Space-time Finite Element Formulation Research Articles

A cut finite element method for two-phase flows with insoluble surfactants

We propose a new unfitted finite element method for simulation of two-phase flows in presence of insoluble surfactant. The key features of the method are 1) discrete conservation of surfactant mass; 2) the possibility of having meshes that do not conform to the evolving interface separating the immiscible fluids; 3) accurate approximation of quantities with weak or strong discontinuities across evolving geometries such as the velocity field and the pressure. The new discretization of the incompressible Navier-Stokes equations coupled to the convection-diffusion equation modeling the surfactant transport on evolving surfaces is based on a space-time cut finite element formulation with quadrature in time and a stabilization term in the weak formulation that provides function extension. The proposed strategy utilizes the same computational mesh for the discretization of the surface Partial Differential Equation (PDE) and the bulk PDEs and can be combined with different techniques for representing and evolving the interface, here the level set method is used. Numerical simulations in both two and three space dimensions are presented including simulations showing the role of surfactant in the interaction between two drops.

Spline-based space-time finite element approach for fluid-structure interaction problems with a focus on fully enclosed domains

Non-Uniform Rational B-Spline (NURBS) surfaces are commonly used within Computer-Aided Design (CAD) tools to represent geometric objects. When using isogeometric analysis (IGA), it is possible to use such NURBS geometries for numerical analysis directly. Analyzing fluid flows, however, requires complex three-dimensional geometries to represent flow domains. Defining a parametrization of such volumetric domains using NURBS can be challenging and is still an ongoing topic in the IGA community.With the recently developed NURBS-enhanced finite element method (NEFEM), the favorable geometric characteristics of NURBS are used within a standard finite element method. This is achieved by enhancing the elements touching the boundary by using the NURBS geometry itself. In the current work, a new variation of NEFEM is introduced, which is suitable for three-dimensional space-time finite element formulations. The proposed method makes use of a new mapping which results in a non-Cartesian formulation suitable for fluid-structure interaction (FSI).This is demonstrated by combining the method with an IGA formulation in a strongly-coupled partitioned framework for solving FSI problems. The framework yields a fully spline-based representation of the fluid-structure interface through a single NURBS.The coupling conditions at the fluid-structure interface are enforced through a Robin-Neumann type coupling scheme. This scheme is particularly useful when considering incompressible fluids in fully Dirichlet-bounded and curved problems, as it satisfies the incompressibility constraint on the fluid for each step within the coupling procedure.The accuracy and performance of the introduced spline-based space-time finite element approach and its use within the proposed coupled FSI framework are demonstrated using a series of two- and three-dimensional benchmark problems.

