Showing papers on "Discretization published in 2009"

Nonlinear Finite Element Methods

Abstract: Nonlinear Phenomena.- Basic Equations of Continuum Mechanics.- Spatial Discretization Techniques.- Solution Methods for Time Independent Problems.- Solution Methods for Time Dependent Problems.- Stability Problems.- Adaptive Methods.- Special Structural Elements.- Special Finite Elements for Continua.- Contact Problems.- Automation of the Finite Element Method by J. Korelc.

Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments

Abstract: This paper presents a modified regularized formulation of the Ambrosio–Tortorelli type to introduce the crack non-interpenetration condition in the variational approach to fracture mechanics proposed by Francfort and Marigo [1998. Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46 (8), 1319–1342]. We focus on the linear elastic case where the contact condition appears as a local unilateral constraint on the displacement jump at the crack surfaces. The regularized model is obtained by splitting the strain energy in a spherical and a deviatoric parts and accounting for the sign of the local volume change. The numerical implementation is based on a standard finite element discretization and on the adaptation of an alternate minimization algorithm used in previous works. The new regularization avoids crack interpenetration and predicts asymmetric results in traction and in compression. Even though we do not exhibit any gamma-convergence proof toward the desired limit behavior, we illustrate through several numerical case studies the pertinence of the new model in comparison to other approaches.

Finite element exterior calculus: from Hodge theory to numerical stability

Abstract: This article reports on the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computation, the other from topology and geometry. In it we consider the numerical discretization of partial differential equations that are related to differential complexes so that de Rham cohomology and Hodge theory are key tools for the continuous problem. After a brief introduction to finite element methods, the discretization methods we consider, we develop an abstract Hilbert space framework for analyzing stability and convergence. In this framework, the differential complex is represented by a complex of Hilbert spaces and stability is obtained by transferring Hodge theoretic structures from the continuous level to the discrete. We show stable discretization is achieved if the finite element spaces satisfy two hypotheses: they form a subcomplex and there exists a bounded cochain projection from the full complex to the subcomplex. Next, we consider the most canonical example of the abstract theory, in which the Hilbert complex is the de Rham complex of a domain in Euclidean space. We use the Koszul complex to construct two families of finite element differential forms, show that these can be arranged in subcomplexes of the de Rham complex in numerous ways, and for each construct a bounded cochain projection. The abstract theory therefore applies to give the stability and convergence of finite element approximations of the Hodge Laplacian. Other applications are considered as well, especially to the equations of elasticity. Background material is included to make the presentation self-contained for a variety of readers.

Large-Eddy Simulation for compressible flows

Abstract: LES Governing Equations.- Compressible Turbulence Dynamics.- Functional Modeling.- Explicit Structural Modeling.- Relation Between SGS Model and Numerical Discretization.- Boundary Conditions for Large-Eddy Simulation of Compressible Flows.- Subsonic Applications with Compressibility Effects.- Supersonic Applications.- Supersonic Applications with Shock-Turbulence Interaction.

Implementation of semi-discrete, non-staggered central schemes in a colocated, polyhedral, finite volume framework, for high-speed viscous flows

Abstract: We describe the implementation of a computational fluid dynamics solver for the simulation of high-speed flows. It comprises a finite volume (FV) discretization using semi-discrete, non-staggered central schemes for colocated variables prescribed on a mesh of polyhedral cells that have an arbitrary number of faces. We describe the solver in detail, explaining the choice of variables whose face interpolation is limited, the choice of limiter, and a method for limiting the interpolation of a vector field that is independent of the coordinate system. The solution of momentum and energy transport in the Navier-Stokes equations uses an operator-splitting approach: first, we solve an explicit predictor equation for the convection of conserved variables, then an implicit corrector equation for the diffusion of primitive variables. Our solver is validated against four sets of data: (1) an analytical solution of the one-dimensional shock tube case; (2) a numerical solution of two dimensional, transient, supersonic flow over a forward-facing step; (3) interferogram density measurements of a supersonic jet from a circular nozzle; and (4) pressure and heat transfer measurements in hypersonic flow over a 25-55 biconic. Our results indicate that the central-upwind scheme of Kurganov, Noelle and Petrova (SIAM J. Sci. Comput. 2001; 23:707-740) is competitive with the best methods previously published (e.g. piecewise parabolic method, Roe solver with van Leer limiting) and that it is inherently simple and well suited to a colocated, polyhedral FV framework. Copyright © 2009 John Wiley & Sons, Ltd.

