Equation Solver Research Articles

Exergy analysis of ejector‐enhanced dual‐evaporator cycle using effective temperature method

AbstractThis study compares the exergy of an ejector‐based two evaporator cycle (EB‐TEC) with a conventional two evaporator cycle (C‐TEC). The analysis utilizes a modified Gouy–Stodola equation, which provides a more accurate insight of the system irreversibility compared to the standard Gouy–Stodola formulation. Furthermore, the comparison includes three working fluids, that is, R134a, R1234ze, and R600 in both the cycles. The study examines the effects of varying evaporators and condenser temperatures and the dryness fraction at the exit of Evaporator 1. The data is analyzed using an Engineering Equation Solver. The findings indicate that increasing the temperature of the low‐temperature evaporator leads to a drop in exergy losses and enhancement in exergy efficiency in both the cycles. When the temperature of Evaporator 1 is increased, the total exergy of the EB‐TEC is decreased but for the C‐TEC, it is increased. Furthermore, increasing the condenser temperature results in higher exergy destruction in both EB‐TEC and C‐TEC. Notably, the maximum exergy destruction is 49.44 kW for R600, whereas the minimum exergy destruction is 14.42 kW for R1234ze in the EB‐TEC.

Solving forward and inverse problems of contact mechanics using physics-informed neural networks

This paper explores the ability of physics-informed neural networks (PINNs) to solve forward and inverse problems of contact mechanics for small deformation elasticity. We deploy PINNs in a mixed-variable formulation enhanced by output transformation to enforce Dirichlet and Neumann boundary conditions as hard constraints. Inequality constraints of contact problems, namely Karush–Kuhn–Tucker (KKT) type conditions, are enforced as soft constraints by incorporating them into the loss function during network training. To formulate the loss function contribution of KKT constraints, existing approaches applied to elastoplasticity problems are investigated and we explore a nonlinear complementarity problem (NCP) function, namely Fischer–Burmeister, which possesses advantageous characteristics in terms of optimization. Based on the Hertzian contact problem, we show that PINNs can serve as pure partial differential equation (PDE) solver, as data-enhanced forward model, as inverse solver for parameter identification, and as fast-to-evaluate surrogate model. Furthermore, we demonstrate the importance of choosing proper hyperparameters, e.g. loss weights, and a combination of Adam and L-BFGS-B optimizers aiming for better results in terms of accuracy and training time.

Accurate Description of Coulombic Interactions in Organic Field‐Effect Transistors Enabled by Efficient 3D Poisson's Equation Solver with Mixed Boundary Conditions

AbstractThe rational development of organic electronic devices can greatly benefit from a molecular‐level description, where an accurate description of the Coulombic interactions among charge carriers holds paramount importance. However, the molecular‐level organic field‐effect transistor (OFET) simulations do not fully include the short‐range charge‐carrier Coulombic interactions, thus limiting their accuracy. Here, an efficient solution is demonstrated to the 3D Poisson's equation with mixed boundary conditions, optimized for integration into the simulations of organic electronic devices. It leverages the finite difference method for discretization, the Krylov subspace iterative methods for solving the linear system, the algebraic multigrid algorithm for preconditioning, and OpenMP and CUDA parallelization for acceleration. For the organic field‐effect transistors of micrometer sizes or smaller dimensions, obtaining the electric potential within the device takes less than 0.1 s, which is sufficiently efficient for molecular‐level modeling. As a result, both short‐ and long‐range electrostatic interactions are successfully incorporated in the molecular‐level modeling of OFET devices. This integration significantly enhances the robustness of OFET modeling for future applications.

LFT: Neural Ordinary Differential Equations With Learnable Final-Time.

