- Email: [email protected]
Optimal structure formation using a chaotic self-organisational algorithm
Comput. & Graphics, Vol. 21, No. 5, pp. 685488. 1997 0 1997 ElsevierScienceLtd. All rights reserved Printed in Great Britain 0097-8493/97 $17.00+ 0.00
PII: !3009743493(97)ooo44-7
Chaos & Graphics
OPTIMAL
STRUCTURE FORMATION USING A CHAOTIC SELF-ORGANISATIONAL ALGORITHM W. M. PAYTEN’? and B. BEN-NISSAN’
‘Australian Nuclear Science and Technology Organisation, (ANSTO), Advanced Materials, Private Mail Bag 1, Menai, NSW 2234, Australia e-mail: [email protected] ‘University of Technology Sydney, Materials Science, PO Box 123, Broadway, NSW 2007, Australia Abstract--Optimal engineering shape design is becoming increasingly important as the efficient utilisation of material can account for significant cost savings during production. Traditional optimisation techniques based on finite element analysis using functional or variational calculus find this task difficult, as the design landscape is one in which the optimisation problem can be non-convex and hence the solution may have many local optima. The approach adopted in this work is based on a selforganisational system governed by a local state operator that defines the state of each element in the finite element mesh at each time step. This operator has the form of a nonlinear differential equation based on the use of strain energy density and is an extension of an expression originally developed for predicting morphogenetic density distributions in natural human bone. 0 1997 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION
The literature suggests [l] that the evolution of many natural physicalsystems,with complexsolutions,can result from low-dimensionalchaotic dynamic systemsgovernedby logistic maps.Thesesystemscan, under the right conditions, display emergentbehaviour or structure that selforganisesasa resultof the underlying dynamic behaviour of the algorithm. Use of suchtechniquescan simplify exceedinglydifficult engineeringtasks. An example has beenthe use of adaptive systemsfor the optimisation of piping and truss systemsvia genetic algorithms. In layout or topology designthe aim is to minimisethe massof a structureunder a particular constraint suchasstress. At the same time the remaining structural mass shouldbe so positionedas to maximisethe stiffness for that level of mass.Under the aboveconditionsthe topology of the design domain can be radically altered, as long as a load path is available between the displacementand load positions. Although a number of techniquessuch as the homogenisation method [2] to an extent solve this problem, they are however, exceedinglycomplex from a mathematical standpoint and require the useof traditional mathematical minimisationof the optimisationfunctional.
derivative expressionbasedon a bone adaptation algorithm [6, 71which has beenfound to be locally chaotic [8, 91is usedto evolve the structure basedon a state operator of density p. The density at each iteration and spatial location is related to the elasticity tensor, E, through a power law relationship. In its discreteform the solution is unstable,the strain energy density acting as an attractor with positive feedback. That is as the density in an element increasesit attracts more strain energy density which increasesthe density and so on. Coupled to ,a global feedbackderivative by the use of a relaxation parameterthat monitors the global constraint, it is possibleto keepthe solutionbounded but not ‘superstable’and hencethe dynamicsof the systemresult in a positive feedbackmechanismthat drives the solution to a segregatedsingle density optimal shape. The local itate operator can be describedby the following rel.ationships,where pi is the ith element density, psis the upper bound materialdensity, UTis the total elementstrain energy, Ulimis the deviatoric strain energy and U, is the volumetric strain energy: Ulim/($r + T(cvm - %m)) + J& PS
1.1. The program
‘Self-Opt’
The program SelfOpt [3,4] is basedon an adaptive approach.At eachstepa finite elementsolver(NISA with II) [5] is used to calculate the displacementand stressesof the designdomain. A modified discrete
Ulim
=
Ifv2Y2 6E
PS
I>(1) (2)
where v represents the Poisson’s ratio and Y representsthe stresscriteria. The global feed back
’ Author for correspondence 685
W. M. Payten and B. Bzn-Nissan
Fig. 1. The spatial temporal development of an optimal Michell truss: Event = 1.
derivative tit+ t is described by the term $t + ~(%ll-- Utim), where clim describes the design constraint stress in the final shape, evm is the maximum global von Mises stress at each step and z indicates a relaxation parameter. The magnitude of the adjustment parameter, tit is not known a priori and relates to the global proximity of the system to the prescribed limit stress. The derivative is integrated using discrete time intervals, calculated by a generalised Euler method. The solution is highly unstable in this form thus the following conditions are also defined based on the global limit stress at the current and preceding time step:
Fig. 3. The spatial temporal development of an optimal Michell truss: Event = 50.
