Isotropic Material Design
Department of Structural Mechanics and Computer Aided Engineering
Warsaw University of Technology
Al. Armii Ludowej 16, 00-637 Warsaw, Poland
E-mail: s.czarnecki@il.pw.edu.pl
Received:
Received: 24 October 2014; revised: 19 April 2015; accepted: 20 April 2015; published online: 02 June 2015
DOI: 10.12921/cmst.2015.21.02.001
Abstract:
The paper deals with optimal distribution of the bulk and shear moduli minimizing the compliance of an inhomogeneous isotropic elastic 3D body transmitting a given surface loading to a given support. The isoperimetric condition is expressed by the integral of the trace of the Hooke tensor being a linear combination of both moduli. The problem thus formulated is reduced to an auxiliary 3D problem of minimization of a certain stress functional over the stresses being statically admissible. The integrand of the auxiliary functional is a linear combination of the absolute value of the trace and norm of the deviator of the stress field. Thus the integrand is of linear growth. The auxiliary problem is solved numerically by introducing element-wise polynomial approximations of the components of the trial stress fields and imposing satisfaction of the variational equilibrium equations. The under-determinate system of these equations is solved numerically thus reducing the auxiliary problem to an unconstrained problem of nonlinear programming.
Key words:
compliance minimization, free material design, isotropic elasticity, topology optimization
References:
[1] M.P. Bendsøe, Optimization of Structural Topology, Shape, and Material, Springer, Berlin 1995.
[2] A.V. Cherkaev, Variational methods for Structural Optimization, Springer, New York 2000.
[3] G. Allaire, Shape optimization by the homogenization method, Springer, New York, 2002.
[4] A. Cherkaev, G. Dzierz ̇ anowski, Three-phase plane composites of minimal elastic stress energy: High-porosity structures, International Journal of Solids and Structures, 50, 4145-4160 (2013).
[5] M.P. Bendsøe, J.M. Guedes, R.B. Haber, P. Pedersen, J.E. Taylor, An analytical model to predict optimal material properties in the context of optimal structural design, J Appl Mech Trans
ASME, 61, 930-937 (1994).
[6] R. Werner, Free material optimization. PhD thesis, Institute of Applied Mathematics II, University of Erlangen-Nuerenberg, Germany (2000).
[7] J. Haslinger, M. Koˇ cvara, G. Leugering, M. Stingl, Multidisciplinary free material optimization, SIAMJ Appl Math 70, 2709-2728 (2010).
[8] S. Czarnecki, T. Lewi ́ nski, A stress-based formulation of the free material design problem with the trace constraint and single loading condition, Bull Pol Acad Sci Tech Ser 60, 191-204 (2012).
[9] S. Czarnecki, T. Lewi ́ nski, The free material design in linear elasticity, Topology Optimization in Structural and Continuum Mechanics. CISM International Centre for Mechanical Sciences 549. Courses and Lectures, G.I.N. Rozvany, T. Lewi ́ nski (eds.) (Springer Wien Heidelberg New York Dordrecht London, CISM, Udine 2014), 213-257 (2014).
[10] S. Czarnecki, T. Lewi ́ nski, A stress-based formulation of the free material design problem with the trace constraint and multiple load conditions, Struct Multidisc Optim 49(5), 707-731 (2014).
[11] A.N. Norris, The isotropic material closest to a given anisotropic material, Journal of Mechanics of Materials and Structures 1, 2 (2006).
[12] G. Duvaut, J.L. Lions, Inequalities in Mechanics and Physics, Springer, Berlin 1976.
[13] K.W. Wojciechowski, Remarks on “Poisson Ratio beyond the Limits of the Elasticity Theory”, Journal of the Physical Society of Japan 72(7), 1819-1820 (2003).
[14] S. Czarnecki, P. Wawruch, The emergence of auxetic material as a result of optimal isotropic design, submitted, DOI:10.1002/pssb.201451733
[15] R. Dwivedi, S. Zekovic, R. Kovacevic, Field feature detection and morphing-based process planning for fabrication of geometries and composition control for functionally graded materials, Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture 220, 1647-1661 (2006).
[16] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C. The Art of Scientific Computing, Cambridge University Press 1992.
[17] G. Dzier z ̇ anowski, On the comparison of material interpolation schemes and optimal composite properties in plane shape optimization, Struct Multidisc Optim 46(5), 693-710, 2012.
[18] M.P. Bendsøe, O. Sigmund, Topology Optimization. Theory, Methods and Applications, Springer, Berlin 2003.
[19] R. Kutyłowski, B. Rasiak, Incremental method of Young’s modulus updating procedure in topology optimization, 19th International Conference on Computer Methods in Mechanics, 9-12
May 2011, Warsaw, Poland, 8 pages on CD, 2011.
[20] R. Kutyłowski, M. Szwechłowicz, Special kind of multimaterial topology optimization, Archives of Civil and Mechanical Engineering, Vol. 13, Issue 3, 334-344 (2013).
[21] R. Kutyłowski, B. Rasiak, Application of topology optimization to bridge girder design, Structural Engineering and Mechanics 51(1), 39-66 (2014).
[22] R. Kutyłowski, B. Rasiak, The use of topology optimization in the design of truss and frame bridge girders, Structural Engineering and Mechanics 51(1), 67-88 (2014).
