Discrete Space Research Articles

Novel designs of blade mixer impellers from the discrete element method and topology optimization

Granular mixers are essential for manufacturing products that require bulk materials as precursors. The modeling of mixers is aided by the discrete element method (DEM), which can simulate the flow of bulk material. DEM is applied in mixer design studies by varying a few design parameters and analyzing the predicted performance, but systematic design via optimization methods has yet to be performed. In this work, we utilized a combined DEM and topology optimization approach to design blade mixer impellers for improved mixing and reduced energy consumption. A DEM model was formulated from a blade mixer setup and validated with experimental particle flow data. The computational cost of the model was then reduced by employing larger particle sizes in the simulations. A binary genetic optimization algorithm was used afterwards to find the tilt angle, number, and shape of impeller blades that would improve mixing or reduce the required driving power. The high mixing impellers have four blades that each have cavities at the center and the tip. The increased blade count promotes convective mixing, while the cavities enhance shear mixing. Meanwhile, the low energy impellers have two blades that both have cavities at the center and less material at the blade tips. These minimize particle-impeller collisions, which in turn reduce the power required to drive the impeller. This work demonstrates the use of optimization methods to design the shape of impeller blades – a variable often overlooked in conventional DEM studies. Our approach may be used to systematically design other mixer types as well.

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 modeling 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.

Extreme value statistics of jump processes.

We investigate extreme value statistics (EVS) of general discrete time and continuous space symmetric jump processes. We first show that for unbounded jump processes, the semi-infinite propagator G_{0}(x,n), defined as the probability for a particle issued from zero to be at position x after n steps whilst staying positive, is the key ingredient needed to derive a variety of joint distributions of extremes and times at which they are reached. Along with exact expressions, we extract universal asymptotic behaviors of such quantities. For bounded, semi-infinite jump processes killed upon first crossing of zero, we introduce the strip probability μ_{0,[under x]̲}(n), defined as the probability that a particle issued from zero remains positive and reaches its maximum x on its nth^{} step exactly. We show that μ_{0,[under x]̲}(n) is the essential building block to address EVS of semi-infinite jump processes, and obtain exact expressions and universal asymptotic behaviors of various joint distributions.

Porous structures inspired by porcupine quill: multiscale design optimization approach

This paper presents a novel approach for designing a freeform bending-resistant structure from the combination of explicit discrete component-based topology optimization (TO) and the porcupine quill-inspired features. To embed the porcupine quill’s features into the TO formulations, the method involves constructing discrete components at various scales to imitate features including solid shell, stochastically distributed pores, and graded stiffeners. The components are iteratively updated, and the optimization process allows for the grading of quill-inspired features while achieving optimal structural compliance under bending loads. The proposed approach is demonstrated to be effective through the resolution of Messershmitt–Bolkow–Blohm (MBB) beam designs, parameterized studies of geometric parameters, and numerical validation of long-span and short-span quill-inspired beam designs. By examining the von Mises stress distribution, the study highlights the mitigation of material yielding at the shell region brought by the geometric features of porcupine quills, leading to the potential theory support for the bending resistance. The optimized MBB beams are manufactured using the material extrusion technique, and three-point bending tests are conducted to explore the failure mitigation capability of the quill-inspired beam under large deformation. Consequently, the study concludes that the proposed quill-inspired component-based TO approach can design a structure with excellent bending resistance according to the improved energy absorption as well as increased deformation after reaching 75% peak load.

Ice Shape Convergence in Multistep Ice Accretion Simulations

Local errors in the geometrical description of the ice front are amplified in multistep simulations over conformal meshes due to the coupling of aerodynamics, water impingement, ice accretion, and grid deformation. Small perturbations in the initial phase of ice formation possibly result in dramatically different ice shapes, which can hinder the stability of the multistep procedure. This problem is analyzed by investigating the combined effects of space and time discretization on the ice growth over airfoils and three-dimensional wings. We propose an automatic procedure for selecting the time interval for the update of the aerodynamics and particle impingement. A local growth limiter λ is introduced here to bound the local ice thickness growth to be comparable to the local grid spacing on the surface, resulting in an automatic adaptive time step to be used in the multistep simulation. Examples are provided for three-dimensional cases under both rime and glaze conditions over straight and swept wings. These examples highlight the different accretion mechanisms of the two ice regimes and preliminarily indicate that ice-shape convergence can be achieved for low values of λ.

