FEM/CFD Lab (ME-505) - Unit 2 Short Notes

UNIT 2: ADVANCED FEM/CFD APPLICATIONS & WORKFLOW

Based on the standard progression from introductory concepts, this unit covers meshing strategies, advanced analysis types, solver configuration, and result validation for both FEM and CFD. The focus is on practical implementation and critical interpretation of simulation results.

A. ADVANCED FINITE ELEMENT METHOD (FEM) APPLICATIONS & ANALYSIS

2D & 3D Element Formulation & Selection

• 2D Elements:

  • Triangular (Tri3, Tri6): Tri3 (linear) is robust for complex geometries but less accurate. Tri6 (quadratic) offers better stress gradient capture.

  • Quadrilateral (Quad4, Quad8): Quad4 (linear) can suffer from shear locking. Quad8 (quadratic) is preferred for accuracy in regular domains.

  • Selection Rule: Use quads for regular shapes, tris for complex boundaries. Quadratic elements reduce needed mesh density.

• 3D Elements:

  • Tetrahedral (Tet4, Tet10): Tet4 (linear) is automatic meshing-friendly but stiff. Tet10 (quadratic) is standard for complex 3D geometries.

  • Hexahedral (Hex8, Hex20): "Brick" elements offer superior accuracy and computational efficiency for regular geometries but are difficult to generate automatically.

  • Key Trade-off: Hex > Tet in accuracy/efficiency for same node count, but Tet wins in meshing automation for complex parts.

[!TIP] Exam Focus: Be prepared to justify element choice for a given geometry (e.g., "Why use Tet10 over Hex20 for an engine block?").

Mesh Quality & Refinement Techniques

• Primary Quality Metrics:

  • Aspect Ratio: Ratio of longest to shortest edge. Should be close to 1. High ratios cause solution errors.

  • Warpage: Deviation of a quadrilateral/hex element from planar. > 0.1 is poor.

  • Skewness: Deviation from ideal shape angle. < 0.85 is acceptable.

  • Jacobian Determinant: Measures element distortion. Negative Jacobian = Invalid element. Must be positive.

• Refinement Methods:

  • h-method: Decrease element size (mesh refinement). Most common.

  • p-method: Increase polynomial order of element shape functions. No remeshing needed.

  • Mesh Convergence Study: Systematically refine mesh until key results (e.g., max stress) change by < 5%. \boxed{\text{Converged Solution}} is one independent of mesh density.

[!TIP] Common Pitfall: A "fine" mesh is not necessarily a "good" mesh. A poor-quality fine mesh can be worse than a coarse high-quality mesh.

Complex Boundary & Loading Conditions

• Non-uniform Loads: Pressure varying with coordinates (e.g., \( p = p_0 + kx \)), must be applied via tabular or functional input.

• Inertia Relief: Used for free-body analysis (e.g., an aircraft in flight). Applies acceleration loads to counteract rigid body motion, allowing static analysis of floating structures.

• Symmetry/Anti-symmetry: Exploits geometric and loading symmetry to model 1/2, 1/4, etc.

  • Symmetry: Displacement normal to plane = 0.

  • Anti-symmetry: Displacement tangential to plane = 0.

• Coupled Field Problems: E.g., Thermo-structural: Temperature field from heat transfer analysis is imported as a thermal load (strain = \( \alpha \Delta T \)) for stress analysis.

Non-linear Static Analysis

• Material Non-linearity:

  • Plasticity: von Mises (ductile metals) vs. Tresca (conservative). Requires yield stress \( \sigma_y \) and hardening rule (isotropic, kinematic).

  • Hyperelasticity: For rubbers/elastomers (Neo-Hookean, Mooney-Rivlin models).

• Geometric Non-linearity: Large deformation / stress stiffening. Equilibrium equations are formulated in deformed configuration. Essential when displacements > 10% of dimension or for buckling.

• Contact Analysis:

  • Types: Surface-to-surface (more accurate) vs. node-to-surface.

  • Friction: Frictionless (default), Coulomb friction (\( F_f = \mu F_n \)).

  • Key Parameters: Contact stiffness, penetration tolerance.

[!TIP] Exam Tip: Distinguish the three non-linearities. A problem with both large deformation and plasticity is geometrically and materially non-linear.

Modal & Harmonic Analysis

• Modal Analysis: Solves eigenvalue problem \( [K] - \omega^2[M] = 0 \) for natural frequencies \( \omega \) and mode shapes.

  • Extraction Methods: Lanczos (fast, default for large models), Subspace (accurate for few modes).

  • Pre-stressed Modal: "Stresses stiffen" or "soften" the structure. Must run a static pre-load step first.

• Harmonic Analysis: For steady-state sinusoidal loading at a range of frequencies. Input is complex force amplitude. Output is complex displacement/response.

  • Relation to Modal: Results can be synthesized from modal results (mode superposition) for efficiency.

B. ADVANCED COMPUTATIONAL FLUID DYNAMICS (CFD) SIMULATIONS

Turbulence Modeling

• RANS (Reynolds-Averaged Navier-Stokes) Models: Most common for industry.

  • k-ε (Standard): Robust, good for free shear flows. Requires wall functions for near-wall treatment. Poor for adverse pressure gradients.

  • k-ω (Standard): Better for low-Re flows and adverse pressure gradients. Sensitive to free-stream values.

  • k-ω SST: Hybrid model. Uses k-ω near walls and k-ε in free flow. Industry standard for aerospace, turbomachinery. Uses enhanced wall treatment (blends wall functions & low-Re).

• Scale-Resolving Simulations (Brief):

  • LES (Large Eddy Simulation): Resolves large scales, models small scales. Very expensive, used for fundamental research.

