Showing papers on "Discretization published in 2005"

Introduction to Computational Fluid Dynamics

Abstract: Introduction to Computational Fluid Dynamics is a textbook for advanced undergraduate and first year graduate students in mechanical, aerospace and chemical engineering. The book emphasizes understanding CFD through physical principles and examples. The author follows a consistent philosophy of control volume formulation of the fundamental laws of fluid motion and energy transfer, and introduces a novel notion of 'smoothing pressure correction' for solution of flow equations on collocated grids within the framework of the well-known SIMPLE algorithm. The subject matter is developed by considering pure conduction/diffusion, convective transport in 2-dimensional boundary layers and in fully elliptic flow situations and phase-change problems in succession. The book includes chapters on discretization of equations for transport of mass, momentum and energy on Cartesian, structured curvilinear and unstructured meshes, solution of discretised equations, numerical grid generation and convergence enhancement. Practising engineers will find this particularly useful for reference and for continuing education.

Physically Based Deformable Models in Computer Graphics.

Abstract: Physically based deformable models have been widely embraced by the Computer Graphics community. Many problems outlined in a previous survey by Gibson and Mirtich [ GM97] have been addressed, thereby making these models interesting and useful for both offline and real-time applications, such as motion pictures and video games. In this paper, we present the most significant contributions of the past decade, which produce such impressive and perceivably realistic animations and simulations: finite element/difference/volume methods, mass-spring systems, meshfree methods, coupled particle systems and reduced deformable models based on modal analysis. For completeness, we also make a connection to the simulation of other continua, such as fluids, gases and melting objects. Since time integration is inherent to all simulated phenomena, the general notion of time discretization is treated separately, while specifics are left to the respective models. Finally, we discuss areas of application, such as elastoplastic deformation and fracture, cloth and hair animation, virtual surgery simulation, interactive entertainment and fluid/smoke animation, and also suggest areas for future research.

A variational discretization concept in control constrained optimization: the linear-quadratic case

Abstract: A new discretization concept for optimal control problems with control constraints is introduced which utilizes for the discretization of the control variable the relation between adjoint state and control. Its key feature is not to discretize the space of admissible controls but to implicitly utilize the first order optimality conditions and the discretization of the state and adjoint equations for the discretization of the control. For discrete controls obtained in this way an optimal error estimate is proved. The application to control of elliptic equations is discussed. Finally it is shown that the new concept is numerically implementable with only slight increase in program management. A numerical test confirms the theoretical investigations.

A family of mimetic finite difference methods on polygonal and polyhedral meshes

Abstract: A family of inexpensive discretization schemes for diffusion problems on unstructured polygonal and polyhedral meshes is introduced. The material properties are described by a full tensor. The theoretical results are confirmed with numerical experiments.

Chemical reaction rates from ring polymer molecular dynamics.

Abstract: We show how the ring-polymer molecular dynamics method can be adapted to calculate approximate Kubo-transformed flux-side correlation functions, and hence rate coefficients for condensed phase reactions. An application of the method to the standard model for a chemical reaction in solution—a quartic double-well potential linearly coupled to a bath of harmonic oscillators—is found to give results of comparable accuracy to those of the classical Wigner model and the centroid molecular dynamics method. However, since the present method does not require that one evaluate the Wigner transform of a thermal flux operator or that one perform a separate path integral calculation for each molecular dynamics time step, we believe it will prove easier to apply to more general problems than either of these alternative techniques. We also present a (logarithmic) discretization scheme for the Ohmic bath in the system-bath model that gives converged results with just nine bath modes—a surprisingly small number for a model of a condensed phase reaction. Finally, we present some calculations of the transmission through an Eckart barrier which show that the present method provides a satisfactory (although not perfect) description of the deep quantum tunneling regime. Part of the reason for the success of the method is that it gives the exact quantum-mechanical rate constant for the transmission through a parabolic barrier, as we demonstrate analytically in the Appendix.

