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1. Identification of the 3D crystallographic orientation using 2D deformations

   https://doi.org/10.1177/03093247211043107  

   Goenezen, Sevan; Kotecha, Maulik_C; Reddy, Junuthula_N (September 2021, The Journal of Strain Analysis for Engineering Design)

   Polycrystalline materials consist of grains (crystals) oriented at different angles resulting in a heterogeneous and anisotropic mechanical behavior at that micro-length scale. In this study, a novel method is proposed for the first time to determine the [Formula: see text] crystal orientations of grains in a [Formula: see text] domain, using solely [Formula: see text] deformation fields. The grain boundaries are assumed to be unknown and delineated from the reconstructed changes in the crystallographic orientation. Further, the constitutive equations that describe the mechanical behavior of the domain in [Formula: see text] under plane stress conditions are derived, assuming that the material is transversely isotropic in 3D. Finite element based algorithms are utilized to discretize the inverse problem. The in-house written inverse problem solver is coupled with Matlab-based optimization scripts to solve for the mechanical property distributions. The performance of this method is tested at different noise levels with synthetic displacements that were used as measured data. The reconstructions deteriorate as the noise level is increased. This work presents a first milestone in the verification of this novel technology with synthetic data. 

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2. A minimization principle for incompressible fluid mechanics

   https://doi.org/10.1063/5.0175959  

   Taha, Haithem; Gonzalez, Cody; Shorbagy, Mohamed (December 2023, Physics of Fluids)

   Most variational principles in classical mechanics are based on the principle of least action, which is only a stationary principle. In contrast, Gauss' principle of least constraint is a true minimum principle. In this paper, we apply Gauss' principle to the mechanics of incompressible flows, thereby discovering the fundamental quantity that Nature minimizes in most flows encountered in everyday life. We show that the magnitude of the pressure gradient over the domain is minimum at every instant of time. We call it the principle of minimum pressure gradient (PMPG). It turns a fluid mechanics problem into a minimization one. We demonstrate this intriguing property by solving four classical problems in fluid mechanics using the PMPG without resorting to Navier–Stokes' equation. In some cases, the PMPG minimization approach is not any more efficient than solving Navier–Stokes'. However, in other cases, it is more insightful and efficient. In fact, the inviscid version of the PMPG allowed solving the long-standing problem of the aerohydrodynamic lift over smooth cylindrical shapes where Euler's equation fails to provide a unique answer. The PMPG transcends Navier–Stokes' equations in its applicability to non-Newtonian fluids with arbitrary constitutive relations and fluids subject to arbitrary forcing (e.g., electromagnetic). 

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3. Ensemble physics informed neural networks: A framework to improve inverse transport modeling in heterogeneous domains

   https://doi.org/10.1063/5.0150016  

   Aliakbari, Maryam; Soltany Sadrabadi, Mohammadreza; Vadasz, Peter; Arzani, Amirhossein (May 2023, Physics of Fluids)

   Modeling fluid flow and transport in heterogeneous systems is often challenged by unknown parameters that vary in space. In inverse modeling, measurement data are used to estimate these parameters. Due to the spatial variability of these unknown parameters in heterogeneous systems (e.g., permeability or diffusivity), the inverse problem is ill-posed and infinite solutions are possible. Physics-informed neural networks (PINN) have become a popular approach for solving inverse problems. However, in inverse problems in heterogeneous systems, PINN can be sensitive to hyperparameters and can produce unrealistic patterns. Motivated by the concept of ensemble learning and variance reduction in machine learning, we propose an ensemble PINN (ePINN) approach where an ensemble of parallel neural networks is used and each sub-network is initialized with a meaningful pattern of the unknown parameter. Subsequently, these parallel networks provide a basis that is fed into a main neural network that is trained using PINN. It is shown that an appropriately selected set of patterns can guide PINN in producing more realistic results that are relevant to the problem of interest. To assess the accuracy of this approach, inverse transport problems involving unknown heat conductivity, porous media permeability, and velocity vector fields were studied. The proposed ePINN approach was shown to increase the accuracy in inverse problems and mitigate the challenges associated with non-uniqueness. 

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4. Stress-constrained optimization of multiscale structures with parameterized microarchitectures using machine learning

   https://doi.org/10.1007/s00158-024-03821-y  

   Black, Nolan; Najafi, Ahmad (June 2024, Structural and Multidisciplinary Optimization)

   Abstract A multiscale topology optimization framework for stress-constrained design is presented. Spatially varying microstructures are distributed in the macroscale where their material properties are estimated using a neural network surrogate model for homogenized constitutive relations. Meanwhile, the local stress state of each microstructure is evaluated with another neural network trained to emulate second-order homogenization. This combination of two surrogate models — one for effective properties, one for local stress evaluation — is shown to accurately and efficiently predict relevant stress values in structures with spatially varying microstructures. An augmented lagrangian approach to stress-constrained optimization is then implemented to minimize the volume of multiscale structures subjected to stress constraints in each microstructure. Several examples show that the approach can produce designs with varied microarchitectures that respect local stress constraints. As expected, the distributed microstructures cannot surpass density-based topology optimization designs in canonical volume minimization problems. Despite this, the stress-constrained design of hierarchical structures remains an important component in the development of multiphysics and multifunctional design. This work presents an effective approach to multiscale optimization where a machine learning approach to local analysis has increased the information exchange between micro- and macroscales. 

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5. Gaussian Process to Identify Hydrogel Constitutive Model

   Wang J., Li T. (December 2021, Challenges in Mechanics of Time Dependent Materials, Mechanics of Biological Systems and Materials & Micro-and Nanomechanics, Volume 2.)

   Unlike traditional structural materials, soft solids can often sustain very large deformation before failure, and many exhibit nonlinear viscoelastic behavior. Modeling nonlinear viscoelasticity is a challenging problem for a number of reasons. In particular, a large number of material parameters are needed to capture material response and validation of models can be hindered by limited amounts of experimental data available. We have developed a Gaussian Process (GP) approach to determine the material parameters of a constitutive model describing the mechanical behavior of a soft, viscoelastic PVA hydrogel. A large number of stress histories generated by the constitutive model constitute the training sets. The low-rank representations of stress histories by Singular Value Decomposition (SVD) are taken to be random variables which can be modeled via Gaussian Processes with respect to the material parameters of the constitutive model. We obtain optimal material parameters by minimizing an objective function over the input set. We find that there are many good sets of parameters. Further the process reveals relationships between the model parameters. Results so far show that GP has great potential in fitting constitutive models. 

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