Secondary Bifurcation Point Research Articles

Numerical studies on stretch-induced and shear-induced wrinkles of hyperelastic membranes based on a uniformly-valid asymptotic plate theory

In this paper, we study the stretch-induced and shear-induced wrinkles of rectangular hyperelastic membranes numerically based on the weak forms of the uniformly-valid asymptotic plate theory. To trigger wrinkling bifurcation, we impose two small and different forms of imperfection loads on the top surface of the membrane when the membrane is subjected to stretching or shearing, respectively. For the case of stretching, we obtain the first and secondary critical bifurcation points and wrinkled configurations. The effects of microstructure in membrane are also analyzed numerically. For the case of shearing, the numerical solutions can be obtained easily for various pretension displacements including the nearly slack and completely slack membranes. The numerical results show that the pretension displacements could affect the critical shear displacement, the amplitude and the number of wrinkles. The numerical results for both cases are consistent with those in the literature.

Drifting Faraday patterns under localised driving

Physical systems undergoing spontaneous pattern formation are governed by intrinsic length scales that may compete with extrinsic ones, resulting in exceptional spatiotemporal behaviour. In this work, we report experimental and theoretical evidence that spatial nonuniformity sets Faraday-wave patterns in motion, which are noticeable in the zigzag and drift dynamics exhibited by their wave crests. We provide a minimal theoretical model that succeeds in characterising the growth of localised patterns under nonuniform parametric driving. Furthermore, the derivation accounts for symmetry-breaking nonlinear gradients that we show are the source of the drift mechanism, which comes into play right after the system has crossed a secondary bifurcation point. Numerical solutions of the governing equations match our experimental findings and theoretical predictions. Our results advance the understanding of pattern behaviour induced by nonuniformity in generic nonlinear extended systems far from equilibrium.

All global bifurcation diagrams of stationary solutions to a phase field model

<p style='text-indent:20px;'>We are concerned with bifurcation diagrams of stationary solutions to a phase field model proposed by Fix and followed by Caginalp. We show all the global bifurcation diagrams of stationary solutions to the model in the 1-dimension case. We see that bifurcation diagrams are surprisingly rich in variety depending on the latent heat and the initial total enthalpy. For instance, bifurcation diagrams include the secondary bifurcation point where symmetric breaking occurs, and curves which connect a limit of boundary layer solutions to the other limit of internal layer solutions.</p>

Rayleigh–Bénard flow for a Carreau fluid in a parallelepiped cavity

A continuation method is used to study Rayleigh–Bénard convection in a non-Newtonian fluid inside a parallelepiped cavity. The cavity has its length equal to twice the side of the square cross-section. Shear-thinning and shear-thickening Carreau fluids are considered. The focus is put on the two stable branches which exist in the Newtonian case, a stable primary branch of transverse rolls $B_1$ and a primary branch of longitudinal roll $B_2$ , stabilized beyond a secondary bifurcation point $S_2$ . Although the primary bifurcation points are unchanged, the non-Newtonian properties strongly modify the bifurcation diagram. Indeed, for a shear-thinning fluid, the stable solutions can exist at much smaller Rayleigh numbers ${Ra}$ , on subcritical branches beyond saddle-node points $SN_1$ and $SN_2$ , and small perturbations can be sufficient to reach them. In agreement with Bouteraa et al. (J. Fluid Mech., vol. 767, 2015, pp. 696–734), the change of the primary bifurcations from supercritical to subcritical occurs at given values of what they define as the degree of shear-thinning parameter $\alpha$ . Moreover, the value of the Rayleigh number at the saddle-node points can be approximated by a simple expression, as proposed by Jenny et al. (J. Non-Newtonian Fluid Mech., vol. 219, 2015, pp. 19–34). In the case of a shear-thickening fluid, the branches remain supercritical, but the secondary point $S_2$ is strongly moved towards larger ${Ra}$ , making it more difficult to reach the longitudinal roll solution. Energy analyses at the bifurcations $SN_1$ , $SN_2$ and $S_2$ show that the changes of the corresponding critical thresholds ${Ra}_c$ are connected with the changes of the viscous properties, but also with changes of the buoyancy effect.

