Equation-based Formulation Research Articles

(Invited) (Digital Presentation) Development of Temperature-Dependent Degradation Models for Large-Format Lithium-Ion Batteries

Advancing electric transportation technologies is critically dependent on maximizing the performance and lifetime of Electric Vehicle (EV) batteries, which is directly related to the economic case for EVs. Large-format cells are achieving increasing popularity for EVs to maximize volumetric and gravimetric energy densities. However, due to the large format size cells that are currently used for applications within the electrification portfolio, challenges have arisen that are tied to thermal management. Spatial variations in current density and temperature negatively affect overall cell utilization and lifetime. Identifying these temperature-dependent degradation mechanisms often relies on post-mortem analysis of cells, while model systems may be required to characterize spatial non-uniformities in degradation adequately. These challenges, along with the need for a comprehensive understanding of the degradation mechanisms, have led to the development of rigorous electrochemical models for battery degradation. These physics-based models serve as a powerful complementary tool in obtaining a better understanding of such phenomena. Simulating performance scenarios can lead to significant improvements in cell design and experimentation while also reducing time and cost commitments for the same.The primary goal of this work is the development and validation of a physicochemical degradation model for automotive lithium-ion batteries to predict the degradation in the large-format cell in the presence of temperature gradients using the given set of cell parameters with temperature dependence. It was realized that the degradation trajectories are a result of a combination of both LLI (loss of lithium inventory) and LAM (loss of active material) fade mechanisms at both anode and cathode. The state-of-the-art electrochemical models for lithium-ion cells (p2D models) were implemented in both in-house codes and equation-based formulation in COMSOL Multiphysics with weak form implementation. The codes were validated against experimental cycling data at different operating temperatures at each stage of the model development. The validated codes possess the ability to estimate local degradation rates and consequent impact on cycling performance for EV batteries at a range of operating conditions. Thus, the model is a means of identifying dominant degradation mechanisms in practically relevant operating conditions, helping improve overall lifetime, and enhancing value by helping identify improvements in battery operations. The errors between the model and data can provide information on the suitability of a given model and the relative prevalence of a given degradation mechanism, with useful, practical benefits.

Full Equiphase Characteristic Mode Solution to Lossless Composite Metallic-Dielectric Problems

We present a new surface integral equation-based formulation for the characteristic modes of lossless composite metallic–dielectric objects. Compared with the existing formulations, the proposed formulation has a more compact form and is easier to be implemented for complex structures, especially for multilayer structures. The full modal electric and magnetic currents are obtained simultaneously, and both modal electric and magnetic currents are equiphase. It should be mentioned that the derivation of the proposed formulation completely avoids the commonly used process of current elimination to obtain most of the existing formulations. Numerical results validate that the proposed formulation can provide accurate modal solutions to lossless composite metallic–dielectric problems.

Identifiability-Guaranteed Simplex-Structured Post-Nonlinear Mixture Learning via Autoencoder

This work focuses on the problem of unraveling nonlinearly mixed latent components in an unsupervised manner. The latent components are assumed to reside in the probability simplex, and are transformed by an unknown post-nonlinear mixing system. This problem finds various applications in signal and data analytics, e.g., nonlinear hyperspectral unmixing, image embedding, and nonlinear clustering. Linear mixture learning problems are already ill-posed, as identifiability of the target latent components is hard to establish in general. With unknown nonlinearity involved, the problem is even more challenging. Prior work offered a function equation-based formulation for provable latent component identification. However, the identifiability conditions are somewhat stringent and unrealistic. In addition, the identifiability analysis is based on the infinite sample (i.e., population) case, while the understanding for practical finite sample cases has been elusive. Moreover, the algorithm in the prior work trades model expressiveness with computational convenience, which often hinders the learning performance. Our contribution is threefold. First, new identifiability conditions are derived under largely relaxed assumptions. Second, comprehensive sample complexity results are presented -- which are the first of the kind. Third, a constrained autoencoder-based algorithmic framework is proposed for implementation, which effectively circumvents the challenges in the existing algorithm. Synthetic and real experiments corroborate our theoretical analyses.

