# Discrete Surface Evolution and Mesh Deformation for Aircraft Icing Applications

Abstract: Robust, automated mesh generation for problems with deforming geometries, such as ice accreting on aerodynamic surfaces, remains a challenging problem. Here we describe a technique to deform a discrete surface as it evolves due to the accretion of ice. The surface evolution algorithm is based on a smoothed, face-offsetting approach. We also describe a fast algebraic technique to propagate the computed surface deformations into the surrounding volume mesh while maintaining geometric mesh quality. Preliminary results presented here demonstrate the ecacy of the approach for a sphere with a prescribed accretion rate, a rime ice accretion, and a more complex glaze ice accretion.

## Summary (4 min read)

### NASA Glenn Research Center, Cleveland, OH, 44315, USA

- Robust, automated mesh generation for problems with deforming geometries, such as ice accreting on aerodynamic surfaces, remains a challenging problem.
- There are several technical challenges associated with mesh generation for simulating ice accretion on a three-dimensional configuration.
- Any mesh generation strategy will necessarily require the ability to handle such complex geometries if it is to represent a viable, long-term solution.
- ¶Assistant Research Professor, Center for Advanced Vehicular Systems, PO Box 5405, Member.
- The authors are developing a solver-neutral interface propagation tool that computes the position of a discrete surface as it evolves under an accretion rate map specified by LEWICE3D.

### II. Background: Surface Evolution

- One of the challenges associated with evolving a faceted, discrete surface is that the normal at a node is not unique.
- One approach that has shown promise for evolving a surface mesh while conserving volume is the method developed by Jiao.2,3 Jiao employs a singular value decomposition (SVD) to solve a least square problem and then applies an eigenvalue/eigenvector analysis at each node to resolve its normal motion, which generates the surface geometry, and its tangential motion, which maintains mesh quality.
- (1) Here, each row of the system corresponds to one of the m faces that are incident on the node and elements of a are the offset distances for each incident face.
- The solution to Equation 2 represents an advective motion in which the resulting surface is the intersection of the propagated face planes.
- A simple solution is to assume the direction of the displacement will not change and adjust the displacement to satisfy the required offset.

### III. Approach

- Generating an ice shape for a specified accretion time tice is accomplished by performing a series of quasi-static, loosely-coupled, ice accretion/flow simulation steps.
- The authors term each of these quasi-static steps an “ice accretion step” with an associated time interval ∆t.
- As noted below, each time interval ∆t may be further subdivided into subintervals ∆ts.
- For each ice accretion step: (1) a CFD simulation is performed to compute the flow field about the current ice shape, (2) a LEWICE3D computation is performed to determine the new ice shape, or alternatively, the accretion rate map, and (3) the surface and volume meshes are evolved based on the ice accretion rate using iceSurf and gridMover, respectively.
- Deform the volume mesh by projecting the surface deformations into the volume mesh using gridMover 3 of 20 American Institute of Aeronautics and Astronautics Each of these processes is described in the sections that follow.

### III.A. Generate Accretion Rate Map using Lofting

- Currently, LEWICE3D does not provide an accretion rate map.
- Two different strategies were employed to circumvent this shortcoming.
- The icing rate for each surface element is calculated assuming that no evaporation or runback occurred and is given by dvice dt =.
- A two-dimensional coordinate system (S,T ) is employed for the loftings for which S is the axial coordinate and T is either the spanwise coordinate for wing-type lofts or the circumferential angle for body of revolution-type lofts .
- The local coordinates systems generated for the surface point and icing cut points are used to transform the local surface points into the local ice cut point coordinate system for the interpolation of the ice thickness .

### III.B. Surface Mesh Evolution

- The authors have developed a computational tool (iceSurf ) that employs elements of Jiao’s algorithm2,3 to evolve a discrete surface mesh in response to the ice accretion rate.
- Like Jiao’s algorithm, the algorithm used in iceSurf emphasizes conservation of the accreted volume.
- IceSurf uses the offset direction in the primary space defined by Jiao’s method as the initial nodal displacement direction and then employs global and local smoothing algorithms to maintain mesh quality.
- B.1. Identify Geometrical Features iceSurf provides special treatment for the nodes associated with these features.
- If the angle between the two faces that share a given edge is greater than a threshold, then this edge is considered to be a geometrical edge.

### III.B.2. Define Nodal Offset Direction

- The next step in the process is to generate an initial nodal offset direction.
- This temporarily circumvents the need for knowledge of the face displacement.
- An additional change is that the weight matrix W in Jiao’s algorithm (see Equation 3) is based on the face areas of the triangles incident on the node under consideration while their weight matrix is based on the included angles of the faces incident to the node.
- Once the primary direction is defined, it is held fixed throughout the remainder of the process.

