Semarang is located in the northern part of Java Island where two tectonic plates meet, making it seismically active and causing the formation of faults. Kaligarang river valley is estimated to be an active fault. The research used FHD and SVD derivative methods, and FFD as a gravity continuation method in the Semarang City area to determine the structure of the Kaligarang fault based on GGMplus satellite data. The derivative analysis was obtained from the residual anomaly. CBA contour map is performed to obtain regional and residual anomaly contour map with moving average method. Furthermore, 2D forward modeling was carried out to describe the layers of geological structures by determining the depth using spectrum analysis. The subsurface structure in the study area consists of 2 types of formations in each of the Line 1, 2 and 3 models, namely the Kaligetas Formation which has an average density of 1.92-2.40 gr/cm 3 . Then in the second layer is the Kalibeng Formation with an average density range of 2.00-2.21 gr/cm 3 . Meanwhile, the line 4 modeling is dominated by the Kerek Formation with an average density of 2.21-2.55 gr/cm 3 . The fault structure boundaries identified based on FHD analysis are located around coordinates 9215536.4 UTM Y on Line 1, 9217402.4 UTM Y on Line 2, 9220103.6 UTM Y on Line 3, and 9221429.6 UTM Y on Line 4. Based on the SVD analysis, the type of faults that can be identified along Kaligarang River is a thrust fault.

Abstract This study introduced a combined structure of a genetic algorithm and continuation method to solve a nonlinear rotordynamic problem known as multiple coexisting solutions. Here, the chromosomes consist of dynamic state elements of a rotor-bearing system, and the genetic objective is the completion of periodicity of the responses in the nonlinear system during the consecutive generations. For the pilot implementation, two exemplary nonlinear models, such as Duffing and Van der Pol, were tested to ensure the capability and compared with the shooting method. Then, a Jeffcott-type rotor supported on two identical six-pad tilting pad journal bearings was employed as a rotor-bearing application. The bearing has rocker-back type pivots, and the fluid film force on each pad is calculated using the finite element method. To find a multiple coexisting solution, 200 initial conditions composed of the dynamic states of the journal and pad were randomly created as the first generation. The optimal results were selected by evaluating the fitness for each generation, and the offspring were created through the processes of crossover, mutation, and elitism up to 15 generations. Then Newton-Raphson method takes a role to refine the final candidates to examine whether they meet the convergent criteria. The branches of the solutions were extended using the arc-length continuation method to obtain a broader perspective of nonlinear behaviors. As a result, multiple coexisting responses and various bifurcation events, such as periodic doubling, saddle-node, could be discovered by the developed combined numerical scheme.

Abstract This paper addresses the problem of recovering the spatial profile of the source in the complex Ginzburg–Landau equation (CGLE) from regional observation data at fixed times. We establish two types of sufficient measurements for the unique solvability of the inverse problem. The first is to determine the source term by using whole data at one fixed instant. Conditional stability is established by using the eigenfunction expansion argument. Next, using the analytic continuation method, both uniqueness and a stability estimate for recovering the unknown source can be established from local data at two instants. Finally, to effectively handle the complex-valued solutions of the CGLE, we propose a novel complex physics-informed neural networks (C-PINNs) framework. This approach designs complex-valued layers that inherently respect the complex structure of the governing equation, overcoming limitations of standard real-valued PINNs for such dissipative systems. Numerical experiments demonstrate the accuracy and efficiency of our C-PINNs algorithm in recovering the source term.

ABSTRACT Time‐varying properties of mechanical systems can lead to parametrically excited oscillations. An example is the time‐varying stiffness of gear teeth meshing in the analysis of gearbox dynamics. A mechanical model can be used to represent the rotational motion of the gear pair. The torsional vibrations occurring around the rigid body rotation can be simplified to a single degree of freedom oscillator. In addition to an assessment of the amplitudes of vibrations, an assessment of the system stability is a crucial aspect to consider. For stability analysis of such a mechanical system, stability charts, known as Ince‐Strutt diagrams, are utilized. These graphical representations of stability regions can be created for various combinations of system parameters to predict whether the system will be stable or unstable. In order to gain an understanding of the stability behavior, a range of methodologies can be employed. The Floquet 's theory serves as the basis for the methodology presented in this study. Stability charts are created by solving the differential equation for discrete parameter values and their combinations. Stability statements are then mapped into the Ince‐Strutt diagram. However, the stability boundaries are of primary interest. This contribution introduces an approach to determine the boundaries between stable and unstable regions without the need to calculate parameter values within the region of interest. Expanding the stability analysis to include a continuation method should make this feasible. It will be shown that the proposed approach is suitable to generate stability charts for the analyzed parametric oscillator by tracing stability boundaries.

