- Email: [email protected]
Topology design for multiple loading conditions of continuum structures using isolines and isosurfaces
ARTICLE IN PRESS Finite Elements in Analysis and Design 46 (2010) 229–237
Contents lists available at ScienceDirect
Finite Elements in Analysis and Design journal homepage: www.elsevier.de/locate/finel
Topology design for multiple loading conditions of continuum structures using isolines and isosurfaces Mariano Victoria a, Osvaldo M. Querin b,, Pascual Martı´ a a b
Structures and Construction Department, Technical University of Cartagena, Campus Muralla del Mar, 30202 Cartagena (Murcia), Spain School of Mechanical Engineering, The University of Leeds, Leeds LS2 9JT, UK
a r t i c l e in fo
abstract
Article history: Received 28 July 2008 Received in revised form 3 August 2009 Accepted 28 September 2009 Available online 2 November 2009
Isolines topology design (ITD) is an iterative algorithm for the topological design of two-dimensional continuum structures using isolines. This paper presents an extension to this algorithm for topology design under multiple load cases of two/three-dimensional continuum structures. The topology and the shape of the design depend on an iterative algorithm, which continually adds and removes material depending on the shape and distribution of the contour isolines/isosurfaces of the required structural behaviour. In this study the von Mises stress was investigated. Several examples are presented to show the effectiveness of the algorithm, which provides very detailed contours without the need to interpret the topology in order to obtain a final design. The ITD algorithm demonstrates how the use the multiple loading conditions can produces more stable and realistic designs with a little additional complexity. & 2009 Elsevier B.V. All rights reserved.
Keywords: Topology design Multiple load cases Three-dimensional continuum Isosurfaces Isolines Fixed grid
1. Introduction Structures used in real environments are subject to multiple loading conditions and in many cases need to be modelled in three-dimensions. Therefore, to obtain more realistic structures, these should be designed for multiple load cases, be threedimensional, and be feasible, among other aspects. Diaz and Bendsøe [1] and Allaire, et al. [2] extended the Homogenization method to multiple load cases (MLC). Xie and Steven [3,4] took into account the highest element stress level over all load cases for identifying the extreme efficiency of material usage. Those elements with the least efficiency under all load conditions are progressively removed until the remaining elements become relativity efficient for at least one of all the load cases. Chu et al. [5] presented a simple evolutionary procedure based on finite element analysis (FEA) to minimize the weight of structures while satisfying stiffness requirements. A wide range of problems including those with multiple displacements conditions, moving loads, and multiple load cases are considered. Young et al. [6] successfully extended bi-directional evolutionary structural optimization (BESO) to MLC for two-dimensional (2D) and threedimensional (3D) problems. Li et al. [7] showed how the ESO
Corresponding author.
E-mail addresses: [email protected] (M. Victoria), [email protected] (O.M. Querin), [email protected] (P. Martı´). 0168-874X/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.finel.2009.09.003
method can be used to achieve a multiple criterion design for a ¨ et al. [8] presented a study structure in thermal problems. Akgun whose main objective was to demonstrate the efficiency of the adjoint method under MLC with and without constraints lumping. Bendsøe and Sigmund [9] extended solid isotropic microstructures with penalty (SIMP) method to MLC, where the objective function is formulated as a minimization of a weighted average of the compliances for each of the load cases. Allaire and Jouve [10] extended the level set (LS) method for shape and topology optimization for 2D and 3D problems to new objective functions such as eigenfrequencies and multiple loads. Cervera and Trevelyan [11] presented and ESO approach based on boundary element method (BEM) and non-uniform rational B-splines (NURBS). To solve multiple load cases problem, a logical AND/OR algorithm is used, where the material is selected for the removal process only if it is low-stressed in all load cases, and it is selected for the additional process if it is high-stressed in any of the load cases. Aguilar et al. [12] developed a computational model for multiobjective optimization problems and it is based on genetic algorithms (GA). Zhou and Li [13] presented a method to optimize the topology of structures under multiple load cases with stress constraints. Fiber-reinforced orthotropic composite is employed as the material model to simulate the constitutive relation of trusslike continua. The first work where the homogenization method was modified to account for 3D effects can be found in Bendsøe [14] and Diaz and Lipton [15]. Olhoff et al. [16] used optimum 3D microstructures for
ARTICLE IN PRESS 230
M. Victoria et al. / Finite Elements in Analysis and Design 46 (2010) 229–237
topology optimization of linearly elastic 3D continuum structures subject to a single case of static loading. To visualize topology designs, a penalization technique was applied. Borrvall and Petersson [17] considered large-scale topology optimization of elastic continua in 3D using the regularized intermediate density control. In order to deal with problems of large size, parallel computing is used in combination with domain decomposition. Hsu et al. [18] presented an automated process for interpreting 3D topology optimization results into a smooth CAD model. The topology optimization is solved by the SIMP method. Koguchi and Kikuchi [19] developed a surface reconstruction algorithm, which consisted of three parts: (1) an enclosed isosurface geometry from which the topologically optimized model was generated, (2) features detected, and (3) the parametric CAD solid model reconstructed as bi-quartic surfaces splines. The novelty of this work is that it extends the isolines topology design (ITD) algorithm [20] to problems experiencing multiple load cases and which are modelled with three dimensional finite elements. The method of determining the isosurfaces within fixed-grid finite elements (FG-FE) is given, together with a brief explanation of the fixed-grid finite elements analysis method (FG-FEA). Several examples are presented to show the effectiveness of the algorithm. The usefulness of the algorithm lies in that it is capable of providing quality solutions with very detailed contours, without the need to interpret the topology or of using a tuning process in order to obtain a final design.
2. Fixed grid finite element analysis The fixed grid method was first introduced by Garcia and Steven [21] as a tool for numerical estimation of two-dimensional elasticity problems. The application of FG-FEA to two-dimensional problems has been a research topic during the last years [22–25]. A lot of works have been also done with 3D structures like Suzuki et al. [26], Garcia et al. [27,28], etc. The benefits of using FG-FEA over conventional FEA in this work are that: (1) FG does not need a fitted mesh to discretize the analysis domain; (2) the boundary of the design is disassociated from the mesh [21]; (3) designs using FE-FEA do not contain checkerboard patterns, making the design more reliable for manufacture [29]; (4) solution time is significantly reduced [30]. In FG-FEA, the elements are in a fixed position and have the real design superimposed on them. This means that there are elements which lie Inside (I), Outside (O), or on the Boundary (B) of the design. The elemental stiffness matrix ðKe Þ is given by (1). 8 > < KI K ¼ KO > e e : KB ¼ KI x þ ð1 x ÞKO e
e
if x ¼ 1 e
if x ¼ 0
ð1Þ
e
if 0 o x o 1
ðeÞ
where x is the design fraction inside the element, KI element stiffness matrix for an element inside, KO is the element stiffness matrix for an element outside, and KB for an element boundary. Normally KO r104 KI . In this work, the value of the criteria in each element (se) is calculated using (2). PnG
wk sek k ¼ 1 wk
se ¼ Pk n¼G 1
ð2Þ
where nG is the number of the Gauss points in the elements, and w is the weighting factor for each Gauss point.
The criteria value at the ith node of an element ðsni Þ is determined by (3). PN e
sni ¼
e¼1
sni
N
ð3Þ
where i is the ith node number of an element; seni is the nodal criterion value at node i for each element surrounding that node, N is the number of elements connected to that node. The nodal value is determined from the criteria values at each Gauss point extrapolated to the nodes using the shape functions of the element.
