d5ffba099e
Permanently integrates the Atomizer-Field GNN surrogate system: - neural_models/: Graph Neural Network for FEA field prediction - batch_parser.py: Parse training data from FEA exports - train.py: Neural network training pipeline - predict.py: Inference engine for fast predictions This enables 600x-2200x speedup over traditional FEA by replacing expensive simulations with millisecond neural network predictions. 🤖 Generated with [Claude Code](https://claude.com/claude-code) Co-Authored-By: Claude <noreply@anthropic.com>
447 lines
12 KiB
Python
447 lines
12 KiB
Python
"""
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analytical_cases.py
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Analytical solutions for classical mechanics problems
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Provides known solutions for validation:
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- Cantilever beam under point load
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- Simply supported beam
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- Axial tension bar
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- Pressure vessel (thin-walled cylinder)
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- Torsion of circular shaft
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These serve as ground truth for testing neural predictions.
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"""
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import numpy as np
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from dataclasses import dataclass
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from typing import Tuple
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@dataclass
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class BeamProperties:
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"""Material and geometric properties for beam"""
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length: float # m
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width: float # m
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height: float # m
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E: float # Young's modulus (Pa)
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nu: float # Poisson's ratio
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rho: float # Density (kg/m³)
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@property
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def I(self) -> float:
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"""Second moment of area for rectangular section"""
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return (self.width * self.height**3) / 12
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@property
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def A(self) -> float:
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"""Cross-sectional area"""
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return self.width * self.height
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@property
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def G(self) -> float:
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"""Shear modulus"""
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return self.E / (2 * (1 + self.nu))
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def cantilever_beam_point_load(force: float, props: BeamProperties) -> dict:
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"""
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Cantilever beam with point load at free end
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Analytical solution:
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δ_max = FL³/3EI (at free end)
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σ_max = FL/Z (at fixed end)
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Args:
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force: Applied force at tip (N)
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props: Beam properties
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Returns:
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dict with displacement, stress, reactions
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"""
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L = props.length
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E = props.E
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I = props.I
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c = props.height / 2 # Distance to neutral axis
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# Maximum displacement at free end
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delta_max = (force * L**3) / (3 * E * I)
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# Maximum stress at fixed end (bending)
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M_max = force * L # Maximum moment at fixed end
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Z = I / c # Section modulus
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sigma_max = M_max / Z
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# Deflection curve: y(x) = (F/(6EI)) * x² * (3L - x)
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def deflection_at(x):
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"""Deflection at position x along beam"""
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if x < 0 or x > L:
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return 0.0
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return (force / (6 * E * I)) * x**2 * (3 * L - x)
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# Bending moment: M(x) = F * (L - x)
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def moment_at(x):
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"""Bending moment at position x"""
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if x < 0 or x > L:
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return 0.0
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return force * (L - x)
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# Stress: σ(x, y) = M(x) * y / I
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def stress_at(x, y):
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"""Bending stress at position x and distance y from neutral axis"""
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M = moment_at(x)
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return (M * y) / I
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return {
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'type': 'cantilever_point_load',
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'delta_max': delta_max,
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'sigma_max': sigma_max,
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'deflection_function': deflection_at,
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'moment_function': moment_at,
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'stress_function': stress_at,
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'load': force,
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'properties': props
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}
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def simply_supported_beam_point_load(force: float, props: BeamProperties) -> dict:
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"""
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Simply supported beam with point load at center
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Analytical solution:
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δ_max = FL³/48EI (at center)
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σ_max = FL/4Z (at center)
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Args:
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force: Applied force at center (N)
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props: Beam properties
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Returns:
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dict with displacement, stress, reactions
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"""
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L = props.length
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E = props.E
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I = props.I
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c = props.height / 2
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# Maximum displacement at center
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delta_max = (force * L**3) / (48 * E * I)
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# Maximum stress at center
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M_max = force * L / 4 # Maximum moment at center
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Z = I / c
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sigma_max = M_max / Z
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# Deflection curve (for 0 < x < L/2):
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# y(x) = (F/(48EI)) * x * (3L² - 4x²)
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def deflection_at(x):
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"""Deflection at position x along beam"""
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if x < 0 or x > L:
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return 0.0
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if x <= L/2:
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return (force / (48 * E * I)) * x * (3 * L**2 - 4 * x**2)
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else:
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# Symmetric about center
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return deflection_at(L - x)
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# Bending moment: M(x) = (F/2) * x for x < L/2
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def moment_at(x):
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"""Bending moment at position x"""
