Incorporation of Topology Optimization Capability in MSC/NASTRAN GaoWen YE Manager, Technical Department MSC Japan, Ltd. 2-39, Akasaka 5-Chome, Minato-ku, Tokyo 07-0052, Japan Keizo ISHII President & CEO Quint Corporation -4- Fuchu-cho, Fuchu, Tokyo 83, Japan ABSTRACT In recent years, CAE based optimization applications have gained a wide acceptance in various structure or component design. In MSC/NASTRAN, various design sensitivity and optimization capabilities, such as sizing and shape optimization, have been introduced and continually enhanced. In order to provide all MSC/NASTRAN users with a more complete spectrum of design optimization analysis capabilities, including topology or layout optimization capability used in the very beginning of design process, the topology optimization optimizer function of OPTISHAPE, a topology optimization program developed by Quint Corporation, has been successfully incorporated into MSC/NASTRAN by the joint efforts of MSC Japan, Ltd. and Quint Corporation. This paper describes the new topology optimization capability in MSC/NASTRAN, and includes several application examples.
Introduction Design optimization is used to produce a design that possesses some optimal characteristics, such as minimum weight, maximum first natural frequency, or minimum noise levels. Design optimization is available in MSC/NASTRAN SOL 200, in which a structure can be optimized considering simultaneous static, normal modes, buckling, transient response, frequency response, aeroelastic, and flutter analyses. In SOL 200, both sizing parameters (the dimension of cross-section of beam elements like the height and width, or the thickness of shell elements) and shape (grid coordinates related) parameters can be used as design variables. However, these optimization capabilities are basically only usable to improve the design of structures or parts in the detailed design process. In the very beginning of the conceptual design process, another category of optimization capability, so-called topology or layout optimization is necessary. In order to provide all MSC/NASTRAN users with a more complete spectrum of design optimization analysis capabilities, the topology optimization optimizer function of OPTISHAPE, the first commercial code and the most famous topology optimization program developed by Quint Corporation, has been successfully incorporated into MSC/NASTRAN, and will be called MSC/NASTRAN-OPTISHAPE. The joint development of MSC/NASTRAN-OPTISHAPE by MSC Japan, Ltd. and Quint Corporation started in the Summer of 998 and is now commercially available through MSC Japan, Ltd. describe MSC/NASTRAN-OPTISHAPE, and will include various application examples. This paper will 2 Topology Optimization Based on Homogenization Method 2. Basic Concept MSC/NASTRAN-OPTISHAPE is based on a structural topology optimization approach, using the homogenization method which was introduced by Bendsoe and Kikuchi [] in 988 as the theory of optimal design of material distribution in the design domain. This theory has been extended and applied to various kinds of problems such as static [2] and normal modes [3]. In topology optimization of elastic structures based on this theory, the design domain is assumed to be composed of infinitely periodic microstructures. In the case of two-dimensional problems, each micro-structure has a rectangular hole as shown in Fig.. In the optimization process, the hole sides and angle of rotation, a, b and θ, respectively, are taken as design variables, which are to be determined by minimizing/maximizing the objective function subject to volume constraint and boundary conditions as mentioned below. Since 2
each element hole is allowed to possess a different size and angle of rotation, uniformly distributed porous material in the initial stage will have a different size of element holes at the end of optimization as shown in Fig.. Therefore, if the domain is viewed in a global sense, an obviously different resultant topology can be obtained at the end of design process. x2 Design domain Ω y2 a θ b y x Figure Concept of Topology Optimization of Using the Homogenization Method 2.2 The Homogenization Method In the topology optimization approach based on finite element analysis, material properties of porous material with various hole sizes are needed for both structural analysis and sensitivity analysis. In MSC/NASTRAN-OPTISHAPE, the homogenized material constants of porous material is calculated by the homogenization method. The homogenization method is promising in that this method gives homogenized material constants of the composite material without any empirical assumptions, if