Showing papers on "Discretization published in 2008"

Dynamic Spectrum Management: Complexity and Duality

Abstract: Consider a communication system whereby multiple users share a common frequency band and must choose their transmit power spectral densities dynamically in response to physical channel conditions. Due to co-channel interference, the achievable data rate of each user depends on not only the power spectral density of its own, but also those of others in the system. Given any channel condition and assuming Gaussian signaling, we consider the problem to jointly determine all users' power spectral densities so as to maximize a system-wide utility function (e.g., weighted sum-rate of all users), subject to individual power constraints. For the discretized version of this nonconvex problem, we characterize its computational complexity by establishing the NP-hardness under various practical settings, and identify subclasses of the problem that are solvable in polynomial time. Moreover, we consider the Lagrangian dual relaxation of this nonconvex problem. Using the Lyapunov theorem in functional analysis, we rigorously prove a result first discovered by Yu and Lui (2006) that there is a zero duality gap for the continuous (Lebesgue integral) formulation. Moreover, we show that the duality gap for the discrete formulation vanishes asymptotically as the size of discretization decreases to zero.

A novel pole figure inversion method: specification of the MTEX algorithm

Abstract: A novel algorithm for ODF (orientation density function) estimation from diffraction pole figures is presented which is especially well suited for sharp textures and high-resolution pole figures measured with respect to arbitrarily scattered specimen directions, e.g. by area detectors. The estimated ODF is computed as the solution of a minimization problem which is based on a model of the diffraction counts as a Poisson process. The algorithm applies discretization by radially symmetric functions and fast Fourier techniques to guarantee smooth approximation and high performance. An implementation of the algorithm is freely available as part of the texture analysis software MTEX.

Isogeometric fluid-structure interaction: theory, algorithms, and computations

Abstract: We present a fully-coupled monolithic formulation of the fluid-structure interaction of an incompressible fluid on a moving domain with a nonlinear hyperelastic solid. The arbitrary Lagrangian–Eulerian description is utilized for the fluid subdomain and the Lagrangian description is utilized for the solid subdomain. Particular attention is paid to the derivation of various forms of the conservation equations; the conservation properties of the semi-discrete and fully discretized systems; a unified presentation of the generalized-α time integration method for fluid-structure interaction; and the derivation of the tangent matrix, including the calculation of shape derivatives. A NURBS-based isogeometric analysis methodology is used for the spatial discretization and three numerical examples are presented which demonstrate the good behavior of the methodology.

A Multiplicative Calderon Preconditioner for the Electric Field Integral Equation

Abstract: In this paper, a new technique for preconditioning electric field integral equations (EFIEs) by leveraging Calderon identities is presented. In contrast to all previous Calderon preconditioners, the proposed preconditioner is purely multiplicative in nature, applicable to open and closed structures, straightforward to implement, and easily interfaced with existing method of moments (MoM) code. Numerical results demonstrate that the MoM EFIE system obtained using the proposed preconditioning converges rapidly, independently of the discretization density.

A generalized gradient smoothing technique and the smoothed bilinear form for galerkin formulation of a wide class of computational methods

Abstract: This paper presents a generalized gradient smoothing technique, the corresponding smoothed bilinear forms, and the smoothed Galerkin weakform that is applicable to create a wide class of efficient numerical methods with special properties including the upper bound properties. A generalized gradient smoothing technique is first presented for computing the smoothed strain fields of displacement functions with discontinuous line segments, by "rudely" enforcing the Green's theorem over the smoothing domain containing these discontinuous segments. A smoothed bilinear form is then introduced for Galerkin formulation using the generalized gradient smoothing technique and smoothing domains constructed in various ways. The numerical methods developed based on this smoothed bilinear form will be spatially stable and convergent and possess three major important properties: (1) it is variationally consistent, if the solution is sought in a Hilbert space; (2) the stiffness of the discretized model will be reduced comp...

Discrete Differential Geometry: Integrable Structure

Abstract: Classical differential geometry Discretization principles. Multidimensional nets Discretization principles. Nets in quadrics Special classes of discrete surfaces Approximation Consistency as integrability Discrete complex analysis. Linear theory Discrete complex analysis. Integrable circle patterns Foundations Solutions of selected exercises Bibliography Notations Index.

