S10-1NAS105, Section 10, May 2005 SECTION 10 OPTIMIZATION

S10-2NAS105, Section 10, May 2005

S10-3NAS105, Section 10, May 2005 TABLE OF CONTENTS SectionPage WHAT IS DESIGN OPTIMIZATION?………………………………………………………….10-7 BASIC FEATURE IMPLEMENTED IN MSC.NASTRAN…………………………………… STRENGTHES OF MSC.NASTRAN STRUCTURE OPTIMIATION……………………… EXTENT OF MSC.NASTRAN OPTIMIZATION CAPABLITIES…………………………… CONCEPT OF THE DESIGN MODEL………………………………………………………… DESIGN MODEL DEFINITION PROCEDURE……………………………………………… CHALLENGES IN DESIGN MODELING……………………………………………………… SOME STRUCTURAL RESPONSES………………………………………………………… RESPONSES FOR COMPOSITE MATERIALS……………………………………………… RESPONSE QUANTITIES – CONSTRAINTS/OBJECTIVE FUNCTION………………… OBJECTIVE AND CONSTRAINTS…………………………………………………………… DESIGN MODELING INPUT DATA…………………………………………………………… SAMPLE PROBLEM…………………………………………………………………………… DESVAR……………………………………………………………………………………………10-24 DVPREL1………………………………………………………………………………………….10-25

S10-4NAS105, Section 10, May 2005 TABLE OF CONTENTS SectionPage DRESP1……………………………………………………………………………………………10-28 DESOBJ……………………………………………………………………………………………10-39 DCONSTR…………………………………………………………………………………………10-40 DESGLB……………………………………………………………………………………………10-44 ANALYSIS…………………………………………………………………………………………10-45 DESSUB………………………………………………………………………………………… SAMPLE OPT1 – CONSTANT CROSS -SECTION CANTILEVER BEAM…………………10-47 SAMPLE OPT2 – VARIABLE CROSS - SECTION CANTILEVER BEAM………………… WORKSHOPS…………………………………………………………………………………… AVANCED FEATURES OF OPTIMIZATION IN MSC.NASTRAN………………………… TYPE-2 STRUCTURAL RESPONSES…………………………………………………………10-78 TYPE-2 RESPONSE FORMULATION…………………………………………………………10-79 RESTRICTIONS IN FORMING SYNTHETIC RESPONSES……………………………… DEQATN………………………………………………………………………………………… DTABLE……………………………………………………………………………………………10-86 DRESP2……………………………………………………………………………………………10-87

S10-5NAS105, Section 10, May 2005 TABLE OF CONTENTS SectionPage DRESP2 BULK DATA ENTRY………………………………………………………………… DOPTPRM…………………………………………………………………………………………10-95 DYNAMIC RESPONSE OPTIMIZATION……………………………………………………… SAMPLE PROBLEM XXX – MINIMIZATION OF DRIVER RESPONSE TO A ROTATING IMBALANCE FOR THE CAR MODEL……………………………… INPUT FOR OPTIMIZATION…………………………………………………………………… SHAPE OPTIMIZATION………………………………………………………………………… HOW DO I CREATE SHAPE DESIGN VARIABLES?……………………………………… MANUAL GRID VARIATION…………………………………………………………………… INPUT FILE STRUCTURE: MANUAL GRID VARIATION………………………………… INPUT FOR SHAPE OPTIMIZATION………………………………………………………… OPTIMIZATION INPUT………………………………………………………………………… SHAPE-OPTIMIZATION RESULTS…………………………………………………………… OPTMIZATION SUMMARY……………………………………………………………………

S10-6NAS105, Section 10, May 2005

S10-7NAS105, Section 10, May 2005 WHAT IS DESIGN OPTIMIZATION? Automated modifications of the analysis model parameters to achieve a desired objective while satisfying specified requirements WHAT ARE THE POSSIBLE APPLICATIONS? n Structural design improvements (optimization) n Generation of feasible designs from infeasible designs n Model matching to produce similar structural responses n System parameter identification n Configuration evaluations n Others(depends on designers creativity)

S10-8NAS105, Section 10, May 2005

S10-9NAS105, Section 10, May 2005 BASIC FEATURE IMPLEMENTED IN MSC.NASTRAN n Easy access to design synthesis capabilities u Concept of design model n Flexibility for design model representation u User-supplied equation interpretation capability n Efficient solution for problems of any size u Number of finite element analysis as the measure of efficiency

S10-10NAS105, Section 10, May 2005 STRENGTHES OF MSC.NASTRAN STRUCTURE OPTIMIATION n Efficient performance for small- to large-scale problems n Reliable convergence characteristics n Flexible user interface and user-defined equations n Full implementation of approximation concepts n Continuous enhancements n Results dependent on the proven reliability of MSC.NASTRAN analysis n Commercial level support as a part of MSC.NASTRAN n Access to the familiar analysis tools in MSC.NASTRAN

S10-11NAS105, Section 10, May 2005 EXTENT OF MSC.NASTRAN OPTIMIZATION CAPABLITIES Design Sensitivity and Optimization n Each SUBCASE may reference a different solution type by using the ANALYSIS command u For example, for one static SUBCASE and one normal modes SUBCASE in SOL 200: SUBCASE 1 ANALYSIS = STATIC LOAD = 100 DISP = all SUBCASE 2 ANALYSIS = MODES METHOD = 1 ESE = ALL * Including Acoustics

S10-12NAS105, Section 10, May 2005 CONCEPT OF THE DESIGN MODEL Given an initial design Find X, that will minimize a scalar function F(X) while satisfying the following: Hardware or Drawings Finite Element Analysis Model Predicted Structural Responses Design Model 1. Design Variables 2. Objective Function 3. Constraints

S10-13NAS105, Section 10, May 2005 CONCEPT OF THE DESIGN MODEL (Cont) n Design constraints: u Deformations within limits u Local web buckling criteria n Structural responses u Displacement at grids u Stress computed at grids Design Model Design Variables Analysis Model Parameters Analysis Model For example:

