Nonlinear Perturbation Equations Research Articles

Nonlinear analysis of the effect of viscoelasticity on ferroconvection

AbstractThis paper concerns a nonlinear analysis of the effects of viscoelasticity on convection in ferroliquids. We consider the Oldroyd model for the constitutive equation of the liquid. The linear stability analysis yields the critical value of the Rayleigh number for the onset of oscillatory convection in Maxwell and Jeffrey ferroliquids. The use of a minimal mode double Fourier series in the nonlinear perturbation equations yields a Khayat–Lorenz model for the ferromagnetic liquid, and that is scaled further to get the classical Lorenz model as a limiting case. The scaled Khayat–Lorenz model thus obtained is solved numerically and the solution is used to compute the time‐dependent Nusselt number, which quantifies the heat transport. The results are analyzed for the dependence of the time‐averaged Nusselt number on different parameters.

Stability Near Hydrostatic Equilibrium to the 2D Boussinesq Equations Without Thermal Diffusion

This paper furthers our studies on the stability problem for perturbations near hydrostatic equilibrium of the 2D Boussinesq equations without thermal diffusion and solves some of the problems left open in Doering et al. (Physica D 376(377):144–159, 2018). We focus on the periodic domain to avoid the complications due to the boundary. We present several results at two levels: the linear stability and the nonlinear stability levels. Our linear stability results state that the velocity field $$\mathbf {u}$$ associated with any initial perturbation converges uniformly to 0 and the temperature $$\theta $$ converges to an explicit function depending only on y as t tends to infinity. In addition, we obtain an explicit algebraic convergence rate for the velocity field in the $$L^2$$-sense. Our nonlinear stability results state that any initial velocity small in $$L^2$$ and any initial temperature small in $$L^2$$ lead to a stable solution of the full nonlinear perturbation equations in large time. Furthermore, we show that the temperature is eventually stratified and converges to a function depending only on y if we know it admits a certain uniform-in-time bound. An explicit decay rate for the velocity in $$L^2$$ is also ensured if we make assumption on the high-order norms of $$\mathbf {u}$$ and $$\theta $$.

A zonal noise prediction method for trailing-edge noise with a porous model

In this contribution, a hybrid zonal simulation tool with volumetric inflow turbulence forcing is applied to trailing-edge noise of a NACA0012 airfoil with and without a porous insert at representative Mach and Reynolds number of 0.1118 and 1.0 × 106, respectively. The governing equations constitute the non-linear perturbation equations with viscous terms (i.e., the full Navier–Stokes equations), in which the porous material is modelled by a volume-averaged approach. Generic simulations with a single vortex passing the trailing edge revealed the expected noise reduction as well as an additional noise source. This new source originates from the turbulent flow passing the transition from solid to porous surface and was shown to increase significantly with increasing permeability. 3D simulations with a solid and porous trailing-edge showed good agreement with experimental aerodynamic and aeroacoustic validation data. The application of porous material to trailing-edge noise confirms the results reported in literature and underlines the validity of the porous model as well as it illustrates possible applications.

Error Analysis of a Fully Discrete Morley Finite Element Approximation for the Cahn–Hilliard Equation

This paper proposes and analyzes the Morley element method for the Cahn–Hilliard equation. It is a fourth order nonlinear singular perturbation equation arises from the binary alloy problem in materials science, and its limit is proved to approach the Hele-Shaw flow. If the $$L^2(\Omega )$$ error estimate is considered directly as in paper [14], we can only prove that the error bound depends on the exponential function of $$\frac{1}{\epsilon }$$ . Instead, this paper derives the error bound which depends on the polynomial function of $$\frac{1}{\epsilon }$$ by considering the discrete $$H^{-1}$$ error estimate first. There are two main difficulties in proving this polynomial dependence of the discrete $$H^{-1}$$ error estimate. Firstly, it is difficult to prove discrete energy law and discrete stability results due to the complex structure of the bilinear form of the Morley element discretization. This paper overcomes this difficulty by defining four types of discrete inverse Laplace operators and exploring the relations between these discrete inverse Laplace operators and continuous inverse Laplace operator. Each of these operators plays important roles, and their relations are crucial in proving the discrete energy law, discrete stability results and error estimates. Secondly, it is difficult to prove the discrete spectrum estimate in the Morley element space because the Morley element space intersects with the $$C^1$$ conforming finite element space but they are not contained in each other. Instead of proving this discrete spectrum estimate in the Morley element space, this paper proves a generalized coercivity result by exploring properties of the enriching operators and using the discrete spectrum estimate in its $$C^1$$ conforming relative finite element space, which can be obtained by using the spectrum estimate of the Cahn–Hilliard operator. The error estimate in this paper provides an approach to prove the convergence of the numerical interfaces of the Morley element method to the interface of the Hele-Shaw flow.