Representation formulas of solutions and bifurcation sheets to a nonlocal Allen-Cahn equation

We are interested in the Neumann problem of a 1D stationary Allen-Cahn equation with a nonlocal term. In our previous papers [ 4 ] and [ 5 ], we obtained a global bifurcation branch, and showed the existence and uniqueness of secondary bifurcation point. At this point, asymmetric solutions bifurcate from a branch of odd-symmetric solutions. In this paper, we give representation formulas of all solutions on the secondary bifurcation branch, and a bifurcation sheet which consists of bifurcation curves with heights.

Analysis and computations of a non-local thin-film model for two-fluid shear driven flows.

This paper is concerned with analysis and computations of a non-local thin-film model developed in Kalogirou & Papageorgiou (J. Fluid Mech. 802, 5-36, 2016) for a perturbed two-layer Couette flow when the thickness of the more viscous fluid layer next to the stationary wall is small compared to the thickness of the less viscous fluid. Travelling wave solutions and their stability are determined numerically, and secondary bifurcation points are identified in the process. We also determine regions in parameter space where bistability is observed with two branches being linearly stable at the same time. The travelling wave solutions are mathematically justified through a quasi-solution analysis in a neighbourhood of an empirically constructed approximate solution. This relies in part on precise asymptotics of integrals of Airy functions for large wave numbers. The primary bifurcation about the trivial state is shown rigorously to be supercritical, and the dependence of bifurcation points, as a function of Reynolds number R and the primary wavelength 2πν -1/2 of the disturbance, is determined analytically.

Resonances between fundamental frequencies for lasers with large delayed feedbacks.

High-order frequency locking phenomena were recently observed using semiconductor lasers subject to large delayed feedbacks. Specifically, the relaxation oscillation (RO) frequency and a harmonic of the feedback-loop round-trip frequency coincided with the ratios 1:5 to 1:11. By analyzing the rate equations for the dynamical degrees of freedom in a laser subject to a delayed optoelectronic feedback, we show that the onset of a two-frequency train of pulses occurs through two successive bifurcations. While the first bifurcation is a primary Hopf bifurcation to the ROs, a secondary Hopf bifurcation leads to a two-frequency regime where a low frequency, proportional to the inverse of the delay, is resonant with the RO frequency. We derive an amplitude equation, valid near the first Hopf bifurcation point, and numerically observe the frequency locking. We mathematically explain this phenomenon by formulating a closed system of ordinary differential equations from our amplitude equation. Our findings motivate experiments with particular attention to the first two bifurcations. We observe experimentally (1) the frequency locking phenomenon as we pass the secondary bifurcation point and (2) the nearly constant slow period as the two-frequency oscillations grow in amplitude. Our results analytically confirm previous observations of frequency locking phenomena for lasers subject to a delayed optical feedback.

Global dynamics and integrity of a micro-plate pressure sensor

This paper investigates the complex bifurcation scenario of electrically-actuated circular micro-plates subjected to differential pressure. Our analysis deals with the primary, secondary and ultimate saddle-node bifurcation points responsible for the device snap-through and addresses the pressure range in which the robustness of the two main stable configurations is undermined by minor coexisting attractors. By making use of basins of attraction and integrity profiles, safe dynamical regions of motion are evaluated with respect to the applied DC voltage. It is found that period-doubling bifurcations are accountable for a sensible reduction in the dynamical integrity, small variations of the DC voltage largely modify the response of the pressure sensor.

Secondary bifurcation for a nonlocal Allen–Cahn equation

This paper studies the Neumann problem of a nonlocal Allen–Cahn equation in an interval. A main result finds a symmetry breaking (secondary) bifurcation point on the bifurcation curve of solutions with odd-symmetry. Our proof is based on a level set analysis for the associated integral map. A method using the complete elliptic integrals proves the uniqueness of secondary bifurcation point. We also show some numerical simulations concerning the global bifurcation structure.

Rigorous continuation of bifurcation points in the diblock copolymer equation

We develop general methods for rigorously computing continuous branches of bifurcation points of equilibria, specifically focusing on fold points and on pitchfork bifurcations which are forced through \begin{document}${\mathbb{Z}}_2$\end{document} symmetries in the equation. We apply these methods to secondary bifurcation points of the one-dimensional diblock copolymer model.

