Nonlinear Terms Research Articles

Experimentally-efficient data-driven rational feedforward control for multivariable systems

Feedforward control with task flexibility for MIMO systems is essential to meet the growing demands on throughput and accuracy of high-tech systems. The aim of this paper is to develop an experimentally efficient framework for data-driven tuning of rational feedforward controllers for general non-commutative MIMO systems. In the developed approach, the nonlinear terms in the non-convex optimisation problem of iterative learning control (ILC) are iteratively circumvented via approximation. This leads to a series of convex optimisation problems that can be solved offline to obtain the parameters of the rational feedforward controller. In addition, by limiting the number of offline iterations an experimentally intensive algorithm is derived, which could be beneficial in the case of severe model mismatch. A simulation study shows that the experimentally efficient approach converges fast through offline iterations and has improved convergence properties through the use of regularisation.

Prediction of wall geometry forcold-metal-transfer-based wire-arc additive manufacturing

Wire-arc additive manufacturing (WAAM) is an advanced technique for fabricating large metal components through layer-by-layer material deposition using arc welding methods. This study focused on optimizing the WAAM process by employing machine learning models to predict and control bead geometries, specifically bead height (BH) and bead width (BW), while ensuring consistent height increments in multibead walls. Based on CMT technology in cold metal transfer experiments, linear regression models achieved high accuracy in predicting BH and BW. Analysis of variance results highlighted the considerable influence of voltage (V) and travel speed (TS) on bead geometries. For multibead wall characteristics, polynomial regression models incorporating non-linear terms, such as travel speed (TS²) and dwell time (Dt²), were developed to predict height (H) and waviness (W). Various optimization metrics were employed to balance the trade-offs between H and W for identifying optimal welding conditions that achieved the target H while minimizing W. A notable innovation of this research is the optimization of dwell time (Dt) for each layer to achieve a linear incremental H profile, minimizing W and ensuring consistent layer quality.

Unconditional error analysis of weighted implicit-explicit virtual element method for nonlinear neutral delay-reaction–diffusion equation

In this paper, we develop a virtual element method in space for the nonlinear neutral delay-reaction–diffusion equation, while a weighted implicit-explicit scheme is utilized in time. The nonlinear term is adopted by using the Newton linearized method. The calculation efficiency is improved using an implicit scheme to analyze linear terms and an explicit scheme to deal with nonlinear terms. Then, the time–space error splitting technique and G-stability are creatively combined to rigorously analyze the unconditionally optimal convergence results of the numerical scheme without any restrictions on the mesh ratio. Finally, numerical examples on a set of polygonal meshes are provided to confirm the theoretical results. In particular, when the weighted parameter is taken θ=12 (or θ=1), the method degenerates into the Crank–Nicolson (or second-order backward differential formula (BDF2)) scheme.

Characteristics of solution to singular fractional differential equation with two Riemann-Stieltjesdintegral boundary value conditions

The purpose of this paper is to study unique solution and iterative sequence of approximate solution for uniformly approaching unique solution to a new class of singular fractional differential equations with two kinds of Riemann-Stieltjes integral boundary value conditions by using some fixed point theorems. Because of different properties of the nonlinear terms and complexity of the boundary conditions in equations, we first probe several fixed point theorems of sum-type operators which expand many existing works in this research area. It is essential to point out that some conditions in our works greatly simplify the proof process of fixed point theorems. By applying the operator conclusions obtained in this paper, some sufficient conditions that guarantee the existence and uniqueness of solutions to singular differential equations are obtained, two iterative schemes that uniformly converges to the unique solution are given which provide computational methods of approximating solutions. As applications, some examples are provided to illustrate our main results.