E = E&
+ dp];
Where EO is the initial exponent.
modulus
(5) and y is the
2. RESULTS
The elasticity tensor of each element, E, at each iteration can then be updated and rewritten to the NISAII run file for the next static analysis. This is performed by the following power law relationship:
2.1. hfichell truss This case involves the generalised layout problem of a Michell truss which represents a complex problem in engineering topology design. The model is constructed using a rectangular domain with edges 50 units in length, a displacement fixed along a circular domain and a single vertical midpoint load on the opposing side of 50 units, modelled using 3000 4-node isoparametric plain stress elements. This force must be optimally transferred to the circular fixed support by constructing a truss like system of elements having a shape similar in form to the Michell solution. EO is 421, a density of 7.8 g cmp3 is used with 1 equal to 3, time step At= 100 and relaxation parameter 7 =0.5. The initial stress for the design domain is 130 MPa with the constraint stress set equal to 140 MPa, thus allowing the cantilever an increase of 10 MPa during the adaption process. Figures l-6 (starting from a random density distribution) show the spatial temporal development of an optimal Michell truss. The algorithm both
Fig. 2. The spatial temporal development of an optimal Michell truss: Event = 10.
Fig. 4. The spatial temporal development of an optimal Michell truss: Event = 105.
I ti,,
if
SI/I < 0 A 60 > 0,
Optimal
Fig.
5. The
spatial temporal development Michell truss: Event = 130.
structure
of an optimal
reabsorbs and grows to form the resulting truss system. This is in contrast to other methods where the material is ‘killed’ off rather than grown [lo, 111. The solution shows the optimal layout generates a Michell truss-like system and is similar to that calculated by other more traditional methods [l l131. These structures, like bone [9], can be considered to be chaotically ordered, a fractal with characteristics of optimal mechanical resistance, minimal mass and microstructure on infinitely many length scales. Using increasingly fine mesh densities and a variation of the Hausdoff dimension it can be shown that the evolved shapes do appear to display fractal behaviour producing structures that are statistically self similar with power law scaling [6]. 2.2.
MBB
formation
Fig.
6. The
spatial temporal Michell truss:
development Event = 200.
of an optimal
holes equally spaced at the centre axis of the beam. Thus the problem can be stated as a long centre loaded beam fixed at one end and free at the opposite end. Due to symmetry only half the beam is modelled. The beam has dimensions of 1200 length, 400 height and 4 mm width. The final design (Fig. 7) using SelfOpt results in a structure, with a deflection of 5.4 mm and a volume reduction of 68%. significantly lighter, stiffer and stronger than the current design used by Airbus. 3. CONCLUSIONS
Self-organisational algorithms based on simple rules of the type described here appear to be extremely s,uited to efficient design of optimal engineering shapes.
beam
The second example involves the optimisation of an MBB beam (Messerschmitt-Bolkow-Blohm GmbbH, Munchen, BRD) that is used to carry the floor of an Airbus passenger carrier. This problem has been analysed by various authors [l 1, 141. The current design of the MBB beam includes six round
Fig. 7. SelfOpt
solution
REFERENCES 1. May, R., Simple mathematical complicated dynamics. Nature, 2. Bendsoe, M. P. and Kikuchi, topologies in structural design method. Comnuter Methodr Engineeri,rg, 1988, 71, 197-224.
of a half symmetry
model
of a MBB
beam.
