Micromechanical Analysis of the Trabecular Bone Stress State at the Interface with Metallic Biomedical Devices*

An iterative procedure for the stress analysis at interfaces between dissimilar materials is presented. The problem is specialised to the case of biomaterial interfaces with particular reference to materials which are characterised by tiny microstructures. The procedure is based on a recursive analysis of small size problems defined upon subdomains obtained by partitioning the whole structural domain. The kinematic boundary conditions are iteratively adjusted by using appropriate preconditioners. The numerical example reported in this paper shows that the procedure is effective regardless of the degree of material heterogeneity, in contrast with the results obtained by using a coarse mesh for the whole domain. The procedure seems to be a promising one for determining the structural strength of interfaces between trabecular bone and metal implants requiring accurate evaluation of stress at the scale level of the single microstructure exhibited by the bone.     

An effective approach for modeling fluid flow in heterogeneous media using numerical manifold method

A major challenge of modeling fluid flow in heterogeneous media is to model the material interfaces, which may be arbitrarily oriented or intersected with Dirichlet, Neumann, or other boundaries, making it difficult to mesh and accurately satisfy the boundary constraints. In order to solve these problems, we derived a new continuous approach in the numerical manifold method (NMM). NMM is an ideal method to handle boundaries, considering its flexibility and efficiency with fixed mathematical mesh and its integration precision. With the two‐cover‐meshing system, we construct physical covers containing gradient jump terms defined as extended degrees of freedom to realize the refraction law across material interfaces. In the global equilibrium equations, the jump terms are naturally considered with the energy‐work seepage model. In this approach, high accuracy is expected from the newly constructed jump function together with simplex integration. Moreover, high mesh efficiency is realized by fixed triangular mathematical mesh with algorithms fully considering interfaces intersecting with Dirichlet, Neumann, or other boundaries and simplex integration on elements in arbitrary shapes. The new approach was coded into our NMM fluid flow model. We calculated examples involving fluid flow through a domain including (1) a single interface, (2) an idealized fault represented by multiple material interfaces, (3) intersected interfaces, and (4) an octagonal inclusion. We compared the simulated results to analytical solutions or results with denser mesh to test precision and efficiency and thereby proved that the new approach is accurate, efficient, and flexible, especially when considering intense geometric change or intersections. Copyright © 2015 John Wiley & Sons, Ltd.     

A technique of flux reconstruction at the interfaces of nonconforming grids for aeroacoustic simulations

A flux reconstruction technique is presented to perform aeroacoustic computations using implicit high-order spatial schemes on multiblock structured grids with nonconforming interfaces. The use of such grids, with mesh spacing discontinuities across the block interfaces, eases local mesh refinements, simplifies the mesh generation process, and thus facilitates the computation of turbulent flows. In this work, the spatial discretization consists of sixth-order finite-volume implicit schemes with low-dispersion and low-dissipation properties. The flux reconstruction is based on the combination of noncentered schemes with local interpolations to define ghost cells and compute flux values at the grid interfaces. The flow variables in the ghost cells are calculated from the flow field in the grid cells using a meshless interpolation with radial basis functions. In this study, the flux reconstruction is applied to both plane and curved nonconforming interfaces. The performance of the method is first evaluated by performing two-dimensional simulations of the propagation of an acoustic pulse and of the convection of a vortex on Cartesian and wavy grids. No significant spurious noise is produced at the grid interfaces. The applicability of the flux reconstruction to a three-dimensional computation is then demonstrated by simulating a jet at a Mach number of 0.9 and a diameter-based Reynolds number of 4×10⁵ on a Cartesian grid. The nonconforming grid interface located downstream of the jet potential core does not appreciably affect the flow development and the jet sound field, while reducing the number of mesh points by a factor of approximately two.     

Three-dimensional mesh embedding for the Navier–Stokes equations using upwind control volumes

A numerical model for the compressible Navier–Stokes equations using local mesh embedding is presented. The model solves for three-dimensional turbulent flow using an algebraic mixing length model of turbulence. The technique of control volume upwinding is used to produce a novel treatment, whereby the hanging nodes on the mesh interfaces are left with null control volumes. This yields an efficient discretization scheme which ensures second-order accuracy, flux conservation and stability at the mesh interfaces, whilst retaining a simple interpolative treatment for the hanging nodes. The discrete flow equations are solved using the semi-implicit pressure correction method. The accuracy of the embedded mesh solver is demonstrated by modelling the three-dimensional flow through a cascade of turbine vanes at design and off-design conditions. Mesh embedding gives a saving of 48% in the number of nodes. The embedded mesh solutions compare well with fine structured mesh solutions and experimental measurements. The capability of the embedded mesh solver to perform solution adaptive calculations is demonstrated using a two-dimensional mid-height section of the cascade at the off-design flow conditions.     

