FEM/CFD Lab (ME-505) - Unit 5 Short Notes

ME-505 FEM/CFD Lab - UNIT 5: Advanced Simulation & Analysis Workflow

5.1 Introduction to Advanced/Integrated Analysis

• 5.1.1 When to Use Coupled/Multi-Physics: Required when physical phenomena interact and cannot be solved independently.

  • Examples: Fluid-Structure Interaction (FSI - pressure deforms structure, deformation changes flow), Thermo-Mechanical (heat causes expansion/stress), Electro-Thermal (Joule heating).

• 5.1.2 Solution Approaches:

  • Sequential/One-Way: Solve physics A, pass results (e.g., pressure, temperature) as load to physics B. Assumes weak coupling.

  • Fully-Coupled: Solve all governing equations simultaneously in a monolithic system. More accurate for strong interaction, computationally expensive.

• 5.1.3 Software Capabilities:

  • ANSYS Workbench: System coupling for FSI, conjugate heat transfer.

  • COMSOL Multiphysics: Native multiphysics interfaces, fully-coupled solvers.

  • OpenFOAM + preCICE: Open-source coupling library for partitioned solvers.

[!TIP] Exam often asks: "Differentiate between sequential and fully-coupled approaches. Give one example where sequential might fail."

Answer: Sequential solves one physics at a time. It fails in strongly coupled problems (e.g., vortex-induced vibration of a very flexible structure) where the feedback loop is critical and requires iteration within each time step.

5.2 Advanced Meshing Strategies for Complex Geometries

• 5.2.1 Mesh Quality Metrics (Critical for Accuracy & Convergence):

  | Metric | Definition | Ideal Value | Problem if Poor | | :--- | :--- | :--- | :--- | | Skewness | Deviation from equi-angular shape | ~0 (0-0.85 acceptable) | Solution accuracy degrades, convergence slows. | | Aspect Ratio | Ratio of longest to shortest edge/side | ~1 ( <5-10 for CFD) | Numerical diffusion, false diffusion in flow. | | Orthogonal Quality | Measure of cell alignment with flow direction | ~1 ( >0.1 acceptable) | Poor flux calculation, high discretization error. | | Jacobian Ratio | For curved cells, measure of mapping distortion | ~1 | Negative Jacobian causes solver failure. |

• 5.2.2 Mesh Independence Study (MIS) Methodology:

  1. Perform simulations with systematically refined meshes (e.g., coarse, medium, fine).

  2. Extract key quantitative results (e.g., max stress, drag coefficient, pressure drop).

  3. Plot result vs. a mesh size metric (e.g., number of elements, element volume).

  4. Conclusion: Solution is independent when further refinement changes result by less than a chosen tolerance (e.g., <1-2%).

  Key Formula: % Change = $$\displaystyle \frac{|Result_{fine} - Result_{medium}|}{Result_{fine}} \times 100\% $$

• 5.2.3 Adaptive Meshing:

  • Error-Based: Solver estimates discretization error and refines mesh where error is high (common in CFD, COMSOL).

  • Solution-Based: Refine based on solution gradients (e.g., high velocity gradient, high temperature gradient).

• 5.2.4 Meshing for Moving/Deforming Domains:

  • CFD (FSI): Use dynamic meshing (smoothing, remeshing, layering). Key parameter: Courant Number $$\displaystyle Co = \frac{u \Delta t}{\Delta x} $$ must be controlled (<1-5) for stability.

  • FEM (Contact): Use contact element meshing with fine, matched meshes at expected contact surfaces to avoid penetration.

5.3 Solver Settings, Convergence, and Stability

• 5.3.1 FEM - Nonlinear & Dynamic:

  • Newton-Raphson: Iterative method. Convergence criteria: Residual force norm $$\displaystyle < \epsilon $$ (e.g., $$\displaystyle 10^{-3} $$) and displacement correction norm $$\displaystyle < \delta $$.

  • Arc-Length Method: For snap-through or post-buckling problems (load-controlled fails).

  • Time Stepping: For dynamic explicit (central difference) vs. implicit (Newmark-$\beta$). Stability limit for explicit: $$\displaystyle \Delta t < \frac{2}{\omega_{max}} $$ (where $$\displaystyle \omega_{max} $$ is highest natural frequency).