Porting Wire-Grid MoM Framework to Reconfigurable Computing Technology

In this letter, a CPU/field-programmable gate array (FPGA) coprocessing implementation of the matrix assembly phase of the method of moments (MoM) is presented. The approach combines the frequency-domain integral equation-based formulation with piecewise-linear (triangular) basis function and method of images to evaluate the radiation pattern of the localizer antenna system. To benefit from the given hardware architecture, the code executed on the device runs only one single work-item OpenCL kernel. Using the Nallatech 385 A device featuring one Intel Arria 10 FPGA, a speedup ratio of about 2.08× is reported when compared to the reference single-core CPU approach, and about 1.11× when compared to the two-core CPU implementation.

Unified Implementation and Cross-Validation of the Integral Equation-Based Formulations for the Characteristic Modes of Dielectric Bodies

Unified Implementation and Cross-Validation of the Integral Equation-Based Formulations for the Characteristic Modes of Dielectric Bodies

Shortcut-based optimization of distillation-based processes by a novel reformulation of the feed angle method

The conceptual design of distillation-based separation processes is a well known and thoroughly investigated research subject in chemical engineering science. Nevertheless, there are only few shortcut methods available that are derived without simplifying assumptions, such as constant relative volatilities or constant molar overflow, allowing for a thermodynamically sound evaluation of the minimum energy demand for the separation of non-ideal mixtures. The current article presents a novel reformulation of the previously introduced feed angle method, which represents a geometrical shortcut method that is based on a direct calculation of characteristic pinch points. The reformulation retains the beneficial properties of the original formulation, providing high accuracy and an equation-based formulation that can be embedded in an optimization problem, while being numerically simpler and requiring significantly less case-specific modifications. The benefit of the so-called FAM+ formulation and the applicability to complex design problems are illustrated on several case studies, including the optimization of a closed-loop flowsheet and a superstructure with two reactors and several distillation columns.

A Fractional-Order Variational Framework for Retinex: Fractional-Order Partial Differential Equation-Based Formulation for Multi-Scale Nonlocal Contrast Enhancement with Texture Preserving.

This paper discusses a novel conceptual formulation of the fractional-order variational framework for retinex, which is a fractional-order partial differential equation (FPDE) formulation of retinex for the multi-scale nonlocal contrast enhancement with texture preserving. The well-known shortcomings of traditional integer-order computation-based contrast-enhancement algorithms, such as ringing artefacts and staircase effects, are still in great need of special research attention. Fractional calculus has potentially received prominence in applications in the domain of signal processing and image processing mainly because of its strengths like long-term memory, nonlocality, and weak singularity, and because of the ability of a fractional differential to enhance the complex textural details of an image in a nonlinear manner. Therefore, in an attempt to address the aforementioned problems associated with traditional integer-order computation-based contrast-enhancement algorithms, we have studied here, as an interesting theoretical problem, whether it will be possible to hybridize the capabilities of preserving the edges and the textural details of fractional calculus with texture image multi-scale nonlocal contrast enhancement. Motivated by this need, in this paper, we introduce a novel conceptual formulation of the fractional-order variational framework for retinex. First, we implement the FPDE by means of the fractional-order steepest descent method. Second, we discuss the implementation of the restrictive fractional-order optimization algorithm and the fractional-order Courant-Friedrichs-Lewy condition. Third, we perform experiments to analyze the capability of the FPDE to preserve edges and textural details, while enhancing the contrast. The capability of the FPDE to preserve edges and textural details is a fundamental important advantage, which makes our proposed algorithm superior to the traditional integer-order computation-based contrast enhancement algorithms, especially for images rich in textural details.

Entropy-guided switching trimmed mean deviation-boosted anisotropic diffusion filter

This report describes the experimental analysis of a proposed switching filter-anisotropic diffusion hybrid for the filtering of the fixed value (salt and pepper) impulse noise (FVIN). The filter works well at both low and high noise densities though it was specifically designed for high noise density levels. The filter combines the switching mechanism of decision-based filters and the partial differential equation-based formulation to yield a powerful system capable of recovering the image signals at very high noise levels. Experimental results indicate that the filter surpasses other filters, especially at very high noise levels. Additionally, its adaptive nature ensures that the performance is guided by the metrics obtained from the noisy input image. The filter algorithm is of both global and local nature, where the former is chosen to reduce computation time and complexity, while the latter is used for best results.

