Steady-state Problem Research Articles

A pseudo-transient pressure gradient method for solving the incompressible Navier–Stokes equations

The fundamental problem behind the numerical solutions of incompressible viscous flow equations lies in the lack of an update equation for the pressure field. In essence, there is no explicit relation that can be utilized to update the pressure field from a given velocity field. Following the principle of the artificial compressibility method and considering that it is the pressure gradient that drives incompressible flows, this study introduces a new primitive variables method to solve the incompressible Navier–Stokes equations. This method ensures that the continuity condition is rigorously satisfied at every grid point within the flow domain. Its principle lies in modifying the continuity equation to include a pseudo-transient pressure gradient term that vanishes upon convergence, which allows the incompressibility constraint to be satisfied to machine zero. This method eliminates the numerical difficulties associated with pressure-based solvers and improves the convergence of all dependent variables. Numerical solutions for three steady-state validation problems, namely fully developed channel flow, lid-driven cavity flow (2D and 3D), and flow in a constricted channel, are obtained to validate the proposed method. The results show good agreement with solutions available in the literature and demonstrate substantially improved convergence.

Bayesian Inference for Estimating Heat Sources Through Temperature Assimilation

Abstract This paper utilizes a Bayesian inference framework to address the two-dimensional (2D) steady-state heat conduction problem, focusing on the estimation of unknown distributed heat sources in a thermally conducting medium with uniform conductivity. The goal is to infer the locations, strength, and shape of heaters by assimilating temperature data in Euclidean space, employing a Fourier series to represent each heater's shape. The Markov Chain Monte Carlo (MCMC) method, incorporating the random-walk Metropolis–Hasting (MH) algorithm and parallel tempering, is utilized for posterior distribution exploration in both unbounded and wall-bounded domains. It is found that multiple solutions arise in cases where the number of temperature sensors is less than the number of unknown states. Moreover, smaller heaters introduce greater uncertainty in estimated strength. To address the challenge of estimating the heater's strength and shape simultaneously due to their strong correlation, our method incorporates sharp priors on one to ensure accurate and feasible solutions of the other. The diffusive nature of heat conduction smooths out any deformations in the temperature contours, especially in the presence of multiple heaters positioned near each other, impacting convergence. In wall-bounded domains with Neumann boundary conditions, the inference of heater parameters tends to be more accurate than in unbounded domains.

A novel optimization framework for natural gas transportation pipeline networks based on deep reinforcement learning

Natural gas is an emerging and reliable energy source in transition to a low-carbon economy. The natural gas transportation pipeline network systems are crucial when transporting natural gas from the production endpoints to processing or consuming endpoints. Optimizing the operational efficiency of compressor stations within pipeline networks is an effective way to reduce energy consumption and carbon emissions during transportation. This paper proposes an optimization framework for natural gas transportation pipeline networks based on deep reinforcement learning (DRL). The mathematical simulation model is derived from mass balance, hydrodynamics principles of gas flow, and compressor characteristics. The optimization control problem in steady state is formulated into a one-step Markov decision process (MDP) and solved by DRL. The decision variables are selected as the discharge ratio of each compressor. By the comprehensive comparison with dynamic programming (DP) and genetic algorithm (GA) in three typical element topologies (a linear topology with gun-barrel structure, a linear topology with branch structure, and a tree topology), the proposed method can obtain 4.60% lower power consumption than GA, and the time consumption is reduced by 97.5% compared with DP. The proposed framework could be further utilized for future large-scale network optimization practices.