Robust and conservative dynamical low-rank methods for the Vlasov equation via a novel macro-micro decomposition

Dynamical low-rank (DLR) approximation has gained interest in recent years as a viable solution to the curse of dimensionality in the numerical solution of kinetic equations including the Boltzmann and Vlasov equations. These methods include the projector-splitting and Basis-update & Galerkin (BUG) DLR integrators, and have shown promise at greatly improving the computational efficiency of kinetic solutions. However, this often comes at the cost of conservation of charge, current and energy. In this work we show how a novel macro-micro decomposition may be used to separate the distribution function into two components, one of which carries the conserved quantities, and the other of which is orthogonal to them. We apply DLR approximation to the latter, and thereby achieve a clean and extensible approach to a conservative DLR scheme which retains the computational advantages of the base scheme. Moreover, our approach requires no change to the mechanics of the DLR approximation, so it is compatible with both the BUG family of integrators and the projector-splitting integrator which we use here. We describe a first-order integrator which can exactly conserve charge and either current or energy, as well as an integrator which exactly conserves charge and energy and exhibits second-order accuracy on our test problems. To highlight the flexibility of the proposed macro-micro decomposition, we implement a pair of velocity space discretizations, and verify the claimed accuracy and conservation properties on a suite of plasma benchmark problems.

Correlated Equilibria for Mean Field Games with Progressive Strategies

In a discrete space and time framework, we study the mean field game limit for a class of symmetric N-player games based on the notion of correlated equilibrium. We give a definition of correlated solution that allows us to construct approximate N-player correlated equilibria that are robust with respect to progressive deviations. We illustrate our definition by way of an example with explicit solutions. Funding: O. Bonesini acknowledges financial support from Engineering and Physical Sciences Research Council [Grant EP/T032146/1]. M. Fischer acknowledges partial support through the University of Padua [Research Project BIRD229791 “Stochastic mean field control and the Schrödinger problem”].

A Multi-Agent RL Algorithm for Dynamic Task Offloading in D2D-MEC Network with Energy Harvesting.

Delay-sensitive task offloading in a device-to-device assisted mobile edge computing (D2D-MEC) system with energy harvesting devices is a critical challenge due to the dynamic load level at edge nodes and the variability in harvested energy. In this paper, we propose a joint dynamic task offloading and CPU frequency control scheme for delay-sensitive tasks in a D2D-MEC system, taking into account the intricacies of multi-slot tasks, characterized by diverse processing speeds and data transmission rates. Our methodology involves meticulous modeling of task arrival and service processes using queuing systems, coupled with the strategic utilization of D2D communication to alleviate edge server load and prevent network congestion effectively. Central to our solution is the formulation of average task delay optimization as a challenging nonlinear integer programming problem, requiring intelligent decision making regarding task offloading for each generated task at active mobile devices and CPU frequency adjustments at discrete time slots. To navigate the intricate landscape of the extensive discrete action space, we design an efficient multi-agent DRL learning algorithm named MAOC, which is based on MAPPO, to minimize the average task delay by dynamically determining task-offloading decisions and CPU frequencies. MAOC operates within a centralized training with decentralized execution (CTDE) framework, empowering individual mobile devices to make decisions autonomously based on their unique system states. Experimental results demonstrate its swift convergence and operational efficiency, and it outperforms other baseline algorithms.

An optimization method for ITER radial x-ray camera line-of-sight design basing on Bayesian uncertainty analysis

This paper presents a novel uncertainty optimization algorithm for the design of line-of-sight (LOS) systems used in tomographic inversion. By extending Gaussian process tomography from discrete pixel space to continuous function space through Bayesian inference, we introduce an uncertainty function and analyze its typical distributions. We develop an algorithm to minimize the uncertainty, which is then applied to optimize the LOS configuration of the internal camera in the ITER project. Uncertainty analysis and phantom testing are conducted to validate the effectiveness of the proposed algorithm. The results demonstrate improved accuracy and stability in tomographic reconstructions. This study contributes to the advancement of LOS design for tomographic inversion, offering a practical solution for optimizing diagnostic systems in complex experimental settings.

