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"# 14 · Bending Stiffness: a Mass-Spring Beam that Resists Bending\n",
"\n",
"Tutorial 11 built a mass-spring **cable** — springs only resist *stretch*,\n",
"so pinned at both ends it sags into a catenary and pinned at one end it just\n",
"flops down. Here we add a **bending** energy so the beam resists *curvature*,\n",
"clamp it at one end (a cantilever), and watch it go from floppy to rigid as the\n",
"bending stiffness grows. Then we refine the mesh and watch the equilibrium\n",
"converge to a single shape.\n",
"\n",
"| step | what we do |\n",
"|---|---|\n",
"| **1** | springs alone can't hold a shape (clamp one end, gravity on) |\n",
"| **2** | bending energy = penalize the hinge *turning angle* |\n",
"| **3** | sweep stiffness: the beam flops when soft, holds its shape when stiff |\n",
"| **4** | refine the mesh → the equilibrium converges to one curve |"
]
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"%matplotlib inline\n",
"import numpy as np\n",
"import scipy as sp\n",
"import matplotlib.pyplot as plt\n",
"import simkit\n",
"import simkit.energies as energies\n",
"from simkit.solvers import newton_solver\n",
"import utils"
]
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"## The cantilever and its potential\n",
"\n",
"A horizontal chain of `N` nodes: consecutive nodes are joined by **springs**\n",
"(stretch) and every interior node carries a **hinge** that measures the turning\n",
"angle between its two segments (bend). We **clamp the root** by pinning the two\n",
"leftmost nodes — fixing two nodes fixes both the position *and* the tangent\n",
"of the built-in end, which is what makes it a cantilever rather than a free\n",
"pivot.\n",
"\n",
"With flattened positions $x$ the total potential is\n",
"\n",
"$$\n",
"\\Phi(x) = \\underbrace{E_\\text{stretch}(x)}_{\\text{springs}}\n",
" + \\underbrace{E_\\text{bend}(x)}_{\\text{hinges}}\n",
" - \\underbrace{f_g^\\top x}_{\\text{gravity}}\n",
" + \\underbrace{\\tfrac12 x^\\top Q x + b^\\top x}_{\\text{clamp}} .\n",
"$$\n",
"\n",
"The bending term is simkit's unified `bending_energy_x(x, C, theta0, kappa)`:\n",
"each hinge stores $\\tfrac12\\,\\kappa\\,(\\theta-\\theta_0)^2$ where $\\theta$\n",
"is the signed turning angle (`dihedral_angles`) and $\\theta_0=0$ for a straight\n",
"rest beam. We pick the physical constants so the discretization is\n",
"**mesh-independent** (so step 4's refinement converges): for a continuum beam of\n",
"length $L$, bending modulus $EI$, axial stiffness $EA$ and segment\n",
"length $l=L/(N-1)$,\n",
"\n",
"* bending: $\\kappa = EI/l$ (from $\\tfrac12 EI\\!\\int\\!\\kappa^2\\,ds$ with turning angle $\\theta\\approx \\kappa\\,l$),\n",
"* stretch: spring stiffness $EA/l$ (set `ym = EA`, quadrature weight `vol = l`), with $EA \\gg EI$ so the beam is near-inextensible,\n",
"* mass/gravity: a lumped mass matrix that distributes the fixed total mass $\\rho L$ regardless of `N`."
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"## A small, readable simulator\n",
"\n",
"`BendingBeam` assembles the cantilever as a sum of named term helpers from\n",
"`utils` — a mass-spring **stretch** energy, a fixed **pin** clamp, and\n",
"**gravity** — plus an inline **bending** term that calls simkit's unified\n",
"`bending_*_x` on the hinge triples. `EI = 0` disables bending (springs only).\n",
"The static `energy / gradient / hessian` sum those terms **one per line**;\n",
"`step()` adds the backward-Euler `inertia` term and Newton-solves, exactly like\n",
"every other dynamic tutorial."