• 5.3.2 CFD - Pressure-Velocity Coupling:

  • SIMPLE (Semi-Implicit Method for Pressure-Linked Equations): Standard for steady-state. Uses under-relaxation factors (URF) for pressure & velocity (e.g., 0.3-0.7).

  • PISO (Pressure Implicit with Splitting of Operators): For transient, performs multiple pressure corrections per time step.

  • Discretization Schemes: First-Order Upwind (stable, diffusive) vs. Second-Order (accurate, may need URF reduction).

• 5.3.3 Diagnosing Convergence Issues:

  | Symptom | Likely Cause | Action | | :--- | :--- | :--- | | Divergence | Large time step, poor mesh, high URF | Reduce $\Delta t$, improve mesh, lower URF (0.1-0.3). | | Stagnation | Under-relaxation too low, poor initial guess | Increase URF gradually, provide better guess. | | Oscillations | Central differencing without URF, mesh too coarse | Switch to upwind, refine mesh. |

• 5.3.4 Solver Choice:

  • Direct (e.g., MUMPS, PARDISO): Robust for small-medium problems, no convergence tolerance.

  • Iterative (e.g., CG, GMRES): Memory efficient for large problems, requires convergence tolerance & preconditioner.

  • Implicit vs. Explicit: Implicit (unconditionally stable) for static/slow dynamics. Explicit (conditionally stable) for high-speed, short-duration events (crash, explosion).

5.4 Boundary Conditions, Loads, and Material Models for Realism

• 5.4.1 Advanced BCs:

  • CFD Inlet: Specify turbulence intensity (I) and hydraulic diameter ($$\displaystyle D_H $$) for k-ε models. $$\displaystyle I = \frac{u'}{U_{avg}} \approx 0.16(Re)^{-1/8} $$ for fully-developed pipe flow.

  • Symmetry vs. Periodic: Symmetry (mirror plane), Periodic (repeating pattern - requires same flow at inlet/outlet).

  • FEM Radiation: Use Stefan-Boltzmann law for surface-to-ambient: $$\displaystyle q = \epsilon \sigma (T^4 - T_{\infty}^4) $$.

• 5.4.2 Realistic Material Models:

  • Nonlinear Stress-Strain: Bilinear Isotropic Hardening (plasticity): Yield stress $$\displaystyle \sigma_y $$, tangent modulus $$\displaystyle E_t $$.

  • Hyperelasticity (Rubbers): Mooney-Rivlin or Ogden models: $$\displaystyle W = \sum_{i=1}^{N} C_i(\bar{I}_1 - 3)^i + \sum_{k=1}^{M} D_k (J-1)^{2k} $$ (strain energy density).

  • Non-Newtonian Fluids: Power-Law: $$\displaystyle \mu = K \dot{\gamma}^{n-1} $$. Carreau: $$\displaystyle \mu = \mu_{\infty} + (\mu_0 - \mu_{\infty})[1 + (\lambda \dot{\gamma})^2]^{(n-1)/2} $$.

• 5.4.3 Load Application: Apply distributed pressure on surfaces, body forces (gravity, centrifugal) on volumes. Avoid point loads in continuum models (stress singularity).

5.5 Post-Processing, Result Validation, and Critical Interpretation

• 5.5.1 Advanced Visualization:

  • Pathlines/Streamlines: Show instantaneous flow paths.

  • Particle Tracing: Track massless/particle-laden flows with time.

  • Deformation Animation: Essential for FSI, large deflection.

• 5.5.2 Quantitative Extraction:

  • Point Probes: Monitor time-series at a location.

  • Surface/Volume Integrals: Calculate total force, heat flux, volume-averaged velocity.

  • Report Generation: Automate table creation for parametric studies.

• 5.5.3 Validation Fundamentals:

  • Compare with: Analytical solutions (simplified cases), experimental data (benchmark tests), published benchmarks (e.g., NACA airfoil drag, lid-driven cavity flow).

  • Key Principle: Validation is for the entire model (geometry, mesh, physics, BCs), not just the solver.

• 5.5.4 Error & Uncertainty Analysis:

  • Sources: Modeling error (simplifications), Discretization error (mesh), Iterative/convergence error (solver tolerance).

  • Sensitivity: Vary key input parameters (e.g., inlet velocity, material property) to see effect on output. Identify dominant parameters.

• 5.5.5 Critical Assessment - "Sanity Checks":

  • Conservation: Check global mass/energy balance (CFD: $$\displaystyle \dot{m}_{in} \approx \dot{m}_{out} $$).

