Cuboid Elements Research Articles

Mathematical model of dry stack structural elements with geometric imperfections under a cyclic bending moment

Dry stack structural elements are characterized by nonlinear stiffness that arises from geometric imperfections of their components and the absence of any bonding between them. Moreover, such elements dissipate energy under cyclic loading because of their internal structure. The authors considered dry stack structural elements loaded with a bending moment to propose a relatively simple mathematical model of dry stacks composed of only three elements. The model consists of a linear spring, a nonlinear spring, and a spring with hysteresis in series. In this model, the first element describes the idealized properties of a dry stack element, while the second and third elements correspond to the influence of geometric imperfections and the behaviour of dry joints. Furthermore, the authors described a procedure for determining the parameters of the model based on test results. The proposed solution was verified via experimental studies of temporary support structures consisting of a stack of cuboid elements and a hydraulic jack typically used in the process of building rectification. This study showed that the proposed model adequately describes both the nonlinearity and the energy dissipation under a cyclic bending moment.

High‐gain wideband all‐dielectric digital metasurface reflectarray antenna for millimeter‐wave applications

AbstractThis paper presents an additive manufacturing technique to design and prototype a reflective metasurface antenna for Ka‐band applications. The dielectric cuboidal element with an air‐filled void is designed to act as a unit cell for designing a three‐dimensional (3D)‐printed phase‐graded metasurface reflector. The designed phase‐graded metasurface reflector is a 3‐bit quantized highly reflective surface, designed by rigorous unit cell optimizations. The quantized unit cells are distributed to provide a 45° phase gradient and wideband reflections operating from 24 to 38 GHz. The designed reflector is excited by a wideband 3D‐printed horn antenna placed at the optimized focal point. The simulated and measured results indicate the designed antenna achieves a maximum gain of 23.66 dBi with a gain enhancement of 60%. The designed antenna is found to have a 1 dB stable gain bandwidth of 4 GHz, from 28 to 32 GHz with a maximum gain of 23.66 dBi at 30 GHz.

Quantum annealing for microstructure equilibration with long-range elastic interactions

We demonstrate the use and benefits of quantum annealing approaches for the determination of equilibrated microstructures in shape memory alloys and other materials with long-range elastic interaction between coherent grains and their different martensite variants and phases. After a one dimensional illustration of the general approach, which requires to formulate the energy of the system in terms of an Ising Hamiltonian, we use distant dependent elastic interactions between grains to predict the variant selection for different transformation eigenstrains. The results and performance of the computations are compared to classical algorithms, demonstrating that the new approach can lead to a significant acceleration of the simulations. Beyond a discretization using simple cuboidal elements, also a direct representation of arbitrary microstructures is possible, allowing fast simulations with currently up to several thousand grains.

A Novel Rigid Finite Element Formulation for Dynamic Analysis of Flexible Plates

This paper introduces a new rigid finite element method (RFEM) formulation for dynamic analysis of plates. In RFEM, a flexible body is divided into several rigid elements which are interconnected by spring and damping elements. Mainly, RFEM has been used to model systems with beam-like slender components, such as ropes and cables, and few RFEM formulations were developed to model plates and shells. In this study however, by means of the Timoshenko beam theory and cuboid rigid elements, a novel RFEM formulation is developed to model flexible flat plates with generic geometries of the surface, considering large deformation. For this purpose, an RFEM formulation is presented for straight beams with rectangular cross-section by means of the Timoshenko beam theory. Next, this formulation is expanded to model flexible rectangular plates with uniform thickness and then, it is elaborated on how to model non-rectangular plates with uniform thickness using only cuboid elements and consequently, the development of the proposed RFEM formulation is completed. After investigating various case studies, the competence of the proposed formulation is evaluated thoroughly. The results of RFEM came on a proper agreement with the FEM results in a way that the maximum difference between these results in dynamic analysis was about 3%.