The critical mass phenomenon in vortex-induced vibration at low

Vortex-induced vibration of a circular cylinder with low mass ratio ($0.05\leqslant m^{\ast }\leqslant 10.0$) is investigated, via a stabilized space–time finite element formulation, in the laminar flow regime where$m^{\ast }$is defined as the ratio of the mass of the oscillating structure to the mass of the fluid displaced by it. Computations are carried out over a wide range of reduced speed,$U^{\ast }$, which is defined as$U/f_{N}D$, where$U$is the free-stream speed,$f_{N}$the natural frequency of the spring mass system in vacuum and$D$the diameter of the cylinder. In particular, the situation where the lock-in regime extends up to infinite reduced speed is explored. Studies at large$Re$, in the past, have shown that the normalized amplitude of cylinder oscillation at infinite reduced speed,$A_{\infty }^{\ast }$, exhibits a sharp increase when$m^{\ast }$is reduced below the critical mass ratio ($m_{crit}^{\ast }$). This jump signifies a shift from desynchronized response to lock-in state. In this work it is shown that in the laminar regime, a jump in$A_{\infty }^{\ast }$occurs only beyond a certain$Re$($=Re_{j}\sim 108$). For$Re<Re_{j}$, the response increases smoothly with decrease in$m^{\ast }$with no discernible jump. In this situation, therefore, the identification of$m_{crit}^{\ast }$based on jump in response at$U^{\ast }=\infty$is not possible. The difference in the$A^{\ast }-m^{\ast }$variation on the two sides of$Re=Re_{j}$, is attributed to the difference in the transition between the lower branch of cylinder response and desynchronization regime. This transition is brought out more clearly by plotting$A^{\ast }$with$f_{v_{o}}/f$, where$f_{v_{o}}$is the vortex shedding frequency for the flow past a stationary cylinder and$f$is the cylinder vibration frequency. In the$A^{\ast }-f_{v_{o}}/f$plane, the response data as well as other quantities related to free vibrations, for different$m^{\ast }$, collapse on a curve. Unlike at high$Re$, the collapsed curves show a dependence on$Re$in the laminar regime. The transition between the lock-in and desynchronized state, as seen from the collapsed curves, is qualitatively different for$Re$on either side of$Re_{j}$. The collapsed curves, at a certain$Re$, are utilized to estimate$A^{\ast }$for the limiting case of$(U^{\ast },m^{\ast })=(\infty ,0)$. Interestingly, unlike at large$Re$, this limit value is found to be lower than the peak amplitude of cylinder vibration at a given$Re$. Hysteresis in the cylinder response, near the higher-$U^{\ast }$end of the lock-in regime, is explored. It is observed that the range of$U^{\ast }$with hysteretic response increases with decrease in$m^{\ast }$. Interestingly, for a certain range of$m^{\ast }$, the response is hysteretic from a finite$U^{\ast }$up to$U^{\ast }=\infty$. We refer to this phenomenon as hysteresis forever. It occurs because of the existence of multiple response states of the system at$U^{\ast }=\infty$, for a certain range of$m^{\ast }$. The study brings out the significant differences in the response of the fluid–structure system associated with the critical mass phenomenon between the low- and high-$Re$regime.

A space-time discontinuous Galerkin method for Boussinesq-type equations

In this work we propose a new high order accurate, fully implicit space-time discontinuous Galerkin (DG) method for advection-diffusion-dispersion equations, i.e. for so-called Korteweg-de-Vries-type equations. In particular, we focus on Boussinesq-type models for free surface flows, which are used for the modeling of water waves that travel in deep water, where the classical shallow water equations are not valid any more. Our method follows the ideas of the local DG method (LDG) for dispersion equations proposed by Yan and Shu (2002), who used an explicit Runge–Kutta method to integrate their scheme in time. However, such explicit time integrators applied to dispersive equations imply a very severe restriction on the time step, which has to be taken proportional to the cube of the mesh spacing, and which therefore can make even one-dimensional computations prohibitively expensive on fine grids. For the scalar case and with some simplifying assumptions, the scheme presented in this paper can be proven to be unconditionally stable in L2 norm. Furthermore, our method is based directly on a space-time finite element formulation, which also provides a natural way to discretize third order dispersive terms that contain a mixed space-time derivative. Such terms appear frequently in the context of Boussinesq-type models for free surface flows. We will show numerical convergence studies for linear scalar advection-diffusion-dispersion equations and furthermore, we will also study the convergence of our method using solitary wave solutions of the nonlinear Boussinesq-type model of Madsen, Murray and Sørensen (1991).

Free vibrations of a cylinder: 3-D computations at Re=1000

Vortex-induced vibrations of a circular cylinder of low non-dimensional mass (m⁎=10.0) are investigated at Re=1000. A stabilized space–time finite element formulation is utilized to solve the incompressible flow equations in primitive variables in three dimensions. Computations are carried out for reduced velocities in the range 4.2≤U⁎≤6.2 for, both, increasing and, decreasing U⁎. Three branches are identified for the cylinder response: initial, upper and lower. The cylinder response and force coefficients exhibit beats in the initial branch. The initial–upper branch transition is hysteretic. The maximum amplitude of transverse oscillations is 0.7D. This value is larger than that observed in the laminar regime and smaller than that reported earlier for large Re. Intermittency is observed in the transition between upper and lower branches. During the transition, the fraction of time spent by the cylinder in lower branch increases with U⁎. Soft-lock-in is observed on the upper branch. Although the frequency of vortex shedding and the cylinder vibration match, they are different than the natural frequency of the spring-mass system. The mode of vortex shedding is 2S on the initial branch while it is 2P along the lower branch. Perhaps, for the first time, 2S mode of vortex shedding is found along the upper branch. This is confirmed via investigation of phase difference between the lift and cylinder response.

