Full Nonlinear Simulation Research Articles

Nonlinear simulations of GAEs in NSTX-U

A set of nonlinear simulations has been performed in order to study the nonlinear evolution of unstable global Alfvén eigenmodes in the National Spherical Torus Experiment-Upgrade (NSTX-U). Results of the single toroidal mode number, n, simulations are compared with a full nonlinear simulation (all toroidal harmonics included). In single-n simulations, the conservation of two integrals of motion of a particle in a cyclotron resonance with a monochromatic wave is demonstrated, resulting in a one-dimensional evolution of the particle distribution in (E,μ,pϕ) phase-space. Nonlinear simulations (both single-n and full nonlinear) show a significant redistribution of the resonant fast ions, especially in the pitch parameter. Thus, the changes in the resonant particle's parallel and perpendicular energies can be several times larger than the total particle energy change, with only a small fraction transferring into the excitation of the mode itself. This implies that even a relatively small amplitude mode can significantly modify the beam distribution in the resonant region. For the NSTX-U case considered, the single-n simulation results are close to full nonlinear simulation only for the most unstable mode, in which case the saturation amplitudes and changes in the fast ion distribution are comparable. In contrast, peak amplitudes of subdominant modes in all-n simulations are smaller by a factor of 3–10 compared to single-n runs due to the flattening of the beam ion distribution by the fastest growing mode.

An analytical linear two-dimensional actuator disc model and comparisons with computational fluid dynamics (CFD) simulations

Abstract. The continuous up-scaling of wind turbines enabled by more lightweight and flexible blades in combination with coning has challenged the assumptions of a plane disc in the commonly used blade element momentum (BEM)-type aerodynamic codes for the design and analysis of wind turbines. The objective with the present work is thus to take a step back relative to the integral 1-dimensional (1-D) momentum theory solution in the BEM model in order to study the actuator disc (AD) flow in more detail. We present an analytical, linear solution for a two-dimensional (2-D) AD flow with one equation for the axial velocity and one for the lateral velocity, respectively. Although it is a 2-D model, we show in the paper that there is a good correlation with axis-symmetric and three-dimensional (3-D) computational fluid dynamics (CFD) simulations on a circular disc. The 2-D model has thus the potential to form the basis for a simple and consistent rotor induction model. For a constant loading, the axial velocity distribution at the disc is uniform as in the case of the classical momentum theory for an AD. However, an important observation of the simulated flow field is that immediately downstream of the disc the axial velocity profiles change rapidly to a shape with increased induction towards the edges of the disc and less induction on the central part. This is typically what is seen at the disc in full non-linear CFD AD simulations, which is what we compare with in the paper. By a simple coordinate rotation the analytical solution is extended to a yawed disc with constant loading. Again, a comparison with CFD, now with a 3-D simulation on a circular disc in yaw, confirms a good performance of the analytical 2-D model for this more complicated flow. Finally, a further extension of the model to simulate a coned disc is obtained using a simple superposition of the solution of two yawed discs with opposite yaw angles and positioned so the two discs just touch each other. Now the validation of the model is performed with results from axis-symmetric CFD simulations of an AD with a coning of both 20 and −20∘. In particular, for the disc coned in the downwind direction there is a very good correlation between the simulated normal velocity to the disc, whereas some deviations are seen for the upwind coning. The promising correlation of the results for the 2-D model in comparison with 3-D simulations of a circular disc with CFD for complicated inflow like what occurs at yaw and coning indicates that the 2-D model could form the basis for a new, consistent rotor induction model. The model should be applied along diagonal lines on a rotor and coupled to an angular momentum model. This application is sketched in the outlook and is a subject for future research.

Buckling resistance of star-battened angle members made of high-strength steel

Recent upwards developments of the power grid requests lead to use of high strength steels (HSS) for the design and construction of higher and heavier loaded steel lattice transmission towers, or to strengthen the existing ones. Usually star-battened sections are preferred for the strengthening of individual members or in case of very high compression loads. Thus, the investigations focus here on the bearing capacity of a star-battened member made of two S460 L250x250x28 angles. The objective was to use a star-battened member made of two S460 L300x300x35 angles with a member length of 4486 mm that will be used in a specific project of 240 m high power transition tower. However, as the request in terms of length capacity cannot be met by the testing labs in Belgium, the studies have concentrated on the profile made of L250x250x28 angles.Firstly, a critical review of the current European normative documents (EN 50341, EN1993-1–3, prEN1993-3) has been made and the buckling resistance of the above-mentioned profile has been evaluated. Then, an experimental compression test has been performed for this member. The test has been complemented by a full non-linear finite element simulation by means of ANSYS software. Experimental, numerical and analytical results have been compared and discussed. Finally, conclusions were drawn considering the design of S460 star-battened members in compression. These studies are part of an ongoing project entitled “New steel” funded by Elia and ArcelorMittal and involving the University of Liège.

