and volumetric forces."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 3,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"rho = 131 # [kg.m-3]\n",
|
||||
"young = 3e8 # [Pa] \n",
|
||||
"poisson = 0.2\n",
|
||||
"force_mass = np.zeros(shape = (3,))\n",
|
||||
"force_mass[2] = - rho * 9.81\n",
|
||||
"\n",
|
||||
"# minX = np.min(v[:, 0])\n",
|
||||
"# pin_idx = np.arange(v.shape[0])[v[:, 0] < minX + 0.2*aabb[0]]\n",
|
||||
"minZ = np.min(v[:, 2])\n",
|
||||
"pin_idx = np.arange(v.shape[0])[v[:, 2] < minZ + 0.1*aabb[2]]\n",
|
||||
"\n",
|
||||
"# ee = LinearElasticEnergy(young, poisson)\n",
|
||||
"ee = NeoHookeanElasticEnergy(young, poisson)\n",
|
||||
"solid = ElasticSolid(v, t, ee, rho=rho, pin_idx=pin_idx, f_mass=force_mass)\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Deform the mesh\n",
|
||||
"\n",
|
||||
"This should now involve elastic forces computation."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 4,
|
||||
"metadata": {
|
||||
"scrolled": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"application/vnd.jupyter.widget-view+json": {
|
||||
"model_id": "7fabee312a0d47a08145c7eca8b09306",
|
||||
"version_major": 2,
|
||||
"version_minor": 0
|
||||
},
|
||||
"text/plain": [
|
||||
"Renderer(camera=PerspectiveCamera(children=(DirectionalLight(color='white', intensity=0.6, position=(-3.974938…"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"3"
|
||||
]
|
||||
},
|
||||
"execution_count": 4,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"v_def = v.copy()\n",
|
||||
"v_def[:, 0] *= 2.\n",
|
||||
"solid.update_def_shape(v_def)\n",
|
||||
"\n",
|
||||
"p = mp.plot(solid.v_def @ rot.T, solid.t, shading=shadingOptions)\n",
|
||||
"p.add_points(solid.v_def[solid.pin_idx, :] @ rot.T, shading={\"point_color\":\"black\", \"point_size\": 0.1 * length_scale})\n",
|
||||
"forcesScale = 2 * np.max(np.linalg.norm(solid.f_ext, axis=1))\n",
|
||||
"p.add_lines(solid.v_def @ rot.T, (solid.v_def + solid.f_ext / forcesScale) @ rot.T)\n",
|
||||
"p.add_edges(v @ rot.T, be, shading={\"line_color\": \"blue\"})\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Find equilibrium\n",
|
||||
"\n",
|
||||
"We compare different methods: number of steps, computation time."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 5,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"n_steps = 1000\n",
|
||||
"thresh = 1.\n",
|
||||
"v_init = v.copy()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Gradient descent"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stderr",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"/opt/notebooks/assignment_2_3/notebook/../src/elasticenergy.py:133: RuntimeWarning: invalid value encountered in log\n",
|
||||
" j = np.log(np.linalg.det(jac))\n",
|
||||
"/opt/notebooks/assignment_2_3/notebook/../src/elasticenergy.py:148: RuntimeWarning: invalid value encountered in log\n",
|
||||
" self.logJ = np.log(np.linalg.det(jac))\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"%run ../src/Utils.py\n",
|
||||
"\n",
|
||||
"report_GD = equilibrium_convergence_report_GD(solid, v_init, n_steps, 1e-8, thresh=thresh)\n",
|
||||
"energies_el_GD = report_GD['energies_el']\n",
|
||||
"energies_ext_GD = report_GD['energies_ext']\n",
|
||||
"energy_GD = energies_el_GD + energies_ext_GD\n",
|
||||
"residuals_GD = report_GD['residuals']\n",
|
||||
"times_GD = report_GD['times']\n",
|
||||
"idx_stop_GD = report_GD['idx_stop']\n",
|
||||
"v_def_GD = solid.v_def.copy()\n",
|
||||
"\n",
|
||||
"# Lastly, plot the resulting shape\n",
|
||||
"p = mp.plot(v_def_GD @ rot.T, solid.t, shading=shadingOptions)\n",
|
||||
"forcesScale = 2 * np.max(np.linalg.norm(solid.f_ext, axis=1))\n",
|
||||
"p.add_lines(v_def_GD @ rot.T, (solid.v_def + solid.f_ext / forcesScale) @ rot.T)\n",
|
||||
"p.add_points(v_def_GD[pin_idx, :] @ rot.T, shading={\"point_color\":\"black\", \"point_size\": 0.1 * length_scale})\n",
|
||||
"p.add_edges(v @ rot.T, be, shading={\"line_color\": \"blue\"})"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## BFGS"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"report_BFGS = equilibrium_convergence_report_BFGS(solid, v_init, n_steps, 1e-8, thresh=thresh)\n",
|
||||
"energies_el_BFGS = report_BFGS['energies_el']\n",
|
||||
"energies_ext_BFGS = report_BFGS['energies_ext']\n",
|
||||
"energy_BFGS = energies_el_BFGS + energies_ext_BFGS\n",
|
||||
"residuals_BFGS = report_BFGS['residuals']\n",
|
||||
"times_BFGS = report_BFGS['times']\n",
|
||||
"idx_stop_BFGS = report_BFGS['idx_stop']\n",
|
||||
"v_def_BFGS = solid.v_def.copy()\n",
|
||||
"\n",
|
||||
"# Lastly, plot the resulting shape\n",
|
||||
"p = mp.plot(v_def_BFGS @ rot.T, solid.t, shading=shadingOptions)\n",
|
||||
"forcesScale = 2 * np.max(np.linalg.norm(solid.f_ext, axis=1))\n",
|
||||
"p.add_lines(v_def_BFGS @ rot.T, (solid.v_def + solid.f_ext / forcesScale) @ rot.T)\n",
|
||||
"p.add_points(v_def_BFGS[pin_idx, :] @ rot.T, shading={\"point_color\":\"black\", \"point_size\": 0.1 * length_scale})\n",
|
||||
"p.add_edges(v @ rot.T, be, shading={\"line_color\": \"blue\"})"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Newton-CG"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"report_NCG = equilibrium_convergence_report_NCG(solid, v_init, n_steps, thresh=thresh)\n",
|
||||
"energies_el_NCG = report_NCG['energies_el']\n",
|
||||
"energies_ext_NCG = report_NCG['energies_ext']\n",
|
||||
"energy_NCG = energies_el_NCG + energies_ext_NCG\n",
|
||||
"residuals_NCG = report_NCG['residuals']\n",
|
||||
"times_NCG = report_NCG['times']\n",
|
||||
"idx_stop_NCG = report_NCG['idx_stop']\n",
|
||||
"v_def_NCG = solid.v_def.copy()\n",
|
||||
"\n",
|
||||
"# Lastly, plot the resulting shape\n",
|
||||
"p = mp.plot(v_def_NCG @ rot.T, solid.t, shading=shadingOptions)\n",
|
||||
"forcesScale = 2 * np.max(np.linalg.norm(solid.f_ext, axis=1))\n",
|
||||
"p.add_lines(v_def_NCG @ rot.T, (solid.v_def + solid.f_ext / forcesScale) @ rot.T)\n",
|
||||
"p.add_points(v_def_NCG[pin_idx, :] @ rot.T, shading={\"point_color\":\"black\", \"point_size\": 0.1 * length_scale})\n",
|
||||
"p.add_edges(v @ rot.T, be, shading={\"line_color\": \"blue\"})"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Compare the different algorithms"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"cmap = plt.get_cmap('viridis')\n",
|
||||
"colors = cmap(np.linspace(0., 1., 4))\n",
|
||||
"\n",
|
||||
"gs = gridspec.GridSpec(nrows=1, ncols=1, width_ratios=[1], height_ratios=[1])\n",
|
||||
"fig = plt.figure(figsize=(10, 6))\n",
|
||||
"\n",
|
||||
"axTmp = plt.subplot(gs[0, 0])\n",
|
||||
"axTmp.plot(times_GD[:idx_stop_GD+1], residuals_GD[:idx_stop_GD+1], c=colors[0], \n",
|
||||
" label=\"Gradient Descent ({:}its)\".format(idx_stop_GD))\n",
|
||||
"axTmp.plot(times_BFGS[:idx_stop_BFGS+1], residuals_BFGS[:idx_stop_BFGS+1], c=colors[1], \n",
|
||||
" label=\"BFGS ({:}its)\".format(idx_stop_BFGS))\n",
|
||||
"axTmp.plot(times_NCG[:idx_stop_NCG+1], residuals_NCG[:idx_stop_NCG+1], c=colors[2], \n",
|
||||
" label=\"Newton-CG ({:}its)\".format(idx_stop_NCG))\n",
|
||||
"y_lim = axTmp.get_ylim()\n",
|
||||
"axTmp.set_title(\"Residuals as optimization goes\", fontsize=14)\n",
|
||||
"axTmp.set_xlabel(\"Computation time [s]\", fontsize=12)\n",
|
||||
"axTmp.set_ylabel(\"Force residuals [N]\", fontsize=12)\n",
|
||||
"axTmp.set_ylim(y_lim)\n",
|
||||
"plt.legend(fontsize=12)\n",
|
||||
"plt.grid()\n",
|
||||
"plt.show()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"gs = gridspec.GridSpec(nrows=1, ncols=1, width_ratios=[1], height_ratios=[1])\n",
|
||||
"fig = plt.figure(figsize=(10, 6))\n",
|
||||
"\n",
|
||||
"axTmp = plt.subplot(gs[0, 0])\n",
|
||||
"axTmp.plot(times_GD[:idx_stop_GD+1], energy_GD[:idx_stop_GD+1], c=colors[0], \n",
|
||||
" label=\"Gradient Descent ({:}its)\".format(idx_stop_GD))\n",
|
||||
"axTmp.plot(times_BFGS[:idx_stop_BFGS+1], energy_BFGS[:idx_stop_BFGS+1], c=colors[1], \n",
|
||||
" label=\"BFGS ({:}its)\".format(idx_stop_BFGS))\n",
|
||||
"axTmp.plot(times_NCG[:idx_stop_NCG+1], energy_NCG[:idx_stop_NCG+1], c=colors[2], \n",
|
||||
" label=\"Newton-CG ({:}its)\".format(idx_stop_NCG))\n",
|
||||
"y_lim = axTmp.get_ylim()\n",
|
||||
"axTmp.set_title(\"Total energy as optimization goes\", fontsize=14)\n",
|
||||
"axTmp.set_xlabel(\"Computation time [s]\", fontsize=12)\n",
|
||||
"axTmp.set_ylabel(\"Energy [J]\", fontsize=12)\n",
|
||||
"axTmp.set_ylim(y_lim)\n",
|
||||
"plt.legend(fontsize=12)\n",
|
||||
"plt.grid()\n",
|
||||
"plt.show()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": []
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"kernelspec": {
|
||||
"display_name": "Python 3 (ipykernel)",
|
||||
"language": "python",
|
||||
"name": "python3"
|
||||
},
|
||||
"language_info": {
|
||||
"codemirror_mode": {
|
||||
"name": "ipython",
|
||||
"version": 3
|
||||
},
|
||||
"file_extension": ".py",
|
||||
"mimetype": "text/x-python",
|
||||
"name": "python",
|
||||
"nbconvert_exporter": "python",
|
||||
"pygments_lexer": "ipython3",
|
||||
"version": "3.9.7"
|
||||
},
|
||||
"toc": {
|
||||
"base_numbering": 1,
|
||||
"nav_menu": {},
|
||||
"number_sections": true,
|
||||
"sideBar": true,
|
||||
"skip_h1_title": false,
|
||||
"title_cell": "Table of Contents",
|
||||
"title_sidebar": "Contents",
|
||||
"toc_cell": false,
|
||||
"toc_position": {},
|
||||
"toc_section_display": true,
|
||||
"toc_window_display": false
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 4
|
||||
}
|
||||
@@ -0,0 +1,279 @@
|
||||
{
|
||||
"cells": [
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Description\n",
|
||||
"\n",
|
||||
"This notebook intends to gather all the functionalities you'll have to implement for assignment 2.2.\n",
|
||||
"\n",
|
||||
"# Load libraries"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 4,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"import numpy as np\n",
|
||||
"import igl\n",
|
||||
"import meshplot as mp\n",
|
||||
"import time\n",
|
||||
"\n",
|
||||
"import sys as _sys\n",
|
||||
"_sys.path.append(\"../src\")\n",
|
||||
"from elasticsolid import *\n",
|
||||
"from elasticenergy import *\n",
|
||||
"from matplotlib import gridspec\n",
|
||||
"import matplotlib.pyplot as plt\n",
|
||||
"\n",
|
||||
"shadingOptions = {\n",
|
||||
" \"flat\":True,\n",
|
||||
" \"wireframe\":False, \n",
|
||||
"}\n",
|
||||
"\n",
|
||||
"rot = np.array(\n",
|
||||
" [[1, 0, 0 ],\n",
|
||||
" [0, 0, 1],\n",
|
||||
" [0, -1, 0 ]]\n",
|
||||
")"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Load mesh\n",
|
||||
"\n",
|
||||
"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."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 5,
|
||||
"metadata": {
|
||||
"scrolled": false
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"v, _, _, t, _, _ = igl.read_obj(\"../data/beam.obj\")\n",
|
||||
"aabb = np.max(v, axis=0) - np.min(v, axis=0)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Linear/Non-Linear Elastic Solid\n",
|
||||
"\n",
|
||||
"## Instantiation\n",
|
||||
"\n",
|
||||
"We first specify the elasticity model to use for the elastic solid, as well as pinned vertices, and volumetric forces."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 12,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"rho = 131 # [kg.m-3]\n",
