2D Examples Research Articles

Boundary Knots Method with ghost points for high-order Helmholtz-type PDEs in multiply connected domains

This paper proposes the Boundary Knot Method with ghost points (BKM-G), which enhances the performance of the BKM for solving 2D (3D) high-order Helmholtz-type partial differential equations in domains with multiple cavities. The BKM-G differs from the conventional BKM by relocating the source points from the boundary collocation nodes to a random region, such as a circle (sphere) encompassing the original domain in 2D (3D). Compared with classical BKM, this modification improves accuracy without sacrificing simplicity and efficiency. Moreover, this paper investigates and analyzes the effect of the ghost circle/sphere’s radius R in BKM-G for solving various high-order Helmholtz-type PDEs. Numerous 2D and 3D numerical examples illustrate that the BKM-G outperforms the BKM for a wide range of R. The effectiveness of the proposed effective condition number (ECN) approach in finding the optimal R has also been demonstrated. Furthermore, the economic ECN (EECN) is studied to significantly improve the efficiency of ECN.

Constraints, Conserved Charges and Extended BRST Algebra for a 3D Field-Theoretic Example for Hodge Theory

We perform the constraint analysis of a three (2 + 1)-dimensional (3D) field-theoretic example for Hodge theory (i) at the classical level within the ambit of Lagrangian formulation, and (ii) at the quantum level within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism. We derive the conserved charges corresponding to the six continuous symmetries of our present theory. These six continuous summery transformations are the nilpotent (anti-)BRST and (anti-)co-BRST symmetries, a unique bosonic symmetry and the ghost-scale symmetry. It turns out that the Noether conserved (anti-)BRST charges are found to be non-nilpotent even though they are derived from the off-shell nilpotent versions of the continuous and infinitesimal (anti-)BRST symmetry transformations. We obtain the nilpotent versions of the (anti-)BRST charges from the non-nilpotent Noether (anti-)BRST charges and discuss the physicality criteria w.r.t. the latter to demonstrate that the operator forms of the first-class constraints (of the classical gauge theory) annihilate the physical states at the quantum level. This observation is consistent with Dirac's quantization conditions for the systems that are endowed with the constraints. We lay emphasis on the existence of a single (anti-)BRST invariant Curci-Ferrari (CF) type restriction in our theory and derive it from various theoretical angles.

A new linearized ADI compact difference method on graded meshes for a nonlinear 2D and 3D PIDE with a WSK

In this work, a new linearized alternating direction implicit (ADI) compact difference method (CDM) is proposed for solving nonlinear two-dimensional (2D) and three-dimensional (3D) partial integrodifferential equation (PIDE) with a weakly singular kernel (WSK). The time derivative is treated by Crank-Nicolson (CN) method and the Riemann-Liouville (R-L) integral by product integration (PI) rule on graded meshes. The linear interpolation combining with Taylor formula is applied in time to deal with nonlinear term v∇v in interval (t0,t1), and linear interpolation concerning two previous time points is employed to deal with v∇v in intervals (tn−1,tn),n≥2. A linearized semi-discrete scheme is obtained, which can achieve second-order convergence in time. Then via introducing two kinds of compact difference operators to discretize the spatial derivatives. To improve the computing efficiency, we construct a ADI compact difference method. It is the first time that the ADI compact difference method is applied for the nonlinear 2D and 3D PIDE with a WSK. The advantage of our proposed scheme is that it not only has second-order accuracy in time and fourth-order accuracy in space, but also fast computational speed, just by solving the linear coupled equations for tridiagonal matrices. In addition, we prove the existence, uniqueness and convergence of 2D scheme. Four numerical examples in 2D and 3D are present to demonstrate our proposed method.

Nucleophilic Displacement Reactions of Silver-Based Metal-Organic Chalcogenolates.

