# Atomizer Future Development: Fast Solver Technologies ## Document Information - **Created:** December 2024 - **Updated:** December 28, 2025 (L-BFGS implementation) - **Purpose:** Technical reference for integrating fast solver technologies into Atomizer - **Status:** Partially Implemented (L-BFGS complete, ROM/meshfree planned) - **Priority:** Phase 2+ (after core parametric optimization is stable) --- ## Executive Summary This document captures research on alternative solver technologies that could dramatically accelerate Atomizer's optimization capabilities. Two main approaches are explored: 1. **Meshfree/Immersed FEA** - Eliminates meshing bottleneck entirely (Intact Solutions, Gridap.jl) 2. **Reduced Order Models (ROM/POD)** - Accelerates existing Nastran by 1000x for parametric problems **Recommendation:** - **Short-term:** Implement ROM/POD for parametric optimization (2-4 weeks, 1000x speedup) - **Medium-term:** Prototype with Gridap.jl for topology optimization capability (2-3 months) - **Long-term:** Evaluate Intact licensing or productionize custom meshfree solver --- ## Part 1: Meshfree / Immersed FEA Methods ### 1.1 The Problem with Traditional FEA Traditional FEA requires **body-fitted meshes** that conform to geometry: ``` Traditional FEA Pipeline: CAD Geometry → Clean/Heal → Defeature → Mesh Generation → Solve → Post-process ↑ BOTTLENECK - 80% of time - Breaks on complex geometry - Must remesh for topology opt ``` **Pain points:** - Meshing complex geometry (lattices, AM parts) often fails - Small features require fine meshes → huge DOF count - CAD errors (gaps, overlaps) break meshing - Topology optimization requires remeshing every iteration - Automation breaks when meshing fails ### 1.2 The Meshfree Solution: Immersed Methods **Core idea:** Don't mesh the geometry. Mesh the SPACE. ``` Immersed FEA Pipeline: CAD Geometry → Convert to Implicit → Immerse in Grid → Solve → Post-process (any format works) (regular grid) (always works) ``` **How it works:** ``` Step 1: Background Grid Step 2: Classify Cells ╔══╤══╤══╤══╤══╗ ╔══╤══╤══╤══╤══╗ ║ │ │ │ │ ║ ║○ │○ │○ │○ │○ ║ ○ = Outside ╠══╪══╪══╪══╪══╣ ╠══╪══╪══╪══╪══╣ ● = Inside ║ │ │ │ │ ║ + Geometry → ║○ │◐ │● │◐ │○ ║ ◐ = Cut by boundary ╠══╪══╪══╪══╪══╣ (implicit) ╠══╪══╪══╪══╪══╣ ║ │ │ │ │ ║ ║○ │◐ │● │◐ │○ ║ ╚══╧══╧══╧══╧══╝ ╚══╧══╧══╧══╧══╝ Step 3: Special Integration for Cut Cells Full cell: Standard Gauss quadrature Cut cell: Moment-fitting quadrature (custom weights that integrate correctly over the irregular "inside" portion) ``` ### 1.3 Implicit Geometry Representation **Signed Distance Function (SDF):** - φ(x) < 0 → inside geometry - φ(x) > 0 → outside geometry - φ(x) = 0 → on boundary (the surface) ``` Example: Plate with hole φ > 0 (outside) ↓ ───────────────────── │ · │ · │ │ -0.3 │ +0.2 │ Numbers show signed distance │─────────┼───────────│ ← φ = 0 (surface) │ -0.5 │ -0.1 │ │ · │ · │ ───────────────────── ↑ φ < 0 (inside) ``` **Why implicit is powerful:** - Boolean operations are trivial: union = min(φ_A, φ_B), intersection = max(φ_A, φ_B) - Works with ANY input: CAD, STL, CT scans, procedural geometry - Resolution-independent - VDB format stores efficiently (sparse, only near surface) ### 1.4 Key Technical Challenges #### Challenge 1: Cut Cell Integration When boundary cuts through