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The paper deals with the identification of optimal structural morphologies through evolutionary procedures. …

The paper deals with the identification of optimal structural morphologies through evolutionary procedures.

Two main approaches are considered. The first one simulates the Biological Growth (BG) of natural structures like the bones and the trees. The second one, called Evolutionary Structural Optimization (ESO), removes material at low stress level. Optimal configurations are addressed by proper optimality indexes and by a monitoring of the structural response. Design graphs suitable to this purpose are introduced and employed in the optimization of a pylon carrying a suspended roof and of a bridge under multiple loads.

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- 1. Structural Morphology Optimization by Evolutionary Procedures A. Baseggio1 , F. Biondini2 , F. Bontempi3 , M. Gambini1 , P.G. Malerba4 ABSTRACT The paper deals with the identification of optimal structural morphologies through evolutionary procedures. Two main approaches are considered. The first one simulates the Biological Growth (BG) of natural structures like the bones and the trees. The second one, called Evolutionary Structural Optimization (ESO), removes material at low stress level. Optimal configurations are addressed by proper optimality indexes and by a monitoring of the structural response. Design graphs suitable to this purpose are introduced and employed in the optimization of a pylon carrying a suspended roof and of a bridge under multiple loads. 1. INTRODUCTION One of the most promising research field which has been recently applied to the identification of optimal structural morphology deals with evolutionary procedures which operate on the basis of some analogies with the growing and the evolutionary processes of natural systems. Such methods are based on the simple concept that by slowly removing and/or reshaping regions of inefficient material, belonging to a given over-designed structure, its shape and topology evolve toward an optimum configuration. Two main approaches among those proposed in literature are here considered. In the first one, the structural morphology is modified by simulating the Biological Growth (BG) of natural structures like the bones and the trees (Mattheck&Burkhardt 1990, Mattheck&Moldenhauer 1990). In the second one, called Evolutionary Structural Optimization (ESO), material at low stress level is removed by degrading its constitutive properties (Xie and Steven 1993, 1994). The basic steps of these procedures should be repeated until optimal configurations appear. However, to this regards, no well established convergence criteria exist. In this work, the better structural solutions emerging from the evolutionary process are identified on the basis of proper optimality indexes and by monitoring the actual structural response. In particular, design graphs suitable to this purpose are firstly introduced. Subsequently, both BG and ESO methods are briefly recalled and these graphs are usefully employed in the selection of the optimal morphology of a pylon carrying a suspended roof and of a bridge type structure under a multiple load condition. These structures are considered to be in plane stress and made of linear elastic material having symmetric or non symmetric behavior in tension and in compression. The structural analyses needed during the evolutionary process are carried out by a LST-based (Linear Strain Triangle) finite element technique (Baseggio 1999, Gambini 2000). 2. OPTIMALITY INDEXES AND DESIGN GRAPHS As mentioned, at each step of the evolutionary process the present structure is modified in such a way that a better configuration with respect to given evolutionary criteria is hopefully achieved. However, the optimality of such solutions needs often to be judged with respect to design criteria which are not necessarily coincident with those which regulate the evolution. In this work, design criteria are synthesized by one or more optimality indexes able to measure the quality of the present solution with respect to the initial one. It is generally recognized that Nature tends to build structures in such a way that the internal strain energy, or the external work done by the applied loads, is minimum. Based on such consideration, a proper optimality index may be represented by the following Performance Structural Index (Zhao et al. 1997): WV WV PSI ⋅ ⋅ = 00 1 Structural Engineers, Milan, Italy (marco.gambini@tiscalinet.it, ambrogio.baseggio@infostrada.it). 