Quasi-static Solution Research Articles

Free and forced vibration analysis of piezolaminated plates via an isogeometric layerwise finite element

Isogeometric layerwise finite element (L-IGA) formulation is a recent state-of-the-art approach integrating Non-Uniform Rational B-spline (NURBS) basis functions into the quasi-static solution process of piezolaminated composite plates. This study extends the application of the L-IGA framework to encompass free, forced vibration, and displacement control analyses of laminated composite plates with straight/curvilinear fibers and piezoelectric layers. To this end, the NURBS basis functions, utilized in geometry definition, are employed to solve electromechanically coupled differential equations following Hamilton’s variational principle. The adoption of high-order continuous NURBS shape functions throughout the IGA discretization span both in-plane and through-thickness laminate dimensions. This effectively facilitates precise geometry representation directly from Computer-Aided Design (CAD). Besides, such a discretization accelerates the convergence of displacement and electric potential solution fields toward exact results. Various benchmark problems have been solved to verify the robustness and high accuracy of the proposed dynamic L-IGA method. These include comparative analyses between L-IGA dynamic solutions (i.e. employing the Newmark-Beta method), analytical solutions, and ANSYS-Solid 226 finite element results. All the results are compared across various span-to-thickness ratios, mechanical-potential loading scenarios, and fiber orientation angles. Remarkably, the L-IGA method attains almost excellently accurate time response of various fields (displacement, stress, electric potential) and modal results, with considerably fewer mesh elements than Solid 226 solutions. Overall, such an outcome reveals the high potential and practical merits of the proposed L-IGA formulation as a proficient finite element approach for the dynamic analysis of piezolaminated plates.

Probabilistic maximization of time-dependent capacities in a gas network

AbstractThe determination of free technical capacities belongs to the core tasks of a gas network owner. Since gas loads are uncertain by nature, it makes sense to understand this as a probabilistic problem provided that stochastic modeling of available historical data is possible. Future clients, however, do not have a history or they do not behave in a random way, as is the case, for instance, in gas reservoir management. Therefore, capacity maximization becomes an optimization problem with uncertainty-related constraints which are partially of probabilistic and partially of robust (worst case) type. While previous attempts to solve this problem were devoted to models with static (time-independent) gas flow, we aim at considering here transient gas flow subordinate to the isothermal Euler equations. The basic challenge addressed in the manuscript is two-fold: first, a proper way of formulating probabilistic constraints in terms of the differential equations has to be provided. This will be realized on the basis of the so-called spherical-radial decomposition of Gaussian random vectors. Second, a suitable characterization of the worst-case load behaviour of future customers has to be found. It will be shown, that this is possible for quasi-static flow and can be transferred to the transient case. The complexity of the problem forces us to constrain ourselves in this first analysis to simple pipes or to a V-like structure of the network. Numerical solutions are presented and show that the differences between quasi-static and transient solutions are small, at least in these elementary examples.

Aether Scalar Tensor (AeST) theory: quasistatic spherical solutions and their phenomenology

ABSTRACT There have been many efforts in the last three decades to embed the empirical Modified Newtonian Dynamics (MOND) programme into a robust theoretical framework. While many such theories can explain the profile of galactic rotation curves, they usually cannot explain the evolution of the primordial fluctuations and the formation of large-scale structures in the Universe. The Aether Scalar Tensor theory seems to have overcome this difficulty, thereby providing the first compelling example of an extension of general relativity able to successfully challenge the particle dark matter hypothesis. Here, we study the phenomenology of this theory in the quasistatic weak-field regime and specifically for the idealized case of spherical isolated sources. We find the existence of three distinct gravitational regimes, that is, Newtonian, MOND, and a third regime characterized by the presence of oscillations in the gravitational potential which do not exist in the traditional MOND paradigm. We identify the transition scales between these three regimes and discuss their dependence on the boundary conditions and other parameters in the theory. Aided by analytical and numerical solutions, we explore the dependence of these solutions on the theory parameters. Our results could help in searching for interesting observable phenomena at low redshift pertaining to galaxy dynamics as well as lensing observations, however, this may warrant proper N-body simulations that go beyond the idealized case of spherical isolated sources.

