Separation-bubble flow solution using Euler/Navier-Stokes zonal approach with downstream compatibility conditions

An Euler/Navier-Stokes zonal scheme is developed to numerically simulate the two-dimensional flow over a blunt leading-edge plate. The computational domain has been divided into inner and outer regions where the Navier-Stokes and Euler equations are used, respectively. On the downstream boundary, compatibility conditions derived from the boundary-layer equations are used. The grid is generated by using conformal mapping and the problem is solved by using a compressible Navier-Stokes code, which has been modified to treat Euler and Navier-Stokes regions. The accuracy of the solution is determined by the reattachment location. Bench-mark solutions have been obtained using the Navier-Stokes equations throughout the optimum computational domain and size. The problem is recalculated with sucessive decrease of the computational domain from the downstream side where the compatibility conditions are used, and with successive decrease of the Navier-Stokes computational region. The results of the zonal scheme are in excellent agreement with those of the benchmark solutions and the experimental data. The CPU time saving is about 15%.     

Domain decomposition for near-wall turbulent flows

Modeling near-wall high-Reynolds-number turbulent flows is a time-consuming problem. A domain decomposition approach is developed to overcome the problem. The original computational domain is split into a near-wall (inner) subdomain and an outer subdomain. The developed approach is applied to a model 2D equation simulating major peculiarities of near-wall high-Reynolds-number flows. On the base of the Calderon–Ryaben’kii potential theory it is possible to consider the near-wall (inner) problem independently on the outer problem. The influence of the inner problem can exactly be represented by a pseudo-differential equation formulated on the intermediate boundary. In a 1D case, it leads to the wall functions represented by Robin boundary conditions, which can be determined either analytically or numerically. It is important that the wall functions (or boundary conditions) are mesh independent and can be realized in a separate routine. Thus, the original problem can only be solved in the outer domain with some specific nonlocal boundary conditions called nonlocal wall functions. The technique can be extended to 3D problems straightforward.     

A zonal RANS-LES method to determine the flow over a high-lift configuration

High Reynolds number flows are particularly challenging problems for large-eddy simulations (LES) since small-scale structures in thin and often transitional boundary layers are to be resolved. The range of the turbulent scales is enormous, especially when high-lift configuration flows are considered. For this reason, the prediction of high Reynolds number flow over the entire airfoil using LES requires huge computer resources. To remedy this problem a zonal RANS-LES method for the flow over an airfoil in high-lift configuration at Re_c=1.0×10⁶ is presented. In a first step, a 2D RANS solution is sought, from which boundary conditions are formulated for an embedded LES domain, which comprises the flap and a sub-part of the main airfoil. The turbulent fluctuations in the boundary layers at the inflow region of the LES domain are generated by controlled forcing terms, which use the turbulent shear stress profiles obtained from the RANS solution. The comparison with an LES solution for the full domain and with experimental data shows likewise results for the velocity profiles and wall pressure distributions. The zonal RANS-LES method reduces the computational effort of a full domain LES by approx. 50%.     

High-order unified symplectic FDTD scheme for the metamaterials

A high-order unified symplectic finite-difference time-domain (US-FDTD) method, which is energy conserved, for modeling the metamaterials is proposed. The lossless Drude dispersive model is taken into account in US-FDTD scheme, and the detailed formulations of the proposed US-FDTD method are also provided. The high-order split perfectly matched layers (SPML) are used as the absorbing boundary conditions (ABCs) to terminate the computational domain. The analysis of Courant stability and numerical dispersion demonstrate that US-FDTD scheme is more efficient than the traditional time domain numerical methods. Focusing and refocusing of the electromagnetic wave in target detection is validated using the normal incident Gaussian beam with a matched slab. Oblique incidence results associated with the inverse Snell effect and the phase compensation effect of the composite slab further demonstrated the efficiency of the method. Numerical results for a more realistic structure are also included. All the results agree well with the theoretical prediction. The method proposed here can be directly put into using as a time-domain full-wave simulation tool for applications in metamaterials.     

