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The strip method for shape derivatives
Abstract A major challenge in shape optimization is the coupling of finite element method (FEM) codes in a way that facilitates efficient computation of shape derivatives. This is particularly difficult with multiphysics problems involving legacy codes, where the costs of implementing and maintaining shape derivative capabilities are prohibitive. The volume and boundary methods are two approaches to computing shape derivatives. Each has a major drawback: the boundary method is less accurate, while the volume method is more invasive to the FEM code. We introduce the strip method , which computes shape derivatives on a strip adjacent to the boundary. The strip method makes code coupling simple. Like the boundary method, it queries the state and adjoint solutions at quadrature nodes, but requires no knowledge of the FEM code implementations. At the same time, it exhibits the higher accuracy of the volume method. As an added benefit, its computational complexity is comparable to that of the boundary method, that is, it is faster than the volume method. We illustrate the benefits of the strip method with numerical examples.
A Poisson–Nernst–Planck Model of Ion Transport and Interface Segregation in Metal–Insulator–Semiconductor Structures and Solar Cells
A numerical model that describes the transport of mobile ionic species in metal–insulator–semiconductor (MIS) and photovoltaic (PV) devices subject to temperature and voltage stress is presented. The finite element method (FEM) is used to solve the Nernst–Planck equation while imposing Poisson's equation self-consistently as a restriction for the electrostatic potential. This allows the contribution of the ionic species to the potential to be taken into account. Using a variational formulation eases the implementation of diverse boundary conditions, including the incorporation of segregation kinetics at the device interfaces. Segregation across the dielectric–semiconductor interface is relevant to modeling the electronic device degradation in systems where contamination reaches the semiconductor. The model in closed systems with no-flux boundary conditions is validated first. In the limiting case of low contamination levels with respect to the gate bias, the FEM solution matches analytically derived approximations. Then, the implementation is broadened to include an open boundary at the dielectric–semiconductor interface to account for leakage of ions. The predicted time dependence of the flatband voltage in Na-contaminated MIS test structures agrees well with measurements. This model successfully captures the role of long-range ion transport at concentrations of relevance to electronic and PV device instability and neuromorphic computing.
Enhancing the cooling performance of thermocouples: a power-constrained topology optimization procedure
Abstract Heat pumping through thermoelectric devices has many advantages over traditional cooling. However, their current efficiency is a limiting factor in their implementation. In this paper, we approach the non-convex topology optimization of thermoelectrical elements for cooling applications through the method of moving asymptotes (MMA) to improve their cooling capabilities per watt usage. The optimization problem is defined for a given power budget, aiming for the minimum temperature with a known heat pumping need. The introduction of power as a constraint justifies the introduction of the voltage gradient across the thermocouple as a design variable to maintain the thermoelectrical device in its optimum power-to-heat extraction ratio. To better understand the convergence of this non-convex problem, we present a two-variable analytical thermoelectric optimization model. This example provides information on how to select the penalty parameters used to scale the three material coefficients involved in the problem to obtain lower objective values and better convergence using MMA. The analytical model shows the non-convexity of the problem and provides the recommendation to use penalization coefficients of the form $$p_k=p_{\sigma }>p_{\alpha }=1$$ p k = p σ > p α = 1 for the thermal conductivity, electrical conductivity, and Seebeck coefficients. We tested these penalization coefficients through optimizations of a model based on the 1MC10-031 commercial thermoelectric-cooler (TEC) using the finite element method (FEM). These penalization coefficients provided local minima without the need for volume constraints. With this procedure, we found designs that provided temperatures close to 10 degrees lower using 60% less semiconductor material volume compared to the initial design.
