Structural Computation

Structural Computation

1. Computational Structural Design Philosophy

The Unity of Form and Force

Structure is not an afterthought applied to a completed form. In the computational paradigm, structure is form. The geometry of a building, a bridge, a canopy -- each is a direct expression of the forces acting upon it. When we allow force to generate geometry rather than merely verifying geometry against force, we unlock an entirely different class of architectural expression: one that is simultaneously more efficient, more beautiful, and more materially honest.

This represents a fundamental epistemological shift: from analysis-after-design to analysis-as-design. The former treats structure as a constraint to be checked; the latter treats structure as a generative engine. In computational structural design, the distinction between "architect" and "engineer" dissolves. The designer works directly with force, curvature, material flow, and stress trajectories as primary design media.

Performance-Driven Geometry

Performance-driven geometry emerges when structural performance metrics (stress utilization, deflection, weight, embodied carbon) become the objective functions of a generative process. The designer defines boundary conditions, loads, material palettes, and performance targets. The computation produces geometry. This is not a loss of authorship -- it is an expansion of the design space far beyond what manual intuition can explore.

Key principles of performance-driven geometry:

• Axial action over bending -- Funicular forms carry load through compression or tension, avoiding costly bending. A catenary cable and an inverted catenary arch are the purest examples.
• Curvature as stiffness -- A flat plate is structurally weak; a curved shell resists load through membrane action. Double curvature provides bidirectional stiffness. The egg is the canonical example.
• Material where stress exists -- Topology optimization removes material where stress is absent and concentrates it along principal stress trajectories. The result resembles trabecular bone.
• Hierarchy and redundancy -- Efficient structures distribute load through hierarchical branching (trees, Gothic vaults, diagrid systems) with multiple load paths for robustness.

Historical Lineage

• Gaudi (1852-1926) -- Hanging chain models for the Colonia Guell chapel: physical form-finding producing pure compression vaults. The method is exact for catenary forms under self-weight.
• Isler (1926-2009) -- Hanging cloth models (fabric + plaster, inverted). Thin concrete shells at Deitingen, Norwich, Sicli: 40m+ spans at 70-90mm thickness. Physical analogue of dynamic relaxation.
• Frei Otto (1925-2015) -- Soap film experiments for minimal surface form-finding. Munich Olympic Stadium (1972), Mannheim Multihalle (1975). Founded the Institute for Lightweight Structures (IL) at Stuttgart.
• Philippe Block (b. 1980) -- Thrust Network Analysis (TNA): computational framework for compression-only funicular surfaces. Armadillo Vault (2016), NEST HiLo (2021).
• Contemporary -- Real-time structural feedback (Karamba3D, Kangaroo Physics), multi-objective optimization coupling structure with daylighting/thermal/fabrication, ML surrogates enabling population-based search over thousands of variants.

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2. Finite Element Analysis Fundamentals

Overview

Finite Element Analysis (FEA) discretizes a continuous structure into a mesh of finite-sized elements, each governed by simple constitutive equations. By assembling element stiffness matrices into a global system, FEA solves for displacements, from which strains and stresses are derived. FEA is the universal method for structural verification in AEC.

Element Types

| Element Type | Dimensionality | DOF per Node | Captures | Typical Use | |---|---|---|---|---| | Truss | 1D (axial only) | 2 (2D) or 3 (3D) translations | Axial force only | Trusses, cables, bracing | | Beam | 1D (axial + bending) | 6 (3 translations, 3 rotations) | Axial, shear, bending, torsion | Frames, columns, beams | | Shell | 2D (membrane + bending) | 5-6 per node | In-plane + out-of-plane | Floors, walls, shells, slabs | | Plate | 2D (bending only) | 3 (1 translation, 2 rotations) | Out-of-plane bending | Floor slabs | | Solid | 3D | 3 translations | Full 3D stress state | Connections, nodes, foundations | | Cable | 1D (tension only) | 3 translations | Tension, large displacements | Cables, tendons | | Spring | 0D/1D | variable | Stiffness in specific DOFs | Supports, connections |