3. Design using isosurfaces The use of isolines/isosurfaces in two/three-dimensions respectively to obtain the optimum design of a structure have been used in some recent studies, e.g. Woon et al. [31], Cui et al. [32], De Ruiter and Van Keulen [33], Hsu and Hsu [34], Koguchi et al. [19], etc. The aim of this work has been to extend the ITD algorithm [20] to problems experiencing multiple load cases and which are modelled with three dimensional finite elements. The ITD is an iterative algorithm which redistributes (adds and removes) material inside of a design domain until it reaches a desired volume fraction. The redistribution process consists of the following four steps: (1) Obtain the design criteria distribution within the design domain; (2) Determine the minimum criteria level (MCL), where its intersection with the design criteria distribution produces the new structural boundary, shown for a 2D continuum in Fig. 1; (3) Eliminate all regions from the design domain where the criteria distribution is lower than the MCL; (4) This design modification requires the re-evaluation of the remaining structure in order to recalculate the design criteria distribution. The MCL is calculated at each iteration and depends on the distribution of the design criterion and on the volume of the design domain in that iteration, given by (4). Vi ¼ V0
ni i i þ Vf ni ni
ð4Þ
where i is the ith iteration, V0 is the initial volume of the design domain; Vf is the final volume desired for the design; ni is the total number of iterations to use for the ITD to design the structure. Once the criterion has been calculated for each element in the design domain, these are arranged in decreasing order of criterion value. An element by element volume summation of the ordered list is carried out until a volume is reached which is as close as possible to the target volume given by (4), where the level of error between the summed and target volume depends on the size of the elements. The criteria value of the next element in the ordered list is then used as the value for the MCL. For multiple load cases, the same process is applied for each individual load case. The final design then consists of the superposition of the design for each load case. 3.1. Criterion selection The design criterion used in this work was the von Mises stress, which for a three-dimensional continuum is given by (5). qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 sVM ¼ pffiffiffi ðs1 s2 Þ2 þ ðs2 s3 Þ2 þ ðs1 s3 Þ2 ð5Þ 2 where, s1, s2, and s3 are the principal stresses.
ARTICLE IN PRESS M. Victoria et al. / Finite Elements in Analysis and Design 46 (2010) 229–237
231
Fig. 1. Structural boundary is defined by the intersection of the MCL with the criteria distribution.
Fig. 2. Look-up table for the MC algorithm showing the 16 different topologic states.
3.2. Combination of individual load case designs to produce the global design
BESO like methods [3,4,6,7,11]. This is achieved by the 9 steps given below:
The multiple load case global design is obtained by the superposition of the designs obtained for each individual load case, which is analogous to the AND/OR method applied to ESO/
1. Determine the volume ðVi Þ of the design domain in the current (ith) iteration using (4). Þ. 2. For each load case calculate the average (or mean) criterion ðsmean l
ARTICLE IN PRESS 232
M. Victoria et al. / Finite Elements in Analysis and Design 46 (2010) 229–237
3. Calculate the required volume for every load case design domain ðVil Þ using (6).
b V Vil ¼ PlNLC i
ð6Þ
bl
l¼1
where i is the ith iteration, l is the lth load case, NLC is the total number of load cases, bl is the volume control weighting factor. It’s calculated using the ratio of the average criterion for the lth load case with the maximum average criterion amongst all g and is given by (7). load cases maxl ¼ 1;NLC fsmean l
bl ¼
smean l maxl ¼ 1;NLC fsmean g l
ð7Þ
4. Use the individual load case design volume from (6) to calculate the MCL for each load case ðsMCLl Þ. 5. For each load case, calculate shape of the design domain using that load case MCL ðsMCLl Þ. 6. Use the shape of the design domain to calculate the volume e fraction of each finite element for all of the load cases ðxl Þ. e 7. The volume fraction for an element ðx Þ is then given by the maximum volume for that element from all of the load cases (8).
xe ¼ max fxel g
ð8Þ
l ¼ 1;NLC
8. Calculate the superimposed volume of the design domain using (9). Visuper ¼
N X
Ve x
e
ð9Þ
cell. The basic assumption of this algorithm is that a contour (MCL isosurface) can only pass through a finite element in a limited number of ways. This algorithm requires the value of the MCL as well as the value of the criteria at each node, and consists of two basic steps: 1. Identify from Fig. 2 the topological state of each element; 2. Determine the shape of the contour of the MCL isosurface through each element. The interaction of an isosurface through a cubic element can have a maximum of 256 different topological states. But, since a cube has double symmetry, the maximum number of states can be reduced to 16, Fig. 2. This shows all different states where a black circle at a node means that the value of the criteria at that node is less than the MCL (i.e. outside of the design). When only one of the nodes in an edge of an element is marked with a black circle it indicates that the MCL isosurface intersects that edge, which is the case for topological states 2–15. To find that intersection point, linear interpolation can be used. To construct facets from the intersection points, Delanauy triangulation [40] can be used. The shape of the MCL isosurface through the element is then obtained by connecting these facets as shown in Fig. 2.
3.4. Structural boundary stabilization
e¼1
where e is the eth finite element number, Ve is the volume of the eth element. 9. Calculate the volume error between the target volume ðVi Þ and the superimposed volume ðVisuper Þ using (10).
DV ¼ Visuper Vi
ð10Þ
If the absolute value of the volume error is less than the volume of an element ðjDVj rVe Þ, then the process can be terminated. If the volume error is negative and greater than the volume of an element, then (11) is used to modify the required volume for every load case design domain ðVil Þ and steps 4 through 9 are repeated. Vil ¼ Vil þ bl Ve
DV% ¼
Vi Vi1 100 Vi1
ð13Þ
This iterative process only requires a few iterations, although the exact number depends on the value of the volume of design domain at the ith iteration ðVi Þ determined by (4).
ð11Þ
If the volume error is positive and greater than the volume of an element, then (12) is used to modify the required volume for every load case design domain ðVil Þ and steps 4 through 9 are repeated. Vil ¼ Vil bl Ve
When the MCL is modified, the structural boundary changes and this affects the criterion distribution. Therefore, before the new iteration is started, an iterative process of reanalysis and material redistribution is carried out until the change in the domain volume between successive boundary adjustments is less than a minimum volume change limit ðDV%Þ. Typical value are DV% ¼ 0:5–2.5%.
ð12Þ
Note that since this process has been implemented using a fixed grid, then all elements have the same volume (Ve). 3.3. Minimum criteria level extraction There are several approaches to generation of a 3D surface, e.g. Keppel [35], Herman et al. [36], Farrell [37], Shen and Jhonson [38], Koguchi et al. [19], etc. The procedure to generate the structural boundary in 3D designs depends on the determination of the MCL isosurface. In order to determine the line segments that produce the profile of the boundary, the contouring algorithm called Marching Cubes [39] was implemented. The Marching Cubes (MC) method uses a divide and conquer approach, treating each finite element independently as a cube
4. The topology design algorithm for multiple load cases The procedure for implementing the ITD method for multiple load cases is as follows: 1. Define the design and non-design domains, supports, loads, and material properties. 2. Specify the size of fixed grid mesh. 3. Specify the final design volume Vf, the total number of iterations ni, and the minimum volume change limit DV%. 4. Specify the design criterion to use: von Mises stress, etc. 5. Carry out a FG-FEA of the design domain for each load case. 6. Using the method from Section 3.2 determine the structural boundary for the design. 7. If the percentage volume change is greater than the minimum volume change limit ðDV%Þ, go to step 5, else go to step 8. 8. If the total number of iterations ni has been reached, go to step 9, else increment the iteration number i by 1 and go to step 5. 9. Stop the design process. This process can be viewed in the flow chart of Fig. 3.
