{ "cells": [ { "cell_type": "markdown", "id": "8fae77e0", "metadata": {}, "source": [ "# 7 · Time Integration: Forward Euler, Backward Euler, BDF2\n", "\n", "To animate motion we march the equations of motion in time. The **integrator**\n", "trades accuracy against stability and damping. We compare three on the *same*\n", "cantilever beam:\n", "\n", "| scheme | type | order | character |\n", "|---|---|---|---|\n", "| **Forward Euler** | explicit | 1 | cheap, but explodes unless dt is tiny |\n", "| **Backward Euler** | implicit | 1 | rock-solid, but adds numerical damping |\n", "| **BDF2** | implicit | 2 | second-order accurate, barely damps |\n", "\n", "Each implicit step minimizes the potential plus an **inertia** term\n", "$\\tfrac{c}{2h^2}\\lVert x - \\tilde x\\rVert_M^2$ pulling $x$ toward where\n", "inertia says it should go ($\\tilde x$). We write the three steppers explicitly\n", "so the only thing that changes is $\\tilde x$ and $c$." ] }, { "cell_type": "code", "execution_count": 1, "id": "3cca93a5", "metadata": { "execution": { "iopub.execute_input": "2026-06-09T05:30:04.598798Z", "iopub.status.busy": "2026-06-09T05:30:04.598654Z", "iopub.status.idle": "2026-06-09T05:30:05.144376Z", "shell.execute_reply": "2026-06-09T05:30:05.144136Z" } }, "outputs": [], "source": [ "%matplotlib inline\n", "import numpy as np\n", "import scipy as sp\n", "import matplotlib.pyplot as plt\n", "import simkit\n", "from simkit.integrators import backward_euler, bdf2, forward_euler\n", "import utils" ] }, { "cell_type": "markdown", "id": "a3956922", "metadata": {}, "source": [ "## The beam and a readable potential\n", "\n", "`BeamPotential` is the static potential (elastic + pinned left edge + gravity).\n", "It composes the standard `utils` term helpers and sums them **one per line** in\n", "`energy / gradient / hessian` (gravity is linear, so it has no Hessian term).\n", "Each integrator layers its own inertia term on top internally." ] }, { "cell_type": "code", "execution_count": 2, "id": "894c8229", "metadata": { "execution": { "iopub.execute_input": "2026-06-09T05:30:05.145528Z", "iopub.status.busy": "2026-06-09T05:30:05.145443Z", "iopub.status.idle": "2026-06-09T05:30:05.152085Z", "shell.execute_reply": "2026-06-09T05:30:05.151847Z" } }, "outputs": [], "source": [ "X, T = utils.triangulated_grid(nx=14, ny=4, width=2.0, height=0.3)\n", "n, dim = X.shape\n", "pin_idx = np.where(X[:, 0] <= X[:, 0].min() + 1e-6)[0]\n", "\n", "class BeamPotential:\n", " \"\"\"Static potential: elastic + pinned left edge + gravity (no inertia).\"\"\"\n", "\n", " def __init__(self, X, T):\n", " self.p = utils.precompute(X, T, ym=500.0, pr=0.4, rho=1.0, gravity=-9.8)\n", " self.psi = utils.make_material(\"Neo-Hookean\")\n", " self.pin = utils.PenaltySpring(self.p.n, self.p.dim, 1e7).set(pin_idx, X[pin_idx])\n", " self.gravity = utils.Gravity(self.p.f_g)\n", "\n", " def energy(self, x):\n", " E_elastic = self.psi.energy(x, self.p)\n", " E_pin = self.pin.energy(x)\n", " E_gravity = self.gravity.energy(x)\n", " return E_elastic + E_pin + E_gravity\n", "\n", " def gradient(self, x):\n", " g_elastic = self.psi.gradient(x, self.p)\n", " g_pin = self.pin.gradient(x)\n", " g_gravity = self.gravity.gradient(x)\n", " return g_elastic + g_pin + g_gravity\n", "\n", " def hessian(self, x):\n", " H_elastic = self.psi.hessian(x, self.p)\n", " H_pin = self.pin.hessian(x)\n", " return H_elastic + H_pin\n", "\n", "pot = BeamPotential(X, T)\n", "M = pot.p.M" ] }, { "cell_type": "markdown", "id": "3af43602", "metadata": {}, "source": [ "## The three steppers\n", "\n", "Each takes the current state and returns the next, delegating the actual update\n", "to `simkit.integrators`. We carry an explicit velocity `V` (physical, and what\n", "the explicit method needs). The library integrators take a *position history*,\n", "so each wrapper just converts between the two:\n", "\n", "* **Forward Euler** — `forward_euler` is explicit; it takes `(U, V)` and\n", " returns `(U_next, V_next)` directly.