Superposition Rule Research Articles

Determination of hydrodynamic derivatives of an ocean vehicle using cfd analyses of synthetic standard dynamic tests

A synthetic method of dynamic test for identifying hydrodynamic coefficients (HCs) of an ocean vehicle has been proposed. The methodology consists of simulating various planar motion mechanism (PMM) tests simultaneously in a single numerical analysis. As a technical method, a stationary numerical domain has been used. Therefore, non-inertial forms of the fluid flow governing equations have been solved for simulating sinusoidal tests in a stationary numerical domain. Consequently, the quality of the grids has been preserved during the analyses. With considering of the well-known underwater geometry of DARPA-SUBOFF, validation of the numerical simulation results has been done with the computing of the pure sway and the pure yaw tests. The PMM maneuvers were induced to the numerical analyses by using a user-defined function augmented to the solver. The current results are in good agreement with the experimental ones reported in the literature. Based on the superposition rule which can be used for combining dynamic tests, a synthetic test has been designed consisted of the pure sway and the pure yaw tests. It has been developed to form more advanced synthetic tests consisted of the various PMM tests as well. The test has the capability to use for deriving non-linear HCs as well as the liner ones. The efficiency of the proposed method has been compared with the other methods, which shows it is comparable to the other approaches.

Multiscale simulation of a polymer melt flow between two coaxial cylinders under nonisothermal conditions

We successfully extend a multiscale simulation (MSS) method to nonisothermal well-entangled polymer melt flows between two coaxial cylinders. In the multiscale simulation, the macroscopic flow system is connected to a number of microscopic systems through the velocity gradient tensor, stress tensor and temperature. At the macroscopic level, in addition to the momentum balance equation, we consider the energy balance equation, where heat generation plays an important role not only in the temperature distribution but also in the flow profile. At the microscopic level, a dual slip-link model is employed for well-entangled polymers. To incorporate the temperature effect into the microscopic systems, we used the time-temperature superposition rule for the slip-link model, in which the temperature dependence of the parameters is not known; on the other hand, the way to take into account the temperature effect in the macroscopic equations has been well established. We find that the extended multiscale simulation method is quite effective in revealing the relation between nonisothermal polymeric flows for both steady and transient cases and the microscopic states of polymer chains expressed by primitive paths and slip-links. It is also found that the temperature-dependent reptation-time-based Weissenberg number is a suitable measure for understanding the extent of the polymer chain deformation in the range of the shear rate used in this study.

Nonlinear Description of Quantum Dynamics. N-level quantum systems

In this work a nonlinear evolution of pure states of a finite dimensional quantum system is introduced, in particular a Riccati evolution equation. It is shown how this class of dynamics is actually a Hamiltonian dynamics in the complex projective space. In this projective space it is shown that there is a nonlinear superposition rule, consistent with its linear counterpart in the Hilbert space. As an example, the developed nonlinear formalism is applied to the semiclassical Jaynes–Cummings model.

Twisted space-frequency and space-time partially coherent beams

We present partially coherent sources that are statistically twisted in the space-frequency and space-time domains. Beginning with the superposition rule for genuine partially coherent sources, we derive source plane expressions for the cross-spectral density (CSD) and mutual coherence functions (MCFs) for twisted space-frequency and space-time Gaussian Schell-model (GSM) beams. Using the Fresnel approximation to the free-space Green’s function, we then paraxially propagate the CSD and MCF to any plane z > 0. We discuss the beams’ behavior as they propagate, with particular emphasis on how the beam shape rotates or tumbles versus z. To validate our analysis, we simulate the generation and subsequent propagation of twisted space-frequency and space-time GSM beams. We compare the simulated moments to the corresponding theoretical predictions and find them to be in excellent agreement. Lastly, we describe how to physically synthesize twisted space-frequency and space-time partially coherent sources.

Thermo-rheological effects on successful 3D printing of biodegradable polyesters

Biodegradable polybutylene succinate (PBS) and poly (butylene succinate-ran-adipate) (PBSA) were characterized to find the thermo-rheological bases for 3D printing by Fused Filament Fabrication (FFF). In dynamic viscoelastic measurements, the samples fulfilled time-temperature superposition and Cox Merz rule. The viscosity results were linked to the excellent filaments obtained and the observed good flow in the printer nozzle. Using specific tearing experiments, outstanding welding of the printed layers was obtained. Results were discussed considering the values of the entanglements density obtained by small amplitude oscillatory shear (SAOS) measurements. The main difference between both polymers was observed in the final production of 3D printed parts because the high crystallinity of PBS produced significant warpage, which prevented its use for practical purposes. On the contrary, the less crystalline PBSA random copolymer showed excellent performance during FFF. Thus, dimensionally stable and ductile printed objects were obtained, opening new processing routes for semi-crystalline biodegradable polyesters.