ARTICLE IN PRESS M. Victoria et al. / Finite Elements in Analysis and Design 46 (2010) 229–237
233
Start Specify: Design and FG domain, non-design region, loads and boundary condition, topology parameters
FG-FEA for each load case Determine the target volume Vi Calculate the MCL for each load case Calculate shape of design domain for each load case Update target volume for each load case Vil
Update target volume for each load case Vil
Calculate elemental volume fraction for each load case Calculate superimposed volume of design domain
No
Superimposed and Target volume equal
Stabilization Process Yes Stop design process
No
Yes End Fig. 3. Flow chart showing the ITD process for structures with multiple load cases.
Table 1 ITD parameters.
F1 = 10 kN
Example
ni
Vf =V0
ðDV%Þ
NLC
Square under torsion Beam with a roller support Short cantilever Electric mast
50 50 100 50
0.300 0.200 0.150 0.025
1.5 1.5 1.5 1.5
2 3 2 2
F2 = 10 kN F1 = 10 kN
F2 = 10 kN
1m
0.2 m 5. Examples To illustrate the ITD algorithm, four structures were studied and are presented here: (1) A square under torsion, (2) a beam with a roller support, (3) a short cantilever, and (4) an electric mast. For all the examples the elastic modulus of I elements is 210 GPa, the Poisson’s ratio is 0.3, and elastic modulus of O elements is 0.0210 MPa. The FE used for the 2D examples is the four-node plane
F1 = 10 kN
F2 = 10 kN
Thickness : 0.1 m F2 = 10 kN
F1 = 10 kN
Fig. 4. Square under torsion.
ARTICLE IN PRESS 234
M. Victoria et al. / Finite Elements in Analysis and Design 46 (2010) 229–237
Fig. 5. Final designs for the square under torsion: (a) ITD algorithm; (b) Diaz and Bendsøe [1].
10m
3m Thickness : 0.1 m F1 = 10 kN 2.5m
2.5m
F2 = 10 kN 2.5m
F3 = 10 kN 2.5m
Fig. 6. Beam with a roller support.
Fig. 8. Optimum topologies [9]: (a) all loads in one load case; (b) multiple loading cases.
1m F1 = 10 kN
Fig. 7. Final designs for the beam with a roller support: (a) all loads act simultaneously; (b) multiple load cases.
1m
stress quadrilateral element with four Gauss integration points [41], and for the 3D example is the eight-node isoparametric with eight Gauss integration points [41]. The design criteria used was the von Mises stress. Table 1 shows the ITD parameters used.
Thickness : 0.1 m 5.1. Square under torsion The design domain is a square area of 1 1 m with a centrally located square hole where all displacements are restricted. The
F2 = 10, 50, 100 kN Fig. 9. Short cantilever.
ARTICLE IN PRESS M. Victoria et al. / Finite Elements in Analysis and Design 46 (2010) 229–237
235
Fig. 10. Designs for the short cantilever beam: (a) F1 =F2 = 10 kN; (b) F1 = 10 kN, F2 = 50 kN; (c) F1 = 10 kN, F2 = 100 kN.
2.5 m 2.5 m
5m 5m F2 = 10 kN F1 = 10 kN
5m Fig. 11. Designs for the short cantilever beam. Volume fraction: Vf =V0 ¼ 0:35: (a) ITD algorithm; (b) Li et al. [42]. F1 = 10 kN F2 = 10 kN
20 m 10 m
10 m
10 m
5m
Fig. 12. Design for the short cantilever beam with F1 = 10 kN and F2 = 55.5 kN. Fig. 13. Electric mast.
mesh used has 250 250 elements. The structure is subject to two different load cases, represented by loads F1 and F2, Fig. 4. The designs generated with ITD algorithm (Fig. 5a) reveals an excellent agreement with optimum topology obtained (Fig. 5b) using homogenization method [1].
5.2. Beam with a roller support The design domain is a rectangular area of 10 3 m. The mesh used has 160 48 elements. It has a fixed support in the bottom left-hand corner and a roller support in the bottom right-hand corner. Three identical vertical downward loads F1, F2, and F3 are applied at the bottom edge, Fig. 6. The resulting designs for multiple load cases, where F1, F2, and F3 represent each load case, and the single load case, where F1, F2, and F3 act simultaneously, are given in Fig. 7.
Similar solutions to this example (Fig. 8) were obtained by Bendsøe and Sigmund [9]. It is interesting to note that the solution for a single load case (Figs. 7a and 8a) in which all three loads acts simultaneously produces a truss-like structure that is not rigid. Hence for this problem, the single load case arrangement produces an unstable structure based on squared frames whereas the multiple load cases problem produces (Figs. 7b and 8b) a stable structure based on triangular frames.
5.3. Short cantilever The dimensions of the cantilever beam are 1 1 m. The design domain was discretized using a mesh of 200 200 FE, fully clamped along the left edge. Two vertical loads are applied, F1 ¼ 10 kN at the upper right-hand corner and F2 at lower free end with three different values 10, 50, and 100 kN, Fig. 9.
ARTICLE IN PRESS 236
M. Victoria et al. / Finite Elements in Analysis and Design 46 (2010) 229–237
Fig. 14. Electric mast. Final designs for: (a) single load case; (b) multiple load case.
The resulting MLC designs for different ratios between F1 and F2 are given in Fig. 10. For the case of F1 ¼ F2 the resulting design is symmetric (Fig. 10a) and when F1 aF2 the design is asymmetric, Fig. 10b and c. Fig. 11a (for a volume fraction of 0.35) shows great agreement with the work of Li et al. [42] with the ESO method, Fig. 11b. Fig. 12 shows the design for the ratio between forces of F2 =F1 ¼ 5:5. For ratios higher than this, the optimal topology always converged to the design of Fig. 10c. 5.4. Electric mast The design domain is the T-shaped box of Fig. 13. Two symmetric vertical loads F1 and F2 are applied in the middle of the lower edges of the horizontal part of the T-section and represent the loads exerted by the wires on the mast. Fixed supported boundary conditions are applied at the corners of the base of the T-shaped box. A one-quarter FEA model using a 90 150 15 mesh and symmetry conditions was used to generate the design. Fig. 14a is the case where all loads act simultaneously and Fig. 14b shows the resulting design for multiple load cases where F1 and F2 represent each load case. The ITD algorithm produces a truss-like design that evolves topology of real electric masts. The number of generated bars or truss elements which emerge depends on the mesh density and size of the fixed grid domain. A full scale real industrial application would require a much finer mesh, and a larger domain in the vertical direction.
6. Conclusions This paper presents an enhancement to the ITD algorithm, where the multiple load cases are considered in design process. By using the fixed-grid FEA method of analysis, changes in the topology of the structure are easily and efficiently handled and stress or other criteria isosurfaces are easily generated and used to generate the emerging structural topology. The ITD is an iterative algorithm where the generation of new contours allows for the removal and redistribution of material. The use of the isosurfaces of the desired structural performance has several benefits: (1) the process works globally although the
design criteria may be local, such as the von Mises stress; (2) the produced structure has smooth boundaries and needs no further interpretation, enhancement or processing; (3) the external shape of the designs are not dependent on the fixed-grid FE mesh density, although the internal features of the topology are. Therefore, if more internal features are desired to be present in the final design, a progressively denser FE mesh must be used. In practise, most of designs are usually subjected to multiple load cases, and in many cases, structures designed for multiple loads are more stable and robust than designs subject to a single load case. Four examples of optimum topology design of 2D/3D continuum structures subject to multiple load cases were presented to demonstrate the applicability and effectiveness of ITD algorithm in this work. The main conclusion of this work is that the ITD algorithm is useful design method for 2D and 3D structures with single and multiple loading conditions. Acknowledgements This work has been partially supported by the CARM (Consejeria de Educacio´n, Ciencia e Investigacio´n de la Regio´n de Murcia) and the Technical University of Cartagena. Its support is greatly appreciated. Travelling funds for the second named author were provided by the School of Mechanical Engineering at the University of Leeds. References [1] A.R. Diaz, M.P. Bendsøe, Shape optimization of structures for multiple loading conditions using a homogenization method, Structural Optimization 4 (1992) 17–22. [2] G. Allaire, Z. Belhachmi, F. Jouve, The homogenization method for topology and shape optimization. Single and multiple loads case, European Journal of Finite Elements 5 (1996) 649–672. [3] Y.M. Xie, G.P. Steven, Optimal design of multiple load case structures using an evolutionary procedure, Engineering Computations 11 (1994) 295–305. [4] Y.M. Xie, G.P. Steven, Evolutionary Structural Optimization, Springer, Berlin, Heidelberg, New York, 1997. [5] D.N. Chu, Y.M. Xie, A. Hira, G.P. Steven, Evolutionary structural optimization for problems with stiffness constraints, Finite Elements in Analysis and Design 21 (1996) 239–251. [6] V. Young, O.M. Querin, G.P. Steven, 3D and multiple load case bi-directional evolutionary structural optimization (BESO), Structural Optimization 18 (1999) 183–192.