Numerical homotopy continuation methods are known to be accurate and fast for obtaining roots of univariate polynomials with random coefficients. Due to a result of Kac (1943), which was extended by Edelman and Kostlan (1995), we know that the roots of such polynomials tend to be uniformly distributed on the unit circle, and due to the low condition numbers of such roots, offer a "best case" scenario for testing numerical root-finding algorithms. This paper considers the accuracy and computation cost of homotopy methods of average case polynomials such as the low degree Mandelbrot polynomials, and polynomials generated from random roots. For a worst case polynomial, we look at the Wilkinson polynomial with all positive roots. We take a novel approach in studying both numerical pseudozeros of the target polynomial, and the exact pseudozeros given by the homotopy. We confirm the practitioner's expectation that accuracy of high-speed homotopy methods are highly dependent on how well the start system is scaled to fit the target roots. Thus, the so-called Bézout start system used to find roots on the unit circle is nearly ideal. We show how to adapt these insights to work with other polynomials, including changing from the monomial basis to the Lagrange basis.

This scientific publication presents an interesting applied problem of numerical research into the optimal design of a spatial thin-walled structure under thermo-mechanical loading. Optimal design of load-bearing structures can be divided into four main types: parametric, topological, shape optimization, and creation of composite materials for a specific task. There may be combinations of two or more types of optimization on a single object under study. This scientific publication considers two types of optimal design on a single object under study. A minimal surface shell is a thin-walled spatial structure with a given contour and a certain specified height, which was subsequently constructed using the parameter continuation method. The parameter extension method makes it possible to construct the optimal shape of the future minimal surface shell, which minimizes the internal forces, which in turn reduce the Mises stress, which is decomposed into normal and tangential stresses, leading to a reduction in the thickness of the shell. Optimal design of load-bearing structures into four main types: parametric, topological, shape optimization, creation of composite materials for a specific task. There may be combinations of two or more types of optimization on a single object under study. Multi-criteria parametric optimization of the minimum surface shell is performed in an automated mode in the Femap with Nastran software package and connected proprietary software, which was developed for a specific area of scientific and technical activity within the framework of applied research. A theoretical formulation of the relationship between the loss of stability of the minimum surface shell on a circular plane is presented, taking into account geometric nonlinearity under thermo-mechanical loading. After optimization calculation based on Figures 2-11, the first form of stability loss is λ=1, which corresponds to the minimum limit before stability loss. Graph 13 shows the change in target functions. We managed to reduce the weight by 8.8 tons, which is 23%, while the coefficient λ decreased from 5.36 to 1.01, which is actually 6 times less. This calculation allows for the minimum thickness of the shell of the minimum surface before loss of stability.

We introduce a mathematical model to study tumor–immune interactions and analyze the effects of strong and weak Allee dynamics to capture the cooperative behavior inherent in tumor growth. In this work, we observe that the presence of Allee dynamics delays tumor proliferation at low cell densities, significantly influencing the critical thresholds that regulate tumor cell cooperation. We examine key dynamical behaviors through bifurcation analysis, including saddle-node, Hopf, Bautin, and Bogdanov–Takens (BT) bifurcations, using detailed analytical expressions and numerical continuation methods. Our model demonstrates that a strong Allee effect raises the critical threshold for tumor cell cooperation, leading to greater tumor control, as a higher antigenicity value is required to maintain equilibrium thresholds. In contrast, a weak Allee effect has a less pronounced impact on the model compared to a strong Allee effect. We also explore the case where the weak Allee effect corresponds to the parameter representing the half-saturation cancer clearance level, revealing a key dynamic behavior of the system under this condition. In this case, we prove the existence of BT bifurcation of co-dimension-three-type cusp. These findings could provide valuable insights into the transitions between elimination, equilibrium, and escape phases in the cancer immunoediting process. Moreover, our results could serve as a valuable tool for precisely calibrating these effects in the development of more effective therapeutic strategies.

Modeling the stochastic reaction dynamics is a significant task to explain the modern measurements of non-equilibrium processes at mesoscopic scales. Marcus's transition-state theory describes the reaction rate of a single-step reaction event, but how it can enlighten a multi-step stochastic reaction process in a continuous chemical-state space remains elusive. In this paper, we develop a stochastic Marcus state model with continuation methods for different reaction systems. The time-resolved evolutions of the probability density functions are expressed via Fokker-Planck equations where the drift and diffusion coefficients are determined by the free-energy functions and reorganization energy. In a system with infinitesimal-reaction transitions, the model allows a scale-invariant transform that preserves its generic form, and the equation of motion describes the over-damped Langevin dynamics space that follows the fluctuation-dissipation theorem in the chemical-state. We also prove that the Onsager reciprocal relation can be retrieved as long as the free energy obeys Schwarz's theorem, which reveals its generality in classical closed near-equilibrium systems.

