First vorticity–velocity–pressure numerical scheme for the Stokes problem

We consider the bidimensional Stokes problem for incompressible fluids and recall the vorticity, velocity and pressure variational formulation, which was previously proposed by one of the authors, and allows very general boundary conditions. We develop a natural implementation of this numerical method and we describe in this paper the numerical results we obtain. Moreover, we prove that the low degree numerical scheme we use is stable for Dirichlet boundary conditions on the vorticity. Numerical results are in accordance with the theoretical ones. In the general case of unstructured meshes, a stability problem is present for Dirichlet boundary conditions on the velocity, exactly as in the stream function-vorticity formulation. Finally, we show on some examples that we observe numerical convergence for regular meshes or embedded ones for Dirichlet boundary conditions on the velocity.     

Multiscale simulation of particle-reinforced elastic–plastic adhesives at small strains

Many aerospace, aircraft or automotive mechanical components are joined together by using a structural adhesive. Adherend-to-adherend joint performance is usually carried out by a thin adhesive layer such that loads are transferred through this region, being then a critical point in the design. In order to ensure a proper behaviour of the adhesive under dynamical, mechanical, thermal or rheological loads, they are typically reinforced with a second phase stiffer material in addition to the adhesive matrix. Due to the intrinsic nature of the matrix, it may be approached using an elastic–plastic behaviour. Under these circumstances the adhesive inherently shows a heterogeneous microstructure whereas the loads are applied at the macroscopic adherend scale. In this work, a multiscale formulation is developed to analyze particle-reinforced adhesive joints. The adherend and the adhesive region, which is modelled using cohesive elements, stand macroscopically. On the other hand, the macroscopic adhesive behaviour is obtained by a direct analysis of the two-distinguished phases interaction at the microscopic level, using micromechanics and homogenization. The presented approach provides macroscopic as well as microscopic information about load distribution avoiding phenomenological lab fitting, case to case, of the overall macroscopic behaviour of the adhesive.     

Efficient formulation of the Gibbs–Appell equations for constrained multibody systems

Multibody System Dynamics - In this study, we present explicit equations of motion for general mechanical systems exposed to holonomic and nonholonomic constraints based on the Gibbs-Appell...     

Lyapunov–Sylvester computational method for numerical solutions of a mixed cubic-superlinear Schrödinger system

Engineering with Computers - An upwind skewed radial basis function (USRBF)-based solution scheme is presented for stabilized solutions of convection-dominated problems over meshfree nodes. The...     

An inverse analysis procedure for the material parameter identification of elastic–plastic free-standing foils

A model calibration technique is considered for the estimation of material parameters in free-standing thin foils. The experimental apparatus is inspired by bursting strength testers for paper, textile fabrics and polymer coatings such as geo-membranes. The procedure referred to herein consists of the following phases. A controlled fluid pressure is applied to the foil specimen placed on an horizontal plane with a suitably shaped hole. The induced out-of-plane displacements are measured by a laser profilometer. The material parameters are then inferred from these measurements through inverse analysis, by simulation of the test and minimisation of a suitable norm which defines the discrepancy between measured and computed displacements. Potentialities and limitations of the proposed method are assessed on the basis of computer-generated “pseudo-experimental” data, where modelling errors are ruled out. The identifiability of some industrially meaningful material parameters is established.     

A Lagrange–Eulerian formulation of an axially moving beam based on the absolute nodal coordinate formulation

In the scope of this paper, a finite-element formulation for an axially moving beam is presented. The beam element is based on the absolute nodal coordinate formulation, where position and slope vectors are used as degrees of freedom instead of rotational parameters. The equations of motion for an axially moving beam are derived from generalized Lagrange equations in a Lagrange–Eulerian sense. This procedure yields equations which can be implemented as a straightforward augmentation to the standard equations of motion for a Bernoulli–Euler beam. Moreover, a contact model for frictional contact between an axially moving strip and rotating rolls is presented. To show the efficiency of the method, simulations of a belt drive are presented.     

Modelling of thermally affected elastic wave propagation within rotating Mori–Tanaka-based heterogeneous nanostructures

Microsystem Technologies - In this article, wave dispersion responses of a temperature-dependent functionally graded (FG) nanobeam undergoing rotation and exposed to thermal load are presented...     

