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Purpose This paper aims to discuss inverse problems that arise in a variety of practical thermal processes and systems. It presents some of the approaches that may be used to obtain results that lie within a small region of uncertainty. Therefore, the non-uniqueness of the solution is reduced so that the final design and boundary conditions may be determined. Optimization methods that may be used to reduce the uncertainty and to select locations for experimental data and for minimizing the error are presented. A few examples of thermal systems are given to illustrate the applicability of these methods and the challenges that must be addressed in solving inverse problems. Design/methodology/approach In most analytical and numerical solutions, the basic equations that describe the process, as well as the relevant and appropriate boundary conditions, are known. The interest lies in obtaining a unique solution that satisfies the equations and boundary conditions. This may be termed as a direct or forward solution. However, there are many problems, particularly in practical systems, where the desired result is known but the conditions needed for achieving it are not known. These are generally known as inverse problems. In manufacturing, for instance, the temperature variation to which a component must be subjected to obtain desired characteristics is prescribed, but the means to achieve this variation are not known. An example of this circumstance is the annealing, tempering or hardening of steel. In such cases, the boundary and initial conditions are not known and must be determined by solving the inverse problem to obtain the desired temperature variation in the component. The solutions, thus, obtained are generally not unique. This is a review paper, which discusses inverse problems that arise in a variety of practical thermal processes and systems. It presents some of the approaches or strategies that may be used to obtain results that lie within a small region of uncertainty. It is important to realize that the solution is not unique, and this non-uniqueness must be reduced so that the final design and boundary conditions may be determined with acceptable accuracy and repeatability. Optimization techniques are often used for minimizing the error. This review presents several methods that may be applied to reduce the uncertainty and to select locations for experimental data for the best results. A few examples of thermal systems are given to illustrate the applicability of these methods and the challenges that must be addressed in solving inverse problems. By considering a variety of systems, the paper also shows the importance of solving inverse problems to obtain results that may be used to model and design thermal processes and systems. Findings The solution of inverse problems, which frequently arise in thermal processes, is discussed. Different strategies to obtain the conditions that lead to the desired result are given. The goal of these approaches is to reduce uncertainty and obtain essentially unique solutions for different circumstances. The error of the method can be checked against known conditions to see if it is acceptable for the given problem. Several examples are given to illustrate the use of these methods. Originality/value The basic strategies presented here for solving inverse problems that arise in thermal processes and systems, as well as the optimization techniques used to reduce the domain of uncertainty, are fairly original. They are used for certain challenging problems that have not been considered in detail earlier. Several methods are outlined for considering different types of problems.

Purpose The purpose of this paper is to apply two optimization methods to the issue of sensible energy store design. Design/methodology/approach This paper is a comparison of topology optimization and genetic algorithms. Findings Genetic algorithms are prone to converge to local maxima while requiring significantly longer convergence times compared to topology optimization. Topology optimization resulted in structures representing parallel sheets, which are as thin as the grid allows. These configurations can maintain the maximum surface area between the low and high conductivity materials at high refinement, resulting in the best performance. Practical implications Time required for 99 per cent store discharge is decreased by 70 per cent using a 50 × 50 optimization grid at a loading of 10 Vol.%. Originality/value These approaches have not been compared nor applied to this specific problem before. Value is in the key finding that maximization of surface area is only possible with fins/sheets and not tree structures. This dictates the optimal solution for dynamic behaviour.

Purpose The purpose of this paper is to highlight the advantages of a simplified algorithm to solve the problem of heat and mass transfer in porous medium by reducing the number of partial differential equations from four to three. Design/methodology/approach The approach of the present paper is to develop a simplified algorithm to reduce the number of equations involved in conjugate heat transfer in porous medium. Findings Developed algorithm/method has many advantages over conventional method of solution for conjugate heat transfer in porous medium. Research limitations/implications The current work is applicable to conjugate heat transfer problem. Practical implications The developed algorithm is useful in reducing the number of equations to be solved, thus reducing the computational resources required. Originality/value Development of simplified algorithm and comparison with conventional method.

PurposeThe presented paper aims to consider algorithm for optimal design of multilayer thermal insulation.Design/methodology/approachDeveloped algorithm is based on a sequential quadratic programming method.Findings2D mathematical model of heat transfer in thermal protection was considered in frame of thermal design of spacecraft. The sensitivity functions were used to estimate the Jacobean of the object functions.Research limitations/implicationsDesign of distributed parameter systems and shape optimization may be thought of as geometrical inverse problems, in which the positions of free boundaries are determined along with the spatial variables. In such problems, the missing data (i.e. the position of boundaries) are compensated for by the presence of the so-called inverse problem additional conditions. In the case under consideration, such conditions are constrains on the temperature values at the discrete points of the system.Practical implicationsResults are presented how to apply the algorithm suggested for solving a practical problem – thickness sampling for a thermal protection system of advanced solar probe.Originality/valueThe procedure proposed in the paper to solve a design problem is based on the method of quadratic approximation of the initial problem statement as a Lagrange formulation. This has allowed to construct a rather universal algorithm applicable without modification for solving a wide range of thermal design problems.

