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choelzl
2022-04-07 18:46:57 +02:00
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commit 15e7120d6d
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{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Description\n",
"\n",
"This notebook intends to gather all the functionalities you'll have to implement for assignment 2.1. You will have to generate an elastic solid, deform it, compute the associated Jacobian of the deformation map $\\phi$, and implement pinning constraints. You will also visualize the eigenvectors and eigenvalues of the metric tensor, given a prescribed deformation.\n",
"\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",
"\n",
"import sys as _sys\n",
"_sys.path.append(\"../src\")\n",
"from elasticsolid import *\n",
"from eigendecomposition_metric import *\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": "33e8252ccc4d4c9298094b0dc3675ee4",
"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",
"\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",
"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": [
"# Manipulate elastic solids\n",
"\n",
"## Instanciation\n",
"\n",
"The rest shape matrices $D_m$ and their inverse matrices $B_m$ are computed during instanciation."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [],
"source": [
"rho = 131 # [kg.m-3]\n",
"solid = ElasticSolid(v, t, rho=rho)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Deform the mesh\n",
"\n",
"This part involves Jacobian computation which relies on deformed shape matrices $D_s$."
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"scrolled": false
},
"outputs": [
{
"data": {
"application/vnd.jupyter.widget-view+json": {
"model_id": "02c94de8642645d8aab800877f8ddbce",
"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": [
"<meshplot.Viewer.Viewer at 0x7fa7dc789070>"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v_def = v.copy()\n",
"v_def[:, 2] *= 2.\n",
"solid.update_def_shape(v_def)\n",
"\n",
"mp.plot(solid.v_def @ rot.T, solid.t, shading=shadingOptions)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Visualize some properties of the metric tensor\n",
"\n",
"The metric tensor measures how stretched and sheared directions in the undeformed space are under the deformation $\\phi$. It is defined from the Jacobian of the deformation $\\mathbf{F}$ as follow (see the handout for a derivation):\n",
"\n",
"$$\\mathbf{M} = \\mathbf{F}^T \\mathbf{F}$$\n",
"\n",
"We intend to plot the eigenvectors coloured by the corresponding eigenvalues in the next cell."
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"data": {
"application/vnd.jupyter.widget-view+json": {
"model_id": "6ff42798692c4a29ad4a37f40ca77a84",
"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": [
"# We limit ourselves to stretching the mesh in the z direction\n",
"# Feel free to experiment with other kinds of deformations!\n",
"\n",
"v_def = v.copy()\n",
"v_def[:, 2] *= 2.0\n",
"solid.update_def_shape(v_def)\n",
"\n",
"squared_eigvals, eigvecs = compute_eigendecomposition_metric(solid.F)\n",
"plot_eigendecomposition_metric(solid, squared_eigvals, eigvecs, rot, scale=0.05)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Pin vertices of the mesh\n",
"\n",
"Pass a `pin_idx` to the constructor, compute the mask for deformations."
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [],
"source": [
"maxZ = np.max(solid.v_rest[:, 2])\n",
"pin_idx = np.arange(solid.v_rest.shape[0])[solid.v_rest[:, 2] > maxZ - 0.1 * aabb[2]]\n",
"\n",
"v_def = v.copy()\n",
"v_def[:, 2] -= 0.1 * aabb[2]\n",
"solid.update_def_shape(v_def)\n",
"\n",
"solid_pinned = ElasticSolid(v, t, rho=rho, pin_idx=pin_idx)\n",
"solid_pinned.update_def_shape(v_def)"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
"application/vnd.jupyter.widget-view+json": {
"model_id": "13f958a33ec945ca979aca15d7381e01",
"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": [
"1"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"p = mp.plot(solid_pinned.v_def @ rot.T, t, shading=shadingOptions)\n",