a cell, standard Gauss quadrature doesn't work. **Solution: Moment Fitting** Find quadrature weights w_i such that polynomial integrals are exact: ``` ∫ xᵃ yᵇ zᶜ dV = Σ wᵢ · xᵢᵃ yᵢᵇ zᵢᶜ for all a+b+c ≤ p Ω_cut i ``` #### Challenge 2: Boundary Condition Enforcement Grid doesn't have nodes ON the boundary. **Solution: Nitsche's Method (Weak Enforcement)** Add penalty term to variational form: ``` a(u,v) = ∫ ε(u):σ(v) dΩ + ∫ β/h (u-g)·(v) dΓ ← penalty term - ∫ σ(u)·n·v dΓ ← consistency - ∫ σ(v)·n·u dΓ ← symmetry ``` #### Challenge 3: Small Cut Cells (Ill-Conditioning) Tiny slivers cause numerical instability. **Solution: Cell Aggregation (AgFEM)** Merge small cuts with neighboring full cells: ``` Before: After (aggregated): ┌───┬───┐ ┌───┬───┐ │ │░░░│ ← tiny sliver │ │▓▓▓│ ← merged with neighbor └───┴───┘ └───┴───┘ ``` ### 1.5 Intact Solutions: Commercial Implementation **Company:** Intact Solutions, Inc. (Madison, WI) **Technology:** Immersed Method of Moments (IMM) **Funding:** DARPA, NSF, NIST **History:** 10+ years (started with Scan&Solve for Rhino) **Products:** | Product | Platform | Use Case | |---------|----------|----------| | Intact.Simulation for Grasshopper | Rhino | Design exploration | | Intact.Simulation for nTop | nTopology | Lattice/AM parts | | Intact.Simulation for Synera | Synera | Automotive | | Intact.Simulation for Automation | Headless/API | Batch processing | | PyIntact | Python API | Custom workflows | | LevelOpt | All platforms | Topology optimization | **Capabilities:** - Physics: Linear static, thermal, modal - Geometry: STL, VDB, BREP, CT scans, G-code, hybrid - Materials: Isotropic, orthotropic - Multi-load case support - Assembly handling (bonded) **Performance (from their case study):** ``` Steering knuckle topology optimization: - 5,600 simulations - 1,400 design iterations - 4 load cases - ~200,000 DOF each - Total time: 45 hours - Per simulation: ~30 seconds - No meshing, no manual intervention ``` **Limitations (inferred):** - No nonlinear mechanics (yet?) - No dynamic response (frequency response, random vib) - Not certified for aerospace (Nastran still required for certification) - Proprietary (black box) ### 1.6 Open-Source Alternative: Gridap.jl **Language:** Julia **Repository:** github.com/gridap/Gridap.jl, github.com/gridap/GridapEmbedded.jl **License:** MIT **Maturity:** Research+ (active development, used in publications) **Why Gridap:** - Native AgFEM support (handles cut cells properly) - Level set geometry built-in - Clean mathematical API - Good performance (Julia compiles to native code) - Active community **Installation:** ```julia using Pkg Pkg.add("Gridap") Pkg.add("GridapEmbedded") ``` **Basic Example:** ```julia using Gridap using GridapEmbedded # 1. Background mesh domain = (-1, 1, -1, 1, -1, 1) partition = (20, 20, 20) bgmodel = CartesianDiscreteModel(domain, partition) # 2. Implicit geometry (level set) R = 0.7 geo = sphere(R) # Built-in primitive # 3. Cut the mesh cutgeo = cut(bgmodel, geo) Ω = Triangulation(cutgeo, PHYSICAL) # Inside domain Γ = EmbeddedBoundary(cutgeo) # Boundary # 4. FE space with aggregation reffe = ReferenceFE(lagrangian, VectorValue{3,Float64}, 2) V = TestFESpace(Ω, reffe, conformity=:H1) U = TrialFESpace(V) # 5. Weak form (linear elasticity) const E, ν = 210e9, 0.3 const λ = E*ν / ((1+ν)*(1-2ν)) const μ = E / (2*(1+ν)) ε(u) = 0.5 * (∇(u) + transpose(∇(u))) σ(u) = λ * tr(ε(u)) * one(ε(u)) + 2*μ * ε(u) a(u,v) = ∫(ε(v) ⊙ σ(u)) * dΩ l(v) = ∫(v ⋅ VectorValue(0,0,-1e6)) * dΓ # 6. Solve op = AffineFEOperator(a, l, U, V) uh = solve(op) # 7. Export writevtk(Ω, "results", cellfields=["u"=>uh]) ``` **Parameterized Solver for Optimization:** ```julia function solve_bracket(hole_radius, thickness, load) # Background mesh domain = (0, 1, 0, 0.5, 0, thickness) partition = (40, 20, 4) bgmodel = CartesianDiscreteModel(domain, partition) # Parameterized geometry function φ(x) d_box = max(max(-x[1], x[1]-1), max(-x[2], x[2]-0.5), max(-x[3], x[3]-thickness)) d_hole = sqrt((x[1]-0.5)^2 + (x[2]-0.25)^2) - hole_radius return max(d_box, -d_hole) end geo = AnalyticalGeometry(φ) cutgeo = cut(bgmodel, geo) Ω = Triangulation(cutgeo, PHYSICAL) # ... FE setup and solve ... return (max_stress=σ_max, max_disp=u_max, mass=m) end # Use in optimization loop for r in 0.05:0.01:0.2 result = solve_bracket(r, 0.1, 1e6) println("r=$r, σ_max=$(result.max_stress/1e6) MPa") end ``` ### 1.7 Other Open-Source Options | Library | Language | Immersed Support | Notes | |---------|----------|------------------|-------| | **FEniCSx** | Python/C++ | Experimental | Most mature FEM, but CutFEM needs work | | **MFEM** | C++ | Manual | Very fast, DOE-backed, shifted boundary | | **deal.II** | C++ | Partial | German powerhouse, some cut cell support | | **scikit-fem** | Python | DIY | Simple, hackable, but build from scratch | **Recommendation:** Start with Gridap.jl for prototyping. It has the best out-of-box immersed support. --- ## Part 2: Reduced Order Models (ROM/POD) ### 2.1 Concept **Problem:** FEA is slow because we solve huge systems (100k+ DOFs). **Insight:** For parametric variations, solutions live in a low-dimensional subspace. **Solution:** Find that subspace, solve there instead. ``` Full Order Model: Reduced Order Model: K u = F K_r u_r = F_r K: 100,000 × 100,000 K_r: 20 × 20 u: 100,000 unknowns u_r: 20 unknowns Solve time: minutes Solve time: milliseconds ``` ### 2.2 How POD Works #### Step 1: Collect Snapshots (Offline) Run N full FEA simulations with different parameters: ```python snapshots = [] for params in sample_parameter_space(n=100): result = nastran_solve(geometry(params), loads, bcs) snapshots.append(result.displacement_field) U = np.array(snapshots) # Shape: (n_samples, n_dofs) ``` #### Step 2: SVD to Find Basis (Offline) ```python from scipy.linalg import svd U_centered = U - U.mean(axis=0) Phi, S, Vt = svd(U_centered, full_matrices=False) # Keep top k modes k = 20 # Usually 10-50 captures 99%+ variance basis = Vt[:k, :] # Shape: (k, n_dofs) ``` The singular values decay rapidly - most "information" is in first few modes: ``` Singular Value Spectrum (typical): S │ 100 │██ │██ 50 │██░░ │██░░░░ 10 │██░░░░░░░░░░░░░░░░░░░░░░ └───────────────────────── 1 5 10 15 20 ... Mode index ``` #### Step 3: Project and Solve (Online - FAST) ```python def rom_solve(K, F, basis): """ Solve in reduced space. Args: K: (n_dofs, n_dofs) stiffness matrix F: (n_dofs,) force vector basis: (k, n_dofs) POD modes Returns: u: (n_dofs,) approximate displacement """ # Project to reduced space K_r = basis @ K @ basis.T # (k, k) F_r = basis @ F # (k,) # Solve tiny system u_r = np.linalg.solve(K_r, F_r) # Instant! # Project back u = basis.T @ u_r # (n_dofs,) return u ``` ### 2.3 Complete ROM Implementation for Atomizer ```python # atomizer/solvers/rom_solver.py import numpy as np from scipy.linalg import svd from scipy.sparse import issparse from typing import List, Dict, Callable, Tuple import pickle class ROMSolver: """ Reduced Order Model solver for fast parametric optimization. Usage: # Training (offline, one-time) rom = ROMSolver() rom.train(parameter_samples, nastran_solver, n_modes=20) rom.save("bracket_rom.pkl") # Inference (online, fast) rom = ROMSolver.load("bracket_rom.pkl") u, error_est = rom.solve(new_params) """ def __init__(self, n_modes: int = 20): self.n_modes = n_modes self.basis = None # (n_modes, n_dofs) self.singular_values = None # (n_modes,) self.mean_solution = None # (n_dofs,) self.parameter_samples = None self.K_func = None # Function to build stiffness self.F_func = None # Function to build force def train(self, parameter_samples: List[Dict], solver_func: Callable, K_func: Callable = None, F_func: Callable = None, verbose: bool = True): """ Train ROM from full-order snapshots. Args: parameter_samples: List of parameter dictionaries solver_func: Function(params) -> displacement field (n_dofs,) K_func: Function(params) -> stiffness matrix (for online solve) F_func: Function(params) -> force vector (for online solve) """ self.parameter_samples = parameter_samples self.K_func = K_func self.F_func = F_func # Collect snapshots if verbose: print(f"Collecting {len(parameter_samples)} snapshots...") snapshots = [] for i, params in enumerate(parameter_samples): u = solver_func(params) snapshots.append(u) if verbose and (i+1) % 10 == 0: print(f" Snapshot {i+1}/{len(parameter_samples)}") U = np.array(snapshots) # (n_samples, n_dofs) # Center the data self.mean_solution = U.mean(axis=0) U_centered = U - self.mean_solution # SVD if verbose: print("Computing SVD...") _, S, Vt = svd(U_centered, full_matrices=False) # Keep top modes self.basis = Vt[:self.n_modes, :] self.singular_values = S[:self.n_modes] # Report energy captured total_energy = (S**2).sum() captured_energy = (self.singular_values**2).sum() if verbose: print(f"Captured {captured_energy/total_energy*100:.2f}% of variance " f"with {self.n_modes} modes") return self def solve(self, params: Dict) -> Tuple[np.ndarray, float]: """ Fast solve using ROM. Args: params: Parameter dictionary Returns: u: Approximate displacement field error_estimate: Estimated relative error """ if self.K_func is None or self.F_func is None: raise ValueError("K_func and F_func required for online solve") # Build matrices for this parameter set K = self.K_func(params) F = self.F_func(params) # Convert sparse to dense if needed (for small reduced system) if issparse(K): # Project while sparse K_r = self.basis @ K @ self.basis.T else: K_r = self.basis @ K @ self.basis.T F_r = self.basis @ F # Solve reduced system u_r = np.linalg.solve(K_r, F_r) # Project back u = self.basis.T @ u_r + self.mean_solution # Error estimate (residual-based) residual = F - K @ u error_estimate = np.linalg.norm(residual) / np.linalg.norm(F) return u, error_estimate def solve_interpolated(self, params: Dict) -> np.ndarray: """ Even faster: interpolate in coefficient space. Requires parameter_samples to be stored