In modern computer algebra system softwares, a variety of symbolic computational capabilities are provided to automate and accelerate the tedious analytical manipulations and procedures often performed in computational physics applications. However, most toolsoffer limited control over their manipulations and internal processes, make unexpected evaluations during operation, and are often uninformed of the broader mathematical contexts behind their inputs. As a result, many available symbolic computational functions lack the precision and robustness needed to be reliably deployed for advanced capabilities inmathematical physics research, such as automating analytical derivations for physics-basedmodel development. Although in-house and ad hoc functions have been developed by researchers throughout the years to address such shortcomings, the underlying algorithmicstructures enabling these capabilities have mostly remained obscure. In this work, we detailan approach for creating symbolic computational functions capable of making fundamentalanalytical manipulations in a sophisticated manner. The proposed framework allowsfunctions to systematically obtain and utilize the operative properties (e.g., tensor ranksand dependencies) and operative definitions (i.e., possible mathematical concepts and operations that could be applied to an expression) of their inputs through expression treetraversal tactics. As a result, functions remain informed during manipulation and can bereliably deployed in sequential manners to automate step-by-step mathematical derivationsand procedures. The particular algorithms presented in this work are foundational to Symbolica, an automated symbolic deduction code for deriving macroscopic equations through homogenization theory via multiple scale expansions with minimal human interaction. Due to its autonomy and robustness, Symbolica has made direct contributions to understanding multiscale, multiphysical couplings in a number of computational physics applications. By disseminating the proposed methodology, we aim to encourage the sharing of alternative algorithms that give rise to other reliable symbolic computational tools, as well as facilitate a broader employment of symbolic computation in new applications of applied mathematical and computational physics research.

Modeling mass and electrochemical coupled processes plays a crucial role in predicting lithium-ion batteries' performance, guiding material designs, and reducing safety hazards. The intrinsic multiscale and multiphysics nature of batteries leads to challenges for modeling their behavior accurately and efficiently [1]. On one hand, this requires to connect physical-electrochemical processes across 6 orders of magnitude in length scales (from sub-micron to centimeter scale) and 10 orders of magnitude in temporal scales (from milliseconds to months). On the other hand, microstructure heterogeneity has a great impact on mass transport and electrochemical processes, which brings more challenges to accurate modeling [2]. Such heterogeneity of physical-electrochemical properties cannot be ignored particularly under high charging/discharging rates [3]. To predict batteries' electrochemical performance efficiently, classical reduced complexity electrochemical models, such as the Doyle-Fuller-Newman model, the single particle model (SPM), the enhanced-SPM, and their generalizations, are used widely. However, as requirements on battery performance become pressing and the operation conditions are more extreme (e.g. high charging/discharging rates), the accuracy and predictivity of such models have been questioned [4]. In addition, although many other studies have been done on developing reduced-complexity electrochemical models via upscale techniques, the challenge of a rigorous model validated under extreme operation conditions remains. In this work, we establish a multiscale model at the electrode level to predict the mass and charge transport with extended applicability to scenarios with rapid surface reactions. We rigorously upscale Planck-Nernst equations to a dual-continua mass and charge transport model at the electrode level with macroscopic representations accounting for local surface reactions under extreme operation conditions, such as high charging/discharging rates. We verify that our proposed models can accurately approximate the pore-scale dynamics through three two-dimensional numerical experiments, including periodic spheres, periodic arbitrary structures, and a half-cell simulation. The dispersions between the pore-scale and our upscaled model are within a priori error estimate provided by the generalized homogenization method. Due to the generality of the proposed model, this work will enable accurate and efficient prediction of electrochemical performance under more general operation conditions with novel electrode microstructure designs. [1] Franco, A. A. Multiscale Modelling and Numerical Simulation of Rechargeable Lithium Ion Batteries: Concepts, Methods and Challenges. RSC Advances 2013 , 3 (32), 13027. [2] Xu, H.; Zhu, J.; Finegan, D. P.; Zhao, H.; Lu, X.; Li, W.; Hoffman, N.; Bertei, A.; Shearing, P.; Bazant, M. Z. Guiding the Design of Heterogeneous Electrode Microstructures for Li‐Ion Batteries: Microscopic Imaging, Predictive Modeling, and Machine Learning. Advanced Energy Materials 2021 , 11 (19), 2003908. [3] Lu, X.; Bertei, A.; Finegan, D. P.; Tan, C.; Daemi, S. R.; Weaving, J. S.; O’Regan, K. B.; Heenan, T. M. M.; Hinds, G.; Kendrick, E.; Brett, D. J. L.; Shearing, P. R. 3D Microstructure Design of Lithium-Ion Battery Electrodes Assisted by X-Ray Nano-Computed Tomography and Modelling. Nature Communications 2020 , 11 (1), 2079. [4] Arunachalam, H.; Onori, S.; Battiato, I. On Veracity of Macroscopic Lithium-Ion Battery Models. Journal of The Electrochemical Society 2015 , 162 (10), A1940–A1951.