A moving mesh BGK scheme for multi‐material flows

In this paper, a moving mesh BGK scheme (MMBGK) for multi‐material flow computations is proposed. The basic idea of constructing the MMBGK is to couple the Lagrangian method, which tracks material interfaces and keeps the interfaces sharp, with a remapping‐free ALE‐type kinetic method within each single material region, where the kinetic method is based on the BGK (Bhatnagar–Gross–Krook) model. Within each single material region, a numerical flux formulation is developed on moving meshes from motion of microscope particles, and the mesh velocity is determined by requiring both mesh adaptation for accuracy and robustness, such that the grids are moving towards to the regions with high flow gradients in a way of diffusive mechanism (velocity) to adjust the distances between neighboring cells, thus increasing the numerical accuracy. To keep the sharpness of material interfaces, the Lagrangian velocity and flux are constructed at the interfaces only. Consequently, a BGK‐scheme‐based ALE‐type method (i.e., the MMBGK scheme) for multi‐material flows is constructed. Numerical examples in one and two dimensions are presented to illustrate the accuracy and robustness of the MMBGK scheme. Copyright © 2011 John Wiley & Sons, Ltd.     

Mesh regularization for an ALE code based on the limitation of the Lagrangian mesh velocity

The Lagrangian approach is usually used for the simulation of flow with strong shock waves. Moreover, this approach is particularly well suited to treatment of material interfaces in the case of multimaterial flows.Unfortunately, this formulation leads to very large deformations in the mesh. The arbitrary Lagrangian‐Eulerian method overcomes this drawback by using a mesh regularization that is based on an analysis of cell geometry. The regularization step may be considered as a method used to correct the nonconvex and potentially tangled cells that constitute the mesh. In this paper, we present a new approach to mesh regularization. Instead of using a purely geometric criterion, we propose that the mesh evolution is computed on the basis of the flow vorticity. This approach is called the large Eddy limitation method, and it is aimed here to be used in finite volume direct arbitrary Lagrangian‐Eulerian methods. The large Eddy limitation method is general, which means that it is not restricted to applications in the finite volume framework dedicated to fluid flow simulation; for instance, it could also be naturally applied to the finite element framework.     

A New modelling strategy for phase-change heat transfer in turbulent interfacial two-phase flow

The paper presents a modelling strategy for phase-change heat transfer in turbulent interfacial two-phase flow. The computational framework is based on interface tracking ITM (level set approach), combined with large-scale prediction of turbulence, a new methodology known as Large-Eddy & Interface Simulation (LEIS), where super-grid scale turbulence and interfaces are directly solved, whereas the sub-scale parts are modelled. Because steady-state flow conditions are difficult to attain, recourse is made of the Very Large-Eddy Simulation (V-LES) instead of LES, where the flow-dependent cut-off filter is larger and independent from the grid. The computational approach is completed by a DNS-based interfacial phase-change heat transfer model built within the Surface Divergence (SD) theory. The original SD model is found to return better results when modified to account for scale separation, i.e. to segregate low-Re from high-Re number flow portions in the same flow. The model was first validated for an experiment involving a smooth to wavy turbulent, stratified steam-water flow in a 2D channel (Lim et al., 1984, Condensation measurement of horizontal concurrent steam-water flow, ASME J. Heat Transfer 106, 425–432.), revealing that the original SD model performs better for high interfacial shear rates. This screening phase also demonstrated that the most critical issue is the accurate prediction of the interfacial shear using ITM. The model was then applied successfully to predict condensing steam in the event of emergency core cooling in a Pressurized Water Reactor (PWR), where water is injected into the cold leg during a postulated loss-of-coolant-accident. The simulation results agree fairly well with the COSI (short for COndensation at Safety Injections) data (Janicot and Bestion, 1993, Condensation modelling for ECC injection, Nucl. Eng. Des. 145, 37–45).     

