FLAME-Conditioned Deformation Branch

• FLAME-Conditioned Deformation Branch is a framework that couples real-time combustion data (e.g., temperature, char mass) with mechanical deformation to simulate fire-induced structural failure.
• It employs a two-phase simulation pipeline alternating between Eulerian combustion updates and Lagrangian deformation solvers using position-based dynamics.
• The approach leverages adaptive sparse voxel grids and efficient numerical solvers to predict structural sag, collapse, and material degradation in fire-exposed assemblies.

The FLAME-Conditioned Deformation Branch refers to a class of models and algorithmic integrations where the combustion state of a charring or non-charring structure—the spatial distributions of temperature, char and volatile mass fractions, and material phase—dynamically conditions the downstream mechanical response of that structure. These models are crucial for simulating coupled thermal–mechanical phenomena such as the sag, collapse, or deformation of combusting wooden beams, bridges, or other assemblies, and have been formalized in advanced multiphysics simulators such as those described in "FlameForge: Combustion of Generalized Wooden Structures" (Liu et al., 2024). Within this context, the deformation solver is explicitly parameterized by local combustion variables, with their influence propagating through changes in mass, stiffness, and insulating properties, which in turn alters the physical motion and stability of the structure.

1. Mathematical Formalism of Coupled Dynamics

The coupling is established through a two-way exchange: each time the combustion simulation step is completed on a voxelized domain, updated scalar fields are produced—material temperature $T_m(x)$, relative char and volatile mass fractions $M_c(x)$ and $M_v(x)$, and a char-insulation coefficient $c(x)$. For each rod- or edge-based element $e$ in the mechanical graph, two critical quantities are recomputed:

• Mass per Segment: $$M_e = \sum_{v \in V_e} (M_c(v)+M_v(v)) \cdot \rho_0 \cdot \mathrm{VoxelVolume}$$
• Effective Young’s Modulus: $$E_e = (1-\mu_e) E_\mathrm{wood} + \mu_e E_\mathrm{char}$$, with $\mu_e$ the local char fraction.

These are mapped into constraint stiffnesses used in position-based dynamics (PBD):

• Stretch constraint stiffness: $k_e^s = \alpha_s E_e A_e / L_e^0$
• Bend constraint stiffness: $k_i^b = \alpha_b E_e I_e$, $I_e \propto A_e r_e^2$

Such tight integration ensures that the loss of mass and modulus from combustion directly modulates mechanical failure and deformation—flame-conditioned deformation—across the full simulation cycle (Liu et al., 2024).

2. Algorithmic Pipeline for Flame-Conditioned Deformation

The simulation alternates between Eulerian (combustion) and Lagrangian (deformation) phases per time step as follows:

1. Combustion Update
   • Solve flow, temperature, pyrolysis, and update $M_c, M_v$.
   • Update char-insulation and surface geometry with a signed distance field (SDF).
2. Data Synchronization
   • Update VDB masks and rebuilt SDF for efficient geometric queries.
3. Conditioning Mechanical State
   • For each rod/edge, sample overlapping voxels; compute updated $M_e, \mu_e, E_e, A_e$.
   • Update PBD constraint stiffnesses.
4. PBD Dynamics Solve
   • Integrate external forces, iterate constraint projections with new stiffnesses.
   • Convergence is reached when rod positions adjust to altered physical properties.
5. Render and Repeat
   • Visualization and post-processing for each generated configuration.

This modular alternation ensures feedback between combustion-induced degradation and geometric or topological modification of the mechanical structure (Liu et al., 2024).

3. Data Structures and Memory Efficiency

The combustion state is held in adaptive, sparse voxel grids (OpenVDB::FloatGrid), with scalar fields ($T_a, T_m, M_v, M_c, \dots$) indexed via multi-resolution trees for scalability. Char thickness and insulation rely on a synchronized SDF updated with a fast-sweeping eikonal solver. The mechanical side employs a graph of Cosserat rods or linear PBD edges, each maintaining pointers to their overlapping voxels for rapid state sampling. OpenVDB’s hierarchical tree enables efficient memory utilization, with computation focused near the evolving geometry while ignoring empty voxels (Liu et al., 2024).