|
||||
"young = 10e6 # [Pa] \n",
|
||||
"poisson = 0.2\n",
|
||||
"force_mass = np.zeros(shape = (3,))\n",
|
||||
"force_mass[2] = - rho * 9.81\n",
|
||||
"\n",
|
||||
"minX = np.min(v[:, 0])\n",
|
||||
"pin_idx = np.arange(v.shape[0])[v[:, 0] < minX + 0.2*aabb[0]]\n",
|
||||
"\n",
|
||||
"ee = LinearElasticEnergy(young, poisson)\n",
|
||||
"neo_ee = NeoHookeanElasticEnergy(young, poisson)\n",
|
||||
"solid = ElasticSolid(v, t, ee, rho=rho, pin_idx=pin_idx, f_mass=force_mass)\n",
|
||||
"neo_solid = ElasticSolid(v, t, neo_ee, rho=rho, pin_idx=pin_idx, f_mass=force_mass)\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Validate Elastic Energy Gradient (Elastic Forces) using Finite Differences on Elastic Energy"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 13,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": "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 truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light"
|
||||
},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": "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 truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light"
|
||||
},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"fd_validation_elastic(solid)\n",
|
||||
"fd_validation_elastic(neo_solid)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Validate External Energy Gradient (External Forces) using Finite Differences on External Energy"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 14,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light"
|
||||
},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": "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 truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light"
|
||||
},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"fd_validation_ext(solid)\n",
|
||||
"fd_validation_ext(neo_solid)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Validate Elastic Energy Force Differentials (Elastic Forces) using Finite Differences on Elastic Forces"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 10,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stderr",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"/opt/notebooks/assignment_2_3/src/Utils.py:402: RuntimeWarning: divide by zero encountered in double_scalars\n",
|
||||
" errors.append(norm_error/norm_an_df)\n",
|
||||
"/opt/notebooks/assignment_2_3/notebook/../src/elasticenergy.py:134: RuntimeWarning: invalid value encountered in log\n",
|
||||
" j = np.log(np.linalg.det(jac))\n",
|
||||
"/opt/notebooks/assignment_2_3/notebook/../src/elasticenergy.py:149: RuntimeWarning: invalid value encountered in log\n",
|
||||
" self.logJ = np.log(np.linalg.det(jac))\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": "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\n",
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light"
|
||||
},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"%run ../src/Utils.py\n",
|
||||
"\n",
|
||||
"v_def = v.copy()\n",
|
||||
"v_def[:, 0] *= 2.\n",
|
||||
"solid.update_def_shape(v_def)\n",
|
||||
"neo_solid.update_def_shape(v_def)\n",
|
||||
"\n",
|
||||
"fd_validation_elastic_differentials(solid)\n",
|
||||
"fd_validation_elastic_differentials(neo_solid)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": []
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": []
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"kernelspec": {
|
||||
"display_name": "Python 3 (ipykernel)",
|
||||
"language": "python",
|
||||
"name": "python3"
|
||||
},
|
||||
"language_info": {
|
||||
"codemirror_mode": {
|
||||
"name": "ipython",
|
||||
"version": 3
|
||||
},
|
||||
"file_extension": ".py",
|
||||
"mimetype": "text/x-python",
|
||||
"name": "python",
|
||||
"nbconvert_exporter": "python",
|
||||
"pygments_lexer": "ipython3",
|
||||
"version": "3.9.7"
|
||||
},
|
||||
"toc": {
|
||||
"base_numbering": 1,
|
||||
"nav_menu": {},
|
||||
"number_sections": true,
|
||||
"sideBar": true,
|
||||
"skip_h1_title": false,
|
||||
"title_cell": "Table of Contents",
|
||||
"title_sidebar": "Contents",
|
||||
"toc_cell": false,
|
||||
"toc_position": {},
|
||||
"toc_section_display": true,
|
||||
"toc_window_display": false
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 4
|
||||
}
|
||||
@@ -0,0 +1,254 @@
|
||||
{
|
||||
"cells": [
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Description\n",
|
||||
"\n",
|
||||
"This notebook shows a way to solve an inverse problem parameterized with a single quantity.\n",
|
||||
"\n",
|
||||
"# Load libraries"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 2,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"import numpy as np\n",
|
||||
"import igl\n",
|
||||
"import meshplot as mp\n",
|
||||
"import time\n",
|
||||
"\n",
|
||||
"import sys as _sys\n",
|
||||
"_sys.path.append(\"../src\")\n",
|
||||
"from elasticsolid import *\n",
|
||||
"from elasticenergy import *\n",
|
||||
"from stretch_utils import *\n",
|
||||
"from matplotlib import gridspec\n",
|
||||
"import matplotlib.pyplot as plt\n",
|
||||
"\n",
|
||||
"shadingOptions = {\n",
|
||||
" \"flat\":True,\n",
|
||||
" \"wireframe\":False, \n",
|
||||
"}\n",
|
||||
"\n",
|
||||
"rot = np.array(\n",
|
||||
" [[1, 0, 0 ],\n",
|
||||
" [0, 0, 1],\n",
|
||||
" [0, -1, 0 ]]\n",
|
||||
")"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Load mesh\n",
|
||||
"\n",
|
||||
"We will use the ball for the current experiment."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 3,
|
||||
"metadata": {
|
||||
"scrolled": false
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"v, _, _, t, _, _ = igl.read_obj(\"../data/ball.obj\")\n",
|
||||
"\n",
|
||||
"be = igl.edges(igl.boundary_facets(t))\n",
|
||||
"bv = np.unique(igl.boundary_facets(t))\n",
|
||||
"e = igl.edges(t)\n",
|
||||
"\n",
|
||||
"aabb = np.max(v, axis=0) - np.min(v, axis=0)\n",
|
||||
"length_scale = np.mean(aabb)\n",
|
||||
"\n",
|
||||
"# p = mp.plot(v @ rot.T, t, shading=shadingOptions)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Linear/Non-Linear Elastic Solid\n",
|
||||
"\n",
|
||||
"## Instantiation\n",
|
||||
"\n",
|
||||
"We first specify the elasticity model to use for the elastic solid, as well as pinned vertices, and volumetric forces."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 4,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"rho = 131 # [kg.m-3]\n",
|
||||
"young = 7e5 # [Pa] \n",
|
||||
"poisson = 0.2\n",
|
||||
"force_mass = np.zeros(shape = (3,))\n",
|
||||
"force_mass[2] = - rho * 9.81\n",
|
||||
"\n",
|
||||
"# Determines the pinned indices and the lowest pinned indext\n",
|
||||
"maxZ = np.max(v[:, 2])\n",
|
||||
"pin_idx = np.arange(v.shape[0])[v[:, 2] > maxZ - 0.1*aabb[2]]\n",
|
||||
"lowest_pin_idx = np.argsort(v[:, 2])[-pin_idx.shape[0]-1]\n",
|
||||
"\n",
|
||||
"ee = NeoHookeanElasticEnergy(young, poisson)\n",
|
||||
"solid = ElasticSolid(v, t, ee, rho=rho, pin_idx=pin_idx, f_mass=force_mass)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Manually search for a solution"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 5,
|
||||
"metadata": {
|
||||
"scrolled": true
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"n_search = 50\n",
|
||||
"n_steps = 20\n",
|
||||
"thresh = 1.\n",
|
||||
"\n",
|
||||
"v_rest_init = v.copy()\n",
|
||||
"v_rest_tmp = v.copy()\n",
|
||||
"v_target = v.copy()[bv]\n",
|
||||
"\n",
|
||||
"stretches = np.linspace(0.55, 1.1, n_search)\n",
|
||||
"\n",
|
||||
"list_v_rest, list_v_eq, target_closeness = report_stretches(solid, v_rest_init, bv, v_target, stretches, lowest_pin_idx)\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 6,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light"
|
||||
},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"plt.plot(stretches, target_closeness)\n",
|
||||
"plt.title(\"Closeness to target\", fontsize=14)\n",
|
||||
"plt.xlabel(\"Stretch along $z$\", fontsize=12)\n",
|
||||
"plt.ylabel(\"Objective\", fontsize=12)\n",
|
||||
"plt.grid()\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Plot best solution"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 7,
|
||||
"metadata": {
|
||||
"scrolled": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"application/vnd.jupyter.widget-view+json": {
|
||||
"model_id": "665a8bf8e742437bb4b51c4b3d384c4c",
|
||||
"version_major": 2,
|
||||
"version_minor": 0
|
||||
},
|
||||
"text/plain": [
|
||||
"Renderer(camera=PerspectiveCamera(children=(DirectionalLight(color='white', intensity=0.6, position=(0.0020764…"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"3"
|
||||
]
|
||||
},
|
||||
"execution_count": 7,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"min_stretch = stretches[np.argmin(target_closeness)]\n",
|
||||
"\n",
|
||||
"stretch_diag = np.array([1., 1., min_stretch])\n",
|
||||
"v_rest_opt = list_v_rest[np.argmin(target_closeness)]\n",
|
||||
"v_eq_opt = list_v_eq[np.argmin(target_closeness)]\n",
|
||||
"\n",
|
||||
"p = mp.plot(v_eq_opt @ rot.T, t, shading=shadingOptions)\n",
|
||||
"p.add_points(v_eq_opt[solid.pin_idx, :] @ rot.T, shading={\"point_color\":\"black\", \"point_size\": 0.1 * length_scale})\n",
|
||||
"p.add_edges(v_rest_opt @ rot.T, be, shading={\"line_color\": \"blue\"})\n",
|
||||
"p.add_edges(v_target @ rot.T, be, shading={\"line_color\": \"red\"})"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": []
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"kernelspec": {
|
||||
"display_name": "Python 3 (ipykernel)",
|
||||
"language": "python",
|
||||
"name": "python3"
|
||||
},
|
||||
"language_info": {
|
||||
"codemirror_mode": {
|
||||
"name": "ipython",
|
||||
"version": 3
|
||||
},
|
||||
"file_extension": ".py",
|
||||
"mimetype": "text/x-python",
|
||||
"name": "python",
|
||||
"nbconvert_exporter": "python",
|
||||
"pygments_lexer": "ipython3",
|
||||
"version": "3.9.7"
|
||||
},
|
||||
"toc": {
|
||||
"base_numbering": 1,
|
||||
"nav_menu": {},
|
||||
"number_sections": true,
|
||||
"sideBar": true,
|
||||
"skip_h1_title": false,
|
||||
"title_cell": "Table of Contents",
|
||||
"title_sidebar": "Contents",
|
||||
"toc_cell": false,
|
||||
"toc_position": {},
|
||||
"toc_section_display": true,
|
||||
"toc_window_display": false
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 4
|
||||
}
|
||||
@@ -0,0 +1,467 @@
|
||||
{
|
||||
"cells": [
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Description\n",
|
||||
"\n",
|
||||
"This notebook intends to gather all the functionalities you'll have to implement for assignment 2.3.\n",
|
||||
"\n",
|
||||
"# Load libraries"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 1,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"import numpy as np\n",
|
||||
"import igl\n",
|
||||
"import meshplot as mp\n",
|
||||
"import time\n",
|
||||
"\n",
|
||||
"import sys as _sys\n",
|
||||
"_sys.path.append(\"../src\")\n",
|
||||
"from elasticsolid import *\n",
|
||||
"from elasticenergy import *\n",
|
||||
"from matplotlib import gridspec\n",
|
||||
"import matplotlib.pyplot as plt\n",
|
||||
"\n",
|
||||
"shadingOptions = {\n",
|
||||
" \"flat\":True,\n",
|
||||
" \"wireframe\":False, \n",
|
||||
"}\n",
|
||||