Three-dimensional quasi-static general solution for isotropic chemoelastic materials and their application

Based on potential theory, the three-dimensional quasi-static general solution for isotropic chemoelastic materials is presented in this work. Through the three-dimensional general solution, the Green’s function for an isotropic chemoelastic material subjected to dynamic point loads is derived. This can serve as theoretical guidance for future engineering practices. Four functions constitute the expressions of the general solution that satisfy the harmonic functions and the quasi-static transport equation, respectively. The Green’s function for an isotropic chemoelastic material subjected to dynamic point loads is derived by combining the general solution with the chemical balance boundary conditions at infinity. It can be expressed in terms of the error function and elementary functions. Finally, the numerical results are provided, as shown in the contours. These results can be used to analyze the variation law in the coupling fields of isotropic chemoelastic materials. The corresponding analysis can provide a theoretical basis for elucidating the mechanism of the chemoelastic coupling problem in further work.

Analysis of layered soil under general time-varying loadings by fractional-order viscoelastic model

Time-varying loading is a frequently encountered loading type in geotechnical engineering. As the deformation of viscoelastic soil is related to its loading history, studying the viscoelastic problems under time-varying loads has important practical engineering significance. In this paper, the stress and displacement of a layered soil with fractional-order viscoelastic model under time-varying loads were solved using the complex variable method and the corresponding principle. Under the assumption of quasi-static and linear elasticity, this paper derives the quasi-static elastic solutions of a layered soil under time-varying strip loads. By introducing the fractional-order viscoelastic model and using the corresponding principle, the Laplace-domain analytical solutions are obtained. Finally, numerical methods are utilized to perform the Laplace inverse transform and obtain the solutions in the physical domain. The correctness of this paper is validated by comparing the numerical results with the ANSYS software, as well as by comparison with previous literature. Based on the experimental data, the material parameters of frozen soil at two temperatures are fitted. The settlement patterns of soil under two engineering loads were analyzed through case studies. By controlling variables, the influence of parameters of the fractional viscoelastic model on elastic aftereffect was investigated.

An improved implicit non-ordinary state-based peridynamics model for modeling crack propagation and coalescence problems

A quasi-static solution for crack propagation and coalescence can provide an efficient way to predict both structure failure load and fracture pattern morphology. This paper proposes an improved implicit static non-ordinary state-based peridynamics (IINOSB-PD) solution procedure to study crack propagation problems through incorporating a stress-based criterion into the model. The comprehensive discretized processes and the approach to mitigate numerical oscillation are presented in this manuscript. A parameter sensitivity analysis is first performed to evaluate the effects of uncertain parameters in this procedure on numerical output. Subsequently, four numerical examples are investigated using the proposed IINOSB-PD to validate this procedure. The research findings indicate that the proposed IINOSB-PD can not only accurately predict material crack propagation and coalescence processes but also reduce the programming difficulties and avoid the traditional singular problem. These results provide valuable insights for the application of the IINOSB-PD method in related fields.

Dynamical theory of topological defects I: the multivalued solution of the diffusion equation

Point-like topological defects are singular configurations that manifest in and out of various equilibrium systems with two-dimensional orientational order. Because they are associated with a nonzero circuitation condition, the presence of defects induces a long-range perturbation of the orientation landscape around them. The effective dynamics of defects is thus generally described in terms of quasi-particles interacting via the orientation field they produce, whose evolution in the simplest setting is governed by the diffusion equation. Because of the multivalued nature of the orientation field, its expression for a defect moving with an arbitrary trajectory cannot be determined straightforwardly and is often evaluated in the quasi-static approximation. Here, we instead derive the exact expression for the orientation created by multiple moving defects, which we find to depend on their past trajectories and thus to be nonlocal in time. Performing various expansions in relevant regimes, we demonstrate how improved approximations with respect to the quasi-static defect solution can be obtained. Moreover, our results lead to so far unnoticed structures in the orientation field of moving defects, which we discuss in light of existing experimental results.

“Implicit” vs “Explicit” gradient plasticity models: Do they always remove mesh dependence in softening materials?

Finite element quasi-static solutions of rate-independent softening plasticity models are known to depend on the mesh size (i.e., are unreliable) due to loss of ellipticity of the governing partial differential equations, which allows for the development of non-smooth solutions, such as shear bands of zero thickness. A “regularization” scheme that has been proposed is the use of “non-local” (Bažant et al., 1984) or “strain-gradient” theories (Aifantis, 1984). We examine in detail the families of “explicit” and “implicit” gradient plasticity models and show that use of such theories does not always eliminate mesh dependence of finite element solutions in rate-independent softening materials. Depending on the value of the non-local hardening modulus, the mathematical problem may still lose ellipticity and allow for the development of solutions with discontinuous spatial velocity gradients, which are responsible for the mesh dependence of numerical solutions. As an example, a rate-independent “implicit” gradient version of the von Mises yield condition with the associated flow rule is implemented in ABAQUS, and the formation of shear bands in plane strain tension is analyzed numerically; it is shown that, unless the material parameters in the non-local model are such that ellipticity is retained, the finite element solutions depend on the mesh size. The example of isotropic porous metal plasticity with a hardening matrix material is considered (e.g., the Gurson (1977) model); it is shown that regularization is achieved if an implicit non-local model is used, in which the local porosity f is replaced by its non-local (neighborhood average) counterpart f̄ in the yield condition. In all non-local plasticity models, dimensional consistency requires the introduction of one or more “material lengths” ℓi, which multiply the additional terms with the higher-order spatial derivatives introduced by the non-local models. The boundary layers associated with the singular perturbation problem when the material lengths ℓi→0 are examined. A linear stability analysis is carried out to study the stability of homogeneous problems under small harmonic perturbations; it is shown that the stability range of the non-local models is always larger than that of the corresponding local models.

