Investigation of fluid–structure interactions using a velocity‐linked P2/P1 finite element method and the generalized‐ α method

A velocity‐linked algorithm for solving unsteady fluid–structure interaction (FSI) problems in a fully coupled manner is developed using the arbitrary Lagrangian–Eulerian method. The P2/P1 finite element is used to spatially discretize the incompressible Navier–Stokes equations and structural equations, and the generalized‐ α method is adopted for temporal discretization. Common velocity variables are employed at the fluid–structure interface for the strong coupling of both equations. Because of the velocity‐linked formulation, kinematic compatibility is automatically satisfied and forcing terms do not need to be calculated explicitly. Both the numerical stability and the convergence characteristics of an iterative solver for the coupled algorithm are investigated by solving the FSI problem of flexible tube flows. It is noteworthy that the generalized‐ α method with small damping is free from unstable velocity fields. However, the convergence characteristics of the coupled system deteriorate greatly for certain Poisson's ratios so that direct solvers are essential for these cases. Furthermore, the proposed method is shown to clearly display the advantage of considering FSI in the simulation of flexible tube flows, while enabling much larger time‐steps than those adopted in some previous studies. This is possible through the strong coupling of the fluid and structural equations by employing common primitive variables. Copyright © 2012 John Wiley & Sons, Ltd.     

A 3D beam‐to‐beam contact finite element for coupled electric–mechanical fields

In this paper the formulation of an electric–mechanical beam‐to‐beam contact element is presented. Beams with circular cross‐sections are assumed to get in contact in a point‐wise manner and with clean metallic surfaces. The voltage distribution is influenced by the contact mechanics, since the current flow is constricted to small contacting spots. Therefore, the solution is governed by the contacting areas and hence by the contact forces. As a consequence the problem is semi‐coupled with the mechanical field influencing the electric one. The electric–mechanical contact constraints are enforced with the penalty method within the finite element technique. The virtual work equations for the mechanical and electric fields are written and consistently linearized to achieve a good level of computational efficiency with the finite element method. The set of equations is solved with a monolithic approach. Copyright © 2005 John Wiley & Sons, Ltd.     

Non‐weak/strong solutions in gas dynamics: a C11 p‐version STLSFEF in Lagrangian frame of reference using ρ, u,p primitive variables

In this paper, we present computations of the non‐weak solutions of class C¹¹ of transient Navier–Stokes equations for compressible flow in Lagrangian frame of reference using space‐time least squares finite element formulation with primitive variables ρ, u, p. For high speed compressible flows the solutions reported here possess the same orders of continuity as the governing differential equations. The role of diffusion i.e. viscosity (physical or artificial) and thermal conductivity on shock structure is demonstrated. Compression of air in a rigid cylinder by a rigid, massless and frictionless piston is used as a model problem. True time evolutions of class C¹¹ are reported beginning with the first time step until steady shock conditions are achieved. Comparisons with analytical solutions are presented when possible. Copyright © 2001 John Wiley & Sons, Ltd.     

Eulerian elasto‐visco‐plastic formulations for residual stress prediction

A stabilized, Galerkin finite element formulation for modeling the elasto‐visco‐plastic response of quasi‐steady‐state processes, such as welding, laser surfacing, rolling and extrusion, is presented in an Eulerian frame. The mixed formulation consists of four field variables, such as velocity, stress, deformation gradient and internal variable, which is used to describe the evolution of the material's resistance to plastic flow. The streamline upwind Petrov–Galerkin method is used to eliminate spurious oscillations, which may be caused by the convection‐type of stress, deformation gradient and internal variable evolution equations. A progressive solution strategy is introduced to improve the convergence of the Newton–Raphson solution procedure. Two two‐dimensional numerical examples are implemented to verify the accuracy of the Eulerian formulation. Copyright © 2008 John Wiley & Sons, Ltd.     