Coupling methods between finite element–based Boussinesq‐type wave and particle‐based free‐surface flow models

We develop one‐way coupling methods between a Boussinesq‐type wave model based on the discontinuous Galerkin finite element method and a free‐surface flow model based on a mesh‐free particle method to strike a balance between accuracy and computational cost. In our proposed model, computation of the wave model in the global domain is conducted first, and the nonconstant velocity profiles in the vertical direction are reproduced by using its results. Computation of the free‐surface flow is performed in a local domain included within the global domain with interface boundaries that move along the reproduced velocity field in a Lagrangian fashion. To represent the moving interfaces, we used a polygon wall boundary model for mesh‐free particle methods. Verification and validation tests of our proposed model are performed, and results obtained by the model are compared with theoretical values and experimental results to show its accuracy and applicability.     

An implicit mixed enthalpy–temperature method for phase-change problems

A finite element procedure for phase-change problems is presented. Enthalpy and temperature are interpolated separately and subsequently linked via the appropriate relation in the nodes of the mesh during the solution phase. A novel technique is here used where, depending on the characteristics of the problem, either temperature or enthalpy may be considered as primary variable. The resulting algorithm is both efficient and robust and is further easy to implement and generalize to arbitrary finite elements. The capabilities of the method are illustrated by the solution both isothermal and non-isothermal phase-change problems.     

A high‐precision unstructured adaptive mesh technique for gas–liquid two‐phase flows

Adaptive mesh techniques are used widely in the numerical simulations of fluid flows, and the simulation results with high accuracies are obtained by appropriate mesh adaptations. However, gas–liquid two‐phase flows are still difficult to be simulated on adaptive meshes, especially on unstructured adaptive meshes, because the physical phenomena near gas–liquid interfaces are highly complicated and in general, not modeled appropriately on adaptive meshes. In this paper, a high‐precision unstructured adaptive mesh technique for gas–liquid two‐phase flows is developed and verified/validated. In the unstructured adaptive mesh technique, the PLIC algorithm is employed to simulate interfacial dynamic behaviors and, therefore, the reconstruction method for the interfaces in refined cells is developed, which satisfies the gas and liquid volume conservations and geometrical conservations of interfaces. In addition, the physics‐based consideration is performed on the momentum calculations near interfaces, and the calculation method with gas and liquid momentum conservations is developed. For verification, the slotted‐disk revolution problem is solved. As a result, the unstructured adaptive mesh technique succeeds in reproducing the slotted‐disk shape accurately and well maintaining the shape after one full‐revolution. The dam‐break problem is also simulated and the momentum conservative calculation method succeeds in providing physically appropriate results, which show good agreements with experimental data. Therefore, it is confirmed that the developed unstructured adaptive mesh technique is very efficient to simulate gas–liquid two‐phase flows accurately. Copyright © 2010 John Wiley & Sons, Ltd.     

Validation of adaptive unstructured hexahedral mesh computations of flow around a wind turbine airfoil

In this paper we investigate local adaptive refinement of unstructured hexahedral meshes for computations of the flow around the DU91 wind turbine airfoil. This is a 25% thick airfoil, found at the mid‐span section of a wind turbine blade. Wind turbine applications typically involve unsteady flows due to changes in the angle of attack and to unsteady flow separation at high angles of attack. In order to obtain reasonably accurate results for all these conditions one should use a mesh which is refined in many regions, which is not computationally efficient. Our solution is to apply an automated mesh adaptation technique. In this paper we test an adaptive refinement strategy developed for unstructured hexahedral meshes for steady flow conditions. The automated mesh adaptation is based on local flow sensors for pressure, velocity, density or a combination of these flow variables. This way the mesh is refined only in those regions necessary for high accuracy, retaining computational efficiency. A validation study is performed for two cases: attached flow at an angle of 6° and separated flow at 12°. The results obtained using our adaptive mesh strategy are compared with experimental data and with results obtained with an equally sized non‐adapted mesh. From these computations it can be concluded that for a given computing time, adapted meshes result in solutions closer to the experimental data compared to non‐adapted meshes for attached flow. Finally, we show results for unsteady computations. Copyright © 2005 John Wiley & Sons, Ltd.     