Lithium-ion batteries are sensitive to temperature. Low-temperature or high-temperature operation may lead to lithium plating and dendrites, SEI decomposition, and separator collapse. Such phenomena may cause internal short circuits, rapid heat generation and, subsequently, may trigger thermal runaway, resulting in fire or explosion [1][2]. Hence, accurate modeling and prediction of the thermal response of battery cells inside the pack is critical to preventing safety hazards as well as optimizing battery performance. Battery thermal management system (BTMS) plays a significant role in maintaining an optimal operation temperature and ensuring thermal safety for electric vehicles [3]. However, the large number of battery cells inside one module makes it challenging for BTMS algorithms to realize real-time simulations based on fully resolved physics-based models. The conventional lumped capacitance (LC) thermal model achieves reduced computation costs under the hypothesis that temperature is spatially uniform. Nevertheless, such a model works better for small Biot number conditions, usually under low C-rate operation and moderate cooling scenarios, and it lacks justification under a broad range of battery operating conditions and materials [4]. To overcome such limitations, rigorous multiscale models, based on mathematical coarse-graining theories, have been developed to predict the thermal response of batteries at the module scale [5] while not compromising on computational cost. In this work, we investigate the accuracy of the classical LC thermal model for twenty potential battery modules constructed with commercial batteries and packing materials. We show that the classical LC thermal model may be inaccurate under a broad range of design conditions. In the context of multiscale modeling, we identify which effective models should be instead used, and analyze the impact of SOC and the relative strength of heat insulation and heat dissipation on model selection and accuracy. This work provides guidance, based on first principles and rigorous multiscale theory, on best practices for model selection in practical applications involving thermal behavior and thermal runaway in battery cells at the module scale. References: [1] Qin, Y., Xu, Z., Wu, Y., Lu, L., Han, X., Du, J., & Ouyang, M. (2022). Temperature distribution of lithium ion battery module with inconsistent cells under pulsed heating method. Applied thermal engineering , 212 , 118529. [2] Feng, X., Ren, D., He, X., & Ouyang, M. (2020). Mitigating thermal runaway of lithium-ion batteries. Joule , 4 (4), 743-770. [3] Noelle, D. J., Wang, M., Le, A. V., Shi, Y., & Qiao, Y. (2018). Internal resistance and polarization dynamics of lithium-ion batteries upon internal shorting. Applied energy , 212 , 796-808. [4] Mahamud, R., & Park, C. (2013). Spatial-resolution, lumped-capacitance thermal model for cylindrical Li-ion batteries under high Biot number conditions. Applied Mathematical Modelling , 37 (5), 2787-2801. [5] Pietrzyk, K., Bucci, G., Behandish, M., & Battiato, I. (2023). Automated upscaling via symbolic computing for thermal runaway analysis in Li-ion battery modules. Journal of Computational Science , 74 , 102134.

Rigorous upscaling techniques offer accurate and computationally-efficient strategies for modeling the average behaviors of multi-physical, multiscale phenomena in geological porous media. However, such techniques often rely on a variety of methodological assumptions that prohibit their rigorous application to practical systems (e.g., systems involving heterogeneous porous media, system-scale boundary conditions, and fine-scale dynamics that are not diffusion-dominant). In this work, we aim to formulate an upscaling methodology with few methodological assumptions to provide high levels of model generality and foster the utilization of rigorously-derived upscaled models in practice. In particular, we introduce the Method of Finite Averages (MoFA), a novel upscaling methodology for rigorously modeling heterogeneous porous media and system-scale boundary conditions. We detail MoFA’s implementation for the advective-diffusive transport of a single species and compare the methodology with classic numerical techniques, as well as other rigorous upscaling techniques, to highlight MoFA’s unique combination of rigor and generality. We then validate the derived model while demonstrating its benefits in three numerical experiments. The results suggest that (1.) the applicability and a priori error guarantees of MoFA models do not directly depend on system geometry, (2.) a model’s applicability and error guarantees can be can arbitrarily expanded and reduced, respectively, with further computational expense, and (3.) downscaling with MoFA provides an efficient strategy for generating accurate pore-scale solutions from upscaled results. Ultimately, the results evidence that upscaled models can be rigorously derived for heterogeneous porous media systems and resolved in a fraction of the time it takes to perform the equivalent pore-scale simulations.

