Showing papers on "Discretization published in 2003"

Finite Element Methods for Flow Problems

Abstract: Preface. 1. Introduction and preliminaries. Finite elements in fluid dynamics. Subjects covered. Kinematical descriptions of the flow field. The basic conservation equations. Basic ingredients of the finite element method. 2. Steady transport problems. Problem statement. Galerkin approximation. Early Petrov-Galerkin methods. Stabilization techniques. Other stabilization techniques and new trends. Applications and solved exercises. 3. Unsteady convective transport. Introduction. Problem statement. The methods of characteristics. Classical time and space discretization techniques. Stability and accuracy analysis. Taylor-Galerkin Methods. An introduction to monotonicity-preserving schemes. Least-squares-based spatial discretization. The discontinuous Galerkin method. Space-time formulations. Applications and solved exercises. 4. Compressible Flow Problems. Introduction. Nonlinear hyperbolic equations. The Euler equations. Spatial discretization techniques. Numerical treatment of shocks. Nearly incompressible flows. Fluid-structure interaction. Solved exercises. 5. Unsteady convection-diffusion problems. Introduction. Problem statement. Time discretization procedures. Spatial discretization procedures. Stabilized space-time formulations. Solved exercises. 6. Viscous incompressible flows. Introduction Basic concepts. Main issues in incompressible flow problems. Trial solutions and weighting functions. Stationary Stokes problem. Steady Navier-Stokes problem. Unsteady Navier-Stokes equations. Applications and Solved Exercices. References. Index.

Multiclass spectral clustering

Abstract: We propose a principled account on multiclass spectral clustering Given a discrete clustering formulation, we first solve a relaxed continuous optimization problem by eigen-decomposition We clarify the role of eigenvectors as a generator of all optimal solutions through orthonormal transforms We then solve an optimal discretization problem, which seeks a discrete solution closest to the continuous optima The discretization is efficiently computed in an iterative fashion using singular value decomposition and nonmaximum suppression The resulting discrete solutions are nearly global-optimal Our method is robust to random initialization and converges faster than other clustering methods Experiments on real image segmentation are reported

A method for computing horizontal pressure-gradient force in an oceanic model with a nonaligned vertical coordinate

Abstract: [1] Discretization of the pressure-gradient force is a long-standing problem in terrain-following (or σ) coordinate oceanic modeling. When the isosurfaces of the vertical coordinate are not aligned with either geopotential surfaces or isopycnals, the horizontal pressure gradient consists of two large terms that tend to cancel; the associated pressure-gradient error stems from interference of the discretization errors of these terms. The situation is further complicated by the nonorthogonality of the coordinate system and by the common practice of using highly nonuniform stretching for the vertical grids, which, unless special precautions are taken, causes both a loss of discretization accuracy overall and an increase in interference of the component errors. In the present study, we design a pressure-gradient algorithm that achieves more accurate hydrostatic balance between the two components and does not lose as much accuracy with nonuniform vertical grids at relatively coarse resolution. This algorithm is based on the reconstruction of the density field and the physical z coordinate as continuous functions of transformed coordinates with subsequent analytical integration to compute the pressure-gradient force. This approach allows not only a formally higher order of accuracy, but it also retains and expands several important symmetries of the original second-order scheme to high orders [Mellor et al., 1994; Song, 1998], which is used as a prototype. It also has built-in monotonicity constraining algorithm that prevents appearance of spurious oscillations of polynomial interpolant and, consequently, insures numerical stability and robustness of the model under the conditions of nonsmooth density field and coarse grid resolution. We further incorporate an alternative method of dealing with compressibility of seawater, which escapes pressure-gradient errors associated with interference of the nonlinear nature of equation of state and difficulties to achieve accurate polynomial fits of resultant in situ density profiles. In doing so, we generalized the monotonicity constraint to guarantee nonnegative physical stratification of the reconstructed density profile in the case of compressible equation of state. To verify the new method, we perform traditional idealized (Seamount) and realistic test problems.

