Nonlinear Steady-state Problems Research Articles

Thermomechanical analysis of the effect of contact forces in heat conduction of composite materials through multiscale contact finite element modeling

The present study proposes a thermomechanical surface contact finite element formulation for studying interfacial heat conductance in correlation to micromechanical contact forces. The proposed methodology is applied for modeling heat surface conductivity of Carbon Nanotubes (CNT). The atomic lattice of CNTs is initially modeled using the molecular structural mechanics approach and reduced to an equivalent 2D continuum element. The equivalent 2D continuum element is used as a robust and efficient surrogate model for the construction of full length CNTs in contact. Coupled structural and heat pde’s have been used to describe the corresponding thermomechanical problem which are discretized using 2D quadrilateral finite elements, along with surface contact elements based on the Node to Node and Node to Segment formulations. The problem has been treated as a nonlinear steady state problem and solved in the frame of a Newton-Raphson incremental-iterative scheme. Extensive sensitivity analysis was performed with respect to several influencing parameters, such as initial contact angle, boundary conditions and CNT overlapping. Based on this analysis, the effect of overlapping and approaching angle on the total heat exchange was quantified. It has been demonstrated that in most cases, increasing the total contact force can produce an almost linear increase in heat exchange. However, in cases of small approaching angles between CNTs an extensive contact force could heavily deform the contacting geometries producing a non-linear heat exchange mechanism.

Harmonic balance-boundary element and continuation methods for steady-state wave scattering by interior and surface-breaking cracks with contact acoustic nonlinearity

A numerical method for nonlinear steady-state wave scattering due to a crack with contact acoustic nonlinearity (CAN) is developed in this article. The in-plane scattering problem is addressed here. The two systems studied are composed of an unbounded elastic solid that includes an interior crack and an elastic half plane that includes a surface-breaking crack. A time-harmonic plane wave is incident, and clapping and rubbing motions on the crack faces are induced as nonlinear phenomena. A harmonic balance-boundary element method (HB-BEM) is utilized to solve the nonlinear steady-state problem. The system of equations obtained by the HB-BEM consists of an elastostatic boundary integral equation (BIE) and frequency-domain elastodynamic BIEs of several different frequencies, which are coupled by nonlinear boundary conditions. To solve the system numerically, the nonlinear boundary conditions are regularized. In the regularized boundary conditions, traction is expressed by a smooth single-valued function with respect to crack-opening displacement. The solution of the steady-state system is tracked with a parameter continuation method to investigate bifurcations. The parameters that vary in the continuation method are the incident frequency and static field variables. To determine the stability of the steady-state solutions, a stability analysis method based on Hill’s infinite determinant is proposed.The stable solutions obtained by the proposed method agree with the conventional time-domain solutions after enough time has elapsed. A 1/2-order subharmonic resonance is observed with bifurcations for particular parameters. The interaction between the local resonant properties due to the crack and the CAN causes subharmonic resonance.

A Multilayer Nonlinear Elimination Preconditioned Inexact Newton Method for Steady-State Incompressible Flow Problems in Three Dimensions

We develop a multilayer nonlinear elimination preconditioned inexact Newton method for a nonlinear algebraic system of equations, and a target application is the three-dimensional steady-state incompressible Navier--Stokes equations at high Reynolds numbers. Nonlinear steady-state problems are often more difficult to solve than time-dependent problems because the Jacobian matrix is less diagonally dominant, and a good initial guess from the previous time step is not available. For such problems, Newton-like methods may suffer from slow convergence or stagnation even with globalization techniques such as line search. In this paper, we introduce a cascadic multilayer nonlinear elimination approach based on feedback from intermediate solutions to improve the convergence of Newton iteration. Numerical experiments show that the proposed algorithm is superior to the classical inexact Newton method and other single layer nonlinear elimination approaches in terms of the robustness and efficiency. Using the proposed nonlinear preconditioner with a highly parallel domain decomposition framework, we demonstrate that steady solutions of the Navier--Stokes equations with Reynolds numbers as large as 7,500 can be obtained for the lid-driven cavity flow problem in three dimensions without the use of any continuation methods.

