Discrete Solution Research Articles

Kernel Principal Component Analysis for Allen–Cahn Equations

Different researchers haveanalyzed effective computational methods that maintain the precision of Allen–Cahn (AC) equations and their constant security. This article presents a method known as the reduced-order model technique by utilizing kernel principle component analysis (KPCA), a nonlinear variation of traditional principal component analysis (PCA). KPCA is utilized on the data matrix created using discrete solution vectors of the AC equation. In order to achieve discrete solutions, small variations are applied for dividing up extraterrestrial elements, while Kahan’s method is used for temporal calculations. Handling the process of backmapping from small-scale space involves utilizing a non-iterative formula rooted in the concept of the multidimensional scaling (MDS) method. Using KPCA, we show that simplified sorting methods preserve the dissipation of the energy structure. The effectiveness of simplified solutions from linear PCA and KPCA, the retention of invariants, and computational speeds are shown through one-, two-, and three-dimensional AC equations.

Accelerating the convergence of Newton’s method for nonlinear elliptic PDEs using Fourier neural operators

It is well known that Newton’s method can have trouble converging if the initial guess is too far from the solution. Such a problem particularly occurs when this method is used to solve nonlinear elliptic partial differential equations (PDEs) discretized via finite differences. This work focuses on accelerating Newton’s method convergence in this context. We seek to construct a mapping from the parameters of the nonlinear PDE to an approximation of its discrete solution, independently of the mesh resolution. This approximation is then used as an initial guess for Newton’s method. To achieve these objectives, we elect to use a Fourier neural operator (FNO). The loss function is the sum of a data term (i.e., the comparison between known solutions and outputs of the FNO) and a physical term (i.e., the residual of the PDE discretization). Numerical results, in one and two dimensions, show that the proposed initial guess accelerates the convergence of Newton’s method by a large margin compared to a naive initial guess, especially for highly nonlinear and anisotropic problems, with larger gains on coarse grids.

A Novel k-Means Framework via Constrained Relaxation and Spectral Rotation.

Owing to its simplicity, the traditional k - means (Lloyd heuristic) clustering method plays a vital role in a variety of machine-learning applications. Disappointingly, the Lloyd heuristic is prone to local minima. In this article, we propose k - mRSR, which converts the sum-of-squared error (SSE) (Lloyd) into a combinatorial optimization problem and incorporates a relaxed trace maximization term and an improved spectral rotation term. The main advantage of k - mRSR is that it only needs to solve the membership matrix instead of computing the cluster centers in each iteration. Furthermore, we present a nonredundant coordinate descent method that brings the discrete solution infinitely close to the scaled partition matrix. Two novel findings from the experiments are that k - mRSR can further decrease (increase) the objective function values of the k - means obtained by Lloyd (CD), while Lloyd (CD) cannot decrease (increase) the objective function obtained by k - mRSR. In addition, the results of extensive experiments on 15 datasets indicate that k - mRSR outperforms both Lloyd and CD in terms of the objective function value and outperforms other state-of-the-art methods in terms of clustering performance.

Discretely Nonlinearly Stable Weight-Adjusted Flux Reconstruction High-Order Method for Compressible Flows on Curvilinear Grids