Numerical bifurcation for the capillary Whitham equation

The so-called Whitham equation arises in the modeling of free surface water waves, and combines a generic nonlinear quadratic term with the exact linear dispersion relation for gravity waves on the free surface of a fluid with finite depth.In this work, the effect of incorporating capillarity into the Whitham equation is in focus. The capillary Whitham equation is a nonlocal equation similar to the usual Whitham equation, but containing an additional term with a coefficient depending on the Bond number which measures the relative strength of capillary and gravity effects on the wave motion.A spectral collocation scheme for computing approximations to periodic traveling waves for the capillary Whitham equation is put forward. Numerical approximations of periodic traveling waves are computed using a bifurcation approach, and a number of bifurcation curves are found. Our analysis uncovers a rich structure of bifurcation patterns, including subharmonic bifurcations, as well as connecting and crossing branches. Indeed, for some values of the Bond number, the bifurcation diagram features distinct branches of solutions which intersect at a secondary bifurcation point. The same branches may also cross without connecting, and some bifurcation curves feature self-crossings without self-connections.

Postbuckling and Mode Jumping Analysis of Deep Hygrothermally Buckled Angle-Ply Laminated Plates

An analytical investigation is performed to study the secondary instability and dynamic aspects of the mode jumping in hygrothermally buckled angle-ply laminated plates. The governing partial differential equations (PDEs) and constitution relations are derived rigorously from an asymptotically correct, geometrically nonlinear theory. A novel and relatively simpler solution approach is developed to solve the two coupled fourth-order PDEs, namely, the compatibility equation and the dynamic governing equation. The von Kármán plate equation, namely, the coupled nonlinear governing equation, is reduced to a system of nonlinear ordinary differential equations (ODEs) by expressing the transverse deflection as a series of linear buckling modes. The ODEs, combined with the nonlinear algebraic constraint equations obtained from in-plane boundary conditions, are then solved numerically under the parametric variation of the temperature and moisture. The comparison between the present method and the FEA shows that the secondary bifurcation point of the hygrothermally loaded plate is far beyond the primary buckling point, and the jump behavior cannot be predicted correctly without sufficient assumed modes, while the present method has the capability of exploring deeply into the post-secondary buckling realm and capture the mode jumping phenomenon for various combinations of plate configurations boundary conditions. Furthermore, by monitoring free vibration along the stable primary postbuckling and the jumped equilibrium paths, we find that a mode shifting phenomenon (the exchange of vibration modes) exists in the primary post-buckling regime.

Study on shear postbuckling failure of composite sandwich plates by a quasi-conforming finite element method

Abstract A quasi-conforming triangular sandwich plate element with 7 degrees of freedom (DOF) per node is established based on a simplified Zig-Zag model and a transverse shear deformation theory. The accuracy of the proposed element has been validated to be equivalent to the element based on the complete Zig-Zag model (11 DOF per note). The path-following method, proposed by Huang and Atluri, is improved and applied to follow complex postbuckling paths with secondary bifurcation points and progressive failure. A finite element (FE) modeling is completed to study the shear postbuckling behavior and failure of composite sandwich plates, based on the energy criterion of stability and the strength criterion of each layer. The basic characteristics of the shear postbuckling and failure are illustrated. The complexity and computation difficulty caused by secondary bifurcations and multi-paths are discussed. Numerical examples show that the proposed method is efficient in improving the convergence of iterations.

Bifurcation analysis of steady natural convection in a tilted cubical cavity with adiabatic sidewalls

AbstractNatural convection in an inclined cubical cavity heated from two opposite walls maintained at different temperatures and with adiabatic sidewalls is investigated numerically. The cavity is inclined by an angle $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\theta $ around a lower horizontal edge and the isothermal wall set at the higher temperature is the lower wall in the horizontal situation ($\theta = 0^\circ $). A continuation method developed from a three-dimensional spectral finite-element code is applied to determine the bifurcation diagrams for steady flow solutions. The numerical technique is used to study the influence of ${\theta }$ on the stability of the flow for moderate Rayleigh numbers in the range $\mathit{Ra} \leq 150\, 000$, focusing on the Prandtl number $\mathit{Pr} = 5.9$. The results show that the inclination breaks the degeneracy of the stable solutions obtained at the first bifurcation point in the horizontal cubic cavity: (i) the transverse stable rolls, whose rotation vector is in the same direction as the inclination vector ${\boldsymbol{\Theta}}$, start from $\mathit{Ra} \to 0$ forming a leading branch and becoming more predominant with increasing $\theta $; (ii) a disconnected branch consisting of transverse rolls, whose rotation vector is opposite to ${\boldsymbol{\Theta}}$, develops from a saddle-node bifurcation, is stabilized at a pitchfork bifurcation, but globally disappears at a critical inclination angle; (iii) the semi-transverse stable rolls, whose rotation axis is perpendicular to ${\boldsymbol{\Theta}}$ at $\theta \to 0^\circ $, develop from another saddle-node bifurcation, but the branch also disappears at a critical angle. We also found the stability thresholds for the stable diagonal-roll and four-roll solutions, which increase with $\theta $ until they disappear at other critical angles. Finally, the families of stable solutions are presented in the $\mathit{Ra}-\theta $ parameter space by determining the locus of the primary, secondary, saddle-node, and Hopf bifurcation points as a function of $\mathit{Ra}$ and $\theta $.

