Edgardo Taroco

Topological sensitivity analysis

The so-called topological derivative concept has been seen as a powerful framework to obtain the optimal topology for several engineering problems. This derivative characterizes the sensitivity of the problem when a small hole is created at each point of the domain. However, the greatest limitation of this methodology is that when a hole is created it is impossible to build a homeomorphic map between the domains in study (because they have not the same topology). Therefore, some specific mathematical framework should be developed in order to obtain the derivatives. This work proposes an alternative way to compute the topological derivative based on the shape sensitivity analysis concepts. The main feature of this methodology is that all the mathematical procedure already developed in the context of shape sensitivity analysis may be used in the calculus of the topological derivative. This idea leads to a more simple and constructive formulation than the ones found in the literature. Further, to point out the straightforward use of the proposed methodology, it is applied for solving some design problems in steady-state heat conduction.

THE TOPOLOGICAL DERIVATIVE FOR THE POISSON'S PROBLEM

The Topological Derivative has been recognized as a powerful tool in obtaining the optimal topology of several engineering problems. This derivative provides the sensitivity of a problem when a small hole is created at each point of the domain under consideration. In the present work the Topological Derivative for Poissons problem is calculated using two different approaches: the Domain Truncation Method and a new method based on Shape Sensitivity Analysis concepts. By comparing both approaches it will be shown that the novel approach, which we call Topological-Shape Sensitivity Method, leads to a simpler and more general methodology. To point out the general applicability of this new methodology, the most general set of boundary conditions for Poissons problem, Dirichlet, Neumann (both homogeneous and nonhomogeneous) and Robin boundary conditions, is considered. Finally, a comparative analysis of these two methodologies will also show that the Topological-Shape Sensitivity Method has an additional advantage of being easily extended to other types of problems.

Shape sensitivity analysis in linear elastic fracture mechanics

Shape sensitivity analysis of an elastic solid in equilibrium with a known load system applied over its boundary is presented in this work. The domain and boundary integral expressions of the first- and second-order shape derivatives of the total potential energy are established, by using an arbitrary change of the domain characterized by a velocity field defined over the initial body configuration. In these expressions we recognize free divergence tensors that are denoted in this paper as energy shape change tensors. Next, shape sensitivity analysis is applied to cracked bodies. For that purpose, a suitable velocity distribution field is adopted to simulate the crack advance of a unit length in a two-dimensional body. Finally, the corresponding domain and the equivalent path-independent integral expressions of the first- and second-order potential energy release rate of fracture mechanics are also derived.

Design sensitivity of post-buckling states including material constraints

This paper reports theoretical studies on the design sensitivity of problems with geometrically non-linear behavior leading to buckling and post-buckling of thin-walled structural members. For the class of problems considered, buckling occurs in the form of a stable bifurcation, and it is assumed that changes in the design parameters do not break the bifurcation behavior. The specific focus of the research is the first yield or first failure of the material as part of the sensitivity study of equilibrium states along the post-critical path. The investigation employs a discrete model of a structure in terms of generalized coordinates (suitable for finite element analysis) and a single load parameter; and perturbation techniques to classify the critical state and to approximate the post-critical path. The problem of material behavior is modeled by means of constraints on the post-critical path, based on a yield criterion. For simplicity, the presentation uses the von Mises yield criterion, but other more complex criteria, such as those employed in composite materials (first-ply failure) can also be represented. Two forms of the constraints are formulated, and the problem of sensitivity with respect to changes in a design parameter is discussed. A simple example of a circular plate is presented to illustrate the use of the formulation for the sensitivity with respect to a single design parameter.

Fracture mechanics via sensitivity analysis and numerical evaluation of J integral in cracked sheets

Abstract In this paper the formulation of the J integral is derived from the shape sensitivity analysis for plane cracked domains. Numerical evaluation of the J integral is performed using the results obtained from a finite element calculation. Finally, three examples of cracked sheets are analyzed and the results compared to available values in the literature.

Configurational Derivative as a Tool for Image Segmentation

The introduction to medicine of techniques coming from areas like Computational Fluid Dynamics, Structural Analysis, and Inverse Problems, made the use of imaging data such us Computed Tomography (CT), Magnetic Resonance Imaging (MRI), Single Photon Emission Tomography (SPECT), Positron Emission Tomography (PET) and Ultrasound (US) mandatory in order to apply this techniques to patient specific data. The process of identification of different tissues and organs, called segmentation, is a maior concern in this analysis. This process can be tedious and time consuming when done by hand, so its been an early concern in image processing to automatize it. Many contributions have been made to the area since the introduction of the Mumford and Shah functional. This functional is endowed to quantify the cost associated to a specific segmentation.

Topological Derivative Applied to Image Enhancement

Medical imaging techniques like Computed Tomography (CT), Magnetic Resonance Imaging (MRI), Single Photon Emission Tomography (SPECT), Positron Emission Tomography (PET) and Ultrasound (US) have introduced a formidably powerful tool in medicine. The poor quality or high signal-to-noise ratio (SNR) of this kind of images is maior limitation for image analysis. For this reason image enhancement takes an important roll in the segmentation and analysis process. Although imaging techniques (e.g., contrast agents, biological markers) should improve the image quality this is not always enough to give a good result. Much effort has been put into the area of image enhancement. Our aim in this paper is to present a method for medical image enhancement based on the well established concept of topological sensitivity analysis and borrowing image processing techniques like anisotropic diffusion. More specifically, an appropriate functional F is associated to the image indicating the cost endowed to an specific image. Let us assume that the image being segmented is characterized by a scalar field u representing the image data. Then, the segmentation algorithm can be cast as: given the evolution equation,

Second‐order sensitivity analysis in vibration and buckling problems

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