Zonal Overset-LES with stochastic volume forcing

In this contribution, a hybrid Zonal LES tool is presented that uses a stochastic volume forcing for “inflow” turbulence generation. The here used hybrid approach is based on an implementation of the Non-Linear Perturbation Equations extended with fluctuating viscous terms, denoted as “Overset” since a perturbation analysis is performed on top of a background flow. A turbulent flat plate is considered to investigate the generation of inflow turbulence and how this artificial turbulence is seeded into the computational domain. At the inflow region, turbulence is stochastically generated (in three-dimensional space and time) with the Fast Random Particle-Mesh method and subsequently injected in the Overset-LES domain with the eddy-relaxation source term. This eddy-relaxation source term is shown to pilot the hydrodynamics while leaving the acoustics untouched (i.e., transparent for acoustic). The application of the Overset-LES method to a NACA0012 trailing edge at zero angle-of-attack is furthermore presented. Mean flow and turbulence characteristics in the wake show a good agreement with experimental data. Finally, a possible way to couple the turbulent sound sources into a CAA simulation is described. Comparison between directivities obtained with this method and reference data revealed good agreement.

Fully nonlinear and exact perturbations of the Friedmann world model: non-flat background

We extend the fully non-linear and exact cosmological perturbation equations in a Friedmann background universe to include the background curvature. The perturbation equations are presented in a gauge ready form, so any temporal gauge condition can be adopted freely depending on the problem to be solved. We consider the scalar, and vector perturbations without anisotropic stress. As an application, we analyze the equations in the special case of irrotational zero-pressure fluid in the comoving gauge condition. We also present the fully nonlinear formulation for a minimally coupled scalar field.

Fully non-linear and exact perturbations of the Friedmann world model

In 1988 Bardeen has suggested a pragmatic formulation of cosmological perturbation theory which is powerful in practice to employ various fundamental gauge conditions easily depending on the character of the problem. The perturbation equations are presented without fixing the temporal gauge condition and are arranged so that one can easily impose fundamental gauge conditions by simply setting one of the perturbation variables in the equations equal to zero. In this way one can use the gauge degrees of freedom as an advantage in handling problems. Except for the synchronous gauge condition, all the other fundamental gauge conditions completely fix the gauge mode, and consequently, each variable in such a gauge has a unique gauge invariant counterpart, so that we can identify the variable as the gauge-invariant one. Here, we extend Bardeen's linear formulation to fully nonlinear order in perturbations, with the gauge advantage kept intact. Derived equations are exact, and from these we can easily expand to higher order perturbations in a gauge-ready form. We consider scalar- and vector-type perturbations of an ideal fluid in a flat background; we also present the multiple components of ideal fluid case. As applications we present fully nonlinear density and velocity perturbation equations in Einstein's gravity in the zero-pressure medium, vorticity generation from pure scalar-type perturbation, and fluid formulation of a minimally coupled scalar field, all in the comoving gauge. We also present the equation of gravitational waves generated from pure scalar- and vector-type perturbations.

Static and dynamic consistent perturbation analysis for nonlinear inextensible planar frames

An asymptotically exact method for static and dynamic analysis of geometrically nonlinear planar frames is illustrated. The method is based on an integration of the nonlinear equations for the beam, carried out via a perturbation method, aiming to express the forces at the ends as series expansion of the displacements at the ends and of the distributed loads. Since the beams are assumed to be inextensible and unshearable, also reactive stresses appear among the unknowns, while compatibility conditions must be appended to the equilibrium equations. The element state-relations are assembled for the frame, and discrete, nonlinear perturbation equations are derived. Examples are worked out and relevant results compared with purely numerical solutions.

A low order flow/acoustics interaction method for the prediction of sound propagation using 3D adaptive hybrid grids

A low-order flow/acoustics interaction method for the prediction of sound propagation and diffraction in unsteady subsonic compressible flow using adaptive 3-D hybrid grids is investigated. The total field is decomposed into the flow field described by the Euler equations, and the acoustics part described by the Nonlinear Perturbation Equations. The method is shown capable of predicting monopole sound propagation, while employment of acoustics-guided adapted grid refinement improves the accuracy of capturing the acoustic field. Interaction of sound with solid boundaries is also examined in terms of reflection, and diffraction. Sound propagation through an unsteady flow field is examined using static and dynamic flow/acoustics coupling demonstrating the importance of the latter.