Fast heuristics for the time‐constrained immobile server problem

AbstractWe propose easy‐to‐implement heuristics for time‐constrained applications of a problem referred to in the literature as the facility location problem with immobile servers, stochastic demand, and congestion, the service system design problem, or the immobile server problem (ISP). The problem is typically posed as one of allocating capacity to a set of M/M/1 queues to which customers with stochastic demand are assigned with the objective of minimizing a cost function composed of a fixed capacity‐acquisition cost, a variable customer‐assignment cost, and an expected‐waiting‐time cost. The expected‐waiting‐time cost results in a nonlinear term in the objective function of the standard binary programming formulation of the problem. Thus, the solution approaches proposed in the literature are either sophisticated linearization or relaxation schemes, or metaheuristics. In this study, we demonstrate that an ensemble of straightforward, greedy heuristics can rapidly find high‐quality solutions. In addition to filling a gap in the literature on ISP heuristics, new stopping criteria for an existing cutting plane algorithm are proposed and tested, and a new mixed‐integer linear model requiring no iterating algorithm is developed. In many cases, our heuristic approach finds solutions of the same or better quality than those found by exact methods implemented with expensive, state‐of‐the‐art mathematical programming software, in particular a commercial nonlinear mixed‐integer linear programming solver, given a five‐minute time limit.

Mathematical Modelling of Spatially Inhomogeneous Non-Stationary Interaction of Pests with Transgenic and Non-Modified Crops Considering Taxis

Introduction. This paper addresses a unified spatially inhomogeneous, non-stationary model of interaction between genetically modified crop resources (corn) and the corn borer pest, which is also present on a relatively small section of non-modified corn. The model assumes that insect pests influence both types of crops and are capable of independent movement (taxis) towards the gradient of plant resources. It also considers diffusion processes in the dynamics of all components of the unified model, biomass growth, genetic characteristics of both types of plant resources, processes of crop consumption, phenomena of growth and degradation, diffusion, and mutation of pests. The model allows for predictive calculations aimed at reducing crop losses and increasing the resistance of transgenic crops to pests by slowing down the natural mutation rate of the pest. Materials and Methods. The mathematical model is an extension of Kostitsin’s model and is formulated as an initial-boundary value problem for a nonlinear system of convection-diffusion equations. These equations describe the spatiotemporal dynamics of biomass density changes in two types of crops — transgenic and non-modified — as well as the specific populations (densities) of three genotypes of pests (the corn borer) resulting from mutations. The authors linearized the convection-diffusion equations by applying a time-lag method on the time grid, with nonlinear terms from eachequation taken from the previous time layer. The terms describing taxis are presented in a symmetric form, ensuring the skew-symmetry of the corresponding continuous operator and, in the case of spatial grid approximation, the finite-difference operator. Results. A stable monotonic finite-difference scheme is developed, approximating the original problem with second-order accuracy on a uniform 2D spatial grid. Numerical solutions of model problems are provided, qualitatively corresponding to observed processes. Solutions are obtained for various ratios of modified and non-modified sections of the field. Discussion and Conclusion. The obtained results regarding pest behavior, depending on the type of taxis, could significantly extend the time for pests to acquire Bt resistance. The concentration dynamics of pests moving in the direction of the food gradient differs markedly from the concentration of pests moving towards a mate for reproduction.

Modeling of the Black Sea circulation using equations of heat and salt advection–diffusion having discrete nonlinear invariants

In this work the accuracy of reconstructing the Black Sea circulation by using new approximations of nonlinear terms in the transport equations, ensuring the conservation of temperature and salinity to a power greater than two, is analyzed based on the results of forecast calculations. Three numerical experiments with differences in the schemes for calculating temperature and salinity are carried out. In the first experiment – traditional schemes are used to conserve of temperature and salinity in the first and second degrees; in the second one – the temperature is conserved in the first and fifth degrees, salinity in the first and third; in the third one – the temperature in the first and third, salinity in the first and fifth degrees. Calculations are performed on the basis of the MHI model with a resolution of 1.6 km and taking into account realistic atmospheric forcing for 2016. Validation of the results is carried out based on comparison of model fields with in-situ and satellite measurements of temperature and salinity in 2016. Analysis of mean and root mean square errors showed that new schemes for the advection-diffusion equations of heat and salt, ensuring the conservation of predictive parameters to a power greater than two, improve the accuracy of reconstructing the salinity in the Black Sea upper 100m layer throughout the year compared with traditional approximation. The root mean square errors in the salinity field are reduced by 15–20%, the thickness of the upper mixed layer in winter and the depth of the upper boundary of the thermocline layer in summer in the central part of the sea are modeled approximately 10% more accurately. Based on the results of three experiments, the smallest deviations from observational data are obtained when using approximations that ensure the conservation of temperature to the third power and salinity to the fifth power.