Effect of Multi-Variant Thermal Treatment on Microstructure Evolution and Mechanical Properties of AlSi10Mg Processed by Direct Metal Laser Sintering and Casting.

This article presents a study on the influence of temperature and time of multi-variant heat treatment on the structure and properties of materials produced in direct metal laser sintering (DMLS) and casting technology. The materials were manufactured in the form of cuboidal elements with a cross-section of 1.5 mm × 15 mm and a length of 60 mm. The samples prepared in this way had a similar volume, but due to the production technology the metal crystallization took place at different rates and directions. In the cast, the direction of heat transfer was toward the mold, and the DMLS was directed locally layer by layer. The small thickness of the cast material allowed reaching conditions similar to the DMLS cooling process. Both DMLS and cast samples show similar mechanical properties (hardness) achieved after long ageing time, i.e., 16 h at 170 °C. The maximum hardness was observed for 8 h. In the DMLS samples, in contrast to cast samples, no lamellar precipitates of silicon were observed, which indicates their better resistance to cracking

Parametric Study on Strength Characteristics of Two-Dimensional Ice Beam Using Discrete Element Method

Strength characteristics of a two-dimensional ice beam were studied using a discrete element method (DEM). The DEM solver was implemented by the open-source discrete element method libraries. Three-point bending and uniaxial compressive tests of the ice beam were simulated. The ice beam consisted of an assembly of disk-shaped particles with a particular thickness. The connection of the ice particles was modelled using a cuboid element, which represents a bond. If the stress acting on the bond exceeded the bond strength criterion, the bond started to break, explaining the cracking of the ice beam. To find out the effect of the local parameters of the contact and bond models on the ice fracture, we performed numerical simulations for various bond Young‘s modulus of the particles, the bond strength, and the relative particle size ratio.

Manufacturing Elements with Small Cross-Sections of 17-4 PH Steel (1.4542) with the Application of the DMLS Additive Manufacturing Method.

The application of direct metal laser sintering renders it possible to manufacture models with complex geometries. However, there are certain limits to the application of this method connected with manufacturing thin-walled cuboidal elements, as well as cylinders and holes with small diameters. The principal objective of the research was to determine the accuracy of manufacturing geometries with small cross-sections and the possibility of application in heat exchangers that are radiators with radially arranged ribs. To that end, four specimens were designed and manufactured; their geometries of representations assumed for the purpose of research (analysis) changed dimensions within the following scope: 10–0.1 mm. The specimens to be applied in the research were manufactured with 17-4 PH stainless steel (1.4542) with the application of 3D-DMLS printing and an EOS M270 printer. The measurement of accuracy was performed with the application of an optical stereomicroscope (KERN OZL-466). In addition to that, research into the chemical composition of the material, as well as the size of spherical agglomerates, was conducted with the application of a scanning electron microscope. The analysis of the chemical composition was conducted as well (after the sintering process). The analysis of the results based on the values received by means of measurements of the manufactured geometries was divided into three parts. Based on this, it is possible to conclude that the representation of models manufactured with the application of DMLS was comparable with the assumptions, and that the deviations between a nominal dimension and that received in the course of the research were within the following scope: 0–0.1 mm. At the final stage of research and based on the received results, two heat exchangers were manufactured.

Effect of Eccentricity of Applied Force and Geometrical Imperfections on Stiffness of Stack of Cuboidal Steel Elements.