the material properties of all the constitutive materials are known. In this method it is usually assumed that the composite material is locally formed by very small, periodical microstructures compared with the overall macroscopic dimensions of the structure of interest. In such cases, the material properties are periodical functions of microscopic variables when the period of the microstructures is very small compared with the macroscopic variables. In the structural topology optimization using the homogenization method, the periodical microstructure of porous material is usually called a unit cell. In MSC/NASTRAN-OPTISHAPE, three kinds of unit cells are provided corresponding to both two-dimensional problems and three-dimensional problems as shown in Fig. 2. 3
2D-Shell Composite Shell 3D-SOLID Figure 2 Unit Cells Used in MSC/NASTRAN-OPTISHAPE 2.3 Optimization Problem Statements In MSC/NASTRAN-OPTISHAPE, the optimal topology of a structure with the highest stiffness or the highest/desired eigenvalues is calculated by changing the hole sizes as expressed by the following optimization problem statements. 2.3. Static Problem Minimize the mean compliance: Φ = 2 Ω Minimize Φ T H ε D ε dω s. t. Ω ρ dω V C 2.3.2 Eigenvalue Problem Case Maximize the mean eigenfrequencies: Λ = m i= w Maximize Λ i m i= w i λ i s. t. Ω ρ dω V C 2.3.3 Eigenvalue Problem Case 2 Minimize the distance between the desired eigenfrequencies and the calculated ones: Λ = m i= Minimize Λ s. t. m 2 2 2 {( λ λ ) λ } ( λ ) i 0i Ω 0i ρ dω V i= C 0i 4
3 Basic Design Notes In the first phase of this development, the most commonly used features of OPTISHAPE, both static and normal modes topology optimization analyses for either 2D or 3D problems, are introduced into MSC/NASTRAN. In 2D problem analysis, the composite shell design capability of OPTISHAPE is also implemented, in which only the 2 outer plies of 3-ply composite are going to be designed with the base middle ply remaining the same. In order to develop an efficient and fully integrated solution sequence for the topology optimization analysis, the optimizer function of OPTISHAPE is extracted and rewritten as the topology optimization optimizer function modules of MSC/NASTRAN, and a new solution sequence, TOPOPT, is developed by modifying the DMAP of both SOL and SOL 3 and adding some special new modules necessary for the purpose of topology optimization and efficiency. Figure 3 shows the conceptual flowchart of the TOPOPT, where TOPOPT is the solution sequence name, and TOP, TOP2, TOPDTI, TOPSDR, TOPMPT, TOPEID and TOPDEN are newly created function modules. Either static or normal modes topology optimization can be performed by this sequence. MSC/NASTRAN TOPOPT Initial data pre-processing and datablocks preparation TOPDTI, TOPEID, TOP Prepare special topology optimization datablocks & modify initial material table TOPMPT, EMG, EMA,,SSGi, SDRi, ELDFDR Form stiffness & mass matrices and solve equations for disp., stress,... TOPSDR, TOP2, TOPDEN Sensitivity calculation & update the material table tabletable Yes Converged / Max design cycles cyc Exit Figure 3 Conceptual Flowchart of TOPOPT 5
4 Additional Data Input Description For the purpose of performing static or normal modes topology optimization, a few additional parameters or data, such as constraint volume, move limit, design domain, etc., are necessary in addition to the common MSC/NASTRAN static or normal modes analysis bulk data. In this version, these additional data are provided by using MSC/NASTRAN DTI (Direct Table Input) entry as described below. This entry can be inserted into MSC/NASTRAN bulk data file manually with some text editor like unix vi, or within MSC/PATRAN by using MSC/NASTRAN-OPTISHAPE preference. In most static topology optimization cases, only 4 (if without composite material design) or 6 (if with composite material design) additional data lines are sufficient if one has the associated static analysis data deck. Please also refer to application example below to have a better understanding. Table DTI Input of TOPOL Data Block DTI Format $ 0 record, control scalar variables 2 3 4 5 6 7 8 9 0 DTI TOPOL 0 TBFLG KANALY IOPT ITERO ITERX CVOL XCI OPTCOV OCYCLE KOBJ MULTIE $ record, design elements by property or element list DTI TOPOL POEFLG POEIDi POEIDj POEIDk ENDREC $ 2 record, shell/plate base layer thickness (k=2 if TBFLG=) DTI TOPOL k TBi TBj TBk ENDREC $ 3 record, normal modes topology optimization inputs (k=2 if TBFLG=0, k=3 otherwise) DTI TOPOL k MODEi EIGRi WGTi MODEj EIGRj WGTj MODEk EIGRk WGTk ENDREC 6
