{ "cells": [ { "cell_type": "code", "execution_count": 1, "id": "c794a91a", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:40:49.239389Z", "iopub.status.busy": "2026-10-02T14:40:49.239208Z", "iopub.status.idle": "2026-10-02T14:40:49.244101Z", "shell.execute_reply": "2026-10-02T14:40:49.243193Z" }, "hide_input": true, "papermill": { "duration": 0.007545, "end_time": "2026-10-02T14:40:49.244636+00:00", "exception": false, "start_time": "2026-10-02T14:40:49.237091+00:00", "status": "completed" }, "tags": [ "active-ipynb", "remove-input", "remove-output" ] }, "outputs": [], "source": [ "try:\n", " from openmdao.utils.notebook_utils import notebook_mode # noqa: F401\n", "except ImportError:\n", " !python -m pip install openmdao[notebooks]" ] }, { "cell_type": "markdown", "id": "276a781c", "metadata": { "papermill": { "duration": 0.001081, "end_time": "2026-10-02T14:40:49.247022+00:00", "exception": false, "start_time": "2026-10-02T14:40:49.245941+00:00", "status": "completed" }, "tags": [] }, "source": [ "# Optimizing the Thickness Distribution of a Cantilever Beam Using the Adjoint Method\n", "\n", "In this example, we optimize the thickness (height) distribution of a cantilever beam\n", "using the adjoint method to compute the gradient. We use Euler--Bernoulli beam theory and assume a rectangular section.\n", "\n", "## Background\n", "\n", "The optimization problem is:\n", "\n", "$$\n", " \\begin{array}{r c l}\n", " \\text{minimize} & & f^T d \\\\\n", " \\text{with respect to} & & h \\\\\n", " \\text{subject to} & & \\text{sum}(h) b L_0 = \\text{volume} \\\\\n", " \\end{array}\n", "$$\n", "\n", "where $f$ is the vector of forces, $h$ is the vector of beam heights, and $L_0$ is the length of a single beam element.\n", "\n", "The displacements vector $d$ is given by\n", "\n", "$$\n", " K d = f \n", "$$\n", "\n", "where $K$ is the stiffness matrix. However, in practice, we augment the linear system with Lagrange multipliers to apply the boundary constraints at the first node.\n", "\n", "Since our model contains a system of equations, we use the adjoint method to compute the gradient of the objective with respect to the beam height vector. The model is shown below." ] }, { "cell_type": "code", "execution_count": 2, "id": "cbba88a5", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:40:49.250176Z", "iopub.status.busy": "2026-10-02T14:40:49.249987Z", "iopub.status.idle": "2026-10-02T14:40:53.347453Z", "shell.execute_reply": "2026-10-02T14:40:53.346956Z" }, "papermill": { "duration": 4.099999, "end_time": "2026-10-02T14:40:53.348131+00:00", "exception": false, "start_time": "2026-10-02T14:40:49.248132+00:00", "status": "completed" }, "tags": [ "remove-input" ] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[1790952051.634841] [runnervm8df0l:6103 :0] ib_iface.c:1269 UCX ERROR mana_0: iface 0x5576494342b0 failed to create UD QP TX wr:256 sge:6 inl:64 resp:0 RX wr:4096 sge:1 resp:0 failed: Operation not supported\n", "[1790952051.635091] [runnervm8df0l:6103 :0] ucp_worker.c:1412 UCX ERROR uct_iface_open(ud_verbs/mana_0:1) failed: Input/output error\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ "[runnervm8df0l:06103] pml_ucx.c:313 Error: Failed to create UCP