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.     

Solving linear and non‐linear space–time fractional reaction–diffusion equations by the Adomian decomposition method

In this paper, we consider linear and non‐linear space–time fractional reaction–diffusion equations (STFRDE) on a finite domain. The equations are obtained from standard reaction–diffusion equations by replacing a second‐order space derivative by a fractional derivative of order β∈(1, 2], and a first‐order time derivative by a fractional derivative of order α∈(0, 1]. We use the Adomian decomposition method to construct explicit solutions of the linear and non‐linear STFRDE. Finally, some examples are given. Copyright © 2007 John Wiley & Sons, Ltd.     

Arbitrary discontinuities in space–time finite elements by level sets and X‐FEM

An enriched finite element method with arbitrary discontinuities in space–time is presented. The discontinuities are treated by the extended finite element method (X‐FEM), which uses a local partition of unity enrichment to introduce discontinuities along a moving hyper‐surface which is described by level sets. A space–time weak form for conservation laws is developed where the Rankine–Hugoniot jump conditions are natural conditions of the weak form. The method is illustrated in the solution of first order hyperbolic equations and applied to linear first order wave and non‐linear Burgers' equations. By capturing the discontinuity in time as well as space, results are improved over capturing the discontinuity in space alone and the method is remarkably accurate. Implications to standard semi‐discretization X‐FEM formulations are also discussed. Copyright © 2004 John Wiley & Sons, Ltd.     

Energy–momentum consistent finite element discretization of dynamic finite viscoelasticity

This paper is concerned with energy–momentum consistent time discretizations of dynamic finite viscoelasticity. Energy consistency means that the total energy is conserved or dissipated by the fully discretized system in agreement with the laws of thermodynamics. The discretization is energy–momentum consistent if also momentum maps are conserved when group motions are superimposed to deformations. The performed approximation is based on a three‐field formulation, in which the deformation field, the velocity field and a strain‐like viscous internal variable field are treated as independent quantities. The new non‐linear viscous evolution equation satisfies a non‐negative viscous dissipation not only in the continuous case, but also in the fully discretized system. The initial boundary value problem is discretized by using finite elements in space and time. Thereby, the temporal approximation is performed prior to the spatial approximation in order to preserve the stress objectivity for finite rotation increments (incremental objectivity). Although the present approach makes possible to design schemes of arbitrary order, the focus is on finite elements relying on linear Lagrange polynomials for the sake of clearness. The discrete energy–momentum consistency is based on the collocation property and an enhanced second Piola–Kirchhoff stress tensor. The obtained coupled non‐linear algebraic equations are consistently linearized. The corresponding iterative solution procedure is associated with newly proposed convergence criteria, which take the discrete energy consistency into account. The iterative solution procedure is therefore not complicated by different scalings in the independent variables, since the motion of the element is taken into account for solving the viscous evolution equation. Representative numerical simulations with various boundary conditions show the superior stability of the new time‐integration algorithm in comparison with the ordinary midpoint rule. Both the quasi‐rigid deformations during a free flight, and large deformations arising in a dynamic tensile test are considered. Copyright © 2009 John Wiley & Sons, Ltd.     

Robust and provably second‐order explicit–explicit and implicit–explicit staggered time‐integrators for highly non‐linear compressible fluid–structure interaction problems

An explicit–explicit staggered time‐integration algorithm and an implicit–explicit counterpart are presented for the solution of non‐linear transient fluid–structure interaction problems in the Arbitrary Lagrangian–Eulerian (ALE) setting. In the explicit–explicit case where the usually desirable simultaneous updating of the fluid and structural states is both natural and trivial, staggering is shown to improve numerical stability. Using rigorous ALE extensions of the two‐stage explicit Runge–Kutta and three‐point backward difference methods for the fluid, and in both cases the explicit central difference scheme for the structure, second‐order time‐accuracy is achieved for the coupled explicit–explicit and implicit–explicit fluid–structure time‐integration methods, respectively, via suitable predictors and careful stagings of the computational steps. The robustness of both methods and their proven second‐order time‐accuracy are verified for sample application problems. Their potential for the solution of highly non‐linear fluid–structure interaction problems is demonstrated and validated with the simulation of the dynamic collapse of a cylindrical shell submerged in water. The obtained numerical results demonstrate that, even for fluid–structure applications with strong added mass effects, a carefully designed staggered and subiteration‐free time‐integrator can achieve numerical stability and robustness with respect to the slenderness of the structure, as long as the fluid is justifiably modeled as a compressible medium. Copyright © 2010 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 novel numerical method of high‐order accuracy for flow in unsaturated porous media