  • Physics: Do results follow expected trends? (e.g., pressure drops along flow direction, stress concentration at geometric discontinuities).

  • Boundary Layers: In CFD, check velocity/temperature profiles at walls.

[!TIP] Common Pitfall: Accepting a "converged" solution without checking physical plausibility. A solver can converge to a non-physical steady-state (e.g., due to wrong BCs or turbulence model). Always visualize flow fields and check balances.

5.6 Typical Integrated/Coupled Case Studies (Lab Exercise Framework)

• 5.6.1 Fluid-Structure Interaction (FSI):

  • Problem: Flow over a flexible cantilever (flag, heart valve).

  • Workflow:

    1. CFD: Solve steady/unsteady flow, extract pressure ($p$) and wall shear stress ($$\displaystyle \tau_w $$) on structure.

    2. FEM: Map $p$, $$\displaystyle \tau_w $$ as pressure loads on structural model. Solve for displacement ($u$) and stress.

    3. Mesh Update: Use displacement ($u$) to deform CFD mesh (dynamic meshing).

    4. Iterate: Repeat for next time step until coupled solution stabilizes.

  • Key Challenge: Time step synchronization and data mapping between non-matching meshes.

• 5.6.2 Thermo-Mechanical Analysis:

  • Problem: Hot fluid flows over a solid component, causing thermal expansion and stress.

  • Workflow:

    1. CFD: Solve conjugate heat transfer or extract convection coefficient ($h$) and bulk temperature ($$\displaystyle T_{\infty} $$).

    2. FEM: Apply $h$ and $$\displaystyle T_{\infty} $$ as convective boundary condition on solid. Solve heat transfer to get temperature field $T(x,y,z)$.

    3. FEM (Structural): Use $T(x,y,z)$ as thermal load to solve for thermal stress/strain ($$\displaystyle \sigma_{thermal} = E \alpha \Delta T $$).

• 5.6.3 Conjugate Heat Transfer (CHT):

  • Problem: Heat exchanger with solid walls separating hot/cold fluids.

  • Workflow: Single simulation domain with multiple regions (Fluid A, Solid, Fluid B). Solves Navier-Stokes + Energy in fluids, Fourier conduction in solid. Continuity of temperature & heat flux at interfaces. (e.g., ANSYS Fluent CHT, COMSOL Heat Transfer Module).

5.7 Reporting and Documentation of Simulation Projects

• 5.7.1 Essential Report Components:

  1. Objective & Scope

  2. Geometry & Assumptions (2D/3D, symmetry, neglected physics)

  3. Material Properties (with sources/justification)

  4. Mesh Details (type, quality metrics, size, independence study results)

  5. Solver Settings (governing equations, schemes, URF, convergence criteria)

  6. Boundary & Initial Conditions (clearly labeled diagram)

  7. Results & Discussion (visuals + quantitative data)

  8. Validation (comparison with what? agreement level?)

  9. Conclusions & Limitations

  10. Recommendations for Improvement

• 5.7.2 Documentation Best Practices:

  • Version Control: Note software version, solver version.

  • Reproducibility: Provide full setup details so another user can replicate.

  • Traceability: Link specific results to specific model setups (e.g., "Figure 5 shows stress for Case B (hyperelastic model)").

• 5.7.3 Presenting Results:

  • Use contour plots for fields, vector plots for direction, line plots for profiles.

  • Tables for parametric results (e.g., drag vs. Reynolds number).

  • Always label axes with units.

• 5.7.4 Discussing Limitations:

  • Example: "2D simulation neglects tip vortices," "Linear elastic model fails beyond yield," "Turbulence model (k-ε) may not capture separation accurately." Suggest future work (e.g., 3D simulation, different material model, LES).

[!TIP] Exam Question Pattern: "You are asked to report on a CFD simulation of a pipe bend. List 5 essential items you MUST include in your report to make it credible."

Answer: 1) Mesh independence study plot/table. 2) Boundary condition details (inlet velocity profile, turbulence parameters). 3) Residual history plot showing convergence. 4) Validation (e.g., comparison with pressure drop correlation like $$\displaystyle \Delta P = f \frac{L}{D} \frac{\rho U^2}{2} $$). 5) Mass balance check ($$\displaystyle \dot{m}_{in} - \dot{m}_{out} $$).