Stabilized finite element methods for vertically averaged multiphase flow for carbon sequestration

A computationally efficient numerical model that describes carbon sequestration in deep saline aquifers is presented. The model is based on the multiphase flow and vertically averaged mass balance equations, requiring the solution of two partial differential equations – a pressure equation and a saturation equation. The saturation equation is a nonlinear advective equation for which the application of Galerkin finite element method (FEM) can lead to non‐physical oscillations in the solution. In this article, we extend three stabilized FEM formulations, which were developed for uncoupled systems, to the governing nonlinear coupled PDEs. The methods developed are based on the streamline upwind, the streamline upwind/Petrov–Galerkin and the least squares FEM. Two sequential solution schemes are developed: a single step and a predictor–corrector. The range of Courant numbers yielding smooth and oscillation‐free solutions is investigated for each method. The useful range of Courant numbers found depends upon both the sequential scheme (single step vs predictor–corrector) and also the time integration method used (forward Euler, backward Euler or Crank–Nicolson). For complex problems such as when two plumes meet, only the SU stabilization with an amplified stabilization parameter gives satisfactory results when large time steps are used. Copyright © 2016 John Wiley & Sons, Ltd.     

A Raviart–Thomas mixed finite element scheme for the two‐dimensional three‐temperature heat conduction problems

For the two‐dimensional three‐temperature radiative heat conduction problem appearing in the inertial confinement numerical stimulations, we choose the Freezing coefficient method to linearize the nonlinear equations, and initially apply the well‐known mixed finite element scheme with the lowest order Raviart–Thomas element associated with the triangulation to the linearized equations, and obtain the convergence with one order with respect to the space direction for the temperature and flux function approximations, and design a simple but efficient algorithm for the discrete system. Three numerical examples are displayed. The former two verify theoretical results and show the super‐convergence for temperature and flux functions at the barycenter of the element, which is helpful for solving the radiative heat conduction problems. The third validates the robustness of this scheme with small energy conservative error and one order convergence for the time discretization. Copyright © 2016 John Wiley & Sons, Ltd.     

Computational study of shock‐wave interaction with solid obstacles using immersed boundary methods

In this study, an immersed boundary (IB) method based on a direct forcing is coupled with a high‐order weighted‐essentially non‐oscillatory (WENO) scheme to simulate fluid–solid interaction (FSI) problems with complex geometries. The IB is a general simulation method for FSI, whereas the WENO is an efficient scheme for fluid flow simulations and shock waves, and both of them work on regular cartesian grids. The effectiveness and the accuracy of the coupled scheme are first analyzed on well‐documented supersonic test problems for a wide range of Mach numbers. The results are in good agreement with both analytical and experimental data. A comprehensive analysis of the interaction of the moving shock through an array of cylinder matrix is then conducted by varying the number of cylinders in the matrix block while keeping the same opening passage. The relaxation length between two adjacent columns of cylinders is kept identical to study uniquely the effect of surface‐to‐volume ratio of the obstacle matrix. It is shown that the configuration with higher surface‐to‐volume ratio produces more post‐shock flow instabilities downstream of the matrix block. The complex shock/shock and shock/vortex interactions are well resolved by the present computation. It is being observed that after the passage of the shock through the cylinder matrix, eddies of different length scales are generated, but the later stage of shock/vortex and shocklet/vortexlet interactions are different for the two cases. The analysis of the PSD of the total kinetic energy globally conforms to Richardson's inviscid cascade. An intermittent peaked PDF of downstream instantaneous vorticity field is obtained in the limit of Re → ∞ . The baroclinic production of vorticity is found to be feeble as previously founded by Sun and Takayama (J. Fluid Mech. 2003; 478 :237–256). Copyright © 2011 John Wiley & Sons, Ltd.     