New solution method for homogenization analysis and its application to the prediction of macroscopic elastic constants of materials with periodic microstructures

A new solution method is proposed for the homogenization analysis of materials with periodic microstructures. A homogeneous integral equation is derived to replace the conventional inhomogeneous integral equation related to the microscopic mechanical behavior in the basic unit cell by introducing a new characteristic function. Based on the new solution method, the computational problem of the characteristic function subject to initial strains and periodic boundary conditions is reduced to a simple displacement boundary value problem without initial strains, which simplifies the computational process. Applications to the predication of macroscopic elastic constants of materials with various two-dimensional and three-dimensional periodic microstructures are presented. The numerical results are compared with previous results obtained from the Hapin-Tsai equations, Mori–Tanaka method and conventional homogenization calculations, which proves that the present method is valid and efficient for prediction of the macroscopic elastic constants of materials with various periodic microstructures.     

On the viscous–viscous and the viscous–inviscid interactions in Computational Fluid Dynamics

We consider here the exterior boundary value problem for compressible viscous flow around airfoils. In a first approximation, the viscosity effects are neglected at some distance to the airfoil. The unbounded domain is decomposed by an artificial boundary into a bounded computational domain (near field) and an associated far field. The complete system of conservation laws, modelling viscous flow in the near field is coupled with simplified models for inviscid flow in the far field. The use of the heterogeneous domain decomposition method including physically and mathematically justified transmission conditions at the artificial interface provides one with a quite accurate approximate solution, modelling the viscous–inviscid interaction between the two model zones. However, such a solution does not take into account the viscosity in the far field and does not satisfy the natural transmission conditions at the artificial interface (i.e. continuity of the solution and of the normal flux). In order to get some information for the a-posteriori improvement of this solution, we introduce one-dimensional transmission-boundary value problems, obtained by an appropriate dimensional reduction of the coupled problems from CFD. The one-dimensional problems are analyzed in the framework of singular perturbation theory. We consider formal asymptotic expansions to construct appropriate boundary layer corrections of the coupled problem modelling the viscous–inviscid interaction. Our one-dimensional analysis seems to allow an extension to higher dimensions and therefore could be used in the computation of the solution to the compressible Navier–Stokes problem by updating the solution of the approximation by a (degenerate) Navier–Stokes/Euler problem with boundary layer viscosity correction terms. Received: 22 February 1999 / Accepted: 17 June 1999     

Parallelization of a two-dimensional flood inundation model based on domain decomposition

Flood modelling often involves prediction of the inundated extent over large spatial and temporal scales. As the dimensionality of the system and the complexity of the problems increase, the need to obtain quick solutions becomes a priority. However, for large-scale problems or situations where fine resolution data is required, it is often not possible or practical to run the model on a single computer in a reasonable timeframe. This paper presents the development and testing of a parallelized 2D diffusion-based flood inundation model (FloodMap-Parallel) which enables large-scale simulations to be run on distributed multi-processors. The model has been applied to three locations in the UK with different flow and topographical boundary conditions. The accuracy of the parallelized model and its computational efficiency have been tested. The predictions obtained from the parallelized model match those obtained from the serialized simulations. The computational performance of the model has been investigated in relation to the granularity of the domain decomposition, the total number of cells and the domain decomposition configuration pattern. Results show that the parallelized model is more effective with simulations of low granularity and a large number of cells. The large communication overhead associated with the potential load-imbalance between sub-domains is a major bottleneck in utilizing this approach with higher domain granularity.     