2 PhD, Department of Structural Engineering, Technical University of Milan, Italy (biondini@stru.polimi.it). 3 Professor, Dept. of Structural and Geotechnical Engineering, University of Rome "La Sapienza", Italy (bontempi@scilla.ing.uniroma1.it). 4 Professor, Department of Civil Engineering, University of Udine, Italy (piergiorgio.malerba@dic.uniud.it). 4th International Colloquium on Structural Morphology August 17–19, 2000, Delft, The Netherlands, 264-271
- 2. being W the external work per unit of volume V and where 0 denotes the initial configuration. This formulation implicitly refers to a single load condition, but it can be easily extended to account for multiple loads, for example by a weighted average of the contributions i PSI of each load condition i=1,…,NC: ∑ ∑ ∑ = = = ⋅= ⋅ = NC i i iNC i i NC i i i PSI w PSIw PSI 1 1 1 ω Of course, depending on the specific problem to be examined, additional indexes may be introduced. For example, structures made of material having low tensile strength, like stone or concrete, should be designed by limiting the amount of tensioned material. Thus, by denoting cV the portion of the volume V which is compression dominated (see Fig. 4), the following percentage of Compressed Material Volume: V V CMV c = appears to be as well a meaningful optimality index. Moreover, sometimes may be useful to optimize not only the mechanical behavior, but also some geometrical properties of the structure. A measure of the present free Perimeter Γ of the structural boundary with respect to the initial one Γ0: 0 2 Γ Γ =P gives for instance an idea about the advantages in terms of cost of formworks and structural durability. In addition, being related to the weight of the structure, the percentage of Removed Material Volume: 0 0 V VV RMV − = may be itself an important indicator about the total structural cost. After some optimality indexes are selected and eventually grouped in a single averaged index, hierarchical arrangements of the solutions explored during the evolutionary process become possible. However, some additional design constraints on the structural response, for example in terms of maximum displacement and maximum stress level, are usually needed to assure the feasibility of the solution which seems to appear optimal. Thus, the best morphology requires to be identified by a monitoring of both the optimality indexes and the structural response. To this aim, design graphs which contemporarily describe the evolution of all such quantities, for example versus the RMV index as shown in Figure 1, are introduced. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 10 20 30 40 50 60 70 80 90 100 % Removed Material Volume - RMV OPTIMALITYINDEX 0 1 2 3 4 5 6 7 8 9 10 ADIMENSIONALSTRUCTURALRESPONSE OPTIMALITY RANGE ADIMENSIONAL MAX STRESS σσ//σσ00 ADIMENSIONAL MAX DISPLACEMENT S/S0 STRUCTURAL RESPONSE LIMIT OPTIMALITY INDEX INITIAL DOMAIN Figure 1. A typical design graph for evolutionary procedures. PSI
- 3. 3. BG PROCEDURES (Biological Growth) The evolutionary procedures considered in the following work by simulating the Biological Growth (BG) of natural structures (Mattheck &Burkhardt 1990, Mattheck &Moldenhauer 1990). Such structures are known to evolve by adapting themselves to the applied loads according to the axiom of uniform stress, which states that in the optimal configuration the stress field distribution tends to be fairly regular over the structure (Mattheck 1998). Thus, structural shape and topology are gradually modified in such a way that material is added in the zones with high stress concentrations and removed from under-loaded zones (swelling). The simplest form of a swelling law able to regulate such modifications is assumed as follows: DFK dt dV V n REF n SW =−⋅== )( 1 σσε& being: SWε& the swelling strain rate; V the evolutionary time-dependent volume V=V(t); K an artificial constant; σ the actual von Mises stress and REFσ its reference, or far-field, value; n a