worker\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "Optimization terminated successfully (Exit mode 0)\n", " Current function value: 23762.153678368355\n", " Iterations: 147\n", " Function evaluations: 429\n", " Gradient evaluations: 147\n", "Optimization Complete\n", "-----------------------------------\n" ] }, { "data": { "text/html": [ "\n", " \n", " " ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "\n", "import openmdao.api as om\n", "from openmdao.test_suite.test_examples.beam_optimization.beam_group import BeamGroup\n", "\n", "if __name__ == '__main__':\n", "\n", " E = 1.\n", " L = 1.\n", " b = 0.1\n", " volume = 0.01\n", "\n", " num_elements = 50\n", "\n", " prob = om.Problem(model=BeamGroup(E=E, L=L, b=b, volume=volume, num_elements=num_elements),\n", " driver=om.ScipyOptimizeDriver(optimizer='SLSQP', tol=1e-9, disp=True))\n", " prob.setup()\n", " prob.run_driver()\n", "\n", " om.n2(prob)" ] }, { "cell_type": "code", "execution_count": 3, "id": "45de7a24", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:40:53.352599Z", "iopub.status.busy": "2026-10-02T14:40:53.352230Z", "iopub.status.idle": "2026-10-02T14:40:53.355261Z", "shell.execute_reply": "2026-10-02T14:40:53.354862Z" }, "papermill": { "duration": 0.006095, "end_time": "2026-10-02T14:40:53.355976+00:00", "exception": false, "start_time": "2026-10-02T14:40:53.349881+00:00", "status": "completed" }, "tags": [ "remove-input", "remove-output" ] }, "outputs": [], "source": [ "assert np.linalg.norm(prob['h'] - [\n", " 0.14915754, 0.14764328, 0.14611321, 0.14456715, 0.14300421, 0.14142417,\n", " 0.13982611, 0.13820976, 0.13657406, 0.13491866, 0.13324268, 0.13154528,\n", " 0.12982575, 0.12808305, 0.12631658, 0.12452477, 0.12270701, 0.12086183,\n", " 0.11898809, 0.11708424, 0.11514904, 0.11318072, 0.11117762, 0.10913764,\n", " 0.10705891, 0.10493903, 0.10277539, 0.10056526, 0.09830546, 0.09599246,\n", " 0.09362243, 0.09119084, 0.08869265, 0.08612198, 0.08347229, 0.08073573,\n", " 0.07790323, 0.07496382, 0.07190453, 0.06870925, 0.0653583, 0.06182632,\n", " 0.05808044, 0.05407658, 0.04975295, 0.0450185, 0.03972912, 0.03363155,\n", " 0.02620192, 0.01610863\n", "]) < 1e-4" ] }, { "cell_type": "markdown", "id": "c5e21ff5", "metadata": { "papermill": { "duration": 0.001415, "end_time": "2026-10-02T14:40:53.358855+00:00", "exception": false, "start_time": "2026-10-02T14:40:53.357440+00:00", "status": "completed" }, "tags": [] }, "source": [ "## Implementation: list of components\n", "\n", "There are 5 components that compute:\n", "\n", "1. Moment of inertia for each element\n", "2. Local stiffness matrix for each element\n", "3. Displacements from solution of the $Kd=f$ linear system augmented with the Lagrange multipliers\n", "4. Compliance\n", "5. Volume" ] }, { "cell_type": "code", "execution_count": 4, "id": "f0127280", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:40:53.363300Z", "iopub.status.busy": "2026-10-02T14:40:53.363117Z", "iopub.status.idle": "2026-10-02T14:40:53.366357Z", "shell.execute_reply": "2026-10-02T14:40:53.365853Z" }, "papermill": { "duration": 0.006385, "end_time": "2026-10-02T14:40:53.366816+00:00", "exception": false, "start_time": "2026-10-02T14:40:53.360431+00:00", "status": "completed" }, "tags": [] }, "outputs": [], "source": [ "class MomentOfInertiaComp(om.ExplicitComponent):\n", "\n", " def initialize(self):\n", " self.options.declare('num_elements', types=int)\n", " self.options.declare('b')\n", "\n", " def setup(self):\n", " num_elements = self.options['num_elements']\n", "\n", " self.add_input('h', shape=num_elements)\n", " self.add_output('I', shape=num_elements)\n", "\n", " def setup_partials(self):\n", " rows = cols = np.arange(self.options['num_elements'])\n", " self.declare_partials('I', 'h', rows=rows, cols=cols)\n", "\n", " def compute(self, inputs, outputs):\n", " outputs['I'] = 1./12. * self.options['b'] * inputs['h'] ** 3\n", "\n", " def compute_partials(self, inputs, partials):\n", " partials['I', 'h'] = 1./4. * self.options['b'] * inputs['h'] ** 2" ] }, { "cell_type": "code", "execution_count": 5, "id": "be74eb7a", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:40:53.370160Z", "iopub.status.busy": "2026-10-02T14:40:53.369990Z", "iopub.status.idle": "2026-10-02T14:40:53.373898Z", "shell.execute_reply": "2026-10-02T14:40:53.373498Z" }, "papermill": { "duration": 0.006468, "end_time": "2026-10-02T14:40:53.374598+00:00", "exception": false, "start_time": "2026-10-02T14:40:53.368130+00:00", "status": "completed" }, "tags": [] }, "outputs": [], "source": [ "class LocalStiffnessMatrixComp(om.ExplicitComponent):\n", "\n", " def initialize(self):\n", " self.options.declare('num_elements', types=int)\n", " self.options.declare('E')\n", " self.options.declare('L')\n", "\n", " def setup(self):\n", " num_elements = self.options['num_elements']\n", " E = self.options['E']\n", " L = self.options['L']\n", "\n", " self.add_input('I', shape=num_elements)\n", " self.add_output('K_local', shape=(num_elements, 4, 4))\n", "\n", " L0 = L / num_elements\n", " coeffs = np.empty((4, 4))\n", " coeffs[0, :] = [12, 6 * L0, -12, 6 * L0]\n", " coeffs[1, :] = [6 * L0, 4 * L0 ** 2, -6 * L0, 2 * L0 ** 2]\n", " coeffs[2, :] = [-12, -6 * L0, 12, -6 * L0]\n", " coeffs[3, :] = [6 * L0, 2 * L0 ** 2, -6 * L0, 4 * L0 ** 2]\n", " coeffs *= E / L0 ** 3\n", "\n", " self.mtx = np.zeros((num_elements, 4, 4, num_elements))\n", " for ind in range(num_elements):\n", " self.mtx[ind, :, :, ind] = coeffs\n", "\n", " self.declare_partials('K_local', 'I',\n", " val=self.mtx.reshape(16 * num_elements, num_elements))\n", "\n", " def compute(self, inputs, outputs):\n", " outputs['K_local'] = 0\n", " for ind in range(self.options['num_elements']):\n", " outputs['K_local'][ind, :, :] = self.mtx[ind, :, :, ind] * inputs['I'][ind]" ] }, { "cell_type": "code", "execution_count": 6, "id": "2cf5dcc3", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:40:53.378488Z", "iopub.status.busy": "2026-10-02T14:40:53.378351Z", "iopub.status.idle": "2026-10-02T14:40:53.385561Z", "shell.execute_reply": "2026-10-02T14:40:53.385187Z" }, "papermill": { "duration": 0.009871, "end_time": "2026-10-02T14:40:53.386020+00:00", "exception": false, "start_time": "2026-10-02T14:40:53.376149+00:00", "status": "completed" }, "tags": [] }, "outputs": [], "source": [ "from scipy.sparse import coo_matrix\n", "from scipy.sparse.linalg import splu\n", "\n", "class StatesComp(om.ImplicitComponent):\n", "\n", " def