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Table 1 Scenarios of immersed geometries leading to different intersection patterns—the geometric complexity increases from the top to the bottom of the table

From: A robust algorithm for implicit description of immersed geometries within a background mesh

Scenario Visualization Possible approximation
Planar intersection In case the immersed boundary is planar within an intersected element, the cutting plane can be determined to exactly represent this boundary
Slight curvatures In case the orientation of the immersed boundary is slightly changing, an exact representation of the boundary is not possible anymore but a cutting plane can be easily determined to minimize the approximation error
Sharp edges A sharp edge located within an element cannot be described by a single cutting plane. Best approximation by a cutting plane is through a chamfered edge
Several boundaries Since an element per definition can just have one cutting plane, the only possible approximation in case several boundaries are intersecting, is a plane that replaces all boundaries by a single one following a fitting criterion
Incomplete intersection In this case, a local approximation of the immersed boundary using four signed distance values is impossible. Hence this case must be ignored. Note, however, that the corresponding approximation error is inherent to any immersed approach and due to the limited mesh resolution. It can be well reduced by refinement strategies