Transient Equation Research Articles

An information-theoretic Machine Learning for upscaling and downscaling of uncertainty stochastic reservoir

This work brings together theoretical formulation and computational strategies for upscaling random heterogeneous media. In this context, heterogeneous means multiphysics models at different scales and correlated random parameters at different locations in the physical domain, such as Stokes-Brinkman flow and fractures at the microscale and heterogeneous Darcy flows at the macroscale when the target problem is hydrocarbon reservoir simulation, or a matrix with random inclusions on the rich scale and a homogeneous matrix on the poor scale when the target problem is solid mechanics. The approach uses information-theoretic machine learning methods to extract relevant probabilistic information from the rich scale and adaptively control the stochastic distance between the responses of the two scales. A goal-oriented upscaling procedure is defined to ensure equivalence between poor targets and rich scales for specific outcomes and its generalization to a multi-goal-oriented mathematically sound response, where users control distinct accuracies of specific target responses. Preliminary applications for elliptic and transient equations are presented with their respective results comparisons. Applications use a series of realizations of rich-scale parameter distributions, producing a reduced number of equivalent poor-scale realizations as required by the user's desired accuracy. The full formulation requires solving a difficult optimization problem for an extended Lagrangian formulation which is solved with a new Parallel Deterministic Annealing. The mathematical formulation is such that parallelization is done over realizations, so the overall cost of calculating stochastic upscaling is of the same order as deterministic. Furthermore, a regression with Tikhonov regularization in linear Machine Learning was used to interpolate and extrapolate data from Parallel Deterministic Annealing in order to find an optimal rich scale.

Numerical Simulation Study on Reverse Source Tracing for Heating Pipeline Network Leaks Based on Adjoint Equations

In order to identify the leak source in complex heating pipeline networks, a timely and effective simulation of the leakage process was conducted. The open-source computational fluid dynamics software OpenFOAM 5.0 was combined with the PISO algorithm to simulate the pressure during the leakage in water supply networks, transforming the reverse source tracing problem into the solution of an adjoint equation. The validation of the transient adjoint equation for single-phase flow was completed through simulation, and the pressure wave change graph at the moment of the network leakage was solved, which was consistent with the experimental results. Using the open-source finite element analysis software OpenFOAM 5.0, the positioning accuracy of pipeline leak points can be controlled within the range from 92% to 96%. Based on the pressure wave change graph, the position of the leak source in the complex network was determined using the reverse source tracing method combined with the second correlation theory. The results show that the calculation speed of the PISO algorithm combined with the adjoint equation is significantly better than that of the individual SIMPLE and PISO algorithms, thereby proving the superiority of the adjoint method.

Multiscale concurrent topology optimization of transient thermoelastic structures

Previous multiscale concurrent topology optimization methods for thermoelastic structures were primarily based on static loading and steady-state heat transfer conditions, which do not account for transient effects associated with time-dependent loads. To address this limitation, this paper establishes a novel generic multiscale concurrent topology optimization method that incorporates transient thermoelastic coupling based on transient heat conduction and structural dynamics. In this study, first, a transient multiscale thermoelastic sensitivity equation is innovatively derived through adjoint sensitivity analysis. The effectiveness of this equation is then demonstrated through comparative cases involving transient heat conduction, structural dynamics, and transient thermoelastic (including multimaterial and 3D problems) optimization. Furthermore, the research finds that the topology optimization of transient thermoelastic structures also presents transient effects at microscale. This method demonstrates good versatility and applicability across various optimization cases. The method has great potential in the integrated design of materials and structures involving coupling between time-dependent thermal loads and time-dependent mechanical loads.