Experimental tests were performed on a stack of cuboidal steel elements and its mathematical model was developed. Each cuboidal element was made of rolled profiles. Such stacks are the part of temporary supports that can be used to eliminate deflections of buildings. Stacks were loaded eccentrically through the inaccurate position of a jack. Moreover, two types of geometrical imperfections could be noticed. They included inaccurate contact between the stack elements and initial relative displacements of profiles that formed cuboidal elements. A mathematical model was developed to describe deformations of the stack and its parameters were determined by analysing test results. The eccentricity of the applied force had a slight impact on the stack stiffness, which was considerably reduced by geometrical imperfections. The imperfection covering initial relative displacements of rolled profiles inside cuboidal elements had the greatest impact on the stiffness. It could cause even a 10-fold drop in the stack stiffness when compared with the theoretical stiffness resulting from the stiffness of the stack section, and the stiffness dropped by ca. 3.5 times when the imperfection included the inaccurate contact between the cuboidal elements. Finally, the occurrence of both types of geometrical imperfections generated the real stiffness more than ten times lower than the theoretical stiffness that did not take into account imperfections.

Effects of the roughness height in turbulent boundary layers over rod- and cuboid-roughened walls

Direct numerical simulations (DNSs) of spatially developing turbulent boundary layers (TBLs) over two-dimensional (2D) rod-roughened walls and three-dimensional (3D) cuboid-roughened walls are conducted to investigate the effects of the roughness height on the flow characteristics in the outer layer. The rod elements are periodically aligned along the downstream direction with a pitch of px/θin = 12, and the cuboid elements are periodically staggered with a pitch of px/θin = 12 and pz/θin = 3, where px and pz are correspondingly the streamwise and spanwise pitches of the roughness and θin is the momentum thickness at the inlet. The first surface roughness is placed 80θin downstream from the inlet, leading to a step change from a smooth to rough surface. The rod and cuboid roughness height (k) is varied in the range of 0.1 ≤ k/θin ≤ 1.8 (13 ≤ δ/k ≤ 285), respectively (δ is the boundary layer thickness), and the Reynolds number based on the momentum thickness (θ) is varied in the range of Reθ = 300 ~ 1400. For each case, the self-preservation form of the velocity-defect and the turbulent Reynolds stresses is achieved along the downstream direction. As the roughness height increases, the roughness function (ΔU+) extracted from the mean velocity profiles increases, although the velocity-defect profiles for the rough-wall cases show good agreement with the profile from the smooth-wall case. The magnitude of the Reynolds stresses in the outer layer increases with an increase of k/δ. The outer layer similarity between the flows over the rough and smooth-walls is found when δ/k ≥ 250 and 100 for the 2D rod and 3D cuboid, respectively. The continuous increase of the Reynolds stresses in the outer layer with an increase of k/δ is explained by a large population of very long structures over the rough-wall flows. Because the characteristic width of the structures increases continuously with an increase of k/δ for the rod and cuboid roughness, a wide width of the structures leads to frequent spanwise merging between adjacent structures. The active spanwise merging events with an increase of k/δ increase the streamwise coherence of the structures with the appearance of significant meandering.

A structured method to generate conformal FE mesh for realistic textile composite micro-geometry

A procedure is developed to generate a conformal finite element mesh of a textile composite unit cell with a complex micro-geometry with the aim of improving the accuracy of micro-mechanics analysis. A realistic micro-geometry of a textile composite unit cell is initially generated by using a fiber level dynamic relaxation approach. Yarn surfaces are then discretized into triangle elements. An algorithm is established to remove inter-yarn interferences and gaps induced by numerical errors. The unit cell is first divided into a uniform cuboid element mesh, which is later modified and converted to a conformal finite element (FE) mesh through of a process of node shifting and element splitting. In the conformal mesh, the element boundary perfectly matches the yarn-to-yarn interface. Compatibility between elements is ensured. The quality of each element is examined. The mesh can be input to commercial FEM softwares for composite stress analysis.

