Incompressible Behavior Research Articles

An error indicator for parameter identification with stabilized mixed tetrahedrals

AbstractThe identification of parameters in constitutive laws considering inhomogeneous states of stress and strain is realized by iteratively minimizing a least squares functional. In each iterative step of this optimization problem a finite element analysis is carried out which results in a significant higher numerical cost than a single finite element analysis. Consequently, an efficient discretization is required to keep the numerical cost low. To address this problem an adaptive mesh refinement is considered which is based on a posteriori error indicators [1] leading to refinements appropriate to the parameter identification problem. The goal is to apply the error indicators to the finite element method for tetrahedral elements of low order which are preferable for adaptive mesh refinements and in addition reduce computational effort. Additional stabilization terms in the element formulation [4, 6] reduce volume locking effects making the elements suitable for (nearly) incompressible material behavior. Numerical examples illustrate the progress on this work. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

REDUCTION OF COMPRESSIBILITY AND PARALLEL TRANSFER BY LANDAU DAMPING IN TURBULENT MAGNETIZED PLASMAS

Three-dimensional numerical simulations of decaying turbulence in a magnetized plasma are performed using a so-called FLR-Landau fluid model which incorporates linear Landau damping and finite Larmor radius (FLR) corrections. It is shown that compared to simulations of compressible Hall-MHD, linear Landau damping is responsible for significant damping of magnetosonic waves, which is consistent with the linear kinetic theory. Compressibility of the fluid and parallel energy cascade along the ambient magnetic field are also significantly inhibited when the beta parameter is not too small. In contrast with Hall-MHD, the FLR-Landau fluid model can therefore correctly describe turbulence in collisionless plasmas such as the solar wind, providing an interpretation for its nearly incompressible behavior.

Tuning a lattice-Boltzmann model for applications in computational hemodynamics

AbstractThe interest in lattice-Boltzmann models in the computational hemodynamics realm has increased in recent years. In this context, the correct choice of numerical parameters for the appropriate simulation of blood flows in major arteries is a crucial aspect. For this reason, we present three parameter-tuning strategies that allow us to reproduce correctly the pulsatile time-dependent flow of an incompressible fluid under physiological regimes. These strategies are studied for a model based on a single-relaxation-time approach in combination with second order boundary conditions for both velocity and pressure, and proper equilibrium distributions that take care of the incompressible behavior exhibited by the fluid. The implementation is validated with the three-dimensional Womersley flow benchmark. As well, the simulation of blood flows in a curved artery, in an anastomosed vessel, in a patient specific vertebral artery and in an aneurysmal region are presented in order to show how the method and the setting of the numerical parameters are applied to different realistic hemodynamics problems.

Acousto-spinodal decomposition of compressible polymer solutions: Early stage analysis

The structure and dynamics of early stage kinetics of pressure-induced phase separation of compressible polymer solutions via spinodal decomposition is analyzed using a linear Euler-Cahn-Hilliard model and the modified Sanchez Lacombe equation of state. The integrated density wave and Cahn-Hilliard equations combine the kinetic and structural characteristics of spinodal decomposition with density waves arising from pressure-induced couplings. When mass transfer rate is slower that acoustic waves, concentration gradients generate density waves that cycle back into the spinodal decomposition dynamics, resulting in oscillatory demixing. The wave attenuation increases with increasing mass transfer rates eventually leading to nonoscillatory spinodal demixing. The novel aspects of acousto-spinodal decomposition arise from the coexistence of stable oscillatory density dynamics and the unstable monotonic concentration dynamics. Scaling laws for structure and dynamics indicate deviations from incompressible behavior, with a significant slowing down of demixing due to couplings with density waves. Partial structure factors for density and density-concentration reflect the oscillatory nature of acousto-spinodal modes at lower wave vectors, while the single maximum at a constant wave vector reflects the presence of a dominant mode in the linear regime. The computed total structure factor is in qualitative agreement with experimental data for a similar polymer solution.

Parametric Study of a Pitching Flat Plate at Low Reynolds Numbers

In this paper we simulate the unsteady, incompressible, and laminar flow behavior over a flat plate with round leading and trailing edges. A pressure-Poisson method is used to solve the incompressible Navier-Stokes equations. Both convection and diffusion terms are discretized using a second-order accurate central difference method. A second-order accurate split-step scheme with an Adam’s predictor corrector time-stepping method is adopted for the time integration. An overlapping moving grid approach is employed to dynamically update the grid due to the plate motion. The effects of the pitch rate, Reynolds number, location of pitch axis, and computational domain size are investigated. Comparisons are made with experimental results.

