Continuous Section Field (CSF)

A Python framework for axially graded non-uniform structural members

Key Features • Installation & setup • Worked Example • Validation: Cylinder • Case Study: NREL 5-MW

📚 CSF Building Blocks Fundamentals - CSF Geometry, Sections, Polygon Tags, and Weight Laws Guide.

Motivation

CSF provides a modelling infrastructure based on a continuous function along the member axis, in which each region hosts two independent longitudinal fields governing its axial-flexural and shear-torsional participation.

It represents variable geometry and material distribution as continuous functions of the longitudinal coordinate $z$, from which sectional properties and the associated material composition can be evaluated at any station along the member.

The resulting data can be exported in CSV and YAML formats for integration with beam solvers such as OpenSees and SAP2000, or with section-analysis tools such as csf_sp and sp_csf.

CSF–CUF structural analysis

CSF can be used as a continuous provider of cross-sectional geometry and material properties for Carrera Unified Formulation (CUF) beam models.

The repository includes an open-source CUF implementation designed to keep the structural formulation modular. Cross-sectional geometry and material fields are provided by CSF, while the transverse expansion and longitudinal shape functions are supplied independently and can be replaced without modifying the CSF model.

See the CSF–CUF documentation for details on the formulation, implementation, and available examples.

cd cuf/tutorials/variable_material_t_section/t_section/models/
csf-actions t_noprismatic_csf.yaml action.yaml   # inspect the non-prismatic T-section

cd ../cases
csf-cuf bending_halfwave_legendre_N08.yaml       # run a CUF bending analysis

Illustrative example

Three-dimensional representation of the tapered pole with localized degradation. Gray regions indicate the non-degraded portions. Weight in this model is the elastic modulus.

Full reproducible example: Tapered prestressed degraded concrete pole

CSF combines:

• Geometry field: arbitrary polygonal sections defined at stations (S0, S1) with continuous interpolation for tapered or varying shapes.

• Axial/bending weight field: per-polygon participation factor w_i(z) along the longitudinal axis, scaling each region’s effective contribution to area, axial, and bending-related section properties.

• Shear/torsion weight field: per-polygon participation factor shear_w_i(z) along the longitudinal axis, allowing shear/torsion-related contributions to vary independently from w_i(z).

  See Ekofisk Jacket Platform - Foundation Piles for a localized corrosion case.

➡️ CSF–CUF framework and documentation

CSF accepts two input files: geometry.yaml (cross-section definition) and actions.yaml (post-processing pipeline).

geometry.yaml file

# geometry.yaml
# Defines the geometry and material participation laws
# of the structural member.
#
# - sections: end cross-sections (S0, S1) with polygon vertices at each
#   z-station.
# - polygons: 2D closed polygons, defined by ordered vertices, that compose
#   each cross-section.
# - vertices: polygon coordinates in the local section plane; vertices should
#   be given in counter-clockwise (CCW) order for positive area.
# - weight: base participation factor assigned to each polygon.
# - weight_laws: longitudinal participation laws w(z) that scale each polygon's
#   contribution along the member axis, independently of geometric interpolation.
# - shear_weight_laws: longitudinal shear/torsional participation laws shear_w(z),
#   used to scale shear- and torsion, related contributions independently from w(z).
# - iso(nu): isotropic shortcut that derives shear participation from weight
#   using G = E / [2 * (1 + nu)].

CSF:
  sections:
    S0:
      z: 0.0
      polygons:
        startsection:
          weight: 1.0
          vertices:
            - [-0.4, -0.4]
            - [0.4, -0.4]
            - [0.4, 0.4]
            - [-0.4, 0.4]
    S1:
      z: 10.0
      polygons:
        endsection:
          weight: 1.0
          vertices:
            - [-0.2, -0.2]
            - [0.2, -0.2]
            - [0.2, 0.2]
            - [-0.2, 0.2]
  # shear_weight_laws: longitudinal shear participation laws shear_w(z),
  # used to scale torsion/shear-related contributions independently from w(z)
  shear_weight_laws:
  # isotropic shortcut: sets shear_w(z) from w(z) using G = E / [2 * (1 + nu)]
  - 'startsection,endsection: iso(0.2)'
  weight_laws:
  # parabolic increase: 72% at base (z=0), full section at top (z=10)
  - 'startsection,endsection:1.0 - 0.28 * (1 - (z / 10.0)**2)'                 

actions.yaml file

# actions.yaml
# Defines the post-processing operations to run on the CSF model.
# - stations: named z-coordinate sets used as analysis points
# - actions: ordered list of operations (3D plots, section analysis, property export, etc.)

