Double-well Function Research Articles

Distance-regularized level set inversion of magnetic data

Inversion is one of the key and difficult issues in potential field data processing and interpretation, and high-precision inversion has long been a popular research topic. We develop the distance-regularized level set inversion of the magnetic data (DRLSI-M) method. A double-well function is introduced into the objective function of the level set inversion, and we take advantage of its mathematical properties to minimize the gradient of the level set function so that it reaches a minimum value at points 0 and 1. As a result, the distance-regularized term not only replaces the costly reinitialization process so that the level set function maintains a signed distance property but also stabilizes the evolution of the level set function. When solving for the minimum value of the objective function, we transform the optimization problem into an initial-boundary value problem and solve it with the finite-difference method. The distance-regularized level set inversion method and the reinitialization level set inversion method are tested using three models: a single-level-set double-inclined plate model, a double-level-set double-inclined plate model, and a multiple-level-set three-cuboid model. Through model tests, the soundness of our method is verified and compared with the reinitialization level set inversion method. Our method avoids the limitations such as low efficiency and instability caused by the reinitialization of the level set inversion method. Furthermore, we apply our method for the inversion of aeromagnetic data from the Jinchuan copper-nickel mine. The results are consistent with the known geologic information, thus validating the practicality and effectiveness of DRLSI-M. The distance-regularized level set inversion method provides a theoretical basis for deep mine exploration.

Colloid dynamics near phase transition: A model for the relaxation of concentrated layers

The dynamics of concentrated colloidal dispersion close to a phase transition are challenging to predict. The presence of attractive interactions leads to hindered motions through the change in transport properties such as collective diffusion. We propose a continuum mechanical model to describe the dynamics of the relaxation of concentrated colloids near a phase transition. The model relies on a specific description of the equation of state of colloids, the osmotic pressure deriving from a free energy double-well function, that induces a decrease in collective diffusion close to the phase transition. The implementation of such transport properties in Computational Fluid Dynamics (CFD) codes enabled the description of a fast expansion of accumulated gels near an interface (induced for instance by evaporation or filtration processes) followed by a relaxation stage, where particles are progressively released from the layer toward the dilute bulk region. The model proposed in this work captures the increase of the relaxation time when approaching the transition, a feature of interest for engineering colloidal mixtures with concentration gradients.

Binary-fluid–solid interaction based on the Navier–Stokes–Korteweg equations

We consider a computational model for binary-fluid–solid interaction based on an arbitrary Lagrangian–Eulerian formulation of the Navier–Stokes–Korteweg equations, and we assess the predictive capabilities of this model. Due to the presence of two distinct fluid components, the stress tensor in the binary-fluid exhibits a capillary component in addition to the pressure and viscous-stress components. The distinct fluid–solid surface energies of the fluid components moreover lead to preferential wetting at the solid substrate. Compared to conventional FSI problems, the dynamic condition coupling the binary-fluid and solid subsystems incorporates an additional term associated with the binary-fluid–solid surface tension. We consider a formulation of the Navier–Stokes–Korteweg equations in which the free energy associated with the standard van-der Waals equation of state is replaced by a polynomial double-well function to provide better control over the diffuse-interface thickness and the surface tension. For the solid subsystem, we regard a standard hyperelastic model. We explore the main properties of the binary-fluid–solid interaction problem and establish a dissipation relation for the aggregated system. In addition, we present numerical results based on a fully monolithic approach to the complete nonlinear system. To validate the computational model, we consider the elasto-capillary interaction of a sessile droplet on a soft solid substrate and compare the numerical results with a corresponding solid model with fabricated fluid loads and with experimental data.

Binary image deblurring with automatic binary value estimation

We propose a total variation-based variational model for nonblind binary image deblurring. The binary constraint is considered using the double-well function as the penalty term. We show the existence of a minimizer for the proposed model. By using operator splitting and alternating split Bregman, we get an effective numerical algorithm for the proposed model. Different from the existing methods in which the binary values are assumed to be known, our method can estimate the binary values automatically in the iteration process. Numerical results and comparisons demonstrate that the proposed algorithm is promising.

Nonlocal phase transitions in homogeneous and periodic media

We discuss some results related to a phase transition model in which the potential energy induced by a double-well function is balanced by a fractional elastic energy. In particular, we present asymptotic results (such as $\Gamma$-convergence, energy bounds and density estimates for level sets), flatness and rigidity results, and the construction of planelike minimizers in periodic media. Finally, we consider a nonlocal equation with a multiwell potential, motivated by models arising in crystal dislocations, and we construct orbits exhibiting symbolic dynamics, inspired by some classical results by Paul Rabinowitz.

