Time-dependent Boundary Conditions Research Articles

Numerical Reconstruction of Time-Dependent Boundary Conditions to 2D Heat Equation on Disjoint Rectangles at Integral Observations

In this paper, two-dimensional (2D) heat equations on disjoint rectangles are considered. The solutions are connected by interface Robin’s-type internal conditions. The problem has external Dirichlet boundary conditions that, in the forward (direct) formulation, are given functions. In the inverse problem formulation, the Dirichlet conditions are unknown functions, and the aim is to be reconstructed upon integral observations. Well-posedness both for direct and inverse problems is established. Using the given 2D integrals of the unknown solution on each of the domains and the specific interface boundary conditions, we reduce the 2D inverse problem to a forward heat 1D one. The resulting 1D problem is solved using the explicit Saul’yev finite difference method. Numerical test examples are discussed to illustrate the efficiency of the approach.

Formation Rate and Energy Efficiency of Ice Plug in Pipelines Driven by the Cascade Utilization of Cold Energy

Ice plug technology is an effective method for isolating the pipeline system, which are promising methods utilized in the nuclear, chemical, and power industries. To reduce the cold energy consumption and temperature stress, the multi-stage (1–10) of time-dependent thermal boundary conditions was proposed for the formation of ice plug, while the gradient cooling wall temperature of multi-stage was applied. A numerical model considering the liquid–solid phase change, heat transfer, and time-dependent thermal boundary condition has been established. The effects of the ratio of length and diameter of the cooling wall lc/d (1–9) on the formation rate and heat flux of ice plug in the pipe has been investigated. The fastest formation rate of ice plug with 800 mm in the axial direction (7.47 cm3/s) was observed in the pipe with the lc/d of 5. The formation rate of ice plug and the ice formation volume under unit energy consumption VE under various stages (1–10) of cooling wall temperature have been compared. The VE of eight temperature stages (1.45 cm3/kJ) was 1.16 times than the VE of one temperature stage, which satisfied the freezing rate at the same time. This investigation provides insight for proposing an energy-saving system for the formation of ice plug.

A new analytical method for transient heat conduction in composite disks applied to thermal plug and abandonment of oil wells

A new hybrid method for transient heat conduction problems is developed and applied to simulate a Plug and Abandonment (P&A) operation of oil wells. An oil well is approximated by concentric disks in a one-dimensional configuration, which allows for use of polar coordinates. The application of the Separation of Variables Method (SVM) is used as the analytical framework for the solution of the conductive heat transfer arising from a volumetric heat source located in the center of the disks. The SVM is able to solve only time-independent boundary conditions. However, using the Duhamel's theorem, the solution determined with the SVM can be used to achieve the solution when both time-dependent boundary conditions and internal heat generation are prescribed. Lastly, for verification purposes, a commercial software that solves the transient temperature field by means of numerical procedures, providing reliability to the analytical method proposed in this work.

Convective thermal losses of long-term underground hot water storage

A case of underground long-term hot water storage is investigated numerically. The study is based on the unsteady two-dimensional Navier-Stokes equations in Boussinesq approximation applied to a closed cavern with time-dependent temperature boundary conditions on the walls. The problem formulated in a vorticity-stream function statement is solved by finite difference method (FDM) for high values of the Rayleigh number and for the Prandtl number of water. Streamlines, velocity and temperature fields are presented graphically for given moments of time. The evolution of the thermocline thickness in the mid-section of the cavern is slow and illustrates that the hot water zone occupies more than the half of the cavern even after 6 months period. The Nusselt number on the walls shows that the convective thermal losses are small and after certain period of time tend to decrease due to the diminished temperature difference at the walls. The influence of the fluid convection on the thermal losses is evaluated quantitatively, showing high seasonal thermal efficiency of the insulated hot water storage.