A rheological interface model and its space–time finite element formulation for fluid–structure interaction

AbstractThis contribution discusses extended physical interface models for fluid–structure interaction problems and investigates their phenomenological effects on the behavior of coupled systems by numerical simulation. Besides the various types of friction at the fluid–structure interface the most interesting phenomena are related to effects due to additional interface stiffness and damping. The paper introduces extended models at the fluid–structure interface on the basis of rheological devices (Hooke, Newton, Kelvin, Maxwell, Zener). The interface is decomposed into a Lagrangian layer for the solid‐like part and an Eulerian layer for the fluid‐like part. The mechanical model for fluid–structure interaction is based on the equations of rigid body dynamics for the structural part and the incompressible Navier–Stokes equations for viscous flow. The resulting weighted residual form uses the interface velocity and interface tractions in both layers in addition to the field variables for fluid and structure. The weak formulation of the whole coupled system is discretized using space–time finite elements with a discontinuous Galerkin method for time‐integration leading to a monolithic algebraic system. The deforming fluid domain is taken into account by deformable space–time finite elements and a pseudo‐structure approach for mesh motion. The sensitivity of coupled systems to modification of the interface model and its parameters is investigated by numerical simulation of flow induced vibrations of a spring supported fluid‐immersed cylinder. It is shown that the presented rheological interface model allows to influence flow‐induced vibrations. Copyright © 2010 John Wiley & Sons, Ltd.

A numerical verification for an unconditionally stable FEM for elastodynamics

Numerical results for a time-discontinuous Galerkin space–time finite element formulation for second-order hyperbolic partial differential equations are presented. Discontinuities are allowed at finite, but not fixed, time increments. A method for h-adaptive refinement of the space–time mesh is proposed and demonstrated. Numerical results are presented for linear elastic problems in one space dimension. Numerical verification of unconditional stability, as proven in [7], is rendered. Comparison is made with analytic solutions when available. It is shown that the accuracy of the numerical solution can be increased without a major penalty on computational cost by using an adaptively refined mesh. Results are presented for a type of solid–solid dynamic phase transition problem where the trajectory of a moving surface of discontinuity is tracked.

Effect of blockage on free vibration of a circular cylinder at low Re

AbstractThe effect of the blockage on vortex‐induced vibrations of a circular cylinder of low non‐dimensional mass (m*=10) in the laminar flow regime is investigated in detail. A stabilized space–time finite element formulation is utilized to solve the incompressible flow equations in primitive variables form in two dimensions. The transverse response of the cylinder is found to be hysteretic at both ends of synchronization/lock‐in region for 5% blockage. However, for the 1% blockage hysteresis occurs only at the higher Re end of synchronization/lock‐in region. Computations are carried out at other blockages to understand its effect on the hysteretic behavior. The hysteresis loop at the lower Re end of the synchronization decreases with decrease in blockage and is completely eliminated for blockage of 2.5% and less. On the other hand, hysteresis persists for all values of blockage at the higher Re end of synchronization/lock‐in. Although the peak transverse oscillation amplitude is found to be same for all blockage (∼0.6D), the peak value of the aerodynamic coefficients vary significantly with blockage. The r.m.s. values show lesser variation with blockage. The effect of streamwise extent of computational domain on hysteretic behavior is also studied. The phase between the lift force and transverse displacement shows a jump of almost 180° at, approximately, the middle of the synchronization region. This jump is not hysteretic and is independent of blockage. Copyright © 2008 John Wiley & Sons, Ltd.

