Three-dimensional Problems Research Articles

Improved Smoothed Particle Hydrodynamics for the Three-Dimensional Solid Impact Problems

Total Lagrangian-Smoothed Particle Hydrodynamics (TL-SPH) method is an important method to improve the tensile instability of traditional SPH, but it still faces some defects, such as insufficient accuracy at boundary region, difficulty in obtaining damage and applying interface conditions. Therefore, the high-order TL-SPH is applied for the elastic–plastic solid. A damage model is proposed by adaptively transforming the type of kernel function. Furthermore, a Riemann contact algorithm is introduced for different solids. The advantages of the methods in improving tensile instability and simulating damage are verified by the cases of the plates and the notched beams with an impact load.

A boundary element analysis of quasi-potential inviscid incompressible flow in multiply connected airfoil wing

AbstractThe quasi-potential flow in a doubly linked airfoil wing with a sharp trailing edge can be solved using a high-order boundary-element method (BEM). A bicubic Hermite spline interpolation is used to express the unknown, with node derivatives given by finite difference method. The revolutionary traits apply to bleeding-edge settings; specifically, challenges relating to the three-dimensional problem of multiply connected domains are investigated. Numerical validation examples were utilized to assess the implemented BEM approach.

Multiphysics performance of surrogate models on skewed pole surface-mounted permanent magnet motors

Purpose Evaluating the multiphysics performance of an electric motor can be a computationally intensive process, especially where several complex subsystems of the motor are coupled together. For example, evaluating acoustic noise requires the coupling of the electromagnetic, structural and acoustic models of the electric motor. Where skewed poles are considered in the design, the problem becomes a purely three-dimensional (3D) multiphysics problem, which could increase the computational burden astronomically. This study, therefore, aims to introduce surrogate models in the design process to reduce the computational cost associated with solving such 3D-coupled multiphysics problems. Design/methodology/approach The procedure involves using the finite element (FE) method to generate a database of several skewed rotor pole surface-mounted permanent magnet synchronous motors and their corresponding electromagnetic, structural and acoustic performances. Then, a surrogate model is fitted to the data to generate mapping functions that could be used in place of the time-consuming FE simulations. Findings It was established that the surrogate models showed promising results in predicting the multiphysics performance of skewed pole surface-mounted permanent magnet motors. As such, such models could be used to handle the skewing aspects, which has always been a major design challenge due to the scarcity of simulation tools with stepwise skewing capability. Originality/value The main contribution involves the use of surrogate models to replace FE simulations during the design cycle of skewed pole surface-mounted permanent magnet motors without compromising the integrity of the electromagnetic, structural, and acoustic results of the motor.

VSHPIC: A Particle-In-Cell algorithm based on vector spherical harmonics expansion

Abstract The Particle-in-Cell (PIC) simulation has been a widely used method for studying plasma physics. However, fully three-dimensional PIC simulations always require huge computational resources. For problems with near azimuthal symmetry, recent work has shown that expanding all the quantities defined on the grid in azimuthal harmonics and truncating the expansion can improve the code efficiency. In this paper, we describe a novel parallel algorithm for efficiently simulating three-dimensional near-spherical symmetry problems. Our approach expands all physical quantities in the θ and φ directions in spherical coordinates using vector spherical harmonics. The code is capable of simulating three-dimensional asymmetric scenarios by accurately tracking the evolution of distinct individual modes while preserving the charge conservation law. The fundamental dispersion relation of EM waves in the plasma has been obtained using VSHPIC simulation results. The code also shows a well strong scalability up to more than 1000 cores.