Numerical search for the effective thermal conductivity of cracked media

Spacecraft parts accumulate damage during operation and defects that are invariably present even in new designs may grow. This leads to changes in the behavior of individual parts of the space vehicle and, consequently, to the risk of fracture. A more accurate assessment of spacecraft safety requires internal defects to be included in the material models under consideration. One of the main hazardous effects on space objects is multiple temperature heating and cooling due to periodic action of solar rays. This paper presents a study of thermal conduction of media containing cracks. It is carried out with the help of a technique developed by the authors to determine the effective thermal conductivity of materials and based on approximate numerical solution of the steady-state thermal conduction problem for a three-dimensional medium with cracks by the boundary element method. This technique allows to obtain the distribution of the temperature field and heat flux density at any point of the body under consideration, as well as to calculate the effective parameters of materials with high accuracy at relatively low calculation time using ordinary personal computers of average power. The basis of the numerical method presented in this paper is the decomposition of the desired solution into a series of some pre-calculated analytical solutions of the heat conduction equations. The dependence of the effective thermal conductivity on the density of thermally insulated cracks was considered. The formula of this dependence is proposed. Verification of the proposed methodology was carried out by comparing the numerical results of a number of problems with the results of other authors.

A thermodynamics-consistent spatiotemporally-nonlocal model for microstructure-dependent heat conduction

The spatiotemporally-nonlocal phenomena in heat conduction become significant but challenging for metamaterials with artificial microstructures. However, the microstructure-dependent heat conduction phenomena are captured under the hypothesis of spatiotemporally local equilibrium. To capture the microstructure-dependent heat conduction phenomena, a generalized nonlocal irreversible thermodynamics is proposed by removing both the temporally-local and spatially-local equilibrium hypotheses from the classical irreversible thermodynamics. The generalized nonlocal irreversible thermodynamics has intrinsic length and time parameters and thus can provide a thermodynamics basis for the spatiotemporally-nonlocal law of heat conduction. To remove the temporally-local equilibrium hypothesis, the generalized entropy is assumed to depend not only on the internal energy but also on its first-order and high-order time derivatives. To remove the spatially local equilibrium hypothesis, the thermodynamics flux field in the dissipation function is assumed to relate not only to the thermodynamics force at the reference point but also to the thermodynamics force of the neighboring points. With the developed theoretical framework, the thermodynamics-consistent spatiotemporally-nonlocal models can then be developed for heat transfer problems. Two examples are provided to illustrate the applications of steady-state and transient heat conduction problems.

A hybrid interpolating element-free Galerkin method for 3D steady-state convection diffusion problems

This study investigates a hybrid interpolating element-free Galerkin (HIEFG) method for solving 3D convection diffusion problems. The HIEFG approach divides a 3D solution domain into a sequence of interconnected 2D sub-domains, and in these 2D sub-domains, the interpolating element-free Galerkin (IEFG) method is applied to form the discretized equations. The improved interpolating moving least-squares (IMLS) method is used to obtain the shape function of the IEFG method for 2D problems. The finite difference method is employed to combine the discretized equations in 2D sub-domains in the splitting direction. Then, the HIEFG method's formulas are derived for steady-state convection diffusion problems in 3D solution domain. Three numerical examples are used to discuss the impacts of the number of nodes, the number of split layers, and the scaling parameters of the influence domain on the computational precision and CPU time of the HIEFG technique. Imposing boundary conditions directly and the dimension splitting technique in this method significantly improves the computational speed greatly.

Modeling catalyst effectiveness factor with space-fractional derivative using Haar wavelet collocation method

Abstract Mass transfer limitations may considerably affect the rate of a heterogeneous catalytic process. The catalyst effectiveness factor is a quantitative measure of the impact of the diffusion process inside a catalyst particle. The effectiveness factor is derived from the solution of the steady-state reaction-diffusion problem. Herein, we simulate the steady-state reaction-diffusion equation with space-fractional derivative and linear reaction kinetics. The solution to the problem is obtained numerically using the Haar wavelet collocation method. The effect of the anomalous diffusion exponent on the catalyst effectiveness factor and process parameters, e.g. reactor volume and catalyst mass, is demonstrated. We anticipate that the process efficiency will be notably improved by changing the diffusion regime from standard to superdiffusive.