We report nucleophilic displacement reactions that can increase the dimensionality or coordination number of silver-based metal-organic chalcogenolates (MOChas). MOChas are crystalline ensembles containing one-dimensional (1D) or two-dimensional (2D) inorganic topologies with structures and properties defined by the choice of metal, chalcogen, and ligand. MOChas can be readily prepared from a variety of small-molecule ligands and metals or metal ions. Although MOChas offer ligand diversity, most reported examples use relatively small ligands, typically involving short alkyl chains, aryl rings, or molecular cages. This is because larger, more complex molecules often yield poor product morphologies with indeterminate structures. In this study, we overcame this limitation by employing a ligand exchange strategy whereby a 1D MOCha, silver(I) methyl 2-mercaptobenzoate (2MMB), is used as a silver source for preparing 2D examples. The reaction proceeds generally toward products composed of the stronger nucleophile. We show that the reaction prefers displacing 1D topologies to yield 2D ones and replacing thiolates with selenolates. We performed a study to characterize the mechanism by which organic chalcogenols and dichalcogenides exchange with MOChas. The collected data and product analysis support a proposed mechanism of nucleophilic substitution, explaining how both organic chalcogenols and dichalcogenides can displace ligands in MOChas. This work provides a new synthetic route that will enable the preparation of more elaborate MOChas and heterostructures thereof. This approach enabled the preparation of previously inaccessible oligophenyl MOChas, which were successfully solved via small-molecule serial femtosecond crystallography (smSFX) at the SPring-8 Ångström Compact Free Electron LAser (SACLA) facility.

Deep learning-based topology optimization for multi-axis machining

This paper presents a novel framework that integrates topology optimization (TO) and deep learning (DL) to generate high-performance structures suitable for multi-axis machining. Within the proposed framework, DL is built on the pix2pix network, with the conditional channel used to determine the tool shape and feed direction in multi-axis machining. This DL model will be trained using our own generated dataset on TO for multi-axis machining. Then, users can customize tool dimensions and machining orientations of the multi-axis machining operation and specify the design boundary and loading conditions as input. The DL model will rapidly generate a near-optimized structure, which subsequently serves as the starting point for further optimization. Ultimately, a topology-optimized structure that meets the tailored requirements is apt for multi-axis machining and can be finalized with only a few iterations. 2D and 3D numerical examples for heat conduction problems are studied to prove the effectiveness of the proposed method, validating improved structural performance and optimization efficiency compared to conventional TO for multi-axis machining.

Robust estimation of structural orientation parameters and 2D/3D local anisotropic Tikhonov regularization

Understanding the orientation of geologic structures is crucial for analyzing the complexity of the earths’ subsurface. For instance, information about geologic structure orientation can be incorporated into local anisotropic regularization methods as a valuable tool to stabilize the solution of inverse problems and produce geologically plausible solutions. We introduce a new variational method that uses the alternating direction method of multipliers within an alternating minimization scheme to jointly estimate orientation and model parameters in 2D and 3D inverse problems. Specifically, our approach adaptively integrates recovered information about structural orientation, enhancing the effectiveness of anisotropic Tikhonov regularization in recovering geophysical parameters. The paper also discusses the automatic tuning of algorithmic parameters to maximize the new method’s performance. Our algorithm is tested across diverse 2D and 3D examples, including structure-oriented denoising and trace interpolation. The results indicate that the algorithm is robust in solving the considered large and challenging problems, alongside efficiently estimating the associated tilt field in 2D cases and the dip, strike, and tilt fields in 3D cases. Synthetic and field examples show that our anisotropic regularization method produces a model with enhanced resolution and provides a more accurate representation of the true structures.