Hierarchical Matrices: A Means to Efficiently Solve Elliptic Boundary Value Problems

Abstract: Hierarchical matrices are an efficient framework for large-scale fully populated matrices arising, e.g., from the finite element discretization of solution operators of elliptic boundary value problems. In addition to storing such matrices, approximations of the usual matrix operations can be computed with logarithmic-linear complexity, which can be exploited to setup approximate preconditioners in an efficient and convenient way. Besides the algorithmic aspects of hierarchical matrices, the main aim of this book is to present their theoretical background. The book contains the existing approximation theory for elliptic problems including partial differential operators with nonsmooth coefficients. Furthermore, it presents in full detail the adaptive cross approximation method for the efficient treatment of integral operators with non-local kernel functions. The theory is supported by many numerical experiments from real applications.

Analytical and numerical modelling of the direct metal deposition laser process

Abstract: The direct metal deposition (DMD) laser process is a novel technique, well adapted for aeronautical applications, that allows the building of complex 3D geometries through the interaction between a powder nozzle system and a continuous laser beam. A three-step analytical and numerical approach was carried out to predict the shapes of manufactured structures and thermal loadings induced by the DMD process. First, powder temperature was calculated using a recent analytical model, then the geometry of walls was predicted by a combined numerical + analytical modelling using a discretization of the physical interaction domain, and finally, a finite element calculation was carried out on COMSOL 3.3 Multiphysics software to describe thermal behaviour during DMD of a titanium alloy.Our thermal model takes into account the moving interface during metal deposition with a specific function κ (t, x, y, z) allowing the conductivity front to move simultaneously with the moving laser source (with an appropriate spatial energy distribution), thus representing rather precisely the DMD process. This allowed us to provide an adequate representation of temperatures near the melt-pool, and to reproduce with a good accuracy thermal cycles and melt-pool dimensions during the construction of up to 25-layer walls. This was confirmed by comparisons with experimental thermocouple data T = f(t), and fast camera melt-pool recording.

Discrete Differential Geometry

Abstract: Discretization of Surfaces: Special Classes and Parametrizations.- Surfaces from Circles.- Minimal Surfaces from Circle Patterns: Boundary Value Problems, Examples.- Designing Cylinders with Constant Negative Curvature.- On the Integrability of Infinitesimal and Finite Deformations of Polyhedral Surfaces.- Discrete Hashimoto Surfaces and a Doubly Discrete Smoke-Ring Flow.- The Discrete Green's Function.- Curvatures of Discrete Curves and Surfaces.- Curves of Finite Total Curvature.- Convergence and Isotopy Type for Graphs of Finite Total Curvature.- Curvatures of Smooth and Discrete Surfaces.- Geometric Realizations of Combinatorial Surfaces.- Polyhedral Surfaces of High Genus.- Necessary Conditions for Geometric Realizability of Simplicial Complexes.- Enumeration and Random Realization of Triangulated Surfaces.- On Heuristic Methods for Finding Realizations of Surfaces.- Geometry Processing and Modeling with Discrete Differential Geometry.- What Can We Measure?.- Convergence of the Cotangent Formula: An Overview.- Discrete Differential Forms for Computational Modeling.- A Discrete Model of Thin Shells.

Model for Cyclic Inelastic Buckling of Steel Braces

Abstract: The paper presents a model for the inelastic buckling behavior of steel braces. The model consists of a force-based frame element with distributed inelasticity and fiber discretization of the cross section. With this approach, the response of the element can be derived by integration of the uniaxial stress-strain relation of the fibers and can account for kinematic and isotropic hardening as well as the Bauschinger effect of the material. The interaction between axial force and bending moment is thus accounted for. Even though the element only accounts for small deformations in the basic system, large displacement geometry is included in the nonlinear transformation of the force-deformation relation of the basic element following the concept of the corotational formulation. With this approach, two elements for each brace suffice to yield results that match an extensive set of experimental data of braces with different cross sections and slenderness ratios. Even though the model does not account for the effect of local buckling, this phenomenon does not seem to appreciably affect the global response of braces with compact sections.

Computational acoustics of noise propagation in fluids : finite and boudary element methods

Abstract: Part I FEM: Numerical Aspects 1 Dispersion, Pollution, and Resolution 2 Different Types of Finite Elements 3 Multifrequency Analysis using Matrix Pad'e-via-Lanczos 4 Computational Aeroacoustics based on Lighthill's Acoustic Analogy Part II FEM: External Problems 5 Computational Absorbing Boundaries 6 PerfectlyMatched Layers 7 Infinite Elements 8 Efficient Infinite Elements based on Jacobi Polynomials Part III FEM: Related Problems 9 Fluid-Structure Acoustic Interaction 10 Energy Finite Element Method Part IV BEM: Numerical Aspects 11 Discretization Requirements 12 Fast Solution Methods 13 Multi-domain Boundary Element Method in Acoustics 14 Waveguide Boundary Spectral Finite Elements Part V BEM: External Problems 15 Treating the Phenomenon of Irregular Frequencies 16 A Galerkin-type BE-formulation 17 Acoustical Radiation and Scattering above an Impedance Plane 18 Time Domain BEM Part VI BEM: Related Problems 19 Coupling a Fast BEM with a FE-Formulation for Fluid-Structure Interaction 20 Inverse BE-Techniques for the Holographic Identification of Vibro-Acoustic Source Parameters