Topography optimisation using a reduced-dimensional model for transient conjugate heat transfer between fluid channels and solid plates with volumetric heat source

Consideration of transient effects is important for industrial applications of heat transfer structure optimisation studies; however, the huge computational cost associated with transient problems is a pressing concern. This paper proposes an extension of a previous reduced-dimensional model to transient conjugate heat transfer between a fluid flow and solid-heated plates in a plate heat exchanger. The extended reduced-dimensional model introduces the temperature field of the plate governed by the heat conduction equation, which is coupled to the temperature field of the fluid, governed by the convection-diffusion equation, through the heat flux balance equation at the contact surface. The model is based on assumptions of fully developed flow and constant temperature profile, reducing the three-dimensional problem to a planar problem and significantly reducing computational costs. The accuracy of the model for the simulation of transient heat transfer is verified by comparison with a three-dimensional model. In this paper, the topography of the heat exchanger plate is optimised for both steady-state and transient conditions by applying the reduced-dimensional model. The effectiveness of the optimised design was demonstrated by the cross-check of both the reduced-dimensional and full three-dimensional models. Furthermore, this work considers the effect of time-independent boundary conditions and time-dependent boundary conditions on transient optimisation. The transient and steady-state optimised designs are analysed and compared for both conditions, and the necessity of transient optimisation is discussed.

Solving reaction-diffusion problems with explicit Runge-Kutta exponential methods without order reduction

Abstract. In this paper a technique is given to recover the classical order of the method when explicit exponential Runge-Kutta methods integrate reaction-diffusion problems. In the literature, methods of high enough stiff order for problems with vanishing boundary conditions have been constructed, but that implies restricting the coefficients and thus, the number of stages and the computational cost may significantly increase with respect to other methods without those restrictions. In contrast, the technique which is suggested here is cheaper because it just needs, for any method, to add some terms with information only on the boundaries. Moreover, time-dependent boundary conditions are directly tackled here.

Perfect plasticity versus damage: an unstable interaction between irreversibility and Γ-convergence through variational evolutions

This paper addresses the question of the interplay between relaxation and irreversibility through quasi-static evolutions in damage mechanics, by inquiring the following question: can the quasistatic evolution of an elastic material undergoing a rate-independent process of plastic deformation be derived as the limit model of a sequence of quasi-static brittle damage evolutions? This question is motivated by the static analysis performed in [J.-F. Babadjian et al. Commun. Pure Appl. Math. 74 (2021) 1803–1854], where the authors have shown how the brittle damage model introduced by Francfort and Marigo (see [G.A. Francfort and J.-J. Marigo, Eur. J. Mech. A Solids 12 (1993) 149–189, G.A. Francfort and J.-J. Marigo, J. Mech. Phys. Solids 46 (1998) 1319–1342]) can lead to a model of Hencky perfect plasticity. Problems of damage mechanics being rather described through evolution processes, it is natural to extend this analysis to quasi-static evolutions, where the inertia is neglected.We consider the case where the medium is subjected to time-dependent boundary conditions, in the one-dimensional setting. The idea is to combine the scaling law considered in [J.-F. Babadjian, et al. Commun. Pure Appl. Math. 74 (2021) 1803–1854] with the quasi-static brittle damage evolution introduced in [G.A. Francfort and A. Garroni, Arch. Rational Mech. Anal. 182 (2006) 125–152] by Francfort and Garroni, and try to understand how the irreversibility of the damage process will be expressed in the limit evolution. Surprisingly, the interplay between relaxation and irreversibility is not stable through time evolutions. Indeed, depending on the choice of the prescribed Dirichlet boundary condition, the effective quasi-static damage evolution obtained may not be of perfect plasticity type.

Semi-analytical models for two-dimensional multispecies transport of sequentially degradation products influenced by rate-limited sorption subject to arbitrary time-dependent inlet boundary conditions

