2. Keywords Topology optimization, stress constraints, Raghava criterion, Ishai criterion.

Transcription

1 oology Otimization with Different Stress Limits in ension and Comression P. Duysinx Robotics and Automation, Institute of Mechanics, C3, University of Liège, 1 Rue Solvay, B-4 Liège, Belgium, el: , Fax: , P.Duysinx@ulg.ac.be 1. Abstract his reort resents new rogresses in toology otimization of continuum structures with stress constraints. One rincial contribution consists in the consideration of uivalent stress criteria which can generalization of von Mises criterion and which are able to take into account non ual stress limits in tension and comression. A literature review led us to consider Raghava and Ishai criteria, which include a contribution of hydrostatic ressure. With the hel of these criteria toology otimization can redict more realistic designs in which structural members are able to withstand better tension loads than comression loads, or vice-versa, as it is sometimes encountered in civil engineering or in comosite material design. he imlementation and sensitivity analysis asects of Raghava and Ishai criteria in the Finite Element context are resented. We also resent recent advanced develoments to the solution of toology roblems with stress constraints like the stress constraint relaxation technique and the numerical otimization rocedure based on convex aroximations and dual otimizers. Finally numerical alications will show the original character of the stress based toology designs ad versus comliance designs when there are unual stress limits or when there are more than one load case.. Keywords oology otimization, stress constraints, Raghava criterion, Ishai criterion. 3. Introduction and formulation of the roblem Since the foundation work of Bendsøe and Kikuchi [1], most of the toology otimization work has been based on comliance tye arguments. Recently a coule of researches (e.g. [], [3]) led to the successful solution of the stress constrained toology roblem. his aer continues along this work and resents new rogresses in toology otimization of continuum structures. U to now toology otimization with stress constraints was based on the quadratic von Mises criterion. his criterion is roblem is very usual, because it redicts very recisely failure for metals which are usually used in mechanical engineering. However there are many other cases where von Mises criterion is unable to redict real-life designs. Structures made of materials with unual stress limits are good examles of this. Engineers know a lot of materials that have different behaviors in tension and comression: e.g. concrete, rocks, chalk, comosite materials, etc. One also can remember that thin structural members, like cables or thin sheets, are also not able to sustain high comressive loads because of buckling. An indirect rocedure to take into account this buckling in the reliminary design hase consists in restricting the comressive loads by reducing the stress limit in comression. From a ractical oint of view these different behaviors in tension and comression have been mastered for a long time and resulted in quite secific designs. For examle since Roman time the arch technique allows the use of the high comressive resistance of bricks to build high buildings and bridges. On other hand civil engineers use cables, which have nearly no load bearing caability in comression for examle by choosing susension bridges where they work in a ure tensile stress state. Desite the fact that the henomenon of unual stress limits was quite well integrated in classical building techniques, toology otimization was nearly unable to take into account this effect and thus to redict realistic designs in this case, which means designs that stick to existing results that roved their efficiency. In order to give an aroriate answer to this roblem one has be able to consider stress-constrained designs and to consider stress criteria able to coe with unual stress limits. his articular tye of roblem is considered here. he general roblem we consider is the following: to find the structural toology design that suorts the alied loads using a minimum amount of material while avoiding the material failure everywhere in the structure. min ρ( dx ρ ( 1 (1) s. t. σ ( ρ(, if ρ( > Ω ρ ( is the material density at oint x. σ ( ρ(, is an uivalent stress criterion which redicts the failure of the material at oint x while is the stress limit. U to now Von Mises was the only criterion that was considered, but other criteria are going to be introduced in the next aragrah. In this study a ure elastic regime is assumed, so that the criterion redicts the end of the elastic behavior everywhere in the material. In order to solve numerically the roblem, the design domain is divided into finite elements and a density variable is attached to each element. he discrete valued /1 roblem is avoided and the density is allowed to vary continuously between void and solid, so comosite materials of intermediate densities are included in the design. he continuous formulation resents the advantage to allow the use of sensitivity analysis and mathematical rogramming algorithms to solve the roblem in an efficient way. Unfortunately the discretized roblem is ill-osed and its numerical solutions are