models with very 1976, 261, 459467. N. Generating optimal using a homogenization in Applied Mechanical
688
W. M. Payten and B. Ben-Nissan
3. Payten, W. M., Mercer, D. J. and Ben-Nissan, B., A non-linear self organisational approach to the solution of optimal structure topology. In Compurational Techniques and Applications, eds. R. May and A. Easton. World Scientific, Singapore, 1996, 61 I-618. 4. Payten, W. M., Integrated computer aided design, finite element analysis and bone remodelling of a modular ceramic knee prosthesis. PhD Thesis, University of Technology, Sydney, NSW, Australia, 1996. 5. EMRC, NISAII User Manual. Engineering Mechanical Research Corporation, Troy, MI, 1996. 6. Cowin, S. C. and Hegedus, D. H., Bone remodelling I: A theory of adaptive elasticity. J. Elasticity, 1976, 6, 313-326. 7. Huiskes, R., Weinans, H., Grootenboer, J., Dalstra, M., Fudala, B. and Slooff, T. J., Adaptive boneremodelling theory applied to prosthetic-design analysis. J. Biomechanics, 1987, 20, 1135-l 150. 8. Cowin, S. C., Arramon, Y. P., Luo, G. M. and Sadegh, A. M., Chaos in the discrete-time algorithm for bonedensity remodelling rate equations. J. Biomechanics, 1993, 26. 1077-1089. 9. Weinans, H., Huiskes, R. and Grootenboer, H. J. The
behaviour of adaptive bone-remodelling simulation rrodels. J. Biomechanics, 1992, 25, 1425-1441. 10. Mattheck, C., Baumgartner, L., Kriechbaum, R. and Walther, F., Computer methods for the understanding 01‘ biological optimization mechanisms. Computional Materials Science, 1993, 1, 302-312. 11. Steven, G. P. and Xie, Y. M., Evolutionary optimization with FEA. In Computational mechanics: from concepts to computations: proceedings of the second Asian-Pacific Conference on Computational Mechanics. Sydney, NSW, Australia, eds. S. Valliappan, V. A. Pulmano and F. Tin-Loi. Rotterdam, 1993, pp. 27-34. 12. Suzuki, K. and Kikuchi, N., A homogenization method for shape and topology optimization. Computer Merhads in Applied Mechanical Engineering, 1991, 93, 291318. 13. Yang, R. J. and Chuang, C. H., Optimal topology design using linear programming. Computer and Structures, 1994, 52, 265-275. 14. Olhoff, N., Bendsoe, M. P. and Rasmussen, J., On CAD-integrated structural topology and design optimzation. Compuier Methods in Applied Mechanical Engineering, 199 1, 89, 259-279.
PII: !3009743493(97)ooo44-7
Chaos & Graphics
OPTIMAL
STRUCTURE FORMATION USING A CHAOTIC SELF-ORGANISATIONAL ALGORITHM W. M. PAYTEN’? and B. BEN-NISSAN’
‘Australian Nuclear Science and Technology Organisation, (ANSTO), Advanced Materials, Private Mail Bag 1, Menai, NSW 2234, Australia e-mail: [email protected] ‘University of Technology Sydney, Materials Science, PO Box 123, Broadway, NSW 2007, Australia Abstract--Optimal engineering shape design is becoming increasingly important as the efficient utilisation of material can account for significant cost savings during production. Traditional optimisation techniques based on finite element analysis using functional or variational calculus find this task difficult, as the design landscape is one in which the optimisation problem can be non-convex and hence the solution may have many local optima. The approach adopted in this work is based on a selforganisational system governed by a local state operator that defines the state of each element in the finite element mesh at each time step. This operator has the form of a nonlinear differential equation based on the use of strain energy density and is an extension of an expression originally developed for predicting morphogenetic density distributions in natural human bone. 0 1997 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION
The literature suggests [l] that the evolution of many natural physicalsystems,with complexsolutions,can result from low-dimensionalchaotic dynamic systemsgovernedby logistic maps.Thesesystemscan, under the right conditions, display emergentbehaviour or structure that selforganisesasa resultof the underlying dynamic behaviour of the algorithm. Use of suchtechniquescan simplify exceedinglydifficult engineeringtasks. An example has beenthe use of adaptive systemsfor the optimisation of piping and truss systemsvia genetic algorithms. In layout or topology designthe aim is to minimisethe massof a structureunder a particular constraint suchasstress. At the same time the remaining structural mass shouldbe so positionedas to maximisethe stiffness for that level of mass.Under the aboveconditionsthe topology of the design domain can be radically altered, as long as a load path is available between the displacementand load positions. Although a number of techniquessuch as the homogenisation method [2] to an extent solve this problem, they are however, exceedinglycomplex from a mathematical standpoint and require the useof traditional mathematical minimisationof the optimisationfunctional.