ARTICLE IN PRESS M. Victoria et al. / Finite Elements in Analysis and Design 46 (2010) 229–237
[7] Q. Li, G.P. Steven, O.M. Querin, Structural topology design with multiple thermal criteria, Engineering Computations 17 (6) (2000) 715–734. ¨ [8] M.A. Akgun, R.T. Haftka, K. Chauncey, J.L. Walsh, Efficient structural optimization for multiple load cases using adjoint sensitivities, AIAA Journal 39 (3) (2001) 511–516. [9] M.P. Bendsøe, O. Sigmund, Topology Optimization. Theory, Methods and Applications, Springer -Verlag, New York, 2003. [10] G. Allaire, F. Jouve, A level-set method for vibration and multiple loads structural optimization, Computer Methods Applied Mechanics and Engineering 194 (2005) 3269–3290. [11] E. Cervera, J. Trevelyan, Evolutionary structural optimisation based on boundary representation of NURBS. Part I: 2D algorithms, Computers and Structures 83 (2005) 1902–1916. [12] J.F. Aguilar, H. Rodrigues, H. Pina, Multi-objective optimization of structures topology by genetic algorithms, Advances in Engineering Software 36 (2005) 21–28. [13] K. Zhou, X. Li, Topology optimization of structures under multiple load cases using a fiber-reinforced composite material model, Computational Mechanics 38 (2006) 163–170. [14] M.P. Bendsøe, Optimal of Structural Topology, Shape, and Material, Springer, New York, 1995. [15] A.R. Diaz, R. Lipton, Optimal material layout for 3D elastic structures, Structural Optimization 13 (1997) 60–64. [16] N. Olhoff, E. Rønholt, J. Scheel, Topology optimization of three-dimensional structures using optimum microstructures, Structural Optimization 16 (1998) 1–18. [17] T. Borrvall, J. Petersson, Large-scale topology optimization in 3D using parallel computing, Computer Methods Applied Mechanics and Engineering 190 (2001) 6201–6229. [18] Y.L. Hsu, M.S. Hsu, C.T. Chen, Interpreting results from topology optimization using density contours, Computer and Structures 79 (2001) 1049–1058. [19] A. Koguchi, N. Kikuchi, A surface reconstruction algorithm for topology optimization, Engineering with Computers 20 (2006) 1–10. [20] M. Victoria, P. Martı´, O.M. Querin, Topology design of two-dimensional continuum structures using isolines, Computer and Structures 87 (2009) 101–109. [21] M.J. Garcia, G.P. Steven, Fixed grid finite elements in elasticity problems, Engineering Computations 16 (2) (1999) 145–164. [22] H. Kim, M.J. Garcia, O.M. Querin, G.P. Steven, Y.M. Xie, Introduction of fixed grid in evolutionary structural optimization, Engineering Computations 17 (4) (2000) 427–439. [23] M.J. Garcia, C.A. Gonzalez, Shape optimization of continuum structures via evolution strategies and fixed grid finite element analysis, Structural and Multidisciplinary Optimization 26 (2004) 92–98. [24] Y. Liu, F. Jin, Q. Li, S. Zhou, A fixed-grid bidirectional evolutionary structural optimization method and its applications in tunneling engineering, International Journal for Numerical Methods in Engineering 73 (2008) 1788–1810. [25] Y. Liu, J. Feng, L. Qing, A strength-based multiple cutout optimization in composite plates using fixed grid finite element method, Composite Structures 73 (2006) 403–412.
237
[26] K. Suzuki, H. Ohtsubo, K. Terada, The analysis of 3D solid using multi-scale voxel data, in: Proceedings of the IV World Congress on Computational Mechanics, Buenos Aires, Argentina, 1998. [27] M.J. Garcia, O.E. Ruiz, L.M. Ruiz, O.M. Querin, Fixed grid finite element analysis for 3D linear elastic structures, Computational Mechanics (WCCM VI), China, 2004. [28] M.J. Garcia, M. Henao, O.E. Ruiz, Fixed grid finite element analysis for 3D structural problems, International Journal of Computational Methods 2 (4) (2005) 569–585. [29] F.S. Maan, O.M. Querin, D.C. Barton, Extension of the fixed grid finite element method to eigenvalue problems, Advances in Engineering Software 38 (2007) 607–617. [30] M.J. Garcı´a, G.P. Steven, Fixed grid finite element analysis in structural design and optimisation, in: Second ISSMO/AIAA Internet Conference on Approximations and Fast Reanalysis in Engineering Optimization, May 25–June 2, 2000. [31] S.Y. Woon, L. Tong, O.M. Querin, G.P. Steven, Optimising topologies through a multi-GA System, in: The Fifth World Congress of Structural and Multidisciplinary Optimization (WCSMO 5), Lido di Jesolo, Italy, 2003, pp. 229–230. [32] C. Cui, H. Ohmori, M. Sasaki, Computational morphogenesis of 3D structures by extended ESO method, Journal of IASS 44 (141) (2003) 51–61. [33] M.J. De Ruiter, F. Van Keulen, Topology optimization using a topology description function, Structural and Multidisciplinary Optimization 26 (2004) 406–416. [34] M.H. Hsu, Y.L. Hsu, Interpreting three-dimensional structural topology optimization results, Computer and Structures 83 (2005) 327–337. [35] E. Keppel, Approximating complex surfaces by triangulation of contour lines, IBM Journal of Research and Development 19 (1975) 2–11. [36] G.T. Herman, J.K. Udupa, Display of 3D digital images: computational foundations and medical applications, IEEE Computer Graphics and Application 3 (5) (1983) 39–46. [37] E.J. Farrell, Color display and interactive interpretation of threedimensional data, IBM Journal of Research and Development 27 (4) (1983) 356–366. [38] H.W. Shen, C.R. Jhonson, Seeping simplifies: a fast isosurface extraction algorithm for unstructured grids, in: Proceedings of Visualization, 1995, pp. 143–151. [39] W.E. Lorensen, H.E. Cline, Marching Cubes: a high resolution 3D surface construction algorithm, Computer Graphics 21 (4) (1987) 163–169. [40] B.N. Delaunay, Sur la sphe re vide. Izvestia Akademii Nauk SSSR, Otdelenie Matematicheskikh i Estestvennykh Nauk 7 (1934) 793–800. [41] O.C. Zienkiewicz, R.L. Taylor, J.Z. Zhu, The Finite Element Method, Its Basis & Fundamentals, Elsevier Butterworth Heinemann, 2005. [42] Q. Li, G.P. Steven, Y.M. Xie, On equivalence between stress criterion and stiffness criterion in evolutionary structural optimization, Structural Optimization 18 (1999) 67–73.