Coupled quintessence and the halo mass function

A sufficiently light scalar field slowly evolving in a potential can account for the dark energy that presently dominates the Universe. This quintessence field is expected to couple directly to matter components, unless some symmetry of a more fundamental theory protects or suppresses it. Such a coupling would leave distinctive signatures in the background expansion history of the Universe and on cosmic structure formation, particularly at galaxy cluster scales. Using semianalytic expressions for the cold dark matter (CDM) halo mass function, we make predictions for halo abundance in models where the quintessence scalar field is coupled to cold dark matter, for a variety of quintessence potentials. We evaluate the linearly extrapolated density contrast at the redshift of collapse using the spherical collapse model and we compare this result to the corresponding prediction obtained from the nonlinear perturbation equations in the Newtonian limit. For all the models considered in this work, if there is a continuous flow of energy from the quintessence scalar field to the CDM component, then the predicted number of CDM haloes can only lie below that of $\ensuremath{\Lambda}\mathrm{CDM}$, when each model shares the same cosmological parameters today. In the last stage of our analysis we perform a global MCMC fit to data to find the best fit values for the cosmological model parameters. We find that for some forms of the quintessence potential, coupled dark energy models can offer a viable alternative to $\ensuremath{\Lambda}\mathrm{CDM}$ in light of the recent detections of massive high-$z$ galaxy clusters, while other models of coupled quintessence predict a smaller number of massive clusters at high redshift compared to $\ensuremath{\Lambda}\mathrm{CDM}$.

Ultrasound-modulated optical tomography: recovery of amplitude of vibration in the insonified region from boundary measurement of light correlation

We address a certain inverse problem in ultrasound-modulated optical tomography: the recovery of the amplitude of vibration of scatterers [p(r)] in the ultrasound focal volume in a diffusive object from boundary measurement of the modulation depth (M) of the amplitude autocorrelation of light [φ(r,τ)] traversing through it. Since M is dependent on the stiffness of the material, this is the precursor to elasticity imaging. The propagation of φ(r,τ) is described by a diffusion equation from which we have derived a nonlinear perturbation equation connecting p(r) and refractive index modulation [Δn(r)] in the region of interest to M measured on the boundary. The nonlinear perturbation equation and its approximate linear counterpart are solved for the recovery of p(r). The numerical results reveal regions of different stiffness, proving that the present method recovers p(r) with reasonable quantitative accuracy and spatial resolution.

Self-equilibration of inverse system in damage detection for dynamic structures

In this study, a modal method coupled with mathematical programming is presented for the assessment of error propagation in damage detection for linear dynamic structural systems. Finite element method is applied to inverse problems in which the structural modifications to the baseline system are sought for the characteristic changes of the perturbed structure. Unlike structural optimization, system identification and damage detection require a unique inverse system. When the exact modal responses are available, accurate structural changes can be obtained through iterative procedures with nonlinear perturbation equations. In practice, however, the modal data may contain numerical errors and consequently yield incorrect inverse solutions that are physically infeasible or quite different from the real perturbed system. Due to the intrinsic instability of eigenproblem, minor errors tend to cause significant element deviations. Hence, mathematical programming techniques are applied to search for a self-equilibrating inverse solution in the close vicinity of the measured data. The proposed approach may provide a breakthrough in the effort to alleviate the numerical difficulty of error propagation in the modal method for the inverse problem of damage detection. The linear perturbation equation is used to obtain an initial approximation, which can be improved through nonlinear iterations.

A SYMPLECTIC ALGORITHM FOR THE STABILITY ANALYSIS OF NONLINEAR PARAMETRIC EXCITED SYSTEMS

A symplectic numerical approach to the stability analysis of nonlinear parametric excited systems with multi-degree-of-freedom is developed and the stability of a nonlinear vibration system depends on the dynamic behavior of its perturbation. The nonlinear parameter-varying differential equations for the perturbation motion are expressed in the form of Hamiltonian equations with time-varying Hamiltonian, and are converted further into the classical Hamiltonian equations with extended time-invariant Hamiltonian by augmenting the state variables. The solution to the augmented Hamiltonian equations has the symplectic structure in terms of the symplectic transformation. Then the difference equations of the symplectic Runge-Kutta algorithm with a sufficient condition are constructed, which is proved to preserve the intrinsic symplectic structure of the original solution. In particular, the symplectic Gauss-Runge-Kutta algorithm with stage 2 and order 4 is proposed and applied to the stability analysis of a nonlinear system. Unstable regions based on the nonlinear periodic-parameter perturbation equation are obtained by using the symplectic Gauss-Runge-Kutta algorithm, analytical solution method and non-symplectic conventional Runge-Kutta algorithm to verify the higher accuracy of the proposed algorithm. Unstable regions based on the nonlinear perturbation are given to illustrate the improvement over those based on the linear perturbation. The developed symplectic approach to the stability analysis can preserve the symplectic structure of the original system and is applicable to nonlinear parametric excited systems with multi-degree-of-freedom.