with coefficients. """ # Find nearest neighbors in parameter space # Interpolate their ROM coefficients # This avoids even building K and F! raise NotImplementedError("TODO: Implement RBF interpolation") def estimate_error(self, u_rom: np.ndarray, K: np.ndarray, F: np.ndarray) -> float: """Residual-based error estimate.""" residual = F - K @ u_rom return np.linalg.norm(residual) / np.linalg.norm(F) def should_fallback_to_full(self, error_estimate: float, threshold: float = 0.05) -> bool: """Check if ROM error is too high, suggesting full solve needed.""" return error_estimate > threshold def save(self, filepath: str): """Save trained ROM to file.""" data = { 'n_modes': self.n_modes, 'basis': self.basis, 'singular_values': self.singular_values, 'mean_solution': self.mean_solution, } with open(filepath, 'wb') as f: pickle.dump(data, f) @classmethod def load(cls, filepath: str) -> 'ROMSolver': """Load trained ROM from file.""" with open(filepath, 'rb') as f: data = pickle.load(f) rom = cls(n_modes=data['n_modes']) rom.basis = data['basis'] rom.singular_values = data['singular_values'] rom.mean_solution = data['mean_solution'] return rom class AdaptiveROMSolver(ROMSolver): """ ROM with automatic fallback to full solver when error is high. """ def __init__(self, n_modes: int = 20, error_threshold: float = 0.05): super().__init__(n_modes) self.error_threshold = error_threshold self.full_solver = None def set_full_solver(self, solver_func: Callable): """Set the full-order solver for fallback.""" self.full_solver = solver_func def solve_adaptive(self, params: Dict) -> Tuple[np.ndarray, float, bool]: """ Solve with automatic fallback. Returns: u: Displacement field error: Error estimate (or 0 if full solve) used_full: Whether full solver was used """ # Try ROM first u_rom, error = self.solve(params) if error < self.error_threshold: return u_rom, error, False # Fallback to full if self.full_solver is not None: u_full = self.full_solver(params) return u_full, 0.0, True else: # Return ROM result with warning print(f"Warning: ROM error {error:.3f} exceeds threshold, " f"but no full solver available") return u_rom, error, False ``` ### 2.4 Integration with Atomizer Optimization Loop ```python # atomizer/optimization/rom_optimization_runner.py from atomizer.solvers.rom_solver import AdaptiveROMSolver from atomizer.solvers.nastran_solver import NastranSolver import optuna class ROMOptimizationRunner: """ Optimization using ROM for speed, Nastran for validation. """ def __init__(self, problem_config: dict): self.config = problem_config self.rom = None self.nastran = NastranSolver() def train_rom(self, n_samples: int = 100, n_modes: int = 20): """Train ROM on problem-specific data.""" # Sample parameter space samples = self._latin_hypercube_sample(n_samples) # Run Nastran for each sample def nastran_solve(params): result = self.nastran.solve( geometry=self._build_geometry(params), loads=self.config['loads'], bcs=self.config['bcs'] ) return result['displacement_field'] # Train ROM self.rom = AdaptiveROMSolver(n_modes=n_modes) self.rom.train(samples, nastran_solve) self.rom.set_full_solver(nastran_solve) return self def optimize(self, n_trials: int = 500): """Run optimization using ROM.""" def objective(trial): # Sample parameters params = {} for name, bounds in