Enriched goal-oriented error estimation for fracture problems solved by continuum-based shell extended finite element method简

An enriched goal-oriented error estimation method with extended degrees of freedom is developed to estimate the error in the continuum-based shell extended finite element method. It leads to high quality local error bounds in three-dimensional fracture mechanics simulation which involves enrichments to solve the singularity in crack tip. This enriched goal-oriented error estimation gives a chance to evaluate this continuum- based shell extended finite element method simulation. With comparisons of reliability to the stress intensity factor calculation in stretching and bending, the accuracy of the continuum-based shell extended finite element method simulation is evaluated, and the reason of error is discussed.     

A posteriori error estimation and anisotropy detection with the dual‐weighted residual method

In this work we develop a new framework for a posteriori error estimation and detection of anisotropies based on the dual‐weighted residual (DWR) method by Becker and Rannacher. The common approach for anisotropic mesh adaptation is to analyze the Hessian of the solution. Eigenvalues and eigenvectors indicate dominant directions and optimal stretching of elements. However, this approach is firmly linked to energy norm error estimation. Here, we extend the DWR method to anisotropic finite elements allowing for the direct estimation of directional errors with regard to given output functionals. The resulting meshes reflect anisotropic properties of both the solution and the functional. For the optimal measurement of the directional errors, the coarse meshes need some alignment with the dominant anisotropies. Numerical examples will demonstrate the efficiency of this method on various three‐dimensional problems including a well‐known Navier–Stokes benchmark. Copyright © 2009 John Wiley & Sons, Ltd.     

Segmented multigrid domain decomposition procedure for incompressible viscous flows

The application of grid stretching or grid adaptation is generally required in order to optimize the distribution of nodal points for fluid-dynamic simulation. This is necessitated by the presence of disjoint high gradient zones, that represent boundary or free shear layers, reversed flow or vortical flow regions, triple deck structures, etc. A domain decomposition method can be used in conjunction with an adaptive multigrid algorithm to provide an effective methodology for the development of optimal grids. In the present study, the Navier-Stokes (NS) equations are approximated with a reduced Navier-Stokes (RNS) system, that represents the lowest-order terms in an asymptotic Re expansion. This system allows for simplified boundary conditions, more generality in the location of the outflow boundary, and ensures mass conservation in all subdomain grid interfaces, as well as at the outflow boundary. The higher-order (NS) diffusion terms are included through a deferred corrector, in selected subdomains, when necessary. Adaptivity in the direction of refinement is achieved by grid splitting or domain decomposition in each level of the multigrid procedure. Normalized truncation error estimates of key derivatives are used to determine the boundaries of these subdomains. The refinement is optimized in two co-ordinate directions independently. Multidirectional adaptivity eliminates the need for grid stretching so that uniform grids are specified in each subdomain. The overall grid consists of multiple domains with different meshes and is, therefore, heavily graded. Results and computational efficiency are discussed for the laminar flow over a finite length plate and for the laminar internal flow in a backward-facing step channel.     

A study of properties of adaptive quadratic grids for three-dimensional near-surface field computations

Curved geometries and the corresponding near-surface fields typically require a large number of linear computational elements. High-order numerical solvers have been primarily used with low-order meshes. There is a need for curved, high-order computational elements. Typical near-surface meshes consist of hexahedral and/or prismatic elements. The present work studies the employment of quadratic meshes that are relatively coarse for field simulations. Directionally quadratic high-order elements are proposed for the near-surface field regions. The quadratic meshes are compared with the conventional low-order ones in terms of accuracy and efficiency. The cases considered include closed surface volume calculations, as well as computation of gradients of several analytic fields. A special method of adaptive local quadratic meshes is proposed and evaluated. Truncation error analysis for quadratic grids yields comparison with the conventional linear hexahedral/prismatic meshes, which are subject to typical distortions such as stretching, skewness, and torsion.     

Stability analysis of a polymer film casting problem

The polymer cast film process consists of stretching a molten polymer film between a flat die and a drawing roll. Drawing instabilities are often encountered and represent a drastic limitation to the process. Newtonian fluid film stretching stability is investigated using two numerical strategies. The first one is a ‘tracking’ method, which consists of solving Stokes equations in the whole fluid area (extrusion die and stretching path) by finite elements. The interface is determined to satisfy a kinematic equation. A domain decomposition meshing technique is used in order to account for a flow singularity resulting from the change in the boundary conditions between the die flow region and the stretching path region. A linear stability method is then applied to this transient kinematic equation in order to investigate the stability of the stationary solution. The second method is a direct finite element simulation in an extended area including the fluid and the surrounding air. The time‐dependent interface is captured by solving an appropriate level‐set function. The agreement between the two methods is fair. The influence of the stretching parameters (Draw ratio and drawing length) is investigated. For a long stretching distance, a critical Draw ratio around 20 delimitating stable and unstable drawing conditions is obtained, and this agrees well with the standard membrane models, which have been developed 40 years ago. When decreasing the stretching distance, the membrane model is no longer valid. The 2D models presented here point out a significant increase of the critical Draw ratio, and this is consistent with experimental results. Copyright © 2015 John Wiley & Sons, Ltd.     