Contents lists available at ScienceDirect
Finite Elements in Analysis and Design journal homepage: www.elsevier.de/locate/finel
Topology design for multiple loading conditions of continuum structures using isolines and isosurfaces Mariano Victoria a, Osvaldo M. Querin b,, Pascual Martı´ a a b
Structures and Construction Department, Technical University of Cartagena, Campus Muralla del Mar, 30202 Cartagena (Murcia), Spain School of Mechanical Engineering, The University of Leeds, Leeds LS2 9JT, UK
a r t i c l e in fo
abstract
Article history: Received 28 July 2008 Received in revised form 3 August 2009 Accepted 28 September 2009 Available online 2 November 2009
Isolines topology design (ITD) is an iterative algorithm for the topological design of two-dimensional continuum structures using isolines. This paper presents an extension to this algorithm for topology design under multiple load cases of two/three-dimensional continuum structures. The topology and the shape of the design depend on an iterative algorithm, which continually adds and removes material depending on the shape and distribution of the contour isolines/isosurfaces of the required structural behaviour. In this study the von Mises stress was investigated. Several examples are presented to show the effectiveness of the algorithm, which provides very detailed contours without the need to interpret the topology in order to obtain a final design. The ITD algorithm demonstrates how the use the multiple loading conditions can produces more stable and realistic designs with a little additional complexity. & 2009 Elsevier B.V. All rights reserved.
Keywords: Topology design Multiple load cases Three-dimensional continuum Isosurfaces Isolines Fixed grid
1. Introduction Structures used in real environments are subject to multiple loading conditions and in many cases need to be modelled in three-dimensions. Therefore, to obtain more realistic structures, these should be designed for multiple load cases, be threedimensional, and be feasible, among other aspects. Diaz and Bendsøe [1] and Allaire, et al. [2] extended the Homogenization method to multiple load cases (MLC). Xie and Steven [3,4] took into account the highest element stress level over all load cases for identifying the extreme efficiency of material usage. Those elements with the least efficiency under all load conditions are progressively removed until the remaining elements become relativity efficient for at least one of all the load cases. Chu et al. [5] presented a simple evolutionary procedure based on finite element analysis (FEA) to minimize the weight of structures while satisfying stiffness requirements. A wide range of problems including those with multiple displacements conditions, moving loads, and multiple load cases are considered. Young et al. [6] successfully extended bi-directional evolutionary structural optimization (BESO) to MLC for two-dimensional (2D) and threedimensional (3D) problems. Li et al. [7] showed how the ESO
Corresponding author.
E-mail addresses: [email protected] (M. Victoria), [email protected] (O.M. Querin), [email protected] (P. Martı´). 0168-874X/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.finel.2009.09.003
method can be used to achieve a multiple criterion design for a ¨ et al. [8] presented a study structure in thermal problems. Akgun whose main objective was to demonstrate the efficiency of the adjoint method under MLC with and without constraints lumping. Bendsøe and Sigmund [9] extended solid isotropic microstructures with penalty (SIMP) method to MLC, where the objective function is formulated as a minimization of a weighted average of the compliances for each of the load cases. Allaire and Jouve [10] extended the level set (LS) method for shape and topology optimization for 2D and 3D problems to new objective functions such as eigenfrequencies and multiple loads. Cervera and Trevelyan [11] presented and ESO approach based on boundary element method (BEM) and non-uniform rational B-splines (NURBS). To solve multiple load cases problem, a logical AND/OR algorithm is used, where the material is selected for the removal process only if it is low-stressed in all load cases, and it is selected for the additional process if it is high-stressed in any of the load cases. Aguilar et al. [12] developed a computational model for multiobjective optimization problems and it is based on genetic algorithms (GA). Zhou and Li [13] presented a method to optimize the topology of structures under multiple load cases with stress constraints. Fiber-reinforced orthotropic composite is employed as the material model to simulate the constitutive relation of trusslike continua. The first work where the homogenization method was modified to account for 3D effects can be found in Bendsøe [14] and Diaz and Lipton [15]. Olhoff et al. [16] used optimum 3D microstructures for
ARTICLE IN PRESS 230
M. Victoria et al. / Finite Elements in Analysis and Design 46 (2010) 229–237
topology optimization of linearly elastic 3D continuum structures subject to a single case of static loading. To visualize topology designs, a penalization technique was applied. Borrvall and Petersson [17] considered large-scale topology optimization of elastic continua in 3D using the regularized intermediate density control. In order to deal with problems of large size, parallel computing is used in combination with domain decomposition. Hsu et al. [18] presented an automated process for interpreting 3D topology optimization results into a smooth CAD model. The topology optimization is solved by the SIMP method. Koguchi and Kikuchi [19] developed a surface reconstruction algorithm, which consisted of three parts: (1) an enclosed isosurface geometry from which the topologically optimized model was generated, (2) features detected, and (3) the parametric CAD solid model reconstructed as bi-quartic surfaces splines. The novelty of this work is that it extends the isolines topology design (ITD) algorithm [20] to problems experiencing multiple load cases and which are modelled with three dimensional finite elements. The method of determining the isosurfaces within fixed-grid finite elements (FG-FE) is given, together with a brief explanation of the fixed-grid finite elements analysis method (FG-FEA). Several examples are presented to show the effectiveness of the algorithm. The usefulness of the algorithm lies in that it is capable of providing quality solutions with very detailed contours, without the need to interpret the topology or of using a tuning process in order to obtain a final design.
2. Fixed grid finite element analysis The fixed grid method was first introduced by Garcia and Steven [21] as a tool for numerical estimation of two-dimensional elasticity problems. The application of FG-FEA to two-dimensional problems has been a research topic during the last years [22–25]. A lot of works have been also done with 3D structures like Suzuki et al. [26], Garcia et al. [27,28], etc. The benefits of using FG-FEA over conventional FEA in this work are that: (1) FG does not need a fitted mesh to discretize the analysis domain; (2) the boundary of the design is disassociated from the mesh [21]; (3) designs using FE-FEA do not contain checkerboard patterns, making the design more reliable for manufacture [29]; (4) solution time is significantly reduced [30]. In FG-FEA, the elements are in a fixed position and have the real design superimposed on them. This means that there are elements which lie Inside (I), Outside (O), or on the Boundary (B) of the design. The elemental stiffness matrix ðKe Þ is given by (1). 8 > < KI K ¼ KO > e e : KB ¼ KI x þ ð1 x ÞKO e
e
if x ¼ 1 e
if x ¼ 0
ð1Þ
e
if 0 o x o 1
ðeÞ
where x is the design fraction inside the element, KI element stiffness matrix for an element inside, KO is the element stiffness matrix for an element outside, and KB for an element boundary. Normally KO r104 KI . In this work, the value of the criteria in each element (se) is calculated using (2). PnG
wk sek k ¼ 1 wk
se ¼ Pk n¼G 1
ð2Þ
where nG is the number of the Gauss points in the elements, and w is the weighting factor for each Gauss point.
The criteria value at the ith node of an element ðsni Þ is determined by (3). PN e
sni ¼
e¼1
sni
N
ð3Þ
where i is the ith node number of an element; seni is the nodal criterion value at node i for each element surrounding that node, N is the number of elements connected to that node. The nodal value is determined from the criteria values at each Gauss point extrapolated to the nodes using the shape functions of the element.