The nonlinear effects of spinning on the dynamics of a pitching cantilever

We investigate the structural dynamics of a pitching cantilever as it spins about a fixed axis, emulating the motion of the rotor blades of wind turbines or rotorcraft to some extent. From first principles, we derive a system of equations consisting of a nonlinear partial differential equation (PDE) coupled with a nonlinear ordinary differential equation (ODE) that describes the motion of the system in bending and pitch as it spins according to a prescribed rotor velocity and acceleration. The equations of motion consider the interaction between dynamics of a flexible body in bending, rigid body motion in pitch, and the nonlinear effects of spinning, concurrently. This simplified model of a rotor blade permits an intuitive understanding of the various nonlinear terms that arise in the equations of motion, and their relative influence on the system dynamics. In a series of numerical experiments, we study the effects of the coupling between the bending and pitch degrees of freedom, the influence of rotor velocity, and rotor acceleration on the system dynamics. We compare the full nonlinear simulation results to those of the underlying linear system, obtained by imposing a small angle assumption in pitch and excluding all remaining nonlinear terms. The numerical experiments are performed by non-dimensionalizing the equations, discretizing the non-dimensionalized equations in space by applying the Galerkin method, discretizing the resulting system of ODEs in time using the Houbolt method and finally solving the resulting system of implicit algebraic equations using Newton’s method.

IBEM-FEM coupling method for full process nonlinear ground motion simulation of near-fault sedimentary basins

Based on the concept of domain reduction, a two-step method combining the indirect boundary element method (IBEM) and the finite element method (FEM) was developed to simulate the full-process ground motion of a 3D (three-dimensional) near-fault complex site. In the first step, the seismic response of dislocation motion in semi-infinite space crust model was accurately solved using the IBEM, which can significantly reduce the dimensionality of the numerical model and easily achieve a computational frequency of 5 Hz on an ordinary workstation. In the second step, considering the inhomogeneity and nonlinearity of soft soils, the seismic wave scattering from a 3D local site was finely simulated using a FEM. The external seismic response obtained through IBEM is converted into equivalent load and transferred to the internal finite element domain, while the PML boundary is applied to absorb the pseudo reflected wave on the truncated boundary. Then, the reliability and accuracy of the method was demonstrated by comparing with the results of existing studies and the traditional IBEM. Further, the proposed coupled method was used to investigate the seismic response of a large near-faulted sedimentary basin. The numerical investigation focuses on four aspects: acceleration histories and response spectra, wave-field snapshot, duration of ground motion, and peak magnitudes and response spectrum of velocity. The results highlight the influence of local site soil non-linearity and medium impedance ratios on the analysis of near-fault seismic response. This research is essential for seismic zoning of complex sedimentary basin sites near faults, and also establishes a “bridge” for seismic design from faults to structures.

Experimental investigation of doubly-symmetric built-up I–girders subjected to high moment gradient

Recent analytical studies have shown that the AISC 360-16 Specification overpredicts the flexural resistance of certain built-up steel I-girders. The largest overpredictions were observed in I‑girders subjected to a high moment gradient having unstiffened webs with a large web slenderness ratio. The objectives of the current research are to provide experimental validation of these strength overpredictions, validate the accuracy of full-nonlinear shell finite element analysis (FEA) solutions, and further investigate the behavior of these susceptible built-up I-girders. To accomplish the objectives, six largescale experimental tests targeted web slenderness ratios ranging from 77 to 192 and a moment gradient factor, Cb, of 1.74 (single curvature (SC) bending) and 2.31 (reverse curvature (RC) bending), calculated using common design equations. Additionally, full nonlinear shell FEA simulations were conducted to evaluate the correlation with the experiments. The tests showed flexural strengths smaller than the predicted resistances from AISC 360, with the predicted resistances ranging from 4% to 16% unconservative for SC members and 9% to 32% unconservative for RC members. Furthermore, the FEA simulation strengths were similar to the strengths measured in the experiments. The results indicate that there are two main phenomena leading to the overpredictions by AISC 360: web distortion effects exacerbated by web shear and the direct scaling of the uniform LTB strength curve by Cb up to the plateau resistance, overestimating the strength gains from moment gradient due to insufficient accounting for the onset of yielding.

