Struct Multidisc Optim 24, 441–448 Springer-Verlag 2003 Digital Object Identifier (DOI) 10.1007/s00158-002-0257-4
Improving efficiency of evolutionary structural optimization by implementing fixed grid mesh H. Kim, O.M. Querin, G.P. Steven and Y.M. Xie
Abstract Evolutionary structural optimization (ESO) has been shown through much published research to be a simple and yet effective method for structural shape and topology optimization. However, attention has been drawn to shortcomings in the method related to the computational efficiency of the algorithm as well as the jagged edge representation of the Finite Element optimal solutions. In this paper a fixed grid (FG) mesh is implemented and an improved ESO methodology is introduced in order to address these shortcomings. The examples show a significant improvement in the solution time as well as eliminating the jagged edges and checkerboard patterns that may appear in current solution topologies. In addition, FG is applied to both stress based and stiffness optimization. This paper demonstrates a simple implementation of FG and the consequent improvement in the efficiency and practicality of the FG ESO formulation. Key words topology optimization, ESO, fixed grid, checkerboard pattern
1 Introduction The competitive and rapidly developing environment of the engineering design industry demands faster and more Received November 14, 2000 H. Kim1 , O.M. Querin2 , G.P. Steven3 and Y.M. Xie4 1
Department of Mechanical Engineering, University of Bath, Bath BA2 7AY, UK e-mail:
[email protected] 2 School of Mechanical Engineering, University of Leeds, Leeds LS2 9JT, UK 3 School of Engineering, University of Durham, Durham DH1 3LE UK 4 School of the Built Environment, Victoria University, PO Box 14428, MCMC VIC 8001, Australia Extended version of a paper presented at ASMO-2, held in Swansea, UK, 2000
practical topology optimization methods. Evolutionary structural optimization (ESO) has been introduced as a simple and effective method for size, shape and topology optimization. The wide range of ESO applications have become proof of its versatility and its potential as a design tool (Xie and Steven 1997). However, the disadvantages of ESO have become apparent in high solution time and the appearance of the optimum topology output. Due to the iterative and slow nature of the ESO process a typical solution time is often excessive for practical problems. The other unfavourable feature is the finite element representation of the final optimum topology, identified by the profile of the remaining elements’ edges. Whilst this format can be viewed satisfactorily on a computer screen, it takes a significant time and effort to extract the boundary in a format that most CAD/CAM systems recognize. Furthermore, the boundaries are represented by the jagged edges of the finite elements, which require some smoothing or image filtering processing in order to manufacture a smooth topology. This paper addresses these problems by incorporating Fixed Grid (FG) into ESO. FG has been used to model problems where the geometry or the physical properties of the domain change with time (Voller et al. 1990). The advantage of FG is that simple modifications enable the existing numerical formulation and solution to be adapted to a changing environment. Garc´ıa and Steven (2000) applied FG to elasticity problems. The study showed that the reduction in solution time outweighs the loss of accuracy in the analysis. The accuracy of a solution was found to be adequate for the initial stage of the design process, where a design is subjected to frequent changes. Also, despite the accuracy reduction, the form of the stress field is correct thereby making the ESO logic work correctly. Kim et al. (2000a) have made the implementation of a FG into ESO, where a FG mesh represented the initial design domain and ESO was applied to remove elements with low stress values. This study clearly demonstrated the benefit to be gained in time reduction. It also proposed an algorithm, which converts a FG representation to a boundary representation of a topology. However, optimization is still based on the presence or absence of
442 elements, and a conversion algorithm must be applied to obtain a boundary representation of the topology. This paper presents an improved FG ESO method, which optimises the given design domain by modifying and adapting the boundaries. This new method not only results in a reduction of solution time, but also determines an optimum topology with a more favourable boundary representation form at every iteration, without the typical finite elements’ “jagged-edges”. The following sections briefly review the ESO methodology and the FG formulation for elasticity problems; some examples are presented followed by some concluding remarks.
