Untangling High-Order Meshes Based on Signed Angles

One challenge in the generation of high-order meshes is that mesh tangling can occur as a consequence of moving the new boundary nodes to the true curved boundary. In this paper, we propose a new optimization-based method that uses signed angles to untangle invalid second- and third-order triangular meshes. Our proposed method consists of two passes. In the first pass, we loop over each high-order interior edge node and minimize an objective function based on the signed angles of the pair of triangles that share the node. In the second pass, we loop over face nodes and move them to the mean of the high-order nodes of the triangle to which the face node belongs. We present several numerical examples in two dimensions with second- and third-order elements that demonstrate the capabilities of our method for untangling invalid meshes.