Transverse liquid composite moulding processes for advanced composites material manufacturing

ABSTRACT The hydrodynamic compaction behaviour in transverse (through-thickness) impregnation variants of Liquid Composite Moulding (LCM) results in a non-homogeneous fibre volume fraction distribution. A fully coupled simulation model of the entire family of transverse resin infusion/compression processes is presented, which simulates the hydrodynamic compaction behaviour and predicts the process time, fluid pressure, effective stress, and fibre volume fraction distribution. The model is applied to the compression of a saturated preform stack, resin injection into a dry preform, and Compression Resin Transfer Moulding. The simulations are verified using analytic solutions and validated against experimental results from the literature. Special forms of the constitutive equations are constructed to lead to a constant diffusion coefficient, and the resulting model is solved semi-analytically. The results are used to verify the full non-linear simulation model. Additionally, it is shown that assumptions of quasi-steady flow and uniform deformation introduce significant errors in processes involving direct piston compaction of the fibre preform.

Powertrain shuffle-mode resonance suppression by means of flywheel mounted torsichrone centrifugal pendulum vibration absorbers

A flywheel mounted centrifugal pendulum absorber (CPVA) is designed to completely suppress the low frequency shuffle-mode resonance of a heavy-duty truck powertrain by exploiting the eigenfrequency veering property of the CPVA. The shuffle-mode is excited by the half-order torque of a five-cylinder engine in a downspeeded powertrain during cruising conditions. It is shown that the design of the CPVA may be performed by linear analysis of the CPVA and powertrain system. The linear design is then validated by full nonlinear simulations of the powertrain. A complete suppression of the shuffle-mode resonance is achieved with relatively small total mass of the pendula which makes it highly possible to implement in practice. Downspeeding and downsizing of combustion engines, without sacrificing the power output, are methods to reduce emissions, although, resulting in increased noise and vibration using conventional vibration reduction technology. The CPVA is an order tuned device that can help to reduce the increased vibrations and thus help to reduce the environmental impact of heavy-duty vehicles.

Mooring system stiffness: A six-degree-of-freedom closed-form analytical formulation

A mooring system project is an iterative process, involving design and analysis with numerous parameters. In this context, analytical tools can help to reduce the use of expensive and time consuming full nonlinear numerical simulations based on high-order hierarchical models such as Finite Element models, mainly during early stages of the design. By employing Analytical Mechanics approaches, the present article brings an analytical and closed-mathematical-form formulation to determine the six-degree-of-freedom (DOF) mooring system stiffness matrix at any generic position of the floating body, provided mooring line forces are modeled as conservative ones. In order to demonstrate the use of these tools, the international benchmark of the OC4-DeepCwind platform is considered as a case study. The influence of the vessel mean position and design parameters in the stiffness, natural periods and oscillation modes is studied, revealing interesting coupling effects. Results are benchmarked against the commercial code OrcaFlexTM showing good agreement, observing that the analytical result requires much less computational cost.

A simplified model of oscillating electrolytes.

Unstable electrophoretic transport leading to oscillations in concentration profiles occur in certain electrolyte systems known as oscillating electrolytes whose eigenmobilities are complex valued. The study of the nonlinear behavior of such systems is of great interest but is constrained due to a high degree of complexity in the governing equations. Here we present a simplified model of unstable electrophoretic transport in a binary system that reduces the governing equations to two partial differential equations only and does away with other equations that characterize acid-base dissociation reactions and electroneutrality. We present analytical expressions for electromigration fluxes and validate the model with full nonlinear simulations. The model exhibits similar nonlinear behavior as the actual unstable electrophoretic system under various initial disturbances. For comparison, we also show that similar modeling for a stable system predicts concentration profiles that quantitatively agree with its nonoscillating dynamics. Moreover, the unique feature of electromigration flux in oscillating electrolytes that unfolds from the modeling led us to find an elegant explanation of the instability mechanism. Our theory gives a qualitative understanding of the existence and growth of large oscillation patterns in oscillating electrolytes.

