Standard Numerical Solution Research Articles

In Silico Design of Freeform Solar Cell Structures from High‐Throughput Artificial Intelligence‐Generated Configurations

Recent advances in artificial intelligence‐generated configurations (AIGC) have transformed various fields in science and technology domains and enabled the inverse design of materials and structures for enhanced performance with minimal human input. The patterned micro‐/nanostructures are widely adopted to increase power conversion efficiencies in thin‐film organic solar cells (OSCs), making these structures highly suitable for AIGC applications. Although the computational cost of traditional numerical simulations is a barrier to making AIGC high throughput, this issue is addressed by integrating the high‐speed automated machine learning (AutoML) Analyzer with genetic algorithm‐based topology optimization algorithms. Compared to standard numerical solutions, the AutoML Analyzer in the proposed system predicts outcomes ≈22 700 times faster, resulting in a 98.47% reduction in computation costs, while maintaining high average accuracies of ≈99%. By evaluating an extensive dataset of 600 000 AIGCs, an optimized PM6:Y6 OSC device design with a power conversion efficiency of 17.58% is identified, significantly outperforming the 14.70% baseline efficiency of the OSC device without the poly methyl methacrylate scattering and substrate layers. Results underscore the potential of AIGC techniques for efficiently enhancing the performance of photovoltaic devices in renewable energy applications.

Semi-analytical solution of the Reynolds equation considering cavitation

The numerical effort of rotordynamic simulations with hydrodynamic bearings is often dominated by the solution of the Reynolds equation. Because of the nonlinear fluid–structure interaction, this equation needs to be solved in every time step. Although computationally efficient analytical approximations and look-up table techniques exist, these approaches often do not provide the necessary modeling depth and accuracy. Thus, in many cases, numerical methods are preferred despite their computational cost.In recent studies, a semi-analytical solution of the Reynolds equation has been developed based on the scaled boundary finite element method (SBFEM) with the objective of improving the numerical efficiency. The developed approach assumed Gümbel conditions, which means that cavitation was handled in a highly simplified manner. In this work, the SBFEM solution is for the first time combined with a more complex, nonlinear cavitation model based on the Elrod algorithm. Because of the semi-analytical technique, this requires a simplification of the switch function that defines the locations of the pressure and cavitation zones. However, as demonstrated in this paper, the resulting bearing forces are still in good agreement with a standard numerical reference solution based on the finite volume method (FVM). Since the SBFEM model uses only a one-dimensional discretization, its high-order formulation is very straightforward, even in case of a non-equidistant grid with varying shape functions that take the smooth and non-smooth regions of the solution into account. As a result, the numerical effort is reduced significantly in comparison to the standard FVM.

Numerical Computation of SEIR Model for the Zika Virus Spreading

The purpose of this study is to present the numerical performances and interpretations of the SEIR nonlinear system based on the Zika virus spreading by using the stochastic neural networks based intelligent computing solver. The epidemic form of the nonlinear system represents the four dynamics of the patients, susceptible patients S(y), exposed patients hospitalized in hospital E(y), infected patients I(y), and recovered patients R(y), i.e., SEIR model. The computing numerical outcomes and performances of the system are examined by using the artificial neural networks (ANNs) and the scaled conjugate gradient (SCG) for the training of the networks, i.e., ANNs-SCG. The correctness of the ANNs-SCG scheme is observed by comparing the proposed and reference solutions for three cases of the SEIR model to solve the nonlinear system based on the Zika virus spreading dynamics through the knacks of ANNs-SCG procedure based on exhaustive experimentations. The outcomes of the ANNs-SCG algorithm are found consistently in good agreement with standard numerical solutions with negligible errors. Moreover, the procedure’s constancy, dependability, and exactness are perceived by using the values of state transitions, error histogram measures, correlation, and regression analysis.

