Combined Analytical-numerical Approach Research Articles

Design of a crash energy absorber for a composite aircraft fuselage using a combined analytical–numerical approach

Design of a crash energy absorber for a composite aircraft fuselage using a combined analytical–numerical approach

Contribution of rolling resistance to the drag coefficient of spheres freely rolling on a rough inclined surface

The drag coefficient (CD) of a sphere freely rolling without slipping on a rough plane is presented in this study. Increasing panel roughness has been found to increase CD, although lubrication theory predicts that the larger gap imposed by the rougher panel should yield a smaller drag. We propose that this increase in drag is due to the effects of rolling resistance, which increases with panel roughness. The total drag on a sphere is decomposed into fluid drag and drag due to rolling resistance, where the fluid drag is predicted using a combined analytical–numerical approach. It is shown that rolling resistance can be modeled as a resistive torque opposing the sphere motion, generated by the offset contact normal force from the sphere center plane. This coefficient of rolling resistance (μr) can be predicted using the root mean square roughness (Rq) of the panel. Additionally, μr is observed to increase with sphere down-slope velocity and an empirical relationship between μr, Rq, and non-dimensional velocity (U∗) is given. A comparison of the drag predicted by the proposed model with measured data indicates good agreement for all the four panels considered. Consistent with previous literature, a non-linear relationship between μr, Rq, and U∗ is proposed. Although increasing panel roughness leads to a smaller fluid drag due to the larger gap imposed by rougher panels, the drag due to rolling resistance increases more rapidly. This leads to an increase in total drag with increase in the panel roughness. Additionally, increasing panel roughness is observed to have a significant effect on the sphere wake, leading to irregular wake shedding and increase in the Strouhal number.

Optimal control methods for quantum batteries

We investigate the optimal charging processes for several models of quantum batteries, finding how to maximize the energy stored in a given battery with a finite-time modulation of a set of external fields. We approach the problem using advanced tools of optimal control theory, highlighting the universality of some features of the optimal solutions, for instance the emergence of the well-known bang-bang behavior of time-dependent external fields. The technique presented here is general, and we apply it to specific cases in which the energy is either pumped into the battery by external forces (direct charging) or transferred into it from an external charger (mediated charging). In this paper we focus on particular systems that consist of coupled qubits and harmonic oscillators, for which the optimal charging problem can be explicitly solved using a combined analytical-numerical approach based on our optimal control techniques. However, our approach can be applied to more complex setups, thus fostering the study of many-body effects in the charging process.

Analysing regular nonlinear vibrations of nano/micro plates based on the nonlocal theory and combination of reduced order modelling and multiple scale method

We study the geometrically nonlinear vibrations of rectangular micro/nanoplates with an account of small-scale effects in a framework of the nonlocal elasticity theory. Hamilton’s principle yields the governing system of nonlinear partial differential equations (PDEs) based on the Kirchhoff-Love hypotheses and geometric nonlinearity introduced through the von Kármán theory.Next, we employed a strategy to get coupled nonlinear and non-autonomous system of ordinary differential equations (ODEs) due to the concept of both reduced order modelling (ROM) truncated to the double-mode approximation of the plate deflection and the Bubnov-Galerkin procedure. The latter one allowed for the transformation of the initial problem to that with separated time and position functions and the used approach was validated.Then we have studied the reduced model of second-order nonlinear ODEs with coupled geometric nonlinearities through the multiple scales method (MSM) in time domain. Both resonant and non-resonant vibrations have been considered with emphasis put on the small-scale effects and the approximation accuracy. In spite of numerous novel results regarding parametric analysis carried out by combined analytical-numerical approach, we have detected ambiguous and unambiguous back-bone curves which allow one to predict and possibly avoid the pull-in phenomena related to the primary problem governed by the nonlinear vibrations of micro/nanoplates.