High Order Strong Stability Preserving Time Discretizations

Abstract: Strong stability preserving (SSP) high order time discretizations were developed to ensure nonlinear stability properties necessary in the numerical solution of hyperbolic partial differential equations with discontinuous solutions SSP methods preserve the strong stability properties--in any norm, seminorm or convex functional--of the spatial discretization coupled with first order Euler time stepping This paper describes the development of SSP methods and the connections between the timestep restrictions for strong stability preservation and contractivity Numerical examples demonstrate that common linearly stable but not strong stability preserving time discretizations may lead to violation of important boundedness properties, whereas SSP methods guarantee the desired properties provided only that these properties are satisfied with forward Euler timestepping We review optimal explicit and implicit SSP Runge---Kutta and multistep methods, for linear and nonlinear problems We also discuss the SSP properties of spectral deferred correction methods

An Integer Projected Fixed Point Method for Graph Matching and MAP Inference

Abstract: Graph matching and MAP inference are essential problems in computer vision and machine learning. We introduce a novel algorithm that can accommodate both problems and solve them efficiently. Recent graph matching algorithms are based on a general quadratic programming formulation, which takes in consideration both unary and second-order terms reflecting the similarities in local appearance as well as in the pairwise geometric relationships between the matched features. This problem is NP-hard, therefore most algorithms find approximate solutions by relaxing the original problem. They find the optimal continuous solution of the modified problem, ignoring during optimization the original discrete constraints. Then the continuous solution is quickly binarized at the end, but very little attention is put into this final discretization step. In this paper we argue that the stage in which a discrete solution is found is crucial for good performance. We propose an efficient algorithm, with climbing and convergence properties, that optimizes in the discrete domain the quadratic score, and it gives excellent results either by itself or by starting from the solution returned by any graph matching algorithm. In practice it outperforms state-or-the art graph matching algorithms and it also significantly improves their performance if used in combination. When applied to MAP inference, the algorithm is a parallel extension of Iterated Conditional Modes (ICM) with climbing and convergence properties that make it a compelling alternative to the sequential ICM. In our experiments on MAP inference our algorithm proved its effectiveness by significantly outperforming [13], ICM and Max-Product Belief Propagation.

Nonlinear simulations of solid tumor growth using a mixture model: invasion and branching

Abstract: We develop a thermodynamically consistent mixture model for avascular solid tumor growth which takes into account the effects of cell-to-cell adhesion, and taxis inducing chemical and molecular species. The mixture model is well-posed and the governing equations are of Cahn-Hilliard type. When there are only two phases, our asymptotic analysis shows that earlier single-phase models may be recovered as limiting cases of a two-phase model. To solve the governing equations, we develop a numerical algorithm based on an adaptive Cartesian block-structured mesh refinement scheme. A centered-difference approximation is used for the space discretization so that the scheme is second order accurate in space. An implicit discretization in time is used which results in nonlinear equations at implicit time levels. We further employ a gradient stable discretization scheme so that the nonlinear equations are solvable for very large time steps. To solve those equations we use a nonlinear multilevel/multigrid method which is of an optimal order O(N) where N is the number of grid points. Spherically symmetric and fully two dimensional nonlinear numerical simulations are performed. We investigate tumor evolution in nutrient-rich and nutrient-poor tissues. A number of important results have been uncovered. For example, we demonstrate that the tumor may suffer from taxis-driven fingering instabilities which are most dramatic when cell proliferation is low, as predicted by linear stability theory. This is also observed in experiments. This work shows that taxis may play a role in tumor invasion and that when nutrient plays the role of a chemoattractant, the diffusional instability is exacerbated by nutrient gradients. Accordingly, we believe this model is capable of describing complex invasive patterns observed in experiments.

