SOLVING TRANSIENT INVERSE HEAT TRANSFER PROBLEMS IN COMPLEX GEOMETRIES USING PHYSICS-GUIDED NEURAL NETWORKS (PGNN)

A fixed grid based accurate phase-field method for dendritic solidification in complex geometries

An efficient computational model for deep low-enthalpy geothermal systems

A stable and efficient hybrid scheme for viscous problems in complex geometries

In most issues representing physical problems, the complex geometry cannot be represented by a Cartesian grid. The multi-block grid technique allows artificially reducing the complexity of the geometry by breaking down the real domain into a number of sub-domains with simpler geometry. The main aim of this article is to show the usefulness of simple solvers in complex geometry problems, when using curvilinear coordinates combined with multi-block grids. This requires adapted solvers to a nine nodes computational cell instead of the five nodes computational cell used with Cartesian coordinates for two-dimensional cases. These developments are presented for the simple iterative methods Jacobi and Gauss-Seidel and also for the incomplete factorization method strongly implicit procedure (SIP). These adapted solvers are tested in two cases: a simple geometry (heat transfer in a circular cross-section) and a complex geometry (solidification case). Results of the simple geometry case show that all the adapted solvers have good performance with a slight advantage for the SIP solver. For increasing the complexity of the geometry, the results showed that Jacobi and Gauss-Seidel solvers are not suitable. However, the SIP method has a reasonable performance. A conclusion could be drawn that the SIP method could be used in complex geometry problems using multi-block grid technique when high precision results are not required.