  • DES (Detached Eddy Simulation): Blends RANS near walls with LES in separated regions. Compromise for high-Re external flows.

[!TIP] Rule of Thumb: For most external aerodynamic/industrial flows, start with k-ω SST.

Advanced Boundary Conditions & Physical Models

• Periodic Boundaries: For repeating sections (e.g., nozzle guide vanes). Flow exiting one plane enters the opposite plane identically.

• Porous Media Model: Simulates flow through a porous region (e.g., filter, heat exchanger) via momentum sink: \( S_i = -(\frac{\mu}{\alpha} + C_2 \frac{1}{2} \rho |v|)v_i \).

• Species Transport: For multi-component mixing, combustion. Solves convection-diffusion equation for each species mass fraction \( Y_i \).

• Multiphase Flow:

  • VOF (Volume of Fluid): Tracks sharp interface between immiscible fluids (e.g., free surface).

  • Eulerian (Mixture): For dispersed phases (e.g., particles, bubbles).

• Conjugate Heat Transfer (CHT): Couples fluid convection with solid conduction. Requires defining both fluid and solid domains and their interface.

Solver Settings & Numerical Schemes

• Pressure-Velocity Coupling:

  • SIMPLE: Standard for steady-state. Can be slow to converge.

  • SIMPLEC: Variant of SIMPLE, often converges faster.

  • PISO: For transient flows. Corrects pressure within a timestep.

• Discretization Schemes (Order of Accuracy):

  • First Order: Upwind. Very stable but high numerical diffusion.

  • Second Order: Power Law, QUICK (for quadrature). Recommended default for accuracy.

  • High-Resolution: MUSCL, TVD. For sharp gradients (shocks, interfaces).

• Under-Relaxation Factors (URFs): Control solution update per iteration (0 < URF < 1). Low URF (0.3-0.7) = stable but slow. High URF (0.8-1.0) = fast but may diverge. Adjust during initialization.

Post-Processing & Flow Visualization

• Pathlines/Streamlines: Pathlines (time-dependent), Streamlines (instantaneous).

• Contours/Iso-Surfaces: Visualize scalar fields (Pressure, Temperature, Turbulent Kinetic Energy k).

• Vectors: Arrows showing direction/magnitude of vector fields (Velocity).

• Derived Quantities:

  • Force Coefficients: \( C_D = \frac{F_D}{\frac{1}{2}\rho V^2 A} \), \( C_L = \frac{F_L}{\frac{1}{2}\rho V^2 A} \).

  • Mass Flow Rate: \( \dot{m} = \rho \cdot A \cdot V_{avg} \).

  • Heat Transfer Coefficient (HTC): \( h = \frac{q}{T_{wall} - T_{fluid}} \).

C. INTEGRATED FEM/CFD WORKFLOW & VALIDATION

Problem Definition & Pre-processing Strategy

1. Physics Selection: Choose correct governing equations (e.g., laminar vs. turbulent, incompressible vs. compressible, structural vs. thermal).

2. Material Model: Define properties (E, ν, ρ for FEM; μ, ρ, Cp for CFD). Temperature-dependent? Non-linear?

3. Geometry & Mesh Strategy:

   • FEM: Structured hex mesh ideal for solids. Avoid high-aspect-ratio elements.

   • CFD: Inflation layers (y+ ~ 1 for low-Re, y+ ~ 30-300 for wall functions) are critical near walls for turbulence.

Solution Monitoring & Convergence

• FEM (Non-linear): Monitor equilibrium residual and contact status. Convergence when residual < tolerance (e.g., 1e-3) and values stabilize.

• CFD:

  • Residuals: Should drop 2-3 orders of magnitude. Do not rely solely on residuals.

  • Monitor Points: Track key integrated values (e.g., drag coefficient, outlet mass flow, point velocity). Must be steady.

  • Global Balances: Mass, momentum, energy imbalances should be < 0.1% for a converged steady-state solution.

• Common Issues & Fixes:

  • Divergence: Reduce under-relaxation factors, check initial conditions, improve mesh quality.

  • Oscillating Monitors: May indicate unsteady physics, need transient solver, or poor discretization scheme.

Result Validation & Critical Interpretation

• Validation Hierarchy:

  1. Analytical Solution: For simplified cases (e.g., cantilever tip deflection, Poiseuille flow).

  2. Experimental/Benchmark Data: Compare with published data (e.g., NACA airfoil lift/drag).

  3. Grid Independence: Ensure results are mesh-independent.

  4. Code/Model Verification: Ensure solver is correctly implementing the chosen model.

• Critical Checks:

  • FEM: Stress concentrations at sharp corners? Deformation direction correct? Reaction forces balance applied loads?

  • CFD: Separation points? Pressure recovery? Expected velocity profiles? Turbulence quantities physically plausible?

  • "Garbage In, Garbage Out": Poor mesh, wrong model, or bad BCs will give a precisely wrong answer.

Documentation & Reporting

• Lab Report Structure:

  1. Objective & Problem Statement

  2. Assumptions & Simplifications (e.g., "Steady, incompressible, turbulent flow")

  3. Methodology: Software, element types, mesh details (size, quality metrics), solver settings (URFs, schemes), models used (k-ω SST).

  4. Results: Plots (contours, vectors, graphs of monitor points), tables of key values.

  5. Discussion: Compare with expected trends/benchmarks. Explain discrepancies. State mesh convergence status.

  6. Conclusion: Summary of findings and confidence level.

• Plot Best Practices: Label axes with units, use clear legends, include geometry reference, state view direction.

[!TIP] Ultimate Exam Rule: You will be judged on your engineering judgment, not just your ability to click buttons. Always ask: "Does this result make physical sense?" and "What are the limitations of my model?"