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This paper aims to propose a semi-analytical benchmarking framework for enthalpy-based methods used in problems involving phase change with latent heat. The benchmark is based on a class of semi-analytical solutions of spatially symmetric Stefan problems in an arbitrary spatial dimension. Via a public repository this study provides a finite element numerical code based on the FEniCS computational platform, which can be used to test and compare any method of choice with the (semi-)analytical solutions. As a particular demonstration, this paper uses the benchmark to test several standard temperature-based implementations of the enthalpy method and assesses their accuracy and stability with respect to the discretization parameters.

The class of spatially symmetric semi-analytical self-similar solutions to the Stefan problem is found for an arbitrary spatial dimension, connecting some of the known results in a unified manner, while providing the solutions’ existence and uniqueness. For two chosen standard semi-implicit temperature-based enthalpy methods, the numerical error assessment of the implementations is carried out in the finite element formulation of the problem. This paper compares the numerical approximations to the semi-analytical solutions and analyzes the influence of discretization parameters, as well as their interdependence. This study also compares accuracy of these methods with other traditional approach based on time-explicit treatment of the effective heat capacity with and without iterative correction.

This study shows that the quantitative comparison between the semi-analytical and numerical solutions of the symmetric Stefan problems can serve as a robust tool for identifying the optimal values of discretization parameters, both in terms of accuracy and stability. Moreover, this study concludes that, from the performance point of view, both of the semi-implicit implementations studied are equivalent, for optimal choice of discretization parameters, they outperform the effective heat capacity method with iterative correction in terms of accuracy, but, by contrast, they lose stability for subcritical thickness of the mushy region.

The proposed benchmark provides a versatile, accessible test bed for computational methods approximating multidimensional phase change problems. The supplemented numerical code can be directly used to test any method of choice against the semi-analytical solutions.

While the solutions of the symmetric Stefan problems for individual spatial dimensions can be found scattered across the literature, the unifying perspective on their derivation presented here has, to the best of the authors’ knowledge, been missing. The unified formulation in a general dimension can be used for the systematic construction of well-posed, reliable and genuinely multidimensional benchmark experiments.

When modeling heat propagation in biological bodies, a non-negligible relaxation time (typically between 15-30 s) is required for the thermal waves to accumulate and transfer, i.e. thermal waves propagate at a finite velocity. To accommodate for this feature that is characteristic to heat transfer in biological bodies, the classical Fourier's law has to be modified resulting in the thermal-wave model of bio-heat transfer. The purpose of the paper is to retrieve the space-dependent blood perfusion coefficient in such a thermal-wave model of bio-heat transfer from final time temperature measurements.

The non-linear and ill-posed blood perfusion coefficient identification problem is reformulated as a non-linear minimization problem of a Tikhonov regularization functional subject to lower and upper simple bounds on the unknown coefficient. For the numerical discretization, an unconditionally stable direct solver based on the Crank–Nicolson finite difference scheme is developed. The Tikhonov regularization functional is minimized iteratively by the built-in routine lsqnonlin from the MATLAB optimization toolbox. Both exact and numerically simulated noisy input data are inverted.

The reconstruction of the unknown blood perfusion coefficient for three benchmark numerical examples is illustrated and discussed to verify the proposed numerical procedure. Moreover, the proposed algorithm is tested on a physical example which consists of identifying the blood perfusion rate of a biological tissue subjected to an external source of laser irradiation. The numerical results demonstrate that accurate and stable solutions are obtained.

Although previous studies estimated the important thermo-physical blood perfusion coefficient, they neglected the wave-like nature of heat conduction present in biological tissues that are captured by the more accurate thermal-wave model of bio-heat transfer. The originalities of the present paper are to account for such a more accurate thermal-wave bio-heat model and to investigate the possibility of determining its space-dependent blood perfusion coefficient from temperature measurements at the final time.

