Tip Velocity Research Articles

A consistent finite element approach for dynamic crack propagation with explicit time integration

The concept of the partition of unity (PU) enabled the development of nodal enrichment strategies, such as the Extended Finite Element Method (XFEM) and Generalised Finite Element Method (GFEM), for realistic simulations of structural behaviour with applications to both static and dynamic problems. Nonetheless, the majority of existing methodologies still inherit instability issues for arbitrary discontinuity geometries when using an explicit time integration scheme. To address this, the discrete strong discontinuity approach is herein developed for the simulation of dynamic crack propagation. The formulation is fully variationally consistent and a new mapping approach is introduced to embed the rigid body movements associated with discontinuities while keeping the critical time step bounded. Multiple discontinuities within a single element are also considered for the accurate modelling of crack branching. The stability of the new technique is first verified in one- and two-dimensional elements. Next, the accuracy and efficiency are validated with structural examples including tensile and mixed-mode loadings. A concrete L-specimen, where different loading rates produce significant variations in failure patterns and strength, is also considered. Results show the good overall agreement with experimental data and other numerical studies available in the literature. The new formulation, however, is able to capture complex crack propagation phenomena, such as crack branching, without any specific additional criterion (e.g. based on crack tip velocity). The formulation presented in this paper is, to the best of the authors’ knowledge, the first PU-based consistent finite element with intrinsically bounded critical time step for explicit time integration.

Hysteresis compensation-based robust output feedback control for long-stroke piezoelectric actuators at high frequency

Piezoelectric actuators (PEAs) are the key element for many vital applications and they play a big role in modern industries due to their operational characteristics. However, parametric uncertainties along with nonlinearities due to hysteresis and creep effects certainly impact severely the high precision tracking control of PEAs. This paper presents a robust output feedback controller for PEAs with hysterical dynamics compensated. In addition, a high gain observer is utilized to estimate the PEA’s tip velocity in order to facilitate the operation of the control algorithm. The proposed controller is designed to handle the unknown parameters and uncertainties associated with the nonlinear behavior of the PEAs. It is also tested on a commercial long-stroke PEA using different high frequency profiles. Experimental results show the effectiveness and robustness of the proposed controller compared to existing controllers.

Influence of Wave Shaper Position on Jet Formation and Penetration Depth

The article presents the influence of the wave shaper position on the jet formation and penetration depth of a shaped charge, which uses the liner of the PG-7VM warhead by numerical method and experimental method. The results show that, when moving the plexiglass wave shaper position towards the liner apex, the jet tip velocity and the jet velocity gradient increase. In addition, the results of the study also pointed out the existence of the wave shaper position at which the penetration depth is greatest. They indicate a match between the simulation and experimental results. The differences between simulated and experimental penetration depth are less than 15%. The results of the study are the basis for the evaluation of the influence of the wave shaper position on the liner collapse and the reasonable use of the wave shaper in the structure of the shaped charge.

Hydrodynamic computations of high-power laser drives generating metal ejecta jets from surface grooves

Understanding dynamic fragmentation in shock-loaded metals and predicting properties of the resulting ejecta are of considerable importance for both basic and applied science. The nature of material ejection has been shown to change drastically when the free surface melts on compression or release. In this work, we present hydrodynamic simulations of laser-driven microjetting from micron-scale grooves on a tin surface. We study microjet formation across a range of shock strengths from drives that leave the target solid after release to drives that induce shock melting in the target. The shock-state particle velocity (Up) varies from 0.3 to 3 km/s and the shock breakout pressure is 3–120 GPa. The microjet tip velocity is 1–8 km/s and the free-surface velocity varies from 0.1 to 5 km/s. Two tin equations of state are examined: a “soft” model (LEOS 501) where the target melts for Up>1 km/s and a more detailed multiphase model (SESAME 2161) that melts for Up>1.4 km/s. We use these two models to examine the influence of phase change and the choice of the material model on microjet formation and evolution. We observe in our computational results that jet formation can be classified into three regimes: a low-energy regime where material strength affects jet formation, a moderate-energy regime dominated by the changing phase of tin material, and a high-energy regime where results are insensitive to the material model and jet formation is described by an idealized steady-jet theory. Using an ensemble of 2D simulations, we show that these trends hold across a wide range of drive energies and groove angles.

