# Convergence of Cell Based Finite Volume Discretizations for Problems of Control in the Conduction Coefficients

Anton Evgrafov; Misha Marie Gregersen; Mads Peter Sørensen

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

- Volume: 45, Issue: 6, page 1059-1080
- ISSN: 0764-583X

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topEvgrafov, Anton, Gregersen, Misha Marie, and Sørensen, Mads Peter. "Convergence of Cell Based Finite Volume Discretizations for Problems of Control in the Conduction Coefficients." ESAIM: Mathematical Modelling and Numerical Analysis 45.6 (2011): 1059-1080. <http://eudml.org/doc/276352>.

@article{Evgrafov2011,

abstract = {
We present a convergence analysis of a cell-based finite volume (FV)
discretization scheme applied to a problem of control in the
coefficients of a generalized Laplace equation modelling, for
example, a steady state heat conduction.
Such problems arise in applications dealing with geometric optimal
design, in particular shape and topology optimization, and are most
often solved numerically utilizing a finite element approach.
Within the FV framework for control in the coefficients problems
the main difficulty we face is the need to analyze the convergence
of fluxes defined on the faces of cells, whereas the
convergence of the coefficients happens only with respect to the
“volumetric” Lebesgue measure.
Additionally,
depending on whether the stationarity conditions are stated for the
discretized or the original continuous problem, two distinct
concepts of stationarity at a discrete level arise.
We provide characterizations of limit points, with respect to FV
mesh size, of globally optimal solutions and two types of
stationary points to the discretized problems.
We illustrate the practical behaviour of our cell-based FV
discretization algorithm on a numerical example.
},

author = {Evgrafov, Anton, Gregersen, Misha Marie, Sørensen, Mads Peter},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Topology optimization; finite volume methods; topology optimization; nonlinear elliptic equation; convergence; Laplace equation; steady state heat conduction; numerical example},

language = {eng},

month = {6},

number = {6},

pages = {1059-1080},

publisher = {EDP Sciences},

title = {Convergence of Cell Based Finite Volume Discretizations for Problems of Control in the Conduction Coefficients},

url = {http://eudml.org/doc/276352},

volume = {45},

year = {2011},

}

TY - JOUR

AU - Evgrafov, Anton

AU - Gregersen, Misha Marie

AU - Sørensen, Mads Peter

TI - Convergence of Cell Based Finite Volume Discretizations for Problems of Control in the Conduction Coefficients

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2011/6//

PB - EDP Sciences

VL - 45

IS - 6

SP - 1059

EP - 1080

AB -
We present a convergence analysis of a cell-based finite volume (FV)
discretization scheme applied to a problem of control in the
coefficients of a generalized Laplace equation modelling, for
example, a steady state heat conduction.
Such problems arise in applications dealing with geometric optimal
design, in particular shape and topology optimization, and are most
often solved numerically utilizing a finite element approach.
Within the FV framework for control in the coefficients problems
the main difficulty we face is the need to analyze the convergence
of fluxes defined on the faces of cells, whereas the
convergence of the coefficients happens only with respect to the
“volumetric” Lebesgue measure.
Additionally,
depending on whether the stationarity conditions are stated for the
discretized or the original continuous problem, two distinct
concepts of stationarity at a discrete level arise.
We provide characterizations of limit points, with respect to FV
mesh size, of globally optimal solutions and two types of
stationary points to the discretized problems.
We illustrate the practical behaviour of our cell-based FV
discretization algorithm on a numerical example.

LA - eng

KW - Topology optimization; finite volume methods; topology optimization; nonlinear elliptic equation; convergence; Laplace equation; steady state heat conduction; numerical example