Vortex-induced vibrations of a circular cylinder at low Reynolds numbers

Results are presented for a numerical simulation of vortex-induced vibrations of a circular cylinder of low non-dimensional mass (m* = 10) in the laminar flow regime (60 < Re < 200). The natural structural frequency of the oscillator, fN, matches the vortex shedding frequency for a stationary cylinder at Re = 100. This corresponds to fND2/ν = 16.6, where D is the diameter of the cylinder and ν the coefficient of viscosity of the fluid. A stabilized space–time finite element formulation is utilized to solve the incompressible flow equations in primitive variables form in two dimensions. Unlike at high Re, where the cylinder response is known to be associated with three branches, at low Re only two branches are identified: ‘initial’ and ‘lower’. For a blockage of 2.5% and less the onset of synchronization, in the lower Re range, is accompanied by an intermittent switching between two modes with vortex shedding occurring at different frequencies. With higher blockage the jump from the initial to lower branch is hysteretic. Results from free vibrations are compared to the data from experiments for forced vibrations reported earlier. Excellent agreement is observed for the critical amplitude required for the onset of synchronization. The comparison brings out the possibility of hysteresis in forced vibrations. The phase difference between the lift force and transverse displacement shows a jump of almost 180° at, approximately, the middle of the synchronization region. This jump is not hysteretic and it is not associated with any radical change in the vortex shedding pattern. Instead, it is caused by changes in the location and value of the maximum suction on the lower and upper surface of the cylinder. This is observed clearly by comparing the time-averaged flow for a vibrating cylinder for different Re. While the mean flow for Re beyond the phase jump is similar to that for a stationary cylinder, it is associated with a pair of counter-rotating vortices in the near wake for Re prior to the phase jump. The phase jump appears to be one of the mechanisms of the oscillator to self-limit its vibration amplitude.

Effect of blockage on vortex-induced vibrations at low Reynolds numbers

There have been quite a few studies in the past to investigate the effect of blockage on flow past a stationary cylinder, but very few for the case when the cylinder is vibrating. Compared to a stationary cylinder, a vibrating cylinder is associated with a wider wake and therefore the blockage is expected to play an even more significant role. The effect of blockage on the vortex-induced vibrations of a cylinder at low Re (Re⩽150) is investigated numerically via a stabilized space–time finite element formulation. The cylinder of low nondimensional mass (m*=10) is free to vibrate in both transverse and in-line directions. Two sets of computations are carried out for each of the cases with 1% and 5% blockage. In the first set of computations the reduced velocity, U*(=U/fnD, where fn is the natural frequency of the oscillator, U the free-stream speed, and D the cylinder diameter) is fixed to 4.92 and the effect of Re is studied. In the second set of computations, both Re and U* are varied. Lock-in is observed for a range of Re. A hysteretic behavior of the cylinder response close to the lower and upper limits of the synchronization/lock-in region is observed for the case with 5% blockage. The flow is associated with a different arrangement of vortices in the wake depending on whether one is on the “increasing Re” or “decreasing Re” branch. However, for the case with 1% blockage, the hysteretic behavior is completely eliminated near the lower Re range of the lock-in. The solutions for the decreasing as well as increasing Re branch are very similar. They are both associated with intermittent switching of the vortex shedding frequency between the structural frequency and the vortex shedding frequency for stationary cylinder. The hysteretic behavior for a range of Re close to the upper limit of the lock-in region is observed for both the low and high blockage.

Elastic Wave Propagation in Laminated Composites Using the Space-Time Least-Squares Formulation in h,p,k Framework

In this paper, numerical simulation of one-dimensional elastic wave propagation in periodically laminated composites using space-time least-squares finite element processes in h, p, k mathematical framework in [1, 2, 3, 4] is presented. It is demonstrated that in space-time coupled processes, if one uses time marching procedure to simulate the evolution in which the solution is computed for a space-time strip for one increment of time, then time marched, the space-time local approximations of class C0 and degree p in time yield global differentiability of order p in time for the space-time strip. However, the inter-strip global differentiability in time remains limited by the global differentiability of the local approximations in time. Four different space-time least-squares finite element formulations are presented here and their merits and drawbacks, convergence rates, and their ability to incorporate single material physics as well as bi-material interface macro physics of wave propagation, transmission, reflection and interaction in laminated composites are discussed. Numerical results are presented to illustrate mathematical features of each formulation for wave propagation in a single medium as well as in domains containing bi-material interfaces. Numerical results are also computed for an impact experiment and compared with experimental data.