3. Design using isosurfaces The use of isolines/isosurfaces in two/three-dimensions respectively to obtain the optimum design of a structure have been used in some recent studies, e.g. Woon et al. [31], Cui et al. [32], De Ruiter and Van Keulen [33], Hsu and Hsu [34], Koguchi et al. [19], etc. The aim of this work has been to extend the ITD algorithm [20] to problems experiencing multiple load cases and which are modelled with three dimensional finite elements. The ITD is an iterative algorithm which redistributes (adds and removes) material inside of a design domain until it reaches a desired volume fraction. The redistribution process consists of the following four steps: (1) Obtain the design criteria distribution within the design domain; (2) Determine the minimum criteria level (MCL), where its intersection with the design criteria distribution produces the new structural boundary, shown for a 2D continuum in Fig. 1; (3) Eliminate all regions from the design domain where the criteria distribution is lower than the MCL; (4) This design modification requires the re-evaluation of the remaining structure in order to recalculate the design criteria distribution. The MCL is calculated at each iteration and depends on the distribution of the design criterion and on the volume of the design domain in that iteration, given by (4). Vi ¼ V0
ni i i þ Vf ni ni
ð4Þ
where i is the ith iteration, V0 is the initial volume of the design domain; Vf is the final volume desired for the design; ni is the total number of iterations to use for the ITD to design the structure. Once the criterion has been calculated for each element in the design domain, these are arranged in decreasing order of criterion value. An element by element volume summation of the ordered list is carried out until a volume is reached which is as close as possible to the target volume given by (4), where the level of error between the summed and target volume depends on the size of the elements. The criteria value of the next element in the ordered list is then used as the value for the MCL. For multiple load cases, the same process is applied for each individual load case. The final design then consists of the superposition of the design for each load case. 3.1. Criterion selection The design criterion used in this work was the von Mises stress, which for a three-dimensional continuum is given by (5). qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 sVM ¼ pffiffiffi ðs1 s2 Þ2 þ ðs2 s3 Þ2 þ ðs1 s3 Þ2 ð5Þ 2 where, s1, s2, and s3 are the principal stresses.
ARTICLE IN PRESS M. Victoria et al. / Finite Elements in Analysis and Design 46 (2010) 229–237
231
Fig. 1. Structural boundary is defined by the intersection of the MCL with the criteria distribution.
Fig. 2. Look-up table for the MC algorithm showing the 16 different topologic states.
3.2. Combination of individual load case designs to produce the global design
BESO like methods [3,4,6,7,11]. This is achieved by the 9 steps given below:
The multiple load case global design is obtained by the superposition of the designs obtained for each individual load case, which is analogous to the AND/OR method applied to ESO/
1. Determine the volume ðVi Þ of the design domain in the current (ith) iteration using (4). Þ. 2. For each load case calculate the average (or mean) criterion ðsmean l
ARTICLE IN PRESS 232
M. Victoria et al. / Finite Elements in Analysis and Design 46 (2010) 229–237
3. Calculate the required volume for every load case design domain ðVil Þ using (6).
b V Vil ¼ PlNLC i
ð6Þ
bl
l¼1
where i is the ith iteration, l is the lth load case, NLC is the total number of load cases, bl is the volume control weighting factor. It’s calculated using the ratio of the average criterion for the lth load case with the maximum average criterion amongst all g and is given by (7). load cases maxl ¼ 1;NLC fsmean l
bl ¼
smean l maxl ¼ 1;NLC fsmean g l
ð7Þ
4. Use the individual load case design volume from (6) to calculate the MCL for each load case ðsMCLl Þ. 5. For each load case, calculate shape of the design domain using that load case MCL ðsMCLl Þ. 6. Use the shape of the design domain to calculate the volume e fraction of each finite element for all of the load cases ðxl Þ. e 7. The volume fraction for an element ðx Þ is then given by the maximum volume for that element from all of the load cases (8).
xe ¼ max fxel g
ð8Þ
l ¼ 1;NLC
8. Calculate the superimposed volume of the design domain using (9). Visuper ¼
N X
Ve x
e
ð9Þ
cell. The basic assumption of this algorithm is that a contour (MCL isosurface) can only pass through a finite element in a limited number of ways. This algorithm requires the value of the MCL as well as the value of the criteria at each node, and consists of two basic steps: 1. Identify from Fig. 2 the topological state of each element; 2. Determine the shape of the contour of the MCL isosurface through each element. The interaction of an isosurface through a cubic element can have a maximum of 256 different topological states. But, since a cube has double symmetry, the maximum number of states can be reduced to 16, Fig. 2. This shows all different states where a black circle at a node means that the value of the criteria at that node is less than the MCL (i.e. outside of the design). When only one of the nodes in an edge of an element is marked with a black circle it indicates that the MCL isosurface intersects that edge, which is the case for topological states 2–15. To find that intersection point, linear interpolation can be used. To construct facets from the intersection points, Delanauy triangulation [40] can be used. The shape of the MCL isosurface through the element is then obtained by connecting these facets as shown in Fig. 2.
3.4. Structural boundary stabilization
e¼1
where e is the eth finite element number, Ve is the volume of the eth element. 9. Calculate the volume error between the target volume ðVi Þ and the superimposed volume ðVisuper Þ using (10).
DV ¼ Visuper Vi
ð10Þ
If the absolute value of the volume error is less than the volume of an element ðjDVj rVe Þ, then the process can be terminated. If the volume error is negative and greater than the volume of an element, then (11) is used to modify the required volume for every load case design domain ðVil Þ and steps 4 through 9 are repeated. Vil ¼ Vil þ bl Ve
DV% ¼
Vi Vi1 100 Vi1
ð13Þ
This iterative process only requires a few iterations, although the exact number depends on the value of the volume of design domain at the ith iteration ðVi Þ determined by (4).
ð11Þ
If the volume error is positive and greater than the volume of an element, then (12) is used to modify the required volume for every load case design domain ðVil Þ and steps 4 through 9 are repeated. Vil ¼ Vil bl Ve
When the MCL is modified, the structural boundary changes and this affects the criterion distribution. Therefore, before the new iteration is started, an iterative process of reanalysis and material redistribution is carried out until the change in the domain volume between successive boundary adjustments is less than a minimum volume change limit ðDV%Þ. Typical value are DV% ¼ 0:5–2.5%.
ð12Þ
Note that since this process has been implemented using a fixed grid, then all elements have the same volume (Ve). 3.3. Minimum criteria level extraction There are several approaches to generation of a 3D surface, e.g. Keppel [35], Herman et al. [36], Farrell [37], Shen and Jhonson [38], Koguchi et al. [19], etc. The procedure to generate the structural boundary in 3D designs depends on the determination of the MCL isosurface. In order to determine the line segments that produce the profile of the boundary, the contouring algorithm called Marching Cubes [39] was implemented. The Marching Cubes (MC) method uses a divide and conquer approach, treating each finite element independently as a cube
4. The topology design algorithm for multiple load cases The procedure for implementing the ITD method for multiple load cases is as follows: 1. Define the design and non-design domains, supports, loads, and material properties. 2. Specify the size of fixed grid mesh. 3. Specify the final design volume Vf, the total number of iterations ni, and the minimum volume change limit DV%. 4. Specify the design criterion to use: von Mises stress, etc. 5. Carry out a FG-FEA of the design domain for each load case. 6. Using the method from Section 3.2 determine the structural boundary for the design. 7. If the percentage volume change is greater than the minimum volume change limit ðDV%Þ, go to step 5, else go to step 8. 8. If the total number of iterations ni has been reached, go to step 9, else increment the iteration number i by 1 and go to step 5. 9. Stop the design process. This process can be viewed in the flow chart of Fig. 3.