Approximation of Uncoupled Quasi-Static Thermoelasticity Solutions Based on Gaussians

A fast approximation method to three dimensional equations in quasi-static uncoupled thermoelasticity is proposed. We approximate the density via Gaussian approximating functions introduced in the method approximate approximations. In this way the action of the integral operators on such functions is presented in a simple analytical form. If the density has separated representation, the problem is reduced to the computation of one-dimensional integrals which admit efficient cubature procedures. The comparison of the numerical and exact solution shows that these formulas are accurate and provide the predicted approximation rate 2,4,6 and 8.

Spacecraft charging with EMA3D Charge

Spacecraft are exposed to harsh radiation environments. These dynamic radiation environments present significant threat to mission performance when charge accumulation and dissipation is not adequately mitigated. EMA3D Charge is a new simulation platform tailored to the study of surface charging, deep dielectric charging, and radiation hardening of spacecraft and their components. Charge implements multiple solvers capable of co-simulating: finite element electromagnetics, boundary element method surface charging, particle transport and particle-in-cell. The multiple solvers facilitate model reuse and enable powerful physical analysis not possible with separate, individual solvers. The solvers include a three-dimensional particle transport tool, a surface charge balance solver, and a finite-element-method electromagnetic solver that supports both quasi-static and full-wave (or fully electrodynamic) solutions. This article summarizes the solvers, example applications, and validation compared to accepted numerical solutions for spacecraft in the space plasma environment.

Using a poroelastodynamic model to investigate the dynamic behaviour of articular cartilage

Background and objectiveThere is still a few studies about the poroelastic model that performed dynamic behaviour, especially for the case of the poroelastic cartilage model. Therefore, this study is aimed to use the poroelastodynamic model to simulate the dynamic behaviour of cartilage. MethodsThe governing equations of the poroelastodynamic model is firstly established. The validation of the model is initialised by modifying the equations into the static poroelastic model. The modified equations are then discretised using the finite element method. Mandel's problem is used to validate the discretised equations. The numerical solution calculated using FreeFEM++ is validated with the analytical solution for the quasi-static state and compared with the results generated using COMSOL Multiphysics software. Finally, the quasi-static solution is compared with the dynamic solution to discuss the difference in pore pressure and displacement variations of the poroelastic cartilage model. ResultsThe dynamic solution showed transient behaviour at the beginning of the excitation. When the compressive force acts on the cartilage, there are obvious fluctuations during the initial stage and then the dynamic numerical solution gradually approaches the quasi-static value over a period of time. The deduced results of the analytical solution were approximately the same as the numerical simulation results. ConclusionThis study was able to use the poroelastodynamics equation to simulate the dynamic behaviour of the poroelastic cartilage model. The comparison between the result coming from poroelastodynamics equation with that of the validated numerical solution was satisfactorily compared. The approximate similarity between the results of quasi-static and dynamic solutions underscored the importance of performing the dynamic solution for a more realistic simulation. This dynamic solution can be further used for the analysis of vibration or stress waves in future research.

A simplified approach for fatigue life prediction of aquaculture nets under waves and currents

With increasing trends to move fish farms to more exposed locations engineers need quick and accurate tools for creating initial design proposals. In this paper, a simplified analytical model for estimating the axial force in the ropes supporting the fish net is presented. Hydrodynamic loads acting on the net are calculated using a state of the art screen model. The simplified analytical model is a quasi-static solution based on the principle of virtual displacements. The assumed mode shape in the calculations is a simplification of the actual quasi-static displacement shape under combined wave and current loading in view of the high natural frequencies of pre-tensioned ropes compared to wave frequencies. The results from the simplified calculations are compared to numerical simulations performed in RIFLEX. The simplified method is extremely efficient with regard to computation time compared to the numerical simulations and is well suited for preliminary design of fish nets.

