3D Problems Research Articles

3D structure–2D plate–1D rod interaction problem

In this paper, we consider the equilibrium problem of interaction of three elastic bodies of different elastic properties. The main body is the unit cube. On top of it, there is a thin layer/cuboid of thickness of material whose stiffness is of order that in the middle contains another cuboid which is of width and thickness that is made of material with elasticity coefficients of order for . We show that the family of solutions of linearized elasticity problems, when tends to zero, converges to a solution of a problem that is posed only on the unit cube with possibly additional elastic terms on the boundary related to the plate/rod energy of the thin elastic parts. It turns out that there are five different regimes related to different values of with different limit problems. We further formulate a model posed on the unit cube that has the same asymptotics when tends to zero as the full 3D problem posed on the union of the unit cube and thin cuboids. This model then can be used as the approximating model in all regimes.

Synthetic Data Generation for Automatic Segmentation of X-ray Computed Tomography Reconstructions of Complex Microstructures.

The greatest challenge when using deep convolutional neural networks (DCNNs) for automatic segmentation of microstructural X-ray computed tomography (XCT) data is the acquisition of sufficient and relevant data to train the working network. Traditionally, these have been attained by manually annotating a few slices for 2D DCNNs. However, complex multiphase microstructures would presumably be better segmented with 3D networks. However, manual segmentation labeling for 3D problems is prohibitive. In this work, we introduce a method for generating synthetic XCT data for a challenging six-phase Al-Si alloy composite reinforced with ceramic fibers and particles. Moreover, we propose certain data augmentations (brightness, contrast, noise, and blur), a special in-house designed deep convolutional neural network (Triple UNet), and a multi-view forwarding strategy to promote generalized learning from synthetic data and therefore achieve successful segmentations. We obtain an overall Dice score of 0.77. Lastly, we prove the detrimental effects of artifacts in the XCT data on achieving accurate segmentations when synthetic data are employed for training the DCNNs. The methods presented in this work are applicable to other materials and imaging techniques as well. Successful segmentation coupled with neural networks trained with synthetic data will accelerate scientific output.

3D joist perception, detection, and climbing for hexapod robot

Avoiding obstacles is challenging for autonomous robotic systems. In this work, we examine obstacle avoidance for legged hexapods, as it relates to climbing over randomly placed wooden joists. We formulate the task as a 3D joist detection problem and propose a detect-plan-act pipeline using a SLAM algorithm to generate a pointcloud and a grid map to expose high obstacles such as joists. A line detector is applied on the grid map to extract parametric information of the joist, such as height, width, orientation, and distance; based on this information the hexapod plans a sequence of leg movements to either climb over the joist or move sideways. We show that our perception and path planning module work well on the real-world joists with different heights and orientations.

Implementation of Flux Limiters in Simulation of External Aerodynamic Problems on Unstructured Meshes

The study is dedicated to the peculiarities of implementing the flux limiter of the flow quantity gradient when solving 3D aerodynamic problems using the system of Navier–Stokes equations on unstructured meshes. The paper describes discretisation of the system of Navier–Stokes equations on a finite-volume method and a mathematical model including Spalart–Allmaras turbulence model and the Advection Upstream Splitting Method (AUSM+) computational scheme for convective fluxes that use a second-order approximation scheme for reconstruction of the solution on a facet. A solution of problems with shock wave structures is considered, where, to prevent oscillations at discontinuous solutions, the order of accuracy is reduced due to the implementation of the limiter function of the gradient. In particular, the Venkatakrishnan limiter was chosen. The study analyses this limiter as it impacts the accuracy of the results and monotonicity of the solution. It is shown that, when the limiter is used in a classical formulation, when the operation threshold is based on the characteristic size of the cell of the mesh, it facilitates suppression of non-physical oscillations in the solution and the upgrade of its monotonicity. However, when computing on unstructured meshes, the Venkatakrishnan limiter in this setup can result in the occurrence of the areas of its accidental activation, and that influences the accuracy of the produced result. The Venkatakrishnan limiter is proposed for unstructured meshes, where the formulation of the operation threshold is proposed based on the gas dynamics parameters of the flow. The proposed option of the function is characterized by the absence of parasite regions of accidental activation and ensures its operation only in the region of high gradients. Monotonicity properties, as compared to the classical formulation, are preserved. Constants of operation thresholds are compared for both options using the example of numerical solution of the problem with shock wave processes on different meshes. Recommendations regarding optimum values of these quantities are provided. Problems with a supersonic flow in a channel with a wedge and transonic flow over NACA0012 airfoil were selected for the examination of the limiter functions applicability. The computation was carried out using unstructured meshes consisting of tetrahedrons, truncated hexahedrons, and polyhedrons. The region of accidental activation of the Venkatakrishnan limiter in a classical formulation, and the absence of such regions in case a modified option of the limiter function, is implemented. The analysis of the flow field around a NACA0012 indicates that the proposed improved implementation of the Venkatakrishnan limiter enables an increase in the accuracy of the solution.