A three-dimensional phase-field model for studying the orientation-dependent interface evolution in stress-induced martensitic phase transformation

In this study, we address the challenging task of predicting the motion of interfaces in stress-driven martensitic phase transformations within shape memory alloys. The novelty of our approach lies in the introduction of a monolithically solved thermodynamic-based phase-field method, specifically tailored for large strain conditions at the nanoscale for three-dimensional problems. To achieve this, we have developed a custom finite element software seamlessly integrated into the FEniCS open-source framework, providing a sophisticated and efficient tool for the examination of nanostructure evolution. Our investigation delves into five distinct and complex scenarios under diverse loading conditions for two– and three-dimensional problems, each presenting unique challenges: (i) a straightforward uniaxial tension scenario featuring a square domain with a pre-existing martensitic nucleus; (ii) the evolution of interfaces in a square sample incorporating a circular central nanovoid under biaxial tensile stress; (iii) the analysis of a rectangular beam subjected to horizontal and vertical compressive loads; (iv) the assessment of an initially voided rectangular beam experiencing mixed loading conditions; and (v) the three-dimensional simulations of cubic-to-tetragonal phase transformation using various orientation of the habit plane. In contrast to prior studies, our analysis not only explores the standard factors of habit plane reorientation and strain transformation values but also delves into the impact of nanovoids on austenite–martensite interface evolution.

Space–time computations of exactly time-periodic flows past hydrofoils

The computation of periodic flows is typically conducted over multiple periods. First, a number of periods is used to develop periodic characteristics, and afterwards statistics are collected from averages over multiple periods. As a consequence, it is uncertain whether the numerical results are exactly time-periodic, and additionally, the time domain might be needlessly long. In this article, we circumvent these concerns by using a time-periodic function space. Consequently, the boundary conditions and solutions are exactly periodic. We employ the isogeometric analysis framework to achieve higher-order smoothness in both space and time. The discretization is performed using residual-based variational multiscale modelling and weak boundary conditions are adopted to enhance the accuracy near the moving boundaries of the computational domain. We enforce the time-periodic boundary condition within the isogeometric discretization spaces, which converts the two-dimensional time-dependent problem into a three-dimensional boundary value problem. Furthermore, we determine the boundary velocities of moving hydrofoils directly from the computational mesh and use a conservation methodology for force extraction. Application of the computational setup to heaving and pitching hydrofoils displays very accurate and exactly periodic results for lift and drag.

A generalization of B-differentiable equations method for elastoplastic contact problems with dual mortar discretization

In this study, a novel method based on the B-differentiable equations (BDEs) is proposed to solve the three-dimensional (3D) static and dynamic elastoplastic contact problems with friction. The contact conditions are formulated as the BDEs system, which can precisely satisfy Coulomb’s law of friction. Based on the mortar segment-to-segment contact model, the contact constraints are enforced by dual Lagrange multipliers for nonconforming meshes of the contact boundary. The non-smooth and nonlinear system equations composed of equilibrium and contact equations are simultaneously solved by the B-differentiable damping Newton method (BDNM), which has global convergence property. In each iteration, firstly, some contact displacements and contact forces, called substituted contact variables, are expressed by others according to the different contact states and contact conditions. The equilibrium equation is then solved, where the stiffness matrix is updated by substituting the contact variables. At last, the substituted contact variables will be restored. In the proposed method, the contact flexibility matrix is no longer needed, which saves the time for calculating the contact flexibility matrix due to changes in the stiffness matrix resulting from the update of stresses in the iteration step. By comparing with the results obtained by finite element commercial software ANSYS, the accuracy of the proposed algorithm is verified, and the property of convergence and efficiency are demonstrated for the elastoplastic contact problems by several numerical examples.