An analysis of the time integration algorithms for the finite element solutions of incompressible Navier–Stokes equations based on a stabilised formulation

This work is concerned with the analysis of time integration procedures for the stabilised finite element formulation of unsteady incompressible fluid flows governed by the Navier–Stokes equations. The stabilisation technique is combined with several different implicit time integration procedures including both finite difference and finite element schemes. Particular attention is given to the generalised-α method and the linear discontinuous in time finite element scheme. The time integration schemes are first applied to two model problems, represented by a first order differential equation in time and the one dimensional advection–diffusion equation, and subjected to a detailed mathematical analysis based on the Fourier series expansion. In order to establish the accuracy and efficiency of the time integration schemes for the Navier–Stokes equations, a detailed computational study is performed of two standard numerical examples: unsteady flow around a cylinder and flow across a backward facing step. It is concluded that the semi-discrete generalised-α method provides a viable alternative to the more sophisticated and expensive space–time methods for simulations of unsteady flows of incompressible fluids governed by the Navier–Stokes equations.     

A fully discrete scheme for the pressure–stress formulation of the time-domain fluid–structure interaction problem

We propose an implicit Newmark method for the time integration of the pressure–stress formulation of a fluid–structure interaction problem. The space Galerkin discretization is based on the Arnold–Falk–Winther mixed finite element method with weak symmetry in the solid and the usual Lagrange finite element method in the acoustic medium. We prove that the resulting fully discrete scheme is well-posed and uniformly stable with respect to the discretization parameters and Poisson ratio, and we provide asymptotic error estimates. Finally, we present numerical tests to confirm the asymptotic error estimates predicted by the theory.     

Development of an Efficient 3–D CFD Software to Simulate and Visualize the Scavenging of a Two-Stroke Engine

In this paper we want to describe in detail how the task of numerically solving the flow through a two-stroke engine with moving parts is solved in an efficient way. The mathematical model behind the scenes is illuminated and the used numerical schemes are specified. First, the computation of the convective flux function is carried out by the AUSMDV Riemann solver, which has been proven to be very efficient in comparison to other schemes. Then the introduction of the temperature dependency of the material properties of the fluid has augmented the realistic setting within the compression and expansion of the hot gas within the cylinder. This temperature dependency of the heat capacity causes a change in the equation of state. The gas is not polytropic any more but calorically imperfect. Thus, the use of a relaxation method is necessary in order to retain our Riemann solver. To account for the complex geometry, it was necessary to realize a special mesh treatment. The computational domain can be assembled by different meshes that are connected in a mass conservative way. Furthermore, the piston and crankshaft motion is obtained by very efficient algorithms. In order to speed up the computation of the numerical solution, different strategies have been followed. Adaptive local time-stepping has been implemented in a time consistent manner. Additionally, a dynamic local mesh adaption with hanging knots is used to reach a better resolution in critical areas. A further reduction in computational time has been obtained by the parallelization of the numerical scheme and the mesh routines. To handle this parallelization of the mesh treatment, an extended partitioning for the dynamic load balancing has been implemented. Finally, a simulation of flow through a real-world geometry of an existing two-stroke engine has been performed, the results have been validated with measured pressure data for this engine, and the flow has been qualitatively and quantitatively studied.     

The numerical integration scheme for a fast Petrov–Galerkin method for solving the generalized airfoil equation

The main purpose of this paper is to give the numerical integration scheme for a fast Petrov–Galerkin method for solving the generalized airfoil equation, considered in a recent paper (Cai, J. Complex. 25:420–436, 2009). This scheme leads to a fully discrete sparse linear system. We show that it requires a nearly linear computational cost to get this system, and the approximate solution of the resulting linear system preserves the optimal convergent order. Numerical experiments are presented to confirm the theoretical estimates.     

An upwind compact difference scheme for solving the streamfunction–velocity formulation of the unsteady incompressible Navier–Stokes equation

In this paper, an upwind compact difference method with second-order accuracy both in space and time is proposed for the streamfunction–velocity formulation of the unsteady incompressible Navier–Stokes equations. The first derivatives of streamfunction (velocities) are discretized by two type compact schemes, viz. the third-order upwind compact schemes suggested with the characteristic of low dispersion error are used for the advection terms and the fourth-order symmetric compact scheme is employed for the biharmonic term. As a result, a five point constant coefficient second-order compact scheme is established, in which the computational stencils for streamfunction only require grid values at five points at both $\left(n\right)$th and $(n+1)$th time levels. The new scheme can suppress non-physical oscillations. Moreover, the unconditional stability of the scheme for the linear model is proved by means of the discrete von Neumann analysis. Four numerical experiments involving a test problem with the analytic solution, doubly periodic double shear layer flow problem, lid driven square cavity flow problem and two-sided non-facing lid driven square cavity flow problem are solved numerically to demonstrate the accuracy and efficiency of the newly proposed scheme. The present scheme not only shows the good numerical performance for the problems with sharp gradients, but also proves more effective than the existing second-order compact scheme of the streamfunction–velocity formulation in the aspect of computational cost.     