Nonlinear Flight Dynamics and Control of a Fixed-Wing Micro Air Vehicle: Numerical, System Identification and Experimental Investigations

The flight dynamics of Micro Air Vehicles (MAVs) exhibit significant nonlinear characteristics, which cannot be ignored in simulation or analysis. In this study, a full nonlinear simulation of the flight dynamics characteristics of a fixed-wing MAV is performed. To strengthen the developed nonlinear mathematical modeling, the MAV’s mass properties and propulsion characteristics are experimentally investigated. Moreover, the aerodynamic characteristics of the designed fixed-wing MAV are experimentally measured in a wind tunnel. The experimental aerodynamics investigation includes the propeller wash effect and the same flight conditions of the MAV. These measured data are fed to the nonlinear flight dynamics model to improve its accuracy and ensure the nonlinear aerodynamics effect on the flight dynamics. The enhanced model is then validated against response experimental measurements of the MAV in the wind tunnel that is free to pitch at different control inputs and initial conditions. Furthermore, it is compared to the response of the system identification model. The nonlinear simulation and dynamic testing investigations indicate many nonlinear phenomena, such as the appearance of the limit cycle in the longitudinal flight. This paper shows that the aerodynamic center of MAV with a low aspect ratio in the low Reynolds number regime of flow can move as a response to flap deflection. The validated nonlinear mathematical model was used to evaluate the MAV’s dynamics, design and evaluate a PID controller in flight conditions similar to the MAV’s actual flying mission. Moreover, the presented model can be used for flapping-wing MAVs using time-dependent experimental measurements.

Dynamic models and control system design for heated channels in a Canadian SCWR

The proposed Canadian SCWR is based on the concept of the CANDU reactor. However, the dynamic characteristics of the SCWR significantly differ from the CANDU reactor because of the special heat transfer phenomenon of the supercritical water in the fuel bundle. Thus, it is important to design a control system to maintain the SCWR operating at the required operating point when it is subjected to disturbances. The transient CFD simulations of the heated channels in a Canadian SCWR are conducted in order to obtain the dynamic relationship between the inputs and outputs of the SCWR. Then, the linear dynamic control models are constructed by the system identification technique based on the dynamic relationship between the inputs and outputs obtained from the CFD simulations. The linear dynamic models are validated using the results from the full scale non-linear CFD simulations. Based on the linear dynamic models, a closed-loop control system, which consists two PID controllers, is designed. The performance evaluation of the proposed control system is also carried in this work.

Optimization of mooring systems in the context of an integrated design methodology

This work describes an enhanced mooring optimization procedure, oriented towards recent floating production systems (FPS) for oil & gas exploitation in ultra-deep-water scenarios, which may present a large number of risers in an asymmetric layout. Acknowledging that the risers are the key component of an FPS, the optimization procedure is associated to an integrated mooring-riser design methodology; thus, instead of simply minimizing the platform offsets and/or the costs of the mooring system itself, one of the main objectives is to obtain a mooring configuration that ensures the integrity of the risers. Other highlights of the optimization procedure include the following aspects: Enhancements in the modeling of the optimization problem (including the definition of design variables, objective function and constraints that are relevant for such actual applications); The use of the PSO optimization algorithm associated to the ε-constrained method to efficiently handle the constraints; Enhancements in the evaluation of candidate solutions, by full nonlinear time-domain dynamic Finite Element simulations with coupled models; and the implementation in a parallel computing environment to deal with the high associated computational costs. A case study considering an FPS representative of actual applications in deepwater scenarios is presented to illustrate the practical application of the optimization tool.