"p.add_points(solid_pinned.v_def[pin_idx, :] @ rot.T, shading={\"point_color\":\"black\", \"point_size\": 0.1 * length_scale})"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 1.2.4 Derivation"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Volume of tetrahedron ($X_1, X_2, X_3, X_4$) is given by $\\frac{1}{6}|det(D_m)|$\n",
"\n",
"We can decompose the provided formula:\n",
"\n",
"$Vol = \\frac{1}{6}|det(D_m)| = \\frac{1}{6}|D_{m1}^T \\cdot (D_{m2} \\times D_{m3})|$\n",
"\n",
"\n",
"Next, we build 3 vectors:\n",
"\n",
"$V_0 = X_1 - X_4 = \\begin{pmatrix}X_{x1}-X_{x4} \\\\ Y_{x1}-Y_{x4} \\\\ Z_{x1}- Z_{x4} \\end{pmatrix}$\n",
"\n",
"$V_1 = X_2 - X_4 = \\begin{pmatrix}X_{x2}-X_{x4} \\\\ Y_{x2}-Y_{x4} \\\\ Z_{x2}- Z_{x4} \\end{pmatrix}$\n",
"\n",
"$V_2 = X_3 - X_4 = \\begin{pmatrix}X_{x3}-X_{x4} \\\\ Y_{x3}-Y_{x4} \\\\ Z_{x3}- Z_{x4} \\end{pmatrix}$\n",
"\n",
"We then compute the base as:\n",
"\n",
"$B = V_1 \\times V_2$\n",
"\n",
"And also the height as:\n",
"\n",
"$H = V_0$\n",
"\n",
"From this we can compute the volume:\n",
"\n",
"$Vol = \\frac{1}{6} \\cdot B \\cdot H = \\frac{1}{6} V_0 \\cdot (V_1 \\times V_2)$\n",
"\n",
"We can note the similarity between both fomulas. We can note the equality between both formulas: \n",
"\n",
"$V_0 = D_{m1}, V_1 = D_{m2}, V_2 = D_{m3}$\n",
"\n",
"Note that the absolute value is not directly taken into account since its absolute volume and not signed volume.\n",
"\n",
"\n",
"\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 1.2.7 Derivation"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Show that: $Vol(x_1, x_2, x_3, x_4)/Vol(X_1, X_2, X_3, X_4) = |det(F)|$\n",
"\n",
"Since we know that $F = D_s D_m^{-1}$, and that: $V = \\frac{1}{6}|det(D_m)|$\n",
"\n",
"We can drag the det and abs into it:\n",
"\n",
"$|det(F)| = \\frac{|det(D_s)|}{|det(D_m)|} = \\frac{\\frac{1}{6}|det(D_s)|}{\\frac{1}{6}|det(D_m)|} = \\frac{Vol(x_1, x_2, x_3, x_4)}{Vol(X_1, X_2, X_3, X_4)}$\n",
"\n",
"\n",
"\n"
]
},
{
"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
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"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.9.7"
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"toc": {
"base_numbering": 1,
"nav_menu": {},
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"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
}

268
cs457-gc/assignment_2_1/notebook/tutorial_higher_order_arrays.ipynb Normal file
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{
"cells": [
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"import time"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Tutorial: Higher Order Array Manipulation\n",
"\n",
"For the sake of simplicity, we define $n$-th order tensors as arrays of dimension $n$. A $0$-th order array is a scalar, a $1$-st order array is a vector in $\\mathbb{R}^{d_1}$, and a $2$-nd order array is a matrix in $\\mathbb{R}^{d_1\\times d_2}$. Going further, a $n$-th order array is an element of $\\mathbb{R}^{d_1\\times...\\times d_n}$ for some dimensions $(d_i)_{i\\in[n]}$.\n",
"\n",
"## Declaration"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"# Declare a third order array \n",
"d1, d2, d3 = 2, 3, 5\n",
"A = np.random.rand(d1, d2, d3)\n",
"\n",
"print(\"The shape of A is {}\".format(A.shape))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Indexing\n",
"\n",
"Say we have a $3$-rd order array $\\mathbf{A}\\in\\mathbb{R}^{d_1\\times d_2\\times d_3}$. Indexing and slicing works as for lower order arrays:"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"print(\"A[0] has shape {}\".format(A[0].shape))\n",
"print(\"A[:, 1:, :] has shape {}\".format(A[:, 1:, :].shape))\n",
"print(\"A[:, 1, 2:4] has shape {}\".format(A[:, 1, 2:4].shape))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can also use a different indexing array $\\mathbf{b}$ to index $\\mathbf{A}$. This indexing operates on the first dimension of $\\mathbf{A}$, meaning that if $\\mathbf{b}\\in\\mathbb{R}^{l_1\\times l_2}$, then `A[b]` will have shape $l_1\\times l_2\\times d_2\\times d_3$."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"b = np.array([0, 0, 1, 0])\n",
"print(\"If A has shape {}, b has shape {}, then A[b] has shape {}.\".format(A.shape, b.shape, A[b].shape))\n",