Most of the semi-analytical and analytical models employed to depict multidimensional, multispecies transport of sequentially degrading reaction products are built upon solving a set of coupled advection-dispersion equations (ADEs). Within these equations, sorption is considered as being equilibrium-controlled. However, it has been demonstrated that more realistic predictions of the transport of contaminants in the groundwater could be obtained by the use of a rate-limited sorption process instead of an equilibrium sorption assumption. This study is thus designed to devise semi-analytical models for the two-dimensional multispecies transport of a chemical mixture comprised of a parent compound and its degradation-daughter products which is influenced by rate-limited sorption subject to arbitrary time-dependent inlet boundary conditions. Three integral transforms are applied to generate a set of linear algebraic equations (AEs) resulting from the reductions of the ADEs. The contaminant concentration of each species is calculated by solving these AEs and then retransforming the solutions back to the original time-space domain. The simulation results obtained with our newly developed semi-analytical model are nearly identical to those generated using a numerical model based on the Laplace transform finite difference (LTFD) method. This confirms the validity of the new semi-analytical model. An investigation of the effect of rate-limited sorption on the migration of the contaminant plume is carried out using various time-dependent inlet boundary conditions. The sorption rate values vary from low to high, specifically, 0.05, 0.5, 5, and 50 years−1. The results show that the predicted concentrations of all contaminants within in the decay chain decrease as the sorption rate constant increases, for both constant and exponentially time-dependent boundary sources. However, under pulse loading boundary conditions, the concentrations of the later degradation products tend to increase with the sorption rate constant. The semi-analytical models created in this study, allow for different inlet boundary conditions, so can be utilized to simulate the transport of sequentially degrading reaction products. This greater accuracy makes them useful for many applications.

Analysis of Multilayer Cylindrical Thermal Conduction With a Time-Varying Convective Boundary Condition

Abstract Heat transfer in a multilayer body plays a key role in design and optimization of several engineering systems. While the analysis of simple multilayer problems is quite straightforward, realistic scenarios such as time-dependent boundary conditions result in significant complications in analysis. This work presents thermal analysis of a one-dimensional heat-generating multilayer cylinder with time-varying convective heat transfer at the boundary. Such a scenario may occur in applications such as nuclear reactors, jet impingement cooling, turbine blade heat transfer, as well as casting and related manufacturing processes. Analysis is presented for both annular and solid cylinders. A derivation for the temperature distribution is carried out, using a shifting function to split the time-dependent boundary condition into two parts, followed by appropriate mathematical substitution. For particular special cases, the analytical results derived here are shown to reduce exactly to results from past work. Good agreement of the theoretical results with numerical simulations is also demonstrated. Thermal response to various realistic time-dependent boundary conditions is analyzed. This work contributes towards the design of realistic multilayer problems and may facilitate the optimization of engineering systems where multilayer thermal conduction plays a key role.

Temperature Response of Double-Layer Steel Truss Bridge Girders

Double-layer steel truss continuous girders are prone to significant temperature stress, deviation, torsion, and warping, thus causing adverse temperature structural responses, and also affecting the safety and durability of bridge structures. This paper presents an investigation on time-dependent characteristics in the temperature field and temperature response of double-layer steel truss continuous bridge girders, fully considering the shielding effect subjected to different solar radiation angles during the high-temperature season. The time-dependent thermal boundary conditions and support conditions provided for the steel truss bridge structure were determined. Subsequently, a thermal analysis model for the entire structure of double-layer steel truss continuous girders was established to attain the temperature distribution law. The research results show that significant differences occur in the position and temperature difference of temperature gradients exhibited in the vertical, horizontal, and longitudinal directions in the double-layer steel truss bridge structure. The temperature distribution pattern within the chord section is mainly influenced by the environmental temperature and solar radiation intensity, along with the heat exchange between different panels. Thereafter, a validated temperature gradient formula for the component section has been proposed. The time-dependent laws in structural displacement, stress, and rotation angle under daily temperature cycling conditions have been revealed, thereby providing a theoretical basis for the life cycle construction and safety maintenance of double-layer steel truss structure bridges.