suitable exponent (n=1 for a stress-based and n=2 for a energy-based criterion); DF the Driving Force of the evolutionary process. Based on this law, the numerical simulation of the growth mechanism is obtained through three steps (Figure 2). F BASIC STEP SWELLING STEP UPDATE STEP SW VM(xi ) u SW(xi) xi,k+1=xi,k+C u SW(xi) P0 P0 PSW:∆εSW SV,SW SV,0 SV,0 L0=DESIGN CONSTRAINT L0=DESIGN CONSTRAINT L0=DESIGN CONSTRAINT Figure 2. Fundamental steps of the BG evolutionary process. (1) Basic Step. A finite element analysis is performed to obtain the stress distribution σ over the structure. (2) Swelling Step. The Driving Forces are firstly computed. In particular, for structure in plane state the following isotropic swelling strain increment vector T 2 1 ]011[SWSW ε∆=∆e is considered for the time increment ∆t. Based on such strain distribution, the load vector SW f∆ equivalent to swelling is derived and the corresponding incremental displacement vector SWu∆ is evaluated as follows: ∫ ∆= V SW T SW dVeDBfÄ ⇒ SWsw fuK ∆=∆ ⇒ SWsw fKu ∆=∆ −1 being B the compatibility matrix of the finite element, D the constitutive matrix of the material and K the stiffness matrix of the structure. It is worth noting that, in this work, additional geometrical design constraints are accounted directly by replacing the actual boundary conditions of the swelling model in such a way that swelling displacements which violate the constraints are not allowed. This concept is shown in Figure 2, where the cantilever beam is forced to maintain its initial length during the evolution. (3) Update Step. The location ki,x of each node i=1,…,N of the finite element model at the current generation k is updated according to the swelling displacements SWu∆ just obtained as follows:
- 4. SWuxx ∆+=+ Ci,kki 1, being C a suitable extrapolation factor which implicitly contains the constant K. Such factor may be either fixed at the first generation and then considered time-independent, or varied during the evolution. In any case, its value should be chosen to assure noticeable shape variations and progressively decreasing driving forces (Mattheck & Moldenhauer 1990). Such BG procedure is applied to the shape optimization of a pylon carrying a suspended roof. The geometry of the initial structure and the load condition are shown in Figure 3.a. Since the distance between the supports is retained, the swelling model in Figure 3.b is adopted. The design graph in Figure 3.d shows the progressive convergence of the evolutionary process towards higher level of the optimality index PSI and lower level of the structural response, while the structural volume remains practically the same. Noteworthy the end of the pylon tends to lie along the line of action of the resultant of the applied loads. Finally, Figure 3.c allows us to compare the maps of the von Mises stress corresponding, respectively, to the initial structure and the optimal one, and to appreciate how the latter present a nearly uniform distribution of stress having lower maximum intensity. 1000010000 1500 15001500 9000 8571.41428.610000 2PP 375 375 1000 30° 45° 0.0 28.1 0.0 3.2 Mpa Mpa N=0 V/V0=1.00 PSI=1.00 N=125 V/V0=0.97 PSI=32.9 σVMσVM Swelling Model L0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0 25 50 75 100 125 150 N Adimensionalstructuralresponse 0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 PerformanceStructuralIndex-PSI σσmean σσmax Smax PSI Volume N=30 N=50 N=90 N=125 Figure 3. Evolutionary shape optimization of a pylon carrying a suspended roof by a BG procedure. (a) (c) (b) (d)
- 5. 4. ESO PROCEDURES (Evolutionary Structural Optimization) The Evolutionary Structural Optimization (ESO) procedures modify the topology of a given over- designed structure by slowly removing regions of inefficient material (Xie and Steven 1993, 1994). The initial domain is subdivided in finite elements and a structural analysis is carried out. A representative quantity of the structural response, say the von Mises stress, is then evaluated at the element level and compared with a portion RR (Rejection Ratio) of a reference value, for instance the maximum stress over the whole structure. If such lower limit is not reached (Criterion 1 in Table 1), the material inside the corresponding elements is considered to be inefficient