Theorem-Proving Analysis of Digital Control Logic Interacting with Continuous Dynamics

This work outlines an equation-based formulation of a digital control program and transducer interacting with a continuous physical process, and an approach using the Coq theorem prover for verifying the performance of the combined hybrid system. Considering thermal dynamics with linear dissipation for simplicity, we focus on a generalizable, physically consistent description of the interaction of the real-valued temperature and the digital program acting as a thermostat. Of interest in this work is the discovery and formal proof of bounds on the temperature, the degree of variation, and other performance characteristics. Our approach explicitly addresses the need to mathematically represent the decision problem inherent in an analog-to-digital converter, which for rare values can take an arbitrarily long time to produce a digital answer (the so-called Buridan's Principle); this constraint ineluctably manifests itself in the verification of thermostat performance. Furthermore, the temporal causality constraints in the thermal physics must be made explicit to obtain a consistent model for analysis. We discuss the significance of these findings toward the verification of digital control for more complex physical variables and fields.

Stochastic analysis of complex reaction networks using binomial moment equations

The stochastic analysis of complex reaction networks is a difficult problem because the number of microscopic states in such systems increases exponentially with the number of reactive species. Direct integration of the master equation is thus infeasible and is most often replaced by Monte Carlo simulations. While Monte Carlo simulations are a highly effective tool, equation-based formulations are more amenable to analytical treatment and may provide deeper insight into the dynamics of the network. Here, we present a highly efficient equation-based method for the analysis of stochastic reaction networks. The method is based on the recently introduced binomial moment equations [Barzel and Biham, Phys. Rev. Lett. 106, 150602 (2011)]. The binomial moments are linear combinations of the ordinary moments of the probability distribution function of the population sizes of the interacting species. They capture the essential combinatorics of the reaction processes reflecting their stoichiometric structure. This leads to a simple and transparent form of the equations, and allows a highly efficient and surprisingly simple truncation scheme. Unlike ordinary moment equations, in which the inclusion of high order moments is prohibitively complicated, the binomial moment equations can be easily constructed up to any desired order. The result is a set of equations that enables the stochastic analysis of complex reaction networks under a broad range of conditions. The number of equations is dramatically reduced from the exponential proliferation of the master equation to a polynomial (and often quadratic) dependence on the number of reactive species in the binomial moment equations. The aim of this paper is twofold: to present a complete derivation of the binomial moment equations; to demonstrate the applicability of the moment equations for a representative set of example networks, in which stochastic effects play an important role.

Multi-scale modeling of fatigue crack propagation applied to random sequence of clustered loading

Fatigue crack propagation in marine structures is obviously governed by mechanics of the considerably different four levels of multi-scale problems. Problems of structural response to environmental loads have length scale of several hundred meters, whereas possible detectable size of cracks from initial defects in a weld is of the order of millimeters. Once a fatigue crack initiates, crack tip plasticity is of the order of several grain sizes, while the resulting fatigue crack growth in each load cycle is of the order of nanometers. In our previous work, the first author and their associates have developed the so-called CP-System, which can treat the first two multi-level problems as an integrated system. Furthermore, we have incorporated the third level of mechanics by using the stress intensity range corresponding to the repeated tensile plastic deformation ahead of the crack tip. In the present paper, we shall discuss a more rational integral equation-based formulation in order to integrate the third and fourth levels of micro-mechanics to the first two levels of continuum mechanics. The method is then applied to fatigue crack propagation under the effects of random sequence of clustered loading. As an example of the random sequence of clustered load, we shall use the so-called “storm model”. In the crack propagation simulation, we have to take into account of the plastic wake on the crack surfaces, whose thicknesses are influenced by the material parameters involved in the crack growth model. These parameters are first identified by the fatigue tests under combined constant and random loading using a CT specimen. Then, fatigue crack growth is investigated by numerical simulation and fatigue tests for various random sequences of clustered loading. The experimental and numerical results agree quite well with each other, and fatigue crack propagation is found to be considerably retarded under random sequence loading, so that the conventional equivalent stress approach may provide rather conservative results to the real seaway loading.

