Nonlinear Analytical Solution Research Articles

Accurate non-linear bending analytical solutions setting new benchmarks for rectangular thin plates with four simply supported edges

In the field of nonlinear structural analysis, the ability to provide accurate and reliable analytical solutions is of paramount importance. This paper presents more accurate nonlinear analytical solutions for rectangular plates with four simply supported edges, establishing a new benchmark that replaces and complements Lévy’s manual analytical solutions. These accurate solutions can serve as a reference for validating various nonlinear numerical and approximate analytical solutions, thereby providing a robust basis for engineering design. Notably, the central deflection error in this study is less than 0.005%, while the central stress error remains below 0.4%. The accuracy of the present solutions provided significantly exceeds that found in the existing literature.

Improving Realism of Facial Interpolation and Blendshapes with Analytical Partial Differential Equation-Represented Physics

How to create realistic shapes by interpolating two known shapes for facial blendshapes has not been investigated in the existing literature. In this paper, we propose a physics-based mathematical model and its analytical solutions to obtain more realistic facial shape changes. To this end, we first introduce the internal force of elastic beam bending into the equation of motion and integrate it with the constraints of two known shapes to develop the physics-based mathematical model represented with dynamic partial differential equations (PDEs). Second, we propose a unified mathematical expression of the external force represented with linear and various nonlinear time-dependent Fourier series, introduce it into the mathematical model to create linear and various nonlinear dynamic deformations of the curves defining a human face model, and derive analytical solutions of the mathematical model. Third, we evaluate the realism of the obtained analytical solutions in interpolating two known shapes to create new shape changes by comparing the shape changes calculated with the obtained analytical solutions and geometric linear interpolation to the ground-truth shape changes and conclude that among linear and various nonlinear PDE-based analytical solutions named as linear, quadratic, and cubic PDE-based interpolation, quadratic PDE-based interpolation creates the most realistic shape changes, which are more realistic than those obtained with the geometric linear interpolation. Finally, we use the quadratic PDE-based interpolation to develop a facial blendshape method and demonstrate that the proposed approach is more efficient than numerical physics-based facial blendshapes.

Mechanical response of tunnels crossing active fault zones under normal or reverse faulting: A refined nonlinear analytical approximate solution

Tunnels crossing active fault zones will be severely damaged under normal or reverse faulting. The tunnel-stratum interaction is nonlinear, and the fault zone width significantly affects the tunnel response. Nevertheless, the existing analytical models neglect the effect of nonlinear tunnel-stratum interaction or fault zone width and adopt unrealistic assumptions. This work proposes a refined nonlinear solution, considering the nonlinear tunnel-stratum interaction and fault zone width, and some widely employed assumptions are removed. Specifically, a tunnel under normal or reverse faulting is simplified to an elastic beam acting on a nonlinear foundation, the nonlinear tunnel-stratum interaction is considered by setting a series of nonlinear axial and vertical springs, and the fault zone width is considered by reducing the spring stiffness. Additionally, the nonlinear axial force and geometrical nonlinearity terms are considered in the governing equations. The comparison results indicate that the results obtained from the proposed model agree well with those from model tests and numerical models and have a better accuracy than the existing analytical models. Finally, the effects of fault, stratum, and lining parameters on the response of tunnels crossing normal or reverse faults are discussed regarding stress distribution and failure scope.

Magneto-thermo-elastic coupled free vibration and nonlinear frequency analytical solutions of FGM cylindrical shell

The magneto-thermo-elastic coupled free vibration of functionally graded materials (FGM) cylindrical shell is investigated. Based on the physical neutral surface theory and Kirchhoff-Love theory, the expressions of internal force and internal torque are obtained by considering the constitutive relationship that includes thermal stress. According to the electromagnetic elasticity theory, the model of Lorentz forces of the shell in magnetic field are derived. The expressions of potential energy, kinetic energy and its variational operators are given by introducing geometric nonlinearity, respectively. Based on Hamilton principle, the magneto-thermal-elastic coupled vibration equation of FGM cylindrical shell in multi-physical field is established. The displacement functions under simply supported boundary conditions are solved, which are combined with Galerkin method for derivation of nonlinear ordinary differential equations. The second-order approximate analytical solution of natural frequency is obtained by applying the multi-scale method. The characteristic curves of natural frequency with different parameters are drawn through numerical examples. The results show that reducing the volume fraction index and increasing the initial amplitude can lead to the increase of the natural frequency under transient conditions. With the increase of temperature and magnetic induction intensity, the natural frequency decreases. In addition, the natural frequency is also influenced by the shell size and volume fraction index.