Surface texture transfer in skin-pass rolling under mixed lubrication

This paper presents a novel integrated approach to modeling surface texture transfer in skin-pass rolling under mixed elasto-plasto-hydrodynamic lubrication (EPHL). The innovation lies in combining discrete fast Fourier transform (DC-FFT) for precise characterisation of elastically deformed asperities on the roll surface, dynamic explicit finite element analysis (FEA) for capturing cross-scale deformations, and a transient average Reynolds equation for governing the lubrication flow. By integrating these methods, the model addresses the complex interplay between elastic roll deformation, microscale asperity-lubricant interactions, and elastoplastic strip deformation, providing a more comprehensive understanding of texture transfer mechanisms. In addition, the model predictions are validated by experimental results. Furthermore, this study investigates the effects of rolling speed and surface pattern orientation, revealing that higher speeds reduce texture transfer while surface patterns aligned with the rolling direction enhance it. These insights demonstrate the potential of this integrated modeling approach for advancing the field of skin-pass rolling.

Thermal management enhancement of building-integrated photovoltaic systems using coupled heat pipe and evaporative porous clay cooler

The building-integrated photovoltaic (BIPV) panel systems often lead to a notable decline in PV panels efficiency and lifetime due to their temperature rise because of inadequate cooling. This study investigates the performance of innovative cooling strategy of using a coupled heat pipes and porous clay cooler and compare it with other related three BIPV cooling systems configurations including a conventional BIPV cooling with and without air gap and a pure evaporative porous clay cooling. The aim is to improve the PV panel cooling, ultimately reduce the PV temperature and enhance the overall performance of the BIPV system as well as reducing the building indoor temperature and boosting energy efficiency within the building. The system model, comprising sets of transient equations for the different system configurations, was solved using MATLAB and confirmed with previous experimental findings. The results indicate that comparing with the traditional BIPV system, using BIPV/Clay cooling systems and the hybrid BIPV/Clay-heat pipe cooling systems achieved peak PV temperature reductions of up to 14 °C and 14.7 °C, respectively. Additionally, these cooling methods led to a maximum interior room temperature reduction of approximately 14 °C and an average decrease of 8 °C. Moreover, the hybrid BIPV/Clay-heat pipe cooling system demonstrates superior performance compared to traditional BIPV system, achieving the highest improvements in PV electrical efficiency, output power, and exergy efficiency, with gains of 7.8 %, 6.4 %, and 8.4 %, respectively. Further, when the hybrid BIPV/Clay-heat pipe cooling system was employed, the clay cooling efficiency and clay exergy efficiency values improved on average by 30.2 % and 29.7 %, respectively, compared to the BIPV/Clay cooling system in addition to the increase of the lifetime of the PV panels due to isolating it from the contact with the water/water vapor of the clay cooling system. However, this hybrid approach resulted in a rise in electricity production costs from 0.077 $/kWh to roughly 0.138 $/kWh and extended the payback time from 7.38 years to 13.6 years. But, despite its initial economic drawbacks, the hybrid BIPV/Clay-HP cooling system demonstrates considerable effectiveness and long lifetime compared to other cooling configurations.

A Novel Approach Integrating the Method of Characteristics with Large Time-Step Scheme (MOC-LTS) for Efficient Transient Flow Simulation in Liquid Pipelines

Water hammer incidents pose a significant risk to the safety and stability of pipeline operations. Therefore, rapid and precise transient flow simulation is essential for efficiently developing scientific water hammer control strategies. Nevertheless, the prevailing transient flow simulation methods for liquid pipelines predominantly employ explicit schemes to solve transient flow control equations, necessitating adherence to the stability criterion of Courant-Friedrichs-Lewy (CFL) ≤ 1. This results in limited time step sizes, which in turn constrains computational efficiency, particularly when simulating hydraulic behavior in large-scale pipeline networks. In this paper, a novel approach integrating the method of characteristics (MOC) with a large time-step scheme (LTS) is proposed to enable rapid and accurate simulation of transient flow in liquid pipelines. The proposed approach discretizes the computational domain into contiguous control volumes, ategorizing them as either boundary or internal control volumes depending on whether they are affected by boundary conditions within a large time step. For internal control volumes, an LTS scheme based on the first-order Godunov format is employed to improve computational efficiency. For boundary control volumes, MOC is applied iteratively to accurately capture boundary dynamics until the simulated time matches that of the internal control volumes. Quantitative analysis is conducted to verify the performance of the proposed approach. The results confirm that the proposed approach outperforms the MOC in computational efficiency while maintaining minimal accuracy loss.