self.config['parameters'].items(): params[name] = trial.suggest_float(name, bounds[0], bounds[1]) # Fast ROM solve u, error, used_full = self.rom.solve_adaptive(params) # Compute objective (e.g., max stress) stress = self._compute_stress(u) max_stress = stress.max() # Log trial.set_user_attr('rom_error', error) trial.set_user_attr('used_full_solver', used_full) return max_stress # Run optimization study = optuna.create_study(direction='minimize') study.optimize(objective, n_trials=n_trials) # Validate best designs with full Nastran return self._validate_top_designs(study, n_top=10) def _validate_top_designs(self, study, n_top: int = 10): """Run full Nastran on top candidates.""" top_trials = sorted(study.trials, key=lambda t: t.value)[:n_top] validated = [] for trial in top_trials: result = self.nastran.solve( geometry=self._build_geometry(trial.params), loads=self.config['loads'], bcs=self.config['bcs'] ) validated.append({ 'params': trial.params, 'rom_prediction': trial.value, 'nastran_result': result['max_stress'], 'error': abs(trial.value - result['max_stress']) / result['max_stress'] }) return validated ``` ### 2.5 When ROM Works Well vs. Poorly ``` WORKS GREAT: WORKS POORLY: ✓ Parametric variations ✗ Topology changes (thickness, radii, positions) (holes appear/disappear) ✓ Smooth parameter dependence ✗ Discontinuous behavior (contact, buckling) ✓ Similar geometry family ✗ Wildly different geometries ✓ Linear problems ✗ Highly nonlinear ✓ Fixed mesh topology ✗ Mesh changes between solves ``` **For Atomizer's current parametric optimization: ROM is ideal.** --- ## Part 3: Comparison of Approaches ### 3.1 Speed Comparison | Method | Typical Solve Time | Speedup vs Nastran | Training Cost | |--------|-------------------|-------------------|---------------| | **Nastran (baseline)** | 10 min | 1x | None | | **Intact IMM** | 30 sec | 20x | None | | **ROM/POD** | 10 ms | 60,000x | 100 Nastran runs | | **Neural Surrogate (MLP)** | 10 ms | 60,000x | 50+ FEA runs | | **L-BFGS on Surrogate** | <1 sec (20-50 evals) | 100-1000x vs TPE | Trained surrogate | ### 3.1.1 L-BFGS Gradient Optimization (IMPLEMENTED) **Status:** ✓ Implemented in `optimization_engine/gradient_optimizer.py` L-BFGS exploits the differentiability of trained MLP surrogates for ultra-fast local refinement: ```python from optimization_engine.gradient_optimizer import run_lbfgs_polish # Polish top candidates from surrogate search results = run_lbfgs_polish(study_dir, n_starts=20, n_iterations=100) ``` **Performance:** | Method | Evaluations to Converge | Time | |--------|------------------------|------| | TPE (surrogate) | 200-500 | ~30 min | | CMA-ES (surrogate) | 100-300 | ~15 min | | **L-BFGS (surrogate)** | **20-50** | **<1 sec** | **When to use:** After Turbo mode identifies promising regions, before FEA validation. **Full documentation:** `docs/protocols/system/SYS_14_NEURAL_ACCELERATION.md` (Section: L-BFGS Gradient Optimizer) ### 3.2 Capability Comparison | Capability | Nastran | Intact | ROM | Neural | |------------|---------|--------|-----|--------| | Parametric opt | ✓ (slow) | ✓ | ✓✓ | ✓✓ | | Topology opt | ✗ | ✓✓ | ✗ | ? | | New geometry | ✓ | ✓ | ✗ (retrain) | ✗ (retrain) | | Certification | ✓✓ | ✗ | ✗ | ✗ | | Nonlinear | ✓ | ? | Limited | Limited | | Dynamic | ✓ | ? | Limited | Limited | ### 