S10-14NAS105, Section 10, May 2005 DESIGN MODEL DEFINITION PROCEDURE 1. Select design variables. – DESVAR entry 2. Describe relations between design variables and analysis model parameters. – DVPREL1 and DVPREL2 entries 3. Define response to be used as an objective function. – DRESP1 and DRESP2 entries 4. Define responses which are to be constrained. – DRESP1 and DRESP2 entries 5. Specify bounds on constrained responses. – DCONSTR entry 6. Select design process control parameters, if necessary (I.e., DSCREEN, DOPTPRM)

S10-15NAS105, Section 10, May 2005 CHALLENGES IN DESIGN MODELING n How to describe the relations between the design variables and the analysis model parameters? n How to describe the relations between the structural responses computed by MSC.NASTRAN and the design objective and constraints? n How to handle large-scale problems effectively? u Several hundred design variables u Thousands of constraints The complexities required for the design model description are far more demanding than for the analysis model description. Therefore, no fixed data structure will be sufficiently general.

S10-16NAS105, Section 10, May 2005 SOME STRUCTURAL RESPONSES n Displacement u Specific displacement degree of freedom, identified by the grid ID and component ID n The stress/strain/internal force u The stress/strain/force component (described in Appendix A of the MSC.NASTRAN Quick Reference Guide) is identified by the component ID, and either the element ID or the property ID. u If identified by a property ID, all elements that make reference to that property are covered. n Buckling load factor u Always associated with the first static subcase, designated by the buckling mode ID (lowest mode = 1) n Natural vibration eigenvalue or frequency u Eigenvalue or frequency, identified by the mode ID (lowest mode = 1) n Weight, Volume

S10-17NAS105, Section 10, May 2005 RESPONSES FOR COMPOSITE MATERIALS n CSTRESS and CSTRAIN u These responses are the same as STRESSES or STRAIN, but are imposed on specific ply laminas. The ATTB field of the DRESP1 entry is used to identify the specific laminas. n CFAILURE u One of the four failure theories selected by the PCOMP entry is applied. The response is normalized so that if the response value is less than 1.0, the laminas do not fail.

S10-18NAS105, Section 10, May 2005 RESPONSE QUANTITIES FOR CONSTRAINTS/OBJECTIVE FUNCTION n Weight n Volume n Eigenvalues or natural frequencies n Buckling load factors n Displacements, Velocities, Accelerations n Stresses n Strains n Forces n Failure indices for composites n Lamina stresses for composites n Lamina strains for composites n Trim n Stability Derivatives n Flutter n Analytic equations supplied by the user

S10-19NAS105, Section 10, May 2005 OBJECTIVE AND CONSTRAINTS n Constraints u Select the responses. u Provide the lower and upper bounds using DCONSTR entries. (Note: If possible, do not use 0.0 for the lower or upper bounds.) u Select constraint sets in Case Control using DESGLB for global constraints and/or DESSUB for subcase-dependent constraints. n Objective Using the DESOBJ Case Control Command: u Select a response. u Choose whether this response is to be minimized or maximized (Default: Minimize).

S10-20NAS105, Section 10, May 2005 DESIGN MODELING INPUT DATA n Design variables u DESVARDesign variable definition* u DLINKDefinition of dependent design variable n Relation between design variables and analysis model parameters u DVPREL1Linear relations* u DVPREL2Nonlinear relations n Definition of structural responses u DRESP1Responses computed directly by analysis* u DRESP2Synthesized responses for design problems _____________________________________________________ *Presented in this section.

S10-21NAS105, Section 10, May 2005 DESIGN MODELING INPUT DATA (Cont.) n Definition of objective and constraint functions u DESOBJ Case Control CommandObjective function definition* u DCONSTRConstraint functions* u DCONADDConstraint Set Combinations u DESSUB Case Control CommandSelection of Subcase-Dependent Constraints* u DESGLB Case Control CommandSelection of Global Constraints* n Optimization control parameters and constants u DSCREENMeasures of constraint screening u DOPTPRMOptimization process control u DTABLEConstants n User equation input u DEQATNUser-defined equation _____________________________________________________ *Presented in this section.

S10-22NAS105, Section 10, May 2005 SAMPLE PROBLEM n For an example, let us look at a cantilever beam with a tip load u Properties: L=100. P=10. Initial dimensions: 2 x 2 square BAR E = 10,000,000. =.3 =.1 WTMASS = n We wish to minimize the weight, while keeping the tip deflection less than.01 (disp for initial design =.0333) and stress less than Let us use a PBARL to define the cross section Let us try two different approaches: u Constant cross-section u we will allow the depth of the section to vary (In order to do this properly, we will use a different PBARL for each element – not recommended for large models, but it makes for a nice sample problem).

S10-23NAS105, Section 10, May 2005 SAMPLE PROBLEM (Cont.) n Basic Model with one PBARL – file cantbeam.dat n Let us now look at the entries need for the optimizer PARAM GRDPNT 0 PARAM WTMASS PARAM POST 0 PARAM AUTOSPC YES $ GRID GRID GRID GRID GRID GRID GRID GRID GRID $ CBAR CBAR CBAR CBAR CBAR CBAR CBAR CBAR $ SPC $ PBARL,1,1,,BAR,2.,2. $ MAT $

S10-24NAS105, Section 10, May 2005 DESVAR Defines a design variable for design optimization Format: Example: FieldContents IDUnique design variable identification number. (Integer > 0). LABELUser-supplied name for printing purposes. (Character). XINITInitial value. (Real, XLB < XINIT < XUB). XLBLower bound. (Real, default = -1.0E+20). XUBUpper bound. (Real, default = +1.0E+20). DELXVFractional change allowed for the design variable during approximate optimization. (Real > 0.0, for default see Remark 2). Remarks: 1. DELXV can be used to control the change in the design variable during one optimization cycle. 2. If DELXV is blank, the default is taken from the specification of the DELX parameter on the DOPTPRM entry. If DELX is not specified then the default is 1.0.