Scalar field–perfect fluid correspondence and non-linear perturbation equations

The properties of dynamical dark energy (DE) and, in particular, the possibility that itcan form or contribute to stable inhomogeneities have been widely debated inrecent literature, and also in association with a possible coupling betweenDE and dark matter (DM). In order to clarify this issue, in this paper wepresent a general framework for the study of the non-linear phases of structureformation, showing the equivalence of two possible descriptions of DE: a scalar fieldϕ self-interactingthrough a potential V (ϕ) and a perfect fluid with an assigned negative equation of statew(a). This enables us to show that, in the presence of coupling, the mass of DE quanta mayincrease where large DM condensations are present, with the result that also DE may beinvolved in the clustering process.

Computation of aeolian tones from twin-cylinders using immersed surface dipole sources

Efficient numerical method is developed for the prediction of aerodynamic noise generation and propagation in low Mach number flows such as aeolian tone noise. The proposed numerical method is based on acoustic/viscous splitting techniques of which acoustic solvers use simplified linearised Euler equations, full linearised Euler equations and nonlinear perturbation equations as acoustic governing equations. All of acoustic equations are forced with immersed surface dipole model which is developed for the efficient computation of aerodynamic noise generation and propagation in low Mach number flows in which dipole source, originating from unsteady pressure fluctuation on a solid surface, is known to be more efficient than quadrupole sources. Multi-scale overset grid technique is also utilized to resolve the complex geometries. Initially, aeolian tone from single cylinder is considered to examine the effects that the immersed surface dipole models combined with the different acoustic governing equations have on the overall accuracy of the method. Then, the current numerical method is applied to the simulation of the aeolian tones from twin cylinders aligned perpendicularly to the mean flow and separated 3 diameters between their centers. In this configuration, symmetric vortices are shed from twin cylinders, which leads to the anti-phase of the lift dipoles and the in-phase of the drag dipoles. Due to these phase differences, the directivity of the fluctuating pressure from the lift dipoles shows the comparable magnitude with that from the drag dipoles at 10 diameters apart from the origin. However, the directivity at 100 diameters shows that the lift-dipole originated noise has larger magnitude than, but still comparable to, that of the drag-dipole one. Comparison of the numerical results with and without mean flow effects on the acoustic wave emphasizes the effects of the sheared background flows around the cylinders on the propagating acoustic waves, which is not generally considered by the classic acoustic analogy methods. Through the comparison of the results using the immersed surface dipole models with those using point sources, it is demonstrated that the current methods can allow for the complex interactions between the acoustic wave and the solid wall and the effects of the mean flow on the acoustic waves.

Diffuse optical tomography through solving a system of quadratic equations: theory and simulations

This paper discusses the iterative solution of the nonlinear problem of optical tomography. In the established forward model-based iterative image reconstruction (MOBIIR) method a linear perturbation equation containing the first derivative of the forward operator is solved to obtain the update vector for the optical properties. In MOBIIR, the perturbation equation is updated by recomputing the first derivative after each update of the optical properties. In the method presented here a nonlinear perturbation equation, containing terms up to the second derivative, is used to iteratively solve for the optical property updates. Through this modification, reconstructions with reasonable contrast recovery and accuracy are obtained without the need for updating the perturbation equation and therefore eliminating the outer iteration of the usual MOBIIR algorithm. To improve the performance of the algorithm the outer iteration is reintroduced in which the perturbation equation is recomputed without re-estimating the derivatives and with only updated computed data. The system of quadratic equations is solved using either a modified conjugate gradient descent scheme or a two-step linearized predictor–corrector scheme. A quick method employing the adjoint of the forward operator is used to estimate the derivatives. By solving the nonlinear perturbation equation it is shown that the iterative scheme is able to recover large contrast variations in absorption coefficient with improved noise tolerance in data. This ability has not been possible so far with linear algorithms. This is demonstrated by presenting results of numerical simulations from objects with inhomogeneous inclusions in absorption coefficient with different contrasts and shapes.