Prediction of the oil injection time of FDBs in an HDD spindle motor with a tied shaft by utilizing Kirchhoff’s pressure law

We propose a method to predict the oil injection time of fluid dynamic bearings (FDBs) with a tied shaft by applying Kirchhoff’s pressure law. Since the oil is injected by capillary phenomenon, the volume flow rate can be calculated by utilizing Kirchhoff’s pressure law. Then, we calculated the oil injection time of the FDBs with a tied shaft by dividing the volume flow rate by the clearance volumes of the journal bearing, the thrust bearing, and the recirculation channel (RC), respectively. We generated simulation models of the FDBs used in a 2.5″ HDD spindle motor with a tied shaft. The total oil injection times of the FDBs with and without a RC were 0.302 and 0.335 s, respectively. Also, we verified the proposed method by measuring the oil injection time of FDBs with a RC. We applied the proposed method to predict and improve the oil injection time of the FDBs with a tied shaft due to the variation of major parameters affecting the oil injection time.     

An efficient recursive rotational-coordinate-based formulation of a planar Euler–Bernoulli beam

Multibody System Dynamics - A recursive rotational-coordinate-based formulation of a planar Euler–Bernoulli beam is developed, where large displacements, deformations, and rotations are...     

The numerical simulation of periodic solutions for a predator–prey system

In this article, the existence and global stability of periodic solutions for a semi-ratio-dependent predator–prey system with Holling IV functional response and time delays are investigated. Using coincidence degree theory and Lyapunov method, sufficient conditions for the existence and global stability of periodic solutions are obtained. A numerical simulation is given to illustrate the results.     

A meshless method to the numerical solution of an inverse reaction–diffusion–convection problem

In this paper, the determination of the source term in a reaction–diffusion convection problem is investigated. First with suitable transformations, the problem is reduced, then a new meshless method based on the use of the heat polynomials as basis functions is proposed to solve the inverse problem. Due to the ill-posed inverse problem, the Tikhonov regularization method with a generalized cross-validation criterion is employed to obtain a numerical stable solution. Finally, some numerical examples are presented to show the accuracy and effectiveness of the algorithm.     

On pressure–velocity coupled time-integration of incompressible Navier–Stokes equations using direct inversion of Stokes operator or accelerated multigrid technique

Two fully pressure–velocity coupled approaches to time-integration of two- and three-dimensional Navier–Stokes equations discretized by finite volume method are proposed and verified. The first approach utilizes a direct sparse matrix solver to inverse the Stokes operator. In the second approach a multigrid iterative solver is accelerated by a modification of CLGS smoother that allows for derivation of an analytical solution for velocity and pressure corrections belonging to a whole row or column of finite volumes. Both approaches are tested by two- and three-dimensional natural convection benchmark problems. It is concluded that the analytical solution accelerated CLGS technique (ASA-CLGS) can be considered as a promising tool for solution of time-dependent three-dimensional fluid dynamics problems.     

Tunable optofluidic microlens through active pressure control of an air–liquid interface

We demonstrate a tunable in-plane optofluidic microlens with a 9× light intensity enhancement at the focal point. The microlens is formed by a combination of a tunable divergent air–liquid interface and a static polydimethylsiloxane lens, and is fabricated using standard soft lithography procedures. When liquid flows through a straight channel with a side opening (air reservoir) on the sidewall, the sealed air in the side opening bends into the liquid, forming an air–liquid interface. The curvature of this air–liquid interface can be conveniently and predictably controlled by adjusting the flow rate of the liquid stream in the straight channel. This change in the interface curvature generates a tunable divergence in the incident light beam, in turn tuning the overall focal length of the microlens. The tunability and performance of the lens are experimentally examined, and the experimental data match well with the results from a ray-tracing simulation. Our method features simple fabrication, easy operation, continuous and rapid tuning, and a large tunable range, making it an attractive option for use in lab-on-a-chip devices, particularly in microscopic imaging, cell sorting, and optical trapping/manipulating of microparticles.     

Asymptotic properties of the space–time adaptive numerical solution of a nonlinear heat equation

We consider the fully adaptive space–time discretization of a class of nonlinear heat equations by Rothe’s method. Space discretization is based on adaptive polynomial collocation which relies on equidistribution of the defect of the numerical solution, and the time propagation is realized by an adaptive backward Euler scheme. From the known scaling laws, we infer theoretically the optimal grids implying error equidistribution, and verify that our adaptive procedure closely approaches these optimal grids.