ARTICLE IN PRESS M. Victoria et al. / Finite Elements in Analysis and Design 46 (2010) 229–237
233
Start Specify: Design and FG domain, non-design region, loads and boundary condition, topology parameters
FG-FEA for each load case Determine the target volume Vi Calculate the MCL for each load case Calculate shape of design domain for each load case Update target volume for each load case Vil
Update target volume for each load case Vil
Calculate elemental volume fraction for each load case Calculate superimposed volume of design domain
No
Superimposed and Target volume equal
Stabilization Process Yes Stop design process
No
Yes End Fig. 3. Flow chart showing the ITD process for structures with multiple load cases.
Table 1 ITD parameters.
F1 = 10 kN
Example
ni
Vf =V0
ðDV%Þ
NLC
Square under torsion Beam with a roller support Short cantilever Electric mast
50 50 100 50
0.300 0.200 0.150 0.025
1.5 1.5 1.5 1.5
2 3 2 2
F2 = 10 kN F1 = 10 kN
F2 = 10 kN
1m
0.2 m 5. Examples To illustrate the ITD algorithm, four structures were studied and are presented here: (1) A square under torsion, (2) a beam with a roller support, (3) a short cantilever, and (4) an electric mast. For all the examples the elastic modulus of I elements is 210 GPa, the Poisson’s ratio is 0.3, and elastic modulus of O elements is 0.0210 MPa. The FE used for the 2D examples is the four-node plane
F1 = 10 kN
F2 = 10 kN
Thickness : 0.1 m F2 = 10 kN
F1 = 10 kN
Fig. 4. Square under torsion.
ARTICLE IN PRESS 234
M. Victoria et al. / Finite Elements in Analysis and Design 46 (2010) 229–237
Fig. 5. Final designs for the square under torsion: (a) ITD algorithm; (b) Diaz and Bendsøe [1].
10m
3m Thickness : 0.1 m F1 = 10 kN 2.5m
2.5m
F2 = 10 kN 2.5m
F3 = 10 kN 2.5m
Fig. 6. Beam with a roller support.
Fig. 8. Optimum topologies [9]: (a) all loads in one load case; (b) multiple loading cases.
1m F1 = 10 kN
Fig. 7. Final designs for the beam with a roller support: (a) all loads act simultaneously; (b) multiple load cases.
1m
stress quadrilateral element with four Gauss integration points [41], and for the 3D example is the eight-node isoparametric with eight Gauss integration points [41]. The design criteria used was the von Mises stress. Table 1 shows the ITD parameters used.
Thickness : 0.1 m 5.1. Square under torsion The design domain is a square area of 1 1 m with a centrally located square hole where all displacements are restricted. The
F2 = 10, 50, 100 kN Fig. 9. Short cantilever.
ARTICLE IN PRESS M. Victoria et al. / Finite Elements in Analysis and Design 46 (2010) 229–237
235
Fig. 10. Designs for the short cantilever beam: (a) F1 =F2 = 10 kN; (b) F1 = 10 kN, F2 = 50 kN; (c) F1 = 10 kN, F2 = 100 kN.
2.5 m 2.5 m
5m 5m F2 = 10 kN F1 = 10 kN
5m Fig. 11. Designs for the short cantilever beam. Volume fraction: Vf =V0 ¼ 0:35: (a) ITD algorithm; (b) Li et al. [42]. F1 = 10 kN F2 = 10 kN
20 m 10 m
10 m
10 m
5m
Fig. 12. Design for the short cantilever beam with F1 = 10 kN and F2 = 55.5 kN. Fig. 13. Electric mast.
mesh used has 250 250 elements. The structure is subject to two different load cases, represented by loads F1 and F2, Fig. 4. The designs generated with ITD algorithm (Fig. 5a) reveals an excellent agreement with optimum topology obtained (Fig. 5b) using homogenization method [1].
5.2. Beam with a roller support The design domain is a rectangular area of 10 3 m. The mesh used has 160 48 elements. It has a fixed support in the bottom left-hand corner and a roller support in the bottom right-hand corner. Three identical vertical downward loads F1, F2, and F3 are applied at the bottom edge, Fig. 6. The resulting designs for multiple load cases, where F1, F2, and F3 represent each load case, and the single load case, where F1, F2, and F3 act simultaneously, are given in Fig. 7.
Similar solutions to this example (Fig. 8) were obtained by Bendsøe and Sigmund [9]. It is interesting to note that the solution for a single load case (Figs. 7a and 8a) in which all three loads acts simultaneously produces a truss-like structure that is not rigid. Hence for this problem, the single load case arrangement produces an unstable structure based on squared frames whereas the multiple load cases problem produces (Figs. 7b and 8b) a stable structure based on triangular frames.
5.3. Short cantilever The dimensions of the cantilever beam are 1 1 m. The design domain was discretized using a mesh of 200 200 FE, fully clamped along the left edge. Two vertical loads are applied, F1 ¼ 10 kN at the upper right-hand corner and F2 at lower free end with three different values 10, 50, and 100 kN, Fig. 9.
ARTICLE IN PRESS 236
M. Victoria et al. / Finite Elements in Analysis and Design 46 (2010) 229–237
Fig. 14. Electric mast. Final designs for: (a) single load case; (b) multiple load case.
The resulting MLC designs for different ratios between F1 and F2 are given in Fig. 10. For the case of F1 ¼ F2 the resulting design is symmetric (Fig. 10a) and when F1 aF2 the design is asymmetric, Fig. 10b and c. Fig. 11a (for a volume fraction of 0.35) shows great agreement with the work of Li et al. [42] with the ESO method, Fig. 11b. Fig. 12 shows the design for the ratio between forces of F2 =F1 ¼ 5:5. For ratios higher than this, the optimal topology always converged to the design of Fig. 10c. 5.4. Electric mast The design domain is the T-shaped box of Fig. 13. Two symmetric vertical loads F1 and F2 are applied in the middle of the lower edges of the horizontal part of the T-section and represent the loads exerted by the wires on the mast. Fixed supported boundary conditions are applied at the corners of the base of the T-shaped box. A one-quarter FEA model using a 90 150 15 mesh and symmetry conditions was used to generate the design. Fig. 14a is the case where all loads act simultaneously and Fig. 14b shows the resulting design for multiple load cases where F1 and F2 represent each load case. The ITD algorithm produces a truss-like design that evolves topology of real electric masts. The number of generated bars or truss elements which emerge depends on the mesh density and size of the fixed grid domain. A full scale real industrial application would require a much finer mesh, and a larger domain in the vertical direction.
6. Conclusions This paper presents an enhancement to the ITD algorithm, where the multiple load cases are considered in design process. By using the fixed-grid FEA method of analysis, changes in the topology of the structure are easily and efficiently handled and stress or other criteria isosurfaces are easily generated and used to generate the emerging structural topology. The ITD is an iterative algorithm where the generation of new contours allows for the removal and redistribution of material. The use of the isosurfaces of the desired structural performance has several benefits: (1) the process works globally although the
design criteria may be local, such as the von Mises stress; (2) the produced structure has smooth boundaries and needs no further interpretation, enhancement or processing; (3) the external shape of the designs are not dependent on the fixed-grid FE mesh density, although the internal features of the topology are. Therefore, if more internal features are desired to be present in the final design, a progressively denser FE mesh must be used. In practise, most of designs are usually subjected to multiple load cases, and in many cases, structures designed for multiple loads are more stable and robust than designs subject to a single load case. Four examples of optimum topology design of 2D/3D continuum structures subject to multiple load cases were presented to demonstrate the applicability and effectiveness of ITD algorithm in this work. The main conclusion of this work is that the ITD algorithm is useful design method for 2D and 3D structures with single and multiple loading conditions. Acknowledgements This work has been partially supported by the CARM (Consejeria de Educacio´n, Ciencia e Investigacio´n de la Regio´n de Murcia) and the Technical University of Cartagena. Its support is greatly appreciated. Travelling funds for the second named author were provided by the School of Mechanical Engineering at the University of Leeds. References [1] A.R. Diaz, M.P. Bendsøe, Shape optimization of structures for multiple loading conditions using a homogenization method, Structural Optimization 4 (1992) 17–22. [2] G. Allaire, Z. Belhachmi, F. Jouve, The homogenization method for topology and shape optimization. Single and multiple loads case, European Journal of Finite Elements 5 (1996) 649–672. [3] Y.M. Xie, G.P. Steven, Optimal design of multiple load case structures using an evolutionary procedure, Engineering Computations 11 (1994) 295–305. [4] Y.M. Xie, G.P. Steven, Evolutionary Structural Optimization, Springer, Berlin, Heidelberg, New York, 1997. [5] D.N. Chu, Y.M. Xie, A. Hira, G.P. Steven, Evolutionary structural optimization for problems with stiffness constraints, Finite Elements in Analysis and Design 21 (1996) 239–251. [6] V. Young, O.M. Querin, G.P. Steven, 3D and multiple load case bi-directional evolutionary structural optimization (BESO), Structural Optimization 18 (1999) 183–192.