A mathematical basis for strain-gradient plasticity theory—Part I: Scalar plastic multiplier

Strain-gradient plasticity theories are reviewed in which some measure of the plastic strain rate is treated as an independent kinematic variable. Dislocation arguments are invoked in order to provide a physical basis for the hardening at interfaces. A phenomenological, flow theory version of gradient plasticity is constructed in which stress measures, work-conjugate to plastic strain and its gradient, satisfy a yield condition. Plastic work is also done at internal interfaces and a yield surface is postulated for the work-conjugate stress quantities at the interface. Thereby, the theory has the potential to account for grain size effects in polycrystals. Both the bulk and interfacial stresses are taken to be dissipative in nature and due attention is paid to ensure that positive plastic work is done. It is shown that the mathematical structure of the elasto-plastic strain-gradient theory has similarities to conventional rigid-plasticity theory. Uniqueness and extremum principles are constructed for the solution of boundary value problems.     

Dynamical energy analysis on mesh grids: A new tool for describing the vibro-acoustic response of complex mechanical structures

We present a new approach for modelling noise and vibration in complex mechanical structures in the mid-to-high frequency regime. It is based on a dynamical energy analysis (DEA) formulation which extends standard techniques such as statistical energy analysis (SEA) towards non-diffusive wave fields. DEA takes into account the full directionality of the wave field and makes sub-structuring obsolete. It can thus be implemented on mesh grids commonly used, for example, in the finite element method (FEM). The resulting mesh based formulation of DEA can be implemented very efficiently using discrete flow mapping (DFM) as detailed in Chappell et al. (2013) and described here for applications in vibro-acoustics. A mid-to-high frequency vibro-acoustic response can be obtained over the whole modelled structure. Abrupt changes of material parameter at interfaces are described in terms of reflection/transmission matrices obtained by solving the wave equation locally. Two benchmark model systems are considered: a double-hull structure used in the ship-building industry and a cast aluminium shock tower from a Range Rover. We demonstrate that DEA with DFM implementation can handle multi-mode wave propagation effectively, taking into account mode conversion between shear, pressure and bending waves at interfaces, and on curved surfaces.     

Comparison of Three FE-FV Numerical Schemes for Single- and Two-Phase Flow Simulation of Fractured Porous Media

We benchmark a family of hybrid finite element–node-centered finite volume discretization methods (FEFV) for single- and two-phase flow/transport through porous media with discrete fracture representations. Special emphasis is placed on a new method we call DFEFVM in which the mesh is split along fracture–matrix interfaces so that discontinuities in concentration or saturation can evolve rather than being suppressed by nodal averaging of these variables. The main objective is to illustrate differences among three discretization schemes suitable for discrete fracture modeling: (a) FEFVM with volumetric finite elements for both fractures and porous rock matrix, (b) FEFVM with lower dimensional finite elements for fractures and volumetric finite elements for the matrix, and (c) DFEFVM with a mesh that is split along material discontinuities. Fracture discontinuities strongly influence single- and multi-phase fluid flow. Continuum methods, when used to model transport across such interfaces, smear out concentration/saturation. We show that the new DFEFVM addresses this problem producing significantly more accurate results. Sealed and open single fractures as well as a realistic fracture geometry are used to conduct tracer and water-flooding numerical experiments. The benchmarking results also reveal the limitations/mesh refinement requirements of FE node-centered FV hybrid methods. We show that the DFEFVM method produces more accurate results even for much coarser meshes.     

Numerical simulation of shock wave interaction with a water column

The paper describes a scheme which is based on the CIP scheme and modified to properly describe a gas/liquid interface without smearing the density jump across the interface. This was achieved by calculating the density separately for each phase. The density at each grid point was determined by using a density function in a similar fashion as CIP. As a result a sharp density gradient was obtainable throughout the flow field and the scheme could handle properly gas/liquid interfaces having a large density ratio. Shock wave interaction with a cylindrical water column was simulated. The numerical results were compared with appropriate interferograms. Good agreement was found between the two. The results obtained for the cylindrical water column were compared with a similar solid cylinder case. The comparison reveled that even 40 s after shock impingement some differences were found between a liquid column and the solid cylinder. Received 11 July 2000 / Accepted 28 March 2001     

Effect of viscous plane stagnation flow on the freezing of fluid

This article investigates the phase-change problem from liquid to solid in the viscous plane stagnation flow. The solution at the initial stage of freezing is obtained by expanding it in powers of time, and the final equilibrium state is determined from the steady-state governing equations. The effect of the stagnation flow on the pure conduction problem can be clearly seen from the explicit analytic solutions, and the characteristics of the growth of solid and the transient heat transfer for all the dimensionless parameters are elucidated.     

