# Mitigation Methods for B-Rep Tolerance Impacts on Mesh Generation and Adaptation

**Abstract:** The control of discretization error is critical to obtaining reliable simulation results. Recent progress has matured anisotropic mesh adaptation, which automates discretization error control for complex geometries. However, the meshing process can fail when geometry model Boundary Representation (BREP) tolerances are larger than the surface mesh spacing requirements. BREP tolerances are strained by three factors: the inherent complexity of the model (e.g., a broad range of scales and complex topology), the numerical difficulties arising from surface/surface intersections, and the omission of crucial data during export and translation. Manual preparation of geometry is commonly employed to enable expert-guided mesh generation, which severely inhibits workflow automation. Accommodation of loose BREP tolerances is required for automated mesh processes because barrier issues may not be detected until well into the solution process. To raise awareness of this class of geometry issues and their impact on simulation, we present examples of these barrier issues with some mitigation techniques.

# Introduction

Significant progress has been made to provide adequate linkage to geometry, which is a bottleneck identified in the CFD Vision 2030 Study. The key concept in adequate linkage to geometry is the interoperability of the geometry model with the CFD analysis process because mesh adaptation is often a consumer of geometry created in an external system, such as a modern MCAD system. Market forces combine with the desire to vertically integrate proprietary MCAD systems with proprietary Computer-Aided Engineering (CAE) analysis packages, dissuading investment in developing seamless translation capabilities.

BREP topology underpins the geometry-modeling kernels embedded in all industrial MCAD platforms. BREP is also a natural fit to describe a closed region for CFD meshing techniques. Details of the BREP can directly impact the reliability, robustness, and repeatability of rapid mesh generation processes. Discrete mesh tools must be aware of BREP tolerance because it can vary by many orders of magnitude across the model or a single topological edge within the model.

BREP repair is invariably an interactive process, making it inappropriate for automation. Ambiguity in the topology can emerge as the tolerances grow larger than the size of topological features. Educating the geometry creation pipeline to avoid the creation of these artifacts and loose BREP tolerances is an attractive alternative to repair. Figure 1 shows the components of unstructured mesh adaptation. Starting with an initial mesh, a flow solution is computed.

*Figure 1. Solution-based mesh adaptation process.*

Geometry creation workflows should also be examined for strategies to prevent the introduction of these artifacts and complement accommodation strategies. The mesh adaptation process must accept typical BREP tolerances and MCAD construction artifacts because the tolerances in the geometry definition are rarely tight enough to satisfy the smallest mesh size that the adaptive mesh may request, and these artifacts are ubiquitous.

# Illustrative Examples of the Effect of BREP Tolerances

Here are some examples that graphically illustrate the effect of BREP tolerances. They range from simple, contrived examples to models obtained from CFD prediction workshops.

**A. Simple Examples**

Figure 2 (left) shows the BREP tolerance of the union of two intentionally offset cylinders to illustrate an extreme case of a gap opening in a BREP. The gap is a combination of slip and shear displacement that vary periodically around the circumference. The BREP tolerance of the upper and lower end caps (light) is tighter than the union with the offset (dark). This example intends to show the perspective of a much smaller gap with a tight viscous spacing request.

*Figure 2. Smoothed BREP tolerance, measured in diameters, for two simple geometries.*

Figure 2 (right) shows the meridian, or seam, of a sphere. The sphere’s surface wraps around and joins along a line with oscillating BREP tolerance. The oscillation (i.e., alternating light and dark patterns along the meridian) is due to a different parameterization for the edge. The largest BREP tolerance is about 10−5 diameters. The surface is continuous on the back side of the sphere, which is indicated by the low BREP tolerance extending to the rear from the pole (i.e., the white stripe).

**B. High lift version of the Common Research Model (CRM-HL)**

Figure 3 shows the smoothed BREP tolerances of the high lift version of the Common Research Model (CRM-HL) for the upper and lower wing surfaces. The upper wing surface has hot spots of large BREP tolerance at the inboard flap end, between the main upper wing faces, and the Wing Under Slat Surface (WUSS). The lower wing surface has large BREP tolerance at the forward flap brackets and the nacelle pylon.

*Figure 3. CRM-HL smoothed BREP tolerances (inches).*

# Mitigation Techniques

The implementation of different mitigation techniques is described below, including variations of surrogate and virtual topology.

**A. Fidelity Pointwise - Quilts**

The Fidelity™ Pointwise® meshing software supports manifold and nonmanifold BREP solid models. The software uses quilts (virtual topology) to organize and control the surface mesh at the conceptual component level. The software performs several foreign entity processing functions to ensure consistency and validity of the BREP model. For example, the curves defining a closed loop for trimming a surface must be oriented properly and joined within suitable tolerance at shared vertices.