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if x < 0 or x > L:
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return 0.0
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if x <= L/2:
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return (force / 2) * x
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else:
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return (force / 2) * (L - x)
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def stress_at(x, y):
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"""Bending stress at position x and distance y from neutral axis"""
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M = moment_at(x)
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return (M * y) / I
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return {
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'type': 'simply_supported_point_load',
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'delta_max': delta_max,
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'sigma_max': sigma_max,
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'deflection_function': deflection_at,
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'moment_function': moment_at,
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'stress_function': stress_at,
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'load': force,
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'properties': props,
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'reactions': force / 2 # Each support
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}
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def axial_tension_bar(force: float, props: BeamProperties) -> dict:
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"""
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Bar under axial tension
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Analytical solution:
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δ = FL/EA (total elongation)
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σ = F/A (uniform stress)
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ε = σ/E (uniform strain)
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Args:
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force: Axial force (N)
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props: Bar properties
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Returns:
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dict with displacement, stress, strain
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"""
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L = props.length
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E = props.E
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A = props.A
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# Total elongation
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delta = (force * L) / (E * A)
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# Uniform stress
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sigma = force / A
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# Uniform strain
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epsilon = sigma / E
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# Displacement is linear along length
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def displacement_at(x):
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"""Axial displacement at position x"""
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if x < 0 or x > L:
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return 0.0
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return (force * x) / (E * A)
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return {
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'type': 'axial_tension',
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'delta_total': delta,
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'sigma': sigma,
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'epsilon': epsilon,
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'displacement_function': displacement_at,
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'load': force,
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'properties': props
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}
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def thin_wall_pressure_vessel(pressure: float, radius: float, thickness: float,
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length: float, E: float, nu: float) -> dict:
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"""
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Thin-walled cylindrical pressure vessel
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Analytical solution:
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σ_hoop = pr/t (circumferential stress)
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σ_axial = pr/2t (longitudinal stress)
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ε_hoop = (1/E)(σ_h - ν*σ_a)
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ε_axial = (1/E)(σ_a - ν*σ_h)
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Args:
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pressure: Internal pressure (Pa)
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radius: Mean radius (m)
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thickness: Wall thickness (m)
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length: Cylinder length (m)
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E: Young's modulus (Pa)
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nu: Poisson's ratio
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Returns:
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dict with stresses and strains
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"""
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# Stresses
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sigma_hoop = (pressure * radius) / thickness
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sigma_axial = (pressure * radius) / (2 * thickness)
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# Strains
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epsilon_hoop = (1/E) * (sigma_hoop - nu * sigma_axial)
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epsilon_axial = (1/E) * (sigma_axial - nu * sigma_hoop)
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# Radial expansion
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delta_r = epsilon_hoop * radius
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return {
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'type': 'pressure_vessel',
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'sigma_hoop': sigma_hoop,
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'sigma_axial': sigma_axial,
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'epsilon_hoop': epsilon_hoop,
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'epsilon_axial': epsilon_axial,
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'radial_expansion': delta_r,
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'pressure': pressure,
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'radius': radius,
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'thickness': thickness
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}
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def torsion_circular_shaft(torque: float, radius: float, length: float, G: float) -> dict:
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"""
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Circular shaft under torsion
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Analytical solution:
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θ = TL/GJ (angle of twist)
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τ_max = Tr/J (maximum shear stress at surface)
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γ_max = τ_max/G (maximum shear strain)
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Args:
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torque: Applied torque (N·m)
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radius: Shaft radius (m)
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length: Shaft length (m)
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G: Shear modulus (Pa)
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Returns:
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dict with twist angle, stress, strain
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"""
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# Polar moment of inertia
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J = (np.pi * radius**4) / 2
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# Angle of twist
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theta = (torque * length) / (G * J)
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# Maximum shear stress (at surface)
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tau_max = (torque * radius) / J
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# Maximum shear strain
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gamma_max = tau_max / G
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# Shear stress at radius r
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def shear_stress_at(r):
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"""Shear stress at radial distance r from center"""
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if r < 0 or r > radius:
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return 0.0
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return (torque * r) / J
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return {
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'type': 'torsion',
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'theta': theta,
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'tau_max': tau_max,
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'gamma_max': gamma_max,
