Kinematic Boundary Conditions Research Articles

Nonlinear three-dimensional modeling for encapsulated microbubble dynamics subject to ultrasound

Encapsulated microbubbles (EMBs) stabilized by thin coatings have been used as contrast agents for ultrasound sonography as well as having been demonstrated as a promising new technology for targeted drug delivery. The dynamics of EMBs is three-dimensional (3D) because EMBs within micro-vessels inevitably interact with boundaries, but the theoretical and numerical studies are limited to spherical, weakly non-spherical, and/or axisymmetric EMBs. Here, we have developed physical, mathematical, and numerical models for nonlinear 3D EMB dynamics. The liquid flow is evaluated using the boundary integral method. The EMB coating is modeled as a thin viscoelastic shell including stretching, bending, and shear effects and simulated using the finite element method. These models are coupled through the kinematic and dynamic boundary conditions at the interface. The model is in good agreement with the Hoff equation for spherical EMBs and the asymptotic theory for weakly non-spherical deformation of EMBs. Using this model, a numerical study for EMB dynamics near a rigid boundary subject to an ultrasonic wave is performed. The migration, non-spherical oscillation, resonant oscillation, and jetting of EMBs are displayed and analyzed systematically. If the ultrasound wave is strong, a high-speed liquid jet forms at the final stage of the collapse, orientated between the directions of the wave and toward the wall. The EMB jet is weaker and slower and has less momentum, as the non-spherical deformation of the coating and the jetting are suppressed by the viscoelastic property of the coating. If the ultrasound is not strong, the EMB remains spherical for many cycles of oscillation but the EMB undergoes resonant oscillation and becomes significantly non-spherical after several oscillation cycles, when the wave frequency is equal to its natural frequency. The numerical capability has the potential to be developed for the optimization of sonography or drug delivery.

Analysis of the mechanical behavior of porous materials containing two populations of voids under dynamic spherical loading

A computational homogenization analysis is performed on three-dimensional representative volume elements (RVE) that contain two distinct populations of voids, to demonstrate the influence of the interaction between cavities. RVEs are constructed as cubic elastic perfectly-plastic matrices embedding two families of spherical voids, and subjected to dynamic loading under homogeneous kinematic boundary conditions. Multiple microstructure models are considered by varying the number, position, and size of voids to evaluate the micro-inertia contribution to the overall macroscopic stress. Velocity fields within numerical RVEs are investigated to reveal the interaction of voids and their key role in the dynamic macroscopic response of the porous material. Based on numerical simulation results, a homogenization analytical model considering void interaction is proposed to describe the mechanical behavior under dynamic loadings. This model relies on adjusting the strain rate level at the boundary of unit cells composing the porous material. Comparison between our numerical and analytical results with those obtained using the classical Taylor homogenization scheme highlights the limitations of the Taylor model in the case of porous materials under dynamic loading.

Multi-layer non-hydrostatic free-surface flow model with kinematic seafloor for seismic tsunami generation

Earthquake induced tsunamis pose devastating threat to coastal communities worldwide. Accurate description of tsunami generation by kinematic fault rupture is of importance to investigate earthquake events and evaluate tsunami impacts. The paper develops a multi-layer non-hydrostatic model with a kinematic bottom boundary condition and conducts comprehensive validation to examine the model’s capability in resolving seismic tsunami generation. The two-dimensional governing equations of the multi-layer non-hydrostatic free-surface flow system equipped with kinematic seafloor displacement is first introduced in Cartesian coordinates. A combined finite difference and finite volume scheme is utilized to discretize the governing equations with flow variables arranged on a staggered Arakawa C-grid. Application of pressure correction technique to the discretized formulations yields Poisson-type equations from which the non-hydrostatic pressure is solved for the next time step to complete the temporal integration. Based on existing analytical solutions, four groups of tsunami generation cases considering broad ranges of source parameters, including horizontal scale, rise time, and rupture velocity, are designed to demonstrate performance of the proposed non-hydrostatic model with one-, two-, and three-layers as well as the hydrostatic one. Comparison of 59 generation cases including extreme scenarios indicates the non-hydrostatic model performs better than the hydrostatic model in reproducing the entire waveform and predicting the maximum wave amplitude. High modeling accuracy can be achieved through incorporation of more layers. The proposed multi-layer non-hydrostatic model is a powerful tool for investigating earthquake source mechanisms and evaluating coastal tsunami hazards.