The global mechanical response and local contact in multilevel helical structures under axial tension

Multilevel helical structures are widely applied in engineering and biological areas with complicated local contact of interwire and global mechanical behaviors. Based on the thin rod theory, we propose a theoretical model which can simultaneously calculate the global mechanical properties and the local contact behaviors of the multilevel helical structure. By considering the contact deformation and the periodic kinematic parameters (the curvatures and twist) of the high order helix, the quasi-static solution of the multilevel helical structure is derived under axial tension, and two types of contact forces including line and point are analyzed. The multilevel contact transmission law is proposed to reveal the relationship between the contact forces with different types and levels, and the tendency of two contact forces with pitch can be obtained. Compared with the Costello model, the deviation of global mechanical responses and finite element (FE) simulations is reduced for the high-level helical structure. The local contact forces from FE simulations are in agreement with the predictions of the present model. The effect of the structural parameters on the global mechanical responses as well as the local contact forces are analyzed according to the present model. Finally, based on an engineering practical cable, the global mechanical responses and local contact behaviors of the simplified cable structure are predicted under axial tension. This study contributes to the understanding of the transmission mode and mechanism of contact forces in the multilevel helical structure.

Structural stability of a lightsail for laser-driven interstellar flight

The structural stability of a lightsail under the intense laser flux necessary for interstellar flight is studied analytically and numerically. A sinusoidal perturbation is introduced into a two-dimensional thin-film sail to determine if the sail remains stable or if the perturbations grow in amplitude. A perfectly reflective sail material that gives specular reflection of the laser illumination is assumed in determining the resulting loading on the sail, although other reflection models can be incorporated as well. The quasi-static solution of the critical point between shape stability and instability is found by equating the bending moments induced on the sail due to radiation pressure with the restoring moments caused by the strength of the sail material and the tension applied at the edges of the sail. From this quasi-static solution, analytical expressions for the critical value of elastic modulus and boundary tension magnitude are found as a function of sail properties (e.g., thickness) and the amplitude and wave number of the initial sinusoidal perturbation. These same expressions are also derived from a more formal variational energy (virtual work) approach. A numerical model of the complete lightsail dynamics is developed by discretizing the lightsail into rectangular finite elements. By introducing torsional and rectilinear springs between the elements into the numerical model, a hierarchy of models is produced that can incorporate the effects of bending and applied tension. The numerical models permit the transient dynamics of a perturbed lightsail to be compared to the analytic results of the quasi-static analysis, visualized as stability maps that show the rate of perturbation growth as a function of sail thickness, elastic modulus, and applied tension. The analytic theory is able to correctly predict the stability boundary found in the numerical simulations. The stiffness required to make a thin lightsail stable against uncontrolled perturbation growth appears to be unfeasible for known materials, however, a relatively modest tensioning of the sail (e.g., via an inflatable structure or spinning of the sail) is able to maintain the sail shape under all wavelengths and amplitudes of perturbations.

A theoretical calculation of the cosmological constant based on a mechanical model of vacuum

Lord Kelvin believed that the electromagnetic aether must also generate gravity. Presently, we have no methods to determine the density of the electromagnetic aether, or we say the Ω(1) substratum. Thus, we also suppose that vacuum is filled with another kind of continuously distributed substance, which may be called the Ω(2) substratum. Based on a theorem of V. Fock on the mass tensor of a fluid, the contravariant energy-momentum tensors of the Ω(1) and Ω(2) substrata are established. Quasi-static solutions of the gravitational field equations in vacuum are obtained. Based on an assumption, relationships between the contravariant energy-momentum tensors of the Ω(1) and Ω(2) substrata and the contravariant metric tensor are obtained. Thus, the cosmological constant is calculated theoretically. The Ω(1) and Ω(2) substrata may be a possible candidate of the dark energy. According to the theory of vacuum mechanics, only those energy-momentum tensors of discrete or continuously distributed sinks in the Ω(0) substratum are permitted to act as the source terms in the generalized Einstein's equations . Thus, the zero-point energy of electromagnetic fields is not qualified for a source term in the generalized Einstein's equations. Some people believed that all kinds of energies should act as source terms in the Einstein's equations. It may be this unwarranted belief that leads to the cosmological constant problem. The mass density of theΩ(1) and Ω(2) substrata is equivalent to that of around 3 protons contained in a box with a volume of 1 cubic metre.