Variational damage model: A novel consistent approach to fracture

The computational modeling of fractures in solids using damage mechanics faces challenge when dealing with complex crack topologies. One effective approach to address this challenge is by reformulating damage mechanics within a variational framework. In this paper, we present a novel variational damage model that incorporates a threshold value to prevent damage initiation at low energy levels. The proposed model defines fracture energy density (ϕ˜) and damage field (s) based on the energy density (ϕ), crack energy release rate (Gc), and crack length scale (ℓ). Specifically, if ϕ≤Gc2ℓ, then ϕ˜=ϕ and s=0; otherwise, ϕ˜=−Gc24ℓ21ϕ+Gcℓ and s=1−Gc2ℓ1ϕ. Furthermore, we extend the model with a threshold value to a higher-order version. Utilizing this functional, we derive the governing equation for fractures that evolve automatically with ease. The formulation can be seamlessly integrated into conventional finite element methods for elastic solids with minimal modifications. The proposed formulation offers sharper crack interfaces compared to phase field methods using the same mesh density. We demonstrate the capabilities of our approach through representative numerical examples in both 2D and 3D, including static fracture problems, cohesive fractures, and dynamic fractures. The open-source code is available on GitHub via the link https://github.com/hl-ren/vdm.

Incorporating interface effects into multi-material topology optimization by improving interface configuration: An energy-based approach

Interfaces between structural multi-materials generally exhibit asymmetric resistance to tension and compression. Given this interface behavior, this work suggests an energy-based approach to improve the interface configuration for multi-material topology optimization. Based on the strain spectral decomposition, we decompose the structural elastic strain energy into tensile and compressive portions. In the density-based topology optimization framework, we use the gradient-based method to track the interface between multiple materials. Then, we construct an interface-associated scalar field to penalize the tensile portion of the strain energy, causing a pseudo-degradation of the strain energy at the interface region. Finally, within limited material usages and by minimizing the linear weighted structural strain energy and its pseudo-degradation, multi-material topology optimization with improved interface configuration is achieved. Several 2D and 3D numerical examples are investigated, by which the effectiveness and robustness of the suggested approach are fairly validated.

Stress-related discrete variable topology optimization with handling non-physical stress concentrations

The accuracy of stress calculation with a fixed mesh significantly affects the stress-based topology optimization, due to potential non-physical stress concentrations in voxel-based topology descriptions. This paper proposes a novel problem-independent machine learning enhanced high-precision stress calculation method (PIML-HPSCM) to address this challenge. As an immersed analysis method, PIML-HPSCM combines the high efficiency of fixed mesh with the accuracy of body-fitted mesh, without complex integration schemes of other immersed methods. PIML-HPSCM utilizes the extended multiscale finite element method to depict the material heterogeneity within high-resolution boundary elements. The accurate stress field can then be calculated conveniently by establishing stress evaluation matrices of high-resolution boundary elements. Moreover, the PIML is independent of problem settings and is applicable for various problems with the same governing equation type. Invoking the offline-trained neural network online can enhance stress calculation efficiency by 10–20 times. The stress-based discrete variable topology optimization, which naturally avoids singular stress phenomenon, is efficiently addressed by the sequential approximate integer programming method with PIML-HPSCM. Results from 2D and 3D examples demonstrate that the stresses calculated by PIML-HPSCM are consistent with those by body-fitted mesh, and optimized designs effectively eliminate stress concentrations of initial designs and have uniform stress distributions.

A unified FFT framework with coupled gradient damage model and phase-field method for the failure analysis of composites

A unified FFT framework coupling the Gradient Damage Model (GDM) and the Variational Phase-Field Model (PFM) for failure analysis of composites is proposed. GDM and PFM are used to simulate the failure of the matrix and the interphase, respectively. Based on the similarity of the damage evolution equations (elliptic partial differential equations) of GDM and PFM, a unified iterative solution and numerical implementation based on fast Fourier transform (FFT) are derived, and simultaneously applied to a computational model. In addition, the high computational efficiency of the FFT method is used to solve the mechanical equilibrium problem. The complete form of the PFM is introduced to avoid erroneous damage diffusion at the different phase interfaces of the composites, and a stress decomposition-based PFM is formulated. 2D and 3D computational examples are established to investigate the mesh insensitivity of the damage models used and the erroneous damage diffusion at interfaces. The applicability of the stress decomposition-based PFM at interphase is discussed in the context of single-edge notched specimens under tension and fiber-reinforced composites with interfacial phases under transverse tension. Finally, the performance of the unified computational framework is evaluated using two simple numerical examples, and the mechanical behavior of unidirectional CF/PEEK composites under transverse tensile loading is simulated based on the unified framework. The results are compared with experiments to evaluate the computational accuracy and practical applicability of the proposed framework.