Characterization Of Cryogenic Sloshing Via Image Velocimetry, Free Surface Tracking And Data Assimilation

Cryogenic fuels such as liquid hydrogen (LH_2 ) and liquified natural gas (LNG) are promising energy carriers for the future. Their thermal management is significantly challenged by the liquid sloshing induced by external acceleration because sloshing strongly impacts the evaporation rate and, thus, the reservoir’s pressure control. This work presents a unique experimental characterization of liquid sloshing in cryogenic conditions combining interface tracking, image velocimetry and physics-constrained data assimilation. The experiments were conducted in a reduced-scale cryogenic tank partly filled with liquid nitrogen undergoing lateral harmonic sloshing. A combination of floating and neutrally buoyant seeding particles was used to simultaneously carry out interface tracking using a moving-window Radon transform and image velocimetry using Particle Tracking Velocimetry (PTV). The data assimilation was carried out using constrained radial basis function (RBF) regression, imposing kinematic boundary conditions at the walls and interface. The assimilation allows for unprecedented super-resolution of the velocity field.

RVE determination and developement of an anisotropic elastic model for auxetic sheet metal

This article deals with the development of an elastic tetragonal model for the 2D auxetic rotating units structures in the framework of orthogonal transformations. The existing anisotropy in the structure was first determined by numerical simulations on the samples with different pattern orientation angles. A suitable representative volume element (RVE), which correctly represents the mechanical properties of the whole structure both in macroscale and in microscale, was then proposed by implementation of the kinematic periodic boundary conditions. In the next step, with the help of the orthogonal transformations relations, an anisotropic elastic model was developed, which correctly reflects the present tetragonal symmetry in the structure. Finally, the model parameters were identified and validated with the help of the corresponding experiments

Linear Spectral Method for Simulating the Generation of Regular Waves by a Moving Bottom in a 3-dimensional Space

In this study, we introduce a linear spectral method capable of simulating wave generation and transformation caused by a moving bottom in a 3-dimensional space. The governing equations are linear dynamic free-surface boundary conditions and linear kinematic free-surface boundary conditions, which are solved in Fourier space. Solved velocity potential and free-surface displacement should satisfy continuity equation and kinematic bottom boundary condition. For numerical analysis, a 4th order Runge-Kutta method was utilized to analyze the time integral. The results obtained in Fourier space can be converted into velocity potential and free-surface displacement in a real space using inverse Fourier transform. Regular waves generated by various types of moving bottoms were simulated with the linear spectral method. Additionally, obliquely generated regular waves using specified bottom movements were simulated. The results obtained from the spectral method were compared to analytical solutions, showing good agreement between the two.

A model for modifying the S‐N curve considering the effect of boundary conditions on the fatigue crack growth of welded components

AbstractThe present study proposes a novel model to modify master S‐N curves of components according to their load redistribution capability reflected in different boundary conditions (BCs) based on the fracture mechanics analysis. To that end, a comprehensive numerical study was conducted on a Single Edge Notch Bend (SENB) specimen constrained with different kinematic BCs using discrete fatigue crack growth (FCG) simulation. It was observed that BCs indeed can have a significant effect on the crack growth behavior and consequently on the resulting fatigue life under the same nominal loading conditions. The proposed model was applied to the S‐N curve of a T‐welded joint, and the predicted fatigue life was validated against 3D FCG simulations. Finally, FCG tests were conducted on SENB specimens to experimentally corroborate the effect of BCs on the FCG rate.