"\n",
"b = np.array([[0, 0, 1, 0], [1, 1, 0, 1]])\n",
"print(\"If A has shape {}, b has shape {}, then A[b] has shape {}.\".format(A.shape, b.shape, A[b].shape))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"... This works provided the indexing array $\\mathbf{b}$ has integer values comprised between $0$ and $d_{1}-1$ (included)."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"try:\n",
" b = np.array([0, 0, 2, 0])\n",
" A[b]\n",
"except Exception as e:\n",
" print(\"We have an out-of bound indexing: d_1=1 but max b=2\")\n",
" print(\"The exception is: {}\".format(e))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Operations\n",
"\n",
"Imagine now that we have a batch of $1000$ $d\\times d$ matrices: $(\\mathbf{a}_i)_{i\\in[1000]}$, for which we want to compute the trace. We could loop over the matrices and compute the traces separately."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"d = 2\n",
"\n",
"ais = [np.random.rand(d, d) for i in range(1000)]\n",
"\n",
"start = time.time()\n",
"traces = ...\n",
"end = time.time()\n",
"\n",
"print(\"Elapsed time: {:.2e}s.\".format(end - start))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Alternatively, we could vectorize this operation using a three dimensional array $\\mathbf{A}\\in\\mathbb{R}^{3\\times d\\times d}$ that contains the stacked matrices."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"A = np.stack(ais, axis=0)\n",
"\n",
"print(\"A has shape {}\".format(A.shape))\n",
"\n",
"start = time.time()\n",
"traces = ...\n",
"end = time.time()\n",
"\n",
"print(\"Elapsed time: {:.2e}s.\".format(end - start))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"And we reduced the computation time by an order of magnitude! A different option that we will use extensively during part 3 is to use [Einstein summation](https://en.wikipedia.org/wiki/Einstein_notation). For the traces computation this would be written: $\\mathbf{A}_{i,j,j}$. This can be done with Numpy with the method [`np.einsum`](https://numpy.org/doc/stable/reference/generated/numpy.einsum.html)."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"start = time.time()\n",
"traces = ...\n",
"end = time.time()\n",
"\n",
"print(\"Elapsed time: {:.2e}s.\".format(end - start))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"As efficient as the trace method! Also, `np.einsum` is highly flexible. It can compute the transpose of a batch of arrays, or various kinds of matrix multiplications."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"ais = [np.random.rand(2, 3) for i in range(1000)]\n",
"\n",
"# Transpose each stacked matrices\n",
"A = np.stack(ais, axis=0)\n",
"print(\"A has shape {}\".format(A.shape))\n",
"AT = ...\n",
"print(\"A^T has shape {}\".format(AT.shape))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Below we show how to compute $\\mathbf{a}_i^T\\mathbf{a}_i$ for some matrices $\\mathbf{a}_i\\in\\mathbb{R}^{2\\times 3}$ using `np.einsum`."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"print(\"A has shape {}\".format(A.shape))\n",
"product_As = ...\n",
"print(\"Stacked ai^T.ai has shape {}\".format(product_As.shape))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"To scale each $\\mathbf{a}_i$ by a weight $w_i$, we can still use `np.einsum`. Define the vector containing all the weights $\\mathbf{w}=(w_i)_i$, we have:"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"w = np.random.rand(A.shape[0])\n",
"weighted_A = ...\n",
"print(\"Weighted and stacked ai has shape {}\".format(weighted_A.shape))"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
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},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
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"version": "3.7.10"
},
"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
}

96
cs457-gc/assignment_2_1/src/eigendecomposition_metric.py Normal file
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import numpy as np
import meshplot as mp
import igl
def compute_eigendecomposition_metric(jac):
'''
Input:
- jac : a np array of shape (#t, 3, 3) containing the stacked jacobians
Output:
