Numerical solutions of systems with (p,δ)‐structure using local discontinuous Galerkin finite element methods

In this paper, we present LDG methods for systems with (p,δ)‐structure. The unknown gradient and the nonlinear diffusivity function are introduced as auxiliary variables and the original (p,δ) system is decomposed into a first‐order system. Every equation of the produced first‐order system is discretized in the discontinuous Galerkin framework, where two different nonlinear viscous numerical fluxes are implemented. An a priori bound for a simplified problem is derived. The ODE system resulting from the LDG discretization is solved by diagonal implicit Runge–Kutta methods. The nonlinear system of algebraic equations with unknowns the intermediate solutions of the Runge–Kutta cycle is solved using Newton and Picard iterative methodology. The performance of the two nonlinear solvers is compared with simple test problems. Numerical tests concerning problems with exact solutions are performed in order to validate the theoretical spatial accuracy of the proposed method. Further, more realistic numerical examples are solved in domains with non‐smooth boundary to test the efficiency of the method. Copyright © 2014 John Wiley & Sons, Ltd.     

An Euler system source term that develops prototype Z‐pinch implosions intended for the evaluation of shock‐hydro methods

In this paper, a phenomenological model for a magnetic drive source term for the momentum and total energy equations of the Euler system is described. This body force term is designed to produce a Z‐pinch like implosion that can be used in the development and evaluation of shock‐hydrodynamics algorithms that are intended to be used in Z‐pinch simulations. The model uses a J × B Lorentz force, motivated by a 0‐D analysis of a thin shell (or liner implosion), as a source term in the equations and allows for arbitrary current drives to be simulated. An extension that would include the multi‐physics aspects of a proposed combined radiation hydrodynamics (rad‐hydro) capability is also discussed. The specific class of prototype problems that are developed is intended to illustrate aspects of liner implosions into a near vacuum and with idealized pre‐fill plasma effects. In this work, a high‐resolution flux‐corrected‐transport method implemented on structured overlapping meshes is used to demonstrate the application of such a model to these idealized shock‐hydrodynamic studies. The presented results include an asymptotic solution based on a limiting‐case thin‐shell analytical approximation in both (x, y) and (r, z). Additionally, a set of more realistic implosion problems that include density profiles approximating plasma pre‐fill and a set of perturbed liner geometries that excite a hydro‐magnetic like Rayleigh–Taylor instability in the implosion dynamics are demonstrated. Finally, as a demonstration of including and evaluating multiphysics effects in the Euler system, a simple radiation model is included and self‐convergence results for two types of (r, z) implosions are presented. Copyright © 2008 John Wiley & Sons, Ltd.     

A semi‐implicit scheme for 3D free surface flows with high‐order velocity reconstruction on unstructured Voronoi meshes

In this paper, we present a computationally efficient semi‐implicit scheme for the simulation of three‐dimensional hydrostatic free surface flow problems on staggered unstructured Voronoi meshes. For each polygonal control volume, the pressure is defined in the cell center, whereas the discrete velocity field is given by the normal velocity component at the cell faces. A piecewise high‐order polynomial vector velocity field is then reconstructed from the scalar normal velocities at the cell faces by using a new high‐order constrained least‐squares reconstruction operator. The reconstructed high‐order piecewise polynomial velocity field is used for trajectory integration in a semi‐Lagrangian approach to discretize the nonlinear convective terms in the governing PDE. For that purpose, a high‐order Taylor method is used as ODE integrator. The resulting semi‐implicit algorithm is extensively validated on a large set of different academic test problems with exact analytical solution and is finally applied to a real‐world engineering problem consisting of a curved channel upstream of two micro‐turbines of a hydroelectric power plant. For this realistic case, some experimental reference data are available from field measurements. Copyright © 2012 John Wiley & Sons, Ltd.     

Application of simple,periodic homogenization techniques to non-linear heat conduction problems in non-periodic,porous media

Often, detailed simulations of heat conduction in complicated, porous media have large runtimes. Then homogenization is a powerful tool to speed up the calculations by preserving accurate solutions at the same time. Unfortunately real structures are generally non-periodic, which requires unpractical, complicated homogenization techniques. We demonstrate in this paper, that the application of simple, periodic techniques to realistic media, that are just close to periodic, gives accurate, approximative solutions. In order to obtain effective parameters for the homogenized heat equation, we have to solve a so called “cell problem”. In contrast to periodic structures it is not trivial to determine a suitable unit cell, which represents a non-periodic media. To overcome this problem, we give a rule of thumb on how to choose a good cell. Finally we demonstrate the efficiency of our method for virtually generated foams as well as real foams and compare these results to periodic structures.     