Vorticity formulations for the incompressible Navier‐Stokes equations have certain advantages over primitive‐variable formulations including the fact that the number of equations to be solved is reduced through the elimination of the pressure variable, identical satisfaction of the incompressibility constraint and the continuity equation, and an implicitly higher‐order approximation of the velocity components. For the most part, vorticity methods have been used to solve exterior isothermal problems. In this research, a vorticity formulation is used to study the natural convection flows in differentially‐heated enclosures. The numerical algorithm is divided into three steps: two kinematic steps and one kinetic step. The kinematics are governed by the generalized Helmholtz decomposition (GHD) which is solved using a boundary element method (BEM) whereas the kinetics are governed by the vorticity equation which is solved using a finite element method (FEM). In the first kinematic step, vortex sheet strengths are determined from a novel Galerkin implementation of the GHD. These vortex sheet strengths are used to determine Neumann boundary conditions for the vorticity equation. (The thermal boundary conditions are already known.) In the second kinematic step, the interior velocity field is determined using the regular (non‐Galerkin) form of the GHD. This step, in a sense, linearizes the convective acceleration terms in both the vorticity and energy equations. In the third kinetic step, the coupled vorticity and energy equations are solved using a Galerkin FEM to determine the updated values of the vorticity and thermal fields. Two benchmark problems are considered to show the robustness and versatility of this formulation including natural convection in an 8×1 differentially‐heated enclosure at a near critical Rayleigh number.

This paper aims to develop a new 3-level implicit numerical method of order 2 in time and 4 in space based on half-step cubic polynomial approximations for the solution of 1D quasi-linear hyperbolic partial differential equations. The method is derived directly from the consistency condition of spline function which is fourth-order accurate. The method is directly applied to hyperbolic equations, irrespective of coordinate system, and fourth-order nonlinear hyperbolic equation, which is main advantage of the work.

In this method, three grid points for the unknown function w(x,t) and two half-step points for the known variable x in spatial direction are used. The methodology followed in this paper is construction of a cubic spline polynomial and using its continuity properties to obtain fourth-order consistency condition. The proposed method, when applied to a linear equation is shown to be unconditionally stable. The technique is extended to solve system of quasi-linear hyperbolic equations. To assess the validity and accuracy, the method is applied to solve several benchmark problems, and numerical results are provided to demonstrate the usefulness of the method.

The paper provides a fourth-order numerical scheme obtained directly from fourth-order consistency condition. In earlier methods, consistency conditions were only second-order accurate. This brings an edge over other past methods. In addition, the method is directly applicable to physical problems involving singular coefficients. Therefore, no modification in the method is required at singular points. This saves CPU time, as well as computational costs.

There are no limitations. Obtaining a fourth-order method directly from consistency condition is a new work. In addition, being an implicit method, this method is unconditionally stable for a linear test equation.

Physical problems with singular and nonsingular coefficients are directly solved by this method.

The paper develops a new fourth-order implicit method which is original and has substantial value because many benchmark problems of physical significance are solved in this method.

Presents recent advances obtained by the authors in the development of enhanced strain finite elements for finite deformation problems. Discusses two options, both involving simple modifications of the original enhancement strategy of the deformation gradient as proposed in previous works. The first new strategy is based on a full symmetrization of the original enhanced interpolation fields; the second involves only the transposed part of these fields. Both modifications lead to a significant improvement of the performance in problems involving high compressive stresses, showing in particular a mode‐free response, while maintaining a simple and efficient (strain driven) numerical implementation. Demonstrates these properties with a number of numerical benchmark simulations, including a complete modal analysis of the elements.

In this paper the effects of choosing dependent variables and cell face velocities on convergence of the SIMPLE algorithm are discussed. Using different velocity components as either dependent variables or cell‐face velocities, both convergent rate and calculation accuracy of the algorithm are compared and studied. A novel method, named “cross‐correction”, is developed to improve the convergence of the algorithm of using non‐orthogonal grids. Cases with benchmark and analytical solutions are used for numerical experiments and validation. The results show that, although different velocity components are employed as either dependent variables or cell face velocities, there is no obvious difference in both the convergent rates and numerical solutions. Moreover, the “cross‐correction” method is validated by computations with several first‐order and high‐order convection schemes; and the generality of convergence improvement achieved by the method is shown in the paper.

This paper aims to propose a new smoothed particle hydrodynamics (SPH)-finite element (FE) algorithm to study fluid–structure interaction (FSI) problems.

The fluid domain is discretized based on the theory of SPH), and solid part is solved through FE method, similar to other SPH-FE methods in the previous studies. Instead of master-slave technique, the interpolating (kernel) functions of immersed boundary method are implemented to couple fluid and solid domains. The procedure of modeling completely follows the classic IB framework where forces and velocities are transferred between interacting parts. Three benchmark FSI problems are simulated and the results are compared with those of similar numerical and experimental works.