S10-25NAS105, Section 10, May 2005 DVPREL1 Defines the relation between an analysis model property and design variables. Format: Example: FieldContents IDUnique identification number. (Integer > 0). TYPEName of a property entry, such as PBAR, PBEAM, etc. (Character). PIDProperty entry identification number. (Integer > 0). FIDField position of the property entry, or word position in the element property table of the analysis model. (Integer 0). PMINMinimum value allowed for this property. If FID references a stress recovery location, then the default value for PMIN is – PMIN must be explicitly set to a negative number for properties that may be less than zero (for example, field Z0 on the PCOMP entry). (Real; default = 0.001). PMAXMaximum value allowed for this property. (Real; Default = 1.0E20). COConstant term of relation. (Real; Default = 0.0) DVIDiDESVAR entry identification number. (Integer > 0). COEFiCoefficient of linear relation. (Real).

S10-26NAS105, Section 10, May 2005 DVPREL1 (Cont.) Remarks: 1. The relationship between the analysis model property and design variables is given by: 2. The continuation entry is required. 3. PTYPE = PBEND is not supported, either directly through FIDs or indirectly via word positions in the element property table. 4. FID may be either a positive or a negative number. If FID > 0, it identifies the field position on a property entry. If FID < 0, it identifies the word position of an entry in the element property table. For example, to specify the area of a PBAR, either FID = +4 or FID = -3 can be used. However, if PTYPE = PBEAM, FID must be negative. See the following element property table for the word positions for PBEAM.

S10-27NAS105, Section 10, May 2005 DVPREL1 (Cont.) n An analysis model property A j is related to design variables as: n Dependent and Independent design variables are treated equally n FID is the field position in the entry if positive. For example, the second field of the third continuation entry has a field ID equal to 22. FID is the entry position in the element property table (see MSC.NASTRAN Programmers Manual), if negative. For example, the cross-sectional area of a PBAR entry may be designated by FID=-3. PBEAM must use negative FID.

S10-28NAS105, Section 10, May 2005 DRESP1 Defines a set of structural responses that is used in the design either as constraints or as an objective. Format: Example: FieldContents IDUnique entry identifier. (Integer > 0). LABELUser-defined label. (Character). RTYPEResponse type. See table below. (Character). PTYPEElement flag (PTYPE = ELEM) or property entry name. Used with element type responses (stress, strain, force, etc.) to identify the property type, since property entry IDs are not unique across property types. (Character: ELEM, PBAR, PSHELL, etc.). REGIONRegion identifier for constraint screening. See Remark 10 for defaults. (Integer > 0). ATTA, ATTB, ATTiResponse attributes. See Table 1. (Integer > 0 or blank). (Continued)

S10-29NAS105, Section 10, May 2005 DRESP1 (Cont.) Table 1. Design Sensitivity Response Attributes. (Continued)

S10-30NAS105, Section 10, May 2005 DRESP1 (Cont.) Table 1. Design Sensitivity Response Attributes. (Cont.)

S10-31NAS105, Section 10, May 2005 DRESP1 (Cont.) n Remarks: 1.Stress, strain, and force item codes can be found in Item Codes in Appendix A, in the MSC.Nastran Quick Reference Guide. For stress or strain item codes that have dual meanings, such as von Mises or maximum shear, the option specified in the Case Control Section will be used; i.e., STRESS(VONM) or STRESS(MAXS). (Continued)

S10-32NAS105, Section 10, May 2005 DRESP1 (Cont.) 2. RTYPE = CSTRESS, CSTRAIN, and CFAILURE are used only with the PCOMP entry. CSTRESS, CSTRAIN and CFAILURE are described in Appendix A in the MSC.Nastran Quick Reference Guide. Only force item codes that refer to failure indices of direct stress and interlaminar shear stress are valid. The CFAILURE response type requires the following specifications on the applicable entries: a. Failure theory in the FT field on PCOMP entry b. Allowable bonding shear stress in the SB field on PCOMP entry c. Stress limits in the ST, SC, and SS fields on the MATi entries 3. ATTB is used only for responses of composite laminae, dynamics, complex eigenvalue, and stability derivatives. For other responses, this field must be blank. 4. All grids associated with a DRESP1 entry are considered to be in the same region for screening purposes. Only up to NSTR displacement constraints (see DSCREEN entry) per group per load case will be retained in the design optimization phase. 5.DRESP1 identification numbers must be unique with respect to DRESP2 identification numbers. 6. If PTYPE = ELEM, the ATTi correspond to element identification numbers. 7. If RTYPE = DISP, TDISP, TVELO, TACCL or TSPC, multiple component numbers (any unique combination of the digits 1 through 6 with no embedded blanks) may be specified on a single entry. Multiple response components may not be used on any other response types. (Continued)

S10-33NAS105, Section 10, May 2005 DRESP1 (Cont.) 8. If RTYPE = FRDISP, FRVELO, FRACCL, or FRSPC only one component number may be specified in the ATTA field. Numbers 1 through 6 correspond to real (or magnitude) components and 7 through 12 imaginary (or phase) components. If more than one component for the same grid is desired, then a separate entry is required. 9.Real/imaginary representation is the default for complex response types. Magnitude/phase representation must be requested by the corresponding Case Control command; e.g., DISP(PHASE) = ALL. 10. REGION is used for constraint screening. The NSTR field on DSCREEN entries gives the maximum number of constraints retained for each region per load case. If RTYPE = WEIGHT, VOLUME, LAMA, EIGN or FREQ, no REGION identification number should be specified. For all other responses, if the REGION field is left blank, the default specified in Table 2 is used. Usually, the default value is appropriate. If the REGION field is not blank, all the responses on this entry as well as all responses on other DRESP1 entries that have the same RTYPE and REGION identification number will be grouped into the same region.

