Nonconservative Field Research Articles

Simulation of Melt Viscosity Effect on the Rate of Solidification in Polymer

Phase field model has been successfully derived from ordinary metal phase field equation to simulate the behavior of semi-crystalline polymer solidification phenomenon. To obtain the polymer phase field model, a non-conserved phase field equation can be expanded to include the unique polymer parameters, which do not exist in metals, for example, polymer melt viscosity and diffusion coefficient. In order to expand this model, we include free energy density and non-local free energy density based on Harrowel-Oxtoby and Ginzburg-Landau theorem for polymers. The expansion principle for a higher order of binary phase field parameter was employed to obtain fully modified phase field equation. To optimize the final properties of the products, the solidification phenomenon in polymers is very important. Here, we use our modified equation to investigate the effect of melt viscosity on the rate of solidification by employing ordinary differential equation numerical methods. It was found that the rate of solidification is related to the melting temperature and the kinetic coefficient.

Near-field optical trapping in a non-conservative force field

The force-field generated by a near-field optical trap is analyzed. A C-shaped engraving on a gold film is considered as the trap. By separating out the conservative component and the solenoidal component of the force-field using Helmholtz-Hodge decomposition, it was found that the force is non-conservative. Conventional method of calculating the optical potential from the force-field is shown to be inaccurate when the trapping force is not purely conservative. An alternative method is presented to accurately estimate the potential. The positional statistics of a trapped nanoparticle in this non-conservative field is calculated. A model is proposed that relates the position distribution to the conservative component of the force. The model is found to be consistent with numerical and experimental results. In order to show the generality of the approach, the same analysis is repeated for a plasmonic trap consisting of a gold nanopillar. Similar consistency is observed for this structure as well.

Image enhancement with PDEs and nonconservative advection flow fields.

We propose a new method for the progressive enhancement of images degraded by noise and blur using shock filters. The originality of our approach lies in the iterative exploitation of nonconservative edge-based force fields normally designed for segmentation to determine both locations and amplitudes of the sharpening action. The proposed nonconservative shock filter (NCSF) produces images with strong discontinuities that can be used as an additional pre-processing step to facilitate higher level tasks such as edge detection or segmentation. NCSF is especially useful when coupled with an anisotropic diffusion equation in noisy configurations. Using the proposed method, edges that are not well defined prior to filtering can be iteratively recovered. We show processing results on various types of images, focusing on medical transrectal ultrasound and brain PET imaging.

Comparison study on the different dynamics between the Allen–Cahn and the Cahn–Hilliard equations

We perform a comparison study on the different dynamics between the Allen–Cahn (AC) and the Cahn–Hilliard (CH) equations. The AC equation describes the evolution of a non-conserved order field during anti-phase domain coarsening. The CH equation describes the process of phase separation of a conserved order field. The AC and the CH equations are second-order and fourth-order nonlinear parabolic partial differential equations, respectively. Linear stability analysis shows that growing and decaying modes for both the equations are the same. While the growth rates are monotonically decreasing with respect to the modes for the AC equation, the growth rates for the CH equation are increasing and then decreasing with respect to the modes. We perform various numerical tests using the Fourier spectral method to highlight the different evolutionary dynamics between the AC and the CH equations.

Integrable Systems on the Tangent Bundle of a Multi-Dimensional Sphere

This paper contains a systematic exposition of some results on the equations of motion of a dynamically symmetric n-dimensional rigid body in a nonconservative field of forces. Similar bodies are considered in the dynamics of actual rigid bodies interacting with a resisting medium under the conditions of jet flow past the body with a nonconservative following force acting on the body in such a way that its characteristic point has a constant velocity, which means that the system has a nonintegrable servo-constraint.

Low-Dimensional and Multi-Dimensional Pendulums in Nonconservative Fields. Part 2

In this review, we discuss new cases of integrable systems on the tangent bundles of finite-dimensional spheres. Such systems appear in the dynamics of multidimensional rigid bodies in non-conservative fields. These problems are described by systems with variable dissipation with zero mean. We found several new cases of integrability of equations of motion in terms of transcendental functions (in the sense of the classification of singularities) that can be expressed as finite combinations of elementary functions.

Sharp Phase Field Method.

Phase field modeling offers an extremely general framework to predict microstructural evolutions in complex systems. However, its computational implementation requires a discretization scheme with a grid spacing small enough to preserve the continuous character of the theory. We present here a new formulation, which is intrinsically discrete, in which the interfaces are resolved with essentially one grid point with no pinning on the grid and an accurate rotational invariance, improving drastically the numerical capabilities of the method. We show that interfacial kinetic properties are reproduced with a high accuracy. Finally, we apply the model to a situation where conserved and nonconserved fields are coupled.

Low-Dimensional and Multi-Dimensional Pendulums in Nonconservative Fields. Part 1

In this review, we discuss new cases of integrable systems on the tangent bundles of finitedimensional spheres. Such systems appear in the dynamics of multidimensional rigid bodies in nonconservative fields. These problems are described by systems with variable dissipation with zero mean. We found several new cases of integrability of equations of motion in terms of transcendental functions (in the sense of the classification of singularities) that can be expressed as finite combinations of elementary functions.