Characteristic basis function method: A new technique for efficient solution of method of moments matrix equations

Abstract: In this paper, we introduce a novel approach for an efficient solution of matrix equations arising in the method of moments (MoM) formulation of electromagnetic scattering problems. This approach is based on the characteristic basis functions (CBFs), which are used to substantially reduce the matrix size because these bases are not bound by the conventional λ/20 domain discretization. As a result, it is possible to electrically solve large problems with much fewer unknowns than those needed when using conventional RWG basis functions. The accuracy and efficiency of the CBFs are demonstrated in a variety of scattering problems to illustrate the versatility of the approach. © 2002 Wiley Periodicals, Inc. Microwave Opt Technol Lett 36: 95–100, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.10685

Dimension-adaptive tensor-product quadrature

Abstract: We consider the numerical integration of multivariate functions defined over the unit hypercube. Here, we especially address the high-dimensional case, where in general the curse of dimension is encountered. Due to the concentration of measure phenomenon, such functions can often be well approximated by sums of lower-dimensional terms. The problem, however, is to find a good expansion given little knowledge of the integrand itself.The dimension-adaptive quadrature method which is developed and presented in this paper aims to find such an expansion automatically. It is based on the sparse grid method which has been shown to give good results for low- and moderate-dimensional problems. The dimension-adaptive quadrature method tries to find important dimensions and adaptively refines in this respect guided by suitable error estimators. This leads to an approach which is based on generalized sparse grid index sets. We propose efficient data structures for the storage and traversal of the index sets and discuss an efficient implementation of the algorithm.The performance of the method is illustrated by several numerical examples from computational physics and finance where dimension reduction is obtained from the Brownian bridge discretization of the underlying stochastic process.

Higher-Order Finite Element Methods

Abstract: INTRODUCTION Finite Elements Orthogonal Polynomials A One-Dimensional Example HIERARCHIC MASTER ELEMENTS OF ARBITRARY ORDER De Rham Diagram H^1-Conforming Approximations H(curl)-Conforming Approximations H(div)-Conforming Approximations L^2-Conforming Approximations HIGHER-ORDER FINITE ELEMENT DISCRETIZATION Projection-Based Interpolation on Reference Domains Transfinite Interpolation Revisited Construction of Reference Maps Projection-Based Interpolation on Physical Mesh Elements Technology of Discretization in Two and Three Dimensions Constrained Approximation Selected Software-Technical Aspects HIGHER-ORDER NUMERICAL QUADRATURE One-Dimensional Reference Domain K(a) Reference Quadrilateral K(q) Reference Triangle K(t) Reference Brick K(B) Reference Tetrahedron K(T) Reference Prism K(P) NUMERICAL SOLUTION OF FINITE ELEMENT EQUATIONS Direct Methods for Linear Algebraic Equations Iterative Methods for Linear Algebraic Equations Choice of the Method Solving Initial Value Problems for ordinary Differential Equations MESH OPTIMIZATION, REFERENCE SOLUTIONS, AND hp-ADAPTIVITY Automatic Mesh Optimization in One Dimension Adaptive Strategies Based on Automatic Mesh Optimization Goal-Oriented Adaptivity Automatic Goal-Oriented h-, p-, and hp-Adaptivity Automatic Goal-Oriented hp-Adaptivity in Two Dimensions

A mixed multiscale finite element method for elliptic problems with oscillating coefficients

Abstract: The recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the large-scale structure of the solutions without resolving all the fine-scale structures. Motivated by the numerical simulation of flow transport in highly heterogeneous porous media, we propose a mixed multiscale finite element method with an over-sampling technique for solving second order elliptic equations with rapidly oscillating coefficients. The multiscale finite element bases are constructed by locally solving Neumann boundary value problems. We provide a detailed convergence analysis of the method under the assumption that the oscillating coefficients are locally periodic. While such a simplifying assumption is not required by our method, it allows us to use homogenization theory to obtain the asymptotic structure of the solutions. Numerical experiments are carried out for flow transport in a porous medium with a random log-normal relative permeability to demonstrate the efficiency and accuracy of the proposed method.

Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems

Abstract: We study the entropy stability of difference approximations to nonlinear hyperbolic conservation laws, and related time-dependent problems governed by additional dissipative and dispersive forcing terms. We employ a comparison principle as the main tool for entropy stability analysis, comparing the entropy production of a given scheme against properly chosen entropy-conservative schemes.To this end, we introduce general families of entropy-conservative schemes, interesting in their own right. The present treatment of such schemes extends our earlier recipe for construction of entropy-conservative schemes, introduced in Tadmor (1987b). The new families of entropy-conservative schemes offer two main advantages, namely, (i) their numerical fluxes admit an explicit, closed-form expression, and (ii) by a proper choice of their path of integration in phase space, we can distinguish between different families of waves within the same computational cell; in particular, entropy stability can be enforced on rarefactions while keeping the sharp resolution of shock discontinuities.A comparison with the numerical viscosities associated with entropy-conservative schemes provides a useful framework for the construction and analysis of entropy-stable schemes. We employ this framework for a detailed study of entropy stability for a host of first- and second-order accurate schemes. The comparison approach yields a precise characterization of the entropy stability of semi-discrete schemes for both scalar problems and systems of equations.We extend these results to fully discrete schemes. Here, spatial entropy dissipation is balanced by the entropy production due to time discretization with a suffciently small time-step, satisfying a suitable CFL condition. Finally, we revisit the question of entropy stability for fully discrete schemes using a different approach based on homotopy arguments. We prove entropy stability under optimal CFL conditions.