Application of shifted Jacobi pseudospectral method for solving (in)finite-horizon min–max optimal control problems with uncertainty

The difficulty of solving the min–max optimal control problems (M-MOCPs) with uncertainty using generalised Euler–Lagrange equations is caused by the combination of split boundary conditions, nonlinear differential equations and the manner in which the final time is treated. In this investigation, the shifted Jacobi pseudospectral method (SJPM) as a numerical technique for solving two-point boundary value problems (TPBVPs) in M-MOCPs for several boundary states is proposed. At first, a novel framework of approximate solutions which satisfied the split boundary conditions automatically for various boundary states is presented. Then, by applying the generalised Euler–Lagrange equations and expanding the required approximate solutions as elements of shifted Jacobi polynomials, finding a solution of TPBVPs in nonlinear M-MOCPs with uncertainty is reduced to the solution of a system of algebraic equations. Moreover, the Jacobi polynomials are particularly useful for boundary value problems in unbounded domain, which allow us to solve infinite- as well as finite and free final time problems by domain truncation method. Some numerical examples are given to demonstrate the accuracy and efficiency of the proposed method. A comparative study between the proposed method and other existing methods shows that the SJPM is simple and accurate.     

Local artificial boundary conditions for Schrödinger and heat equations by using high-order azimuth derivatives on circular artificial boundary

The aim of the paper is to design high-order artificial boundary conditions for the Schrödinger equation on unbounded domains in parallel with a treatment of the heat equation. We first introduce a circular artificial boundary to divide the unbounded definition domain into a bounded computational domain and an unbounded exterior domain. On the exterior domain, the Laplace transformation in time and Fourier series in space are applied to achieve the relation of special functions. Then the rational functions are used to approximate the relation of the special functions. Applying the inverse Laplace transformation to a series of simple rational function, we finally obtain the corresponding high-order artificial boundary conditions, where a sequence of auxiliary variables are utilized to avoid the high-order derivatives in respect to time and space. Furthermore, the finite difference method is formulated to discretize the reduced initial–boundary value problem with high-order artificial boundary conditions on a bounded computational domain. Numerical experiments are presented to illustrate the performance of our method.     

Computationally efficient nonlinear Min–Max Model Predictive Control based on Volterra series models—Application to a pilot plant

The mathematical model used in Min–Max MPC (MMMPC) to predict the future trajectory of the system explicitly considers disturbances and uncertainties. Based on the future trajectory, the control sequence is computed minimizing the worst case cost with respect to all possible trajectories of the disturbances and uncertainties. This approach leads to a more robust control performance but also complicates the practical implementation of MMMPC due to the high computational burden required to solve the optimization problem. This computational burden is even worse if a nonlinear prediction model is used. In fact, to the best of the authors’ knowledge, there have not yet been reported any applications of nonlinear MMMPC to real processes. In this paper a nonlinear MMMPC strategy based on a second order Volterra series model is presented. The particular structure of the used prediction model allows to obtain an explicit formulation of the worst case cost and its computation in polynomial time. Real time applications with typical prediction and control horizons are possible because of the reduced complexity of the proposed control strategy. Furthermore, input-to-state practical stability for the proposed control strategy is guaranteed under certain conditions. The MMMPC strategy is implemented and validated in experiments with a continuous stirred tank reactor whose temperature dynamics are approximated by a second order Volterra series model. The control performance of the proposed MMMPC strategy is illustrated by the obtained experimental results.     

Concurrent finite element analysis of periodic boundary value problems

A parallel finite element approach for analyzing micromechanical problems with periodic unit cells is discussed. The method uses a direct solution strategy so that general periodic boundary conditions can be treated using a two-step domain decomposition strategy. The speedup results show a good performance of the method on coarse-grained problems, i.e. for cases where the computational work done on the substructures that are treated in parallel is relatively large compared to the total amount of computational work. Application examples using crystal-plasticity on an array of planar crystals and a metal matrix composite are used to show that the overall response of these materials is rather strongly dependent on the constraint imposed on the unit cell so that a correct treatment of the periodic boundary conditions is required to accurately predict the macroscopic response of a periodic material even though a unit cell with a large number of grains or fibers is used.     