- squared_eigvals : np array of shape (#t, 3) containing the square root of the eigenvalues of the metric tensor
- eigvecs : np array of shape (#t, 3, 3) containing the eigenvectors of the metric tensor
'''
jact = np.transpose(jac,(0,2,1))
smt = jact @ jac
eigvals, eigvecs = np.linalg.eigh(smt)
return np.sqrt(eigvals), eigvecs #/ np.linalg.norm(eigvecs)
def plot_eigendecomposition_metric(solid, squared_eigvals, eigvecs, rot, scale=0.05):
'''
Input:
- solid : an ElasticSolid object
- squared_eigvals : np array of shape (#t, 3) containing the square root of the eigenvalues of the metric tensor
- eigvecs : np array of shape (#t, 3, 3) containing the eigenvectors of the metric tensor
- rot : a rotation matrix for plotting purposes
- scale : scaling for plotting purposes
'''
scaled_eigvecs = scale * np.einsum('ik, ijk -> ijk', squared_eigvals, eigvecs)
start_plot0 = (solid.def_barycenters - scaled_eigvecs[..., 0]) @ rot.T
start_plot1 = (solid.def_barycenters - scaled_eigvecs[..., 1]) @ rot.T
start_plot2 = (solid.def_barycenters - scaled_eigvecs[..., 2]) @ rot.T
end_plot0 = (solid.def_barycenters + scaled_eigvecs[..., 0]) @ rot.T
end_plot1 = (solid.def_barycenters + scaled_eigvecs[..., 1]) @ rot.T
end_plot2 = (solid.def_barycenters + scaled_eigvecs[..., 2]) @ rot.T
# Get boundary edges
be = igl.edges(igl.boundary_facets(solid.t))
p = mp.plot(solid.v_def @ rot.T, be, shading={"line_color": "black"})
p.add_points(solid.def_barycenters @ rot.T, shading={"point_color":"black", "point_size": 0.2})
# In tension
tens0 = np.argwhere(squared_eigvals[:, 0]>1. + 1e-6)
tens1 = np.argwhere(squared_eigvals[:, 1]>1. + 1e-6)
tens2 = np.argwhere(squared_eigvals[:, 2]>1. + 1e-6)
if tens0.shape[0] != 0:
p.add_lines(start_plot0[tens0, :],
end_plot0[tens0, :],
shading={"line_color": "#182C94"})
if tens1.shape[0] != 0:
p.add_lines(start_plot1[tens1, :],
end_plot1[tens1, :],
shading={"line_color": "#182C94"})
if tens2.shape[0] != 0:
p.add_lines(start_plot2[tens2, :],
end_plot2[tens2, :],
shading={"line_color": "#182C94"})
# In compression
comp0 = np.argwhere(squared_eigvals[:, 0]<1. - 1e-6)
comp1 = np.argwhere(squared_eigvals[:, 1]<1. - 1e-6)
comp2 = np.argwhere(squared_eigvals[:, 2]<1. - 1e-6)
if comp0.shape[0] != 0:
p.add_lines(start_plot0[comp0, :],
end_plot0[comp0, :],
shading={"line_color": "#892623"})
if comp1.shape[0] != 0:
p.add_lines(start_plot1[comp1, :],
end_plot1[comp1, :],
shading={"line_color": "#892623"})
if comp2.shape[0] != 0:
p.add_lines(start_plot2[comp2, :],
end_plot2[comp2, :],
shading={"line_color": "#892623"})
# Neutral
# In compression
neut0 = np.argwhere(abs(squared_eigvals[:, 0]-1.) < 1e-6)
neut1 = np.argwhere(abs(squared_eigvals[:, 1]-1.) < 1e-6)
neut2 = np.argwhere(abs(squared_eigvals[:, 2]-1.) < 1e-6)
if neut0.shape[0] != 0:
p.add_lines(start_plot0[neut0, :],
end_plot0[neut0, :],
shading={"line_color": "#027337"})
if neut1.shape[0] != 0:
p.add_lines(start_plot1[neut1, :],
end_plot1[neut1, :],
shading={"line_color": "#027337"})
if neut2.shape[0] != 0:
p.add_lines(start_plot2[neut2, :],
end_plot2[neut2, :],
shading={"line_color": "#027337"})

175
cs457-gc/assignment_2_1/src/elasticsolid.py Normal file
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import numpy as np
from numpy import linalg
from scipy import sparse
import igl
# -----------------------------------------------------------------------------
# -----------------------------------------------------------------------------
# ELASTIC SOLID CLASS
# -----------------------------------------------------------------------------
# -----------------------------------------------------------------------------
class ElasticSolid(object):
def __init__(self, v_rest, t, rho=1, pin_idx=[]):
'''
Input:
- v_rest : position of the vertices of the mesh (#v, 3)
- t : indices of the element's vertices (#t, 4)
- rho : mass per unit volume [kg.m-3]
- pin_idx : list of vertex indices to pin
'''
self.v_rest = v_rest.copy()
self.v_def = v_rest.copy()
self.t = t
self.rho = rho
self.pin_idx = pin_idx
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.make_free_indices_and_pin_mask()
self.update_rest_shape(self.v_rest)
self.update_def_shape(self.v_def)
