# Topology optimization using the finite volume method

Allan Gersborg-Hansen, Martin P. Bendsøe, Ole Sigmund

Research output: Contribution to conferenceConference abstract for conferenceResearchpeer-review

### Abstract

Computational procedures for topology optimization of continuum problems using a material distribution method are typically based on the application of the finite element method (FEM) (see, e.g. [1]). In the present work we study a computational framework based on the finite volume method (FVM, see, e.g. [2]) in order to develop methods for topology design for applications where conservation laws are critical such that element--wise conservation in the discretized models has a high priority. This encompasses problems involving for example mass and heat transport. The work described in this presentation is focused on a prototype model for topology optimization of steady heat diffusion. This allows for a study of the basic ingredients in working with FVM methods when dealing with topology optimization problems. The FVM and FEM based formulations differ both in how one computes the design derivative of the system matrix $\mathbf K$ and in how one computes the discretized version of certain objective functions. Thus for a cost function for minimum dissipated energy (like minimum compliance for an elastic structure) one obtains an expression $c = \mathbf u^\T \tilde{\mathbf K} \mathbf u$, where $\tilde{\mathbf K}$ is different from $\mathbf K$; in a FEM scheme these matrices are equal following the principle of virtual work. Using a staggered mesh and averaging procedures consistent with the FVM the checkerboard problem is eliminated. Two averages are compared to FE solutions, being the arithmetic and harmonic average with the latter being the well known Reuss lower bound. [1] Bendsøe, MP and Sigmund, O 2004: Topology Optimization - Theory, Methods, and Applications. Berlin Heidelberg: Springer Verlag [2] Versteeg, HK and Malalasekera, W 1995: An introduction to Computational Fluid Dynamics: the Finite Volume Method. London: Longman Scientific Technical
Original language English 2005 Published - 2005 M.I.T. conference - Boston, USADuration: 1 Jan 2005 → …Conference number: 3

### Conference

Conference M.I.T. conference 3 Boston, USA 01/01/2005 → …

Keynote lecture

### Cite this

Gersborg-Hansen, A., Bendsøe, M. P., & Sigmund, O. (2005). Topology optimization using the finite volume method. Abstract from M.I.T. conference, Boston, USA, .
Gersborg-Hansen, Allan ; Bendsøe, Martin P. ; Sigmund, Ole. / Topology optimization using the finite volume method. Abstract from M.I.T. conference, Boston, USA, .
title = "Topology optimization using the finite volume method",
abstract = "Computational procedures for topology optimization of continuum problems using a material distribution method are typically based on the application of the finite element method (FEM) (see, e.g. [1]). In the present work we study a computational framework based on the finite volume method (FVM, see, e.g. [2]) in order to develop methods for topology design for applications where conservation laws are critical such that element--wise conservation in the discretized models has a high priority. This encompasses problems involving for example mass and heat transport. The work described in this presentation is focused on a prototype model for topology optimization of steady heat diffusion. This allows for a study of the basic ingredients in working with FVM methods when dealing with topology optimization problems. The FVM and FEM based formulations differ both in how one computes the design derivative of the system matrix $\mathbf K$ and in how one computes the discretized version of certain objective functions. Thus for a cost function for minimum dissipated energy (like minimum compliance for an elastic structure) one obtains an expression $c = \mathbf u^\T \tilde{\mathbf K} \mathbf u$, where $\tilde{\mathbf K}$ is different from $\mathbf K$; in a FEM scheme these matrices are equal following the principle of virtual work. Using a staggered mesh and averaging procedures consistent with the FVM the checkerboard problem is eliminated. Two averages are compared to FE solutions, being the arithmetic and harmonic average with the latter being the well known Reuss lower bound. [1] Bends{\o}e, MP and Sigmund, O 2004: Topology Optimization - Theory, Methods, and Applications. Berlin Heidelberg: Springer Verlag [2] Versteeg, HK and Malalasekera, W 1995: An introduction to Computational Fluid Dynamics: the Finite Volume Method. London: Longman Scientific Technical",
author = "Allan Gersborg-Hansen and Bends{\o}e, {Martin P.} and Ole Sigmund",
note = "Keynote lecture; M.I.T. conference ; Conference date: 01-01-2005",
year = "2005",
language = "English",

}

Gersborg-Hansen, A, Bendsøe, MP & Sigmund, O 2005, 'Topology optimization using the finite volume method' M.I.T. conference, Boston, USA, 01/01/2005, .

