Fictitious Points Research Articles

A new spline method on graded mesh for fourth-order time-dependent PDEs: application to Kuramoto-Sivashinsky and extended Fisher-Kolmogorov equations

In this research, we introduce a two-tier non-polynomial spline approach with graded mesh discretization for addressing fourth-order time-dependent partial differential equations, which find applications in various physical scenarios like the nonlinear Kuramoto-Sivashinsky equation and extended Fisher-Kolmogorov equation. Our method involves considering three spatial points at each time step for scheme development, achieving spatial accuracy of three and temporal accuracy of two. Notably, our approach offers the advantage of directly tackling singular biharmonic problems without the need for discretizing boundary conditions or incorporating fictitious points. It demonstrates unconditional stability when tested on fourth-order problems and outperforms previous methods, as evidenced by comparative analyses. Additionally, we present 2D and 3D plots illustrating numerical solutions for different benchmark scenarios, including the depiction of the chaotic nature of the Kuramoto-Sivashinsky equation under Gaussian initial conditions at various time levels.

Autonomous data detection and inspection with a fleet of UAVs

Consider an area of interest A, where a set of nsites lie. Two kinds of information can be captured from each site: light and heavy information. A fleet of m homogeneous UAVs, each one equipped with a battery B, is available at a common depot, where the flight mission of each UAV starts and finishes. The problem we consider focuses on a single flight of the fleet of UAVs and aims at collecting their light information from all sites (that can be retrieved, not necessarily passing over each site, but simply “close” to it). At the same time, the fleet will have to select a limited number of sites from which to collect their heavy information. Flying among sites and acquiring information from them (both light and heavy) has a battery cost. On the other hand, a profit is associated with the action of acquiring heavy information from a site. We refer to the extraction of light and heavy information from a site as to weakly or strongly cover the site. The aim of the problem consists of retrieving light information from all sites while maximizing the overall profit, keeping the battery consumption of each UAV within B. In this paper, we model this real-life situation as a new combinatorial optimization problem that we call m3DIP, for which we provide a mixed integer programming model. Given the high degree of complexity of the problem, in this way we are not able to provide a solution in a reasonable time. To address larger instances we propose a matheuristic in which we exploit a path-based algorithm filled with only a subset of feasible cycles (paths) provided by different heuristics. The output indicates which path to select and the set of nodes to be strongly and weakly covered by each trip. We compare our matheuristic with the results obtained by every single heuristic on a large set of instances, showing that the matheuristic strongly outperforms them. An interesting insight is that even paths provided by a heuristic with very bad performances can be useful if combined with paths provided by other heuristics and if the coverage decisions are reoptimized by the matheuristic. We also show the benefit of adding fictitious additional points that UAVs can visit to weakly cover a subset of sites, without actually visiting none of them.

Control of a Wheeled Robot on a Plane with Obstacles

The work proposes an algorithm for controlling a wheeled robot in an environment with static and dynamic obstacles. A wheeled robot (WR) consists of a platform, two wheels with a differential drive and one roller, which is used solely for the stability of the structure and does not affect the dynamics of the system. The robot’s motion algorithm assumes its movement from the starting point to the final point in an environment with obstacles. The robot’s motion program is specified through servo-constraints, and the algorithm that implements the motion program is based on the potential field method. In the case of a dynamic obstacle, a repulsive field of a shape elongated in the direction of movement of the obstacle is constructed, allowing the robot to safely go around it. It is possible to change the geometric dimensions of the field using the entered numerical parameters. An algorithm for overcoming a potential hole by a robot is presented, according to which the robot is taken out of the potential hole and directed to a global goal by an introduced fictitious point located outside the critical region (local minimum region) and having its own attractive field. The paper presents the results of numerical simulation of the robot’s movement both in an environment with static and dynamic obstacles, as well as the results of a numerical experiment with overcoming the region of a potential well. Graphs of the required mechanical parameters are presented. The results of numerical simulation confirm the effectiveness of the proposed algorithms.