Impulsive impact of a twin hull

An impulsively starting motion of two cylindrical bodies floating on a free liquid surface is considered. The shape of the cross-section of each body and the distance between them are arbitrary. The integral hodograph method is advanced to derive the complex velocity potential defined in a rectangular parameter region in terms of the elliptic quasi-doubly periodic Jacobi theta functions. A system of singular integral equations in the velocity magnitude on the free surface and in the slope of the wetted part of each body is derived using the kinematic boundary condition, which is then solved numerically. The velocity field, the pressure impulse on the bodies and the added mass coefficients of each body immediately after the impact are determined in a wide range of distances between the bodies and for cross-sectional shapes such as the flat plate and half-circle.

Wind noise due to flow induced around a windscreen in large-scale turbulence

Turbulent wind noise on microphones is a ubiquitous problem for sound measurement and recording outdoors. Windscreens, such as open-cell foam spheres, are effective at reducing wind noise. The unsteady pressure sources induced on the windscreen surface must contribute less noise on average than the stagnation pressure on a bare microphone, or windscreens would not be effective. To quantify this noise reduction mechanism, a model is presented for the unsteady pressure within an impermeable windscreen of arbitrary shape immersed in an initially isotropic turbulent flow. Only turbulent scales larger than the windscreen are considered, which contribute the most intense low-frequency band of the noise spectrum in most practical situations. The inhomogeneous velocity field around the windscreen is modeled in terms of an induced irrotational blocking field that meets the kinematic boundary condition. The interior pressure is expressed as a convolution of the surface pressure field with a filter function in wavevector space. The wind noise wavevector spectrum is then shown to be only a function of the inflow isotropic velocity spectrum and the windscreen shape. Solutions are obtained for the pressure frequency spectrum within a spherical windscreen for a von Kármán turbulence model and compared with prior results in literature.

Numerical modeling of surface subsidence due to compaction of soil with fine inclusions

A mathematical model of filtration consolidation of an inhomogeneous soil mass was formed taking into account the change in the size of the area during the compaction process. The inhomogeneity is considered as the presence of fine inclusions (geobarriers) the physical and mechanical characteristics of which differ from those of the main soil. From a mathematical viewpoint, the model is described by a one-phase Stefan problem that has a kinematic boundary condition on the upper moving boundary as its component. The purpose of the research is to find out the effect of fine inclusion on the dynamics of subsidence of the soil surface in the process of compaction. The change in the dimensions of the solution area is physically determined by the change in the volume of the pores of the porous medium in the process of dissipating excess pressure. If the permeability of the geobarrier is low, it affects the dynamics of consolidation processes and, accordingly, the magnitude of subsidence. Finite element solutions of the initial-boundary value problem for the nonlinear parabolic equation in the heterogeneous region with the conjugation condition of non-ideal contact were found. Numerical time discretization methods, a method for determining the change in the position of the upper boundary at discrete moments of time, and an algorithm for determining the physical and mechanical characteristics of a porous medium depending on the degree of consolidation are given. A number of test examples were considered, and the effect of a thin inclusion on the dynamics of the change in the position of the upper boundary of the problem solution area was investigated.

Stability analysis of a finite element approximation for the Navier-Stokes equation with free surface

In this work we study the numerical approximation of incompressible Navier-Stokes equations with free surface. The evolution of the free surface is driven by the kinematic boundary condition, and an Arbitrary Lagrangian Eulerian (ALE) approach is used to derive a (formal) weak formulation which involves three fields, namely, velocity, pressure, and the function describing the free surface. This formulation is discretised using finite elements in space and a time-advancing explicit finite difference scheme in time. In fact, the domain tracking algorithm is explicit: first, we solve the equation for the free surface, then move the mesh according to the sigma transform, and finally we compute the velocity and pressure in the updated domain. This explicit strategy is built in such a way that global conservation can be proven, which plays a pivotal role in the proof of stability of the discrete problem. The well-posedness and stability results are independent of the viscosity of the fluid, but while the proof of stability for the velocity is valid for all time steps, and all geometries, the stability for the free surface requires a CFL condition. The performance of the current approach is presented via numerical results and comparisons with the characteristics finite element method.