Topology optimization using the finite volume method. / Gersborg-Hansen, Allan; Bendsøe, Martin P.; Sigmund, Ole.

2005. Abstract from M.I.T. conference, Boston, USA, .

Research output: Contribution to conferenceConference abstract for conferenceResearchpeer-review

TY - ABST

T1 - Topology optimization using the finite volume method

AU - Gersborg-Hansen, Allan

AU - Bendsøe, Martin P.

AU - Sigmund, Ole

N1 - Keynote lecture

PY - 2005

Y1 - 2005

N2 - Computational procedures for topology optimization of continuum problems using a material distribution method are typically based on the application of the finite element method (FEM) (see, e.g. [1]). In the present work we study a computational framework based on the finite volume method (FVM, see, e.g. [2]) in order to develop methods for topology design for applications where conservation laws are critical such that element--wise conservation in the discretized models has a high priority. This encompasses problems involving for example mass and heat transport. The work described in this presentation is focused on a prototype model for topology optimization of steady heat diffusion. This allows for a study of the basic ingredients in working with FVM methods when dealing with topology optimization problems. The FVM and FEM based formulations differ both in how one computes the design derivative of the system matrix $\mathbf K$ and in how one computes the discretized version of certain objective functions. Thus for a cost function for minimum dissipated energy (like minimum compliance for an elastic structure) one obtains an expression $c = \mathbf u^\T \tilde{\mathbf K} \mathbf u$, where $\tilde{\mathbf K}$ is different from $\mathbf K$; in a FEM scheme these matrices are equal following the principle of virtual work. Using a staggered mesh and averaging procedures consistent with the FVM the checkerboard problem is eliminated. Two averages are compared to FE solutions, being the arithmetic and harmonic average with the latter being the well known Reuss lower bound. [1] Bendsøe, MP and Sigmund, O 2004: Topology Optimization - Theory, Methods, and Applications. Berlin Heidelberg: Springer Verlag [2] Versteeg, HK and Malalasekera, W 1995: An introduction to Computational Fluid Dynamics: the Finite Volume Method. London: Longman Scientific Technical

AB - Computational procedures for topology optimization of continuum problems using a material distribution method are typically based on the application of the finite element method (FEM) (see, e.g. [1]). In the present work we study a computational framework based on the finite volume method (FVM, see, e.g. [2]) in order to develop methods for topology design for applications where conservation laws are critical such that element--wise conservation in the discretized models has a high priority. This encompasses problems involving for example mass and heat transport. The work described in this presentation is focused on a prototype model for topology optimization of steady heat diffusion. This allows for a study of the basic ingredients in working with FVM methods when dealing with topology optimization problems. The FVM and FEM based formulations differ both in how one computes the design derivative of the system matrix $\mathbf K$ and in how one computes the discretized version of certain objective functions. Thus for a cost function for minimum dissipated energy (like minimum compliance for an elastic structure) one obtains an expression $c = \mathbf u^\T \tilde{\mathbf K} \mathbf u$, where $\tilde{\mathbf K}$ is different from $\mathbf K$; in a FEM scheme these matrices are equal following the principle of virtual work. Using a staggered mesh and averaging procedures consistent with the FVM the checkerboard problem is eliminated. Two averages are compared to FE solutions, being the arithmetic and harmonic average with the latter being the well known Reuss lower bound. [1] Bendsøe, MP and Sigmund, O 2004: Topology Optimization - Theory, Methods, and Applications. Berlin Heidelberg: Springer Verlag [2] Versteeg, HK and Malalasekera, W 1995: An introduction to Computational Fluid Dynamics: the Finite Volume Method. London: Longman Scientific Technical

M3 - Conference abstract for conference

ER -

Gersborg-Hansen A, Bendsøe MP, Sigmund O. Topology optimization using the finite volume method. 2005. Abstract from M.I.T. conference, Boston, USA, .