A double-phase field method for mixed mode crack modelling in 3D elasto-plastic solids with crack-direction-based strain energy decomposition

Crack-direction-based decomposition of elastic strain energy could effectively control the propagation of tensile and shear cracks in a phase field modelling context. The objective of the proposed double-phase field model is to extend the crack-direction-based decomposition strategy from 2D brittle fracture to complex mixed-mode crack modelling in a 3D setting, with plastic deformation incorporated. Both effective (undamaged) stress and plastic strain are split in the crack-orientation-based coordinate system. The decomposed tensile/shear stress contribute to tensile/shear damage evolution, respectively. The plastic contribution is coupled by relating the decomposed tensile/shear plastic strain to the corresponding tensile/shear crack energy release rates. Crack surface normal direction, represented by two orientation variables in 3D spatial domains, is determined by the F-criterion. The proposed model is implemented via ABAQUS subroutines with a staggered scheme for two phase field variables and crack direction. The simulation of a single-edge notch specimen under shear loading demonstrates that the ratio between shear and tensile crack energy release rates plays a significant role in the crack mode and mechanical response. Numerical results of a group of uniaxial tension, simple shear and tension shear specimens show good agreement with the experimental data in terms of the force–displacement curve and crack path, exhibiting the validity of the proposed model for capturing different crack modes. This model has also been proven effective for complex 3D problems via the third Sandia Challenge example.

A wide-band based weighted-sum-of-gray-gases model for participating media: Application to [formula omitted]-[formula omitted] mixtures with or without soot

The presence of participating gases and soot in combustion processes often makes the thermal radiation the main mechanism of heat transfer due to the elevated temperatures involved. The accurate solution of the radiative transfer equation (RTE) through the line-by-line (LBL) integration in engineering applications is in general still impracticable due to the high computational effort required to account for the millions of spectral lines that characterize the absorption coefficient of a gas. Global spectral models, such as the weighted-sum-of-gray-gases (WSGG) model, are attempts to replace the LBL integration with low computational cost and satisfactory accuracy. In the present study, an improvement of the WSGG model through a partition of the spectrum is proposed. The spectrum is divided into a set of ranges in which the standard methodology of the WSGG model is applied to determine the gray gases coefficients in these bands. This is the central aspect of what is named here a wide-band based WSGG (WBW) model, which consists firstly of determining the emittance of each band from the HITEMP 2010 database, then obtaining the pressure-absorption coefficients and the temperature-dependent coefficients. The total radiative heat flux and radiative heat source are obtained by the summation of the contribution of each band individually. For soot, a gray gas approach is considered for each band via Planck-mean absorption coefficients. A set of 1D problems consisting of two infinite parallel black walls, filled with a non-isothermal and non-homogeneous gaseous mixture of water vapor, carbon dioxide and soot, is solved. The results show that, especially in cases where the thermal radiation of soot is dominant, the maximum deviations obtained with the proposed method in relation to the LBL solution reduce by almost three times compared to other global WSGG models in the literature, in addition to decreasing by 40% the number of equations that need to be solved.