Generalized thermo-mechanical framework for heterogeneous materials through asymptotic homogenization
Abstract A fundamental understanding of the interaction between microstructure and underlying physical mechanisms is essential, especially for developing more accurate multi-physics models for heterogeneous materials. Effects of microstructure on the material response at the macroscale are modeled by using the generalized thermomechanics. In this study, strain gradient theory is employed as a higher-order theory on the macroscale with thermodynamics modeled as a first-order theory on the microscale. Hence, energy depends only on the temperature such that we circumvent an extension of Fourier’s law and analyze the “simplest” thermo-mechanical model in strain gradient elasticity. Developing multiphysics models for heterogeneous materials is indeed a challenge and even this “simplest” model in generalized thermomechanics creates dozens of parameters to be determined. We develop a thermo-mechanical framework, in which microstructure is modeled as a periodic structure and through asymptotic homogenization approach, higher-order parameters at macroscopic scale are calculated. To illustrate the importance of higher-order parameters in overall thermo-mechanical response of a heterogeneous materials, finite element method (FEM) is employed with the aid of open-source codes (FEniCS). Verification example of a bulk system and several case studies of porous structures demonstrate how such numerical framework can be beneficial in the design of materials with tailored microstructures.
Dirichlet absorbing boundary conditions for classical and peridynamic diffusion-type models
Diffusion-type problems in (nearly) unbounded domains play important roles in various fields of fluid dynamics, biology, and materials science. The aim of this paper is to construct accurate absorbing boundary conditions (ABCs) suitable for classical (local) as well as nonlocal peridynamic (PD) diffusion models. The main focus of the present study is on the PD diffusion formulation. The majority of the PD diffusion models proposed so far are applied to bounded domains only. In this study, we propose an effective way to handle unbounded domains both with PD and classical diffusion models. For the former, we employ a meshfree discretization, whereas for the latter the finite element method (FEM) is employed. The proposed ABCs are time-dependent and Dirichlet-type, making the approach easy to implement in the available models. The performance of the approach, in terms of accuracy and stability, is illustrated by numerical examples in 1D, 2D, and 3D.
Peri-Net-Pro: the neural processes with quantified uncertainty for crack patterns
Abstract This paper develops a deep learning tool based on neural processes (NPs) called the Peri-Net-Pro, to predict the crack patterns in a moving disk and classifies them according to the classification modes with quantified uncertainties. In particular, image classification and regression studies are conducted by means of convolutional neural networks (CNNs) and NPs. First, the amount and quality of the data are enhanced by using peridynamics to theoretically compensate for the problems of the finite element method (FEM) in generating crack pattern images. Second, case studies are conducted with the prototype microelastic brittle (PMB), linear peridynamic solid (LPS), and viscoelastic solid (VES) models obtained by using the peridynamic theory. The case studies are performed to classify the images by using CNNs and determine the suitability of the PMB, LBS, and VES models. Finally, a regression analysis is performed on the crack pattern images with NPs to predict the crack patterns. The regression analysis results confirm that the variance decreases when the number of epochs increases by using the NPs. The training results gradually improve, and the variance ranges decrease to less than 0.035. The main finding of this study is that the NPs enable accurate predictions, even with missing or insufficient training data. The results demonstrate that if the context points are set to the 10th, 100th, 300th, and 784th, the training information is deliberately omitted for the context points of the 10th, 100th, and 300th, and the predictions are different when the context points are significantly lower. However, the comparison of the results of the 100th and 784th context points shows that the predicted results are similar because of the Gaussian processes in the NPs. Therefore, if the NPs are employed for training, the missing information of the training data can be supplemented to predict the results.