Fictitious Point Technique Based on Finite-Difference Method for 2.5D Direct-Current Resistivity Forward Problem

With the widespread application of the direct-current resistivity method, searching for accurate and fast-forward algorithms has become the focus of research for geophysicists and engineers. Three-dimensional forward modeling can be the best way to identify geo-electrical anomalies but are hampered by computational limitations because of the large amount of data. A practical compromise, or even alternative, is represented by 2.5D modeling characterized using a 3D source in a 2D medium. Thus, we develop a 2.5D direct-current resistivity forward modeling algorithm. The algorithm incorporates the finite-difference approximation and fictitious point technique that can improve the efficiency and accuracy of numerical simulation. Firstly, from the boundary value problem of the electric potential generated by the point source, the discrete expressions of the governing equation are derived from the finite-difference approach. The numerical solutions of the discrete electric potential are calculated after the approximate treatment of the boundary conditions with a finite-difference method based on a fictitious point scheme. Secondly, through the simulation of a homogeneous half-space model and a one-dimensional model, and compared with the analytical results, the correctness and stability of the finite-difference forward algorithm are verified. Lastly, through the numerical simulation for a two-dimensional model, 2.5D direct-current sounding responses are summarized, which can provide a qualitative interpretation of field data.

Coupled total- and semi-Lagrangian peridynamics for modelling fluid-driven fracturing in solids

This paper presents a novel computational approach for modelling fluid-driven fracturing in quasi-brittle solids using peridynamics. The approach leverages a rigorous coupling of the total- and semi-Lagrangian formulations of peridynamics. Specifically, the total-Lagrangian formulation, whch is based on classical peridynamics theory used for modelling fractures in solids, is combined with the semi-Lagrangian formulation to solve the Navier-Stokes equation using a non-local differential operator for weakly compressible fluid flow. The proposed approach offers a unified peridynamics-based framework that enables simulations of a wide range of fluid-driven fracturing problems in solids. The framework can model the solid as either an ordinary or non-ordinary material and quantify the fluid with different equations of state. To prevent unphysical inter-penetration, a fluid-solid interaction scheme that assumes fictitious fluid points in the solid domain is proposed to quantify the forces at the fluid-solid interface. The proposed computational approach is validated through simulations of several classic problems, including the dam break and the Kristianovich-Geertsma-de Klerk (KGD) problem. The predictive capability of the proposed approach is further demonstrated by numerical examples of fractures induced by fluid injection in sandstone, which reasonably capture the fracture patterns compared to experimental results. The presented approach offers an alternative to explicit modelling of fluid-driven fracturing processes and may find wide and useful applications to a variety of industrial and geophysical processes.

A Novel ANN-Based Radial Basis Function Collocation Method for Solving Elliptic Boundary Value Problems

Elliptic boundary value problems (BVPs) are widely used in various scientific and engineering disciplines that involve finding solutions to elliptic partial differential equations subject to certain boundary conditions. This article introduces a novel approach for solving elliptic BVPs using an artificial neural network (ANN)-based radial basis function (RBF) collocation method. In this study, the backpropagation neural network is employed, enabling learning from training data and enhancing accuracy. The training data consist of given boundary data from exact solutions and the radial distances between exterior fictitious sources and boundary points, which are used to construct RBFs, such as multiquadric and inverse multiquadric RBFs. The distinctive feature of this approach is that it avoids the discretization of the governing equation of elliptic BVPs. Consequently, the proposed ANN-based RBF collocation method offers simplicity in solving elliptic BVPs with only given boundary data and RBFs. To validate the model, it is applied to solve two- and three-dimensional elliptic BVPs. The results of the study highlight the effectiveness and efficiency of the proposed method, demonstrating its capability to deliver accurate solutions with minimal data input for solving elliptic BVPs while relying solely on given boundary data and RBFs.