The Interaction between Grains during Columnar-to-Equiaxed Transition in Laser Welding: A Phase-Field Study

A phase-field model was applied to study CET (columnar-to-equiaxed transition) during laser welding of an Al-Cu model alloy. A parametric study was performed to investigate the effects of nucleation undercooling for the equiaxed grains, nucleation density and location of the first nucleation seed ahead of the columnar front on the microstructure of the fusion zone. The numerical results indicated that nucleation undercooling significantly influenced the occurrence and the time of CET. Nucleation density affected the occurrence of CET and the size of equiaxed grains. The dendrite growth behavior was analyzed to reveal the mechanism of the CET. The interactions between different grains were studied. Once the seeds ahead of the columnar dendrites nucleated and grew, the columnar dendrite tip velocity began to fluctuate around a value. It did not decrease until the columnar dendrite got rather close to the equiaxed grains. The undercooling and solute segregation profile evolutions of the columnar dendrite tip with the CET and without the CET had no significant difference before the CET occurred. Mechanical blocking was the major blocking mechanism for the CET. The equiaxed grains formed first were larger than the equiaxed grains formed later due to the decreasing of undercooling. The size of equiaxed grain decreased from fusion line to center line. The numerical results were basically consistent with the experimental results obtained by laser welding of a 2A12 Al-alloy.

Breakup characteristics of high speed liquid jets from a single-hole injector

In this study, spray characteristics from a single-hole gasoline direct injection injector with nozzle diameters of 0.14 mm and 0.2 mm were investigated. High-speed macroscopic and microscopic imaging techniques were applied to record the overall spray structure and the initial fuel spray behavior. The injection pressure ranged from 50 bar to 350 bar, and the ambient pressure was kept constant at 1 bar. Several semi-empirical models for predicting spray penetration at non-evaporating conditions were investigated, including models by Dent, Hiroyasu and Arai, and Naber and Siebers. Results showed that the Naber and Siebers equation demonstrated the best match with the experimental data. Both the initial and global spray penetration were well predicted. The spray penetration can be divided into two regions based on the relationship between spray penetration and injection time, namely the regions before and after breakup, and the transitional point was defined as the breakup point. Higher fuel pressures resulted in shorter breakup time and length for the injector with the smaller nozzle diameter, and the injector with larger nozzle diameter produced shorter breakup time and longer breakup length. A linear relationship was identified between the initial spray penetration and injection time at all fuel pressures, which indicates the initial spray tip velocity can be treated as constant at each fuel pressure. The ratio between the slope of experimental initial spray penetration and average flow velocity (based on static flow rate measurements) was around 0.9 at all fuel pressures for the injector with 0.14 mm nozzle diameter. Consequently, the average flow velocity obtained based on the static flow rate at each fuel pressure can be used to accurately predict the initial jet velocity.

Experimental statistics of micrometer-sized water droplet deformation and breakup behavior in continuous air jet flow

Fundamental study on breakup of the supercooled large droplet is important for understanding the microphysical phenomenon in aircraft icing. The deformation and breakup processes of water droplets with diameters in the range of 700 to 1000 microns injected into continuous air jet flow are investigated experimentally. Evolutions of the bag, the bag and stamen and the dual breakup morphology are recorded through the high-speed photography. Experiments are repeated 319 times to determine the statistics based dynamic behavior, including the mean deformation rate, velocity and duration. As concluded, minimum value of the droplet deformation rate in the x-direction increases from 0.46 to 0.72 when Weber number increases from 18 to 56. When the initial droplet breaks, the velocity of droplet tip is more than two times of the velocity of droplet terminal. Duration of the deformation stage accounts for about half of the total duration. Besides, a new interpretation of physical mechanism, combining effects of the aerodynamic drag, motion hysteresis and inertia, is proposed to explain formation and development of different deformation and breakup morphologies. This work is meaningful for the development of numerical simulation and theoretical models.

