Friction Boundary Conditions Research Articles

A three-step subgrid stabilized Oseen iterative method for Navier–Stokes type variational inequality

This work is devoted to developing a two-grid subgrid stabilized Oseen iterative finite element method for the convection dominated Navier–Stokes problem with friction boundary conditions whose weak form is the variational inequality of the second kind. This method inherits the best algorithmic advantages of each and involves three steps. Specifically, in the first step, a nonlinear Navier–Stokes type variational inequality is solved by applying m steps Oseen iterations to approximate the low frequencies of the solution on a coarse grid (H mesh), and then in the second and third steps, two subgrid stabilized and Newton-linearized Navier–Stokes type variational inequalities are solved to attain the high frequencies on a fine grid (h mesh). The developed method is easily implementable since there have identical stiffness matrices with only various right-hand sides in the second and third steps. Error bounds of the approximate velocity and pressure with regard to the mesh sizes h and H, stabilization parameters αH and αh and iteration step m are estimated, and new termination condition for nonlinear iterations is derived. The scalings both for the coarse grid and fine grid sizes as well as fine grid stabilization parameter obtained by our present method are improved in comparison with the developed method that only stays in the second step. Performance and effectiveness of the developed method are illustrated through a series of numerical experiments, which show that the accuracy of the calculated velocity and pressure of each step is higher than that of the previous step.

Navier-Stokes-ω model with slip and friction boundary conditions at high Reynolds numbers

Navier-Stokes-ω model with slip and friction boundary conditions at high Reynolds numbers

Analysis of a Parallel Grad-Div Stabilized Method for the Navier–Stokes Problem with Friction Boundary Conditions

Analysis of a Parallel Grad-Div Stabilized Method for the Navier–Stokes Problem with Friction Boundary Conditions

Variational–hemivariational system for contaminant convection–reaction–diffusion model of recovered fracturing fluid

Abstract This work is devoted to study the convection–reaction–diffusion behavior of contaminant in the recovered fracturing fluid which flows in the wellbore from shale gas reservoir. First, we apply various constitutive laws for generalized non-Newtonian fluids, diffusion principles, and friction relations to formulate the recovered fracturing fluid model. The latter is a partial differential system composed of a nonlinear and nonsmooth stationary incompressible Navier-Stokes equation with a multivalued friction boundary condition, and a nonlinear convection–reaction–diffusion equation with mixed Neumann boundary conditions. Then, we provide the weak formulation of the fluid model which is a hemivariational inequality driven by a nonlinear variational equation. We establish existence of solutions to the recovered fracturing fluid model via a surjectivity theorem for multivalued operators combined with an alternative iterative method and elements of nonsmooth analysis.

Numerical analysis of a steady Oseen flow problem with frictional type boundary conditions

Numerical analysis of a steady Oseen flow problem with frictional type boundary conditions

Stability for variable-coefficient wave equation with local frictional damping and memory type dynamic boundary conditions

Stability for variable-coefficient wave equation with local frictional damping and memory type dynamic boundary conditions

Elastic Behavior of Transversely Isotropic Cylindrical Rock Samples under Uniaxial Compression Considering Ideal and Frictional Boundary Conditions

Conceptualizing sophisticated measurement set-ups as well as testing and evaluation procedures for laboratory experiments on anisotropic rocks requires a basic understanding of the potential specimen behavior. The focus of the present work was therefore to investigate the influence of different transversely isotropic parameters and their ratios on the elastic behavior of cylindrical rock samples in uniaxial compression tests. Parameter sets corresponding to soft anisotropic rocks were chosen based on naturally observed ranges for the five elastic transversely isotropic constants. Analytical results for the radial and vertical strain distributions around the sample circumference and a comparison with finite element simulations are presented. Further, the effect of interface friction between samples and loading platens was analyzed within the numerical models. The results suggest that radial strains around cylindrical anisotropic samples are rarely uniform except for specific combinations of parameters and isotropy plane inclinations. The effect of interface friction was found to have a clear influence on the developing elastic stress and strain distributions for samples with inclined isotropy planes. Nevertheless, no significant influence of frictional boundary conditions on the back-calibrated values of the elastic parameters could be identified, suggesting that friction-reducing measures in uniaxial compression tests on transversely isotropic samples with predominantly linear behavior are not required.

