Adaptive Block Coordinate Descent for Distortion Optimization

Abstract

We present a new algorithm for optimizing geometric energies and computing positively oriented simplicial mappings. Our major improvements over the state‐of‐the‐art are: (i) introduction of new energies for repairing inverted and collapsed simplices; (ii) adaptive partitioning of vertices into coordinate blocks with the blended local‐global strategy for more efficient optimization and (iii) introduction of the displacement norm for improving convergence criteria and for controlling block partitioning. Together these improvements form the basis for the Adaptive Block Coordinate Descent (ABCD) algorithm aimed at robust geometric optimization. ABCD achieves state‐of‐the‐art results in distortion minimization, even under hard positional constraints and highly distorted invalid initializations that contain thousands of collapsed and inverted elements. Starting with an invalid non‐injective initial map, ABCD behaves as a modified block coordinate descent up to the point where the current mapping is cleared of invalid simplices. Then, the algorithm converges rapidly into the chosen iterative solver. Our method is very general, fast‐converging and easily parallelizable. We show over a wide range of 2D and 3D problems that our algorithm is more robust than existing techniques for locally injective mapping.

Globally Injective Geometry Optimization with Non‐Injective Steps

“…Next, we consider free‐boundary problems where we seek a non‐inverting solution from an inverting initializer. The most recent work in this domain is ABCD [NZZ20], which we compare GINI to in this section. Table 3 shows the time to a non‐inverting solution using the data set provided by [NZZ20] which contains both 3D (wrench, bar12k, bar30k) and 2D examples.…”

“…The most recent work in this domain is ABCD [NZZ20], which we compare GINI to in this section. Table 3 shows the time to a non‐inverting solution using the data set provided by [NZZ20] which contains both 3D (wrench, bar12k, bar30k) and 2D examples. We show one such example in Fig.…”

“…For the higher resolution armadillo of the Dillo‐Anim testbed (121,190 elements), we pin the hands in place and run GINI until the mesh is injective. Then, we switch to ABCD's core projected Newton solver and optimize until the convergence criteria is met, which is the combination of the characteristic gradient [ZBK18] and a displacement norm (see [NZZ20] for details). With ABCD, we give it the same initializer as GINI and let the method run until the same convergence criterion is met.…”

“…We compare to TLC in Section 8.1 and show that our method is as robust for most 3D problems, but much faster. One of the few methods that does not require fixed boundaries is ABCD [NZZ20] which introduces a modified distortion energy to repair injective elements. We also compare to ABCD in Section 8.2 and show our method scales better with increased complexity.…”

“…For these problems, maintenance‐based solvers must be preceded by an initial phase that repairs the input mesh until it is feasible. This task is addressed by a much smaller class of methods, referred to as “map fixers” [NZZ20]. For large and complex domains, this map‐fixing phase can have a substantial additional cost.…”