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Author: National Aeronautics and Space Administration (NASA) Publisher: Createspace Independent Publishing Platform ISBN: 9781722163440 Category : Languages : en Pages : 38
Book Description
In this research effort, the usefulness of hp-version finite elements and adaptive solution-refinement techniques in generating numerical solutions to optimal control problems has been investigated. Under NAG-939, a general FORTRAN code was developed which approximated solutions to optimal control problems with control constraints and state constraints. Within that methodology, to get high-order accuracy in solutions, the finite element mesh would have to be refined repeatedly through bisection of the entire mesh in a given phase. In the current research effort, the order of the shape functions in each element has been made a variable, giving more flexibility in error reduction and smoothing. Similarly, individual elements can each be subdivided into many pieces, depending on the local error indicator, while other parts of the mesh remain coarsely discretized. The problem remains to reduce and smooth the error while still keeping computational effort reasonable enough to calculate time histories in a short enough time for on-board applications. Hodges, Dewey H. and Warner, Michael S. Unspecified Center NAG1-1435...
Author: National Aeronautics and Space Administration (NASA) Publisher: Createspace Independent Publishing Platform ISBN: 9781722163440 Category : Languages : en Pages : 38
Book Description
In this research effort, the usefulness of hp-version finite elements and adaptive solution-refinement techniques in generating numerical solutions to optimal control problems has been investigated. Under NAG-939, a general FORTRAN code was developed which approximated solutions to optimal control problems with control constraints and state constraints. Within that methodology, to get high-order accuracy in solutions, the finite element mesh would have to be refined repeatedly through bisection of the entire mesh in a given phase. In the current research effort, the order of the shape functions in each element has been made a variable, giving more flexibility in error reduction and smoothing. Similarly, individual elements can each be subdivided into many pieces, depending on the local error indicator, while other parts of the mesh remain coarsely discretized. The problem remains to reduce and smooth the error while still keeping computational effort reasonable enough to calculate time histories in a short enough time for on-board applications. Hodges, Dewey H. and Warner, Michael S. Unspecified Center NAG1-1435...
Author: National Aeronautics and Space Adm Nasa Publisher: Independently Published ISBN: 9781792083686 Category : Languages : en Pages : 28
Book Description
The purpose of this research effort was to begin the study of the application of hp-version finite elements to the numerical solution of optimal control problems. Under NAG-939, the hybrid MACSYMA/FORTRAN code GENCODE was developed which utilized h-version finite elements to successfully approximate solutions to a wide class of optimal control problems. In that code the means for improvement of the solution was the refinement of the time-discretization mesh. With the extension to hp-version finite elements, the degrees of freedom include both nodal values and extra interior values associated with the unknown states, co-states, and controls, the number of which depends on the order of the shape functions in each element. One possible drawback is the increased computational effort within each element required in implementing hp-version finite elements. We are trying to determine whether this computational effort is sufficiently offset by the reduction in the number of time elements used and improved Newton-Raphson convergence so as to be useful in solving optimal control problems in real time. Because certain of the element interior unknowns can be eliminated at the element level by solving a small set of nonlinear algebraic equations in which the nodal values are taken as given, the scheme may turn out to be especially powerful in a parallel computing environment. A different processor could be assigned to each element. The number of processors, strictly speaking, is not required to be any larger than the number of sub-regions which are free of discontinuities of any kind. Hodges, Dewey H. and Warner, Michael S. NASA-CR-192220, NAS 1.26:192220 NAG1-1435...
Author: Wenbin Liu Publisher: Alpha Science International Limited ISBN: 9781842657157 Category : Mathematics Languages : en Pages : 197
Book Description
Summary: "This book emphasizes the discussions of some unique issues from the adaptive finite element approximation of optimal control. The main idea used in the approximation error analysis (both a priori and a posteriori) is to first combine convex analysis and interpolation error estimations of suitable interpolators, which much depend on the structure of the control constraints, to derive the error estimates for the control via the variational inequalities in the optimality conditions, and then to apply the standard techniques to derive the error estimates for the state equations. The need, the framework and the techniques of using multi adaptive meshes in developing efficient numerical algorithms for optimal control have been emphasized throughout the book. The book starts from several typical examples of optimal control problems and then discusses existence and optimality conditions for some optimal control problems. It is believed that these discussions are especially useful for the researchers and students who first entered this area. Then the finite element approximation schemes for several typical optimal control problems are set up, their a priori and a posteriori error estimates are derived following the main idea mentioned, and their computational methods are studied."-- Publisher website, viewed 13th July, 2012.