Построение сферы по мнимым точкам

Euclidean spaces of various dimensions do not contain imaginary images and objects by definition, but are inextricably linked with them through special cases, and this leads to the need to expand the field in geometry into the region of imaginary values [1, 19, 26]. Such an extension, i.e. adding to the field of real coordinates spaces of different dimensions, the field of imaginary coordinates leads to different variants of spaces of different dimensions, depending on the chosen axiomatics. Earlier in a number of articles, examples of solving some actual problems of geometry using imaginary geometric images and objects were shown [4, 5, 6, 13, 21, 22, 29]. 
 The article provides constructions for constructing a sphere from four predetermined points, of which one pair or both pairs of points can be imaginary complex conjugate. The construction is carried out on combined diagrams by the methods of descriptive geometry by analogy with the well-known problem of constructing a sphere from four real points. The construction of a sphere is based on seven auxiliary constructions for constructing a circle from points that can be imaginary conjugates. Both 3D problems of constructing spheres for given points and methods of 2D construction problems for determining the required imaginary points are considered. A method for calculating the parameters of the obtained sphere is described. The application of the method to other problems of descriptive geometry, for example, to the problems of finding geometric places of points, is considered. equidistant from two given surfaces. Recently, this issue has been intensively studied, for example, in the works [5, 6].

Discontinuous Galerkin spectral element method for shock capturing with summation by parts properties

This paper presents a computational methodology developed for a high-order approximation of compressible fluid dynamics equations with discontinuities. The methodology is based on a discontinuous Galerkin spectral-element method (DGSEM) built upon a split discretization framework with summation-by-parts (SBP) property, which mimics the integration-by-parts operation in a discrete sense. To extend the split DGSEM framework to discontinuous cases, we implement a shock capturing method based on the entropy viscosity formulation. The developed high-order split-form DGSEM with shock-capturing methodology is subject to a series of evaluation on both one-dimensional and two-dimensional, continuous and discontinuous cases. Convergence of the method is demonstrated both for smooth and shocked cases that have analytical solutions. The 2D Riemann problem tests illustrate an accurate representation of all the relevant flow phenomena, such as shocks, contact discontinuities, and rarefaction waves. All test cases are able to run with a polynomial order of 7 or higher. The values of the tunable parameters related to the entropy viscosity are robust for both 1D and 2D test problems. We also show that higher-order approximations yield smaller errors than lower-order approximations, for the same number of total degrees of freedom.

A New Numerical Implementation for Solar Coronal Modeling by an HLL Generalized Riemann Problem Solver

In this paper, we employ a Harten–Lax–van Leer (HLL) generalized Riemann problem (HLL-GRP) solver within the framework of a finite volume method to model 3D solar coronal structures for the first time. Based on the rotational invariance of magnetohydrodynamics (MHD) equations, the HLL-GRP solver is successfully implemented into 3D MHD simulations. To constrain the divergence of the magnetic field, the locally divergence-free weighted-least-squares-based essentially nonoscillatory reconstruction and the properly discretized Godunov–Powell source term are applied. To keep density and pressure positive, a positivity-preserving limiter is added to the reconstructed polynomials of density and pressure. We first test a 3D blast wave problem to preliminarily validate the effectiveness of the proposed scheme on Cartesian structured grid. Then, we further run our code on a six-component grid to numerically study the steady-state coronal structures of Carrington rotation 2218 during the solar minimum phase. A comparison with the two-stage Runge–Kutta scheme is performed for both the 3D blast wave problem and solar coronal problem. Numerical results of large-scale solar coronal structures are basically consistent with the observational characteristics, indicating the robustness of the proposed model.