Algebraically stable SDIRK methods with controllable numerical dissipation for first/second-order time-dependent problems

In this paper, a family of four-stage singly diagonally implicit Runge-Kutta methods are proposed to solve first-/second-order time-dependent problems, exhibiting the following numerical properties: fourth-order accuracy in time, unconditional stability, controllable numerical dissipation, and adaptive time step selection. The BN-stability condition is employed as a constraint to optimize parameters in the Butcher table, having significant benefits, and hence is recommended for nonlinear dynamics problems in contrast to existing methods. Numerical examples involving both first- and second-order linear/nonlinear dynamics problems validate the proposed method, and numerical results reveal that the proposed methods are free from the order reduction phenomenon when applied to nonlinear dynamics problems. The performance of adaptive time-stepping using the embedded scheme is further illustrated by the phase-field modeling problem. Additionally, the advantages and disadvantages of three-stage third-order accurate algebraically stable methods are discussed. The proposed high-order time integration can be readily integrated into high-order spatial discretization methods, such as the high-order spectral element method employed in this paper, to obtain high-order discretization in space and time dimensions.

Many-option collective decision making: discrete collective estimation in large decision spaces

Collective consensus forming in spatially distributed systems is a challenging task. In previous literature, multi-option consensus-forming scenarios, with the number of options being smaller or equal to the number of agents, have been well studied. However, many well-performing decision-making strategies on a few options suffer from scalability when the number of options increases, especially for many-option scenarios with significantly more options than agents. In this paper, we investigate the viabilities of discrete decision-making strategies with ranked voting (RV) and belief fusion (DBBS) decision mechanisms in many-option scenarios with large decision spaces compared to the number of agents. We test the investigated strategies on an expanded discrete collective estimation scenario where the decision space can be expanded using two factors: a higher number of environmental features and/or finer decision space discretization. We have used a continuous collective consensus forming strategy, linear consensus protocol (LCP), as a baseline. Our experimental results have shown that, although susceptible to environmental influences, discrete decision-making strategies can reliably outperform those of LCP in terms of error and convergence time at the tested sizes of decision space. We have also shown that the two factors that lead to the expansion of the decision space have different impacts on performances for both RV and DBBS strategies, due to differences in the correlations between the discrete options. When facing a higher number of features, both discrete strategies experience a smaller error and a significant increase in decision time, while a finer decision space discretization has a negative influence on all considered metrics.

Optimal control of an evolutionary variational–hemivariational inequality model with new results in numerical analysis and control of the dynamic frictional contact problem

In this paper, we are concerned with numerical analysis on optimal control of a dynamic contact model. Our work builds on some existing results of this model. We derive a weak formulation, and present an approximate problem for the weak formulation by discretization in space. We prove that the approximate problem has a unique solution, which converges to the unique solution of the original problem. The existence of optimal solutions to the control problem is established based on a known continuous dependence result. We consider the numerical solutions of the optimal control problem and prove a convergence result of numerical solutions.

On the product of almost discrete Grothendieck spaces

A topological space X is called almost discrete, if it has precisely one nonisolated point. In this paper, we get that for a countable product X=∏Xi of almost discrete spaces Xi the space Cp(X) of all continuous real-valued functions with the topology of pointwise convergence is a μ-space if, and only if, X is a weak q-space if, and only if, t(X)=ω if, and only if, X is functionally generated by the family of all its countable subspaces.This result makes it possible to solve Archangel'skii's problem on the product of Grothendieck spaces. It is proved that in the model of ZFC, obtained by adding one Cohen real, there are Grothendieck spaces X and Y such that X×Y is not weakly Grothendieck space. In (PFA): the product of any countable family almost discrete Grothendieck spaces is a Grothendieck space.