Numerical investigation on tornado-like flows and immersed bodies using vortex models

A numerical investigation to evaluate tornado flows with immersed bodies is presented in this work, where vortex profile models are adopted to generate the flow field based on time-dependent boundary conditions. In order to reproduce flow conditions obtained from field data and experimental analyses, a parametric study is performed considering the main model parameters influencing the tornado flow characteristics. A Taylor–Galerkin finite element model is used for flow simulation, where eight-node hexahedral elements with one-point integration are adopted. Vortex flow fields are generated numerically using the Modified Rankine Combined Vortex Method (MRCVM) and a fully three-dimensional approach is utilized for tornado-like flows considering the Vastistas vortex model. Turbulence is analyzed using Large Eddy Simulation (LES) with Smagorinsky's sub-grid scale modeling. The vortex formulations are verified considering a circular cylinder submitted to vortex flows, where aerodynamic force coefficients are obtained. A tornado flow field generated experimentally is reproduced utilizing the vortex models proposed here and a cubic building subject to different tornado flow conditions is also analyzed. Results demonstrate that velocity profile models are able to reproduce tornado flow fields and aerodynamic forces on immersed bodies satisfactorily since the boundary conditions and vortex model parameters are calibrated properly.

Nonlinear two-dimensional transient thermoelastic analysis of functionally graded composite plates subjected to localised cooling loads

In this study, for the first time, localized thermal cooling is considered for the transient thermoelastic response of an axisymmetric plate, addressing the underexplored area of localized low-temperature thermoelasticity. Functionally graded composite plates are analyzed which consist of a mixture of stainless steel (SUS304) and low-carbon steel (AISI1020). A two-dimensional transient heat conduction equation is employed to capture localized cooling effects, featuring various adaptive time-dependent thermal boundary conditions which mirror both loading and unloading scenarios, effectively simulating real-world phenomena. Novel parameters are introduced to facilitate a comprehensive understanding of various aspects of thermal load handling and their impact on thermoelastic responses. The heat conduction equations are solved using the Generalized Differential Quadrature Method (GDQM) and the Crank-Nicolson scheme. The nonlinear governing equations, incorporating geometrical nonlinearity through the von Kármán assumption within the First Shear Deformation Theory (FSDT) framework, are solved using the GDQM and the Picard's technique. The model is validated using Finite Element Method (FEM) through Abaqus. A comprehensive analysis is provided that considers influence of the ratio of thermally affected and unaffected plate surface, thermal load magnitude, rapidity of thermal loading and unloading, duration of the cooling load, geometrical nonlinearity, and temperature dependence of material. The study shows that the maximum von Mises stress within the structure remains consistent regardless of the duration of the cooling load, as long as the rate at which the cooling load is applied stays the same. Additionally, the findings reveal that even minor localized thermal loads can produce substantial stress, especially at the intersection of thermally affected and non-affected zones on the exposed surface in some situations.

A Computationally Efficient Method for the Thermo-Structural Analysis of a Turbocharger Turbine Wheel Under Engine Transient Operation

Abstract The accurate evaluation of thermally-induced stresses is of particular importance for life-cycle analysis of modern turbochargers. Conjugated heat transfer (CHT), steady-state calculations represent the state-of-the-art, whose temperature distribution is usually imported into a structural solver as a thermal load for carrying out thermo-structural analyses. In the case of transient analyses, however, issues arise. Indeed, due to the different timescales of convective and conductive heat-transfer mechanisms, unsteady CHT simulations are burdened by high computational costs. Moreover, in the specific case of turbocharged engines, the timescales of engine load transients and turbocharger response may differ by a factor of up to 105. The present study proposes an approach for carrying out transient thermo-structural analyses of the solid domain only, characterized by a physical time in the order of 101 s, without the need for unsteady fluid flow calculations, which would imply the use of a solver time-step in the order of 10−5 s. In the proposed methodology, the effects of the fluid flow in the heat-transfer process are modeled by imposing adequate boundary conditions to wet surfaces of the solid domain with minimum computational effort. Indeed, by performing two steady-state CHT calculations and one steady-state computational fluid dynamics (CFD) calculation, time-dependent boundary conditions can be derived in terms of convective heat transfer coefficient and near-wall fluid temperature. The case study used for model development is a high-performance turbocharger turbine for automotive applications, responding to engine load transients of tenths of seconds. Initially, steady-state CFD and CHT models of the turbine stage are validated against experimental data. Then, transient analyses on the turbine wheel are carried out, and their results are imported as thermal loads into an unsteady structural solver, allowing for accurate transient thermo-structural analyses at sustainable computational cost.