S10-34NAS105, Section 10, May REGION is valid only among the same type of responses. Responses of different types will never be grouped into the same region, even if they are assigned the same REGION identification number by the user. (Continued) DRESP1 (Cont.) Table 2. Default Regions for Design Sensitivity Response Types.

S10-35NAS105, Section 10, May 2005 DRESP1 (Cont.) 12. If RTYPE = WEIGHT or VOLUME, field ATTi = ALL implies total weight/volume of all superelements except external superelements. 13. RTYPE = STABDER identifies a stability derivative response. ATTB is the restraint flag for the stability derivative. ATTB = 0 means unrestrained and ATTB = 1 means restrained. For example, ATTA = 4000, ATTB = 0, and ATT1 = 3, references the unrestrained C z derivative for the AESTAT (or AESURF) entry ID = RTYPE = FLUTTER identifies a set of damping responses. The set is specified by ATTi: ATT1 = identification number of a SET1 entry that specifies a set of modes. ATT2 = identification number of an FLFACT entry that specifies a list of densities. ATT3 = identification number of an FLFACT entry that specifies a list of Mach numbers. ATT4 = Identification number of an FLFACT entry that specifies a list of velocities. If the flutter analysis is type PKNL, it is necessary to put PKNL in the PTYPE field of this entry. 15. For RTYPE = FRDISP, FRVELO, FRACCL, FRSPC, FRFORC, and FRSTRE, a real value for ATTB specifies a frequency value in cycles per unit time. If ATTB is specified, then the responses are evaluated at the closest frequency selected by the OFREQ command. The default for ATTB is all frequencies selected by the OFREQ command. See Remark 20 for additional ATTB options. 16. For RTYPE = TDISP, TVELO, TACCL, TSPC, TFORC, and TSTRE, ATTB specifies a time value. If ATTB is specified, then the responses are evaluated at the closest time selected by the OTIME command. The default for ATTB is all time steps selected by the OTIME command.

S10-36NAS105, Section 10, May 2005 DRESP1 (Cont.) 17. Intermediate station responses on CBAR elements due to PLOAD1 and / or CBARAO entries may not be defined on the DRESP1 entry. 18. RTYPE = EIGN refers to normal modes response in terms of eigenvalue (radian / time)**2 while RTYPE = FREQ refers to normal modes response in terms of natural frequency or units of cycles per unit time. 19. For RTYPE = LAMA, EIGN or FREQ, the response approximation used for optimization can be individually selected using the ATTB field. (Approximation Code = 1 = direct linearization, = 2 = Inverse Linearization). 20. Character input for ATTB is available for RTYPE of FRDISP, FRVELO, FRACCL, FRSPCF, FRSTRE, FRFORC, TDISP, TVELO, TACCL, TSPCF, TSTRE and TFORC. The character input represents a mathematical function and the options for character input are SUM, AVG, SSQ, RSS,MAX and MIN. The expression of mathematical function is shown as follows:

S10-37NAS105, Section 10, May 2005 DRESP1 (Cont.) MAX (X 1, X 2, …, X n ) = Largest value among x i (i=1 to n) MIN (X 1, X 2, …, X n ) = Smallest value among x i (i=1 to n) where X i is the response for a forcing frequency. For example DRESP1,10,DX1,FRSTRE,ELEM,,3,AVG,10 yields a response which is equal to the average stress for element 10. The average is done by first adding up the component 3 stress of element 10 for all forcing frequencies (all time steps for transient response), and then dividing by the number of forcing frequencies. Note that the response computed is considered as type 2 response. Therefore, if referenced on a DRESP2, the ID of such DRESP1 (ATTB with character input) must be listed following DRESP2 keyword. 21. Element strain energy item codes can be found under Table A-5 in Element Strain Energy Item Codes on page 1571 in Appendix A. Only element strain energy and element strain energy density can be referenced on a DRESP1 entry. 22. For RTYPE=CEIG, the allowable character inputs are ALPHA and OMEGA with ALPHA being the default. 23. For RTYPE=RMSDISP, RMSVELO, or RMSACCL the ATTB specifies the appropriate RANDPS ID.

S10-38NAS105, Section 10, May 2005 DRESP1 (Cont.) 24. Input other than 1 or 7 of ATTA field, acoustic pressure component, for PRES response type will be reset to 1 (if less than 7) or 7 (if greater than 7 and less than 13) 25. Design response weight is obtained from Grid Point Weight Generator for a reference point GRDPNT (see parameter GRDPNT). If GRDPNT is either not defined, equal to zero, or not a defined grid point, the reference point is taken as the origin of the basic coordinate system. Fields ATTA and ATTB refer to the row and column numbers of the rigid body weight matrix, which is partitioned as follows: 6x6 The default values of ATTA and ATTB are 3, which specifies weight in the Z direction. Field ATT1 = ALL implies total weight of all superelements except external superelements. SEIDi refers to a superelement identification number. SEIDi = 0 refers to the residual superelement. The default of ATT1 is blank which is equivalent to ALL.

S10-39NAS105, Section 10, May 2005 DESOBJ Selects the DRESP1 or DRESP2 entry to be used as the design objective. Format: Examples: DESOBJ = 10 DESO = 25 DescriberMeaning MINSpecifies that the objective is to be minimized. MAXSpecifies that the objective is to be maximized. n Set identification of a DRESP1 or DRESP2 Bulk Data entry (Integer > 0). Remarks: 1. A DESOBJ command is required for a design optimization task and is optional for a sensitivity task. 2. If the DESOBJ command is specified within a SUBCASE, the identified DRESP1 Bulk Data entry uses responses only from that subcase. If DESOBJ appears above all SUBCASE commands and there are multiple subcases, it uses global responses. 3. The referenced DRESP1 entry must define a scalar response.