To achieve genuine predictive capability, an algorithm must consistently deliver accurate results over prolonged temporal integration periods, avoiding the unwarranted growth of aliasing errors that compromise the discrete solution. Provable nonlinear stability bounds the discrete approximation and ensures that the discretization does not diverge. Nonlinear stability is accomplished by satisfying a secondary conservation law, namely for compressible flows; the second law of thermodynamics. For high-order methods, discrete nonlinear stability and entropy stability, have been successfully implemented for discontinuous Galerkin (DG) and residual distribution schemes, where the stability proofs depend on properties of L2-norms. In this paper, nonlinearly stable flux reconstruction (NSFR) schemes are developed for three-dimensional compressible flow in curvilinear coordinates. NSFR is derived by merging the energy stable flux reconstruction (ESFR) framework with entropy stable DG schemes. NSFR is demonstrated to use larger time-steps than DG due to the ESFR correction functions, at the cost of larger error levels at design order convergence for equivalent degrees of freedom, while preserving discrete nonlinear stability. NSFR differs from ESFR schemes in the literature since it incorporates the FR correction functions on the volume terms through the use of a modified mass matrix. We also prove that discrete kinetic energy stability cannot be preserved to machine precision for quadrature rules where the surface quadrature is not a subset of the volume quadrature. This result stems from the inverse mapping from the kinetic energy variables to the conservative variables not existing for the kinetic energy projected variables. This paper also presents the NSFR modified mass matrix in a weight-adjusted form. This form reduces the computational cost in curvilinear coordinates because the dense matrix inversion is approximated by a pre-computed projection operator and the inverse of a diagonal matrix on-the-fly and exploits the tensor product basis functions to utilize sum-factorization. The nonlinear stability properties of the scheme are verified on a nonsymmetric curvilinear grid for the inviscid Taylor-Green vortex problem and the correct orders of convergence were obtained on a curvilinear mesh for a manufactured solution. Lastly, we perform a computational cost comparison between conservative DG, overintegrated DG, and our proposed entropy conserving NSFR scheme, and find that our proposed entropy conserving NSFR scheme is computationally competitive with the conservative DG scheme.

Explicit A Posteriori Error Representation for Variational Problems and Application to TV-Minimization

AbstractIn this paper, we propose a general approach for explicit a posteriori error representation for convex minimization problems using basic convex duality relations. Exploiting discrete orthogonality relations in the space of element-wise constant vector fields as well as a discrete integration-by-parts formula between the Crouzeix–Raviart and the Raviart–Thomas element, all convex duality relations are transferred to a discrete level, making the explicit a posteriori error representation –initially based on continuous arguments only– practicable from a numerical point of view. In addition, we provide a generalized Marini formula that determines a discrete primal solution in terms of a given discrete dual solution. We benchmark all these concepts via the Rudin–Osher–Fatemi model. This leads to an adaptive algorithm that yields a (quasi-optimal) linear convergence rate.

A Q-learning multi-objective grey wolf optimizer for the distributed hybrid flowshop scheduling problem

Most existing research focuses on a single objective for the distributed hybrid flowshop scheduling problem (DHFSP). This article focuses on a multi-objective DHFSP with sequence-dependent set-up time (DHFSP-SDST). A Q-learning multi-objective grey wolf optimizer (QMOGWO) is designed to optimize the makespan, total energy consumption and total tardiness. A mathematical model for DHFSP-SDST is established. Several initialization strategies and a random method are introduced to improve the quality of the initial population. The new individual is developed by the discrete solution updating mechanism of QMOGWO. Based on the Q-learning, local search strategies are designed to avoid local optima. To verify the performance of the proposed QMOGWO, different scales of instances are tested in various factories and at different stages, and the simulation results show that the QMOGWO outperforms the comparison methods.

Optimal [formula omitted] error estimates of mass- and energy- conserved FE schemes for a nonlinear Schrödinger–type system

In this paper, we present an implicit Crank–Nicolson finite element (FE) scheme for solving a nonlinear Schrödinger–type system, which includes Schrödinger–Helmholz system and Schrödinger–Poisson system. In our numerical scheme, we employ an implicit Crank–Nicolson method for time discretization and a conforming FE method for spatial discretization. The proposed method is proved to be well-posedness and ensures mass and energy conservation at the discrete level. Furthermore, we prove optimal L2 error estimates for the fully discrete solutions. Finally, some numerical examples are provided to verify the convergence rate and conservation properties.

Adapting to climate change: the ultimate challenge for the next half-century of local government?

ABSTRACT Climate change will have a disproportionate and asymmetric impact on cities and urban areas, and some of their most vulnerable residents will be at particular risk. Studies have found that some municipalities have done far more to adapt to it than others, but there has been a general lack of funding, implementation and engagement with marginalised groups to help them prepare. We suggest that the unpredictable and evolving nature of climate impacts means that adaptation represents a defining public policy challenge for local governments in the coming decades. We set out the broad epistemological, practical and justice issues that this challenge presents for the practice and study of local government, and argue that addressing it will require new approaches that go beyond discrete and familiar solutions.