[23] P. Szczepaniak, Topology optimization as a design tool, MSc thesis. Department of Structural Mechanics and Computer Aided Engineering, Warsaw University of Technology and Technical University of Cartagena, Structures and Construction Department, June 2012.
The paper deals with optimal distribution of the bulk and shear moduli minimizing the compliance of an inhomogeneous isotropic elastic 3D body transmitting a given surface loading to a given support. The isoperimetric condition is expressed by the integral of the trace of the Hooke tensor being a linear combination of both moduli. The problem thus formulated is reduced to an auxiliary 3D problem of minimization of a certain stress functional over the stresses being statically admissible. The integrand of the auxiliary functional is a linear combination of the absolute value of the trace and norm of the deviator of the stress field. Thus the integrand is of linear growth. The auxiliary problem is solved numerically by introducing element-wise polynomial approximations of the components of the trial stress fields and imposing satisfaction of the variational equilibrium equations. The under-determinate system of these equations is solved numerically thus reducing the auxiliary problem to an unconstrained problem of nonlinear programming.
Key words:
compliance minimization, free material design, isotropic elasticity, topology optimization
References:
[1] M.P. Bendsøe, Optimization of Structural Topology, Shape, and Material, Springer, Berlin 1995.
[2] A.V. Cherkaev, Variational methods for Structural Optimization, Springer, New York 2000.
[3] G. Allaire, Shape optimization by the homogenization method, Springer, New York, 2002.
[4] A. Cherkaev, G. Dzierz ̇ anowski, Three-phase plane composites of minimal elastic stress energy: High-porosity structures, International Journal of Solids and Structures, 50, 4145-4160 (2013).
[5] M.P. Bendsøe, J.M. Guedes, R.B. Haber, P. Pedersen, J.E. Taylor, An analytical model to predict optimal material properties in the context of optimal structural design, J Appl Mech Trans
ASME, 61, 930-937 (1994).
[6] R. Werner, Free material optimization. PhD thesis, Institute of Applied Mathematics II, University of Erlangen-Nuerenberg, Germany (2000).
[7] J. Haslinger, M. Koˇ cvara, G. Leugering, M. Stingl, Multidisciplinary free material optimization, SIAMJ Appl Math 70, 2709-2728 (2010).
[8] S. Czarnecki, T. Lewi ́ nski, A stress-based formulation of the free material design problem with the trace constraint and single loading condition, Bull Pol Acad Sci Tech Ser 60, 191-204 (2012).
[9] S. Czarnecki, T. Lewi ́ nski, The free material design in linear elasticity, Topology Optimization in Structural and Continuum Mechanics. CISM International Centre for Mechanical Sciences 549. Courses and Lectures, G.I.N. Rozvany, T. Lewi ́ nski (eds.) (Springer Wien Heidelberg New York Dordrecht London, CISM, Udine 2014), 213-257 (2014).
[10] S. Czarnecki, T. Lewi ́ nski, A stress-based formulation of the free material design problem with the trace constraint and multiple load conditions, Struct Multidisc Optim 49(5), 707-731 (2014).
[11] A.N. Norris, The isotropic material closest to a given anisotropic material, Journal of Mechanics of Materials and Structures 1, 2 (2006).
[12] G. Duvaut, J.L. Lions, Inequalities in Mechanics and Physics, Springer, Berlin 1976.
[13] K.W. Wojciechowski, Remarks on “Poisson Ratio beyond the Limits of the Elasticity Theory”, Journal of the Physical Society of Japan 72(7), 1819-1820 (2003).
[14] S. Czarnecki, P. Wawruch, The emergence of auxetic material as a result of optimal isotropic design, submitted, DOI:10.1002/pssb.201451733
[15] R. Dwivedi, S. Zekovic, R. Kovacevic, Field feature detection and morphing-based process planning for fabrication of geometries and composition control for functionally graded materials, Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture 220, 1647-1661 (2006).
[16] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C. The Art of Scientific Computing, Cambridge University Press 1992.
[17] G. Dzier z ̇ anowski, On the comparison of material interpolation schemes and optimal composite properties in plane shape optimization, Struct Multidisc Optim 46(5), 693-710, 2012.
[18] M.P. Bendsøe, O. Sigmund, Topology Optimization. Theory, Methods and Applications, Springer, Berlin 2003.
[19] R. Kutyłowski, B. Rasiak, Incremental method of Young’s modulus updating procedure in topology optimization, 19th International Conference on Computer Methods in Mechanics, 9-12
May 2011, Warsaw, Poland, 8 pages on CD, 2011.
[20] R. Kutyłowski, M. Szwechłowicz, Special kind of multimaterial topology optimization, Archives of Civil and Mechanical Engineering, Vol. 13, Issue 3, 334-344 (2013).
[21] R. Kutyłowski, B. Rasiak, Application of topology optimization to bridge girder design, Structural Engineering and Mechanics 51(1), 39-66 (2014).
[22] R. Kutyłowski, B. Rasiak, The use of topology optimization in the design of truss and frame bridge girders, Structural Engineering and Mechanics 51(1), 67-88 (2014).
[23] P. Szczepaniak, Topology optimization as a design tool, MSc thesis. Department of Structural Mechanics and Computer Aided Engineering, Warsaw University of Technology and Technical University of Cartagena, Structures and Construction Department, June 2012.