A New Asymptotic Preserving Scheme Based on Micro-Macro Formulation for Linear Kinetic Equations in the Diffusion Limit

Abstract: We propose a new numerical scheme for linear transport equations. It is based on a decomposition of the distribution function into equilibrium and nonequilibrium parts. We also use a projection technique that allows us to reformulate the kinetic equation into a coupled system of an evolution equation for the macroscopic density and a kinetic equation for the nonequilibrium part. By using a suitable time semi-implicit discretization, our scheme is able to accurately approximate the solution in both kinetic and diffusion regimes. It is asymptotic preserving in the following sense: when the mean free path of the particles is small, our scheme is asymptotically equivalent to a standard numerical scheme for the limit diffusion model. A uniform stability property is proved for the simple telegraph model. Various boundary conditions are studied. Our method is validated in one-dimensional cases by several numerical tests and comparisons with previous asymptotic preserving schemes.

Newton-GMRES Preconditioning for Discontinuous Galerkin Discretizations of the Navier-Stokes Equations

Abstract: We study preconditioners for the iterative solution of the linear systems arising in the implicit time integration of the compressible Navier-Stokes equations. The spatial discretization is carried out using a discontinuous Galerkin method with fourth order polynomial interpolations on triangular elements. The time integration is based on backward difference formulas resulting in a nonlinear system of equations which is solved at each timestep. This is accomplished using Newton's method. The resulting linear systems are solved using a preconditioned GMRES iterative algorithm. We consider several existing preconditioners such as block Jacobi and Gauss-Seidel combined with multilevel schemes which have been developed and tested for specific applications. While our results are consistent with the claims reported, we find that these preconditioners lack robustness when used in more challenging situations involving low Mach numbers, stretched grids, or high Reynolds number turbulent flows. We propose a preconditioner based on a coarse scale correction with postsmoothing based on a block incomplete LU factorization with zero fill-in (ILU0) of the Jacobian matrix. The performance of the ILU0 smoother is found to depend critically on the element numbering. We propose a numbering strategy based on minimizing the discarded fill-in in a greedy fashion. The coarse scale correction scheme is found to be important for diffusion dominated problems, whereas the ILU0 preconditioner with the proposed ordering is effective at handling the convection dominated case. While little can be said in the way of theoretical results, the proposed preconditioner is shown to perform remarkably well for a broad range of representative test problems. These include compressible flows ranging from very low Reynolds numbers to fully turbulent flows using the Reynolds averaged Navier-Stokes equations discretized on highly stretched grids. For low Mach number flows, the proposed preconditioner is more than one order of magnitude more efficient than the other preconditioners considered.

A mass spring model for hair simulation

Abstract: Our goal is to simulate the full hair geometry, consisting of approximately one hundred thousand hairs on a typical human head. This will require scalable methods that can simulate every hair as opposed to only a few guide hairs. Novel to this approach is that the individual hair/hair interactions can be modeled with physical parameters (friction, static attraction, etc.) at the scale of a single hair as opposed to clumped or continuum interactions. In this vein, we first propose a new altitude spring model for preventing collapse in the simulation of volumetric tetrahedra, and we show that it is also applicable both to bending in cloth and torsion in hair. We demonstrate that this new torsion model for hair behaves in a fashion similar to more sophisticated models with significantly reduced computational cost. For added efficiency, we introduce a semi-implicit discretization of standard springs that makes them truly linear in multiple spatial dimensions and thus unconditionally stable without requiring Newton-Raphson iteration. We also simulate complex hair/hair interactions including sticking and clumping behavior, collisions with objects (e.g. head and shoulders) and self-collisions. Notably, in line with our goal to simulate the full head of hair, we do not generate any new hairs at render time.

A multiplicative Calderón preconditioner for the electric field integral equation

Abstract: A new technique for preconditioning electric field integral equations (EFIEs) by leveraging Calderon identities is presented. In contrast to all previous Calderon EFIE preconditioners, the proposed preconditioner is purely multiplicative in nature, applicable to open and closed structures, straightforward to implement, and easily interfaced with existing method of moments codes. Numerical results demonstrate that the method of moments (MoM) matrix equations obtained using the proposed preconditioner converge rapidly, independently of the discretization density.

