A nodal based evolutionary structural optimisation algorithm

Transcription

1 Computer Aided Optimum Design in Engineering IX 55 A dal based evolutionary structural optimisation algorithm Y.-M. Chen 1, A. J. Keane 2 & C. Hsiao 1 1 ational Space Program Office (SPO), Taiwan 2 Computational Engineering & Design Centre, UK Abstract This paper proposes a simple and interesting algorithm dealing with structural shape, topology and thickness optimisation simultaneously. The proposed approach is based on an interesting idea of migrating boundary des in an iterative manner. An intuitive dal-based evolutionary structural algorithm drives the optimisation process. Finite element analysis is required at each stage to reveal the relative stress distribution of the evolving structure, from which the lowly stressed edge des are identified as design variables and shift towards the higher stressed areas within the design domain during optimisation. Migrating the geometry boundary des directly changes the structural shape and thus perform shape optimisation. By introducing a fixed shape circular cavity into the evolving structure during optimisation initiates topology optimisation. Further shifting the cavity boundary des during optimisation gradually reveals the structural topology. In addition, dal thickness is gradually decreased to enhance the reduction of structural weight. One feature of this dal based approach is that by employing boundary dal coordinates as design variables the interior finite element mesh becomes irrelevant to the shape definition of structures, thus unstructured mesh may be used. This enhances the compatibility on integrating the proposed algorithm with auto-meshing capability which is incorporated in most commercial finite element software today. A couple of benchmark problems, including the classical MBB beam, are used for illustration purposes. Keywords: dal based approach, structural optimisation, evolutionary optimisation, finite element, topology optimisation.

2 56 Computer Aided Optimum Design in Engineering IX 1 Introduction Predicting ideal topologies is an active research subject within the field of structural optimization. Currently, the discretised optimality criteria method (DCOC) of Zhou and Rozvany [1,2] is considered to be highly efficient in handling a structure with a very large number of elements and active stress constraints. The DCOC method works well in multipurpose generalized shape optimization problems by making use of the so-called SIMP method (Solid Isotropic Microstructure with Penalty for intermediate densities) (see [3] and [4]). The SIMP approach models the material properties as the relative material density raised to some power times the material properties of the solid material [4] producing solid-empty (SE) solutions showing very good approximations to kwn exact solutions. However, the approach has been criticized because the solutions obtained can lack easy physical interpretation. Bendsoe and Sigmund [5] suggests a physical interpretation to the power-law approach by giving a few simple conditions that need to be satisfied. In contrast, the so-called evolutionary structural optimization (ESO) method [6] is based on repeatedly removing low stress elements from the design model during an evolutionary process. The evolutionary approach has been much criticized: Zhao et al. [7] te that the classical stress based rejection ratio and evolutionary rate were the main weaknesses of the ESO approach. As a result, the use of dal displacement has been suggested to replace the classical ESO stress based criteria. In addition, Zhou and Rozvany [8] show that the ESO rejection criteria may result in a highly n-optimal design through a simple test example. Researchers within the field of topology optimization have investigated various ways of finding the optimum distribution of the selected elements or materials within a design domain. There is a commonality shared by most of the existing topology optimization algorithms, i.e., the use of repetitive (periodic) elements (microstructure) to discretise the design domain. The optimization is then based on stresses, displacements, material properties (densities) or mesh-densities. As a result, the solution converged to is represented by an optimum distribution of elements within the domain. In other words, the final geometry is made up of building blocks of the elements chosen. However, the objectives of structural optimization are usually to reduce structural weight and at the same time maintain boundary smoothness. Most topology optimization approaches are capable of obtaining impressive solutions that satisfy functional requirements, but the boundary smoothness criterion is usually satisfied only by image post processing or interpolations (see [5] and [9]). In a previous paper [10], the author proposed a method which combined the idea of moving boundary des and evolutionary optimization together, producing the odal-based Evolutionary Structural Optimization (ESO) approach. In addition, a parallel optimization algorithm was introduced which extended the capabilities of the ESO method to topology optimization. Here, a refined and extended algorithm is presented. ESO algorithm that integrates shape, topology and thickness optimization together is presented in this paper. odal migration strategy has been published (see [10]); details will t be