"\n",
|
||||
"rot = np.array(\n",
|
||||
" [[1, 0, 0 ],\n",
|
||||
" [0, 0, 1],\n",
|
||||
" [0, -1, 0 ]]\n",
|
||||
")"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Load mesh\n",
|
||||
"\n",
|
||||
"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."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 2,
|
||||
"metadata": {
|
||||
"scrolled": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"application/vnd.jupyter.widget-view+json": {
|
||||
"model_id": "bc63cdd8cfe14603b4e54d871b4206d7",
|
||||
"version_major": 2,
|
||||
"version_minor": 0
|
||||
},
|
||||
"text/plain": [
|
||||
"Renderer(camera=PerspectiveCamera(children=(DirectionalLight(color='white', intensity=0.6, position=(-1.987469…"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"v, _, _, t, _, _ = igl.read_obj(\"../data/dinosaur.obj\")\n",
|
||||
"# v, _, _, t, _, _ = igl.read_obj(\"../data/beam.obj\")\n",
|
||||
"\n",
|
||||
"# t = np.array([\n",
|
||||
"# [0, 1, 2, 3],\n",
|
||||
"# [1, 2, 3, 4]\n",
|
||||
"# ])\n",
|
||||
"# v = np.array([\n",
|
||||
"# [0., 0., 0.],\n",
|
||||
"# [1., 0., 0.],\n",
|
||||
"# [0., 1., 0.],\n",
|
||||
"# [0., 0., 1.],\n",
|
||||
"# [2/3, 2/3, 2/3]\n",
|
||||
"# ])\n",
|
||||
"\n",
|
||||
"be = igl.edges(igl.boundary_facets(t))\n",
|
||||
"e = igl.edges(t)\n",
|
||||
"\n",
|
||||
"aabb = np.max(v, axis=0) - np.min(v, axis=0)\n",
|
||||
"length_scale = np.mean(aabb)\n",
|
||||
"\n",
|
||||
"p = mp.plot(v @ rot.T, t, shading=shadingOptions)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Linear/Non-Linear Elastic Solid\n",
|
||||
"\n",
|
||||
"## Instantiation\n",
|
||||
"\n",
|
||||
"We first specify the elasticity model to use for the elastic solid, as well as pinned vertices, and volumetric forces."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 3,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"rho = 131 # [kg.m-3]\n",
|
||||
"young = 3e8 # [Pa] \n",
|
||||
"poisson = 0.2\n",
|
||||
"force_mass = np.zeros(shape = (3,))\n",
|
||||
"force_mass[2] = - rho * 9.81\n",
|
||||
"\n",
|
||||
"# minX = np.min(v[:, 0])\n",
|
||||
"# pin_idx = np.arange(v.shape[0])[v[:, 0] < minX + 0.2*aabb[0]]\n",
|
||||
"minZ = np.min(v[:, 2])\n",
|
||||
"pin_idx = np.arange(v.shape[0])[v[:, 2] < minZ + 0.1*aabb[2]]\n",
|
||||
"\n",
|
||||
"# ee = LinearElasticEnergy(young, poisson)\n",
|
||||
"ee = NeoHookeanElasticEnergy(young, poisson)\n",
|
||||
"solid = ElasticSolid(v, t, ee, rho=rho, pin_idx=pin_idx, f_mass=force_mass)\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Deform the mesh\n",
|
||||
"\n",
|
||||
"This should now involve elastic forces computation."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 4,
|
||||
"metadata": {
|
||||
"scrolled": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"application/vnd.jupyter.widget-view+json": {
|
||||
"model_id": "b6c6920e14be4332b4a96e814c9236da",
|
||||
"version_major": 2,
|
||||
"version_minor": 0
|
||||
},
|
||||
"text/plain": [
|
||||
"Renderer(camera=PerspectiveCamera(children=(DirectionalLight(color='white', intensity=0.6, position=(-3.974938…"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"3"
|
||||
]
|
||||
},
|
||||
"execution_count": 4,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"v_def = v.copy()\n",
|
||||
"v_def[:, 0] *= 2.\n",
|
||||
"solid.update_def_shape(v_def)\n",
|
||||
"\n",
|
||||
"p = mp.plot(solid.v_def @ rot.T, solid.t, shading=shadingOptions)\n",
|
||||
"p.add_points(solid.v_def[solid.pin_idx, :] @ rot.T, shading={\"point_color\":\"black\", \"point_size\": 0.1 * length_scale})\n",
|
||||
"forcesScale = 2 * np.max(np.linalg.norm(solid.f_ext, axis=1))\n",
|
||||
"p.add_lines(solid.v_def @ rot.T, (solid.v_def + solid.f_ext / forcesScale) @ rot.T)\n",
|
||||
"p.add_edges(v @ rot.T, be, shading={\"line_color\": \"blue\"})\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Find equilibrium\n",
|
||||
"\n",
|
||||
"We compare different methods: number of steps, computation time."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 5,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"n_steps = 1000\n",
|
||||
"thresh = 1.\n",
|
||||
"v_init = v.copy()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Gradient descent"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 6,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"application/vnd.jupyter.widget-view+json": {
|
||||
"model_id": "fe18307f9c24439db1ed8fc58928c284",
|
||||
"version_major": 2,
|
||||
"version_minor": 0
|
||||
},
|
||||
"text/plain": [
|
||||
"Renderer(camera=PerspectiveCamera(children=(DirectionalLight(color='white', intensity=0.6, position=(-1.987469…"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"3"
|
||||
]
|
||||
},
|
||||
"execution_count": 6,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"%run ../src/Utils.py\n",
|
||||
"\n",
|
||||
"report_GD = equilibrium_convergence_report_GD(solid, v_init, n_steps, 1e-8, thresh=thresh)\n",
|
||||
"energies_el_GD = report_GD['energies_el']\n",
|
||||
"energies_ext_GD = report_GD['energies_ext']\n",
|
||||
"energy_GD = energies_el_GD + energies_ext_GD\n",
|
||||
"residuals_GD = report_GD['residuals']\n",
|
||||
"times_GD = report_GD['times']\n",
|
||||
"idx_stop_GD = report_GD['idx_stop']\n",
|
||||
"v_def_GD = solid.v_def.copy()\n",
|
||||
"\n",
|
||||
"# Lastly, plot the resulting shape\n",
|
||||
"p = mp.plot(v_def_GD @ rot.T, solid.t, shading=shadingOptions)\n",
|
||||
"forcesScale = 2 * np.max(np.linalg.norm(solid.f_ext, axis=1))\n",
|
||||
"p.add_lines(v_def_GD @ rot.T, (solid.v_def + solid.f_ext / forcesScale) @ rot.T)\n",
|
||||
"p.add_points(v_def_GD[pin_idx, :] @ rot.T, shading={\"point_color\":\"black\", \"point_size\": 0.1 * length_scale})\n",
|
||||
"p.add_edges(v @ rot.T, be, shading={\"line_color\": \"blue\"})"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## BFGS"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 7,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"application/vnd.jupyter.widget-view+json": {
|
||||
"model_id": "0621481088744a34af8662b30613ceac",
|
||||
"version_major": 2,
|
||||
"version_minor": 0
|
||||
},
|
||||
"text/plain": [
|
||||
"Renderer(camera=PerspectiveCamera(children=(DirectionalLight(color='white', intensity=0.6, position=(-0.297981…"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"3"
|
||||
]
|
||||
},
|
||||
"execution_count": 7,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"report_BFGS = equilibrium_convergence_report_BFGS(solid, v_init, n_steps, 1e-8, thresh=thresh)\n",
|
||||
"energies_el_BFGS = report_BFGS['energies_el']\n",
|
||||
"energies_ext_BFGS = report_BFGS['energies_ext']\n",
|
||||
"energy_BFGS = energies_el_BFGS + energies_ext_BFGS\n",
|
||||
"residuals_BFGS = report_BFGS['residuals']\n",
|
||||
"times_BFGS = report_BFGS['times']\n",
|
||||
"idx_stop_BFGS = report_BFGS['idx_stop']\n",
|
||||
"v_def_BFGS = solid.v_def.copy()\n",
|
||||
"\n",
|
||||
"# Lastly, plot the resulting shape\n",
|
||||
"p = mp.plot(v_def_BFGS @ rot.T, solid.t, shading=shadingOptions)\n",
|
||||
"forcesScale = 2 * np.max(np.linalg.norm(solid.f_ext, axis=1))\n",
|
||||
"p.add_lines(v_def_BFGS @ rot.T, (solid.v_def + solid.f_ext / forcesScale) @ rot.T)\n",
|
||||
"p.add_points(v_def_BFGS[pin_idx, :] @ rot.T, shading={\"point_color\":\"black\", \"point_size\": 0.1 * length_scale})\n",
|
||||
"p.add_edges(v @ rot.T, be, shading={\"line_color\": \"blue\"})"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Newton-CG"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"Energy: -127985.65419881282 Force norm: 287742.8969935862 Line search Iters: 0\n",
|
||||
"Energy: -178892.14400109363 Force norm: 182952.00531737506 Line search Iters: 0\n",
|
||||
"Energy: -206901.42653635694 Force norm: 151950.01953001413 Line search Iters: 0\n",
|
||||
"Energy: -227792.85782151608 Force norm: 135158.66283929092 Line search Iters: 0\n",
|
||||
"Energy: -244838.85692066932 Force norm: 123433.77675577633 Line search Iters: 0\n",
|
||||
"Energy: -259318.48388755036 Force norm: 114389.35466936302 Line search Iters: 0\n",
|
||||
"Energy: -271937.0843840749 Force norm: 107190.15322070086 Line search Iters: 0\n",
|
||||
"Energy: -283160.0740997393 Force norm: 101411.77988596291 Line search Iters: 0\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"report_NCG = equilibrium_convergence_report_NCG(solid, v_init, n_steps, thresh=thresh)\n",
|
||||
"energies_el_NCG = report_NCG['energies_el']\n",
|
||||
"energies_ext_NCG = report_NCG['energies_ext']\n",
|
||||
"energy_NCG = energies_el_NCG + energies_ext_NCG\n",
|
||||
"residuals_NCG = report_NCG['residuals']\n",
|
||||
"times_NCG = report_NCG['times']\n",
|
||||
"idx_stop_NCG = report_NCG['idx_stop']\n",
|
||||
"v_def_NCG = solid.v_def.copy()\n",
|
||||
"\n",
|
||||
"# Lastly, plot the resulting shape\n",
|
||||
"p = mp.plot(v_def_NCG @ rot.T, solid.t, shading=shadingOptions)\n",
|
||||
"forcesScale = 2 * np.max(np.linalg.norm(solid.f_ext, axis=1))\n",
|
||||
"p.add_lines(v_def_NCG @ rot.T, (solid.v_def + solid.f_ext / forcesScale) @ rot.T)\n",
|
||||
"p.add_points(v_def_NCG[pin_idx, :] @ rot.T, shading={\"point_color\":\"black\", \"point_size\": 0.1 * length_scale})\n",
|
||||
"p.add_edges(v @ rot.T, be, shading={\"line_color\": \"blue\"})"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Compare the different algorithms"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"cmap = plt.get_cmap('viridis')\n",
|
||||
"colors = cmap(np.linspace(0., 1., 4))\n",
|
||||
"\n",
|
||||
"gs = gridspec.GridSpec(nrows=1, ncols=1, width_ratios=[1], height_ratios=[1])\n",
|
||||
"fig = plt.figure(figsize=(10, 6))\n",
|
||||
"\n",
|
||||
"axTmp = plt.subplot(gs[0, 0])\n",
|
||||
"axTmp.plot(times_GD[:idx_stop_GD+1], residuals_GD[:idx_stop_GD+1], c=colors[0], \n",
|
||||
" label=\"Gradient Descent ({:}its)\".format(idx_stop_GD))\n",
|
||||
"axTmp.plot(times_BFGS[:idx_stop_BFGS+1], residuals_BFGS[:idx_stop_BFGS+1], c=colors[1], \n",
|
||||
" label=\"BFGS ({:}its)\".format(idx_stop_BFGS))\n",
|
||||
"axTmp.plot(times_NCG[:idx_stop_NCG+1], residuals_NCG[:idx_stop_NCG+1], c=colors[2], \n",
|
||||
" label=\"Newton-CG ({:}its)\".format(idx_stop_NCG))\n",
|
||||
"y_lim = axTmp.get_ylim()\n",
|
||||
"axTmp.set_title(\"Residuals as optimization goes\", fontsize=14)\n",
|
||||
"axTmp.set_xlabel(\"Computation time [s]\", fontsize=12)\n",
|
||||
"axTmp.set_ylabel(\"Force residuals [N]\", fontsize=12)\n",
|
||||
"axTmp.set_ylim(y_lim)\n",
|
||||
"plt.legend(fontsize=12)\n",
|
||||
"plt.grid()\n",
|
||||
"plt.show()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"gs = gridspec.GridSpec(nrows=1, ncols=1, width_ratios=[1], height_ratios=[1])\n",
|
||||
"fig = plt.figure(figsize=(10, 6))\n",
|
||||
"\n",
|
||||
"axTmp = plt.subplot(gs[0, 0])\n",
|
||||
"axTmp.plot(times_GD[:idx_stop_GD+1], energy_GD[:idx_stop_GD+1], c=colors[0], \n",
|
||||
" label=\"Gradient Descent ({:}its)\".format(idx_stop_GD))\n",
|
||||
"axTmp.plot(times_BFGS[:idx_stop_BFGS+1], energy_BFGS[:idx_stop_BFGS+1], c=colors[1], \n",
|
||||
" label=\"BFGS ({:}its)\".format(idx_stop_BFGS))\n",
|
||||
"axTmp.plot(times_NCG[:idx_stop_NCG+1], energy_NCG[:idx_stop_NCG+1], c=colors[2], \n",
|
||||
" label=\"Newton-CG ({:}its)\".format(idx_stop_NCG))\n",
|