Author: Ekkehard Ramm Publisher: John Wiley & Sons ISBN: 9780471496502 Category : Technology & Engineering Languages : en Pages : 426
Book Description
Finite Element Methods are used for numerous engineering applications where numerical solutions of partial differential equations are needed. As computers can now deal with the millions of parameters used in these methods, automatic error estimation and automatic adaptation of the utilised method (according to this error estimation), has become a hot research topic. This text offers comprehensive coverage of this new field of automatic adaptation and error estimation, bringing together the work of eight outstanding researchers in this field who have completed a six year national research project within the German Science Foundation. The result is a state-of-the-art work in true reference style. Each chapter is self-contained and covers theoretical, algorithmic and software presentations as well as solved problems. A main feature consists of several carefully elaborated benchmarks of 2D- and 3D- applications. First book to go beyond the Finite Element Method in itself Covers material from a new research area Presents benchmarks of 2D- and 3D- applications Fits with the new trend for genetic strategies in engineering
Author: Leszek Demkowicz Publisher: CRC Press ISBN: 1420011685 Category : Mathematics Languages : en Pages : 428
Book Description
Offering the only existing finite element (FE) codes for Maxwell equations that support hp refinements on irregular meshes, Computing with hp-ADAPTIVE FINITE ELEMENTS: Volume 1. One- and Two-Dimensional Elliptic and Maxwell Problems presents 1D and 2D codes and automatic hp adaptivity. This self-contained source discusses the theory and implementat
Author: Stefan Klaus Wilhelm Henneking Publisher: ISBN: Category : Languages : en Pages : 382
Book Description
In this dissertation, we present a scalable parallel version of hp3D—a finite element (FE) software for analysis and discretization of complex three-dimensional multiphysics applications. The developed software supports hybrid MPI/OpenMP parallelism for large-scale computation on modern manycore architectures. The focus of the effort lies on the development and optimization of the parallel software infrastructure underlying all distributed computation. We discuss the challenges of designing efficient data structures for isotropic and anisotropic hp-adaptive meshes with tetrahedral, hexahedral, prismatic, and pyramidal elements supporting discretization of the exact sequence energy spaces. While the code supports standard Galerkin methods, special emphasis is given to systems arising from discretization with the discontinuous Petrov–Galerkin (DPG) method. The method guarantees discrete stability by employing locally optimal test functions, and it has a built-in error indicator which we exploit to guide mesh adaptivity. In addition to interfacing with third-party packages for various tasks, we have developed our own tools including a parallel nested dissection solver suitable for scalable FE computation of waveguide geometries. We present weak-scaling results with up to 24576 CPU cores and numerical simulations with more than one billion degrees of freedom. The new software capabilities enable solution of challenging wave propagation problems with important applications in acoustics, elastodynamics, and electromagnetics. These applications are difficult to solve in the high-frequency regime because the FE discretization suffers from significant numerical pollution errors that increase with the wavenumber. It is critical to control these errors to obtain a stable and accurate method. We study the pollution effect for waveguide problems with more than 8000 wavelengths in the context of robust DPG FE discretizations for the time-harmonic Maxwell equations. We also discuss adaptive refinement strategies for multi-mode fiber waveguides where the propagating transverse modes must be resolved sufficiently. Our study shows the applicability of the DPG error indicator to this class of problems. Finally, we present both modeling and computational advancements to a unique three-dimensional DPG FE model for the simulation of laser amplification in a fiber amplifier. Fiber laser amplifiers are of interest in communication technology, medical applications, military defense capabilities, and various other fields. Silica fiber amplifiers can achieve high-power operation with great efficiency. At high optical intensities, multi-mode amplifiers suffer from undesired thermal coupling effects which pose a major obstacle in power-scaling of such devices. Our nonlinear 3D vectorial model is based on the time-harmonic Maxwell equations, and it incorporates both amplification via an active dopant and thermal effects via coupling with the heat equation. The model supports co-, counter-, and bi-directional pumping configurations, as well as inhomogeneous and anisotropic material properties. The high-fidelity simulation comes at the cost of a high-order FE discretization with many degrees of freedom per wavelength. To make the computation more feasible, we have developed a novel longitudinal model rescaling, using artificial material parameters with the goal of preserving certain quantities of interest. Numerical tests demonstrate the applicability and utility of this scaled model in the simulation of an ytterbium-doped, step-index fiber amplifier that experiences laser amplification and heating
Author: Wolfgang Bangerth Publisher: Springer Science & Business Media ISBN: 9783764370091 Category : Mathematics Languages : en Pages : 222
Book Description
The key issues are a posteriori error estimation and it automatic mesh adaptation. Besides the traditional approach of energy-norm error control, a new duality-based technique, the Dual Weighted Residual method for goal-oriented error estimation, is discussed in detail. This method aims at economical computation of arbitrary quantities of physical interest by properly adapting the computational mesh. This is typically required in the design cycles of technical applications. For example, the drag coefficient of a body immersed in a viscous flow is computed, then it is minimized by varying certain control parameters, and finally the stability of the resulting flow is investigated by solving an eigenvalue problem. `Goal-oriented' adaptivity is designed to achieve these tasks with minimal cost. At the end of each chapter some exercises are posed in order to assist the interested reader in better understanding the concepts presented. Solutions and accompanying remarks are given in the Appendix.
Author: National Aeronautics and Space Administration (NASA)
Author: National Aeronautics and Space Administration (NASA)
Author: National Aeronautics and Space Adm Nasa
Author: Wenbin Liu
Author: Dewey H. Hodges
Author: 
Author: 
Author: Ekkehard Ramm
Author: Leszek Demkowicz
Author: Stefan Klaus Wilhelm Henneking
Author: Wolfgang Bangerth