Unconditionally superconvergence error analysis of an energy-stable finite element method for Schrödinger equation with cubic nonlinearity

In this paper, the unconditionally superconvergence analysis is studied for the cubic Schrödinger equation with an energy-stable finite element method. A different approach is proposed to obtain the unconditionally superclose error estimate in H1-norm firstly without using the time splitting technique required in the previous literature. The key to the analysis is to use a priori boundedness of the numerical solution in energy norm and control the nonlinear terms rigorously by two cases, i.e., τ≤h2 and τ≥h2, where τ denotes the temporal size and h is the spatial size. Subsequently, the global superconvergence error estimate in H1-norm is derived by an effective interpolation post-processing approach. Finally, some numerical experiments are carried out to confirm the theoretical findings.

A novel dimensionality reduction iterative method for the unknown coefficient vectors in TGFECN solutions of unsaturated soil water flow problem

The main purpose of this paper is to reduce the dimensionality of unknown coefficient vectors in finite element (FE) solutions of two-grid FE Crank-Nicolson (CN) (TGFECN) format for the unsaturated soil water flow (USWF) problem with two strong nonlinear terms by using proper orthogonal decomposition (POD). For this purpose, our first step involves designing a time semi-discrete CN (TSDCN) scheme for the USWF problem and demonstrate the existence, boundedness, and error estimations of TSDCN solutions. Thereafter, we discretize the TSDCN scheme using the two-grid FE method to create a new TGFECN format and prove the existence, boundedness, and error estimations of TGFECN solutions. The primary focus should be on reducing the dimension of unknown coefficient vectors of TGFECN solutions through the utilization of the POD technique in creating a novel format, referred to as dimension reduction iterative TGFECN (DRITGFECN) format, while establishing the existence, boundedness, and error estimations for DRITGFECN solutions. Lastly, we use two sets of numerical tests to exhibit the advantage of the DRITGFECN format. Due to the presence of two highly nonlinear terms in the unsaturated soil flow problem, the development and analysis of DRITGFECN format pose greater challenges and necessitates more advanced technical skills compared to previous studies. However, the significance and broad applications of this research make it a valuable subject for investigation.

High sound pressure level sound absorption in acoustic resonant metamaterials: adapting a mass-spring model for nonlinear effects

In aerospace, conventional materials as micro-perforated panels backed by cavities are used to absorb sound at high sound pressure levels. These solutions are not optimal at low frequencies compared to resonant acoustic metamaterials, which present an acoustic absorption peak with a wavelength-to-thickness ratio greater than conventional materials, i.e., a ratio>>5. Acoustic resonant metamaterials are mainly studied at low sound pressure levels (<100dB). However, they are sensitive to increasing the amplitude of acoustic excitation. The studied metamaterial is composed of a compact array of thin single-perforation panels spaced by thin cylindrical air cavities. In a previous study, a mass-spring model was developed for low sound pressure levels. The perforations are modeled by equivalent masses, whereas cavities by equivalent springs. In the perforations, high values of the acoustic velocity lead to an increase in the acoustic resistance of the material. This effect is considered by the Forchheimer parameter in the mass-spring model. To determine this parameter and to study how it is impacted by the metamaterial geometry, the computational fluid dynamic method is used. The mass-spring model is adapted with a nonlinear term. The adapted model agrees well with simulations by finite element method at sound pressure levels up to 140dB.