Physical discovery in representation learning via conditioning on prior knowledge

Recent advances in electron, scanning probe, optical, and chemical imaging and spectroscopy yield bespoke data sets containing the information of structure and functionality of complex systems. In many cases, the resulting data sets are underpinned by low-dimensional simple representations encoding the factors of variability within the data. The representation learning methods seek to discover these factors of variability, ideally further connecting them with relevant physical mechanisms. However, generally, the task of identifying the latent variables corresponding to actual physical mechanisms is extremely complex. Here, we present an empirical study of an approach based on conditioning the data on the known (continuous) physical parameters and systematically compare it with the previously introduced approach based on the invariant variational autoencoders. The conditional variational autoencoder (cVAE) approach does not rely on the existence of the invariant transforms and hence allows for much greater flexibility and applicability. Interestingly, cVAE allows for limited extrapolation outside of the original domain of the conditional variable. However, this extrapolation is limited compared to the cases when true physical mechanisms are known, and the physical factor of variability can be disentangled in full. We further show that introducing the known conditioning results in the simplification of the latent distribution if the conditioning vector is correlated with the factor of variability in the data, thus allowing us to separate relevant physical factors. We initially demonstrate this approach using 1D and 2D examples on a synthetic data set and then extend it to the analysis of experimental data on ferroelectric domain dynamics visualized via piezoresponse force microscopy.

A 3D field-theoretic example for Hodge theory

We focus on the continuous symmetry transformations for the three (2 + 1)-dimensional (3D) system of a combination of the free Abelian 1-form and 2-form gauge theories within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism. We establish that this combined system is a tractable field-theoretic model of Hodge theory. The symmetry operators of our present system provide the physical realizations of the de Rham cohomological operators of differential geometry at the algebraic level. Our present investigation is important in the sense that, for the first time, we are able to establish an odd dimensional (i.e., D = 3) field-theoretic system to be an example for Hodge theory (besides earlier works on a few interesting (0 + 1)-dimensional (1D) toy models as well as a set of well-known SUSY quantum mechanical systems of physical interest). For the sake of brevity, we have purposely not taken into account the 3D Chern-Simon term for the Abelian 1-form gauge field in our theory which allows the mass as well as the gauge invariance to co-exist together.

3D Surrealism and Examples in Visual Communication Design

The main purpose of this research is to discuss the concept of 3D surrealism, which has become an important trend in the field of visual communication design in recent years, and some related visual works. With its ability to evoke subconscious emotions, surrealism is treated as a theme in various fields. However, here the topic of surrealism is limited to three-dimensional design and animation examples in the field of visual communication design. Document analysis is the preferred research method. A literature review is carried out using digital resources on the internet on the subject. In addition, examples of surreal images and animations created in 3D design software are included. The research concludes that the audience’s interaction with the product takes longer when visual communication products use 3D surrealism. Elements such as the visualization of subconscious elements, including objects with different functions in the same composition, and the uncertainty between the real and the virtual effectively create this time-based difference.