Phase-field Simulation of Lithium Ion Diffusion in Solid Electrolyte Interphase

Fracture occurs in silicon-based electrodes due to large volume change during the lithiation/delithiation in lithium-ion batteries. A stable solid electrolyte interphase (SEI) can effectively prevent fractures. However, lithium-ion diffusion inside SEI layer has not been fully understood. In this study, a phase-field model is developed to simulate two-dimensional formation of SEI layer and lithium-ion diffusion inside SEI layer formed on silicon electrode. The details of SEI species are confirmed by XPS experiments reported in the literature. Double-well function is applied to represent the total free energy density considering lithium-ion concentration inside the SEI layer. Simulation results show lithium-ion diffusion coefficients between 298K and 318K are 2.021-2.211(10-16) cm2/s, 7.146-8.038(10-13) cm2/s, and 1.448-1.585(10-15) cm2/s for compact, porous, and multilayered SEI layers, respectively. Resistances of the aforementioned SEI layers between 298K and 318K are 1.484-2.609 Ω cm2, 4.175-6.372 Ω cm2, and 5.171-7.007 Ω cm2.

Diffuse interface tracking of immiscible fluids: Improving phase continuity through free energy density selection

Diffuse interface (DI) tracking methods frequently adopt the double-well energy density function to describe the free energy variation across an interface, leading to phase interpenetration and spontaneous drop shrinkage when applied to immiscible two-phase systems. While the observed continuity losses can be limited by constraints placed on the interfacial width and mobility parameter, the associated increase in computational cost and mesh requirements has limited DI methods to 2D planar and axi-symmetric flow. Using the proposed temperature-variant simplified energy density function (TVSED), the effect of the metastable thermodynamic region on phase continuity is examined using a reduced temperature parameter, T R . The solutions obtained by the proposed DI implementation and a volume-of-fluid (VOF) based solver are discussed for two common benchmark simulations (collapse of a column of water, and droplet deformation/relaxation in simple shear). While comparable solutions are obtained from the T R = 0 and VOF simulations; the large metastable region of the more commonly used double-well function ( T R = 1) promoted inter-mixing in the bulk phases, diluting the inertia transferred between impacting fluids and under-predicting deformation in shear. The fundamental mechanisms responsible for spontaneous drop shrinkage are eliminated at T R = 0, allowing for constraints on the interfacial width and mobility to be relaxed. The comparable performance of T R = 0 and VOF simulations is a promising indication of the potential for application to complex 3D flows at reduced mesh resolution relative to existing DI methods.

Exploiting Benefits of a Periodically-Forced Nonlinear Oscillator for Energy Harvesting from Ambient Vibrations

In this paper, we present an experimental and theoretical study of a nonlinear oscillator for energy harvesting applications. The collected energy comes from ambient vibrations as external energy sources. Specifically, we consider both periodic (e.g. 50Hz or 60Hz power line couplings) and stochastic sources (e.g. human-made devices, automobiles, train, etc.). The proposed mechanism is a double-well oscillator with a large amplitude response over a broad range of frequencies. Such system improves proposed scavengers based on linear mechanical principles, which only give appreciable response if the dominant ambient vibration frequency is close to the mechanical resonance. At the same time, it improves nonlinear harvesters (e.g. bi-stable) proposed in the literature by decreasing the vibration amplitude required to exceed the potential barrier of the double-well function through the benefit of the additive sinusoid. Modeling, numerical simulations and experiments are here reported to show a good accordance with expectations.

An evolutionary double-well problem

We establish the existence theorem and study the long time behaviour of the following PDE problem:(0.1){ut−div∇W(∇u)−f(x)=0in Ω×(0,−∞),∇W(∇u)⋅n|∂Ω×(0,∞)=0,u(x,0)=u0(x)in Ω where W is a specially given quasiconvex double-well function and f∈L2(Ω) is a given function independent of time t. In particular, the existence theorem is established for general given source term f, the long time behaviour is analyzed under the assumption that ∫Ωf(x)dx=0.The system is an evolutionary quasimonotone system. We believe that the existence of solutions established here is stronger than the usual Young Measure solution and is the first of its kind. The existence of a compact ω-limit set as t→∞ is also established under some non-restrictive conditions.

Non-linear mechanical model of DNA dynamics

A non-linear mechanical model is constructed in order to study the dynamics of DNA double helix. The potential energy of the system is found to be a double-well function with respect to the first derivative of twist angle of the helix. The Lagrangian contains also the second derivative of the twist angle. Due to these facts the evolution equations of the system have kink and pulse solitary-wave solutions. The physical interpretation of our solutions is an untwisting of DNA during transcription of messenger RNA.