Kwang-Yeon Kim

A Posteriori Error Analysis for Locally Conservative Mixed Methods

In this work we present a theoretical analysis for a residual-type error estimator for locally conservative mixed methods. This estimator was first introduced by Braess and Verfurth for the Raviart-Thomas mixed finite element method working in mesh-dependent norms. We improve and extend their results to cover any locally conservative mixed method under minimal assumptions, in particular, avoiding the saturation assumption made by Braess and Verfurth. Our analysis also takes into account discontinuous coefficients with possibly large jumps across interelement boundaries. The main results are applied to the P1 nonconforming finite element method and the interior penalty discontinuous Galerkin method as well as the mixed finite element method.

Store Separation Analysis of a Fighter Aircraft\'s External Fuel Tank

The repetitive vibrating action of an aerodynamic load causes an external fuel tank’s horizontal fin to experience a shorter life cycle than its originally predicted one. Store separation analysis is needed to redesign the fin of an external fuel tank. In this research, free-drop tests were conducted using 15% scaled models in a subsonic wind tunnel in order to analyze the store separation characteristics of an external fuel tank. The store separation trajectory based on grid tests was also obtained to verify the results of the free-drop tests. The results acquired from the free-drop tests correlated well with the grid tests in regards to the trajectories and behavior of the stores separated from the aircraft. This agreement was especially noted in the early stages of the store separation.

Guaranteed a posteriori error estimator for mixed finite element methods of elliptic problems

Abstract In this work we analyze a posteriori error estimator for a general class of mixed finite element methods of elliptic problems which guarantees an upper bound on the vector error, thus extending the recent result obtained for the lowest order triangular Raviart–Thomas mixed finite element method in [M. Ainsworth, A posteriori error estimation for lowest order Raviart–Thomas mixed finite elements, SIAM J. Sci. Comput. 30 (2007/08) 189–204]. The error estimator is constructed through a variant of Stenberg’s postprocessing procedure, and the guaranteed upper bound is readily established by making use of the argument similar to the hypercircle method. However, the proof of the lower bound given in the above reference does not seem to apply to other kinds of mixed elements. So we employ a different technique using the discrete Friedrichs inequality to establish the lower bound of the error estimator.

A posteriori error estimators for nonconforming finite element methods of the linear elasticity problem

In this work we derive and analyze a posteriori error estimators for low-order nonconforming finite element methods of the linear elasticity problem on both triangular and quadrilateral meshes, with hanging nodes allowed for local mesh refinement. First, it is shown that equilibrated Neumann data on interelement boundaries are simply given by the local weak residuals of the numerical solution. The first error estimator is then obtained by applying the equilibrated residual method with this set of Neumann data. From this implicit estimator we also derive two explicit error estimators, one of which is similar to the one proposed by Dorfler and Ainsworth (2005) [24] for the Stokes problem. It is established that all these error estimators are reliable and efficient in a robust way with respect to the Lame constants. The main advantage of our error estimators is that they yield guaranteed, i.e., constant-free upper bounds for the energy-like error (up to higher order terms due to data oscillation) when a good estimate for the inf-sup constant is available, which is confirmed by some numerical results.

Fully computable a posteriori error estimates for the Stokes equation without the global inf-sup constant

In this paper we consider the a posteriori error estimates for the Stokes equation which provide computable upper bounds on the actual errors. It is known that such error estimates typically involve the global inf-sup constant which can be quite tricky to find and lead to extra difficulty in numerical computations. To resolve this difficulty, we propose a new error estimate which relies only upon the inf-sup constants local to the subdomains forming a partition of the original domain. Hence our new error estimate is fully computable whenever the subdomains are simple enough to make the local inf-sup constants readily available, and moreover, provides a sharper upper bound than the previous estimate when these local constants are bigger than the global constant. Application to the Crouzeix-Raviart and Fortin-Soulie nonconforming finite elements is presented along with some effective minimization technique for further improvement of the upper bound on the error. Finally, numerical experiments are carried out to investigate the performance of the new error estimate.

Guaranteed A Posteriori Error Estimator for Mixed Finite Element Methods of Linear Elasticity with Weak Stress Symmetry

In this paper we propose an a posteriori error estimator for the mixed finite element methods of the linear elasticity problem with the symmetry condition weakly imposed on the stress tensor. The error estimator is constructed by making a proper decomposition of the stress error and using an argument similar to the hypercircle method. It is shown that the resulting estimator yields a guaranteed upper bound on the stress error which relies on computable upper bounds of the constants in the first and second Korn inequalities. We also establish the local lower bound by using the discrete Friedrichs inequality. Our approach is equivalent to the Helmholtz decomposition of the stress error but requires assumptions neither on the regularity of the solution nor the geometry of the domain. Numerical results are provided to illustrate the efficiency of our error estimator.

Algorithm for Pairwise Collision Detection and Avoidace in 3-D

This paper presents the development of a real-time algorithm for collision detection, collision avoidance and guidance to way-point. Three-dimensional point-mass aircraft models are used. For collision detection, time of closest point of approach(CPA) and distance at CPA are compared to threshold values. For collision avoidance, optimal acceleration input which maximizes the terminal relative distance is calculated based on optimal control theory. For guidance to way-point, proportional navigation guidance, the well-known method, is used. Two scenarios of encounter situation are illustrated to validate performance of proposed algorithm.

A Probabilistic Algorithm for Multi-aircraft Collision Detection and Resolution in 3-D

This paper presents a real-time algorithm for collision detection, collision avoidance and guidance. Three-dimensional point-mass aircraft models are used. For collision detection, conflict probability is calculated by using the Monte-Carlo Simulation. Time at the closest point of approach(CPA) and distance at CPA are needed to determine the collision probability, being compared to certain threshold values. For collision avoidance, one of possible maneuver options is chosen to minimize the collision probability. For guidance to a designated way-point, proportional navigation guidance law is used. Two scenarios on encounter situation are studied to demonstrate the performance of proposed algorithm.