An Example of a Variational Approach to Mechanics Using “Different” Rules of Algebra

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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