UR - http://eudml.org/doc/276352

ER -

## References

top- . URIhttp://www.openfoam.com
- N. Aage, T.H. Poulsen, A. Gersborg-Hansen and O. Sigmund, Topology optimization of large scale Stokes flow problems. Struct. Multidisc. Optim.35 (2008) 175–180. Zbl1273.76094
- S. Agmon, Lectures on elliptic boundary value problems. Van Nostrand, Princeton, N.J. (1965). Zbl0142.37401
- G. Allaire, Conception optimale de structures, Mathématiques et Applications58. Springer (2007).
- L. Ambrosio and G. Buttazzo, An optimal design problem with perimeter penalization. Calc. Var. Partial Differential Equations1 (1993) 55–69. Zbl0794.49040
- C.S. Andreasen, A.R. Gersborg and Ole Sigmund, Topology optimization of microfluidic mixers. Int. J. Numer. Methods Fluids61 (2008) 498–513. Zbl1172.76014
- H. Attouch, G. Buttazzo and G. Michaille, Variational analysis in Sobolev and BV spaces: applications to PDEs and optimization. SIAM (2006) 648. ISBN 9780898716009. Zbl1095.49001
- M.S. Bazaraa, H.D. Sherali and C.M. Shetty, Nonlinear Programming. John Wiley & Sons, Inc, New York (1993).
- M.P. Bendsøe and N. Kikuchi, Generating optimal topologies in structural design using a homogenization method. Comput. Methods Appl. Mech. Engrg.71 (1988) 197–224. CODEN CMMECC. ISSN 0045-7825. Zbl0671.73065
- M.P. Bendsøe and O. Sigmund, Topology Optimization: Theory, Methods, and Applications. Springer-Verlag, Berlin (2003). 370. ISBN 3-540-42992-1. Zbl1059.74001
- J.F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems. Springer-Verlag, New York (2000), p. 601. ISBN 0-387-98705-3. Zbl0966.49001
- T. Borrvall and J. Petersson, Topology optimization of fluids in Stokes flow. Int. J. Numer. Methods Fluids41 (2003) 77–107. CODEN IJNFDW. ISSN 0271-2091. Zbl1025.76007
- B. Dacorogna, Direct methods in the calculus of variations, Applied Mathematical Sciences78. Springer-Verlag, Berlin (1989). x+308 ISBN 3-540-50491-5. Zbl0703.49001
- D.A. Di Pietro and A. Ern, Discrete functional analysis tools for Discontinuous Galerkin methods with application to the incompressible Navier-Stokes equations. Math. Comput.79 (2010) 1303–1330. Zbl05776268
- L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. CRC Press (1992). Zbl0804.28001
- A. Evgrafov, On the limits of porous materials in the topology optimization of Stokes flows. Appl. Math. Optim.52 (2005) 263–267. Zbl1207.49004
- A. Evgrafov, Topology optimization of slightly compressible fluids. Z. Angew. Math. Mech.86 (2005) 46–62. Zbl1176.76113
- A. Evgrafov, G. Pingen and K. Maute, Topology optimization of fluid problems by the lattice Boltzmann method, in IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials: Status and Perspectives, edited by M.P. Bendsøe, N. Olhoff and O. Sigmund. Springer, Netherlands (2006) 559–568.
- A. Evgrafov, G. Pingen and K. Maute, Topology optimization of fluid domains: Kinetic theory approach. Z. Angew. Math. Mech.88 (2008) 129–141. Zbl1290.76140
- A. Evgrafov, K. Maute, R.G. Yang and M.L. Dunn, Topology optimization for nano-scale heat transfer. Int. J. Numer. Methods Engrg.77 (2009) 285. ISSN 00295981. Zbl1257.80005
- R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Handbook of Numerical Analysis, edited by P.G. Ciarlet and J.L. Lions 7. North Holland (2000) 713–1020. Zbl0981.65095
- R. Eymard, T. Gallouët and R. Herbin, A cell-centred finite-volume approximation for anisotropic diffusion operators on unstructured meshes in any space dimension. IMA J. Numer. Anal26 (2006) 326–353. . Zbl1093.65110URIhttp://imajna.oxfordjournals.org/cgi/content/abstract/26/2/326
- R. Eymard, T. Gallouët, R. Herbin and J.-C. Latche, Analysis tools for finite volume schemes. Acta Math. Univ. ComenianaeLXXVI (2007) 111–136. Zbl1133.65062
- P. Fernandes, J.M. Guedes and H. Rodrigues, Topology optimization of three-dimensional linear elastic structures with a constraint on “perimeter”. Comput. Struct.73 (1999) 583–594. CODEN CMSTCJ. ISSN 0045-7949. Zbl0992.74058
- T. Gallouët, R. Herbin and M.H. Vignal, Error estimates on the approximate finite volume solution of convection diffusion equations with general boundary conditions. SIAM J. Numer. Anal.37 (2000) 1935–1972. . Zbl0986.65099URIhttp://link.aip.org/link/?SNA/37/1935/1
