## 1 Introduction  
Delaunay mesh generation of non-smooth domains such as piecewise smooth surfaces and complexes is a difficult challenge. Aided by recent developments in sampling theory and computational topology, [PERSON]'s furthest point strategy [1; 2] (Delaunay refinement) has been applied to generate Delaunay meshes for smooth surfaces with provable guarantees [3; 4]. The lack of global smoothness poses two main difficulties in extending these methods to non-smooth domains. First, the sampling theory developed for smooth surfaces breaks down for non-smooth surfaces. Secondly, small input angles possibly present at non-smooth regions pose problems for the termination of Delaunay refinement [5]. [PERSON] and [PERSON] [6] gave a provable algorithm for a class of non-smooth surfaces called Lipschitz-surfaces. They showed that their algorithm for smooth surface meshing [3] extends to this class only if input angles are sufficiently large. Unfortunately, this approach failed to admit small input angles which limited the input class. [PERSON] and [PERSON] [7] implemented an algorithm for meshing volumes bounded by piecewise smooth surfaces; their approach also suffers from an angle constraint. Recently [PERSON], [PERSON] and [PERSON] [8] proposed an approach that completely removed any constraint on input angles. As a result their algorithm could accommodate large class of input domains called piecewise smooth complexes (PSCs). This class includes polyhedral domains, smooth and piecewise smooth surfaces, and even non-manifolds. The algorithm of [8] uses the idea of protecting non-smooth curves and vertices in the input complex with balls. This idea already gave good results in the polyhedral case [9; 10; 11]. A novelty introduced by [PERSON] _et al._ is that they turn the balls into weighted points and then carry out the mesh refinement in the weighted Delaunay triangulation [12; 13]. Notwithstanding its theoretical success, practical relevance of this algorithm remains questionable. It employs some expensive numerical computations that are hard to implement. Unless these computations are removed, one cannot expect a practical solution for Delaunay mesh generation of PSCs. The goal of this paper is to design a refinement strategy without expensive predicates so that it becomes implementation-friendly and practical. **Bottleneck. ** Consider a smooth surface given by an implicit equation. To generate a meaningful mesh for this surface, one needs to sample it at a scale that captures its smallest local variations. One approach would be to _guess_ a scale and sample the surface with it. If the guess is right, sampling with the furthest point strategy gives a mesh with provable guarantees as shown by [PERSON] and [PERSON] [3]. A second approach would be to compute a version of the local length scales and mesh the surface with those scales. The algorithm of [PERSON] _et al._[8] works on this latter principle. It computes how a curve or a surface varies normal-wise around a point. It also computes separation distances between different elements (vertices, curves, and surfaces) to capture separation feature size (gap size) in the sense of [9; 14]. These assumed powerful numerical primitives allow [PERSON] _et al._ to determine the size of the protecting balls with certain desirable properties and allow them to sample surface patches at appropriate length scales. Unfortunately, these computations are expensive and are hard to implement which makes the algorithm impractical. **Solution. ** To circumvent the problem we follow the guessing approach. We would like to guarantee that even if the guess is incorrect, the algorithm terminates and outputs a mesh which approximates the input complex at a coarse level. A main difficulty in this approach is to formulate a unified refinement strategy that captures the input topology correctly when the guessed scale is right and outputs a mesh with some reasonable properties all the time.