ABSTRACT Downward continuation of gravity data is a critical challenge in the practical applications of airborne gravity, such as global geopotential modeling and geophysical interpretation. This study introduces a machine learning algorithm to address this classical problem in physical geodesy. We first constructed a U-Net-based neural network, termed DWC-Unet, specifically designed for gravity anomaly downward continuation. The model was trained using gravity anomalies simulated by the prior global gravity model. Subsequently, the trained DWC-Unet was used on airborne gravity anomalies to calculate the downward continued gravity anomalies at an arbitrary altitude surface. We applied the proposed method to the airborne gravity data in Colorado, Iowa, and regions in the southeast coast of the United States. The experiment results confirm that the DWC-Unet method shows better performance than the analytic continuation algorithm. The standard deviation of downward continuation errors reached 4.39 mGal in Colorado and approximately 3 mGal in other regions when compared to terrestrial gravity data. This study demonstrates the good performance of applying machine learning to the calculation of gravity data continuation, serving as a reference for the application of artificial intelligence in the field of gravity data processing.

Nonlinear vibration analysis of circular multilayer graphene-based NEMS sensors using harmonic balance and pseudo-arclength continuation methods

ACTest: A testing toolkit for analytic continuation methods and codes

Optimal design plays an important role in the approach to and formation of building structures. This stage of structural calculation has been little studied and is virtually unused by practicing engineers in the field of mechanical resistance and stability. The active development of optimal design began in the mid-20th century with the development of computers. The first works were performed on proprietary software in the Fortran language. Over more than half a century, computers have been modernized into PCs, and software complexes such as Femap and Ansys have been created, which include sections on optimal design. For the optimal design of fairly unique structures, it is not enough to have a basic functional set in these calculation complexes, so for interesting target functions such as stability and weight, it is necessary to add this part to the optimization functionality. The object of study is thin-walled shells of minimal surfaces. The essence of these spatial structures lies in the uniqueness of their surface, which has been optimized using the parameter continuation method based on the defined contour and height of the future shell of the minimal surface. After integration, the optimal shape for the future structure is derived, which has a minimum area and minimum internal forces, which are compensated by the geometric shape of the thin-walled shell of the minimal surface. Optimal design in construction and applied mechanics is divided into four types. The first type is shape optimization. The second type is parametric optimization. The third type is topological optimization. The fourth type is optimization of physical and mechanical characteristics. To study multi-criteria parametric optimization of a minimum surface shell, taking into account geometric nonlinearity, a special additional optimizer module created by the authors is used, which is linked to the Femap with Nastran calculation complex [12]. This scientific article reveals the essence of four types of optimal design and an approach to the optimal design scheme. A numerical study of multi-criteria parametric optimization of the stability and weight of a minimal surface shell with a square plan consisting of two straight lines and two semicircles was performed. The thickness of the shell after optimization calculation ranges from 40 to 1 mm under thermo-mechanical loading. A graph of the target functions of weight and stability loss coefficient λ has been constructed. The weight on the target function graph decreased from 31 tons to 25.5 tons, which is 17.75% in percentage terms, while the stability loss coefficient decreased from 2.12 to 1.08, which means the maximum use of structural material in this optimization calculation.

The solution of the Dyson equation for the small-gap systems can be plagued by large non-converging iterations. In addition to the convergence issues, due to a high non-linearity, the Dyson equation may have multiple solutions. We apply the homotopy continuation approach to control the behavior of iterations. We used the homotopy continuation to locate multiple fully self-consistent GW solutions for the NdNiO2 solid and to establish the corresponding Hartree-Fock limits. Some of the solutions found are qualitatively new and help to understand the nature of electron correlation in this material. We show that there are multiple low-energy charge-transfer solutions leading to the formation of charge-density waves. Our results qualitatively agree with the experimental conductivity measurements. To rationalize the structure of solutions, we compare the k-point occupations and generalize the concept of natural difference orbitals for correlated periodic solids.

This paper considers cylindrical composite sandwich shells that consist of two thin outer layers and one thick honeycomb filler. The outer layers are made of a composite orthotropic material, for example, carbon-reinforced plastic. The honeycomb filler is made of an orthotropic plastic, for example, PLA, using additive technologies. To obtain a mathematical model of nonlinear structural vibrations, the honeycomb core is homogenized into a uniform orthotropic solid layer using a finite-element simulation in ANSYS. Parametric vibrations of the cylindrical shell under the action of a longitudinal load are considered. Each layer of the structure is described by a higher-order shear theory, which uses five generalized displacements (three displacements projections onto the axes and two rotation angles of the middle surface normal). The displacement projections are continuous at the layer interfaces. The assumed mode method is used to obtain a system of nonlinear ordinary differential equations in the generalized coordinates. The method uses the kinetic and the potential energy of the structure. The shooting technique and the parameter continuation method are used jointly to analyze nonlinear vibrations, their stability and bifurcations. The multipliers are calculated to estimate the vibration stability. The stability and bifurcations of periodic oscillations are shown in the frequency responses, which describe the structure dynamics in the principal parametric resonances. As shown by the numerical analysis, standing waves are observed in the cylindrical shell. As a result of the bifurcations, the standing waves are transformed into travelling ones, which are described by a loop in the frequency response.