An Integrated Simulation of Multiple-Pass U-10Mo Alloy Hot Rolling and Static Recrystallization
To achieve a desired microstructure and minimize the thickness variation in rolled foils, researchers must understand the effects of foil fabrication process variables on microstructure evolution. We developed an integrated simulation of deformation and recrystallization that employs the finite element method (FEM) and the kinetic Monte Carlo (KMC) Potts model, respectively, to investigate microstructure evolution during multiple-pass hot rolling and heat treatment in polycrystalline U-10Mo fuel. Scanning electron microscopy and electron backscatter diffraction images of microstructures were directly used as input in FEM calculation of deformation, and the calculated strains were used to determine the driving force of nucleation and growth of recrystallized grains in the Potts model. Grain structures predicted by the Potts model were used to update the grain structure and material properties for FEM. Simulation alternated between FEM and the Potts model to simulate grain structure evolution during multiple rolling and heat treatments. The initial model parameters were determined by benchmarking the recrystallization kinetics against experimental data. Then, the model was applied to predict the grain structure evolution. Results showed that our model can capture the coupling between deformation and recrystallization and can quantitatively reproduce the observed U-10Mo recrystallization and grain growth kinetics. The simulation results demonstrated that the developed model can predict U-10Mo grain structures as a function of initial microstructure and foil fabrication parameters.
Enhanced accuracy through ensembling of randomly initialized auto-regressive models for dynamical systems
Computational mechanics simulations using traditional finite element methods (FEM) require prohibitively expensive computational resources for real-time engineering applications, design optimization, and digital twin implementations. While machine learning (ML) surrogate models offer significant computational speedups, autoregressive ML models for time-dependent mechanical systems suffer from error accumulation that compromises long-term prediction reliability - a critical concern for engineering applications where accuracy over extended time horizons is essential for safety and performance assessments. Here, we propose a deep ensemble framework specifically designed to address this challenge in computational mechanics applications, where multiple ML surrogate models with random weight initializations are trained in parallel and their predictions aggregated during inference. This approach leverages statistical diversity to maximize information gain from a fixed set of training data and to mitigate error propagation, while maintaining the computational efficiency that makes ML surrogates attractive for engineering practice. We validate the framework on three representative problems spanning critical areas of computational mechanics: stress field evolution in heterogeneous microstructures under complex loading (relevant to advanced materials design and composite analysis), planetary-scale shallow water dynamics (applicable to environmental and geotechnical engineering), and Gray-Scott reaction-diffusion systems (relevant to mass transport and chemical process engineering). Across all test cases, the ensemble approach demonstrates consistent error reduction of 15-33% compared to individual models. The codes for this work are available on GitHub (https://github.com/Graham-Brady-Research-Group/AutoregressiveEnsemble_SpatioTemporal_Evolution).
Utilizing computer vision and artificial intelligence algorithms to predict and design the mechanical compression response of direct ink write 3D printed foam replacement structures
Additive Manufacturing (AM) of porous polymeric materials, such as foams, recently became a topic of intensive research due their unique combination of low density, impressive mechanical properties, and stress dissipation capabilities. Conventional methods for fabricating foams rely on complex and stochastic processes, making it challenging to achieve precise architectural control of structured porosity. In contrast, AM provides access to a wide range of printable materials, where precise spatial control over structured porosity can be modulated during the fabrication process enabling the production of foam replacement structures (FRS). Current approaches for designing FRS are based on intuitive understanding of their properties or an extensive number of finite element method (FEM) simulations. These approaches, however, are computationally expensive and time consuming. As such, in this work, we present a novel methodology for determining the mechanical compression response of direct ink write (DIW) 3D printed FRS using a simple cross-sectional image. By obtaining measurement data for a relatively small number of samples, an artificial neural network (ANN) was trained, and a computer vision algorithm was used to make inferences about foam compression characteristics from a single cross-sectional image. Finally, a genetic algorithm (GA) was used to solve the inverse design problem, generating the AM printing parameters that an engineer should use to achieve a desired compression response from a DIW printed FRS. The methods developed herein present an avenue for entirely autonomous design and analysis of additively manufactured structures using artificial intelligence.