initialize(self):\n", " self.options.declare('num_elements', types=int)\n", " self.options.declare('force_vector', types=np.ndarray)\n", "\n", " def setup(self):\n", " num_elements = self.options['num_elements']\n", " num_nodes = num_elements + 1\n", " size = 2 * num_nodes + 2\n", "\n", " self.add_input('K_local', shape=(num_elements, 4, 4))\n", " self.add_output('d', shape=size)\n", "\n", " cols = np.arange(16*num_elements)\n", " rows = np.repeat(np.arange(4), 4)\n", " rows = np.tile(rows, num_elements) + np.repeat(np.arange(num_elements), 16) * 2\n", "\n", " self.declare_partials('d', 'K_local', rows=rows, cols=cols)\n", " self.declare_partials('d', 'd')\n", "\n", " def apply_nonlinear(self, inputs, outputs, residuals):\n", " force_vector = np.concatenate([self.options['force_vector'], np.zeros(2)])\n", "\n", " self.K = self.assemble_CSC_K(inputs)\n", " residuals['d'] = self.K.dot(outputs['d']) - force_vector\n", "\n", " def solve_nonlinear(self, inputs, outputs):\n", " force_vector = np.concatenate([self.options['force_vector'], np.zeros(2)])\n", "\n", " self.K = self.assemble_CSC_K(inputs)\n", " self.lu = splu(self.K)\n", "\n", " outputs['d'] = self.lu.solve(force_vector)\n", "\n", " def linearize(self, inputs, outputs, jacobian):\n", " num_elements = self.options['num_elements']\n", "\n", " self.K = self.assemble_CSC_K(inputs)\n", " self.lu = splu(self.K)\n", "\n", " i_elem = np.tile(np.arange(4), 4)\n", " i_d = np.tile(i_elem, num_elements) + np.repeat(np.arange(num_elements), 16) * 2\n", "\n", " jacobian['d', 'K_local'] = outputs['d'][i_d]\n", "\n", " jacobian['d', 'd'] = self.K.toarray()\n", "\n", " def solve_linear(self, d_outputs, d_residuals, mode):\n", " if mode == 'fwd':\n", " d_outputs['d'] = self.lu.solve(d_residuals['d'])\n", " else:\n", " d_residuals['d'] = self.lu.solve(d_outputs['d'])\n", "\n", " def assemble_CSC_K(self, inputs):\n", " \"\"\"\n", " Assemble the stiffness matrix in sparse CSC format.\n", "\n", " Returns\n", " -------\n", " ndarray\n", " Stiffness matrix as dense ndarray.\n", " \"\"\"\n", " num_elements = self.options['num_elements']\n", " num_nodes = num_elements + 1\n", " num_entry = num_elements * 12 + 4\n", " ndim = num_entry + 4\n", "\n", " data = np.zeros((ndim, ), dtype=inputs._get_data().dtype)\n", " cols = np.empty((ndim, ))\n", " rows = np.empty((ndim, ))\n", "\n", " # First element.\n", " data[:16] = inputs['K_local'][0, :, :].flat\n", " cols[:16] = np.tile(np.arange(4), 4)\n", " rows[:16] = np.repeat(np.arange(4), 4)\n", "\n", " j = 16\n", " for ind in range(1, num_elements):\n", " ind1 = 2 * ind\n", " K = inputs['K_local'][ind, :, :]\n", "\n", " # NW quadrant gets summed with previous connected element.\n", " data[j-6:j-4] += K[0, :2]\n", " data[j-2:j] += K[1, :2]\n", "\n", " # NE quadrant\n", " data[j:j+4] = K[:2, 2:].flat\n", " rows[j:j+4] = np.array([ind1, ind1, ind1 + 1, ind1 + 1])\n", " cols[j:j+4] = np.array([ind1 + 2, ind1 + 3, ind1 + 2, ind1 + 3])\n", "\n", " # SE and SW quadrants together\n", " data[j+4:j+12] = K[2:, :].flat\n", " rows[j+4:j+12] = np.repeat(np.arange(ind1 + 2, ind1 + 4), 4)\n", " cols[j+4:j+12] = np.tile(np.arange(ind1, ind1 + 4), 2)\n", "\n", " j += 12\n", "\n", " data[-4:] = 1.0\n", " rows[-4] = 