S10-40NAS105, Section 10, May 2005 DCONSTR Defines design constraints Format: Example: Field Contents DCIDDesign constraint set identification number (Integer > 0) RID DRESP1 entry identification number (Integer > 0) LALLOWLower bound on the response quantity (Real, Default = -1E20) UALLOWUpper bound on the response quantity (Real, Default = 1E20) LOWFQLow end of frequency range in Hertz (Real >=0.0, Default=0.0). See Remark 8 HIGHFQHigh end of frequency range in Hertz (Real >=LOWFQ, Default=1.E20). Remarks: 1. The DCONSTR entry may be selected in Case Control by the DESSUB or SESGLB command 2. DCID may be referenced by the DCONNAD Bulk Data entry

S10-41NAS105, Section 10, May 2005 DCONSTR (Cont) 3. For a given DCID, the associated RID can be referenced only once 4. The units of LALLOW and UALLOW must be consistent with the referenced response defined on the DRESP1 entry. If a RID refers to an eigenvalue response, the the imposed bounds must be expressed in units of eigenvalue, (radian/time) 2 5. LALLOW and UALLOW are unrelated to the stress limits specified on the MATI entry 6. Constraints are computed as follows: g = (LALLOW – r) / GNORM for lower bound constraints g = (r – UALLOW) / GNORM for upper bound constraints where r is the response defined on the DRESPi entry and GSCAL is specified on the DOPTPRM entry (Default = 0.001). 7. As Remark 6 indicates, small values of UALLOW and LALLOW require special processing and should be avoided. Bounds of exactly zero are particularly troublesome. This can be avoided by using a DRESP2 entry that offsets the constrained response from zero.

S10-42NAS105, Section 10, May 2005 DCONSTR (Cont) 8. L OWFQ and HIGHFQ fields are functional only for RTYPE with FR prefix, e.g., FRDISP. The bounds provided in LALLOW and UALLOW are applicable to a response only when the value of forcing frequency of the response falls between the LOQFQ and HIGHFQ. If the ATTB field of the DRESP1 entry is not blank, LOQFQ and HIGHFQ are ignored.

S10-43NAS105, Section 10, May 2005 DCONSTR (Cont) n It is possible that many constraints are generated by one DECONSTR entry The user should be aware of this structure when describing the design model. n Equality constraints are not supported directly at this time. If necessary, provide equivalent lower and upper bounds.

S10-44NAS105, Section 10, May 2005 DESGLB Selects the design constraints to be applied at the global level in design optimization task. Format: DESGLB = n Example: DESGLB = 10 DESG = 25 DescriberMeaning n Set identification of a DCONSTR or DCONADD Bulk Data entry identification number (Integer > 0). Remarks: 1. A DESGLB command is optional and invokes constraints that are to be applied independent of a particular subcase. These constraints could be based on responses that are independent of subcases (e.g., WEIGHT or VOLUME). 2. The DESGLB command can be used to invoke constraints that are not a function of DRESP1 entries; e.g., DRESP2 responses that are not functions of DRESP1 responses are subcase independent.

S10-45NAS105, Section 10, May 2005 ANALYSIS Specifies the type of analysis being performed for the current subcase Format: ANALYSIS = type Examples: ANALYSIS = STATICS ANAL = MODES DescriberMeaning typeAnalysis type. Allowable values and application solution sequences (Character): STATICS Statics MODES Normal Modes BUCK Buckling DFREQDirect Frequency MFREQ Modal Frequency MTRAN Modal Transient SAERO Static Aeroelastic FLUTTER Flutter DCEIG Direct Complex Eigenvalue MCEIG Modal Complex Eigenvalue DIVERGE Static Aeroelastic Divergence HEAT Heat Transfer Analysis(SOLs 153 and 159 only) STRUCURE Structural Analysis (SOLs 153 and 159 only) Remarks: In SOL 200, all subcases must be assigned by an ANALYSIS command. Also, all subcases assigned by ANALYSIS=MODES must contain a DESSUB request. (SOL 200 only)

S10-46NAS105, Section 10, May 2005 DESSUB Select the design constraints to be used in design optimization task for the current subcase Format: DESSUB = n Examples: DESSUB = 10 DESS = 25 DescriberMeaning n Set Identification of DCONTR or DCONADD Bulk Data entry Identification number (Integer > 0) Remarks: 1. A DESSUB command is required for every subcase for which constraints are to be applied. 2. The DESSUB command in a given subcase is the default for all subsequent subcases. In order to override the default, specify a new DESSUB = n or DESSUB = 0 if no responses are desired.

S10-47NAS105, Section 10, May 2005 SAMPLE OPT1 – CONSTANT CROSS- SECTION CANTILEVER BEAM n Let us now create the optimization input. n First define the response quantities which will be input for the optimizer. u We need to measure the tip displacement, the stress at the root, and the weight of the beam. $ create design response entries $ $ total weight = objective will be to minimize this $ DRESP1,1,TOTW,WEIGHT $ $ tip displacement – we need to keep this less than.01 $ DRESP1,2000,tipdisp,DISP,,,2,,9 $ $ Stress at the root – select the element, rather than the property $ select the maximum stress at the start of the BAR $ DRESP1,3000,rootstr,STRESS,ELEM,,8,,1 $

S10-48NAS105, Section 10, May 2005 SAMPLE OPT1 – CONSTANT CROSS- SECTION CANTILEVER BEAM n Now let us define the design variables for the optimizer. n We want to vary the depth of the element, so we use DVPREL1 entries to tie the DESVAR entry to the fields on the PBARL. $ $ Create design variable $ $ allow height to vary from.5 to 8.0 $ DESVAR,2,Height,2.,.5,8.0 $ $ Relate Design Variables to the Fields on the PBARL $ DVPREL1,110,PBARL,1,13,.5,8.0,2,1.0 $