Purpose The purpose of this study is to show that the methods of the numerical simulation can be a very effective tool for a proper choice of control parameters of artificial hyperthermia. An electromagnetic field induced by two external electrodes and a temperature field resulting from electrodes action in a 3D domain of biological tissue is considered. An important problem is the appropriate directing of heat in the region of tumor, so as to avoid damaging healthy cells surrounding the tumor. Recently, to concentrate the heat on the tumor, magnetic nanoparticles, which are introduced into the tumor, were used. The nanoparticles should be made of material that ensures appropriate magnetic properties and has a high biocompatibility with the biological tissue. External electric field causes the heat generation in the tissue domain. Design/methodology/approach The distribution of electric potential in the domain considered is described by the Laplace system of equations, while the temperature field is described by the Pennes’ system of equations. These problems are coupled by source function being the additional component in the Pennes’ equation and resulting from the electric field action. The boundary element method is applied to solve the coupled problem connected with the heating of biological tissues. Findings The aim of investigations is to determine an electric potential of external electrodes and the number of nanoparticles introduced to a tumor region to obtain the artificial hyperthermia state. The tests performed showed that the proposed tool to solve the inverse problem provides correct results. Research limitations/implications In the paper the steady state bioheat transfer problem is considered, so the thermal damage is a function of the temperature only. Therefore, the solution can be considered as the maximum ablation zone of cancer. Additionally, the choice of appropriate parameters will be affected on the position and shape of the tumor and the electrodes. Originality/value In the paper the inverse problem has been solved using the evolutionary algorithm, gradient method and hybrid algorithm which is a combination of the two previous.

Purpose The purpose of this paper is to present the multicriteria identification method used for solving the microscale heat transfer problem. The thin film exposed to ultrashort laser pulse is modeled using the finite difference method. The parameters of the model are tuned on the basis of experimental data. The multicriteria identification of the numerical model parameters is performed for subsets of experimental data. Design/methodology/approach The multicriteria identification method is used in the paper. The Pareto front for two criterions is created. The two-temperature model of heat transfer in microscale is used in the numerical model. Findings The multicriteria identification for two subsets of experimental data leads to different results. The obtained Pareto front allows to choose the most suitable set of numerical model parameters. Originality/value The multicriteria identification method was used for the first time to solve the microscale heat transfer problem.

Purpose This paper aims to present development and application of the Bayesian inverse approach for retrieving parameters of non-linear diffusion coefficient based on the integral information. Design/methodology/approach The Bayes formula was used to construct posterior distribution of the unknown parameters of non-linear diffusion coefficient. The resulting aposteriori distribution of sought parameters was integrated using Markov Chain Monte Carlo method to obtain expected values of estimated diffusivity parameters as well as their confidence intervals. Unsteady non-linear diffusion equation was discretised with the Global Radial Basis Function Collocation method and solved in time using Crank–Nicholson technique. Findings A number of manufactured analytical solutions of the non-linear diffusion problem was used to verify accuracy of the developed inverse approach. Reasonably good agreement, even for highly correlated parameters, was obtained. Therefore, the technique was used to compute concentration dependent diffusion coefficient of water in paper. Originality/value An original inverse technique, which couples efficiently meshless solution of the diffusion problem with the Bayesian inverse methodology, is presented in the paper. This methodology was extensively verified and applied to the real-life problem.

Purpose The purpose of this paper is to use the variational iteration method (VIM) for studying boundary value problems (BVPs) characterized with dual solutions. Design/methodology/approach The VIM proved to be practical for solving linear and nonlinear problems arising in scientific and engineering applications. In this work, the aim is to use the VIM for a reliable treatment of nonlinear boundary value problems characterized with dual solutions. Findings The VIM is shown to solve nonlinear BVPs, either linear or nonlinear. It is shown that the VIM solves these models without requiring restrictive assumptions and in a straightforward manner. The conclusions are justified by investigating many scientific models. Research limitations/implications The VIM provides convergent series solutions for linear and nonlinear equations in the same manner. Practical implications The VIM is practical and shows more power compared to existing techniques. Social implications The VIM handles linear and nonlinear models in the same manner. Originality/value This work highlights a reliable technique for solving nonlinear BVPs that possess dual solutions. This paper has shown the power of the VIM for handling BVPs.