Propagation, Observation, and Control of Waves Approximated by Finite Difference Methods

Abstract: This paper surveys several topics related to the observation and control of wave propagation phenomena modeled by finite difference methods. The main focus is on the property of observability, corresponding to the question of whether the total energy of solutions can be estimated from partial measurements on a subregion of the domain or boundary. The mathematically equivalent property of controllability corresponds to the question of whether wave propagation behavior can be controlled using forcing terms on that subregion, as is often desired in engineering applications. Observability/controllability of the continuous wave equation is well understood for the scalar linear constant coefficient case that is the focus of this paper. However, when the wave equation is discretized by finite difference methods, the control for the discretized model does not necessarily yield a good approximation to the control for the original continuous problem. In other words, the classical convergence (consistency + stability) property of a numerical scheme does not suffice to guarantee its suitability for providing good approximations to the controls that might be needed in applications. Observability/controllability may be lost under numerical discretization as the mesh size tends to zero due to the existence of high-frequency spurious solutions for which the group velocity vanishes. This phenomenon is analyzed and several remedies are suggested, including filtering, Tychonoff regularization, multigrid methods, and mixed finite element methods. We also briefly discuss these issues for the heat, beam, and Schrodinger equations to illustrate that diffusive and dispersive effects may help to retain the observability/controllability properties at the discrete level. We conclude with a list of open problems and future subjects for research.

Algorithms for iterative decoding in the presence of strong phase noise

Abstract: We present two new iterative decoding algorithms for channels affected by strong phase noise and compare them with the best existing algorithms proposed in the literature. The proposed algorithms are obtained as an application of the sum-product algorithm to the factor graph representing the joint a posteriori probability mass function of the information bits given the channel output. In order to overcome the problems due to the presence in the factor graph of continuous random variables, we apply the method of canonical distributions . Several choices of canonical distributions have been considered in the literature. Well-known approaches consist of discretizing continuous variables or treating them as jointly Gaussian, thus obtaining a Kalman estimator. Our first new approach, based on the Fourier series expansion of the phase probability density function, yields better complexity/performance tradeoff with respect to the usual discretized-phase method. Our second new approach, based on the Tikhonov canonical distribution, yields near-optimal performance at very low complexity and is shown to be much more robust than the Kalman method to the placement of pilot symbols in the coded frame. We present numerical results for binary LDPC codes and LDPC-coded modulation, with particular reference to some phase-noise models and coded-modulation formats standardized in the next-generation satellite Digital Video Broadcasting (DVB-S2). These results show that our algorithms achieve near-coherent performance at very low complexity without requiring any change to the existing DVB-S2 standard.

Space-time finite element techniques for computation of fluid-structure interactions

Abstract: We describe the space–time finite element techniques we developed for computation of fluid–structure interaction (FSI) problems. Among these techniques are the deforming-spatial-domain/stabilized space–time (DSD/SST) formulation and its special version, and the mesh update methods, including the solid-extension mesh moving technique (SEMMT). Also among these techniques are the block-iterative, quasi-direct and direct coupling methods for the solution of the fully discretized, coupled fluid and structural mechanics equations. We present some test computations for the mesh moving techniques described. We also present numerical examples where the fluid is governed by the Navier– Stokes equations of incompressible flows and the structure is governed by the membrane and cable equations. Overall, we demonstrate that the techniques we have developed have increased the scope and accuracy of the methods used in computation of FSI problems. � 2005 Elsevier B.V. All rights reserved.

Reduced-Order Models for MEMS Applications

Abstract: We review the development of reduced-order models for MEMS devices. Based on their implementation procedures, we classify these reduced-order models into two broad categories: node and domain methods. Node methods use lower-order approximations of the system matrices found by evaluating the system equations at each node in the discretization mesh. Domain-based methods rely on modal analysis and the Galerkin method to rewrite the system equations in terms of domain-wide modes (eigenfunctions). We summarize the major contributions in the field and discuss the advantages and disadvantages of each implementation. We then present reduced-order models for microbeams and rectangular and circular microplates. Finally, we present reduced-order approaches to model squeeze-film and thermoelastic damping in MEMS and present analytical expressions for the damping coefficients. We validate these models by comparing their results with available theoretical and experimental results.