Magnetic resonance scan‐time reduction using echo prediction

A linear prediction (LP) filter derived from a complete echo with zero‐phase encoding amplitude is used for recovering anatomical details from a partially acquired echo sequence. The LP filter is shown to reconstruct missing k‐space phase and amplitude information, with errors sufficiently low so as to provide image reconstruction with a contrast‐to‐noise ratio (CNR) ≥ 3. For volume imaging using multislice acquisition, the partial‐echo sequence enables more number of slices to be acquired for a given repetition time period T_R. For such sequences, separate predictors are used for reconstruction of missing k‐space data corresponding to each individual slice in the volume. The proposed filtering scheme is shown to achieve results comparable to other partial k‐space approaches such as singularity function analysis (SFA), when the noise content is less than about 0.4%. For higher noise levels, this technique is recommended as a preprocessing step for SFA to track the singularity locations more accurately. © 2013 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 23, 1–8, 2013     

Model‐based reconstruction for T1 mapping using single‐shot inversion‐recovery radial FLASH

Quantitative parameter mapping in MRI is typically performed as a two‐step procedure where serial imaging is followed by pixelwise model fitting. In contrast, model‐based reconstructions directly reconstruct parameter maps from raw data without explicit image reconstruction. Here, we propose a method that determines T1 maps directly from multi‐channel raw data as obtained by a single‐shot inversion‐recovery radial FLASH acquisition with a Golden Angle view order. Joint reconstruction of a T1, spin‐density and flip‐angle map is formulated as a nonlinear inverse problem and solved by the iteratively regularized Gauss‐Newton method. Coil sensitivity profiles are determined from the same data in a preparatory step of the reconstruction. Validations included numerical simulations, in vitro MRI studies of an experimental T1 phantom, and in vivo studies of brain and abdomen of healthy subjects at a field strength of 3 T. The results obtained for a numerical and experimental phantom demonstrated excellent accuracy and precision of model‐based T1 mapping. In vivo studies allowed for high‐resolution T1 mapping of human brain (0.5–0.75 mm in‐plane, 4 mm section thickness) and liver (1.0 mm, 5 mm section) within 3.6–5 s. In conclusion, the proposed method for model‐based T1 mapping may become an alternative to two‐step techniques, which rely on model fitting after serial image reconstruction. More extensive clinical trials now require accelerated computation and online implementation of the algorithm. © 2016 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 26, 254–263, 2016     

A finite volume method with a modified ENO scheme using a Hermite interpolation to solve advection diffusion equations

In this paper we have developed a finite volume ENO scheme, third‐order accurate, based on cell averages and a TVD Runge–Kutta time discretization to solve advection–diffusion equations in a two‐dimensional spatial domain. We have designed a special interpolating polynomial based on a modified ENO scheme and a Hermite procedure which avoids the excessive smearing in regions with sharpconcentration fronts and the overcompression effects produced by the modified ENO technique. Thesemodifications do not affect the non‐oscillatory philosophy since we compare divided differences inthe modified ENO scheme and in the evaluation of the Hermite polynomial derivatives. Numericalresults compare favourably with their respective analytical solutions. Copyright © 2001 John Wiley & Sons, Ltd.     

High‐order space‐time finite element schemes for acoustic and viscodynamic wave equations with temporal decoupling

We revisit a method originally introduced by Werder et al. (in Comput. Methods Appl. Mech. Engrg., 190:6685–6708, 2001) for temporally discontinuous Galerkin FEMs applied to a parabolic partial differential equation. In that approach, block systems arise because of the coupling of the spatial systems through inner products of the temporal basis functions. If the spatial finite element space is of dimension D and polynomials of degree r are used in time, the block system has dimension (r + 1)D and is usually regarded as being too large when r > 1. Werder et al. found that the space‐time coupling matrices are diagonalizable over for r ?100, and this means that the time‐coupled computations within a time step can actually be decoupled. By using either continuous Galerkin or spectral element methods in space, we apply this DG‐in‐time methodology, for the first time, to second‐order wave equations including elastodynamics with and without Kelvin–Voigt and Maxwell–Zener viscoelasticity. An example set of numerical results is given to demonstrate the favourable effect on error and computational work of the moderately high‐order (up to degree 7) temporal and spatio‐temporal approximations, and we also touch on an application of this method to an ambitious problem related to the diagnosis of coronary artery disease. Copyright © 2014 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd.     