We construct finite volume schemes of very high order of accuracy in space and time for solving the nonlinear Richards equation (RE). The general scheme is based on a three‐stage predictor–corrector procedure. First, a high‐order weighted essentially non‐oscillatory (WENO) reconstruction procedure is applied to the cell averages at the current time level to guarantee monotonicity in the presence of steep gradients. Second, the temporal evolution of the WENO reconstruction polynomials is computed in a predictor stage by using a global weak form of the governing equations. A global space–time DG FEM is used to obtain a scheme without the parabolic time‐step restriction caused by the presence of the diffusion term in the RE. The resulting nonlinear algebraic system is solved by a Newton–Krylov method, where the generalized minimal residual method algorithm of Saad and Schulz is used to solve the linear subsystems. Finally, as a third step, the cell averages of the finite volume method are updated using a one‐step scheme, on the basis of the solution calculated previously in the space–time predictor stage. Our scheme is validated against analytical, experimental, and other numerical reference solutions in four test cases. A numerical convergence study performed allows us to show that the proposed novel scheme is high order accurate in space and time. Copyright © 2011 John Wiley & Sons, Ltd.     

A new hybrid velocity integration method applied to elastic wave propagation

We present a novel space–time Galerkin method for solutions of second‐order time‐dependent problems. By introducing the displacement–velocity relationship implicitly, the governing set of equations is reformulated into a first‐order single field problem with the unknowns in the velocity field. The resulting equation is in turn solved by a time‐discontinuous Galerkin approach (Int. J. Numer. Anal. Meth. Geomech. 2006; 30 :1113–1134), in which the continuity between time intervals is weakly enforced by a special upwind flux treatment. After solving the equation for the unknown velocities, the displacement field quantities are computed a posteriori in a post‐processing step. Various numerical examples demonstrate the efficiency and reliability of the proposed method. Convergence studies with respect to the h‐ and p‐refinement and different discretization techniques are given. Copyright © 2007 John Wiley & Sons, Ltd.     

Space–time approach to numerical analysis of a string with a moving mass

Inertial loading of strings, beams and plates by mass travelling with near‐critical velocity has been a long debate. Typically, a moving mass is replaced by an equivalent force or an oscillator (with ‘rigid’ spring) that is in permanent contact with the structure. Such an approach leads to iterative solutions or imposition of artificial constraints. In both cases, rigid constraints result in serious computational problems. A direct mass matrix modification method frequently implemented in the finite element approach gave reasonable results only in the range of relatively low velocities. In this paper we present the space–time approach to the problem. The interaction of the moving mass/supporting structure is described in a local coordinate system of the space–time finite element domain. The resulting characteristic matrices include inertia, Coriolis and centrifugal forces. A simple modification of matrices in the discrete equations of motion allows us to gain accurate analysis of a wide range of velocities, up to the velocity of the wave speed. Numerical examples prove the simplicity and efficiency of the method. The presented approach can be easily implemented in the classic finite element algorithms. Copyright © 2008 John Wiley & Sons, Ltd.     

A space–time discontinuous Galerkin method for the solution of the wave equation in the time domain

In recent years, the focus of research in the field of computational acoustics has shifted to the medium frequency regime and multiscale wave propagation. This has led to the development of new concepts including the discontinuous enrichment method. Its basic principle is the incorporation of features of the governing partial differential equation in the approximation. In this contribution, this concept is adapted for the simulation of transient problems governed by the wave equation. We present a space–time discontinuous Galerkin method with Lagrange multipliers, where the shape approximation in space and time is based on solutions of the homogeneous wave equation. The use of hierarchical wave‐like basis functions is enabled by means of a variational formulation that allows for discontinuities in both the spatial and the temporal discretizations. Numerical examples in one space dimension demonstrate the outstanding performance of the proposed method compared with conventional space–time finite element methods. Copyright © 2008 John Wiley & Sons, Ltd.     