||||
"y_lim = axTmp.get_ylim()\n",
|
||||
"axTmp.set_title(\"Total energy as optimization goes\", fontsize=14)\n",
|
||||
"axTmp.set_xlabel(\"Computation time [s]\", fontsize=12)\n",
|
||||
"axTmp.set_ylabel(\"Energy [J]\", fontsize=12)\n",
|
||||
"axTmp.set_ylim(y_lim)\n",
|
||||
"plt.legend(fontsize=12)\n",
|
||||
"plt.grid()\n",
|
||||
"plt.show()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": []
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"kernelspec": {
|
||||
"display_name": "Python 3 (ipykernel)",
|
||||
"language": "python",
|
||||
"name": "python3"
|
||||
},
|
||||
"language_info": {
|
||||
"codemirror_mode": {
|
||||
"name": "ipython",
|
||||
"version": 3
|
||||
},
|
||||
"file_extension": ".py",
|
||||
"mimetype": "text/x-python",
|
||||
"name": "python",
|
||||
"nbconvert_exporter": "python",
|
||||
"pygments_lexer": "ipython3",
|
||||
"version": "3.9.7"
|
||||
},
|
||||
"toc": {
|
||||
"base_numbering": 1,
|
||||
"nav_menu": {},
|
||||
"number_sections": true,
|
||||
"sideBar": true,
|
||||
"skip_h1_title": false,
|
||||
"title_cell": "Table of Contents",
|
||||
"title_sidebar": "Contents",
|
||||
"toc_cell": false,
|
||||
"toc_position": {},
|
||||
"toc_section_display": true,
|
||||
"toc_window_display": false
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 4
|
||||
}
|
||||
@@ -0,0 +1,289 @@
|
||||
{
|
||||
"cells": [
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Description\n",
|
||||
"\n",
|
||||
"This notebook intends to gather all the functionalities you'll have to implement for assignment 2.2.\n",
|
||||
"\n",
|
||||
"# Load libraries"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 13,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"import numpy as np\n",
|
||||
"import igl\n",
|
||||
"import meshplot as mp\n",
|
||||
"import time\n",
|
||||
"\n",
|
||||
"import sys as _sys\n",
|
||||
"_sys.path.append(\"../src\")\n",
|
||||
"from elasticsolid import *\n",
|
||||
"from elasticenergy import *\n",
|
||||
"from matplotlib import gridspec\n",
|
||||
"import matplotlib.pyplot as plt\n",
|
||||
"\n",
|
||||
"shadingOptions = {\n",
|
||||
" \"flat\":True,\n",
|
||||
" \"wireframe\":False, \n",
|
||||
"}\n",
|
||||
"\n",
|
||||
"rot = np.array(\n",
|
||||
" [[1, 0, 0 ],\n",
|
||||
" [0, 0, 1],\n",
|
||||
" [0, -1, 0 ]]\n",
|
||||
")"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Load mesh\n",
|
||||
"\n",
|
||||
"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."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 14,
|
||||
"metadata": {
|
||||
"scrolled": false
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"v, _, _, t, _, _ = igl.read_obj(\"../data/beam.obj\")\n",
|
||||
"aabb = np.max(v, axis=0) - np.min(v, axis=0)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Linear/Non-Linear Elastic Solid\n",
|
||||
"\n",
|
||||
"## Instantiation\n",
|
||||
"\n",
|
||||
"We first specify the elasticity model to use for the elastic solid, as well as pinned vertices, and volumetric forces."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 15,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"rho = 131 # [kg.m-3]\n",
|
||||
"young = 10e6 # [Pa] \n",
|
||||
"poisson = 0.2\n",
|
||||
"force_mass = np.zeros(shape = (3,))\n",
|
||||
"force_mass[2] = - rho * 9.81\n",
|
||||
"\n",
|
||||
"minX = np.min(v[:, 0])\n",
|
||||
"pin_idx = np.arange(v.shape[0])[v[:, 0] < minX + 0.2*aabb[0]]\n",
|
||||
"\n",
|
||||
"ee = LinearElasticEnergy(young, poisson)\n",
|
||||
"neo_ee = NeoHookeanElasticEnergy(young, poisson)\n",
|
||||
"solid = ElasticSolid(v, t, ee, rho=rho, pin_idx=pin_idx, f_mass=force_mass)\n",
|
||||
"neo_solid = ElasticSolid(v, t, neo_ee, rho=rho, pin_idx=pin_idx, f_mass=force_mass)\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Validate Elastic Energy Gradient (Elastic Forces) using Finite Differences on Elastic Energy"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 16,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAYEAAAD8CAYAAACRkhiPAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjQuMywgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy/MnkTPAAAACXBIWXMAAAsTAAALEwEAmpwYAABKzklEQVR4nO29d3ib53no/bsBAuAC9xSpTU1LXpJlx5NOnMROrTi7ttORxI2/tMfpGU1Pk7Zfk9NTf26b03QkOU2d2nHS1nYdN42XvB1aHrItyUN7UJOkuPcESOD5/ngBECABEiQAkiDv33XpMvHO5zbB937vLcYYFEVRlKWJbb4XoCiKoswfqgQURVGWMKoEFEVRljCqBBRFUZYwqgQURVGWMKoEFEVRljAZ872AeBCRncBOt9v91fXr18/6OoODg+Tk5CRvYfPEYpEDVJaFyGKRA1SWIPv37+8wxpRG25cWSiBIeXk5+/btm/X5dXV11NbWJm9B88RikQNUloXIYpEDVJYgInIu1r60cAcZY54yxtydn58/30tRFEVZVKSFEhCRnSJyf29v73wvRVEUZVGRFkpALQFFUZTUkBZKQC0BRVGU1JAWSkAtAUVRlNSQFkpALQFFUZTUkBZKINmWwKGmXrxj/qRcS1EUJZ1JCyWQTNr7Pez8wev8wc8/QGcpKIqy1EkLJZBMd9D5riGMgac+uMCDb5xNfHGKoihpTFoogWS6g1p6RwDYVJnH/7frKO+c6Ur4moqiKOlKWiiBZNLcOwzA/b+5jZVF2fzev71La9/IPK9KURRlflhySuBCzwhZDjvVhVn86De30THg4ef7GuZ7WYqiKPPCklMCLX3DVBZkIiKsL3fjdmXQOeid72UpiqLMC2mhBJIZGL7QM0Jlfmboc362g96h0YSvqyiKko6khRJIZmC4uXeYyvys0OeCbAc9w6oEFEVZmqSFEkgWY35DW7+HZWGWQEGWk54hdQcpirI0WVJKoNdjMAYqwiyB/Cy1BBRFWbrM62QxEbkO+GJgHZuNMVen8n5dI1aFcGVBZEygT5WAoihLlFlbAiLyoIi0icihCdtvFpHjIlIvIt+c6hrGmNeMMV8DngZ+Otu1xEtQCSwLjwlkOegZGtUWEoqiLEkScQc9BNwcvkFE7MAPgVuAzcAdIrJZRLaKyNMT/pWFnXon8EgCa4mLaJZAQbaDMb9h0OtL9e0VRVEWHLN2BxljdovIqgmbdwD1xpjTACLyKHCbMeY+4NZo1xGRFUCvMaYvxv67gbvBGjRfV1c32yXT2uch0y7s3/M6IgJAS6PlCnruld2UZM1MJ9b3+OgaMeyomFuv2sDAQEL/HxYSKsvCY7HIASpLPCT76VUFhJffNgJXTnPOXcBPYu00xtwvIs3ATrfbva22tnbWi/v+e89RXZzFjTfeENrmOdzCg4f2s/HibWypmlkK6v/9pz2839DDV3deT2GOM+oxb57q4Psv1/MXn97C2tLcWa89nLq6OhL5/7CQUFkWHotFDlBZ4iHZ2UESZduUznZjzLeNMW9Oc0xS6gS6RkxEoRhYMQGA3hkGh0dGfbzf0IN3zM9/vNsY9RhjDPftOsae05189h/fZP+57tktXFEUJUUkWwk0AsvDPlcDFxK9aLIqhqMqgWzrDb5nQtXwXz57jP/YH/3hDoQUQLbTzsPvnI8aWH7zVCcHm3r5vdq1FGQ5+OI/v8VLR1oTkkFRFCWZJFsJ7AXWichqEXECtwNPJvkes8I75qfPYyKqhcEKDAP0DEcWjD389jn+7uUTMbOG3j7dhQj8wcc2cLp9MGpL6h+9eoqSXBe//5F1PP67V7Oh3M3X/nU/bdq1VFGUBUIiKaKPAHuADSLSKCJ3GWPGgHuA54GjwGPGmMOJLjIZ7qDWvhEMsKwg0hLID7iDwi2BkVEffSNjNHQN80FjdOvj7TOdbKrI484dK3BnZvDwO+cj9h9q6uW1kx3cde1qMh12SnJd/OHHNzLmN5zuGJy1HIqiKMlk1krAGHOHMabSGOMwxlQbYx4IbN9ljFlvjFlrjLk3GYtMhjuoOTBMZqIlkOmwk+mwRcQE2vs9oZ+f/mCyN8s75ufd891cuaaILKedz1xWxbMHW+gO60b6o1dP4XZl8MWrVoS2VeS7AHR+gaIoC4a0aBuRDEsgOExmYkwAJvcPCj6k87McPH2gGb8/0iV0oLGHkVE/V64uBuCOK1fg9VkB4t6hUfac6mTXwWbuvGoFeZmO0Hnleda9g9PNFEVR5pu0UAJJtQQKsibtK8h2RLiD2gKWwB07VtDSN8K+CVk9bwf8/ztWFwGwsSKPy1YU8BfPHOWSP3+BO378FpkOO3ddszriPHemg1xXRmgtiqIo88289g6KF2PMU8BT27dv/+psr9HcM0xWBuS6Joucn+WIcAcFA7d37ljBQ2+e4ekDF0IPfIC3TneyodxNUVhtwLd3XsST71+gMj+TqsIsLq7OpyxvstVRnudSd5CiKAuGtFACIrIT2FlTUzPrazT3jlCUGa2MwbIEznUOhT639XvIsAnVhVl8ZGM5uw4282e3bibDbmPU52f/uW4+t6064hqXLi/g0uUF066jIj+TFlUCiqIsENLCHZScmMAIRZnRxbViApHuoJJcFzabsPOSSjoGvLx12nIBHWzqZcjrC8UDZkpFXhat6g5SFGWBkBaWQDJo7h3mooLolkB+tiOiTqCt30NZnpXJU7uhjBynne88dZhbL64MBXXD3UMzoSLfRWu/B5/fYLdFX4+iKMpckRaWQKKBYc+Yj44Bb0x3UH6Wg5FRPyOjVifRtr4RytyWEsh02PmLT2/BlWHj718+yaN7G1hXlktpYP9MqcjLxOc3dA54pj9YURQlxaSFJZBoYLi113rgThUTAKt/UKbDTnu/h8tXFob2f/qyaj59WTW9w6O8e76b6igZRvESShPtG4kaOFYURZlL0sISSBSbDT5zeRXL3bFjAmBVDXvH/HQOekOWQDj5WQ5u3FDGunL3rNcSLFaLJ03UGMNP3zxLU8/wrO+nKIoyFUtCCVQXZvO9L1zKqnx71P2h/kFDXjoCbpoyd2re0stnUDV8pmOQbz95mEfePj/tsYqiKLMhLdxBqSbUP2h4FFd/UAnMzuc/HSU5LjJsElfVcLBI7URrf0rWoiiKkhaWQLJaScciFBMYGg0VigWzg5KNzSaU58VXK7D/rKUETrYNJHUNPr9hwDOW1GsqipKepIUSSNZQmViEZgoMe0MtI1LlDgKrajgeS2D/eUsJnOscDGUuJYMfvXqK2u/W4R3zJ+2aiqKkJ2mhBFJNjtNOhk3oGRqlrd+DCJTkRh8XmQziqRruGfJS3zbA5so8/AZOtSfPGth3touOAQ8fNPYk7ZqKoqQnqgQAEbGayA2P0t4/QnGOiwx76v7XlOdlRlQNt/aNcMvfv8bR5r7QtuAoyjuutFpRn2yNrQS8Y35GffG/1R9rsWIMb9R3zGjdiqIsPuZVCYjIChF5UkQeFJFvzudagk3k2vo8KQsKB6nMz2TQ66N/xGpV8eKRVo429/HA62dCx+w7102GTbjt0mVk2CRmcPi1k+186L6X+W///n5c9+4Z8obSU9+s70xMEEVR0p5EJos9KCJtInJowvabReS4iNTH8WBfDzxjjPkKsHm2a0kGBdlOeodGae0fSVlQOMjEuQJvnrLeyJ8+cIG+gGLYf66bi5blkZfpYHVJDicmWAI+v+F7L57gtx58hyGvj2cONMflMgpaARsr3LzX0M2QVwPEirKUScQSeAi4OXyDiNiBHwK3YD3U7xCRzSKyVUSenvCvDHgPuF1EXgF+lcBaEqYgy+ofNBeWQEVY1bDfb3jzVCcXLctjZNTPE+9fwDvm54OGHrattPoTrS93c7It0hK45+F3+YeXT/Lpy6p44b9fjzPDFmFJxOJ4QAncde1qRn0m6mxkRVGWDrOuEzDG7BaRVRM27wDqjTGnAUTkUeA2Y8x9wK0TryEi3wC+HbjW48BPohxzN3A3QHl5OXV1dbNdMgMDAzHPH+nzcKHLR/eIYbirNaH7TEfbkOW/r3v7fU4dsdEzNMoXamz099v48ctH8LbU4xnzkzV4gbq6NjKGvJzvHOX5l3+Fyy6c7Rjg2UODfGK1g52l3Zw68A5XVdj4+d7zXJXTQZ4zdmO6Vw55yHVAXk89doFHf/U+NKcuCD4dU/1O0o3FIstikQNUlnhIdrFYFdAQ9rkRuHKK458DviMidwJnox1gjLlfRJqBnW63e1ttbe2sF1dXV0es83f3H+GNC9ab9BVb11P7oVWzvs90jIz6+J+7n6OgchWeDBtwjN+59VqWH2nl//3lIQ6MFAEX+K1PXEt5XiaDRc08cepdqjZezpaqfP7oJy8CXr75uWtZUZwNQNWmfj76t7s5Y6vmv9aui3nvvzv8BluX2/n4TVex7dQeGjxj1NZelzJZp2Oq30m6sVhkWSxygMoSD8kODEd7BTVRtlk7jDlkjPmcMeZrxphvTHFcSusEYLxgDFJXLRwk02GnMNtBS98Ir9d3sK4sl7K8TG67dBlZDjtPvH+B6sKsUOxgfXkuMF45vL91jM2VeSEFALCu3M2NG0r5l7fOxqwp8PsNx1v62Vhp9T66Zm0JR5r76B70Rj1eUZTFT7KVQCOwPOxzNXAh0YumumIYxltHAHPS3bM8L5OG7mH2nu3impoSAPIyHdx6cSUA28K6mK4qycFhF060DtDaN0J9j59btlRMuuZXr1tDx4CX7z5/nJ/tOcvfvXSCl460hvaf7xpieNTHpoo8AK6pKcYY2HNas4QUZamSbHfQXmCdiKwGmoDbgTuTfI+UMJeWAFhpoq/XdzDqMyElAFZdwM/3N0ZMLnPYbawuyeFkaz8vHG4B4Jatk5XAh9YWc3F1fkSAOMdp5+0/uYlcVwbHWqw6hA0VliVwyfICcpx23qjv4BNbK1Mip6IoC5tEUkQfAfYLine truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light"
|
||||
},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAXwAAAEBCAYAAAB7Wx7VAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjQuMywgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy/MnkTPAAAACXBIWXMAAAsTAAALEwEAmpwYAABAOklEQVR4nO3dd3yb1b348c+RLFneeySOEzt7hyQmYeO0UKBl3FJoWf21hZaWlq7bBe1tS29vy23puC1Qxm0plzJCoLRlb0wYgSwSsrdjO473HrI1zu8PDS/ZlizZeix/369XXkSPHz06BzlfHX3Peb5Haa0RQggR+0zRboAQQoiJIQFfCCGmCAn4QggxRUjAF0KIKUICvhBCTBES8IUQYoqQgC+EEFOEBHwhhJgiJizgK6UWKaXuVUo9qZS6aaJeVwghhEdQAV8p9YBSqk4ptXvQ8QuVUgeUUoeVUreMdA2t9T6t9VeATwMlY2+yEEKIsQh2hP8gcGH/A0opM3A3cBGwGLhaKbVYKbVMKfXsoD+53udcCrwNvBaxHgghhAiKCraWjlKqCHhWa73U+/h04Dat9QXex7cCaK1vD+Jaz2mtPzHMz24EbgRISEhYXVhYGFT7BnO73ZhMsTFFIX0xnljpB0hfjCjcfhw8eLBBa50z+HhcGG0qACr7Pa4C1g53slKqFLgciAeeH+48rfX9wP0AJSUleuvWrWNqXFlZGaWlpWN6rtFIX4wnVvoB0hcjCrcfSqnjgY6HE/BVgGPDfl3QWpcBZUFdWKlLgEvmzp07poYJIYQYKpzvPlVA/3zLDKA6vOZ4aK2f0VrfmJaWFonLCSGEILyAvwWYp5QqVkpZgauApyPRKKXUJUqp+1tbWyNxOSGEEAS/LPMxYBOwQClVpZS6QWvtBG4GXgL2ARu01nsi0SgZ4QshROQFlcPXWl89zPHnGWECdqwkhy+EEJFnyPVLMsIXQojIM2TAlxy+EGKqcrk1b59w4HJHfr9xQwZ8GeELIaaqP752iD/v6qXsQF3Er23IgC8jfCHEVPTGgTr++PohziqI4yMLcyN+fUMGfBnhCyGmmsqmLr61fgcL81P57GIrSgW6tzU8hgz4QggxldgdLr76yHbcWnPPtauIN0c+2IMEfCGEiLqfPbOXXSda+e2VKyjKThq31zFkwJccvhBiqnhiayWPba7gptI5fGxJ/ri+liEDvuTwhRBTwd7qNv7jn7s5fXYW3zl//ri/niEDvhBCxLrWbgc3PbKN9EQLf7x6JXHm8Q/H4ZRHFkIIMQZut+Y7G3Zyormbx798Gjkp8RPyuoYc4UsOXwgRy+7deIRX99Xyo08sYvWszAl7XUMGfMnhCyFi1buHG/jNSwe4ePk0Pn9G0YS+tiEDvhBCxKKaVjtff+wDZuck86tPLR+Xm6tGIjl8IYSYAL1ON199ZBvdDhePX7eKpPiJD78S8IUQYgLc/sI+tle0cOfVK5mbmxKVNhgypSOTtkKIWPLMzmr++k45XziziEtWTI9aOwwZ8GXSVggRKw7XtfODv3/I6lkZ3HrRoqi2xZABXwghYkFHj5Mv/20bCRYzd1+zCmtcdEOu5PCFEGIcaK255e8fcqyhk4e/uJb8NFu0myQjfCGEGA9/faecZz88yfcuWMgZc7Kj3RxAAr4QQkTc1vImfvn8Ps5fnMdXzp0d7eb4ScAXQogIqm/v4WuPbqcgI4HfXLliwm+uGsmEBnylVJJSaptS6uKJfF0hhJgITpebbzz2AS1dDu65djVpCZZoN2mAoAK+UuoBpVSdUmr3oOMXKqUOKKUOK6VuCeJSPwA2jKWhQghhdL995SCbjjbyi08uY/H01Gg3Z4hgV+k8CNwFPOQ7oJQyA3cD5wNVwBal1NOAGbh90POvB5YDe4HoT1ULIUSEvbynhnvKjnD1mplcsXpGtJsTkNJaB3eiUkXAs1rrpd7HpwO3aa0v8D6+FUBrPTjY+57/CyAJWAx0A5/UWrsDnHcjcCNAXl7e6vXr14fYJY+Ojg6Sk5PH9Fyjkb4YT6z0A6QvkVDb6ea2Td3kJZr44Vob1jA3IQ+3H+vWrdumtS4Z8gOtdVB/gCJgd7/HVwB/7vf4s8BdQVzn88DFwbzm6tWr9Vi98cYbY36u0UhfjCdW+qG19CVcXT1OfcHv39TLb3tJVzR2RuSa4fYD2KoDxNRwbrwK9BE26tcFrfWDo15YqUuAS+bOnTuGZgkhxMTQWvMf/9zNgdp2Hvj8qRRmJka7SSMKZ5VOFVDY7/EMoDq85gghxOSxfkslf99exdc/Mo91C3Kj3ZxRhRPwtwDzlFLFSikrcBXwdCQapaV4mhDC4D6sauGn/9rD2fOy+eZH50W7OUEJdlnmY8AmYIFSqkopdYPW2gncDLwE7AM2aK33RKJRUh5ZCGFkLV293PTwdrKTrfzhqpWYTca5uWokQeXwtdZXD3P8eeD5iLbIc91ngGdKSkq+FOlrCyFEONxuzbce30Fdu50nvnIGmUnWaDcpaFJaQQghQnDXG4cpO1DPTy5ZwimF6dFuTkgMGfAlpSOEMKKNB+v5/asH+eTKAq5bOzPazQmZIQO+TNoKIYzmREs331z/AfNzU/jFJ5caqihasAwZ8GWEL4Qwkh6ni68+sh2nS3PvZ1eTaJ2ce0cZMuDLCF8IYSQ/f3YvOytbuOPKFRRnJ0W7OWNmyIAvhBBG8dT2Kh5+r4IvnzObC5fmR7s5YTFkwJeUjhDCCPbXtPHDf+xibXEm37tgQbSbEzZDBnxJ6Qghoq3N7uCmh7eTarNw5zUriTMbMlyGZPL3IATHGjr5y9vHot0MIYTBaa353hM7qWjq4u5rV5GbEhvbeEypgH/fm0f4+bN7aezoiXZThBAG9r9vHeWlPbXcetFCTi3KjHZzIsaQAX88cvhaazYerAc862mFECKQ94428qsXD/DxZfnccFZxtJsTUYYM+OHm8LXWtPUMLM1/pL6D6lY7AFXNAwN+j9PFcx+e9G3QIoSYoura7Nz86AfMykrkV59aPilvrhqJIQN+uL6zYSe3b+4eEMDfPNjg/3tlU9eA81/YVcPXHt3O+8eaJqyNQghjcbjc3PzoB3T2OLn3utWk2CzRblLExWTAP31OFic7NZuONvqPvXWontnZSaTa4oaM8I/WdwDw/lEJ+EJMVb96YT+by5v4708tY35eSrSbMy5iMuBfsmI6SRZ45L0KAOwOF+8dbeSc+TnMyEikqnngCP9Yo+fx5vLGIdcSQsS+53ed5M9vH+PzZxRx2SkF0W7OuJmcBSFGYbOYOWt6HC/tqaGu3c7Bmg7sDjfnzM/mZGs3R+s7B5xf3uB5vO14M71ON9a4mPwcFEIEcKS+g+89sZOVM9P54ccXRbs548qQkS0Sq3TWzbTgdGs2bKlk46F6rGYTp83O8o7w+/L7WmvKGzrJS43H7nCzu1ru7hViqujqdXLTw9uIt5j507WrYn6wZ8jeReJO2/wkE2fOzeLR9ysoO1BHSVEGidY4ZmQk0O1w0dTZC0BjZy/tPU6uWD0DkDy+EFOF1ppb/r6Lw3Ud/PGqlUxLS4h2k8adIQN+pFy3dhbVrXYO1nZwzvwcAGZkJAJ9SzN96ZySWZnMyUli8zHJ4wsxFfztveM8vbOa73xsAWfNy452cyZETAf88xbnkZsSD8DZ3jd0RobnU9wX8I95A35RdhJrirPYWt6Myy3r8YWIZdsrmvn5s3v56MJcbjp3TrSbM2FiOuBbzCZuKp3DKYXpLMpPBfoCfqV3pU55Yydmk2JGRgJrizNp73Gy72Rb1NoshBhfjR09fO2R7eSn2fjdp0/BZIqtm6tGEtMBH+ALZxbzz6+d6X9TU2wW0hMt/qWZ5Q1dFGYkYDGbWFPsqZmxWW7AEiImudyab67fQWNnL/dcu5q0xNi7uWokMR/wA5mRkTAgpVPk3cFmenoChZkJvC95fCFi0v+8epC3DzfwX5ctZWnB1Cu/PmEBXylVqpR6Syl1r1KqdKJeN5AZ6X1LM8sbOynK6tuybE1RFpuPNY1aV+dvm8qHlGgQQhjXa/tqufP1w1x1aiGfPrUw2s2JiqACvlLqAaVUnVJq96DjFyqlDiilDiulbhnlMhroAGxA1diaGxmeEX4Xde09dPW6BuxRubY4k+YuB4fqOoZ9fm2bnR//aw+Pbq6YiOYKIcJU0djFtx/fwdKCVG67dEm0mxM1wd5p+yBwF/CQ74BSygzcDZyPJ4BvUUo9DZiB2wc9/3rgLa31m0qpPOB3wLXhNX3sZmQkYHe42Xa8GcCf0gE41ZvH31LeNGw9Dd+k7vHGzoA/F0IYh93h4qZHtgHwp2tWY7OYo9yi6FHBlgRWShUBz2qtl3ofnw7cprW+wPv4VgCt9eBgP/g6VuBRrfUVw/z8RuBGgLy8vNXr168PrieDdHR0kJycHPBnO+qc/M/2HkoL4yirdPLrcxLITfR82dFa8803ulmSZeLLKwLvcvP8sV42HHAwK9XEz84Y/5s1RurLZBMrfYmVfkDs9+WB3T1srHLyrVXxnJI7OarJhPuerFu3bpvWumTw8XB6XwBU9ntcBawd7mSl1OXABUA6nm8LAWmt7wfuBygpKdGlpaVjalxZWRnDPXdaTTv/s30jRzutxJlcXH5B6YD9Ks+q3sbOytZhn/907Q7gBI09Js4999xxr5k9Ul8mm1jpS6z0A2K7L49vqWBj1S6+/pG5fOtjk2cT8vF6T8KZtA0U5Yb9uqC1fkpr/WWt9We01mUjXngcdrzqr8C7Fr+iqYuZmYlDNicumZXJiZZuqofZGWt/TTsAHT1OGr0lGoQQxrL7RCs//tcezpqbzbfOmx/t5hhCOAG/Cug/1T0DqA6vORMjOT6ODO/62/75e581/fL4gzlcbg7XdbDAm98fKY9/oqV7SClmIcT4a+1ycNMj28hKsvKHq07BPIVurhpJOAF/CzBPKVXszctfBTwdiUZFonjLine truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light"
|
||||
},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"fd_validation_elastic(solid)\n",
|
||||
"fd_validation_elastic(neo_solid)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Validate External Energy Gradient (External Forces) using Finite Differences on External Energy"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 17,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light"
|
||||
},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": "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 truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light"
|
||||
},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"fd_validation_ext(solid)\n",
|
||||
"fd_validation_ext(neo_solid)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Validate Elastic Energy Force Differentials (Elastic Forces) using Finite Differences on Elastic Forces"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 18,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light"
|
||||
},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"name": "stderr",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"/opt/notebooks/assignment_2_3/notebook/../src/elasticenergy.py:145: RuntimeWarning: invalid value encountered in log\n",
|
||||
" self.logJ = np.log(np.linalg.det(jac))\n",
|
||||
"/opt/notebooks/assignment_2_3/notebook/../src/elasticenergy.py:131: RuntimeWarning: invalid value encountered in log\n",
|
||||
" j = np.log(np.linalg.det(jac))\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light"
|
||||
},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"%run ../src/Utils.py\n",
|
||||
"\n",
|
||||
"v_def = v.copy()\n",
|
||||
"v_def[:, 0] *= 2.\n",
|
||||
"solid.update_def_shape(v_def)\n",
|
||||
"neo_solid.update_def_shape(v_def)\n",
|
||||
"\n",
|
||||
"fd_validation_elastic_differentials(solid)\n",
|
||||
"fd_validation_elastic_differentials(neo_solid)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": []
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": []
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"kernelspec": {
|
||||
"display_name": "Python 3 (ipykernel)",
|
||||
"language": "python",
|
||||
"name": "python3"
|
||||
},
|
||||
"language_info": {
|
||||
"codemirror_mode": {
|
||||
"name": "ipython",
|
||||
"version": 3
|
||||
},
|
||||
"file_extension": ".py",
|
||||
"mimetype": "text/x-python",
|
||||
"name": "python",
|
||||
"nbconvert_exporter": "python",
|
||||
"pygments_lexer": "ipython3",
|
||||
"version": "3.9.7"
|
||||
},
|
||||
"toc": {
|
||||
"base_numbering": 1,
|
||||
"nav_menu": {},
|
||||
"number_sections": true,
|
||||
"sideBar": true,
|
||||
"skip_h1_title": false,
|
||||
"title_cell": "Table of Contents",
|
||||
"title_sidebar": "Contents",
|
||||
"toc_cell": false,
|
||||
"toc_position": {},
|
||||
"toc_section_display": true,
|
||||
"toc_window_display": false
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 4
|
||||
}
|
||||
@@ -0,0 +1,403 @@
|
||||
import numpy as np
|
||||
import time
|
||||
import math
|
||||
|
||||
def conjugate_gradient(L_method, b):
|
||||
'''
|
||||
Finds an inexact Newton descent direction using Conjugate Gradient (CG)
|
||||
Solves partially L(x) = b where A is positive definite, using CG.
|
||||
The method should be take care of checking whether the added direction is
|
||||
an ascent direction or not, and whether the residuals are small enough.
|
||||
Details can be found in the handout.
|
||||
|
||||
Input:
|
||||
- L_method : a method that computes the Hessian vector product. It should
|
||||
take an array of shape (n,) and return an array of shape (n,)
|
||||
- b : right hand side of the linear system (n,)
|
||||
|
||||
Output:
|
||||
- p_star : np array of shape (n,) solving the linear system approximately
|
||||
'''
|
||||
N = b.shape[0]
|
||||
fshape = (3*N,N)
|
||||
z = np.zeros(fshape)
|
||||
r = np.zeros(fshape)
|
||||
d = np.zeros(fshape)
|
||||
alpha = np.zeros(3*N)
|
||||
beta = np.zeros(3*N)
|
||||
r[0] = -b
|
||||
d[0] = -r[0]
|
||||
ck = 0
|
||||
for k in range(0,3*N - 1):
|
||||
Ldk = L_method(d[k])
|
||||
if (d[k].T @ Ldk) <= 0:
|
||||
if (k==0):
|
||||
return b
|
||||
else:
|
||||
return z[k]
|
||||
alpha[k] = (r[k].T @ r[k])/(d[k] @ Ldk)
|
||||
z[k+1] = z[k] + alpha[k] * d[k]
|
||||
r[k+1] = r[k] + alpha[k] * Ldk
|
||||
va = np.linalg.norm(r[k+1])
|
||||
vb = math.sqrt(np.linalg.norm(b))
|
||||
if va < min(0.5,vb) * np.linalg.norm(b):
|
||||
break
|
||||
|
||||
beta[k+1] = (r[k].T @ r[k+1])/(r[k].T @ r[k])
|
||||
d[k+1] = -r[k+1] + beta[k+1]*d[k]
|
||||
ck = k
|
||||
|
||||
return z[ck+1]
|
||||
|
||||
|
||||
def compute_inverse_approximate_hessian_matrix(sk, yk, invB_prev):
|
||||
'''
|
||||
Input:
|
||||
- sk : previous step x_{k+1} - x_k, shape (n, 1)
|
||||
- yk : grad(f)_{k+1} - grad(f)_{k}, shape (n, 1)
|
||||
- invB_prev : previous Hessian estimate Bk, shape (n, n)
|
||||
|
||||
Output:
|
||||
- invB_new : previous Hessian estimate Bk, shape (n, n)
|
||||
'''
|
||||
invB_new = invB_prev.copy()
|
||||
invB_new += (sk.T @ yk + yk.T @ invB_prev @ yk) / ((sk.T @ yk) ** 2) * (sk @ sk.T)
|
||||
prod = (invB_prev @ yk) @ sk.T
|
||||
invB_new -= (prod + prod.T) / (sk.T @ yk)
|
||||
return invB_new
|
||||
|
||||
def equilibrium_convergence_report_GD(solid, v_init, n_steps, step_size, thresh=1e-3):
|
||||
'''
|
||||
Finds the equilibrium by minimizing the total energy using gradient descent.
|
||||
|
||||
Input:
|
||||
- solid : an elastic solid to optimize
|
||||
- v_init : the initial guess for the equilibrium position
|
||||
- n_step : number of optimization steps
|
||||
- step_size : scaling factor of the gradient when taking the step
|
||||
- thresh : threshold to stop the optimization process on the gradient's magnitude
|
||||
|
||||
Ouput:
|
||||
- report : a dictionary containing various quantities of interest
|
||||
'''
|
||||
|
||||
solid.update_def_shape(v_init)
|
||||
|
||||
energies_el = np.zeros(shape=(n_steps+1,))
|
||||
energies_ext = np.zeros(shape=(n_steps+1,))
|
||||
residuals = np.zeros(shape=(n_steps+1,))
|
||||
times = np.zeros(shape=(n_steps+1,))
|
||||
step_sizes = np.zeros(shape=(n_steps,))
|
||||
|
||||
energies_el[0] = solid.energy_el
|
||||
energies_ext[0] = solid.energy_ext
|
||||
residuals[0] = np.linalg.norm((solid.f + solid.f_ext)[solid.free_idx, :])
|
||||
idx_stop = n_steps
|
||||
|
||||
energy_tot_prev = energies_el[0] + energies_ext[0]
|
||||
|
||||
t_start = time.time()
|
||||
for i in range(n_steps):
|
||||
## TODO: Find the descent direction
|
||||
## descent_dir has shape (#v, 3)
|
||||
descent_dir = (solid.f + solid.f_ext)
|
||||
|
||||
|
||||
step_size_tmp = step_size
|
||||
max_l_iter = 20
|
||||
for l_iter in range(max_l_iter):
|
||||
step_size_tmp *= 0.5
|
||||
solid.displace(step_size_tmp * descent_dir)
|
||||
|
||||
## TODO: Check if the armijo rule is satisfied
|
||||
## energy_tot_tmp is the current total energy
|
||||
## armijo is a boolean that says whether the condition is satisfied
|
||||
energy_tot_tmp = solid.energy_ext + solid.energy_el
|
||||
grad_tmp = -descent_dir
|
||||
armijo = energy_tot_tmp < energy_tot_prev + thresh * step_size_tmp * ((descent_dir*grad_tmp).sum())
|
||||
|
||||
if armijo or l_iter == max_l_iter-1:
|
||||
break
|
||||
else:
|
||||
solid.displace(-step_size_tmp * descent_dir)
|
||||
step_sizes[i] = step_size_tmp
|
||||
|
||||
# Measure the force residuals
|
||||
energies_el[i+1] = solid.energy_el
|
||||
energies_ext[i+1] = solid.energy_ext
|
||||
residuals[i+1] = np.linalg.norm((solid.f + solid.f_ext)[solid.free_idx, :])
|
||||
energy_tot_prev = energy_tot_tmp
|
||||
|
||||
if residuals[i+1] < thresh:
|
||||
energies_el[i+1:] = energies_el[i+1]
|
||||
energies_ext[i+1:] = energies_ext[i+1]
|
||||
residuals[i+1:] = residuals[i+1]
|
||||
idx_stop = i
|
||||
break
|
||||
|
||||
times[i+1] = time.time() - t_start
|
||||
|
||||
report = {}
|
||||
report['energies_el'] = energies_el
|
||||
report['energies_ext'] = energies_ext
|
||||
report['residuals'] = residuals
|
||||
report['times'] = times
|
||||
report['idx_stop'] = idx_stop
|
||||
report['step_sizes'] = step_sizes
|
||||
|
||||
return report
|
||||
|
||||
|
||||
def equilibrium_convergence_report_BFGS(solid, v_init, n_steps, step_size, thresh=1e-3):
|
||||
'''
|
||||
Finds the equilibrium by minimizing the total energy using BFGS.