Oncolytic viral therapies and the delicate balance between virus-macrophage-tumour interactions: a mathematical approach

The success of oncolytic virotherapies depends on the tumour microenvironment, which contains a large number of infiltrating immune cells. In this theoretical study, we derive an ODE model to investigate the interactions between breast cancer tumour cells, an oncolytic virus (Vesicular Stomatitis Virus), and tumour-infiltrating macrophages with different phenotypes which can impact the dynamics of oncolytic viruses. The complexity of the model requires a combined analytical-numerical approach to understand the transient and asymptotic dynamics of this model. We use this model to propose new biological hypotheses regarding the impact on tumour elimination/relapse/persistence of: (i) different macrophage polarisation/re-polarisation rates; (ii) different infection rates of macrophages and tumour cells with the oncolytic virus; (iii) different viral burst sizes for macrophages and tumour cells. We show that increasing the rate at which the oncolytic virus infects the tumour cells can delay tumour relapse and even eliminate tumour. Increasing the rate at which the oncolytic virus particles infect the macrophages can trigger transitions between steady-state dynamics and oscillatory dynamics, but it does not lead to tumour elimination unless the tumour infection rate is also very large. Moreover, we confirm numerically that a large tumour-induced M1$\to$M2 polarisation leads to fast tumour growth and fast relapse (if the tumour was reduced before by a strong anti-tumour immune and viral response). The increase in viral-induced M2$\to$M1 re-polarisation reduces temporarily the tumour size, but does not lead to tumour elimination. Finally, we show numerically that the tumour size is more sensitive to the production of viruses by the infected macrophages.

Hydrodynamics of active particles confined in a periodically tapered channel

Active particles in diverse circumstances encounter confined channels with asymmetric bounding walls. In the present work, employing the squirmer model, we analyze the trajectory of a single and a pair of active particles in a two-dimensional periodically tapered channel with asymmetric bounding walls through a combined analytical-numerical approach. Assuming Stokes equations for the flow inside the channel, both puller and pusher types of squirmers are treated. We illustrate through phase diagrams how for different projection angles of the squirmer the associated swimming trajectories are non-trivially altered for various tapering angles of the channel. The phase diagram characterizes the trajectory of the squirmer as trapped or escaped depending on these angles. It is observed that for a fixed projection angle, the swimmer exhibits a transition in the swimming state at a critical tapering of the channel. Correspondingly, the combination of the projection and tapering angles may serve as a control mechanism guiding the swimmer for relevant applications in micro-fluidic systems. We further investigate the stability of the individual squirmer trajectory in the presence of a second squirmer, which hints at the development of parallel or coordinated swimming motion inside the channel. The results indicate that the tapering of the channel acts as a decisive parameter in the mutual attraction or repulsion and navigates the collective swimming state of the squirmers.

Stable asymmetric spike equilibria for the Gierer–Meinhardt model with a precursor field

AbstractPrecursor gradients in a reaction-diffusion system are spatially varying coefficients in the reaction kinetics. Such gradients have been used in various applications, such as the head formation in the Hydra, to model the effect of pre-patterns and to localize patterns in various spatial regions. For the 1D Gierer–Meinhardt (GM) model, we show that a non-constant precursor gradient in the decay rate of the activator can lead to the existence of stable, asymmetric and two-spike patterns, corresponding to localized peaks in the activator of different heights. These stable, asymmetric patterns co-exist in the same parameter space as symmetric two-spike patterns. This is in contrast to a constant precursor case, for which asymmetric spike patterns are known to be unstable. Through a determination of the global bifurcation diagram of two-spike steady-state patterns, we show that asymmetric patterns emerge from a supercritical symmetry-breaking bifurcation along the symmetric two-spike branch as a parameter in the precursor field is varied. Through a combined analytical-numerical approach, we analyse the spectrum of the linearization of the GM model around the two-spike steady state to establish that portions of the asymmetric solution branches are linearly stable. In this linear stability analysis, a new class of vector-valued non-local eigenvalue problem is derived and analysed.

Near-wall hydrodynamic slip triggers swimming state transition of micro-organisms

The interaction of motile micro-organisms with a nearby solid substrate is a well-studied phenomenon. However, the effects of hydrodynamic slippage on the substrate have received little attention. In the present study, within the framework of the squirmer model, we impose a tangential velocity at the swimmer surface as a representation of the ciliary propulsion, and subsequently obtain an exact solution of the Stokes equation based on a combined analytical–numerical approach. We illustrate how the near-wall swimming velocities are non-trivially altered by the interaction of wall slip and hydrodynamic forces. We report a characteristic transition of swimming trajectories for both puller- and pusher-type microswimmers by hydrodynamic slippage if the wall slip length crosses a critical value. In the case of puller microswimmers that are propelled by a breaststroke-like action of their swimming apparatus ahead of their cell body, the wall slip can cause wall-bound trapping swimming states, as either periodic or damped periodic oscillations, which would otherwise escape from a no-slip wall. The associated critical slip length has a non-monotonic dependence on the initial orientation of the swimmer, which is represented by novel phase diagrams. Pushers, which get their propulsive thrust from posterior flagellar action, also show similar swimming state transitions, but in this case the wall-slip-mediated reorientation dynamics and the swimming modes compete in a different fashion from that of the pullers. Although neutral swimmers lack a sufficient reorientation torque to exhibit any wall-bound trajectory, their detention time near the substrate can be significantly increased by tailoring the extent of hydrodynamic slippage at the nearby wall. The present results pave the way for understanding the motion characteristic of biological microswimmers near confinements with hydrophobic walls or strategize the design of microfluidic devices used for sorting and motion rectification of artificial swimmers by tailoring their surface wettability.