Evaluation of adsorption and mechanical strength of 13X zeolite mixtures with phyllosilicate binders using molecular dynamics simulation and positron annihilation spectroscopy
There is growing interest in developing zeolites with novel internal structures that have optimal adsorptive capacity and high mechanical strength, while offering advantages, such as being light weight. We integrate computational and experimental methods to explore the effect of binder/zeolite types, and weight percentages on the mechanical strength of 13X zeolite and adsorption capacities of N 2 , H 2 O, and CO 2 for additive manufacturing (AM) applications with the goal of maximizing both adsorption and strength. Zeolite 13X mixtures and phyllosilicate binders (either bentonite or kaolin) are combined using molecular dynamics (MD) simulations to create structures with various binder/zeolite weight percentages. Further, adsorption capabilities and mechanical strength are assessed using the grand canonical Monte Carlo (GCMC) and ReaxFF modules, respectively. Our modeling shows that an optimized zeolite/binder ratio for N 2 adsorption is around 15 wt% for kaolin and roughly 10 wt% for bentonite. The resulting parameters can be applied to facilitate macro-scale computational fluid dynamics (CFD) and finite element method (FEM) simulations of an AM zeolite structure. We also performed Positron Annihilation Lifetime Spectroscopy (PALS) measurements on zeolite samples to explore the effect of changes in the internal volume. The results show an inverse relationship between the free volume and the solid loading and adding a binder changes the morphology of the zeolite-binder compound and decreases open-volume area significantly.
A subdivision-stabilized B-spline mixed material point method
Subjected to external loadings, polymeric materials, e.g., biological tissues, hydrogels, and elastomers, may undergo extreme, nearly incompressible, (self-)contact deformations. For numerical modeling employing mesh-based techniques such as the finite element method (FEM), these deformations pose significant challenges due to large distortions in the deformed geometry, accuracy issues stemming from volumetric locking effects, and increased computational cost from complex contact searches. As an alternative to mesh-based methods, the material point method (MPM), a continuum-based particle technique, is gaining attention for its ability to handle extreme distortions and capture no-slip contact without added cost. For nearly incompressible material behaviors, while mixed formulations can address locking effects by treating displacements and pressure as independent fields, they can suffer from numerical instabilities close to the incompressibility limit due to the violation of the inf-sup condition, leading to inaccurate nodal pressure solutions. Here we propose an efficient and stable mixed B-spline material point method with highest achievable regularity for quasi-compressible polymeric materials. Using the two-scale relation of B-splines, we introduce a subdivision-stabilization for the two-field mixed MPM and obtain numerically stable, oscillation-free nodal solutions with equal-order interpolations with optimal regularity. Building on the Eulerian-Lagrangian nature of MPM, a previously-converged solution framework is adopted to mitigate issues related to cell-crossing and numerical fracture artifact present in standard MPM. We assess the stability and accuracy of the developed mixed MPM at large deformations for soft materials through the benchmark Cook’s membrane problem. Additionally, we test the robustness of the proposed MPM by modeling several examples, including the compression and indentation of a circular block into a quasi-compressible substrate and the twisting deformation of a rectangular block. The findings demonstrate the MPM’s capabilities for modeling practical soft material applications.
Simulation toolkit for digital material characterization of large image-based microstructures
In this paper, an efficient image-based simulation toolkit for material characterization is presented, which is scalable to work from personal computers to workstations. The effective thermal conductivity, elasticity, and permeability are evaluated employing a computational homogenization framework based on the Finite Element Method (FEM). Two complementary open-source packages are presented: one developed in Python, which can convert digital images into voxel meshes (pyTomoviewer); the other developed in Julia, that can run numerical simulations to compute effective material properties (chpack). Also, a CUDA C version of chpack is provided (chfem_gpu). They were designed to deal with large multi-phase models, so strategies were devised to minimize their memory footprint, while avoiding a high toll on execution time. The voxel-based approach significantly simplifies the FEM meshes and allows efficient matrix-free implementations. In that sense, to handle large linear systems of equations, the element-by-element (EBE) technique is adopted, in conjunction with a low-memory implementation of the Preconditioned Conjugate Gradient (PCG) method. Finally, the code was thoroughly tested on an artificial geometry made of a square array of cylinders, for which analytical solutions exist, as well as on a real micro-tomographic reconstruction of FiberForm TM , a carbon preform commonly used in thermal protection systems.