Since the last decade, deep neural networks have shown remarkable capability in learning representations. The recently proposed neural ordinary differential equations (NODEs) can be viewed as the continuous-time equivalence of residual neural networks. It has been shown that NODEs have a tremendous advantage over the conventional counterparts in terms of spatial complexity for modeling continuous-time processes. However, existing NODEs methods entail their final time to be specified in advance, precluding the models from choosing a desirable final time and limiting their expressive capabilities. In this article, we propose learnable final-time (LFT) NODEs to overcome this limitation. LFT rebuilds the NODEs learning process as a final-time-free optimal control problem and employs the calculus of variations to derive the learning algorithm of NODEs. In contrast to existing NODEs methods, the new approach empowers the NODEs models to choose their suitable final time, thus being more flexible in adjusting the model depth for given tasks. Additionally, we analyze the gradient estimation errors caused by numerical ordinary differential equations (ODEs) solvers and employ checkpoint-based methods to obtain accurate gradients. We demonstrate the effectiveness of the proposed method with experimental results on continuous normalizing flows (CNFs) and feedforward models.

Optimizing Infragravity Wave Attenuation to Improve Coral Reef Restoration Design for Coastal Defense

Coral reefs are effective natural flood barriers that protect adjacent coastal communities. As the need to adapt to rising sea levels, storms, and environmental changes increases, reef restoration may be one of the best tools available to mitigate coastal flooding along tropical coastlines, now and in the future. Reefs act as a barrier to incoming short-wave energy but can amplify low-frequency infragravity waves that, in turn, drive coastal flooding along low-lying tropical coastlines. Here, we investigate whether the spacing of reef restoration elements can be optimized to maximize infragravity wave energy dissipation while minimizing the number of elements—a key factor in the cost of a restoration project. With this goal, we model the hydrodynamics of infragravity wave dissipation over a coral restoration or artificial reef, represented by a canopy of idealized hemispherical roughness elements, using a three-dimensional Navier–Stokes equations solver (OpenFOAM). The results demonstrate that denser canopies of restoration elements produce greater wave dissipation under larger waves with longer periods. Wave dissipation is also frequency-dependent: dense canopies remove wave energy at the predominant wave frequency, whereas sparse canopies remove energy at higher frequencies, and hence are less efficient. We also identify an inflection point in the canopy density–energy dissipation curve that balances optimal energy losses with a minimum number of canopy elements. Through this work, we show that there are an ideal number of restoration elements per across-shore meter of coral reef flat that can be installed to dissipate infragravity wave energy for given incident heights and periods. These results have implications for designing coral reef restoration projects on reef flats that are effective both from a coastal defense and costing standpoint.

Zeroing Neural Network With Coefficient Functions and Adjustable Parameters for Solving Time-Variant Sylvester Equation.

To solve the time-variant Sylvester equation, in 2013, Li et al. proposed the zeroing neural network with sign-bi-power function (ZNN-SBPF) model via constructing a nonlinear activation function. In this article, to further improve the convergence rate, the zeroing neural network with coefficient functions and adjustable parameters (ZNN-CFAP) model as a variation in zeroing neural network (ZNN) model is proposed. On the basis of the introduced coefficient functions, an appropriate ZNN-CFAP model can be chosen according to the error function. The high convergence rate of the ZNN-CFAP model can be achieved by choosing appropriate adjustable parameters. Moreover, the finite-time convergence property and convergence time upper bound of the ZNN-CFAP model are proved in theory. Computer simulations and numerical experiments are performed to illustrate the efficacy and validity of the ZNN-CFAP model in time-variant Sylvester equation solving. Comparative experiments among the ZNN-CFAP, ZNN-SBPF, and ZNN with linear function (ZNN-LF) models further substantiate the superiority of the ZNN-CFAP model in view of the convergence rate. Finally, the proposed ZNN-CFAP model is successfully applied to the tracking control of robot manipulator to verify its practicability.

Power plant systems performance assessment with applications of combined pinch and exergy analysis

Steam power plants forms an effective share of global electricity production. Thus, improving such plants performance is crucial to minimize its environmental effects and to enhance operational economy. Al-Mosyab thermal power station in Iraq is considered as a case study of this paper, with an objective of achieving best system performance. A combined pinch & exergy analysis (CPEA) is conducted under actual operating conditions. Further analytical attempt is varying pressure (165, 175, 185) bar, and temperature (450, 550, 620) °C at selected cycle location. Examining thereby the impact of each variable on the system power output, exergy destruction and 2nd low efficiency. The Engineering Equation Solving program (EES) is used for the simulation of each system components under these various operating conditions. The yielded results showed that the maximum mechanical & thermal output are (36.75 & 94.88) % of fuel energy respectively, while, turbine, boiler and condenser irreversibility and 2nd law efficiency are (157.76, 1086.18, 303.19) kJ/kg) and (88.36, 68.97, 28.68) % respectively. The analytical part of the analysis however showed that the high-pressure turbine network output increases by about (1%) with pressure variation, while such increase rise up to (70%) with temperature variation. Final concluded results predicted that best performance could be obtained at maximum pressure and temperature of (175) bar and (620) °C respectively.