Rivlin's Representation Formula is Ill-Conceived for the Determination of Response Functions via Biaxial Testing

The experimental determination of a strain energy function W for a rubber specimen must address departures from an elastic ideal in a rational fashion. Herein, such a rational experimental method is developed for biaxial stretching experiments and applied to rubber data in the literature. It is shown that Rivlin's representation formula is experimentally ill-conceived because experimental error is magnified to the extent that error obscures trends in the response function plots. Upon developing direct tensor expressions for the response function calculations, we show that Rivlin's representation formula (or any such constitutive law that has high covariance amongst the response terms) magnifies experimental error greatly. By “high covariance”, we mean the inner product amongst the response terms in the constitutive law is nearly equal to the maximum possible value - i.e., the product of their magnitudes. Moreover, we show that the second partials of W with respect to I ₁ and I ₂ should approach infinity as the strain decreases. Using an alternate set of invariants with minimal covariance (i.e., a null inner product amongst the response terms), a W for rubber can be determined forthwith.     

Flows induced by power-law stretching surface motion modulated by transverse or orthogonal surface shear

Boundary-layer solutions to Banks' problem for the flow induced by power-law stretching of a plate are obtained for two generalizations that include arbitrary transverse plate shearing motion. In one extension an arbitrary transverse shearing motion is the product of the power-law stretching. In the other extension the streamwise stretching coordinate is added to an arbitrary transverse shearing and together raised to the power of stretching. In addition we find that Banks' power law stretching may be accompanied by orthogonal power-law shear. In all cases, the original boundary-value problem of Banks [1] is recovered. Results are illustrated with velocity profiles both at the plate and at fixed height in the fluid above the plate.     

Flow and heat transfer over hyperbolic stretching sheets

The boundary layer flow and heat transfer analysis of an incompressible viscous fluid for a hyperbolically stretching sheet is presented. The analytical and numerical results are obtained by a series expansion method and a local non-similarity (LNS) method, respectively. The analytical and numerical results for the skin friction and the Nusselt number are calculated and compared with each other. The significant observation is that the momentum and the thermal boundary layer thickness decrease as the distance from the leading edge increases. The well-known solution of linear stretching is found as the leading order solution for the hyperbolic stretching.     

Double stratified radiative flow of an Oldroyd-B nanofluid with nonlinear convection

The nonlinear convective flow of an Oldroyd-B fluid due to a nonlinear stretching sheet with varying thickness is examined. The salient features of the random movement and thermophoresis are described. Formulation is made with the nonlinear thermal radiation and heat generation/absorption. Further, the convective conditions and double stratification are taken into account. The resulting flow problems are tackled by the optimal homotopy analysis method(OHAM). The resulting nonlinear problems are solved for the velocity, temperature, and concentration fields. The temperature and concentration gradients are numerically discussed. The total residual error is calculated.The Nusselt number is an increasing function of the radiation parameter. The Sherwood number increases with the increase in the solutal stratification or the Schmidt number.The main outcomes are presented in conclusions. This study has a wide range of applications such as thermal stratification of oceans, reservoirs, and rivers, density stratification of atmosphere, hydraulic lifts, and polymer processing.     

Analytical solution to stagnation-point flow and heat transfer over a stretching sheet based on homotopy analysis

This paper is concerned with two-dimensional stagnation-point steady flow of an incompressible viscous fluid towards a stretching sheet whose velocity is proportional to the distance from the slit. The governing system of partial differential equations is first transformed into a system of dimensionless ordinary differential equations. Analytical solutions of the velocity distribution and dimensionless temperature profiles are obtained for different ratios of free stream velocity and stretching velocity, Prandtl number, Eckert number and dimensionality index in series forms using homotopy analysis method（HAM）. It is shown that a boundary layer is formed when the free stream velocity exceeds the stretching velocity, and an inverted boundary layer is formed when the free stream velocity is less than the stretching velocity. Graphs are presented to show the effects of different parameters.     