Investigating Instabilities in the Mammalian Cochlea Using a Stochastic Uncertainty Model

We examine the problem of mean-square stability in a dynamical model of the mammalian cochlea with stochastically uncertain parameters. The cochlea is a mechanical spectrum analyzer with an adaptive gain amplification mechanism that gives it a very large dynamic range as an acoustic sensor. This adaptive gain mechanism has been conjectured to be responsible for occasional instabilities that can be clinically significant. We model the cochlea as a spatio-temporal dynamical system made up of a spatially distributed array of coupled oscillators incorporating an adaptive amplification mechanism. We consider settings where the cochlear amplifier has spatially and temporally varying stochastic parameters. It is shown that relatively small parameter variations (few orders of magnitude smaller than the nominal values) are sufficient to destabilize the dynamics and induce spontaneous oscillations. This extreme sensitivity of the cochlear dynamics appears to be due to a combination of the local cochlear amplification mechanism, as well as the spatial coupling of the distributed resonators. The analysis technique used in this work allows for a simulation-free prediction of the stability thresholds and the statistics of the spontaneous oscillations. Theoretical predictions are verified using full nonlinear stochastic simulations that demonstrate a good agreement with the theoretical predictions.

Nuclear Fuel Assembly Deformation, Reduced Mechanical Model Dedicated to FSI Simulation

In the pressure vessel of a PWR the core is constituted of a number of fuel assemblies. Under operation, they all suffer from various phenomena, among them, initial trapped efforts, thermal expansion, irradiation growth, creeping, tightening relaxation and rod slipping. Each fuel assembly may develop contacts with its neighbors and strongly interacts with the fluid. It then undergoes vertical as well as lateral fluid forces, which are partly coupled with its instantaneous shape. As a consequence the fuel assembly tends to get distorted during and still after operation. The first concern is economic: at the end of a cycle, when changing the assembly position in the core, the contact interactions with its neighbors due to its distorted shape sometimes delay the handling. The second concern is that the shutdown rods have to slide into the deformed shape of the guide tube and might be slowed due to excessive friction forces. In order to precisely assess these situations it is necessary to simulate numerically a complete core, taking into account all of the previously mentioned phenomena. For that purpose, we propose a light but relevant reduced mechanical model of a fuel assembly, partially based on a POD analysis performed on a detailed fuel assembly model. We then show that a rather simple model, still based on classical finite elements can be used and simplifies the integration of the local nonlinear constitutive equations. We then compare both of the detailed and reduced model on typical situations and show excellent agreement. The CPU time ratio between them, for a full nonlinear simulation of a single fuel assembly under operation, is over 1000, which is competitive against separation of variables based reduced order methods such as POD, PGD and even APHR. So, yet still with a simplified FSI approach, the drastic memory and CPU reductions obtained now enable us to complete a 3D core simulation representing at least 4 years under operation (4 cycles), with as much as 241 individual fuel assemblies, in a reasonable time of a few hours.

Linear instability of shear thinning pressure driven channel flow

We study theoretically pressure driven planar channel flow of shear thinning viscoelastic fluids. Combining linear stability analysis and full nonlinear simulation, we study the instability of an initially one-dimensional base state to the growth of two-dimensional perturbations with wavevector in the flow direction. We do so within three widely used constitutive models: the microscopically motivated Rolie-Poly model, and the phenomenological Johnson-Segalman and White-Metzner models. In each model, we find instability when the degree of shear thinning exceeds some level characterised by the logarithmic slope of the flow curve at its shallowest point, n=dlogΣ/dlogγ˙|min. Specifically, we find instability for n < n*, with n* ≈ 0.21, 0.11 and 0.30 in the Rolie-Poly, Johnson-Segalman and White-Metzner models respectively. Within each model, we show that the critical pressure drop for the onset of instability obeys a criterion expressed in terms of this degree of shear thinning, n, together with the derivative of the first normal stress with respect to shear stress. Both shear thinning and rapid variations in first normal stress across the channel are therefore key ingredients driving the instability. In the Rolie-Poly and Johnson-Segalman models, the underlying mechanism appears to involve the destabilisation of a quasi-interface that exists in each half of the channel, across which the normal stress varies rapidly. (The flow is not however shear banded in any parameter regime that we consider.) In the White-Metzner model, no such quasi-interface exists, but the criterion for instability nonetheless appears to follow the same form as in the Rolie-Poly and Johnson-Segalman models. This presents an outstanding puzzle concerning any possibly generic nature of the instability mechanism. We finally make some brief comments on the Giesekus model, which is rather different in its predictions from the other three.