]
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"default cantilever: |grad| = 2.9e-05 tip = [ 3.83 -1.064]\n"
]
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"source": [
"L, EA, RHO, G = 4.0, 2.0e4, 0.2, -9.8 # fixed continuum beam + load\n",
"PIN_PEN = 1e10\n",
"\n",
"\n",
"class Bending:\n",
" \"\"\"Inline bending term: 1/2 kappa (theta - theta0)^2 over the hinge triples,\n",
" calling simkit's unified bending_*_x (theta = signed turning angle).\"\"\"\n",
" def __init__(self, C, theta0, kappa):\n",
" self.C, self.theta0, self.kappa = C, theta0, kappa\n",
"\n",
" def energy(self, x): return energies.bending_energy_x(x, self.C, self.theta0, self.kappa)\n",
" def gradient(self, x): return energies.bending_gradient_x(x, self.C, self.theta0, self.kappa)\n",
" def hessian(self, x): return energies.bending_hessian_x(x, self.C, self.theta0, self.kappa, psd=True)\n",
"\n",
"\n",
"class BendingBeam:\n",
" \"\"\"Cantilever potential = springs + bending + gravity + clamp.\n",
" Static `solve()` minimizes it; `step()` adds inertia for dynamics.\"\"\"\n",
"\n",
" def __init__(self, N=21, EI=50.0, h=0.02):\n",
" self.X = np.zeros((N, 2)); self.X[:, 0] = np.linspace(0, L, N)\n",
" self.E = np.array([[i, i + 1] for i in range(N - 1)]) # springs\n",
" self.C = np.array([[i, i + 1, i + 2] for i in range(N - 2)]) # hinge triples\n",
" self.n, self.dim = self.X.shape\n",
" l_seg = L / (N - 1)\n",
"\n",
" # stretching (mass-springs, as in tutorial 11): stiffness EA/l <- ym=EA, vol=l\n",
" Be = simkit.edge_displacement_jacobian(self.X, self.E)\n",
" J = sp.sparse.kron(Be, sp.sparse.eye(self.dim)).tocsc()\n",
" l0 = simkit.edge_lengths(self.X, self.E).reshape(-1, 1)\n",
" ym = np.full((self.E.shape[0], 1), EA); vol = l0.copy() # vol = l_seg\n",
" self.spring = utils.SpringEnergy(J, ym, vol, l0)\n",
"\n",
" # bending (unified energy; kappa = EI/l, theta0 = 0 for a straight beam)\n",
" theta0 = simkit.dihedral_angles(self.X, self.C)\n",
" kappa = np.full((self.C.shape[0], 1), EI / l_seg)\n",
" self.bending = Bending(self.C, theta0, kappa)\n",
"\n",
" # mass, gravity, clamp (pin the two leftmost nodes), inertia\n",
" self.M = sp.sparse.kron(simkit.massmatrix(self.X, self.E, rho=RHO),\n",
" sp.sparse.eye(self.dim)).tocsc()\n",
" f_g = simkit.gravity_force(self.X, self.E, a=G, rho=RHO).reshape(-1, 1)\n",
" self.gravity = utils.Gravity(f_g)\n",
" self.pin = utils.PenaltySpring(self.n, self.dim, PIN_PEN).set([0, 1], self.X[[0, 1]])\n",
" self.inertia = utils.Inertia(self.M, h)\n",
" self.h = h\n",
"\n",
" # ---- static potential Phi(x) ----\n",
" def energy(self, x):\n",
" E_spring = self.spring.energy(x)\n",
" E_bending = self.bending.energy(x)\n",
" E_gravity = self.gravity.energy(x)\n",
" E_pin = self.pin.energy(x)\n",
" return E_spring + E_bending + E_gravity + E_pin\n",
"\n",
" def gradient(self, x):\n",
" g_spring = self.spring.gradient(x)\n",
" g_bending = self.bending.gradient(x)\n",
" g_gravity = self.gravity.gradient(x)\n",
" g_pin = self.pin.gradient(x)\n",
" return g_spring + g_bending + g_gravity + g_pin\n",
"\n",
" def hessian(self, x):\n",
" H_spring = self.spring.hessian(x)\n",
" H_bending = self.bending.hessian(x)\n",
" H_pin = self.pin.hessian(x)\n",
" return H_spring + H_bending + H_pin\n",
"\n",