In porous media theory, upscaling techniques are fundamental to deriving rigorous Darcy‐scale models for flow and reactive transport in subsurface systems. Due to limitations in classical techniques, a number of ad hoc approaches have been proposed to address physical regimes in which reactive time scales are similar to, or faster than, diffusive time scales. In Part 1 of this two part series, we present a strategy for expanding the applicability of classical homogenization theory by generalizing the assumed closure form. We detail the implementation of this strategy on two reactive mass transport problems with moderately reactive physics. The strategy produces nontrivial homogenized models with emergent terms and effective parameters that couple reactive, diffusive, and advective transport. The differences in equation forms between the macroscopic and pore‐scale descriptions advise caution to further studies where the forms of macroscopic equations are assumed, as opposed to rigorously derived. Numerical validation is provided for each problem to show that the error estimates of homogenization theory are satisfied, and to justify the implemented strategy. In Part 2, the presented strategy is automated using symbolic computing to expedite its implementation.

Despite the availability of advanced numerical capabilities and computational power, complex multi-physical, multiscale systems continue to challenge modeling efforts due to their elusive behaviors and nontrivial couplings. High-fidelity simulations have paved the way for simulating such systems, but the associated computational costs prohibit such tools from being used in iterative routines for engineering design and optimization. While numerical strategies for accelerating complex at-scale multi-physical simulations exist, they often concede predictive accuracy and lack rigorous justification. Alternatively, methods for developing rigorous multiscale models (e.g., upscaling techniques) offer a priori modeling error guarantees, but are primarily used in academic settings due to their requirements in time and specialized expertise for handling analytically intractable derivations. In our previous work, we developed an automated upscaling engine, Symbolica, that uses symbolic computation to avoid these overheads and rapidly develop multiscale models via rigorous homogenization theory. In this work, we extend Symbolica’s model development capabilities to accommodate multi-domain structures and demonstrate Symbolica’s generality in model development by applying it to a real-world problem; namely, to analyze thermal runaway in Li-ion battery modules. We obtain homogenized models that capture the sharp spatiotemporal gradients associated with thermal runaway to within the guaranteed modeling error. Upon numerically validating the models, we demonstrate how Symbolica can be used to reduce the level of effort in rigorous multiscale modeling and model implementation for real-world applications.

In this second part of the two paper series, we detail an algorithmic procedure for systematically implementing the generalized closure form strategy presented in Part 1. This strategy extends the applicability of homogenized models with respect to classical homogenization theory, as demonstrated in Part 1 where upscaled models are rigorously derived in moderately reactive physical regimes. After encoding the algorithm into Symbolica, an automated upscaling framework, we upscale two reactive mass transport problems and numerically validate the resulting nonlinear homogenized models by showing the absolute error estimates predicted by homogenization theory are satisfied. In both problems, nontrivial closure forms and closure problems are automatically formulated using the encoded strategy with no human interaction, nor prior knowledge regarding the closure required for the systems. We hope these demonstrations spark further interest in automated analytical frameworks for multiscale modeling, as such capabilities are invaluable for generating rigorous multiscale models of complex phenomena in porous media.

Predicting part quality for additive manufacturing (AM) processes requires high-fidelity numerical simulation of partial differential equations (PDEs) governing process multiphysics on a scale of minimum manufacturable features. This makes part-scale predictions computationally demanding, especially when they require many small-scale simulations. We consider drop-on-demand liquid metal jetting (LMJ) as an illustrative example of such computational complexity. A model describing droplet coalescence for LMJ may include coupled incompressible fluid flow, heat transfer, and phase change equations. Numerically solving these equations becomes prohibitively expensive when simulating the build process for a full part consisting of thousands to millions of droplets. Reduced-order models (ROMs) based on neural networks (NN) or k-nearest neighbor (kNN) algorithms have been built to replace the original physics-based solver and are computationally tractable for part-level simulations. However, their quick inference capabilities often come at the expense of accuracy, robustness, and generalizability. We apply an operator learning (OL) approach to learn a mapping between initial and final states of the droplet coalescence process for enabling rapid and accurate part-scale build simulation. Preliminary results suggest that OL requires order-of-magnitude fewer data points than a kNN approach and is generalizable beyond the training set while achieving similar prediction error.

