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Edgardo Taroco - One of the best experts on this subject based on the ideXlab platform.
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Shape Sensitivity Analysis and the energy momentum tensor for the kinematic and static models of torsion
International Journal of Solids and Structures, 2006Co-Authors: Edgardo Taroco, Raúl A. FeijóoAbstract:Abstract The aim of this paper is to bridge Shape Sensitivity Analysis and configurational mechanics by means of a widespread use of the Shape derivative concept. This technique will be applied as a systematic procedure to obtain the Eshelby’s energy momentum tensor associated to the problem under consideration. In order to highlight special features of this procedure and without loss of generality, we focus our attention in the application of Shape Sensitivity Analysis to the problem of twisted straight bars within the framework of linear elasticity. Kinematic and static variational formulations as well as the direct method of Sensitivity Analysis are used to perform Shape derivatives of both models. Integral expressions of first and second order Shape derivatives of the total potential energy and the complementary potential energy with respect to an arbitrary transverse cross-section Shape change, are achieved. These integral expressions put in evidence the relationship between Shape Sensitivity Analysis and the first and second order Eshelby’s energy momentum tensors. Also, the null divergence property of these tensors is easily proved by comparing, in each case, the domain and boundary integral Shape derivative arrived at. Finally, an example with a known exact solution, corresponding to an elastic bar with elliptical transverse cross-section submitted to twist, is presented in order to illustrate the usefulness of these tensors to compute the corresponding Shape derivatives.
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Adaptivity in linear elastic fracture mechanics based on Shape Sensitivity Analysis
Computer Methods in Applied Mechanics and Engineering, 2005Co-Authors: Roberto Saliba, Edgardo Taroco, Claudio Padra, Marcelo J. Vénere, Raúl A. FeijóoAbstract:Abstract If crack growth of an elastic body is viewed as a Shape change we can use the well known concept of Shape Sensitivity Analysis to compute the energy release rate. To do this, we adopt as cost function the total potential energy and as state equation the equilibrium equation. The Shape derivative of the total potential energy stored in the cracked body Π ˙ depends on the displacement field u and on the Shape change velocity field V which characterize the crack growth. Following this procedure the present paper deals with the derivation of a novel a posteriori error estimator which is an upper bound of the global error | Π ˙ - Π ˙ h | . This error estimator has been specifically designed to evaluate the energy release rate in mesh refinement or re-meshing procedures so as to obtain improved meshes for which the optimal rate of convergence is recovered even in a case of singularities. This novel estimator is capable to capture all source of errors for the energy release rate Π ˙ : the ones from stress concentration and the errors from the Sensitivity of the solution to Shape changes due to crack growth. Finally, well known three-dimensional examples of un-cracked and cracked body are considered in order to illustrate the potentiality of the proposed methodology.
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A unified approach for Shape Sensitivity Analysis of elastic shells
Structural and Multidisciplinary Optimization, 2004Co-Authors: Edgardo Taroco, Raúl A. FeijóoAbstract:A Shape Sensitivity Analysis is presented of a shell loaded along its boundary, within the framework of a linear elastic approach that takes into account the effect of transverse shear deformation. By making an analogy with continuum mechanics and taking advantage of the concept of the total (time) derivative in its spatial description, an explicit general expression for the Shape derivative of the total potential energy is obtained in terms of the strain–stress state and the shell Shape change velocity field. For this purpose, a suitable velocity distribution over the middle surface of the shell, decomposed on a local basis given by the unit normal and the tangent plane of the shell, is adopted. This approach and the direct method of Shape Sensitivity Analysis leads to a very useful expression for the Shape Sensitivity of the total potential energy. This expression is an explicit function of the tangent and normal components of the known velocity field that characterizes the Shape change of the shell.
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Shape Sensitivity Analysis OF ELASTIC SHELLS WITH CRACKS
Journal of Theoretical and Applied Mechanics, 2003Co-Authors: Edgardo Taroco, Raúl A. FeijóoAbstract:This study concerns the application of Shape Sensitivity Analysis as a systematic methodology to determine the energy relase rate of cracked shells, within the framework of a linear elastic approach that takes into account the effect of transverse shear deformation. This methodology and the direct method of Shape Sensitivity Analysis is applied to shells with an arbitrary middle surface and leads to an explicit general expression for the Shape Sensitivity of the total potential strain energy. In elastic shells with cracks, crack initiation is simulated by a change of Shape characterized by a suitable tangential velocity distribution over the middle surface of the shell. In this case, a useful expression of energy release rate is expressed in terms of the strain-stress state and the adopted Shape change velocity field. Finally, Shape Sensitivity Analysis is applied to the circular cylindrical shell and thus the condition of null divergence of the corresponding Eshelby tensor is verified.