Neural Networks with Local Converging Inputs (NNLCI) for Solving Conservation Laws, Part I: 1D Problems

Neural Networks with Local Converging Inputs (NNLCI) for Solving Conservation Laws, Part I: 1D Problems

A COMPARATIVE HEAT TRANSFER STUDY OF WATER AND LIQUID GALLIUM IN A SQUARE ENCLOSURE UNDER NATURAL CONVECTION

A comprehensive comparison of fluid flow and heat transfer characteristics inside a square enclosure with water or liquid gallium as fluids under natural convection has been studied numerically in the laminar flow regime. The left and right walls of the square cavity are treated as hot and cold isothermal walls, respectively, while the other walls are kept adiabatic. A 2D problem is considered, and the buoyancy term in the momentum equation is represented by the Boussinesq approximation. The governing equations are solved using finite difference methods. The convective terms are evaluated by using sixth-order compact schemes. Results are shown in the form of isotherms, streamlines, temperature profiles, average Nusselt number, average heat transfer coefficient values, and average skin friction values for various Grashof numbers (Gr) ranging from 10<sup>3</sup> to 10<sup>6</sup>. The present solver is validated against numerical and experimental work published in the literature. From the study, it is observed that the flow and heat transfer characteristics of conventional fluids like water differ significantly from low Prandtl number fluids like liquid gallium. Results show that the diffusion mechanism is dominant compared to convection in liquid gallium as a fluid for Gr ≤ 10<sup>4</sup> but the convection phenomenon is dominant in water as a fluid for all Grashof numbers. The comparison shows that at a Grashof number of 10<sup>3</sup> liquid gallium has 25.61 times greater heat transfer than water. Finally, a correlation for the average Nusselt number is developed with liquid gallium as fluid for a Grashof number ranging from 10<sup>3</sup> to 10<sup>6</sup>.

Rapid adsorption enthalpy surface sampling (RAESS) to characterize nanoporous materials.

Molecular adsorption in nanoporous materials has many large-scale industrial applications ranging from separation to storage. To design the best materials, computational simulations are key to guiding the experimentation and engineering processes. Because nanoporous materials exist in a plethora of forms, we need to speed up the existing simulation tools to be able to screen databases of hundreds of thousands of structures. Here, we describe a new algorithm that quickly calculates adsorption enthalpies by sampling the surface of the material instead of the whole porous space. This surface sampling has been tested on the CoRE MOF 2019 database and has been proven to be more than 2 orders of magnitude faster than the gold standard method (Widom insertion), with an acceptable level of error on an enthalpy value of 0.34 kJ mol-1, and is therefore proposed as a valuable addition to the high-throughput screening toolbox.

A Dimension-Splitting Variational Multiscale Element-Free Galerkin Method for Three-Dimensional Singularly Perturbed Convection-Diffusion Problems

By introducing the dimensional splitting (DS) method into the multiscale interpolating element-free Galerkin (VMIEFG) method, a dimension-splitting multiscale interpolating element-free Galerkin (DS-VMIEFG) method is proposed for three-dimensional (3D) singular perturbed convection-diffusion (SPCD) problems. In the DS-VMIEFG method, the 3D problem is decomposed into a series of 2D problems by the DS method, and the discrete equations on the 2D splitting surface are obtained by the VMIEFG method. The improved interpolation-type moving least squares (IIMLS) method is used to construct shape functions in the weak form and to combine 2D discrete equations into a global system of discrete equations for the three-dimensional SPCD problems. The solved numerical example verifies the effectiveness of the method in this paper for the 3D SPCD problems. The numerical solution will gradually converge to the analytical solution with the increase in the number of nodes. For extremely small singular diffusion coefficients, the numerical solution will avoid numerical oscillation and has high computational stability.