S10-49NAS105, Section 10, May 2005 SAMPLE OPT1 – CONSTANT CROSS- SECTION CANTILEVER BEAM n Now we need to define the design constraints (stress and displacement constraints). $ Define Design Constraints $ $ limit tip displacement $ DCONSTR,200,2000,–.01,.01 $ $ limit stress at root $ DCONSTR,200,3000,–1000.,1000. $

S10-50NAS105, Section 10, May 2005 SAMPLE OPT1 – CONSTANT CROSS- SECTION CANTILEVER BEAM n Now we need to supply the executive and case control SOL 200 TIME 5 CEND TITLE = Practical Dynamics Seminar Sample Problem 1 SUBTITLE = Cantilever Beam LABEL = Perform Model Checks DISP = ALL SUBCASE 1 SPC = 1 load = 100 ANALYSIS = STATICS DESOBJ = 1000 $ minimize weight DESSUB = 200 $ apply constraints in the optimizer $

S10-51NAS105, Section 10, May 2005 SAMPLE OPT1 – CONSTANT CROSS- SECTION CANTILEVER BEAM n The complete input file SOL 200 CEND TITLE = Practical Dynamics Seminar Sample Problem 1 SUBTITLE = Cantilever Beam LABEL = Perform Model Checks DISP = ALL SUBCASE 1 SPC = 1 load = 100 ANALYSIS = STATICS DESOBJ = 1000 $ minimize weight DESSUB = 200 $ apply constraints in the optimizer BEGIN BULK FORCE,100,9,,10.,0.,–1.,0. include cantbeam.dat DRESP1,1000,TOTW,WEIGHT DRESP1,2000,tipdisp,DISP,,,2,,9 DRESP1,3000,rootstr,STRESS,ELEM,,8,,1 DESVAR,2,Height,2.,.5,4.0 DVPREL1,110,PBARL,1,13,.5,8.0,2,1.0 DCONSTR,200,2000,–.01,.01 DCONSTR,200,3000,–1000.,1000. ENDDATA

S10-52NAS105, Section 10, May 2005 OUTPUT FROM OPT1

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S10-55NAS105, Section 10, May 2005 OUTPUT FROM OPT1

S10-56NAS105, Section 10, May 2005 OUTPUT FROM OPT1

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S10-58NAS105, Section 10, May 2005 OUTPUT FROM OPT1

S10-59NAS105, Section 10, May 2005 OUTPUT FROM OPT1

S10-60NAS105, Section 10, May 2005 OUTPUT FROM OPT1

S10-61NAS105, Section 10, May 2005 OUTPUT FROM OPT1

S10-62NAS105, Section 10, May 2005 SAMPLE OPT2 – VARIABLE CROSS- SECTION CANTILEVER BEAM n Now let us allow the thickness to vary along the length. n For this, we will need to define a different property for each BAR element and allow the properties of each to vary. n This is done in file cantbeam2. dat (on the following page). n Measured response quantities and constraints will be unchanged. n Only the properties, design variables, and DVPREL1 entries will change.

S10-63NAS105, Section 10, May 2005 SAMPLE OPT2 – VARIABLE CROSS- SECTION CANTILEVER BEAM (Cont)

S10-64NAS105, Section 10, May 2005 SAMPLE OPT2 – VARIABLE CROSS- SECTION CANTILEVER BEAM (Cont) n Let us define the new design variables for the optimizer. n We want to be able to vary the depth of the elements, so we use DVPREL1 entries to tie the DESVAR entry to the fields on the BARL. DESVAR,21,Height1,2.,.5,8.0 DESVAR,22,Height2,2.,.5,8.0 DESVAR,23,Height3,2.,.5,8.0 DESVAR,24,Height4,2.,.5,8.0 DESVAR,25,Height5,2.,.5,8.0 DESVAR,26,Height6,2.,.5,8.0 DESVAR,27,Height7,2.,.5,8.0 DESVAR,28,Height8,2.,.5,8.0 $ $ Relate Design Variables to the Fields on the PBARL DVPREL1,111,PBARL,1,13,.5,8.0,21,1.0 DVPREL1,112,PBARL,2,13,.5,8.0,22,1.0 DVPREL1,113,PBARL,3,13,.5,8.0,23,1.0 DVPREL1,114,PBARL,4,13,.5,8.0,24,1.0 DVPREL1,115,PBARL,5,13,.5,8.0,25,1.0 DVPREL1,116,PBARL,6,13,.5,8.0,26,1.0 DVPREL1,117,PBARL,7,13,.5,8.0,27,1.0 DVPREL1,118,PBARL,8,13,.5,8.0,28,1.0

S10-65NAS105, Section 10, May 2005 SAMPLE OPT2 – VARIABLE CROSS- SECTION CANTILEVER BEAM (Cont) $ file opt2. at – cantilever beam static optimization with variable $ cross–section SOL 200 CEND TITLE = Practical Dynamics Seminar Sample Problem 1 opt2. dat SUBTITLE = Cantilever Beam with variable cross–section LABEL = Perform Model Checks DISP = ALL STRESS = ALL SUBCASE 1 SPC = 1 load = 100 ANALYSIS = STATICS DESOBJ = 1000 $ minimize weight DESSUB = 200 $ apply constraints in the optimizer $ BEGIN BULK FORCE,100,9,,10.,0.,–1.,0. include cantbeam2. dat $ create design response entries $ total weight = objective will be to minimize this $ DRESP1,1000,TOTW,WEIGHT $ $ tip displacement – we need to keep this less than.01 $ DRESP1,2000,tipdisp,DISP,,,2,,9 $ Stress at the root – select the element, rather than the property $ select the maximum stress at the start of the BAR $ DRESP1,3000,rootstr,STRESS,ELEM,,8,,1 The complete input file

