Hasan Metin Aktulga

Bryce Palmer, Hasan Metin Aktulga, Tong Gao

This paper reformulates complementarity-based time-stepping for frictionless nonsmooth contact between smooth rigid bodies as a recursively generated linear complementarity problem (ReLCP), involving a sequence of LCPs of increasing dimension. Starting from a classical single-constraint shared-normal signed-distance (SNSD) LCP, the method adds unilateral constraints only when the discrete-time update predicted by the current contact set would violate nonpenetration of the underlying smooth surfaces. The resulting procedure acts directly on smooth geometry, enforces nonpenetration to a prescribed tolerance, and avoids the oversampling inherent to proxy-surface contact models such as tessellations or multi-sphere decompositions, for which improved geometric fidelity can drive rapid growth in constraint count and cost. For strictly convex bodies, we prove that an initially overlap free configuration with sufficiently small timestep sizes, imply finite termination of the adaptive augmentation, and yield a unique discrete-time velocity update. In the small timestep limit and for any fixed overlap-free discrete state with a fixed geometric overlap tolerance, we prove that the recursion terminates after the initial solve, reducing the method to the classical single-constraint SNSD LCP and retaining the usual consistency of complementarity time-stepping with the underlying differential variational inequality. Numerical tests on colliding ellipsoids, compacting ellipsoid suspensions, growing bacterial colonies, and taut chainmail networks demonstrate stable large-timestep behavior, bounded interpenetration without discretization-induced surface roughness, and substantial reductions in both active constraint counts and runtime relative to representative discrete-surface complementarity formulations.

Doga Dikbayir, Abdel Alsnayyan, Vishnu Naresh Boddeti et al.

The inverse scattering problem is of critical importance in a number of fields, including medical imaging, sonar, sensing, non-destructive evaluation, and several others. The problem of interest can vary from detecting the shape to the constitutive properties of the obstacle. The challenge in both is that this problem is ill-posed, more so when there is limited information. That said, significant effort has been expended over the years in developing solutions to this problem. Here, we use a different approach, one that is founded on data. Specifically, we develop a deep learning framework for shape reconstruction using limited information with single incident wave, single frequency, and phase-less far-field data. This is done by (a) using a compact probabilistic shape latent space, learned by a 3D variational auto-encoder, and (b) a convolutional neural network trained to map the acoustic scattering information to this shape representation. The proposed framework is evaluated on a synthetic 3D particle dataset, as well as ShapeNet, a popular 3D shape recognition dataset. As demonstrated via a number of results, the proposed method is able to produce accurate reconstructions for large batches of complex scatterer shapes (such as airplanes and automobiles), despite the significant variation present within the data.