3 Computer Aided Optimum Design in Engineering IX 57 discussed here again. The subject of this paper is to present the ESO algorithm which is outlined in the subsequent sections. 2 odal based evolutionary structural optimisation algorithm The ESO algorithm is an infinite loop that drives the optimization. The idea behind the ESO algorithm is to move low stress des using the ESO method iteratively. These low stress des are identified by comparing the dal stress values with the maximum dal stress value σ max in the design domain. VM Creation of an initial oversize design domain Stop FEA Upper limit of optimum ratio (OR) reached? Any external edge or cavity des with VM max stress less than σ OR? OR=OR+δOR External edge movement and Cavity edge movement FEA Any internal ineffective points? Cavity Formation Algorithm Thickness Optimisation Algorithm FEA Has the minimum dal stress of structure VM max edges reached σ OR? Figure 1: ESO algorithm.

4 58 Computer Aided Optimum Design in Engineering IX All the edge des with dal stress values less than the product of VM σ max OR are classified as ineffective des. Ineffective des are allowed to participate in the de shifting process. As the low stress des gradually shift towards the high stress locations within the design domain, the stress level increases. When there are more ineffective des, i.e., more des with stress values less than the product of VM σ max OR the OR value is increased by a small amount δor until ineffective des are found, and then the optimization continues. The ESO algorithm terminates when OR=1. This criterion is never attained because the minimum stress at the structure edges is never going to reach VM σ max OR. This allow the optimization continues indefinitely and allows designers to choose the appropriate end point rather than simply relying on experience to prescribe a terminating criterion for each optimization. The ESO algorithm combines both shape and topology optimizations in the following way: for topology optimization, a cavity formation algorithm is used within the ESO algorithm to create circular cavities in the internal design domain during the optimization. In addition, after cavities are created within the domain, the cavity edge des are treated as shape design variables and qualify for the de-shifting process. This allows freedom to these des to explore the surrounding low stress areas and to evolve into a better cavity shape. Therefore, within the ESO algorithm both the external boundary des and the cavity des are treated as shape design variables. The ESO algorithm is depicted in fig. 1. The cavity formation algorithm and thickness optimisation algorithm within the ESO algorithm is used for topology optimization and is discussed separately in the subsequent sections. 2.1 Cavity formation algorithm The cavity formation algorithm lies at the heart of the ESO method in topology optimization. This algorithm is used to determine the cavity locations within the unstructured meshed design domain during the evolutionary optimization. During the optimization, circular cavities are inserted into the domain. The initial radius of each new circular cavity is a user defined variable and needs to be small eugh relative to the internal ineffective (low stress) regions within the domain. Here, the ineffective point is defined to be a point inside the design domain that has a dal stress that is less than the minimum dal stress along the external boundary des. The ineffective regions (groups of ineffective points) within the domain are identified by searching through the internal domain with dense grid points in both X- and Y- directions. Decreasing the point step size increases the resolution of the ineffective regions. When the domain has been searched with sufficiently small search step points, ineffective points are often found just inside the external boundaries and near existing cavities within the domain. The areas covered by these low stress points are generally very small and they usually disappear after the next few iterations of structure edge and/or cavity edge movement. Hence, we can state that cavities should be introduced within the domain only as long as their locations are neither near existing cavities or adjacent to the structure boundary.

5 Computer Aided Optimum Design in Engineering IX 59 Delete ineffective points For each ineffective point D edge > D edge min D cavity > D cavity min Delete ineffective points Record all qualified ineffective point coordinates While the ineffective point list is t empty For each remaining ineffective point: cavity cavity D > D min Find the lowest dal stress ineffective point Remove all points in group from ineffective point list Group all points within a radius R from the lowest stress ineffective points Determine the cavity centre for this group of ineffective points Create a cavity Figure 2: Cavity formation algorithm. Therefore, it is better to allow the existing external edges or cavities to explore these nearby ineffective regions in later iterations rather than to introduce a cavity at these points. A cavity formation criterion is therefore used to decide where cavities should be introduced within the domain during the evolutionary design process. To avoid cavities being introduced near edges or cavities during an evolutionary process, we use two control parameters: edge D min