Analysis of a hybrid process for manufacturing sheet metal-polymer structures using a conceptual tool design and an analytical-numerical modelling

Abstract This paper presents an experimental and numerical study on the hybrid process of polymer injection forming (PIF) developed to manufacture sheet metal-polymer structures in one single operation. Despite the wide potential application of the PIF process, several challenges have prevented conducting an in-depth analysis of the simultaneous injection/forming condition that occurs during this hybrid process. One such challenge is the lack of a special feature in the regular injection molding machine for an independent application and control of the blank holder force (BHF) from the preset clamping force. To enable such, a new concept-to-design tool is proposed in this work. Using this specialized setup, the influence of the BHF, injection rate and their interactions are experimentally determined focusing on the transient PIF process variables and the quality of the final hybrid product. The use of the fluid-structure interaction (SFI) method to perform a multi-physics simulation on the PIF process incurs such computational costs that it is unreasonable to conduct multiple simulations to investigate the effects of several process parameters and their interactions. To achieve both efficiency and accuracy, a new combined analytical-numerical approach is presented to enable fundamental understanding of the PIF process physics considering the interaction of filling, stretching and drawing mechanisms. The feasibility of the proposed numerical and experimental methodologies is demonstrated through a comparative analysis of the deformation of the AA1100 sheet during the injection of a Polypropylene compound with different injection rates and BHF settings.

Aerothermal Postflight Analysis of the Sharp Edge Flight Experiment-II

The aerothermal instrumentation of the Sharp Edge Hypersonic Flight Experiment SHEFEX-II, which was launched from Andøya Rocket Range on top of a two-stage rocket configuration, provided very useful flight data. Complementary to the pressure data, heat flux rate and surface temperature were measured at selected locations on the payload along the complete ascent and descent trajectory. Measured heat flux rate and pressure fluctuations show excellent correlation with angle-of-attack variations. Calculated heat flux evolution using a hypersonic boundary-layer code along the trajectory is in reasonable agreement with measured heat flux data during flight. Boundary-layer transition was measured at a boundary-layer edge Reynolds number of approximately for both ascent and descent phases. The complementary analytical and numerical study for the flight rebuilding at selected flight points provided very useful data and allowed better understanding of the physical phenomena. Finally, a combined analytical–numerical approach was validated and applied for the computation of the heat flux rate along the flight trajectory. This verified method can be used for the design and analysis of future hypersonic flights of slender vehicles.

Predator-driven elemental cycling: the impact of predation and risk effects on ecosystem stoichiometry.

Empirical evidence is beginning to show that predators can be important drivers of elemental cycling within ecosystems by propagating indirect effects that determine the distribution of elements among trophic levels as well as determine the chemical content of organic matter that becomes decomposed by microbes. These indirect effects can be propagated by predator consumptive effects on prey, nonconsumptive (risk) effects, or a combination of both. Currently, there is insufficient theory to predict how such predator effects should propagate throughout ecosystems. We present here a theoretical framework for exploring predator effects on ecosystem elemental cycling to encourage further empirical quantification. We use a classic ecosystem trophic compartment model as a basis for our analyses but infuse principles from ecological stoichiometry into the analyses of elemental cycling. Using a combined analytical-numerical approach, we compare how predators affect cycling through consumptive effects in which they control the flux of nutrients up trophic chains; through risk effects in which they change the homeostatic elemental balance of herbivore prey which accordingly changes the element ratio herbivores select from plants; and through a combination of both effects. Our analysis reveals that predators can have quantitatively important effects on elemental cycling, relative to a model formalism that excludes predator effects. Furthermore, the feedbacks due to predator nonconsumptive effects often have the quantitatively strongest impact on whole ecosystem elemental stocks, production and efficiency rates, and recycling fluxes by changing the stoichiometric balance of all trophic levels. Our modeling framework predictably shows how bottom-up control by microbes and top-down control by predators on ecosystems become interdependent when top predator effects permeate ecosystems.