Purpose – The purpose of this paper is to propose a computational model for thin layers, for problems of linear time-dependent heat conduction. The thin layer is replaced by a zero-thickness interface. The advantage of the new model is that it saves the need to construct and use a fine mesh inside the layer and in regions adjacent to it, and thus leads to a reduction in the computational effort associated with implicit or explicit finite element schemes. Design/methodology/approach – Special asymptotic models have been proposed for linear heat transfer and linear elasticity, to handle thin layers. In these models the thin layer is replaced by an interface with zero thickness, and specific jump conditions are imposed on this interface in order to represent the special effect of the layer. One such asymptotic interface model is the first-order Bövik-Benveniste model. In a paper by Sussmann et al., this model was incorporated in a FE formulation for linear steady-state heat conduction problems, and was shown to yield an accurate and efficient computational scheme. Here, this work is extended to the time-dependent case. Findings – As shown here, and demonstrated by numerical examples, the new model offers a cost-effective way of handling thin layers in linear time-dependent heat conduction problems. The hybrid asymptotic-FE scheme can be used with either implicit or explicit time stepping. Since the formulation can easily be symmetrized by one of several techniques, the lack of self-adjointness of the original formulation does not hinder an accurate and efficient solution. Originality/value – Most of the literature on asymptotic models for thin layers, replacing the layer by an interface, is analytic in nature. The proposed model is presented in a computational context, fitting naturally into a finite element framework, with both implicit and explicit time stepping, while saving the need for expensive mesh construction inside the layer and in its vicinity.

Purpose – In an earlier paper (Turkyilmazoglu, 2011a), the author introduced a new optimal variational iteration method. The idea was to insert a parameter into the classical variational iteration formula in an aim to prevent divergence or to accelerate the slow convergence property of the classical approach. The purpose of this paper is to approve the superiority of the proposed method over the traditional one on several physical problems treated before by the classical variational iteration method. Design/methodology/approach – A sufficient condition theorem with an upper bound for the error is also presented to further justify the convergence of the new variational iteration method. Findings – The optimal variational iteration method is found to be useful for heat and fluid flow problems. Originality/value – The optimal variational iteration method is shown to be convergent under sufficient conditions. A novel approach to obtain the optimal convergence parameter is introduced.

Purpose – The purpose of this paper is to propose a parallel partition of unity method to solve the time-dependent convection-diffusion equations. Design/methodology/approach – This paper opted for the time-dependent convection-diffusion equations using the finite element method and the partition of unity method. Findings – This paper provides one efficient parallel algorithm which reaches the same accuracy as the standard Galerkin method (SGM) but saves a lot of computational time. Originality/value – In this paper, a parallel partition of unity method is proposed for the time-dependent convection-diffusion equations. At each time step, the authors only need to solve a series of independent local sub-problems in parallel instead of one global problem.

Purpose – The purpose of this paper is to investigate numerically the effect of rheology on the leveling of thin fluid films on horizontal solid substrates. Design/methodology/approach – A mathematical model based on the lubrication approximation which defines non-Newtonian rheology using a Power-law model is presented. The rheology is described by two parameters: the consistency factor and the flow behavior index. The resulting highly non-linear coupled set of equations is discretized using Finite-Difference and the resulting algebraic system is solved via an efficient Multigrid algorithm. Findings – Importantly, the non-dimensionalization process leads to a pair of Partial Differential Equations which depends on one parameter only, the flow behavior index. The authors show that the consistency factor only affects the time scale of the leveling process, hence stretching or contracting the time line. Results for the leveling of sinusoidal perturbations of the fluid film highlights important differences between the leveling of shear-thinning and shear-thickening fluids. In a normalized time frame, the onset of leveling occurs earlier for the shear-thinning fluid than for the shear-thickening one. However, the dimensionless leveling rate is higher for the shear-thickening fluid than the shear-thinning one. This results in a “threshold thickness” which delimits two regimes: the shear-thinning fluid levels to a thickness above this threshold faster than the shear-thickening fluid but the opposite is true for a film thickness below this threshold. An important aspect of this study is the verification of the numerical implementation using the Method of Manufactured Solutions (MMS), a first in the context of thin film studies. The paper also highlights differences between the leveling of two-dimensional and three-dimensional thickness perturbations. Originality/value – The study of the leveling of disturbances at the free surface of a liquid film using a Power-law rheological model does not appear to have been covered in the literature. Also, the paper uses the MMS to test the validity of the implementation. This appears to be the first time it has been used in the context of the lubrication approximation. Finally, unlike most prior studies, the work does away with the planar assumption.

– The first purpose of this paper is to design more accurate, efficient and robust gradient smoothing methods (GSMs) for spatial derivative approximations for computational fluid dynamics (CFD) application. The second purpose is to design an adaptive GSM-CFD solver for the compressible turbulent flows, with special focus on the shock-wave boundary layer interactions. , – A new integration scheme is proposed for the node-associated GSM to improve the accuracy and robustness of the previous versions. A matrix-based algorithm and corresponding data structures are devised to improve the computational efficiency of GSM. The GSM-CFD solver is coupled with a mixed solution-based adaptive mesher to form a functional adaptive GSM-CFD solver., – The improved GSMs are insensitive to mesh qualities, and can achieve high accuracy on all kinds of hybrid meshes. The adaptive GSM-CFD solver can better capture the shock wave. , – The matrix-based GSM and its corresponding data structure for improved GSM, and the development of the adaptive GSM-CFD solver for compressible turbulent flows are newly presented in this paper.