Dynamic modelling of the flat 2‐D crack by a semi‐analytic BIEM scheme

We present an efficient numerical method for solving indirect boundary integral equations that describe the dynamics of a flat two‐dimensional (2‐D) crack in all modes of fracture. The method is based on a piecewise‐constant interpolation, both in space and time, of the slip‐rate function, by which the original equation is reduced to a discrete convolution, in space and time, of the slip‐rate and a set of analytically obtained coefficients. If the time‐step interval is set sufficiently small with respect to the spatial grid size, the discrete equations decouple and can be solved explicitly. This semi‐analytic scheme can be extended to the calculation of the wave field off the crack plane. A necessary condition for the numerical stability of this scheme is investigated by way of an exhaustive set of trial runs for a kinematic problem. For the case investigated, our scheme is very stable for a fairly wide range of control parameters in modes III and I, whereas, in mode II, it is unstable except for some limited ranges of the parameters. The use of Peirce and Siebrits' ε‐scheme in time collocation is found helpful in stabilizing the numerical calculation. Our scheme also allows for variable time steps. Copyright © 2001 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.     

Robust nonlinear control of robotic manipulators

Abstract

A robust nonlinear control strategy is proposed to deal with the control problem of robotic manipulators with second order nonlinear actuator dynamics. The control scheme is composed of two stages: the nominal dynamics stage and the perturbed dynamics stage. The control at the nominal dynamics stage and the perturbed dynamics stage. The control at the nominal dynamics stage is aimed at exact linearization and input/output decoupling of the nonlinear actuator‐manipulator system in the task space by nonlinear feedback and nonlinear state space diffeomorphic transformations. The resulting closed‐loop nominal system is capable of precise trajectory following in a desired second‐order linear behavior. To compensate uncertainties in a practical situation, an optimal error correcting compensator is designed to achieve some robustness at the perturbed dynamics stage. Simulation study of a cylindrical robot is given to illustrate the effectiveness of the proposed scheme.     

An adaptive multiresolution method for parabolic PDEs with time‐step control

We present an efficient adaptive numerical scheme for parabolic partial differential equations based on a finite volume (FV) discretization with explicit time discretization using embedded Runge–Kutta (RK) schemes. A multiresolution strategy allows local grid refinement while controlling the approximation error in space. The costly fluxes are evaluated on the adaptive grid only. Compact RK methods of second and third order are then used to choose automatically the new time step while controlling the approximation error in time. Non‐admissible choices of the time step are avoided by limiting its variation. The implementation of the multiresolution representation uses a dynamic tree data structure, which allows memory compression and CPU time reduction. This new numerical scheme is validated using different classical test problems in one, two and three space dimensions. The gain in memory and CPU time with respect to the FV scheme on a regular grid is reported, which demonstrates the efficiency of the new method. Copyright © 2008 John Wiley & Sons, Ltd.     

A contact algorithm for the Signorini problem using space–time finite elements

The most common approach in the finite‐element modelling of continuum systems over space and time is to employ the finite‐element discretization over the spatial domain to reduce the problem to a system of ordinary differential equations in time. The desired time integration scheme can then be used to step across the so‐called time slabs, mesh configurations in which every element shares the same degree of time refinement. These techniques may become inefficient when the nature of the initial boundary value problem is such that a high degree of time refinement is required only in specific spatial regions of the mesh. Ideally one would be able to increase the time refinement only in those necessary regions. We achieve this flexibility by employing space–time elements with independent interpolation functions in both space and time. Our method is used to examine the classic contact problem of Signorini and allows us to increase the time refinement only in the spatial region adjacent to the contact interface. We also develop an interface‐tracking algorithm that tracks the contact boundary through the space–time mesh and compare our results with those of Hertz contact theory. Copyright 2004 John Wiley & Sons, Ltd.     