Asymptotic analysis of particle cluster formation in the presence of anchoring sites

The aggregation or clustering of proteins and other macromolecules plays an important role in the formation of large-scale molecular assemblies within cell membranes. Examples of such assemblies include lipid rafts, and postsynaptic domains (PSDs) at excitatory and inhibitory synapses in neurons. PSDs are rich in scaffolding proteins that can transiently trap transmembrane neurotransmitter receptors, thus localizing them at specific spatial positions. Hence, PSDs play a key role in determining the strength of synaptic connections and their regulation during learning and memory. Recently, a two-dimensional (2D) diffusion-mediated aggregation model of PSD formation has been developed in which the spatial locations of the clusters are determined by a set of fixed anchoring sites. The system is kept out of equilibrium by the recycling of particles between the cell membrane and interior. This results in a stationary distribution consisting of multiple clusters, whose average size can be determined using an effective mean-field description of the particle concentration around each anchored cluster. In this paper, we derive corrections to the mean-field approximation by applying the theory of diffusion in singularly perturbed domains. The latter is a powerful analytical method for solving two-dimensional (2D) and three-dimensional (3D) diffusion problems in domains where small holes or perforations have been removed from the interior. Applications range from modeling intracellular diffusion, where interior holes could represent subcellular structures such as organelles or biological condensates, to tracking the spread of chemical pollutants or heat from localized sources. In this paper, we take the bounded domain to be the cell membrane and the holes to represent anchored clusters. The analysis proceeds by partitioning the membrane into a set of inner regions around each cluster, and an outer region where mean-field interactions occur. Asymptotically matching the inner and outer stationary solutions generates an asymptotic expansion of the particle concentration, which includes higher-order corrections to mean-field theory that depend on the positions of the clusters and the boundary of the domain. Motivated by a recent study of light-activated protein oligomerization in cells, we also develop the analogous theory for cluster formation in a three-dimensional (3D) domain. The details of the asymptotic analysis differ from the 2D case due to the contrasting singularity structure of 2D and 3D Green’s functions.Graphical abstract2D model of diffusion-based protein cluster formation in the presence of anchoring cites and particle recycling. a A set of N anchoring sites at positions xj\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{x}}_j$$\\end{document}, j=1,…,N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$j=1,\\ldots ,N$$\\end{document}, in a bounded domain Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document}. b Diffusing particles accumulate at the anchoring sites resulting in the formation of particle aggregates or clusters Uj\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {U}}}_j$$\\end{document}. c The clusters are dynamically maintained by a combination of lateral diffusion outside the clusters and particle recycling

Analytical determination of a quasi-stationary electromagnetic field created by magnetic moments and eddy currents in conducting half-space

Aim. Study of the distribution of a three-dimensional alternating quasi-stationary electromagnetic field at the surface of conducting half-space with strong skin-effect, the source of which is an arbitrarily oriented magnetic moment. Methodology. The expressions for non-uniform electromagnetic field with strong skin effect are used for the analysis, which is based on the found exact analytical solution of the general three-dimensional problem and the use of expansion into asymptotic series with respect to a small parameter that is proportional to the ratio of the field penetration depth to the distance between the sources of the external field and the surface of body. Specific expressions at the surface are completely determined by the known field of external sources. In this work, the external magnetic moment field is used. Results. For strong skin effect, expressions for the electric and magnetic field strength are obtained separately for the components of the magnetic moment oriented perpendicularly and parallel to the flat surface between the dielectric and conducting areas. The features of the electromagnetic field distribution are analyzed depending on the value of introduced small parameter. The results are presented for the module and phase shift of the field strength with respect to the phase of the external field source. Originality. The expressions found for the electromagnetic field appear to be more general than the use of closed contours with alternating current, since they extend types of external field sources and allow the use of the superposition method instead of integration over the entire contour. Practical value. The found specific analytical expressions of the electromagnetic field at the surface for the external field of magnetic moments significantly simplify the solution of the problems, since they do not require additional solution of the field equations. References 20, figures 8.