CSF_ACTIONS:
  stations:
    station_edge:
      - 0
      - 10
  actions:  
    - plot_volume_3d:
        params:
          title: "Not prismatic"          
    - plot_weight:
        output:
          - stdout
    - plot_properties:
        properties: [A,Ix,Iy,Ip]
    - plot_section_2d:
        stations:
          - station_edge
    - section_selected_analysis:
        stations:
          - station_edge
        properties: [A, Cx, Cy, Ix, Iy, Ixy, Ip, I1, I2, rx, ry, Wx, Wy,J_s_vroark, J_s_vroark_fidelity]
    - volume:
        stations:
          - station_edge
        output:
          - stdout
          - volume_report.txt
                                   

then run

linux / Mac

python3 -m venv venv
source venv/bin/activate
pip install csfpy
csf-actions geometry.yaml actions.yaml

Windows

py install 3.12
py -V:3.12 -m venv venv
.\venv\Scripts\Activate.ps1
pip install csfpy
csf-actions geometry.yaml actions.yaml

Minimal working CSF model: two sections, one polygon per section, custom weight laws, isotropic shear_weight_laws, and automatic evaluation of continuous section properties along the member axis.

CSF Geometry, Sections, Polygon Tags, and Weight Laws Guide

Example list Complete worked example covering all available actions on a tapered rectangular section

Using CSF as a Python Library

CSF can also be imported and used directly as a Python library to evaluate section fields, access sectional properties, and integrate CSF with custom workflows or external solvers.

Full Python API example - Tapered T-beam

Programmer Guide

Key Features

• Polygon-based section representation (algebraic composition): The element is geometrically defined by its end cross-sections, each represented as an algebraic composition of 2D polygons; intermediate sections are generated from these definitions along z. Curved outlines (e.g., circular shells/towers) are represented through discretized polygons with user-selected vertex count.

• Per-polygon longitudinal weight laws: independent analytical or lookup-based laws govern each polygon’s axial–bending and shear–torsional participation along the member. Custom Weight Laws

Section vertices can be generated from any CAD tool or script that can sample points along a curve and export their coordinates.

Geometric scope and limitations
CSF is not a finite element solver. It provides a continuous geometric and material representation of non-prismatic members along the longitudinal coordinate and evaluates the corresponding sectional properties and derived stiffness fields for beam-based analyses or external solvers. Curved boundaries are represented through polygonal discretization; the approximation accuracy can be increased by refining the number of polygon sides.

CSFStacked module (segmented ContinuousSectionField stacking)

CSFStacked is a container that assembles multiple ContinuousSectionField segments along the global z coordinate and dispatches each query to the corresponding segment.

It provides a single global interface for segmented CSF geometries, allowing sections and section properties to be queried anywhere along the assembled member through methods such as section(z) and section_full_analysis(z).

The individual CSF segments may also have different transverse positions. By enforcing geometric continuity at their junctions, they can therefore be assembled into globally non-straight, piecewise-defined members.