2 ESO methodology The concept of ESO states that by slowly removing inefficient materials from a structure, a structure evolves towards an optimum (Xie and Steven 1997). The inefficiency of any portion of the material (i.e. a finite element) is determined by low sensitivity number which is measured against the optimality objective function. For stress based optimization, an element’s von Mises stress is commonly used as the sensitivity number (Xie and Steven 1997). This is the original form of ESO where it is said that a reliable sign of inefficient material use is low stress, and an optimum design is where, for a single load case, every part of a structure is near a constant stress level, i.e. a fully stressed design. Chu (1997) derived a sensitivity number for compliance or stiffness design. This compliance sensitivity number, si indicates the change in the compliance as a result of removing element i, as defined in (1). si is also referred to as an element contribution to the structure’s total compliance, and the sum of si over all elements equates to the compliance of the structure. Removing elements with low compliance sensitivity number minimizes the increase in compliance as the volume is reduced, leading towards a minimum compliance design. Note that although compliance will always increase when an element is removed the endeavour is to minimize the specific compliance, equivalent to maximizing the specific stiffness, si =
1 i T i i u K u , 2
(2)
where smax = maximum elemental sensitivity number RR = rejection ratio = a0 + a1 · SS , (Xie and Steven 1997) .
Specify design problem; SS = 0; Do while (optimum is not reached), Perform FEA; Remove elements according to (2); If (number of removed elements = 0), SS = SS + 1; Evaluate RR using (3); End if ; End do; Fig. 1 Summary of ESO process
3 Fixed grid for elasticity problems A FG is generated by superimposing a rectangular grid of equal sized elements on the given structure instead of generating a mesh to fit the structure. Some of these elements are inside the structure (I), some are outside (O) and some are on the boundary, namely Neither-In-norOut (NIO) elements as illustrated in Fig. 2. An O element is given a material property significantly less than an I element, resulting in a bi-material problem.
(1)
where si = sensitivity number of element i; u = nodal displacement vector; K i = stiffness matrix of element i. ESO removes the material slowly. The slowness of the removal is ensured by the rejection ratio. The material is removed when the stress satisfies the ESO inequality, (2). si ≤ RR · smax ,
The rejection ratio increases as the structure is evolved and its function is to delay the element removal process so that the design does not change significantly after each iteration. The steady state number, SS is a positive integer counter which increases as the structure evolves. Figure 1 summarizes the ESO methodology.
(3)
Fig. 2 FG mesh
A NIO element is partially inside the structure and its material property value is not constant nor continuous over the element. Such an element is approximated into a homogeneous isotropic element, with its material property defined by (4), D(NIO)i = α D(I)i ,
(4)
443 i where D(NIO) = elemental material property of a NIO element; D(I)i = elemental material property of element i, α = AI /Ae ,
(5)
where α = area ratio; AI = area inside of the structure within NIO element i; Ae = total area of element i. As the sizes of all elements are identical, the stiffness matrix entries are linearly proportional to their elemental area ratio. This greatly reduces the time taken in the stiffness matrix generation and regeneration. For FEA the preconditioned conjugate gradient solver is implemented as it is known to be fast and efficient (Garcia 1999; Garc´ıa and Steven 2000). In addition, the solution vector from the previous optimization iterations offers an excellent initial solution vector for reanalysis, reducing the solution time even further.
4 Methodology 4.1 Boundary modification The following explanation of the method uses stress as the optimization criterion as ESO in its original format is stress based. However, it should be noted that replacing stress with compliance sensitivity number of (1) achieves compliance based optimization. Unlike the standard ESO formation the stress along the boundaries is considered separately from that of a non-boundary region. The optimization algorithm firstly examines stress on a boundary, which is determined by a linear interpolation of the two adjacent nodal stresses. If this boundary stress is lower than the deletion criterion, i.e. the ESO inequality of (6) is satisfied, the boundary is modified to be the contour line of the deletion criterion. Therefore stress along the modified boundary is either equal to or higher than the deletion criterion. This removes material with low stress of any shape and size. Hence a boundary is no longer restricted to right angles, but can be defined at any angle, σb < σdel ,
(6)
where σb = stress of a boundary point; σdel = deletion criterion. After modifying all existing boundaries, stresses of the non-boundary nodes are examined. If a non-boundary nodal stress satisfies the ESO inequality of (6), the stress values of the adjacent nodes are examined for a new boundary, i.e. a cavity. If stress of an adjacent node does not satisfy the ESO inequality, there exists a point between the two nodes, where its stress equates the deletion criterion. This point is again obtained by a linear interpolation of the two nodal stresses, and becomes a boundary point. The optimization algorithm then searches for