Boussinesq-type equations of hydroelastic waves in shallow water

Accurate computation of hydroelastic waves in shallow water is critical because many hydroelastic wave applications are nearshores, such as sea-ice and floating infrastructures. In this paper, Boussinesq assumptions for shallow water are employed to derive nonlinear Boussinesq-type equations of hydroelastic waves, in which non-uniform distribution of structural stiffness and varying water depth are considered rigorously. Application of Boussinesq assumptions enables complicated three-dimensional problems to be reduced and formulated on the two-dimensional horizontal plane, therefore the proposed Boussinesq-type models are straightforward and versatile for a wide range of hydroelastic wave applications. Two configurations, a floating plate and a submerged plate, are studied. The first-order linear governing equations are solved analytically with periodic conditions assuming constant depth and uniform stiffness, and the linear dispersion relations are subsequently derived for both configurations. For flexural-gravity waves of a floating plate, unique behaviours of flexural-gravity waves different from shallow-water waves are discussed, and a generalized solitary wave solution is investigated. A nonlinear numerical solver is developed, and nonlinear flexural-gravity waves are found to have smaller wavelength and celerity than their linear counterparts. For hydroelastic waves of a submerged plate, dual-mode analytical solutions are discovered for the first time. Numerical computation has demonstrated that a plate with decreasing submerged depth is able to transfer wave energy from the deeper water to the surface layer.

An Algorithm for Solving Three-dimensional Assignment Problem

This article presents a algorithm for solving Three-dimensional assignment problem. Firstly, decompose the three-dimensional cubic matrix corresponding to the three-dimensional assignment problem into multiple two-dimensional planar matrices, and obtain that the assignment problems corresponding to these two-dimensional planar matrices have the same feasible solution as the original three-dimensional assignment problem. Then, the leading principal submatrix algorithm is used to solve each two-dimensional assignment problem. The characteristic of the leading principal submatrix algorithm is that each operation only needs to consider the local (leading principal submatrix) of the assignment matrix of the two-dimensional assignment matrix, without considering the entire assignment matrix. Starting from the first-order leading principal submatrix of the assignment matrix, Through the same solution transformation, the row minimum element of the leading principal submatrix of each order of the assignment matrix is found step by step, and the optimal solution of the two-dimensional assignment problems are obtained. Finally, by comparing the optimal solutions of these two-dimensional assignment problems, the optimal solutions of the original three-dimensional assignment problems are obtained. This algorithm can find the optimal solution for three-dimensional assignment problems in a patterned manner, facilitating computer programming and handling assignment problems with a large number of people and tasks.

Accelerated construction of projection-based reduced-order models via incremental approaches

We present an accelerated greedy strategy for training of projection-based reduced-order models for parametric steady and unsteady partial differential equations. Our approach exploits hierarchical approximate proper orthogonal decomposition to speed up the construction of the empirical test space for least-square Petrov–Galerkin formulations, a progressive construction of the empirical quadrature rule based on a warm start of the non-negative least-square algorithm, and a two-fidelity sampling strategy to reduce the number of expensive greedy iterations. We illustrate the performance of our method for two test cases: a two-dimensional compressible inviscid flow past a LS89 blade at moderate Mach number, and a three-dimensional nonlinear mechanics problem to predict the long-time structural response of the standard section of a nuclear containment building under external loading.

Cross-sectional analysis of initially curved composite beams with rectangular and circular sections using an asymptotic meshless method

An analytical meshless cross-sectional dimensional reduction method is developed to perform structural analysis of initially curved composite beams. By introducing the Pascal polynomials in the Cartesian coordinates system for beams with rectangular cross-sections and in the polar coordinates system for the beams with circular cross-sections to the warping field, a three-dimensional beam problem can be split into a two-dimensional cross-sectional analysis and a one-dimensional along the reference line of the beam. The obtained cross-sectional stiffness matrix based on higher-order of the strain energy can be incorporated into a one-dimensional finite element solution for initially curved composite beams, for the purpose of strain and stress analysis. Finally, the three-dimensional strain through the thickness of the beam can be achieved by the recovery analysis. The present method takes advantage of Pascal polynomials so that the solution procedure will be more simplified and computationally more efficient compared to three-dimensional finite element (3D FE) method. The proposed method for different isotropic and anisotropic beams is examined and compared with the literature, 3D FE, and Variational Asymptotic Beam Sectional (VABS) package which is a finite element based cross-sectional analysis tool for composite beams.