|
||||
|
||||
Input:
|
||||
- solid : an elastic solid to optimize
|
||||
- v_init : the initial guess for the equilibrium position
|
||||
- n_step : number of optimization steps
|
||||
- step_size : scaling factor of the direction when taking the step
|
||||
- thresh : threshold to stop the optimization process on the gradient's magnitude
|
||||
|
||||
Ouput:
|
||||
- report : a dictionary containing various quantities of interest
|
||||
'''
|
||||
|
||||
solid.update_def_shape(v_init)
|
||||
|
||||
energies_el = np.zeros(shape=(n_steps+1,))
|
||||
energies_ext = np.zeros(shape=(n_steps+1,))
|
||||
residuals = np.zeros(shape=(n_steps+1,))
|
||||
times = np.zeros(shape=(n_steps+1,))
|
||||
energies_el[0] = solid.energy_el
|
||||
energies_ext[0] = solid.energy_ext
|
||||
## TODO: Collect free vertex positions
|
||||
## grad_tmp is the current flattened gradient of the total energy
|
||||
## with respect to the free vertices
|
||||
grad_tmp = - (solid.f + solid.f_ext)[solid.free_idx,:].reshape(-1,1)
|
||||
residuals[0] = np.linalg.norm(grad_tmp)
|
||||
idx_stop = n_steps
|
||||
|
||||
energy_tot_prev = energies_el[0] + energies_ext[0]
|
||||
|
||||
|
||||
## TODO: Collect free vertex positions
|
||||
## v_tmp are the current flattened free vertices
|
||||
v_tmp = solid.v_def[solid.free_idx,:].reshape(-1,1)
|
||||
dir_zeros = np.zeros_like(solid.v_def)
|
||||
invB_prev = np.eye(v_tmp.shape[0])
|
||||
|
||||
t_start = time.time()
|
||||
for i in range(n_steps):
|
||||
|
||||
dir_tmp = - invB_prev @ grad_tmp
|
||||
dir_zeros[solid.free_idx, :] = dir_tmp.reshape(-1, 3)
|
||||
|
||||
step_size_tmp = step_size
|
||||
max_l_iter = 20
|
||||
for l_iter in range(max_l_iter):
|
||||
step_size_tmp *= 0.5
|
||||
solid.displace(step_size_tmp * dir_zeros)
|
||||
|
||||
## TODO: Check if the armijo rule is satisfied
|
||||
## energy_tot_tmp is the current total energy
|
||||
## armijo is a boolean that says whether the condition is satisfied
|
||||
energy_tot_tmp = solid.energy_el + solid.energy_ext
|
||||
armijo = energy_tot_tmp < energy_tot_prev + thresh * step_size_tmp*((dir_tmp*grad_tmp).sum())
|
||||
|
||||
if armijo or l_iter == max_l_iter-1:
|
||||
break
|
||||
else:
|
||||
solid.displace(-step_size_tmp * dir_zeros)
|
||||
|
||||
## TODO: Update all quantities
|
||||
## v_new are the new flattened free vertices
|
||||
## grad_new is the new flattened gradient of the total energy
|
||||
## with respect to the free vertices
|
||||
v_new = solid.v_def[solid.free_idx,:].reshape(-1,1)
|
||||
grad_new = - (solid.f+solid.f_ext)[solid.free_idx,:].reshape(-1,1)
|
||||
invB_prev = compute_inverse_approximate_hessian_matrix(v_new - v_tmp,
|
||||
grad_new - grad_tmp,
|
||||
invB_prev)
|
||||
v_tmp = v_new.copy()
|
||||
grad_tmp = grad_new.copy()
|
||||
|
||||
energies_el[i+1] = solid.energy_el
|
||||
energies_ext[i+1] = solid.energy_ext
|
||||
residuals[i+1] = np.linalg.norm(grad_tmp)
|
||||
|
||||
if residuals[i+1] < thresh:
|
||||
residuals[i+1:] = residuals[i+1]
|
||||
energies_el[i+1:] = energies_el[i+1]
|
||||
energies_ext[i+1:] = energies_ext[i+1]
|
||||
idx_stop = i
|
||||
break
|
||||
|
||||
times[i+1] = time.time() - t_start
|
||||
|
||||
report = {}
|
||||
report['energies_el'] = energies_el
|
||||
report['energies_ext'] = energies_ext
|
||||
report['residuals'] = residuals
|
||||
report['times'] = times
|
||||
report['idx_stop'] = idx_stop
|
||||
|
||||
return report
|
||||
|
||||
def equilibrium_convergence_report_NCG(solid, v_init, n_steps, thresh=1e-3):
|
||||
'''
|
||||
Finds the equilibrium by minimizing the total energy using Newton CG.
|
||||
|
||||
Input:
|
||||
- solid : an elastic solid to optimize
|
||||
- v_init : the initial guess for the equilibrium position
|
||||
- n_step : number of optimization steps
|
||||
- thresh : threshold to stop the optimization process on the gradient's magnitude
|
||||
|
||||
Ouput:
|
||||
- report : a dictionary containing various quantities of interest
|
||||
'''
|
||||
|
||||
solid.update_def_shape(v_init)
|
||||
|
||||
energies_el = np.zeros(shape=(n_steps+1,))
|
||||
energies_ext = np.zeros(shape=(n_steps+1,))
|
||||
residuals = np.zeros(shape=(n_steps+1,))
|
||||
times = np.zeros(shape=(n_steps+1,))
|
||||
energies_el[0] = solid.energy_el
|
||||
energies_ext[0] = solid.energy_ext
|
||||
residuals[0] = np.linalg.norm((solid.f + solid.f_ext)[solid.free_idx, :])
|
||||
idx_stop = n_steps
|
||||
|
||||
t_start = time.time()
|
||||
for i in range(n_steps):
|
||||
# Take a Newton step
|
||||
solid.equilibrium_step()
|
||||
|
||||
# Measure the force residuals
|
||||
energies_el[i+1] = solid.energy_el
|
||||
energies_ext[i+1] = solid.energy_ext
|
||||
residuals[i+1] = np.linalg.norm((solid.f + solid.f_ext)[solid.free_idx, :])
|
||||
|
||||
if residuals[i+1] < thresh:
|
||||
residuals[i+1:] = residuals[i+1]
|
||||
energies_el[i+1:] = energies_el[i+1]
|
||||
energies_ext[i+1:] = energies_ext[i+1]
|
||||
idx_stop = i
|
||||
break
|
||||
|
||||
times[i+1] = time.time() - t_start
|
||||
|
||||
report = {}
|
||||
report['energies_el'] = energies_el
|
||||
report['energies_ext'] = energies_ext
|
||||
report['residuals'] = residuals
|
||||
report['times'] = times
|
||||
report['idx_stop'] = idx_stop
|
||||
|
||||
return report
|
||||
|
||||
import matplotlib.pyplot as plt
|
||||
def fd_validation_ext(solid):
|
||||
epsilons = np.logspace(-9,-3,100)
|
||||
perturb_global = np.random.uniform(-1e-3, 1e-3, size=solid.v_def.shape)
|
||||
solid.displace(perturb_global)
|
||||
v_def = solid.v_def.copy()
|
||||
perturb = np.random.uniform(-1, 1, size=solid.v_def.shape)
|
||||
errors = []
|
||||
for eps in epsilons:
|
||||
# Back to original
|
||||
solid.update_def_shape(v_def)
|
||||
grad = np.zeros(solid.f_ext.shape)
|
||||
grad[solid.free_idx] = -solid.f_ext.copy()[solid.free_idx]
|
||||
an_delta_E = (grad*perturb).sum()
|
||||
|
||||
# One step forward
|
||||
solid.displace(perturb * eps)
|
||||
E1 = solid.energy_ext.copy()
|
||||
|
||||
# Two steps backward
|
||||
solid.displace(-2*perturb * eps)
|
||||
E2 = solid.energy_ext.copy()
|
||||
|
||||
# Compute error
|
||||
fd_delta_E = (E1 - E2)/(2*eps)
|
||||
errors.append(abs(fd_delta_E - an_delta_E)/abs(an_delta_E))
|
||||
plt.loglog(epsilons, errors)
|
||||
plt.grid()
|
||||
plt.show()
|
||||
|
||||
def fd_validation_elastic(solid):
|
||||
epsilons = np.logspace(-9,-3,100)
|
||||
perturb_global = np.random.uniform(-1e-3, 1e-3, size=solid.v_def.shape)
|
||||
solid.displace(perturb_global)
|
||||
|
||||
v_def = solid.v_def.copy()
|
||||
solid.make_elastic_forces()
|
||||
|
||||
perturb = np.random.uniform(-1, 1, size=solid.v_def.shape)
|
||||
errors = []
|
||||
for eps in epsilons:
|
||||
# Back to original
|
||||
solid.update_def_shape(v_def)
|
||||
solid.make_elastic_forces()
|
||||
grad = np.zeros(solid.f.shape)
|
||||
grad[solid.free_idx] = -solid.f.copy()[solid.free_idx]
|
||||
an_delta_E = (grad*perturb).sum()
|
||||
|
||||
# One step forward
|
||||
solid.displace(perturb * eps)
|
||||
E1 = solid.energy_el.copy()
|
||||
|
||||
# Two steps backward
|
||||
solid.displace(-2*perturb * eps)
|
||||
E2 = solid.energy_el.copy()
|
||||
|
||||
# Compute error
|
||||
fd_delta_E = (E1 - E2)/(2*eps)
|
||||
errors.append(abs(fd_delta_E - an_delta_E)/abs(an_delta_E))
|
||||
solid.displace(perturb * eps)
|
||||
|
||||
plt.loglog(epsilons, errors)
|
||||
plt.grid()
|
||||
plt.show()
|
||||
|
||||
def fd_validation_elastic_differentials(solid):
|
||||
epsilons = np.logspace(-9, 3,500)
|
||||
perturb_global = 1e-3*np.random.uniform(-1., 1., size=solid.v_def.shape)
|
||||
solid.displace(perturb_global)
|
||||
|
||||
v_def = solid.v_def.copy()
|
||||
|
||||
perturb = np.random.uniform(-1, 1, size=solid.v_def.shape)
|
||||
errors = []
|
||||
for eps in epsilons:
|
||||
# Back to original
|
||||
solid.update_def_shape(v_def)
|
||||
perturb_0s = np.zeros_like(perturb)
|
||||
perturb_0s[solid.free_idx] = perturb[solid.free_idx]
|
||||
an_df = solid.compute_force_differentials(perturb_0s)[solid.free_idx, :]
|
||||
an_df_full = np.zeros(solid.f.shape)
|
||||
an_df_full[solid.free_idx] = an_df.copy()
|
||||
|
||||
# One step forward
|
||||
solid.displace(perturb * eps)
|
||||
f1 = solid.f[solid.free_idx, :]
|
||||
f1_full = np.zeros(solid.f.shape)
|
||||
f1_full[solid.free_idx] = f1
|
||||
|
||||
# Two steps backward
|
||||
solid.displace(-2*perturb * eps)
|
||||
f2 = solid.f[solid.free_idx, :]
|
||||
f2_full = np.zeros(solid.f.shape)
|
||||
f2_full[solid.free_idx] = f2
|
||||
|
||||
# Compute error
|
||||
fd_delta_f = (f1_full - f2_full)/(2*eps)
|
||||
norm_an_df = np.linalg.norm(an_df_full)
|
||||
norm_error = np.linalg.norm(an_df_full - fd_delta_f)
|
||||
errors.append(norm_error/norm_an_df)
|
||||
|
||||
plt.loglog(epsilons, errors)
|
||||
plt.grid()
|
||||
plt.show()
|
||||
@@ -0,0 +1,163 @@
|
||||
import numpy as np
|
||||
from numpy.core.einsumfunc import einsum
|
||||
from numpy.core.fromnumeric import swapaxes
|
||||
|
||||
class ElasticEnergy:
|
||||
def __init__(self, young, poisson):
|
||||
'''
|
||||
Input:
|
||||
- young : Young's modulus [Pa]
|
||||
- poisson : Poisson ratio
|
||||
'''
|
||||
self.young = young
|
||||
self.poisson = poisson
|
||||
self.lbda = young * poisson / ((1 + poisson) * (1 - 2 * poisson))
|
||||
self.mu = young / (2 * (1 + poisson))
|
||||
|
||||
self.psi = None
|
||||
self.E = None
|
||||
self.P = None
|
||||
|
||||
self.dE = None
|
||||
self.dP = None
|
||||
|
||||
def make_energy_density(self, jac):
|
||||
'''
|
||||
This method computes the energy density at each tetrahedron (#t,),
|
||||
and stores the result in self.psi
|
||||
|
||||
Input:
|
||||
- jac : jacobian of the deformation (#t, 3, 3)
|
||||
|
||||
Updated attributes:
|
||||
- psi : energy density per tet (#t,)
|
||||
'''
|
||||
|
||||
print("Please specify the kind of elasticity model.")
|
||||
raise NotImplementedError
|
||||
|
||||
def make_strain_tensor(self, jac):
|
||||
'''
|
||||
This method computes the strain tensor (#t, 3, 3), and stores it in self.E
|
||||
|
||||
Input:
|
||||
- jac : jacobian of the deformation (#t, 3, 3)
|
||||
|
||||
Updated attributes:
|
||||
- E : strain induced by the deformation (#t, 3, 3)
|
||||
'''
|
||||
|
||||
print("Please specify the kind of elasticity model.")
|
||||
raise NotImplementedError
|
||||
|
||||
def make_piola_kirchhoff_stress_tensor(self, jac):
|
||||
'''
|
||||
This method computes the stress tensor (#t, 3, 3), and stores it in self.P
|
||||
|
||||
Input:
|
||||
- jac : jacobian of the deformation (#t, 3, 3)
|
||||
|
||||
Updated attributes:
|
||||
- P : stress tensor induced by the deformation (#t, 3, 3)
|
||||
'''
|
||||
|
||||
print("Please specify the kind of elasticity model.")
|
||||
raise NotImplementedError
|
||||
|
||||
def make_differential_strain_tensor(self, jac, dJac):
|
||||
'''
|
||||
This method computes the differential of strain tensor (#t, 3, 3),
|
||||
and stores it in self.dE
|
||||
|
||||
Input:
|
||||
- jac : jacobian of the deformation (#t, 3, 3)
|
||||
- dJac : differential of the jacobian of the deformation (#t, 3, 3)
|
||||
|
||||
Updated attributes:
|
||||
- dE : differential of the strain tensor (#t, 3, 3)
|
||||
'''
|
||||
|
||||
print("Please specify the kind of elasticity model.")