a series of new boundary points which define an enclosed boundary of a low stress region. This region of material is removed to create a cavity or a new boundary. One iteration of optimization is completed when all boundary and nodal stresses are examined. The area ratios of the modified elements are computed and the elemental stiffness values are updated according to the area ratios for FEA of the modified topology. 4.2 Calculation of deletion criterion The ESO’s definition of deletion criterion is a percentage of the maximum value of the criterion. When the deletion criterion is too small, it is increased by a pre-specified step until the deletion criterion value is large enough to remove elements. This sometimes requires a series of calculations of the deletion criterion and checking all elemental criteria before the deletion criterion is finally large enough. However, the minimum value of the criterion can be determined and the deletion criterion can thus be computed to be always greater than the minimum criterion. This ensures that a modification is made at every iteration and increases the efficiency of ESO. Here, a new definition of deletion criterion is proposed: σdel = σmin + aσmean ,
(7)
where σmin = minimum value of the criteria; σmean = mean value of the criteria; a = evolutionary rate constant, a positive real value which ensures that deletion criterion is slightly greater than the minimum criterion value, typically 0.01 deletion criterion. The value of the evolutionary rate constant, a defines how much greater σdel is relative to σmin . The larger the value of a, the more material ESO removes during an iteration and hence a faster optimization. However increasing the rate of optimization leads to a less refined optimum solution. 4.3 Definition of steady state In standard ESO, a steady state is reached when no elements have criteria less than the deletion criterion. When a topology reaches a steady state, the deletion criterion is increased to further optimize the structure if desired. For FG ESO, a steady state would be equivalent to a state where no more modifications are made. However, unlike standard ESO, FG ESO allows almost infinitely small amount of material removal. A small boundary modification still adjusts the nodal stress values, which leads to another small modification. Therefore FG ESO often continues to remove or add a small percentage of material, e.g. 0.0001% of the total volume. In such cases, it is said that a steady state is reached and a new deletion criterion is calculated to obtain a significant modification
444 of the topology. Thus a topology is said to have reached a steady state if the total volume change during an iteration is less than the elemental volume, (8), ∆V < Velem ,
(8)
where ∆V = volume of removed material during an iteration; Velem = volume of an element. 4.4 Merging two boundaries As a topology is optimized and material is removed, two separate boundaries may merge into one. FG ESO identifies such a case by examining each element that contains some boundary. When two boundaries lie on one element as shown in Fig. 3a, the two boundaries, A and B become one boundary. The assumption here is that an area smaller than the element width is negligible, as it would be impractical to manufacture. An example of Fig. 3 is used here in order to display the boundary merge mechanism more clearly. Element e with two boundaries A and B of Fig. 3a is enlarged in Fig. 3b. The shaded areas represent the structure and the clear areas represent the outside or voids. As boundaries are modified, FG ESO checks all edges of the boundary elements. When modifying element e of boundary A, FG ESO checks for another boundary on the element and would find that boundary B also lies on element e. Boundary A is then modified such that boundary B becomes a part of A, as shown in Fig. 3c. This therefore reduces the total number of boundaries or cavities by one.
Fig. 3 Merging two boundaries
4.5 FG ESO formulation The standard formulation of ESO removes material by elements, where the existence of an element becomes the design variable. However, FG ESO removes a region of material with low stress values, and the design variable becomes the area ratio, α of each element. Therefore, the mathematical representation of FG ESO is modified to reflect this change, (9), n σi αi νi minimize f (x) = i=1 FL subject to V ≤ V ∗ , σ − (σmin + aσmean) ≥ 0 . (9) The objective function and the finite element formulation include αi to incorporate the use of FG elements. The second constraint represents the ESO inequality which ensures stress in the domain is always greater than the deletion criterion, (6). The following step-by-step procedure gives an overview of the boundary based FG ESO algorithm and has been summarized in Fig. 4. 1. The user is required to define an optimization problem by defining the maximum domain, the design environment and optimization parameters. Von Mises stress is usually specified as the optimization criterion for a fully stressed design, hence it will be used as the optimization criterion here. However, it should be noted that other criteria such as compliance and frequency sensitivity numbers could be used instead. 2. FG mesh is generated. 3. FEA is conducted to determine displacement and stress at all nodes. 4. The minimum stress value is determined and the deletion criterion is calculated, (7). 5. Using the nodal stress values, the stress along the boundary is examined. If the stress on a boundary is less than the deletion criterion, a contour line of the deletion criterion becomes the new boundary. This boundary is obtained by a linear interpolation of two nodal stresses. 6. The stress values of the non-boundary nodes are examined, and a new boundary, ie. a cavity along the deletion criterion contour is initiated if it exists. 7. While modifying the boundaries, if two boundaries pass through a single element, they merge and become one boundary. 8. If an optimum is reached, the optimization process is terminated. 9. Otherwise area ratios of the elements are obtained and the new stiffness matrix is generated. Another FEA is carried out and the nodal stress values are determined. 10. If a steady state is reached according to (8), the process is repeated from step 5 to compute a new deletion criterion. If a steady state is not reached, the process is repeated from step 6 and continues to modify the structure.