A simultaneous solution procedure for strong interactions of generalized Newtonian fluids and viscoelastic solids at large strains

AbstractThe paper presents a methodology for numerical analyses of coupled systems exhibiting strong interactions of viscoelastic solids and generalized Newtonian fluids. In the monolithic approach, velocity variables are used for both solid and fluid, and the entire set of model equations is discretized with stabilized space–time finite elements. A viscoelastic material model for finite deformations, which is based on the concept of internal variables, describes the stress‐deformation behaviour of the solid. In the generalized Newtonian approach for the fluid, the viscosity depends on the shear strain rate, leading to common non‐Newtonian fluid models like the power‐law. The consideration of non‐linear constitutive equations for solid and fluid documents the capability of the monolithic space–time finite element formulation to deal with complex material models. The methodology is applied to fluid‐conveying cantilevered pipes in order to determine the influence of material non‐linearities on stability characteristics of coupled systems. Copyright © 2005 John Wiley & Sons, Ltd.

Unstructured Finite Element Method for Transient Heat Conduction of Moving Heat Source

We develop a space-time finite element formulation of heat transfer involving a moving heat source so that small time steps can be used in area of large time rates of change of temperature. The weighted residual process will be used to formulate a finite element method in a space and time domain based upon the continuous Galerkin method. A mesh refinement algorithm which will be on adaptively controlling the time step is developed and implemented for one-dimensional moving heat source simulation. A moving heat source will produce steep gradients of the temperature within and near the region of moving source. The space-time domain is divided into time-slabs and the mesh generator produces a triangular mesh that has small elements close to the front of moving source and relatively large elements away from the front

Performance analysis of ventricular assist devices using finite element flow simulation

AbstractNumerical simulation plays an increasingly important role in the design of medical devices in general, and ventricular assist devices in particular. Modelling of flow in blood pumps has the potential to shorten the design cycle and give the designers important insights into causes of blood damage and suboptimal performance. We present a set of flow simulations designed to reproduce previously obtained experimental data relating flow volume generated by the pump to the pressure head imposed between the outflow and inflow cannulas. These performance curves are obtained using a stabilized space–time finite element formulation of the Navier–Stokes equations, with a shear‐slip mesh update used to accommodate the movement of the impeller with respect to non‐axisymmetric housing. Copyright © 2004 John Wiley & Sons, Ltd.

The multi-domain method for computation of the aerodynamics of a parachute crossing the far wake of an aircraft

We present the multi-domain method (MDM) for computation of unsteady flow past a cargo aircraft and around a parachute crossing the aircraft's far wake. The base computational methods used here are the stabilized semi-discrete and space-time finite element formulations developed earlier. In the MDM, the computational domain is divided into an ordered sequence of overlapping subdomains. The flow field computed over Subdomain-1, which contains the aircraft, supplies the inflow boundary conditions for Subdomain-2, which is used for computing the long-wake flow. Subdomain-3 contains the parachute, and moves across Subdomain-2. The boundary conditions for Subdomain-3 are extracted from the flow field computed over Subdomain-2, at locations corresponding to the positions of the boundaries of Subdomain-3 as it crosses Subdomain-2. The computation over Subdomain-1, which contains a complex but fixed object, is based on a general-purpose implementation of the semi-discrete formulation. The computation over Subdomain-2, which contains no objects, is based on a special-purpose implementation that exploits the simplicity of the mesh to increase the computational speed. The computation over Subdomain-3, which contains a complex and moving object, is based on a general-purpose implementation of the space-time formulation. With a numerical example, we show that different methods can be brought together in the context of the MDM to address the computational challenges involved in the aerodynamics of a parachute crossing the far wake of an aircraft.

Stabilized space‐time finite element formulations for free‐surface flows

AbstractAspects of a method for 3D finite element computation of unsteady, incompressible free‐surface flow are presented. The approach is based on the deformable‐spatial‐domain/stabilized space‐time (DSD/SST) finite element formulation, which takes automatically into account the deformation of the elements in response to the motion of the free surface. The free‐surface elevation is governed by a kinematic free‐surface condition, which is also solved with a stabilized formulation. A new governing equation and stabilized formulation is derived for cases where the channel walls are not vertical. The parallel implementation based on MPI message‐passing standard is fully portable, and have been demonstrated to be scalable on a range of architectures. A 3D computation of a flow past a spillway of a dam is shown as an example application. Copyright © 2001 John Wiley & Sons, Ltd.

