{
  "id": "2010.13935",
  "version": "v1",
  "published": "2020-10-26T22:45:39.000Z",
  "updated": "2020-10-26T22:45:39.000Z",
  "title": "A discretize-then-map approach for the treatment of parameterized geometries in model order reduction",
  "authors": [
    "Tommaso Taddei",
    "Lei Zhang"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "We propose a new general approach for the treatment of parameterized geometries in projection-based model order reduction. During the offline stage, given (i) a family of parameterized domains $\\{ \\Omega_{\\mu}: \\mu \\in \\mathcal{P} \\} \\subset \\mathbb{R}^D$ where $\\mu \\in \\mathcal{P} \\subset \\mathbb{R}^P$ denotes a vector of parameters, (ii) a parameterized mapping $\\underline{\\Phi}_{\\mu}$ between a reference domain $\\Omega$ and the parameter-dependent domain $\\Omega_{\\mu}$, and (iii) a finite element triangulation of $\\Omega$, we resort to an empirical quadrature procedure to select a subset of the elements of the grid. During the online stage, we first use the mapping to \"move\" the nodes of the selected elements and then we use standard element-wise residual evaluation routines to evaluate the residual and possibly its Jacobian. We discuss how to devise an online-efficient reduced-order model and we discuss the differences with the more standard \"map-then-discretize\" approach (e.g., Rozza, Huynh, Patera, ACME, 2007); in particular, we show how the discretize-then-map framework greatly simplifies the implementation of the reduced-order model. We apply our approach to a two-dimensional potential flow problem past a parameterized airfoil, and to the two-dimensional RANS simulations of the flow past the Ahmed body.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-26T22:45:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65N30",
      "41A45",
      "35J57"
    ],
    "keywords": [
      "model order reduction",
      "parameterized geometries",
      "discretize-then-map approach",
      "two-dimensional potential flow problem past",
      "reduced-order model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}