A finite volume‐based high‐order,Cartesian cut‐cell method for wave propagation

Computational aeroacoustics requires numerical techniques capable of yielding low artificial dispersion and dissipation to preserve the amplitude and the frequency characteristics of the physical processes. Furthermore, for engineering applications, the techniques need to handle irregular geometries associated with realistic configurations. We address these issues by developing an optimized prefactored compact finite volume (OPC‐fv) scheme along with a Cartesian cut‐cell technique. The OPC‐fv scheme seeks to minimize numerical dispersion and dissipation while satisfying the conservation laws. The cut‐cell approach treats irregularly shaped boundaries using divide‐and‐merge procedures for the Cartesian cells while maintaining a desirable level of accuracy. We assess these techniques using several canonical test problems, involving different levels of physical and geometric complexities. Richardson extrapolation is an effective tool for evaluating solutions of no high gradients or discontinuities, and is used to evaluate the performance of the solution technique. It is demonstrated that while the cut‐cell method has a modest effect on the order of accuracy, it is a robust method. The combined OPC‐fv scheme and the Cartesian cut‐cell technique offer good accuracy as well as geometric flexibility. Copyright © 2008 John Wiley & Sons, Ltd.     

On pressure separation algorithms (PSepA) for improving the accuracy of incompressible flow simulations

We investigate a special technique called ‘pressure separation algorithm’ (PSepA) (see Applied Mathematics and Computation 2005; 165 :275–290 for an introduction) that is able to significantly improve the accuracy of incompressible flow simulations for problems with large pressure gradients. In our numerical studies with the computational fluid dynamics package FEATFLOW ( www.featflow.de ), we mainly focus on low‐order Stokes elements with nonconforming finite element approximations for the velocity and piecewise constant pressure functions. However, preliminary numerical tests show that this advantageous behavior can also be obtained for higher‐order discretizations, for instance, with Q₂/P₁ finite elements. We analyze the application of this simple, but very efficient, algorithm to several stationary and nonstationary benchmark configurations in 2D and 3D (driven cavity and flow around obstacles), and we also demonstrate its effect to spurious velocities in multiphase flow simulations (‘static bubble’ configuration) if combined with edge‐oriented, resp., interior penalty finite element method stabilization techniques. Copyright © 2008 John Wiley & Sons, Ltd.     

The oscillatory squeeze flow rheometer: comprehensive theory and a new experimental facility

In this paper, we give detailed attention to a relatively recent method for the determination of the linear dynamic properties of viscoelastic systems, namely, the so-called oscillatory squeeze flow (OSF) technique. We provide a comprehensive theory for the OSF rheometer, which includes a full discussion of the influence of fluid inertia. In the process, it is argued that, fortuitously perhaps, fluid inertia is more easily accommodated in the OSF rheometer than in the corresponding torsional-flow techniques. A new version of the OSF rheometer is described and experimental results on a set of viscoelastic systems are used to demonstrate the versatility of the technique. In the process, the potential use of the instrument within an industrial quality control environment is stressed.     

Model reduction using L1‐norm minimization as an application to nonlinear hyperbolic problems

We are interested in the model reduction techniques for hyperbolic problems, particularly in fluids. This paper, which is a continuation of an earlier paper of Abgrall et al, proposes a dictionary approach coupled with an L¹ minimization approach. We develop the method and analyze it in simplified 1‐dimensional cases. We show in this case that error bounds with the full model can be obtained provided that a suitable minimization approach is chosen. The capability of the algorithm is then shown on nonlinear scalar problems, 1‐dimensional unsteady fluid problems, and 2‐dimensional steady compressible problems. A short discussion on the cost of the method is also included in this paper.     