## 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.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 Dm 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 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)
## 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
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, 3) containing the 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()
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)

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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 eigendecomposition_metric import *
eps = 1E-6
with open('test_data1.json', 'r') as infile:
homework_datas = json.load(infile)
@pytest.mark.timeout(0.5)
@pytest.mark.parametrize("data", homework_datas[0])
def test_rest_barycenters(data):
v, t, rest_barycenters_gt = data
es = ElasticSolid(np.array(v), np.array(t))
es.make_rest_barycenters()
rest_barycenters_student = es.rest_barycenters
if(len(v) > 5000):
return
assert np.linalg.norm(rest_barycenters_gt - rest_barycenters_student) < eps
@pytest.mark.timeout(0.5)
@pytest.mark.parametrize("data", homework_datas[1])
def test_rest_shape_matrices(data):
v, t, Dm_gt, Bm_gt = data
es = ElasticSolid(np.array(v), np.array(t))
es.make_rest_shape_matrices()
Dm, Bm = es.Dm, es.Bm
if(len(v) > 5000):
return
assert np.linalg.norm(Dm - Dm_gt) < eps
assert np.linalg.norm(Bm - Bm_gt) < eps
@pytest.mark.timeout(0.5)
@pytest.mark.parametrize("data", homework_datas[2])
def test_def_barycenters(data):
v, t, v_def, def_barycenters_gt = data
es = ElasticSolid(np.array(v), np.array(t))
es.v_def = np.array(v_def)
es.make_def_barycenters()
def_barycenters_student = es.def_barycenters
if(len(v) > 5000):
return
assert np.linalg.norm(def_barycenters_gt - def_barycenters_student ) < eps
@pytest.mark.timeout(0.5)
@pytest.mark.parametrize("data", homework_datas[3])
def test_def_shape_matrices(data):
v, t, v_def, Ds_gt = data
es = ElasticSolid(np.array(v), np.array(t))
es.v_def = np.array(v_def)
es.make_def_shape_matrices()
Ds = es.Ds
if(len(v) > 5000):
return
assert np.linalg.norm(Ds - Ds_gt) < eps
@pytest.mark.timeout(0.5)
@pytest.mark.parametrize("data", homework_datas[4])
def test_update_rest_shape(data):
v, t, v_update, rest_barycenters_gt, W0_gt, F_gt = data
es = ElasticSolid(np.array(v), np.array(t))
es.update_rest_shape(np.array(v_update))
if(len(v) > 5000):
return
assert np.linalg.norm(es.rest_barycenters - rest_barycenters_gt) < eps
assert np.linalg.norm(es.W0 - W0_gt) < eps
assert np.linalg.norm(es.F - F_gt) < eps
@pytest.mark.timeout(0.5)
@pytest.mark.parametrize("data", homework_datas[5])
def test_update_def_shape(data):
v, t, v_def, def_barycenters_gt, W_gt, F_gt = data
es = ElasticSolid(np.array(v), np.array(t))
es.update_def_shape(np.array(v_def))
if(len(v) > 5000):
return
assert np.linalg.norm(es.def_barycenters - def_barycenters_gt) < eps
assert np.linalg.norm(es.W - W_gt) < eps
assert np.linalg.norm(es.F - F_gt) < eps
@pytest.mark.timeout(0.5)
@pytest.mark.parametrize("data", homework_datas[6])
def test_jacobians(data):
v, t, v_def, Ds_gt, Bm_gt, F_gt = data
es = ElasticSolid(np.array(v), np.array(t))
es.v_def, es.Ds, es.Bm = np.array(v_def), np.array(Ds_gt), np.array(Bm_gt)
es.make_jacobians()
if(len(v) > 5000):
return
assert np.linalg.norm(es.F - F_gt) < eps
@pytest.mark.timeout(0.5)
@pytest.mark.parametrize("data", homework_datas[7])
def test_pinning(data):
v, t, pin_idx, free_idx_gt, pin_mask_gt = data
es = ElasticSolid(np.array(v), np.array(t), pin_idx = np.array(pin_idx))
if(len(v) > 5000):
return
assert np.linalg.norm(es.free_idx - free_idx_gt) < eps
assert np.linalg.norm(es.pin_mask.astype(float) - np.array(pin_mask_gt).astype(float)) < eps
@pytest.mark.timeout(0.5)
@pytest.mark.parametrize("data", homework_datas[8])
def test_eig(data):
F, eigvals_gt, eigvecs_gt = data
eigvals, eigvecs = compute_eigendecomposition_metric(np.array(F))
if(len(F) > 5000):
return
assert np.linalg.norm(eigvals - eigvals_gt) < eps
assert np.linalg.norm(abs(np.diagonal(np.einsum('ijk, ijl -> ikl', eigvecs, eigvecs_gt), axis1 = 1, axis2 = 2)) - 1) < eps

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