Purpose– The purpose of this paper is to present an improved computational method for determining the friction factor for turbulent flow in pipes.Design/methodology/approach– Given that the absolute pipe roughness is generally constant in most systems, and that there are few changes to the pipe diameter, the proposed method uses a simplified equation for systems with a specific relative pipe roughness. The accuracy of the estimation of the friction factor using the proposed method is compared to the values obtained using the implicit Colebrook-White equation while the computational efficiency is determined by comparing the time taken to perform 300 million calculations.Findings– The proposed method offers a significant improvement in computational efficiency for its accuracy and is compared 28 of the explicit equations currently in use.Practical implications– This method enables a simplified equation to offer a significant improvement in computational efficiency for its accuracy and is easier to code, enabling engineers to more efficiently calculate frictional pressure loss for flow in pipes.Originality/value– Due to the complexities in flow regime and pipe roughness, there is a limit to the scope for further computational efficiency through simplification of the explicit equation. This paper presents a new method and simplified equation which combined are able to deliver results with similar accuracy to less computationally efficient explicit equations.

Purpose – The purpose of this paper is concerned with a reliable treatment of the classical Stephan problem. The Adomian decomposition method (ADM) is used to carry out the analysis, Moreover, the authors extend the work to examine the Stefan problem with variable latent heat. The study confirms the accuracy and efficiency of the employed method. Design/methodology/approach – The new technique, as presented in this paper in extending the applicability of the ADM, has been shown to be very efficient for solving the Stefan problem. Findings – The Stefan problem with variable latent heat was examined as well. The ADM was effectively used for analytic treatment of the Stefan problem with and without variable latent heat. Originality/value – The paper presents a new solution algorithm for the Stefan problem.

Purpose – The purpose of this paper is to develop new types of direct expansion method of moments (DEMM) by using the n/3th moments for simulating nanoparticle Brownian coagulation in the free molecule regime. The feasibilities of new proposed DEMMs with n/3th moments are investigated to describe the evolution of aerosol size distribution, and some of the models will be applied to further simulation of physical processes. Design/methodology/approach – The accuracy and efficiency of some kinds of methods of moments are mainly compared including the quadrature method of moments (QMOM), Taylor-expansion method of moments (TEMOM), the log-normal preserving method of moments proposed by Lee (LMM) and the derived DEMM in this paper. QMOM with 12 quadrature approximation points is taken as a reference to evaluate other methods. Findings – The newly derived models, namely DEMM(4/3,4) and DEMM(2,6), as well as the previous DEMM(2,4), are considered to be qualified models due to their high accuracy and efficiency. They are confirmed to be valid and alternative models to describe the evolution of aerosol size distribution for particle dynamical process involving the n/3th moments. Originality/value – The n/3th moments, which have clear physical interpretations when n stands for first several integers, are first introduced in the DEMM method for simulating nanoparticle Brownian coagulation in the free molecule regime.

Purpose – Multiple length and time scales arise in a wide variety of practical and fundamental problems. It is important to obtain accurate and validated numerical simulation results, considering the different scales that exist, in order to predict, design and optimize the behavior of practical thermal processes and systems. The purpose of this paper is to present modeling at the different length scales and then addresses the question of coupling the different models to obtain the overall model for the system or process. Design/methodology/approach – Both numerical and experimental methods to obtain results at the different length scales, particularly at micro and nanoscales, are considered. Even though the paper focusses on length scales, multiple time scales lead to similar concerns and are also considered. The two circumstances considered in detail are multiple length scales in different domains and those in the same domain. These two cases have to be modeled quite differently in order to obtain a model for the overall process or system. The basic considerations involved in such a modeling are discussed. A wide range of thermal processes are considered and the methods that may be used are presented. The models employed must be validated and the accuracy of the simulation results established if the simulation results are to be used for prediction, control and design. Findings – Of particular interest are concerns like verification and validation, imposition of appropriate boundary conditions, and modeling of complex, multimode transport phenomena in multiple scales. Additional effects such as viscous dissipation, surface tension, buoyancy and rarefaction that could arise and complicate the modeling are discussed. Uncertainties that arise in material properties and in boundary conditions are also important in design and optimization. Large variations in the geometry and coupled multiple regions are also discussed. Research limitations/implications – The paper is largely focussed on multiple-scale considerations in thermal processes. Both numerical modeling/simulation and experimentation are considered, with the latter being used for validation and physical insight. Practical implications – Several examples from materials processing, environmental flows and electronic systems, including data centers, are given to present the different techniques that may be used to achieve the desired level of accuracy and predictability. Originality/value – Present state of the art and future needs in this interesting and challenging area are discussed, providing the impetus for further work. Different methods for treating multiscale problems are presented.