A Benchmark for Neutron Kinetics Methods in Hexagonal Geometry

A new benchmark solution has been developed to aid in the development of neutron kinetics solvers for hexagonal geometries, such as those in water-water energetic reactors. This benchmark problem is based on the two-dimensional, two-group, International Atomic Energy Agency–Hex steady-state benchmark problem. Two transient problems are presented: a ramp and a step transient. To create a benchmark-quality solution to this transient problem, a basic neutron kinetics model was added to the computer program LUPINE (Liquid metal–cooled fast reactor Utility for Physics Informed Nuclear Engineering). LUPINE solves neutron kinetics equations in general unstructured mesh. First, the LUPINE kinetics solvers are verified using the TWIGL benchmark problems. Then the methods in LUPINE are used to perform a spatiotemporal convergence analysis to ensure that the solutions are sufficiently converged. Finally, Richardson extrapolation is performed to obtain the reference solutions for these new kinetics benchmark problems.

A novel finite-difference converged ENO scheme for steady-state simulations of Euler equations

The high-order shock-capturing scheme is critical for compressible fluid simulations, in particular for cases where strong shockwaves are present. However, for the steady-state solution of Euler equations, the high-order shock-capturing schemes usually suffer from the difficulty that the numerical residual hangs at a relatively high level instead of converging to machine zero. In this paper, a new paradigm of finite-difference converged ENO (CENO) scheme for steady-state simulations of Euler equations is proposed. Different from the existing methods dedicating to eliminating the slight post-shock oscillation, the unsteady effect of shock-induced numerical instability on the nonlinear weighting process as well as the resultant variation of the numerical formula for the spatially fixed cell interface flux at different time instants are demonstrated to be the core contributors to the non-convergence of high-order shock-capturing schemes. Unlike classical ENO/WENO/TENO schemes, which generate the final reconstruction based only on spatial information, evolutionary information is also exploited by a novel converged stencil selection strategy in the present scheme. With this strategy, the unsteady effect of shock-induced numerical instability on the nonlinear weighting process is incorporated and eliminated by a tailored adaptive scale-separation algorithm, while shock-capturing capabilities are not sacrificed. A set of well-established and newly designed challenging benchmark simulations involving strong shockwaves, contact discontinuities and rarefaction waves demonstrates that the new fifth-order CENO5 scheme features superior convergence properties and achieves the essentially non-oscillation property better when compared to the robust TENO5 scheme and the state-of-the-art WENO-type schemes designed for steady-state problems.

On Loss Functionals for Physics-Informed Neural Networks for Steady-State Convection-Dominated Convection-Diffusion Problems

On Loss Functionals for Physics-Informed Neural Networks for Steady-State Convection-Dominated Convection-Diffusion Problems

The steady-state response of a three-phase elliptical inhomogeneity with interface slip and diffusion under an edge dislocation in the matrix

Abstract We study the steady-state response of a three-phase elliptical inhomogeneity in which the internal elliptical elastic inhomogeneity is bonded to the surrounding infinite matrix through an interphase layer with two confocal elliptical interfaces permitting simultaneous interface slip and diffusion. The matrix is subjected to an edge dislocation at an arbitrary position and uniform remote in-plane stresses. An analytical solution to the steady-state problem is derived using Muskhelishvili’s complex variable formulation. The effect of the edge dislocation and remote loading on the elastic fields in the inhomogeneity and the interphase layer is exhibited through a single loading parameter. More specifically, when divided by this loading parameter, the expressions for the stresses and strains in the inhomogeneity and the interphase layer are uninfluenced by the specific loading applied in the matrix. After excluding a particular common factor, the stresses and strains in the inhomogeneity and the interphase layer are also unaffected by the mismatch in shear moduli between the interphase layer and the matrix.