Field Contents TBFLG = if TBi are present. = 0 if TBi are not present. KANALY Kind of analysis. = : static topology optimization. = 2 : eigenvalue topology optimization. IOPT Kind of elements to be optimized. = 2 : shell elements ( CTRIA3, CQUAD4 ). = 3 : solid elements ( CTETRA, CPENTA, CHEXA ). ITER0 Start cycle ( = in this version) ITERX Maximum allowable number of design cycles to be performed. CVOL Constraint volume ( 0.0 < cvol < 0.99, default = 0.5 ). XCI Maximum move limit imposed ( 0.0 < xci < 0.5, default = 0.3). OPTCOV Relative criterion to detect convergence (default = 0.00). OCYCLE Output element volume density at every n-th cycle. (default = -, only output the element volume density at the last cycle.) KOBJ MULTIE POEFLG POEIDi TBi If OCYCLE>0, then the element volume density will be output at first cycle; at every design cycle that is a multiple of OCYCLE; and the last design cycle. Kind of objective function for normal modes topology optimization. =, maximize mean eigenfrequencies. = 2, minimize distances between input eigenfrequencies and computed ones. Total number of eigenfrequencies to be considered. poeflg=0, specify design elements with element ID list. poeflg=, specify design elements with property ID list. Element or property list of design elements. Base layer thickness if TBFLG=. If provided, the same number of TBi as POEIDi must be given. MODEi Mode number to be considered. EIGRi Desired frequencies (not necessary in case of KOBJ= ). WGTi Weighting factors (default =.0). 5 MSC/PATRAN MSC/NASTRAN-OPTISHAPE Preference As described above, in addition to the common MSC/NASTRAN data deck, only a few data lines should be modified and added for the topology optimization. In order to do this with MSC/PATRAN, a special 7
MSC/PATRAN preference, MSC/NASTRAN-OPTISHAPE preference, is developed by adding some topology optimization specific data pre and post processing part to the existing MSC/PATRAN MSC/NASTRAN preference. The major functions of the preference are as follows: ) Define MSC/NASTRAN-OPTISHAPE analysis parameters, specify design domain for optimization and generate MSC/NASTRAN-OPTISHAPE specific data based on an original MSC/NASTRAN analysis job and analysis bulk data deck. 2) Read existing OPTISHAPE analysis deck into database. 3) Read MSC/NASTRAN-OPTISHAPE specific results (element volume density) into database. 4) Postprocess MSC/NASTRAN-OPTISHAPE specific results (element volume density). To efficiently define the design domain (i.e. to define the elements that are to be optimized), the preference will use property set names to indicate design elements indirectly, instead of specifying design elements directly. Figure 4 shows several main menus of the newly developed MSC/NASTRAN-OPTISHAPE preference. Figure 4 Several Main Menus of MSC/NASTRAN-OPTISHAPE Preference 8
6 Application Examples In order to highlight and illustrate the topology optimization features of MSC/NASTRAN-OPTISHAPE, several examples are provided here. Although not all capabilities are demonstrated with these examples, these examples cover some important features that are used. 6. A 2D Plate under 3 Concentrated Forces The first example is a very simple 2D plate under 3 concentrated loading cases as shown in Fig. 5. constraint volume is 0.3, which means 70% of the material in the design domain is going to be removed. The design domain is modeled with 800 CQUAD4 elements, and the 3 concentrated loading forces are applied to the model with 3 subcases. Table 2 shows part of input bulk data deck. The By looking at the bulk data deck, one can see that there are 3 changes/modifications besides the common static analysis data as follows: ) assign userfile='s2dl3.inp',unit=3,form=formatted,status=unknown to assign a file for element volume density ratios output; 2) sol topopt to select topology optimization sequence; 3) dti topol 0 2 30.3 0.5.00 5 dti topol endrec to provide the topology optimization parameters, define the design domain. etc.. L=400.0 W=200.0 Y X t=.0 Design Domain E=2.90E+7 v=0.32 Case Case2 Case3 Nodal Force Figure 5 A 2D Plate under 3 Concentrated Loading Forces 9
Table 2 Input Bulk Data ID TEST2D, S2D800 assign userfile='s2dl3.inp',unit=3,form=formatted,status=unknown sol topopt CEND SPC = SUBCASE LOAD = SUBCASE 2 LOAD = 2 SUBCASE 3 LOAD = 3 $ BEGIN BULK PARAM POST - PARAM AUTOSPC YES $ADDITIONAL BULK DATA FOR STATIC TOPOLOGY OPTIMIZATION dti topol 0 2 30.3 0.5.00 5 dti topol endrec $ PSHELL. CQUAD4 2 43 42 all other data to define element connection, grid points, boundary and loading forces etc. FORCE 3 3 0. 0. -. 0. ENDDATA Figure 6 shows the resultant topology obtained from MSC/NASTRAN-OPTISHAPE. In this figure, these elements with element volume density ratios smaller than 0.3 are hidden. Figure 6 Resultant Topology of a 2D Plate under 3 Loading Cases 0