A combined BEM–FEM model for the velocity–vorticity formulation of the Navier–Stokes equations in three dimensions

This paper describes a combined boundary element and finite element model for the solution of velocity–vorticity formulation of the Navier–Stokes equations in three dimensions. In the velocity–vorticity formulation of the Navier–Stokes equations, the Poisson type velocity equations are solved using the boundary element method (BEM) and the vorticity transport equations are solved using the finite element method (FEM) and both are combined to form an iterative scheme. The vorticity boundary conditions for the solution of vorticity transport equations are exactly obtained directly from the BEM solution of the velocity Poisson equations. Here the results of medium Reynolds number of up to 1000, in a typical cubic cavity flow are presented and compared with other numerical models. The combined BEM–FEM model are generally in fairly close agreement with the results of other numerical models, even for a coarse mesh.     

Low‐Reynolds‐number flow around an oscillating circular cylinder using a cell viscousboundary element method

Flow fields from transversely oscillating circular cylinders in water at rest are studied by numerical solutions of the two‐dimensional unsteady incompressible Navier–Stokes equations adopting a primitive‐variable formulation. These findings are successfully compared with experimental observations. The cell viscous boundary element scheme developed is first validated to examine convergence of solution and the influence of discretization within the numerical scheme of study before the comparisons are undertaken. A hybrid approach utilising boundary element and finite element methods is adopted in the cell viscous boundary element method. That is, cell equations are generated using the principles of a boundary element method with global equations derived following the procedures of finite element methods. The influence of key parameters, i.e. Reynolds number Re, Keulegan–Carpenter number KC and Stokes' number β, on overall flow characteristics and vortex shedding mechanisms are investigated through comparisons with experimental findings and theoretical predictions. The latter extends the study into assessment of the values of the drag coefficient, added mass or inertia coefficient with key parameters and the variation of lift and in‐line force results with time derived from the Morison's equation. The cell viscous boundary element method as described herein is shown to produce solutions which agree very favourably with experimental observations, measurements and other theoretical findings. Copyright © 2001 John Wiley & Sons, Ltd.     

Non‐weak/strong solutions in gas dynamics: a C11 p‐version STLSFEF in Eulerian frame of reference using ρ, u,p primitive variables

This paper presents a space–time least squares finite element formulation of one‐dimensional transient Navier–Stokes equations (governing differential equations: GDE) for compressible flow in Eulerian frame of reference using ρ, u, p as primitive variables with C¹¹ type p‐version hierarchical interpolations in space and time. Time marching procedure is utilized to compute time evolutions for all values of time. For high speed gas dynamics the C¹¹ type interpolations in space and time possess the same orders of continuity in space and time as the GDE. It is demonstrated that with this approach accurate numerical solutions of Navier–Stokes equations are possible without any assumptions or approximations. In the approach presented here SUPG, SUPG/DC, SUPG/DC/LS operators are neither used nor needed. Time accurate numerical simulations show resolution of shock structure (i.e. shock speed, shock relations and shock width) to be in excellent agreement with the analytical solutions. The role of diffusion i.e. viscosity (physical or artificial) and thermal conductivity on shock structure is demonstrated. Riemann shock tube is used as a model problem. True time evolutions are reported beginning with the first time step until steady shock conditions are achieved. In this approach, when the computed error functionals become zero (computationally), the computed non‐weak solutions have characteristics as those of the strong solutions of the gas dynamics equations. Copyright © 2001 John Wiley & Sons, Ltd.     

Analysis of transient flow in pipelines with fluid–structure interaction using method of lines

A mathematical model is presented for transient flow in a pipeline with fluid–structure interaction. Water hammer theory and equations of axial motion for the pipeline are employed and the Poisson, junction and transient shear stress couplings are taken into account, which give rise to four coupled non‐linear, first‐order hyperbolic partial differential equations governing the fluid flow and pipe motion. These equations are discretized in space using the Keller box scheme and the method of lines is employed to reduce the partial differential equations to a system of ordinary differential equations. The resulting system is solved using a backward differentiation formulation method. The effect of transient shear stress on transient flow is investigated and the mechanisms underlying this effect are explored. The results revealed that the influence of transient shear stress can be significant and varies considerably, depending on the boundary conditions, viz, valve closure time. Copyright © 2005 John Wiley & Sons, Ltd.     