Explicit Isogeometric Topology Optimization Method with Suitably Graded Truncated Hierarchical B-Spline

This work puts forward an explicit isogeometric topology optimization (ITO) method using moving morphable components (MMC), which takes the suitably graded truncated hierarchical B-Spline based isogeometric analysis as the solver of physical unknown (SGTHB-ITO-MMC). By applying properly basis graded constraints to the hierarchical mesh of truncated hierarchical B-splines (THB), the convergence and robustness of the SGTHB-ITOMMC are simultaneously improved and the tiny holes occurred in optimized structure are eliminated, due to the improved accuracy around the explicit structural boundaries. Moreover, an efficient computational method is developed for the topological description functions (TDF) of MMC under the admissible hierarchical mesh, which consists of reducing the dimensionality strategy for design space and the locally computing strategy for hierarchical mesh. We apply the above SGTHB-ITO-MMC with improved efficiency to a series of 2D and 3D compliance design problems. The numerical results show that the proposed SGTHB-ITO-MMC method outperforms the traditional THB-ITO-MMC method in terms of convergence rate and efficiency. Therefore, the proposed SGTHB-ITO-MMC is an effective way of solving topology optimization (TO) problems.

Global regularity to the 3D Cauchy problem of inhomogeneous magnetic Bénard equations with vacuum

<abstract><p>This paper deals with the Cauchy problem of 3D inhomogeneous incompressible magnetic Bénard equations. Through some time-weighted a <italic>priori</italic> estimates, we prove the global existence of strong solution provided that the upper boundedness of initial density and initial magnetic field satisfy some smallness condition. Furthermore, we also obtain large time decay rates of the solution.</p></abstract>

A Methodology for the Analysis of Tunnel Intersections Using Two-dimensional Numerical Modeling

Intersections of tunnels are an integral component of most underground applications. The design and construction of tunnel intersections can be the most technically challenging element of a tunnel project. Yet, there are relatively few publications and methods regarding tunnel intersection design. Tunnel intersections are essentially a three-dimensional problem. However, 3D models require significantly longer time for solving and interpreting, greater computer resources, and higher user expertise. Hence, there is a great incentive to develop simpler and quicker analysis methods for tunnel intersections. In this paper, a methodology is developed so that a 3D tunnel intersection problem can be modelled using an equivalent 2D numerical model. For this purpose, it is proposed that the main tunnel cross section is modelled with an additional horizontal expansion component that represents the cross tunnel. Multiple 3D and 2D models were computed for the purpose of finding the expansion distances in the 2D models that yield similar deformations to the 3D models. Based on the results, regression analysis was employed to derive a formula relating the ratio of equivalent diameters of the main to cross tunnel to the proper expansion distance. This formula provides a useful and practical tool for preliminary tunnel intersection design. The current work relies on a number of assumptions, mainly that the rock mass is elastic. More work can be carried out to extend and modify the proposed method for more complex conditions.

MOMENT FINITE ELEMENT FOR SOLVING 3D ELASTICITY PROBLEMS

The implementation of a new 8-node finite element for solving 3D elasticity static problems is considered. The idea of this finite element is based on the projection of a rare mesh FEM scheme into a space of lower dimension. The resulting scheme of the 8-node hexahedral finite element has a number of features compared to the scheme of the classical 8-node polyline element. This is one integration point in the element compared to eight in the classical element, as well as the presence of four parameters that allow you to adjust the convergence of the computational process. In a given finite element, the stresses as well as their moments (three bending and one torsional) are assumed to be constant within the element. Also, the continuity of displacement fields is preserved only in the nodes of finite elements. This circumstance is not a disadvantage and is characteristic of many new numerical FEM schemes (suffice it to mention the recently actively developed direction of the discontinuous Galerkin method). The issues of using this finite element in solving the well-known problem of increased shear stiffness (shear locking) of a number of well-known FEM schemes are discussed. The problem of hourglass instability, which is typical for FEM schemes with reduced integration and a number of other numerical schemes, in particular, the Wilkins scheme, which is popular in solving dynamic elasticity and plasticity problems, is also considered. The implementation of a technique for the numerical solution of three-dimensional static problems of the theory of elasticity on the basis of a given finite element within the framework of the traditional FEM technique using a vector-matrix notation is described. The results of solving test static problems of elasticity theory are presented.