Composite high resolution localized relaxation scheme based on upwinding for hyperbolic conservation laws

In this work we present an upwind‐based high resolution scheme using flux limiters. Based on the direction of flow we choose the smoothness parameter in such a way that it leads to a truly upwind scheme without losing total variation diminishing (TVD) property for hyperbolic linear systems where characteristic values can be of either sign. Here we present and justify the choice of smoothness parameters. The numerical flux function of a high resolution scheme is constructed using wave speed splitting so that it results into a scheme that truly respects the physical hyperbolicity property. Bounds are given for limiter functions to satisfy TVD property. The proposed scheme is extended for non‐linear problems by using the framework of relaxation system that converts a non‐linear conservation law into a system of linear convection equations with a non‐linear source term. The characteristic speed of relaxation system is chosen locally on three point stencil of grid. This obtained relaxation system is solved using composite scheme technique, i.e. using a combination of proposed scheme with the conservative non‐standard finite difference scheme. Presented numerical results show higher resolution near discontinuity without introducing spurious oscillations. Copyright © 2008 John Wiley & Sons, Ltd.     

Incompressible multiphase flow and encapsulation simulations using the moment‐of‐fluid method

A moment‐of‐fluid method is presented for computing solutions to incompressible multiphase flows in which the number of materials can be greater than two. In this work, the multimaterial moment‐of‐fluid interface representation technique is applied to simulating surface tension effects at points where three materials meet. The advection terms are solved using a directionally split cell integrated semi‐Lagrangian algorithm, and the projection method is used to evaluate the pressure gradient force term. The underlying computational grid is a dynamic block‐structured adaptive grid. The new method is applied to multiphase problems illustrating contact‐line dynamics, triple junctions, and encapsulation in order to demonstrate its capabilities. Examples are given in two‐dimensional, three‐dimensional axisymmetric (R–Z), and three‐dimensional (X–Y–Z) coordinate systems. Copyright © 2015 John Wiley & Sons, Ltd.     

Positivity preserving pointwise implicit schemes with application to turbulent compressible flat plate flow

A family of positivity preserving pointwise implicit schemes applicable to source term dominated problems is constructed, where the minimum order of spatial accuracy is one and the maximum is three. It is designed for achieving steady state numerical solutions and is constructed through the analysis of appropriate model problems, where the convective fluxes for the higher‐order members are prescribed by the Chakravarthy–Osher family of total variation diminishing (TVD) schemes. Multidimensionality is facilitated by operator splitting. Numerical experimentation confirms the stability, convergence, accuracy, positivity, and computational efficiency associated with the proposed schemes. These schemes are ideally suited to solving the low‐Reynolds number turbulent k–ϵ equations for which the positivity of k and ϵ and the presence of stiff source terms are critical issues. Hence, using a finite volume formulation of these schemes, the low‐Reynolds number Chien k–ϵ turbulence model is implemented for a flat plate geometry and a series of turbulent flow (steady state) computations are carried out to demonstrate the positivity, robustness, and reliability of the algorithm. The free‐stream and initial k and ϵ values are specified in a very simple manner. Algorithm convergence acceleration is achieved using Multigrid techniques. The k–ϵ model flow predictions are shown to be in agreement with empirical profiles. Copyright © 2001 John Wiley & Sons, Ltd.     

Feature‐based adaptive mesh refinement for wingtip vortices

Feature‐based solution‐adaptive mesh refinement is an attractive strategy when it is known a priori that the resolution of certain key features is critical to achieving the objectives of a simulation. In this paper, we apply vortex characterization techniques, which are typically employed to visualize vortices, to identify regions of the computational domain for mesh refinement. We investigate different refinement strategies that are facilitated by these vortex characterization techniques to simulate the flow past a wing in a wind tunnel. Our results, which we compare with experimental data, indicate that it is necessary to refine the region within and near the vortex extent surface to obtain an accurate prediction. Application of the identified mesh refinement strategy also produced observed improvement in the results predicted for a spinning missile with deflected canards. Copyright © 2010 John Wiley & Sons, Ltd.     

Relaxation of the Navier–Stokes–Korteweg equations for compressible two‐phase flow with phase transition

The Navier–Stokes–Korteweg (NSK) system is a classical diffuse‐interface model for compressible two‐phase flow. However, the direct numerical simulation based on the NSK system is quite expensive and in some cases even not possible. We propose a lower‐order relaxation of the NSK system with hyperbolic first‐order part. This allows applying numerical methods for hyperbolic conservation laws and removing some of the difficulties of the original NSK system. To illustrate the new ansatz, we first present a local discontinuous Galerkin method in one and two spatial dimensions. It is shown that we can compute initial boundary value problems with realistic density ratios and perform stable computations for small interfacial widths. Second, we show that it is possible to construct a semi‐discrete finite‐volume scheme that satisfies a discrete entropy inequality. Copyright © 2015 John Wiley & Sons, Ltd.     