3.3 Implementation Effort | Approach | Effort | Risk | Payoff | |----------|--------|------|--------| | ROM/POD | 2-4 weeks | Low | Immediate 1000x speedup | | Gridap prototype | 2-3 months | Medium | Topology opt capability | | Intact license | $$ + 2-4 weeks | Low | Production meshfree | | AtomizerField | Ongoing | Medium | Field predictions | --- ## Part 4: Recommended Atomizer Architecture ### 4.1 Three-Tier Fidelity Stack ``` ┌─────────────────────────────────────────────────────────────────────────┐ │ ATOMIZER SOLVER STACK │ ├─────────────────────────────────────────────────────────────────────────┤ │ │ │ TIER 1: AtomizerField / ROM (Screening) │ │ ────────────────────────────────────────── │ │ Speed: ~10 ms per evaluation │ │ Accuracy: ~90-95% │ │ Use case: Design space exploration, optimization inner loop │ │ Output: Promising candidates │ │ │ │ TIER 2: Intact IMM / Gridap (Refinement) [FUTURE] │ │ ────────────────────────────────────────────────── │ │ Speed: ~30 sec per evaluation │ │ Accuracy: ~98% │ │ Use case: Topology optimization, complex geometry │ │ Output: Optimized designs │ │ │ │ TIER 3: NX Nastran (Validation) │ │ ──────────────────────────────── │ │ Speed: ~10 min per evaluation │ │ Accuracy: 100% (reference) │ │ Use case: Final validation, certification │ │ Output: Certified results │ │ │ └─────────────────────────────────────────────────────────────────────────┘ Workflow: Design Space ──▶ Tier 1 ──▶ Candidates ──▶ Tier 2 ──▶ Best ──▶ Tier 3 (1M points) (1 sec) (1000 pts) (8 hrs) (10) (2 hrs) │ └── Skip if parametric only **Current Implemented Workflow (Turbo + L-BFGS):** ``` ┌─────────────────────────────────────────────────────────────────────────┐ │ 1. FEA Exploration │ 50-100 Nastran trials │ │ └─► Train MLP surrogate on FEA data │ ├─────────────────────────────────────────────────────────────────────────┤ │ 2. Turbo Mode │ 5000+ NN predictions (~10ms each) │ │ └─► Find promising design regions │ ├─────────────────────────────────────────────────────────────────────────┤ │ 3. L-BFGS Polish │ 20 multi-start runs (<1 sec total) [NEW] │ │ └─► Precise local optima via gradient descent │ ├─────────────────────────────────────────────────────────────────────────┤ │ 4. FEA Validation │ Top 3-5 candidates verified with Nastran │ │ └─► Final certified results │ └─────────────────────────────────────────────────────────────────────────┘ ``` ### 4.2 Implementation Phases ``` PHASE 1: NOW (Q4 2024 - Q1 2025) ──────────────────────────────── [✓] Core optimization engine (Optuna/CMA-ES/NSGA-II) [✓] NX/Nastran integration [✓] Dashboard [✓] MLP Surrogate + Turbo Mode [✓] L-BFGS Gradient Optimizer ← IMPLEMENTED 2025-12-28 [ ] ROM/POD implementation [ ] AtomizerField training pipeline PHASE 2: MEDIUM-TERM (Q2-Q3 2025) ──────────────────────────────────── [ ] Gridap.jl prototype [ ] Evaluate Intact licensing [ ] Topology optimization capability [ ] Multi-fidelity optimization PHASE 3: FUTURE (Q4 2025+) ────────────────────────── [ ] Production meshfree solver (Intact or custom) [ ] Full AtomizerField deployment [ ] Cloud/HPC scaling [ ] Certification workflow integration ``` --- ## Part 5: Key Resources ### 5.1 Intact Solutions - Website: https://intact-solutions.com - PyIntact docs: https://intact-solutions.com/intact_docs/PyIntact.html - Contact