Consistent solutions for simultaneous dynamic water, solute, stable isotope and radon balances to estimate average exchange fluxes between surface water bodies and groundwater

Many papers show that water, conservative solute, stable isotope and radon balances can be applied to the study of surface water – groundwater interaction. When undertaken simultaneously, such analysis provides an even more powerful method for estimating groundwater inflows and outflows, i.e. the exchange fluxes that for most people are hardest to estimate. The overarching objective of this paper is to provide practitioners with a consistent conceptual and simulation modelling framework with which to achieve a rapid initial understanding of surface water – groundwater interaction, as well as to guide the efficient and targeted acquisition of field data. This paper makes several important contributions. First, after an introduction that explains the complexities of groundwater flow patterns near surface water bodies, a set of general balance equations is developed using consistent terminology and notation. Second, after focusing on a simple conceptual model with steady groundwater inflow I, groundwater outflow O, precipitation P and ever-present evaporation E, analytical solutions are developed for transient and steady state conditions: one water balance solution is found to apply in all situations; solutions for the solute balance equation are developed for 23 different subcases; a transient balance equation for stable isotopes written in terms of 1+δ is found to apply in all situations; and a new transient solution for radon is found in terms of the upper incomplete gamma function. On the path towards developing the isotope balance equation, we present a rigorous approach, based on coupled nonlinear balance equations for abundant and rare isotope pairs (1H and 2H, then 16O and 18O), using isotope ratios and the fractions of each isotope. Third, we describe a Lake Water Balance Calculator (LWBC) written using GoldSim simulation software, a tool to help others to conduct the simultaneous analyses that we promote; we use GoldSim’s numerical methods to solve the general balance equations, we implement the analytical solutions to demonstrate that both GoldSim’s numerical methods and the analytical solutions are correct, and we also show that the 1+δ approximation is equivalent to the rigorous coupled equations, at least for heavy stable isotopes with very low abundance. Fourth, we describe field studies: we provide three examples of applications of LWBC to short-term studies of Perry Lakes East, Lake Jasper and Georgetown Billabong in Australia; we then describe WSIBal, a simulation model based on the same balance equations and describe its application to Lake Jasper over a 19-year period; finally, we describe R-WSIBal, which evolved from WSIBal, and explain its application to 2500 km of the Murray River in Australia.

Least-squares pressure recovery in reduced order methods for incompressible flows

In this work, we introduce a method to recover the reduced pressure for Reduced Order Models (ROMs) of incompressible flows. The pressure is obtained as the least-squares minimum of the residual of the reduced velocity with respect to a dual norm. We prove that this procedure provides a unique solution whenever the full-order pair of velocity-pressure spaces is inf-sup stable. We also prove that the proposed method is equivalent to solving the reduced mixed problem with reduced velocity basis enriched with the supremizers of the reduced pressure gradients. Optimal error estimates for the reduced pressure are obtained for general incompressible flow equations and specifically, for the transient Navier-Stokes equations. We also perform some numerical tests for the flow past a cylinder and the lid-driven cavity flow which confirm the theoretical expectations, and show an improved convergence with respect to other pressure recovery methods.

Impact of Interpolation on Numerical Properties of the Method of Characteristics Used for Solution of the Transient Pipe Flow Equations

Impact of Interpolation on Numerical Properties of the Method of Characteristics Used for Solution of the Transient Pipe Flow Equations

Transient thermal convective and radiative diffusion–reaction of magneto‐metallic Brinkman‐type nanofluid flow with isothermal and ramped temperature impacts