Purpose – This paper aims to test the inverse analysis, based on the standard least-square method, which will finally lead to find the appropriate parameters of modelling of the brain cooling process. Design/methodology/approach – To test the presented in this paper method of inverse analysis the numerical simulations of the bioheat transfer process in the neonatal body were performed. To model the bioheat transfer the Pennes bioheat equation and the modified Fiala model were applied. Findings – The performed tests of the inverse analysis proved that it is possible to estimate the proper parameters of the process using this tool, but always with the small mistake. Research limitations/implications – The presented method still requires a lot of tests. The test with the data from real measurements can be very valuable. Practical implications – The determination of the proper parameters of the bioheat transfer in the neonatal body can finally be used to perform the numerical simulations of the brain cooling process. Social implications – The performance of the numerical simulations of the brain cooling process in the proper way can finally helps protect newborns’ health and life. Originality/value – In the paper the attempt of the inverse analysis in order to determine the parameters of bioheat transfer in the newborn's body is made.

Purpose – Both the importance of the natural convection in science and engineering and the shortage of publications in the field of numerical features of time-stepping schemes for the simulation of coupled heat and fluid flow problems motivate the present work. The paper aims to discuss these issues. Design/methodology/approach – The paper presents the unconditionally stable time-stepping scheme for simulation of coupled problems of mass and heat transport. The paper is divided into two parts. The first part concerns the mathematical formulation of the scheme and discusses its implementation. The second part focuses on the numerical simulation and its results. A detailed investigation of the temporal order of the scheme with respect to the L2-norms of the errors of the pressure, velocity, temperature and divergence of velocity fields has also been given. Findings – The work shows that it is possible to formulate a numerical scheme which is unconditionally stable with respect to the time step size. Moreover, application of the spectral element method for the spatial discretization results in a high order of approximation in space and very good overall accuracy. Furthermore, the investigation of the numerical features of the scheme showed that the formal temporal order of the scheme (formally second order) has been deferred very slightly and the order of 1.8-1.9 is achieved for all unknown fields. Originality/value – The paper presents a new unconditionally stable scheme for simulation of unsteady flows with bidirectional coupling of heat transfer and the fluid flow. It also carefully investigates the numerical behaviour of the method.

PurposeThe authors' objective in this paper is to find the numerical solutions of obstacle, unilateral and contact second‐order boundary‐value problems.Design/methodology/approachTo achieve this, the authors formulate a spatially adaptive grid refinement scheme following Galerkin's finite element method based on a weighted‐residual. A residual based a‐posteriori error estimation scheme has been utilized for checking the approximate solutions for various finite element grids. The local element balance has been considered as an error assessment criterion. The approach utilizes piece‐wise linear approximations utilizing linear Langrange polynomials. Numerical experiments indicate that local errors are large in regions where the gradients are large.FindingsA comparison of the spatially adaptive grid refinement with that of uniform meshing for second order obstacle boundary value problems confirms the superiority of the scheme without increasing the number of unknown coefficients.Originality/valueThe authors believe the work has merit not only in terms of the approach but also of the problem solved in the paper.

PurposeThe purpose of this paper is to investigate a large‐eddy simulation, using low order numerical discretization and upwinding schemes on unstructured grids, for a turbulent free jet at Mach number 0.95. The accuracy and stability performance is discussed for the finite element/volume upwinding numerical code used.Design/methodology/approachThis code is equipped with a self‐adaptive upwinding method which has been previously developed to reduce the numerical dissipation of applied low order flux calculation on unstructured elements using Roe's scheme. Herein, this method is used to numerically investigate a high Reynolds, compressible turbulent free jet and compare the results with a recently published set of experimental data. The effect of grid size is also investigated. A reasonable good agreement with the experimental measurements is obtained.FindingsBased on the results, it is concluded that the developed self‐adaptive upwinding scheme provides a considerably better emulation of the flow regime in comparison to the full‐upwinding scheme. Different case studies have been carried out to assess the performance of self‐adaptive upwinding method and the effect of the subgrid model.Originality/valueThis paper presents an original research on self‐adaptive upwinding scheme and the effect of the subgrid model on a compressible turbulent free jet.