A multidimensional HLL‐Riemann solver for non‐linear hyperbolic systems

This article presents a numerical model that enables to solve on unstructured triangular meshes and with a high order of accuracy, Riemann problems that appear when solving hyperbolic systems. For this purpose, we use a MUSCL‐like procedure in a ‘cell‐vertex’ finite‐volume framework. In the first part of this procedure, we devise a four‐state bi‐dimensional HLL solver (HLL‐2D). This solver is based upon the Riemann problem generated at the barycenter of a triangular cell, from the surrounding cell‐averages. A new three‐wave model makes it possible to solve this problem, approximately. A first‐order version of the bi‐dimensional Riemann solver is then generated for discretizing the full compressible Euler equations. In the second part of the MUSCL procedure, we develop a polynomial reconstruction that uses all the surrounding numerical data of a given point, to give at best third‐order accuracy. The resulting over determined system is solved by using a least‐square methodology. To enforce monotonicity conditions into the polynomial interpolation, we use and adapt the monotonicity‐preserving limiter, initially devised by Barth (AIAA Paper 90‐0013, 1990). Numerical tests and comparisons with competing numerical methods enable to identify the salient features of the whole model. Copyright © 2010 John Wiley & Sons, Ltd.     

Advances in light microscope stereo vision

The increasing research focus on small-scale mechanical systems has generated a need for deformation and strain measurement systems for microscale applications. Optical measurement systems, such as digital image correlation, present an obvious choice due to their non-contacting nature. However, the transfer of measurement technology developed for macroscale applications to the microscale presents unique challenges due to the differences in the required highmagnification optics. In this paper we illustrate the problems involved in calibrating a stereo microscope using traditional techniques and present a novel methodology for acquiring accurate, three-dimensional surface shape and deformation data on small-scale specimens. Experimental results demonstrate that stereo microscope systems can be accurately and reliably calibrated using a priori distortion estimation techniques in combination with traditional bundle-adjustment. The unique feature of the present methodology is that it does not require a precision calibration target but relies solely on point correspondences obtained by image correlation. A variety of experiments illustrate the measurement performance of a stereo microscope system. It is shown that the surface strains obtained from the full-field, three-dimensional measurements on tensile specimens undergoing large rigid-body motions are within ±50 microstrain of strain gage measurements for strains ranging from 0 to 2000 microstrain. H. W. Schreier was a PhD Student from Ecole Mines des Albi in France     

Remarks on the links between low‐order DG methods and some finite‐difference schemes for the Stokes problem

In this paper we demonstrate that some well‐known finite‐difference schemes can be interpreted within the framework of the local discontinuous Galerkin (LDG) methods using the low‐order piecewise solenoidal discrete spaces introduced in (SIAM J. Numer. Anal. 1990; 27 (6): 1466–1485). In particular, it appears that it is possible to derive the well‐known MAC scheme using a first‐order Nédélec approximation on rectangular cells. It has been recently interpreted within the framework of the Raviart–Thomas approximation by Kanschat (Int. J. Numer. Meth. Fluids 2007; published online). The two approximations are algebraically equivalent to the MAC scheme, however, they have to be applied on grids that are staggered on a distance h/2 in each direction. This paper also demonstrates that both discretizations allow for the construction of a divergence‐free basis, which yields a linear system with a ‘biharmonic’ conditioning. Both this paper and Kanschat (Int. J. Numer. Meth. Fluids 2007; published online) demonstrate that the LDG framework can be used to generalize some popular finite‐difference schemes to grids that are not parallel to the coordinate axes or that are unstructured. Copyright © 2008 John Wiley & Sons, Ltd.     