Methods for 3D computation of fluid–object interactions in spatially periodic flows

We present computational methods for 3D simulation of fluid–object interactions in spatially periodic flows. These methods include a stabilized space-time finite element formulation for incompressible flows with spatial periodicity, automatic mesh generation and update techniques for fluid–object mixtures with spatial periodicity, and parallel implementations. The methods can be applied to uni-periodic (i.e., periodic in one direction), bi-periodic, or tri-periodic flows. The methods are described here in the context of tri-periodic flows with fluid–object interactions, and are applied to the simulation of sedimentation of particles in a fluid. We present several case studies where the results obtained provide notable insight into the behavior of fluid–particle mixtures during sedimentation.

Finite elements in space and time for generalized viscoelastic maxwell model

The generalization of a new numerical approach with simultaneous space–time finite element discretization for viscoelastic problems developed in the papers by Buch et al. (1999) and Idesman et al. (2000) is presented for the case of the generalized viscoelastic Maxwell model. New non-symmetric variational and discretized formulations are derived using the continuous Galerkin method (CGM) and discontinuous Galerkin method (DGM). Viscoelastic behaviour described by the generalized Maxwell model is represented by means of internal variables. It allows to use only differential equations for the constitutive equations instead of integrodifferential ones. The variational formulation reduces to two types of equations with total displacements and internal displacements (internal variables) as unknowns, namely to the equilibrium equation and the evolution equations for the internal displacements which are fulfilled in the weak form. Using continuous test functions in space and time, a continuous space–time finite element formulation is obtained with simultaneous discretization in space and time. Subdividing the total observation time interval into appropriate time slabs and introducing discontinuous trial functions, being continuous within time slabs and allowing jumps across time interfaces, a more general discontinuous finite element formulation is obtained. The difference between these two formulations for one time slab consists in the satisfaction of initial conditions which are fulfilled exactly for the continuous formulation and in a weak form for the discontinuous case.

Numerical Analysis of Space–Time Finite Element Formulations for Miscible Displacements

A finite element formulation is proposed to approximate a nonlinear system of partial differential equations, composed by an elliptic subsystem for the pressure–velocity and a transport equation (convection–diffusion) for the concentration, which models the incompressible miscible displacement of one fluid by another in a rigid porous media. The pressure is approximated by the classical Galerkin method and the velocity is calculated by a post-processing technique. Then, the concentration is obtained by a Galerkin/least-squares space–time (GLS/ST) finite element method. A numerical analysis is developed for the concentration approximation. Then, stability, convergence and numerical results are presented confirming the a priori error estimates.

Continuous and discontinuous Galerkin methods with finite elements in space and time for parallel computing of viscoelastic deformation

New non-symmetric variational and discretized formulations (with space-time finite elements) are proposed for viscoelastic problems based on the continuous Galerkin method (CGM) and discontinuous Galerkin method (DGM). Viscoelastic behaviour is described by the three-parameter Malvern model, which is represented by means of internal variables. It allows to use only differential equations for the constitutive equations instead of integrodifferential ones. The variational formulation reduces to two types of equations for total displacements and internal displacements (internal variables), namely to the equilibrium equation and the evolution equation for the internal displacements, which are fulfilled in the weak form. Using continuous trial functions, a continuous space-time finite element formulation is obtained with simultaneous discretization in space and time. Subdividing the total observation time interval into time slabs and introducing discontinuous trial functions, being continuous within time slabs and allowing jumps across interfaces, a more general discontinuous finite element formulation is obtained. The difference between these two formulations for one time slab consists in the satisfaction of initial conditions which are fulfilled exactly for the continuous formulation and in a weak form for the discontinuous case. The proposed approach has some very attractive advantages with respect to semidiscretization methods, regarding the possibility of adaptive space-time refinements and parallel processing on MIMD-parallel computers. The considered numerical examples show the effectiveness of simultaneous space-time finite element calculations and a high convergence rate for adaptive refinement. Numerical efficiency is an advantage of DGM in comparison with CGM for discontinuously changing (e.g. piecewise constant) boundary conditions in time.