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Delft University of Technology

1. Verzijl, C.J.O. On the determination of approximations of first integrals for few-body gravitational problems: with applications to capture trajectories:.

Degree: Electrical Engineering, Mathematics and Computer Science, Mathematical Physics, 2007, Delft University of Technology

On the determination of approximations of first integrals using the method of integrating vectors for ODE systems, as applied to few-body gravitational problems. Considers the Jacobi 3-body problem, the circular restricted 3-body problem and a 4-body model for ballistic lunar capture. Also discusses the application of these techniques to numerical solutions of the ODE systems using methods designed to preserve exact and approximate first integrals, such as those developed using the method of integrating vectors. Advisors/Committee Members: Van Horssen, W.T., Noomen, R..

Subjects/Keywords: integral; conservative; approximation; astrodynamics; gravitational

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Verzijl, C. J. O. (2007). On the determination of approximations of first integrals for few-body gravitational problems: with applications to capture trajectories:. (Masters Thesis). Delft University of Technology. Retrieved from http://resolver.tudelft.nl/uuid:dacac69b-b5f8-4216-a143-825a0e3bc2ea

Chicago Manual of Style (16th Edition):

Verzijl, C J O. “On the determination of approximations of first integrals for few-body gravitational problems: with applications to capture trajectories:.” 2007. Masters Thesis, Delft University of Technology. Accessed December 08, 2019. http://resolver.tudelft.nl/uuid:dacac69b-b5f8-4216-a143-825a0e3bc2ea.

MLA Handbook (7th Edition):

Verzijl, C J O. “On the determination of approximations of first integrals for few-body gravitational problems: with applications to capture trajectories:.” 2007. Web. 08 Dec 2019.

Vancouver:

Verzijl CJO. On the determination of approximations of first integrals for few-body gravitational problems: with applications to capture trajectories:. [Internet] [Masters thesis]. Delft University of Technology; 2007. [cited 2019 Dec 08]. Available from: http://resolver.tudelft.nl/uuid:dacac69b-b5f8-4216-a143-825a0e3bc2ea.

Council of Science Editors:

Verzijl CJO. On the determination of approximations of first integrals for few-body gravitational problems: with applications to capture trajectories:. [Masters Thesis]. Delft University of Technology; 2007. Available from: http://resolver.tudelft.nl/uuid:dacac69b-b5f8-4216-a143-825a0e3bc2ea


Delft University of Technology

2. Verzijl, C.J.O. On the integral-conservative numerical solution of few-body gravitational problems: with applications to capture trajectories:.

Degree: Aerospace Engineering, Astrodynamics and Satellite Systems, 2007, Delft University of Technology

On the design and implementation of integral-conservative numerical integration schemes for few-body problems in astrodynamics. Focuses on exact and approximate energy and angular-momentum integrals in the Jacobi 3-body problem, and related Jacobi-type integrals in the circular restricted 3-body problem and a 4-body model for ballistic lunar capture. Includes a self-contained discussion of necessary astrodynamics and mathematics background, as well as a discussion of the application of these techniques to ballistic lunar capture trajectories for small satellites. Advisors/Committee Members: Noomen, R., van Horssen, W.T..

Subjects/Keywords: astrodynamics; few-body; integral; conservative; approximation

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Verzijl, C. J. O. (2007). On the integral-conservative numerical solution of few-body gravitational problems: with applications to capture trajectories:. (Masters Thesis). Delft University of Technology. Retrieved from http://resolver.tudelft.nl/uuid:922fcd8d-bd5a-446a-ad27-6827e3aeba1d

Chicago Manual of Style (16th Edition):

Verzijl, C J O. “On the integral-conservative numerical solution of few-body gravitational problems: with applications to capture trajectories:.” 2007. Masters Thesis, Delft University of Technology. Accessed December 08, 2019. http://resolver.tudelft.nl/uuid:922fcd8d-bd5a-446a-ad27-6827e3aeba1d.

MLA Handbook (7th Edition):

Verzijl, C J O. “On the integral-conservative numerical solution of few-body gravitational problems: with applications to capture trajectories:.” 2007. Web. 08 Dec 2019.

Vancouver:

Verzijl CJO. On the integral-conservative numerical solution of few-body gravitational problems: with applications to capture trajectories:. [Internet] [Masters thesis]. Delft University of Technology; 2007. [cited 2019 Dec 08]. Available from: http://resolver.tudelft.nl/uuid:922fcd8d-bd5a-446a-ad27-6827e3aeba1d.