Secondary instability and mode jumping analysis of deep hygrothermally buckled cross-ply laminated plates

An analytical investigation is performed to study the secondary instability and dynamic aspects of the mode jumping in hygrothermally buckled cross-ply laminated plates. The governing partial differential equations (PDEs) and constitution relations are derived rigorously from an asymptotically correct, geometrically nonlinear theory. A novel and relatively simpler solution approach is developed to solve the two coupled fourth-order PDEs, namely, the compatibility equation and the dynamic governing equation. The von Kàmàn plate equation, namely, the coupled nonlinear governing equation, is reduced to a system of nonlinear ordinary differential equations (ODEs) by expressing the transverse deflection as a series of linear buckling modes. The ODEs, combined with the non-linear algebraic constraint equations obtained from in-plane boundary conditions, are then solved numerically under the parametric variation of the temperature and moisture. The comparison between the present method and the FEA shows that the secondary bifurcation point of the hygrothermally loaded plate is far beyond the primary buckling point, and the jump behavior cannot be predicted correctly without sufficient assumed modes, while the present method has the capability of exploring deeply into the post-secondary buckling realm and capturing the mode jumping phenomenon for various combinations of plate configurations boundary conditions. Furthermore, by monitoring free vibration along the stable primary postbuckling and the jumped equilibrium paths, we find that a mode shifting phenomenon (the exchange of vibration modes) exists in the primary post-buckling regime.

Three-dimensional continuation study of convection in a tilted rectangular enclosure

A continuation method developed from a three-dimensional spectral finite element code is used to study natural convection in a tilted rectangular cavity. The cavity has its length equal to two times the side of its square cross section and it contains a fluid with a Prandtl number Pr = 1. A detailed bifurcation diagram is first obtained in the case without inclination in order to get the sequence of the different branches of solutions and determine the stable solutions. The focus is then put on the stable solutions in the inclined cavity, when the tilt occurs around its longest axis. The subtle changes induced by the tilt on the convective system are clarified. Three different stable solutions are obtained: the longitudinal roll L- solution (with the same sense of rotation as the inclination angle), which develops smoothly from zero Rayleigh number on the leading branch; the longitudinal roll L+ solution (with a sense of rotation opposite to the inclination angle), which is on a disconnected branch and is stabilized beyond a secondary bifurcation point; the oblique roll O ± solutions (corresponding to transverse roll solutions perturbed by the longitudinal flow induced by the tilt), which quickly appear beyond saddle-node points on new disconnected branches. The domain of existence of these stable solutions is eventually obtained and described in the Rayleigh number-inclination parameter space. Finally, the Nusselt number is determined as a function of the inclination at a constant Rayleigh number for the different stable solutions. The Nusselt number is maximum at an inclination of 49.55° for the leading longitudinal roll L- solution.

Primary and secondary bifurcations of a compressible hyperelastic layer: Asymptotic model equations and solutions