[1] Smits et al. [2011], hereafter referred to as S11, presented a study of the coupled vertical transfer of moisture and heat in a bare soil that included both laboratory measurements of transient flow and implementation of physically-based numerical models corresponding to these measurements. In the laboratory experiment a 1.1 m long vertical cylindrical column of uniformly packed medium-coarse textured sand was initially saturated at room temperature (23 ± 1°C) with the water table maintained at the top of the column and with a plastic sheet covering the upper surface to prevent evaporation. The bottom of the column was then sealed (preventing drainage) and the soil surface uncovered and exposed to infrared heating with a controller that kept the surface temperature nearly constant (56–60°C) for 32 day. Evaporation of moisture from the soil surface then occurred without forced convection due to an artificial wind source and was measured by monitoring column weight. Relative humidity of the air at the soil surface was measured by placing a sensor in “very good contact with the soil grains” at the top of the column. Soil temperature, volumetric water content, and matric potential were measured at 0.1 m intervals along the column starting at a depth of 0.1 m and down to 1 or 1.1 m. These measurements were used to test moisture and heat transfer theory that considers both liquid and vapor phases for water within the soil and associated mass flow of dry air. Two versions of the theory were considered, the first making the usual assumption that the vapor density in the soil pores is equal to the thermodynamic equilibrium value associated with the bound liquid phase and the second relaxing this assumption and providing an explicit expression for the local rate of evaporation. The authors claim that the theory with the nonequilibrium assumption fits the measurements better than the equilibrium case although their final conclusions are less definite, suggesting that “transport in the gas phase is better suited to be modeled with nonequilibrium liquid/gas phase change for highly transient field conditions where the thermal conditions at the land-atmosphere interface are constantly changing” (Abstract) and that the “nonequilibrium model provides an excellent alternative to the equilibrium approach when knowledge of the evaporation rate is not readily available” (Conclusions). Given that the validity of the equilibrium assumption has long been standard doctrine within soil physics (with some well-understood exceptions) and of particular importance for coupled heat and moisture flow the results claimed in this paper are unexpected and need careful examination. [5] Calculating the fractional degree of nonequilibrium with equation (6) is problematic mainly because Aep varies strongly with water content in a complex manner [Costanza-Robinson and Brusseau, 2002; Culligan et al., 2004] and, for a soil drying by evaporation, Seis significantly nonzero only in a narrow zone (the “drying front,” typically less than 2 mm wide) whose location varies dynamically near the soil surface and which separates the nearly air-dry surface layer and the wetter underlying soil [Fuchs and Tanner, 1967; van de Griend and Owe, 1994; Yamanaka et al., 1998; Novak, 2010]. This behavior is not discussed explicitly by S11 but the nonequilibrium vapor density curves in their Figures 3b, 3d, and 12 are consistent with it. The drying front apparently penetrates nearly 30 cm by the end of the experiment which perhaps is consistent with the artificially high evaporation rates (1.7 mm h−1 initially), initial saturation of the soil, prevention of drainage, and high saturated hydraulic conductivity (3744 mm h−1). According to equation (6), equilibrium is expected below the drying front (for ) because is nearly zero or even negative (condensation). Similar reasoning does not apply for the dry layer above the drying front because Aep also approaches zero at low water contents [Costanza-Robinson and Brusseau, 2002; Culligan et al., 2004]. Nonequilibrium in that layer though has little effect on overall evaporation since evaporation within it is small, the dominant process being diffusion of water vapor from the underlying evaporation zone to the atmosphere. The moderate sensitivity of evaporation to parameter b shown in Figure 3c of S11 is expected since the vapor density at the drying front partly controls the gradient driving water vapor through the dry layer (Figure 3d). [6] S11 infers nonequilibrium by comparing theory and measurements. To do so accurately requires isolating a measurement that is strongly dependent on the degree of nonequilibrium. S11 used the cumulative evaporation curve. This curve also depends critically on soil hydraulic properties and imposed boundary conditions and to a lesser extent on the soil thermal regime. S11 independently measured the soil water retention curve, saturated hydraulic conductivity, and soil thermal conductivity and fitted these to well-known empirical expressions but how well these represented the measurements and how well the samples used for these measurements were representative of the soil in the column is not discussed. The dependence of hydraulic conductivity on was not measured, therefore the accuracy of the van Genuchten-Mualem relative permeability model they used is not known.Hanks and Gardner [1965] showed conclusively that the dependence of the wet end of the hydraulic diffusivity curve strongly determines cumulative evaporation in a drying soil (and not the dry surface “mulch” layer usually considered limiting). Furthermore, according to S11 cumulative evaporation is more sensitive to the parameter a in the expression for , the vapor enhancement factor, than the nonequilibrium parameter b. As recognized by S11 the enhancement factor is controversial and according to their Figures 3a and 3d a lower value would require a lower value of b which reduces nonequilibrium. Therefore, the case for nonequilibrium presented by S11 is not proved unambiguously. [7] S11's description of surface boundary conditions for moisture transport and some of their theoretical results can be questioned, as follows: [8] 1. The nonequilibrium theory requires their transport equations (1) and (6), or alternately (1) and (10) or (6) and (10), and therefore correspondingly two upper boundary conditions for moisture flow, but only one was discussed. [9] 2. For the equilibrium/nonequilibrium comparison a surface resistance boundary condition, equation (11), was assumed. The surface resistance in large part represents the resistance to the diffusion of water vapor from the evaporation zone through the dry surface layer to the atmosphere [Fuchs and Tanner, 1967; van de Griend and Owe, 1994]. This resistance is already accounted for implicitly in the moisture flow equations and imposing such a resistance in the boundary condition amounts to “double counting”. Indeed the theory can be used to calculate the surface resistance at any time during the experiment. This is especially evident in S11 since the van de Griend and Owe [1994] expression they used was developed from typical field situations for which the depth of the drying layer is cm whereas apparently it was as much as 30 cm in the S11 experiment. Furthermore, the description of equation (11) is vague, referring to vapor concentrations immediately above (measured time-dependent relative humidity/temperature supposedly at the soil surface) and below (calculation depth unspecified) the soil surface. The best way to compare cumulative evaporation rate between equilibrium and nonequilibrium models would have been to impose the measured time-dependent vapor concentration as a Dirichlet boundary condition (as done for the heat equation with measured surface temperature). [10] 3. According to section 3.2, the surface resistance boundary condition, equation (11), was only used for the equilibrium/nonequilibrium comparison discussed in section 4.2. According to section 3.3, for the comparison of the nonequilibrium model with experimental data (section 4.1) either time-dependent vapor flux or concentration may be used, both apparently from measurements. The authors indicate that both boundary conditions yielded identical results. But if measured (evaporative) flux (from column weight) was imposed as a boundary condition then why do Figures 3a, 3c, and 6 show differences between calculated and measured cumulative evaporation? Indeed it was by minimizing these differences that parametersa and b were determined. Furthermore, it is difficult to comprehend the statements that the results were unchanged for vapor flux and vapor concentration boundary conditions. [11] 4. In section 4.3 and the Conclusions the authors indicate strongly that choosing whether to use an equilibrium or nonequilibrium model depends on which type of boundary condition is to be used and that the nonequilibrium model is more suited to transient field conditions. This is incorrect both mathematically and physically. Concentration, flux, or mixed boundary conditions (linear or nonlinear) may be applied to either model (e.g., Novak [2010], which utilizes an equilibrium model similar to S11's but with strongly nonlinear energy balance/aerodynamic transfer boundary conditions to simulate field measurements). It is not true that the equilibrium model cannot be driven with a surface vapor concentration or any other type of boundary condition. Furthermore, the equilibrium assumption is either valid or not valid in a soil and there is not any question of choice although, as discussed above, the author expects that if and when nonequilibrium occurs in the dry surface layer the consequences for evaporation are likely small. [12] 5. S11 neglected gas phase resistance in the equilibrium model for the equilibrium/nonequilibrium comparison. As it turns out advective vapor flux was 5 orders of magnitude smaller than diffusive flux in the nonequilibrium model (Figure 8) and so the assumption had little effect. Physically, advective gas fluxes must have been similar for both models because cumulative evaporation was similar and soil temperatures were likely similar for both models. The advection of air in the theory was due to expansion associated with heating and the replacement of evaporated water. [13] In conclusion, the authors of S11 have not shown beyond a doubt that nonequilibrium occurred in their laboratory column experiment. In this paper, the author argued above that the nonequilibrium term in their theory, equation (9), is unphysical and incomplete. The description of how the surface boundary condition for moisture was implemented in their models is often unclear. Interested readers would benefit greatly if in their reply to this comment all boundary conditions and input parameters used in the models and not described fully in Sll would be reported in a table (moisture b.c.’s, expressions for soil thermal properties, water retention curve, water and gas permeabilities, radial heat loss) and graphically (measured surface temperature). Explicit presentation of all model variables that would allow readers to repeat the calculations should be standard for theoretical studies of coupled moisture and heat transfer but unfortunately this has been overlooked in a number of such recent publications.

