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The purpose of this paper is to provide a brief introduction of the finite difference based parabolic equation (PE) modeling to the advanced engineering students and academic researchers.

A three-dimensional parabolic equation (3DPE) model is developed from the ground up for modeling wave propagation in the tunnel via a rectangular waveguide structure. A discussion of vector wave equations from Maxwell’s equations followed by the paraxial approximations and finite difference implementation is presented for the beginners. The obtained simulation results are compared with the analytical solution.

It is shown that the alternating direction implicit finite difference method (FDM) is more efficient in terms of accuracy, computational time and memory than the explicit FDM. The reader interested in maximum details of individual contributions such as the latest achievements in PE modeling until 2021, basic PE derivation, PE formulation’s approximations, finite difference discretization and implementation of 3DPE, can learn from this paper.

For the purpose of this paper, a simple 3DPE formulation is presented. For simplicity, a rectangular waveguide structure is discretized with the finite difference approach as a design problem. Future work could use the PE based FDM to study the possibility of utilization of meteorological techniques, including the effects of backward traveling waves as well as making comparisons with the experimental data.

The proposed work is directly applicable to typical problems in the field of tunnel propagation modeling for both national commercial and military applications.

We describe a novel mathematical approach to deriving and solving covolume models of the incompressible 2‐D Navier‐Stokes flow equations. The approach integrates three technical components into a single modelling algorithm: 1. Automatic Grid Generation. An algorithm is described and used to automatically discretize the flow domain into a Delaunay triangulation and a dual Voronoi polygonal tessellation. 2. Covolume Finite Difference Equation Generation. Three covolume discretizations of the Navier‐Stokes equations are presented. The first scheme conserves mass over triangular control volumes, the second scheme over polygonal control volumes and the third scheme conserves mass over both. Simple consistent finite difference equations are derived in terms of the primitive variables of velocity and pressure. 3. Dual Variable Reduction. A network theoretic technique is used to transform each of the finite difference systems into equivalent systems which are considerably smaller than the original primitive finite difference system.

We present a new nonlinear axisymmetric finite element model for heat transfer and powder deposition in rotational molding. Arbitrary Lagrangian Eulerian techniques are employed to track the gradual growth of the plastic layer. Results using this approach compare well with earlier 1‐D models and with experimental data. Using the model to study the effects of locally enhanced heat transfer on part wall thickness, we find that controlling the relative magnitudes of radial and circumferential heat transfer is crucial in order to obtain desired wall thickness profiles.

The purpose of this paper is to suggest a new approach to the numerical simulation of shallow‐water flows both in plane domains and on the sphere.

The approach involves the technique of splitting of the model operator by geometric coordinates and by physical processes. Specially chosen temporal and spatial approximations result in one‐dimensional finite difference schemes that conserve the mass and the total energy. Therefore, the mass and the total energy of the whole two‐dimensional split scheme are kept constant too.

Explicit expressions for the schemes of arbitrary approximation orders in space are given. The schemes are shown to be mass‐ and energy‐conserving, and hence absolutely stable because the square root of the total energy is the norm of the solution. The schemes of the first four approximation orders are then tested by simulating nonlinear solitary waves generated by a model topography. In the analysis, the primary attention is given to the study of the time‐space structure of the numerical solutions.

The approach can be used for the numerical simulation of shallow‐water flows in domains of both Cartesian and spherical geometries, providing the solution adequate from the physical and mathematical standpoints in the sense of keeping its mass and total energy constant even when fully discrete shallow‐water models are applied.

In this paper, finite element techniques are used to model laser spot welding of both metallic and ceramic materials. The materials modelled were aluminum and alumina, a typical ceramic. In the formulation, the steady state mass, momentum, and energy conservation equations are considered with surface tension and buoyancy forces driving the fluid flow. The results indicate that fluid flow must be considered in order to accurately model molten pool characteristics for ceramics since the coupling of fluid flow to heat flow is more important for ceramics than for metals.

The purpose of this study is twofold: first, to derive a consistent model free-surface runup flow problems over deformable beds. The authors couple the nonlinear one-dimensional shallow water equations, including friction terms for the water free-surface and the two-dimensional second-order solid elastostatic equations for the bed deformation. Second, to develop a robust hybrid finite element/finite volume method for solving free-surface runup flow problems over deformable beds. The authors combine the finite volume for free-surface flows and the finite element method for bed elasticity.

The authors propose a new model for wave runup by static deformation on seabeds. The model consists of the depth-averaged shallow water system for the water free-surface coupled to the second-order elastostatic formulation for the bed deformation. At the interface between the water flow and the seabed, transfer conditions are implemented. Here, hydrostatic pressure and friction forces are considered for the elastostatic equations, whereas bathymetric forces are accounted for in the shallow water equations. As numerical solvers, the authors propose a well-balanced finite volume method for the flow system and a stabilized finite element method for elastostatics.

The developed coupled depth-averaged shallow water system and second-order solid elastostatic system is well suited for modeling wave runup by deformation on seabeds. The derived coupling conditions at the interface between the water flow and the bed topography resolve well the condition transfer between the two systems. The proposed hybrid finite volume element method is accurate and efficient for this class of models. The novel technique used for wet/dry treatment accurately captures the moving fronts in the computational domain without generating nonphysical oscillations. The presented numerical results demonstrate the high performance of the proposed methods.