Isogeometric analysis of nearly incompressible solids

AbstractThis paper describes the use of isogeometric methods to solve problems in finite deformation solid mechanics in which compressible and nearly incompressible behavior may be encountered. The work is based on a three‐field variational structures in which displacements, mean stress and volume effects are independently approximated. Using this approach and tensor product NURBS approximations it is shown how the solution can be conducted in a standard approach, where main variables are organized at nodes (control points). This leads to an efficient procedure to solve large problems of a general form without the need to develop special solution strategies. The approach is demonstrated on some standard test problems. Copyright © 2010 John Wiley & Sons, Ltd.

A model reduction technique for laminated solids of revolution with a curved cross-section

This paper presents an extension of the numerical reduction method, which has been proposed in Lejeunes et al. (Arch Appl Mech, 76:311–326, 2006), for modeling curved laminated structures of revolution such as for instance rubber bearings. This method based on high-order finite elements is developed in the context of nearly incompressible hyperelastic behavior. The displacement is approximated with a sum of independent functions, leading to a separation of variables. Therefore, a one-dimensional finite element can be formulated, which represents a 3-dimensional solid in a general loading case. Comparisons with classical finite element models are provided and show the reliability of this model reduction. An important decrease in the model size and a greatly reduced computing time, compared to standard models, is observed.

Equations of state for postperovskite phases in the MgSiO 3–FeSiO 3–FeAlO 3 system

Equations of state and axial compressions are investigated for the Mg 0.9Fe 0.1SiO 3 and Mg 0.85Fe 0.15Al 0.15Si 0.85O 3 postperovskite phases in the MgSiO 3–FeSiO 3–FeAlO 3 system by high-pressure in situ X-ray diffraction measurements using a laser-heated diamond anvil cell. The b-axis is found to be the most compressible axis for both phases, while the c-axis is slightly less compressible. The incompressible behavior of the c-axis is likely to be attributable to repulsion among oxygen atoms arranged along the c-axis. Using the Tsuchiya gold pressure scale, the Birch–Murnaghan equation of state yields isothermal bulk moduli of 224(11) GPa and 220(13) GPa, and zero-pressure unit-cell volumes of 164.1(15) Å 3 and 166.5(20) Å 3 for these FeSiO 3- and FeAlO 3-bearing postperovskites, respectively. The incorporation of FeSiO 3 and FeAlO 3 into the MgSiO 3 postperovskite results in a marked decrease in the isothermal bulk modulus. The bulk sound velocity is also confirmed to decrease considerably at the perovskite–postperovskite transition, even in the FeSiO 3- and FeAlO 3-bearing systems. The velocity thus decreases not only by the postperovskite transition but also by enrichment in the FeSiO 3 and FeAlO 3 components. Chemical heterogeneity of the FeSiO 3 and FeAlO 3 components may therefore represent an additional mechanism of velocity variation in the D″ layer of the lower mantle.

Homogenized Euler equation: a model for compressible velocity gradient dynamics

Along the lines of the restricted Euler equation (REE) for incompressible flows, we develop homogenized Euler equation (HEE) for describing turbulent velocity gradient dynamics of an isentropic compressible calorically perfect gas. Starting from energy and state equations, an evolution equation for pressure Hessian is derived invoking uniform (homogeneous) velocity gradient assumption. Behaviour of principal strain rates, vorticity vector alignment and invariants of the normalized velocity gradient tensor is investigated conditioned on dilatation level. The HEE results agree very well with the known behaviour in the incompressible limit. Indeed, at zero dilatation HEE reproduces the incompressible anisotropic pressure Hessian behaviour very closely. When compared against compressible direct numerical simulation results, the HEE accurately captures the strain rate behaviour at different dilatation levels. The model also recovers the fixed point behaviour of pressure-released (high-Mach-number limit) Burgers turbulence.

Generalized finite element method for modeling nearly incompressible bimaterial hyperelastic solids

An extension of the generalized finite element method to the class of mixed finite element methods is presented to tackle heterogeneous systems with nearly incompressible non-linear hyperelastic material behavior. In particular, heterogeneous systems with large modulus mismatch across the material interface undergoing large strains are investigated using two formulations, one based on a continuous deformation map, the other on a discontinuous one. A bimaterial patch test is formulated to assess the ability of the two formulations to reproduce constant stress fields, while a mesh convergence study is used to examine the consistency of the formulations. Finally, compression of a model heterogeneous propellant pack is simulated to demonstrate the robustness of the proposed discontinuous deformation map formulation.

A 4-node 3D-shell element to model shell surface tractions and incompressible behavior

We present in this paper a shell element that models the three-dimensional (3D) effects of surface tractions, like needed when a shell is confined between other solid media. The element is the widely used MITC4 shell element enriched by the use of a fully 3D stress–strain description, appropriate through-the-thickness displacements to model surface tractions, and pressure degrees of freedom for incompressible analyses. The element formulation avoids instabilities and ill-conditioning. Various example solutions are presented to illustrate the capabilities of the element.