Tool-Center-Point Control of a Concrete Pump Using Constrained Quadratic Optimization

Tool-center-point (TCP) control increases the operating comfort of concrete pumps by shifting the control to the task space. The operation is accompanied by various requirements and constraints from the mechanical structure, the hydraulic system, and the operator. They must be taken into account for the efficient and error-free operation of the concrete pump. In this publication, a TCP control algorithm based on constrained quadratic optimization (cQP) is presented, which addresses the requirements and constraints of a mobile concrete pump. The features include boom deformation compensation, multiple operation modes, configuration control, compliant constraints on the joint movement, and hydraulic flow rate constraints. The methods are verified by simulation for a flexible link six degrees of freedom concrete pump. The results include deformation-compensated 2-D/3-D movements of the TCP with different operation modes and configuration control under joint and hydraulic constraints. Note to Practitioners —For effective concrete placement, the tip of the concrete pump needs to move linearly in the task space. Tool-center-point (TCP) control allows the operator to directly specify the tip velocity in the task space instead of separately controlling the axes of the manipulator. In this article, the control task is posed as an inverse kinematic problem, which is solved by quadratic optimization (cQP). The framework of cQP enables the direct inclusion of system constraints while being computational efficient for online implementation on the electronic control unit of the truck. The approach addresses concrete pump specific problems and requirements of the tool center point movement and actuator constraints. Restrictions in the task space are not treated.

Investigation of Explosion Simulation on Jet Forming of ZrCuNiAlAg Amorphous Alloy Liner with Eccentric Sub-hemisphere Structure

Structure and material are the two key factors affecting the jet forming of the liner. To investigate the influence of new structure and new materials on the jet forming of the liner, a shaped charge model with eccentric sub-hemisphere structure based on ZrCuNiAlAg amorphous energetic alloy was proposed. The jet forming law of eccentric sub-hemispherical liner is studied by fixing other parameters and changing the structure parameters of liner in turn by means of AUTODYN numerical simulation software. The simulation results show that the eccentric sub-hemispherical liner of ZrCuNiAlAg amorphous alloy can form a shaped jet very well by using gathering energy effect. The speed of the shaped jet decreases gradually with time, and the shape of the jet changes from short and thick to long and thin. The curvature radius, eccentricity and wall thickness have a significant effect on the jet forming performance of ZrCuNiAlAg amorphous alloy. As the curvature radius or wall thickness increase, the tip velocity and the length diameter ratio (LDR) of the jet will decrease gradually, while with the increase of the eccentricity of the liner, the jet tip velocity and the LDR will increase gradually. The analysis of simulation data shows that the change of eccentricity has the greatest influence on the jet LDR. When the eccentricity increases from 25 mm to 39 mm, the jet LDR increases by 74% at 60μs. The wall thickness has the greatest influence on the tip velocity of the jet. When the wall thickness increases from 1.4 mm to 3.8 mm, the tip velocity of the jet at 60μs decreases by 29.77%. The results of this paper can be used for scientific reference in the structural design of this kind of liner.

