The topological derivative of stress-based cost functionals in anisotropic elasticity. (May 2015)

• Record Type: Journal Article
• Authors: Delgado, Gabriel; Bonnet, Marc
• Is Part Of: Computers & mathematics with applications. Volume 69:issue 10(2015)
• Page Start: 1144
• Page End: 1166
• Publication Date: 2015-05
• Subjects: Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2015.03.010
• Languages: English
• ISSNs: 0898-1221

Abstract

<abstract xml:lang="en" abstract-type="author" id="a000005"> <title id="st000005">Abstract</title> <sec> <p id="sp000125">The topological derivative of cost functionals <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgjhm3sgqs" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="si32.gif" display="inline" overflow="scroll" id="d13e615" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>J</mml:mi></mml:math></alternatives></inline-formula> that depend on the stress (through the displacement gradient, assuming a linearly elastic material behavior) is considered in a quite general 3D setting where both the background and the inhomogeneity may have arbitrary anisotropic elastic properties. The topological derivative <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgjhm3dx6c" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="si33.gif" display="inline" overflow="scroll" id="d13e619" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mstyle mathvariant="normal"><mml:mi>D</mml:mi></mml:mstyle><mml:mi>J</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mstyle mathvariant="bold-italic"><mml:mi>z</mml:mi></mml:mstyle><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> of <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgjhm3fjxc" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="si34.gif" display="inline" overflow="scroll" id="d13e634" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>J</mml:mi></mml:math></alternatives></inline-formula> quantifies the asymptotic behavior of <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgjhm3qkxf" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="si35.gif" display="inline" overflow="scroll" id="d13e638" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>J</mml:mi></mml:math></alternatives></inline-formula> induced by the nucleation in the background elastic medium of a small anisotropic inhomogeneity of characteristic radius <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgjhm3ht89" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="si36.gif" display="inline" overflow="scroll" id="d13e642" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>a</mml:mi></mml:math></alternatives></inline-formula> at a specified location <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgjhm3dz9d" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="si37.gif" display="inline" overflow="scroll" id="d13e647" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mstyle mathvariant="bold-italic"><mml:mi>z</mml:mi></mml:mstyle></mml:math></alternatives></inline-formula>. The fact that the strain perturbation inside an elastic inhomogeneity remains finite for arbitrarily small <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgjhm3n61s" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="si38.gif" display="inline" overflow="scroll" id="d13e652" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>a</mml:mi></mml:math></alternatives></inline-formula> makes the small-inhomogeneity asymptotics of stress-based cost functionals quite different than that of the more usual displacement-based functionals.</p> <p id="sp000130">The asymptotic perturbation of <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgjhm3m84v" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="si39.gif" display="inline" overflow="scroll" id="d13e658" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>J</mml:mi></mml:math></alternatives></inline-formula> is shown to be of order <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgjhm3kpq0" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="si40.gif" display="inline" overflow="scroll" id="d13e662" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> for a wide class of stress-based cost functionals having smooth densities. The topological derivative of <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgjhm3g69p" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="si41.gif" display="inline" overflow="scroll" id="d13e678" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>J</mml:mi></mml:math></alternatives></inline-formula>, i.e. the coefficient of the <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgjhm3nwfn" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="si42.gif" display="inline" overflow="scroll" id="d13e682" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> perturbation, is established, and computational procedures then discussed. The resulting small-inhomogeneity expansion of <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgjhm3pmf7" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="si43.gif" display="inline" overflow="scroll" id="d13e698" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>J</mml:mi></mml:math></alternatives></inline-formula> is mathematically justified (i.e. its remainder is proved to be of order <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgjhm3dvx8" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="si44.gif" display="inline" overflow="scroll" id="d13e703" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>o</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>). Several 2D and 3D numerical examples are presented, in particular demonstrating the proposed formulation of <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgjhm3sq9z" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="si45.gif" display="inline" overflow="scroll" id="d13e719" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mstyle mathvariant="normal"><mml:mi>D</mml:mi></mml:mstyle><mml:mi>J</mml:mi></mml:math></alternatives></inline-formula> on cases involving anisotropic elasticity and non-quadratic cost functionals.</p> </sec> </abstract>