Evaluating the bond strength and fracture mechanisms of cold-sprayed zinc coating on AZ91 magnesium substrate via a combined experimental and computational approach
Magnesium (Mg) alloys are ideal candidates for automotive applications due to their high strength to weight ratio, castability, recyclability etc., however, they lack corrosion and oxidation resistance. Solid-state deposition techniques, such as cold spray, have been demonstrated to enhance their corrosion resistance as it relies on the severe plastic deformation of powder particles upon impact with the substrate to form a metallurgical bond with the substrate and within the coating. At cold sprayed interfaces, a heterogeneous microstructure is formed that includes some porosity, oxides and intermetallics which can significantly affect coating performance. Thus, establishing a direct correlation between the interface microstructure and its properties can aid in designing optimal cold spray parameters. In this study, we investigated the microstructure and mechanical properties of a zinc (Zn) coating deposited on a high pressure die cast (HPDC) AZ91 Mg substrate via high resolution scanning transmission electron microscopy, in situ micro-tensile testing, and finite element method (FEM) modeling. Micro-tensile pillars fabricated using the plasma focused ion beam (PFIB) successfully isolates the coating-substrate interface within the gauge length. The average bond strength of Zn-Mg interface was determined to be ∼140 MPa with failure occurring partially at the interface and mostly into the coatings. A detailed microstructural characterization revealed evidence of a strong metallurgical bonding at the Zn-Mg interface and formation of the C14 MgZn 2 laves phase interlayer resulting in a mixed mode of fracture during the micro-tensile experiments. FEM modeling reveals the stress distribution along the interfaces and suggests that a MgZn 2 layer thickness between 200–400 nm is optimum to increase the bond strength and minimize the triaxiality. Such a site-specific interfacial analysis with correlative computational modeling provides crucial insight into the overall performance of cold spray interfaces.
A multiscale model to understand the interface chemistry, contacts, and dynamics during lithium stripping
A reversible Li-metal electrode, paired with a solid electrolyte, is critical for attaining higher energy density and safer batteries beyond the current lithium-ion cells. A stable stripping process may be even harder to attain as the stripping process will remove Li-atoms from the surface, and naturally reduce surface contact area, if not self-corrected by other mechanisms, such as diffusion and plastic deformation under an applied external stack pressure. Here, we capture these mechanisms occurring at multiple length- and time- scales, i.e., interface interactions, vacancy hopping, and plastic deformation, by integrating density functional theory (DFT) simulations, kinetic Monte Carlo (KMC), and continuum finite element method (FEM). By assuming the self-affine nature of multiscale contacts, we predict the steady-state contact area as a function of stripping current density, interface wettability, and stack pressure. We further estimate the exponential increase of overpotential due to contact area loss to maintain the same stripping current density. We demonstrate that a lithiophilic interface requires less stack pressure to reach the same steady-state contact area fraction than a lithiophobic interface. A “tolerable steady-state” contact area loss for maintaining stable stripping is estimated at 20 %, corresponding to a 10 % increase in overpotential. To constrain contact loss within the tolerance, the required stack pressure is 0.1, 0.5, and 2 times the yield strength of lithium metal for three distinct interfaces, lithiophilic Li/lithium oxide(Li2O), Li/lithium lanthanum zirconium oxide(LLZO), and lithiophoblic Li/lithium fluoride(LiF), respectively. The modeling results agree with experiments on the impact of the stack pressure quantitatively, while the discrepancy in stripping rate sensitivity is attributed to the simplifying interface interaction in our simulations. Overall, this multiscale simulation framework demonstrates the importance of electrochemical-mechanical coupling in understanding the dynamics of the Li/SE interface during stripping.