Dendritic crystallization from the undercooled melts: effect of tiny amount of impurity

The kinetics of dendrite crystal growth in pure Ni and diluted Ni–B melts is analyzed. Using a model for anisotropic dendrites rapidly growing from undercooled melts, numerical solutions for dendrite tip velocity V and tip radius R are presented. The influence of boron content on tip velocity V and tip radius R of the dendrite growing in Ni-B alloys is shown in comparison with the growth kinetics of pure Ni–dendrites. Particularly, we show that there is a well-known regime of growth in which the impurity (boron) decreases the velocity of the alloying Ni–B–dendrite regarding the nickel dendrite velocity. In addition to this well-established regime, we also show quantitatively that there is a tiny amount of boron which enhances the dendrite velocity in comparison with the velocity of pure nickel dendrite. Explanations of this non-linearity are given using the results of calculations for dendrite tip radii and dendrite growth velocity for alloying Ni–B and pure Ni systems.

Numerical study of the effects of laminar flame speed on flame dynamics in the early stages of flame propagation in two-dimensional half open tubes

The effect of laminar flame speed on the flame dynamics in two-dimensional half open tubes is investigated through a two-dimensional laminar combustion model. Several single-step reaction mechanisms are established to alter the corresponding laminar flame speed. The simulation results show that the similarity of flame propagation only holds during the acceleration phase. In the flame acceleration phase, the flame shape and flame tip position corresponding to each dimensionless moment are almost the same. This similarity gradually disappears after the flame touches the wall. When the flame decelerates, the periodic oscillation of the flame tip velocity is related to the laminar flame speed. The higher the laminar flame speed, the longer the corresponding dimensionless oscillation period. The destruction of this similarity is also reflected in the correlation between the flame surface area and the growth rate of burnt gas volume. In addition, the dimensionless evolution correlation of the flame surface area during the acceleration phase is given in this paper. The moving speed of the flame skirt at different laminar flame speeds and the time interval of flame inversion in the two-dimensional half open tubes are also presented.

Velocity and mass density of the ejecta produced from sinusoidal grooves in laser shock-loaded tin

When a shock wave of several tens of GPa breaks out at a free surface, a material is ejected ahead of this surface. The amount and velocity of such ejecta depend on the breakout pressure, state of the released material (solid, liquid, or mixed), whether the shockwave is supported or unsupported, and the initial geometrical perturbation (or roughness) of the free surface. If surface defects consist of small grooves, pits, or scratches, material ejection occurs in the form of jets breaking up into tiny particles (so-called microjetting), with jet tip velocities up to several times higher than the free surface velocity. The laser-based experiments presented in this paper focus on microjetting in shock-melted tin with periodic surface perturbations. Several complementary diagnostics are combined to measure the velocity and mass of ejecta during the early stages of the jetting process. One relevant advancement is the use of ps-laser x-ray radiography to probe the density of the ejecta in distinct jets a few tens of μm-wide. The effects of the depth and wavelength of the initial perturbation are investigated in both linear and near-linear growth regimes. The results are compared with predictions derived from the Richtmyer–Meshkov Instability theory.

Bubble formation in viscous fluids by a microfluidic flow-focusing junction: a computational study

A three-dimensional (3D) numerical study of bubble formation process in viscous fluids in a microfluidic flow-focusing junction is carried out in the present study. The numerical approach is first validated by the experimental data and shows a very good agreement, and a reciprocal relationship can be derived between the normalized bubble length and liquid fraction. Afterwards, the numerical approach is further applied to study bubble formation in fluids with different viscosities ranging from 5.6 to 400 mPa s, and the viscosity ratio of liquid phase to gas phase (μl/μg) is up to 2.2 × 104. Two different interpolation treatments including the Geo-Reconstruction method and the CICSAM are implemented when dealing with the situation for high viscosity, and the results are compared with the experimental data, indicating a better performance of the Geo-Reconstruction method. The numerical results of the bubble shapes are also compared with the experimental visualization results, showing good agreements. The evolution of the gas tip is further investigated quantitatively, and the liquid film is found to play a very important role in affecting the velocity of the gas tip. Finally, the hydrodynamic behaviors of the Taylor bubble formed in the junction including velocity and pressure distributions are studied in detail, and the contribution of pressure drop within the channel is also discussed.