Effect of random variation in input parameter on cracked orthotropic plate using extended isogeometric analysis (XIGA) under thermomechanical loading

Abstract This research introduces the Stochastic Extended Isogeometric Analysis (XIGA) method to investigate fracture behavior of isotropic and orthotropic materials under mechanical, thermal, and thermomechanical loads. Employing knot spans from Isogeometric Analysis (IGA) for domain discretization, the study utilizes identical basis functions for geometry construction and solution discretization. Utilizing Extended Finite Element Method (XFEM) enrichment functions, accurate crack face displacement discontinuity and tip singularity within the stress field are characterized. Additionally, employing a second-order perturbation technique within XIGA framework, the research derives mean and coefficient of variance values for mixed-mode Stress Intensity Factors (SIF). Stochastic variations in material elastic properties, crack length, and crack angle are considered in this computation. Credibility and robustness of the study are confirmed through comparative analyses against available literatures and Monte Carlo Simulations (MCS). The observed exceptional agreement validates the precision and reliability of the proposed stochastic XIGA method for fracture analysis in orthotropic material systems under thermomechanical loading conditions.

Casting Light on Degeneracies: A Comprehensive Study of Lightcurve Variations in Microlensing Events OGLE-2017-BLG-0103 and OGLE-2017-BLG-0192

This study investigates orbital parallax in gravitational microlensing events, focusing on OGLE-2017-BLG-0103 and OGLE-2017-BLG-0192. For events with timescales ≤60 days, a Jerk-Parallax degeneracy arises due to high Jerk velocity ( vj˜ ), causing a fourfold continuous parallax degeneracy. OGLE-2017-BLG-0103, after incorporating orbital parallax, reveals four discrete degenerate parallax solutions, while OGLE-2017-BLG-0192 exhibits four discrete solutions without degeneracy. The asymmetric lightcurve of OGLE-2017-BLG-0103 suggests a more probable model where Xallarap is added to the parallax model, introducing tension. The galactic model analysis predicts a very low-mass stellar lens for OGLE-2017-BLG-0192. For OGLE-2017-BLG-0103, degenerate solutions suggest a low-mass star or a darker lens in the disk, while the Xallarap+Parallax model also predicts a stellar lens in the bulge, with the source being a solar-type star orbited by a dwarf star. This study presents five degenerate solutions for OGLE-2017-BLG-0103, emphasizing the potential for confirmation through high-resolution Adaptive Optics observations with Extremely Large Telescopes in the future. The complexities of degenerate scenarios in these microlensing events underscore the need to analyze special single-lens events in the Roman Telescope Era.

An adaptive hybridizable discontinuous Galerkin method for Darcy–Forchheimer flow in fractured porous media

In this paper, we consider an adaptive hybridizable discontinuous Galerkin (HDG) method based on the discrete fracture model for approximation of a single-phase flow in fractured porous media. We are interested in the case that the flow rate in fracture is large enough to justify the use of Forchheimer’s law for modeling flow within the fracture, while Darcy’s law is applied to the surrounding matrix. The HDG method could be designed to simulate the flow in porous media with reduced fractures which consist of many straight lines or planes. More specifically, we use piecewise polynomials of degree [Formula: see text] to approximate the velocity and pressure in fracture and surrounding porous media. The existence and uniqueness of discrete solutions are proved by the Brouwer fixed point theorem, and an efficient and reliable a posteriori error estimator is obtained with respect to an energy norm. Moreover, the HDG scheme, the existence and uniqueness of discrete solutions, and the a posteriori error estimates are also extended to the problem with non-planar, embedded, and intersecting fractures. Finally, several numerical examples are provided to validate the performance of the obtained a posteriori error estimator.

Morphogenesis analysis of icosahedral and prismatic tensegrity modules

Morphogenesis is a critical aspect of tensegrity system design. This paper explores icosahedral and prismatic tensegrity modules' pattern formation and foundational theories. First, a material consumption functional and self-strain energy functional are introduced from an engineering perspective, along with a variational principle involving independent dual-variable in the morphogenesis process of tensegrity structures. Next, a Symbol Parsing Method is proposed and successfully applied to the optimal morphology analysis of the icosahedral tensegrity module. Due to memory limitations in personal computers, the Singular Value Decomposition of the symbolic equilibrium matrix for the prismatic tensegrity module is constrained. As an alternative, a discrete numerical solution, the semi-symbolic parsing method, is obtained by directly utilizing a numerical equilibrium matrix. This method was recently employed to analyze and optimize a practical tensegrity footbridge designed by the authors. Lastly, the paper delves into the stability of self-equilibrium states in icosahedral and classical prismatic tensegrity modules by employing a second-order variational analysis of the self-strain energy functional. The findings presented are expected to contribute to future research on the morphogenesis of tensegrity structures.