Axisymmetric motion of a spherical porous particle perpendicular to two parallel plates with slip surfaces

A combined analytical–numerical approach to the problem of the low Reynolds number motion of a porous sphere normal to one of two infinite parallel plates at an arbitrary position between them in a viscous fluid is investigated. The clear fluid motion governed by the Stokes equation and the Darcy–Brinkman equation is used to model the flow inside the porous material. The motion in each of the homogeneous regions is coupled with the continuity of the velocity components, the continuity of the normal stress, and the tangential stress jump condition. The fluid is allowed to slip at the surface of the walls. A general solution for the field equations in the clear region is constructed from the superposition of the fundamental solutions in both cylindrical and spherical coordinate systems. The collocation solutions for the hydrodynamic interactions between the porous sphere and the walls are calculated with good convergence for various values of the slip coefficient of the walls, the separation between the porous sphere and the walls, the stress jump coefficient, and a coefficient that is proportional to the permeability. For the special cases of a solid sphere, our drag results show excellent agreement with the available solutions in the literature for all relative particle-to-wall spacing.

Crowdion–solute interactions: Analytical modelling and stochastic simulation

The interactions between point defects such as vacancies and self-interstitial atoms with solute atoms plays an important role in the evolution of the radiation damage microstructure. Alloys can suffer less void swelling, and exhibit reduced prismatic loop growth, compared with elemental metals, and this can be attributed to the solute atoms inhibiting point defect mobility. However, the microstructural evolution takes place on timescales inaccessible to electronic or atomistic methods, and high temperatures increase the complexity of Monte Carlo simulations, as more transitions become available, and more rates must be calculated. Here we present a combined analytical–numerical approach to model the stochastic evolution of the microstructure, focussing on the interactions between 〈111〉 crowdions and solute atoms in tungsten.

Influence of inclined electric fields on the effective fracture toughness of piezoelectric ceramics

Electric fields effectuate the fracture behavior of ferroelectrics on different scales. On the atomic scale, forces in covalent bonds are influenced. On the mesoscopic scale, the ferroelectric domain wall motion shields the crack tip or leads to an additional loading. Based on classical approaches, on the macroscopic scale, the electric field has a major impact on the energy release rate and the electric displacement intensity factor; however, it scarcely influences the mechanical stress intensity factors. The situation is different if Maxwell stresses at crack faces, as a consequence of the jump of dielectric properties, are incorporated in the crack tip loading analysis. First, general considerations deal with the question how a fracture criterion has to be formulated, incorporating all three scales. On the macroscopic scale, the influence of an arbitrarily inclined electric field on the critical mechanical crack loading is investigated. The extended model, in particular, reveals the significance of electric fields parallel to the crack faces. A combined analytical–numerical approach is applied to solve the coupled two-domains problem of solid and crack dielectric, supporting the well-established capacitor approach even for inclined electric fields.

Comparison of Finite-Element-Based State-Space Models for PM Synchronous Machines

An interior permanent-magnet (PM) motor is modeled by a combined analytical–numerical approach, in which the relationships between the stator currents and flux linkages are identified with static finite-element (FE) analysis. In addition to the previous approaches using the current space vector as the state variable, new models are also developed using the flux-linkage space vector, which leads to more convenient time-integration of the voltage equations. In order to account for the zero-sequence effects in delta connection, the models also include either the zero-sequence flux or current as an additional state variable. Finally, the possibilities of deriving the required quantities as partial derivatives of the magnetic field energy are discussed. The energy-based approaches avoid inaccuracies related to torque computation and thus allow better satisfying the power balance in the state-space model. We show the ability of the developed state-space models to predict the currents and torque equally to a nonlinear time-stepping FE model with much less computational burden. The results are validated by means of measurements for a prototype machine in both star and delta connections. In addition, we also demonstrate the effect of the zero-sequence current on the torque ripple in case of a delta-connected stator winding.