Joint magnetic resonance imaging artifacts and noise reduction on discrete shape space of images

Magnetic resonance (MR) images can be corrupted by artifacts and noise, potentially leading to misinterpretation of the images. In this paper, we propose a novel approach based on the discrete shape space of images (DSSI) to jointly reduce artifacts and noise in MR images. The proposed method restores MR images in multiple domains based on the distinct generation mechanisms of noise and artifacts. The images in multiple domains are analyzed in a non-Euclidean space. The DSSI is constructed as a Riemannian manifold to measure the intrinsic properties of images. Images are considered shapes from a geometric perspective, and the impact of similarity transformations (e.g., rotation, scaling, and translation) on image analysis is eliminated. The patch-based rank-ordered difference (PROD) detector is defined in k-space within the framework of DSSI to detect and remove sparse outliers that cause artifacts. In addition, a novel similarity function for images is defined using the DSSI and be used to design the improved filter. Finally, the convergence of the improved filter is theoretically analyzed, indicating that our method offers an effective estimator of the ideal image. The experimental results of various MR images demonstrate that the proposed approach outperforms classical and state-of-the-art methods for artifact correction and noise removal, both qualitatively and quantitatively.

Deep learning-aided topology design of metasurfaces for Rayleigh waves

Abstract Metasurfaces can effectively attenuate Rayleigh waves propagating in soil, safeguarding structures from ambient vibrations or earthquakes. However, there remains a lack of efficient approaches for designing metasurfaces that can isolate Rayleigh waves within desired frequency ranges under different site conditions. This study presents a deep learning (DL)-based topology optimization method for designing metasurfaces isolating Rayleigh waves in a target frequency range, which has potential applications for surface wave control. The proposed DL model employs a variational autoencoder to transform high-dimensional and discrete topologies into low-dimensional and continuous latent vectors, reducing the design difficulty. On this basis, the conditional tandem neural network is constructed to optimize latent vectors under different soil conditions, improving the design efficiency and verifying the universality of the method. The design reliability of the DL model is validated through 100 tests with determination coefficients more than 0.99. In addition, the generations of different metasurfaces for the same target are explored, providing designers with more choices. The insulation capabilities of the designed metasurfaces against Metro-induced Rayleigh waves are demonstrated from the time- and frequency-domain responses. The presented DL-aided design method provides a novel insight into the customization of metasurfaces for manipulating Rayleigh waves.

Optimizing large-scale hydrogen storage: A novel hybrid genetic algorithm approach for efficient pipeline network design

As hydrogen production technologies continue to advance, efficient and high-capacity hydrogen storage solutions become increasingly crucial. To address the complex optimization challenges in the design of large-scale hydrogen storage pipeline networks, the study introduces a mixed integer nonlinear optimization model inspired by hydrogen pipeline design optimization theory. The model not only focuses on optimizing network design parameters but also aims to minimize investment costs. To achieve this, the study proposes an innovative hybrid genetic algorithm that combines the Modified Feasible Directions Method (MFDM) and Genetic Algorithm Theory (GAT), specifically targeting the optimization of pipeline diameter's discrete distribution challenges. Comparative analyses with SUMT and GA algorithms demonstrate the superiority of the approach. In continuous space, the algorithm achieves a lower convergence cost of 232.4437 × 104 Yuan, while in discrete space, the cost further decreases to 212.1636 × 104 Yuan. Additionally, the study delves into the sensitivity of the HGA-MFDM algorithm to ambient temperature and wellhead temperature in hydrogen storage facilities. The analysis reveals that wellhead temperature has a more significant impact on pipeline network investment, whereas ambient temperature exhibits a more pronounced effect on iteration curves. The findings provide valuable insights for the precise adjustment and optimization of hydrogen storage pipeline networks in practical applications.