A hybrid discontinuous in space and time Galerkin method for wave propagation problems

Medium‐frequency regime and multi‐scale wave propagation problems have been a subject of active research in computational acoustics recently. New techniques have attempted to overcome the limitations of existing discretization methods that tend to suffer from dispersion. One such technique, the discontinuous enrichment method, incorporates features of the governing partial differential equation in the approximation, in particular, the solutions of the homogeneous form of the equation. Here, based on this concept and by extension of a conventional space–time finite element method, a hybrid discontinuous Galerkin method (DGM) for the numerical solution of transient problems governed by the wave equation in two and three spatial dimensions is described. The discontinuous formulation in both space and time enables the use of solutions to the homogeneous wave equation in the approximation. In this contribution, within each finite element, the solutions in the form of polynomial waves are employed. The continuity of these polynomial waves is weakly enforced through suitably chosen Lagrange multipliers. Results for two‐dimensional and three‐dimensional problems, in both low‐frequency and medium‐frequency regimes, show that the proposed DGM outperforms the conventional space–time finite element method. Copyright © 2014 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.     

Space–time meshfree collocation method: Methodology and application to initial‐boundary value problems

A novel space–time meshfree collocation method (STMCM) for solving systems of non‐linear ordinary and partial differential equations by a consistent discretization in both space and time is proposed as an alternative to established mesh‐based methods. The STMCM belongs to the class of truly meshfree methods, i.e. the methods that do not have any underlying mesh, but work on a set of nodes only without any a priori node‐to‐node connectivity. Instead, the neighbouring information is established on‐the‐fly. The STMCM is constructed using the Interpolating Moving Least‐squares technique, which allows a simplified implementation of boundary conditions due to fulfillment of the Kronecker delta property by the kernel functions, which is not the case for the major part of other meshfree methods. The method is validated by several examples ranging from interpolation problems to the solution of PDEs, whereas the STMCM solutions are compared with either analytical or reference ones. Copyright © 2009 John Wiley & Sons, Ltd.     

Time‐domain soil–structure interaction analysis in two‐dimensional medium based on analytical frequency‐dependent infinite elements

A direct method for soil–structure interaction analysis in two‐dimensional medium is presented in time domain, which is based on the transformation of the analytical frequency‐dependent dynamic stiffness matrix. The present dynamic stiffness matrix for the far‐field region is constructed by assembling stiffness matrices of the analytical frequency‐dependent dynamic infinite elements, so that the equation of motion can be analytically transformed into the time‐domain equation. An efficient procedure is devised to evaluate the dynamic responses in time domain. Verification of the present formulation is carried out by comparing the compliances for a strip foundation on a homogeneous and layered half‐spaces with those obtained by other methods. Numerical analyses are also carried out for the transient responses of an elastic block and tunnel in a homogeneous and a layered half‐space. The comparisons with those by other approaches indicate that the proposed time‐domain method for soil–structure interaction analysis gives good solutions. Copyright © 2000 John Wiley & Sons, Ltd.     

Segregated Runge–Kutta methods for the incompressible Navier–Stokes equations

In this work, we propose Runge–Kutta time integration schemes for the incompressible Navier–Stokes equations with two salient properties. First, velocity and pressure computations are segregated at the time integration level, without the need to perform additional fractional step techniques that spoil high orders of accuracy. Second, the proposed methods keep the same order of accuracy for both velocities and pressures. The segregated Runge–Kutta methods are motivated as an implicit–explicit Runge–Kutta time integration of the projected Navier–Stokes system onto the discrete divergence‐free space, and its re‐statement in a velocity–pressure setting using a discrete pressure Poisson equation. We have analysed the preservation of the discrete divergence constraint for segregated Runge–Kutta methods and their relation (in their fully explicit version) with existing half‐explicit methods. We have performed a detailed numerical experimentation for a wide set of schemes (from first to third order), including implicit and IMEX integration of viscous and convective terms, for incompressible laminar and turbulent flows. Further, segregated Runge–Kutta schemes with adaptive time stepping are proposed. Copyright © 2015 John Wiley & Sons, Ltd.     

Extended space–time finite elements for landslide dynamics

The paper introduces a methodology for numerical simulation of landslides experiencing considerable deformations and topological changes. Within an interface capturing approach, all interfaces are implicitly described by specifically defined level‐set functions allowing arbitrarily evolving complex topologies. The transient interface evolution is obtained by solving the level‐set equation driven by the physical velocity field for all three level‐set functions in a block Jacobi approach. The three boundary‐coupled fluid‐like continua involved are modeled as incompressible, governed by a generalized non‐Newtonian material law taking into account capillary pressure at moving fluid–fluid interfaces. The weighted residual formulation of the level‐set equations and the fluid equations is discretized with finite elements in space and time using velocity and pressure as unknown variables. Non‐smooth solution characteristics are represented by enriched approximations to fluid velocity (weak discontinuity) and fluid pressure (strong discontinuity). Special attention is given to the construction of enriched approximations for elements containing evolving triple junctions. Numerical examples of three‐fluid tank sloshing and air–water‐liquefied soil landslides demonstrate the potential and applicability of the method in geotechnical investigations. Copyright © 2012 John Wiley & Sons, Ltd.     