6 60 Computer Aided Optimum Design in Engineering IX and cavity edge D min. D min is the distance measured from the external boundary edge. Any ineffective points lie at a point within the design domain with a distance to edge the closest external boundary less than the D min is automatically deleted. The retained ineffective points then need to be checked for the distance to any existing cavities within the domain. To do this, D cavity min is used to define a distance measured from the cavity edges. For each retained ineffective point, if cavity the distance to the nearest cavity de is greater than D min then such a point is retained for cavity formation; otherwise it is deleted. When the domain is searched with small step sizes, ineffective points that are t near any existing cavities or external boundaries often appear in groups and the center of each cavity is then taken as the mean of the X- and Y- coordinates of the ineffective points for each group. 2.2 Thickness algorithm Following the typical evolutionary process where the starting point for each optimisation is to create an initial oversized design domain, the thickness optimisation also starts by assigning each mesh point with a large thickness value. Initially, every mesh point has the same thickness value, i.e., a flat geometry. During optimisation dal thickness is gradually decreased (representing the trimming off surface materials). With re-meshing constantly invoked during optimisation, a potential difficulty arises in accumulating the dal thickness iteratively. One way to overcome such a difficulty is via interpolation: the dal thickness distribution is firstly recorded before initiating re-meshing. After re-meshing, and based on the new mesh point locations (X and Y coordinates) within the design domain, new thickness values for each de are interpolated from the recorded dal thickness distribution. The major advantage of interpolating thickness values for previous mesh data is that smooth surface variation can be maintained during iterative optimisation. In addition, dal thickness interpolation is independent of dal movement or cavity insertion in the plane since every de in the mesh always has a thickness value. The thickness algorithm is discussed next. The thickness optimisation algorithm starts by identifying the low stress des within the design domain, any des with dal stress σ i less or equal to the mean dal stress ( σ VMmean ) of the design domain being treated as thickness design variables. The dal thickness of these des is then reduced. For each qualifying thickness design variable, the dal thickness is changed by an amount calculated using eqn. (1): t i σ = t 1 i i T, i = 1,2,... VMmean d σ (1) where t i is the dal thickness at de i and T d is a user prescribed unit magnitude for thickness reduction. eqn. (1) reduces dal thickness based on the stress ratio. If the dal stress of de i is relatively small compared with the

7 Computer Aided Optimum Design in Engineering IX 61 mean stress of the design domain, then eqn. (1) tends to decrease the dal thickness with a larger magnitude. Conversely, if the dal stress of de i is VMmean closer to σ, the dal thickness is reduced by a smaller amount. t LB = n% max{ t} (2) Identify des with dal stress VM max σ i < σ Apply the thickness reduction equation Any dal thickness less than the thickness lower bound t LB? Apply thickness scaling equation Figure 3: Thickness optimisation algorithm. By reducing the dal thickness iteratively using equ. (1) during optimisation the dal thickness may reach zero. To overcome this problem, a lower bound t LB (equ. (2)) is imposed. The thickness lower bound (t LB ) is treated as the minimum acceptable dal thickness value during optimisation and is a user prescribed value. The lower bound t LB can be a prescribed value, say 10% of the initial thickness value. Here, the lower bound is set to be a percentage of the maximum thickness in the thickness distribution list. Equ. (2) has the advantage of automatically calculating the lower bound based on a prescribed percentage (n) of the maximum thickness value found among the mesh points during optimisation. During iterative dal thickness reduction, if any dal thickness reaches the lower bound t LB the dal thickness distribution is then scaled according to equ. (3). { t} { t new} = (max{ t} t LB ) + t LB (3) max{ t} Equ. (3) makes sure the dal thickness distribution is within the range between the dal thickness lower bound t LB and the maximum dal thickness value