Convergence rates for direct transcription of optimal control problems using collocation at Radau points

Abstract: We present convergence rates for the error between the direct transcription solution and the true solution of an unconstrained optimal control problem. The problem is discretized using collocation at Radau points (aka Gauss-Radau or Legendre-Gauss-Radau quadrature). The precision of Radau quadrature is the highest after Gauss (aka Legendre-Gauss) quadrature, and it has the added advantage that the end point is one of the abscissas where the function, to be integrated, is evaluated. We analyze convergence from a Nonlinear Programming (NLP)/matrix algebra perspective. This enables us to predict the norms of various constituents of a matrix that is "close" to the KKT matrix of the discretized problem. We present the convergence rates for the various components, for a sufficiently small discretization size, as functions of the discretization size and the number of collocation points. We illustrate this using several test examples. This also leads to an adjoint estimation procedure, given the Lagrange multipliers for the large scale NLP.

A Priori Error Estimates for Space-Time Finite Element Discretization of Parabolic Optimal Control Problems Part I: Problems Without Control Constraints

Abstract: In this paper we develop a priori error analysis for Galerkin finite element discretizations of optimal control problems governed by linear parabolic equations. The space discretization of the state variable is done using usual conforming finite elements, whereas the time discretization is based on discontinuous Galerkin methods. For different types of control discretizations we provide error estimates of optimal order with respect to both space and time discretization parameters. The paper is divided into two parts. In the first part we develop some stability and error estimates for space-time discretization of the state equation and provide error estimates for optimal control problems without control constraints. In the second part of the paper, the techniques and results of the first part are used to develop a priori error analysis for optimal control problems with pointwise inequality constraints on the control variable.

Recursive Trust-Region Methods for Multiscale Nonlinear Optimization

Abstract: A class of trust-region methods is presented for solving unconstrained nonlinear and possibly nonconvex discretized optimization problems, like those arising in systems governed by partial differential equations. The algorithms in this class make use of the discretization level as a means of speeding up the computation of the step. This use is recursive, leading to true multilevel/multiscale optimization methods reminiscent of multigrid methods in linear algebra and the solution of partial differential equations. A simple algorithm of the class is then described and its numerical performance is shown to be numerically promising. This observation then motivates a proof of global convergence to first-order stationary points on the fine grid that is valid for all algorithms in the class.

Benchmark on Discretization Schemes for Anisotropic Diffusion Problems on General Grids

Abstract: We present here a number of test cases and meshes which were designed to form a benchmark for finite volume schemes and give a summary of some of the results which were presented by the participants to this benchmark. We address a two-dimensional anisotropic diffusion problem, which is discretized on general, possibly non-conforming meshes. In most cases, the diffusion tensor is taken to be anisotropic, and at times heterogeneous and/or discontinuous. The meshes are either triangular or quadrangular, and sometimes quite distorted. Several methods were tested, among which finite element, discontinous Galerkin, cell centred and vertex centred finite volume methods, discrete duality finite volume methods, mimetic methods. The results given by the participants to the benchmark range from the number of unknowns, the errors on the fluxes or the minimum and maximum values and energy, to the order of convergence (when available).

Iterative real-time path integral approach to nonequilibrium quantum transport

Abstract: We have developed a numerical approach to compute real-time path integral expressions for quantum transport problems out of equilibrium. The scheme is based on a deterministic iterative summation of the path integral for the generating function of the nonequilibrium current. Self-energies due to the leads, being nonlocal in time, are fully taken into account within a finite memory time, thereby including non-Markovian effects, and numerical results are extrapolated both to vanishing (Trotter) time discretization and to infinite memory time. This extrapolation scheme converges except at very low temperatures, and the results are then numerically exact. The method is applied to nonequilibrium transport through an Anderson dot.

Proper orthogonal decomposition for optimality systems

Abstract: Proper orthogonal decomposition (POD) is a powerful technique for model reduction of non-linear systems. It is based on a Galerkin type discretization with basis elements created from the dynamical system itself. In the context of optimal control this approach may suffer from the fact that the basis elements are computed from a reference trajectory containing features which are quite different from those of the optimally controlled trajectory. A method is proposed which avoids this problem of unmodelled dynamics in the proper orthogonal decomposition approach to optimal control. It is referred to as optimality system proper orthogonal decomposition (OS-POD).