\n", "* **Backward Euler** — `backward_euler` wants the last two positions, so we\n", " pass `U` and `U - hV` (whose backward difference is exactly `V`).\n", "* **BDF2** — `bdf2` reconstructs *two* velocities from a four-level\n", " position history; we synthesize the two older positions that encode the\n", " carried `V` and `V_prev`." ] }, { "cell_type": "code", "execution_count": 3, "id": "6be74ecc", "metadata": { "execution": { "iopub.execute_input": "2026-06-09T05:30:05.153153Z", "iopub.status.busy": "2026-06-09T05:30:05.153093Z", "iopub.status.idle": "2026-06-09T05:30:05.155193Z", "shell.execute_reply": "2026-06-09T05:30:05.155014Z" } }, "outputs": [], "source": [ "NEWTON_ITERS = 30 # max Newton iterations for the implicit solves\n", "\n", "def fe_step(U, V, h):\n", " \"\"\"Forward (explicit) Euler via simkit.integrators.forward_euler.\"\"\"\n", " U_next, V_next = forward_euler(U.reshape(-1, 1), V.reshape(-1, 1), pot.gradient, M, h)\n", " U_next = U_next.reshape(n, dim); V_next = V_next.reshape(n, dim)\n", " U_next[pin_idx] = X[pin_idx]; V_next[pin_idx] = 0.0 # hard-clamp the pins\n", " return U_next, V_next\n", "\n", "def be_step(U, V, h):\n", " \"\"\"Backward Euler via simkit.integrators.backward_euler.\n", " Pass U and (U - hV): their backward difference is exactly the velocity V.\"\"\"\n", " U_next = backward_euler(U.reshape(-1, 1), (U - h * V).reshape(-1, 1),\n", " pot.energy, pot.gradient, pot.hessian, M, h,\n", " max_iter=NEWTON_ITERS, do_line_search=True).reshape(n, dim)\n", " V_next = (U_next - U) / h\n", " return U_next, V_next\n", "\n", "def bdf2_step(U, V, U_prev, V_prev, h):\n", " \"\"\"BDF2 via simkit.integrators.bdf2. The integrator reconstructs both\n", " velocities from a 4-level position history with the 2nd-order backward\n", " difference, so we synthesize the two older positions that encode V (at U)\n", " and V_prev (at U_prev).\"\"\"\n", " U_prev2 = 2 * h * V - 3 * U + 4 * U_prev # velocity_bdf2(U, U_prev, U_prev2) = V\n", " U_prev3 = 2 * h * V_prev - 3 * U_prev + 4 * U_prev2 # velocity_bdf2(U_prev, U_prev2, U_prev3) = V_prev\n", " U_next = bdf2(U.reshape(-1, 1), U_prev.reshape(-1, 1),\n", " U_prev2.reshape(-1, 1), U_prev3.reshape(-1, 1),\n", " pot.energy, pot.gradient, pot.hessian, M, h,\n", " max_iter=NEWTON_ITERS, do_line_search=True).reshape(n, dim)\n", " V_next = (3 * U_next - 4 * U + U_prev) / (2 * h)\n", " return U_next, V_next" ] }, { "cell_type": "markdown", "id": "ebc14c04", "metadata": {}, "source": [ "## Drivers that snapshot at fixed times\n", "\n", "So we can compare runs with very different timesteps on the same time axis." ] }, { "cell_type": "code", "execution_count": 4, "id": "6a4880c0", "metadata": { "execution": { "iopub.execute_input": "2026-06-09T05:30:05.156077Z", "iopub.status.busy": "2026-06-09T05:30:05.156019Z", "iopub.status.idle": "2026-06-09T05:30:05.158295Z", "shell.execute_reply": "2026-06-09T05:30:05.158141Z" } }, "outputs": [], "source": [ "def simulate_fe(h, T_total, n_frames=60):\n", " sample = np.linspace(0, T_total, n_frames)\n", " U, V, t, si, frames = X.copy(), np.zeros_like(X), 0.0, 0, [X.copy()]\n", " while si < n_frames - 1:\n", " U, V = fe_step(U, V, h); t += h\n", " if not np.isfinite(U).all() or np.abs(U).max() > 1e3: # exploded\n", " frames += [U.copy()] * (n_frames - len(frames)); break\n", " while si < n_frames - 1 and t >= sample[si + 1]:\n", " si += 1; frames.append(U.copy())\n", " return frames\n", "\n", "def simulate_be(h, T_total, n_frames=60):\n", " sample = np.linspace(0, T_total, n_frames)\n", " U, V, t, si, frames = X.copy(), np.zeros_like(X), 0.0, 0, [X.copy()]\n", " while si < n_frames - 1:\n", " U, V = be_step(U, V, h); t += h\n", " while si < n_frames - 1 and t >= sample[si + 1]:\n", " si += 1; frames.append(U.copy())\n", " return frames\n", "\n", "def simulate_bdf2(h, T_total, n_frames=60):\n", " sample = np.linspace(0, T_total, n_frames)\n", " U, V = X.copy(), np.zeros_like(X)\n", " U_next, V_next = be_step(U, V, h) # bootstrap one BE step\n", " U_prev, V_prev, U, V = U, V, U_next, V_next\n", " t, si, frames = h, 0, [X.copy()]\n", " while si < n_frames - 1:\n", " U_next, V_next = bdf2_step(U, V, U_prev, V_prev, h)\n", " U_prev, V_prev, U, V = U, V, U_next, V_next; t += h\n", " while si < n_frames - 1 and t >= sample[si + 1]:\n", " si += 1; frames.append(U.copy())\n", " return frames" ] }, { "cell_type": "markdown", "id": "56bf2878", "metadata": {}, "source": [ "## Order of accuracy, measured on *our* beam\n", "\n", "No toy oscillator: we run the beam to a fixed time and compare each scheme to a\n", "**fine-timestep BDF2 reference** on the same mesh. To measure order cleanly we\n", "integrate with **exactly** `round(T/h)` steps so every timestep lands at the\n", "*same* final time (the frame-sampling drivers above would each overshoot $T$ by\n", "up to one step). On a log-log plot the error of an order-$p$ method is a line\n", "of slope $p$: **Backward Euler tracks $O(h)$; BDF2 tracks $O(h^2)$**.\n", "(Forward Euler is excluded — it needs a far smaller dt just to stay stable,\n", "as we'll see next.)" ] }, { "cell_type": "code", "execution_count": 5, "id": "c8707e7b", "metadata": { "execution": { "iopub.execute_input": "2026-06-09T05:30:05.159112Z", "iopub.status.busy": "2026-06-09T05:30:05.159067Z", "iopub.status.idle": "2026-06-09T05:30:07.988703Z", "shell.execute_reply": "2026-06-09T05:30:07.988497Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Backward Euler order ~ 0.73\n", "BDF2 order ~ 1.71\n" ] }, { "data": { "image/png": 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", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "def integrate_exact(kind, h, T_total):\n", " \"\"\"Step EXACTLY round(T_total / h) times so every dt ends at the same time.\"\"\"\n", " steps = int(round(T_total / h))\n", " U, V = X.copy(), np.zeros_like(X)\n", " if kind == \"Backward Euler\":\n", " for _ in range(steps):\n", " U, V = be_step(U, V, h)\n", " else: # BDF2, bootstrapped with one BE step\n", " U_next, V_next = be_step(U, V, h)\n", " U_prev, V_prev, U, V = U, V, U_next, V_next\n", " for _ in range(2, steps + 1):\n", " U_next, V_next = bdf2_step(U, V, U_prev, V_prev, h)\n", " U_prev, V_prev, U, V = U, V, U_next, V_next\n", " return U\n", "\n", "T_total = 0.5\n", "reference = integrate_exact(\"BDF2\", 0.0003125, T_total) # fine BDF2 reference\n", "hs = np.array([0.02, 0.01, 0.005, 0.0025])\n", "errors = {\n", " \"Backward Euler\": np.array([np.linalg.norm(integrate_exact(\"Backward Euler\", h, T_total) - reference) for h in hs]),\n", " \"BDF2\": np.array([np.linalg.norm(integrate_exact(\"BDF2\", h, T_total) - reference) for