A combined necking and shear localization analysis for aluminum sheets under biaxial stretching conditions

A combined necking and shear localization analysis is adopted to model the failures of two aluminum sheets, AA5754 and AA6111, under biaxial stretching conditions. The approach is based on the assumption that the reduction of thickness or the necking mode is modeled by a plane stress formulation and the final failure mode of shear localization is modeled by a generalized plane strain formulation. The sheet material is modeled by an elastic-viscoplastic constitutive relation that accounts for the potential surface curvature, material plastic anisotropy, material rate sensitivity, and the softening due to the nucleation, growth, and coalescence of microvoids. Specifically, the necking/shear failure of the aluminum sheets is modeled under uniaxial tension, plane strain tension and equal biaxial tension. The results based on the mechanics model presented in this paper are in agreement with those based on the forming limit diagrams (FLDs) and tensile tests. When the necking mode is suppressed, the failure strains are also determined under plane strain conditions. These failure strains can be used as guidances for estimation of the surface failure strains on the stretching sides of the aluminum sheets under plane strain bending conditions. The estimated surface failure strains are higher than the failure strains of the forming limit diagrams under plane strain stretching conditions. The results are consistent with experimental observations where the surface failure strains of the aluminum sheets increase significantly on the stretching sides of the sheets under bending conditions. The results also indicate that when a considerable amount of necking is observed for a sheet metal under stretching conditions, the surface failure strains on the stretching sides of the sheet metal under bending conditions can be significantly higher.     

On the numerical solution for the flow and heat transfer in a thin liquid film over an unsteady stretching sheet in a saturated porous medium in the presence of thermal radiation

The problem of the flow and heat transfer over an unsteady stretching sheet embedded in a porous medium in the presence of thermal radiation is studied theoretically and numerically. The continuity, momentum, and energy equations, which are coupled nonlinear partial differential equations, are reduced to a set of two nonlinear ordinary differential equations. Special attention is given to study the convergence of the proposed method. The error estimation is also given. The effects of various parameters, such as the Darcy parameter, the radiation parameter, and the Prandtl number, on the flow and temperature profiles, as well as on the local skin-friction coefficient and the local Nusselt number are presented and discussed. The results obtained agree very well with the data obtained by the Runge-Kutta method coupled with the shooting technique.     

Heating analysis of a droplet on stretchable hydrophilic surface

Heating of a droplet on a stretchable hydrophilic surface is investigated and fluid dynamics in the droplet under the heating load is assessed. Elastomer wafers are considered as the sample material and the fixture is designed and manufactured to assure uniform stretching of the droplet located elastomer surface. Droplet adhesion and possible slipping/sliding of the droplet are evaluated during stretching of the sample surface. Numerical simulations are carried out to predict thermal and flow response of the droplet fluid before and after stretching. The effect of droplet volume on heating enhancement is also included in the numerical simulations. Experiments are carried out using a high-speed recording system towards comparing the flow predictions. Findings reveal that predictions are in agreement with their counterparts of experiments. Stretching of sample surface increases wetting area and lowers height of the droplet while influencing thermal flow structures in the fluid. The Nusselt and the Bond numbers increase with enlarging stretching, which becomes more visible for large droplet volume (80 µl). Hence, stretching corresponding to 80% extension of elastomer surface gives rise to 60% improvement in the Nusselt number.     

An Anisotropic A posteriori Error Estimator for CFD

In this article, a robust anisotropic adaptive algorithm is presented, to solve compressible-flow equations using a stabilized CFD solver and automatic mesh generators. The association includes a mesh generator, a flow solver, and an a posteriori error-estimator code. The estimator was selected among several choices available (Almeida et al. (2000). Comput. Methods Appl. Mech. Engng , 182 , 379-400; Borges et al. (1998). "Computational mechanics: new trends and applications". Proceedings of the 4th World Congress on Computational Mechanics , Bs.As., Argentina) giving a powerful computational tool. The main aim is to capture solution discontinuities, in this case, shocks, using the least amount of computational resources, i.e. elements, compatible with a solution of good quality. This leads to high aspect-ratio elements (stretching). To achieve this, a directional error estimator was specifically selected. The numerical results show good behavior of the error estimator, resulting in strongly-adapted meshes in few steps, typically three or four iterations, enough to capture shocks using a moderate and well-distributed amount of elements.     

A new look at extensional rheology of low-density polyethylene

The nonlinear rheology of three selected commercial low-density polyethylenes (LDPE) is measured in uniaxial extensional flow. The measurements are performed using three different devices including an extensional viscosity fixture (EVF), a homemade filament stretching rheometer (DTU-FSR) and a commercial filament stretching rheometer (VADER-1000). We show that the measurements from the EVF are limited by a maximum Hencky strain of 4, while the two filament stretching rheometers are able to probe the nonlinear behavior at larger Hencky strain values where the steady state is reached. With the capability of the filament stretching rheometers, we show that LDPEs with quite different linear viscoelastic properties can have very similar steady extensional viscosity. This points to the potential for independently controlling shear and extensional rheology in certain rate ranges.     