Reducing the complexity of chemical networks via interpretable autoencoders

In many astrophysical applications, the cost of solving a chemical network represented by a system of ordinary differential equations (ODEs) grows significantly with the size of the network and can often represent a significant computational bottleneck, particularly in coupled chemo-dynamical models. Although standard numerical techniques and complex solutions tailored to thermochemistry can somewhat reduce the cost, more recently, machine learning algorithms have begun to attack this challenge via data-driven dimensional reduction techniques. In this work, we present a new class of methods that take advantage of machine learning techniques to reduce complex data sets (autoencoders), the optimization of multiparameter systems (standard backpropagation), and the robustness of well-established ODE solvers to to explicitly incorporate time dependence. This new method allows us to find a compressed and simplified version of a large chemical network in a semiautomated fashion that can be solved with a standard ODE solver, while also enabling interpretability of the compressed, latent network. As a proof of concept, we tested the method on an astrophysically relevant chemical network with 29 species and 224 reactions, obtaining a reduced but representative network with only 5 species and 12 reactions, and an increase in speed by a factor 65.

Resilient Adaptive Parallel sImulator for griD (RAPID): An Open Source Power System Simulation Toolbox

This paper describes an open source power system simulation toolbox: Resilient Adaptive Parallel sImulator for griD (RAPID), a package of Python codes that implements an advanced power system dynamic simulation framework. The main function of RAPID is time-domain simulation, and it contains details of the solution process, including the quasi-steady-state power flow analysis, to solve nonlinear differential algebraic equations describing the detailed representation of network and transient stability models of dynamic devices in a computationally efficient manner. In particular, RAPID utilizes and incorporates emerging solution techniques; a novel “parallel-in-time” (Parareal) algorithm, adaptive model reduction, and semi-analytical solution methods as well as the standard numerical solution methods. Moreover, the whole simulation process for the transmission network has been coupled with OpenDSS, a widely used open-source distribution system simulator, to enable the co-simulation of integrated transmission and distribution systems. Basic structure and features of RAPID are presented to provide an easy-to-understand guideline for the usage of the tool and illustrate its unique capabilities. This paper also presents simulation results for a number of test scenarios and demonstrates RAPID’s values for advancing power systems research in dynamic modeling and simulation techniques toward researchers as well as educators and students.

Approximate analytical solution for transient heat and mass transfer across an irregular interface

Motivated by practical applications in heat conduction and contaminant transport, we consider heat and mass diffusion across a perturbed interface separating two finite regions of distinct diffusivity. Under the assumption of continuity of the solution and diffusive flux at the interface, we use perturbation theory to develop an asymptotic expansion of the solution valid for small perturbations. Each term in the asymptotic expansion satisfies an initial–boundary value problem on the unperturbed domain subject to interface conditions depending on the previously determined terms in the asymptotic expansion. Demonstration of the perturbation solution is carried out for a specific, practically-relevant set of initial and boundary conditions with semi-analytical solutions of the initial–boundary value problems developed using standard Laplace transform and eigenfunction expansion techniques. Results for several choices of the perturbed interface confirm the perturbation solution is in good agreement with a standard numerical solution.

New Semi-Analytical Solutions for Advection–Dispersion Equations in Multilayer Porous Media

A new semi-analytical solution to the advection-dispersion-reaction equation for modelling solute transport in layered porous media is derived using the Laplace transform. Our solution approach involves introducing unknown functions representing the dispersive flux at the interfaces between adjacent layers, allowing the multilayer problem to be solved separately on each layer in the Laplace domain before being numerically inverted back to the time domain. The derived solution is applicable to the most general form of linear advection-dispersion-reaction equation, a finite medium comprising an arbitrary number of layers, continuity of concentration and dispersive flux at the interfaces between adjacent layers and transient boundary conditions of arbitrary type at the inlet and outlet. The derived semi-analytical solution extends and addresses deficiencies of existing analytical solutions in a layered medium, which consider analogous processes such as diffusion or reaction-diffusion only and/or require the solution of complicated nonlinear transcendental equations to evaluate the solution expressions. Code implementing our semi-analytical solution is supplied and applied to a selection of test cases, with the reported results in excellent agreement with a standard numerical solution and other analytical results available in the literature.