Large deflection deformation behavior of a Zr-based bulk metallic glass for compliant spinal fixation application

A novel compliant spinal fixation designed based on the concept of compliant mechanisms can reduce the stress-shielding effect and adjacent segment degeneration (ASD) effectively, but propose higher requirements for the properties of the used materials. Bulk metallic glasses (BMGs), as a kind of young biomaterials, exhibiting excellent comprehensive properties, which are attractive for compliant spinal fixation. Here, according to the practical service condition of the basic elements in compliant spinal fixation, large deflection deformation behaviors of Zr61Ti2Cu25Al12 (at.%, ZT1) BMG beam, including elastic, yielding and plastic were investigated systematically. It was shown that the theoretical nonlinear analytical solution curve as the benchmark not only with the capacity to predict the nonlinear load-deflection relation within the elastic deformation regime, but also assists to capture the yielding event roughly, which can be used as a powerful design tool for engineers. To capture the beginning of the yielding event exactly, bending proof strength (σp,0.05%) accompanied with tiny permanent strain of 0.05% was proposed and determined for BMGs in biomedical implant applications, which is of significance for setting the allowable operating limits of the basic flexible elements. By approach of interrupted loading-unloading cycles, plastic deformation driven by the bending moment can be classified into two typical stages: the initial stage which mainly characterized by the nucleation and intense interaction of abundant shear bands when the plastic strain below the critical value, and the second stage which dominated by the progressive propagation of shear bands and coupled with the emergence of shear offsets on tensile side. The plasticity of BMG beam structures depends on the BMG's inherent plastic zone size (rp). When the half beam thickness less than that of the rp, the plastic deformation of BMGs will behave in a stable manner, which can be acted as the margin of safety effectively.

Nonlinear analytical solution for radiation stress of higher-order Stokes waves on a flat bottom

The radiation stress of water surface waves plays an important role in many aspects of marine hydrodynamics. The nonlinear effects of highly nonlinear waves on the radiation stress were investigated on a flat bottom under finite water depth and deep water conditions. Analytical solutions for radiation stress were derived based on different order nonlinear Stokes wave theory, respectively. The results indicate that the nonlinear radiation stresses are significantly greater than those based on the linear wave, and they increase as the nonlinearity strengthens. Through an inter-comparison of the expressions of nonlinear radiation stress of various orders of Stokes waves, it turned out that the results based on the second-order Stokes wave overestimated the radiation stress, while the expressions based on Fenton's fifth-order Stokes wave were more reasonable in reflecting the nonlinear effect of waves. In the case of extreme wave steepness, the fifth-order Stokes wave radiation stresses Sxx5 was approximately 17% larger than that of the linear wave. Finally, using the expressions of the fifth-order Stokes wave radiation stress, bound infragravity waves excited by the bichromatic wave groups formed by the superposition of two fifth-order Stokes waves were obtained.

Nonlinear analytical solution of a slender reinforced concrete element under axial tension

Nonlinear analytical solution of a slender reinforced concrete element under axial tension

Perturbation analysis of nonlinear partial reflected wave on a sloping bottom

In this paper, we use the perturbation method to develop a new mathematical derivation to describe nonlinear partial standing wave over uniformly sloping bottoms. In the Lagrangian coordinate system, the particle trajectories are obtained as a function of the nonlinear order parameter ɛ and the bottom slope α to the second order of the perturbation. The setups and mean return flow for an arbitrary bottom in the Lagrangian framework are also found as part of the solutions. The direct influences of reflection coefficient, wave steepness and bottom slope on the surface profiles of partially reflected waves are also derived. This nonlinear analytical solution is verified by reduction to the classical Stokes solution of progressive waves in both the deep water and constant depth limit. This solution allows to describe the successive deformation of partial reflected wave profiles and water particle trajectories prior to wave breaking to be described. The dynamic properties including mass transport, and Lagrangian mean level and radiation stress for nonlinear partially reflected waves on various sloping bottoms are investigated.