4. Numerical Stability and Time Integration

The design ensures robust stability across rapid or highly localized combustion events:

• Combustion: Uses second-order MacCormack for advection (stable at large Courant–Friedrichs–Lewy numbers), implicit central differencing for diffusion (unconditionally stable), and AMGCL multigrid for the Navier–Stokes projection.
• Deformation: PBD enforces constraints with Gauss–Seidel projection, remaining unconditionally stable for stretch constraints despite large, spatially varying stiffnesses.
• Coupling: Mechanical updates are performed after the combustion step; the constraint that $(\Delta t) \cdot \max(k_e / M_e) \lesssim 1$ guarantees rapid convergence (3–5 iterations/frame sufficient). No substepping is necessary, which is critical for handling the severe local weakening characteristic of charring transitions (Liu et al., 2024).

5. Pseudocode and Implementation Example

A simplified schematic for the deformation branch conditioned on recent combustion data is as follows:

// After combustion has computed M_v, M_c, T_m, and updated the SDF: for each rod_edge e between nodes i,j do // Gather overlapping voxels M_sum = 0 M_c_sum = 0 for v in V_e do M_rel = M_v[v] + M_c[v] M_sum += M_rel * ρ0 * voxelVolume M_c_sum += M_c[v] * ρ0 * voxelVolume end μ_e = M_c_sum / max(M_sum, ε) M_e = M_sum A_e = A0 * (M_sum / M_initial_per_segment) E_e = (1−μ_e) * E_wood + μ_e * E_char k_e^s = α_s * E_e * A_e / L0_e I_e = A_e * (radius_e)^2 / 4 k_e^b = α_b * E_e * I_e end // Position-Based Dynamics solve follows with updated k^s, k^b.

Per-step recomputation of constraints based on local combustion ensures that the simulated object sags, fails, or deforms where material properties have been degraded. This framework captures the continuous emergence of sag and collapse patterns as a direct result of fire progression, which is central to predictive virtual combustion (Liu et al., 2024).

6. Generalization and Nonlinear Branch Behavior

A formal definition of "deformation branch"—in a more abstract context—appears in the study of wrinkled premixed flames, where the nonlinear response of flame fronts to dynamic forcing and stretch leads to a multiplicity of solution branches. Here, the governing equation for the flame front $h(x,t)$ includes instability, curvature, geometric nonlinearity, and mechanical stretch:

$$\partial_t h + S(x,t) \partial_x h + \frac{1}{2} (\partial_x h)^2 - \nu \partial_{xx} h + \mathcal{I}[\partial_x h] = F(x,t)$$

Deformation branches are defined as continuous families of periodic solutions and mean speed increments $V_\text{av}$, parameterized by forcing amplitude and phase. Bifurcation analysis reveals regimes with unique, bistable, or fast-jumping branches, corresponding to qualitatively distinct mechanical response patterns under varying combustion-induced stresses and forcing (Joulin et al., 2019). A plausible implication is that, in realistic simulations, such branching dynamics could manifest as abrupt or state-dependent transitions in structural integrity or failure, depending on local combustion progression.

7. Significance, Applications, and Broader Impact

The FLAME-Conditioned Deformation Branch methodology provides the foundation for accurate and efficient simulation of coupled combustion–mechanics problems, directly enabling the virtual prototyping and failure prediction of fire-exposed structures. Applications include civil engineering safety analysis (e.g., bridge or building collapse under fire), fire dynamics studies in forestry or wildland management, and special effects for virtual environments. By explicitly tying local loss of mass and mechanical rigidity to combustion physics within a sparse, scalable framework, such algorithms enable predictive and physically grounded assessments of real-world fire scenarios (Liu et al., 2024). The generalization of deformation branches to nonlinear, forced regimes also links these approaches to foundational studies of bifurcation phenomena in reacting flow systems (Joulin et al., 2019).