Experimental and numerical modelling of the three-dimensional incompressible flow behaviour in the near wake of circular cylinders

An experimental investigation was carried out to study the structure of the flow field around three-dimensional circular cylinders. The study of the flow field around an obstacle was performed in a wind tunnel using a particle image velocimetry (PIV) system. The flow of a fluid around an obstacle with a different velocity to the oncoming flow was examined. The results showed the dependence of the flow structure around the obstacle on its Reynolds number, and the spacing between a pair of obstacles. Detailed quantitative information of turbulence parameters in the vicinity of the obstacle was attained. Extensive wind tunnel experimental results are presented and compared with numerical simulation. A three-dimensional numerical model with Reynolds stress model (RSM) turbulence and a non-uniform grid system were used to examine the effects of a single cylinder and two cylinders in tandem on the flow. The principal objective was to analyse three-dimensional flow past a single cylinder and two circular cylinders placed in tandem by combining the application of a PIV experimental technique and an RSM turbulence model. For the case of two cylinders in tandem, the flow patterns are characterized in the gap region as a function of the distance between the cylinders. A good level of agreement was found between the experimental results of flow and numerical simulation.

Hyperelastic Muscle Simulation

Biological soft tissues like muscles and cartilages are anisotropic, inhomogeneous, and nearly incompressible. The incompressible material behavior may lead to some difficulties in numerical simulation, such as volumetric locking and solution divergence. Mixed u-P formulations can be used to overcome incompressible material problems. The hyperelastic materials can be used to describe the biological skeletal muscle behavior. In this study, experiments are conducted to obtain the stress-strain behavior of a solid silicone rubber tube. It is used to emulate the skeletal muscle tensile behavior. The stress-strain behavior of silicone is compared with that of muscles. A commercial finite element analysis package ABAQUS is used to simulate the stress-strain behavior of silicone rubber. Results show that mixed u-P formulations with hyperelastic material model can be used to successfully simulate the muscle material behavior. Such an analysis can be used to simulate and analyze other soft tissues that show similar behavior.

인공위성 카메라 주반사경 지지부에 적용되는 접착제의 전단 특성 연구

The optical performance of the mirror for satellite camera is highly dependent on the adhesive properties between the mirror and its support. Therefore, in order to design a mirror with high optical performance, the mechanical properties of adhesives should be well defined. In this research, the mechanical properties of three kinds of space adhesives are studied. In case of the materials which show nearly incompressible behavior such as space adhesives, it is important to measure shear modulus which governs deviatoric stress components. Also the experiment should be performed in circumstances similar to real manufacturing process of mirror, because extra factors such as size effects, the adhesion effects of primer and reactions between adhesive and primer affect the properties of adhesive regions. In this research shear moduli of the adhesives are determined by using a single lap adhesively bonded joint. For the shear tests, several temperatures have been selected from -20℃ to 55℃ which is operating temperature range of the adhesive. In the case of linear behavior materials, shear moduli are calculated through a linear curve fitting. Shear stress-strain relation is obtained by using an exponential curve fitting for material which shows nonlinear behavior. The shear modulus of each adhesive is expressed as a function of temperature. Characteristics and adaptability of the adhesives are discussed regarding their temperature sensitivity.

On the use of element-free Galerkin Method for problems involving incompressibility

The improvement of the performance of numerical methods, namely in the presence of incompressible deformations, have been, in the recent years, an important issue of discussion. However, incompressible or near incompressible behaviour has been a source of problems for numerical analysis of solid mechanics as the methods tend to present a locking response, or alternatively, spurious modes of deformation. The element-free Galerkin method (EFGM) is one of these methods, which exhibits some problems, namely, the so-called volumetric locking and hourglass. The first one is related with an inability to deform in some conditions. The hourglass is related with the apparition of non-physical eigenvectors. In this publication a new vision based in vector spaces theory is presented which can explain why these problems appear in the EFGM context.

Finite element formulation for modeling particle debonding in reinforced elastomers subjected to finite deformations

Interfacial damage nucleation and evolution in reinforced elastomers is modeled using a three-dimensional updated Lagrangian finite element formulation based on the perturbed Petrov–Galerkin method for the treatment of nearly incompressible behavior. The progressive failure of the particle–matrix interface is modeled by a cohesive law accounting for mode mixity. The meso-scale is characterized by a unit cell, which contains particles dispersed in a homogenized blend. A new, fully implicit and efficient finite element formulation, including consistent linearization, is presented. The proposed finite element model is capable of predicting the non-homogeneous meso-fields and damage nucleation and propagation along the particle–matrix interface. Simple deformations involving an idealized solid rocket propellant are considered to demonstrate the algorithm.