" # ---- dynamic potential = static potential + backward-Euler inertia ----\n",
" def dyn_energy(self, x):\n",
" E_inertia = self.inertia.energy(x)\n",
" return E_inertia + self.energy(x)\n",
"\n",
" def dyn_gradient(self, x):\n",
" g_inertia = self.inertia.gradient(x)\n",
" return g_inertia + self.gradient(x)\n",
"\n",
" def dyn_hessian(self, x):\n",
" H_inertia = self.inertia.hessian(x)\n",
" return H_inertia + self.hessian(x)\n",
"\n",
" def solve(self, iters=120):\n",
" x = self.X.flatten().astype(float).reshape(-1, 1)\n",
" x = newton_solver(x, self.energy, self.gradient, self.hessian,\n",
" max_iter=iters, do_line_search=True)\n",
" return x.reshape(self.n, self.dim)\n",
"\n",
" def step(self, x_n, v_n, iters=20):\n",
" self.inertia.update(x_n, v_n)\n",
" return newton_solver(x_n.copy(), self.dyn_energy, self.dyn_gradient,\n",
" self.dyn_hessian, max_iter=iters, do_line_search=True)\n",
"\n",
"\n",
"def simulate(beam, n_steps=160, n_frames=60):\n",
" \"\"\"Backward-Euler release from the flat rest shape; sample n_frames states.\"\"\"\n",
" n, dim, M, h = beam.n, beam.dim, beam.M, beam.h\n",
" x = beam.X.flatten().astype(float).reshape(-1, 1)\n",
" v = np.zeros_like(x)\n",
" states, ke, ts = [beam.X.copy()], [0.0], [0.0]\n",
" sample = np.linspace(0, n_steps, n_frames).astype(int)\n",
" for s in range(1, n_steps + 1):\n",
" x_new = beam.step(x, v)\n",
" v = (x_new - x) / h; x = x_new\n",
" if s in sample:\n",
" states.append(x.reshape(n, dim).copy())\n",
" ke.append(0.5 * (v.T @ (M @ v))[0, 0]); ts.append(s * h)\n",
" return states, np.array(ke), np.array(ts)\n",
"\n",
"\n",
"beam0 = BendingBeam(21, EI=50.0)\n",
"U0 = beam0.solve()\n",
"print(\"default cantilever: |grad| =\", f\"{np.linalg.norm(beam0.gradient(U0.flatten().reshape(-1,1))):.1e}\",\n",
" \" tip =\", np.round(U0[-1], 3))"
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"## 1 · Springs alone can't hold a shape\n",
"\n",
"Turn bending **off** (`EI = 0`) and release the clamped chain under gravity. With\n",
"no resistance to curvature the beam folds and flops straight down — exactly\n",
"the cable behavior from tutorial 11. The kinetic energy keeps sloshing because\n",
"there is nothing to settle the shape into."
]
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"states, ke, ts = simulate(BendingBeam(21, EI=0.0), n_steps=160, n_frames=60)\n",
"fig, anim = utils.animate_springs_energy(\n",
" states, beam0.E, ts, {\"kinetic\": ke}, rest=beam0.X,\n",
" lims=((-0.6, 4.6), (-4.3, 0.6)), xlabel=\"time (s)\", ylabel=\"kinetic energy\",\n",
" scene_title=\"springs only (EI = 0): it flops down\", title=\"kinetic energy\",\n",
" colors={\"kinetic\": utils.ENERGY_COLORS[\"kinetic\"]}, pin_pts=beam0.X[[0, 1]])\n",
"utils.show_video(fig, anim, \"media/14_springs_only.mp4\", fps=20)"
]
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"## 2 · Bending energy penalizes the turning angle\n",
"\n",
"At each interior node the **hinge angle** $\\theta$ is the signed turn between\n",
"the incoming and outgoing segments (`simkit.dihedral_angles`); it is $0$ for a\n",
"straight beam. The bending energy sums $\\tfrac12\\,\\kappa\\,(\\theta-\\theta_0)^2$\n",
"over hinges, so it grows quadratically as we curl the beam into an arc of\n",
"increasing total turning angle $\\Phi$."