Model consistency, even in the presence of large BREP tolerances, is an absolute prerequisite to mesh automation. Automated mesh topology construction involves the agglomeration of quilt shared boundary curves into chains of curves based on turning angle. The quilt boundary is a collection of model space curves that approximate the parametric space curves used in the definition of trimmed surfaces contained in the quilt or quilts sharing the boundary. The model space curve approximation is bounded by the local BREP tolerance.

Meshing robustness issues arise when the local BREP tolerance is of the same order of magnitude (or larger) as the local mesh edge length. Fidelity Pointwise can automatically detect local conditions that require model-space meshing and preserve the robustness (validity) of the mesh through a heuristic assessment of target mesh edge length, local BREP tolerance, and other measures such as surface normal. These detection methods are also required to mesh the quilt boundary curve smoothly and robustly.

*Figure 4. BREP geometry model of the DrivAer vehicle: Colored to show quilt topology (left) and quilt component surface boundaries (right).*

The BREP tolerance can be larger than the local mesh spacing at the quilt boundary. In this case, Fidelity Pointwise will first construct a coarse isotropic “scaffold” mesh using a mixture of parametric and 3D meshing. The resolution of the scaffold mesh is adaptive, based on factors such as geometry curvature and BREP tolerance. The anisotropic mesh can then be produced using the scaffold mesh for support wherever BREP tolerance prevents robust inverse geometry evaluation (point projection). Throughout the remedial action, mesh association to the parametric geometry is maintained. The presented methodology has proven effective in meshing complex geometries with BREP tolerances near or above mesh size requests.

**B. EGADS - Effective Topology**

EGADS is the foundational object-based geometry kernel that supports geometry construction (from either a top-down or bottom-up perspective) and provides a suite of functions that fully support surface meshing. These meshing functions include topological traversal, evaluations (with 1st and 2nd derivatives), inverse evaluations, containment predicates, p-curve handling, computation of arc length, curvature, and more. Interoperability is a design feature where the application of metadata in the form of attributes can be assigned to objects (and gets tracked through construction operations). EGADS is differentiated to support sensitivity analysis in parametric settings. An API is used within EGADS to build both triangle and full quadrilateral watertight discrete representations of the BREP (EGADS tessellation objects).

**C. FEFLO.A - Surrogates**

FEFLO.A is a 2D, 3D, and surface mesh adaptation tool. It uses a combination of generalized standard operators (e.g., insertion, collapse, element swap). Generalized operators are based on recasting the standard operators in a cavity framework. The cavity operator allows a simultaneous application of multiple standard operator combinations. FEFLO.A can support virtual topology indirectly through a discrete surface surrogate.

The input mesh may have been constructed with an external tool that made BREP topology modifications internally without providing a consistent geometry model that matches the surface mesh. For instance, quilting is a common approach to reduce the number of BREP patches where the link to the initial geometry is often lost or difficult to recover from the generated mesh. For these situations, FEFLO.A generates a high-order (cubic) discrete surface grid as a surrogate geometry model. When geometry is provided, FEFLO.A relies on EGADS and EGADSlite to perform standard BREP forward evaluations, inverse evaluations, and topology queries.

**D. refine - BREP Closures**

refine accesses BREP model topology and evaluates geometry via EGADS. EGADSlite, a lightweight ANSI-C version of the EGADS functions required by the initial and adaptive mesh process, supports high-performance computing applications. While work is underway to refine EGADS Effective Topology, this description focuses on EGADS and EGADSlite without Effective Topology. Two strategies, Displacement Map and Surrogate are used for reducing the BREP tolerances exposed to the adaptive mesh mechanics in refine. The former computes a smooth displacement field defined for edges and faces, which reduces the distance between face p-curves and edge curves (same for edge curve endpoints and points). The latter replaces BREP surfaces and curves with a discrete surrogate, which can be constructed as watertight to machine precision.

# Conclusion

Unstructured mesh adaptation aims to automate the labor-intensive expert-guided mesh generation process. To attain this automation goal, the adaptive mesh modification mechanics must accommodate typical BREP tolerances unless upstream geometry creation processes can be improved. Poorly shaped elements or failure of the meshing process can occur when solution-adaptive error estimates create mesh sizing requests at or below local BREP tolerances. Applying mesh adaptation to “CFD-ready” geometry requires accommodation in most of these applications.

The mitigation and accommodation techniques described above significantly advance automation and the reduction of manual intervention. However, a completely automated solution to meshing these classes of MCAD input geometries is not universally available.

**Reference**

- Park, Michael A., Haimes, R., Wyman, Nicholas J., Baker, Patrick A., and Loseille, A., “Boundary Representation Tolerance Impacts on Mesh Generation and Adaptation,” AIAA paper no. 2021-2992, August 2021.

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