35-EFT.WP.EDX.OrientedTension v1.0 | Chapter 3 Geometry & Orientation: Fields, Fibers & Distributions

Chapter 3 Geometry & Orientation: Fields, Fibers & Distributions

I. Abstract & Scope
This chapter provides the minimal description of orientation geometry: the unit orientation vector field n_hat(r,t), the orientation distribution function f_orient(n_hat,r,t), and the order tensor Q_ij(r,t)—their definitions, normalization, and mappings. It fixes embedded geometries of fibers/bundles (volume–surface–line domains) and their weighting, and standardizes reference frames and boundary conditions. All symbols use English notation wrapped in backticks; SI units apply. ToA does not appear in this chapter.

II. Dependencies & References

• Layout & numbering: EFT Technical Whitepaper & Memo Template – Comprehensive Checklist v0.1.
• Symbols & dimensions: Chapter 2 (P80-1/4/5/6, Tabs 2-1…2-5).
• Axioms & minimal equations: Chapter 4 (P80-2/3, S80-3/4).
• Metrology & inversion: Chapter 5 (M80-1…4).
• Coupling & transport: Chapter 6 (S80-5/6).
• Energy accounting: Chapter 7 (S80-7/8).
• Numerics & simulation: Chapter 10 (SimStack-OT).

III. Normative Anchors (added in this chapter)

• P80-7 (Orientation Domain & Measure): The domain is the unit sphere S^2 with measure dΩ the intrinsic solid-angle measure; normalization is taken with dΩ.
• P80-8 (Reference Frames & Tags): Report tensors and moments in the co-moving flow frame (_flow) by default; use domain tags (_sheet/_shear) only when layer/shear geometries must be distinguished; units and dimensions are unchanged by tags.
• S80-1 (Orientation Normalization): ∫_{S^2} f_orient(n_hat,r,t) dΩ = 1.
• S80-2 (Order Tensor): Q_ij(r,t) = ⟨ n_i n_j − (1/3) δ_ij ⟩_{f_orient} = ∫_{S^2} ( n_i n_j − δ_ij/3 ) f_orient dΩ.

IV. Body Structure

I. Background & Problem Statement

• Orientation geometry connects directional statistics of microscopic fibers/chains to macroscopic anisotropic response. f_orient carries full statistics; Q_ij provides a second-moment compression; both are used for coupling terms and energy-closure relations.
• Embedded geometries (line/surface/volume) determine volumetric weighting and boundary conditions; unified measures and domain tags are required to avoid ambiguity.

II. Key Equations & Derivations (S-series)

• S80-1 (Normalization): ∫_{S^2} f_orient dΩ = 1, with dΩ the unit-sphere measure.
• S80-2 (Q_ij definition): Q_ij = ∫_{S^2} ( n_i n_j − δ_ij/3 ) f_orient dΩ; Q_ij is symmetric and Tr(Q)=0.
• Fiber weighting (definition, not a minimal equation): volume-averages ⟨·⟩_V = (1/V) ∫_V (·) dV; likewise ⟨·⟩_A and ⟨·⟩_L for surfaces/lines. For mixed embeddings declare weights w_V,w_A,w_L with w_V+w_A+w_L=1.
• Higher moments (note): if fourth moments ⟨n_i n_j n_k n_l⟩ are required, list contraction rules and closures (e.g., Bingham/Maxwell-type) explicitly in the model card or appendix; not mandated here.

III. Methods & Flows (M-series)

1. M80-13 (Discretization & Quadrature on S^2):
   • Choose a spherical grid (equal-area, HEALPix, or Lebedev).
   • Assign weights w_α to directions n_hat^α with ∑_α w_α = 1.
   • Approximate ∫_{S^2} g(n_hat) dΩ ≈ ∑_α w_α g(n_hat^α).
   • Output error estimates and convergence curves.
2. M80-14 (Mapping f_orient → Q_ij):
   • From discrete f_α = f_orient(n_hat^α) compute Q_ij = ∑_α w_α ( n_i^α n_j^α − δ_ij/3 ) f_α.
   • Check Tr(Q)=0 and eigenvalue bounds λ ∈ [−1/3, 2/3].
   • Record SymbolUnitAudit status and numerical tolerances.
3. M80-15 (Embedded Geometry & Boundary Conditions):
   • Declare boundary conditions for line/surface/volume domains (e.g., n_hat·n_b = 0 / free / prescribed).
   • Specify fiber volume fraction φ_f and its constraints on f_orient.
   • Emit geometry–boundary card fields (domain, measure, weights).

IV. Cross-References within/beyond this Volume

• Chapter 4: inputs for mapping Q_ij to oriented tension S80-3/4.
• Chapter 5: inversion from polarimetry/scattering to Q_ij (M80-1…4).
• Chapter 6: Q_ij in coupling terms and transport coefficients (S80-5/6).
• Chapter 10: implementation and benchmarks of S^2 discretization in SimStack-OT.
• Cross-volume: see the companion Energy Filaments chapters for geometric/statistical definitions.

V. Validation, Criteria & Counterexamples

1. Positive criteria:
   • f_orient passes normalization (∫_{S^2} f_orient dΩ = 1); Q_ij is symmetric with Tr(Q)=0.
   • Spherical quadrature meets convergence thresholds; eigenvalues of Q_ij lie in the physical domain.
   • Volume/surface/line weights and boundary conditions are consistent; dimensional audits pass.
2. Negative criteria:
   • Using a non-unit measure or missing weights causing normalization failure.
   • Q_ij eigenvalues out of bounds or non-zero trace.
   • Confusing Q_ij with unrelated quantities (e.g., Q0_Z).
3. Contrasts:
   • {equal-area vs non-equal-area} grid error comparisons.
   • {full second moment vs scalar principal-axis} impacts on coupling and energy closure.
   • {free, prescribed, no-slip} boundary conditions and their effects on interfacial Q_ij.

VI. Deliverables & Figure List

1. Deliverables:
   • S2Grid.json (spherical discretization and weights).
   • QtensorBench.csv (reference Q_ij for standard distributions).
   • GeometryCard.json (domain, measure, boundary, weight fields).
   • UnitsAudit.log (dimensional and normalization audits).
2. Figures/Tables (suggested):
   • Tab. 3-1 Minimal orientation symbols & measures (n_hat, dΩ, f_orient, Q_ij).
   • Fig. 3-1 Spherical discretization schematic and weight distribution.
   • Tab. 3-2 Q_ij analytic/numerical results for typical distributions (isotropic/axial/bimodal).
   • Tab. 3-3 Templates for boundary conditions and weights in line/surface/volume embeddings.