TY - JOUR
T1 - Optimal Shape Design for the p-Laplacian Eigenvalue Problem
AU - Mohammadi, Seyyed Abbas
AU - Bozorgnia, Farid
AU - Voss, Heinrich
N1 - Copyright © Springer Science+Business Media, LLC, part of Springer Nature 2018
PY - 2019/2/15
Y1 - 2019/2/15
N2 - In this paper, a shape optimization problem corresponding to the p-Laplacian operator is studied. Given a density function in a rearrangement class generated by a step function, find the density such that the principal eigenvalue is as small as possible. Considering a membrane of known fixed mass and with fixed boundary of prescribed shape consisting of two different materials, our results determine the way to distribute these materials such that the basic frequency of the membrane is minimal. We obtain some qualitative aspects of the optimizer and then we determine nearly optimal sets which are approximations of the minimizer for specific ranges of parameters values. A numerical algorithm is proposed to derive the optimal shape and it is proved that the numerical procedure converges to a local minimizer. Numerical illustrations are provided for different domains to show the efficiency and practical suitability of our approach.
AB - In this paper, a shape optimization problem corresponding to the p-Laplacian operator is studied. Given a density function in a rearrangement class generated by a step function, find the density such that the principal eigenvalue is as small as possible. Considering a membrane of known fixed mass and with fixed boundary of prescribed shape consisting of two different materials, our results determine the way to distribute these materials such that the basic frequency of the membrane is minimal. We obtain some qualitative aspects of the optimizer and then we determine nearly optimal sets which are approximations of the minimizer for specific ranges of parameters values. A numerical algorithm is proposed to derive the optimal shape and it is proved that the numerical procedure converges to a local minimizer. Numerical illustrations are provided for different domains to show the efficiency and practical suitability of our approach.
KW - Eigenvalue optimization
KW - p-Laplacian
KW - Rearrangement algorithm
UR - http://www.scopus.com/inward/record.url?eid=2-s2.0-85051775812&partnerID=MN8TOARS
U2 - 10.1007/s10915-018-0806-7
DO - 10.1007/s10915-018-0806-7
M3 - Article
SN - 0885-7474
VL - 78
SP - 1231
EP - 1249
JO - Journal of Scientific Computing
JF - Journal of Scientific Computing
IS - 2
ER -