Element Formulations

• Linear (first-order) elements: Triangles (3-node), quadrilaterals (4-node), tetrahedra (4-node), hexahedra (8-node). Linear interpolation of displacements within each element. Require finer meshes but are computationally cheap per element.
• Quadratic (second-order) elements: Mid-side nodes added (6-node triangle, 8-node quad, 10-node tet, 20-node hex). Capture curved geometry and stress gradients more accurately. Preferred for stress analysis near holes, notches, and concentrated loads.
• Reduced integration: Uses fewer Gauss points than full integration (e.g., 1 point for a 4-node quad instead of 4). Faster but can exhibit hourglass modes (zero-energy deformation patterns). Remedied by hourglass control or selective reduced integration.

Mesh Requirements for FEA

Structural FEA meshes differ fundamentally from visualization/rendering meshes:

• Aspect ratio -- Elements should be as equilateral as possible. Aspect ratios above 5:1 degrade accuracy. Target 0.5, warpage 1.0).
• Deflection: Check against limits (L/250 total, L/360 live load typical). See references/fea-fundamentals.md for complete deflection limits table.

Convergence Checking

• h-method -- Refine mesh (halve element sizes) and re-run. If results change by compute internal forces -> compute residuals -> update velocities with damping -> update positions -> check convergence (kinetic energy or residual force iterate.

Kinetic damping (Barnes, 1988) is the standard method: reset all velocities to zero when total kinetic energy peaks. Extremely robust, no parameter tuning. Time step must satisfy Courant condition: dt 0 (positive) | Same sign (both up or both down) | Dome, elliptic paraboloid | | Anticlastic | K 0 = solid, phi = minimum desired feature size.

Penalization Factor

The standard SIMP penalization p = 3 works well for stiffness-based optimization:

• p = 1: No penalization, result is full of gray.
• p = 2: Moderate penalization. Some gray remains.
• p = 3: Standard. Most intermediate densities penalized away.
• p = 4-5: Aggressive penalization. Can cause convergence issues.

Continuation approach: Start with p = 1 and gradually increase to p = 3 over iterations. Improves convergence and can find better optima.

Interpretation of Results

Raw topology optimization output requires post-processing:

• Thresholding -- Set a density cutoff (e.g., rho > 0.5 = solid). Produces a binary solid/void result.
• Smoothing -- Apply Laplacian or Gaussian smoothing to the boundary. Removes staircase artifacts.
• CAD reconstruction -- Fit NURBS surfaces or B-rep geometry to the smoothed result. Manual or semi-automated (e.g., nTopology, Altair Inspire).
• Structural verification -- Re-analyze the interpreted geometry with FEA to confirm performance. The post-processed shape may differ from the optimized result.

2D vs. 3D Topology Optimization

2D optimization is used for:

• Planar components (brackets, gusset plates, flat connection details)
• Cross-sections of beams or walls
• Pedagogical examples and initial exploration
• Computation time: seconds to minutes

3D optimization is used for:

• Spatial structural nodes
• Transfer structures (load paths through a 3D volume)
• Foundation layouts
• Full building volumes
• Computation time: minutes to hours (can be very large for fine meshes)

Tools for Topology Optimization

| Tool | Platform | Method | 2D/3D | Cost | Notes | |---|---|---|---|---|---| | Millipede | Grasshopper | SIMP-like | 2D + 3D | Free | Fast, good GH integration | | Ameba | Grasshopper | BESO | 2D + 3D | Free | Clear results, slower | | TopOpt | Web (DTU) | SIMP | 2D | Free | Educational, interactive | | TOSCA | Abaqus (Dassault) | SIMP, Level-set | 3D | Expensive | Industrial, validated | | Altair Inspire | Standalone | SIMP (OptiStruct) | 3D | Expensive | Intuitive GUI, mfg constraints | | nTopology | Standalone | Lattice + TO | 3D | Expensive | Lattice infill, AM-ready | | ANSYS Topology | ANSYS | SIMP | 3D | Expensive | Integrated with ANSYS FEA |