FEM With Curved Hexahedron Element and Application on Tunnel Integrated Grounding System in High-Speed Railway

With finite element method (FEM), cuboid elements (or tetrahedron elements) are often used in the traditional grounding analysis. In the irregular boundary, smaller and denser elements are usually applied to approximate the real boundary. However, this increases the number of elements in the boundary, leading to an increase in computation time. In order to introduce FEM into tunnel integrated grounding system (IGS) in high-speed railway better, a curved hexahedron element is proposed for tunnel structure. In this paper, the stiffness matrix of curved hexahedral element in space is first derived based on coordinate transformation. Two different stiffness matrices are compared in the condition that the structure of curved hexahedron element is highly similar to that of cuboid element. Later, the number of required elements according to the proposed method is compared with that of the traditional one on the basis of meeting same accuracy requirement. Finally, the results of the proposed method are compared with those of current distribution, electromagnetic fields, grounding and soil structure analysis (CDEGS). Through the above-mentioned three comparison schemes, the curved hexahedron element model is verified by its precision and applicability. From this, the distribution of the electromagnetic field inside the tunnel IGS is obtained.

A fast forward algorithm for three-dimensional magnetic anomaly on undulating terrain

This paper presents a fast and high-precision three-dimensional magnetic anomaly forward algorithm on undulating terrain. Firslty, The forward algorithm for three-dimensional magnetic anomalies on a horizontal observation surface is developed with the Block-Toeplitz-Toeplitz-Block (BTTB) matrix. In the proposed algorithm, the cuboid elements are used to divide the underground space. And the position of the observation point is set as the projection of the central point of the cuboid element onto the observation surface. With the symmetry of the BTTB matrix, the coefficient matrix is compressed and stored to reduces the memory requirements during the computation process. In order to deal with the coefficient matrix and parameter vector computations, the matrix cross-multiplication computation is transformed into a point-multiplication computation with the fast Fourier transform (FFT) method. Then, to solve the problem of undulating terrain, a hierarchical interpolation method is proposed in forward algorithm, in which the undulating terrain is divided into several horizontal observation surfaces. Subsequently, the magnetic field on the undulating terrain is interpolated. Some 3D models with horizontal observation surface and undulating terrain are tested to verify the proposed algorithm. The results show that the forward algorithm has the advantages of fast computation speed, high precision, and low memory requirement. It is suitable for large-scale (ten millions of nodes) three-dimensional magnetic forward modeling.

Thermal Modeling of Hollow Conductors for Direct Cooling of Electrical Machines

A direct cooling design using hollow conductors with the coolant flowing inside can significantly improve the heat dissipation in an electrical machine. To predict the thermal performances of an electrical machine with such cooling configuration, this paper proposes a computationally efficient thermal model of hollow conductors with direct cooling features. The hollow conductor is modeled using four equivalent solid cuboidal elements with a three-dimensional thermal network and internal heat generation. The heat transfer coefficient between the coolant and conductors is determined by an empirical model considering fluid dynamics behaviors. Axial discretization is performed to take into account the nonuniform temperature distribution along the axial direction. Experimental validation is performed with a U-shaped hollow conductor test rig. Compared to computational fluid dynamics analysis, the proposed thermal model is much more computationally efficient, and thus can be incorporated into design optimization process and electrothermal simulations of the electrical machine over a driving cycle.

A moving discontinuous Galerkin finite element method for flows with interfaces

SummaryA moving discontinuous Galerkin finite element method with interface condition enforcement is formulated for flows with discontinuous interfaces. The underlying weak formulation enforces the interface condition separately from the conservation law, so that the residual only vanishes upon satisfaction of both. In this formulation, the discrete grid geometry is treated as a variable, so that, in contrast to the standard discontinuous Galerkin method, this method has both the means to detect interfaces, via interface condition enforcement, and to satisfy, via grid movement, the conservation law and its associated interface condition. The method therefore directly fits interfaces, including shocks, preserving a high‐order representation up to the interface without requiring shock capturing or an upwind numerical flux to achieve stability. It can be generalized to flows with a priori unknown interfaces with nontrivial topology and curved interface geometry as well as to an arbitrary number of spatial dimensions. Unsteady flows are represented in a manner similar to steady flows using a space‐time formulation. In addition to computing flows with interfaces, the method can represent point singularities in a flow field by degenerating cuboid elements. In general, the method works in conjunction with standard local grid operations, including edge collapse, to ensure that degenerate cells are removed. Test cases are presented for up to three‐dimensional flows that provide an initial assessment of the stability and accuracy of the method.