445 Specify design problem; Generate FG mesh; Generate stiffness matrix; Conduct FEA; Evaluate the initial deletion criterion, (7); Do while (optimum is not reached), Modify all existing boundaries; Initiate new boundaries; Update stiffness matrix; Conduct FEA; If (SS is reached), Evaluate deletion criterion, (7); End if ; End do; Fig. 4 Summary of FG ESO process
5 Examples 5.1 MBB beam To demonstrate the method, the well-known MBB beam (Olhoff et al. 1991; Zhou and Rozvany 1991) is optimized using standard ESO with the traditional finite element formulation and FG ESO for stress and compliance. The design environment and the maximum domain are as shown in Fig. 5. As only the qualitative result is of interest, the nondimensional physical parameters are chosen. Due to symmetry, only the left half of the model is optimized with the mesh density of 75 × 25. For stress based standard ESO, the evolutionary rate constants are a1 = 0.0001 and a0 = 0.0, and for FG ESO, a = 0.006 for stress and a = 0.003 for compliance. The computer used for all problems presented in this paper is a Pentium 133 MHz with 32 MB RAM. The optimum solutions from standard ESO; stress based FG ESO; compliance based FG ESO are selected at an equivalent volume level for comparison and they are displayed in Fig. 6, together with the known optimal grillage layout. Similar truss-like topologies are obtained for all cases: The locations and angles of the members as well as the general topology of the solutions compare closely. They also agree favourably with the exact analytical (Fig. 6d, after Lewi´ nski et al. 1994) optimal layout and with dis-
Fig. 6 Optimum solutions of MBB beam problem. (a) Standard ESO solution, (b) stress based FG ESO solution, (c) compliance based FG ESO solution, and (d) exact analytical optimal truss layout (Lewi´ nski et al. 1994)
cretized solutions by other researchers (Zhou and Rozvany 1991; Olhoff et al. 1991; Hassani and Hinton 1998; Sigmund 1994). Table 1 presents the details of the solutions. The advantage of using FG can easily be viewed by comparing the solutions time, where the FG ESO solution times are significantly lower than that of standard ESO: For stress optimization, using FG reduced the solution time by 87%. The mean to maximum stress ratio is an indication of the even stress distribution, and as a topology is optimized, the stress ratio is expected to rise. The relative mean to maximum stress in Table 1 is measured relative to its initial value. Both standard ESO and FG ESO solutions display approximately 20% increase in the relative Table 1 Comparison of MBB beam optimization
ESO
Fig. 5 MBB beam design problem
volume (%) solution time relative mean/max stress ratio maximum displacement
Standard FG ESO
Stress FG ESO
Compliance
65 13:06:24 1.22
65 1:41:49 1.23
61 1:25:04 –
–
41.6
41.8
446 mean to maximum stress ratio values, again indicating the equivalence of the two solutions. As mentioned earlier, stress based and compliance based FG ESO both produced similar solutions. Some differences in their internal structural member arrangements in Fig. 6b and c are primarily due to stress approximation. However the equivalence of these solutions is demonstrated clearly in their maximum displacement values in Table 1. This is in agreement with Li et al. (1999) and Rozvany and K´ arolyi (1999)’s finding, where an optimal stress design is equivalent to a compliance design for single load case problems.