]
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",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"def arc_shape(N, total_turn, L=L):\n",
" \"\"\"Bend a straight beam into a circular arc with given total turning angle.\"\"\"\n",
" l = L / (N - 1)\n",
" dphi = total_turn / (N - 1) # turn added at each segment\n",
" angs = -0.5 * total_turn + dphi * np.arange(N - 1) + dphi / 2 # segment headings\n",
" pts = np.zeros((N, 2))\n",
" for i in range(N - 1):\n",
" pts[i + 1] = pts[i] + l * np.array([np.cos(angs[i]), np.sin(angs[i])])\n",
" return pts\n",
"\n",
"N = 21\n",
"beamN = BendingBeam(N)\n",
"C, theta0 = beamN.C, simkit.dihedral_angles(beamN.X, beamN.C)\n",
"kappa = np.full((C.shape[0], 1), 50.0 / (L / (N - 1)))\n",
"turns = np.linspace(0.0, 2.5, 60)\n",
"Ebend = [energies.bending_energy_x(arc_shape(N, t).flatten().reshape(-1, 1), C, theta0, kappa) for t in turns]\n",
"\n",
"fig, ax = utils.line_plot(turns, {\"bending energy\": Ebend},\n",
" xlabel=r\"total turning angle $\\Phi$ (rad)\", ylabel=\"bending energy\",\n",
" colors={\"bending energy\": utils.ENERGY_COLORS[\"elastic\"]},\n",
" title=r\"Bending energy is quadratic in curvature ($\\propto \\Phi^2$)\")\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"id": "a0483a0d",
"metadata": {},
"source": [
"## 3 · Flop when soft, hold its shape when stiff\n",
"\n",
"Now release the cantilever under gravity at three bending stiffnesses. The soft\n",
"beam **flops around** and droops nearly straight down; as $EI$ grows the beam\n",
"**retains its shape**, barely deflecting from horizontal. (Stretching stays stiff\n",
"throughout — $EA \\gg EI$ — so what changes is purely the\n",
"resistance to *bending*.)"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "4b6f1b4f",
"metadata": {
"execution": {
"iopub.execute_input": "2026-06-09T05:35:46.189975Z",
"iopub.status.busy": "2026-06-09T05:35:46.189915Z",
"iopub.status.idle": "2026-06-09T05:35:49.379593Z",
"shell.execute_reply": "2026-06-09T05:35:49.379314Z"
}
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
""
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"panels = []\n",
"for EI, label in [(2.0, \"soft (EI=2)\"), (50.0, \"medium (EI=50)\"), (2000.0, \"stiff (EI=2000)\")]:\n",
" states, _, _ = simulate(BendingBeam(21, EI=EI), n_steps=160, n_frames=60)\n",
" panels.append({\"states\": states, \"E\": beam0.E, \"title\": label})\n",
"fig, anim = utils.animate_springs_grid(\n",
" panels, lims=((-0.6, 4.6), (-4.3, 0.6)), fps=20, pin_pts=beam0.X[[0, 1]],\n",
" suptitle=\"Same beam, same gravity: stiffer bending = holds its shape\")\n",
"utils.show_video(fig, anim, \"media/14_flop_to_rigid.mp4\", fps=20)"
]
},
{
"cell_type": "markdown",
"id": "0d69c43a",
"metadata": {},
"source": [
"The **static equilibria** tell the same story: the droop shrinks steadily as\n",
"the bending stiffness increases."
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "a6eba623",
"metadata": {
"execution": {
"iopub.execute_input": "2026-06-09T05:35:49.380803Z",
"iopub.status.busy": "2026-06-09T05:35:49.380721Z",
"iopub.status.idle": "2026-06-09T05:35:49.511951Z",
"shell.execute_reply": "2026-06-09T05:35:49.511759Z"
}
},
"outputs": [
{
"data": {
"image/png": 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",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"fig, ax = plt.subplots(figsize=(7.5, 4.5))\n",
"utils.setup_axes(ax, (-0.5, 4.5), (-4.2, 0.6), title=\"Static cantilever sag vs bending stiffness\")\n",
"cmap = plt.get_cmap(\"viridis\"); EIs = [2.0, 20.0, 200.0, 2000.0]\n",
"for j, EI in enumerate(EIs):\n",
" Us = BendingBeam(21, EI=EI).solve()\n",
" ax.plot(Us[:, 0], Us[:, 1], \"-o\", ms=3, lw=2, color=cmap(j / (len(EIs) - 1)), label=f\"EI = {EI:g}\")\n",
"ax.scatter(beam0.X[[0, 1], 0], beam0.X[[0, 1], 1], s=70, marker=\"s\", color=utils.PIN_C, zorder=5, label=\"clamped\")\n",
"ax.legend(fontsize=9, loc=\"lower left\"); plt.show()"
]
},
{
"cell_type": "markdown",
"id": "0522c4d2",
"metadata": {},
"source": [
"## 4 · Refine the mesh → convergence to one shape\n",
"\n",
"Hold the *continuum* beam fixed ($L, EI, EA, \\rho, g$) and only change the\n",
"number of nodes `N`. Because the stiffnesses were scaled to be mesh-independent,\n",
"the discrete equilibria pile up onto a **single limiting curve**: coarse meshes\n",
"are too stiff and under-droop, but each refinement moves less than the last."