Crystal mechanics-based thermo-elastic constitutive modeling of orthorhombic uranium using generalized spherical harmonics and first-order bounding theories
In earlier works, a mathematical procedure for invertible microstructure-property linkages was developed using computationally efficient spectral methods for polycrystalline cubic and hexagonal metals. This paper formulates such invertible microstructure–property linkages for orthorhombic polycrystalline metals relying on the generalized spherical harmonics (GSH) spectral basis. The procedure is used to compute property closures of orthorhombic polycrystals. The closures represent the complete set of theoretically possible combinations of effective properties for a selected material. The procedure relies on the first-order bounding theories and considers orientation distribution functions (ODFs) as the main microstructural descriptor influencing homogenized properties. Numerous examples of these closures involving second-rank thermal expansion and fourth-rank elastic stiffness tensorial properties over a broad range of temperatures are presented for α-uranium (α-U). In doing so, certain key properties of these closures are exploited to facilitate their computation with drastically reduced computational effort. Along with the recently developed GSH-based interpolation procedure for ODFs from coarsely spaced experimental measurement grids to finely spaced finite element mesh resolution grids presented in Barrett et al., the developed computationally efficient ODF-effective property linkages are used to establish a crystal mechanics-based simulation framework coupled with the finite element method (FEM). The ODF dependent thermal expansion and elastic stiffness tensors are efficiently calculated at every integration point and used by the FEM to predict the overall distortion of a hemispherical part made of α-U during heating. In conclusion, it is shown that the developed framework can be used to simulate microstructurally heterogeneous components under thermo-mechanical loadings in a computationally efficient manner.
Microstructure-process relationships in monolithic U-10Mo fuel foil single-pass rolling: A parametric simulation study
In this work, a previously validated coupling of Kinetic Monte Carlo (KMC) Potts Model and finite element method (FEM) simulations was implemented to investigate the effects of microstructural features in as-cast and homogenized monolithic U-10Mo foils on the emergent microstructure after rolling and reheating. Parameters that could potentially affect recrystallization behavior of the rolled U-10Mo foils were considered: grain size distribution, uranium carbide (UC) size distribution, UC volume fraction, spatial distribution of UC, and rolling reduction magnitude. Grain structure and the magnitude of rolling reduction have the strongest influence on recrystallization kinetics and the fabricated grain size distribution. The UC distribution had only a weak effect on the recrystallization kinetics and final microstructures. While particle-stimulated nucleation (PSN) occurred in simulation more frequently as grain size increased, its incidence did not appear to considerably affect the recrystallization kinetics or grain size distribution.
Dynamic data-driven multiscale modeling for predicting the degradation of a 316L stainless steel nuclear cladding material
Here, we have developed a long short-term memory stacked ensemble (LSTM-SE) surrogate modeling approach that can provide rapid predictions of microstructural evolution and the resultant mechanical properties of American Iron and Steel Institute (AISI) 316L series stainless steel (316LSS) fuel cladding under conditions of varying temperature and radiation dose rate. To acquire training data, we developed and implemented a kinetic Monte Carlo (KMC) model to simulate precipitation kinetics of M 23 C 6 , γ', and G phases within SS316L cladding. Experimentally reported precipitation kinetics of SS316L in literature were linked to the kinetic parameters of the simulated precipitation in our KMC model. The model was then used to simulate microstructure evolution under synthetically generated treatments of varying temperature and radiation dose rate, for periods of up to 3000 hours. Changes in volume fraction, number density, and particle size of precipitates were recorded, and particle area fractions were correlated using statistical methods to develop the surrogate model. Simultaneously, the mechanical properties of the simulated microstructures were evaluated using microstructure-based finite element method (FEM) analysis to determine the elastic modulus, yield stress, ultimate tensile strength, and elongation to failure of the aged microstructures. Using this approach, our surrogate model can predict precipitation behavior within 0.25% volume fraction and mechanical properties within 6% relative error from the values predicted by the KMC and FEM models using 50 training simulations as input. The trained recurrent neural network-based model can return estimations of precipitation kinetics and mechanical properties ~1000 times faster than the physics-based codes. This work demonstrates, as a proof of concept, that reactor material service lifetimes under variable service conditions can be predicted for a statistics-based model from a practicably obtainable dataset.