|
||||
raise NotImplementedError
|
||||
|
||||
def make_differential_piola_kirchhoff_stress_tensor(self, jac, dJac):
|
||||
'''
|
||||
This method computes the differential of the stress tensor (#t, 3, 3), and stores it in self.dP
|
||||
|
||||
Input:
|
||||
- jac : jacobian of the deformation (#t, 3, 3)
|
||||
- dJac : differential of the jacobian of the deformation (#t, 3, 3)
|
||||
|
||||
Updated attributes:
|
||||
- dP : differential of the stress tensor (#t, 3, 3)
|
||||
'''
|
||||
|
||||
print("Please specify the kind of elasticity model.")
|
||||
raise NotImplementedError
|
||||
|
||||
class LinearElasticEnergy(ElasticEnergy):
|
||||
def __init__(self, young, poisson):
|
||||
super().__init__(young, poisson)
|
||||
|
||||
def make_energy_density(self, jac):
|
||||
fa = self.mu * np.einsum("ijk,ijk->i",self.E, self.E)
|
||||
fb = self.lbda/2.0 * np.einsum('ijj->i' ,self.E)**2
|
||||
self.psi = fa + fb
|
||||
|
||||
def make_strain_tensor(self, jac):
|
||||
self.E = (jac+np.transpose(jac,axes=(0,2,1)))/2.0 - np.identity(3)
|
||||
|
||||
def make_piola_kirchhoff_stress_tensor(self, jac):
|
||||
fa = 2.0*self.mu*self.E
|
||||
fb = self.lbda* np.einsum('i,jk->ijk',np.einsum('ijj->i',self.E), np.identity(3))
|
||||
self.P = fa + fb
|
||||
|
||||
def make_differential_strain_tensor(self, jac, dJac):
|
||||
self.dE = (dJac+np.transpose(dJac,axes=(0,2,1)))/2.0
|
||||
|
||||
def make_differential_piola_kirchhoff_stress_tensor(self, jac, dJac):
|
||||
fa = 2.0*self.mu*self.dE
|
||||
fb = self.lbda* np.einsum("i,jk->ijk",np.trace(self.dE, axis1=1,axis2=2), np.identity(3))
|
||||
self.dP = fa + fb
|
||||
|
||||
class NeoHookeanElasticEnergy(ElasticEnergy):
|
||||
def __init__(self, young, poisson):
|
||||
super().__init__(young, poisson)
|
||||
self.logJ = None
|
||||
self.Finv = None
|
||||
|
||||
def make_energy_density(self, jac):
|
||||
i1 = np.einsum('ijk,ijk->i',jac,jac)
|
||||
j = np.log(np.linalg.det(jac))
|
||||
fa = self.mu/2.0 * (i1-3 - 2*j)
|
||||
fb = self.lbda/2.0 * j**2
|
||||
self.psi = fa + fb
|
||||
|
||||
def make_strain_tensor(self, jac):
|
||||
pass
|
||||
|
||||
def make_piola_kirchhoff_stress_tensor(self, jac):
|
||||
'''
|
||||
Additional updated attributes:
|
||||
- logJ ; log of the determinant of the jacobians (#t,)
|
||||
- Finv : inverse of the jacobians (#t, 3, 3)
|
||||
'''
|
||||
self.logJ = np.log(np.linalg.det(jac))
|
||||
self.Finv = np.linalg.inv(jac)
|
||||
FinvT = np.transpose(self.Finv,axes=(0,2,1))
|
||||
fa = self.mu * (jac - FinvT)
|
||||
fb = self.lbda * np.einsum("i,ijk->ijk",self.logJ, FinvT)
|
||||
self.P = fa +fb
|
||||
|
||||
def make_differential_strain_tensor(self, jac, dJac):
|
||||
pass
|
||||
|
||||
def make_differential_piola_kirchhoff_stress_tensor(self, jac, dJac):
|
||||
self.logJ = np.log(np.linalg.det(jac))
|
||||
self.Finv = np.linalg.inv(jac)
|
||||
FinvT = np.transpose(self.Finv,axes=(0,2,1))
|
||||
dFT = np.transpose(dJac,axes=(0,2,1))
|
||||
fa = self.mu * dJac
|
||||
fb = np.einsum("i,ijk->ijk",(self.mu - self.lbda * self.logJ),FinvT @ dFT @ FinvT)
|
||||
fc = self.lbda * np.einsum("i,ijk->ijk",np.trace(self.Finv @ dJac, axis1=1,axis2=2),FinvT)
|
||||
self.dP = fa + fb + fc
|
||||
@@ -0,0 +1,404 @@
|
||||
import numpy as np
|
||||
from numpy import linalg
|
||||
from scipy import sparse
|
||||
from Utils import *
|
||||
import igl
|
||||
|
||||
|
||||
# -----------------------------------------------------------------------------
|
||||
# -----------------------------------------------------------------------------
|
||||
# ELASTIC SOLID CLASS
|
||||
# -----------------------------------------------------------------------------
|
||||
# -----------------------------------------------------------------------------
|
||||
|
||||
|
||||
class ElasticSolid(object):
|
||||
|
||||
def __init__(self, v_rest, t, ee, rho=1, pin_idx=[], f_mass=None):
|
||||
'''
|
||||
Input:
|
||||
- v_rest : position of the vertices of the mesh (#v, 3)
|
||||
- t : indices of the element's vertices (#t, 4)
|
||||
- ee : elastic energy object that can be found in elasticenergy.py
|
||||
- rho : mass per unit volume [kg.m-3]
|
||||
- pin_idx : list or np array of vertex indices to pin
|
||||
- f_mass : external force per unit mass (3,) [N.kg-1]
|
||||
'''
|
||||
|
||||
self.v_rest = v_rest.copy()
|
||||
self.v_def = v_rest.copy()
|
||||
self.t = t
|
||||
self.ee = ee
|
||||
self.rho = rho
|
||||
self.pin_idx = np.array(pin_idx)
|
||||
self.f_mass = f_mass.copy()
|
||||
self.free_idx = None
|
||||
self.pin_mask = None
|
||||
|
||||
self.W0 = None
|
||||
self.Dm = None
|
||||
self.Bm = None
|
||||
self.rest_barycenters = None
|
||||
|
||||
self.W = None
|
||||
self.Ds = None
|
||||
self.F = None
|
||||
self.def_barycenters = None
|
||||
|
||||
self.energy_el = None
|
||||
self.energy_ext = None
|
||||
|
||||
self.f = None
|
||||
self.f_vol = None
|
||||
self.f_ext = None
|
||||
|
||||
self.make_free_indices_and_pin_mask()
|
||||
self.update_rest_shape(self.v_rest)
|
||||
self.update_def_shape(self.v_def)
|
||||
|
||||
## Utils ##
|
||||
|
||||
def vertex_tet_sum(self, data):
|
||||
'''
|
||||
Distributes data specified at each tetrahedron to the neighboring vertices.
|
||||
All neighboring vertices will receive the value indicated at the corresponding tet position in data.
|
||||
|
||||
Input:
|
||||
- data : np array of shape (#t,) or (4*#t,)
|
||||
|
||||
Output:
|
||||
- data_sum : np array of shape (#v,), containing the summed data
|
||||
'''
|
||||
i = self.t.flatten('F') # (4*#t,)
|
||||
j = np.arange(len(self.t)) # (#t,)
|
||||
j = np.tile(j, 4) # (4*#t,)
|
||||
|
||||
if len(data) == len(self.t):
|
||||
data = data[j]
|
||||
|
||||
# Has shape (#v, #t)
|
||||
m = sparse.coo_matrix((data, (i, j)), (len(self.v_rest), len(self.t)))
|
||||
return np.array(m.sum(axis=1)).flatten()
|
||||
|
||||
## Precomputation ##
|
||||
|
||||
def make_free_indices_and_pin_mask(self):
|
||||
'''
|
||||
Should list all the free indices and the pin mask.
|
||||
|
||||
Updated attributes:
|
||||
- free_index : np array of shape (#free_vertices,) containing the list of unpinned vertices
|
||||
- pin_mask : np array of shape (#v, 1) containing 1 at free vertex indices and 0 at pinned vertex indices
|
||||
'''
|
||||
self.pin_mask = np.ones((self.v_rest.shape[0],1))
|
||||
self.free_idx = np.arange(0,self.v_rest.shape[0])
|
||||
if(self.pin_idx.shape[0]<=0):
|
||||
return
|
||||
self.pin_mask[self.pin_idx] = 0
|
||||
self.free_idx = np.where(self.pin_mask==1)[0]
|
||||
|
||||
|
||||
|
||||
## Methods related to rest quantities ##
|
||||
|
||||
def make_rest_barycenters(self):
|
||||
'''
|
||||
Construct the barycenters of the undeformed configuration
|
||||
|
||||
Updated attributes:
|
||||
- rest_barycenters : np array of shape (#t, 3) containing the position of each tet's barycenter
|
||||
'''
|
||||
self.rest_barycenters = np.einsum("ijk->ik",self.v_rest[self.t])/4
|
||||
|
||||
def make_rest_shape_matrices(self):
|
||||
'''
|
||||
Construct Ds that has shape (#t, 3, 3), and its inverse Bm
|
||||
|
||||
Updated attributes:
|
||||
- Dm : np array of shape (#t, 3, 3) containing the shape matrix of each tet
|
||||
- Bm : np array of shape (#t, 3, 3) containing the inverse shape matrix of each tet
|
||||
'''
|
||||
tv = self.v_rest[self.t]
|
||||
|
||||
tva = tv[:,:3,:]
|
||||
tvb = tv[:,3,:].reshape((self.t.shape[0],1,3))
|
||||
tvb= np.repeat(tvb, 3, axis=1)
|
||||
|
||||
tvs = tva - tvb
|
||||
tvsf = np.transpose(tvs,(0,2,1))
|
||||
|
||||
self.Dm = tvsf
|
||||
self.Bm = np.linalg.inv(tvsf)
|
||||
|
||||
def update_rest_shape(self, v_rest):
|
||||
'''
|
||||
Updates the vertex position, the shape matrices Dm and Bm, the volumes W0,
|
||||
and the mass matrix at rest
|
||||
|
||||
Input:
|
||||
- v_rest : position of the vertices of the mesh at rest state (#v, 3)
|
||||
|
||||
Updated attributes:
|
||||
- v_rest : np array of shape (#v, 3) containing the position of each vertex at rest
|
||||
- W0 : np array of shape (#t,) containing the signed volume of each tet
|
||||
'''
|
||||
self.v_rest = v_rest
|
||||
self.make_rest_barycenters()
|
||||
self.make_rest_shape_matrices()
|
||||
self.W0 = -np.linalg.det(self.Dm)/6
|
||||
self.update_def_shape(self.v_def)
|
||||
self.make_volumetric_and_external_forces()
|
||||
|
||||
## Methods related to deformed quantities ##
|
||||
|
||||
def make_def_barycenters(self):
|
||||
'''
|
||||
Construct the barycenters of the deformed configuration
|
||||
|
||||
Updated attributes:
|
||||
- def_barycenters : np array of shape (#t, 3) containing the position of each tet's barycenter
|
||||
'''
|
||||
self.def_barycenters = np.einsum("ijk->ik",self.v_def[self.t])/4
|
||||
|
||||
def make_def_shape_matrices(self):
|
||||
'''
|
||||
Construct Ds that has shape (#t, 3, 3)
|
||||
|
||||
Updated attributes:
|
||||
- Ds : np array of shape (#t, 3, 3) containing the shape matrix of each tet
|
||||
'''
|
||||
tv = self.v_def[self.t]
|
||||
|
||||
tva = tv[:,:3,:]
|
||||
tvb = tv[:,3,:].reshape((self.t.shape[0],1,3))
|
||||
tvb= np.repeat(tvb, 3, axis=1)
|
||||
|
||||
tvs = tva - tvb
|
||||
tvsf = np.transpose(tvs,(0,2,1))
|
||||
|
||||
self.Ds = tvsf
|
||||
|
||||
def make_jacobians(self):
|
||||
'''
|
||||
Compute the current Jacobian of the deformation
|
||||
|
||||
Updated attributes:
|
||||
- F : np array of shape (#t, 3, 3) containing Jacobian of the deformation in each tet
|
||||
'''
|
||||
self.F = self.Ds @ self.Bm # <=> np.einsum("lij,ljk->lik",self.Ds,self.Bm)
|
||||
|
||||
def update_def_shape(self, v_def):
|
||||
'''
|
||||
Updates the vertex position, the Jacobian of the deformation, and the
|
||||
resulting elastic forces.
|
||||
|
||||
Input:
|
||||
- v_def : position of the vertices of the mesh (#v, 3)
|
||||
|
||||
Updated attributes:
|
||||
- v_def : np array of shape (#v, 3) containing the position of each vertex after deforming the solid
|
||||
- W : np array of shape (#t,) containing the signed volume of each tet
|
||||
'''
|
||||
self.v_def = self.v_rest
|
||||
self.v_def[self.free_idx] = v_def[self.free_idx]
|
||||
self.make_def_barycenters()
|
||||
self.make_def_shape_matrices()
|
||||
self.W = -np.linalg.det(self.Ds)/6
|
||||
self.make_jacobians()
|
||||
if(self.f_vol is None):
|
||||
self.make_volumetric_and_external_forces()
|
||||
self.make_elastic_forces()
|
||||
self.make_elastic_energy()
|
||||
self.make_external_energy()
|
||||
|
||||
def displace(self, v_disp):
|
||||
'''
|
||||
Displace the whole mesh so that v_def += v_disp
|
||||
|
||||
Input:
|
||||
- v_disp : displacement of the vertices of the mesh (#v, 3)
|
||||
'''
|
||||
self.update_def_shape(self.v_def + v_disp)
|
||||
|
||||
## Energies ##
|
||||
|
||||
def make_elastic_energy(self):
|
||||
'''
|
||||
This updates the elastic energy
|
||||
|
||||
Updated attributes:
|
||||
- energy_el : elastic energy of the system [J]
|
||||
'''
|
||||
self.ee.make_strain_tensor(self.F)
|
||||
self.ee.make_energy_density(self.F)
|
||||
self.energy_el = np.sum(self.W0 * self.ee.psi)
|
||||
|
||||
def make_external_energy(self):
|
||||
'''
|
||||
This computes the external energy potential
|
||||
|
||||
Updated attributes:
|
||||
- energy_ext : postential energy due to external forces [J]
|
||||
'''
|
||||
dx = self.def_barycenters - self.rest_barycenters
|
||||
ifv = np.einsum('i,ij -> ij', self.W0, self.f_vol)
|
||||
self.energy_ext = - np.einsum("i,ij,ij->",self.W0, self.f_vol, dx)
|
||||
|
||||
## Forces ##
|
||||
|
||||
def make_elastic_forces(self):
|
||||
'''
|
||||
This method updates the elastic forces stored in self.f (#v, 3)
|
||||
|
||||
Updated attributes:
|
||||
- f : elastic forces per vertex (#v, 3)
|
||||
- ee : elastic energy, some attributes should be updated
|
||||
'''
|
||||
self.ee.make_strain_tensor(self.F)
|
||||
self.ee.make_piola_kirchhoff_stress_tensor(self.F)
|
||||
tdm = np.transpose(self.Bm,(0,2,1))
|
||||
ptdm = self.ee.P @ tdm
|
||||
f123 = -np.einsum("i,ijk->ikj",self.W0,ptdm )
|
||||
f4 = -f123[:,0] - f123[:,1] - f123[:,2]
|
||||
H = np.hstack((f123,f4[:,None]))
|
||||
|
||||
x = self.vertex_tet_sum(H[:,:,0].flatten('F'))
|
||||
y = self.vertex_tet_sum(H[:,:,1].flatten('F'))
|
||||
z = self.vertex_tet_sum(H[:,:,2].flatten('F'))
|
||||
|
||||
self.f = np.hstack((x[:,None],y[:,None],z[:,None]))
|
||||
|
||||
def make_volumetric_and_external_forces(self):
|
||||
'''
|
||||
Convert force per unit mass to volumetric forces, then distribute
|
||||
the forces to the vertices of the mesh.