An improved method for predicting the shear strength of corroded reinforced concrete beams with experimental and numerical validation

Shear failure of reinforced concrete (RC) beams is a brittle failure without giving sufficient warning. Steel bar corrosion is one of the main deterioration phenomena for concrete structures in marine environments. However, the effect of steel bar corrosion on the shear-transfer mechanism of RC beams is not yet fully understood, and existing findings are insufficient for reasoned and accurate prediction of shear strength. Thus, the primary objective of this study was to propose and validate a revised fictitious-support-point method for better predicting the shear strength of corroded RC beams. The revised method determines the fictitious support points based on a more transparent mechanical model than previously. Finite element simulation confirms the validity of the revised method from the load-bearing mechanism point of perspective. Validation with copious previously-published test data show that the revised method predicts the shear strength of corroded RC beams more accurately and reasonably than the existing method proposed by other researchers. This study contributes to effective and reasonable maintenance of corroded RC structures.

Fictitious finite integration method for solving high order partial differential equations

In this paper, we combine the finite integration method (FIM) with the technique of fictitious points to solve high order partial differential equations (HOPDEs). Two type of fictitious finite integration method (FFIM) are proposed and analyzed. The first type FFIM (FFIM-I) constructs the first order integration matrices by shifted Chebyshev polynomials so that the analytical high order integration matrices can be recursively obtained. Compared with the classical FIM, whose high order integration matrices are approximated directly through low order integration matrices, this FFIM-I leads to higher accuracy. Based on FFIM-I, the second type FFIM (FFIM-II) is proposed in which the fictitious points are located freely either inside or outside the problem domain. The FFIM-II can further improve the performance of FFIM-I in solving HOPDEs. The advantage of this fictitious finite integration method will be demonstrated by making comparisons among FIM, FFIM-I and FFIM-II through several 2D and 3D numerical experiments.

Python Routine for an Easy Visualization of the Influence of Supply Network Characteristics on the Hydraulic Behavior of a Small Closed Loop

A routine coded in Python aiming at a quick detection of the hydraulic behavior of a closed supply network as a function of the variation of the input data is presented. Such variables are lengths and diameters of the sections, roughness coefficients of the pipes, terrain elevations, flow distributed in the section and residual flow, and pressure at the upstream node. After a review of the literature on the subject of hydraulic supply models, the process of the fictitious sectioning point is used, so that the closed circuit network can have its behavior assimilated as that of branched network sections, and its calculation be performed as such. This arrangement is achieved by matching head losses between sections. In order to demonstrate the functionality of the routine, 12 simulations are exposed, 06 in each particular sectioning condition, with different input data and respective influences on the network operating conditions, notably, the positioning of the fictitious sectioning point and the contribution of each stretch to the residual flow node.

Preconscious Ocular Aesthetics: Returning to a Fictitious Point in Photography’s History

Preconscious Ocular Aesthetics: Returning to a Fictitious Point in Photography’s History

On the determination of locating the source points of the MFS using effective condition number

The method of fundamental solutions (MFS) is a highly accurate numerical method for solving homogeneous solutions subject to a properly selection of the sources location. In this work, we choose the effective condition number as a tool for the determination of a good source location of the MFS that leads to highly accurate results with low computational cost. Three approaches for the location of the fictitious source points are considered. An efficient algorithm for the evaluation of the effective condition number is proposed. We also compare the proposed method with the well-known LOOCV (leave-one-out cross validation) and show the advantages and shortcomings of each approach. Five numerical examples with different geometric shapes of the domain for both harmonic and non-harmonic boundary conditions in 2D and 3D are presented.

Manifold learning for global active noise control: Sensor interpolation

To achieve global control, unmeasured frequency responses (FRs) of a control source and preselected control points are required. In this paper, sensor interpolation (SI) approaches based on plane wave decomposition (PWD), kernel ridge regression, deep neural network (DNN) are presented. A broadened control region encompassing a large number of measured and fictitious control points can be obtained using the SI techniques. PWD is a two-stage procedure that begins with trimming down plane wave components, followed by an extraction of the associated complex amplitudes. In kernel ridge regression, a reproducing kernel in the Hilbert space (RKHS) is utilized to interpolate continuous relative transfer functions. The DNN-based SI approach is inspired by the kernel method. Convolutional neural networks and the like are exploited to interpolate the FRs at the fictitious points. Simulations are conducted for a thirty-microphone uniform linear array. From the results, an excellent fit between the FRs at the fictitious points and the ground truth generated by the image source method can be obtained below the spatial aliasing frequency. The efficacy of the interpolated FRs in relation to global control is demonstrated via an example of feedforward active noise control using a linear loudspeaker array.