SPATIAL THINKING ABILITY IN ELEMENTARY SCHOOL

Spatial thinking ability is the ability to mentally manipulate and rotate an object in solving two-dimensional and third-dimensional geometric problems as well as the ability to imagine the shape of an object from a different perspective in solving three-dimensional geometric problems. This spatial ability is very necessary for students because good spatial abilities will also support good understanding of mathematics. Apart from that, students also need spatial abilities to solve everyday problems, such as looking at a map, explaining their position and so on. This spatial ability consists of spatial visualization, spatial rotation and spatial orientation. To improve spatial abilities, teachers must be creative in designing learning by using various media or teaching aids that are appropriate to the material being studied.Keywords: Spatial thinking skills, Geometry, Elementary education

Research on Genetic Algorithm Optimization with Fusion Tabu Search Strategy and Its Application in Solving Three-Dimensional Packing Problems

Symmetry is an important principle and characteristic that is prevalent in nature and artificial environments. In the three-dimensional packing problem, leveraging the inherent symmetry of goods and the symmetry of the packing space can enhance packing efficiency and utilization.The three-dimensional packing problem is an NP-hard combinatorial optimization problem in the field of modern logistics, with high computational complexity. This paper proposes an improved genetic algorithm by incorporating a fusion tabu search strategy to address this problem. The algorithm employs a three-dimensional loading mathematical model and utilizes a wall-building method under residual space constraints for stacking goods. Furthermore, adaptation of fitness variation strategy, chromosome adjustment, and tabu search algorithm are introduced to balance the algorithm’s global and local search capabilities, as well as to enhance population diversity and convergence speed. Through testing on benchmark cases such as Bischoff and Ratcliff, the improved algorithm demonstrates an average increase of over 3% in packing space utilization compared to traditional genetic algorithms and other heuristic algorithms, validating its feasibility and effectiveness. The proposed improved genetic algorithm provides new insights for solving three-dimensional packing problems and optimizing logistics loading schedules, offering promising prospects for various applications.

A fast Bayesian parallel solution framework for large-scale parameter estimation of 3D inverse heat transfer problems

The inverse heat transfer problems (IHTP) have a wide range of applications in the engineering field. Bayesian methods using Markov Chain Monte Carlo (MCMC) have long been considered as a robust and effective method for solving inverse problems. However, the discretization of the problem domain by the spatio-temporal Galerkin skill, i.e., the finite element interpolation also includes the time dimension, making the scale of the unknown parameters extremely difficult for Bayesian calculations. In this paper, a fast Bayesian parallel sampling (FBPS) framework is proposed for large-scale parameter estimation of benchmark three-dimensional inverse heat transfer problems (3D-IHTP). The FBPS we developed achieves a parameter computation scale of 105 magnitude within minutes, through dimensionality reduction of the space-time dependent problem domain. The Hamiltonian Monte Carlo (HMC) sampler, which is proven to be more efficient for high-dimensional parameter estimation, is employed. Through several simulation tests of IHTP, it was confirmed that the solving efficiency of FBPS surpasses that of the traditional Bayesian strategy significantly. Finally, FBPS is successfully developed to estimate the unknown heat flux on the chip heat sink and pack interface effectively, given some simulated high resolution measurement data. The reliability and efficiency show that FBPS has the potential to support efficient prediction techniques for a class of IHTPs in engineering applications.