We consider the Lauricella hypergeometric function F D ( N ) ( a 1 , … , a N ; b , c ; z 1 , … , z N ) and the system of partial differential equations that it satisfies for logarithmic (resonant) relations between the parameters a j , b and c. The paper proposes a method for analytical continuation of the Lauricella function for such relationships between the parameters and studies the case of large in modulo variables. The method is based on analytic continuation formulas previously found by the author and new regularization formulas for sets of basic solutions of the mentioned system of equations in the case of resonant sets of parameters. We give an example of constructing analytical continuation formulas and basis sets in the solution space of the specified Lauricella system of equations. The found basis functions are the N-dimensional analogue of Kummer's solutions, known in the theory of the classical Gauss hypergeometric equation. Canonical solutions for the parameter case under consideration are written in the form of generalized hypergeometric series, including not only powers, but also logarithms of variables. In this regard, we refer to such cases as logarithmic ones.

Analysis of the stability of frames composed of thin-walled beams with open cross-section using a High Order Continuation Method

The method of fundamental solutions and a high order continuation for nonlinear bioheat transfer problems during hyperthermia treatment.

The analysis of Ordinary Differential Equation (ODE) dynamical systems, particularly in applied disciplines such as mathematical biology and neuroscience, often requires flexible computational workflows tailored to model-specific questions. XPPAUT is a widely used tool combining numerical integration and continuation methods. Various XPPAUT toolboxes have emerged to customize analyses; however, they typically rely on summary ‘.dat’ files and cannot parse the more informative ‘.auto’ files, which contain detailed continuation data, e.g. periodic orbits and boundary-value problem solutions. We present XPPLORE, a user-friendly and structured MATLAB toolbox overcoming this limitation through the handling of ‘.auto’ files. This free software enables post-processing of continuation results, facilitates analyses such as manifold reconstruction and averaging, and it supports the creation of high-quality visualizations suitable for scientific publications. This paper introduces the core data structures of XPPLORE and demonstrates the software’s exploration capabilities, highlighting its value as a customizable and accessible extension for researchers working with ODE-based dynamical systems.

First order shape optimization methods, in general, require a large number of iterations until they reach a locally optimal design. While higher order methods can significantly reduce the number of iterations, they exhibit only local convergence properties, necessitating a sufficiently close initial guess. In this work, we present an unregularized shape-Newton method and combine shape optimization with homotopy (or continuation) methods in order to allow for the use of higher order methods even if the initial design is far from a solution. The idea of homotopy methods is to continuously connect the problem of interest with a simpler problem and to follow the corresponding solution path by a predictor-corrector scheme. We use a shape-Newton method as a corrector and arbitrary order shape derivatives for the predictor. Moreover, we apply homotopy methods also to the case of multi-objective shape optimization to efficiently obtain well-distributed points on a Pareto front. Finally, our results are substantiated with a set of numerical experiments.

Abstract The non-Newtonian (NN) hybrid nanofluids (HNF) flow over a porous stretching or shrinking Riga sheet is calculated. The HNF is produced by the scattering of cerium oxide (CeO2) and aluminum oxide (Al2O3) nanoparticles. NN HNF offers a wide variety of uses. For instance, enhanced heat transportation, cooling, maintenance, and reliability in mechanically powered delivery of medicines, increased efficacy in microfluidic devices, advanced material synthesis, and energy-related applications such as storing energy and solar power generation systems are a few of them. For this purpose, the flow phenomena are modeled in the form of nonlinear partial differential equations (PDEs), which are reduced into the dimension-free form through the similarity conversion. The solution is obtained by using the numerical approach parametric continuation method. The stability analysis has also been performed to check which solution is stable and reliable in practice. The results are compared to the numerical outcomes of the published studies. The present findings have shown the best correlation with the previous published studies. The relative error between the published and present study at Pr = 10 (Prandtl number) is 0.00046%, which is gradually reduced up to 0.00202% with the variation of Pr = 0.7. Furthermore, the impact of a viscoelastic factor enhances the velocity field of HNF (Al2O3 and CeO2/SA) for both types of NN fluids (second-grade fluid & Walter’s B fluid) in the case of stretching Riga sheet.