Nonlinear analytical solution for cylindrical shell deformations under asymmetric axial and transverse loads using FSDT

Nonlinear analytical solution for cylindrical shell deformations under asymmetric axial and transverse loads using FSDT

Analytical and numerical solutions to describe water table fluctuations due to canal seepage and time-varying recharge

Abstract Hybrid finite analytic solution (HFAS), Galerkin's method based finite element solution (FES) and fully implicit finite difference solution (FIFDS) of one dimensional nonlinear Boussinesq equation and Analytical solution of Boussinesq equation linearized by Baumann's transformation (analytical solution I, AS I) as well as linearized by Werner's transformation (analytical solution II, AS II) were employed to obtain water table rise in a horizontal unconfined aquifer lying between two canals located at finite distance having different elevations and subjected to various patterns of recharge, i.e. zero recharge, constant recharge, as well as time varying recharge. Considering HFAS as benchmark solution, water table in mid region as obtained from FES followed by FIFDS was observed quite close to that obtained from HFAS and as per L2 and Tchebycheff norms computation, it was ranked at first and second place, respectively. Both AS I and AS II predicted higher water table at t = 5 days but at t = 10 days, AS I predicted lower and AS II predicted higher water table at all distances due to linearization effect. So, analytical solutions of linearized Boussinesq equation were rated lower than numerical solutions of nonlinear Boussinesq equation.

Electric field-induced nonlinear behavior of lead zirconate titanate piezoceramic actuators in bending mode

In this article, analytical solutions of piezoelectric bimorph and unimorph actuator in cantilever configurations are developed using the second-order constitutive equation. Tip deflection ratio (TDR), a dimensionless parameter, is defined to analyze the effect of second-order coefficients of piezoelectric materials on the response of bimorph and unimorph actuators. Four piezoelectric unimorphs of varying geometries are fabricated and tested to verify the nonlinear response under applied electric field. Furthermore, second-order coefficients of lead zirconate titanate (PZT; APC 850) are derived by fitting the experimental data with nonlinear analytical solution. The nonlinear response of all the piezoelectric unimorphs compares well with the experimental results.

Fully nonlinear hydroelastic modeling and analytic solution of large-scale floating photovoltaics in waves

This paper concerns the fully nonlinear fluid–structure interaction (FSI) of Large-scale floating photovoltaics (LFPV) in waves. The Euler Bernoulli–von Kármán beam models the structure while potential flow represents the fluid. A set of coupled dynamical equations is established. The fully analytic solution is sought with the unified Stokes perturbation method. The characteristic equation is derived up to third order, which has not been reported in literature before. The expressions obtained from the solution are applied to two typical cases of a pontoon LFPV and a membrane LFPV, with physical parameters from literature. The comparison with literature demonstrates our methodology for the membrane-type in waters of arbitrary depth and pontoon-type in relatively deep waters.

Mathematical model of brush seals for gas turbine engines: A nonlinear analytical solution