A semi-analytical matrix formalism for stress singularities in anisotropic multi-material corners with frictional boundary and interface conditions

A new general semi-analytic procedure for the characterisation of singular asymptotic elastic states in the vicinity of the apex of linearly elastic anisotropic multi-material corners, including frictional contact, is developed and tested. The corners can consist of any finite number of homogeneous wedges defined by polar sectors. The variability of configurations covered is enormous as frictional contact can be considered on one or more outer boundary surfaces or interfaces, in addition to a large variety of homogeneous boundary conditions and perfect bonding or frictionless sliding interface conditions between wedges in the corner. The Coulomb rate-independent and dry frictional contact law is assumed. One of the novelties is that, in addition to the singularity exponent λ, the angle ω of the friction tangential stress vector on each frictional contact surface is an a priory unknown to be determined by solving a nonlinear corner eigensystem. The procedure, which considers power-law stress singularities, is based on the Stroh formalism of anisotropic elasticity, assuming generalised plane strain (2.5D) conditions, and on the semi-analytic matrix formalism for wedge transfer-matrices and boundary and interface condition matrices. This makes it, firstly, very suitable for computational implementations, secondly, very efficient especially in cases with several perfectly bonded homogeneous wedges, and, thirdly, very accurate due to its fully semi-analytic nature. The code developed is tested by solving a large variety of examples, comparing the present results with those obtained by solving closed-form corner-eigenequations deduced by previous authors for specific useful practical configurations, confirming the extremely high accuracy of the present code in the computation of λ and ω. The differences observed in some cases with anisotropic materials are explained by the fact that some of the previous authors did not take the true 3D Coulomb friction law into account.

Identification of discontinuous parameters in contaminant convection–reaction–diffusion model of recovered fracturing fluid

The primary goal of this paper is to identify the discontinuous parameters in a complicated contaminant convection–reaction–diffusion model of recovered fracturing fluid (RFF model, for short), which is modeled by a nonlinear stationary incompressible Navier–Stokes equation with multivalued and nonsmooth friction boundary condition coupled with a nonlinear convection–reaction–diffusion equation involving mixed Neumann boundary conditions. First, we impose the parameter identification problem which is, exactly, a nonlinear and nonsmooth inverse problem. Then, we show the local boundedness and weakly relative compactness of solution map to RFF model with respect to the discontinuous parameters. Moreover, the generalized continuity of solution map of RFF model is proved. Finally, via employing the theory of nonsmooth analysis and optimization, a sufficiency theorem of existence of solutions to the inverse problem under consideration is established.

Frictional boundary condition for lattice Boltzmann modelling of dense granular flows

Hydrodynamic approaches that treat granular materials as a continuum via the homogenization of discrete flow properties have become viable options for efficient predictions of bulk flow behaviours. However, simplified boundary conditions in computational fluid dynamics are often adopted, which have difficulty in describing the complex stick–slip phenomenon at the boundaries. This paper extends the lattice Boltzmann method for granular flow simulations by incorporating a novel frictional boundary condition. The wall slip velocity is first calculated based on the shear rate limited by the Coulomb friction, followed by the reconstruction of unknown density distribution functions through a modified bounce-back scheme. Validation is performed against a unique plane Couette flow configuration, and the analytical solutions for the flow velocity profile and the wall slip velocity, as functions of the friction coefficient, are reproduced by the numerical model. The transition between no-slip and partial-slip regimes is captured well, but the convergence rate drops from second order to first order when slip occurs. The rheological parameters and the basal friction coefficient are calibrated further against the discrete element simulation of a square granular column collapsing over a horizontal bottom plane. It is found that the calibrated continuum model can predict other granular column collapses with different initial aspect ratios and slope inclination angles, including the basal slip and the complex internal flow structures, without any further adjustments to the model parameters. This highlights the generalization ability of the numerical model, which has a wide range of application in granular flow predictions and controls.