Artificial Protozoa Optimizer (APO): A novel bio-inspired metaheuristic algorithm for engineering optimization

This study proposes a novel artificial protozoa optimizer (APO) that is inspired by protozoa in nature. The APO mimics the survival mechanisms of protozoa by simulating their foraging, dormancy, and reproductive behaviors. The APO was mathematically modeled and implemented to perform the optimization processes of metaheuristic algorithms. The performance of the APO was verified via experimental simulations and compared with 32 state-of-the-art algorithms. Wilcoxon signed-rank test was performed for pairwise comparisons of the proposed APO with the state-of-the-art algorithms, and Friedman test was used for multiple comparisons. First, the APO was tested using 12 functions of the 2022 IEEE Congress on Evolutionary Computation benchmark. Considering practicality, the proposed APO was used to solve five popular engineering design problems in a continuous space with constraints. Moreover, the APO was applied to solve a multilevel image segmentation task in a discrete space with constraints. The experiments confirmed that the APO could provide highly competitive results for optimization problems. The source codes of Artificial Protozoa Optimizer are publicly available at https://seyedalimirjalili.com/projects and https://ww2.mathworks.cn/matlabcentral/fileexchange/162656-artificial-protozoa-optimizer.

Learning to Branch: Generalization Guarantees and Limits of Data-Independent Discretization

Tree search algorithms, such as branch-and-bound, are the most widely used tools for solving combinatorial and non-convex problems. For example, they are the foremost method for solving (mixed) integer programs and constraint satisfaction problems. Tree search algorithms come with a variety of tunable parameters that are notoriously challenging to tune by hand. A growing body of research has demonstrated the power of using a data-driven approach to automatically optimize the parameters of tree search algorithms. These techniques use a training set of integer programs sampled from an application-specific instance distribution to find a parameter setting that has strong average performance over the training set. However, with too few samples, a parameter setting may have strong average performance on the training set but poor expected performance on future integer programs from the same application. Our main contribution is to provide the first sample complexity guarantees for tree search parameter tuning. These guarantees bound the number of samples sufficient to ensure that the average performance of tree search over the samples nearly matches its future expected performance on the unknown instance distribution. In particular, the parameters we analyze weight scoring rules used for variable selection. Proving these guarantees is challenging because tree size is a volatile function of these parameters: we prove that, for any discretization (uniform or not) of the parameter space, there exists a distribution over integer programs such that every parameter setting in the discretization results in a tree with exponential expected size, yet there exist parameter settings between the discretized points that result in trees of constant size. In addition, we provide data-dependent guarantees that depend on the volatility of these tree-size functions: our guarantees improve if the tree-size functions can be well approximated by simpler functions. Finally, via experiments, we illustrate that learning an optimal weighting of scoring rules reduces tree size.

Forest fire spreading: A nonlinear stochastic model continuous in space and time

AbstractForest fire spreading is a complex phenomenon characterized by a stochastic behavior. Nowadays, the enormous quantity of georeferenced data and the availability of powerful techniques for their analysis can provide a very careful picture of forest fires opening the way to more realistic models. We propose a stochastic spreading model continuous in space and time that has the potentiality to use such data in their full power. The state of the forest fire is described by the subprobability densities of the green trees and of the trees on fire that can be estimated thanks to data coming from satellites and earth detectors. The fire dynamics is encoded into a kernel function that can take into account wind conditions, land slope, spotting phenomena, and so on, bringing to a system of integrodifferential equations whose solutions provide the evolution in time of the subprobability densities. That makes the model complementary to models based on cellular automata that furnish single instantiations of the stochastic phenomenon. Moreover, stochastic models based on cellular automata can be derived from the present model by space and time discretization. Existence and uniqueness of the solutions is proved by using Banach's fixed‐point theorem. The asymptotic behavior of the model is analyzed as well. By specifying a particular structure for the kernel, we obtain numerical simulations of the fire spreading under different conditions. For example, in the case of a forest fire evolving toward a river, the simulations show that the probability density of the trees on fire is different from zero beyond the river due to the spotting phenomenon. The kernel could be slightly modified to include firefighters interventions and weather changes.

Viscosity solutions approach to finite-horizon continuous-time Markov decision process

This paper investigates the optimal control problems for the finite-horizon continuous-time Markov decision processes with delay-dependent control policies. We develop compactification methods in decision processes and show that the existence of optimal policies. Subsequently, through the dynamic programming principle of the delay-dependent control policies, the differential-difference Hamilton-Jacobi-Bellman (HJB) equation in the setting of discrete space is established. Under certain conditions, we give the comparison principle and further prove that the value function is the unique viscosity solution to this HJB equation. Based on this, we show that among the class of delay-dependent control policies, there is an optimal one which is Markovian.