In this paper, we study the equilibrium states of a compressible hyperelastic layer under compression after the primary and secondary bifurcations. Starting from the two-dimensional field equations for a compressible hyperelastic material, we use a methodology of coupled series-asymptotic expansions developed earlier to derive two coupled non-linear ordinary differential equations (ODEs) as the model equations. The critical buckling stresses are determined by a linear bifurcation analysis, which are in agreement with the results in the literature. The method of multiple scales is used to solve the model equations to obtain the second-order asymptotic solutions after the primary bifurcations. An analytical formula for the post-buckling amplitudes is derived. Two kinds of numerical solutions are also obtained, the numerical solutions of the model equations by a difference method and those of the two-dimensional field equations by the finite elements method. Comparisons among the analytical solutions, numerical solutions and solutions obtained by the Lyapunov–Schmidt–Koiter (LSK) method in the literature are made and good agreements for the displacements are found. It is also found that at some places the axial strain is tensile, although the layer is under compression. To consider the secondary bifurcation, we superimpose a small deformation on the state after the primary bifurcation. With the analytical solution of the primary bifurcation, we manage to reduce the problem of the secondary bifurcation to one of the first bifurcations governed by a second order variable-coefficient ODE. And, our analysis identifies an explicit function and from the existence/non-existence of its zero one can immediately judge whether a secondary bifurcation can take place or not. The zero corresponds to a turning point of the governing ODE, which leads to non-trivial solutions. Further, by the WKB method the equation (in a very simple form) for determining the critical stress for the secondary bifurcation is derived. We further use AUTO to compute the secondary bifurcation point numerically, which confirms the validity of our analytical results. The numerical solution in the secondary bifurcation branch is also computed by AUTO. It is found that the secondary bifurcation induces a “wave number doubling” phenomenon and also the shape of the layer has a convexity change along the axial direction.

Branch interactions and long-term dynamics for the diblock copolymer model in one dimension

Diblock copolymers are a class of materials formed by the reaction of two linear polymers. The different structures taken on by these polymers grant them special properties, which can prove useful in applications such as the development of new adhesives and asphalt additives. We consider a model for the formation of diblock copolymers first proposed by Ohta and Kawasaki [26]. Their model yields a Cahn-Hilliard-like equation, where a nonlocal term is added to the standard Cahn-Hilliard energy. We study the long-term dynamics of this model on one-dimensional domains through a combination of bifurcation theoretic results and numerical simulations. Our results shed light on how the complicated bifurcation behavior of the diblock copolymer model is related to the better known bifurcation structure of the Cahn-Hilliard equation. In addition, we demonstrate that this knowledge can be used to predict the long-term dynamics of solutions originating close to the homogeneous equilibrium. In particular, we show that the periodicity of the long-term limit of such solutions can be predicted by tracking certain secondary bifurcation points in the bifurcation diagram, and that the long-term limit is in general not given by the global energy minimizer.

Monotonicity of the first eigenvalue and the global bifurcation diagram for the branch of interior peak solutions

Let B⊂Rn (n⩾1) be a unit ball and D⊂Rm be a bounded domain. We study the global branch consisting of interior single-peak solutions of the elliptic Neumann problemΔu+λ(−u+up)=0in B,∂νu=0on ∂B and the monotonicity of the first eigenvalue along the branch for large λ, where1<p<pS:={n+2n−2,n⩾3,∞,n=1,2. When the domain is replaced with B×D, using this monotonicity, we show that the branch of solutions concentrating on {0}×D has secondary bifurcation points. For n=1 and p⩾2 an integer, we determine the global bifurcation diagram by showing the monotonicity of the time-map (the period function). The monotonicity of the first eigenvalue along the whole branch is also proved for n=1 and p=3.

Plastic buckling of thin hemispherical shell subjected to concentrated load at the apex

This study presents the analytical, numerical, and experimental results of thin hemispherical metal shells into the plastic buckling range illustrating the importance of geometry changes on the buckling load. The hemispherical shell is rigidly supported around the base circumference against horizontal and vertical translation and the load is vertically applied by a rigid cylindrical boss at the apex. Kinematics stages of initial buckling and subsequent propagation of plastic deformation for rigid-perfectly plastic shells are formulated on the basis of Drucker–Shield's limited interaction yield condition. The effect of the radius of the boss, used to apply the loading, on the initial and subsequent collapse load is studied. In the numerical model, the material is assumed to be isotropic and linear elastic perfectly plastic without strain hardening obeying the Tresca or Von Mises yield criterion. Both axisymmmetric and 3D models are implemented in the numerical work to verify the presence of non-symmetric deformation modes in the case of thin shells. In the end, the results of the analytical solution are compared and verified with the numerical results using ABAQUS software and experimental findings. Good agreement is observed between the load–deflection curves obtained using three different approaches. A secondary bifurcation point is detected in thin shells in which the deformation degenerates from symmetric to non-symmetric behavior. The bifurcation point depends on the (R/t0) ratio and the material parameters.