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Shape Sensitivity Analysis in linear elastic fracture mechanics
Computer Methods in Applied Mechanics and Engineering, 2000Co-Authors: Edgardo TarocoAbstract: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.
Raúl A. Feijóo - One of the best experts on this subject based on the ideXlab platform.
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Shape Sensitivity Analysis and the energy momentum tensor for the kinematic and static models of torsion
International Journal of Solids and Structures, 2006Co-Authors: Edgardo Taroco, Raúl A. FeijóoAbstract:Abstract The aim of this paper is to bridge Shape Sensitivity Analysis and configurational mechanics by means of a widespread use of the Shape derivative concept. This technique will be applied as a systematic procedure to obtain the Eshelby’s energy momentum tensor associated to the problem under consideration. In order to highlight special features of this procedure and without loss of generality, we focus our attention in the application of Shape Sensitivity Analysis to the problem of twisted straight bars within the framework of linear elasticity. Kinematic and static variational formulations as well as the direct method of Sensitivity Analysis are used to perform Shape derivatives of both models. Integral expressions of first and second order Shape derivatives of the total potential energy and the complementary potential energy with respect to an arbitrary transverse cross-section Shape change, are achieved. These integral expressions put in evidence the relationship between Shape Sensitivity Analysis and the first and second order Eshelby’s energy momentum tensors. Also, the null divergence property of these tensors is easily proved by comparing, in each case, the domain and boundary integral Shape derivative arrived at. Finally, an example with a known exact solution, corresponding to an elastic bar with elliptical transverse cross-section submitted to twist, is presented in order to illustrate the usefulness of these tensors to compute the corresponding Shape derivatives.
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Adaptivity in linear elastic fracture mechanics based on Shape Sensitivity Analysis
Computer Methods in Applied Mechanics and Engineering, 2005Co-Authors: Roberto Saliba, Edgardo Taroco, Claudio Padra, Marcelo J. Vénere, Raúl A. FeijóoAbstract:Abstract If crack growth of an elastic body is viewed as a Shape change we can use the well known concept of Shape Sensitivity Analysis to compute the energy release rate. To do this, we adopt as cost function the total potential energy and as state equation the equilibrium equation. The Shape derivative of the total potential energy stored in the cracked body Π ˙ depends on the displacement field u and on the Shape change velocity field V which characterize the crack growth. Following this procedure the present paper deals with the derivation of a novel a posteriori error estimator which is an upper bound of the global error | Π ˙ - Π ˙ h | . This error estimator has been specifically designed to evaluate the energy release rate in mesh refinement or re-meshing procedures so as to obtain improved meshes for which the optimal rate of convergence is recovered even in a case of singularities. This novel estimator is capable to capture all source of errors for the energy release rate Π ˙ : the ones from stress concentration and the errors from the Sensitivity of the solution to Shape changes due to crack growth. Finally, well known three-dimensional examples of un-cracked and cracked body are considered in order to illustrate the potentiality of the proposed methodology.
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A unified approach for Shape Sensitivity Analysis of elastic shells
Structural and Multidisciplinary Optimization, 2004Co-Authors: Edgardo Taroco, Raúl A. FeijóoAbstract:A Shape Sensitivity Analysis is presented of a shell loaded along its boundary, within the framework of a linear elastic approach that takes into account the effect of transverse shear deformation. By making an analogy with continuum mechanics and taking advantage of the concept of the total (time) derivative in its spatial description, an explicit general expression for the Shape derivative of the total potential energy is obtained in terms of the strain–stress state and the shell Shape change velocity field. For this purpose, a suitable velocity distribution over the middle surface of the shell, decomposed on a local basis given by the unit normal and the tangent plane of the shell, is adopted. This approach and the direct method of Shape Sensitivity Analysis leads to a very useful expression for the Shape Sensitivity of the total potential energy. This expression is an explicit function of the tangent and normal components of the known velocity field that characterizes the Shape change of the shell.