PurposeThe purpose of this paper is to present two efficient immersed boundary methods (IBM) for simulation of thermal flow problems. One method is for given temperature condition (Dirichlet type), while the other is for given heat flux condition (Neumann type). The methods are applied to simulate natural and mixed convection problems to check their performance. The comparison of present results with available data in the literature shows that the present methods can obtain accurate numerical results efficiently.Design/methodology/approachThe paper presents two efficient IBM solvers, in which the effect of thermal boundary to its surrounding fluid is considered through the introduction of a heat source/sink term into the energy equation. One is the temperature correction‐based IBM developed for problems with given temperature on the wall. The other is heat flux correction‐based IBM for problems with given heat flux on the wall. Note that in this solver, the offset of derivative condition is directly used to correct the temperature field.FindingsAs compared with existing solvers, the temperature correction‐based IBM determines the heat source/sink implicitly instead of pre‐calculated explicitly, so that the boundary condition for temperature is accurately satisfied. To the best of the authors' knowledge, the work of heat flux correction‐based IBM is the first endeavour for application of IBM to solve thermal flow problems with Neumann (heat flux) boundary condition. It was found that both methods presented in this work can efficiently obtain accurate numerical results for thermal flow problems.Originality/valueThe two methods presented in this paper are novel. They can effectively solve thermal flow problems with Dirichlet and Neumann boundary conditions.

PurposeSimple methods for the steady‐state analysis of a flow network are readily available, but the dynamic behavior of a large‐scale flow network is difficult to study due to the complex differential‐algebraic equation system resulting from its modeling. It is the aim of this paper to present two simple methods for the dynamic analysis of large‐scale flow networks and to demonstrate their use by examining the dynamics of a self‐similar branching tree network.Design/methodology/approachTwo numerical projection methods are proposed for one‐dimensional dynamic analysis of large piping networks. Both are extensions of that suggested by Chorin for the nonlinear differential‐algebraic system resulting from the Navier‐Stokes equations. Each numerical algorithm is discussed and verified for turbulent flow in a nonlinear, self‐similar, branching tree network with constant friction factor for which an exact solution is available.FindingsThe dynamics of this network are calculated for more realistic friction factors and described as system parameters are varied. Self‐excited oscillations due to laminar‐turbulent transition are found for some parameter values and dynamic component behavior is observed in the network which is not observable in components apart from it.Practical implicationsIt is shown that the dynamics of a flow network can exhibit unexpected behavior, reinforcing the need for simple methods to perform dynamic analysis.Originality/valueThis paper presents two numerical projection schemes for dynamic analysis of large‐scale flow networks to aid in their study and design.

PurposeThe aim of this paper is to present the numerical simulation of the population dynamics model with density‐dependent migrations and the Allee effects using the homotopy perturbation method (HPM).Design/methodology/approachThe paper applied HPM and the results obtained by this method are compared to the exact solution. The results show applicability, accuracy and efficiency of HPM in solving the transient non‐linear advection‐diffusion‐reaction equation.FindingsIn this study, HPM is applied to solve the model of population dynamics with density‐ dependent migrations and the Allee effects. Numerical approximations show a high degree of accuracy. The numerical results we obtained justify the advantage of this methodology, even in the few terms approximation is accurate. Errors can be made smaller by taking new terms of the iteration formulas.Originality/valueIt is confirmed that the work has not been published in any other journal.

PurposeMulti‐point boundary value problems have important roles in the modelling of various problems in physics and engineering. This paper aims to present the solution of ordinary differential equations with multi‐point boundary value conditions by means of a semi‐numerical approach which is based on the homotopy analysis method.Design/methodology/approachThe convergence of the obtained solution is expressed and some typical examples are employed to illustrate validity, effectiveness and flexibility of this procedure. This approach, in contrast to perturbation techniques, is valid even for systems without any small/large parameters and therefore it can be applied more widely than perturbation techniques, especially when there do not exist any small/large quantities.FindingsUnlike other analytic techniques, this approach provides a convenient way to adjust and control the convergence of approximation series. Some applications will be briefly introduced.Originality/valueThe paper shows how an important boundary value problem is solved with a semi‐analytical method.

PurposeThe main aim of this paper is to utilize the different forms of functions for the numerical solution of the two‐dimensional (2‐D) inverse heat conduction problem with temperature‐dependent thermo‐physical properties (TDTPs).Design/methodology/approachThe proposed numerical technique is based on the modified elitist genetic algorithm (MEGA) combined with finite different method (FDM) to simultaneously estimate temperature‐dependent thermal conductivity and heat capacity. In this paper, simulated (noisy and filtered) temperatures are used instead of experimental data. The estimated temperatures are obtained from the direct numerical solution (FDM) of the 2‐D conductive model by using an estimate for the unknown TDTPs and MEGA is used to minimize a least squares objective function containing estimated and simulated (noisy and filtered) temperatures.FindingsThe accuracy of the MEGA is assessed by comparing the estimated and the pre‐selected TDTPs. The results show that the measurement errors do not considerably affect the accuracy of the estimates. In other words, the proposed method provides a practical and confident prediction in simultaneously estimating the temperature‐dependent heat capacity and thermal conductivity. From the results, it is found that the RMS error between estimated and simulated temperatures is smaller for linear simulation and also we found this form convenient for parameters estimations.Research limitations/implicationsFuture approaches should find the optimal design of case study and then apply the proposed method to achieve the best results.Originality/valueApplications of the results presented in this paper can be of value in practical applications in parameter estimation even with one sensor temperature history.