AbstractFluid‐containing nanoparticle research is well‐known for its heat transmission properties due to real‐world applications in various thermal systems. This demonstration shows how time‐dependent magneto‐hydrodynamic (MHD) Brinkman‐type nanofluid can flow through a vertical oscillating absorbent plate immersed in a porous environment. The governing dimensional partial differential system of equations is translated into a non‐dimensional partial differential system with applicable scaling variables. The transient equations governing nanofluid flow's momentum, thermal and mass are solved using the finite semi‐discretization difference line method. The obtained outcomes are validated by comparing them with previous works. Graphical resolutions for momentum, heat, and mass distribution fields are presented to examine essential physical terms with the isothermal and ramped temperature impacts. It was found that magnifying the Brinkman parameter and magnetic parameter brings in a decrement in fluid velocity. In contrast, reversal deportment is exposed with an enlargement of permeability parameters and thermal and solute buoyancy effects. Reynold's number has shown a declining impact in the fluid velocity, but time progress unveiled a contrary tendency. The sprouting radiation parameter deepens the fluid temperature, whereas the rising heat absorption parameter curtails the fluid temperature. The fluid concentration deflates for growing Schmidt number and chemical reactive agent. The wall friction increased with the porosity parameter but has shown a reverse trend with magnetic and Brinkman parameters. Increasing the Prandtl number and heat‐consumption parameter helps to raise the temperature gradient. The concentration gradient lessened on augmentation of chemical reaction and Schmidt number.

Development of a transient analysis code for trans-critical simulation of SCWR

Supercritical water-cooled reactors (SCWR) experience pressure reductions below the critical point during a loss of coolant accident (LOCA). During the reactor depressurization, boiling crises may occur, leading to pressure fluctuations and significant increases in wall temperatures, posing a severe threat to the cladding. In this study, a one-dimensional transient analysis code has been developed to incorporate one-dimensional steady-state flow equations in the fluid domain and transient thermal conductivity equations in the solid domain. The code also includes a wall heat transfer model and a model for the velocity of the moving quench front to simulate transcritical depressurization transient processes. The simulation successfully predicts the typical experimental phenomena. It is reasonable to use the critical temperature as the demarcation point between the dry and wet conditions of the axial wall of the rod bundle under transcritical conditions. By combining the experimental data with the program calculations, the critical interface for the occurrence of boiling crisis under transcritical conditions is determined using mass flow rate, heat flow density and fluid temperature as inputs. The criterion for the occurrence of CHF under transcritical conditions is obtained, which provides a reference to the safety analysis of SCWRs.

Stress Analysis of a Concrete Pipeline in a Semi-Infinite Seabed under the Action of Elliptical Cosine Waves Based on the Seepage Equation

This study aims to investigate the mechanical response of a submarine concrete pipeline under wave action in shallow waters, taking into account factors such as the compressibility of pores and the permeability of the seabed. The control equation of the elliptical cosine wave theory is adopted to simulate the action of waves. In order to simulate the interaction between the solid skeleton and pore fluid, the concept of a “porous medium” is used to establish the transient seepage control equation. Utilizing the stress and displacement conditions at the interface of the ideal fluid media, porous media, and concrete pipeline, the numerical solutions for the internal force and pore pressure of the concrete pipeline buried in a semi-infinite thickness seabed were obtained; meanwhile, the effects of changes in the gas content in pore water and changes in the seabed permeability coefficient on a concrete pipeline were analyzed. The numerical calculation results show that, with the increase in the gas content in the pore water, the amplitude of the pore pressure on the pipeline surface decreases, and both the horizontal and vertical forces acting on the pipeline decrease; the amplitude of the pore pressure on the pipeline surface increases with the increase in seabed permeability and decreases with the enhancement of seabed permeability anisotropy; the improvement of the seabed permeability or enhancement of the permeability anisotropy can increase the horizontal force acting on the pipeline. This study provides a reference for the stability evaluation of submarine concrete pipelines under wave action in shallow water areas.

An artificial neural network based deep collocation method for the solution of transient linear and nonlinear partial differential equations

A combined deep machine learning (DML) and collocation based approach to solve the partial differential equations using artificial neural networks is proposed. The developed method is applied to solve problems governed by the Sine–Gordon equation (SGE), the scalar wave equation and elasto-dynamics. Two methods are studied: one is a space-time formulation and the other is a semi-discrete method based on an implicit Runge–Kutta (RK) time integration. The methodology is implemented using the Tensorflow framework and it is tested on several numerical examples. Based on the results, the relative normalized error was observed to be less than 5% in all cases.

