Application of Metric-Based Mesh Adaptation to Hypersonic Aerothermal Simulations Using US3D

This paper demonstrates that metric-based mesh adaptation using the temperature Hessian enables accurate hypersonic aerothermal simulations of complex geometries, such as atmospheric entry capsules with RCS jets, while achieving surface heating predictions comparable to conventional block-structured methods.

The Gist

Imagine you are trying to predict how hot a spaceship gets as it screams through the atmosphere at hypersonic speeds (over 17,000 mph). The air in front of the ship gets so hot it turns into a plasma, and the heat can melt the vehicle if you don't calculate it perfectly.

For decades, engineers have used a specific type of digital "net" (a mesh) to simulate this. Think of this net as a grid of tiny boxes. To get accurate heat predictions, these boxes had to be perfectly aligned with the shockwave (the invisible wall of compressed air) in front of the ship.

The Problem:
Imagine trying to wrap a gift box with a grid of square wrapping paper. It's easy if the box is a perfect cube. But what if the gift is a weirdly shaped alien artifact with tiny nozzles, bumps, and curves? Trying to force a perfect grid of squares around that shape is a nightmare. You either end up with a messy, inaccurate net, or you have to spend months manually cutting and fitting the paper.

Furthermore, standard computer grids often use "tetrahedra" (pyramid-shaped blocks). While flexible, these are notoriously bad at predicting heat on the front of the ship because they don't line up well with the shockwave, leading to "smudged" and inaccurate results.

The Solution: The "Smart Rubber Sheet"
This paper introduces a new tool called MIMIC (Metric-Informed Mesh Improvement Capability) used with a supercomputer program called US3D.

Instead of manually building a perfect grid, the researchers use a "smart rubber sheet" approach. Here is the analogy:

1. The Initial Sketch: They start with a rough, low-resolution net around the spaceship. It's like a low-poly video game character—blocky and simple.
2. The "Heat" Sensor: The computer runs a quick simulation to see where the temperature is changing rapidly. Think of this as a sensor that feels the "heat map."
3. The Stretching: The MIMIC tool looks at this heat map and says, "Hey, the air is getting super hot right here at the shockwave, and swirling weirdly in the wake behind the ship." It then automatically stretches and reshapes the net.
   • Where the heat is intense, it pulls the mesh nodes closer together, creating a super-dense, high-definition net.
   • Where the air is calm, it lets the net stretch out, saving computer power.
   • Crucially, it stretches the net to align perfectly with the shockwave, just like a tailor stretching fabric to fit a specific curve.

The Two Big Experiments

The paper tests this "smart net" on two scenarios:

1. The Simple Test (The Hemisphere):
They tested a simple half-sphere shape. They compared two types of "boundary layers" (the innermost layer of the net touching the ship):

• Prisms: Like stacking triangular slices of bread.
• Hexahedra: Like stacking square bricks.
• The Result: They found that using the "square bricks" (hexahedra) combined with the smart stretching resulted in a much smoother, more accurate heat map. It proved that you don't need a perfect, hand-crafted grid; you just need the right shape of the inner layer and the ability to stretch the outer layers intelligently.

2. The Real-World Test (The Mars Capsule):
This was the big one. They simulated a capsule entering the Martian atmosphere. This capsule is complex:

• It has a blunt nose.
• It has a curved back.
• The kicker: It has 8 tiny rocket thrusters (RCS jets) on the back.

In the old days, simulating those 8 tiny jets would be a nightmare. You'd have to manually carve out the geometry in the grid, which is incredibly difficult and prone to errors.

• The Old Way (DPLR): Used a rigid, blocky grid. To make it work, they had to ignore the tiny jets and pretend the back of the ship was smooth.
• The New Way (US3D + MIMIC): The "smart net" automatically wrapped itself around the tiny jets, the curves, and the complex backshell. It didn't matter how weird the shape was; the net just stretched to fit.