Pressure transient behavior and flow regimes characteristics of vertical commingled well with vertical combined boundary: A field case in XC gas field of northwest Sichuan Basin, China

The multilayered deposition is the main feature of the XC gas reservoir in the northwestern Sichuan Basin. The boundaries among layers of the XC gas reservoir is different. In this paper, we propose an extended pressure transient analysis model for vertical commingle well, considering different boundary radii and boundary types, to capture the Vertical Combined Boundary (VCB) effect on the pressure transient behavior and flow regime characteristics.The analytical solution is yielded by the Laplace transform, Bessel functions, Duhamel's superposition principle, and Stehfest inversion algorithm, and new flow regimes under different boundary combinations are identified by the pressure derivative curve. Taking the dual-layer VCB model as a type example, nine combinations of boundaries are analyzed. Compared with the current vertical commingled well model with the same boundary radii or single boundary types, this model has novel flow regimes controlled by the VCB.The research results indicate that the pressure derivative type curve during the novel flow regimes stage can diagnosis the VCB type. On the one hand, there are quantitative linear relationships between the pressure derivative feature value and interlayer heterogeneity of the VCB model with the same boundary radii. These linear relationships provide a new perspective to quickly capture interlay heterogeneity through the of the pressure derivative feature value. One the other hand, the pressure behavior and flow regimes of the VCB model with different boundary radii obey the successive superposition rule. This rule provides a method to recognize the VCB type with different boundary radii based on that of the VCB with the same boundary radii.Finally, a field case from the northwestern Sichuan Basin gas reservoir is discussed in detail to exhibit the workflow of our model application and validate the capability of our model.

Relationships between depth-resolved primary radiation damage, irradiation-induced nanostructure and nanoindentation response of ion-irradiated Fe-Cr and ODS Fe-Cr alloys

• Irradiation-induced loops arise from displacement damage and injected interstitials. • Dislocation loops give rise to depth-dependent hardening. • The depth of maximum hardening and the substrate effect are correctly predicted. • The size of the plastic zone and the superposition rule are varied. • The most appropriate model version is material-specific. Ion irradiations are indispensable for exploring radiation effects on materials, for example, radiation hardening. However, the extraction of radiation hardening as function of displacement damage from the nanoindentation (NI) response of self-ion-irradiated metallic alloys is a challenge. In particular, recent attempts suffer from interference with contributions arising from injected self-interstitial atoms. Moreover, instances of available microstructural evidence and NI results reported for the same material and same irradiation are rare. In order to tackle these issues, the depth-dependent irradiated microstructure and the NI response were analyzed for Fe-9Cr and oxide dispersion strengthened Fe-Cr alloys irradiated with 5 MeV iron ions. Cross-sectional transmission electron microscopy indicated the appearance of irradiation-induced dislocation loops but no other types of visible microstructural changes. NI indicated maxima of the radiation hardening as function of contact depth. Links between the depth-resolved primary radiation damage, the observed depth-dependent characteristics of loops and the measured hardening are considered. As a key point, the link between loops and hardening requires the integration of the local hardening contributions over the indentation plastic zone. Calculations and measurements are compared with respect to both the depth position of maximum hardening and the substrate effect. The role of the model assumptions is discussed with special emphasis on the plastic zone size and the superposition of hardening contributions. The latter is found to be material-specific. The model also allows hardening contributions arising from displacement damage and injected interstitials to be separated.

Theoretical study of post-shock stress relaxation and shock wave deformation around a pore in single and poly-crystalline α-RDX

The mechanical properties and shock dynamics of α-hexahydro-1,3,5-trinitro-1,3,5-triazine (α-RDX) are simulated by molecular dynamics (MD). Based on the simulation results, the equation of state, Hugoniot curve and isoentropic curve for α-RDX are calculated, and the microscopic structure of shock wave is investigated, including the wave profile and the shock front deformation. First, we prove that the shock wave profile is determined by the viscoelastic equation of α-RDX, and develop a method to inverse the viscoelastic coefficients from the velocity profile calculated by MD. Two stress relaxation mechanisms are obtained: the boundary scattering and molecular rotation. The boundary scattering mechanism shows that the damping time of stress relaxation is determined by the particle size of polycrystal. Second, we find that the pore collapse induces rarefaction wave after the shock front. The rarefaction wave function is evaluated by using the Riemann invariant method, and the shock front deformation is derived from the superposition rule of shock wave and rarefaction wave. A physical picture to describe the nanoscale pore collapse process for α-RDX is obtained.