h in hs]),\n", "}\n", "for name, e in errors.items():\n", " print(f\"{name:16s} order ~ {np.polyfit(np.log(hs), np.log(e), 1)[0]:.2f}\")\n", "\n", "fig, _ = utils.loglog_plot(hs, errors, xlabel=\"timestep h\", ylabel=\"error vs fine reference\",\n", " colors=utils.INTEGRATOR_COLORS, ref_slopes={\"O(h)\": 1, \"O(h^2)\": 2},\n", " title=\"Order of accuracy on the beam\")\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "821d5986", "metadata": {}, "source": [ "## Forward Euler explodes; Backward Euler doesn't (side by side)\n", "\n", "Explicit integration is only stable below a tiny critical timestep (CFL). At\n", "$h = 0.008$ — a timestep Backward Euler handles without blinking\n", "— **Forward Euler diverges within a few steps** and flies off-screen, while\n", "Backward Euler calmly sags under gravity. This is *why* we use implicit\n", "integrators for stiff elasticity." ] }, { "cell_type": "code", "execution_count": 6, "id": "c8575463", "metadata": { "execution": { "iopub.execute_input": "2026-06-09T05:30:07.989645Z", "iopub.status.busy": "2026-06-09T05:30:07.989585Z", "iopub.status.idle": "2026-06-09T05:30:08.992120Z", "shell.execute_reply": "2026-06-09T05:30:08.991852Z" } }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "h_blow = 0.008\n", "fe_states = simulate_fe(h_blow, 0.3, n_frames=40)\n", "be_states = simulate_be(h_blow, 0.3, n_frames=40)\n", "fig, anim = utils.animate_meshes_grid(\n", " [{\"states\": fe_states, \"T\": T, \"title\": f\"Forward Euler dt={h_blow} (EXPLODES)\"},\n", " {\"states\": be_states, \"T\": T, \"title\": f\"Backward Euler dt={h_blow} (stable)\"}],\n", " lims=((-1.3, 1.3), (-1.1, 0.5)), fps=20,\n", " suptitle=\"Same timestep: explicit blows up, implicit is fine\")\n", "utils.show_video(fig, anim, \"media/07_fe_vs_be.mp4\", fps=20)" ] }, { "cell_type": "markdown", "id": "f31602f7", "metadata": {}, "source": [ "## Numerical damping: the timestep changes perceived stiffness\n", "\n", "Release the beam under gravity and step **Backward Euler** at three (smooth,\n", "stable) timesteps — 0.001, 0.010, 0.1. Larger dt → **more artificial\n", "damping** → the swing dies out faster, so the material *looks* stiffer even\n", "though the physics is identical." ] }, { "cell_type": "code", "execution_count": 7, "id": "6b157b6c", "metadata": { "execution": { "iopub.execute_input": "2026-06-09T05:30:08.993111Z", "iopub.status.busy": "2026-06-09T05:30:08.993038Z", "iopub.status.idle": "2026-06-09T05:30:14.575823Z", "shell.execute_reply": "2026-06-09T05:30:14.575568Z" } }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "T_total = 3\n", "dts = [0.001, 0.01, 0.1]\n", "be_runs = [{\"states\": simulate_be(h, T_total, 60), \"T\": T, \"title\": f\"Backward Euler dt={h}\"}\n", " for h in dts]\n", "fig, anim = utils.animate_meshes_grid(be_runs, lims=((-1.2, 1.2), (-1.9, 0.5)), fps=30,\n", " suptitle=\"Backward Euler: bigger timestep -> more numerical damping\")\n", "utils.show_video(fig, anim, \"media/07_be_damping.mp4\", fps=30)" ] }, { "cell_type": "markdown", "id": "769c742c", "metadata": {}, "source": [ "## BDF2 barely damps\n", "\n", "The same three timesteps with **BDF2**. It keeps swinging at all of them\n", "— second-order accuracy buys far less numerical dissipation than Backward\n", "Euler." ] }, { "cell_type": "code", "execution_count": 8, "id": "d0a344de", "metadata": { "execution": { "iopub.execute_input": "2026-06-09T05:30:14.577103Z", "iopub.status.busy": "2026-06-09T05:30:14.577021Z", "iopub.status.idle": "2026-06-09T05:30:20.227982Z", "shell.execute_reply": "2026-06-09T05:30:20.227718Z" } }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "bdf2_runs = [{\"states\": simulate_bdf2(h, T_total, 60), \"T\": T, \"title\": f\"BDF2 dt={h}\"}\n", " for h in dts]\n", "fig, anim = utils.animate_meshes_grid(bdf2_runs, lims=((-1.2, 1.2), (-1.9, 0.5)), fps=30,\n", " suptitle=\"BDF2: much less numerical damping at the same timesteps\")\n", "utils.show_video(fig, anim, \"media/07_bdf2_damping.mp4\", fps=30)" ] }, { "cell_type": "markdown", "id": "58fe5a25", "metadata": {}, "source": [ "## The damping, quantified: total energy bleeds away faster at bigger dt\n", "\n", "The artificial damping isn't just visual — it removes **mechanical energy**.\n", "We track the total energy (elastic + kinetic + gravitational) over time. For\n", "Backward Euler the bigger the timestep, the **faster the energy drops**; BDF2\n", "(shown for the largest dt) holds onto far more of it." ] }, { "cell_type": "code", "execution_count": 9, "id": "aa3364a7", "metadata": { "execution": { "iopub.execute_input": "2026-06-09T05:30:20.229195Z", "iopub.status.busy": "2026-06-09T05:30:20.229118Z", "iopub.status.idle": "2026-06-09T05:30:24.437724Z", "shell.execute_reply": "2026-06-09T05:30:24.437519Z" } }, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "def total_energy(U, V):\n", " E_elastic = pot.psi.energy(U.reshape(-1, 1), pot.p)\n", " E_kinetic = 0.5 * float(V.flatten() @ (M @ V.flatten()))\n", " E_gravity = pot.gravity.energy(U.reshape(-1, 1))\n", " return E_elastic + E_kinetic + E_gravity\n", "\n", "def be_energy_trace(h, T_total):\n", " U, V = X.copy(), np.zeros_like(X)\n", " ts, Es = [0.0], [total_energy(U, V)]\n", " for s in range(int(T_total / h)):\n", " U, V = be_step(U, V, h); ts.append((s + 1) * h); Es.append(total_energy(U, V))\n", " return np.array(ts), np.array(Es)\n", "\n", "def bdf2_energy_trace(h, T_total):\n", " U, V = X.copy(), np.zeros_like(X)\n", " Un, Vn = be_step(U, V, h); U_prev, V_prev, U, V = U, V, Un, Vn\n", " ts, Es = [0.0, h], [total_energy(X, np.zeros_like(X)), total_energy(U, V)]\n", " for s in range(2, int(T_total / h)):\n", " Un, Vn = bdf2_step(U, V, U_prev, V_prev, h)\n", " U_prev, V_prev, U, V = U, V, Un, Vn\n", " ts.append(s * h); Es.append(total_energy(U, V))\n", " return np.array(ts), np.array(Es)\n", "\n", "fig, ax = plt.subplots(figsize=(7.5, 4.8))\n", "shades = [\"#9ecae1\", \"#4292c6\", \"#08519c\"]\n", "for h, col in zip(dts, shades):\n", " ts, Es = be_energy_trace(h, T_total)\n", " ax.plot(ts, Es, color=col, lw=2, label=f\"Backward Euler dt={h}\")\n", "ts, Es = bdf2_energy_trace(0.1, T_total)\n", "ax.plot(ts, Es, color=\"#2ca02c\", lw=2, ls=\"--\", label=\"BDF2 dt=0.1\")\n", "ax.set_xlabel(\"time (s)\"); ax.set_ylabel(\"total mechanical energy\")\n", "ax.set_title(\"Numerical damping drains energy faster at larger dt\")\n", "ax.grid(True, color=\"0.9\"); ax.legend(fontsize=9); plt.show()" ] }, { "cell_type": "markdown", "id": "4bd0e2a2", "metadata": {}, "source": [ "### Takeaways\n", "* **Backward Euler** is order 1; **BDF2** is order 2 (verified on our own beam).\n", "* **Forward Euler** is explicit and explodes unless dt is tiny — implicit\n", " methods stay stable at the same dt.\n", "* **Backward Euler** damps motion (and *drains total energy*) more as dt grows;\n", " **BDF2** stays lively and conserves energy far better." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "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": "3.13.9" } }, "nbformat": 4, "nbformat_minor": 5 }