We analyze three-dimensional particle-laden, isotropic turbulence to develop an understanding of inertial particle dynamics from a kinetic energy perspective. Data trends implying inhomogeneous sampling of the flow by particles are identified and used to support a proposed particle behavior: particles appear to accumulate in regions of low flow kinetic energy over time because they lose kinetic energy and slow down in such regions, ultimately causing them to spend more time there. To elucidate this behavior, we derive a particle kinetic energy equation from the particle momentum equation, which incorporates inertial effects through the Schiller–Naumann drag correlation. Upon extracting fundamental physics from this equation, hypotheses regarding the role of the Stokes number in the temporal change of particle kinetic energy and the previously proposed particle behavior are evaluated using simulation data considering three Stokes numbers. Finally, a Fokker–Planck equation is used to derive the steady-state probability density function of the particle kinetic energy. The model fits the simulation data well and provides a tool for further investigation into understanding preferential concentration, as well as a reduced order model for predicting particle kinetic energy in turbulent flows.

Macroscopic differential equations that accurately account for microscopic phenomena can be systematically generated using rigorous upscaling methods. However, such methods are time-consuming, prone to error, and become quickly intractable for complex systems with tens or hundreds of equations. To ease these complications, we propose a method of automatic upscaling through symbolic computation. By streamlining the upscaling procedure and derivation of applicability conditions to just a few minutes, the potential for democratization and broad utilization of upscaling methods in real-world applications emerges. We demonstrate the ability of our software prototype, Symbolica, by reproducing homogenized advective–diffusive–reactive (ADR) systems from earlier studies and homogenizing a large ADR system deemed impractical for manual homogenization. Novel upscaling scenarios previously restricted by unnecessarily conservative assumptions are discovered, and numerical validation of the models derived by Symbolica is provided.

Acoustic streaming is the net time-averaged flow that results from the nonlinearities in an oscillating flow. Extensive research has sought to identify different physical mechanisms and types of acoustic streaming in systems of various geometries. While streaming in a channel maintains one of the simplest geometries, dimensional analysis of the governing equations reveals that multiple regimes of streaming may occur within a channel. In this study, a framework is developed for investigating and understanding the physical streaming regimes in a two-dimensional channel. By taking different limits of the dimensionless number ratios found within the framework, streaming models derived in previous works are recovered to demonstrate the different streaming regimes within a channel. The onset of fast streaming is then analyzed with the framework and nonlinear Reynolds numbers, which indicate whether the streaming is slow or fast, are found for the different physical streaming regimes. As a result, the framework provides a base for analyzing fast streaming in a channel and streaming in multi-scale systems while organizing previous streaming models into a physical spectrum for a channel geometry.

Many biological fluids display shear-thinning rheology, where the viscosity decreases with an increasing shear rate. To better understand how this non-Newtonian rheology affects the motion of biological and artificial micro-swimmers, recent efforts have begun to seek answers to fundamental questions about active bodies in shear-thinning fluids. Previous analyses based on a squirmer model have revealed non-trivial variations of propulsion characteristics in a shear-thinning fluid via the reciprocal theorem. However, the reciprocal theorem approach does not provide knowledge about the flow surrounding the squirmer. In this work, we fill in this missing information by calculating the non-Newtonian correction to the flow analytically in the asymptotic limit of small Carreau number. In particular, we investigate the local effect due to viscosity reduction and the non-local effect due to induced changes in the flow; we then quantify their relative importance to locomotion in a shear-thinning fluid. Our results demonstrate cases where the non-local effect can be more significant than the local effect. These findings suggest that caution should be exercised when developing physical intuition from the local viscosity distribution alone around a swimmer in a shear-thinning fluid.

Micro-organisms expend energy moving through complex media. While propulsion speed is an important property of locomotion, efficiency is another factor that may determine the swimming gait adopted by a micro-organism in order to locomote in an energetically favorable manner. The efficiency of swimming in a Newtonian fluid is well characterized for different biological and artificial swimmers. However, these swimmers often encounter biological fluids displaying shear-thinning viscosities. Little is known about how this nonlinear rheology influences the efficiency of locomotion. Does the shear-thinning rheology render swimming more efficient or less? How does the swimming efficiency depend on the propulsion mechanism of a swimmer and rheological properties of the surrounding shear-thinning fluid? In this work, we address these fundamental questions on the efficiency of locomotion in a shear-thinning fluid by considering the squirmer model as a general locomotion model to represent different types of swimmers. Our analysis reveals how the choice of surface velocity distribution on a squirmer may reduce or enhance the swimming efficiency. We determine optimal shear rates at which the swimming efficiency can be substantially enhanced compared with the Newtonian case. The nontrivial variations of swimming efficiency prompt questions on how micro-organisms may tune their swimming gaits to exploit the shear-thinning rheology. The findings also provide insights into how artificial swimmers should be designed to move through complex media efficiently.