8 62 Computer Aided Optimum Design in Engineering IX found within an evolving geometry during optimisation. oted if we substitute equ. (2) into equ. (3) we get So if we used 10% we would get { t new } = t(100% n%) + n% max{ t} (4) { t new } = 90% t + 10% max{ t} (5) ow, as max{t}>t we get t new is greater than t. In this way, the lowest dal thickness value is always greater than the prescribed thickness lower bound. In other words, during iterative cycles of thickness reduction, a relative thickness distribution can be obtained within a prescribed range. The thickness optimisation algorithm is summarized in fig Optimisation In this section, three commonly used examples (examples A, B & C, see figures 4,5,6, respectively) are presented to demonstrate the proposed ESO algorithm in handling both topology (examples A & B 10mx5mx1m) and thickness optimisations (example C-10mx5mx2m). In both cases A and B, plane stress problems are been considered but in example C, the element thickness is taken as the mean of three dal thickness for each planar triangular element during stress calculation. In addition, in example C, it is assumed that a point force is applied at the centre plane (i.e., at the plane z = 1m) and due to symmetry; optimisation of only half of the initial design domain is needed (i.e., dimensions 10mx5mx1m). Figure 4: Initial starting point of example A. Figure 5: Initial starting point of example B. Figure 6: Initial starting point of example C. For all three examples, the magnitude of the applied force is 100. The design domains are meshed with unstructured meshes and each element has the following standard properties: E=210 GPa, ν=0.3, ρ=7800 kg/m 3. The dal thickness lower bound t LB required in example C in set to 10% of the maximum thickness and a brief optimisation history for each example is included for

9 Computer Aided Optimum Design in Engineering IX 63 illustration purposes: The full optimised geometry of example C is visualized by producing a mirror reflection about the centre plane. A brief optimisation histories for examples A, B & C are shown through fig. 7~9, fig. 11~13 & fig. 15~17, respectively. The theoretical truss solution for these examples are shown in figures 10, 14 & 18 for comparison. Figure 7: Optimisation example A: Figure 11: Optimisation example B: Figure 15: Optimisation example C: Figure 8: Optimisation example A: stage 2. Figure 12: Optimisation example B: Figure 16: Optimisation example C: Figure 9: Optimisation example A: stage 3. Figure 13: Optimisation example B: Figure 17: Optimisation example C: Figure 10: Theoretical truss solution of example A. Figure 14: Theoretical truss solution of example B. Figure 18: Theoretical truss solution of example C.

10 64 Computer Aided Optimum Design in Engineering IX 4 Concluding remarks The dal based evolutionary structural optimisation (ESO) algorithm has been presented. The cavity formation algorithm and thickness optimisation strategy that are used for topology and thickness optimisation have been discussed. A few benchmark problems have been used to demonstrate the proposed algorithm in producing optimum geometry from an initial oversized blank. References [1] Zhou, M. & Rozvany, G.I.., An optimality criteria method for large systems Part I: theory, Structural Optimisation, 5, pp , [2] Zhou, M. & Rozvany, G.I.., An optimality criteria method for large systems, Part II: algorithm, Structural Optimisation, 6, pp , [3] Rozvany, G.I.. & Zhou, M. & Birker, T., Generalised shape optimisation without homogenization, Structural Optimisation, 4, pp , [4] Sigmund, O., A 99 line topology optimisation code, Structural Optimisation, 21(2), pp , [5] Bendsoe, M.P & Sigmund, O., Material interpolation schemes in topology optimisation, Archive of Applied Mechanics, 69, pp , [6] Xie, Y.M. & Steven, G.P., Evolutionary Structural Optimisation, Springer Verlag, London, [7] Zhao, C & Steven, G.P. & Xie, Y.M., A generalized evolutionary method for numerical topology optimisation of structures under static loading conditions, Structural Optimisation, 15, pp , [8] Zhou, M. & Rozvany, G.I.., On the validity of ESO type methods in topology optimisation, Structural Optimisation, 21, pp 80 83, [9] Sigmund, O. & Petersson, J., umerical instabilities in topology optimisation: A survey on procedures dealing with checkerboards, meshdependencies and local minima, Structural Optimisation, 16, pp 68 75, [10] Chen, Y.M & Bhaskar, A. & Keane, A., A parallel dal based evolutionary structural optimisation algorithm, Structural and Multidisciplinary Optimisation, 23(3), pp , 2002.