Radiative heat transfer in a complex two-dimensional enclosure with an absorbing, emitting, and isotropically scattering gray medium is investigated by using the finite-volume method. In particular, an implementation of the unstructured polygonal meshes, based on an unstructured triangular system, is introduced, thereby reducing the computation time required for the analysis of thermal radiation. By connecting each center point of the triangular meshes rather than joining the centroids of the triangular elements to the midpoints of the corresponding sides to form a polygonal element, it not only prevents concave shapes of adjacent faces, it also reduces the number of control-volume faces of the polygonal mesh. After a mathematical formulation and corresponding discretization equation for the radiative transfer equation are derived, the final discretization equation is introduced with the conventional finite-volume approaches. The present formulation is then validated by comparing results with those obtained in previous related works. All the results presented in this work show that the present method is accurate and computationally efficient for the analysis of radiative heat transfer problems in complex geometries.

Prediction of radiative heat transfer in 2D irregular geometries using the collocation spectral method based on body-fitted coordinates

While the Anomaly flow was originally motivated by string theory, its zero slope case is potentially of considerable interest in non-Kahler geometry, as it is a flow of conformally balanced metrics whose stationary points are precisely Kahler metrics. We establish its convergence on Kahler manifolds for suitable initial data. We also discuss its relation to some current problems in complex geometry.

