Hexahedral Mesh Repair via Sum‐of‐Squares Relaxation

© 2020 The Author(s) Computer Graphics Forum © 2020 The Eurographics Association and John Wiley & Sons Ltd. Published by John Wiley & Sons Ltd. The validity of trilinear hexahedral (hex) mesh elements is a prerequisite for many applications of hex meshes, such as finite element analysis. A commonly used check for hex mesh validity evaluates mesh quality on the corners of the parameter domain of each hex, an insufficient condition that neglects invalidity elsewhere in the element, but is straightforward to compute. Hex mesh quality optimizations using this validity criterion suffer by being unable to detect invalidities in a hex mesh reliably, let alone fix them. We rectify these challenges by leveraging sum-of-squares relaxations to pinpoint invalidities in a hex mesh efficiently and robustly. Furthermore, we design a hex mesh repair algorithm that can certify validity of the entire hex mesh. We demonstrate our hex mesh repair algorithm on a dataset of meshes that include hexes with both corner and face-interior invalidities and demonstrate that where naïve algorithms would fail to even detect invalidities, we are able to repair them. Our novel methodology also introduces the general machinery of sum-of-squares relaxation to geometry processing, where it has the potential to solve related problems

Toward the Development of Virtual Surgical Tools to Aid Orthopaedic FE Analyses

Computational models of joint anatomy and function provide a means for biomechanists, physicians, and physical therapists to understand the effects of repetitive motion, acute injury, and degenerative diseases. Finite element models, for example, may be used to predict the outcome of a surgical intervention or to improve the design of prosthetic implants. Countless models have been developed over the years to address a myriad of orthopaedic procedures. Unfortunately, few studies have incorporated patient-specific models. Historically, baseline anatomic models have been used due to the demands associated with model development. Moreover, surgical simulations impose additional modeling challenges. Current meshing practices do not readily accommodate the inclusion of implants. Our goal is to develop a suite of tools (virtual instruments and guides) which enable surgical procedures to be readily simulated and to facilitate the development of all-hexahedral finite element mesh definitions