ARTICLE IN PRESS M. Victoria et al. / Finite Elements in Analysis and Design 46 (2010) 229–237
[7] Q. Li, G.P. Steven, O.M. Querin, Structural topology design with multiple thermal criteria, Engineering Computations 17 (6) (2000) 715–734. ¨ [8] M.A. Akgun, R.T. Haftka, K. Chauncey, J.L. Walsh, Efficient structural optimization for multiple load cases using adjoint sensitivities, AIAA Journal 39 (3) (2001) 511–516. [9] M.P. Bendsøe, O. Sigmund, Topology Optimization. Theory, Methods and Applications, Springer -Verlag, New York, 2003. [10] G. Allaire, F. Jouve, A level-set method for vibration and multiple loads structural optimization, Computer Methods Applied Mechanics and Engineering 194 (2005) 3269–3290. [11] E. Cervera, J. Trevelyan, Evolutionary structural optimisation based on boundary representation of NURBS. Part I: 2D algorithms, Computers and Structures 83 (2005) 1902–1916. [12] J.F. Aguilar, H. Rodrigues, H. Pina, Multi-objective optimization of structures topology by genetic algorithms, Advances in Engineering Software 36 (2005) 21–28. [13] K. Zhou, X. Li, Topology optimization of structures under multiple load cases using a fiber-reinforced composite material model, Computational Mechanics 38 (2006) 163–170. [14] M.P. Bendsøe, Optimal of Structural Topology, Shape, and Material, Springer, New York, 1995. [15] A.R. Diaz, R. Lipton, Optimal material layout for 3D elastic structures, Structural Optimization 13 (1997) 60–64. [16] N. Olhoff, E. Rønholt, J. Scheel, Topology optimization of three-dimensional structures using optimum microstructures, Structural Optimization 16 (1998) 1–18. [17] T. Borrvall, J. Petersson, Large-scale topology optimization in 3D using parallel computing, Computer Methods Applied Mechanics and Engineering 190 (2001) 6201–6229. [18] Y.L. Hsu, M.S. Hsu, C.T. Chen, Interpreting results from topology optimization using density contours, Computer and Structures 79 (2001) 1049–1058. [19] A. Koguchi, N. Kikuchi, A surface reconstruction algorithm for topology optimization, Engineering with Computers 20 (2006) 1–10. [20] M. Victoria, P. Martı´, O.M. Querin, Topology design of two-dimensional continuum structures using isolines, Computer and Structures 87 (2009) 101–109. [21] M.J. Garcia, G.P. Steven, Fixed grid finite elements in elasticity problems, Engineering Computations 16 (2) (1999) 145–164. [22] H. Kim, M.J. Garcia, O.M. Querin, G.P. Steven, Y.M. Xie, Introduction of fixed grid in evolutionary structural optimization, Engineering Computations 17 (4) (2000) 427–439. [23] M.J. Garcia, C.A. Gonzalez, Shape optimization of continuum structures via evolution strategies and fixed grid finite element analysis, Structural and Multidisciplinary Optimization 26 (2004) 92–98. [24] Y. Liu, F. Jin, Q. Li, S. Zhou, A fixed-grid bidirectional evolutionary structural optimization method and its applications in tunneling engineering, International Journal for Numerical Methods in Engineering 73 (2008) 1788–1810. [25] Y. Liu, J. Feng, L. Qing, A strength-based multiple cutout optimization in composite plates using fixed grid finite element method, Composite Structures 73 (2006) 403–412.
237
[26] K. Suzuki, H. Ohtsubo, K. Terada, The analysis of 3D solid using multi-scale voxel data, in: Proceedings of the IV World Congress on Computational Mechanics, Buenos Aires, Argentina, 1998. [27] M.J. Garcia, O.E. Ruiz, L.M. Ruiz, O.M. Querin, Fixed grid finite element analysis for 3D linear elastic structures, Computational Mechanics (WCCM VI), China, 2004. [28] M.J. Garcia, M. Henao, O.E. Ruiz, Fixed grid finite element analysis for 3D structural problems, International Journal of Computational Methods 2 (4) (2005) 569–585. [29] F.S. Maan, O.M. Querin, D.C. Barton, Extension of the fixed grid finite element method to eigenvalue problems, Advances in Engineering Software 38 (2007) 607–617. [30] M.J. Garcı´a, G.P. Steven, Fixed grid finite element analysis in structural design and optimisation, in: Second ISSMO/AIAA Internet Conference on Approximations and Fast Reanalysis in Engineering Optimization, May 25–June 2, 2000. [31] S.Y. Woon, L. Tong, O.M. Querin, G.P. Steven, Optimising topologies through a multi-GA System, in: The Fifth World Congress of Structural and Multidisciplinary Optimization (WCSMO 5), Lido di Jesolo, Italy, 2003, pp. 229–230. [32] C. Cui, H. Ohmori, M. Sasaki, Computational morphogenesis of 3D structures by extended ESO method, Journal of IASS 44 (141) (2003) 51–61. [33] M.J. De Ruiter, F. Van Keulen, Topology optimization using a topology description function, Structural and Multidisciplinary Optimization 26 (2004) 406–416. [34] M.H. Hsu, Y.L. Hsu, Interpreting three-dimensional structural topology optimization results, Computer and Structures 83 (2005) 327–337. [35] E. Keppel, Approximating complex surfaces by triangulation of contour lines, IBM Journal of Research and Development 19 (1975) 2–11. [36] G.T. Herman, J.K. Udupa, Display of 3D digital images: computational foundations and medical applications, IEEE Computer Graphics and Application 3 (5) (1983) 39–46. [37] E.J. Farrell, Color display and interactive interpretation of threedimensional data, IBM Journal of Research and Development 27 (4) (1983) 356–366. [38] H.W. Shen, C.R. Jhonson, Seeping simplifies: a fast isosurface extraction algorithm for unstructured grids, in: Proceedings of Visualization, 1995, pp. 143–151. [39] W.E. Lorensen, H.E. Cline, Marching Cubes: a high resolution 3D surface construction algorithm, Computer Graphics 21 (4) (1987) 163–169. [40] B.N. Delaunay, Sur la sphe re vide. Izvestia Akademii Nauk SSSR, Otdelenie Matematicheskikh i Estestvennykh Nauk 7 (1934) 793–800. [41] O.C. Zienkiewicz, R.L. Taylor, J.Z. Zhu, The Finite Element Method, Its Basis & Fundamentals, Elsevier Butterworth Heinemann, 2005. [42] Q. Li, G.P. Steven, Y.M. Xie, On equivalence between stress criterion and stiffness criterion in evolutionary structural optimization, Structural Optimization 18 (1999) 67–73.