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Jacobi Equation (jacobi + equation)

Distribution by Scientific Domains

Engineering	83%

Selected Abstracts

A structural optimization method based on the level set method using a new geometry-based re-initialization scheme

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2010
Shintaro Yamasaki
Abstract Structural optimization methods based on the level set method are a new type of structural optimization method where the outlines of target structures can be implicitly represented using the level set function, and updated by solving the so-called Hamilton,Jacobi equation based on a Eulerian coordinate system. These new methods can allow topological alterations, such as the number of holes, during the optimization process whereas the boundaries of the target structure are clearly defined. However, the re-initialization scheme used when updating the level set function is a critical problem when seeking to obtain appropriately updated outlines of target structures. In this paper, we propose a new structural optimization method based on the level set method using a new geometry-based re-initialization scheme where both the numerical analysis used when solving the equilibrium equations and the updating process of the level set function are performed using the Finite Element Method. The stiffness maximization, eigenfrequency maximization, and eigenfrequency matching problems are considered as optimization problems. Several design examples are presented to confirm the usefulness of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd. [source]

Piecewise constant level set method for structural topology optimization

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2009
Peng Wei
Abstract In this paper, a piecewise constant level set (PCLS) method is implemented to solve a structural shape and topology optimization problem. In the classical level set method, the geometrical boundary of the structure under optimization is represented by the zero level set of a continuous level set function, e.g. the signed distance function. Instead, in the PCLS approach the boundary is described by discontinuities of PCLS functions. The PCLS method is related to the phase-field methods, and the topology optimization problem is defined as a minimization problem with piecewise constant constraints, without the need of solving the Hamilton,Jacobi equation. The result is not moving the boundaries during the iterative procedure. Thus, it offers some advantages in treating geometries, eliminating the reinitialization and naturally nucleating holes when needed. In the paper, the PCLS method is implemented with the additive operator splitting numerical scheme, and several numerical and procedural issues of the implementation are discussed. Examples of 2D structural topology optimization problem of minimum compliance design are presented, illustrating the effectiveness of the proposed method. Copyright © 2008 John Wiley & Sons, Ltd. [source]

A partition-of-unity-based finite element method for level sets

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 10 2008
Stéphane Valance
Abstract Level set methods have recently gained much popularity to capture discontinuities, including their possible propagation. Typically, the partial differential equations that arise in level set methods, in particular the Hamilton,Jacobi equation, are solved by finite difference methods. However, finite difference methods are less suited for irregular domains. Moreover, it seems slightly awkward to use finite differences for the capturing of a discontinuity, while in a subsequent stress analysis finite elements are normally used. For this reason, we here present a finite element approach to solving the governing equations of level set methods. After a review of the governing equations, the initialization of the level sets, the discretization on a finite domain, and the stabilization of the resulting finite element method will be discussed. Special attention will be given to the proper treatment of the internal boundary condition, which is achieved by exploiting the partition-of-unity property of finite element shape functions. Finally, a quantitative analysis including accuracy analysis is given for a one-dimensional example and a qualitative example is given for a two-dimensional case with a curved discontinuity. Copyright © 2008 John Wiley & Sons, Ltd. [source]

An assumed-gradient finite element method for the level set equation

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2005
Hashem M. Mourad
Abstract The level set equation is a non-linear advection equation, and standard finite-element and finite-difference strategies typically employ spatial stabilization techniques to suppress spurious oscillations in the numerical solution. We recast the level set equation in a simpler form by assuming that the level set function remains a signed distance to the front/interface being captured. As with the original level set equation, the use of an extensional velocity helps maintain this signed-distance function. For some interface-evolution problems, this approach reduces the original level set equation to an ordinary differential equation that is almost trivial to solve. Further, we find that sufficient accuracy is available through a standard Galerkin formulation without any stabilization or discontinuity-capturing terms. Several numerical experiments are conducted to assess the ability of the proposed assumed-gradient level set method to capture the correct solution, particularly in the presence of discontinuities in the extensional velocity or level-set gradient. We examine the convergence properties of the method and its performance in problems where the simplified level set equation takes the form of a Hamilton,Jacobi equation with convex/non-convex Hamiltonian. Importantly, discretizations based on structured and unstructured finite-element meshes of bilinear quadrilateral and linear triangular elements are shown to perform equally well. Copyright © 2005 John Wiley & Sons, Ltd. [source]

Tidal dynamics of relativistic flows near black holes

ANNALEN DER PHYSIK, Issue 5 2005
C. Chicone
Abstract We point out novel consequences of general relativity involving tidal dynamics of ultrarelativistic relative motion. Specifically, we use the generalized Jacobi equation and its extension to study the force-free dynamics of relativistic flows near a massive rotating source. We show that along the rotation axis of the gravitational source, relativistic tidal effects strongly decelerate an initially ultrarelativistic flow with respect to the ambient medium, contrary to Newtonian expectations. Moreover, an initially ultrarelativistic flow perpendicular to the axis of rotation is strongly accelerated by the relativistic tidal forces. The astrophysical implications of these results for jets and ultrahigh energy cosmic rays are briefly mentioned. [source]

H, control for nonlinear affine systems: a chain-scattering matrix description approach

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 4 2001
Jang-Lee Hong
Abstract This paper combines an alternative chain-scattering matrix description with (J, J,)-lossless and a class of conjugate (,J, ,J,)-lossless systems to design a family of nonlinear H, output feedback controllers. The present systems introduce a new chain-scattering setting, which not only offers a clearer expression for the solving process of the nonlinear H, control problem but also removes the fictitious signals introduced by the traditional chain-scattering approach. The intricate nonlinear affine control problem thus can be transformed into a simple lossless network and is easy to deal with in a network-theory context. The relationship among these (J, J,) systems, L2 -gain, and Hamilton,Jacobi equations is also given. Block diagrams are used to illustrate the central theme. Copyright © 2001 John Wiley & Sons, Ltd. [source]