### III.B.4. Smooth the Height Field

- In general, the heights for two triangular faces that share an edge will not be equal.
- A height field smoothing that conserves volume is then employed to redistribute the volume.
- Assume that the authors have two triangles, T1 and T2, that share an edge, with heights h1 and h2, respectively, and h1 > h2.
- The resulting accreted volume is then compared to the value obtained by multiplying the accretion rate by the time increment ∆ts.
- The resulting volume residue is converted to a rate and then added or subtracted during the next subinterval step as appropriate.

### III.B.5. Compute Nodal Positions

- The next step is to determine the nodal positions using the smoothed height field.
- In contrast, for wavefront motion, the node should reside on a smooth nonlinear patch.
- Of course, there are multiple faces associated with this node.
- The final nodal displacement is given by a weighted average of the displacement from the faces incident on the node.
- The weighting is based on the included angles of the faces at that node.

### III.B.6. Smooth the Evolved Surface Mesh

- The nodal positions are then smoothed using the null space smoothing described by Jiao.
- Null space smoothing moves nodes in the tangent plane or in the direction of minimum curvature of the surface and therefore, tends to preserve the volume better than other forms of smoothing, such as Laplacian.
- Their approach takes boundary displacements as input and returns a deformed volume mesh.
- The local rotation for a given node is computed by using a least squares fitting to determine the rotation about the node that best matches the displacements of all edges and normals from surface facets that reference the given node.
- This accelerated IDW approach has been shown to be competitive with the considerably more expensive radial basis function (RBF) proposed by deBoer et al.7.

### IV. Results

- The authors now present results that demonstrate the efficacy of the mesh evolution and deformation algorithms.
- The authors then employ the algorithms in a loosely coupled approach utlizing Loci/Chem and LEWICE3D for a rime icing condition and a more challenging glaze icing condition.

### IV.A. Test Cases: Sphere with Prescribed Accretion Map

- A face volume accretion rate was then computed by multiplying the face area by the accretion velocity.
- This was performed by iceSurf using six substeps to produce the surface shown in figure 8.
- The resulting accretion closely matched the expected analytical results.
- The surface evolution algorithm employed in iceSurf produced a valid surface mesh even for a discontinuous rate map and no smoothing.

### IV.B. Rectangular Planform Wing with GLC305 Cross Section

- Loosely-coupled Loci/CHEM-LEWICE3D simulations of ice accretion on a rectangular planform wing with a constant GLC305 airfoil section were performed and the results compared with available LEWICE2D and LEWICE3D simulation results and experimental data.
- A far field boundary condition was also applied at the top, bottom, and outboard 9 of 20 American Institute of Aeronautics and Astronautics side boundaries of the computational domain, all of which were located approximately five chords from the wing.
- A droplet tracking window was specified near the airfoil covering the whole span.
- The effects of employing substeps and height smoothing in iceSurf will now be discussed.
- In the comparisons that follow, the authors note that only single-step ice accretions can be generated by 10 of 20 American Institute of Aeronautics and Astronautics LEWICE3D.

### IV.B.1. Results for Rime Ice Conditions

- The icing conditions considered for this case were a liquid water content (LWC) of 0.405 gm/m3, an ambient temperature and pressure of 257.88 K and 1 atm, respectively, and a relative humidity of 100 percent.
- At this temperature, the water droplets freeze on impact without runback, so the simplified method for estimating the icing rate based on collection efficiency described in Section III.
- In both cases, a single accretion step was used with no smoothing.
- These figures also show a comparison of ice shapes generated by iceSurf and LEWICE3D for two different icing times.
- The authors are currently investigating this discrepancy.

### IV.B.2. Results for Glaze Icing Conditions

- The icing conditions for this case corresponded to Case 072604 in the LEWICE2D validation report.9.
- A comparison of ice shapes generated by iceSurf, LEWICE3D, and LEWICE2D and experimental data is illustrated in figure 13.
- The authors now discuss the ripples that appear in the surface as shown in figure 14.
- Starting from the single-step, 6-min ice accretion, the authors intended to advance the ice shape for a total icing time of 22.5 min using multiple 2-min intervals, which would correspond to Case 072605 in the LEWICE2D validation report.
- These images show that, as the surface mesh evolved, the volume mesh was deformed in a manner that maintains mesh quality up to the point that the calculation failed due to the self-intersection of the surface mesh.

### V. Conclusion

- The authors describe a meshing strategy designed for simulating the accretion of ice on aerodynamic surfaces.
- Additionally, the authors employ a fast algebraic approach to project the surface deformation into the volume mesh.
- Results obtained for the rime ice case, for which the accretion rate map is based on collection efficiency, show similar trends to results predicted using LEWICE3D ; however, discrepancies exist that can be attributed to the manner in which the accretion rate is computed.
- Results for the glaze ice case demonstrate that their approach can handle more complex shapes.
- The authors are concurrently developing tools that employ local quality improvement operations, such as edge swaps, etc., and local mesh regeneration to maintain geometric mesh quality as the accreted surface and volume mesh evolve.

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