Toplogical optimization of structures using Fourier representations

White, Daniel A.; Stowell, Mark L.; Tortorelli, Daniel A.

doi:10.1007/s00158-018-1962-y

Title: Toplogical optimization of structures using Fourier representations

Abstract

The minimization of compliance subject to a mass constraint is the topology optimization design problem of interest. The goal is to determine the optimal configuration of material within an allowed volume. Our approach builds upon the well-known density method in which the decision variable is the material density in every cell in a mesh. In it’s most basic form the density method consists of three steps: 1) the problem is convexified by replacing the integer material indicator function with a volume fraction, 2) the problem is regularized by filtering the volume fraction field to impose a minimum length scale; 3) the filtered volume fraction is penalized to steer the material distribution toward binary designs. The filtering step is used to yield a mesh-independent solution and to eliminate checkerboard instabilities. In image processing terms this is a low-pass filter, and a consequence is that the decision variables are not independent and a change of basis could significantly reduce the dimension of the nonlinear programming problem. Based on this observation, we represent the volume fraction field with a truncated Fourier representation. Furthermore, this imposes a minimal length scale on the problem, eliminates checkerboard instabilities, and also reduces the number of decision variablesmore »« less

Authors:

White, Daniel A. ^[1]; Stowell, Mark L. ^[1]; Tortorelli, Daniel A. ^[1]

Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

Publication Date:: Mon Apr 02 00:00:00 EDT 2018

Research Org.:: Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

Sponsoring Org.:: USDOE National Nuclear Security Administration (NNSA)

OSTI Identifier:: 1479078

Report Number(s):: LLNL-JRNL-736779
Journal ID: ISSN 1615-147X; 889675

Grant/Contract Number:: AC52-07NA27344

Resource Type:: Accepted Manuscript

Journal Name:: Structural and Multidisciplinary Optimization

Additional Journal Information:: Journal Volume: 58; Journal Issue: 3; Journal ID: ISSN 1615-147X

Publisher:: Springer

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; Topology optimization; Fourier analysis

Citation Formats


                    White, Daniel A., Stowell, Mark L., and Tortorelli, Daniel A. Toplogical optimization of structures using Fourier representations.  United States: N. p., 2018. 
Web.  doi:10.1007/s00158-018-1962-y.

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                    White, Daniel A., Stowell, Mark L., & Tortorelli, Daniel A. Toplogical optimization of structures using Fourier representations.  United States.  https://doi.org/10.1007/s00158-018-1962-y

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                    White, Daniel A., Stowell, Mark L., and Tortorelli, Daniel A. Mon .  
"Toplogical optimization of structures using Fourier representations".  United States.  https://doi.org/10.1007/s00158-018-1962-y.  https://www.osti.gov/servlets/purl/1479078.

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@article{osti_1479078,

  title        = {Toplogical optimization of structures using Fourier representations},

  author       = {White, Daniel A. and Stowell, Mark L. and Tortorelli, Daniel A.},

  abstractNote = {The minimization of compliance subject to a mass constraint is the topology optimization design problem of interest. The goal is to determine the optimal configuration of material within an allowed volume. Our approach builds upon the well-known density method in which the decision variable is the material density in every cell in a mesh. In it’s most basic form the density method consists of three steps: 1) the problem is convexified by replacing the integer material indicator function with a volume fraction, 2) the problem is regularized by filtering the volume fraction field to impose a minimum length scale; 3) the filtered volume fraction is penalized to steer the material distribution toward binary designs. The filtering step is used to yield a mesh-independent solution and to eliminate checkerboard instabilities. In image processing terms this is a low-pass filter, and a consequence is that the decision variables are not independent and a change of basis could significantly reduce the dimension of the nonlinear programming problem. Based on this observation, we represent the volume fraction field with a truncated Fourier representation. Furthermore, this imposes a minimal length scale on the problem, eliminates checkerboard instabilities, and also reduces the number of decision variables by over 100 × (two dimensions) or 1000 × (three dimensions).},

  doi          = {10.1007/s00158-018-1962-y},

  journal      = {Structural and Multidisciplinary Optimization},

  number       = 3,

  volume       = 58,

  place        = {United States},

  year         = {Mon Apr 02 00:00:00 EDT 2018},

  month        = {Mon Apr 02 00:00:00 EDT 2018}

}

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Works referencing / citing this record:

A dual mesh method with adaptivity for stress-constrained topology optimization
journal, November 2019

White, Daniel A.; Choi, Youngsoo; Kudo, Jun
Structural and Multidisciplinary Optimization, Vol. 61, Issue 2
DOI: 10.1007/s00158-019-02393-6

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Similar Records

Title: Toplogical optimization of structures using Fourier representations

Abstract

Citation Formats

Incorporating topological derivatives into level set methods journal, February 2004

Achieving minimum length scale in topology optimization using nodal design variables and projection functions journal, August 2004