On the discretization schemes for the CIR (and Bessel squared) processes

Abstract: In this paper, we focus on the simulation of the Cox-Ingersoll-Ross processes and present several discretization schemes of both the implicit and explicit types. We study their strong and weak convergence. We also examine numerically their behaviour and compare them to the schemes already proposed by Deelstra and Delbaen and Diop. Finally, we gather all the results obtained and recommend, in the standard case, the use of one of our explicit schemes.

On High Order Strong Stability Preserving Runge---Kutta and Multi Step Time Discretizations

Abstract: Strong stability preserving (SSP) high order time discretizations were developed for solution of semi-discrete method of lines approximations of hyperbolic partial differential equations. These high order time discretization methods preserve the strong stability properties---in any norm or seminorm--of the spatial discretization coupled with first order Euler time stepping. This paper describes the development of SSP methods and the recently developed theory which connects the timestep restriction on SSP methods with the theory of monotonicity and contractivity. Optimal explicit SSP Runge---Kutta methods for nonlinear problems and for linear problems as well as implicit Runge---Kutta methods and multi step methods will be collected

Computing hierarchical curve-skeletons of 3D objects

Abstract: A curve-skeleton of a 3D object is a stick-like figure or centerline representation of that object. It is used for diverse applications, including virtual colonoscopy and animation. In this paper, we introduce the concept of hierarchical curve-skeletons and describe a general and robust methodology that computes a family of increasingly detailed curve-skeletons. The algorithm is based upon computing a repulsive force field over a discretization of the 3D object and using topological characteristics of the resulting vector field, such as critical points and critical curves, to extract the curve-skeleton. We demonstrate this method on many different types of 3D objects (volumetric, polygonal and scattered point sets) and discuss various extensions of this approach.

Preconditioning Lanczos Approximations to the Matrix Exponential

Abstract: The Lanczos method is an iterative procedure to compute an orthogonal basis for the Krylov subspace generated by a symmetric matrix $A$ and a starting vector $v$. An interesting application of this method is the computation of the matrix exponential $\exp(-\tau A)v$. This vector plays an important role in the solution of parabolic equations where $A$ results from some form of discretization of an elliptic operator. In the present paper we will argue that for these applications the convergence behavior of this method can be unsatisfactory. We will propose a modified method that resolves this by a simple preconditioned transformation at the cost of an inner-outer iteration. A priori error bounds are presented that are independent of the norm of $A$. This shows that the worst case convergence speed is independent of the mesh width in the spatial discretization of the elliptic operator. We discuss, furthermore, a posteriori error estimation and the tuning of the coupling between the inner and outer iteration. We conclude with several numerical experiments with the proposed method.

A linearly conforming point interpolation method (lc-pim) for 2d solid mechanics problems

Abstract: A linearly conforming point interpolation method (LC-PIM) is developed for 2D solid problems. In this method, shape functions are generated using the polynomial basis functions and a scheme for the selection of local supporting nodes based on background cells is suggested, which can always ensure the moment matrix is invertible as long as there are no coincide nodes. Galerkin weak form is adopted for creating discretized system equations, and a nodal integration scheme with strain smoothing operation is used to perform the numerical integration. The present LC-PIM can guarantee linear exactness and monotonic convergence for the numerical results. Numerical examples are used to examine the present method in terms of accuracy, convergence, and efficiency. Compared with the finite element method (FEM) using linear triangle elements and the radial point interpolation method (RPIM) using Gauss integration, the LC-PIM can achieve higher convergence rate and better efficiency.

Analysis of Composite Plates Using a Layerwise Theory and Multiquadrics Discretization

Abstract: A layerwise shear deformation theory for composite laminated plates is discretized using multiquadrics. This new type of meshless method considers radial basis functions as the approximation method for both the differential governing equations and the boundary conditions. The combination of this layerwise theory and the multiquadrics discretization method allows a very accurate prediction of the field variables. Laminated composite and sandwich plates are analyzed.