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Shape Sensitivity Analysis OF ELASTIC SHELLS WITH CRACKS
Journal of Theoretical and Applied Mechanics, 2003Co-Authors: Edgardo Taroco, Raúl A. FeijóoAbstract:This study concerns the application of Shape Sensitivity Analysis as a systematic methodology to determine the energy relase rate of cracked shells, within the framework of a linear elastic approach that takes into account the effect of transverse shear deformation. This methodology and the direct method of Shape Sensitivity Analysis is applied to shells with an arbitrary middle surface and leads to an explicit general expression for the Shape Sensitivity of the total potential strain energy. In elastic shells with cracks, crack initiation is simulated by a change of Shape characterized by a suitable tangential velocity distribution over the middle surface of the shell. In this case, a useful expression of energy release rate is expressed in terms of the strain-stress state and the adopted Shape change velocity field. Finally, Shape Sensitivity Analysis is applied to the circular cylindrical shell and thus the condition of null divergence of the corresponding Eshelby tensor is verified.
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Shape Sensitivity Analysis for energy release rate evaluation and its application to the study of three-dimensional cracked bodies
Computer Methods in Applied Mechanics and Engineering, 2000Co-Authors: Raúl A. Feijóo, Edgardo Taroco, Roberto Saliba, Claudio Padra, Marcelo J. VénereAbstract:The energy release rate is an important parameter for the Analysis of cracked bodies in linear elastic fracture mechanics. This parameter, usually denoted by G, is equivalent to Π, the rate of change with respect to crack change of the energy available for fracture. In this paper, crack growth is simulated by an action of change of the Shape of the body characterized by an appropriate known smooth velocity field v defined over the domain of the body. A general (integral) expression for Π using Shape Sensitivity Analysis based on distributed parameters is also obtained in this paper. Since this expression depends on the displacement field u and on ⊇v, a simple post-processing technique is required for the numerical evaluation of this expression. An adaptive finite element Analysis is performed in order to ensure a good accuracy during the numerical evaluation of Π. Finally, well known three-dimensional examples in fracture mechanics are considered in order to illustrate the potentiality of the proposed methodology.
B. N. Rao - One of the best experts on this subject based on the ideXlab platform.
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Fractal finite element method based Shape Sensitivity Analysis of multiple crack system
Engineering Fracture Mechanics, 2009Co-Authors: B. N. Rao, R.m. ReddyAbstract:This paper presents fractal finite element based continuum Shape Sensitivity Analysis for a multiple crack system in a homogeneous, isotropic, and two dimensional linear-elastic body subjected to mixed-mode (modes I and II) loading conditions. The salient feature of this method is that the stress intensity factors and their derivatives for the multiple crack system can be obtained efficiently since it only requires an evaluation of the same set of fractal finite element matrix equations with a different fictitious load. Three numerical examples are presented to calculate the first-order derivative of the stress intensity factors or energy release rates.
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Fractal finite element method based Shape Sensitivity Analysis of mixed-mode fracture
Finite Elements in Analysis and Design, 2008Co-Authors: R.m. Reddy, B. N. RaoAbstract:In this paper, a new fractal finite element based method for continuum-based Shape Sensitivity Analysis for a crack in a homogeneous, isotropic, and two-dimensional linear-elastic body subject to mixed-mode (modes I and II) loading conditions, is presented. The method is based on the material derivative concept of continuum mechanics, and direct differentiation. Parametric study is carried out to examine the effects of the similarity ratio, the number of transformation terms, and the integration order on the quality of the numerical solutions. Three numerical examples which include both mode-I and mixed-mode problems, are presented to calculate the first-order derivative of the J-integral or stress-intensity factors. The results show that first-order sensitivities of J-integral or stress-intensity factors obtained using the proposed method are in excellent agreement with the reference solutions obtained using the finite-difference method for the structural and crack geometries considered in this study.
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Continuum Shape Sensitivity Analysis of mixed-mode fracture using fractal finite element method
Engineering Fracture Mechanics, 2008Co-Authors: R.m. Reddy, B. N. RaoAbstract:Abstract This paper presents a new fractal finite element based method for continuum-based Shape Sensitivity Analysis for a crack in a homogeneous, isotropic, and two-dimensional linear-elastic body subject to mixed-mode (modes I and II) loading conditions. The method is based on the material derivative concept of continuum mechanics, and direct differentiation. Unlike virtual crack extension techniques, no mesh perturbation is needed in the proposed method to calculate the Sensitivity of stress-intensity factors. Since the governing variational equation is differentiated prior to the process of discretization, the resulting Sensitivity equations predicts the first-order Sensitivity of J -integral or mode-I and mode-II stress-intensity factors, K I and K II , more efficiently and accurately than the finite-difference methods. Unlike the integral based methods such as J -integral or M -integral no special finite elements and post-processing are needed to determine the first-order Sensitivity of J -integral or K I and K II . Also a parametric study is carried out to examine the effects of the similarity ratio, the number of transformation terms, and the integration order on the quality of the numerical solutions. Four numerical examples which include both mode-I and mixed-mode problems, are presented to calculate the first-order derivative of the J -integral or stress-intensity factors. The results show that first-order sensitivities of J -integral or stress-intensity factors obtained using the proposed method are in excellent agreement with the reference solutions obtained using the finite-difference method for the structural and crack geometries considered in this study.