Approximate solutions for propagating light beams in a material with quadratic, exponential, and quartic transverse refractive index profiles

This work develops approximate solutions for the electric field associated with a Gaussian beam moving through a material with a transverse variation in the refractive index. The solutions rely on a Lagrangian problem formulation whereby the path taken by the beam is governed by an ordinary differential equation and the intensity is governed by a continuity equation. Quadratic, exponential, and quartic transverse variations in refractive index are considered. The first (quadratic) case can be handled analytically while the latter two (quartic, exponential) are handled using a new, semi-analytical approach. In each case the approximations are compared to the standard numerical solution.

Mesoscale Electrostatics Driving Particle Dynamics in Nonhomogeneous Dielectrics.

We introduce a density functional-based formalism to compute the electrostatic energy and forces for a mesoscopic system in the condensed phase, described with molecular resolution. The dielectric permittivity is variable in space, and it is dependent on the density fields of the individual particles present in the system. The electrostatic potential is obtained from standard numerical solutions of the generalized Poisson equation. The presence of a particle-dependent varying dielectrics produces the appearance of mesoscopic polarization forces, which are dependent on the local fluctuations of the permittivity, as well as of the electrostatic field. The proposed implementation is numerically robust, with an error on the Coulomb forces that can be systematically controlled by the mesh of spatial grid used for solving the generalized Poisson equation. We show that the method presented here is able to reproduce the concentration-dependent partitioning of an ideal salt in water/oil mixtures, in particular, reproducing the ∝ 1/ϵ dependency of the partition coefficient for the free ions predicted by Born theory. Moreover, this approach reproduces the correct electrostatic features of both dipolar and charged lipid bilayers, with positive membrane and dipole potentials. The sum of both Coulomb and polarization interactions inside the membrane yields a globally repulsive potential of mean force for the ions, independently on their charge. The computational efficiency of the method makes it particularly suitable for the description of large-scale polyelectrolyte soft-matter systems.

Isogeometric analysis for 2D and 3D curl–div problems: Spectral symbols and fast iterative solvers

Alfvén-like operators are of interest in magnetohydrodynamics, which is used in plasma physics to study the macroscopic behavior of plasma. Motivated by this important and complex application, we focus on a parameter-dependent curl–div problem that can be seen as a prototype of an Alfvén-like operator, and we discretize it using isogeometric analysis based on tensor-product B-splines. The involved coefficient matrices can be very ill-conditioned, so that standard numerical solution methods perform quite poorly here. In order to overcome the difficulties caused by such ill-conditioning, a two-step strategy is proposed. First, we conduct a detailed spectral study of the coefficient matrices, highlighting the critical dependence on the different physical and approximation parameters. Second, we exploit such spectral information to design fast iterative solvers for the corresponding linear systems. For the first goal we apply the theory of (multilevel block) Toeplitz and generalized locally Toeplitz sequences, while for the second we use a combination of multigrid techniques and preconditioned Krylov solvers. Several numerical tests are provided both for the study of the spectral problem and for the solution of the corresponding linear systems.

Performance, Thermal Stability and Optimum Design Analyses of Rectangular Fin with Temperature-Dependent Thermal Properties and Internal Heat Generation

In this study, we analysed the thermal performance, thermal stability and optimum design analyses of a longitudinal, rectangular fin with temperature-dependent, thermal properties and internal heat generation under multi-boiling heat transfer using Haar wavelet collocation method. The effects of the key and controlling parameters on the thermal performance of the fin are investigated. The thermal stability criteria and optimum design parameter were established. From the investigation, the study reveals that the performance of the fin is enhanced as the boiling condition parameter or the exponent decreases. It is also established that the optimum fin length (at which Q/ζ reaches a maximum value) increases as the non-linear thermal conductivity term β, increases. Furthermore, the study shows that the optimum value of M can be obtained based on the value of the non-linear term. The computational results obtained in this study were compared with established numerical solutions and is found to be in good agreement with the standard numerical solutions.