2 * num_nodes\n", " rows[-3] = 2 * num_nodes + 1\n", " rows[-2] = 0.0\n", " rows[-1] = 1.0\n", " cols[-4] = 0.0\n", " cols[-3] = 1.0\n", " cols[-2] = 2 * num_nodes\n", " cols[-1] = 2 * num_nodes + 1\n", "\n", " n_K = 2 * num_nodes + 2\n", " return coo_matrix((data, (rows, cols)), shape=(n_K, n_K)).tocsc()\n" ] }, { "cell_type": "code", "execution_count": 7, "id": "2e47f33a", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:40:53.389741Z", "iopub.status.busy": "2026-10-02T14:40:53.389595Z", "iopub.status.idle": "2026-10-02T14:40:53.392548Z", "shell.execute_reply": "2026-10-02T14:40:53.392104Z" }, "papermill": { "duration": 0.005399, "end_time": "2026-10-02T14:40:53.392987+00:00", "exception": false, "start_time": "2026-10-02T14:40:53.387588+00:00", "status": "completed" }, "tags": [] }, "outputs": [], "source": [ "class ComplianceComp(om.ExplicitComponent):\n", "\n", " def initialize(self):\n", " self.options.declare('num_elements', types=int)\n", " self.options.declare('force_vector', types=np.ndarray)\n", "\n", " def setup(self):\n", " num_nodes = self.options['num_elements'] + 1\n", "\n", " self.add_input('displacements', shape=2 * num_nodes)\n", " self.add_output('compliance')\n", "\n", " def setup_partials(self):\n", " num_nodes = self.options['num_elements'] + 1\n", " force_vector = self.options['force_vector']\n", " self.declare_partials('compliance', 'displacements',\n", " val=force_vector.reshape((1, 2 * num_nodes)))\n", "\n", " def compute(self, inputs, outputs):\n", " outputs['compliance'] = np.dot(self.options['force_vector'], inputs['displacements'])" ] }, { "cell_type": "code", "execution_count": 8, "id": "1ba2d4d0", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:40:53.396684Z", "iopub.status.busy": "2026-10-02T14:40:53.396552Z", "iopub.status.idle": "2026-10-02T14:40:53.399400Z", "shell.execute_reply": "2026-10-02T14:40:53.399073Z" }, "papermill": { "duration": 0.005338, "end_time": "2026-10-02T14:40:53.399836+00:00", "exception": false, "start_time": "2026-10-02T14:40:53.394498+00:00", "status": "completed" }, "tags": [] }, "outputs": [], "source": [ "class VolumeComp(om.ExplicitComponent):\n", "\n", " def initialize(self):\n", " self.options.declare('num_elements', types=int)\n", " self.options.declare('b', default=1.)\n", " self.options.declare('L')\n", "\n", " def setup(self):\n", " num_elements = self.options['num_elements']\n", " b = self.options['b']\n", " L = self.options['L']\n", " L0 = L / num_elements\n", "\n", " self.add_input('h', shape=num_elements)\n", " self.add_output('volume')\n", "\n", " self.declare_partials('volume', 'h', val=b * L0)\n", "\n", " def compute(self, inputs, outputs):\n", " L0 = self.options['L'] / self.options['num_elements']\n", "\n", " outputs['volume'] = np.sum(inputs['h'] * self.options['b'] * L0)" ] }, { "cell_type": "code", "execution_count": 9, "id": "26b26fe9", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:40:53.404633Z", "iopub.status.busy": "2026-10-02T14:40:53.404506Z", "iopub.status.idle": "2026-10-02T14:40:53.411899Z", "shell.execute_reply": "2026-10-02T14:40:53.411408Z" }, "papermill": { "duration": 