S10-66NAS105, Section 10, May 2005 SAMPLE OPT2 – VARIABLE CROSS- SECTION CANTILEVER BEAM (Cont) $ Create design variables DESVAR,21,Height1,2.,.5,8.0 DESVAR,22,Height2,2.,.5,8.0 DESVAR,23,Height3,2.,.5,8.0 DESVAR,24,Height4,2.,.5,8.0 DESVAR,25,Height5,2.,.5,8.0 DESVAR,26,Height6,2.,.5,8.0 DESVAR,27,Height7,2.,.5,8.0 DESVAR,28,Height8,2.,.5,8.0 $ Relate Design Variables to the Fields on the PBARL DVPREL1,111,PBARL,1,13,.5,8.0,21,1.0 DVPREL1,112,PBARL,2,13,.5,8.0,22,1.0 DVPREL1,113,PBARL,3,13,.5,8.0,23,1.0 DVPREL1,114,PBARL,4,13,.5,8.0,24,1.0 DVPREL1,115,PBARL,5,13,.5,8.0,25,1.0 DVPREL1,116,PBARL,6,13,.5,8.0,26,1.0 DVPREL1,117,PBARL,7,13,.5,8.0,27,1.0 DVPREL1,118,PBARL,8,13,.5,8.0,28,1.0 $ Define Design Constraints $ limit tip displacement DCONSTR,200,2000,–.01,.01 $ limit stress at root DCONSTR,200,3000,–1000.,1000. DOPTPRM,DESMAX,25 ENDDATA The complete input file (Cont.)

S10-67NAS105, Section 10, May 2005 OUTPUT FROM OPT2

S10-68NAS105, Section 10, May 2005 OUTPUT FROM OPT2

S10-69NAS105, Section 10, May 2005 OUTPUT FROM OPT2

S10-70NAS105, Section 10, May 2005 OUTPUT FROM OPT2

S10-71NAS105, Section 10, May 2005 OUTPUT FROM OPT2

S10-72NAS105, Section 10, May 2005 OUTPUT FROM OPT2

S10-73NAS105, Section 10, May 2005 OUTPUT FROM OPT2

S10-74NAS105, Section 10, May 2005 OUTPUT FROM OPT2

S10-75NAS105, Section 10, May 2005 OUTPUT FROM OPT2

S10-76NAS105, Section 10, May 2005 WORKSHOPS n Part 1: for the optimization example (opt1.dat) add the bar width as a design variable and re-run the problem n Part 2: using a copy of the file from part1, add a requirement that the buckling eigenvalue must be greater that 1.5 n Part 3: using a copy of the input file from part 2, add a concentrated load weight of 10.0 on the end of the beam and then add a requirement that the first natural frequency be greater that 4.0hz (initial value is 4.45hz, but since the program will attempt to reduce the width, the first frequency will drop) n Part 4: try the same thing using opt2. dat as a starting point

S10-77NAS105, Section 10, May 2005 ADVANCED FEATURES OF OPTIMIZATION IN MSC.NASTRAN

S10-78NAS105, Section 10, May 2005 TYPE-2 STRUCTURAL RESPONSES Responses that are obtained directly from MSC.NASTRAN (displacements stress, strain, internal forces, buckling loads, natural frequencies, composite failure criteria, weight, volume Responses that are synthesized from simple responses, design variables, and constants through analytic equations given by user

S10-79NAS105, Section 10, May 2005 TYPE-2 RESPONSE FORMULATION Type 2 Response(s) = Function of: Note: Recall that one designation of type 1 response from the DRESP1 data entry may generate a large number of type1 responses of they are stress, strain or internal forces. If the type1 responses are of that type, one type2 response data entry can generate the same number of type 2 responses.

S10-80NAS105, Section 10, May 2005 RESTRICTIONS IN FORMING SYNTHETIC RESPONSES n Different types of responses (for example, weight and displacement) can be mixed in one equation, however, use caution. Each associated DRESP1 entry must generate a single response only. n Responses cannot be mixed across the subcases in one equation. n Multiple equations must be separated by a semi- colon (;) and no recursive reference is allowed.

S10-81NAS105, Section 10, May 2005 DEQATN Defines one or more equations for use in design sensitivity or p-element analysis Format: Example: FieldContents EQIDUnique equation identification number. (Integer > 0) EQUATIONEquation(s). See Remarks. (Character) Remarks: 1. EQUATION is a single equation, or a set of nested equations, and is specified in fields 3 through 9 on the first entry and may be continued on fields 2 through 9 on the continuation entries. On the continuation entries, no commas can appear in columns 1 through 8. All data in fields 2 through 9 must be specified in columns 9 through 72. The large field format is not allowed A single equation has the following format: Variable1 (x1, x2, …, xn) = expression1

S10-82NAS105, Section 10, May 2005 DEQATN (Cont) A set of nested equations is separated by semi-colons and has the format: variable1 (x1, x2, …, xn) = expression1 variable2 = expression2 variable3 = expression3 etc. variable_m = expression_m Expression_i is a collection of constants, real variables, and real functions, separated by operators, and must produce a single real value. (x1, x2, …, xn) is the list of all the variable names except variable_i, which appears in all expressions. Variable_i may be used in subsequent expressions. The last equation, variable_m = expression_m, provides the value that is returned to the Bulk Data entry that references EQUID; e.g., DRESP2. The example above represents the following mathematical equations: where SIN and PI are intrinsic functions. See Remark 4 2. EQUATION may contain embedded blanks. EQUATION must contain less than 12,500 nonblank characters. This is equivalent to approximately 195 continuation entries.