Enhancing modeling and computations for wave runup problems is at an early stage in the literature, and it is a new and exciting area of research. To the best of our knowledge, solving wave runup problems by static deformation on seabeds using a hybrid finite volume element method is presented for the first time. The results of this research study, and the research methodologies, will have an important influence on a range of other scientists carrying out research in related fields.

Selective laser melting (SLM) is a major additive manufacturing (AM) process in which laser beams are used as the heat source to melt and deposit metals in a layerwise fashion to enable the construction of components of arbitrary complexity. The purpose of this paper is to develop a framework for accurate and fast prediction of the temperature distribution during the SLM process.

A fast computation tool is proposed for thermal analysis of the SLM process. It is based on the finite volume method (FVM) and the quiet element method to allow the development of customized functionalities at the source level. The results obtained from the proposed FVM approach are compared against those obtained from the finite element method (FEM) using a well-established commercial software, in terms of accuracy and efficiency.

The results show that for simulating the SLM deposition of a cubic block with 81,000, 189,000 and 297,000 cells, the computation takes about 767, 3,041 and 7,054 min, respectively, with the FEM approach; while 174, 679 and 1,630 min with the FVM code. This represents a speedup of around 4.4x. Meanwhile, the average temperature difference between the two is below 6%, indicating good agreement between them.

The thermal field for the multi-track and multi-layer SLM process is for the first time computed by the FVM approach. This pioneering work on comparing FVM and FEM for SLM applications implies that a fast and simple computing tool for thermal analysis of the SLM process is within the reach, and it delivers comparable accuracy with significantly higher computational efficiency. The research results lay the foundation for a potentially cost-effective tool for investigating the fundamental microstructure evolution, and also optimizing the process parameters in the SLM process.

Intumescent coatings are nowadays a dominant passive system used to protect structural materials in case of fire. Due to their reactive swelling behaviour, intumescent coatings are particularly complex materials to be modelled and predicted, which can be extremely useful especially for performance-based fire safety designs. In addition, many parameters influence their performance, and this challenges the definition and quantification of their material properties. Several approaches and models of various complexities are proposed in the literature, and they are reviewed and analysed in a critical literature review.

Analytical, finite-difference and finite-element methods for modelling intumescent coatings are compared, followed by the definition and quantification of the main physical, thermal, and optical properties of intumescent coatings: swelled thickness, thermal conductivity and resistance, density, specific heat capacity, and emissivity/absorptivity.

The study highlights the scarce consideration of key influencing factors on the material properties, and the tendency to simplify the problem into effective thermo-physical properties, such as effective thermal conductivity. As a conclusion, the literature review underlines the lack of homogenisation of modelling approaches and material properties, as well as the need for a universal modelling method that can generally simulate the performance of intumescent coatings, combine the large amount of published experimental data, and reliably produce fire-safe performance-based designs.

Due to their limited applicability, high complexity and little comparability, the presented literature review does not focus on analysing and comparing different multi-component models, constituted of many model-specific input parameters. On the contrary, the presented literature review compares various approaches, models and thermo-physical properties which primarily focusses on solving the heat transfer problem through swelling intumescent systems.

The presented literature review analyses and discusses the various modelling approaches to describe and predict the behaviour of swelling intumescent coatings as fire protection for structural materials. Due to the vast variety of available commercial products and potential testing conditions, these data are rarely compared and combined to achieve an overall understanding on the response of intumescent coatings as fire protection measure. The study highlights the lack of information and homogenisation of various modelling approaches, and it underlines the research needs about several aspects related to the intumescent coating behaviour modelling, also providing some useful suggestions for future studies.

The purpose of this paper is to analyze the bulk energy transport processes in the build chamber environment before and during laser sintering (LS) to provide a basis for effective and accurate thermal control for the LS process. This leads to improved mechanical properties and geometrical tolerances for LS products and may be applied to optimize operation cycle times for the LS process.

Computational models with two levels of complexity were built to explore the heat transfer mechanisms in the LS process. In a one-dimensional model (1D), the powder performed as a semi-infinite solid and heater flux to the powder surface was modeled with a heater control law. A two-dimensional (2D) fluid/solid finite element model of the build chamber and powder bins provided insight into the thermal processes in the build chamber.

Numerical 1D simulations were verified with measurements from sensors embedded in the build chamber powder bed. Using a 2D model, computed powder surface temperatures during the warm up and build phases were verified with an infrared camera. Convective currents in the build chamber and non-uniformities in the distribution of temperature over the radiant heater surface were found to be substantial contributors to non-uniformities in the powder bed surface temperature.

Limited heat sources were analyzed. No three-dimensional model was built. Assumptions to decrease the part bed temperature difference were not tested.

These simulation and experimental results may be used to enhance thermal control and operation efficiency during the LS process and to improve LS product mechanical properties.

The accuracy of finite element discretizations modelling one‐dimensional wave propagation problems is presented. The spurious reflections arising from finite/infinite element discretizations for unbounded domain problems are quantified. The error curves, numerically obtained, yield a criterion for rational mesh design. Formulae for minimum discretization ratios are given.