Add notebook for debugging ELM force terms
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experiments/ELM_Debug.ipynb
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experiments/ELM_Debug.ipynb
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{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "2342d5ae-1839-4132-ace8-da8449223a1c",
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"metadata": {},
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"source": [
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"> (O'Brien, 2008)\n",
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"\n",
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"The force acting on node $i$ is given by\n",
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"\n",
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"$$\n",
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"\\mathbf{F}_i = \\sum^N_j \\mathbf{F}_{ij}\n",
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"$$\n",
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"\n",
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"Where $N = 8$ (for the 2D case) is the number of neighbors $j$.\n",
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"The force $\\mathbf{F}_{ij}$ on node $i$ from node $j$ is given by\n",
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"\n",
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"$$\n",
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"\\mathbf{F}_{ij} = K_{ij} (\\mathbf{u}_{ij} \\cdot \\mathbf{x}_{ij}) + \\frac{c \\mathbf{u}_{ij}}{|\\mathbf{x}_{ij}|^2}\n",
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"$$\n",
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"\n",
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"where\n",
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"\n",
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"* $K_{ij}$ is the elastic spring constant\n",
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"* $\\mathbf{u}_{ij} = \\mathbf{u}_i - \\mathbf{u}_j \\in \\mathbb{R}^2$ is the displacement\n",
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"* $\\mathbf{x}_{ij} = \\mathbf{x}_i - \\mathbf{x}_j \\in \\mathbb{R}^2$ is the vector connecting nodes $i$ and $j$ on the undistorted lattice\n",
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"* $c \\in \\mathbb{R}$ is the bond-bending constant\n",
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"\n",
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"The Lamé constants are given by\n",
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"\n",
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"$\\lambda_\\text{2D} = K - \\frac{c}{\\Delta x^2}$\n",
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"\n",
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"and\n",
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"\n",
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"$\\mu_\\text{2D} = K + \\frac{c}{\\Delta x^2}$\n",
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"\n",
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"where $\\Delta x$ is the lattice grid spacing."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 33,
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"id": "05d4e22c-3d7d-4ac2-bf22-59b78aa40b84",
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"metadata": {},
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"outputs": [],
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"source": [
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"import matplotlib.pyplot as plt\n",
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"import numpy as np\n",
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"\n",
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"from ipywidgets import interact\n",
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"from typing import Tuple"
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]
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},
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{
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"cell_type": "markdown",
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"id": "50ba2e84-bb40-44c8-bb86-8804fd26f674",
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"metadata": {},
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"source": [
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"$$\n",
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"\\mathbf{F}_{ij} = K_{ij} (\\mathbf{u}_{ij} \\cdot \\mathbf{x}_{ij}) + \\frac{c \\mathbf{u}_{ij}}{|\\mathbf{x}_{ij}|^2}\n",
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"$$"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 61,
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"id": "82bb4a02-1ecf-4c84-8b9d-b753dee75337",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"j=(0, 0) K_ij=2.0000000000000004 u_ij=array([0.1, 0. ]) x_ij=array([1., 1.])\n"
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]
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},
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{
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"data": {
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"text/plain": [
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"-0.20000000000000007"
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]
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},
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"execution_count": 61,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"def calculate_force_elastic(\n",
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" displacement: np.ndarray,\n",
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" i: Tuple[int, int],\n",
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" j: Tuple[int, int]\n",
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") -> float:\n",
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" u_ij = displacement[i] - displacement[j]\n",
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" x_ij = np.array(i, dtype=float) - np.array(j, dtype=float)\n",
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" K_ij = np.linalg.norm(x_ij) ** 2\n",
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"\n",
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" print(f\"{j=} {K_ij=} {u_ij=} {x_ij=}\")\n",
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"\n",
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" return -K_ij * np.dot(u_ij, x_ij)\n",
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"\n",
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"def calculate_force_bond(\n",
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" displacement: np.ndarray,\n",
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" i: Tuple[int, int],\n",
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" j: Tuple[int, int]\n",
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") -> float:\n",
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" return 0.0\n",
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"\n",
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"def calculate_force(\n",
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" displacement: np.ndarray,\n",
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" i: Tuple[int, int],\n",
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" j: Tuple[int, int]\n",
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") -> float:\n",
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" return calculate_force_elastic(displacement, i, j) + calculate_force_bond(displacement, i, j)\n",
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"\n",
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"displacement = np.array([\n",
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" [[0.0, 0.0], [0.0, 0.0], [0.0, 0.0]],\n",
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" [[0.0, 0.0], [0.1, 0.0], [0.0, 0.0]],\n",
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" [[0.0, 0.0], [0.0, 0.0], [0.0, 0.0]]\n",
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"])\n",
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"calculate_force_elastic(displacement, (1, 1), (0, 0))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 71,
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"id": "f7a019c2-8d20-4e84-9e0c-ef788781a07a",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"application/vnd.jupyter.widget-view+json": {
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"model_id": "89618a87c5514c6698d62e3344092fdf",
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"version_major": 2,
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"version_minor": 0
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},
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"text/plain": [
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"interactive(children=(FloatSlider(value=0.0, description='dp_x', max=1.0, min=-1.0, step=0.05), FloatSlider(va…"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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},
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{
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"data": {
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"text/plain": [
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"<function __main__.do_plot(dp_x: float, dp_y: float) -> None>"
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]
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},
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"execution_count": 71,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"def do_plot(dp_x: float, dp_y: float) -> None:\n",
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" i = (1, 1)\n",
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" displacement[i] = np.array([dp_x, dp_y])\n",
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" \n",
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" width, height, _ = displacement.shape\n",
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"\n",
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" xs = []\n",
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" ys = []\n",
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" elastic_xs = []\n",
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" elastic_ys = []\n",
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" \n",
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" for x in range(width):\n",
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" for y in range(height):\n",
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" j = (x, y)\n",
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" xs.append(x + displacement[j][0])\n",
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" ys.append(y + displacement[j][1])\n",
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"\n",
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" if j != i:\n",
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" F_ij = calculate_force_elastic(displacement, i, j)\n",
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" F_ij_vec = (\n",
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" (i + displacement[i])\n",
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" - (j + displacement[j])\n",
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" ) * F_ij\n",
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" print(f\"{F_ij_vec=}\")\n",
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" elastic_xs.append(F_ij_vec[0])\n",
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" elastic_ys.append(F_ij_vec[1])\n",
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"\n",
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" pi = i + displacement[i]\n",
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" plt.scatter(xs, ys)\n",
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" plt.quiver(\n",
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" [pi[0]] * len(elastic_xs),\n",
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" [pi[1]] * len(elastic_ys),\n",
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" elastic_xs,\n",
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" elastic_ys,\n",
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" scale = 5,\n",
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" pivot = \"tip\",\n",
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" width = 0.005\n",
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" )\n",
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" plt.quiver(\n",
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" pi[0], pi[1],\n",
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" sum(elastic_xs),\n",
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" sum(elastic_ys),\n",
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" pivot = \"tip\",\n",
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" scale = 5,\n",
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" color = \"red\"\n",
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" )\n",
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" \n",
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"\n",
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"interact(\n",
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" do_plot,\n",
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" dp_x = (-1.0, 1.0, 0.05),\n",
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" dp_y = (-1.0, 1.0, 0.05)\n",
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")"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "d4c55b68-903a-48de-8226-9cfbbbca4af6",
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"metadata": {},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3 (ipykernel)",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.10.11"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}
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