Modeling Cardiovascular Flow with Artificial Viscosity: Analyzing Navier-Stokes Solutions and Simulating Cardiovascular Diseases

In this paper, the numerical solutions of the Navier-Stokes equations (NSE) used for modeling the flow in the cardiovascular system are investigated using the Finite Element Method (FEM). A fully discrete solution scheme of the NSE and its stability and error analysis are presented. Artificial viscosity stabilization is added to the fully discrete scheme to better model the real flow structure and to remove non-physical oscillations. Numerical tests are also presented to demonstrate the effectiveness of the resulting scheme. Simulations analyzing the flow structure in the case of cardiovascular diseases such as atherosclerosis and brain aneurysm are presented in detail along with wall shear stress values.

Integrable Discretization and Multi-soliton Solutions of Negative Order AKNS Equation

Integrable Discretization and Multi-soliton Solutions of Negative Order AKNS Equation

A discrete element solution method embedded within a Neural Network

This paper introduces a novel methodology, the Neural Network framework for the Discrete Element Method (NN4DEM), as part of a broader initiative to harness specialised AI hardware and software environments, marking a transition from traditional computational physics programming approaches. NN4DEM enables GPU-parallelised computations by mapping particle data (coordinates and velocities) onto uniform grids as solution fields and computing contact forces by applying mathematical operations that can be found in convolutional neural networks (CNN). Essentially, this framework transforms a DEM problem into a series of layered “images” composed of pixels, using stencil operations to compute the DEM physics, which is inherently local. The method revolves around custom kernels, with operations prescribed by the laws of physics for contact detection and interaction. Therefore, unlike conventional AI methods, it eliminates the need for training data to determine network weights. NN4DEM utilises libraries such as PyTorch for relatively easier programmability and platform interoperability. This paper presents the theoretical foundations, implementation and validation of NN4DEM through hopper test benchmarks. An analysis of the results from random packing cases highlights the ability of NN4DEM to scale to 3D models with millions of particles. The paper concludes with potential research directions, including further integration with other physics-based models and applications across various multidisciplinary fields.

Impact of mixed boundary conditions and nonsmooth data on layer‐originated nonpremixed combustion problems: Higher‐order convergence analysis

AbstractThis work explores the theoretical and computational impacts of mixed‐type flux conditions and nonsmooth data on boundary/interior layer‐originated singularly perturbed semilinear reaction–diffusion problems. Such problems are prevalent in nonpremixed combustion models and catalytic reaction models. The inclusion of arbitrarily small diffusion terms results in boundary layers influenced by specific flux conditions normalized by perturbation parameters. We have demonstrated theoretically that the sharpness of the boundary layer is significantly reduced when this normalization is independent of the diffusion parameter. In addition, the presence of a nonsmooth source function gives rise to an interior layer in the current problem. We show that using upwind discretizations for mixed boundary fluxes achieves nearly second‐order accuracy if the first derivatives are not normalized concerning perturbation parameters. This outcome arises from the bounds of a prior derivative of the continuous solution. Furthermore, it is theoretically shown that nearly second‐order uniform accuracy can be attained for reaction‐dominated semilinear problems using a hybrid scheme at the discontinuity point. To ensure the uniform stability of the discrete solution, a transformation is necessary for the corresponding discrete problem. Theoretical results are supported by various experiments on nonlinear problems, illustrating the pointwise rates and highlighting both linear and higher‐order accuracy at specific points.

Co-Clustering by Directly Solving Bipartite Spectral Graph Partitioning.