Solving linear systems of fractional integro-differential equations by Haar and Legendre wavelets techniques

In this study, we present two highly effective approaches aimed at solving linear systems of equations, specifically focusing on the Fredholm and Volterra equations in fractional integro-differential formula. To tackle these challenging problems, we employ two distinct methodologies: the Haar wavelet technique (HWT) and the Legendre wavelet Technique (LWT). To accurately describe fractional derivatives with the equations, we adopt the Caputo sense, which provides a robust framework for handling fractional calculus. By applying HWT and LWT, we are able to transform the intricate equations of the fractional integro-differential system into algebraic equations. Fredholm and Volterra fractional integro-differential equations serve as benchmarks for evaluating the accuracy and efficiency of LWT and HWT. By comparing the approximate solutions obtained through these wavelet methods with the exact solutions, we are able to provide a comprehensive assessment of their performance. Through detailed numerical computations conducted using Mathematical 8,which demonstrate the effectiveness of both LWT and HWT in solving systems of Fredholm and Volterra equations in fractional integro-differential equations. The results obtained from these computations are presented through graphs, allowing for a clear and comparison between the approximate and exact solutions. The successful utilization of these wavelet methods provides researchers and practitioners with valuable tools for tackling complex mathematical problems in various fields of study. This study contributes to the progress of numerical techniques for solving of fractional integro-differential equations which aid in the development of more accurate and efficient computational approaches for analyzing and solving similar mathematical problems in the future.

Across Time and Space: Senju ’s Approach for Scaling Iterative Stencil Loop Accelerators on Single and Multiple FPGAs

Stencil-based applications play an essential role in high-performance systems as they occur in numerous computational areas, such as partial differential equation solving. In this context, Iterative Stencil Loops (ISLs) represent a prominent and well-known algorithmic class within the stencil domain. Specifically, ISL-based calculations iteratively apply the same stencil to a multi-dimensional point grid multiple times or until convergence. However, due to their iterative and intensive nature, ISLs are highly performance-hungry, demanding specialized solutions. Here, Field Programmable Gate Arrays (FPGAs) represent a valid architectural choice as they enable the design of custom, parallel, and scalable ISL accelerators. Besides, the regular structure of ISLs makes them an ideal candidate for automatic optimization and generation flows. For these reasons, this article introduces Senju , an automation framework for the design of highly parallel ISL accelerators targeting single-/multi-FPGA systems. Given an input description, Senju automates the entire design process and provides accurate performance estimations. The experimental evaluation shows remarkable and scalable results, outperforming single- and multi-FPGA literature approaches under different metrics. Finally, we present a new analysis of temporal and spatial parallelism trade-offs in a real-case scenario and discuss our performance through a single- and novel specialized multi-FPGA formulation of the Roofline Model.

Design of a two-renewable energy source-based system with thermal energy storage and hydrogen storage for sustainable development