Presented herein is a mathematical model employing differential equations formulation for brush seals used in gas turbine engines. These components are used to seal the bearing chamber from the environment and reduce the loss of lubricant in the atmosphere, ensuring a MTBR long enough to have required the change the seals only during the engine overhaul operation. The model assumes a single curved bristle loop in the form of a curved-bridge beam subjected to the influences of complex external loads (static and dynamic). Further, a model for clustered bristles is proposed. Specifically, the static forces acting on the curved-bridge beam include the weight of the oil capillary attached to the beam, the weight of the beam itself, the capillary force developed between the surfaces of the bristles in the brush and the temperature gradient. The dynamic forces include the leakage oil pressure and the rotation of the shaft. This complex loading induces a nonlinear large deflection on the curved-bridge beam. Also, the temperature gradient present on the bristles during the gas turbine engine operation generates a change in the geometry of the beam and in the magnitude of the forces acting on the bristles modeled as beams. In the present model, the weights are assumed as uniformly distributed forces on the surface of the beam while the capillary forces and the force generated by the rotating shaft are considered to be non-uniform. The equation expressing the curvature of the beam under general loading force is developed and one can choose the appropriate method of solving the generated differential equation after the expression of the general force is defined. Hence, the ordinary differential equation describing the nonlinear large deflection of the curved-bridge beam will be derived using general nonlinear elasticity theory.

Periodic Solutions of Nonlinear Relative Motion Satellites

The relative motion of an outline of the rendezvous problem has been studied by assuming that the chief satellite is in circular symmetric orbits. The legitimacy of perturbation techniques and nonlinear relative motion are investigated. The deputy satellite equations of motion with respect to the fixed references at the center of the chief satellite are nonlinear in the general case. We found the periodic solutions of the linear relative motion satellite and for the nonlinear relative motion satellite using the Lindstedt–Poincaré technique. Comparisons among the analytical solutions of linear and nonlinear motions and the obtained solution by the numerical integration of the explicit Euler method for both motions are investigated. We demonstrate that both analytical and numerical solutions of linear motion are symmetric periodic. However, the solutions of nonlinear motion obtained by the Lindstedt–Poincaré technique are periodic and the numerical solutions obtained by integration by using explicit Euler method are non-periodic. Thus, the Lindstedt–Poincaré technique is recommended for designing the periodic solutions. Furthermore, a comparison between linear and nonlinear analytical solutions of relative motion is investigated graphically.

Nonlinear and non-local analytical solution for Darcy–Forchheimer flow through a deformable porous inclusion within a semi-infinite elastic medium

Permeability is a nonlinear and non-local function of the intimate coupling between pore fluid flow and solid deformation in porous media. A class of related problems involves the effect of fluid injection or withdrawal on the transport properties of geomaterials. This paper presents an analytical solution for the nonlinear and non-local problem of fluid flow through a disk-shaped porous elastic inclusion below the free surface of an elastic half-space. The solution accounts for the following nonlinear mechanisms: (i) variations in the permeability coefficient due to the flow-induced deformation of the inclusion; (ii) the inertial losses of the pore fluid flow. The former and latter mechanisms are formulated using the Green's function for a dilatation centre in an elastic half-space and the Darcy–Forchheimer equation for fluid flow through porous media, respectively. An analytical perturbation solution to the considered problem is developed and validated against the numerical finite element solution to the same problem. The described nonlinear mechanisms are represented by two dimensionless parameter groups. The extreme values of these dimensionless groups govern the solution asymptotic behaviours mimicking the special-case solutions in which either mechanism is forced to vanish. The applied aspects of the solution are demonstrated through the wellbore performance index parameter that quantifies the subsurface rock ability to deliver the pore fluid toward or away from a wellbore in a porous reservoir. Unlike the linear models, the presented nonlinear solution captures the observed dependence of the performance index on the wellbore flow rate.

Superharmonic vibrations of sandwich beams with fibre composite core layer based on the multiple scale method

This paper deals with the secondary vibration problem in the superharmonic case near the harmonic excitation of 1/3w1, arising from the vibration nonlinearity that characterizes the slender and less damping laminated beam with composite material core. For this aim the multiple scale method in conjunction with the higher order zigzag theories are used to obtain the resonance responses. In the present work the nonlinear forced vibration problem of sandwich beams under harmonic excitation is solved by the multiples scales method, based by the introduction of an artificial parameter with higher order expansions, to control the nonlinear analytical solutions. The application of this method demonstrates the sensitivity of the sandwich beams with viscoelastic composite layer to the secondary superharmonic vibrations. Following, parametric study is conducted to demonstrate the vulnerability of the laminated structures to the superharmonic vibrations and to reduce as far as possible the amplitude vibrations achieved by more appropriated structural design. The results reveal the effect of the slenderness of the sandwich beams on the hardening changes. In the other hand the results demonstrate the importance of fibre orientation angle to reduce as far as possible the amplitude responses of the sandwich structures in superharmonic vibration case.