Efficient solution techniques for implicit finite element schemes with flux limiters

The algebraic flux correction (AFC) paradigm is equipped with efficient solution strategies for implicit time‐stepping schemes. It is shown that Newton‐like techniques can be applied to the nonlinear systems of equations resulting from the application of high‐resolution flux limiting schemes. To this end, the Jacobian matrix is approximated by means of first‐ or second‐order finite differences. The edge‐based formulation of AFC schemes can be exploited to devise an efficient assembly procedure for the Jacobian. Each matrix entry is constructed from a differential and an average contribution edge by edge. The perturbation of solution values affects the nodal correction factors at neighbouring vertices so that the stencil for each individual node needs to be extended. Two alternative strategies for constructing the corresponding sparsity pattern of the resulting Jacobian are proposed. For nonlinear governing equations, the contribution to the Newton matrix which is associated with the discrete transport operator is approximated by means of divided differences and assembled edge by edge. Numerical examples for both linear and nonlinear benchmark problems are presented to illustrate the superiority of Newton methods as compared to the standard defect correction approach. Copyright © 2007 John Wiley & Sons, Ltd.     

Weak imposition of boundary conditions for the Navier–Stokes equations by a penalty method

We prove convergence of the finite element method for the Navier–Stokes equations in which the no‐slip condition and no‐penetration condition on the flow boundary are imposed via a penalty method. This approach has been previously studied for the Stokes problem by Liakos (Weak imposition of boundary conditions in the Stokes problem. Ph.D. Thesis, University of Pittsburgh, 1999). Since, in most realistic applications, inertial effects dominate, it is crucial to extend the validity of the method to the nonlinear Navier–Stokes case. This report includes the analysis of this extension, as well as numerical results validating their analytical counterparts. Specifically, we show that optimal order of convergence can be achieved if the computational boundary follows the real flow boundary exactly. Copyright © 2008 John Wiley & Sons, Ltd.     

The concept of minimized integrated exponential error for low dispersion and low dissipation schemes

We devise two novel techniques to optimize parameters which regulate dispersion and dissipation effects in numerical methods using the notion that dissipation neutralizes dispersion. These techniques are baptized as the minimized integrated error for low dispersion and low dissipation (MIELDLD) and the minimized integrated exponential error for low dispersion and low dissipation (MIEELDLD) . These two techniques of optimization have an advantage over the concept of minimized integrated square difference error (MISDE) , especially in the case when more than one optimal cfl is obtained, out of which only one of these values satisfy the shift condition. For instance, when MISDE is applied to the 1‐D Fromm's scheme, we have obtained two optimal cfl numbers: 0.28 and 1.0. However, it is known that Fromm's scheme satisfies shift condition only at r=1.0. Using MIELDLD and MIEELDLD , the optimal cfl of Fromm's scheme is computed as 1.0. We show that like the MISDE concept, both the techniques MIELDLD and MIEELDLD are effective to control dissipation and dispersion. The condition ν²>4µ is satisfied for all these three techniques of optimization, where ν and µ are parameters present in the Korteweg‐de‐Vries‐Burgers equation. The optimal cfl number for some numerical schemes namely Lax–Wendroff, Beam–Warming, Crowley and Upwind Leap‐Frog when discretized by the 1‐D linear advection equation is computed. The optimal cfl number obtained is in agreement with the shift condition. Some numerical experiments in 1‐D have been performed which consist of discontinuities and shocks. The dissipation and dispersion errors at some different cfl numbers for these experiments are quantified. Copyright © 2010 John Wiley & Sons, Ltd.     

Local moving least square‐one‐dimensional integrated radial basis function networks technique for incompressible viscous flows

This paper presents a local moving least square‐one‐dimensional integrated radial basis function networks method for solving incompressible viscous flow problems using stream function‐vorticity formulation. In this method, the partition of unity method is employed as a framework to incorporate the moving least square and one‐dimensional integrated radial basis function networks techniques. The major advantages of the proposed method include the following: (i) a banded sparse system matrix which helps reduce the computational cost; (ii) the Kronecker‐ δ property of the constructed shape function which helps impose the essential boundary condition in an exact manner; and (iii) high accuracy and fast convergence rate owing to the use of integration instead of conventional differentiation to construct the local radial basis function approximations. Several examples including two‐dimensional (2D) Poisson problems, lid‐driven cavity flow and flow past a circular cylinder are considered, and the present results are compared with the exact solutions and numerical results from other methods in the literature to demonstrate the attractiveness of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.