derivative expressionbasedon a bone adaptation algorithm [6, 71which has beenfound to be locally chaotic [8, 91is usedto evolve the structure basedon a state operator of density p. The density at each iteration and spatial location is related to the elasticity tensor, E, through a power law relationship. In its discreteform the solution is unstable,the strain energy density acting as an attractor with positive feedback. That is as the density in an element increasesit attracts more strain energy density which increasesthe density and so on. Coupled to ,a global feedbackderivative by the use of a relaxation parameterthat monitors the global constraint, it is possibleto keepthe solutionbounded but not ‘superstable’and hencethe dynamicsof the systemresult in a positive feedbackmechanismthat drives the solution to a segregatedsingle density optimal shape. The local itate operator can be describedby the following rel.ationships,where pi is the ith element density, psis the upper bound materialdensity, UTis the total elementstrain energy, Ulimis the deviatoric strain energy and U, is the volumetric strain energy: Ulim/($r + T(cvm - %m)) + J& PS
1.1. The program
‘Self-Opt’
The program SelfOpt [3,4] is basedon an adaptive approach.At eachstepa finite elementsolver(NISA with II) [5] is used to calculate the displacementand stressesof the designdomain. A modified discrete
Ulim
=
Ifv2Y2 6E
PS
I>(1) (2)
where v represents the Poisson’s ratio and Y representsthe stresscriteria. The global feed back
’ Author for correspondence 685
W. M. Payten and B. Bzn-Nissan
Fig. 1. The spatial temporal development of an optimal Michell truss: Event = 1.
derivative tit+ t is described by the term $t + ~(%ll-- Utim), where clim describes the design constraint stress in the final shape, evm is the maximum global von Mises stress at each step and z indicates a relaxation parameter. The magnitude of the adjustment parameter, tit is not known a priori and relates to the global proximity of the system to the prescribed limit stress. The derivative is integrated using discrete time intervals, calculated by a generalised Euler method. The solution is highly unstable in this form thus the following conditions are also defined based on the global limit stress at the current and preceding time step:
Fig. 3. The spatial temporal development of an optimal Michell truss: Event = 50.
E = E&
+ dp];
Where EO is the initial exponent.
modulus
(5) and y is the
2. RESULTS
The elasticity tensor of each element, E, at each iteration can then be updated and rewritten to the NISAII run file for the next static analysis. This is performed by the following power law relationship:
2.1. hfichell truss This case involves the generalised layout problem of a Michell truss which represents a complex problem in engineering topology design. The model is constructed using a rectangular domain with edges 50 units in length, a displacement fixed along a circular domain and a single vertical midpoint load on the opposing side of 50 units, modelled using 3000 4-node isoparametric plain stress elements. This force must be optimally transferred to the circular fixed support by constructing a truss like system of elements having a shape similar in form to the Michell solution. EO is 421, a density of 7.8 g cmp3 is used with 1 equal to 3, time step At= 100 and relaxation parameter 7 =0.5. The initial stress for the design domain is 130 MPa with the constraint stress set equal to 140 MPa, thus allowing the cantilever an increase of 10 MPa during the adaption process. Figures l-6 (starting from a random density distribution) show the spatial temporal development of an optimal Michell truss. The algorithm both
Fig. 2. The spatial temporal development of an optimal Michell truss: Event = 10.
Fig. 4. The spatial temporal development of an optimal Michell truss: Event = 105.
I ti,,
if
SI/I < 0 A 60 > 0,
Optimal
Fig.
5. The
spatial temporal development Michell truss: Event = 130.
structure
of an optimal
reabsorbs and grows to form the resulting truss system. This is in contrast to other methods where the material is ‘killed’ off rather than grown [lo, 111. The solution shows the optimal layout generates a Michell truss-like system and is similar to that calculated by other more traditional methods [l l131. These structures, like bone [9], can be considered to be chaotically ordered, a fractal with characteristics of optimal mechanical resistance, minimal mass and microstructure on infinitely many length scales. Using increasingly fine mesh densities and a variation of the Hausdoff dimension it can be shown that the evolved shapes do appear to display fractal behaviour producing structures that are statistically self similar with power law scaling [6]. 2.2.