The infrared behavior of tame two-field cosmological models

We study the first order infared behavior of tame hyperbolizable two-field cosmological models, defined as those classical two-field models whose scalar manifold is a connected, oriented and topologically finite hyperbolizable Riemann surface (Σ,G) and whose scalar potential Φ admits a positive and Morse extension to the end compactification of Σ. We achieve this by determining the universal forms of the asymptotic gradient flow of the classical effective potential V with respect to the uniformizing metric G near all interior critical points and ends of Σ, finding that some of the latter act like fictitious but exotic stationary points of the gradient flow. We also compare these results with numerical studies of cosmological orbits. For critical cusp ends, we find that cosmological curves have transient quasiperiodic behavior but are eventually attracted or repelled by the cusp along principal geodesic orbits determined by the extended effective potential. This behavior is approximated in the infrared by that of gradient flow curves near the cusp.

Feedforward active noise global control using a linearly constrained beamforming approach

Global control of noise in a three-dimensional space is generally difficult, given a limited number of discrete sensors. This paper presents a feedforward multichannel active noise control (ANC) approach to overcome this difficulty. In light of the model-matching principle, an underdetermined multichannel inverse filtering (UMIF) system is formulated. Infinite number of solutions exist to yield zero residual noise at the error microphone positions. By incorporating the UMIF system as a design constraint, a multichannel feedforward controller is formulated using the linearly constrained minimum variance (LCMV) approach in which a cost function is introduced to minimize the residual noise energy at a large number of preselected fictitious control points. To implement the ANC system in reverberant environments, a novel sensor interpolation technique based on plane wave decomposition is developed to find the frequency responses between secondary loudspeakers and the fictitious control points. Only a limited number of room frequency response measurements are required in the use of these techniques. Simulations and experiments are undertaken using a uniform linear loudspeaker array to validate the proposed ANC system. The results have demonstrated that the proposed approach yielded substantial noise reduction in a widened and connected control area.

A MODEL OF WELDING THERMAL PROCESSES WITH A LIMITED NUMBER OF FICTITIOUS POINT SOURCES

Analytical methods of calculation of thermal processes at welding enable to estimate operatively the influence of welding modes on parameters of a temperature field and to predict structural changes in a zone of thermal influence. In the mid-twentieth century it was suggested to calculate the heat processes in welded products with parallel boundary planes by summing up the temperature fields of the main and infinite number of fictitious point sources using thereflection method. However, due to computational difficulties, this method could not be brought to a state of wide practical application. Furthermore, since the schematization of heated bodies is conditional, the choice of the body scheme (plate, flat layer, semi-infinite body) and therefore the thermal calculation model is performed intuitively based on the accumulated experience. This uncertainty inevitably leads to discrepancy between the calculated and actual parameters of the thermal welding process. The purpose of this work is to adapt a model of thermal processes with a limited number of fictitious fast moving point sources to the calculation of thermal welding processes. The proposed model, solvable in Mathcad software, contains the equations of temperature field and temperature rate of change with account of the number
 of fictitious heat sources involved in the calculation. This model allows a reasonable limitation of the number of fictitious heat sources for providing the required accuracy of calculations, since as the fictitious heat sources move away from the product and their influence on the product heating decreases. In most cases for practical calculations 2 pairs of fictitious sources are enough for welding of 2 cm thick steel products, of 0.5 cm thick products ‒ 5 pairs, of 0.2 cm thick products ‒ 12 pairs. Comparison of the calculated values of the proposed model with actual temperature field measurements and known effects of cooling rate in the plate, plane layer and semi-infinite body within the accepted limits have confirmed its adequacy. The presented comparative results demonstrate versatility of the proposed model with a limited number of fictitious fast moving point sources, which can be applied sufficiently and successfully for analysis of the thermal welding processes of real industrial products of arbitrary thickness without discriminating into plate, plane layer and semi-infinite body models.

