{
  "id": "2107.03159",
  "version": "v1",
  "published": "2021-07-07T11:37:13.000Z",
  "updated": "2021-07-07T11:37:13.000Z",
  "title": "Phase-Field Methods for Spectral Shape and Topology Optimization",
  "authors": [
    "Harald Garcke",
    "Paul Hüttl",
    "Christian Kahle",
    "Patrik Knopf"
  ],
  "categories": [
    "math.OC",
    "math.AP",
    "math.SP"
  ],
  "abstract": "We optimize a selection of eigenvalues of the Laplace operator with Dirichlet or Neumann boundary conditions by adjusting the shape of the domain on which the eigenvalue problem is considered. Here, a phase-field function is used to represent the shapes over which we minimize. The idea behind this method is to modify the Laplace operator by introducing phase-field dependent coefficients in order to extend the eigenvalue problem on a fixed design domain containing all admissible shapes. The resulting shape and topology optimization problem can then be formulated as an optimal control problem with PDE constraints in which the phase-field function acts as the control. For this optimal control problem, we establish first-order necessary optimality conditions and we rigorously derive its sharp interface limit. Eventually, we present and discuss several numerical simulations for our optimization problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-07T11:37:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35P05",
      "35P15",
      "35R35",
      "49M05",
      "49M41",
      "49K20",
      "49J20",
      "49J40",
      "49Q10",
      "49R05"
    ],
    "keywords": [
      "topology optimization",
      "phase-field methods",
      "spectral shape",
      "optimal control problem",
      "optimization problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}