A SPACE-TIME COUPLED p-VERSION LEAST SQUARES FINITE ELEMENT FORMULATION FOR UNSTEADY TWO-DIMENSIONAL NAVIER–STOKES EQUATIONS

This paper presents a p-version least squares finite element formulation for two-dimensional unsteady fluid flow described by Navier–Stokes equations where the effects of space and time are coupled. The dimensionless form of the Navier–Stokes equations are first cast into a set of first-order differential equations by introducing auxiliary variables. This permits the use of C⁰ element approximation. The element properties are derived by utilizing the p-version approximation functions in both space and time and then minimizing the error functional given by the space–time integral of the sum of squares of the errors resulting from the set of first-order differential equations. This results in a true space–time coupled least squares minimization procedure. The application of least squares minimization to the set of coupled first-order partial differential equations results in finding a solution vector {δ} which makes gradient of error functional with respect to {δ} a null vector. This is accomplished by using Newton's method with a line search. A time marching procedure is developed in which the solution for the current time step provides the initial conditions for the next time step. Equilibrium iterations are carried out for each time step until the error functional and each component of the gradient of the error functional with respect to nodal degrees of freedom are below a certain prespecified tolerance. The space–time coupled p-version approximation functions provide the ability to control truncation error which, in turn, permits very large time steps. What literally requires hundreds of time steps in uncoupled conventional time marching procedures can be accomplished in a single time step using the present space–time coupled approach. The generality, success and superiority of the present formulation procedure is demonstrated by presenting specific numerical examples for transient couette flow and transient lid driven cavity. The results are compared with the analytical solutions and those reported in the literature. The formulation presented here is ideally suited for space–time adaptive procedures. The element error functional values provide a mechanism for adaptive h, p or hp refinements.     

A particle‐in‐cell solution to the silo discharging problem

The problem of flow of a granular material during the process of discharging a silo is considered in the present paper. The mechanical behaviour of the material is described by the use of the model of the elastic–plastic solid with the Drucker–Prager yield condition and the non‐associative flow rule. The phenomenon of friction between the stored material and the silo walls is taken into account—the Coulomb model of friction is used in the analysis. The problem is analysed by means of the particle‐in‐cell method—a variant of the finite element method which enables to solve the pertinent equations of motion on an arbitrary computational mesh and trace state variables at points of the body chosen independently of the mesh. The method can be regarded as an arbitrary Lagrangian–Eulerian formulation of the finite element method, and overcomes the main drawback of the updated Lagrangian formulation of FEM related to mesh distortion. The entire process of discharging a silo can be analysed by this approach. The dynamic problem is solved by the use of the explicit time‐integration scheme. Several numerical examples are included. The plane strain and axisymmetric problems are solved for silos with flat bottoms and conical hoppers. Some results are compared with experimental ones. Copyright © 1999 John Wiley & Sons, Ltd.     

Study of EDL effect on 3‐D developing flow in microchannel with Poisson–Boltzmann and Nernst–Planck models