Effective condition number for the selection of the RBF shape parameter with the fictitious point method

Based on the Uncertainty Principle of radial basis functions (RBFs), it is known that the condition number and the error cannot be both kept small at the same time. In contrast to the traditional condition number, the effective condition number provides a much better estimation of the actual condition number of the resultant matrix system. In this paper, motivated by the Uncertainty Principle of RBFs, we propose to apply the effective condition number as a numerical tool to determine a reasonably good shape parameter value in the context of the Kansa method coupled with the fictitious point method. Six examples for second and fourth order partial differential equations in 2D and 3D are presented to demonstrate the effectiveness of the proposed method.

Application of generalized equations of finite difference method to computation of bent isotropic stretched and/or compressed plates of variable stiffness under elastic foundation

Abstract The computation of bent isotropic plates, stretched and/or compressed, is a topic widely explored in the literature from both experimental and numerical point of view. We expose in this work an application of the generalized equations of Finite difference method to that topic. The strength of the proposed method is the ability to reconstruct the approximate solution with respect of eventual discontinuities involved in the investigated function as well as its first and second derivatives, including the right-hand side of the equilibrium equation. It is worth mentioning that by opposition to finite element methods our method needs neither fictitious points nor a special condensation of grid. Well-known benchmarks are used in this work to illustrate the efficiency of our numerical and the high accuracy of calculation as well. A comparison of our results with those available in the literature also shows good agreement.

High resolution operator compact implicit half-step approximation for 3D quasi-linear hyperbolic equations and ADI method for 3D telegraphic equation on an irrational domain

We propose a novel three-level implicit half-step method of order two in time and four in space using unequal mesh for the solution of three-space dimensional quasi-linear hyperbolic equations on an irrational domain. The proposed method is directly applicable to 3D hyperbolic equation with singular coefficients, which is the key charisma of our scheme and we do not need any fictitious points for computation. The stability analysis of the model problem for unequal mesh has been discussed and it has been shown that the proposed method for Telegraphic equation is unconditionally stable. Alternating direction implicit (ADI) method by the help of a tri-diagonal solver is used to solve 3D linear schemes in three sweeps. The proposed method is scrutinized on several physical problems on dissimilar domain to exhibit the accuracy and effectiveness of the proposed method.

A 4-node quadrilateral element with center-point based discrete shear gap (CP-DSG4)

This work aims at presenting a novel four-node quadrilateral element, which is enhanced by integrating with discrete shear gap (DSG), for analysis of Reissner-Mindlin plates. In contrast to previous studies that are mainly based on three-node triangular elements, here we, for the first time, extend the DSG to four-node quadrilateral elements. We further integrate the fictitious point located at the center of element into the present formulation to eliminate the so-called anisotropy, leading to a simplified and efficient calculation of DSG, and that enhancement results in a novel approach named as "four-node quadrilateral element with center-point based discrete shear gap - CP-DSG4". The accuracy and efficiency of the CP-DSG4 are demonstrated through our numerical experiment, and its computed results are validated against those derived from the three-node triangular element using DSG, and other existing reference solutions.

Market Power in Agricultural Land Markets: Concepts and Empirical Challenges

This paper provides review about challenges and opportunities to assess and quantify market power in agricultural land markets. Measuring land market power is challenging because the characteristics of this production factor hinder the direct application of familiar concepts from commodity markets. Immobility, fixed availability, and large heterogeneity of land and potential users contradict assumptions of fictitious point market for homogeneous goods. Moreover, the use of concentration indicators for policy assessments is hampered by two problems. First, defining the relevant regional size of the market is challenging and concentration indicators are not robust with regard to market size and number of actors. Second, high concentration of land ownership or land operation may point at potential market power, but it may also be the result of an efficient allocation of land due to structural change in agriculture. The aforementioned challenges are illustrated with a case study for the Federal State of Brandenburg in Germany. Using available data for land sales, a regression analysis reveals a negative relationship between land use concentration and farmland prices. This result can be interpreted as an indication of market power on the buyer side in agricultural land markets. However, it is hardly possible to translate this finding into recommendations for land market regulations because the evaluation of the potential misuse of dominant positions in land markets requires a case-specific analysis. Providing evidence for the exertion of market power in land markets is extremely complex and deserves further attention from researchers and politicians.