Numerical optimisation of damage in extrusion processes

AbstractNumerical optimisation is applied to rod extrusion in order to generate optimal parameter sets which result in reduced damage accumulation. A brief overview of academic applications for damage optimisation is given. Their restrictions are reflected upon, leading to the proposed framework utilising the commercial software Abaqus FEA, which enables optimisation of industrial problems with frictional contact. The framework is setup modularly to enable arbitrary choices of design variables, such as geometric parameters, and process parameters like friction coefficients or boundary conditions. The optimisation strategy is applied to forward hollow extrusion. Compared to forward rod extrusion, an additional design variable, that is, the mandrel radius, is introduced into the forming process, which increases the design space.

Modeling surface wave dynamics in upper Delaware Bay with living shorelines

Living shorelines gain increasing attention because they stabilize shorelines and reduce erosion. This study leverages physics-based models and bagged regression tree (BRT) machine learning algorithm to simulate wave dynamics at a living shoreline composed of constructed oyster reefs (CORs) in upper Delaware Bay. The physics-based models consist of coupled Delft3D-FLOW and SWAN in four-level nested domains. The model accuracy converges with increasing mesh resolution. The simulated wave-induced current circulation substantiates the effectiveness of CORs in trapping sediments. The simulated yearly-averaged wave power correlates qualitatively with historical shoreline retreat rates. BRT is adopted to improve the model accuracy, identify key processes responsible for simulation errors in wave height (Hs) and wave period (Tp), and quantify their importance. In the CORs sheltered area, BRT reveals that simulation errors of wind seas mainly arise from wind forcing, wave breaking and wave triad interactions. Wave breaking is seven times more important than wind forcing for simulating Hs, while wind forcing and triad interactions are of equal importance for simulating Tp. Simulation errors of swells mostly stem from bottom friction and offshore wave boundary conditions. Results from this study can help the assessment and adaptive management of CORs-based living shoreline restoration projects under climate change.

Regularized SPH model for soil–structure interaction with generalized frictional boundary

AbstractThis paper proposes a regularized smoothed particle hydrodynamics (SPH) scheme to solve the problems of tensile instability and stress noise in large‐deformation soil–structure interaction. Particle shifting and stress regularization are combined to maintain uniform stresses and particle distributions. A generalized frictional boundary condition is presented to model soil–structure interfaces with abitrary friction coefficients. A modified particle shifting technique is developed to regularize particles near the frictional boundaries. Furthermore, GPU acceleration algorithm is adopted to improve the efficiency. The proposed scheme is compared with the conventional SPH methods using numerical examples, including the collapse of three‐dimensional granular column, the collapse of the cohesive soil, and pipe penetrates into soft clay. The comparison indicates that the proposed regularized SPH model can give highly accurate results for large‐deformation soil–structure interaction problems with arbitrary friction coefficient.

On the Finite Element Approximation of a Semicoercive Stokes Variational Inequality Arising in Glaciology

Stokes variational inequalities arise in the formulation of glaciological problems involving contact. We consider the problem of a two-dimensional marine ice sheet with a grounding line, although the analysis presented here is extendable to other contact problems in glaciology, such as that of subglacial cavitation. The analysis of this problem and its discretization is complicated by the nonlinear rheology commonly used for modelling ice, the enforcement of a friction boundary condition given by a power law, and the presence of rigid modes in the velocity space, which render the variational inequality semicoercive. In this work, we consider a mixed formulation of this variational inequality involving a Lagrange multiplier and provide an analysis of its finite element approximation. Error estimates in the presence of rigid modes are obtained by means of a specially-built projection operator onto the subspace of rigid modes and a Korn-type inequality. These proofs rely on the fact that the subspace of rigid modes is at most one-dimensional. Numerical results are reported to validate the error estimates.

Development and application of the multifunctional transmission roll-sliding friction tester

This paper aims to elaborate on the development and the main features of the multifunctional transmission roll-sliding friction tester. The tester adopts the independent drive for controlled relative movement, and the contacting specimen that sustains the radial loads can be replaced, so it can complete the rolling friction, sliding friction, and roll-sliding friction test. The integrated transmission performance test machine not only has solved the inconsistency between the friction and transmission experimental boundary conditions but also can precede the friction drive research. Piecewise calibration data and interpolating correction algorithm are used to avoid the affecting generated by the spindle friction to the specimen. The monitoring system software is developed by the LabVIEW platform, and the parameters are controlled automatically by the computer. Thus it can simulate actual working conditions very well and achieve automatic real-time data acquisition, processing and showing the results, etc.

