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  "body": "\n## Problem\n\nLet $D \\subseteq \\R^d$ be a bounded open design region with Lipschitz boundary, and let $0 \u003c V \u003c \\abs{D}$ be a volume bound.\nLet $A$ be the elasticity tensor of a given isotropic elastic material, and let $f \\in L^2(D; \\R^d)$ be a load.\nFor a measurable set $\\omega \\subseteq D$ let $\\chi_\\omega$ be its indicator function, and fix a small $\\varepsilon \u003e 0$.\nLet $u_\\omega \\in H^1(D; \\R^d)$ be the displacement that solves the linear elasticity system $-\\operatorname{div}\\big((\\chi_\\omega + \\varepsilon (1 - \\chi_\\omega)) A \\, e(u)\\big) = f$ in $D$, where $e(u) = \\frac12 (\\nabla u + \\nabla u^{T})$.\nThe boundary conditions are $u = 0$ on a part of $\\partial D$ of positive surface measure and zero traction on the rest.\nThe *minimum compliance topology optimization problem* is to find a set $\\omega$ with $\\abs{\\omega} \\le V$ that minimizes the compliance $\\int_D f \\cdot u_\\omega \\, dx$.\n\n## Remarks\n\nThe material occupies $\\omega$, and a very weak material of stiffness $\\varepsilon$ fills the rest of $D$ so that the equation stays elliptic.\nUnlike classical [[shape optimization|WWQJQXJECK]], no topology of $\\omega$ is fixed in advance: holes may appear or disappear.\nIn general this problem has no minimizer: minimizing sequences develop finer and finer mixtures of material and void.\nBendsøe and Kikuchi replaced the sets $\\omega$ by composite materials with a microstructure, given by homogenization, and this relaxed problem has solutions.\nA widely used simpler method replaces $\\chi_\\omega$ by a density $\\rho \\colon D \\to [0, 1]$ and the stiffness by $\\rho^p A$ with $p \u003e 1$, which pushes $\\rho$ toward the values $0$ and $1$.\nThe resulting discrete problems are solved by gradient methods with adjoint sensitivities, as in [[optimal control of partial differential equations|S2BBZPAFDR]].\n\n## Explanations\n\n### L1 human\n\nTopology optimization asks where to put material inside a box so that a structure is as stiff as possible for a given amount of material.\nThe answer may have holes, struts and branches that no designer fixed in advance.\nComputers solve such problems by giving each small cell of the box a density between $0$ and $1$ and improving the densities step by step.\nThe results often look like bones or trees, and they are used to design light parts for aircraft and cars.\n",
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