A simple two-segment zigzag example is available here:

Zigzag element example

🛠 Installation

Linux / Mac

Option A - Install from PyPI (recommended)

python3 -m venv venv
source venv/bin/activate
pip install csfpy

 # you need both geometry.yaml and actions.yaml files
csf-actions geometry.yaml actions.yaml

Option B - Local editable install (development)

python3 -m venv venv
source venv/bin/activate
git clone https://github.com/giovanniboscu/continuous-section-field.git
cd continuous-section-field
pip install -e .

python3 example/nrel_5mw_tower.py
python3 example/cylinder_withcheck.py
python3 example/csf_rotated_validation_benchmark.py

cd actions-examples/stell_degradated_model
mkdir -p out
csf-actions stell_degradated_model_s.yaml stell_degradated_model_action.yaml

Windows

py install 3.12

Option 1 - Install from PyPI (recommended)

py install 3.12
py -3.12 -m venv venv
.\venv\Scripts\Activate.ps1
pip install csfpy

 # you need both geometry.yaml and actions.yaml files
csf-actions geometry.yaml actions.yaml

Option 2 - Local editable install (development)

py install 3.12
py -3.12 -m venv venv
.\venv\Scripts\Activate.ps1
git clone https://github.com/giovanniboscu/continuous-section-field.git
cd continuous-section-field
pip install -e .

python .\example\nrel_5mw_tower.py
python .\example\cylinder_withcheck.py
python .\example\csf_rotated_validation_benchmark.py
python .\example\tsection_lab.py

cd actions-examples\stell_degradated_model
csf-actions stell_degradated_model_s.yaml stell_degradated_model_action.yaml

CSF Validation : Numerical Case Study: NREL 5-MW Reference Wind Turbine Tower

CSF is listed as a third-party tool in the OpenFAST documentation

• NREL 5-MW reference turbine report: Jonkman, J., Butterfield, S., Musial, W., and Scott, G.
  "Definition of a 5-MW Reference Wind Turbine for Offshore System Development,"
  NREL/TP-500-38060, February 2009.
  • Download PDF (NREL)
  • OpenFAST documentation
  • NLR wind research portal

512sided polygons

CSF computes continuous sectional properties along the tower height (z) - (A(z)), (EI(z)), (GJ(z)), etc. - compatible with OpenFAST ElastoDyn/SubDyn distributed property requirements and validated against NREL 5‑MW reference data.

The script example/nrel_5mw_tower.py is used to demonstrate the library's performance on complex, real-world structural members. A full-scale model of the NREL 5-MW Reference Wind Turbine Tower was implemented. The geometry and material properties strictly follow the technical report "Definition of a 5-MW Reference Wind Turbine for Offshore System Development" (NREL/TP-500-38060).

The following tables provide a side-by-side comparison between the Numerical results generated by CSF and the official NREL reference data (Table 6-1 of the official PDF).

Density = 8500 kg/m3

CSF Numerical Results (Generated Model)

Total Calculated Tower Volume: 40.8634 m³
Total Calculated Tower Mass: 347.339 t
 

Official NREL 5-MW Tower Data (Reference Table 6-1, p. 15)

Total NREL 5-MW Ref. Mass: 347.460 t

Relative error (Generated − Reference) / Reference × 100

Validation Summary

• All discrepancies are well below 0.05%.
• Differences are attributable to rounding and numerical precision.
• The generated CSF tower properties faithfully reproduce the NREL 5-MW reference.

OpenFAST Integration - CSF API Example

For a complete example of CSF integrated into the OpenFAST aeroelastic simulation framework - from parametric tower geometry through BModes to ElastoDyn - see the

HISTWIN Example

Two NREL/OpenFAST tower examples are provided:

• Code-only example: example/nrel_5mw_tower.py
• Formal AES reproducibility case: docs/aes/nrel_case/README.md

Worked Example: Continuously Tapered T-Beam

This section demonstrates how to model a structural member where the geometry transitions smoothly between two different T-profiles.