3-D Seismic Fault Detection Using Recurrent Convolutional Neural Networks With Compound Loss

Fault detection is an essential component of seismic interpretation and plays a crucial role in industrial processes. However, it is also one of the main challenges, especially in delineating faults in 3D seismic data. Recently, the rapidly developing technology, deep learning, has proven to be a powerful tool for this task. A number of neural networks have been proposed for this purpose by regarding 3D fault detection as a semantic segmentation task. To further enhance the effectiveness of the deep learning methods, we propose a novel network architecture, named R2SE-Unet, to solve the 3D segmentation problem. In the neural network, we design a recurrent residual-SE convolution unit (RRCU-SE) that integrates the residual learning and Squeeze-Excitation module to store information in 3D seismic data. This component promotes the spread of 3D volumetric information and aids in learning spatial dependencies in 3D images. In addition, to reduce the impact of insufficient spatial resolution resulting from the base architecture of U-net, we add an attention unit between skip connection operations. These two new units enable our R2SE-Unet to exploit semantic information more accurately in the feature maps. After many experiments on region-based loss functions and distribution-based loss functions, we also propose a novel loss function, which takes the advantage of generalized dice (GDice) loss and balanced binary cross entropy (b-BCE) loss, named Gdice-bce, to effectively train R2SE-Unet. Although only synthetic seismic data samples are used to train the network parameters, our R2SE-Unet could produce more reliable fault feature maps on field seismic data than two other conventional fault detection neural networks. Thus, the proposed neural network is easy to train and reliably works for seismic fault interpretation on field seismic data.

A novel numerical method based on a high order polynomial approximation of the fourth order Steklov equation and its eigenvalue problems

<p style='text-indent:20px;'>Based on high order polynomial approximation and dimension reduction technique, we propose a novel numerical method for the fourth order Steklov problems in the circular domain. We first decompose the primal problem into a set of 1D problems via polar coordinate transformation and Fourier basis functions expansion. Then, by introducing a non-uniformly weighed Sobolev space, the variational form and corresponding discrete scheme are derived. Employing the Lax-Milgram lemma and approximation properties of the projection operators, we further prove existence and uniqueness of weak solutions and approximation solutions for each one-dimensional problems, and the error estimation between them, respectively. We also carry out ample numerical experiments which illustrate that the numerical algorithm is efficient and highly accurate.</p>

Towards a Unified Single Analysis Framework Embedded with Multiple Spatial and Time Discretized Methods for Linear Structural Dynamics

We propose a novel computational framework that is capable of employing different time integration algorithms and different space discretized methods such as the Finite Element Method, particle methods, and other spatial methods on a single body sub-divided into multiple subdomains. This is in conjunction with implementing the well known Generalized Single Step Single Solve (GS4) family of algorithms which encompass the entire scope of Linear Multistep algorithms that have been developed over the past 50 years or so and are second order accurate into the Differential Algebraic Equation framework. In the current state of technology, the coupling of altogether different time integration algorithms has been limited to the same family of algorithms such as the Newmark methods and the coupling of different algorithms usually has resulted in reduced accuracy in one or more variables including the Lagrange multiplier. However, the robustness and versatility of the GS4 with its ability to accurately account for the numerical shifts in various time schemes it encompasses, overcomes such barriers and allows a wide variety of arbitrary implicit-implicit, implicit-explicit, and explicit-explicit pairing of the various time schemes while maintaining the second order accuracy in time for not only all primary variables such as displacement, velocity and acceleration but also the Lagrange multipliers used for coupling the subdomains. By selecting an appropriate spatial method and time scheme on the area with localized phenomena contrary to utilizing a single process on the entire body, the proposed work has the potential to better capture the physics of a given simulation. The method is validated by solving 2D problems for the linear second order systems with various combination of spatial methods and time schemes with great flexibility. The accuracy and efficacy of the present work have not yet been seen in the current field, and it has shown significant promise in its capabilities and effectiveness for general linear dynamics through numerical examples.