One-dimensional modelling of a vascular network in space-time variables

Abstract: In this paper a one-dimensional model of a vascular network based on space-time variables is investigated. Although the one-dimensional system has been more widely studied using a space-frequency decomposition, the space-time formulation offers a more direct physical interpretation of the dynamics of the system. The objective of the paper is to highlight how the space-time representation of the linear and nonlinear one-dimensional system can be theoretically and numerically modelled. In deriving the governing equations from first principles, the assumptions involved in constructing the system in terms of area-mass flux (A,Q), area-velocity (A,u), pressure-velocity (p,u) and pressure-mass flux(p,Q) variables are discussed. For the nonlinear hyperbolic system expressed in terms of the (A,u) variables the extension of the single-vessel model to a network of vessels is achieved using a characteristic decomposition combined with conservation of mass and total pressure. The more widely studied linearised system is also discussed where conservation of static pressure, instead of total pressure, is enforced in the extension to a network. Consideration of the linearised system also allows for the derivation of a reflection coefficient analogous to the approach adopted in acoustics and surface waves. The derivation of the fundamental equations in conservative and characteristic variables provides the basic information for many numerical approaches. In the current work the linear and nonlinear systems have been solved using a spectral/hp element spatial discretisation with a discontinuous Galerkin formulation and a second-order Adams-Bashforth time-integration scheme. The numerical scheme is then applied to a model arterial network of the human vascular system previously studied by Wang and Parker (To appear in J. Biomech. (2004)). Using this numerical model the role of nonlinearity is also considered by comparison of the linearised and nonlinearised results. Similar to previous work only secondary contributions are observed from the nonlinear effects under physiological conditions in the systemic system. Finally, the effect of the reflection coefficient on reversal of the flow waveform in the parent vessel of a bifurcation is considered for a system with a low terminal resistance as observed in vessels such as the umbilical arteries.

Immersed boundary technique for turbulent flow simulations

Abstract: The application of the Immersed Boundary ~IB! method to simulate incompressible, turbulent flows around complex configurations is illustrated; the IB is based on the use of non-body conformal grids, and the effect of the presence of a body in the flow is accounted for by modifying the governing equations. Turbulence is modeled using standard Reynolds-Averaged Navier-Stokes models or the more sophisticated Large Eddy Simulation approach. The main features of the IB technique are described with emphasis on the treatment of boundary conditions at an immersed surface. Examples of flows around a cylinder, in a wavy channel, inside a stirred tank and a piston/cylinder assembly, and around a road vehicle are presented. Comparison with experimental data shows the accuracy of the present technique. This review article cites 70 references. @DOI: 10.1115/1.1563627# 1 CONTEXT The continuous growth of computer power strongly encourages engineers to rely on computational fluid dynamics ~CFD! for the design and testing of new technological solutions. Numerical simulations allow the analysis of complex phenomena without resorting to expensive prototypes and difficult experimental measurements. The basic procedure to perform numerical simulation of fluid flows requires a discretization step in which the continuous governing equations and the domain of interest are transformed into a discrete set of algebraic relations valid in a finite number of locations ~computational grid nodes! inside the domain. Afterwards, a numerical procedure is invoked to solve the obtained linear or nonlinear system to produce the local solution to the original equations. This process is simple and very accurate when the grid nodes are distributed uniformly ~Cartesian mesh! in the domain, but becomes computationally intensive for disordered ~unstructured! point distributions. For simple computational domains ~a box, for example! the generation of the computational grid is trivial; the simulation of a flow around a realistic configuration ~a road vehicle in a wind tunnel, for example!, on the other hand, is extremely complicated and time consuming since the shape of the domain must include the wetted surface of the geometry of interest. The first difficulty arises from the necessity to build a smooth surface mesh on the boundaries of the domain ~body conforming grid!. Usually industrially relevant geometries are defined in a CAD environment and must be translated and cleaned ~small details are usually eliminated, overlapping surface patches are trimmed, etc! before a surface grid can be generated. This mesh serves as a starting point to generate the volume grid in the computational domain. In addition, in many industrial applications, geometrical complexity is combined with moving boundaries and high Reynolds numbers. This requires regeneration or deformation of the grid during the simulation and turbulence modeling, leading to a considerable increase of the computational difficulties. As a result, engineering flow simulations have large computational overhead and low accuracy owing to a large number of operations per node and high storage requirements in combination with low order dissipative spatial discretization. Given the finite memory and speed of computers, these simulations are very expensive and time consuming with computational meshes that are generally limited to around one million nodes. In view of these difficulties, it is clear that an alternative numerical procedure that can handle the geometric complexity, but at the same time retains the accuracy and high efficiency of the simulations performed on regular grids, would represent a significant advance in the application of CFD to industrial flows.