Analysis and Approximation of Contact Problems with Adhesion or Damage

Abstract: Preface List of Symbols Modeling and Mathematical Background Basic Equations and Boundary Conditions Physical Setting and Evolution Equations Boundary Conditions Contact Processes with Adhesion Constitutive Equations with Damage Preliminaries on Functional Analysis Function Spaces and Their Properties Elements of Nonlinear Analysis Standard Results on Variational Inequalities and Evolution Equations Elementary Inequalities Preliminaries on Numerical Analysis Finite Difference and Finite Element Discretizations Approximation of Displacements and Velocities Estimates on the Discretization of Adhesion Evolution Estimates on the Discretization of Damage Evolution Estimates on the Discretization of Viscoelastic Constitutive Law Estimates on the Discretization of Viscoplastic Constitutive Law Frictionless Contact Problems with Adhesion Quasistatic Viscoelastic Contact with Adhesion Problem Statement Existence and uniqueness Continuous Dependence on the Data Spatially Semidiscrete Numerical Approximation Fully Discrete Numerical Approximation Dynamic Viscoelastic Contact with Adhesion Problem Statement Existence and Uniqueness Fully Discrete Numerical Approximation Quasistatic Viscoplastic Contact with Adhesion Problem Statement Existence and Uniqueness for the Signorini Problem Numerical Approximation for the Signorini Problem Existence and Uniqueness for the Problem with Normal Compliance Numerical Approximation of the Problem with Normal Compliance Relation between the Signorini and Normal Compliance Problems Contact Problems with Damage Quasistatic Viscoelastic Contact with Damage Problem Statement Existence and Uniqueness Fully Discrete Numerical Approximation Dynamic Viscoelastic Contact with Damage Problem Statement Existence and Uniqueness Fully Discrete Numerical Approximation Quasistatic Viscoplastic Contact with Damage Problem Statement Existence and Uniqueness for the Signorini Problem Numerical Approximation for the Signorini Problem Existence and Uniqueness for the Problem with Normal Compliance Numerical Approximation of the Problem with Normal Compliance Relation between the Signorini and Normal Compliance Problems Notes, Comments, and Conclusions Bibliographical Notes, Problems for Future Research, and Conclusions Bibliographical Notes Problems for Future Research Conclusions References Index

Two dimensional mortar contact methods for large deformation frictional sliding

Abstract: This paper presents a mortar-based formulation for the solution of two dimensional frictional contact problems involving finite deformation and large sliding. As is widely recognized, traditional node-to-surface contact formulations have several drawbacks in solution of deformable-to-deformable contact problems, including lack of general patch test passage, degradation of spatial convergence rates, and robustness issues associated with the faceted representation of contacting surfaces. The mortar finite element method, initially proposed as a technique to join dissimilarly meshed domains, has been shown to preserve optimal convergence rates in tied contact problems (see (Discretization Methods and Iterative Solvers Based on Domain Decomposition, Springer-Verlag, Heidelberg, 2001) for a recent review), and is examined here as an alternative spatial discretization method for large sliding contact. In particular, a novel description for frictional sliding conditions in large deformation mortar formulations is proposed in this work. In recent years, the mortar element method has already been successfully implemented to solve frictional contact problems with linearized kinematics (see (Int. J. Numer. Meth. Engng 1993; 36: 3451)). However, in the presence of large deformations and finite sliding, one must face difficulties associated with the definition and linearization of contact virtual work in the case where the mortar projection has a direct dependence on the tangential relative motion along the interface. In this paper, such a formulation is presented, with particular emphasis on key aspects of the linearization procedure and on the robust description of the friction kinematics. Some novel techniques are proposed to treat the non-smoothness in the contact geometry and the searching required to define mortar segments. A number of numerical examples illustrate the performance and accuracy of the proposed formulation. Copyright © 2004 John Wiley & Sons, Ltd.