The use of dynamic grid adaptation algorithms for the modelling of flow around a circular cylinder in sub‐critical flow regime

In the present study a dynamic grid adaptation (DGA) algorithm is utilized for predicting flow around a circular cylinder in sub‐critical flow regime at a Reynolds number of 1.4×10⁵. The reason for adopting a DGA algorithm is the unsteadiness of the flow field which makes a conventional mesh inefficient. The concept being adopted is to concentrate mesh refinement in regions with high gradients and high turbulent viscosity, while in the region further downstream where the flow is fully developed a coarser mesh will develop and turbulence is modelled with the large eddy simulation (LES) turbulence model. The aim of the study is to present an appropriate variable for mesh refinement, which accomplishes a high rate of mesh refinement in the region with high gradients. The new variable is a product of the local mesh cell size and the rate of strain and includes two additional variables to allow control over the refinement behaviour. The results are compared with experimental data at the corresponding Reynolds number and also with numerical results obtained with conventional mesh. It is demonstrated that DGA algorithms can give results of a very high quality for a mesh that is significantly smaller than for a conventional mesh. Copyright © 2003 John Wiley & Sons, Ltd.     

Newton's method for solving heat transfer,fluid flow and interface shapes in a floating molten zone

Newton's method is applied to the finite volume approximation for the steady state heat transfer, fluid flow and unknown interfaces in a floating molten zone. The streamfunction/vorticity and temperature formulation of the Navier–Stokes and energy equations and their associated boundary conditions are written in generalized curvilinear co-ordinates and conservative law form with the Boussinesq approximation. During Newton iteration the ILU(0) preconditioned GMRES matrix solver is applied for solving the linear system, where the sparse Jacobian matrix is estimated by finite differences. Nearly quadratic convergence of the method is observed. Sample calculations are reported for sodium nitrate, a high-Prandtl-number material (Pr = 9.12). Both natural convection and thermocapillary flow as well as an overall mass balance constraint in the molten zone are considered. The effects of convection and heat input on the flow patterns, zone position and interface shapes are illustrated. After the lens effect due to the molten zone is considered, the calculated flow patterns and interface shapes are compared with the observed ones and are found to be in good agreement.     

Mesh adaption for efficient multiscale implementation of one-dimensional turbulence

One-Dimensional Turbulence (ODT) is a stochastic model for turbulent flow simulation. In an atmospheric context, it is analogous to single-column modeling (SCM) in that it lives on a 1D spatial domain, but different in that it time advances individual flow realizations rather than ensemble-averaged quantities. The lack of averaging enables a physically sound multiscale treatment, which is useful for resolving sporadic localized phenomena, as seen in stably stratified regimes, and sharp interfaces, as observed where a convective layer encounters a stable overlying zone. In such flows, the relevant scale range is so large that it is beneficial to enhance model performance by introducing an adaptive mesh. An adaptive-mesh algorithm that provides the desired performance characteristics is described and demonstrated, and its implications for the ODT advancement scheme are explained.     

Computation of non-equilibrium hypersonic flow with artificially upstream flux vector splitting (AUFS) schemes

The AUFS scheme has been presented for solving the Euler equations [Sun, M., Takayama, K., 2003. An artificially upstream flux vector splitting scheme for the Euler equations. Journal of Computational Physics, 189, 305–329]. An extension of this high resolution scheme-based on upwind numerical methods has been developed to calculate a two-dimensional hypersonic viscous flowfield in thermochemical non-equilibrium. The time-dependent Navier–Stokes governing equations are computed by using a multi-block finite volume technique on a structured mesh. The convective fluxes at the interfaces are evaluated using a flux vector splitting (FVS) method with a second-order reconstruction of the interface values and the viscous terms are discretised by second-order central differences. A better evaluation of aerodynamic parameters are obtained with this AUFS scheme and they are also compared to those obtained by previous works. The freestream flow conditions of these computations correspond to high-enthalpy flows with a Mach number range between 6.4 and 25.9. The obtained numerical results indicate that the AUFS scheme is accurate, robust, and efficient for the calculation of hypersonic flow.