Microchannels have been suggested to be a very effective heat transfer device. However, the electrical double layer (EDL) effect in a microchannel is anticipated to be significant. In this paper, two EDL models coupled with Navier–Stokes equations were used to compute a 3‐D developing microchannel flow. The Poisson–Boltzmann model (PBM) has been proven to be a promising tool in studying the EDL effect for developed microchannel flow, with respect to its accuracy and efficiency. However, it was commended that the assumption of Boltzmann distribution in PBM, for electric ion concentration distribution in particular was questionable in the developing flow. Nernst–Planck model (NPM), with its two extra partial differential equations (PDEs) to predict the ion concentration distribution, was reported to be a more appropriate model for developing microchannel flow though increased RAM and CPU were needed as compared to the PBM. The governing equations for both models were discretized for developing rectangular microchannel flows in Cartesians co‐ordinate. An additional source term, related to the electric potential, resulting from the EDL effect was introduced in the conventional z‐axis momentum equation as a body force, thereby modifying the flow characteristics. A finite‐volume scheme was used to solve the PDEs. The discrepancy in the results predicted by the two models was more dominant in the near‐wall region and not in the mainstream region. However, the performance of the microchannel was significantly affected by the EDL effect. An increase in Schmidt number will lead to decrease in friction coefficient. For the effect of aspect ratios (ARs), the discrepancy of f Re and Nu values for different ARs was obvious. Therefore, a 3‐D analysis was concluded to be important and necessary for investigating the EDL effect in microchannels. Copyright © 2006 John Wiley & Sons, Ltd.     

The enriched space–time finite element method (EST) for simultaneous solution of fluid–structure interaction

The paper introduces a weighted residual‐based approach for the numerical investigation of the interaction of fluid flow and thin flexible structures. The presented method enables one to treat strongly coupled systems involving large structural motion and deformation of multiple‐flow‐immersed solid objects. The fluid flow is described by the incompressible Navier–Stokes equations. The current configuration of the thin structure of linear elastic material with non‐linear kinematics is mapped to the flow using the zero iso‐contour of an updated level set function. The formulation of fluid, structure and coupling conditions uniformly uses velocities as unknowns. The integration of the weak form is performed on a space–time finite element discretization of the domain. Interfacial constraints of the multi‐field problem are ensured by distributed Lagrange multipliers. The proposed formulation and discretization techniques lead to a monolithic algebraic system, well suited for strongly coupled fluid–structure systems. Embedding a thin structure into a flow results in non‐smooth fields for the fluid. Based on the concept of the extended finite element method, the space–time approximations of fluid pressure and velocity are properly enriched to capture weakly and strongly discontinuous solutions. This leads to the present enriched space–time (EST) method. Numerical examples of fluid–structure interaction show the eligibility of the developed numerical approach in order to describe the behavior of such coupled systems. The test cases demonstrate the application of the proposed technique to problems where mesh moving strategies often fail. Copyright © 2007 John Wiley & Sons, Ltd.     

A dual BIE approach for large‐scale modelling of 3‐D electrostatic problems with the fast multipole boundary element method

A dual boundary integral equation (BIE) formulation is presented for the analysis of general 3‐D electrostatic problems, especially those involving thin structures. This dual BIE formulation uses a linear combination of the conventional BIE and hypersingular BIE on the entire boundary of a problem domain. Similar to crack problems in elasticity, the conventional BIE degenerates when the field outside a thin body is investigated, such as the electrostatic field around a thin conducting plate. The dual BIE formulation, however, does not degenerate in such cases. Most importantly, the dual BIE is found to have better conditioning for the equations using the boundary element method (BEM) compared with the conventional BIE, even for domains with regular shapes. Thus the dual BIE is well suited for implementation with the fast multipole BEM. The fast multipole BEM for the dual BIE formulation is developed based on an adaptive fast multiple approach for the conventional BIE. Several examples are studied with the fast multipole BEM code, including finite and infinite domain problems, bulky and thin plate structures, and simplified comb‐drive models having more than 440 thin beams with the total number of equations above 1.45 million and solved on a PC. The numerical results clearly demonstrate that the dual BIE is very effective in solving general 3‐D electrostatic problems, as well as special cases involving thin perfect conducting structures, and that the adaptive fast multipole BEM with the dual BIE formulation is very efficient and promising in solving large‐scale electrostatic problems. Copyright © 2007 John Wiley & Sons, Ltd.     