A renewable energy-based system, including an energy storage subsystem, is designed within the scope of this proposed study to meet the fresh water and electricity demands of a community. The city of Imperial in the state of California is, in this regard, considered for a case study. The energy storage solutions are incorporated into the system to enhance the flexibility and efficiency of energy systems since solar and wind energy resources fluctuate. These energy storage options help offset the mismatch between demand and supply to provide feasible solutions practically. The current system employs storage technologies, such as thermal energy storage (TES) and hydrogen, to efficiently store excess energy for usage during peak demand periods. The parabolic through solar collector (PTSC) and wind turbine in the system are modelled using the System Advisor Model (SAM). Furthermore, the energy and exergy aspects of thermodynamic are considered when performing the thermodynamic analysis of the entire system. The Engineering Equation Solver (EES) is employed to construct the programming codes and perform a thermodynamic analysis of the overall system and its components. The effects of varying numerous parameters on the system performance are studied in this proposed study using a parametric investigation which is carried out through the changes in the ambient temperature, source temperatures, and mass flow rate of the current system. The results of this study show that the annual net amount of electricity generated by the proposed system is 74.90 GWh, which meets 11.7 % of the residential electricity needs of the Imperial City. The desalination unit in the system is also able to produce fresh water from seawater, meeting the needs of 4708 families. The current system further produces a total of 3379.53 tons of fresh water and 226.98 tons of H2 annually. As a result of the energy and exergy analyses performed for the designed system, the energy and exergy efficiencies for the overall system are found to be 47.5 % and 34.3 %, respectively.

An Application-Driven Method for Assembling Numerical Schemes for the Solution of Complex Multiphysics Problems

Within recent years, considerable progress has been made regarding high-performance solvers for partial differential equations (PDEs), yielding potential gains in efficiency compared to industry standard tools. However, the latter largely remains the status quo for scientists and engineers focusing on applying simulation tools to specific problems in practice. We attribute this growing technical gap to the increasing complexity and knowledge required to pick and assemble state-of-the-art methods. Thus, with this work, we initiate an effort to build a common taxonomy for the most popular grid-based approximation schemes to draw comparisons regarding accuracy and computational efficiency. We then build upon this foundation and introduce a method to systematically guide an application expert through classifying a given PDE problem setting and identifying a suitable numerical scheme. Great care is taken to ensure that making a choice this way is unambiguous, i.e., the goal is to obtain a clear and reproducible recommendation. Our method not only helps to identify and assemble suitable schemes but enables the unique combination of multiple methods on a per-field basis. We demonstrate this process and its effectiveness using different model problems, each comparing the resulting numerical scheme from our method with the next best choice. For both the Allen–Cahn and advection equations, we show that substantial computational gains can be attained for the recommended numerical methods regarding accuracy and efficiency. Lastly, we outline how one can systematically analyze and classify a coupled multiphysics problem of considerable complexity with six different unknown quantities, yielding an efficient, mixed discretization that in configuration compares well to high-performance implementations from the literature.

High Order Absolutely Convergent Fast Sweeping Methods with Multi-resolution WENO Local Solvers for Eikonal and Factored Eikonal Equations

High Order Absolutely Convergent Fast Sweeping Methods with Multi-resolution WENO Local Solvers for Eikonal and Factored Eikonal Equations

Utilizing optimal physics-informed neural networks for dynamical analysis of nanocomposite one-variable edge plates

This study presents an innovative Physics-Informed Neural Network (PINN) approach designed to predict the dynamic responses of the One-Variable Edge Plate (OVEP), a unique plate structure characterized by an edge defined by arbitrary mathematical functions. The OVEP is constructed from a nanocomposite material reinforced with graphene nanoplatelets. Utilizing an optimized PINN pipeline, this research successfully predicts the vibration characteristics of the OVEP, encompassing linear, nonlinear vibrations, and beating phenomena. The study also demonstrates that the optimal PINN outperforms conventional neural network in terms of stability, accuracy (achieving above 99% accuracy), and efficiency in predicting long-duration vibrations. Additionally, the computational time required for generating testing results is notably diminished compared to traditional partial differential equation (PDE) solvers (reduced by about 3 to 12 times). To demonstrate model robustness, synthetic noise is intentionally introduced into the training data. The results not only enhance our understanding of the complex dynamics of the OVEP but also highlight the effectiveness of the proposed PINN framework in capturing and forecasting the dynamic behaviors of advanced plate structures. This research presents a promising potential for addressing dynamic problems in the fields of aerospace, civil, and mechanical engineering.