Non-Linear Dynamic Inversion Control Design for Rotorcraft

Flight control design for rotorcraft is challenging due to high-order dynamics, cross-coupling effects, and inherent instability of the flight dynamics. Dynamic inversion design offers a desirable solution to rotorcraft flight control as it effectively decouples the plant model and effectively handles non-linearity. However, the method has limitations for rotorcraft due to the requirement for full-state feedback and issues with non-minimum phase zeros. A control design study is performed using dynamic inversion with reduced order models of the rotorcraft dynamics, which alleviates the full-state feedback requirement. The design is analyzed using full order linear analysis and non-linear simulations of a utility helicopter. Simulation results show desired command tracking when the controller is applied to the full-order system. Classical stability margin analysis is used to achieve desired tradeoffs in robust stability and disturbance rejection. Results indicate the feasibility of applying dynamic inversion to rotorcraft control design, as long as full order linear analysis is applied to ensure stability and adequate modelling of low-frequency dynamics.

A Linearised Analysis for Structures With Synchronized Switch Damping

Synchronized Switch Damping (SSD) is a semi-active damping technology based on piezoelectric materials. It has advantages such as broadband and no need to tune. Despite the nonlinear governing equations, structures with SSD exhibit quasi-linear behaviour such as the resonant frequencies hardly vary with respect to energy level of excitation. Inspired by these phenomena, in this paper we propose a linearisation method for SSD, where the nonlinear force is equivalent to frequency-dependent viscous damping and linear stiffness coefficients. Closed-form expressions of these linearised parameters are given, making the method applicable both for lumped parameter models and finite element (FE) models. In the derivation a general force equation is used, so the linearised method is applicable for several typical variants of SSD, such as SSDS (S for 'on short-circuited'), SSDI (I for `on inductance'), SSDV (V for `on voltage') and SSDNC (NC for `on negative capacitance'). The method is first validated against nonlinear simulations with harmonic and random vibration respectively, then further compared with experimental data in a published paper. Good agreements are found. We show that the proposed method can dramatically accelerate the computational efficiency, which is especially suitable for predicting the dynamic performance of complex structures with SSD. Eventually, a dummy integrally bladed disk with SSD is analysed to illustrate a potential application direction. There are more than 120k DOFs in the FE model, making full nonlinear simulation very time-consuming. However the simulation is finished within seconds by the proposed method and the typical damping characteristics of SSD are well captured.

Design of a fast real-time LPV model predictive control system for semi-active suspension control of a full vehicle

This article is concerned with the control of a Semi-Active suspension system of a 7DOF Full Vehicle model, equipped with four Electro Rheological (ER) dampers, taking into account their incipient dissipativity constraints. Herein, a real-time, fast, advanced control structure is presented within the Model Predictive Control framework for Linear Parameter Varying (LPV) systems. The control algorithm is developed to provide a suitable trade-off between comfort and handling performances of the vehicle in a very limited sampling period (Ts=5ms), in view of a possible realtime implementation on a real vehicle. The control structure is tested and compared to other standard fast control approaches. Full nonlinear realistic simulation results illustrate the overall good operation and behaviour of the proposed control approach.

Secondary flows due to finite aspect ratio in inertialess viscoelastic Taylor–Couette flow

Both in rheometry and in fundamental fluid mechanics studies, the Taylor–Couette geometry is used frequently to investigate viscoelastic fluids. In order to ensure a constant shear rate in the gap between the inner and outer cylinders, such studies are usually restricted to the small-gap limit where the assumption of a linear velocity distribution is well justified. In conjunction with a sufficiently large aspect ratio$\unicode[STIX]{x1D6EC}$(i.e. ratio of cylinder height to gap), the flow is then assumed to be viscometric. Here we demonstrate, using a perturbation technique with the curvature ratio (i.e. ratio of the half-gap to the mid-radius of the cylinders) as the perturbation parameter, full nonlinear simulations using a finite-volume technique, and supporting experiments, that, even in the creeping-flow (inertialess) narrow-gap limit, for viscoelastic fluids end effects due to finite aspect ratio always give rise to a secondary motion. Using the constant-viscosity Oldroyd-B model we are able to show that this secondary motion, as has been observed in related pressure-driven flows with curvature, such as the viscoelastic Dean flow, is solely a consequence of the combination of gradients of the first normal-stress difference and curvature. Our results show that end effects can significantly change the flow characteristics, especially for small aspect ratios, and this may have important consequences in some situations such as the onset criteria for purely elastic instabilities.