Characterization of personal solar ultraviolet radiation exposure using detrended fluctuation analysis

Studies of personal solar ultraviolet radiation (pUVR) exposure are important to identify populations at-risk of excess and insufficient exposure given the negative and positive health impacts, respectively, of time spent in the sun. Electronic UVR dosimeters measure personal solar UVR exposure at high frequency intervals generating large datasets. Sophisticated methods are needed to analyze these data. Previously, wavelet transform (WT) analysis was applied to high-frequency personal recordings collected by electronic UVR dosimeters. Those findings showed scaling behavior in the datasets that changed from uncorrelated to long-range correlated with increasing duration of time spent in the sun. We hypothesized that the WT slope would be influenced by the duration of time that a person spends in continuum outside. In this study, we address this hypothesis by using an experimental study approach. We aimed to corroborate this hypothesis and to characterize the extent and nature of influence time a person spends outside has on the shape of statistical functions that we used to analyze individual UVR exposure patterns. Detrended fluctuation analysis (DFA) was applied to personal sun exposure data. We analyzed sun exposure recordings from skiers (on snow) and hikers in Europe, golfers in New Zealand and outdoor workers in South Africa. Results confirmed validity of the DFA superposition rule for assessment of pUVR data and showed that pUVR scaling is determined by personal patterns of exposure on lower scales. We also showed that this dominance ends at the range of time scales comparable to the maximal duration of continuous exposure to solar UVR during the day; in this way the superposition rule can be used to quantify behavioral patterns, particularly accurate if it is determined on WT curves. These findings confirm a novel way in which large datasets of personal UVR data may be analyzed to inform messaging regarding safe sun exposure for human health.

Effects of MgO additive on microwave dielectric properties of NdNbO4 ceramics

NdNbO4−x MgO (x = 0.5, 1.0, 2.0, 3.0, 4.0) ceramics with high Q × ƒ values were prepared via the conventional solid-state reaction. The effects of MgO additive on the microstructures and microwave dielectric properties of NdNbO4 ceramics were investigated through various approaches. Microscopic analysis showed that the NdNbO4 phase and MgO phase were coexisted, and the crystal structure of the two phases did not changed with the additional amount of MgO. The dielectric constant could be explained by bulk density as a function of sintering temperature and satisfied with the Lichtenecker logarithmic superposition rule. The Q × ƒ value increased quickly when the phase composition changed from single phase to multiphase and was afterwards linked to the grain growth. The change in the temperature coefficient of resonant frequency was not large and was consistent with that in the dielectric constant. When x = 3.0, MgO additive effectively stepped up the Q × ƒ value of NdNbO4 to 1,35,000 GHz, which was fourfold of pure phase.

Quasi-Lie schemes for PDEs

The theory of quasi-Lie systems, i.e. systems of first-order ordinary differential equations that can be related via a generalized flow to Lie systems, is extended to systems of partial differential equations (PDEs) and its applications to obtain [Formula: see text]-dependent superposition rules, and integrability conditions are analyzed. We develop a procedure of constructing quasi-Lie systems through a generalization to PDEs of the so-called theory of quasi-Lie schemes. Our techniques are illustrated with the analysis of Wess–Zumino–Novikov–Witten models, generalized Abel differential equations, Bäcklund transformations, as well as other differential equations of physical and mathematical relevance.

Research on dynamic hardening effect on workpiece’s hardness distribution with depth of cut

Grinding hardening is a kind of attractive processing technology, which utilizes grinding heat to generate the hardening layers. Previous studies have attached the importance on generation mechanism in experiments and simulations. However, grinding hardening is a dynamic hardness improvement process due to the system’s dynamic characteristics, whose hardness becomes different during the dynamic process. To this end, this paper presents a new hybrid model to investigate the dynamic hardness distribution with the system’s dynamic characteristics. When the effect of cutting, rubbing, and plowing is considered, the dynamic grinding force is calculated firstly. Afterwards, coupled with the dynamic grinding force, the dynamic grinding temperature distribution is obtained accordingly. Moreover, in terms of the hardness superposition rule, the dynamic hardness model is established based on the calculated dynamic grinding force and the dynamic temperature distribution. Finally, the model is validated by the measurement of the ground workpiece’s hardness via grinding experiments under different grinding depths. It can be concluded that the system’s dynamic characteristics can result in the time delay of the hardness temporal distribution and the inhomogeneity of the hardness spatial distribution. Consequently, the model is anticipated to be meaningful for both the manufacturing industries and the research fields.