Advancing energy transition with novel biomass-solar based multigeneration energy system using hydrogen and storage options for sustainable cities

An integrated energy system is designed to produce various useful products, such as hydrogen, electricity, heat and cooling. This integrated plant utilizes various subsystems; solar power system, biomass energy system, steam Rankine cycle (SRC), Brayton cycle (BC), organic Rankine cycle (ORC), double-effect absorption system (DEAS), reverse osmosis system, and sonohydrogen production unit. The specific energetic, exergetic, economic and environmental impact assessment studies are carried out to evaluate the system's performance and sustainability. The engineering equation solver (EES) is used for these studies. The energy efficiencies for the BC, ORC and SRC are recorded as 29.69 %, 10.49 %, and 30 %, respectively. In regard to exergy efficiency, the BC, ORC and SRC achieve the following: 46.74 %, 62.73 %, and 53.43 %, respectively. Furthermore, the DEAS attains energetic and exergetic COPs of 1.67 and 0.65, respectively. Additionally, the overall system's energy and exergy efficiencies are found to be 37.68 % and 71.25 %, respectively. The results from the thermal energy storage system indicate that the heat production begins at 7:00 a.m., and the charging of the storage unit commences two hours after and continues until 5:00 p.m., in coinciding with the availability of sunlight. The economic analysis demonstrates that with an initial investment of $53.80 million, the integrated multigeneration system achieves a net present value ranging from $6.87 million to $173.4 million across different hydrogen selling prices, from $7000 to $13,000 per ton up to 2054.

Techno-exergy-economic assessment of humidification-dehumidification/reverse osmosis hybrid desalination system integrated with concentrated photovoltaic/thermal

Desalination is a process used to generate water for human consumption, irrigation, or industrial purposes. Assessment of desalination plants is crucial for freshwater productivity, energy consumption, exergy destruction, and economic feasibility calculations. Therefore, the study aims to conduct a technical, exergy, and economic analysis of a concentrated photovoltaic/thermal hybrid humidification-dehumidification reverse osmosis desalination system. The photovoltaic/thermal provides the electrical and thermal requirements for the hybrid desalination unit. Theoretical models for each unit are developed and integrated using the Engineering Equation Solver software. The sensitivity of system performance parameters was investigated under different climatic and operation factors. The analysis shows that the system produces 860 L per hour at the lowest specific energy consumption and water cost (3.3 kW-hours per cubic meter and 0.916 United States dollars per cubic meter), respectively, with a cost payback period of only 12 years compared to other green-powered desalination systems. The highest destruction portion of 84.6 % is at photovoltaic/thermal, while the largest cost portion of 68 % is at reverse osmosis. Productivity, energy consumption, and water cost are influenced by feed salinity, while destruction is most affected by solar irradiance. Ambient temperature, wind speed, and cooling fluid flow rate slightly affect the techno-exergy-economics of the system.

A machine learning-driven approach to predicting thermo-elasto-hydrodynamic lubrication in journal bearings

Traditional methods of evaluating the performance of journal bearings, for example thermal-elastic-hydrodynamic- lubrication theory, are limited to simplified conditions that often fail to accurately model real-world components. Numerical models that include additional phenomena such as cavitation and fully coupled effects like deformation, temperature, pressure and viscosity can be more accurate but require a large amount of computational overhead, making analysis slower and more costly. To address this limitation, a novel machine learning-driven approach is developed to predict the 2D distribution of surface deformation, film thickness, temperature, and pressure across the bearing surface as a function of design variables such as load and speed. The training dataset, generated using a fully coupled Reynolds’ Equation solver implemented in OpenFOAM, contains a significantly extended range of conditions than in previous studies with approximately 39000000 points encompassing 4925 different test cases. Modelled bearing speeds range from 2000 to 10000rpm, while load values are varied between 1 and 30kN. Predicting surface deformation, film thickness, temperature and pressure across the bearing surface results in a mean absolute percentage error below 0.4% or better. The work also demonstrates that the trained models have a strong ability to generalise the prediction beyond the original training data range with only a 1% error at up to 200% of the highest trained speed. This work also demonstrates that machine learning-based processes are a practical alternative to physics-based numerical modelling, especially in cases where rapid performance evaluation is desired as real-time calculation is possible with significantly reduced computational cost. This has the potential to enable development of rapid design optimisation tools and real-time performance monitoring at high resolution and with low latency. Using consumer hardware, it is found that the neural network-based approach is faster than the existing numerical modelling technique by a factor of over 10000, enabling real-time predictions of lubrication systems.