Higher spins and Finsler geometry

Finsler geometry is a natural generalization of (pseudo-)Riemannian geometry, where the line element is not the square root of a quadratic form but a more general homogeneous function. Parameterizing this in terms of symmetric tensors suggests a possible interpretation in terms of higher-spin fields. We will see here that, at linear level in these fields, the Finsler version of the Ricci tensor leads to the curved-space Fronsdal equation for all spins, plus a Stueckelberg-like coupling. Nonlinear terms can also be systematically analyzed, suggesting a possible interacting structure. No particular choice of spacetime dimension is needed. The Stueckelberg mechanism breaks gauge transformations to a redundancy that does not change the geometry. This creates a serious issue: non-transverse modes are not eliminated, at least for the versions of Finsler dynamics examined in this paper.

Potential of data-driven discovery of nonlinear wave equations

Data-driven partial differential equation discovery techniques have been growing in importance in recent years. This paper discusses its possible applications in the field of finite-amplitude sound propagation and the related phenomena (such as generation of higher harmonics and shock formation). Two cases are investigated, namely the propagation of pressure pulses as travelling waves and the interference of pressure pulses, the former leading to the Westervelt equation and the latter to the Kuznetsov equation. It is shown how representative wave equations can be extracted from data obtained by numerically solving the compressible Navier-Stokes equations (up to the third order smallness in entropy changes) and subsequently employing the sparsity promoting regression techniques. The resulting error of the equation coefficients is about 7% in the nonlinear terms. The limitations of this approach are discussed (e.g., the appropriate capturing of thermal and viscous loss terms).

Interval estimation of sensor fault in rotary steerable drilling tools based on set-membership approach

The rotary steerable drilling tools (RSDTs) are invented for high-precision wellpath control. Fault diagnosis is essential for the RSDT as it can improve the reliability of the drilling processes. To estimate the sensor fault of the RSDT, this paper investigates the interval estimation problem for Lipschitz nonlinear systems with sensor fault via the set-membership approach. Firstly, the RSDT is modeled by a discrete-time Lipschitz nonlinear system with unknown inputs, noises, and parameter uncertainties. Next, based on the ellipsoid bundles, a set-membership fault estimator is presented, where the unknown inputs are decoupled. Additionally, a sufficient condition, which is derived based on the P-radius of the ellipsoid bundles, is put forward to design the parameters of the estimator with Lipschitz nonlinear terms. Then, the interval estimations of the states and fault are obtained via ellipsoid bundles. Finally, the efficiency of the proposed approach is evaluated through numerical simulations and experiments.

A fast compact difference scheme on graded meshes for solving the time-fractional nonlinear KdV equation

The fractional KdV equation has significant physical implications, the study of its efficient method has great scientific significance. A fast compact difference (FCD) scheme on graded meshes is used to address the time-fractional nonlinear KdV equation with an initial singularity. The temporal discretization employs the fast L1 algorithm, while a fourth-order compact scheme is utilized for spatial discretization. The Newton's method is used to approximate the nonlinear term. The fast L1 algorithm can greatly shorten the CPU calculation time without losing the precision of numerical solution. We verify the existence and uniqueness of numerical solutions, while also provide unconditional stability and convergence analysis. Theoretical analysis is validated by numerical results, indicating the FCD scheme on graded meshes has high-accuracy and time-saving performance.

Uniform error analysis of an exponential IMEX-SAV method for the incompressible flows with large Reynolds number based on grad-div stabilization

Based on the grad-div stabilization and scalar auxiliary variable (SAV) methods, a first-order Euler implicit/explicit finite element scheme is studied for the Navier–Stokes equations with large Reynolds number. In the designing of numerical scheme, the nonlinear term is explicitly treated such that one only needs to solve a constant coefficient algebraic system at each time step. Meanwhile, the proposed scheme is unconditionally stable without any condition of the time step τ and mesh size h. In finite element discretization, we use the stable Taylor-Hood element for the approximation of the velocity and pressure. By a rigorous analysis, we derive an uniform L2 error estimate O(τ+h2) of the velocity in which the constant is independent of the viscosity coefficient. Finally, numerical experiments are given to support theoretical results and the efficiency of the proposed scheme.