2 mesh-deendent. o overcome the difficulty, we use here a restriction method of the design sace based on a bound over the erimeter [4], a low-ass filtering scheme [5], or a combination of both. he modelling of the intermediate density roerties is based on the ower-law aroach (also called SIMP model). If the scrit denotes the effective roerties of the orous material and the index is relative to the solid material roerties, the effective Young's modulus E is given in term of the density ρ by: E = ρ E () he factor >1 is introduced to enalize the intermediate densities in order to end u with black and white designs. Moreover to consider stress constraints in continuous toology otimization, one also needs the definition of a relevant stress measure in the orous comosites. Following the aroach develoed in [3] we consider an overall stress measure that controls the stress state in the microstructure. For the SIMP model of stiffness, a careful study [3] showed that a ower-law model with the same ower is a consistent model for the micro-stresses σ : ij ij σ = σ / ρ (3) herefore the overall failure criteria in the orous comosites σ that redicts the first failure in the microstructure is given by: σ ( ρ) = σ / ρ (4) 4. Failure criteria for unual stress limits in tension and comression reating different behaviors in tension and comression ruires articular failure criteria. For brittle materials one can consider a rincial stress criterion. σ I, σ II and σ III are defined as the rincial stresses. and C are the stress limits (in absolute value) resectively in tension and comression. One defines also 's ' the ratio between the stress limits in comression and in tension: s = C /. he material fails when the maximum rincial stress does not satisfy: C max σ (5) K = I, II, II K he rincial stress criterion is quite realistic for brittle materials (like glass), but this criterion does not fit so well to exeriments for ductile materials. Furthermore one would refer to use smooth criteria which render better the hysics of the failure for a lot of materials. We also refer to find a quadratic criterion that can generalize the well-known von Mises theory. Quadratic character is quite interesting, because it insures that the criterion is smooth and convex. Smoothness is very good for differentiability in sensitivity analysis. A review work (see for examle [6] and [7]) was necessary to scan different criteria that are able to cater with unual comressive and tensile stress limits. From the comarison of the different criteria, we selected the Raghava criterion and the Ishai criterion for this study in toology otimization. As the two criteria are functions of the first stress invariant J 1 and the second deviator stress invariant J D we remind the reader with the definition of these two stress invariant numbers: ( σ I σ II ) + ( σii σ III ) + ( σ III σi ) J1 = σi + σ II + σ III JD = (6) 6 he second invariant is directly related to the distortional shear stress τ and to the uivalent von Mises stress σ VM : 3 JD τ = σvm =. his resence of the second invariant is obvious because the criteria have to render the von Mises criterion when the stress limits are ual. he first invariant is related to the hydrostatic ressure J1 = 3σ h and its resence in the criterion is essential since it introduces the deendence uon the sign of the stress state and so the different behaviors in tension and comression. he Raghava criterion [8] is a quadratic failure criterion that is generally used with adhesive materials. In terms of the stress invariants, the Raghava stress criterion and the related uivalent stress can be written as: J1( s 1) + J1 ( s 1) + 1JDs σ RAG = (7) s As it is demonstrated in [6], the Raghava criterion can also be viewed as a articular case of sai-wu criterion of orthotroic materials when one assumes that the stress limits along the 3 orthotroy directions are the same. An interesting alternative to Raghava is the Ishai criterion [9]. With the notations that has been defined, one writes the Ishai uivalent stress and its failure criterion: ( s + 1) 3JD + ( s 1) J1 σ ISH = (8) s ij

3 As shown in [6] this criterion belongs to the Drucker-Prager lasticity criterion family that is of a general use for chalk, rocs and soils materials. he two criteria can be comared in Fig. 1 and where their failure enveloes are lotted in the sace (σ h, τ ) of the hydrostatic ressure and of the distortional shear stress. At first the failure surface of the Raghava aroach is a arabolic curve whereas the Ishai criterion is a hyerbolic one. But from the figures, one also notices that the two theories redict the failure for different maximum hydrostatic ressures and for different ure distortional shear stresses. he ure hydrostatic stress limit redicted by Ishai s theory is twice Raghava s one. We remind the reader that this hydrostatic failure stress is infinite in