High order conservative finite difference WENO scheme for three-temperature radiation hydrodynamics

The three-temperature (3-T) radiation hydrodynamics (RH) equations play an important role in the high-energy-density-physics fields, such as astrophysics and inertial confinement fusion (ICF). It describes the interaction between radiation and high-energy-density plasmas including electron and ion in the assumption that radiation, electron and ion are in their own equilibrium state, which means they can be characterized by their own temperatures. The 3-T RH system consists of the density, momentum and three internal energy (electron, ion and radiation) equations. In this paper, we propose a high order conservative finite difference weighted essentially non-oscillatory (WENO) scheme solving one-dimensional (1D) and two-dimensional (2D) 3-T RH equations respectively. Following our previous paper [7], we introduce the three new energy variables, and then design a finite difference scheme with both the conservative property and arbitrary high order accuracy. Based on the WENO interpolation and the strong stability preserving (SSP) high order time discretizations, taken as an example, we design a class of fifth order conservative finite difference schemes in space and third order in time. Compared with the Lagrangian method we proposed in [7], which can only reach second order accuracy for 2D 3-T RH equations if straight-line edged meshes are used, the finite difference scheme can be easily designed to arbitrary high order accuracy for multi-dimensional 3-T RH equations. The finite difference formulation is also much less expensive in multi-dimensions than finite volume schemes used in [7]. Furthermore, our method can handle fluids with large deformation easily. Numerous 1D and 2D numerical examples are presented to verify the desired properties of the high order finite difference WENO schemes such as high order accuracy, non-oscillation, conservation and adaptation to severely distorted single-material radiation hydrodynamics problems.

LibEMMI_MGFD: A program of marine controlled-source electromagnetic modelling and inversion using frequency-domain multigrid solver

We develop a software package libEMMI_MGFD for 3D frequency-domain marine controlled-source electromagnetic (CSEM) modelling and inversion. It is the first open-source C program tailored for geometrical multigrid (GMG) CSEM simulation. An volumetric anisotropic averaging scheme has been employed to compute effective medium for modelling over uniform and nonuniform grid. The computing coordinate is aligned with acquisition geometry by rotation with the azimuth and dip angles, facilitating the injection of the source and the extraction of data with arbitrary orientations. Efficient nonlinear optimization is achieved using quasi-Newton scheme assisted with bisection backtracking line search. In constructing the modularized Maxwell solver and evaluating the misfit and gradient for 3D CSEM inversion, the reverse communication technique is the key to the compaction of the software while maintaining the computational performance. A number of numeric tests demonstrate the efficiency of the modelling while preserving the solution accuracy. A 3D marine CSEM inversion example has been examined for resistivity imaging. Program summaryProgram Title: libEMMI_MGFDCPC Library link to program files:https://doi.org/10.17632/mpk7xrd38m.1Developer's repository link:https://github.com/yangpl/libEMMI_MGFDLicensing provisions: GNU General Public License v3.0Programming language: C, ShellSoftware dependencies: MPI [1]Nature of problem: 3D Controlled-source electromagnetic modelling and inversion.Solution method: Frequency-domain modelling on staggered grid; Geometrical multigrid (GMG) method as an iterative solver; Quasi-Newton method for nonlinear minimization; Bisection-based backtracking line search.

Solving Mazes: A New Approach Based on Spectral Graph Theory

The use of graph theory for solving labyrinths and mazes is well known, understanding the possible paths as the connections between the nodes that represent the corners or bifurcations. This work presents a new idea: minimizing the length of the optimal path formulated as a topology optimization problem. The maze is mapped with finite elements, and then, through the eigenvalues of the Laplacian matrix of the graph, a constraint is imposed over the connectivity between the input and the output. Several 2D examples are provided to support this approach, allowing for unequivocally finding the shortest path in mazes with multiple connections between entrance and exit, resulting in an nonheuristic algorithm.