PurposeThe Hermite collocation method of discretization can be used to determine highly accurate solutions to the steady‐state one‐dimensional convection‐diffusion equation (which can be used to model the transport of contaminants dissolved in groundwater). This accuracy is dependent upon sufficient refinement of the finite‐element mesh as well as applying upstream or downstream weighting to the convective term through the determination of collocation locations which meet specified constraints. Owing to an increase in computational intensity of the application of the method of collocation associated with increases in the mesh refinement, minimal mesh refinement is sought. Very often this optimization problem is the one where the feasible region is not connected and as such requires a specialized optimization search technique. This paper aims to focus on this method.Design/methodology/approachAn original hybrid method that utilizes a specialized adaptive genetic algorithm followed by a hill‐climbing approach is used to search for the optimal mesh refinement for a number of models differentiated by their velocity fields. The adaptive genetic algorithm is used to determine a mesh refinement that is close to a locally optimal mesh refinement. Following the adaptive genetic algorithm, a hill‐climbing approach is used to determine a local optimal feasible mesh refinement.FindingsIn all cases the optimal mesh refinements determined with this hybrid method are equally optimal to, or a significant improvement over, mesh refinements determined through direct search methods.Research limitationsFurther extensions of this work could include the application of the mesh refinement technique presented in this paper to non‐steady‐state problems with time‐dependent coefficients with multi‐dimensional velocity fields.Originality/valueThe present work applies an original hybrid optimization technique to obtain highly accurate solutions using the method of Hermite collocation with minimal mesh refinement.

PurposeThis paper aims to perform a comparative study between capabilities of two numerical schemes from two main branches of numerical methods for solving hyperbolic conservation equations.Design/methodology/approachThe accuracy and performance of a newly developed high‐resolution central scheme vs a higher‐order Godunov‐based method are evaluated in the context of black‐oil reservoir simulations. Both methods are modified enabling study of applications that are not strictly hyperbolic and exhibit local linear degeneracies in their wave structure.FindingsThe numerical computations show that while both schemes produce results with virtually the same accuracy, the Godunov method reproduces slightly more accurate results at the expense of calculation of eigen‐structures.Research limitations/implicationsThe paper investigates only one dimensional problems, but the idea can be easily extended to multi‐dimensional problems.Practical implicationsThe paper shows the strengths and weaknesses of two practical numerical methods.Originality/valueSuch comparative study has not been published elsewhere and in particular, the performance of numerical methods on simulating hysteresis effect in hydrocarbon reservoirs has not been investigated in detail before.

PurposeThis paper aims to develop and describe an improved process for determining the rate of heat generation in living tissue.Design/methodology/approachPrevious work by the authors on solving the bioheat equation has been updated to include a new localized meshless method which will create a more robust and computationally efficient technique. Inclusion of this technique will allow for the solution of more complex and realistic geometries, which are typical of living tissue. Additionally, the unknown heat generation rates are found through genetic algorithm optimization.FindingsThe localized technique showed superior accuracy and significant savings in memory and processor time. The computational efficiency of the newly proposed meshless solver allows the optimization process to be carried to a higher level, leading to more accurate solutions for the inverse technique. Several example cases are presented to demonstrate these conclusions.Research limitations/implicationsThis work includes only 2D development of the approach, while any realistic modeling for patient‐specific cases would be inherently 3D. The extension to 3D, as well as studies to improve the technique by decreasing the sensitivity to measurement noise and to incorporate non‐invasive measurement positioning, are under way.Practical implicationsAs medical imaging continuously improves, such techniques may prove useful in patient diagonosis, as heat generation can be correlated to the presence of tumors, infections, or other conditions.Originality/valueThis paper describes a new application of meshless methods. Such methods are becoming attractive due to their decreased pre‐processing requirements, especially for problems involving complex geometries (such as patient specific tissues), as well as optimization problems, where geometries may be constantly changing.

PurposeThis paper sets out to implement bounded high‐order (HO) resolution schemes in a hybrid finite volume/finite element method for the solution of the radiative transfer equation.Design/methodology/approachThe hybrid finite volume/finite element method had formerly been developed using the step scheme, which is only first‐order accurate, for the spatial discretization. Here, several bounded HO resolution schemes, namely the MINMOD, CLAM, MUSCL and SMART schemes, formulated using the normalized variable diagram, were implemented using the deferred correction procedure.FindingsThe results obtained reveal an interaction between spatial and angular discretization errors, and show that the HO resolution schemes yield improved accuracy over the step scheme if the angular discretization error is small.Research limitations/implicationsAlthough the HO resolution schemes reduce the spatial discretization error, they do not influence the angular discretization error. Therefore, the global error is only reduced if the angular discretization error is also small.Practical implicationsThe use of HO resolution schemes is only effective if the angular refinement yields low‐angular discretization errors. Moreover, spatial and angular refinement should be carried out simultaneously.Originality/valueThe paper extends a methodology formerly developed in computational fluid dynamics, and aimed at the improvement of the solution accuracy, to the hybrid finite volume/finite element method for the solution of the radiative transfer equation.