von Mises theory. Failure under a ure shear stress state is also different in the two theories. he ure distortional shear stress limit for Raghava s theory is always greater than Ishai s one because s s ( s + 1) Figure 1: Raghava failure criterion Figure : Ishai failure criterion 5. Numerical imlementation and sensitivity analysis of Ishai and Raghava criteria If the imlementation of von Mises criterion in an otimization code couled with a finite element analysis is well known, it may be quite interesting here to describe the guidelines of the imlementation of Ishai s and Raghava s criteria. his is the oortunity to underline the secific character of the two criteria. he uilibrium uation after discretization by finite element writes K q = f where K is the stiffness matrix, q is the generalized dislacement vector and f is the load vector. When writing the overal stress in the finite element in a vector form, it is given by s = q where is the stress matrix of the element. he two first invariants of the stress in the finite element that are ruired to calculate the uivalent stress criteria can be written in matrix form: J 1 ( ) = 1 q V q = 3σ h = w σ = W q and JD = 1 3 σvm 3 (9) As the stress matrix deends linearly uon the stiffness roerties, one can also write = where is the stress matrix of an element of relative density of 1. In the same way the influence of density can be ut in evidence in matrices V W and V: V = ρ and W = ρ W where matrices the V and W are the von Mises stress matrix and the hydrostatic stress matrices of the solid element. herefore one can evaluate easily the effective failure criteria based on Ishai and Raghava theories with the following exressions: s 1 s + 1 σ ISH = σ ISH / ρ = W q + q V q (1) s s ( s 1) W q + ( s 1) ( W q) + 4s q V q σ RAG = σ RAG / ρ = (11) s Sensitivity analysis can also be imlemented efficiently for Ishai and Raghava criteria as it can be done for Von Mises theory. As only the guidelines of the imlementation are resented we give only the main results of the derivation calculus. ρ

4 For Ishai uivalent stress, we get: σish s s q = W + V q (1) ρ i s s q V q ρ i Raghava s criterion is a bit more difficult. After some calculus the derivative of the averaged stress criterion can be written as: σrag s σ 1 RAG q = W + V q (13) ρ i s s W q s q V q s W q s q V q ρ i ( 1) ( ) + 4 ( 1) ( ) + 4 In a direct aroach of the sensitivity analysis one would evaluate all the derivatives of the generalized dislacements q ρ i, which could be a major job in toology otimization. However the number of active stress constraints is generally smaller than the number of design variables and the adjoined method is generally referred because only one additional load case is ruired er active constraint. In this case the adjoined load vector is the vector that is ut between curly braces. 6. Numerical solution of stress-constrained roblems As it has been shown in [1] and [3], the toology otimization with stress constraints is subject to the singularity henomenon. At short the aradox comes from the fact that the otimization rocedure is often unable to remove or to add some vanishing members without violating the stress constraints although one would end u with a erfectly feasible design if they were removed or added. From a mathematical oint of view the classical algorithms are unable to reach some otimum configurations because of the degeneracy of the design sace. In order to turn around the difficulty one has to use a erturbation technique of the stress constraints, generally known as the ε-relaxation technique [1] that results in a relaxation of the stress limits in the low-density regions. We use here the following ε-relaxed formulation of the overall stress criterion that is slightly modified for density variables: 1 σ ρ ε ρ + ε 1 he advantage of this relaxation formulation is that the influence of the erturbation always cancels for ρ = 1 whatever is the value of relaxation arameter ε. herefore one can sto the otimization rocedure with any value of ε with a feasible design for the stress constraints in hands. In addition to this new relaxation formulation, an automatic strategy to reduce the ε arameter is also rovided. he rogram without any interaction of the user makes the reduction of the erturbation arameter. he arameter is reduced as soon as the otimization roblem is solved with a sufficiently small tolerance. A second interesting asect of the rocedure is the use of a mathematical rogramming aroach suited to the solution of large-scale systems. A robust aroximation strategy has been elaborated on the basis on the Method of Moving Asymtotes (MMA) [11] while the core of the solution algorithm relies on a dual algorithm [1]. In fact the searability of the aroximations is one of the key ideas to manage efficiently the solution of large-scale roblems. he MMA algorithm has been customized in order to give rise to small trust regions and avoid constraint violations. he rimaldual otimizer from Fleury [1] showed itself as extremely robust for large-scale roblem solution. Otimal solutions are generally roduced within a comutation time that of the same order of magnitude than the finite element analysis for all our numerical alications. 