Initial condition and parameter estimation in physical ‘on’off’ processes by variational data assimilation

In this paper we investigate the feasibility of using the variational data assimilation method to estimate both the initial condition and parameter in an idealized simple model with parametrized discontinuity.Amethod based on the non-linear perturbation equation (NPE) is applied to calculate the gradient of the cost function with respect to the control variables used in the minimization procedure. The results obtained by this method and by the conventional treatment (ignoring the variation of the switch point due to the perturbation in initial condition, i.e. keeping the switching point in the tangent linear model the same as in the basic state) are compared through numerical experiment. Because the cost function could have artificial minima after time discretization, the optimization algorithm combined with the NPE method is less sensitive to the first guess in retrieving the optimal initial condition and parameter value. It is shown that the gradient computation deserves consideration when there are discontinuities in the model used in variational data assimilation.

Direct Approach in Inverse Problems for Dynamic Systems

A direct and effective approach is presented for the inverse problem of dynamic structural systems, which is related to structural optimization, system identification, and damage detection. The structural modifications are sought for the characteristic changes assigned from the design goals or modal measurements. A finite element method is used for the system analysis and inverse problem. Mathematical programming techniques are applied for the minimization of the deviation of the finite element model from the desired inverse system, along with an objective function of least structural change. The modal method is based on the perturbation equations of a set of selected degrees of freedom and the energy equation associated with the frequency change. The mode shape change is expressed as the sum of the baseline mode shape and complementary vector, which plays a very important role in the search for the inverse solution. The linear perturbation equation is employed to get an initial approximation, which can be improved through iterations with the nonlinear perturbation equation. The proposed method does not involve the system reduction. Therefore, it is believed to be superior to conventional methods, which suffer from the residual error due to transformation of the eigenproblem.

Redesign of Cylindrical Shells by Large Admissible Perturbations

Structural redesign is the process of finding a new design that satisfies a set of performance requirements starting from a poorly performing design. Structural redesign is formulated as a two-state problem where the baseline design exhibits undesirable response characteristics and the objective design satisfies the design requirements. A LargE Admissible Perturbations (LEAP) methodology is developed to formulate and solve the problem of structural redesign of cylindrical shells for modal dynamics. First, the nonlinear perturbation equations of cylindrical shells for modal dynamics are derived relating the baseline to the unknown objective design. The redesign problem is formulated as an optimization problem. Next, an algorithm is developed to solve the nonlinear problem and to identify a locally optimal design that satisfies the given modal dynamics specifications. The developed LEAP algorithm calculates incrementally without trial and error or the repetitive finite-element analyses the structural design variables of the objective design. Numerical applications of cylindrical shell redesign for modal requirements are used to verify the methodology and test the algorithm. The developed methodology identifies incompatible frequency requirements where solutions cannot be achieved. Further, systematic redesign applications show that even for strip uniform shells, modes are linked, making satisfaction of multiple frequency objectives impossible.

Topology and performance redesign of complex structures by large admissible perturbations

A methodology for topology redesign of com- plex structures by LargE Admissible Perturbations (LEAP) is developed. LEAP theory is extended to solve topology redesign problems using 8-node solid elements. The corresponding solution algorithm is developed as well. The redesign problem is defined as a two-state problem. State S1 has undesirable characteristics and/or performance not satisfying certain designer specifica- tions. The unknown State S2 has the desired structural response and locally optimum topology. First, the gen- eral nonlinear perturbation equations relating specific response of States S1 and S2 are derived. Next, a LEAP algorithm is developed which solves successfully two-state problems for large structural changes (on the order of 100%-300%) of State S2, without repetitive finite elem- ent analyses, based on the initial State S1 and the spe- cifications for State S2. The solution algorithm is based on an incremental predictor-corrector method. The op- timization problems formulated in both the predictor and corrector phases are solved using commercial non- linear optimization solvers. Minimum change is used as the optimality criterion. The designer specifications are imposed as constraints on modal dynamic and/or static displacement. The static displacement general perturba- tion equation is improved by static mode compensation thus reducing errors significantly. The moduli of elastic- ity of solid elements are used as redesign variables. The LEAP and optimization solvers are implemented in code RESTRUCT (REdesign of STRUCTures) which postpro- cesses finite element analyses results of MSC-NASTRAN. Several topology redesign problems are solved success- fully by code RESTRUCT to illustrate the methodology and study its accuracy. Performance changes on the order of 3300% with high accuracy are achieved with only 3-5 intermediate finite element analyses (iterations) to arrest