for pricing: info@intact-solutions.com ### 5.2 Gridap.jl - Main repo: https://github.com/gridap/Gridap.jl - Embedded: https://github.com/gridap/GridapEmbedded.jl - Tutorials: https://gridap.github.io/Tutorials/stable/ - Paper: "Gridap: An extensible Finite Element toolbox in Julia" (2020) ### 5.3 ROM/POD References - Benner et al., "Model Reduction and Approximation" (2017) - Quarteroni et al., "Reduced Basis Methods for PDEs" (2016) - pyMOR library: https://pymor.org (Python model order reduction) ### 5.4 Immersed FEM Theory - Burman et al., "CutFEM: Discretizing geometry and PDEs" (2015) - Schillinger & Ruess, "The Finite Cell Method: A Review" (2015) - Main & Scovazzi, "The Shifted Boundary Method" (2018) --- ## Appendix A: Quick Reference - Immersed FEM Math ### Weak Form with Nitsche BCs Standard elasticity: ``` Find u ∈ V such that: a(u,v) = l(v) ∀v ∈ V a(u,v) = ∫_Ω ε(u):C:ε(v) dΩ l(v) = ∫_Ω f·v dΩ + ∫_Γ_N t·v dΓ ``` With Nitsche (weak Dirichlet on Γ_D): ``` a(u,v) = ∫_Ω ε(u):C:ε(v) dΩ - ∫_Γ_D (σ(u)·n)·v dΓ # Consistency - ∫_Γ_D (σ(v)·n)·u dΓ # Symmetry + ∫_Γ_D (β/h) u·v dΓ # Penalty l(v) = ∫_Ω f·v dΩ + ∫_Γ_N t·v dΓ - ∫_Γ_D (σ(v)·n)·g dΓ # Consistency (RHS) + ∫_Γ_D (β/h) g·v dΓ # Penalty (RHS) where β ~ E (penalty parameter), h = cell size, g = prescribed displacement ``` ### Ghost Penalty Stabilization For small cut cells, add: ``` j(u,v) = Σ_F γ h² ∫_F [∂ⁿu/∂nⁿ]·[∂ⁿv/∂nⁿ] dF where F = interior faces, [·] = jump across face, γ = stabilization parameter ``` --- ## Appendix B: Julia/Gridap Cheat Sheet ```julia # Installation using Pkg Pkg.add(["Gridap", "GridapEmbedded", "GridapGmsh"]) # Basic imports using Gridap using GridapEmbedded using GridapGmsh # For external meshes # Geometry primitives sphere(r) # Sphere of radius r disk(r) # 2D disk cylinder(r, h) # Cylinder box(corner1, corner2) # Box # Boolean operations union(geo1, geo2) intersect(geo1, geo2) setdiff(geo1, geo2) # geo1 - geo2 # Custom level set geo = AnalyticalGeometry(x -> my_sdf(x)) # Cutting cutgeo = cut(bgmodel, geo) Ω = Triangulation(cutgeo, PHYSICAL) Γ = EmbeddedBoundary(cutgeo) # FE spaces reffe = ReferenceFE(lagrangian, Float64, 2) # Scalar, quadratic reffe = ReferenceFE(lagrangian, VectorValue{3,Float64}, 2) # 3D vector # Measures dΩ = Measure(Ω, 4) # Quadrature degree 4 dΓ = Measure(Γ, 4) # Weak form a(u,v) = ∫(∇(u)⋅∇(v)) * dΩ l(v) = ∫(f*v) * dΩ # Solve op = AffineFEOperator(a, l, U, V) uh = solve(op) # Export writevtk(Ω, "output", cellfields=["u"=>uh]) ``` --- ## Appendix C: ROM/POD Implementation Checklist ``` □ Data Collection □ Define parameter space bounds □ Latin hypercube sampling (or similar) □ Run Nastran for each sample □ Store displacement fields consistently (same mesh!) □ Basis Construction □ Center snapshot matrix □ SVD decomposition □ Select number of modes (energy criterion: 99%+) □ Store basis and mean □ Online Solver □ Stiffness matrix assembly for new params □ Force vector assembly for new params □ Project to reduced space □ Solve reduced system □ Project back to full space □ Error Estimation □ Residual-based estimate □ Fallback threshold selection □ Adaptive solver wrapper □ Integration □ Save/load trained ROM □ Plug into optimization loop □ Validation against full solver ``` --- *Document maintained by: Atomizer Development Team* *Last updated: December 2024*