5.2 Michell’s beam design A typical Michell type problem (Michell 1904) has been optimized. The design environment and the maximum domain are as shown in Fig. 7. Due to symmetry, only the left half of the model is optimized and the mesh size of 50 × 50 is used. For standard ESO, the evolutionary rate constants are a1 = 0.0001 and a0 = 0.0; a = 0.04 for stress FG ESO; and a = 0.008 for compliance FG ESO. The optimal solutions for all three optimization are obtained at around 40% volume level as shown in Fig. 8. Again, the expected topology of a truss-like structure (Michell 1904) is observed for all cases. The standard stress based ESO solution of Fig. 8a is obtained after 22 hours 55 minutes and 30 seconds whilst stress based FG ESO obtained its solution of Fig. 8b in 2 hours and 22 minutes and 52 seconds, which is almost 90% reduction in time. The compliance FG ESO solution of Fig. 8c required 50 minutes and 40 seconds. However, three cavities are created in the topology by standard ESO while 5 cavities are created in FG ESO’s topology. This demonstrates the benefits of the boundary based optimization where a part of an element may be removed according to the stress patterns. This not only gains a boundary representation of the solution, it leads to a more refined and detailed topology, indicated by the increased number of cavities. A greater number of cavities can be obtained by increasing the mesh density and/or reducing the evolution-
Fig. 7 Michell’s design problem of beam AR2
Fig. 8 Optimum solutions of Michell’s beam problem
ary rate (Kim et al. 2000b). It can therefore be understood that applying standard ESO with a reduced evolutionary rate may gain a more comparable topology with the same number of cavities. ESO is applied again to the same problem with the same sized mesh but with a slower rate of a1 = 1 × 10−5 . After 982 iterations and 24 hours 6 minutes and 24 seconds, the same topology with three cavities of Fig. 8a is obtained again. Therefore, it is induced that an even slower evolutionary rate and a finer mesh density must be applied in order to obtain a topology with 5 cavities by standard ESO. This will increase the solution time even further. Thus, the benefits of using FG ESO can be appreciated from this illustration. The layout of the stress and compliance solutions of Fig. 8b and c, again compare closely, but with difference number of cavities. This is due to the stress approximation procedure, where a nodal stress is approximated by a volume weighted average of the von Mises stresses at the Gauss points of 4 surrounding elements. This introduces a smoothing effect on the stress distribution, similar to the patch smoothing technique (Li et al. 1999), thus reducing the number of cavities (Kim 2000). Therefore, a smaller number of cavities may be obtained in
447 stress based solutions. However, the layout of both solutions display the features of the optimal arrangement determined by Michell (1904), albeit his work related to pin-jointed frame and not continua.
5.3 Bridge A simple bridge model is optimized for compliance. The design environment is as shown in Fig. 9. A deck of 4 unit thickness is placed on the top and is specified as the nondesign domain. A uniformly distributed load is applied on the top deck to simulate traffic. Applying a symmetry condition, only half of the design domain is modelled with a mesh density of 50 × 30. An evolutionary rate constant of a = 0.01 is applied. A continuous optimum of an arch-like support, Fig. 10a is reached at 55% volume and the stiffness sensitivity
Fig. 9 Bridge design problem
Fig. 10 Optimum topologies with uniformly distributed load
Fig. 11 Optimum topology with a point load
values of the boundary are uniform. Thus, applying FG ESO creates the cavities in order to reduce the volume further and this leads to a discrete optimum at 27% volume of Fig. 10b. Stiffness FG ESO is also applied to the same design domain subjected to a point load at the centre of the top deck, instead of the uniformly distributed load, and its solution is displayed in Fig. 11. Due to the nature of the applied load, two diagonal members transmitting the load from its point of application to the fixed supports are obtained as the optimum configuration, at the volume of 32% relative to its initial design domain.