Bipartite spectral graph partitioning (BSGP) method as a co-clustering method, has been widely used in document clustering, which simultaneously clusters documents and words by making full use of the duality between documents and words. It consists of two steps: 1) graph construction and 2) singular value decomposition on the bipartite graph to compute a continuous cluster assignment matrix, followed by post-processing to get the discrete solution. However, the generated graph is unstructured and fixed. It heavily relies on the quality of the graph construction. Moreover, the two-stage process may deviate from directly solving the primal problem. In order to tackle these defects, a novel bipartite graph partitioning method is proposed to learn a bipartite graph with exactly c connected components (c is the number of clusters), which can obtain clustering results directly. Furthermore, it is experimentally and theoretically proved that the solution of the proposed model is the discrete solution of the primal BSGP problem for a special situation. Experimental results on synthetic and real-world datasets exhibit the superiority of the proposed method.

Novel Design and Performance Analysis of 1R1T Remote Center-of-Motion Mechanisms With Partially Decoupled T- and R-Motions

Abstract Remote center-of-motion (RCM) mechanisms provide a way for surgical instruments to pass through a remote center (e.g., skin incision) under geometrical constraints, facilitating safer operations in minimally invasive surgery (MIS). One rotation and one translation (1R1T, pitch and insertion) are the basic requirements for RCM mechanisms. To make the structure simpler and control easier, a novel concept of 1R1T RCM mechanisms with partially decoupled motions, inspired by the double-parallelogram 1R RCM mechanisms, is proposed in this article, by investigating and proving its motion combination principle based on the screw theory. New evolution procedures based on the configuration evolution method have been derived to design 1R1T RCM mechanisms based on two approaches of inserting the T-motion in an original 1R RCM mechanism, resulting in two types of 1R1T RCM mechanisms with partially decoupled motions and base-locating actuators. The kinematic models of one typical proposed mechanism (including the forward and inverse kinematics) and its Jacobian matrix are derived. The performance analysis is presented, including RCM validation, velocity, singularity, and workspace analysis. Then, the dimensional optimization based on the discrete solution method is derived. Finally, a prototype of the proposed mechanism is presented with preliminary experiments performed to verify the feasibility of the synthesized RCM mechanisms. The results show that the RCM mechanism performs the 1R1T partially decoupled motion, and it can be used as the basic element of an active manipulator of an MIS robot.

A Novel Set-Based Discrete Particle Swarm Optimization for Wastewater Treatment Process Effluent Scheduling.

With the escalating severity of environmental pollution caused by effluent, the wastewater treatment process (WWTP) has gained significant attention. The wastewater treatment efficiency and effluent quality are significantly impacted by effluent scheduling that adjusts the hydraulic retention time. However, the sequential batch and continuous nature of the effluent pose challenges, resulting in complex scheduling models with strong constraints that are difficult to tackle using existing scheduling methods. To optimize maximum completion time and effluent quality simultaneously, this article proposes a restructured set-based discrete particle swarm optimization (RS-DPSO) algorithm to address the WWTP effluent scheduling problem (WWTP-ESP). First, an effective encoding and decoding method is designed to effectively map solutions to feasible schedules using temporal and spatial information. Second, a restructured set-based discrete particle swarm algorithm is introduced to enhance the searching ability in discrete solution space via restructuring the solution set. Third, a constraint handling strategy based on violation degree ranking is designed to reduce the waste of computational resources. Fourth, a Sobel filter based local search is proposed to guide particle search direction to enhance search efficiency ability. The RS-DPSO provides a novel method for solving WWTP-ESP problems with complex discrete solution space. The comparative experiments indicate that the novel designs are effective and the proposed algorithm has superior performance over existing algorithms in solving the WWTP-ESP.

H(div)-conforming and discontinuous Galerkin approach for Herschel–Bulkley flow with density-dependent viscosity and yield stress

This paper presents a comprehensive study on Herschel–Bulkley flow, where the flow parameters are dependent on the density. The Herschel–Bulkley model is a generalized power-law model used to simulate viscoplastic fluids defined by a plasticity threshold. We consider the case where the plasticity threshold and the viscosity depend on the shear rate and fluid density. To analyze this model, we use a Huber regularization of the stress and propose an H(div)-conforming and discontinuous Galerkin (DG) numerical approximation for the coupled equations governing the flow. We discuss the stability and existence of discrete solutions and propose a semismooth Newton linearization for the numerical solution of the discretized system. Our numerical scheme is validated through several experiments that explore the behavior of Herschel–Bulkley flow under different conditions. The results demonstrate the robustness of our numerical method.