# --- Core CSF imports used in this minimal example ---
from csf import (
    Pt, Polygon,integrate_volume,section_statical_moment_partial,section_properties, section_derived_properties,Section, ContinuousSectionField,Visualizer,
    section_full_analysis, section_print_analysis,section_full_analysis_keys
)
# ----------------------------------------------------------------------------------
# 1. DEFINE START SECTION (Z = 0)
# ----------------------------------------------------------------------------------
#   GUIDELINES FOR POLYGON CONSTRUCTION:
# - COUNTER-CLOCKWISE POLYGON
# - VERTICES ORDER: You MUST define vertices in COUNTER-CLOCKWISE (CCW) order.
#   This is MANDATORY for the Shoelace/Green's Theorem algorithm to compute a
#   POSITIVE Area and correct Moments of Inertia. Clockwise order will result
#   in negative area values and mathematically incorrect results.
# - WEIGHT: Use 1.0 for solid parts and 0 to define voids/holes.
# - The start section here is a T-shape composed of two non-overlapping polygons:
#   a "flange" (top horizontal) and a "web" (vertical stem).
# ----------------------------------------------------------------------------------

# Flange Definition: Rectangle from (-1, -0.2) to (1, 0.2)
# Order: Bottom-Left -> Bottom-Right -> Top-Right -> Top-Left (CCW)
poly0_start = Polygon(
    vertices=(Pt(-1, -0.2), Pt(1, -0.2), Pt(1, 0.2), Pt(-1, 0.2)),
    weight=1.0,
    name="flange",
)

# Web Definition: Rectangle from (-0.2, -1.0) to (0.2, 0.2)
# Order: Bottom-Left -> Bottom-Right -> Top-Right -> Top-Left (CCW)
poly1_start = Polygon(
    vertices=(Pt(-0.2, -1.0), Pt(0.2, -1.0), Pt(0.2, -0.2), Pt(-0.2, -0.2)),
    weight=1.0,
    name="web",
)

# ----------------------------------------------------------------------------------
# 2. DEFINE END SECTION (Z = 10)
# ----------------------------------------------------------------------------------
# GEOMETRIC CONSISTENCY:
# - To enable linear interpolation (tapering), the end section must contain the
#   same number of polygons with the same names as the start section.
# - The web depth here increases linearly from 1.0 down to 2.5 (negative Y direction),
#   creating a tapered profile along the longitudinal Z-axis.
# ----------------------------------------------------------------------------------

# Flange remains unchanged for this prismatic top part
poly0_end = Polygon(
    vertices=(Pt(-1, -0.2), Pt(1, -0.2), Pt(1, 0.2), Pt(-1, 0.2)),
    weight=1.0,
    name="flange",
)

# Web becomes deeper: Y-bottom moves from -1.0 to -2.5
# MAINTAIN CCW ORDER: Bottom-Left -> Bottom-Right -> Top-Right -> Top-Left
poly1_end = Polygon(
    vertices=(Pt(-0.2, -2.5), Pt(0.2, -2.5), Pt(0.2, -0.2), Pt(-0.2, -0.2)),
    weight=1.0,
    name="web",
)

# ----------------------------------------------------------------------------------
# 3. CREATE SECTIONS WITH Z-COORDINATES
# ----------------------------------------------------------------------------------
# Sections act as containers for polygons at a specific coordinate along the beam axis.
# All polygons defined at Z=0.0 are grouped into s0, and those at Z=10.0 into s1.
# ----------------------------------------------------------------------------------

# Order matters: poly0_start pairs with poly0_end,
# and poly1_start pairs with poly1_end
# because they appear in the same position in their respective sections.

s0 = Section(polygons=(poly0_start, poly1_start), z=0.0)
s1 = Section(polygons=(poly0_end, poly1_end), z=10.0)

# --------------------------------------------------------
# 4. INITIALIZE CONTINUOUS SECTION FIELD
# --------------------------------------------------------
# A linear interpolator is used to generate intermediate
# sections between Z = 0 and Z = 10.
field = ContinuousSectionField(section0=s0, section1=s1)

# --------------------------------------------------------
# 5. PRIMARY SECTION PROPERTIES (Z = 5.0)
# --------------------------------------------------------
# Properties are computed at mid-span.
sec_mid = field.section(5.0)
props = section_properties(sec_mid)

print("\n" + "="*40)
print("PRIMARY PROPERTIES AT Z=5.0 (Centroidal)")
print("="*40)
print(f"A:   {props['A']:.4f}      # Net Cross-Sectional Area")
print(f"Cx:  {props['Cx']:.4f}     # Horizontal Centroid location (Global X)")
print(f"Cy:  {props['Cy']:.4f}     # Vertical Centroid location (Global Y)")
print(f"Ix:  {props['Ix']:.4f}     # Second Moment of Area about Centroidal X-axis")
print(f"Iy:  {props['Iy']:.4f}     # Second Moment of Area about Centroidal Y-axis")
print(f"Ixy: {props['Ixy']:.4f}    # Product of Inertia (Measure of asymmetry)")