Topology optimization of structures using meshless density variable approximants: TOPOLOGY OPTIMIZATION USING MESHLESS METHODS journal, December 2012

Doing Topology Optimization Explicitly and Geometrically—A New Moving Morphable Components Based Framework journal, May 2014

A geometry projection method for the topology optimization of plate structures journal, May 2016

Line Search Filter Methods for Nonlinear Programming: Motivation and Global Convergence journal, January 2005

Structural topology optimization based on non-local Shepard interpolation of density field journal, December 2011

Level-set methods for structural topology optimization: a review journal, March 2013

Optimal design of elastic plates with a constraint on the slope of the thickness function journal, January 1983

Adaptive topology optimization with independent error control for separated displacement and density fields journal, April 2014

On projection methods, convergence and robust formulations in topology optimization journal, December 2010

Multi-resolution multi-scale topology optimization — a new paradigm journal, September 2000

An adaptive refinement approach for topology optimization based on separated density field description journal, February 2013

Optimal shape design as a material distribution problem journal, December 1989

A topological derivative method for topology optimization journal, February 2007

Filters in topology optimization based on Helmholtz-type differential equations journal, December 2010

Heterogeneous material modeling with distance fields journal, March 2004

Filters in topology optimization journal, January 2001

Structural Boundary Design via Level Set and Immersed Interface Methods journal, September 2000

An element removal and reintroduction strategy for the topology optimization of structures and compliant mechanisms journal, January 2003

A geometry projection method for continuum-based topology optimization with discrete elements journal, August 2015

Topology optimization in wavelet space journal, January 2001

Topology optimization in B-spline space journal, October 2013

Aims, scope, methods, history and unified terminology of computer-aided topology optimization in structural mechanics journal, April 2001

Topology optimization of non-linear elastic structures and compliant mechanisms journal, March 2001

Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima journal, August 1998

A new approach to variable-topology shape design using a constraint on perimeter journal, February 1996

An optimal design problem with perimeter penalization journal, March 1993

Application of spectral level set methodology in topology optimization journal, April 2006

On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming journal, April 2005

A level set method for structural topology optimization journal, January 2003

Reducing dimensionality in topology optimization using adaptive design variable fields: REDUCING DIMENSIONALITY IN TOPOLOGY OPTIMIZATION journal, August 2009

A dual mesh method with adaptivity for stress-constrained topology optimization journal, November 2019

Incorporating topological derivatives into level set methods
journal, February 2004

Achieving minimum length scale in topology optimization using nodal design variables and projection functions
journal, August 2004

Topology optimization of structures using meshless density variable approximants: TOPOLOGY OPTIMIZATION USING MESHLESS METHODS
journal, December 2012

Doing Topology Optimization Explicitly and Geometrically—A New Moving Morphable Components Based Framework
journal, May 2014

A geometry projection method for the topology optimization of plate structures
journal, May 2016

Line Search Filter Methods for Nonlinear Programming: Motivation and Global Convergence
journal, January 2005

Structural topology optimization based on non-local Shepard interpolation of density field
journal, December 2011

Level-set methods for structural topology optimization: a review
journal, March 2013

Optimal design of elastic plates with a constraint on the slope of the thickness function
journal, January 1983

Adaptive topology optimization with independent error control for separated displacement and density fields
journal, April 2014

On projection methods, convergence and robust formulations in topology optimization
journal, December 2010

Multi-resolution multi-scale topology optimization — a new paradigm
journal, September 2000

An adaptive refinement approach for topology optimization based on separated density field description
journal, February 2013

Optimal shape design as a material distribution problem
journal, December 1989

A topological derivative method for topology optimization
journal, February 2007

Filters in topology optimization based on Helmholtz-type differential equations
journal, December 2010

Heterogeneous material modeling with distance fields
journal, March 2004

Filters in topology optimization
journal, January 2001

Structural Boundary Design via Level Set and Immersed Interface Methods
journal, September 2000

An element removal and reintroduction strategy for the topology optimization of structures and compliant mechanisms
journal, January 2003

A geometry projection method for continuum-based topology optimization with discrete elements
journal, August 2015

Topology optimization in wavelet space
journal, January 2001

Topology optimization in B-spline space
journal, October 2013

Aims, scope, methods, history and unified terminology of computer-aided topology optimization in structural mechanics
journal, April 2001

Topology optimization of non-linear elastic structures and compliant mechanisms
journal, March 2001

Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima
journal, August 1998

A new approach to variable-topology shape design using a constraint on perimeter
journal, February 1996

An optimal design problem with perimeter penalization
journal, March 1993

Application of spectral level set methodology in topology optimization
journal, April 2006

On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming
journal, April 2005

A level set method for structural topology optimization
journal, January 2003

Reducing dimensionality in topology optimization using adaptive design variable fields: REDUCING DIMENSIONALITY IN TOPOLOGY OPTIMIZATION
journal, August 2009

A dual mesh method with adaptivity for stress-constrained topology optimization
journal, November 2019