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94 changes: 94 additions & 0 deletions
94
chapter5/fsi/fluid-structure_interaction_in_a_flexible_channel.ipynb
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,94 @@ | ||
| { | ||
| "cells": [ | ||
| { | ||
| "cell_type": "markdown", | ||
| "id": "c06958bc", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "# Fluid-Structure Interaction in a Flexible Channel\n", | ||
| "\n", | ||
| "## Mathematical Model\n", | ||
| "\n", | ||
| "Consider the FSI of a flexible channel with fluid flow. The fluid velocity $\\mathbf{v}$ and structural displacement $u$ are coupled through the FSI problem.\n", | ||
| "\n", | ||
| "- Structural Equation:\n", | ||
| "\n", | ||
| "$$\n", | ||
| "\\begin{align*}\n", | ||
| "\\rho_s A_s \\frac{\\partial^2 u}{\\partial t^2} - \\nabla \\cdot \\sigma(u) &= 0 \\quad \\text{in } \\Omega_s \\times (0, T) \\\\\n", | ||
| "u &= 0 \\quad \\text{on } \\Gamma_{\\text{fixed}} \\times (0, T) \\\\\n", | ||
| "\\sigma(u) \\cdot \\mathbf{n} &= \\mathbf{t} \\quad \\text{on } \\Gamma_{\\text{interface}} \\times (0, T)\n", | ||
| "\\end{align*}\n", | ||
| "$$\n", | ||
| "\n", | ||
| "- Fluid Equation:\n", | ||
| "\n", | ||
| "$$\n", | ||
| "\\begin{align*}\n", | ||
| "\\rho_f \\frac{\\partial \\mathbf{v}}{\\partial t} + \\rho_f (\\mathbf{v} \\cdot \\nabla) \\mathbf{v} - \\nabla \\cdot \\sigma(\\mathbf{v}) &= 0 \\quad \\text{in } \\Omega_f \\times (0, T) \\\\\n", | ||
| "\\mathbf{v} &= \\mathbf{0} \\quad \\text{on } \\Gamma_{\\text{inlet}} \\times (0, T) \\\\\n", | ||
| "\\sigma(\\mathbf{v}) \\cdot \\mathbf{n} &= p \\cdot \\mathbf{n} \\quad \\text{on } \\Gamma_{\\text{interface}} \\times (0, T)\n", | ||
| "\\end{align*}\n", | ||
| "$$\n", | ||
| "\n", | ||
| "- Coupling Conditions:\n", | ||
| "\n", | ||
| "$$\n", | ||
| "\\begin{align*}\n", | ||
| "\\mathbf{v}(\\mathbf{x}, t) &= \\mathbf{v}_f(\\mathbf{x}, t) \\quad \\text{on } \\Gamma_{\\text{interface}} \\times (0, T) \\\\\n", | ||
| "\\sigma(u) \\cdot \\mathbf{n} &= \\sigma(\\mathbf{v}) \\cdot \\mathbf{n} \\quad \\text{on } \\Gamma_{\\text{interface}} \\times (0, T)\n", | ||
| "\\end{align*}\n", | ||
| "$$\n", | ||
| "\n", | ||
| "## Weak Formulation\n", | ||
| "\n", | ||
| "Find $u \\in V_s$ and $\\mathbf{v} \\in V_f$ such that\n", | ||
| "\n", | ||
| "$$\n", | ||
| "\\begin{align*}\n", | ||
| "\\int_{\\Omega_s} \\rho_s A_s \\frac{\\partial^2 u}{\\partial t^2} \\phi_s \\, d\\Omega &- \\int_{\\Omega_s} \\nabla \\cdot \\sigma(u) \\cdot \\nabla \\phi_s \\, d\\Omega = 0 \\\\\n", | ||
| "\\int_{\\Omega_f} \\rho_f \\frac{\\partial \\mathbf{v}}{\\partial t} \\cdot \\mathbf{\\phi}_f \\, d\\Omega &+ \\int_{\\Omega_f} \\rho_f (\\mathbf{v} \\cdot \\nabla) \\mathbf{v} \\cdot \\mathbf{\\phi}_f \\, d\\Omega - \\int_{\\Omega_f} \\nabla \\cdot \\sigma(\\mathbf{v}) \\cdot \\nabla \\mathbf{\\phi}_f \\, d\\Omega = 0\n", | ||
| "\\end{align*}\n", | ||
| "$$\n", | ||
| "\n", | ||
| "for all $\\phi_s \\in V_s$ and $\\mathbf{\\phi}_f \\in V_f$.\n", | ||
| "\n", | ||
| "\n", | ||
| "\n", | ||
| "\n", | ||
| "\n", | ||
| "\n", | ||
| "\n" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": null, | ||
| "id": "f77f76ce", | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [] | ||
| } | ||
| ], | ||
| "metadata": { | ||
| "kernelspec": { | ||
| "display_name": ".iga-python", | ||
| "language": "python", | ||
| "name": ".iga-python" | ||
| }, | ||
| "language_info": { | ||
| "codemirror_mode": { | ||
| "name": "ipython", | ||
| "version": 3 | ||
| }, | ||
| "file_extension": ".py", | ||
| "mimetype": "text/x-python", | ||
| "name": "python", | ||
| "nbconvert_exporter": "python", | ||
| "pygments_lexer": "ipython3", | ||
| "version": "3.10.12" | ||
| } | ||
| }, | ||
| "nbformat": 4, | ||
| "nbformat_minor": 5 | ||
| } |
88 changes: 88 additions & 0 deletions