7. Alications wo-bar truss benchmark he first alication is the classic two-bar truss roblem. In fact the two-bar truss roblem is not a trivial examle in the framework of toology otimization with stress constraints, moreover with unual stress limits. Indeed even if the layout of the solution is the obvious two-bar truss for ual stress constraints and the von Mises stress criterion (see Fig. 3 right), one have to be able to overcome the roblem of the singular stress aradox. he art of the design domain that is enclosed inside the two bars remains highly strained and thus highly stressed while all its material vanishes. Without the ε-relaxation technique it is imossible to remove totally this art. In the framework of unual stress limits the second oint of interest is to be able to check the analytical solution for this roblem. he analytical solution has been exhibited by Rozvany [13] while discussing shortcoming of Michell s truss theory. When adoting a tensile stress limit is 3 times higher than the comressive stress limit, the solution is still a two bar truss, but the two bars make an angle of 3 and 6 degrees resectively with the foundation wall. Ishai uivalent stress is adoted to take into account the different stress limits effect. he otimized material distribution is resented in Fig.3 (left). he numerical result totally matches the theoretical rediction, which validates the method. (14)

5 Figure 3: wo-bar truss solutions for ual stress limits (=C) and different stress limits (=3C) Four-bar truss roblem Figure 4: Four-bar truss solutions for ual stress limits (C=) (left), high comressive strength (C=4) (center) and high tension strength (=4C) (right) he second examle stems from an alication that was initially suggested by Swan and Kosaka [1] to demonstrate that ultimate strength otimization can lead to substantially different results from a minimum elastic comliance design. In the resent study the same examle is revisited in the framework of first failure stress constraints and elastic behavior. he design domain is a square (L= 1m) clamed in the four corners. A unit load is alied downwards in the center of the square domain. Using a normalized material (E=1N/m² and ν=.3) one bounds at first the von Mises uivalent stress to =C=1 N/m² (see Fig 4-left). he same otimal toologies could be obtained for minimum comliance as for minimum volume subject to stress constraints. It is cross-like structure that withstands the load in tension (uer members) and comression (lower members). Now if one uses a material that has a better strength limit in comression than in tension (=6 N/m², C=4 N/m²), the otimal structure (Fig 4-center) is an arch that works only in comression. Conversely if the material works better in tension than in comression (=4 N/m², C=6 N/m²) one gets a structure (Fig. 4-right) working exclusively in tension like cables in susension bridges. We have also to remark that the results for unual stress limits have been made with the Ishai criterion, but the same results can be obtained with the Raghava criterion since they give exactly the same strength for members under a one dimensional stress state as here. hree-bar truss roblem he last examle is the three-bar truss roblem [3], which lends itself to demonstrate the secific character of the stressconstrained design when there is more than one load cases. he sizes and material data of the roblem are normalized (L=1 m, W =.5 m, E=1 N/m², ν=.3). he minimum comliance design leads to a two-bar truss design (see Fig. 5 to, left). he von Mises stress constraints (with =C=15 N/m²) give rise to a different layout with a three-bar truss design (Fig. 5 to, right). A osteriori comutation of the stress level in the two-bar truss solution shows that this configuration is over-stressed by nearly a factor 1.5 comared to the three-bar truss, which roves that the 3-bar truss toology is not a local otimum. One may also remark that introducing unual stress limits with Ishai criterion (=15 N/m² and C=45 N/m² in Fig 5 bottom left and =45 N/m² and C=15 N/m² in Fig.5 bottom right) leads again to a two-bar truss configuration. his demonstrates the different nature of minimum comliance design and strength maximization when there are several load cases. 8. Conclusion Although most of structural roblems can be successfully studied with comliance minimization or local von Mises constraints, there are many cases where the effect of unual stress limits is crucial uon structural layout. When the structure is made of materials with different stress limits, toology redictions matching ractical engineering exerience can be made with Raghava and Ishai uivalent stresses. hese quadratic failure criteria generalize von Mises theory in the case of unual stress limits, by introducing a deendence uon the first invariant of stresses. Numerical alications validated the quality of the results. hese criteria are not difficult to imlement in toology otimization codes.