On the importance of wind predictions in wake steering optimization

Abstract. Wake steering is a technique that optimizes the energy production of a wind farm by employing yaw control to misalign upstream turbines with the incoming wind direction. This work highlights the important dependence between wind direction variations and wake steering optimization. The problem is formalized over time as the succession of multiple steady-state yaw control problems interconnected by the rotational constraints of the turbines and the evolution of the wind. Then, this work proposes a reformulation of the yaw optimization problem of each time step by augmenting the objective function by a new heuristic based on a wind prediction. The heuristic acts as a penalization for the optimization, encouraging solutions that will guarantee future energy production. Finally, a synthetic sensitivity analysis of the wind direction variations and wake steering optimization is conducted. Because of the rotational constraints of the turbines, as the magnitude of the wind direction fluctuations increases, the importance of considering wind prediction in a steady-state optimization is empirically demonstrated. The heuristic proposed in this work greatly improves the performance of controllers and significantly reduces the complexity of the original sequential decision problem by decreasing the number of decision variables.

The variational approach to study the mixed convection boundary layer flow over a permeable Riga plate

AbstractThe physical problem of steady state, laminar, mixed convection (Ri) with double‐diffusive (N) in an electrically low conducting fluid past a semi‐infinite electromagnetic () influenced flat plate with internal uniform heat generation (Q) in the presence of suction/injection (H) by considering viscous dissipation (Ec), thermophoresis (Nt) and thermal diffusion effects (Sr) is mathematically modeled as a simultaneous system of nonlinear partial differential equations. To achieve the solution of the problem numerically, Gyarmati's variational principle known as the “Governing Principle of Dissipative Processes” on the basis of nonequilibrium thermodynamic processes in the theory of continua, is adopted. This research work correlates the phenomenon of fluid around submersibles/space vehicles and provides related insights. To estimate the transportation fluid fields within the boundary layer, the appropriate trial polynomials have been employed, and functionals for the integral variational principle are determined. Next, the Euler–Lagrange equations of the functionals are obtained as a system of polynomial equations involving boundary layer thicknesses of momentum, temperature, and concentration. The expressions of local shear stress, local Nusselt, and local Sherwood numbers have been derived and the effects of various physical factors involved in the problem are explored. A comparison with the previously published results in the literature is provided to confirm the validity of the solution procedure. The results depict that injection () and opposing buoyancy () decrease the skin friction about 38% in sea water and 11% in ionized air when compared to impermeable plate for . The aiding buoyancy () plays a dominant role in heat and mass transfers, respectively, with the massive gradients of 340% and 763% in magnitude for the heavier fluid sea water flow while 47% and 3% for the lighter fluid ionized air flow when . The buoyancy parameters (Ri, N) decrease the heat transfer, but increase the mass transfer.

Transmission problem between two Bingham fluids in a tow dimensional thin layer

The paper focuses on studying the asymptotic behavior of the steady-state transmission problem between two Bingham fluids in a two-dimensional thin layer. We are interested in the asymptotic behavior, to this aim we prove some convergence results concerning the velocity and pressure when the thickness tends to zero. The limit problem obtained after transforming the original problem into one posed over a fixed reference domain and the parameter representing the thickness of the layer tend to zero is studied. The lower-dimensional constitutive law and the differential equation satisfied by the limit variables in the non rigid zone are obtained. In addition, the uniqueness of limit solution has been also established. Existence and uniqueness results and a lower-dimensional constitutive law are obtained.

Educational Interface of Load Flow Methods Using Matlab Graphic User’s Interface Software

this paper introduces a new created programapplication built based on MATLAB programing languageto solve power system networks at steady state problems,which include power flow studies with significant approach.The load flow solution uses iterative techniques that arenamely, Newton-Raphson, Gauss-Seidel and Fast Decouple.After obtaining their results, line changes like lineswitching, tap changer and bus changing such as busdeletion or any specific information at the same bus can beeasily achieved. Moreover, the users can enter their ownsystem, and save it to apply the pervious analysis. Thisfriendly interactive program designed to be simple for theusers with accurate results and it can be applied for anynetwork.