Multi-objective robust design optimization (MORDO) of an aeroelastic high-aspect-ratio wing

In this paper, a new approach for multi-objective robust optimization of flutter velocity and maximum displacement of the wing tip are investigated. The wing is under the influence of bending–torsion coupling and its design variables have different levels of uncertainty. In designing and optimizing wings with a high aspect ratio, the optimization process can be done in such a way to increase the flutter velocity, but this can increase the amplitude of the wing tip displacement to a point that leads to the wings damage and structural failure. Therefore, single-objective design optimization may lead to infeasible designs. Thus, for multi-objective optimization, modeling is based on the Euler–Bernoulli cantilever beam model in quasi-steady aerodynamic condition. Using the Galerkin’s techniques, the aeroelastic equations are converted to ODE equations. After validating the results, the system time response is obtained by the numerical solution of the governing equations using 4th Runge–Kutta method and the flutter velocity of the wing is obtained using the theory of eigenvalues. Subsequently, by choosing bending and torsional rigidity and mass per unit wing length as the optimization variables, using Monte Carlo–Latin hypercube (MC-LH) simulation and 4th polynomial chaos expansion (PCE), the effect of uncertainty on these variables is modeled in modeFRONTIER™ software coupled with MATLAB™ and optimization is performed by genetic algorithm. Finally, by plotting the Pareto front, it is observed that with an acceptable increase in flutter velocity, the maximum wing displacement amplitude is reduced as much as possible. The results of the multi-objective robust optimization show more feasible results compared with deterministic optimization.

Numerical Simulations of Air Cavities in Inclined Plunging Jets

The dynamic process of circular water jets plunging into a quiescent pool was analyzed in this study based on the RNG k∼ε turbulence model and VOF method. The effects of jet velocity and inclination angles relative to horizontal on the cavity shapes and sizes were analyzed. The simulation successfully captured the formation, development, pinch-off, and disintegration phenomena of cavities. The shape of the cavity is mainly affected by the impact angle, while the impact velocity mainly affects the size of the cavity. The cavity pinch-off initially appears at a certain point in any direction for vertical jets, while the cavity in the opposite direction of flow pinch-off appears before the cavity in the direction of flow for inclined jets. Before cavity pinch-off, the maximum radial and axial sizes of the cavity generally increase with the impact velocity and the time after impingement. The axial penetration velocity of the cavity tip is approximately half of the impact velocity, which is consistent with previous research. Finally, based on the statistics of the cavity sizes, empirical formulas for predicting the maximum radial and axial sizes of the cavity were established.

A three dimension lattice-spring model with rotational degree of freedom and its application in dynamic crack propagation

Applying parameter mapping theory, this paper establishes a new three dimension lattice-spring model (3D-LSM) containing the rotational degree of freedom. Using the finite element method (FEM) and share grid, the spring stiffness coefficient is obtained with a rigorous mathematical logic. This 3D-LSM contains the nearest neighbor, next nearest neighbor and third nearest neighbor which highly increases the accuracy of the system. Wave velocity in elastic solid and dynamic fracture of concrete l-specimen are calculated by both 3D-LSM and elastic theory. Results indicate that this model can present the behavior of the material in the elastic stage. By introducing the energy balance principle, 3D-LSM can simulate dynamic brittle fracture both in mesoscopy and macroscopy. This model is used to analyze the evolution law of crack propagation path and propagation velocity in brittle materials under dynamic tension of shock wave. The results show that the stress concentration area at the crack tip changes during the crack propagation, showing a "butterfly shape" before the crack initiation and a "swept wing shape" after the crack initiation. After the crack initiation, the crack tip velocity jumps instantly and then increases continuously, corresponding to the expansion of a single crack. Then the tip velocity decreases slightly and increases to the maximum crack propagation velocity, corresponding to the crushing stage before bifurcation. After that, the crack growth rate will oscillate significantly, which corresponds to the bifurcation stage of the crack.