Algorithms for finding attribute value group for binary segmentation of categorical databases

We consider the problem of finding a set of attribute values that give a high quality binary segmentation of a database. The quality of a segmentation is defined by an objective function suitable for the user's objective, such as "mean squared error," "mutual information," or "/spl chi//sup 2/" each of which is defined in terms of the distribution of a given target attribute. Our goal is to find value groups on a given conditional domain that split databases into two segments, optimizing the value of an objective function. Though the problem is intractable for general objective functions, there are feasible algorithms for finding high quality binary segmentations when the objective function is convex, and we prove that the typical criteria mentioned above are all convex. We propose two practical algorithms, based on computational geometry techniques, which find a much better value group than conventional heuristics.     

Simplifying the control of certain large non-linear systems using hierarchical optimisation

In this paper, the recently developed combined pseudo model coordination/prediction principle method is slightly modified in order to simplify the control of certain non-linear systems. The simplification arises because for many systems, it is now possible to have a single point boundary value problem at the first level and this leads to considerable computational savings. The class of subsystems where the simplification occurs is examined and it is seen that virtually all practical systems will have some subsystems in this class. This is illustrated by two examples taken from the literature. The first of these is the problem of reactivity control of a nuclear reactor described by a single group of delayed neutrons and the second of synchronous machine excitation. For both the cases, all the subsystems can be simplified leading to substantial computational savings. This is demonstrated for the machine example where a comparison between the computation times of the overall solution using quasilinearisation and the hierarchical solution shows that the latter requires only a third of the computation time for the overall solution.     

Boundary conditions for subsonic compressible Navier-Stokes calculations

A systematic study of inflow and outflow boundary conditions for the numerical solution of the compressible Navier-Stokes equations is presented. Combinations of several representative inflow and outflow boundary conditions are applied in the solution of subsonic flow over a flat plate in a finite computational domain. These boundary conditions are evaluated in terms of their effect on the accuracy of the solution and the rate of convergence to a steady state. It is shown that errors in the data specified at the inflow boundary can produce significant errors in the computed flow field. It is also shown that a non-reflecting outflow boundary condition can significantly reduce the total computational time required.     

A boundary integral method for the simulation of two-dimensional particle coarsening

A boundary integral method for the solution of a time-dependent free-boundary problem in a two-dimensional, multiply-connected, exterior domain is described. The method is based on an iterative solution of the resulting integral equations at each time step, with the initial guesses provided by extrapolation from previous time steps. The method is related to a technique discussed by Baker for the study of water waves. The discretization is chosen so that the solvability conditions required for the exterior Dirichlet problem do not degrade the convergence rate of the iterative solution procedure. Consideration is given to the question of vectorizing the computation. The method is applied to the problem of the coarsening of two-dimensional particles by volume diffusion.     

Numerical solution of eddy current problems in bounded domains using realistic boundary conditions

The aim of this paper is to analyze a finite element method to solve the low-frequency harmonic Maxwell equations in a bounded domain containing conductors and dielectrics, and using realistic boundary conditions in that they can be easily measured. These equations provide a model for the so-called eddy currents. The problem is formulated in terms of the magnetic field. This formulation is discretized by using Nédélec edge finite elements on a tetrahedral mesh. Error estimates are easily obtained when the curl-free condition is imposed explicitly on the elements in the dielectric domain.A multivalued magnetic scalar potential is introduced then to impose this curl-free condition. The discrete counterpart of this formulation leads to an important saving in computational effort. Problems related to the topology are also considered, more precisely, the possibility of having a non-simply connected dielectric domain is taken into account. Finally, the method is applied to solve two three-dimensional model problems: a test with a known analytical solution and the computation of the electromagnetic field in a metallurgical arc furnace.     

Iterative methods for model reduction by domain decomposition

It is proposed a method to reduce the computational effort to solve a partial differential equation on a given domain. The main idea is to split the domain of interest in two subdomains, and to use different approximation methods in each of the two subdomains. In particular, in one subdomain we discretize the governing equations by a canonical scheme, whereas in the other one we solve a reduced order model of the original problem. Different approaches to couple the low-order model to the usual discretization are presented. The effectiveness of these approaches is tested on numerical examples pertinent to non-linear model problems including Laplace equation with non-linear boundary conditions and compressible Euler equations.     