MBB
formation
Fig.
6. The
spatial temporal Michell truss:
development Event = 200.
of an optimal
holes equally spaced at the centre axis of the beam. Thus the problem can be stated as a long centre loaded beam fixed at one end and free at the opposite end. Due to symmetry only half the beam is modelled. The beam has dimensions of 1200 length, 400 height and 4 mm width. The final design (Fig. 7) using SelfOpt results in a structure, with a deflection of 5.4 mm and a volume reduction of 68%. significantly lighter, stiffer and stronger than the current design used by Airbus. 3. CONCLUSIONS
Self-organisational algorithms based on simple rules of the type described here appear to be extremely s,uited to efficient design of optimal engineering shapes.
beam
The second example involves the optimisation of an MBB beam (Messerschmitt-Bolkow-Blohm GmbbH, Munchen, BRD) that is used to carry the floor of an Airbus passenger carrier. This problem has been analysed by various authors [l 1, 141. The current design of the MBB beam includes six round
Fig. 7. SelfOpt
solution
REFERENCES 1. May, R., Simple mathematical complicated dynamics. Nature, 2. Bendsoe, M. P. and Kikuchi, topologies in structural design method. Comnuter Methodr Engineeri,rg, 1988, 71, 197-224.
of a half symmetry
model
of a MBB
beam.
models with very 1976, 261, 459467. N. Generating optimal using a homogenization in Applied Mechanical
688
W. M. Payten and B. Ben-Nissan
3. Payten, W. M., Mercer, D. J. and Ben-Nissan, B., A non-linear self organisational approach to the solution of optimal structure topology. In Compurational Techniques and Applications, eds. R. May and A. Easton. World Scientific, Singapore, 1996, 61 I-618. 4. Payten, W. M., Integrated computer aided design, finite element analysis and bone remodelling of a modular ceramic knee prosthesis. PhD Thesis, University of Technology, Sydney, NSW, Australia, 1996. 5. EMRC, NISAII User Manual. Engineering Mechanical Research Corporation, Troy, MI, 1996. 6. Cowin, S. C. and Hegedus, D. H., Bone remodelling I: A theory of adaptive elasticity. J. Elasticity, 1976, 6, 313-326. 7. Huiskes, R., Weinans, H., Grootenboer, J., Dalstra, M., Fudala, B. and Slooff, T. J., Adaptive boneremodelling theory applied to prosthetic-design analysis. J. Biomechanics, 1987, 20, 1135-l 150. 8. Cowin, S. C., Arramon, Y. P., Luo, G. M. and Sadegh, A. M., Chaos in the discrete-time algorithm for bonedensity remodelling rate equations. J. Biomechanics, 1993, 26. 1077-1089. 9. Weinans, H., Huiskes, R. and Grootenboer, H. J. The
behaviour of adaptive bone-remodelling simulation rrodels. J. Biomechanics, 1992, 25, 1425-1441. 10. Mattheck, C., Baumgartner, L., Kriechbaum, R. and Walther, F., Computer methods for the understanding 01‘ biological optimization mechanisms. Computional Materials Science, 1993, 1, 302-312. 11. Steven, G. P. and Xie, Y. M., Evolutionary optimization with FEA. In Computational mechanics: from concepts to computations: proceedings of the second Asian-Pacific Conference on Computational Mechanics. Sydney, NSW, Australia, eds. S. Valliappan, V. A. Pulmano and F. Tin-Loi. Rotterdam, 1993, pp. 27-34. 12. Suzuki, K. and Kikuchi, N., A homogenization method for shape and topology optimization. Computer Merhads in Applied Mechanical Engineering, 1991, 93, 291318. 13. Yang, R. J. and Chuang, C. H., Optimal topology design using linear programming. Computer and Structures, 1994, 52, 265-275. 14. Olhoff, N., Bendsoe, M. P. and Rasmussen, J., On CAD-integrated structural topology and design optimzation. Compuier Methods in Applied Mechanical Engineering, 199 1, 89, 259-279.