and it is removed by degrading its constitutive properties, typically the Young modulus. The parameter RR, who determines the portion of material which is removed at each step of the evolutionary process, is usually assumed as follows: SSAASSRR ⋅+= 10)( being SS an integer counter which is added by a unity whenever a Steady State is reached, while A0 and A1 are numerical constants able to assure a gradual evolution. Proper values seems to be A0 ≅0.0 and A1≅0.005. In this work, however, the rate of the process is controlled also by introducing an upper limit on the percentage of removed material volume VREM at each step (Rate of Removed Material): maxRRM V V RRM REM ≤= Criterion 1 is appropriate for materials having good strength in both tension and compression. However, many structures exhibit low strength in tension, like those made of stone or concrete, or in compression, like those subjected to buckling phenomena. To account for such cases, in which the optimal structural morphology should be defined by limiting the amount of material subjected to critical stress states, the concept of tension and compression dominated material has been introduced (Guan et al. 1999). As shown in Figure 4, material is considered tension (compression) dominated if the maximum (minimum) principal stress is of tension (compression) type. Based on this concept, the actual domain Ω is subdivided at each step in two parts, ΩT and ΩC, and in each of them the efficiency of the material is verified by using the absolute values of the principal stresses instead of the von Mises stresses (Criteria 2 and 3 in Table 1). The criteria just introduced implicitly refer to a single load condition, but can they be easily applied to the case of multiple loads by removing material only if the rejection criterion is verified for every load condition. In the basic formulation the minimum portion of removable material is identified with a single finite element. However, it is worth noting that a more general formulation can be achieved if the control of efficiency is performed on a minimum elimination unit formed by a group of elements. Several grouping criteria are clearly possible. By joining for example two adjacent triangular elements, structural solutions characterized by more regular boundaries are usually obtained. Moreover, the discrete nature of some structural types like masonry can be also better modeled, for instance, by building blocks representing one or more bricks. Of course, since in each group a different rejection criteria can be considered, the previous approach also allows to take the case of non homogeneous structures into account. Finally, by introducing a no-rejection criterion, is possible to freeze a sub-region of the initial domain (Non Design sub-domain) which cannot be never removed. This is particularly useful with bridge type structures, where the deck level is usually fixed. Figure 4. Tension and compression dominated material. CRITERION AND MATERIAL TYPE FORMULATION (1) symmetric VMVM SSRR max)( σσ ⋅≤ (2) asymmetric with low tensile strength 0.011 ≥σ and max,2222 )( σσ SSRR≤ (3) asymmetric with low compressive strength 0.022 ≤σ and max,1111 )( σσ SSRR≤ Table 1.Efficiency and rejection criteria for symmetric and asymmetric material.
- 6. The ESO procedure is here applied to the optimization of the structural morphology of a bridge subjected to six load conditions (Ito 1996). The position of both the deck and the supports is assumed to be fixed and a free space for navigation is provided under the deck. The first window of Figure 5 shows the geometric proportions, the design requirements, the load conditions and the initial domain chosen for the procedure. At first, a cable-stayed scheme is searched for by adding two axially rigid pylons to the Non Design domain. DESIGN REQUIREMENTS LOAD CONDITIONS B/4 B/4 B/4 B/4 B/4 LC 1 LC 2 LC 3 LC 4 LC 5 LC 6 B H=0.22B B H=0.22B 0.3 B 0.4 B 0.3 B 0.35H0.65B t DESIGN DOMAIN NON DESIGN DOMAIN EMPTY SPACE FOR SHIPWAY A B H=0.22B 0.3 B 0.4 B 0.3 B 0.35H0.65H 0.04 B 0.04 B CRITERION 1 B B H=0.22B 0.3 B 0.4 B 0.3 B 0.35H0.65H 0.04 B 0.04 B CRITERION 3 C B H=0.22B 0.3 B 0.4 B 0.3 B 0.35H0.65H 0.04 B 0.04 B CRITERION 3 CRITERION 2 Figure 5. Some optimal structural morphologies of a bridge type structure.