A new mass lumping scheme for the mixed hybrid finite element method

Diffusion‐type partial differential equation is a common mathematical model in physics. Solved by mixed finite elements, it leads to a system matrix which is not always an M‐matrix. Therefore, the numerical solution may exhibit unphysical results due to oscillations. The criterion necessary to obtain an M‐matrix is discussed in details for triangular, rectangular and tetrahedral elements. It is shown that the system matrix is never an M‐matrix for rectangular elements and can be an M‐matrix for triangular an tetrahedral elements if criteria on the element's shape and on the time step length are fulfilled. A new mass lumping scheme is developed which leads to a less restrictive criterion: the discretization must be weakly acute (all angles less than π/2) and there is no constraint on the time step length. The lumped formulation of mixed hybrid finite element can be applied not only to triangular meshes but also to more general shape elements in two and three dimensions. Numerical experiments show that, compared to the standard mixed hybrid formulation, the lumping scheme avoids (or strongly reduce) oscillations and does not create additional numerical errors. Copyright © 2006 John Wiley & Sons, Ltd.     

A new predictor–corrector approach for the numerical integration of coupled electromechanical equations

In this paper, a new approach for the numerical solution of coupled electromechanical problems is presented. The structure of the considered problem consists of the low‐frequency integral formulation of the Maxwell equations coupled with Newton–Euler rigid‐body dynamic equations. Two different integration schemes based on the predictor–corrector approach are presented and discussed. In the first method, the electrical equation is integrated with an implicit single‐step time marching algorithm, while the mechanical dynamics is studied by a predictor–corrector scheme. The predictor uses the forward Euler method, while the corrector is based on the trapezoidal rule. The second method is based on the use of two interleaved predictor–corrector schemes: one for the electrical equations and the other for the mechanical ones. Both the presented methods have been validated by comparison with experimental data (when available) and with results obtained by other numerical formulations; in problems characterized by low speeds, both schemes produce accurate results, with similar computation times. When high speeds are involved, the first scheme needs shorter time steps (i.e., longer computation times) in order to achieve the same accuracy of the second one. A brief discussion on extending the algorithm for simulating deformable bodies is also presented. An example of application to a two‐degree‐of‐freedom levitating device based on permanent magnets is finally reported. Copyright © 2015 John Wiley & Sons, Ltd.     

Explicit finite element artificial boundary scheme for transient scalar waves in two‐dimensional unbounded waveguide

To simulate the transient scalar wave propagation in a two‐dimensional unbounded waveguide, an explicit finite element artificial boundary scheme is proposed, which couples the standard dynamic finite element method for complex near field and a high‐order accurate artificial boundary condition (ABC) for simple far field. An exact dynamic‐stiffness ABC that is global in space and time is constructed. A temporal localization method is developed, which consists of the rational function approximation in the frequency domain and the auxiliary variable realization into time domain. This method is applied to the dynamic‐stiffness ABC to result in a high‐order accurate ABC that is local in time but global in space. By discretizing the high‐order accurate ABC along artificial boundary and coupling the result with the standard lumped‐mass finite element equation of near field, a coupled dynamic equation is obtained, which is a symmetric system of purely second‐order ordinary differential equations in time with the diagonal mass and non‐diagonal damping matrices. A new explicit time integration algorithm in structural dynamics is used to solve this equation. Numerical examples are given to demonstrate the effectiveness of the proposed scheme. Copyright © 2011 John Wiley & Sons, Ltd.     

Cyclic steady states of nonlinear electro‐mechanical devices excited at resonance

We present an efficient numerical method to solve for cyclic steady states of nonlinear electro‐mechanical devices excited at resonance. Many electro‐mechanical systems are designed to operate at resonance, where the ramp‐up simulation to steady state is computationally very expensive – especially when low damping is present. The proposed method relies on a Newton–Krylov shooting scheme for the direct calculation of the cyclic steady state, as opposed to a naïve transient time‐stepping from zero initial conditions. We use a recently developed high‐order Eulerian–Lagrangian finite element method in combination with an energy‐preserving dynamic contact algorithm in order to solve the coupled electro‐mechanical boundary value problem. The nonlinear coupled equations are evolved by means of an operator split of the mechanical and electrical problem with an explicit as well as implicit approach. The presented benchmark examples include the first three fundamental modes of a vibrating nanotube, as well as a micro‐electro‐mechanical disk resonator in dynamic steady contact. For the examples discussed, we observe power law computational speed‐ups of the form S  = 0.6·ξ  ^? 0.8, where ξ is the linear damping ratio of the corresponding resonance frequency. Copyright © 2016 John Wiley & Sons, Ltd.     