Aeolian Liquid Vibrations in Conical Tanks with Baffles under Wind Loading with Fuzzy Parameters

The Aeolian liquid vibrations in conical reservoirs caused by low-velocity, steady winds have been under consideration. Both amplitudes and dominant frequencies of wind loadings have been constantly changed, so to adequately describe the vibration process, fuzzy logic methods have been applied. At the first stage, the crisp initial value problem for conical shells with and without baffles has been considered. The liquid inside the reservoirs has been supposed to be an incompressible and ideal one, and its flow induced by the forced harmonic excitation, has been considered as potential. So, there exists a potential to satisfy the Laplace equation. The impermeable condition has been used at wetted surface boundaries of the shell, whereas the dynamic and kinematic boundary conditions have been set on the free liquid surface. A system of singular integral equations has been obtained for values of the velocity potential and the function describing the free surface rise. Its solution has been gained by boundary element methods. The crisp boundary value problem has been reduced to the second-order system of differential equations. After receiving the crisp solution of this system, the initial data have been fuzzified, involving triangular fuzzy numbers, and the fuzzy initial value problem has been formulated. The numerical solution to this problem with uncertain intervals involved has been obtained and analyzed.

Free vibration analysis of functionally graded nanobeams based on surface stress-driven nonlocal model

A surface stress-driven nonlocal model is employed in this manuscript to study the coupled effects of the long-range interaction and surface energy on the free vibrations of nano-beams made of metal-ceramic functionally graded material. The nanobeam theory is formulated based on the Bernoulli-Euler kinematics and surface effects include surface elasticity, surface residual stresses, surface density and rotary inertia. Hamilton’s principle is applied to derive the size-dependent governing equation. The main results of a parametric investigation, carried out considering four different kinematic boundary conditions, i.e. Cantilever, Simply-Supported, Clamped-Pinned and Doubly-Clamped, are also presented and discussed, varying the nonlocal parameter and the material gradient index. The results show how the proposed model is able to capture surface energy effects on the overall dynamic behavior of functionally graded Bernoulli-Euler nanobeams and provides a cost-effective method for the design and the optimization of nano-scaled structures.

Method of solution composition in contact problems with friction of deformable bodies

A new iterative method for solving static contact problems of two deformable bodies is proposed. The method is based on alternately solving the unilateral contact problem for the first body and the linear elasticity problem with natural boundary conditions for the second body. Fulfillment of Coulomb's friction law involves correction of tangential nodal forces in the sliding area and setting kinematic boundary conditions in the sticking area for the contact boundary of the first body. The goal of solving the linear elasticity problem for the second body is to gradually equalize contact loads on the interacting surfaces. The advantages of the method are demonstrated by solving a number of model examples, including unilateral contact of a linear-elastic plate with a solid foundation, bilateral contact of pressing a deformable block into the foundation, the Hertz problem of contact of two deformable cylinders etc. The method can solve problems on flat and curvilinear contact boundaries.

Analysis of structures with arbitrary kinematic boundary conditions by the semi-analytical finite element method

The successful application of FEM to the analysis of structures is largely due to the efficiency of the use of modern software packages, in connection with which the role of program systems that implement the solution process increases. The correct organization of the computing complex, the choice of optimal algorithms for solving systems of linear and nonlinear equations largely determine the possibilities of the method in terms of the structural complexity of the objects under consideration, the accuracy of the results obtained, and the complexity of setting nonlinear problems. Therefore, there is an increased interest in the development of fairly universal computing complexes based on FEM. One of the effective software complexes is the "Strength" system, designed to conduct comprehensive research in the field of mechanics of a deformable solid, the basic principles of construction of which are used in this work in the implementation of a semi-analytical version of the FEM.
 In this work, solutions of a significant number of control problems of deformation of massive and thin-walled prismatic bodies under different boundary conditions and loads are obtained. In the process of solving new problems, the estimation of the convergence of results was carried out on the basis of a sequential increase in finite elements and contained terms of decomposition, an increase in the accuracy of systems of linear and nonlinear equations, and the accuracy of satisfaction with natural boundary conditions was checked. The developed effective method for solving new complex problems of deformation of prismatic bodies is implemented in the form of complex programs and can be used in design and construction practice in construction, mechanical engineering and other fields of technology.