Adaptivity with moving grids

Abstract: In this article we look at the modern theory of moving meshes as part of an r-adaptive strategy for solving partial differential equations with evolving internal structure. We firstly examine the possible geometries of a moving mesh in both one and higher dimensions, and the discretization of partial differential equation on such meshes. In particular, we consider such issues as mesh regularity, equidistribution, variational methods, and the error in interpolating a function or truncation error on such a mesh. We show that, guided by these, we can design effective moving mesh strategies. We then look in more detail as to how these strategies are implemented. Firstly we look at position-based methods and the use of moving mesh partial differential equation (MMPDE), variational and optimal transport methods. This is followed by an analysis of velocity-based methods such as the geometric conservation law (GCL) methods. Finally we look at a number of examples where the use of a moving mesh method is effective in applications. These include scale-invariant problems, blow-up problems, problems with moving fronts and problems in meteorology. We conclude that, whilst r-adaptive methods are still in a relatively new stage of development, with many outstanding questions remaining, they have enormous potential for development, and for many problems they represent an optimal form of adaptivity.

Surface integral formulation for 3D simulations of plasmonic and high permittivity nanostructures

Abstract: Among the most popular approaches used for simulating plasmonic systems, the discrete dipole approximation suffers from poorly scaling volume discretization and limited near-field accuracy. We demonstrate that transformation to a surface integral formulation improves scalability and convergence and provides a flexible geometric approximation allowing, e.g., to investigate the influence of fabrication accuracy. The occurring integrals can be solved quasi-analytically, permitting even rapidly changing fields to be determined arbitrarily close to a scatterer. This insight into the extreme near-field behavior is useful for modeling closely packed particle ensembles and to study "hot spots" in plasmonic nanostructures used for plasmon-enhanced Raman scattering.

Optimal control problems with delays in state and control variables subject to mixed control–state constraints

Abstract: Optimal control problems with delays in state and control variables are studied. Constraints are imposed as mixed control–state inequality constraints. Necessary optimality conditions in the form of Pontryagin's minimum principle are established. The proof proceeds by augmenting the delayed control problem to a nondelayed problem with mixed terminal boundary conditions to which Pontryagin's minimum principle is applicable. Discretization methods are discussed by which the delayed optimal control problem is transformed into a large-scale nonlinear programming problem. It is shown that the Lagrange multipliers associated with the programming problem provide a consistent discretization of the advanced adjoint equation for the delayed control problem. An analytical example and numerical examples from chemical engineering and economics illustrate the results. Copyright © 2008 John Wiley & Sons, Ltd.

Discretization for naive-Bayes learning: managing discretization bias and variance

Abstract: Quantitative attributes are usually discretized in Naive-Bayes learning. We establish simple conditions under which discretization is equivalent to use of the true probability density function during naive-Bayes learning. The use of different discretization techniques can be expected to affect the classification bias and variance of generated naive-Bayes classifiers, effects we name discretization bias and variance. We argue that by properly managing discretization bias and variance, we can effectively reduce naive-Bayes classification error. In particular, we supply insights into managing discretization bias and variance by adjusting the number of intervals and the number of training instances contained in each interval. We accordingly propose proportional discretization and fixed frequency discretization, two efficient unsupervised discretization methods that are able to effectively manage discretization bias and variance. We evaluate our new techniques against four key discretization methods for naive-Bayes classifiers. The experimental results support our theoretical analyses by showing that with statistically significant frequency, naive-Bayes classifiers trained on data discretized by our new methods are able to achieve lower classification error than those trained on data discretized by current established discretization methods.