Boundary corrections on multi-dimensional PDEs

Two new boundary correction techniques are proposed in order to mitigate the order reduction phenomenon associated with the numerical solution of initial boundary value problems for parabolic partial differential equations in arbitrary spatial dimensions with time-dependent Dirichlet boundary conditions. The new techniques are based on the idea of discretizing the PDE problem at the boundary points as similar as possible to that of the interior points of the domain. These new techniques are considered for the time integration with W-methods based on approximate matrix factorization. By suitably modifying the internal stages of the methods on the boundary points, it is illustrated by numerical testing with time-dependent boundary conditions that the new boundary correction techniques are able to keep the same accuracy and order of convergence that the method reaches in the case of homogeneous boundary conditions.

Towards resilient operation of photovoltaic-thermal collector with incorporated organic phase change material: Numerical and experimental investigation

The resilient operation of the novel free-standing photovoltaic-thermal (PVT) collector with an integrated phase change material (PCM) in variable weather conditions was examined by developing the computational fluid dynamics (CFD) 3D model to perform numerical analysis of heat transfer. The developed transient multi-parameter numerical model was directly compared with experimental measurements for the case of variable cloudiness, unstable ambient temperature, and fluctuating wind. Due to volatile weather conditions, the numerical analysis included variable time-dependent boundary conditions implemented using user-defined functions (UDF). The investigation considered a novel multi-parameter numerical approach, together with novel experimental data for unconventional organic PCM. Specifically, the thermal characteristics of unconventional organic PCM, i.e., pork fat, were experimentally determined with the differential scanning calorimetry (DSC) method and embedded in the numerical model. It was found that melting of the crystal phase of pork fat PCM starts at 8.3 °C and ends at 45.2 °C with a latent heat of melting being 45.4 Jg−1. The numerical model was successfully validated with experimental data. The mean absolute percentage deviation of the operating temperature numerical forecast compared with the experiment was from about 2.9% to 4.9%, depending on the considered domain. The findings of this investigation can be used to improve the existing designs of PVT-PCM collectors. Furthermore, a validated numerical model can be used to investigate the impact of various operating parameters on the system performance to enhance the resilience aspect in the early development phase of novel designs.

No pressure? Energy-consistent ROMs for the incompressible Navier-Stokes equations with time-dependent boundary conditions

This work presents a novel reduced-order model (ROM) for the incompressible Navier-Stokes equations with time-dependent boundary conditions. This ROM is velocity-only, i.e. the simulation of the velocity does not require the computation of the pressure, and preserves the structure of the kinetic energy evolution.The key ingredient of the novel ROM is a decomposition of the velocity into a field with homogeneous boundary conditions and a lifting function that satisfies the mass equation with the prescribed inhomogeneous boundary conditions. This decomposition is inspired by the Helmholtz-Hodge decomposition and exhibits orthogonality of the two components. This orthogonality is crucial to preserve the structure of the kinetic energy evolution. To make the evaluation of the lifting function efficient, we propose a novel method that involves an explicit approximation of the boundary conditions with POD modes, while preserving the orthogonality of the velocity decomposition and thus the structure of the kinetic energy evolution.We show that the proposed velocity-only ROM is equivalent to a velocity-pressure ROM, i.e., a ROM that simulates both velocity and pressure. This equivalence can be generalized to other existing velocity-pressure ROMs and reveals valuable insights in their behaviour.Numerical experiments on test cases with inflow-outflow boundary conditions confirm the correctness and efficiency of the new ROM, and the equivalence with the velocity-pressure formulation.