A procedure to obtain accurate solutions for many transient conduction problems in complex geometries using a Galerkin-based integral (GBI) method is presented. The nonhomogeneous boundary conditions are accommodated by the Green’s function solution technique. A Green’s function obtained by the GBI method exhibits excellent large-time accuracy. It is shown that the time partitioning of the Green’s function yields accurate small-time and large-time solutions. In one example, a hollow cylinder with convective inner surface and prescribed heat flux at the outer surface is considered. Only a few terms for both large-time and small-time solutions are sufficient to produce results with excellent accuracy. The methodology used for homogeneous solids is modified for application to complex heterogeneous solids.

The limiting hole pressure in three dimensions

On the Finite Difference-Based Lattice Boltzmann Method in Curvilinear Coordinates

As computational fluid dynamics (CFD) simulations still take a lot of time in industry for design optimization analyses, we propose to reduce the number of required CFD simulations considerably using a new meta-algorithm. The algorithm consists of employing kriging in conjunction with simulated annealing (SA) as a design tool. The method enables effective and fast design optimization for complex fluid flow systems. In this paper, it is applied to a convective heat transfer problem in a channel with periodic heat sources with constant heat flux. We show that the geometry optimization of two dimensional thermo-fluids problems in complex geometries is made more effective and faster using kriging-simulated annealing meta-algorithm.

SummaryThis study documents the first attempt to extend the singular boundary method, a novel meshless boundary collocation method, for the solution of 3D elasticity problems. The singular boundary method involves a coupling between the regularized BEM and the method of fundamental solutions. The main idea here is to fully inherit the dimensionality and stability advantages of the former and the meshless and integration‐free attributes of the later. This makes it particularly attractive for problems in complex geometries and three dimensions. Four benchmark 3D problems in linear elasticity are well studied to demonstrate the feasibility and accuracy of the proposed method. The advantages, disadvantages, and potential applications of the proposed method, as compared with the FEM, BEM, and method of fundamental solutions, are also examined and discussed. Copyright © 2015 John Wiley & Sons, Ltd.

The Earth's crust is an enormous thermal reservoir that releases or absorbs heat energy in response to rapid changes in near surface temperatures. Numerical simulations of basin-scale heat transport which set the lower thermal boundary condition at the sediment-basement interface assume that temperature changes in the near surface have no effect on basement rocks and vice versa. We compare two numerical models of transient fluid flow and heat transport by topographically driven recharge in a foreland basin. In the non-extended model, a constant heat flow boundary condition is set at the sediment-basement contact. In the extended model, the same thermal constraint is set at a depth of 15 km. For rapid changes in temperatures caused by fluid flow ([approximately]1 m.y.), thermal buffering by basement rocks reduces cooling of deep basin sediments near the fold-thrust belt by fresh water recharge and warming of the basin margin by upward discharge of pore fluids from depth. The non-extended model predicts an exaggerated thermal transient or pulse, a temporary warming of basin sediments by hot fluids in excess of steady-state temperatures, in the discharge region of the basin. Exaggerated or spurious temperature predictions also can occur in numerical simulations of heat advection by othermore » geodynamic processes or thermal problems where heat refraction is important. For thermal systems that change slowly ([approximately]100 m.y.) and where heat refraction is minor, the choice of lower thermal boundary condition makes little difference.« less

This paper presents numerical results for the effects of flow development length and thermal boundary conditions on local sCO2 heat transfer and flow performance in horizontal circular tubes. For the development length study, the tubes consist of various lengths of adiabatic sections and were connected by a fixed-length heat transfer section. An adiabatic entry region with lengths between 0 and 1000 mm, followed by a 500 mm long heat transfer section were tested. Boundary conditions including heat flux, mass flux, and inlet temperature were varied. Centerline velocity and local heat transfer rate were compared to investigate the impact of the length of adiabatic development section. To study the effects of thermal boundary conditions on sCO2 flows, a single tube (with and without a surrounding wall) and an idealized tube-in-tube heat exchanger were implemented. A uniform heat flux and uniform temperature boundary condition were applied to the single tubes, while the tube-in-tube heat exchanger had an inherent conjugate boundary condition. The resulting local sCO2 heat transfer coefficients for the different thermal boundary condition cases were compared.