Certifying Ground-State Properties of Many-Body Systems

A ubiquitous problem in quantum physics is to understand the ground-state properties of many-body systems. Confronted with the fact that exact diagonalization quickly becomes impossible when increasing the system size, variational approaches are typically employed as a scalable alternative: Energy is minimized over a subset of all possible states and then different physical quantities are computed over the solution state. Despite remarkable success, rigorously speaking, all that variational methods offer are upper bounds on the ground-state energy. On the other hand, so-called relaxations of the ground-state problem based on semidefinite programming represent a complementary approach, providing lower bounds to the ground-state energy. However, in their current implementation, neither variational nor relaxation methods offer provable bound on other observables in the ground state beyond the energy. In this work, we show that the combination of the two classes of approaches can be used to derive certifiable bounds on the value of any observable in the ground state, such as correlation functions of arbitrary order, structure factors, or order parameters. We illustrate the power of this approach in paradigmatic examples of 1D and 2D spin-1/2 Heisenberg models. To improve the scalability of the method, we exploit the symmetries and sparsity of the considered systems to reach sizes of hundreds of particles at much higher precision than previous works. Our analysis therefore shows how to obtain certifiable bounds on many-body ground-state properties beyond energy in a scalable way. Published by the American Physical Society 2024

Robust design optimization using a non-intrusive second-order approximation of stochastic moments

This paper presents a new formulation of the second-order fourth-moment method (sometimes referred to as second-order perturbation method or second-order method of moments). The method allows to efficiently predict the stochastic moments of a response function and is therefore often used within robust design optimization. The new approach allows a non-intrusive implementation at the same cost as existing, highly intrusive formulations. Therefore, the new approach can be applied to any objective function without significant implementation effort. It is based on a few finite difference steps into special directions and hence is dependent on the corresponding step sizes. An automatic step size procedure is supplied beside a detailed convergence analysis. The advantages of the new formulation are demonstrated by robust design optimizations of a 2D and a 3D example using the geometrically nonlinear finite element method.

On fundamental inconsistencies in a commonly used modification of a fluid model for glow discharge

This work considers the fundamental contradictions in the concept of one of the most well-known and widely used modifications of the fluid model for simulation of a glow discharge (GD), the ‘local mean energy approximation’ (LMEA). In this model, it is proposed to determine the kinetic coefficients in the electron particle and energy balance equations as functions of the electron mean energy (temperature) rather than local electric field, using a one-to-one correspondence between these parameters through the electron Boltzmann equation. It is shown that the scope of applicability of this model, like any other modification of the fluid model, is limited by the local mode of formation of the electron energy distribution function (EEDF). Therefore, as demonstrated by the examples of typical 1D and 2D problems for a GD in argon, its extension to the region of nonlocal EEDF is in no way justified and leads not only to serious errors in the results, but also to a logically intractable situation in attempts to apply the main postulate of the LMEA model to the region of a weak (or even reverse) electric field in a negative glow plasma. At the same time, the apparent reliability of calculations within the framework of the LMEA model for a number of parameters, in our opinion, only slows down progress in modeling of gas discharge plasma.

Application of meshless generalized finite difference method (GFDM) in single-phase coupled heat and mass transfer problem in three-dimensional porous media

This paper achieves effective and precise meshless modeling of three-dimensional (3D) single-phase coupled heat and mass transfer problems based on the generalized finite difference method (GFDM). It utilizes the Taylor formula and the weighted least squares method in the node influence domains to derive a generalized finite difference scheme for spatial derivatives of pressure and temperature. Consequently, a sequential coupled discrete scheme for the pressure diffusion equation and heat convection–conduction equation is formulated, resulting in the determination of pressure and temperature. An example conducts sensitivity analysis with different schemes of node collocation and different radius of influence domains. The calculation results demonstrate that this method exhibits good convergence. Two 3D model examples with regular and irregular boundaries illustrate the advantages of the GFDM in handling complex geometric problems within the computational domain, showcasing its superior flexibility and simplicity. This paper demonstrates the significant potential of GFDM in addressing complex geometric multi-physics field coupling challenges, offering innovative ideas for geothermal resource development, groundwater management, and thermal recovery in oil and gas reservoirs.