Periodic Solutions of Duffing-Type Oscillators Using Continuous Piecewise Linearization Method

A new analytic algorithm called ‘continuous piecewise linearization method’ (CPLM) is developed to obtain periodic solutions of freely vibrating Duffing-type oscillators. This simple analytic algorithm is based on continuous piecewise linearization of the nonlinear stiffness with respect to displacement and was shown to produce very accurate results for few iterations. The algorithm is valid for Duffing-type oscillators possessing strong nonlinearity and/or undergoing large-amplitude oscillations. Studies conducted on Duffing oscillators with cubic, cubic-quintic and trigonometric sine stiffness nonlinearities showed that the CPLM results match standard numerical solutions and is more accurate than the popular energy balance method (EBM). Additionally, the present analysis shows that the CPLM is capable of predicting the quasi-linear behaviour observed in the oscillation history of Duffing-type oscillators with strongly nonlinearity and/or large-amplitude oscillations. This quasi-linear behaviour cannot be predicted by the EBM to which the CPLM is compared.

Intelligent computing to analyze the dynamics of Magnetohydrodynamic flow over stretchable rotating disk model

In this study a novel application of neurocomputing technique is presented for nonlinear fluid mechanics problem arising in the model of the flow over stretchable rotating disk in the presence of strong magnetic field. The scheme comprises of the power of effective modelling of neural networks supported with integrated optimization strength of genetic algorithm and interior-point method. The governing partial differential equation of the system is converted to nonlinear systems of simultaneous ordinary differential equations by incorporating the similarity variables. Neural network based approximate differential equation models are formulated for the transformed system that are used to construct the merit function in mean squared error sense. The networks are trained initially by genetic algorithm for the global search and rapid local refinements is attained through efficient interior point method. The given scheme is applied for dynamical analysis of the system model in terms of radial, tangential, axial velocities and heat effects by varying magnetic interaction parameters, unsteadiness factors, disk stretchable magnitudes, and Prandtl numbers. The statistical performance indices based on error from standard numerical solutions are used to validate the correctness, consistency, robustness and stability of the proposed stochastic solver.

Haar Wavelet Collocation Method for Thermal Analysis of Porous Fin with Temperature-dependent Thermal Conductivity and Internal Heat Generation

In this study, the thermal performance analysis of porous fin with temperature-dependent thermal conductivity and internal heat generation is carried out using Haar wavelet collocation method. The effects of various parameters on the thermal characteristics of the porous fin are investigated. It is found that as the porosity increases, the rate of heat transfer from the fin increases and the thermal performance of the porous fin increases. The numerical solutions by the Haar wavelet collocation method are in good agreement with the standard numerical solutions.

Efficient Closed-Form Solution of the Kinematics of a Tunnel Digging Machine

In this work, the kinematics of a large size tunnel digging machine is investigated. The closed-loop mechanism is made by 13 links and 13 class 1 couplings, seven of which are actuated. This kind of machines are commonly used to perform ground drilling for the placement of reinforcement elements during the construction of tunnels. The direct kinematic solution is obtained using three methods: the first two are based on the numerical solution of the closure equation written using the Denavit–Hartenberg convention, while the third is based on the definition and solution in closed form of an equivalent spherical mechanism. The procedures have been tested and implemented with reference to a real commercial tunnel digging machine. The use of the proposed method for the closed-form solution of direct kinematics allows to obtain a major reduction of the computation time in comparison with the standard numerical solution of the closure equation.

Role of qubit-cavity entanglement for switching dynamics of quantum interfaces in superconductor metamaterials

We study quantum effects of strong driving field applied to dissipative hybrid qubit-cavity system which are relevant for a realization of quantum gates in superconducting quantum metamaterials. We demonstrate that effects of strong and non-stationary drivings have significantly quantum nature and can not be treated by means of mean-field approximation. This is shown from a comparison of steady state solution of the standard Maxwell-Bloch equations and numerical solution of Lindblad equation on a density matrix. We show that mean-field approach provides very good agreement with the density matrix solution at not very strong drivings $f<f^*$ but at $f>f^*$ a growing value of quantum correlations between fluctuations in qubit and photon sectors changes a behavior of the system. We show that in regime of non-adiabatic switching on of the driving such a quantum correlations influence a dynamics of qubit and photons even at weak $f$.