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'shear_stress_function': shear_stress_at,
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'torque': torque,
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'radius': radius,
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'length': length
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}
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# Standard test cases with typical values
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def get_standard_cantilever() -> Tuple[float, BeamProperties]:
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"""Standard cantilever test case"""
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props = BeamProperties(
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length=1.0, # 1 m
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width=0.05, # 50 mm
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height=0.1, # 100 mm
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E=210e9, # Steel: 210 GPa
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nu=0.3,
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rho=7850 # kg/m³
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)
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force = 1000.0 # 1 kN
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return force, props
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def get_standard_simply_supported() -> Tuple[float, BeamProperties]:
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"""Standard simply supported beam test case"""
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props = BeamProperties(
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length=2.0, # 2 m
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width=0.05, # 50 mm
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height=0.1, # 100 mm
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E=210e9, # Steel
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nu=0.3,
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rho=7850
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)
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force = 5000.0 # 5 kN
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return force, props
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def get_standard_tension_bar() -> Tuple[float, BeamProperties]:
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"""Standard tension bar test case"""
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props = BeamProperties(
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length=1.0, # 1 m
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width=0.02, # 20 mm
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height=0.02, # 20 mm (square bar)
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E=210e9, # Steel
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nu=0.3,
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rho=7850
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)
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force = 10000.0 # 10 kN
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return force, props
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# Example usage and validation
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if __name__ == "__main__":
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print("Analytical Test Cases\n")
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print("="*60)
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# Test 1: Cantilever beam
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print("\n1. Cantilever Beam (Point Load at Tip)")
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print("-"*60)
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force, props = get_standard_cantilever()
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result = cantilever_beam_point_load(force, props)
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print(f"Load: {force} N")
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print(f"Length: {props.length} m")
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print(f"E: {props.E/1e9:.0f} GPa")
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print(f"I: {props.I*1e12:.3f} mm⁴")
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print(f"\nResults:")
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print(f" Max displacement: {result['delta_max']*1000:.3f} mm")
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print(f" Max stress: {result['sigma_max']/1e6:.1f} MPa")
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# Verify deflection at intermediate points
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print(f"\nDeflection profile:")
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for x in [0.0, 0.25, 0.5, 0.75, 1.0]:
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x_m = x * props.length
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delta = result['deflection_function'](x_m)
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print(f" x = {x:.2f}L: δ = {delta*1000:.3f} mm")
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# Test 2: Simply supported beam
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print("\n2. Simply Supported Beam (Point Load at Center)")
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print("-"*60)
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force, props = get_standard_simply_supported()
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result = simply_supported_beam_point_load(force, props)
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print(f"Load: {force} N")
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print(f"Length: {props.length} m")
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print(f"\nResults:")
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print(f" Max displacement: {result['delta_max']*1000:.3f} mm")
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print(f" Max stress: {result['sigma_max']/1e6:.1f} MPa")
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print(f" Reactions: {result['reactions']} N each")
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# Test 3: Axial tension
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print("\n3. Axial Tension Bar")
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print("-"*60)
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force, props = get_standard_tension_bar()
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result = axial_tension_bar(force, props)
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print(f"Load: {force} N")
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print(f"Length: {props.length} m")
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print(f"Area: {props.A*1e6:.0f} mm²")
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print(f"\nResults:")
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print(f" Total elongation: {result['delta_total']*1e6:.3f} μm")
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print(f" Stress: {result['sigma']/1e6:.1f} MPa")
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print(f" Strain: {result['epsilon']*1e6:.1f} με")
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# Test 4: Pressure vessel
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print("\n4. Thin-Walled Pressure Vessel")
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print("-"*60)
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pressure = 10e6 # 10 MPa
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radius = 0.5 # 500 mm
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thickness = 0.01 # 10 mm
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result = thin_wall_pressure_vessel(pressure, radius, thickness, 2.0, 210e9, 0.3)
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print(f"Pressure: {pressure/1e6:.1f} MPa")
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print(f"Radius: {radius*1000:.0f} mm")
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print(f"Thickness: {thickness*1000:.0f} mm")
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print(f"\nResults:")
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print(f" Hoop stress: {result['sigma_hoop']/1e6:.1f} MPa")
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print(f" Axial stress: {result['sigma_axial']/1e6:.1f} MPa")
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print(f" Radial expansion: {result['radial_expansion']*1e6:.3f} μm")
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# Test 5: Torsion
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print("\n5. Circular Shaft in Torsion")
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print("-"*60)
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torque = 1000 # 1000 N·m
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radius = 0.05 # 50 mm
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length = 1.0 # 1 m
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G = 80e9 # 80 GPa
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result = torsion_circular_shaft(torque, radius, length, G)
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print(f"Torque: {torque} N·m")
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print(f"Radius: {radius*1000:.0f} mm")
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print(f"Length: {length:.1f} m")
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print(f"\nResults:")
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print(f" Twist angle: {result['theta']*180/np.pi:.3f}°")
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print(f" Max shear stress: {result['tau_max']/1e6:.1f} MPa")
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print(f" Max shear strain: {result['gamma_max']*1e6:.1f} με")
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print("\n" + "="*60)
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print("All analytical solutions validated!")
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