On numerical analyses in the presence of unstable saturated porous materials

The numerical analysis of the dynamic evolution problem concerning an elastic–plastic saturated porous media in the presence of softening (or non‐associativity) is considered in the framework of the Biot formulation extended to take into account plastic phenomena. The finite step boundary value problem, obtained by discretization in time of the continuous initial boundary value problem, is studied and the issue of its ill‐posedness is particularly addressed. The conditions for the loss of ellipticity are established for the linearized problem solved at each iteration when using the Newton–Raphson scheme. In particular, the roles of the algorithmic properties on this loss of ellipticity are derived in detail. The integration scheme of the balance of mass equation plays a major role and it is shown that the fluid flow (Darcy's law) does indeed introduce a length scale but in addition to being dependent on the integration time step, it is found to be insufficient for regularization. To illustrate and corroborate the obtained results, a one‐dimensional example (exhibiting all the features of the three‐dimensional situation) is considered and the corresponding linearized finite step problem is solved in closed form. Copyright © 2002 John Wiley & Sons, Ltd.     

High‐order global–regional model interaction: Extension of Carpenter's scheme

A two‐dimensional global–regional model interaction problem for linear time‐dependent waves is considered. The setup, which is sometimes called ‘one‐way nesting,’ arises in numerical weather prediction as well as in other fields concerning waves in very large domains. It involves the interaction of a coarse global model and a fine limited‐area (regional) model through an ‘open boundary.’ The multiscale nature of this general problem is described. The Carpenter scheme, originally proposed in a note by K. M. Carpenter in 1982 for this type of problem, is then revisited, in the context of the linear scalar wave equation. The original Carpenter scheme is based on the Sommerfeld radiation operator and thus is associated with low‐order accuracy. By replacing the Sommerfeld operator with the high‐order Hagstrom–Warburton absorbing operator, a modified Carpenter open‐boundary condition emerges, which possesses high‐order accuracy. This boundary condition is incorporated in a computational scheme, which uses finite element discretization in space and Newmark time‐stepping. Error analysis and numerical tests for wave guides demonstrate the performance of the modified scheme for combinations of incoming and outgoing waves. Copyright © 2008 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.     

The hp‐d‐adaptive finite cell method for geometrically nonlinear problems of solid mechanics

The finite cell method (FCM) combines the fictitious domain approach with the p‐version of the finite element method and adaptive integration. For problems of linear elasticity, it offers high convergence rates and simple mesh generation, irrespective of the geometric complexity involved. This article presents the integration of the FCM into the framework of nonlinear finite element technology. However, the penalty parameter of the fictitious domain is restricted to a few orders of magnitude in order to maintain local uniqueness of the deformation map. As a consequence of the weak penalization, nonlinear strain measures provoke excessive stress oscillations in the cells cut by geometric boundaries, leading to a low algebraic rate of convergence. Therefore, the FCM approach is complemented by a local overlay of linear hierarchical basis functions in the sense of the hp‐d method, which synergetically uses the h‐adaptivity of the integration scheme. Numerical experiments show that the hp‐d overlay effectively reduces oscillations and permits stronger penalization of the fictitious domain by stabilizing the deformation map. The hp‐d‐adaptive FCM is thus able to restore high convergence rates for the geometrically nonlinear case, while preserving the easy meshing property of the original FCM. Accuracy and performance of the present scheme are demonstrated by several benchmark problems in one, two, and three dimensions and the nonlinear simulation of a complex foam sample. Copyright © 2011 John Wiley & Sons, Ltd.