6 Conclusions This paper has presented the FG ESO algorithm which removes material along the contour line of the deletion criterion. In contrast to element based ESO with its element by element removal, FG ESO removes material in any form. The definition of the deletion criterion is modified so that it always removes a small percentage of materials, which again increases the efficiency of the optimization algorithm. In some structural designs, a large deflection is not favourable and compliance becomes an important consideration. FG ESO was thus extended to compliance based optimization. When applied to single-load problems, the solutions were comparable to the stress based optimization results. This confirms that the compliance design is equivalent to a fully stressed design (Li et al. 1999; Rozvany and K´ arolyi 1999). The “jagged-edges” which are a prominent feature of standard ESO topologies are not observed in the FG ESO topologies, however their boundaries of the solutions presented in the paper are not perfectly smooth. It should be noted that these solutions are depicted by a series of points, simply connected by straight lines. Another factor contributing to the nonsmooth boundaries is the use of linear interpolation in the determination of boundary stresses. However, as the purpose of this study is to appreciate the solution time reduction and the feasibility of the optimization method, the proposed algorithm is deemed adequate to show the efficiency and effectiveness of boundary based FG ESO. Since the output format of the FG ESO results is the boundary points arranged in the anticlockwise manner, a more sophisticated image filtering technique can be applied to the output in order to obtain smooth boundaries. Equivalently a higher order interpolation of the nodal stresses may be applied to better approximate the boundary stress, leading to smooth boundaries. The examples demonstrate that FG ESO is able to produce an optimal topology in agreement with the known solutions. The topologies are not represented by the finite elements’ “jagged-edges” but by a series of boundary points arranged in the anticlockwise direc-
448 tion, and such an output of a topology is produced at every iteration. This greatly simplifies further analysis or further design manipulation or the preparation for manufacturing. However the most significant advantage is the reduction of solution time.
ber of cavities. In: CD-Rom Proc. 8-th AIAA/NASA/USAF/ ISSMO Symp. on Multidisciplinary Analysis and Optimization (held in Long Beach, CA, USA)
References
Li, Q.; Steven, G.P.; Xie, Y.M. 1999: On equivalence between stress criterion and stiffness criterion in evolutionary structural optimisation. Struct. Optim. 18, 67–73
Chu, D.N. 1997: Evolutionary structural optimization method for systems with stiffness and displacement constraints. Ph.D. Thesis, Victoria University, Australia Garc´ıa, M.J. 1999: Fixed grid finite element analysis in structural design and optimisation. Ph.D. Thesis, University of Sydney, Australia
Lewi´ nski, T.; Zhou, M.; Rozvany, G.I.N. 1994: Extended exact least-weight truss layouts. Part II: unsymmetric cantilevers. Int. J. Mech. Sci. 36, 399–419
Michell, A.G.M. 1904: The limits of economy of material in frame-structures. Phil. Mag. 8, 589–597 Olhoff, N.; Bendsøe, M.P.; Rasmussen, J. 1991: On CADintegrated structural topology and design optimization. Comp. Meth. Appl. Mech. Engrg. 89, 259–279
Garc´ıa, M.J.; Steven, G.P. 2000: Fixed grid finite element analysis in structural design and optimization. In: Baranger, T.; van Keulen, F.(eds) Proc. 2nd ASMO/AIAA Internet Conf. Approx. Fast Reanal. Engrg. Optim., http://www-tm.wbmt. tudelft.nl/∼wbtmavk/2aro_conf
Rozvany, G.I.N.; K´ arolyi, G. 1999: New basic topological properties of exact optimal multipurpose place trusses and thrie implications for optimal composites. CD-Rom Proc. WCSMO-3 (held in Buffalo, NY)
Hassani, B.; Hinton, E. 1998: Homogenization and structural topology optimization: theory, practice and software. Berlin Heidelberg New York: Springer
Sigmund, O. 1994: Design of material structures using topology optimisation. Ph.D. Thesis, Technical University of Denmark
Kim, H. 2000: Development of evolutionary structural optimisation for engineering design. Ph.D. Thesis, University of Sydney, Australia
Voller, V.R.; Swaminathan, C.R.; Thomas, B.G. 1990: Fixed grid techniques for phase change problems. Int. J. Num. Meth. Engrg. 30, 875–898
Kim, H.; Garc´ıa, M.J.; Querin, O.M.; Steven, G.P.; Xie, Y.M. 2000a: Introduction of fixed grid in evolutionary structural optimisation. Engrg. Comp. 17, pp. 427–439
Xie, Y.M.; Steven, G.P. 1997: Evolutionary structural optimization. Berlin, Heidelberg, New York: Springer
Kim, H.; Querin, O.M.; Steven, G.P.; Xie, Y.M. 2000b: Determination of an optimal topology with a predefined num-
Zhou, M.; Rozvany, G.I.N. 1991: The COC algorithm, part II: topological, geometry and generalized shape optimization. Comp. Meth. App. Mech. Engrg. 89, 309–336