Parallel defect-correction methods for incompressible flows with friction boundary conditions

We study three parallel defect-correction methods based on finite element approximations for the incompressible Navier–Stokes problem with friction boundary conditions and high Reynolds numbers in this work, where a fully overlapping domain decomposition is considered for parallelization. In the proposed methods, with a global multiscale grid that builds a fine grid around its own subdomain and coarse elsewhere, we iteratively solve an artificial viscosity nonlinear variational inequality problem in a defect step, and then compute the residual by the linearized variational inequality problems in the r-step corrections. The studied methods are easy to implement on the basis of the existing Navier–Stokes solver and possess less communication complexity. We provide a rigorously theoretical derivation for the error estimates of the one-step correction solutions from the proposed methods under some stable conditions, and derive scalings of the algorithmic parameters. We demonstrate by a series of numerical experiments that the velocity and pressure errors computed by our parallel defect-correction methods are comparable to those of the standard defect-correction method, while our present methods reduce the computational cost.

Instability of Boundary Layers with the Navier Boundary Condition

We study the L^{infty } stability of the Navier-Stokes equations in the half-plane with a viscosity-dependent Navier friction boundary condition around shear profiles which are linearly unstable for the Euler equation. The dependence from the viscosity is given in the Navier boundary condition as partial _y u = nu ^{-gamma }u for some gamma in {mathbb {R}}, where u is the tangential velocity. With the no-slip boundary condition, which corresponds to the limit gamma rightarrow +infty , a celebrated result from E. Grenier (Comm. Pure Appl. Math. 53:1067–1091, 2000) provides an instability of order nu ^{1/4}. M. Paddick (Differ. Integral Equ. 27:893–930, 2014) proved the same result in the case gamma =1/2, furthermore improving the instability to order one. In this paper, we extend these two results to all gamma in {mathbb {R}}, obtaining an instability of order nu ^{vartheta }, where in particular vartheta =0 for gamma le 1/2 and vartheta =1/4 for gamma ge 3/4. When gamma ge 1/2, the result denies the validity of the Prandtl boundary layer expansion around the chosen shear profile.

A three-step defect-correction algorithm for incompressible flows with friction boundary conditions

A three-step defect-correction algorithm for incompressible flows with friction boundary conditions

Stokes equations under Tresca friction boundary condition: a truncated approach

Stokes equations under Tresca friction boundary condition: a truncated approach

Finite element simulation of a viscoelastic cell entering a cylindrical channel: Effects of frictional contact

Finite element method has been extensively used to simulate the process of single cells entering a narrow channel under a prescribed pressure. Yet an in-depth understanding on the effects of the frictional boundary condition in such simulations is lacking. The main challenge lies in the continuously evolving contact region between the deformed cell and the channel wall during the entry process, which makes it difficult to apply proper boundary conditions for the friction and pressure. In this work, we present an approach to continuously update the boundary condition according to the contact status during entry. Using this approach and modeling the cell as a visco-hyperelastic sphere, we study how the pressure boundary conditions and the friction coefficient affect the simulated cell entry process. Specifically, we compare two special cases of pressure boundary conditions: positive pressure at the trailing edge of the cell or negative pressure at the leading edge. It is found that the two cases produce the same result in terms of the cell entry length under frictionless contact, but give considerably different results under frictional contact. Effects of whether the pressure boundary is updated are then studied by considering more special cases. All these results are well explained by the pressure superposition principle, and further insights into the effects of pressure boundary condition and boundary update are also acquired. Finally, focusing on the case of positive pressure with boundary updates, we obtain simulation results revealing how the entry length depends on the pressure, time and friction coefficient. In particular, a larger friction coefficient can significantly reduce the entry length under a prescribed pressure or a certain entry time. Our results suggest that it is necessary to implement accurate pressure boundary conditions (e.g., pressure magnitudes at the two edges of cell and continuously updated pressure boundary regions) in order to faithfully capture the frictional contact between the cell and the channel wall. In addition, the simulation approach can be useful in modeling the general process of a viscoelastic sphere entering a cylindrical channel involving frictional contact.