Parallel Algorithms for Numerical Solution of Spatially Three-Dimensional Diffusion-Convection Equations in Coastal Systems Based on Splitting Schemes

Introduction. To prevent the occurrence and mitigate the consequences of hazardous and catastrophic phenomena associated with sediment transport in natural systems, it is necessary to develop operational and scientifically justified forecasts, identify critical states at which the emergence of emergency situations is possible. For these purposes, it is necessary to create an accurate and efficient toolkit, including algorithms for numerical solution of a model problem that takes into account the specifics of natural systems. In this work, parallel algorithms for numerical solution of a spatially three-dimensional diffusion-convection problem of sediment are presented, which allow a significant reduction in computation time (by more than 4 times) compared to calculations conducted using a sequential algorithm.Materials and Methods. For the parallel solution of the spatially three-dimensional diffusion-convection problem, an implicit splitting scheme is constructed, in which the original continuous problem is replaced by a chain of two-dimensional and one-dimensional problems. The splitting schemes proposed in the work are physically justified and take into account the specifics of coastal marine systems, for which the influence of micro-turbulent diffusion and advective transport of substances are comparable, and the Peclet number does not exceed unity when approximating real problems. For the parallel numerical implementation, a method of decomposing the grid domain into two families of vertical planes parallel to the coordinate planes Oxz and Oyz, combined with the Seidel method for solving two-dimensional grid problems in horizontal planes and the tridiagonal matrix algorithm when solving one-dimensional three-point problems in the vertical direction, is used. Within the framework of the parallel computing software implementation, a parallel algorithm is presented that implements the diffusion-convection problem on a computing system using MPI technology.Results. A comparative analysis of parallel and sequential algorithms is obtained using a model problem.Discussion and Conclusions. The developed software allows its practical use for solving specific hydrophysical problems, including as part of a software complex.

Solving the three-dimensional open-dimension rectangular packing problem: A constraint programming model

This paper addresses the three-dimensional open-dimension rectangular packing problem (3D-ODRPP). This problem addresses a set of rectangular boxes of given dimensions and a rectangular container of open dimensions. The objective is to pack all boxes orthogonally into the container while minimizing the container volume. Real-world applications of the 3D-ODRPP arise in production systems with operations of shipping or moving. The literature has presented mainly mixed-integer programming (MIP) formulations and their linearization techniques for the problem allied with general-purpose optimization solvers. To model and solve the 3D-ODRPP, we propose a constraint programming model based on a position-free modeling approach with logic operators. We ran computational experiments to assess the performance of the proposed model compared to the benchmark MIP models from instances of the literature. The results show our approach is competitive in different sets of problem instances in terms of reaching optimality as well as providing satisfactory feasible solutions quickly.

Three-dimensional frictional contact within the framework of couple stress elasticity

This work investigates a three-dimensional size-dependent frictional contact problem of a rigid spherical punch sliding on an elastic half-space. The size-dependent material behaviors are governed by the couple stress theory, and the corresponding frequency response functions, which include two additional length-related elastic parameters, are obtained. Based on the conjugate gradient method and the discrete convolution-fast Fourier transform algorithm, a semi-analytical model is established to predict the frictional size-dependent contact behaviors and then the detailed parameter analysis is performed. The transverse effects of the couple stress, which are induced due to the considerations of the two additional elastic parameters in couple stress theory, are investigated in the contact problems for the first time. And the dependences of the size-dependent contact behavior on the coupling effects of the friction and the two additional elastic parameters are presented.

Employing an operator form of the Rodrigues formula to calculate wavefunctions without differential equations

The factorization method of Schrödinger shows us how to determine the energy eigenstates without needing to determine the wavefunctions in position or momentum space. A strategy to convert the energy eigenstates to wavefunctions is well known for the one-dimensional simple harmonic oscillator by employing the Rodrigues formula for the Hermite polynomials in position or momentum space. In this work, we illustrate how to generalize this approach in a representation-independent fashion to find the wavefunctions of other problems in quantum mechanics that can be solved by the factorization method. We examine three problems in detail: (i) the one-dimensional simple harmonic oscillator; (ii) the three-dimensional isotropic harmonic oscillator; and (iii) the three-dimensional Coulomb problem. This approach can be used in either undergraduate or graduate classes in quantum mechanics.