Minimum-time and minimum-energy trajectory planning under dynamic constraints for a quadrotor

This article proposes an approach for optimizing the trajectory planning of quadrotors in structured environments, taking into consideration the trade-off between task execution time and energy consumption. The constraints taken into account are geometrical (boundary conditions in position and orientation, obstacle avoidance), kinematic (boundary conditions in velocity, linear velocity, and acceleration limits of the system), and dynamic (limit capacities in torque of the actuators, constraints related to the under-actuation of the system). The approach extends the Random Trajectory Profile Approach (RPA) to the case of quadrotors. The results of this approach are validated through experimental tests and compared with those in the literature, demonstrating its effectiveness in offline and online scenarios. The proposed approach offers a promising solution for achieving optimal trajectory planning of quadrotors in structured environments.

Numerical Modeling of Earthquake Cycles Based On Navier‐Stokes Equations With Viscoelastic‐Plasticity Rheology

AbstractVisco‐elastic‐plastic modeling approaches for long‐term tectonic deformation assume that co‐seismic fault displacement can be integrated over 1000s–10,000s years (tens of earthquake cycles) with the appropriate failure law, and that short‐timescale fluctuations in the stress field due to individual earthquakes have no effect on long‐term behavior. Models of the earthquake rupture process generally assume that the tectonic (long‐range) stress field or kinematic boundary conditions are steady over the course of multiple earthquake cycles. This study is aimed to fill the gap between long‐term and short‐term deformations by modeling earthquake cycles with the rate‐and‐state frictional (RSF) relationship in Navier‐Stokes equations. We reproduce benchmarks at the earthquake timescale to demonstrate the effectiveness of our approach. We then discuss how these high‐resolution models degrade if the time‐step cannot capture the rupture process accurately and, from this, infer when it is important to consider coupling of the two timescales and the level of accuracy required. To build upon these benchmarks, we undertake a generic study of a thrust fault in the crust with a prescribed geometry. It is found that lower crustal rheology affects the periodic time of characteristic earthquake cycles and the inter‐seismic, free‐surface deformation rate. In particular, the relaxation of the surface of a cratonic region (with a relatively strong lower crust) has a characteristic double‐peaked uplift profile that persists for thousands of years after a major slip event. This pattern might be diagnostic of active faults in cratonic regions.

Response to comment on “Stability of a moving radial liquid sheet: Time-dependent equations” [Phys. Fluids 35, 099101 (2023)

In a recent comment, Mehring and Gyurkovich show that in our derivation of the time-dependent equations, we have neglected the perturbed term in the interface location while applying the kinematic boundary condition. When the neglected term is incorporated in the analysis, the time-dependent equation for the sinuous mode is different from our original expression. They conclude that the instability predicted by the original model does not exist for their new model. We show that, on the contrary, the new model predicts spatial growth of sinuous disturbance that depends on forcing frequency and the slope of the forcing disturbance.

Embedded boundary conditions for shear-deformable plate bending

An efficient procedure for embedding kinematic boundary conditions of shear-deformable plate bending is based on stabilized variational formulations, obtained by Nitsche’s approach for enforcing surface constraints. Effective use of the plate thickness as scaling in the rotational constraint terms leads to a uniform problem statement with a single stabilization parameter. An alternative form of the plate kinematics precludes the vulnerability of computation based on the standard Reissner–Mindlin formulation to shear locking, yet raises other challenges in the form of higher-order regularity and unconventional boundary conditions. The embedded boundary Nitsche approach provides a framework that easily accommodates these features. The accuracy of the approach for both plate formulations is verified by representative computations with bicubic B-splines, exhibiting optimal rates of convergence, and robust performance with respect to values of the parameters.

Impact of numerical modelling of kinematic and static boundary conditions on stability of cold-formed sigma beam

Impact of numerical modelling of kinematic and static boundary conditions on stability of cold-formed sigma beam