Vibration and nonlinear dynamic analysis of sandwich FG-CNTRC plate with porous core layer

This paper focuses on the influence of CNTs, porosity, mechanical and thermal loading on the vibration and dynamic response of the sandwich functionally graded carbon nanotube-reinforced composite (FG-CNTRC) composite plate. The plate is made by three layers in which the core layer is porous FGM materials, bottom and top surfaces are FG-CNTRC. The motion equations are given based on Hamilton’s principle, TSDT, Galerkin method and the fourth-order Runge–Kutta method. The numerical illustration is shown to examine the influence of various parameters such as porosity distribution, CNTs volume fraction, geometrical parameters, elastic foundations, temperature, mechanical loads on the dynamic behaviors of the plate. Highlights The FG-CNTRC plates with porous core layer The vibration and nonlinear dynamic analysis Elastic foundation and temperature Analytical solutions, Hamilton’s principle with the high-order shear deflection theory is used Galerkin method and the fourth-order Runge–Kutta method are applied

Analysis of nonlinear vibration of nano/microelectromechanical system switch induced by electromagnetic force under zero initial conditions

A significant phenomenon associated with devices used in nano/microelectromechanical systems (N/MEMS) is the dynamic pull-in instability. The basic theme of this manuscript is to study the nonlinear oscillation and dynamic pull-in instability in an N/MEMS switch with a current carrying conductor due to magnetic force. The variational iteration method accompanied by the techniques of the Laplace transform is used to find the approximate nonlinear frequency and approximate analytic solution of the model problem. The critical condition for the pull-in threshold is successfully obtained, which is extremely useful for the optimal design of N/MEMS. The effects of parameters on the frequency and the solution are discussed as well. The present findings exceptionally agree with the numerical results obtained by the Runge-Kutta method of order four and also give better accuracy when compared to other methods.

Nonlinear Analytical Modeling of Vertical Drain-Installed Soft Soil Considering a Varied Discharge Capacity

Discharge capacity is one of the important factors affecting the consolidation behavior of vertical drains (VDs). Both previous experimental tests and field behavior indicated that the discharge capacity of VDs usually decreased with an increase in effective stress. The aim of this study is to develop a nonlinear analytical solution for consolidation analysis of VD-installed soft deposits considering a reduction of discharge capacity, in which a nonlinear distribution of VD’s discharge capacity with depth is adopted. To verify this work, the proposed solution is compared with previous solutions. The analyzed results show that at a given time and depth, the consolidation rate in the case of discharge capacity with a nonlinear distribution is lower than that of a linear and constant distribution. The proposed solution is then applied to a real embankment on the VD-installed thick soft ground in Busan New Port. It is shown that the proposed solution using the nonlinear distribution of VD discharge capacity provides a good agreement with the field data.

Research on a nonlinear quasi-zero stiffness vibration isolator with a vibration absorber.

A negative stiffness mechanism consisting of a spring and cylinder is proposed, and a grounded dynamic vibration absorber is designed based on a quasi-zero stiffness vibration isolator to constitute the vibration isolator with a vibration absorber system. The range of parameters for attaining zero stiffness is derived from static analysis. The dynamic analysis of the vibration isolator with a vibration absorber system is carried out by a multiscale method, and the amplitude-frequency response equation of the system is obtained. The influence of different system parameters on the amplitude-frequency response is analyzed. The amplitude-frequency response of the quasi-zero stiffness vibration isolator is compared with that of the vibration isolator with a vibration absorber, and the linear and nonlinear analytical solutions of the vibration isolator with a vibration absorber system are also compared. The results show that the designed vibration isolator with a vibration absorber is an ideal choice for low-frequency vibration isolation, with no large resonance peak throughout the system and significantly improved reliability of the system.