A modified, hybrid nodal-integral/finite-element method for 3D convection-diffusion problems in arbitrary geometries

A modified, hybrid nodal-integral/finite-element method (NI-FEM) is developed to solve the three-dimensional (3D), steady-state, convection-diffusion problems in arbitrary geometries. The hybrid NI-FEM takes advantage of the high efficiency of the conventional nodal-integral method (NIM) and the flexible mesh generation of the finite element method (FEM), which is applicable to arbitrary geometries. In this method, the computational domain is discretized into, for 3D problems, four-node tetrahedral elements and eight-node cuboid elements. The cuboid elements are used to discretize the interior region and regions adjacent to boundaries that are parallel to planes formed by two of the axes, while the tetrahedral elements are used to discretize the remaining irregular regions. The conventional NIM is used to develop difference equations for the transverse-averaged variables on the interface between two adjacent cuboid elements. The FEM is applied to develop algebraic equations for the node temperatures of tetrahedral elements. On the interface between two different kinds of elements, the transverse-averaged variable for the cuboid element is obtained by averaging the node values of its adjacent tetrahedral elements, while the heat flux for the tetrahedral element is calculated using the corresponding transverse-averaged variables of its adjacent cuboid element. The hybrid NI-FEM is developed to be solved using a matrix formulation for the entire domain rather than an iterative procedure, detailed derivations of which for the 3D case are presented in this paper. Using the NI-FEM developed here, 2D and 3D convection-diffusion test problems are solved, and the numerical results are compared to the exact (manufactured) solutions to determine the order, accuracy and efficiency of the method. Numerical scheme is found to be of, as expected, second order.

Combined Hybrid Finite Element Method Applied in Elastic Thermal Stress Problem

In view of combinative stability, combinative variational principle based on domain decomposition for elastic thermal stress problem is constructed with the merits of avoiding Lax–Babuska–Brezzi (LBB) conditions. Compared with the principle of elasticity problem, new load items from thermal are involved. In addition, combined hybrid finite element is proposed to discretize the new principle and to formulate element stiffness matrix. Energy compatibility is introduced not only to simplify the variational principle and the corresponding element stiffness matrix but also to reduce the error of finite element solutions. On cuboid element, the energy compatible stress mode is given explicitly. The numerical results indicate that combined hybrid element with eight nodes can give almost the same computing accuracy of displacement and better computing accuracy of stress compared with cuboid element with 20 nodes, is not sensitive to mesh distortion and can circumvent Poisson-locking phenomenon.

Modelling turbulent boundary layer flow over fractal-like multiscale terrain using large-eddy simulations and analytical tools

In recent years, there has been growing interest in large-eddy simulation (LES) modelling of atmospheric boundary layers interacting with arrays of wind turbines on complex terrain. However, such terrain typically contains geometric features and roughness elements reaching down to small scales that typically cannot be resolved numerically. Thus subgrid-scale models for the unresolved features of the bottom roughness are needed for LES. Such knowledge is also required to model the effects of the ground surface 'underneath' a wind farm. Here we adapt a dynamic approach to determine subgrid-scale roughness parametrizations and apply it for the case of rough surfaces composed of cuboidal elements with broad size distributions, containing many scales. We first investigate the flow response to ground roughness of a few scales. LES with the dynamic roughness model which accounts for the drag of unresolved roughness is shown to provide resolution-independent results for the mean velocity distribution. Moreover, we develop an analytical roughness model that accounts for the sheltering effects of large-scale on small-scale roughness elements. Taking into account the shading effect, constraints from fundamental conservation laws, and assumptions of geometric self-similarity, the analytical roughness model is shown to provide analytical predictions that agree well with roughness parameters determined from LES.This article is part of the themed issue 'Wind energy in complex terrains'.

