Abstract
The Löwner partial order is taken into consideration in order to define Löwner majorants for a given finite set of symmetric matrices. A special class of Löwner majorants is analyzed based on two specific matrix parametrizations: a two-parametric form and a four-parametric form, which arise in the context of so-called zeroth-order bounds of the effective linear behavior in the field of solid mechanics in engineering. The condensed explicit conditions defining the convex parameter sets of Löwner majorants are derived. Examples are provided, and potential application to semidefinite programming problems is discussed. Open-source MATLAB software is provided to support the theoretical findings and for reproduction of the presented results. The results of the present work offer in combination with the theory of zeroth-order bounds of mechanics a highly efficient approach for the automated material selection for future engineering applications.
Highlights
Introduction e Lowner partial order introduced by [1] is connected to several matrix partial orders
It implies several matrix inequalities and has been widely studied. is matrix partial order is important for linear and nonlinear optimization problems in semidefinite programming since it describes an essential part of the constraints in many real-world optimization problems
A zerothorder bound of the effective linear material behavior of the N-phase solid is a symmetric matrix B ∈ Rn×n which satisfies
Summary
We consider the Lowner partial order of symmetric matrices A, B ∈ Sn (see, e.g., [6]):. Any B ∈ Sn is referred to, in this work, as a (Lowner) majorant of a given A ∈ Sn if A ⪯ B holds. Any B ∈ Sn is referred to as a sharp (Lowner) majorant of a given A ∈ Sn if A ⪯ B and det(B − A) 0 hold, i.e., if B − A ∈ zS+n holds. For given A ∈ Sn, the corresponding majorant set B ⊂ Rn×n and sharp majorant set zB are denoted as. E trivial majorant T ∈ zB of a given A is defined as. A(N) of symmetric matrices, we define the analogous majorant set For a finite set A A(1), . . . , A(N) of symmetric matrices, we define the analogous majorant set
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