6.2 Partial Domain Design of an I Beam Structure The second example is used to illustrate the features when ) only partial structure design is necessary; 2) the design domain can be composed of composite shell with a middle base ply which remains unchanged, which is usually used in the design of a shell structure with distributing reinforcement. The volume constraint is 0.3. Figure 7 shows the model, loading and boundary conditions, and final resultant topology. L=20.0 No Design No Design Nodal force H=5.0 Design Domain E=2000.0 v =0.3 W=4.0 No Design t =.0 t0=0.2 t t0 No Design Figure 7 Partial Domain Design of an I Beam Structure 6.3 A 3D Universal Joint Design The third example is a 3D static topology optimization example for a universal joint. Only a half model is used here due to symmetry condition. In this analysis, 9932 CHEXA and RBE2 elements are used. Figures 8-0 show the analysis model, the design domain and the resultant topology, respectively. Force Rigid Element Force Rigid Element Load case Force2 Load case2 Figure 8 Analysis Model
Non-design domain Design domain Figure 9 Finite Element Mesh and Design Domain Figure 0 Resultant Topology of the Universal Joint 6.4 Normal Modes Topology Optimization of a Copy Machine Stand The forth example is a normal mode topology optimization applied to a copy machine stand in order to find the optimal distributing reinforcement. In this analysis, CONM2, RBE2 and 6852 CQUAD4 elements are used with 2060 CQUAD4 element being designed. The objective function is to maximize the first 6 eigenvalues subject to a volume constraint of 50%, which means 50% of the material in the design domain is to be removed. Figure shows the analysis model. Figure 2 shows both the resultant topology of the copy machine stand and its changing history of the first 6 eigenvalues. Please note that the eigenvalues obtained after the first design cycle are the values of the structure with 50% of the material removed uniformly from all design elements. 2
Figure Normal Modes Topology Optimization Model of a Copy Machine Stand 30 20 0 0 0 20 30 Figure 2 Resultant Topology and History of First 6 Eigenvalues 7 Concluding Remarks In this development, by combining the topology optimization optimizer function of OPTISHAPE into MSC/NASTRAN as functional modules, both static and normal modes topology optimization analysis capabilities are introduced into MSC/NASTRAN as a special solution sequence. The principal features of the new capabilities are as follows: 3
a) Static topology optimization : To minimize mean compliance subject to volume constraint. b) Normal modes topology optimization : To maximize eigenfrequencies in appointed order or minimize the difference between given eigenfrequencies and the calculated ones subject to volume constraint. c) Shell or solid design elements : Either first-order shell or solid elements can be used as design elements. All other elements that can be used in linear static or normal modes analysis can also be used to model the non-design part of the optimization analysis model d)msc/patran integration : A special MSC/PATRAN preference, MSC/NASTRAN OPTISHAPE preference, has been developed. With this preference, all pre and post processing for both static and normal modes topology optimization analyses can be carried out within MSC/PATRAN. e) MSC/NASTRAN bulk data format: All input bulk data are provided in MSC/NASTRAN bulk data format compared with the common MSC/NASTRAN static or normal modes analysis jobs, only a few additional data lines are sufficient to perform the associated static or normal modes topology analysis. These additional data lines can be added by MSC/PATRAN. It is also possible and quite easy to manually insert these additional lines once the common MSC/NASTRAN static or normal modes bulk data are generated by any other preprocessors. f) Efficient solution of very large models: By using MSC/NASTRAN advanced elements and efficient solver engines for static or normal modes analysis, very efficient solution to very large scale topology optimization models can be realized. Acknowledgements The authors would like to thank Mr. Gopal K. Nagendra of MSC India, Mr. Qiwen Liu of DongFeng Motor Corporation, and Mr. JiDong Yang of MSC Japan, Ltd. for their great help in the development of MSC/NASTRAN-OPTISHAPE. References [] Bendsoe, M.P.,Kikuchi,N. : Comput. Methods Appl. Mech. Eng., 7 (988), 97. [2] Suzuki,K., Kikuchi,N. : Comput. Methods Appl. Mech. Eng., 93 (99), 29. [3] Diaz, A.R., Kikuchi,N. : Int. J. Numer. Methods Eng., 35 (992), 487. 4