On a high‐order Newton linearization method for solving the incompressible Navier–Stokes equations

The present study aims to accelerate the non‐linear convergence to incompressible Navier–Stokes solution by developing a high‐order Newton linearization method in non‐staggered grids. For the sake of accuracy, the linearized convection–diffusion–reaction finite‐difference equation is solved line‐by‐line using the nodally exact one‐dimensional scheme. The matrix size is reduced and, at the same time, the CPU time is considerably saved owing to the reduction of stencil points. This Newton linearization method is computationally efficient and is demonstrated to outperform the classical Newton method through computational exercises. Copyright © 2005 John Wiley & Sons, Ltd.     

On time integration in the XFEM

The extended finite element method (XFEM) is often used in applications that involve moving interfaces. Examples are the propagation of cracks or the movement of interfaces in two‐phase problems. This work focuses on time integration in the XFEM. The performance of the discontinuous Galerkin method in time (space–time finite elements (FEs)) and time‐stepping schemes are analyzed by convergence studies for different model problems. It is shown that space–time FE achieve optimal convergence rates. Special care is required for time stepping in the XFEM due to the time dependence of the enrichment functions. In each time step, the enrichment functions have to be evaluated at different time levels. This has important consequences in the quadrature used for the integration of the weak form. A time‐stepping scheme that leads to optimal or only slightly sub‐optimal convergence rates is systematically constructed in this work. Copyright © 2009 John Wiley & Sons, Ltd.     

A combined space–time extended finite element method

The Newmark method for the numerical integration of second order equations has been extensively used and studied along the past fifty years for structural dynamics and various fields of mechanical engineering. Easy implementation and nice properties of this method and its derivatives for linear problems are appreciated but the main drawback is the treatment of discontinuities. Zienkiewicz proposed an approach using finite element concept in time, which allows a new look at the Newmark method. The idea of this paper is to propose, thanks to this approach, the use of a time partition of the unity method denoted Time Extended Finite Element Method (TX‐FEM) for improved numerical simulations of time discontinuities. An enriched basis of shape functions in time is used to capture with a good accuracy the non‐polynomial part of the solution. This formulation allows a suitable form of the time‐stepping formulae to study stability and energy conservation. The case of an enrichment with the Heaviside function is developed and can be seen as an alternative approach to time discontinuous Galerkin method (T‐DGM), stability and accuracy properties of which can be derived from those of the TX‐FEM. Then Space and Time X‐FEM (STX‐FEM) are combined to obtain a unified space–time discretization. This combined STX‐FEM appears to be a suitable technique for space–time discontinuous problems like dynamic crack propagation or other applications involving moving discontinuities. Copyright © 2005 John Wiley & Sons, Ltd.     

An adaptive fixed mesh method for modelling and simulating of unsteady two‐phase flows with sharp physical interfaces

We present in this paper a new computational method for simulation of two‐phase flow problems with moving boundaries and sharp physical interfaces. An adaptive interface‐capturing technique (ICT) of the Eulerian type is developed for capturing the motion of the interfaces (free surfaces) in an unsteady flow state. The adaptive method is mainly based on the relative boundary conditions of the zero pressure head, at which the interface is corresponding to a free surface boundary. The definition of the free surface boundary condition is used as a marker for identifying the position of the interface (free surface) in the two‐phase flow problems. An initial‐value‐problem (IVP) partial differential equation (PDE) is derived from the dynamic conditions of the interface, and it is designed to govern the motion of the interface in time. In this adaptive technique, the Navier–Stokes equations written for two incompressible fluids together with the IVP are solved numerically over the flow domain. An adaptive mass conservation algorithm is constructed to govern the continuum of the fluid. The finite element method (FEM) is used for the spatial discretization and a fully coupled implicit time integration method is applied for the advancement in time. FE‐stabilization techniques are added to the standard formulation of the discretization, which possess good stability and accuracy properties for the numerical solution. The adaptive technique is tested in simulation of some numerical examples. With the test problems presented here, we demonstrated that the adaptive technique is a simple tool for modelling and computation of complex motion of sharp physical interfaces in convection–advection‐dominated flow problems. We also demonstrated that the IVP and the evolution of the interface function are coupled explicitly and implicitly to the system of the computed unknowns in the flow domain. Copyright © 2005 John Wiley & Sons, Ltd.