Numerical modeling of three-dimensional compressible gas flow in microchannnels

Nitrogen gas flow in long microchannels with square cross-sections was simulated numerically with a three-dimensional continuum model with slip and no-slip boundary conditions. The governing equations of the model were solved by a control volume method. The numerical model was validated with the available experimental and numerical results. For incompressible flow, it was found that when Dh was less than 60 µm, a slip boundary condition must be applied. An analytical expression for normalized friction coefficients, C*IC, i.e. the ratio of f Re (slip) to f Re (no-slip), was developed on the basis of incompressible flow behavior. For compressible flow, a parametric study was conducted for Dh = 1 µm, L/Dh = 200 and with varying pressure ratios (PR = 1.5–5.0). It was found that as the pressure ratio increased from 1.5 to 5.0, compressibility effects increased while the rarefaction effects started diminishing. Slip effects also played an important role in the friction characteristics of microchannel flows. An analytical expression for normalized friction coefficients, C*C, i.e. the ratio of f Re (compressible) to f Re (incompressible), was developed on the basis of flow behavior for compressible flow. A comparative study of two-dimensional and three-dimensional flows was also conducted, and it was shown that the two-dimensional assumption for the compressible flow was not valid since it predicted 15–45% higher flow velocities, and 7–12% lower friction factors than those predicted by the 3D models.

On the overall behavior, microstructure evolution, and macroscopic stability in reinforced rubbers at large deformations: II—Application to cylindrical fibers

In Part I of this paper, we presented a general homogenization framework for determining the overall behavior, the evolution of the underlying microstructure, and the possible onset of macroscopic instabilities in fiber-reinforced elastomers subjected to finite deformations. In this work, we make use of this framework to generate specific results for general plane-strain loading of elastomers reinforced with aligned, cylindrical fibers. For the special case of rigid fibers and incompressible behavior for the matrix phase, closed-form, analytical results are obtained. The results suggest that the evolution of the microstructure has a dramatic effect on the effective response of the composite. Furthermore, in spite of the fact that both the matrix and the fibers are assumed to be strongly elliptic, the homogenized behavior is found to lose strong ellipticity at sufficiently large deformations, corresponding to the possible development of macroscopic instabilities [ Geymonat, G., Müller, S., Triantafyllidis, N., 1993. Homogenization of nonlinearly elastic materials, macroscopic bifurcation and macroscopic loss of rank-one convexity. Arch. Rat. Mech. Anal. 122, 231–290]. The connection between the evolution of the microstructure and these macroscopic instabilities is put into evidence. In particular, when the reinforced elastomers are loaded in compression along the long, in-plane axis of the fibers, a certain type of “flopping” instability is detected, corresponding to the composite becoming infinitesimally soft to rotation of the fibers.

Axial effects investigation in fixed-end circular bars under torsion with a finite deformation model based on logarithmic strain

In this paper the torsion problem of a circular bar with fixed ends is solved using a finite deformation constitutive model based on the corotational rates of the logarithmic strain. The logarithmic, Green–Naghdi and Eulerian corotational rates of the logarithmic strain are used in the model. The solution is based on a von Mises type yield function that incorporates isotropic and kinematic hardenings. For the kinematic hardening, a modified Armstrong–Fredrick hardening model with the corotational rate of the logarithmic strain is used. Assuming incompressible behavior, the fixed-end torsion problem is simplified to the simple shear problem. Solving the problem, the stress components are integrated to calculate the torque and axial force. It is qualitatively shown that the results based on the logarithmic corotational rate are in good agreement with the experimental results.

Multiscale modelling of particle debonding in reinforced elastomers subjected to finite deformations

AbstractInterfacial damage nucleation and evolution in reinforced elastomers subjected to finite strains is modelled using the mathematical theory of homogenization based on the asymptotic expansion of unknown variables. The microscale is characterized by a periodic unit cell, which contains particles dispersed in a blend and the particle matrix interface is characterized by a cohesive law. A novel numerical framework based on the perturbed Petrov–Galerkin method for the treatment of nearly incompressible behaviour is employed to solve the resulting boundary value problem on the microscale and the deformation path of a macroscale particle is predefined as in the micro‐history recovery procedure. A fully implicit and efficient finite element formulation, including consistent linearization, is presented. The proposed multiscale framework is capable of predicting the non‐homogeneous micro‐fields and damage nucleation and propagation along the particle matrix interface, as well as the macroscopic response and mechanical properties of the damaged continuum. Examples are considered involving simple unit cells in order to illustrate the multiscale algorithm and demonstrate the complexity of the underlying physical processes. Copyright © 2005 John Wiley & Sons, Ltd.