scholarly journals Shape optimization for an elliptic operator with infinitely many positive and negative eigenvalues

2018 ◽  
Vol 7 (1) ◽  
pp. 49-66 ◽  
Author(s):  
Catherine Bandle ◽  
Alfred Wagner
Keyword(s):  
Eigenvalue Problem ◽  
Shape Optimization ◽  
Elliptic Operator ◽  
Fundamental Frequency ◽  
Isoperimetric Inequalities ◽  
Main Question ◽  
Shape Derivatives ◽  
Negative Eigenvalues ◽  
Domain Variations ◽  
Circular Domains

AbstractThis paper deals with an eigenvalue problem possessing infinitely many positive and negative eigenvalues. Inequalities for the smallest positive and the largest negative eigenvalues, which have the same properties as the fundamental frequency, are derived. The main question is whether or not the classical isoperimetric inequalities for the fundamental frequency of membranes hold in this case. The arguments are based on the harmonic transplantation for the global results and the shape derivatives (domain variations) for nearly circular domains.

scholarly journals Generalized Emden-Fowler equations of subcritical growth

1993 ◽  
Vol 54 (2) ◽  
pp. 254-262 ◽  
Author(s):  
Wolfgang Rother
Keyword(s):  
Eigenvalue Problem ◽  
Elliptic Operator ◽  
Positive Solutions ◽  
Subcritical Growth ◽  
Existence Of Positive Solutions

AbstractThe existence of positive solutions, vanishing at infinity, for the semilinear eigenvalue problem Lu = λ f(x, y) in RN is obtained, where L is a strictly elliptic operator. The function f is assumed to be of subcritical growth with respect to the variable u.

scholarly journals A variational formulation for computing shape derivatives of geometric constraints along rays

2020 ◽  
Vol 54 (1) ◽  
pp. 181-228 ◽  
Author(s):  
Florian Feppon ◽  
Grégoire Allaire ◽  
Charles Dapogny
Keyword(s):  
Velocity Field ◽  
Shape Optimization ◽  
Explicit Knowledge ◽  
Optimization Problems ◽  
Geometric Constraints ◽  
Normal Vector ◽  
Minimum Thickness ◽  
Full Generality ◽  
Structural Shape ◽  
Shape Derivatives

In the formulation of shape optimization problems, multiple geometric constraint functionals involve the signed distance function to the optimized shape Ω. The numerical evaluation of their shape derivatives requires to integrate some quantities along the normal rays to Ω, a challenging operation to implement, which is usually achieved thanks to the method of characteristics. The goal of the present paper is to propose an alternative, variational approach for this purpose. Our method amounts, in full generality, to compute integral quantities along the characteristic curves of a given velocity field without requiring the explicit knowledge of these curves on the spatial discretization; it rather relies on a variational problem which can be solved conveniently by the finite element method. The well-posedness of this problem is established thanks to a detailed analysis of weighted graph spaces of the advection operator β ⋅ ∇ associated to a C1 velocity field β. One novelty of our approach is the ability to handle velocity fields with possibly unbounded divergence: we do not assume div(β) ∈ L∞. Our working assumptions are fulfilled in the context of shape optimization of C2 domains Ω, where the velocity field β = ∇dΩ is an extension of the unit outward normal vector to the optimized shape. The efficiency of our variational method with respect to the direct integration of numerical quantities along rays is evaluated on several numerical examples. Classical albeit important implementation issues such as the calculation of a shape’s curvature and the detection of its skeleton are discussed. Finally, we demonstrate the convenience and potential of our method when it comes to enforcing maximum and minimum thickness constraints in structural shape optimization.

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