Phase-field modeling of Li-insertion kinetics in single LiFePO4-nano-particles for rechargeable Li-ion battery application

We develop a continuum phase-field model for the simulation of diffusion limited solid-solid phase transformations during lithium insertion in LiFePO4-nano-particles. The solid-solid phase boundary between the LiFePO4 (LFP)-phase and the FePO4 (FP)-phase is modeled as a diffuse interface of finite width. The model-description explicitly resolves a single LiFePO4-particle, which is embedded in an elastically soft electrolyte-phase. Furthermore, we explicitly include anisotropic (orthorhombic) and inhomogeneous elastic effects, resulting from the coherency strain, as well as anisotropic (1D) Li-diffusion inside the nano-particle. In contrast to other related research work, we employ an Allen-Cahn-type phase-field approach for the diffuse interface modeling of the solid-solid phase boundary. The model contains an extra non-conserved order parameter field to distinguish the two different phases. The evolution of this order parameter field is controlled by an extra kinetic parameter independent from the Li-diffusion. Further, the effect of the nano-particle’s size on the kinetics of FP to LFP phase transformations is investigated by means of both model. Both models predict a substantial increase in the steady state transformation velocity as the particle-size decreases down to dimensions that are comparable with the width of the interface between the FP and the LFP-phase. However, the extra kinetic parameter of the Allen-Cahn-type description may be used to reduce the strength of the velocity-increase with the decreasing particle size. Further, we consider the influence of anisotropic and inhomogeneous elasticity on the lithiation-kinetics within a rectangularly shaped LiFePO4-particle embedded in an elastically soft electrolyte. Finally, the simulation of equilibrium shapes of LiFePO4-particles is discussed. Within a respective feasibility study, we demonstrate that also the simulation of strongly anisotropic particles with aspect ratios up to 1/5 is possible.

Braneworld gravity within non-conservative gravitational theory

We investigate the braneworld gravity starting from the non-conservative gravitational field equations in a five-dimensional bulk. The approach is based on the Gauss–Codazzi formalism along with the study of the braneworld consistency conditions. The effective gravitational equations on the brane are obtained and the constraint leading to a brane energy-momentum conservation is analyzed.

Action principle for action-dependent Lagrangians toward nonconservative gravity: Accelerating universe without dark energy

In the present work, we propose an Action Principle for Action-dependent Lagrangians by generalizing the Herglotz variational problem for several independent variables. This Action Principle enables us to formulate Lagrangian densities for non-conservative fields. In special, from a Lagrangian depending linearly on the Action, we obtain a generalized Einstein's field equations for a non-conservative gravity and analyze some consequences of their solutions to cosmology and gravitational waves. We show that the non-conservative part of the field equations depends on a constant cosmological four-vector. Depending on this four-vector, the theory displays damped/amplified gravitational waves and an accelerating Universe without dark energy.

Critical initial-slip scaling for the noisy complex Ginzburg–Landau equation

We employ the perturbative fieldtheoretic renormalization group method to investigate the universal critical behavior near the continuous non-equilibrium phase transition in the complex Ginzburg–Landau equation with additive white noise. This stochastic partial differential describes a remarkably wide range of physical systems: coupled nonlinear oscillators subject to external noise near a Hopf bifurcation instability; spontaneous structure formation in non-equilibrium systems, e.g., in cyclically competing populations; and driven-dissipative Bose–Einstein condensation, realized in open systems on the interface of quantum optics and many-body physics, such as cold atomic gases and exciton-polaritons in pumped semiconductor quantum wells in optical cavities. Our starting point is a noisy, dissipative Gross–Pitaevski or nonlinear Schrödinger equation, or equivalently purely relaxational kinetics originating from a complex-valued Landau–Ginzburg functional, which generalizes the standard equilibrium model A critical dynamics of a non-conserved complex order parameter field. We study the universal critical behavior of this system in the early stages of its relaxation from a Gaussian-weighted fully randomized initial state. In this critical aging regime, time translation invariance is broken, and the dynamics is characterized by the stationary static and dynamic critical exponents, as well as an independent ‘initial-slip’ exponent. We show that to first order in the dimensional expansion about the upper critical dimension, this initial-slip exponent in the complex Ginzburg–Landau equation is identical to its equilibrium model A counterpart. We furthermore employ the renormalization group flow equations as well as construct a suitable complex spherical model extension to argue that this conclusion likely remains true to all orders in the perturbation expansion.

Continuum-wise expansiveness for non-conservative or conservative systems

In this paper, we show that a non-conservative vector field is robustly continuum-wise expansive if and only if it satisfies both Axiom A and the quasi-transversality condition. Moreover, a conservative vector field (divergence-free vector field, Hamiltonian system) is robustly continuum-wise expansive if and only if it is Anosov.