0.010744, "end_time": "2026-10-02T14:40:53.412333+00:00", "exception": false, "start_time": "2026-10-02T14:40:53.401589+00:00", "status": "completed" }, "tags": [] }, "outputs": [], "source": [ "class BeamGroup(om.Group):\n", "\n", " def initialize(self):\n", " self.options.declare('E')\n", " self.options.declare('L')\n", " self.options.declare('b')\n", " self.options.declare('volume')\n", " self.options.declare('num_elements', int)\n", "\n", " def setup(self):\n", " E = self.options['E']\n", " L = self.options['L']\n", " b = self.options['b']\n", " volume = self.options['volume']\n", " num_elements = self.options['num_elements']\n", " num_nodes = num_elements + 1\n", "\n", " force_vector = np.zeros(2 * num_nodes)\n", " force_vector[-2] = -1.\n", "\n", " I_comp = MomentOfInertiaComp(num_elements=num_elements, b=b)\n", " self.add_subsystem('I_comp', I_comp, promotes_inputs=['h'])\n", "\n", " comp = LocalStiffnessMatrixComp(num_elements=num_elements, E=E, L=L)\n", " self.add_subsystem('local_stiffness_matrix_comp', comp)\n", "\n", " comp = StatesComp(num_elements=num_elements, force_vector=force_vector)\n", " self.add_subsystem('states_comp', comp)\n", "\n", " comp = ComplianceComp(num_elements=num_elements, force_vector=force_vector)\n", " self.add_subsystem('compliance_comp', comp)\n", "\n", " comp = VolumeComp(num_elements=num_elements, b=b, L=L)\n", " self.add_subsystem('volume_comp', comp, promotes_inputs=['h'])\n", "\n", " self.connect('I_comp.I', 'local_stiffness_matrix_comp.I')\n", " self.connect('local_stiffness_matrix_comp.K_local', 'states_comp.K_local')\n", " self.connect('states_comp.d', 'compliance_comp.displacements',\n", " src_indices=np.arange(2 *num_nodes))\n", "\n", " self.add_design_var('h', lower=1e-2, upper=10.)\n", " self.add_objective('compliance_comp.compliance')\n", " self.add_constraint('volume_comp.volume', equals=volume)" ] }, { "cell_type": "markdown", "id": "dcdbb1ec", "metadata": { "papermill": { "duration": 0.066998, "end_time": "2026-10-02T14:40:53.480814+00:00", "exception": false, "start_time": "2026-10-02T14:40:53.413816+00:00", "status": "completed" }, "tags": [] }, "source": [ "## Implementation: Optimization Script\n", "\n", "Here is the optimization script:" ] }, { "cell_type": "code", "execution_count": 10, "id": "7f944597", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:40:53.485598Z", "iopub.status.busy": "2026-10-02T14:40:53.485255Z", "iopub.status.idle": "2026-10-02T14:40:54.566275Z", "shell.execute_reply": "2026-10-02T14:40:54.565538Z" }, "papermill": { "duration": 1.084229, "end_time": "2026-10-02T14:40:54.566632+00:00", "exception": false, "start_time": "2026-10-02T14:40:53.482403+00:00", "status": "completed" }, "tags": [] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Optimization terminated successfully (Exit mode 0)\n", " Current function value: 23762.153678368355\n", " Iterations: 147\n", " Function evaluations: 429\n", " Gradient evaluations: 147\n", "Optimization Complete\n", "-----------------------------------\n", "[0.14915737 0.14764327 0.14611335 0.14456713 0.14300431 0.14142418\n", " 0.13982608 0.1382096 0.13657418 0.13491879 0.13324264 0.13154515\n", " 0.12982577 0.12808303 0.12631659 0.12452485 0.12270702 0.12086201\n", " 0.11898784 0.11708423 0.11514893 0.11318075 0.11117767 0.1091377\n", " 0.1070589 0.10493896 0.10277536 0.10056519 0.09830517 0.09599247\n", " 0.09362247 0.0911908 0.08869262 0.0861219 0.08347227 0.08073572\n", " 0.07790323 0.074964 0.07190457 0.06870938 0.06535838 0.06182647\n", " 0.05808039 0.0540766 0.04975292 0.04501857 0.03972915 0.03363159\n", " 0.02620189 0.01610863]\n" ] } ], "source": [ "from openmdao.test_suite.test_examples.beam_optimization.beam_group import BeamGroup\n", "\n", "E = 1.