The Outcome
The results were amazing. The new method predicted the heat on the front of the ship just as accurately as the old, labor-intensive method. But on the back of the ship, where the flow is chaotic and the jets are, the new method captured details the old method couldn't even see.

Why This Matters
Think of it like upgrading from a hand-drawn map to Google Maps.

• Old Method: You had to draw the roads yourself. If a new building went up, you had to redraw the whole map. It was slow, rigid, and missed small details.
• New Method: The map updates itself. If there's a traffic jam (a shockwave) or a new construction site (a rocket jet), the map automatically zooms in and redraws the streets to show you exactly what's happening.

In Summary
This paper shows that we can now simulate incredibly complex, real-world space missions with less manual effort and higher accuracy. By letting the computer "stretch" the mesh based on the physics of the flow, engineers can finally simulate the messy, complex parts of a spacecraft (like the back and the thrusters) without spending months trying to force a rigid grid to fit. It's a giant leap toward safer, more efficient space travel.

Technical Summary

1. Problem Statement

Hypersonic flow simulations for atmospheric entry vehicles face significant challenges in accurately predicting surface heating (heat flux), particularly on complex geometries.

• Current Limitations: State-of-the-art codes like DPLR and US3D typically rely on block-structured hexahedral meshes aligned with the bow shock to ensure accurate heat flux predictions. While effective for simple geometries, generating these meshes for complex vehicles (e.g., those with Reaction Control System (RCS) jets, backshell steps, or complex aft-body features) is labor-intensive and often requires geometric simplification.
• Unstructured Mesh Issues: Unstructured meshes (typically tetrahedral) are flexible but historically yield poor surface heating predictions in hypersonic regimes. This is due to the bow shock introducing numerical entropy that convects downstream, polluting the surface heat flux. Furthermore, standard unstructured meshes often fail to resolve complex wake dynamics and shock-wave/boundary-layer interactions.
• The Gap: There is a need for a workflow that combines the geometric flexibility of unstructured meshes with the accuracy of shock-aligned structured meshes, specifically for real-gas hypersonic flows involving complex geometries.

2. Methodology

The paper presents a workflow integrating the Metric-Informed Mesh Improvement Capability (MIMIC) with the unstructured flow solver US3D.

• Solver (US3D): A cell-centered finite volume code capable of simulating multi-species reacting flows (real gas effects) and turbulence (RANS/IDDES). The simulations in this paper utilize a modified Steger-Warming flux vector splitting approach.
• Mesh Adaptation (MIMIC):
  • Metric Tensor: MIMIC computes a metric tensor field based on the Hessian (second-order gradients) of the temperature solution. This acts as a local error indicator to dictate mesh size and directionality.
  • Hybrid Mesh Strategy: The workflow supports a fixed boundary layer mesh (composed of prisms, hexahedra, or pyramids) near the wall, while the volume is filled with anisotropic tetrahedra that are adapted.
  • Adaptation Engine: The open-source library ParMMG performs parallel anisotropic Delaunay mesh adaptation on the tetrahedral volume based on the metric tensor.
  • Iterative Process: The process involves running a flow solution, computing the metric, adapting the mesh, and re-running the simulation. This is repeated for $N$ iterations (typically 5 in this study).
• Key Enhancements:
  • Extension of MIMIC to support quadrilateral surface elements extruded into hexahedral boundary layers (previously limited to triangles/prisms).
  • Handling of real-gas mixtures (e.g., $CO_2$-$N_2$ for Martian atmosphere) and vibration-electronic energy relaxation.
  • Stabilization Strategy: To avoid "carbuncle" effects (numerical instabilities at shocks) during initial adaptation, the authors initially used first-order inviscid fluxes. Once the mesh was sufficiently refined near the shock, they switched to second-order fluxes.