AEC Applications

• Structural nodes -- Topology-optimized steel nodes for space frames and gridshells. 3D-printed in metal. Up to 75% weight reduction vs. conventional welded nodes. Arup's 3D-printed steel nodes for a structural tree.
• Floor plates -- Ribbed slabs with topology-optimized rib patterns. ETH Zurich NEST HiLo project: 3D-printed concrete floor with 70% less material than flat slab.
• Facade brackets -- Connection elements between facade panels and primary structure. Optimize for multiple load cases (wind, self-weight, seismic).
• Foundations -- Material distribution in transfer beams and pile caps. Strut-and-tie models are a classical topology optimization concept.
• Furniture and pavilions -- Increasingly common for one-off or small-series production. Branch Technology, AI Build, MX3D.

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7. Material-Aware Computation

Material Properties Table

| Property | Steel (S355) | Concrete (C40/50) | Timber (GL28h) | Aluminum (6061-T6) | GFRP | CFRP | Bamboo | |---|---|---|---|---|---|---|---| | E (GPa) | 210 | 35 | 12.6 (parallel) | 69 | 25-40 | 70-150 | 15-20 | | fy (MPa) | 355 | - | - | 275 | - | - | - | | fu (MPa) | 510 | - | - | 310 | 400-800 | 600-2000 | 100-200 | | fc (MPa) | - | 40 | 28 | - | 150-250 | 500-1500 | 40-80 | | ft (MPa) | 355 | 3.5 | 22.3 | 275 | 400-800 | 600-2000 | 100-200 | | f_b (MPa) | 355 | - | 28 | 275 | 250-500 | 600-1500 | 80-150 | | Density (kg/m^3) | 7850 | 2500 | 410 | 2700 | 1800-2100 | 1500-1600 | 600-800 | | Poisson's ratio | 0.30 | 0.20 | 0.35 (major) | 0.33 | 0.25-0.35 | 0.25-0.30 | 0.30 | | alpha (10^-6/C) | 12 | 10 | 5 (parallel) | 23 | 6-10 | -1 to 2 | 3-5 |

Anisotropic Materials

Timber: Strongly anisotropic. Properties differ along grain (longitudinal), across grain (radial), and tangential directions. EL : ER : E_T is approximately 20 : 1.6 : 1. Computational models must account for grain direction. CLT (Cross-Laminated Timber) alternates grain direction for quasi-isotropic behavior in-plane.

Fiber composites (GFRP, CFRP): Properties depend on fiber orientation. Unidirectional laminates are strongly anisotropic. Quasi-isotropic layups ([0/+45/-45/90]s) provide balanced in-plane properties but are weaker than aligned laminates in any single direction. Classical Laminate Theory (CLT -- confusingly same acronym) governs composite analysis.

Masonry: Anisotropic due to mortar joints. Different stiffness and strength along bed joints vs. head joints vs. diagonal. Homogenized masonry models treat the assembly as an equivalent anisotropic continuum.

Material Behavior Models

• Linear elastic -- Stress proportional to strain (Hooke's law). Valid for most materials at service load levels. The basis for most FEA in AEC.
• Elasto-plastic -- Linear elastic up to yield, then plastic deformation at constant (or hardening) stress. Required for steel design at ULS. Bilinear or multilinear stress-strain models.
• Viscoelastic -- Time-dependent deformation under sustained load (creep). Critical for concrete and timber. Modeled as spring-dashpot combinations (Maxwell, Kelvin-Voigt, Burgers models).
• Nonlinear elastic -- Stress-strain curve is nonlinear but unloading follows the loading path. Rubber, some polymers.
• Brittle -- No plastic deformation before failure. Concrete in tension, glass, unreinforced masonry. Requires fracture mechanics or damage models.