|
||||
|
||||
Updated attributes:
|
||||
- f_vol : np array of shape (#t, 3) external force per unit volume acting on the tets
|
||||
- f_ext : np array of shape (#v, 3) external force acting on the vertices
|
||||
'''
|
||||
mpv = self.f_mass*self.rho
|
||||
self.f_vol = np.ones((self.W0.shape[0],1)) * mpv
|
||||
|
||||
ifv = np.einsum('i,ij -> ij', self.W0, self.f_vol)
|
||||
|
||||
f_ext_x = self.vertex_tet_sum(ifv[:,0])
|
||||
f_ext_y = self.vertex_tet_sum(ifv[:,1])
|
||||
f_ext_z = self.vertex_tet_sum(ifv[:,2])
|
||||
|
||||
self.f_ext = np.stack((f_ext_x,f_ext_y,f_ext_z),axis=1) / 4.0
|
||||
|
||||
## Force Differentials
|
||||
|
||||
def compute_force_differentials(self, v_disp):
|
||||
'''
|
||||
This computes the differential of the force given a displacement dx,
|
||||
where df = df/dx|x . dx = - K(x).dx. Where K(x) is the stiffness matrix (or Hessian)
|
||||
of the solid. Note that the implementation doesn't need to construct the stiffness matrix explicitly.
|
||||
|
||||
Input:
|
||||
- v_disp : displacement of the vertices of the mesh (#v, 3)
|
||||
|
||||
Output:
|
||||
- df : force differentials at the vertices of the mesh (#v, 3)
|
||||
|
||||
Updated attributes:
|
||||
- ee : elastic energy, some attributes should be updated
|
||||
'''
|
||||
|
||||
# First compute the differential of the Jacobian
|
||||
tv = v_disp[self.t]
|
||||
tva = tv[:,:3,:]
|
||||
tvb = tv[:,3,:].reshape((-1,1,3))
|
||||
|
||||
tvb= np.repeat(tvb, 3, axis=1)
|
||||
|
||||
tvs = tva - tvb
|
||||
tvsf = np.transpose(tvs,(0,2,1))
|
||||
|
||||
dDs = tvsf
|
||||
dF = dDs @ self.Bm
|
||||
|
||||
# Then update differential quantities in self.ee
|
||||
self.ee.make_differential_strain_tensor(self.F,dF)
|
||||
self.ee.make_differential_piola_kirchhoff_stress_tensor(self.F,dF)
|
||||
|
||||
# Compute the differential of the forces
|
||||
tdm = np.transpose(self.Bm,(0,2,1))
|
||||
ptdm = self.ee.dP @ tdm
|
||||
f123 = -np.einsum("i,ijk->ikj",self.W0,ptdm)
|
||||
f4 = -f123[:,0] - f123[:,1] - f123[:,2]
|
||||
H = np.hstack((f123,f4[:,None]))
|
||||
|
||||
x = self.vertex_tet_sum(H[:,:,0].flatten('F'))
|
||||
y = self.vertex_tet_sum(H[:,:,1].flatten('F'))
|
||||
z = self.vertex_tet_sum(H[:,:,2].flatten('F'))
|
||||
|
||||
return np.hstack((x[:,None],y[:,None],z[:,None]))
|
||||
|
||||
def equilibrium_step(self, step_size_init = 2, max_l_iter = 20, c1 = 1e-4):
|
||||
'''
|
||||
This function displaces the whole solid to the next deformed configuration
|
||||
using a Newton-CG step.
|
||||
|
||||
Updated attributes:
|
||||
- LHS : The hessian vector product
|
||||
- RHS : Right hand side for the conjugate gradient linear solve
|
||||
Other than them, only attributes updated by displace(self, v_disp) should be changed
|
||||
'''
|
||||
|
||||
# Define LHS
|
||||
def LHS(dx):
|
||||
'''
|
||||
Should implement the Hessian-Vector Product L(dx), and take care of pinning constraints
|
||||
as described in the handout.
|
||||
'''
|
||||
if len(dx.shape)==1:
|
||||
dx = dx.reshape((-1,3))
|
||||
|
||||
if (dx.shape[0] < self.v_rest.shape[0]):
|
||||
dxf = np.zeros((self.v_rest.shape[0],3))
|
||||
dxf[self.free_idx] = dx
|
||||
dx = dxf
|
||||
|
||||
return -self.compute_force_differentials(dx)[self.free_idx,:].flatten()
|
||||
|
||||
self.LHS = LHS # Save to class for testing
|
||||
|
||||
# Define RHS
|
||||
RHS = (self.f + self.f_ext)[self.free_idx].flatten()
|
||||
|
||||
self.RHS = RHS # Save to class for testing
|
||||
|
||||
# Use conjugate gradient to find a descent direction
|
||||
# (see conjugate_gradient in Utils.py)
|
||||
dx_CG = conjugate_gradient(LHS,RHS).reshape(-1,3) # shape (#v, 3)
|
||||
if (dx_CG.shape[0] < self.v_rest.shape[0]):
|
||||
dxf = np.zeros((self.v_rest.shape[0],3))
|
||||
dxf[self.free_idx] = dx_CG
|
||||
dx_CG = dxf
|
||||
|
||||
# Run line search on the direction
|
||||
step_size = step_size_init
|
||||
ft = self.f + self.f_ext
|
||||
energy_tot_prev = (self.energy_el + self.energy_ext)
|
||||
for l_iter in range(max_l_iter):
|
||||
step_size *= 0.5 # DO NOT CHANGE
|
||||
dx_search = dx_CG * step_size
|
||||
self.displace(dx_search)
|
||||
|
||||
# Current force norm for unpinned vertices
|
||||
f_tmp = (self.f + self.f_ext)[self.free_idx]
|
||||
g = np.linalg.norm(f_tmp)
|
||||
|
||||
energy_tot_tmp = self.energy_el + self.energy_ext
|
||||
|
||||
#Make sure you only use the gradient of the unpinned vertices
|
||||
# USE parameter c1 = 1e-4 (from the function arguments) when you submit
|
||||
armijo = energy_tot_tmp < energy_tot_prev + c1 * step_size*(-dx_CG*dx_CG).sum()
|
||||
|
||||
if armijo or l_iter == max_l_iter-1:
|
||||
print("Energy: " + str(energy_tot_tmp) + " Force norm: " + str(g) + " Line search Iters: " + str(l_iter))
|
||||
break
|
||||
else:
|
||||
self.displace(-dx_search)
|
||||
@@ -0,0 +1,13 @@
|
||||
import numpy as np
|
||||
|
||||
def objective_target_BV(v, vt, bv):
|
||||
'''
|
||||
Input:
|
||||
- v : array of shape (#v, 3), containing the current vertices position
|
||||
- vt : array of shape (#bv, 3), containing the target surface
|
||||
- bv : boundary vertices index (#bv,)
|
||||
|
||||
Output:
|
||||
- objective : single scalar measuring the deviation from the target shape
|
||||
'''
|
||||
return 0.
|
||||
@@ -0,0 +1,51 @@
|
||||
import numpy as np
|
||||
from objectives import *
|
||||
from Utils import *
|
||||
|
||||
def stretch_from_point(v, stretches, v_idx):
|
||||
'''
|
||||
Input:
|
||||
- v : vertices to be deformed (#v, 3)
|
||||
- stretches : np array of shape (3,) giving the stretching coefficients along x, y, z
|
||||
- v_idx : vertex index around which the mesh is stretched
|
||||
|
||||
Output:
|
||||
- v_stretched : deformed vertices (#v, 3)
|
||||
'''
|
||||
v_stretched = v
|
||||
return v_stretched
|
||||
|
||||
def report_stretches(solid, v_rest_init, bv, v_target, stretches, lowest_pin_idx):
|
||||
'''
|
||||
Input:
|
||||
- solid : an elastic solid to deform
|
||||
- v_rest_init : reference vertex position to compress/stretch (#v, 3)
|
||||
- bv : boundary vertices index (#bv,)
|
||||
- v_target : target boundary vertices position
|
||||
- stretches : np array of shape (n_stretches,) containing a stretch factors to try out
|
||||
- lowest_pin_idx : index of the pinned index that has the lowest z coordinate
|
||||
|
||||
Output:
|
||||
- list_v_rest : list of n_stretches rest vertex positions that have been tried out
|
||||
- list_v_eq : list of the corresponding n_stretches equilibrium vertex positions
|
||||
- target_closeness : np array of shape (n_stretches,) containing the objective values for each stretch factor
|
||||
'''
|
||||
list_v_rest = []
|
||||
list_v_eq = []
|
||||
target_closeness = np.zeros(shape=(stretches.shape[0],))
|
||||
v_rest_tmp = v_rest_init.copy()
|
||||
|
||||
for i, stretch in enumerate(stretches):
|
||||
# Compute and update the rest shape of the solid
|
||||
|
||||
# Compute new equilibrium (use the previous equilibrium state as initial guess if available)
|
||||
# You may use equilibrium_convergence_report_NCG to find the equilibrium (20 steps and thresh=1 should do)
|
||||
|
||||
# Update guess for next stretch factor
|
||||
|
||||
# Fill in the
|
||||
list_v_rest.append(solid.v_rest.copy())
|
||||
list_v_eq.append(solid.v_def.copy())
|
||||
target_closeness[i] = objective_target_BV(solid.v_def, v_target, bv)
|
||||
|
||||
return list_v_rest, list_v_eq, target_closeness
|
||||
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|
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@@ -0,0 +1,83 @@
|
||||
import time
|
||||
import pytest
|
||||
import json
|
||||
import sys
|
||||
import igl
|
||||
import numpy as np
|
||||
sys.path.append('../')
|
||||
sys.path.append('../src')
|
||||
from elasticsolid import *
|
||||
from elasticenergy import *
|
||||
eps = 1E-6
|
||||
|
||||
with open('test_data3.json', 'r') as infile:
|
||||
homework_datas = json.load(infile)
|
||||
|
||||
@pytest.mark.timeout(0.5)
|
||||
@pytest.mark.parametrize("data", homework_datas[0])
|
||||
def test_linear_dE(data):
|
||||
young, poisson, F, dF, dE_gt = data
|
||||
ee = LinearElasticEnergy(young, poisson)
|
||||
ee.make_differential_strain_tensor(np.array(F), np.array(dF))
|
||||
assert np.linalg.norm(ee.dE - np.array(dE_gt)) < eps
|
||||
|
||||
@pytest.mark.timeout(0.5)
|
||||
@pytest.mark.parametrize("data", homework_datas[1])
|
||||
def test_linear_dP(data):
|
||||
young, poisson, F, dF, dE, dP_gt = data
|
||||
ee = LinearElasticEnergy(young, poisson)
|
||||
ee.dE = np.array(dE)
|
||||
ee.make_differential_piola_kirchhoff_stress_tensor(np.array(F), np.array(dF))
|
||||
assert np.linalg.norm(ee.dP - np.array(dP_gt)) < eps
|
||||
|
||||
@pytest.mark.timeout(0.5)
|
||||
@pytest.mark.parametrize("data", homework_datas[2])
|
||||
def test_neo_dP(data):
|
||||
young, poisson, F, dF, dP_gt = data
|
||||
ee = NeoHookeanElasticEnergy(young, poisson)
|
||||
ee.logJ = np.log(np.linalg.det(np.array(F)))
|
||||
ee.Finv = np.linalg.inv(np.array(F))
|
||||
ee.make_differential_piola_kirchhoff_stress_tensor(np.array(F), np.array(dF))
|
||||
assert np.linalg.norm(ee.dP - np.array(dP_gt)) < eps
|
||||
|
||||
@pytest.mark.timeout(0.5)
|
||||
@pytest.mark.parametrize("data", homework_datas[3])
|
||||
def test_force_differentials(data):
|
||||
etype, young, poisson, v, t, rho, pin_idx, force_mass, v_def, dx, df_gt = data
|
||||
if etype == "linear":
|
||||
ee = LinearElasticEnergy(young, poisson)
|
||||
else:
|
||||
ee = NeoHookeanElasticEnergy(young, poisson)
|
||||
|
||||
es = ElasticSolid(np.array(v), np.array(t), ee, rho=rho, pin_idx=pin_idx, f_mass=np.array(force_mass))
|
||||
es.update_def_shape(np.array(v_def))
|
||||
|
||||
assert np.linalg.norm(es.compute_force_differentials(np.array(dx)) - np.array(df_gt)) < eps
|
||||
|
||||
@pytest.mark.timeout(1)
|
||||
@pytest.mark.parametrize("data", homework_datas[4])
|
||||
def test_CG(data):
|
||||
A, RHS, dx_gt = data
|
||||
A = np.array(A)
|
||||
|
||||
def LHS(dx_):
|
||||
return A@dx_
|
||||
assert np.linalg.norm(conjugate_gradient(LHS, np.array(RHS)) - np.array(dx_gt)) < eps
|
||||
|
||||
@pytest.mark.timeout(1)
|
||||
@pytest.mark.parametrize("data", homework_datas[5])
|
||||
def test_lhs_rhs(data):
|
||||
etype, young, poisson, v, t, rho, pin_idx, force_mass, v_def, dx, lhsdx_gt, rhs_gt = data
|
||||
if etype == "linear":
|
||||
ee = LinearElasticEnergy(young, poisson)
|
||||
else:
|
||||
ee = NeoHookeanElasticEnergy(young, poisson)
|
||||
|
||||
es = ElasticSolid(np.array(v), np.array(t), ee, rho=rho, pin_idx=pin_idx, f_mass=np.array(force_mass))
|
||||
es.update_def_shape(np.array(v_def))
|
||||
es.equilibrium_step(max_l_iter=0)
|
||||
|
||||
|
||||
assert np.linalg.norm(es.LHS(np.array(dx)) - np.array(lhsdx_gt)) < eps
|
||||
print(es.RHS, rhs_gt)
|
||||
assert np.linalg.norm(es.RHS - rhs_gt) < eps
|
||||
@@ -0,0 +1 @@
|
||||
[[[10000000.0, 0.2, [[[3.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[3.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[3.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[3.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[3.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[3.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[3.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[3.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[3.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[3.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[3.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[3.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[3.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[3.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[3.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[3.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[3.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[3.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]], [[2.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0Line truncated
|
||||
Reference in new issue
Block a user