Biological locomotion in nature is often achieved by the interaction between a flexible body and its surrounding medium. The interaction of a flexible body with granular media is less understood compared with viscous fluids partially due to its complex rheological properties. In this work, we explore the effect of flexibility on granular propulsion by considering a simple mechanical model in which a rigid rod is connected to a torsional spring that is under a displacement actuation using a granular resistive force theory. Through a combined numerical and asymptotic investigation, we characterize the propulsive dynamics of such a flexible flapper in relation to the actuation amplitude and spring stiffness, and we compare these dynamics with those observed in a viscous fluid. In addition, we demonstrate that the maximum possible propulsive force can be obtained in the steady propulsion limit with a finite spring stiffness and large actuation amplitude. These results may apply to the development of synthetic locomotive systems that exploit flexibility to move through complex terrestrial media.

Bacterial contamination of water, for example by Escherichia coli, is a major cause of preventable diseases in rural developing communities. With the lack of infrastructure and experienced laboratory professionals, careful water sampling and analysis becomes inaccessible in these resource-limited environments. There is a need for a low-cost and equipment-free detection platform for detecting pathogens within fluid samples. Latex agglutination assays have been widely used in biology and medicine for its simplicity and specificity. However, conventional agglutination tests have their shortcomings, including the subjectivity in determining the degree of agglutination and the non-quantitative output. We propose a miniaturized device capable of quantifying the E. coli contamination levels in fluid samples. The detection employs the latex agglutination test, where functionalized latex particles form agglutinates with specific antigens in the fluid sample. Channels fabricated using adhesive tapes coated with hydrophilic materials are used to drive the fluid sample through a detection zone, where an image analysis algorithm quantifies the degree of agglutination using images obtained by a smartphone. The use of capillary-driven flows eliminates the needs for an external active pumping system and other equipment typically necessary for interfacing fluidic chips. This aspect of our device could resolve the issues with lack of trained personnel, which is a major and severe constraint in deploying health technology in these regions.

In this paper, a new characterization factor for thermoelectric module design in thermoelectric refrigeration is presented with guidelines for practical design strategy. It has been general practice to optimize the geometric factor (G-factor), the ratio of the area to the leg length of a thermoelectric leg, and the number of leg pairs simultaneously to gain a minimum refrigeration temperature from a module for a refrigeration system. However, the B-factor, which is defined as the ratio between the leg length and the fill factor (ratio of the area filled with thermoelectric materials to the module area), allows for module optimization with only one parameter. To demonstrate, a theoretical model of a module was created with energy conservation equations. While disregarding electrical contact resistance, the number of leg pairs did not affect the obtainable maximum temperature difference or the power consumption of a module when utilizing the B-factor. The effects of contact resistance on the optimum B-factor were also evaluated and avoided when the leg length was increased. It was then found that using fewer legs in a module would produce a temperature difference that was less sensitive to a varying input current. The present theoretical approach was validated with experimental evidence.

Recent studies on improving the thermoelectric figure of merit (ZT) have advanced research into self powered, wearable technologies using thermoelectric generators. However, previous design approaches do not consider structurally practical heat sink and module geometries, the use of a boost converter, or the size constraint of the generator due to aesthetic appeal, all of which lower the overall power output. Additionally, the reduced efficiency in using a boost converter changes the electrical and thermal load matching conditions for maximum power. In this study, the limitations of practicality were considered for a wearable thermoelectric generator that utilizes a state-of-the-art boost converter and an optimized heat sink. Heat sink fin geometries and thermoelectric module geometries were explored to maximize the power output within a 42.0 cm(2) area and a 1.0 cm total height, in order to justify the wearability of the energy harvester. With optimized values of fin and module heights, the system was designed to produce 0.48 mW of electrical power at a boosted output voltage of 3.0 V, enough to power a small heart-rate monitor. (C) 2016 Elsevier Ltd. All rights reserved.