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1. A general two-stage initialization for sag-free deformable simulations

   https://doi.org/10.1145/3528223.3530165  

   Hsu, Jerry; Truong, Nghia; Yuksel, Cem; Wu, Kui (July 2022, ACM Transactions on Graphics)

   Initializing simulations of deformable objects involves setting the rest state of all internal forces at the rest shape of the object. However, often times the rest shape is not explicitly provided. In its absence, it is common to initialize by treating the given initial shape as the rest shape. This leads to sagging, the undesirable deformation under gravity as soon as the simulation begins. Prior solutions to sagging are limited to specific simulation systems and material models, most of them cannot handle frictional contact, and they require solving expensive global nonlinear optimization problems. We introduce a novel solution to the sagging problem that can be applied to a variety of simulation systems and materials. The key feature of our approach is that we avoid solving a global nonlinear optimization problem by performing the initialization in two stages. First, we use a global linear optimization for static equilibrium. Any nonlinearity of the material definition is handled in the local stage, which solves many small local problems efficiently and in parallel. Notably, our method can properly handle frictional contact orders of magnitude faster than prior work. We show that our approach can be applied to various simulation systems by presenting examples with mass-spring systems, cloth simulations, the finite element method, the material point method, and position-based dynamics. 

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2. Shape Optimization of Optical Microscale Inclusions

   https://doi.org/10.1137/23M158262X  

   Bezbaruah, Manaswinee; Maier, Matthias; Wollner, Winnifried (August 2024, SIAM Journal on Scientific Computing)

   This paper describes a class of shape optimization problems for optical metamaterials comprised of periodic microscale inclusions composed of a dielectric, low-dimensional material suspended in a non-magnetic bulk dielectric. The shape optimization approach is based on a homogenization theory for time-harmonic Maxwell's equations that describes effective material parameters for the propagation of electromagnetic waves through the metamaterial. The control parameter of the optimization is a deformation field representing the deviation of the microscale geometry from a reference configuration of the cell problem. This allows for describing the homogenized effective permittivity tensor as a function of the deformation field. We show that the underlying deformed cell problem is well-posed and regular. This, in turn, proves that the shape optimization problem is well-posed. In addition, a numerical scheme is formulated that utilizes an adjoint formulation with either gradient descent or BFGS as optimization algorithms. The developed algorithm is tested numerically on a number of prototypical shape optimization problems with a prescribed effective permittivity tensor as the target. 

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3. Adjoint-based inversion for stress and frictional parameters in earthquake modeling

   https://doi.org/10.1016/j.jcp.2024.113447  

   Stiernström, Vidar; Almquist, Martin; Dunham, Eric M (December 2024, Journal of Computational Physics)

   We present an adjoint-based optimization method to invert for stress and frictional parameters used in earthquake modeling. The forward problem is linear elastodynamics with nonlinear rate-and-state frictional faults. The misfit functional quantifies the difference between simulated and measured particle displacements or velocities at receiver locations. The misfit may include windowing or filtering operators. We derive the corresponding adjoint problem, which is linear elasticity with linearized rate-and-state friction and, for forward problems involving fault normal stress changes, nonzero fault opening, with time-dependent coefficients derived from the forward solution. The gradient of the misfit is efficiently computed by convolving forward and adjoint variables on the fault. The method thus extends the framework of full-waveform inversion to include frictional faults with rate-and-state friction. In addition, we present a space-time dual-consistent discretization of a dynamic rupture problem with a rough fault in antiplane shear, using high-order accurate summation-by-parts finite differences in combination with explicit Runge–Kutta time integration. The dual consistency of the discretization ensures that the discrete adjoint-based gradient is the exact gradient of the discrete misfit functional as well as a consistent approximation of the continuous gradient. Our theoretical results are corroborated by inversions with synthetic data. We anticipate that adjoint-based inversion of seismic and/or geodetic data will be a powerful tool for studying earthquake source processes; it can also be used to interpret laboratory friction experiments. 

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4. From images to properties: A NeRF-driven framework for granular material parameter inversion

   https://doi.org/10.1051/epjconf/202534010017  

   Hsiao, Cheng-Hsi; Kumar, Krishna (January 2025, EPJ Web of Conferences)

   Alam, M; Das, SL; Bose, M; Murthy, TG; Chikkadi, V; Singh, A; Luding, S (Ed.)

   We introduce a novel framework that integrates Neural Radiance Fields (NeRF) with Material Point Method (MPM) simulation to infer granular material properties from visual observations. Our approach begins by generating synthetic experimental data, simulating an plow interacting with sand. The experiment is rendered into realistic images as the photographic observations. These observations include multi-view images of the experiment’s initial state and time-sequenced images from two fixed cameras. Using NeRF, we reconstruct the 3D geometry from the initial multi-view images, leveraging its capability to synthesize novel viewpoints and capture intricate surface details. The reconstructed geometry is then used to initialize material point positions for the MPM simulation, where the friction angle remains unknown. We render images of the simulation under the same camera setup and compare them to the observed images. By employing Bayesian optimization, we minimize the image loss to estimate the best-fitting friction angle. Our results demonstrate that friction angle can be estimated with an error within 2°, highlighting the effectiveness of inverse analysis through purely visual observations. This approach offers a promising solution for characterizing granular materials in real-world scenarios where direct measurement is impractical or impossible. 

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5. Interval-Based Parameter Identification for Structural Static Problems

   3. Xiao, Naijia; Fedele, Francesco; Muhanna, Rafi (July 2016, Reliable computing)

   We present an interval-based approach for parameter identification in structural static problems. Our inverse formulation models uncertainties in measurement data as interval and exploits the Interval Finite Element Method (IFEM) combined with adjoint-based optimization. The inversion consists of a two-step algorithm: first, an estimate of the parameters is obtained by a deterministic iterative solver. Then, the algorithm switches to the interval extension of the previous solver, using the deterministic estimate of the parameters as an initial guess. The formulation is illustrated in solutions of various numerical examples showing how the guaranteed interval enclosures always contain Monte Carlo predictions. 

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