author = {Atwal, Pradeep and Conti, Sergio and Geihe, Benedict and Pach, Martin
	and Rumpf, Martin and Schultz, R{\"{u}}diger},
  title = {On shape optimization with stochastic loadings},
  booktitle = {Constrained Optimization and Optimal Control for Partial Differential
  publisher = {Springer},
  year = {2012},
  editor = {Leugering, G{\"{u}}nter and Engell, Sebastian and Griewank, Andreas
	and Hinze, Michael and Rannacher, Rolf and Schulz, Volker and Ulbrich,
	Michael and Ulbrich, Stefan},
  volume = {160},
  series = {International Series of Numerical Mathematics},
  chapter = {2},
  pages = {215--243},
  address = {Basel},
  abstract = {This article is concerned with different approaches to elastic shape
	optimization under stochastic loading. The underlying stochastic
	optimization strategy builds upon the methodology of two-stage stochastic
	programming. In fact, in the case of linear elasticity and quadratic
	objective functionals our strategy leads to a computational cost
	which scales linearly in the number of linearly independent applied
	forces, even for a large set of realizations of the random loading.
	We consider, besides minimization of the expectation value of suitable
	objective functionals, also two different risk averse approaches,
	namely the expected excess and the excess probability . Numerical
	computations are performed using either a level set approach representing
	implicit shapes of general topology in combination with composite
	finite elements to resolve elasticity in two and three dimensions,
	or a collocation boundary element approach, where polygonal shapes
	represent geometric details attached to a lattice and describing
	a perforated elastic domain. Topology optimization is performed using
	the concept of topological derivatives. We generalize this concept,
	and derive an analytical expression which takes into account the
	interaction between neighboring holes. This is expected to allow
	efficient and reliable optimization strategies of elastic objects
	with a large number of geometric details on a fine scale.},
  doi = {10.1007/978-3-0348-0133-1_12},
  pdf = { 1},
  isbn = {978-3-0348-0133-1},
  keywords = {SHAPE_OPT},
  url = {}