Generalized boundary conditions and effective properties in cracked piezoelectric solids

AbstractIn this paper we present two subjects of our actual research in the field. The first deals with the boundary conditions at the crack faces. The well known model by Hao and Shen gives opportunity to take the finite dielectric permeability of the crack into account, without having to solve the two‐ or three‐dimensional coupled boundary value problem of solid material and crack medium. This approach, however, is based on the assumption of the electric field being perpendicular to the crack faces. We investigate this problem for arbitrary poling and field directions based on a combined analytical‐numerical approach. The second focus of the paper is on the effective properties of piezoelectrics with cracks. Here, homogenization procedures are applied and extended towards coupled field problems including e.g. Maxwell stresses at internal boundaries and interfaces. Effective elastic, dielectric and piezoelectric constants exhibit interesting effects. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Anisotropic plasticity model coupled with Lode angle dependent strain-induced transformation kinetics law

A phenomenological macroscopic plasticity model is developed for steels that exhibit strain-induced austenite-to-martensite transformation. The model makes use of a stress-state dependent transformation kinetics law that accounts for both the effects of the stress triaxiality and the Lode angle on the rate of transformation. The macroscopic strain hardening is due to nonlinear kinematic hardening as well as isotropic hardening. The latter contribution is assumed to depend on the dislocation density as well as the current martensite volume fraction. The constitutive equations are embedded in the framework of finite strain isothermal rate-independent anisotropic plasticity. Experimental data for an anisotropic austenitic stainless steel 301LN is presented for uniaxial tension, uniaxial compression, transverse plane strain tension and pure shear. The model parameters are identified using a combined analytical–numerical approach. Numerical simulations are performed of all calibration experiments and excellent agreement is observed. Moreover, we make use of experimental data from ten combined tension and shear experiments to validate the proposed constitutive model. In addition, punch and notched tension tests are performed to evaluate the model performance in structural applications with heterogeneous stress and strain fields.

Combined analytical-numerical approach to the voltage of a cylindrical coil with pulsed current

This paper presents a combined analytical-numerical approach to solving the pulsed eddy current problem accurately and quickly. Considering the displacement current, the analytical solution to the voltage of a cylindrical coil above a laminated conductor in the complex-frequency domain is deduced by Laplace transform. The time-domain induction voltage values of a cylindrical coil with a pulsed current are calculated by the fourth-order integro-differential FFT-based numerical inversion of Laplace transform. At the same time, the time-domain analytical solution to the induced voltage of a cylindrical coil with a pulsed current above a half-infinite non-ferromagnetic conductor is derived, and has been verified by comparison with Finite Element Method (FEM) simulation results. The calculation results prove that the adopted numerical inversion method of applying Laplace transforms to the pulsed eddy current problem has a high accuracy and fast convergence. The transient voltages produced by a square-wave current excitation when considering the displacement current in the vacuum area are higher than those when ignoring the displacement current, by as much as 27.7% at certain times. The higher the lift-off is, the smaller the voltage peak is and the faster the voltage drops. As the application of this method, the induced voltages are computed in the measurements of metal's thickness and metal coating thickness.

The Turning Factor in the Estimation of Stream‐Aquifer Seepage

A combined analytical-numerical approach is presented to characterize properly the exchange flow between a stream and a hydraulically connected aquifer. It eliminates the need to use a three-dimensional fine grid under and in the vicinity of the river cross section in order to obtain accurate results. Basically the approach matches an analytical solution in a vertical two-dimensional (2D) plane with the numerical description of the aquifer behavior in a 2D horizontal plane. The approach is compared with a finite-difference formulation such as used in MODFLOW.

The local threshold for geographical spread of infectious diseases between farms

We investigated the influence of the spatial pattern of farms on the geographical spread of infectious livestock diseases, such as classical swine fever, foot-and-mouth disease and avian influenza in a combined analytical–numerical approach. Our purpose of this paper is to develop a method to identify the areas in which an infection has the potential to spread in an outbreak. In our model, each infected farm can infect neighbouring farms and the probability of transmission is a function of the inter-farm distance (spatial kernel). Therefore, the density of farms in an area is a good indicator for the probability of a major outbreak. In the epidemiological nomenclature, such density corresponds to a local reproduction ratio and we studied the critical behaviour of both the local density and the local reproduction ratio. We found that a threshold can be defined above which major outbreaks can occur, and the threshold value depends on the spatial kernel. Our expression for the threshold value is derived based on scaling arguments and contains two parameters in the exponents of the equation. We estimated these parameters from numerical results for the spatial spread using one particular mathematical function for the form of the spatial kernel. Subsequently, we show that our expression for the threshold using these estimated parameters agrees very well with numerical results for a number of different other functional forms of the spatial kernel (thus suggesting that we are dealing with universal parameters). As an illustration of the practical relevance of the presented method, we calculated the threshold value for avian influenza in the Netherlands and use it to produce a risk map for this disease.