Composite Action

When two materials act together (steel-concrete beams, timber-concrete floors), transformed section analysis converts to an equivalent single-material section using modular ratio n = E1/E2 (full composite). Partial composite action (interface slip) is modeled with interface springs. Shear connectors (headed studs, screws) transfer horizontal shear.

Material Efficiency Metrics

| Material | Strength/Weight (f/rho, kN.m/kg) | Stiffness/Weight (E/rho, MN.m/kg) | Embodied Carbon (kgCO2e/kg) | |---|---|---|---| | Steel S355 | 45 | 27 | 1.5-2.5 | | Concrete C40 | 16 (compression) | 14 | 0.1-0.2 | | Timber GL28h | 68 | 31 | -1.0 to 0.5 | | Aluminum 6061 | 102 | 26 | 8.0-12.0 | | CFRP | 400-1300 | 47-100 | 20-30 | | Bamboo | 130-250 | 20-30 | 0.5-2.0 |

Timber and bamboo are exceptional: high strength-to-weight, low-to-negative embodied carbon. CFRP is phenomenal structurally but carries enormous environmental cost. Steel and concrete are the workhorses with moderate efficiency.

Digital Material Systems

Emerging paradigm of spatially varying material properties:

• Functionally Graded Materials (FGM) -- Continuous variation of composition (e.g., concrete with graded fiber density)
• Variable-density lattices -- 3D-printed lattice infill matched to stress fields from topology optimization (nTopology, Altair)
• Multi-material printing -- Simultaneous deposition of stiff and flexible materials for spatially tuned stiffness
• Programmable materials -- Shape-memory alloys, 4D printing, materials responding to stimuli (temperature, moisture)

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8. Structural Computation Tools

8.1 Karamba3D (Grasshopper)

Capabilities: Real-time FEA within Grasshopper. Beam and shell elements. Linear and second-order analysis. Cross-section optimization. Utilization checking. Eigen-frequency and buckling analysis. Large deformation analysis (geometrically nonlinear).

Limitations: No material nonlinearity (no concrete cracking, no steel yielding). No dynamic time-history analysis. Not a code-checking tool (no automatic Eurocode/ASCE capacity checks -- requires manual setup). Accuracy depends on mesh quality.

Integration: Fully embedded in Grasshopper. Takes Rhino geometry (lines, meshes) as input. Outputs displaced shapes, stress results, utilization ratios as colored meshes. Connects to Octopus, Galapagos, Wallacei for optimization.

Learning curve: Moderate. Structural concepts required. Well-documented with tutorials.

Typical workflow: Define geometry (lines/mesh) -> assign cross-sections -> define supports -> apply loads -> assemble model -> analyze -> read results -> iterate/optimize.

8.2 Kangaroo Physics (Grasshopper)

Capabilities: Interactive particle-spring solver. Form-finding (cable nets, membranes, inflatables, shells, gridshells). Multi-physics simulation (structural + fabrication constraints simultaneously). Real-time manipulation. Custom goal creation via C# scripting.

Limitations: Not a verified FEA tool. Cannot produce code-compliant stress results. Approximate stiffness (no rigorous element formulations). Not suitable for final structural verification.

Integration: Native Grasshopper component. Kangaroo 2 is the current version (position-based dynamics). Works with any mesh or line network. Combined with Weaverbird, Mesh+, Lunchbox for mesh processing.

Learning curve: Low-to-moderate. Very intuitive for form-finding. Advanced use (custom goals, coupled simulations) requires deeper understanding.

Typical workflow: Create mesh/network -> assign goals (springs, loads, anchors, constraints) -> run solver -> extract equilibrium geometry -> refine.

8.3 Millipede (Grasshopper)

Capabilities: Topology optimization (2D

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Source & license

This open-source skill is cataloged on AgentStack and links to its original source — we do not rehost the code.

• Author: Amanbh997
• Source: Amanbh997/Claude-skills-for-Computational-Designers
• License: MIT

Install and usage instructions live in the source repository linked above.