Using three-dimensional finite element analysis in time domain to model railway-induced ground vibrations

For the prediction of ground vibrations generated by railway traffic, finite element analysis (FEA) appears as a competitive alternative to simulation tools based on the boundary element method: it is largely used in industry and does not suffer any limitation regarding soil geometry or material properties. However, boundary conditions must be properly defined along the domain border so as to mimic the effect of infinity for ground wave propagation. This paper presents a full three-dimensional FEA for the prediction of railway ground-borne vibrations. Non-reflecting boundaries are compared to fixed and free boundary conditions, especially concerning their ability to model the soil wave propagation and reflection. Investigations with commercial FEA software ABAQUS are presented also, with the development of an external meshing tool, so as to automatically define the infinite elements at the model boundary. Considering that ground wave propagation is a transient problem, the problem is formulated in the time domain. The influence of the domain dimension and of the element size is analysed and rules are established to optimise accuracy and computational burden. As an example, the structural response of a building is simulated, considering homogeneous or layered soil, during the passage of a tram at constant speed.     

Finite element methods for the Stokes problem on complicated domains

It is a standard assumption in the error analysis of finite element methods that the underlying finite element mesh has to resolve the physical domain of the modeled process. In case of complicated domains which appear in many applications such as ground water flows this requirement sometimes becomes a bottleneck. The resolution condition links the computational complexity to the number (and size) of geometric details although the accuracy requirements, possibly, are moderate and would allow a (locally) coarse mesh width. Therefore even the coarsest available discretization can lead to a huge number of unknowns. The composite mini element is a remedy to this dilemma because the degrees of freedom are not linked to the number of geometric details. The basic concept for the Stokes problem with uniform no-slip boundary conditions has been introduced and analyzed in [D. Peterseim, S. Sauter, The composite mini element – coarse mesh computation of Stokes flows on complicated domains, SINUM, 46(6) (2008) 3181–3206]. Here, we generalize the composite mini element to slip, leak and Neumann boundary conditions so that it becomes applicable to this much larger and more important problem class. The main results are (a) the algorithmic concept remains unchanged and the new boundary conditions can be implemented as a weighted quadrature rule, (b) the stability and convergence can be proved under very mild assumption on the domain geometries, (c) the analysis is far from trivial and requires the development of substantially new tools compared to the simple case of uniform no-slip boundary conditions.     

Transient fluid–structure interaction with non-matching spatial and temporal discretizations

This paper presents a review of spatial and temporal discretization schemes for unsteady flow interacting with structure. Two types of spatial coupling schemes are analyzed: (i) point-to-element projection and (ii) common-refinement based projection. It is shown that the point-to-element projection schemes may yield inaccurate load transfer from the source fluid mesh to the target solid mesh, leading to a weak instability in the form of spurious oscillations and overshoots in the interface solution. The common-refinement scheme resolves this problem by providing an accurate transfer of discrete interface conditions across non-matching meshes. With respect to the temporal discretization, three coupling techniques are assessed: (i) conventional sequential staggered (CSS); (ii) generalized serial staggered (GSS) and (iii) combined interface boundary condition (CIBC). Traditionally, continuity of velocity and traction along interfaces are satisfied through algebraic interface conditions applied in a sequential fashion, which is often referred to as staggered computation. In existing partitioned staggered procedures, the interface conditions may undermine stability and accuracy of coupled fluid–structure simulations. By utilizing the CIBC technique on the velocity and traction boundary conditions, a staggered coupling procedure can be constructed with similar order of accuracy and stability as standalone computations. The effectiveness of spatial and temporal coupling schemes is investigated with the aid of simple 1D examples and new 2D subsonic flow-shell aeroelastic simulations.