Identifying optimal architectures of physics-informed neural networks by evolutionary strategy

Physics-Informed Neural Networks (PINNs) are artificial neural networks that encode Partial Differential Equations (PDEs) as an integral component of the ML model. PINNs are successfully used nowadays to solve PDEs, fractional equations, and integral–differential equations, including direct and inverse problems. Just as in the case of other kinds of artificial neural networks, the architecture, including the number and sizes of layers, activation functions, and other hyperparameters can significantly influence the network performance. Despite the serious work in this field, there are still no clear directions on how to choose an optimal network architecture in a consistent manner. In practice, expertise is required, with a significant number of manual trial and error cycles. In this paper, we propose PINN/GA (PINN/Genetic Algorithm), a fully automatic design of a PINN by an evolutionary strategy with specially tailored operators of selection, crossover, and mutation, adapted for deep neural network architecture and hyperparameter search. The PINN/GA strategy starts from the population of simple PINNs, adding new layers only if it brings clear accuracy benefits, keeping PINNs in the population as simple as possible. Since the examination of dozens of neural networks through the evolutionary process implies enormous computational costs, it employs a scalable computational design based on containers and Kubernetes batching orchestration. To demonstrate the potential of the proposed approach, we chose two non-trivial direct problems. The first is 1D Stefan transient model with time-dependent Dirichlet boundary conditions, describing the melting process, and the second is the Helmholtz wave equation over a 2D square domain. The authors found that PINNs accuracy gradually improves throughout the evolutionary process, exhibiting better performance and stability than parallel random search and Hyperopt Tree of Parzen Estimators, while keeping the network design reasonably simple.

On a multiple time-scales perturbation analysis of a Stefan problem with a time-dependent Dirichlet boundary condition

In this paper, a classical Stefan problem with a prescribed and small time-dependent temperature at the boundary is studied. By using a multiple time-scales perturbation method, it is shown analytically how the moving boundary profile is influenced by the prescribed temperature at the boundary and the initial conditions. Only a few exact solutions are available for this type of problems and it turns out that the constructed approximations agree very well with these exact solutions. In particular, approximations of solutions for this type of problems, with periodic and decaying temperatures at the boundary, are constructed. Furthermore, these approximations are valid on a long time scale, and seems to be not available in the literature.

An Analytic Solution for 2D Heat Conduction Problems with Space–Time-Dependent Dirichlet Boundary Conditions and Heat Sources

This study proposes a closed-form solution for the two-dimensional (2D) transient heat conduction in a rectangular cross-section of an infinite bar with space–time-dependent Dirichlet boundary conditions and heat sources. The main purpose of this study is to eliminate the limitations of the previous study and add heat sources to the heat conduction system. The restriction of the previous study is that the values of the boundary conditions and initial conditions at the four corners of the rectangular region should be zero. First, the boundary value problem of 2D heat conduction system is transformed into a dimensionless form. Second, the dimensionless temperature function is transformed so that the temperatures at the four endpoints of the boundary of the rectangular region become zero. Dividing the system into two one-dimensional (1D) subsystems and solving them by combining the proposed shifting function method with the eigenfunction expansion theorem, the complete solution in series form is obtained through the superposition of the subsystem solutions. Three examples are studied to illustrate the efficiency and reliability of the method. For convenience, the space–time-dependent functions used in the examples are considered separable in the space–time domain. The linear, parabolic, and sine functions are chosen as the space-dependent functions, and the sine, cosine, and exponential functions are chosen as the time-dependent functions. The solutions in the literature are used to verify the correctness of the solutions derived using the proposed method, and the results are completely consistent. The parameter influence of the time-dependent function of the boundary conditions and heat sources on the temperature variation is also investigated. The time-dependent function includes exponential type and harmonic type. For the exponential time-dependent function, a smaller decay constant of the time-dependent function leads to a greater temperature drop. For the harmonic time-dependent function, a higher frequency of the time-dependent function leads to a more frequent fluctuation of the temperature change.

Instability analysis of double diffusive convection under time dependent solute boundary conditions in the presence of internal heat generator

The present study provides heat and mass transport of double-diffusive convective flow of Newtonian fluid with the presence of internal heat source. Here, we have considered two infinite horizontal plates apart from a constant distance under the solutal modulated boundary condition. To obtain the weakly nonlinear instability analysis, we drive Ginzburg–Landau equation. The effects of physical parameters obtained during the study on the Nusselt number and Sherwood number have been examined graphically. The Prandtl number and Richardson number enhance the rate of heat and mass transport, and Lewis number suppressed it. The major finding of the current study is that the internal heat source has a negligible effect on mass transfer, but a significant effect on the heat transfer.