Inelastic light scattering from a dielectric sphere with a time-varying radius

This work reports on light scattering by a homogeneous dielectric sphere with a periodically time-varying radius. The off-shell inelastic scattering $T$ matrix, which describes the dynamically changing particle, is evaluated using the Floquet method, and some remarkable phenomena, emerging in the strong- and weak-coupling regimes, are discussed. In particular, the limits of validity of the approximate quasistatic solution are established through comparison with the results of fully dynamic calculations, and the scattering in the strong-coupling regime is analyzed in terms of the general behavior of parametrically driven oscillators. Additionally, the influence of damping of the sphere vibrations on the optical spectra is also investigated.

The Stefan Problem With Internal Heat Generation in Spherical Coordinates

Abstract A weakly time-dependent equation for the evolution of the solid–liquid interface in spherical coordinates, driven by internal heat generation, is derived for constant surface temperature boundary conditions. The derivation comes by making an assumption that the interface moves slowly compared to the changes in the temperature so that the technique of separation of variables may be applied for Stefan numbers less than one. Under this approximation, we can separate the nonhomogeneous heat diffusion equation into transient and steady-state terms, and then integrate to get the temperature relations. With the temperature equations in hand, the derivatives are inserted into the interface equation giving a first-order differential equation for the location of the solid–liquid interface as a function of time. The results are compared to a previously derived quasi-static solution and a numerical simulation generated using the method of catching of the front. This method allows for direct tracking of a moving boundary via the calculation of the time it takes to move from node to node in a discretized grid characteristic of classical finite difference methods. Plots of the interface evolution show excellent agreement between the three methods, especially for lower Stefan numbers. The quality of the approximation decreases as the Stefan number increases, but the model is more accurate than the previously studied quasi-static model. For the Stefan numbers St = 1.0 and 10.0, the weakly time-dependent solutions are in better agreement with the numerical results than the quasi-static solutions.

On the Voltage Response of Homogeneous Earth Models in Central Loop Electromagnetic Sounding

An accurate series-form explicit expression is derived for the voltage induced in the small receiving loop of a central loop electromagnetic sounding system positioned above a homogeneous earth model. The solution is obtained by converting the semi-infinite integral representation for the vertical magnetic field induced at the center of the two loops into a series of simpler integrals, amenable to analytical evaluation. The developed formulation is based on replacing the exponential term in the integrand of the field integral with its power series expansion with respect to the difference of the squares of the wavenumbers in air and in the conducting medium. As a result, the vertical magnetic field and the induced voltage are finally expressed as sums of the spherical Hankel functions depending on geometrical parameters and the wavenumbers. The derived solution exhibits advantages in terms of both time cost and accuracy, when compared to the conventional numerical schemes for evaluating the Sommerfeld-type integrals and to the previously published quasistatic solution to the considered problem.

Two-grid based sequential peridynamic analysis method for quasi-static crack propagation

We introduce two-grid based sequential analysis algorithm for implicit peridynamics to find the quasi-static solutions for crack propagation problems. The sequential analysis of implicit peridynamic formulation is capable of studying quasi-static fracturing of brittle materials. The failure model in peridynamics, however, usually allows one to break a single bond between two nodes under a certain damage criterion in tracing equilibrium points, which results in the whole simulation quite lengthy and inefficient. In order to circumvent the efficiency issue, we propose the two-grid based sequential peridynamic analysis algorithm, in which a coarse grid is used to find the equilibrium path with sequential analysis and the converged solution is properly prolongated to the fine grid. The failure model is also implemented in the coarse grid and the computed damage information is properly mapped into the fine grid. In this way, the proposed two-grid based scheme incorporates breaking multiple bonds into one step of sequential analyses. We document the superior behavior of the proposed scheme with respect to the conventional solution procedures of implicit peridynamics in both solution accuracy and computational efficiency for quasi-static crack propagation.

Analytical approach to mesh stiffness modeling of high-speed spur gears

The centrifugal effect is a natural phenomenon in rotor dynamics, significantly changing the deformation responses in high-speed conditions. However, this phenomenon is generally unexplored in high-speed gears. This paper developed an analytical mesh stiffness model to bridge the kinematics and the centrifugal fields of spur gears. Starting from the elasticity of fillet-foundation, this model refined and extended the bi-dimensional formulas to consider gears in simultaneous rotating and meshing. The proposed load-sharing model additionally considered the extended tooth contact and centrifugal expansions, non-iteratively determining the complicated tooth engagement regions. Analytical formulas are further implemented with a fast numerical method, which removed restrictions of classical fillet-foundation formulas and covered both thick-walled and thin-walled gears. Quasi-static and dynamic solutions are compared with existing models, FEM, and experiments, showing the acceptable accuracy of this model and demonstrating noticeable centrifugal stiffening in high-speed conditions. To this end, sensitivities of centrifugal stiffening to multi-parameters are further investigated and characterized.