6 Figure 5. hree-bar truss designs for minimum comliance (to, left) and for stress constraints (to, right), and for unual stress limits: high comressive strength (bottom left) and high tensile strength (bottom right). he numerical alications resented herein lead also to demonstrate the original character of stress-constrained designs comared to comliance distributions. Even if engineers know that minimum comliance designs and stress-constrained designs can differ, most of the examles from literature are cases in which minimum comliance and maximum strength designs are the same. Indeed the examles are generally based on one load case, the same stress limits in tension and comression and one material (and void). Here our examles have demonstrated two imortant cases for which structural toologies are different for maximum stiffness designs and maximum strength design: When one considers unual stress limits (even if one load case); When there are several load cases (even if ual stress limits). New researches will be devoted to toology roblems in which the structure is made of more than one material (and the void) in order to illustrate the third category. 9. Acknowledgements he work resented in this aer was artly suorted by the BRIE-EURAM roject COMPOP (contract BRPR- C96-33 of the Euroean Community). he author would also like to thank Professor Claude Fleury (University of Liège) for roviding the author with the CONLIN otimizer. 1. References [1] Bendsøe, M.P. and Kikuchi N. (1988). Generating Otimal oologies in Structural Design Using a Homogenization Method. Comuter Methods in Alied Mechanics and Engineering. vol. 71, [] Shim P.Y. and Manoocheri S. (1997). Generating Otimal Configurations in Structural Design Using Simulated Annealing. International Journal for Numerical Methods in Engineering. vol. 4, [3] Duysinx P. and M.P. Bendsøe. (1997). oology Otimization of Continuum Structures With Local Stress Constraints. International Journal for Numerical Methods in Engineering. o aear. [4] Haber R.B., C.S. Jog and M.P. Bendsøe. (1996). A New Aroach to Variable-oology Shae Design Using a Constraint on Perimeter. Structural Otimization, vol. 11, [5] Sigmund O. (1997). On the Design of Comliant Mechanisms Using oology Otimization", Mechanics of Machines and Structures, vol. 5, no 4, [6] Laschet G. (1994). Prédiction ar Eléments Finis du Comortement non Linéaire et de la Résistance D assemblage Multimatériaux Collés. Ph.D. thesis. University of Liège (in French). [7] Boresi A.P., Schmidt R.. and O.M. Sidebottom. (1993). Advanced Mechanics of Materials. 5 th Edition. John Wiley and Sons, New York. [8] Raghava R., R.M. Caddel and G.S. Yeh. (1973). he Macroscoic Yield Behaviour of Polymers. Journal of. Material Sciences, vol. 8, 5-3. [9] Geli S., G. Dolex and O. Ishai. (1981). An Effective Stress/Strain Concet in the Mechanical Characterization of the Structural Adhesive Bonding. Int. J. of Adhesion & Adhesives, 1981, [1] Cheng G.D. and X. Guo. (1997). ε-relaxed Aroach in Structural oology Otimization. Structural Otimization, vol. 13, [11] Svanberg K. (1987). he Method of Moving Asymtotes A New Method for Structural Otimization. International Journal for Numerical Methods in Engineering. vol. 4, [1] Fleury C. (1993). Mathematical Programming Methods for Constrained Otimization: Dual Methods. In volume 15 of Progresses in Astronautics and Aeronautics(M.P. Kamat ed.) cha 9, 183-8, AIAA. [13] Rozvany G.I.N (1996). Some Shortcomings in Michell s russ heory. Structural Otimization, vol.1, [14] Swan C.C. and I. Kosaka. (1997). Voigt-Reuss oology Otimization for Structures with Nonlinear Material Behaviors. International Journal for Numerical Methods in Engineering, vol. 4,