Effect of flow modifiers on the fluid flow characteristics in a slab caster tundish

ABSTRACT In a slab caster tundish, controlling liquid steel flow is of paramount importance for clean steel production. A conventional tundish design consists of a pouring box, a weir and a dam. To improve the overall performance of a 40 ton slab caster tundish, three different arrangements by incorporating additional flow modifiers have been examined. A three-dimensional steady state fluid flow model has been developed using a RANS (Reynolds-Averaged Navier-Stokes) approach. Subsequently, transient scalar-transport equations were solved to obtain the C-curve of the tundish. A different approach has been followed to evaluate the inlet boundary condition to mimic the process conditions more accurately. The CFD predictions were validated with the results of the water model. Mean residence time, dead volume fraction and flow structures have been considered for evaluating tundish performance. All the proposed configurations have shown improvements in comparison to the base case. The mean residence time of the fluid was found to be the highest for the case having two pairs of weir and dam. However, the insights derived from the flow structures specifically plug flow in zone 2 and near the tundish outlet region suggest that that having a single weir and two dams is the optimal choice.

Self-detection of thermal ambient parameters for air-conditioned space by learning operation data and self-prediction of indoor temperature and power consumption

For efficient energy use of air conditioning unit (AC), there is a need to find an optimal control method for operation. To deduce an optimal control algorithm, indoor temperature and power consumption need to be predicted according to given operating conditions. Among the key factors, the thermal ambient parameters such as thermal resistances of the operation sites are most difficult to identify. This study proposed a method for an AC to deduce these parameters for itself by learning operation data obtained from the embedded sensors and autonomously predict indoor temperature and power consumption by solving the transient energy conservation equation incorporating refrigeration cycle simulation. A novel aspect of this study is that the AC can autonomously perceive thermal ambient parameters of its surroundings and utilize these parameters to predict indoor temperature variations and power consumption. These data could be fed back to the control software of the AC in real time. To validate the proposed method, experiments were conducted in model house test facility using a real air-conditioner, and the results were compared with the simulation results. It was confirmed that the time to reach the air temperature setpoint was within 3 min and the power consumption could be predicted within 9 %.

Applications of Asymptotic Solutions of the Diffusivity Equation to Infinite Acting Pressure Transient Analysis

Summary Understanding how pressure propagates in a reservoir is fundamental to the interpretation of pressure and rate transient measurements at a well. Unconventional reservoirs provide unique technical challenges as the simple geometries and flow regimes [wellbore storage (WBS) and radial, linear, spherical, and boundary-dominated flow] applied in well test analysis are now replaced by nonideal flow patterns due to complex multistage fracture completions, nonplanar fractures, and the interaction of flow with the reservoir heterogeneity. In this paper, we introduce an asymptotic solution technique for the diffusivity equation applied to pressure transient analysis (PTA), in which the 3D depletion geometry is mapped to an equivalent 1D streamtube. This allows the potentially complex pressure depletion geometry within the reservoir to be treated as the primary unknown in an interpretation, compared with the usual method of interpretation in which the depletion geometry is assumed and parameters of the formation and well are the unknown properties. The construction is based upon the solution to the Eikonal equation, derived from the diffusivity equation in heterogeneous reservoirs. We develop a Green’s function that provides analytic solutions to the pressure transient equations for which the geometry of the flow pattern is abstracted from the transient solution. The analytic formulation provides an explicit solution for many well test pressure transient characteristics such as the well test semi-log pressure derivative (WTD), the depth of investigation (DOI), and the stabilized zone (SZ) (or dynamic drainage area), with new definitions for the limit of detectability (LOD), the transient drainage volume, and the pseudosteady-state (PSS) limit. Generalizations of the Green’s function approach to bounded reservoirs are possible (Wang et al. 2017) but are beyond the scope of the current study. We validate our approach against well-known PTA solutions solved using the Laplace transform, including pressure transients with WBS and skin. Our study concludes with a discussion of applications to unconventional reservoir performance analysis for which reference solutions do not otherwise exist.