# 2. DERIVED PROPERTIES (Radius of Gyration, Principal Axes)
derived = section_derived_properties(props)
print("\n" + "-"*40)
print("DERIVED GEOMETRIC PROPERTIES")
print("-"*40)
print(f"rx:  {derived['rx']:.4f}     # Radius of Gyration about X (sqrt(Ix/A))")
print(f"ry:  {derived['ry']:.4f}     # Radius of Gyration about Y (sqrt(Iy/A))")
print(f"I1:  {derived['I1']:.4f}     # Maximum Principal Moment of Area")
print(f"I2:  {derived['I2']:.4f}     # Minimum Principal Moment of Area")
print(f"Deg: {derived['theta_deg']:.2f}°   # Principal Axis Rotation Angle")

# --------------------------------------------------------
# 7. STATICAL MOMENT OF AREA (Q)
# --------------------------------------------------------
# Computed at the neutral axis (y = Cy).
# Used in shear stress calculations.
Q_na = section_statical_moment_partial(sec_mid, y_cut=props['Cy'])
print("\n" + "-"*40)
print("SHEAR ANALYSIS PROPERTIES")
print("-"*40)
print(f"Q_na: {Q_na:.4f}    # First Moment of Area above Neutral Axis (for shear stress)")

# 8. VOLUMETRIC PROPERTIES
total_vol = integrate_volume(field,0,10)
print("\n" + "="*40)
print("GLOBAL FIELD PROPERTIES (3D)")
print("="*40)
print(f"Total Volume: {total_vol:.4f} # Total material volume of the ruled solid")

# Technical Explanation for the negative Cy
print("\n" + "-"*50)
print("TECHNICAL NOTE ON COORDINATES:")
print("-"*50)
print("The negative value for 'Cy' is physically correct.")
print("In this model, the flange is centered at y=0, while the web")
print("extends downwards (from y=0.2 to y=-1.0 at Z=0, and y=-2.5 at Z=10).")
print("Since the majority of the T-section's mass is located below the")
print("global X-axis (y=0), the centroid MUST have a negative Y-coordinate.")
print("This indicates the geometric center is below the drawing origin.")
print("-"*50)

# --------------------------------------------------------
# 9. VISUALIZATION
# --------------------------------------------------------
# - 2D section plot at Z = 5.0
# - 3D ruled solid visualization
viz = Visualizer(field)

# Generate 2D plot for the specified slice
viz.plot_section_2d(z=5.0)

# Generate 3D plot of the interpolated solid
# line_percent determines the density of the longitudinal ruled lines
viz.plot_volume_3d(line_percent=100.0, seed=1)

import matplotlib.pyplot as plt
plt.show()

How to Cite this Library

@software{boscu_continuous_section_field_2026,
  author    = {Boscu, Giovanni},
  title     = {{continuous-section-field}},
  year      = {2026},
  publisher = {Zenodo},
  version   = {v0.2.3},
  doi       = {10.5281/zenodo.22722439},
  url       = {https://doi.org/10.5281/zenodo.22722439}
}

Development and Validation

This project was developed with assistance from generative artificial intelligence tools.

The author defined the conceptual framework, mathematical formulation, software requirements, validation strategy, and test cases, and reviewed the resulting implementation through analytical benchmarks, numerical comparisons, and independent reference solutions.

AI tools were also used to propose implementation details, numerical methods, and software structures. These components were accepted, modified, or rejected based on their consistency with the intended formulation and the results of the validation process.

Responsibility for the methodology, validation, released software, and conclusions remains with the author.

License

This project is licensed under the GNU General Public License v3.0 - see the LICENSE file for details.