88
chapter5/fsi/flutter_analysis_of_a_flexible_wing_in_fluid_flow.ipynb
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,88 @@ | ||
| { | ||
| "cells": [ | ||
| { | ||
| "cell_type": "markdown", | ||
| "id": "37cc9bbf", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "# Flutter Analysis of a Flexible Wing in Fluid Flow\n", | ||
| "\n", | ||
| "## Mathematical Model\n", | ||
| "\n", | ||
| "Consider the fluid-structure interaction of a flexible wing in a steady airflow. The structural displacement \\(u\\) and fluid velocity \\(\\mathbf{v}\\) are coupled through the FSI problem.\n", | ||
| "\n", | ||
| "- Structural Equation:\n", | ||
| "\n", | ||
| "$$\n", | ||
| "\\begin{align*}\n", | ||
| "\\rho_s A_s \\frac{\\partial^2 u}{\\partial t^2} + c_s \\frac{\\partial u}{\\partial t} - \\nabla \\cdot (\\sigma(u)) &= 0 \\quad \\text{in } \\Omega_s \\times (0, T) \\\\\n", | ||
| "u &= 0 \\quad \\text{on } \\Gamma_{\\text{fixed}} \\times (0, T) \\\\\n", | ||
| "\\sigma(u) \\cdot \\mathbf{n} &= p \\cdot \\mathbf{n} \\quad \\text{on } \\Gamma_{\\text{interface}} \\times (0, T)\n", | ||
| "\\end{align*}\n", | ||
| "$$\n", | ||
| "\n", | ||
| "- Fluid Equation:\n", | ||
| "\n", | ||
| "$$\n", | ||
| "\\begin{align*}\n", | ||
| "\\rho_f \\frac{\\partial \\mathbf{v}}{\\partial t} + \\rho_f (\\mathbf{v} \\cdot \\nabla) \\mathbf{v} - \\nabla \\cdot \\sigma(\\mathbf{v}) &= 0 \\quad \\text{in } \\Omega_f \\times (0, T) \\\\\n", | ||
| "\\mathbf{v} &= \\mathbf{0} \\quad \\text{on } \\Gamma_{\\text{inlet}} \\times (0, T) \\\\\n", | ||
| "\\sigma(\\mathbf{v}) \\cdot \\mathbf{n} &= \\mathbf{t} \\quad \\text{on } \\Gamma_{\\text{wing}} \\times (0, T) \\\\\n", | ||
| "\\end{align*}\n", | ||
| "$$\n", | ||
| "\n", | ||
| "- Coupling Conditions:\n", | ||
| "\n", | ||
| "$$\n", | ||
| "\\begin{align*}\n", | ||
| "\\mathbf{v}(\\mathbf{x}, t) &= \\mathbf{v}_f(\\mathbf{x}, t) \\quad \\text{on } \\Gamma_{\\text{interface}} \\times (0, T) \\\\\n", | ||
| "\\sigma(u) \\cdot \\mathbf{n} &= \\sigma(\\mathbf{v}) \\cdot \\mathbf{n} \\quad \\text{on } \\Gamma_{\\text{interface}} \\times (0, T)\n", | ||
| "\\end{align*}\n", | ||
| "$$\n", | ||
| "\n", | ||
| "## Weak Formulation\n", | ||
| "\n", | ||
| "Find $u \\in V_s$ and $\\mathbf{v} \\in V_f$ such that\n", | ||
| "\n", | ||
| "$$\n", | ||
| "\\begin{align*}\n", | ||
| "\\int_{\\Omega_s} \\rho_s A_s \\frac{\\partial^2 u}{\\partial t^2} \\phi_s \\, d\\Omega &+ \\int_{\\Omega_s} c_s \\frac{\\partial u}{\\partial t} \\phi_s \\, d\\Omega - \\int_{\\Omega_s} \\nabla \\cdot \\sigma(u) \\cdot \\nabla \\phi_s \\, d\\Omega = 0 \\\\\n", | ||
| "\\int_{\\Omega_f} \\rho_f \\frac{\\partial \\mathbf{v}}{\\partial t} \\cdot \\mathbf{\\phi}_f \\, d\\Omega &+ \\int_{\\Omega_f} \\rho_f (\\mathbf{v} \\cdot \\nabla) \\mathbf{v} \\cdot \\mathbf{\\phi}_f \\, d\\Omega - \\int_{\\Omega_f} \\nabla \\cdot \\sigma(\\mathbf{v}) \\cdot \\nabla \\mathbf{\\phi}_f \\, d\\Omega = 0\n", | ||
| "\\end{align*}\n", | ||
| "$$\n", | ||
| "\n", | ||
| "for all $\\phi_s \\in V_s$ and $\\mathbf{\\phi}_f \\in V_f$.\n", | ||
| "\n" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": null, | ||
| "id": "c4278e28", | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [] | ||
| } | ||
| ], | ||
| "metadata": { | ||
| "kernelspec": { | ||
| "display_name": ".iga-python", | ||
| "language": "python", | ||
| "name": ".iga-python" | ||
| }, | ||
| "language_info": { | ||
| "codemirror_mode": { | ||
| "name": "ipython", | ||
| "version": 3 | ||
| }, | ||
| "file_extension": ".py", | ||
| "mimetype": "text/x-python", | ||
| "name": "python", | ||
| "nbconvert_exporter": "python", | ||
| "pygments_lexer": "ipython3", | ||
| "version": "3.10.12" | ||
| } | ||
| }, | ||
| "nbformat": 4, | ||
| "nbformat_minor": 5 | ||
| } |
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