Multisymplectic structures and invariant tensors for Lie systems

A Lie system is a non-autonomous system of differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional Lie algebra of vector fields, a so-called Vessiot–Guldberg Lie algebra. This work pioneers the analysis of Lie systems admitting a Vessiot–Guldberg Lie algebra of Hamiltonian vector fields relative to a multisymplectic structure: the multisymplectic Lie systems. Geometric methods are developed to consider a Lie system as a multisymplectic one. By attaching a multisymplectic Lie system via its multisymplectic structure with a tensor coalgebra, we find multisymplectic and coalgebra methods to derive superposition rules, constants of motion, and invariant tensor fields relative to its evolution. Our results are illustrated with examples occurring in physics and mathematics such as Schwarz equations, control systems, or diffusion-type equations. This also shows the interest of multisymplectic techniques in ordinary differential equations, which represents a not much investigated field of application of multisymplectic geometry.

Microstructural changes and their effect on hardening in neutron irradiated Fe-Cr alloys

A series of Fe-3 to 18 at.% Cr binary ferritic alloys were neutron irradiated side by side, in the Advanced Test Reactor (ATR) at a temperature of ∼320 °C to a dose of ∼1.8 dpa. Three types of features that form under irradiation are relevant: (i) solute clusters; (ii) α′ precipitates; and, (iii) dislocation loops. The size and number density of the precipitates and loops were measured by transmission electron microscopy (TEM) and compared to previous atom probe tomography (APT) observations. The loop density systematically decreases with increasing Cr, while the number of α′ precipitates increase at 9Cr and above. A major objective of this work is to estimate the dispersed barrier obstacle strength factors (αj) for loops, clusters and α′ precipitates, based on the combined microstructural observations and corresponding irradiation hardening measurements. Standard dispersed barrier-hardening models and computationally derived superposition rules were least square fit to determine the αj. The optimized hardening predictions are in very good agreement with experiment if mixed linear sum and root square sum superposition rules are used. Five increments of 168 h isochronal anneals of the 6Cr alloy between 300 and 400 °C coarsened the loops, while 300 h anneals of the 18Cr at 500 and 600 °C coarsened and dissolved the α′ precipitates, respectively, consistent with the Fe-Cr phase diagram.

Prediction of irradiation hardening in austenitic stainless steels: Analytical and crystal plasticity studies

The flow stress in irradiated Austenitic Stainless Steels (ASSs) is analyzed in terms of the microstructure features and the fundamental mechanisms of plastic deformation. A set of constitutive equations is proposed to assess the contributions of dislocation network, grain size effect, dislocation loops and solute clusters, accompanied with an appropriate superposition rule. Using simple equations for the evolution of densities of dislocations and radiation defects, it is possible to integrate numerically the tensile stress-strain curves over a wide range of irradiation doses. It is shown that irradiation hardening can be obtained analytically without adjustable parameters. Simulation results are challenged in comparison with experimental results and found in excellent agreement with all the reported experimental trends in ASSs irradiated at conditions close to those prevailing in pressurized water reactors.

A simplified method for calculating the heat rejection from a rectangle droplet sheet

To calculate the thermal characteristics of the rectangle liquid droplet radiator more accurately, a novel method, which estimated the characteristics of each droplet, was introduced. The shape factors of two and three droplets were theoretically deduced and verified by the Monte Carlo method. It was found that the shape factor of two droplets depends only on the ratio of the distance between the two droplets to the radii of droplets. According to this characteristic, the shape factor of two droplets was calculated and fitted to be a formula, which makes it convenient to calculate the shape factor in the following analysis. Based on the superposition rule of the shape factor, a shape factor matrix was introduced to calculate the effect of surrounding droplets when analyzing the heat transfer of the droplet sheet. The temperature distribution of the droplet sheet was calculated based on the shape factor matrix, which obtained using the combination of theoretical method and Monte Carlo method. The results showed that the temperature difference among the droplets on a same width-thickness plane was negligible. Therefore, a simplified model was developed based on this characteristic. The discrepancies of the temperature distributions obtained by the original model and the simplified model were found to be less than 0.35%, which demonstrates the feasibility of the simplified model to analyze the thermal characteristics of the rectangle droplet radiator. By further analyzing the heat transfer characteristics of the droplet sheet, proposals were made for design of the rectangle liquid droplet radiators.

