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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: 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: 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: Jan-Eric Wurst Publisher: Wurzburg University Press ISBN: 9783958260245 Category : Languages : en Pages : 190
Book Description
Optimal control theory is a versatile mathematical discipline with applications in many fields. It has gained interest over the last decades mainly because increasing computational power allowed to tackle large and complex real life problems numerically. For offering reliable results, a thorough theoretical analysis of solution algorithms, their convergence properties, and approximation quality is inevitable. We follow this need and investigate linear quadratic optimal control problems with elliptic partial differential equations. The discretization with hp-finite elements is embedded in both Newton-type and interior point methods. Different efficient strategies are presented and accompanied by new results on regularity, approximation, and convergence theory.
Author: I. Babuska Publisher: ISBN: Category : Languages : en Pages : 20
Book Description
The finite element method has become the main tool in computational mechanics. The success is manifested by the development of over five hundred user-oriented finite element program systems. The literature on the subject is overwhelming. To date there are over two hundred monographs and conference proceedings and new monographs and proceedings are continuously appearing. Various forms of the finite element method are used in practice for the numerical treatment of elliptic, parabolic, hyperbolic, linear and nonlinear partial differential equations, integral and integrodifferential equations, etc. Any class of problems has its own specific features. This paper deals with the class of partial differential equation of elliptic type. For the sake of simplicity the author elaborates on a characteristic model problem and illustrative results and makes only additional comments of more general nature although the results referred to are general.
Author: National Aeronautics and Space Adm Nasa
Author: National Aeronautics and Space Adm Nasa
Author: National Aeronautics and Space Administration (NASA)
Author: Dewey H. Hodges
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Author: Jan-Eric Wurst
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Author: I. Babuska