PurposeThe aim of this paper is to determine a parallel computational methodology for simultaneously predicting the macro/micro scale phenomena occurring during solidification processing with external electromagnetic stirring.Design/methodology/approachMacro and micro phenomena occurring in an electromagnetically‐stirred solidifying melt are simulated using a numerical model that integrates the finite element methodology for transport phenomena and the Monte‐Carlo cellular‐automata method for microstructure formation. Parallel algorithm is introduced to enhance the computational efficiency.FindingsComputed results show that parallel algorithm can be effective in enhancing the computational efficiency of a combined macro/micro model if it is applied appropriately. Also, electromagnetically induced stirring can have a strong effect on the nucleation and grain growth and hence the final solidification microstructure.Originality/valueThis paper fulfils a need for developing an efficient numerical methodology to simulate complex electromagnetically‐assisted transport phenomena and microstructure formation during solidification processing systems.

PurposeThis paper seeks to develop an adaptive finite volume algorithm, and to present an extensive numerical analysis of it.Design/methodology/approachThe effectiveness of the developed algorithm is demonstrated through practical and computationally challenging problems. The algorithm is tested for a wide range of singularities.FindingsThe convergence of the presented algorithm is independent of the regularity of the problems. It is shown that the our algorithm produces more accurate and well conditioned matrix systems.Research limitations/implicationsThough the presented algorithm works for extreme singularities on rectangular meshes, it may not be as efficient if the underlying meshes are distorted, and it may not converge. Further research is under way for including the multi‐point approximation technique into the algorithm.Practical implicationsAlmost all reservoir simulators use the two‐point method, and this algorithm is based on this method. The algorithm can be easily incorporated into the reservoir simulators. The results show that such an implementation will greatly improve the computational efficiency of the simulators. The work is useful for computational scientists, and especially for the researchers in oil industries. The paper reports the numerical work with practical applications.Originality/valueThe paper develops an adaptive finite volume algorithm. It is shown that adaptive meshes represent the underlying problem more accurately, and matrix systems associated with adaptive meshes are easier to solve compared with matrix systems associated with uniform meshes.

PurposeThe purpose of this paper is to investigate the performance of a specific class of high‐resolution central schemes in conjunction with the black oil models for hydrocarbon reservoir simulation.Design/methodology/approachA generalized black oil model is adopted, in which the solubility of gas in both oil and water and evaporation of oil are considered, leading to a system of equations prone to degeneracy. A computer code is generated and three test cases are solved to evaluate the performance of various schemes in terms of accuracy and discontinuity handling.FindingsIt is shown that, although some of the central schemes are highly sensitive to the choice of Courant‐Friedrich‐Levy (CFL) number and produce overly diffusive results, a certain type of this class is insensitive to the CFL number and can conveniently handle degenerate equations appearing in the reservoir simulation. The obtained results are compared with those available in the literature, showing merits of this class of schemes in complex reservoir simulation models.Research limitations/implicationsThis paper gives the one‐dimensional implementation of the above‐mentioned schemes. Extension to higher dimensional black oil model is currently under development by the authors.Practical implicationsThe specific class of high‐resolution central schemes investigated here presents the same level of accuracy as more complicated numerical methods, yet keeping it much more simple, by avoiding Riemann solvers.Originality/valueThe high‐resolution central scheme used in this work has been newly developed and applied to simple scalar hyperbolic equations. It has been adopted for the black oil for the first time.

PurposeThis paper seeks to introduce an optimization method for maximizing the effectiveness of the micro compact heat exchanger (MHE) under various geometrical parameters.Design/methodology/approachOptimization is realized using the neuro‐genetic methodology which combines the application of artificial neural network (ANN) together with genetic algorithms (GA). The analyses are divided into two main sections; the first being the modeling and prediction using finite element method, the second being the neuro‐genetic optimization. Initial results obtained from the finite element modeling are utilized for training in ANN. Subsequently, optimization is done using GA, once a well trained ANN is achieved.FindingsANN accurately predicts the effectiveness of the MHE and compares well with those obtained from the finite element simulation. Optimization shows a significant improvement in the maximum effectiveness of the MHE achievable for the given range of input parameters. Additionally, computational effort has been minimized and simulation time has been drastically reduced.Research limitations/implicationsThis analysis is valid for constant fluid properties and for steady‐state conditions. Additionally, optimization is limited to the range of the trained input parameters.Practical implicationsThis paper is very useful for practical design of various types of heat exchangers.Originality/valueThis paper will be useful for the design of the MHE where its performance can be analyzed for a given range of geometries with minimal effort. This methodology will also be applicable for other types of heat exchangers.