Research Papers: Design Automation

An Inverse Method for Optimizing Elastic Properties Considering Multiple Loading Conditions and Displacement Criteria

[+] Author and Article Information
Danny Smyl

Department of Mechanical Engineering,
Aalto University,
Espoo 02150, Finland
e-mail: danny.smyl@aalto.fi

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received April 2, 2018; final manuscript received June 29, 2018; published online September 7, 2018. Assoc. Editor: Raymundo Arroyave.

J. Mech. Des 140(11), 111411 (Sep 07, 2018) (8 pages) Paper No: MD-18-1283; doi: 10.1115/1.4040788 History: Received April 02, 2018; Revised June 29, 2018

Significant research effort has been devoted to topology optimization (TO) of two- and three-dimensional structural elements subject to various design and loading criteria. While the field of TO has been tremendously successful over the years, literature focusing on the optimization of spatially varying elastic material properties in structures subject to multiple loading states is scarce. In this article, we contribute to the state of the art in material optimization by proposing a numerical regime for optimizing the distribution of the elastic modulus in structural elements subject to multiple loading conditions and design displacement criteria. Such displacement criteria (target displacement fields prescribed by the designer) may result from factors related to structural codes, occupant comfort, proximity of adjacent structures, etc. In this work, we utilize an inverse problem based framework for optimizing the elastic modulus distribution considering N target displacements and imposed forces. This approach is formulated in a straight-forward manner such that it may be applied in a broad suite of design problems with unique geometries, loading conditions, and displacement criteria. To test the approach, a suite of optimization problems are solved to demonstrate solutions considering N = 2 for different geometries and boundary conditions.

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Grahic Jump Location
Fig. 1

Example 1: Schematic detailing the external forces and structural elements deformed by the design displacement targets (a) Δ1, a cubic polynomial translating the x grid coordinates and (b) Δ2, a pure axial translation along the y-axis

Grahic Jump Location
Fig. 2

Optimized distribution of EΨ for Example 1, (a) EΨ plotted atop U1 and (b) EΨ plotted atop U2. The right-hand side color bar indicates the magnitude of inhomogeneous EΨ in each image.

Grahic Jump Location
Fig. 3

Example 2: Schematic detailing the external forces and structural elements deformed by the design displacement targets (a) Δ1, a quadratic polynomial translating the x grid coordinates and (b) Δ2 a cubic polynomial translating the x grid coordinates

Grahic Jump Location
Fig. 4

Optimized distribution of EΨ for Example 2, (a) EΨ plotted atop U1 and (b) EΨ plotted atop U2. The right-hand side color bar indicates the magnitude of inhomogeneous EΨ in each image.

Grahic Jump Location
Fig. 5

Example 3: Schematic detailing the external forces and structural elements deformed by the design displacement targets Δ1 and Δ2 for (a) a coarse triangular discretization and (b) a refined triangular discretization

Grahic Jump Location
Fig. 6

Optimized distributions of EΨ for Example 3 for (a) the coarse mesh with Ec > 0, (b) the coarse mesh with Ec > 50, (c) the refined mesh with Ec > 0, and (d) the refined mesh with Ec > 50

Grahic Jump Location
Fig. 7

Minimization curves for the cost function Ψ as a function of the iteration number k considering the lower constraint Ec and the discretization



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