\n", "L = 1.\n", "b = 0.1\n", "volume = 0.01\n", "\n", "num_elements = 50\n", "\n", "prob = om.Problem(model=BeamGroup(E=E, L=L, b=b, volume=volume, num_elements=num_elements))\n", "\n", "prob.driver = om.ScipyOptimizeDriver()\n", "prob.driver.options['optimizer'] = 'SLSQP'\n", "prob.driver.options['tol'] = 1e-9\n", "prob.driver.options['disp'] = True\n", "\n", "prob.setup()\n", "\n", "prob.run_driver()\n", "\n", "print(prob['h'])" ] }, { "cell_type": "markdown", "id": "2661d7d1", "metadata": { "papermill": { "duration": 0.001303, "end_time": "2026-10-02T14:40:54.569623+00:00", "exception": false, "start_time": "2026-10-02T14:40:54.568320+00:00", "status": "completed" }, "tags": [] }, "source": [ "![optimized](images/optimized.png)" ] }, { "cell_type": "code", "execution_count": 11, "id": "98df7eaa", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:40:54.573298Z", "iopub.status.busy": "2026-10-02T14:40:54.572809Z", "iopub.status.idle": "2026-10-02T14:40:54.577259Z", "shell.execute_reply": "2026-10-02T14:40:54.576902Z" }, "papermill": { "duration": 0.006902, "end_time": "2026-10-02T14:40:54.577771+00:00", "exception": false, "start_time": "2026-10-02T14:40:54.570869+00:00", "status": "completed" }, "tags": [ "remove-input", "remove-output" ] }, "outputs": [ { "data": { "text/plain": [ "np.float64(9.037037485372181e-07)" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from openmdao.utils.assert_utils import assert_near_equal\n", "\n", "assert_near_equal(prob['h'],\n", " [0.14915754, 0.14764328, 0.14611321, 0.14456715, 0.14300421, 0.14142417,\n", " 0.13982611, 0.13820976, 0.13657406, 0.13491866, 0.13324268, 0.13154528,\n", " 0.12982575, 0.12808305, 0.12631658, 0.12452477, 0.12270701, 0.12086183,\n", " 0.11898809, 0.11708424, 0.11514904, 0.11318072, 0.11117762, 0.10913764,\n", " 0.10705891, 0.10493903, 0.10277539, 0.10056526, 0.09830546, 0.09599246,\n", " 0.09362243, 0.09119084, 0.08869265, 0.08612198, 0.08347229, 0.08073573,\n", " 0.07790323, 0.07496382, 0.07190453, 0.06870925, 0.0653583, 0.06182632,\n", " 0.05808044, 0.05407658, 0.04975295, 0.0450185, 0.03972912, 0.03363155,\n", " 0.02620192, 0.01610863], 1e-4)" ] } ], "metadata": { "celltoolbar": "Tags", "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": "3.13.14" }, "papermill": { "default_parameters": {}, "duration": 6.532729, "end_time": "2026-10-02T14:40:55.094099+00:00", "environment_variables": {}, "exception": null, "input_path": "/home/runner/work/OpenMDAO/OpenMDAO/openmdao/docs/openmdao_book/examples/beam_optimization_example.ipynb", "output_path": "/home/runner/work/OpenMDAO/OpenMDAO/openmdao/docs/_executed_book/examples/beam_optimization_example.ipynb", "parameters": {}, "start_time": "2026-10-02T14:40:48.561370+00:00", "version": "2.7.0" } }, "nbformat": 4, "nbformat_minor": 5 }