Algorithm Developments for Discrete Adjoint Methods

Abstract: This paper presents a number of algorithm developments for adjoint methods using the 'discrete' approach in which the discretisation of the non-linear equations is linearised and the resulting matrix is then transposed. With a new iterative procedure for solving the adjoint equations, exact numerical equivalence is maintained between the linear and adjoint discretisations. The incorporation of strong boundary conditions within the discrete approach is discussed, as well as a new application of adjoint methods to linear unsteady flow in turbomachinery.

Two direct Tustin discretization methods for fractional-order differentiator/integrator

Abstract: This paper deals with fractional calculus and its approximate discretization. Two direct discretization methods useful in control and digital filtering are presented for discretizing the fractional-order differentiator or integrator. Detailed mathematical formulae and tables are given. An illustrative example is presented to show the practically usefulness of the two proposed discretization schemes. Comparative remarks between the two methods are also given.

Anisotropic diffusion of surfaces and functions on surfaces

Abstract: We present a unified anisotropic geometric diffusion PDE model for smoothing (fairing) out noise both in triangulated two-manifold surface meshes in IR3 and functions defined on these surface meshes, while enhancing curve features on both by careful choice of an anisotropic diffusion tensor. We combine the C1 limit representation of Loop's subdivision for triangular surface meshes and vector functions on the surface mesh with the established diffusion model to arrive at a discretized version of the diffusion problem in the spatial direction. The time direction discretization then leads to a sparse linear system of equations. Iteratively solving the sparse linear system yields a sequence of faired (smoothed) meshes as well as faired functions.

Legendre pseudospectral approximations of optimal control problems

Abstract: Summary. We consider nonlinear optimal control problems with mixed statecontrol constraints. A discretization of the Bolza problem by a Legendre pseudospectral method is considered. It is shown that the operations of discretization and dualization are not commutative. A set of Closure Conditions are introduced to commute these operations. An immediate consequence of this is a Covector Mapping Theorem (CMT) that provides an order-preserving transformation of the Lagrange multipliers associated with the discretized problem to the discrete covectors associated with the optimal control problem. A natural consequence of the CMT is that for pure state-constrained problems, the dual variables can be easily related to the D-form of the Lagrangian of the Hamiltonian. We demonstrate the practical advantage of our results by numerically solving a state-constrained optimal control problem without deriving the necessary conditions. The costates obtained by an application of our CMT show excellent agreement with the exact analytical solution.

A fast direct solver for boundary integral equations in two dimensions

Abstract: We describe an algorithm for the direct solution of systems of linear algebraic equations associated with the discretization of boundary integral equations with non-oscillatory kernels in two dimensions. The algorithm is ''fast'' in the sense that its asymptotic complexity is O(n), where n is the number of nodes in the discretization. Unlike previous fast techniques based on iterative solvers, the present algorithm directly constructs a compressed factorization of the inverse of the matrix; thus it is suitable for problems involving relatively ill-conditioned matrices, and is particularly efficient in situations involving multiple right hand sides. The performance of the scheme is illustrated with several numerical examples. rformance of the scheme is illustrated with several numerical examples. ples.

Optimization of a Pressure-Swing Adsorption Process Using Zeolite 13X for CO2 Sequestration

Abstract: A pressure-swing adsorption process, which uses zeolite 13X as an adsorbent to recover and sequester carbon dioxide from mixture gas (nitrogen and carbon dioxide), is investigated through dynamic simulation and optimization. The purpose of this paper is to improve the purity of each component by finding optimal values of decision variables with a given power constraint. Langmuir isotherm parameters are calculated from experimental data of zeolite 13X and a general mathematical model consisting of a set of partial differential and algebraic equations and solved in gPROMS. The method of centered finite differences is adopted for the discretization of the spatial domains, and a reduced space SQP method is used for the optimization. As a result, the optimal conditions at cyclic steady state are obtained.