Design an efficient neural network for solving steady state problems

Design an efficient neural network for solving steady state problems

Quasi-Newton positivity-preserving scheme for the Spalart-Allmaras turbulence model using unstructured grids

A new implicit method for solving the Spalart-Allmaras (SA) turbulence model is proposed using a formal second-order upwind scheme on unstructured grids for steady-state flows. The mean-flow set of equations and the SA model equation are solved in a loosely coupled manner. While the Jacobian of the mean-flow equations is derived from a low-order approximation of the residual (i.e. the defect-correction approach), the SA Jacobian is obtained by modifying a direct linearization of the second-order residual. The modified Jacobian is designed to form an M-matrix without the artificial time derivative commonly used for steady-state problems. Namely, no time step is involved in solving the SA model equation. Thanks to a special design of the M-matrix, the positivity of the SA model working variable and the convergence of the SA model linearized problem (fixed-point iterative problem) is guaranteed. To further enhance the overall stability of the flow solver, the Runge-Kutta implicit algorithm is employed, allowing CFL numbers as high as one to two thousand for the mean-flow equations. A second-order upwind scheme is achieved with a least-square method using a limiter to suppress oscillation in the solution. However, second-order upwind scheme discretization of turbulence models may lack robustness, especially when using unstructured grids. The limiter can hinder the convergence of the non-linear iterations, especially of the turbulence model. A square root transformation is applied to the dependent variable of a baseline SA model to improve the non-linear convergence characteristics. Three variants of the SA model are implemented and tested together with two limiters. Numerical simulations of a broad range of aerodynamic applications are conducted. The transformed variant exhibits the least sensitivity to the limiter type.

Derivative-Based Finite-Volume MR-HWENO Scheme for Steady-State Problems

Derivative-Based Finite-Volume MR-HWENO Scheme for Steady-State Problems

A numerical validation between the neutron transport and diffusion theories for a spatial kinetics problem

In this paper, a comparative analysis of numerical results of the neutron transport and diffusion theories for steady-state and transient multigroup problems is presented. The neutron transport equation is known as the one that best describes the behavior of the neutron population in a nuclear reactor. However, due to the difficulty of working with its complete form, other models are considered as approximations to this equation. One such approximation is the neutron diffusion equation, which uses the Fick's Law. It is well known, however, that the diffusion model may not work well under specific conditions. The objective of this work is to establish a quantitative comparison of numerical results obtained for the K dominant eigenvalue and for the scalar fluxes from the two theories and to analyze the influence of the model on the results.

Dynamic controlled variables based dynamic self-optimizing control

Self-optimizing control is a strategy for selecting controlled variables, where the economic objective guides the selection and design of controlled variables, with the expectation that maintaining the controlled variables at constant values can achieve optimization effects, translating the process optimization problem into a process control problem. Currently, self-optimizing control is widely applied to steady-state optimization problems. However, the development of process systems exhibits a trend towards refinement, highlighting the importance of optimizing dynamic processes such as batch processes and grade transitions. This paper formally introduces the self-optimizing control problem for dynamic optimization, termed the dynamic self-optimizing control problem, extending the original definition of self-optimizing control. A novel concept, ”dynamic controlled variables” (DCVs), is proposed, and an implicit control policy is presented based on this concept. The paper theoretically analyzes the advantages and generality of DCVs compared to explicit control strategies and elucidates the relationship between DCVs and traditional controllers. Moreover, this paper puts forth a data-driven approach to designing self-optimizing DCVs, which considers DCV design as a mapping identification problem and employs deep neural networks to parameterize the variables. Three case studies validate the efficacy and superiority of DCVs in approximating multi-valued and discontinuous functions, as well as their application to dynamic optimization problems with non-fixed horizons, which traditional self-optimizing control methods are unable to address.