Nonlinear free vibrations of a structurally tailored anisotropic pre-twisted thin-walled beam subjected to large deformations

It is essential to understand the nonlinear free vibration behaviour of a thin-walled composite structure as they find their applications in harsher environments prone to large deformations. Further, these structures made of fibrous materials, have directional properties for different fibre orientations resulting in a diversified and a complex nonlinear behaviour. In this regard, present work provides a comprehensive mathematical formulation on nonlinear vibration of pre-twisted thin-walled anisotropic box beam. The mathematical model is derived as coupled (flap-lag-extension-torsion) model considering shear deformation and green's strain tensor for large displacements in order to study the influence of nonlinear couplings on the dynamic behaviour. The derived energy equations are presented depending on degree of nonlinearity. These nonlinear equations those derived are nondimensionalized and solved with in the frame work of energy method using classical Ritz approximation. An iterative method is used to solve the nonlinear equations pertaining displacement terms. This nonlinear behaviour is evaluated for two important types of structural tailoring techniques available for thin-walled composite beams namely Circumferentially Asymmetric Stiffness (CAS) and Circumferentially Uniform Stiffness (CUS). It is found that CAS layup experiences significantly severe nonlinear effects compared to CUS due to the presence of 3rd degree nonlinear coupling terms. The variation of nonlinear frequency ratios over different ply angles showing the influence of nonlinearity on fibres orientation is presented for the first time for the two ply lay ups. The influence of degree of nonlinearity on the accuracy of the nonlinear frequencies is evaluated and presented. Moreover, the impact of nonlinearity on the pre-twisted beam is presented for different pre-twist angles.

The renormalization group for large-scale structure: origin of galaxy stochasticity

The renormalization group equations for large-scale structure (RG-LSS) describe how the bias and stochastic (noise) parameters — both of matter and biased tracers such as galaxies — evolve as a function of the cutoff Λ of the effective field theory. In previous work, we derived the RG-LSS equations for the bias parameters using the Wilson-Polchinski framework. Here, we extend these results to include stochastic contributions, corresponding to terms in the effective action that are higher order in the current J. We derive the general local interaction terms that describe stochasticity at all orders in perturbations,and a closed set of nonlinear RG equations for their coefficients. These imply thata single nonlinear bias term generates all stochastic moments through RG evolution. Further, theevolution is controlled by a different, lower scale than the nonlinear scale. This has implications for the optimal choice of the renormalization scale when comparing the theory with data to obtain cosmological constraints.

Liouville-type theorems for partial trace equations with nonlinear gradient terms

Liouville-type theorems for partial trace equations with nonlinear gradient terms

Environmental management and restoration under unified risk and uncertainty using robustified dynamic Orlicz risk

Environmental management and restoration should be designed such that the risk and uncertainty owing to nonlinear stochastic systems can be successfully addressed. We apply the robustified dynamic Orlicz risk to the modeling and analysis of environmental management and restoration to consider both the risk and uncertainty within a unified theory. We focus on the control of a jump-driven hybrid stochastic system that represents macrophyte dynamics. The dynamic programming equation based on the Orlicz risk is first obtained heuristically, from which the associated Hamilton–Jacobi–Bellman (HJB) equation is derived. In the proposed Orlicz risk, the risk aversion of the decision-maker is represented by a power coefficient that resembles a certainty equivalence, whereas the uncertainty aversion is represented by the Kullback–Leibler divergence, in which the risk and uncertainty are handled consistently and separately. The HJB equation includes a new state-dependent discount factor that arises from the uncertainty aversion, which leads to a unique, nonlinear, and nonlocal term. The link between the proposed and classical stochastic control problems is discussed with a focus on control-dependent discount rates. We propose a finite difference method for computing the HJB equation. Finally, the proposed model is applied to an optimal harvesting problem for macrophytes in a brackish lake that contains both growing and drifting populations.

Banded Preconditioners for Two-Sided Space Variable-Order Fractional Diffusion Equations with a Nonlinear Source Term

Banded Preconditioners for Two-Sided Space Variable-Order Fractional Diffusion Equations with a Nonlinear Source Term