On the locality of local neural operator in learning fluid dynamics

This paper launches a thorough discussion on the locality of local neural operator (LNO), which is the core that enables LNO great flexibility on varied computational domains in solving transient partial differential equations (PDEs). We investigate the locality of LNO by looking into its receptive field and receptive range, carrying a main concern about how the locality acts in LNO training and applications. In a large group of LNO training experiments for learning fluid dynamics, it is found that an initial receptive range compatible with the learning task is crucial for LNO to perform well. On the one hand, an over-small receptive range is fatal and usually leads LNO to numerical oscillation; on the other hand, an over-large receptive range hinders LNO from achieving the best accuracy. We deem rules found in this paper general when applying LNO to learn and solve transient PDEs in diverse fields. Practical examples of applying the pre-trained LNOs in flow prediction are presented to confirm the findings further. Overall, with the architecture properly designed with a compatible receptive range, the pre-trained LNO shows commendable accuracy and efficiency in solving practical cases.

Optimal [formula omitted] error estimates of stabilizer-free weak Galerkin finite element method for the drift-diffusion problem

We continue our effort in Li et al. (2024) to develop a stabilizer-free weak Galerkin finite element method (SFWG-FEM) for the two-dimensional unsteady-state drift-diffusion (DD) model. The transient state equation includes both the first derivative convection and the second derivative diffusion terms, and it couples with a Poisson potential equation. The piecewise Pk(k≥1) polynomials elements in the interior and on the boundaries are utilized for both electron concentration and electric potential functions. The vector function of [Pj(K)]2(j≥j0) for the discrete weak gradient space is used on each element K. The main difficulty is the treatment of the non-linearity and coupling of the system. Based on an introduced Ritz projection and two L2 projection operators, optimal error estimates of O(Δt+hk+1) in the L2 norm and O(Δt+hk) in the energy-like norm are derived. Numerical experiments are presented to illustrate the theoretical analysis.

Nonlinear coupled multi-physics field analysis of permanent magnet synchronous motor combining finite element analysis with optimized meshless method

Aiming at significant disadvantages of finite element analysis (FEA) performed in coupled multiphysics field calculation, including high computing cost and strong dependency on meshing quality. Taking a high-speed permanent magnet synchronous motor (P MSM) as an example, an approach combining FEA with optimized meshless method is proposed to solve the heat transfer problem. Firstly, electromagnetic losses under rated and 1.2 times overloaded operation conditions are calculated using FEA respectively. Secondly, three-dimensional FEA flow models considering the influence of viscosity varying with temperature are established. Thirdly, convective heat transfer coefficients are calculated as the boundary condition, after which the electromagnetic losses as heat sources are mapped to the threedimensional steady-state temperature field calculations using FEA and conventional meshless method respectively. Applying the above two methods, a singular boundary method (SBM) is employed to replace the temporal derivative term of the transient calculation equation. Finally, to mitigate the illconditioned matrix appearing in the conventional meshless method coupled with FEA, an optimized meshless method is proposed to recalculate the heat transfer equation. After that, the calculation results of the above methods are compared with experimental results. This research is beneficial for the multiphysical coupling analysis of electrical equipment with complex structures considering nonlinear factors.

A numerical method for two-dimensional transient nonlinear convection-diffusion equations

In this research, we have developed the half-boundary method (HBM) for nonlinear convection-diffusion equations (CDEs), which hold significant importance in nuclear power engineering. The HBM employs a variable relationship between the nodes within the computational domain and the nodes located on half of the boundaries. This approach offers notable benefits, includingthe reduction ofthe maximum matrix order and the optimization the maximum memory storage for calculations. Moreover, the HBM is an efficient and streamlined approach to directly handle Neumann boundary conditions, thanks to the utilization of mixed variables. We primarily investigate the memory usage and accuracy of the proposed algorithm in the unsteady-state CDEs, in context of material nonlinear CDEs, the Burgers’ equation and the system of Burgers' equations. The numerical results obtained demonstrate the method’s potential in simulating flow and heat transfer phenomena, particularly in situations where convection is dominant.