The use of Stokes' fundamental solution for the boundary only element formulation of the three-dimensional Navier–Stokes equations for moderate Reynolds numbers

This paper presents a boundary element formulation for the permanent Navier–Stokes equations in which the well-known closed-form fundamental solution for the steady Stokes equations is employed. In this way, from the integral representation formulae for the Stokes' equations, an integral equation is found in which the original non-linear convective terms of the Navier–Stokes equations appear as a domain integral. Additionally, the method of dual reciprocity is used to transform the domain integral to boundary integrals (this method is closely related to the method of particular integrals also used in the literature to transform domain integrals to boundary integrals). Numerical results are presented for the three-dimensional internal flow in a cylindrical container with a rotating cover, in which the accuracy of the method is shown.     

Deforming-Spatial-Domain/Stabilized Space–Time (DSD/SST) method in computation of non-Newtonian fluid flow and heat transfer with moving boundaries

This work presents an extension of the Deforming-Spatial-Domain/Stabilized Space–Time (DSD/SST) method to non-Newtonian fluid flow and heat transfer with moving boundaries. In this method, the variational formulation is written over the space–time domain. Three sets of stabilization parameters are used for the continuity, momentum and thermal energy equations. The more efficient solution for highly non-linear problems is achieved by using the Newton–Raphson iterative method for non-linear terms and the generalized minimal residual method for algebraic equations. This work makes the computations feasible with third-order accuracy in time, which is higher then most versions of the FEM. To validate this method, it is used to solve the well-known benchmark problems such as channel-confined flow, lid-driven cavity, flow around a cylinder, and flow in channel with wavy wall, where the non-Newtonian fluid rheological behaviour is incorporated. In particular, the results in terms of the Nusselt number, wall shear stress (WSS), vorticity fields and streamlines are discussed. It shows that the flow and heat transfer characteristics are quite different if the moving boundaries are taken into account. In summary, this work provides an effective extension of the DSD/SST method to hydrodynamics and heat transfer problems involving complex fluids and moving boundaries.     

Unified fractional step method for Lagrangian analysis of quasi‐incompressible fluid and nonlinear structure interaction using the PFEM

The fractional step method (FSM) is an efficient solution technique for the particle finite element method, a Lagrangian‐based approach to simulate fluid–structure interaction (FSI). Despite various refinements, the applicability of the FSM has been limited to low viscosity flow and FSI simulations with a small number of equations along the fluid–structure interface. To overcome these limitations, while incorporating nonlinear response in the structural domain, an FSM that unifies structural and fluid response in the discrete governing equations is developed using the quasi‐incompressible formulation. With this approach, fluid and structural particles do not need to be treated separately, and both domains are unified in the same system of equations. Thus, the equations along the fluid–structure interface do not need to be segregated from the fluid and structural domains. Numerical examples compare the unified FSM with the non‐unified FSM and show that the computational cost of the proposed method overcomes the slow convergence of the non‐unified FSM for high values of viscosity. As opposed to the non‐unified FSM, the number of iterations required for convergence with the unified FSM becomes independent of viscosity and time step, and the simulation run time does not depend on the size of the FSI interface. Copyright © 2016 John Wiley & Sons, Ltd.     

Analysis of orthotropic thick plates by meshless local Petrov–Galerkin (MLPG) method

A meshless local Petrov–Galerkin (MLPG) method is applied to solve static and dynamic problems of orthotropic plates described by the Reissner–Mindlin theory. Analysis of a thick orthotropic plate resting on the Winkler elastic foundation is given too. A weak formulation for the set of governing equations in the Reissner–Mindlin theory with a unit test function is transformed into local integral equations on local subdomains in the mean surface of the plate. Nodal points are randomly spread on the surface of the plate and each node is surrounded by a circular subdomain to which local integral equations are applied. The meshless approximation based on the Moving Least–Squares (MLS) method is employed in the numerical implementation. The present computational method is applicable also to plates with varying thickness. Numerical results for simply supported and clamped plates are presented. Copyright © 2006 John Wiley & Sons, Ltd.     