A structure preserving numerical method for the ideal compressible MHD system

We introduce a novel structure-preserving method in order to approximate the compressible ideal Magnetohydrodynamics (MHD) equations. This technique addresses the MHD equations using a non-divergence formulation, where the contributions of the magnetic field to the momentum and total mechanical energy are treated as source terms. Our approach uses the Marchuk-Strang splitting technique and involves three distinct components: a compressible Euler solver, a source-system solver, and an update procedure for the total mechanical energy. The scheme allows for significant freedom on the choice of Euler's equation solver, while the magnetic field is discretized using a curl-conforming finite element space, yielding exact preservation of the involution constraints. We prove that the method preserves invariant domain properties, including positivity of density, positivity of internal energy, and the minimum principle of the specific entropy. If the scheme used to solve Euler's equation conserves total energy, then the resulting MHD scheme can be proven to preserve total energy. Similarly, if the scheme used to solve Euler's equation is entropy-stable, then the resulting MHD scheme is entropy stable as well. In our approach, the CFL condition does not depend on magnetosonic wave-speeds, but only on the usual maximum wavespeed from Euler's system. To validate the effectiveness of our method, we solve a variety of ideal MHD problems, showing that the method is capable of delivering second-order accuracy in space for smooth problems, while also offering unconditional robustness in the shock hydrodynamics regime as well.

Engineering topological interface states in metal-wire waveguides for broadband terahertz signal processing.

Innovative terahertz waveguides are in high demand to serve as a versatile platform for transporting and manipulating terahertz signals for the full deployment of future six-generation (6G) communication systems. Metal-wire waveguides have emerged as promising candidates, offering the crucial advantage of sustaining low-loss and low-dispersion propagation of broadband terahertz pulses. Recent advances have opened up new avenues for implementing signal-processing functionalities within metal-wire waveguides by directly engraving grooves along the wire surfaces. However, the challenge remains to design novel groove structures to unlock unprecedented signal-processing functionalities. In this study, we report a plasmonic signal processor by engineering topological interface states within a terahertz two-wire waveguide. We construct the interface by connecting two multiscale groove structures with distinct topological invariants, i.e., featuring a π-shift difference in the Zak phases. The existence of this topological interface within the waveguide is experimentally validated by investigating the transmission spectrum, revealing a prominent transmission peak in the center of the topological bandgap. Remarkably, we show that this resonance is highly robust against structural disorders, and its quality factor can be flexibly controlled. This unique feature not only facilitates essential functions such as band filtering and isolating but also promises to serve as a linear differential equation solver. Our approach paves the way for the development of new-generation all-optical analog signal processors tailored for future terahertz networks, featuring remarkable structural simplicity, ultrafast processing speeds, as well as highly reliable performance.

The numerical methods for solving of the one-dimensional anomalous reaction-diffusion equation

The numerical methods for solving of the one-dimensional anomalous reaction-diffusion equation

A Water-to-Water Heat Pump Model with Experimental Validation

An experimental validation of a steady-state model for water-to-water heat pumps is conducted on a 10 kW test bench. The objective of the model is to predict the capacity and the required compressor power, based on the inlet conditions of the secondary fluids in the evaporator and condenser. Detailed manufacturer performance maps based on the AHRI 540-2020 standard are utilized to model the fixed-speed scroll compressor. A new semi-empirical model for the thermostatic expansion valve incorporates condensing temperature effects on superheating prediction. Sub-models for individual components, including detailed representations of the evaporator and condenser, are integrated into a global model, resulting in a set nonlinear equation solved using an equation solver with appropriate guess values. The validation of the model is conducted in an experimental test facility equipped with two precisely controlled secondary fluid loops. The heat pump is instrumented to measure condensation and evaporation pressures, the compressor discharge temperature, compressor power, superheating, and sub-cooling. The results are divided into three sub-sections: the first validates the complete heat pump model by comparing its power consumption and COPs in heating and cooling; the second compares the predicted and measured operational conditions; finally, it is shown how the model can be used to predict the non-operational conditions of the heat pump for specific scenarios.