Council of Science Editors:

Verzijl CJO. On the integral-conservative numerical solution of few-body gravitational problems: with applications to capture trajectories:. [Masters Thesis]. Delft University of Technology; 2007. Available from: http://resolver.tudelft.nl/uuid:922fcd8d-bd5a-446a-ad27-6827e3aeba1d

3. Deklerck, M. Optimization of stiffened panels using a combination of FEM and a predictor-corrector interior point method:.

Degree: 2016, Delft University of Technology

Structural optimization, first introduced by Schmidt in 1960, is a rapid growing factor in the development of new aerospace structures. This growth is established by the increase in numerical modelling techniques, cheaper computer power, the increasing cost of production and competition between companies. The combination of both structural optimization and finite element software allowed for the rise of new and more efficient optimization methods provided that the software can performsensitivity analysis. Many programs used in industry today such as BOSS Quattro , PASCO and VICONOPT restrict themselves to basic optimization methods. The goal now is to develop an optimizer for stiffened panels, using a combination of FEM and a more advanced optimization method. Interior point methods have been proven to be more efficient than primal-dual methods for solving sub-problems. Therefore Mehrotra’s predictor-corrector interior point method is used in the version of Zillober. To reach convergence convex approximations are required. The conservative approximation from Fleury’s ConLin provides the basis of many other more advance approximation methods. Therefore this method is chosen to form the initial optimizer. A 2D The FEM model is established using shell and bar elements for the panel and stiffeners respectively. This allows for easy adjustment of the geometry without the need to change the model itself. The bar element properties are defined by the PBAR card rather than the PBARL card in NASTRAN. This avoids the input of fixed NASTRAN specified cross sections with limited design freedom. The sensitivities with respect to stiffener properties are extracted from NASTRAN. These are then converted to the required sensitivities using analytical equations. With all the necessary information available, the inner loop of the optimization process is initiated. Approximations of the constraints, objective and sensitivities are produced. Based on the approximations, the predictor step establishes a maximum step size, which is then adjusted by the corrector step to a more feasible one. This is done iteratively until the duality gap is below a specified limit. Finally a new outer iteration can start if no convergence is reached. Three goals were achieved by analysing of 11 test cases. First the optimizer shows that it can handle different property sets for the stiffeners within the same panel. Secondly, the optimization works for different cross sections. Finally, when performed for similar panels with a different amount of stiffeners, an optimal number is found. The optimization is performed for minimum weight while limited by stress, buckling and design constraints. The results indicate that for 8 out of 11 cases convergence is reached within 12 cycles. Due oscillatory behaviour two other cases converged relatively slow and one did not converge at all. This happens due to the incapability of the optimizer to consider new buckling modes establishing with the adjustment of the parameters. In the end however all three… Advisors/Committee Members: Abdallah, M.M..

Subjects/Keywords: structural optimization; stiffened panel; interior point method; conservative approximation; sequential convex programming

…required. The conservative approximation from Fleury’s ConLin [6] provides the basis of… …order to reach convergence. Therefore the ConLin conservative approximation scheme is used… …negative set resulting in a conservative approximation. f (x) = f (x 0 )… …4.2.2 Approximation of the primal problem . . . . . . . . . . . . . . . . . . . 41 4.2.3… …8 2.7 Linear and reciprocal approximation methods… 

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Deklerck, M. (2016). Optimization of stiffened panels using a combination of FEM and a predictor-corrector interior point method:. (Masters Thesis). Delft University of Technology. Retrieved from http://resolver.tudelft.nl/uuid:d0715121-d6d3-4816-95b1-4254af5a75c1

Chicago Manual of Style (16th Edition):

Deklerck, M. “Optimization of stiffened panels using a combination of FEM and a predictor-corrector interior point method:.” 2016. Masters Thesis, Delft University of Technology. Accessed December 08, 2019. http://resolver.tudelft.nl/uuid:d0715121-d6d3-4816-95b1-4254af5a75c1.

MLA Handbook (7th Edition):

Deklerck, M. “Optimization of stiffened panels using a combination of FEM and a predictor-corrector interior point method:.” 2016. Web. 08 Dec 2019.

Vancouver:

Deklerck M. Optimization of stiffened panels using a combination of FEM and a predictor-corrector interior point method:. [Internet] [Masters thesis]. Delft University of Technology; 2016. [cited 2019 Dec 08]. Available from: http://resolver.tudelft.nl/uuid:d0715121-d6d3-4816-95b1-4254af5a75c1.

Council of Science Editors:

Deklerck M. Optimization of stiffened panels using a combination of FEM and a predictor-corrector interior point method:. [Masters Thesis]. Delft University of Technology; 2016. Available from: http://resolver.tudelft.nl/uuid:d0715121-d6d3-4816-95b1-4254af5a75c1

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