DG Approximation of Coupled Navier-Stokes and Darcy Equations by Beaver-Joseph-Saffman Interface Condition

Abstract: In this work, we couple the incompressible steady Navier-Stokes equations with the Darcy equations, by means of the Beaver-Joseph-Saffman's condition on the interface. Under suitable smallness conditions on the data, we prove existence of a weak solution as well as some a priori estimates. We establish local uniqueness when the data satisfy additional smallness restrictions. Then we propose a discontinuous Galerkin scheme for discretizing the equations and do its numerical analysis.

Reactive Power and Voltage Control in Distribution Systems With Limited Switching Operations

Abstract: An algorithm based on a nonlinear interior-point method and discretization penalties is proposed in this paper for the solution of the mixed-integer nonlinear programming (MINLP) problem associated with reactive power and voltage control in distribution systems to minimize daily energy losses, with time-related constraints being considered. Some of these constraints represent limits on the number of switching operations of transformer load tap changers (LTCs) and capacitors, which are modeled as discrete control variables. The discrete variables are treated here as continuous variables during the solution process, thus transforming the MINLP problem into an NLP problem that can be more efficiently solved exploiting its highly sparse matrix structure; a strategy is developed to round these variables off to their nearest discrete values, so that daily switching operation limits are properly met. The proposed method is compared with respect to other well-known MINLP solution methods, namely, a genetic algorithm and the popular GAMS MINLP solvers BARON and DICOPT. The effectiveness of the proposed method is demonstrated in the well-known PG&E 69-bus distribution network and a real distribution system in the city of Guangzhou, China, where the proposed technique has been in operation since 2003.

Discontinuous Galerkin solution of the Navier-Stokes equations on deformable domains - eScholarship

Abstract: Discontinuous Galerkin Solution of the Navier-Stokes Equations on Deformable Domains P.-O. Persson a J. Bonet band .J. Peraire c,* Department of Mathematics, University of California, Be'rkeley, Berkeley, CA 94720-3840, USA b School of Engineering, Swansea University, Swansea SA2 8PP, UK C Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Abstract We describe a method for computing time-dependent solutions to the compressible Navier-Stokes equations on variable geometries. We introduce a continuous mapping between a fixed reference configuration and the time varying domain, By writing the Navier-Stokes equations as a conservation law for the independent variables in the reference configuration, the complexity introduced by variable geometry is reduced to solving a transformed conservation law in a fixed reference configuration, The spatial discretization is carried out using the Discontinuous Galerkin method on unstructured meshes of triangles, while the time integration is performed using an explicit Runge-K utta method, For general domain changes, the standard scheme fails to preserve exactly the free-stream solution which leads to some accuracy degradation, especially for low order approximations. This situation is remedied by adding an additional equation for the time evolution of the transformation Jacobian to the original conservation law and correcting for the accumulated metric integration errors. A number of results are shown to illustrate the flexibility of the approach to handle high order approximations on complex geometries. Key words: Discontinuous Galerkin, Deformable domains, Navier-Stokes, Arbitrary Lagrangian-Eulerian, Geometric Conservation 1. Introduction There is a growing interest in high-order methods for fluid problems, largely because of their ability to produce highly accurate solutions with minimum numerical dispersion. The Discontinuous Calerkin (DC) method produces stable discretizations of the convective operator for any order discretization, Moreover, it can be used with unstructured meshes of simplices, which appears to be a requirement for real-world complex geometries. In this paper, we present a high order DC formulation for computing high order solutions to problems with variable geometries. Time varying geometries appear in a number of practical applications such us rotor-stator flows, flapping flight or fluid-structure interactions, In such cases, it is necessary to properly account for the time variation of the solution domain if accurate solutions are to be obtained. For the Navier-Stokes equations, there has been a considerable effort in the development of Arbitrary Lagrangian Eulerian (ALE) methods to deal with * Corresponding author. Tel.: +1-617-25.3-1981: Fax.: +1-617-258-,514:3. Ematl address: peraire~mit .edu (J. Peraire). Preprint SUbJIlitted to Computer l\Jethods in Applied l\Jechanics and Engineering January 2009