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "15dcb3f6",
"metadata": {
"execution": {
"iopub.execute_input": "2026-06-09T05:35:49.512981Z",
"iopub.status.busy": "2026-06-09T05:35:49.512926Z",
"iopub.status.idle": "2026-06-09T05:35:49.759323Z",
"shell.execute_reply": "2026-06-09T05:35:49.759114Z"
}
},
"outputs": [
{
"data": {
"image/png": 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",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"Ns = [5, 9, 17, 33, 65]\n",
"sols = {N: BendingBeam(N, EI=50.0).solve() for N in Ns}\n",
"fig, ax = plt.subplots(figsize=(7.5, 4.5))\n",
"utils.setup_axes(ax, (-0.5, 4.5), (-1.6, 0.4), title=\"Refining the mesh converges to one shape (EI=50)\")\n",
"cmap = plt.get_cmap(\"plasma\")\n",
"for j, N in enumerate(Ns):\n",
" U = sols[N]\n",
" ax.plot(U[:, 0], U[:, 1], \"-o\", ms=3, lw=1.8, color=cmap(j / (len(Ns) - 1)), label=f\"N = {N}\")\n",
"ax.legend(fontsize=9, loc=\"lower left\"); plt.show()"
]
},
{
"cell_type": "markdown",
"id": "c3eb1ed4",
"metadata": {},
"source": [
"Measured against the finest mesh, the tip deflection error falls steadily with\n",
"resolution — first-order convergence (halving the segment size roughly\n",
"halves the error)."
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "6411be4b",
"metadata": {
"execution": {
"iopub.execute_input": "2026-06-09T05:35:49.760408Z",
"iopub.status.busy": "2026-06-09T05:35:49.760349Z",
"iopub.status.idle": "2026-06-09T05:35:50.220638Z",
"shell.execute_reply": "2026-06-09T05:35:50.220429Z"
}
},
"outputs": [
{
"data": {
"image/png": 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",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"ref = BendingBeam(129, EI=50.0).solve()[-1, 1]\n",
"Nc = [5, 9, 17, 33, 65]\n",
"err = [abs(BendingBeam(N, EI=50.0).solve()[-1, 1] - ref) for N in Nc]\n",
"fig, ax = utils.loglog_plot(np.array(Nc), {\"tip error vs finest\": err},\n",
" xlabel=\"number of nodes N\", ylabel=\"|tip_y(N) - tip_y(fine)|\",\n",
" colors={\"tip error vs finest\": \"#d62728\"}, ref_slopes={\"O(1/N)\": -1},\n",
" title=\"Tip deflection converges as the mesh refines\")\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"id": "e3089d5e",
"metadata": {},
"source": [
"## 5 · Aside: the same bending one dimension up\n",
"\n",
"The hinge angle we used here is the **2D** member of one family. Discrete bending\n",
"always lives on a *codimension-1 hinge* shared by two flaps, and the energy\n",
"$\\tfrac12\\,\\kappa\\,(\\theta-\\theta_0)^2$ is identical across dimensions —\n",
"only the geometry of $\\theta$ changes:\n",
"\n",
"| | flaps | shared hinge | connectivity | angle |\n",
"|---|---|---|---|---|\n",
"| **curve in 2D** (this tutorial) | edges | a **vertex** | triple `(A,B,C)` | hinge turning angle |\n",
"| **surface in 3D** (cloth/shells) | triangles | an **edge** | quad `(x0,x1,x2,x3)` | dihedral fold angle |\n",
"\n",
"simkit reflects this: `simkit.dihedral_angles(X, C)` returns the hinge angle for\n",
"`(n,2)` positions with triple connectivity and the dihedral angle for `(n,3)`\n",
"positions with quad connectivity — the dimension is inferred from the data.\n",
"The same `bending_energy_x(x, C, theta0, kappa)` assembles both; the 3D\n",
"`discrete_shells_bending_*` functions are thin wrappers over it. So everything\n",
"above carries over to a bending **cloth** by swapping the connectivity and adding\n",
"a third coordinate."
]
},
{
"cell_type": "markdown",
"id": "ee3d5919",
"metadata": {},
"source": [
"### Takeaways\n",
"* Springs resist stretch; a **bending** energy on the hinge angles is what lets a\n",
" beam resist *curvature* and hold a shape.\n",
"* Increasing bending stiffness $EI$ takes the cantilever from **floppy** (flops\n",
" down under gravity) to **rigid** (holds nearly horizontal).\n",
"* Scaling the discrete stiffnesses to a fixed continuum beam makes the simulation\n",
" **mesh-independent**: refining `N` converges to a single equilibrium shape.\n",
"* The 2D hinge and 3D dihedral bending are the *same* energy one codimension\n",
" apart — unified behind `simkit.dihedral_angles` / `bending_energy_x`."
]
}
],
"metadata": {
"kernelspec": {
"display_name": "simkit",
"language": "python",
"name": "python3"
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"language_info": {
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"name": "ipython",
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"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.13.9"
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