- A. Gersborg-Hansen, M. Bendsøe and O. Sigmund, Topology optimization of heat conduction problems using the finite volume method. Struct. Multidisc. Optim.31 (2006) 251–259. ISSN 1615-147X. Zbl1245.80011
- A. Gersborg-Hansen, O. Sigmund and R.B. Haber, Topology optimization of channel flow problems. Struct. Multidisc. Optim.30 (2005) 181–192. Zbl1243.76034
- M.M. Gregersen, F. Okkels, M.Z. Bazant and H. Bruus, Topology and shape optimization of induced-charge electro-osmotic micropumps. New J. Phys.11 (2009) 075019. . URIhttp://stacks.iop.org/1367-2630/11/i=7/a=075019
- R.B. Haber, M.P. Bendsøe and C.S. Jog, Perimeter constrained topology optimization of continuum structures, in IUTAM Symposium on Optimization of Mechanical Systems (Stuttgart, 1995). Solid Mech. Appl.43. Kluwer Acad. Publ., Dordrecht (1996) 113–120. Zbl0868.73056
- F.R. Klimetzek, J. Paterson and O. Moos, Autoduct: topology optimization for fluid flow, in Proceedings of Konferenz für angewandte Optimierung. Karlsruhe (2006).
- S. Kreissl, G. Pingen, A. Evgrafov and K. Maute, Topology optimization of flexible micro-fluidic devices. Struct. Multidisc. Optim.42 (2010) 495–516. ISSN 1615-147X. . URIhttp://dx.doi.org/10.1007/s00158-010-0526-6
- B. Mohammadi and O. Pironneau, Applied shape optimization for fluids. Numerical Mathematics and Scientific Computation, Oxford University Press, New York (2001) xvi+251. ISBN 0-19-850743-7. Zbl0970.76003
- O. Moos, F.R. Klimetzek and R. Rossmann, Bionic optimization of air-guiding systems, in Proceedings of SAE 2004 World Congress & Exhibition. Detroit, MI, USA, Society of Automotive Engineering, Inc (2004) 95–100.
- F. Okkels and H. Bruus, Design of micro-fluidic bio-reactors using topology optimization. J. Comput. Theoret. Nano.4 (2007) 814–816.
- L.H. Olesen, F. Okkels and H. Bruus, A high-level programming-language implementation of topology optimization applied to steady-state Navier-Stokes flow. Int. J. Numer. Meth. Engrg.65 (2006) 975–1001. Zbl1111.76017
- C. Othmer, A continuous adjoint formulation for the computation of topological and surface sensitivities of ducted flows. Internat. J. Numer. Methods Fluids58 (2008). Zbl1152.76025
- C. Othmer, Th. Kaminski and R. Giering, Computation of topological sensitivities in fluid dynamics: Cost function versatility, in ECCOMAS CFD 2006, Delft (2006).
- J. Outrata, M. Kočvara and J. Zowe, Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer Academic Publishers, Dordrecht (1998) xxii+273. ISBN 0-7923-5170-3. Zbl0947.90093
- J. Petersson, Some convergence results in perimeter-controlled topology optimization. Comput. Methods Appl. Mech. Engrg.171 (1999) 123–140. Zbl0947.74050
- G. Pingen, A. Evgrafov and K. Maute, A parallel Schur complement solver for the solution of the adjoint steady-state lattice Boltzmann equations: application to design optimization. Int. J. Comput. Fluid Dynamics22 (2008) 464–475. Zbl1184.76799
- G. Pingen, A. Evgrafov and K. Maute, Adjoint parameter sensitivity analysis for the hydrodynamic lattice Boltzmann method with applications to design optimization. Comput. Fluids38 (2009) 910–923. Zbl1242.76254
- G. Pingen, M. Waidmann, A. Evgrafov and K. Maute, A parametric level-set approach for topology optimization of flow domains. Struct. Multidisc. Optim.41 (2010) 117–131. ISSN 1615-147X. . Zbl1274.76183URIhttp://dx.doi.org/10.1007/s00158-009-0405-1
- K. Svanberg, The method of moving asymptotes—a new method for structural optimization. Int. J. Numer. Methods Engrg.24 (1987) 359–373. CODEN IJNMBH. ISSN 0029-5981. Zbl0602.73091
- K. Svanberg, A class of globally convergent optimization methods based on conservative convex separable approximations. SIAM J. Optim.12 (2002) 555–573. ISSN 1095-7189. Zbl1035.90088
- A.-M. Toader, Convergence of an algorithm in optimal design. Struct. Optim.13 (1997) 195–198.
- E. Wadbro and M. Berggren, Megapixel topology optimization on a graphics processing unit. SIAM Rev.5 (2009) 707–721. Zbl1179.65079
- E. Zeidler, Applied Functional Analysis: Main Principles and Their Applications, 1st edition. Springer (1995). ISBN 0387944222. Zbl0834.46003

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