- 7. Windows A, B and C of Figure 5 show some layouts obtained during the evolutionary process for different material types. By assuming symmetric material (Criterion 1), a balanced arch scheme emerges instead of the expected one (Figure 5A). To be winning, the cable-stayed scheme should favor tensioned fields and then work on asymmetric material having low compression strength (Criterion 3, Figure 5B). However, such a solution tends to anchor some tensioned elements directly on the lateral supports. A more rational scheme can be achieved if the rejection criteria are differentiated over the structure, for instance by assuming the material under the deck to be asymmetric with low tensile strength (Criteria 2-3, Figure 5C). Despite of the found solutions, the balanced arch scheme initially obtained should be preferred if material having low tension strength is used. Figure 6 shows some of the configurations resulting from the evolutionary process for the case of symmetric material without pylons. The design graph of Figure 7 allows either to appreciate the optimality level of such schemes with reference to several optimality indexes, or to control the corresponding feasibility of the structural response. RMV PSI CMV 83% 0.95 1.21 70% 1.42 1.22 75% 1.25 1.22 50% 1.58 1.16 60% 1.57 1.20 23% 1.24 1.08 STRUCTURAL SCHEME 0% 1.00 1.00 H=0.22B 0.3 B 0.4 B 0.3 B 0.35H0.65H 0.08 B0.08 B CRITERION 1 Figure 6. Optimal balanced arch schemes for a bridge type structure.
- 8. Figure 7. Design graph for the bridge type structure of Figure 6. 5. CONCLUDING REMARKS The Biological Growth (BG) and the Evolutionary Structural Optimization (ESO) have been applied to morphology optimization problems. The swelling step of the BG procedures has been extended to take geometrical constraints into account. A formulation of ESO suitable to deal with asymmetric (tension and compression dominated) materials, fixed geometrical boundaries and alignments (non design domains) and multiple load conditions has been presented. In designing the morphology, F.E. grouping techniques allow us to drive the final configurations towards either smooth profiles or segmented boundaries as in case of masonry structures. Such processes may lead to many final optimal choices, as has been shown by an application searching for the optimal structural layout of a bridge having clearance limitations. Among these choices, the final actual optimum may be judged by using suitable design graphs, with reference to design criteria not necessarily coincident with those which control the evolutionary process. REFERENCES 1. Baseggio A. 1999. A Technique for the Optimization of Structures in Plane Stress. Dissertation. Department of Structural Engineering, Technical University of Milan, Milan, Italy (in Italian). 2. Gambini M. 2000. Identification and Optimization of structural Schemes by Evolutionary Procedures. Dissertation. Dept. of Structural Engineering, Technical University of Milan, Milan, Italy (in Italian). 3. Guan H., Steven G.P., Querin O.M., Xie Y.M. 1999. Optimisation of Bridge Deck Positioning by the Evolutionary Procedure. Structural Engineering and Mechanics 7(6), 551-559. 4. Ito M. 1996. Selection of Bridge Types from a Japanese Experiences. Proc. of IASS International Symposium on Conceptual Design of Structures, University of Stuttgart, Stuttgart, 1, 65-72. 5. Mattheck C., Burkhardt S. 1990. A New Method of Structural Shape Optimization based on Biological Growth. Int. J. of Fatigue. 12(3), 185-190. 6. Mattheck C., Moldenhauer H. 1990. An Intelligent CAD-Method based on Biological Growth. Fatigue Fract. Engng. Mat. Struct. 13(1), 41-51. 7. Mattheck C. 1998. Design in Nature. Learning from Trees. Springer Verlag. 8. Xie Y.M, Steven G.P. 1994. Optimal Design of Multiple Load Case Structures using an Evolutionary Procedure. Engineering Computation, 11, 295-302. 9. Xie Y.M., Steven G.P. 1993. A Simple Evolutionary Procedure for Structural Optimization. Computers & Structures, 49(5), 885-896. 10. Zaho C., Hornby P., Steven G.P., Xie Y.M. 1998. A Generalized Evolutionary Method for Numerical Topology Optimization of Structures under Static Loading Conditions. Structural Optimization, 15, 251-260.

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