Thermomechanical States in Arc Weld Surfaced Steel Elements

AbstractThe paper presents a model of temperature, phase transformation and stresses fields in a steel element during single-pass Gas Metal Arc Weld (GMAW) surfacing.Kinetics of phase transformations during heating is limited by temperature values at the beginning and at the end of austenitic transformation, while the progress of phase transformations during cooling is determined on the basis of TTT-welding diagram and Johnson-Mehl-Avrami and Kolomogorov law for diffusive transformations and Koistinen-Marburger for martensitic transformation. Stress state of a bar subjected to thermo-mechanical loads is described assuming the plane cross section hypothesis and using integral equations of stress equilibrium of a bar as well as simple Hook’s law. Stresses in the elastic-plastic state are determined by iteration using solutions with a variable elastic modulus of elasticity, conditioned by tensile curves. Dependence of stresses on strains is assumed on the basis of tensile curves of particular structures, taking into account the influence of temperature. There were performed calculations of the temperature field, phase transformations, strains and stresses for GMAW surfacing of a cuboid element made of S235 steel. Authors’ programs, made in Borland Delphi, were used for calculations.

Structured Mesh Generation: Open-source automatic nonuniform mesh generation for FDTD simulation

The finite-difference time-domain (FDTD) method is a numerical technique that is widely used to solve Maxwell's differential equations in the time domain [1]. Both space and time are discretized. Space is discretized into rectangular-shaped elements in two-dimensional (2-D) or cuboid elements in three-dimensional (3-D). Cuboid elements, where the electric fields are located on the edges of the cuboid and the magnetic fields are normal to the faces, are called Yee cells and are the fundamental elements of most FDTD methods [2]. By filling up the problem space with these cells, we obtain a 3-D mesh, where neighboring cells share edges and faces. Each cell has three associated electric and magnetic field components, while the other field components belong to adjacent cells. The properties of each cell are adjusted to represent materials such as dielectrics or conductors by adjusting the constitutive parameters used in the field equations in the corresponding cells, hence forming the geometrical structure of the problem to be solved. For example, Figure 1 shows a sphere and a version of the sphere discretized using cubic elements.

Simulation of Plasto-Elastohydrodynamic Lubrication in a Rolling Contact

Surface plastic deformation due to contact (lubricated or dry) widely exists in many mechanical components, as subsurface stress caused by high-pressure concentrated in the contact zone often exceeds the material yielding limit, and the plastic strain accumulates when the load is increased and/or repeatedly applied to the surface in a rolling contact. However, previous plasto-elastohydrodynamic lubrication (PEHL) studies were mainly for the preliminary case of having a rigid ball (or roller) rotating on a stationary elastic–plastic flat with a fixed contact center, for which the numerical simulation is relatively simple. This paper presents an efficient method for simulating PEHL in a rolling contact. The von Mises yield criteria are used for determining the plastic zone, and the total computation domain is discretized into a number of cuboidal elements underneath the contacting surface, each one is considered as a cuboid with uniform plastic strain inside. The residual stress and surface plastic deformation resulted from the plastic strain can be solved as a half-space eigenstrain–eigenstress problem. A combination of three-dimensional (3D) and two-dimensional (2D) discrete convolution and fast Fourier transform (DC-FFT) techniques is used for accelerating the computation. It is observed that if a rigid ball rolls on an elastic–plastic surface, the characteristics of PEHL lubricant film thickness and pressure distribution are different from those of PEHL in the preliminary cases previously investigated. It is also found that with the increase of rolling cycles, the increment of plastic strain accumulation gradually approaches a stable value or drops down to zero, determined by the applied load and the material hardening properties, eventually causing a groove along the rolling direction. Simulation results for different material hardening properties are also compared to reveal the effect of body materials on the PEHL behaviors.