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Fractal Finite-Element Based Continuum Shape Sensitivity Analysis of Cracks
Volume 2: Computer Applications Technology and Bolted Joints, 2007Co-Authors: B. N. Rao, R.m. ReddyAbstract:Probabilistic fracture mechanics (PFM) that blends the theory of fracture mechanics and the probability theory provides a more rational means to describe the actual behavior and reliability of structures. However in PFM, the fracture parameters and their derivatives are often required to predict the probability of fracture initiation and/or instability in cracked structures. The calculation of the derivatives of fracture parameters with respect to load and material parameters, which constitutes size-Sensitivity Analysis, is not unduly difficult. However, the evaluation of response derivatives with respect to crack size is a challenging task, since it requires Shape Sensitivity Analysis. Using a brute-force type finite-difference method to calculate the Shape sensitivities is often computationally expensive, in that numerous repetitions of deterministic finite element Analysis may be required for a complete reliability Analysis. Therefore, an essential need of probabilistic fracture-mechanics is to evaluate the Sensitivity of fracture parameters accurately and efficiently.Copyright © 2007 by ASME
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Continuum Shape Sensitivity Analysis of a mode-I fracture in functionally graded materials
Computational Mechanics, 2005Co-Authors: Sharif Rahman, B. N. RaoAbstract:This paper presents a new method for conducting a continuum Shape Sensitivity Analysis of a crack in an isotropic, linear-elastic, functionally graded material. This method involves the material derivative concept from continuum mechanics, domain integral representation of the J -integral and direct differentiation. Unlike virtual crack extension techniques, no mesh perturbation is needed to calculate the Sensitivity of stress-intensity factors. Since the governing variational equation is differentiated prior to the process of discretization, the resulting Sensitivity equations are independent of approximate numerical techniques, such as the meshless method, finite element method, boundary element method, or others. In addition, since the J -integral is represented by domain integration, only the first-order Sensitivity of the displacement field is needed. Several numerical examples are presented to calculate the first-order derivative of the J -integral, using the proposed method. Numerical results obtained using the proposed method are compared with the reference solutions obtained from finite-difference methods for the structural and crack geometries considered in this study.
Kuang-hua Chang - One of the best experts on this subject based on the ideXlab platform.
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Continuum Shape Sensitivity Analysis and what-if study for two-dimensional multi-scale crack propagation problems using bridging scale decomposition
Structural and Multidisciplinary Optimization, 2015Co-Authors: Yunxiang Wang, Kuang-hua ChangAbstract:This paper presents a Shape Sensitivity Analysis and what-if study for two-dimensional multi-scale crack propagation problems using bridging scale decomposition. The Sensitivity equations are derived in a continuum setting using direct differentiation method based on a continuum variational formulation of the bridging scale. Due to the fact that the crack propagation speed in an atomistic simulation is discrete in design, and cannot be formulated as a continuous function of Shape design variables, we propose a hybrid method that combines analytical Sensitivity Analysis with finite difference approach. The finite difference part of the Sensitivity Analysis is only intended for calculating the Sensitivity of crack growth speed based on the analytically obtained Sensitivity coefficients of structural responses. The theoretical development on Sensitivity formulation in this paper extends the application of the method to irregular-Shaped finite elements and general design velocity fields. Furthermore, we evaluate and compare several performance measures that quantify crack propagation speed based on crack tip locations for Sensitivity Analysis and ultimately for structural optimization. A two-dimensional beam example is used to verify the accuracy of the proposed Sensitivity approach. It is also demonstrated through a what-if study that with an adequate performance measure, the impact of macroscopic Shape changes on microscopic crack propagation speed can be accurately predicted.