Nonlinear optical response of a two-dimensional quantum-dot supercrystal: Emerging multistability, periodic and aperiodic self-oscillations, chaos, and transient chaos

We conduct a theoretical study of the nonlinear optical response of a two-dimensional semi-conductor quantum dot supercrystal subjected to a quasi-resonant continuous wave excitation. Aconstituent quantum dot is odeled as a three-level ladder-like system (comprising the ground, theone-exciton, and the bi-exction states). To study the stationary response of the supercrystal, wepropose an exact linear parametric method of solving the nonlinear steady-state problem, whileto address the supercrystal optical dynamics qualitatively, we put forward a novel method to cal-culate the bifurcation diagram of the system. Analyzing the dynamics, we demonstrate that thesupercrystal can exhibit multistability, periodic and aperiodic self-oscillations, and chaotic behavior,depending on parameters of the supercrystal and excitation conditions. The effects originate fromthe interplay of the intrinsic nonlinearity of quantum dots and the retarded inter-dot dipole-dipoleinteraction. The latter provides a positive feedback which results in the exotic supercrystal opticaldynamics. These peculiarities of the supercrystal optical response open up a possibility for all-opticalapplications and devices. In particular, an all-optical switch, a tunable generator of THz pulses (inself-oscillating regime), a noise generator (in chaotic regime), and a tunable bistable mirror can bedesigned.

A Modified Polynomial Expansion Algorithm for Solving the Steady-State Allen-Cahn Equation for Heat Transfer in Thin Films

Meshfree algorithms offer a convenient way of solving nonlinear steady-state problems in arbitrary plane areas surrounded by complicated boundary shapes. The simplest of these is the polynomial expansion approach. However, it is rarely utilized as a primary tool for this purpose because of its rather ill-conditioned behavior. A well behaved polynomial expansion algorithm is presented in this paper which can be more effectively used to solve the steady-state Allen-Cahn (AC) equation for heat transfer in thin films. In this method, modified polynomial expansion was used to cope with each iteration of the steady-state Allen-Cahn equation to produce nonlinear algebraic equations where multiple scales are automatically determined by the collocation points. These scales can largely decrease the condition number of the coefficient matrix in each nonlinear system, so that the iteration process converges very quickly. The numerical solutions were found to be accurate and stable against moderate noise to better than 7.5%. Computational results verified the method and showed the steady-state Allen-Cahn equation for heat transfer in thin films could easily be resolved for several arbitrary plane domains.

Comparison of Exact Solutions for Heat Transfer in Extended Surfaces of Different Geometries

We consider a steady state problem for heat transfer in fins of various geometries, namely, rectangular, radial, and spherical. The nonlinear steady state problem is linearizable provided that the thermal conductivity is the differential consequence of the term involving the heat transfer coefficient. As such, one is able to construct exact solutions. On the other hand, we employ the Lie point symmetry methods when the problem is not linearizable. Some interesting results are obtained and analyzed. The effects of the parameters such as thermogeometric fin parameter and the exponent on temperature are studied. Furthermore, fin efficiency and heat flux along the fin length of a spherical geometry are also studied.

On the steady-state relations between disturbances above and below a critical level

In studying the behaviour of a density-stratified shear flow difficulties are encountered at the ‘critical’ level where wave velocity equals fluid velocity.Here a stratified shear layer of finite thickness is considered and a two-dimensional nonlinear steady-state problem is studied. It is assumed that blocking creates separate pockets of trapped fluid, each mixed to uniform density. These pockets are not in static equilibrium with the surrounding stratified fluid. They must be supported either by pressures dynamically developed in the curved flow along continuous streamlines outside the pockets or by centrifugal forces resulting from circulation within the pockets. The latter effect is considered only through evaluation of a crude ‘factor of importance’, FR, for the rotational effects and the pockets are assumed to be stagnant in the primary analysis.For small but finite disturbance amplitude FR approaches zero, indicating that no correction of the primary analysis is required. A limiting Richardson number of unity appears. Above this limit the primary analysis gives no solutions and apparently the separate pockets of stagnant fluid merge to form a continuous stagnant insulating layer. This behaviour of the critical level (as a barrier to communication) resembles earlier results from transient linearized investigations although the two analyses have little in common except the existence of a critical level separating two fluid regions.For moderate-to-large disturbance amplitudes the geometry of the flow pattern suggests Kelvin–Helmholtz billows. Rotational effects increase as the amplitude increases and may become significant at this stage. The primary analysis then becomes less accurate and cannot be used to exclude Kelvin–Helmholtz billows at Richardson numbers somewhat greater than unity.