The proposed SPH-FE algorithm with promising and acceptable results can be utilized as a reliable method to simulate FSI problems.

Contrary to most SPH-FE algorithms, the calculation of contact force is not required at interacting boundaries and no iterative process is proposed to calculate forces, velocities and positions at new time step.

Traditional adaptive analysis based on a coarse mesh, using finite element method (FEM) analysis, produces the original solution. Then post-processing the result and figuring out the regions should be refined and these regions refined once. Finally, this new mesh is used to get the solution of first refinement. After several iterations of above procedures, we can achieve the last result that is closer to the true solution, which takes time, making adaptive scheme inpractical to engineering application. The paper aims to discuss these issues.

This paper based on FEM proposes a multi-level refinement strategy with a refinement strategy and an indicator. The proposed indicator uses value of the maximum difference of strain energy density among the elements that associated with one node, and divides all nodes into several categories based on the value. A multi-level refinement strategy is proposed according to which category the node belongs to refine different elements to different times rather than whether refine or not.

Multi-level refinement strategy takes full use of the numerical calculation, resulting in the whole adaptive analysis that only need to iterate twice while other schemes must iterate more times. Using much less times of numerical calculation and approaches, more accurate solution, making adaptive analysis more practical to engineering.

Multi-level refinement strategy takes full use of the numerical calculation, resulting in the whole adaptive analysis only need iterate twice while other schemes must iterate more times. using much less times of numerical calculation and approaches more accurate solution, making adaptive analysis more practical to engineering.

This paper numerically investigates the effect of cavity radiation on the thermal response of hollow aluminium tubes and facade systems subjected to fire.

Finite element simulations were performed using ABAQUS 6.14. The accuracy of the numerical model was established through experimental and numerical results available in the literature. The proposed numerical model was utilised to study the effect of cavity radiation on the thermal response of aluminium hollow tubes and facade system. Different scenarios were considered to assess the applicability of the commonly used lumped capacitance heat transfer model.

The effects of cavity radiation were found to be significant for non-uniform fire exposure conditions. The maximum temperature of a hollow aluminium tube with 1-sided fire exposure was found to be 86% greater when cavity radiation was considered. Further, the time to attain critical temperature under non-uniform fire exposure, as calculated from the conventional lumped heat capacity heat transfer model, was non-conservative when compared to that predicted by the proposed simulation approach considering cavity radiation. A metal temperature of 550 °C was attained about 18 min earlier than what was calculated by the lumped heat capacitance model.

The present study will serve as a basis for the study of the effects of cavity radiation on the thermo-mechanical response of aluminium hollow tubes and facade systems. Such thermo-mechanical analyses will enable the study of the effects of cavity radiation on the failure mechanisms of facade systems.

Cavity radiation was found to significantly affect the thermal response of hollow aluminium tubes and façade systems. In design processes, it is essential to consider the potential consequences of non-uniform heating situations, as they can have a significant impact on the temperature of structures. It was also shown that the use of lumped heat capacity heat transfer model in cases of non-uniform fire exposure is unsuitable for the thermal analysis of such systems.

This is the first detailed investigation of the effects of cavity radiation on the thermal response of aluminium tubes and façade systems for different fire exposure conditions.

Boussinesq approximation is widely used in solving natural convection problems, but it has severe practical limitations. Using Boussinesq approximation, the temperature difference should be less than 28.6 K. The purpose of this study is to get rid of Boussinesq approximation and simulates the natural convection problems using an unsteady compressible Navier-Stokes solver. The gravity force is included in the source term. Three temperature differences are used namely 20 K, 700 K and 2000 K.

The calculations are carried out on the square and sinusoidal cavities. The results of low temperature difference have good agreement with the experimental and previous calculated data. It is found that, the high temperature difference has a significant effect on the density.

Due to mass conservation, the density variation affects the velocity distribution and its symmetry. On the other hand, the density variation has a negligible effect on the temperature distribution.

The present calculation method has no limitations but its convergence is slow. The current study can be used in fluid flow simulations for nuclear power applications in natural convection flows subjected to large temperature differences.