Flow and heat transfer with prescribed skin friction in a rotating fluid

This paper investigates the rotating flow and heat transfer of a viscous fluid induced by a stretching surface. The nonlinear problem subject to a given skin friction at the boundary is solved. Analytic solution is obtained using homotopy analysis method. The velocity, temperature, and stretching velocity is calculated for different values of the rotation parameter (λ). The obtained results are compared with the well known results of rotating flow induced by a stretching surface by using four sets of boundary conditions. Copyright © 2011 John Wiley & Sons, Ltd.     

Linear stability of viscous flow induced by surface stretching

The stability of the two-dimensional flow in a semi-infinite fluid induced by the stretching of a planar surface is investigated by a normal-mode linear stability analysis. The base flow is described by Crane’s exact analytical solution of the Navier–Stokes equation. A finite-difference method and a shooting method are implemented to solve the relevant eigenvalue problem for three-dimensional perturbations that vary sinusoidally with arbitrary wave number in the spanwise direction normal to the plane of the flow. The Crane flow is found to be linearly stable, with the least-damped mode arising in the limit of two-dimensional perturbations. Eigenfunctions for the disturbance velocity are presented and discussed for selected values of the transverse wave number. The numerical results for the dispersion relation corresponding to the least-damped mode for each transverse wave number are described by a simple analytical expression. The results of the stability analysis are contextualized by superposing a planar stagnation-point flow toward the stretching surface. As the ratio of the rate of elongation of the stagnation-point flow to the rate of stretching of the surface tends to zero, the dispersion curves smoothly asymptote to those found for the Crane flow. The results for planar stagnation-point flow toward a stationary surface are recovered in the opposite limit. The analysis is extended to account for suction or injection through a porous surface undergoing in-plane stretching.     

Complex frequency shifted convolution PML for FDTD modelling of elastic waves

The perfectly matched layer (PML) is nowadays considered as the best optimum absorbing boundary condition available. However, the PML with the classical stretching tensor has certain limitations. Strangely, these limitations have rarely been addressed in elastic wave modelling. For example, substantial reflections occur when strong evanescent waves are propagating parallel to the interface. To circumvent problems like this, the complex frequency shifted stretching tensor has been introduced in electromagnetic modelling. In this paper we show that the convolution PML with this stretching tensor as used in electromagnetic modelling can be adapted for elastic wave modelling. Numerical results of a model where the presence of evanescent waves is predominant show that the PML based on the complex frequency shifted stretching tensor can improve the performance of the absorbing boundary layer considerably.     

The nonlinear vibration analysis of a clamped-clamped beam by a reduced multibody method

The nonlinear response characteristics for a dynamic system with a geometric nonlinearity is examined using a multibody dynamics method. The planar system is an initially straight clamped-clamped beam subject to high frequency excitation in the vicinity of its third natural mode. The model includes a pre-applied static axial load, linear bending stiffness and a cubic in-plane stretching force. Constrained flexibility is applied to a multibody method that lumps the beam into N elements for three substructures subjected to the nonlinear partial differential equation of motion and N-1 linear modal constraints. This procedure is verified by d'Alembert's principle and leads to a discrete form of Galerkin's method. A finite difference scheme models the elastic forces. The beam is tuned by the axial force to obtain fourth order internal resonance that demonstrates bimodal and trimodal responses in agreement with low and moderate excitation test results. The continuous Galerkin method is shown to generate results conflicting with the test and multibody method. A new checking function based on Gauss' principle of least constraint is applied to the beam to minimize modal constraint error.     

Experimental investigation of the stress–stretch behavior of EPDM rubber with loading rate effects

The tensile stress–stretch behavior of an ethylene–propylene–diene terpolymer (EPDM) was experimentally investigated, both in a quasi-static stretching rate range (<0.4/s) with a conventional material test machine and in a dynamic stretching rate range (2800/s–3200/s) with a split Hopkinson tension bar (SHTB) technique. Experimental data were then analyzed using the Ogden and Roxburgh’s idealized Mullins effect modeling theory. Results show that the stress–stretch behavior is significantly dependent on stretching rate and the Mullins effect exists under dynamic loading. Furthermore, stretching rate only affects the material properties. The degree of damage in a stretched specimen is a function of only the maximum stretch ratio the specimen experienced.