Integrable Cases in the Dynamics of a Multi-Dimensional Rigid Body in a Nonconservative Field in the Presence of a Tracking Force

This paper is a survey of integrable cases in the dynamics of a five-dimensional rigid body under the action of a nonconservative force field. We review both new results and results obtained earlier. Problems examined are described by dynamical systems with so-called variable dissipation with zero mean. The problem of the search for complete sets of transcendental first integrals of systems with dissipation is quite topical; a large number of works are devoted to it. We introduce a new class of dynamical systems that have a periodic coordinate. Due to the existence of nontrivial symmetry groups of such systems, we can prove that these systems possess variable dissipation with zero mean, which means that on the average for a period with respect to the periodic coordinate, the dissipation in the system is equal to zero, although in various domains of the phase space, either energy pumping or dissipation can occur. Based on the facts obtained, we analyze dynamical systems that appear in the dynamics of a five-dimensional rigid body and obtain a series of new cases of complete integrability of the equations of motion in transcendental functions that can be expressed through a finite combination of elementary functions.

A nucleation algorithm for the coupled conserved–nonconserved phase field model

This paper presents a refinement to the existing nucleation algorithm for a coupled conserved–nonconserved phase field model. In the new method, which offers greater ease of implementation as compared to the existing approach, only the nonconserved order parameter is modified to seed supercritical nuclei (thus termed order-parameter-only seeding). The order-parameter-only seeding method naturally satisfies the conservation law for the conserved order parameter. In addition, the implementation within a finite element framework is described. The evolution of a single nucleus is examined to ensure that the precipitate growth kinetics are not affected by the seeding method. We find that, after a brief initial transient period, order-parameter-only nucleation yields similar precipitate growth characteristics to that of the existing model. The kinetics of a phase transformation exhibiting concurrent nucleation and growth is analyzed in the form of the Avrami equation, and a statistical analysis is performed to determine if mesh and/or time adaptivity affects the simulation results. The statistical analysis indicates that the nucleation algorithm is amenable to adaptive meshing and adaptive time stepping.

Complete list of the first integrals of dynamic equations of a multidimensional solid in a nonconservative field under the assumption of linear damping

The dynamic part of equations of motion is investigated for a dynamically symmetric multidimensional solid in a nonconservative force field in the presence of the following force in the case when the solid is under the action of a pair of forces. In this case, there is additional linear damping in the system. The new case of complete integrability is found in transcendental (in the sense of complex analysis) functions, which are expressed through a finite combination of elementary functions.

Some Classes of Integrable Problems in Spatial Dynamics of a Rigid Body in a Nonconservative Force Field

This paper is a review of some previous and new results on integrable cases in the dynamics of a three-dimensional rigid body in a nonconservative field of forces. These problems are stated in terms of dynamical systems with the so-called zero-mean variable dissipation. Finding a complete set of transcendental first integrals for systems with dissipation is a very interesting problem that has been studied in many publications. We introduce a new class of dynamical systems with a periodic coordinate. Since such systems possess some nontrivial groups of symmetries, it can be shown that they have variable dissipation whose mean value over the period of the periodic coordinate vanishes, although in various regions of the phase space there may be energy supply or scattering. The results obtained allow us to examine some dynamical systems associated with the motion of rigid bodies and find some cases in which the equations of motion can be integrated in terms of transcendental functions that can be expressed as finite combinations of elementary functions.

The Application of Kiuttu’s Formulation to Study Coaxial Flux Compression Generators

A class of flux compression generators (FCGs) is based on the compression of the cross-sectional area of a coaxial geometry where the current flows along the outer conductor and returns through the inner conductor. This compression causes an increase in current since magnetic flux must be conserved. Kiuttu’s inductive electric-field formulation is a powerful tool for the conceptual design of coaxial FCGs. The usefulness of this formulation is demonstrated in this paper for a simplified geometry using a finite-element partial differential equation solver (FlexPDE) for calculation of the inductive electric field. A time-varying applied current or a moving surface creates the nonconservative electric field. Losses due to diffusion of magnetic flux into conducting surfaces can also be accounted for and modeled in this setting. This analytical–computational approach serves as an important step in validating the magnetohydrodynamic (MHD) portion of the complex multiphysics parallel Lawrence Livermore code, Arbitrary Lagrangian-Eulerian (ALE3D). The nonintuitive boundary conditions involved in solving the otherwise straightforward partial differential equations are described in detail and illustrated in a simple model. The physical parameters used in the simulations are not based on a specific design.

New case of complete integrability of dynamics equations on a tangent fibering to a 3D sphere

The paper presents the results of study of the motion equations for a dynamically symmetric 4D-rigid body placed in a certain non-conservative field of forces. The form of the field is taken from the dynamics of actual 2D- and 3D-rigid bodies interacting with the medium in the case when the system contains a non-conservative pair of forces forcing the center of mass of a body to move rectilinearly and uniformly. A new case of integrability is obtained for dynamic equations of body motion in a resisting medium filling a four-dimensional space under presence of a tracking force.

Complete list of first integrals of dynamic equations for a multidimensional solid in a nonconservative field

Complete list of first integrals of dynamic equations for a multidimensional solid in a nonconservative field