3. Key Contributions

1. Hexahedral Boundary Layer Capability: Demonstrated that MIMIC can successfully handle hybrid meshes where the boundary layer consists of hexahedra (extruded from quadrilateral surfaces) rather than just prisms (extruded from triangles).
2. Real-Gas Hypersonic Adaptation: Extended metric-based adaptation to hypersonic regimes with real-gas chemistry, a domain where such methods were previously underutilized due to accuracy concerns.
3. Complex Geometry Handling: Successfully simulated a 70° sphere-cone entry capsule with eight RCS jets on the backshell, a geometry too complex for traditional block-structured meshing without significant simplification.
4. Workflow Validation: Established a robust workflow for automating mesh generation and refinement for aerothermal analysis, reducing the manual effort required for high-fidelity simulations.

4. Results

Case A: Supersonic Flow Past a Hemisphere (Perfect Gas)

• Objective: Compare prismatic vs. hexahedral boundary layers and assess mesh sensitivity.
• Findings:
  • Adaptation Efficacy: Mesh adaptation significantly improved shock definition and reduced noise in surface heat flux contours compared to the initial coarse mesh.
  • Prisms vs. Hexahedra: Hexahedral boundary layers produced smoother, more symmetric heat flux distributions compared to prismatic layers.
  • Layer Thickness Sensitivity: Results were highly sensitive to the thickness of the boundary layer mesh. A mesh with 110 layers of hexahedra ($T^5_{h,110}$) combined with adaptation produced heat flux predictions within $\pm 7.5\%$ of DPLR reference data, matching the accuracy of conventional block-structured methods.
  • Pressure: Pressure profiles were accurately recovered across all topologies after adaptation.

Case B: Hypersonic Flow Past a 70° Sphere-Cone Capsule (Real Gas)

• Conditions: Martian atmosphere ($CO_2$-$N_2$ mixture), $M_\infty = 21.3$, peak heating conditions, with RCS jets included.
• Mesh Evolution:
  • Initial State ($T_0$): Coarse mesh resulted in diffusive solutions, noisy mass fraction contours, and weak carbuncle effects.
  • Adapted State ($T_5$): After five iterations, the mesh aligned perfectly with the bow shock and shear layers. The tetrahedral volume refined anisotropically in the wake and shock regions.
  • Wake Dynamics: The adapted mesh captured complex unsteady interactions, such as the breakup of shear layers and recirculation bubbles behind the backshell step, which are difficult to resolve with single-block structured meshes.
• Surface Heat Flux:
  • Initial simulations showed large streaks in heat flux due to numerical errors.
  • By the fifth iteration, streaks vanished, and the heat flux distribution became nearly symmetric.
  • Comparison with DPLR: The US3D/MIMIC results (using a full domain with RCS jets) showed qualitative agreement with DPLR (using a half-domain with a smooth OML). The US3D solution predicted a slightly shorter shock stand-off distance near the upper shoulder, attributed to higher local mesh resolution.
• RCS Integration: The unstructured approach successfully incorporated the detailed geometry of the RCS thrusters, whereas the DPLR reference required smoothing these features.

5. Significance and Future Work

• Paradigm Shift: This work demonstrates that metric-based mesh adaptation can bridge the gap between the geometric flexibility of unstructured meshes and the accuracy requirements of hypersonic aerothermal simulations. It allows engineers to simulate highly complex geometries (like full vehicles with RCS) without the prohibitive cost of manual block-structured mesh generation.
• Accuracy: With sufficient adaptation and the use of hexahedral boundary layers, unstructured solvers can achieve accuracy comparable to state-of-the-art block-structured codes (DPLR).
• Future Directions:
  • Activating active RCS jet inflows to study plume-wake interactions.
  • Integrating MIMIC with HyperSolve, a hyperbolic Navier-Stokes solver designed to perform better on anisotropic unstructured meshes for heat flux prediction.
  • Further refinement of numerical schemes to minimize entropy production at shocks on unstructured grids.

In conclusion, the paper validates that automated, metric-based mesh adaptation is a viable and powerful tool for next-generation hypersonic aerothermal analysis, enabling high-fidelity simulations of complex entry vehicles that were previously out of reach for unstructured solvers.