Unsupervised neural network model optimized with evolutionary computations for solving variants of nonlinear MHD Jeffery-Hamel problem

A heuristic technique is developed for a nonlinear magnetohydrodynamics (MHD) Jeffery-Hamel problem with the help of the feed-forward artificial neural network (ANN) optimized with the genetic algorithm (GA) and the sequential quadratic programming (SQP) method. The two-dimensional (2D) MHD Jeffery-Hamel problem is transformed into a higher order boundary value problem (BVP) of ordinary differential equations (ODEs). The mathematical model of the transformed BVP is formulated with the ANN in an unsupervised manner. The training of the weights of the ANN is carried out with the evolutionary calculation based on the GA hybridized with the SQP method for the rapid local convergence. The proposed scheme is evaluated on the variants of the Jeffery-Hamel flow by varying the Reynold number, the Hartmann number, and the angles of the walls. A large number of simulations are performed with an extensive analysis to validate the accuracy, convergence, and effectiveness of the scheme. The comparison of the standard numerical solution and the analytic solution establishes the correctness of the proposed designed methodologies.

Closure to “Evaluation of Explicit Numerical Solution Methods of the Muskingum Model” by Ali R. Vatankhah

Closure to “Evaluation of Explicit Numerical Solution Methods of the Muskingum Model” by Ali R. Vatankhah

Negative-saturation approach for (non)-isothermal compositional three-phase flow simulations in porous media

This article describes the extension of the two-phase (non)-isothermal negative-saturation solution approach to the three-phase (non)-isothermal negative-saturation (NegSat3) solution approach. The NegSat3 solution approach solves efficiently any (non)-isothermal compositional flow problem in porous media that involves phase transitions between different phase states when the maximum number of phases is less than four. The solution approach circumvents using different equations and primary variables for single-phase, two-phase, and three-phase regions in porous media. Consequently, the NegSat3 solution approach avoids switches or variable substitutions at phase transitions where the convergence of the Newton-Raphson procedure is hampered by oscillations between m-phase and n-phase states. The NegSat3 solution approach can be implemented efficiently in numerical simulators to deal with modeling issues for thermal recovery processes, CO2 sequestration, and for multicontact miscible gas injection in oil reservoirs if the number of phases is less than four. We illustrate the NegSat3 solution approach by way of example to steam injection in a 1D heavy-oil reservoir. However, the NegSat3 solution approach can be used for any n ≤ 3-dimensional, multicomponent three-phase problems. The example solution is compared with a standard numerical solution that is analytically verified by the method of characteristics and shows excellent agreement. The results show that the oil recovery depends critically on whether the boiling temperature of the volatile oil is around the water boiling temperature, or much below or above it. These boiling-temperature ranges give rise to three different types of wave structures. When the boiling temperature of the volatile oil is near the boiling temperature of water, the striking result is that the speed of the evaporation front is equal or somewhat larger than the speed of the steam-condensation front. Thus, the volatile oil condenses at the location where the steam condenses too, yielding virtually complete oil recovery. Conversely, if the boiling temperature is too high or too low, there is incomplete recovery.

Exactly satisfying initial conditions neural network models for numerical treatment of first Painlevé equation

In this paper, novel computing approach using three different models of feed-forward artificial neural networks (ANNs) are presented for the solution of initial value problem (IVP) based on first Painlevé equation. These mathematical models of ANNs are developed in an unsupervised manner with capability to satisfy the initial conditions exactly using log-sigmoid, radial basis and tan-sigmoid transfer functions in hidden layers to approximate the solution of the problem. The training of design parameters in each model is performed with sequential quadratic programming technique. The accuracy, convergence and effectiveness of the proposed schemes are evaluated on the basis of the results of statistical analyses through sufficient large number of independent runs with different number of neurons in each model as well. The comparisons of these results of proposed schemes with standard numerical and analytical solutions validate the correctness of the design models.