Monte Carlo calculation of view factors between some complex surfaces: Rectangular plane and parallel cylinder, rectangular plane and torus, especially cold-rolled strip and W-shaped radiant tube in continuous annealing furnace

View factor plays an important role in evaluation of radiative heat transfer. This paper gives a detailed account of setting up the Monte Carlo calculation models of view factors between some complex surfaces. Main purpose is to calculate view factor between cold-rolled strip and W-shaped radiant tube in continuous annealing furnace. Given advantages of Monte Carlo method, it has been used to calculate view factor between the strip and radiant tube. Firstly, the Monte Carlo calculation models for view factor between rectangular plane and parallel cylinder and for view factor between rectangular plane and torus are established and verified, respectively. Then, according to reciprocity and superposition rules of view factors, the Monte Carlo calculation model for view factor between the strip and radiant tube is established by the above models. By comparing with analytical expression, or checking reciprocity after performing the Monte Carlo analysis in each direction between surface pairs, the calculation errors of above Monte Carlo models are analyzed and the results show that the Monte Carlo method is effective and reliable in calculating view factors between some complex surfaces.

Effects of ultrasonic surface mechanical attrition treatment on microstructures and mechanical properties of high entropy alloys

Most high entropy alloys (HEAs) are cast to form single phase solid solution. Their hardness and strength at room temperature under the as-cast condition are typically lower than expectation. In the research, the ultrasonic surface mechanical attrition treatment (SMAT) is conducted on the surface of two HEAs, FeCoNiCrMn and FeCoNiCrMn-Al, to upgrade their room temperature surface characteristics. By proper SMAT multiple paths, the grain size can be reduced from ∼50 μm down to ∼0.1–1 μm, the hardness increased from ∼2.5–5.0 GPa up to ∼5.0–8.5 GPa, and the tensile strength and elongation can be nearly doubled. The gradient refined and strengthened surface layers are demonstrated to appreciably upgrade the HEA performance. The strengthening mechanisms and superposition rules are established and are compared well with the experimental measurements.

Lie–Hamilton systems on curved spaces: a geometrical approach

A Lie–Hamilton system is a nonautonomous system of first-order ordinary differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional Lie algebra, a Vessiot–Guldberg Lie algebra, of Hamiltonian vector fields relative to a Poisson structure. Its general solution can be written as an autonomous function, the superposition rule, of a generic finite family of particular solutions and a set of constants. We pioneer the study of Lie–Hamilton systems on Riemannian spaces (sphere, Euclidean and hyperbolic plane), pseudo-Riemannian spaces (anti-de Sitter, de Sitter, and Minkowski spacetimes) as well as on semi-Riemannian spaces (Newtonian spacetimes). Their corresponding constants of motion and superposition rules are obtained explicitly in a geometric way. This work extends the (graded) contraction of Lie algebras to a contraction procedure for Lie algebras of vector fields, Hamiltonian functions, and related symplectic structures, invariants, and superposition rules.

Microstructure characterization and strengthening mechanisms of oxide dispersion strengthened (ODS) Fe-9%Cr and Fe-14%Cr extruded bars

The collaborative study is focused on the relationship between microstructure and yield stress for an ODS Fe-9%Cr-based transformable alloy and an ODS Fe-14%Cr-based ferritic alloy. The contributions to the total room temperature yield stress arising from various strengthening mechanisms are addressed on the basis of a comprehensive description of the microstructures uncovered by means of transmission electron microscopy (TEM), electron backscatter diffraction (EBSD), small-angle neutron scattering (SANS) and atom probe tomography (APT). While these methods provide a high degree of complementarity, a reasonable agreement was found in cases of overlap of information. The derived set of microstructure parameters along with reported strengthening equations was used to calculate the room temperature yield stress. The estimates were critically compared with the measured yield stress for an extended set of alloys including data reported for Fe-Cr model alloys and steels thus covering one order of magnitude or more in grain size, dislocation density, particle density and yield stress. The comparison shows that particle strengthening, dislocation forest strengthening, and Hall-Petch strengthening are the major contributions and that a mixed superposition rule reproduces the measured yield stress within experimental scatter for the whole extended set of alloys. The wide variation of microstructures additionally underpins the conclusions and goes beyond previous work, in which one or few ODS steels and narrow microstructure variations were typically covered.