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Continuum-Based Shape Sensitivity Analysis for 2D Coupled Atomistic/Continuum Simulations Using Bridging Scale Decomposition
Mechanics Based Design of Structures and Machines, 2014Co-Authors: Yunxiang Wang, Kuang-hua ChangAbstract:In this paper, we propose the first attempt to perform Shape Sensitivity Analysis for two-dimensional coupled atomistic and continuum problems using bridging scale decomposition. Based on a continuum variational formulation of the bridging scale, the Sensitivity expressions are derived in a continuum setting using both direct differentiation method and adjoint variable method. To overcome the issue of discontinuity in Shape design due to the discrete nature of the molecular dynamics (MD) simulation, we define design velocity fields in a way that the Shape of the MD region does not change. Another major challenge is that the discrete finite element (FE) mass matrix in bridging scale is not continuous with respect to Shape design variables. To address this issue, we assume an evenly distributed mass density when evaluating the material derivative of the FE mass matrix. In order to support accuracy verification of Sensitivity results using overall finite difference method, we use regular-Shaped finite elemen...
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Shape Sensitivity Analysis for 2D Mixed Mode Fractures Using Extended FEM (XFEM) and Level Set Method (LSM)
Mechanics Based Design of Structures and Machines, 2010Co-Authors: Mangesh Edke, Kuang-hua ChangAbstract:This article presents a Shape Sensitivity Analysis method for calculating gradients of crack growth rate and crack growth direction for 2D structural components under mixed-mode loading using extended finite element method (XFEM) and level set method (LSM). XFEM is a computational technique in which special enrichment functions are used to incorporate the discontinuity of structural responses caused by the crack surfaces and crack tip fields into finite element approximation. The LSM employs level set functions to track the crack during the crack propagation Analysis. As a result, this method does not require highly refined mesh around the crack tip nor remesh to conform to the geometric Shape of the crack when it propagates, which makes the method extremely attractive for crack propagation Analysis. However, Shape Sensitivity Analysis for crack propagation involves calculating derivatives of enrichment functions employed in XFEM that are discontinuous or unsmooth. The proposed Sensitivity Analysis method...
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Shape Sensitivity Analysis and design studies for CAD flume sections
Structural and Multidisciplinary Optimization, 2008Co-Authors: Kuang-hua ChangAbstract:This paper presents Shape design Sensitivity Analysis (DSA) and design studies for recreational waterslides represented in computer-aided design (CAD) environment. The mathematical representations of a number of commonly used flume sections that serve as the building blocks for waterslide configurations are created in CAD tools. Geometric dimensions of the individual sections that affect not only their geometric Shape but also the overall configurations are identified as design variables. These design variables can be varied to search for better design alternatives, for example, safer waterslides. A set of coupled differential equations based on Lagrange’s equations of motion that describe the motion of the riding object are derived. The equations of motion incorporate friction forces between the riding object and the surface of the flume sections. These second-order differential equations are then solved using Mathematica. Based on the equations of motion and design variables identified, a set of differential equations are derived for calculating Shape DSA coefficients. These equations are solved numerically again using Mathematica. The major contribution of the paper are (1) extending waterslide design parameterization and Shape DSA computation to true CAD-based flume sections, which greatly alleviates the design for manufacturing issues previously encountered, (2) incorporating friction forces into Shape DSA computation, and (3) developing a design scenario that includes Sensitivity display and what-if studies for a compromised design that is safer and with a larger acceleration, therefore, higher excitement levels. Incorporating friction forces into the computation supports design for rider’s excitement levels, which are related to accelerations. Waterslide design will not be realistic without including friction forces.
Byung Man Kwak - One of the best experts on this subject based on the ideXlab platform.
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Shape Sensitivity Analysis and Optimization of Linear and Nonlinear Transient Thermal Problems Using BEM
Computational Mechanics ’95, 1995Co-Authors: Dooho Lee, Byung Man KwakAbstract:Coupling of boundary element and Shape Sensitivity Analysis is recognized as a powerful tool in many fields. A Shape Sensitivity formulation for transient thermal systems is found in [1–3]. Nonlinear thermal problems are addressed in [4–6].
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Shape Sensitivity Analysis of thin-shell structures
Finite Elements in Analysis and Design, 1992Co-Authors: Shin Sup Lee, Byung Man KwakAbstract:Abstract This paper presents explicit formulas for Shape Sensitivity Analysis of thin shell structures. The curvature distribution is the design to be determined. The thin-shell theory employed is the general Koiter model in the Cartesian coordinates. For the Shape Sensitivity formulation, both the direct differentiation method and the material derivative concept have been used. The two formulations are shown to be equivalent. A computer program based on these formulations has been developed and applied to examples. The Shape Sensitivity results obtained have been compared to those obtained by finite differencing.
