An example illustrating the potentiality and peculiarities of a variational approach to electrostatic problems

In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave–convex nonlinearities: [Formula: see text] where [Formula: see text] is a smooth bounded domain in [Formula: see text] containing [Formula: see text] in its interior, and [Formula: see text] with [Formula: see text] which may change sign in [Formula: see text]. We use the variational methods and the Nehari manifold decomposition to prove that this problem has at least two positive solutions for [Formula: see text] sufficiently small. The variational approach requires that [Formula: see text] [Formula: see text] [Formula: see text], and [Formula: see text], the latter being the best fractional Hardy constant on [Formula: see text].

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A variational approach of homogenization of heterogeneous materials towards second gradient continua

Mechanics of Materials ◽

10.1016/j.mechmat.2021.103743 ◽

2021 ◽

pp. 103743

Author(s):

J.F. Ganghoffer ◽

H. Reda

Keyword(s):

Variational Approach ◽

Heterogeneous Materials ◽

Second Gradient

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A new variational approach for the thermodynamic topology optimization of hyperelastic structures

Computational Mechanics ◽

10.1007/s00466-020-01949-4 ◽

2020 ◽

Author(s):

Philipp Junker ◽

Daniel Balzani

Keyword(s):

Topology Optimization ◽

Variational Approach ◽

Large Deformations ◽

Mathematical Structure ◽

Detailed Model ◽

Novel Approach ◽

Extremal Principles ◽

Model Derivation ◽

Numerical Discretization ◽

Total Structure

AbstractWe present a novel approach to topology optimization based on thermodynamic extremal principles. This approach comprises three advantages: (1) it is valid for arbitrary hyperelastic material formulations while avoiding artificial procedures that were necessary in our previous approaches for topology optimization based on thermodynamic principles; (2) the important constraints of bounded relative density and total structure volume are fulfilled analytically which simplifies the numerical implementation significantly; (3) it possesses a mathematical structure that allows for a variety of numerical procedures to solve the problem of topology optimization without distinct optimization routines. We present a detailed model derivation including the chosen numerical discretization and show the validity of the approach by simulating two boundary value problems with large deformations.

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