Assessment of three preconditioning schemes for solution of the two‐dimensional Euler equations at low Mach number flows

Three preconditioners proposed by Eriksson, Choi and Merkel, and Turkel are implemented in a 2D upwind Euler flow solver on unstructured meshes. The mathematical formulations of these preconditioning schemes for different sets of primitive variables are drawn, and their eigenvalues and eigenvectors are compared with each other. For this purpose, these preconditioning schemes are expressed in a unified formulation. A cell‐centered finite volume Roe's method is used for the discretization of the preconditioned Euler equations. The accuracy and performance of these preconditioning schemes are examined by computing steady low Mach number flows over a NACA0012 airfoil and a two‐element NACA4412–4415 airfoil for different conditions. The study shows that these preconditioning schemes greatly enhance the accuracy and convergence rate of the solution of low Mach number flows. The study indicates that the preconditioning methods implemented provide nearly the same results in accuracy; however, they give different performances in convergence rate. It is demonstrated that although the convergence rate of steady solutions is almost independent of the choice of primitive variables and the structure of eigenvectors and their orthogonality, the condition number of the system of equations plays an important role, and it determines the convergence characteristics of solutions. Copyright © 2011 John Wiley & Sons, Ltd.     

Stabilized finite element method for viscoplastic flow: formulation with state variable evolution

A stabilized, mixed finite element formulation for modelling viscoplastic flow, which can be used to model approximately steady‐state metal‐forming processes, is presented. The mixed formulation is expressed in terms of the velocity, pressure and state variable fields, where the state variable is used to describe the evolution of the material's resistance to plastic flow. The resulting system of equations has two sources of well‐known instabilities, one due to the incompressibility constraint and one due to the convection‐type state variable equation. Both of these instabilities are handled by adding mesh‐dependent stabilization terms, which are functions of the Euler–Lagrange equations, to the usual Galerkin method. Linearization of the weak form is derived to enable a Newton–Raphson implementation into an object‐oriented finite element framework. A progressive solution strategy is used for improving convergence for highly non‐linear material behaviour, typical for metals. Numerical experiments using the stabilization method with hierarchic shape functions for the velocity, pressure and state variable fields in viscoplastic flow and metal‐forming problems show that the stabilized finite element method is effective and efficient for non‐linear steady forming problems. Finally, the results are discussed and conclusions are inferred. Copyright © 2002 John Wiley & Sons, Ltd.     

A space–time coupled p-version least-squares finite element formulation for unsteady fluid dynamics problems

This paper presents a p-version least-squares finite element formulation for unsteady fluid dynamics problems where the effects of space and time are coupled. The dimensionless form of the differential equations describing the problem are first cast into a set of first-order differential equations by introducing auxiliary variables. This permits the use of C° element approximation. The element properties are derived by utilizing p-version approximation functions in both space and time and then minimizing the error functional given by the space–time integral of the sum of squares of the errors resulting from the set of first-order differential equations. This results in a true space–time coupled least-squares minimization procedure. A time marching procedure is developed in which the solution for the current time step provides the initial conditions for the next time step. The space–time coupled p-version approximation functions provide the ability to control truncation error which, in turn, permits very large time steps. What literally requires hundreds of time steps in uncoupled conventional time marching procedures can be accomplished in a single time step using the present space–time coupled approach. For non-linear problems the non-linear algebraic equations resulting from the least-squares process are solved using Newton's method with a line search. This procedure results in a symmetric Hessian matrix. Equilibrium iterations are carried out for each time step until the error functional and each component of the gradient of the error functional with respect to nodal degrees of freedom are below a certain prespecified tolerance. The generality, success and superiority of the present formulation procedure is demonstrated by presenting specific formulations and examples for the advection–diffusion and Burgers equations. The results are compared with the analytical solutions and those reported in the literature. The formulation presented here is ideally suited for space–time adaptive procedures. The element error functional values provide a mechanism for adaptive h, p or hp refinements. The work presented in this paper provides the basis for the extension of the space–time coupled least-squares minimization concept to two- and three-dimensional unsteady fluid flow.