Experimental investigation on the propagation of flow and flame in a confined combustion chamber equipped with a single-hole perforated plate

The effects of the hole size and perforated plate position on the propagation of flame and flow and pressure oscillation were explored in a constant volume combustion chamber (CVCB) with a single-hole perforated plate. Stoichiometric hydrogen-air mixture was used in the experiments, and the propagation of the flame and jet flow were recorded by high-speed schlieren photography. The results demonstrated that the flame velocity firstly increases and then decreases with the increasing hole size. With the hole size of 13 mm, the flame velocity, peak pressure, and pressure oscillation reached the maximum under the current experimental conditions. Moreover, the influence of the position of the perforated plate was investigated. It was found that the jet flow before the flame front is prominently distinct when the perforated plate is in positions A and B, resulting in an apparent difference in the flame shape behind the perforated plate, which means that the movement of the jet flow plays a leading role in the development of the flame. In Position B, the flame front overlapped with the jet flow and has the same shape. Besides, an interesting phenomenon was captured: with perforated plate in Position B, a secondary flame front was generated with higher flame tip velocity before the primary flame front under the effect of the flame-vortex interaction.

Investigation of the Spray Development Process of Gasoline-Biodiesel Blended Fuel Sprays in a Constant Volume Chamber

This study investigated gasoline–biodiesel blended fuel (GB) subjected to a fuel spray development process on macroscopic and microscopic scales. The four tested fuels were neat gasoline and gasoline containing biodiesel (5%, 20%, and 40% by volume) at three different ratios. The initial spray near the nozzle revealed that the spray penetration and spray tip velocity both decreased with decreasing biodiesel blending ratio. In addition, the different spray tip velocities at the start of spraying result in different atomization regimes between the fuels. The GB fuels with a low biodiesel blending ratio were disadvantaged in terms of spray atomization due to their lower spray penetration and tip velocity. The macroscopic spray penetration changes were similar to those observed in the microscopic spray. The fuel with the lower biodiesel blending ratio had a larger spray cone angle, indicating increased radial spray dispersion.

Kinetic relations and local energy balance for LEFM from a nonlocal peridynamic model

A simple nonlocal field theory of peridynamic type is applied to model brittle fracture. The kinetic relation for the crack tip velocity given by Linear Elastic Fracture Mechanics (LEFM) is recovered directly from the nonlocal dynamics, this is seen both theoretically and in simulations. An explicit formula for the change of internal energy inside a neighborhood enclosing the crack tip is found for the nonlocal model and applied to LEFM.

An efficient Pseudo-spectral based phase field method for dendritic solidification

In the current study, Fourier Pseudo-spectral based dealiasing scheme Random Phase Shift Method (RPSM) using the RK4 temporal scheme is implemented for simulating some benchmark problems of Phase Field based pure metal solidification. This high-resolution scheme is capable of resolving the thin solid-liquid interface accurately without using Adaptive Mesh Refinement (AMR) algorithm even using just adequate number of grid points. Both the conservative and non-conservative forms of the Phase Field equations are studied. It is found that the conservative RPSM scheme gives better computational performances as indicated by the lower values of average CPU time and different error norms calculated for the Phase Field variable. The RPSM based Phase Field model is applied for simulating dendritic solidification and it is found that dendrite shape and dimensionless tip velocity agree well with the existing literature. The RK4 based RPSM scheme is also compared with a 10th order exponential Fourier smoothing Filter in the evolution of four arm dendritic structure and it is observed that the conservative Phase Field equations with filtering scheme display numerical instabilities at later simulation time, while both forms of the RPSM scheme show stable and accurate solutions. Subsequently, the capability of the RPSM based Phase Field model in simulating microstructure evolution consisting of multiple dendrites with random location and orientation is demonstrated. The present study shows that the RPSM based Phase Field model computes dendritic growth accurately and efficiently.