MPEC strategies for optimization of a class of hybrid dynamic systems

With the development and widespread use of large-scale nonlinear programming (NLP) tools for process optimization, there has been an associated application of NLP formulations with complementarity constraints in order to represent discrete decisions. In particular, these constraints arise frequently in equation-based formulations for real-time optimization. Also known as mathematical programs with equilibrium constraints (MPECs), these formulations can be used to model certain classes of discrete events and can be more efficient than a mixed integer formulation, particularly for large systems with many discrete decisions, such as dynamic systems with switches at any point in time. In this study, we consider and extend MPEC formulations for the optimization of a class of hybrid dynamic models, where the differential states remain continuous over time. These include differential inclusions of the Filippov type. Here, particular care is required in the formulation in order to preserve smoothness properties of the dynamic system. Results on three case studies, including process control examples, illustrate the effectiveness and accuracy of the proposed MPEC optimization methodology for a class of hybrid dynamic systems.

Stationary voltage and current excited complex system of multimaterial conductors with BEM

A new, complete, integral equation-based formulation for the computation of stationary current distribution in multimaterial, multibody, massive conductors is proposed and compared with other classical integral formulations. Furthermore, simple equivalence allows this method to treat pure Neumann problems without the need for regularization. Examples of complex bus-bar systems in power transformers are provided, and finite-element method comparisons attest to the strength of this method. Generalization to a fully symmetric formulation is also discussed

Smoothing methods for complementarity problems in process engineering

AbstractComplementarity is central to all constrained optimization problems. However, direct enforcement of complementarity conditions is difficult because of the inherent nondifferentiability associated with them. Here, a class of smoothing methods for solving the complementarity problem by using a continuation algorithm to solve nonlinear equations is studied. The applicability of smoothing methods to approximate complicated nested derivative discontinuities is investigated using simple functions with a single smoothing parameter. In addition, an equation‐based formulation for solving the phase equilibrium problem with complementarity conditions is formulated. This approach can model the appearance and disappearance of phases directly in phase equilibrium problems. Moreover, it is shown how smoothing methods can be used to solve limiting distillation cases, such as dry and vaporless trays, modeled within an equation‐based formulation.

Application of on-surface MEI method on wire antennas

The formulas of the on-surface measured equation of invariance (OSMEI) for wire antennas are derived. The unknowns of each node on the antenna surface are expressed by the vector potential function and surface current density. The unknowns in the vicinity of each node are coupled in a linear equation and the coefficients of the linear equation are determined by the measured equation of invariance (MEI) method. The final impedance matrix obtained by the OSMEI is a highly sparse matrix. It demonstrates that the currents on thin wire antennas may also be solved by a differential equation-based formulation in addition to the conventional integral equations.

Regularized integral equations and curvilinear boundary elements for electromagnetic wave scattering in three dimensions

The boundary integral equations (BIEs), in their original forms, which govern the electromagnetic (EM) wave scattering in three-dimensional space contain at least a hypersingularity (1/R/sup 3/) or a Cauchy-singularity (1/R/sup 2/), usually both. Thus, obtaining reliable numerical solutions using such equations requires considerable care, especially when developing systematic numerical integration procedures for realistic problems. Regularized BIEs for the numerical computation of time-harmonic EM scattering fields due to arbitrarily-shaped scatterers are introduced. Two regularization approaches utilizing an isolation method plus a mapping are presented to remove all singularities prior to numerical integration. Both approaches differ from all existing approaches to EM scattering problems. Both work for integral equations initially containing either hypersingularities or Cauchy-singularities, without the need to introduce surface divergences or other derivatives of the EM fields on the boundary. Also, neither approach is limited to flat surfaces nor flat-element models of curved surfaces. The Muller linear combination of the electric- and magnetic-field integral equations (EFIE) and (MFIE) is used to avoid the resonance difficulty that is usually associated with integral equation-based formulations. Some preliminary numerical results for EM scattering due to single and multiple dielectric spheres are presented and compared with analytical solutions.