S10-83NAS105, Section 10, May 2005 DEQATN (Cont) 3. The syntax of the expression follows FORTRAN language standards. The allowable arithmetic operations are shown in Table 3 in the order of execution precedence. Parenthesis are used to change the order of precedence. Operations within parenthesis are performed first with the usual order of precedence maintained within the parenthesis 4. The expression may contain intrinsic functions. Table 4 contains the format and descriptions of functions which may appear in the expressions. The use of functions that may be discontinuous must be used with caution because they can cause discontinuous derivatives. These are ABS, DIM, MAX, MIN, and MOD. For examples and further details see the MSC.NASTRAN DMAP Programmers Guide. Table 3. DEQATN Entry Operators

S10-84NAS105, Section 10, May 2005 DEQATN (Cont) Table 4. DEQATN Entry Functions

S10-85NAS105, Section 10, May 2005 DEQATN (Cont) 5. If the DEQATN entry is referenced by the: a.DVPREL2 entry, then Xi represents the DVIDj and LABLk fields b.DRESP2 entry then xi represents the DVIDj, LABLk, NRm, Gp, DPIPq, DCICr, DMIMs, DPI2Pt, DCI2Cu, DMI2Mv, and NRRw fields in that order c.GMLOAD, GMBC, or TEMPF entries then X1 represents x in the basic coordinate system, X2 represents y in the basic coordinate system, and X3 represents z in the basic coordinate system d.GMCURV entry then X1 represents line parameter u e.GMSURF entry then X1 represents surface parameter u and X2 represents surface parameter v 6. If the DEQATN entry is referenced by the GMLOAD, GMBC, TEMPF, GMCURV, or GMSURF entries and your computer has a short word length (e.g., 32 bits/word), the EQUATION is processed with double precision and constraints may be specified in double precision; e.g., 1.2DO. If your machine has a long word length (e.g., 64 bits/word) then EQUATION is processed in single precision and constants must be specified in single precision; e.g., 1.2.

S10-86NAS105, Section 10, May 2005 DTABLE Defines a table of real constraints that are used in equations (see DEQATN entry). Format: Example: FieldContents LABLiLabel for the constant. (Character). VALUiValue of the constant. (Real). Remarks: 1. Only one DTABLE entry may be specified in the Bulk Data Section. 2. LABLI are referenced by the LABI on the DVPREL2 or DRESP2 entries.

S10-87NAS105, Section 10, May 2005 DRESP2 Defines equation responses that are used in the design, either as constraints or as an objective. Format:

S10-88NAS105, Section 10, May 2005 DRESP2 (Cont.) Example: FieldContents IDUnique identification number. (Integer > 0). LABELUser-defined label. (Character). EQIDDEQUATN entry identification number. (Integer > 0). FUNCFunction to be applied to the arguments. See Remark 8. (Character) REGIONRegion identifier for constraint screening. See Remark 5. (Integer > 0). DESVARFlag indicating DESVAR entry identification numbers. (Character). DVIDiDESVAR entry identification number. (Integer > 0).

S10-89NAS105, Section 10, May 2005 DRESP2 (Cont.) DTABLEFlag indicating that the labels for the constants in a DTABLE entry follow. (Character) LABLjLabel for a constant in the DTABLE entry. (Character) DRESP1Flag indicating DRESP1 entry identification numbers. (Character) NRkDRESP1 entry identification number. (Integer > 0) DNODEFlag indicating grid point and component identification numbers. (Character) GmGrid point identification number. (Integer > 0) CmComponent number of grid point Gm. (1 0) DVMREL1Flag indicating DVPREL2 entry identification number. (Character) DMIMiDVMREL1 entry identification number. (Integer > 0) DVPREL2Flag indicating DVPREL2 entry identification number. (Character) DPI2PiDVPREL2 entry identification number. (Integer > 0) DVCREL2Flag indicating DVCREL2 entry identification number. (Character) DCI2CiDVCREL2 entry identification number. (Integer > 0)

S10-90NAS105, Section 10, May 2005 Remarks: 1.DRESP2 entries may only reference DESVAR, DTABLE, DRESP1, DNODE, DVPREL1, DVCREL1, DVMREL1, DVPREL2, DVCREL2, and DVMREL2 entries. They may also reference other DRESP2 entries. However, a DRESP2 entry cannot reference itself directly or recursively. 2. Referenced DRESP1 entries cannot span analysis types or superelements. 3.DRESP2 entries must have unique identification number with respect to DRESP1 entries. 4. The DESVAR, DTABLE, DNODE, DVPREL1, DVCREL1 and DVMREL1, DVPREL2, DVCREL2, DVMREL2, and DRESP2 flags in field 2 must appear in the order given above. Any of these words along with the identification numbers associated with them may be omitted if they are not involved in this DRESP2 relationship. However, at least one of these four types of arguments must exist. DRESP2 (Cont.) DVMREL2Flag indicating DVMREL2 entry identification number. (Character) DMI2MiDVMREL2 entry identification number. (Integer > 0) DRESP2Flag indicating other DRESP2 entry identification number. (Character) NRRkDRESP2 entry identification number. (Integer > 0)

S10-91NAS105, Section 10, May 2005 DRESP2 (Cont.) 5. The REGION field follows the same rules as for the DRESP1 entries. DRESP1 and DRESP2 responses will never be contained in the same region, even if they are assigned the same REGION identification number. The default is to put all responses referenced by one DRESP2 entry in the same region. 6. The variables identified by DVIDi, LABLj, NRk, and the Gm, Cm pairs, DPIPi, DCICm, DMIMn, DPI2Po, DCI2Cp, DMI2Mq, and NRRu are assigned (in that order) to the variable names (x1, x2, x3, etc.) specified in the left-hand side of the first equation on the DEQUATN entry, referenced by EQID. In the example below, DESVARs 101 and 3 are assigned to arguments A and B. DTABLEs PI and YM are assigned to arguments C and D. Grid 14, Component 1 is assigned to argument R.

S10-92NAS105, Section 10, May 2005 DRESP2 (Cont.) 7.(Gm, Cm) refer to a any grid component and is no longer limited to a designed grid component. 8. The FUNC attributes can be used in place of the EQID and supports the functions shown in the following table: When EQID has character input, the DEQATN entry is no longer needed. The functions are applied to all arguments on the DRESP2 regardless of the type. 9. The number of arguments of a DEQATN can be more than the number of values defined on the DRESP2 if the DRESP1s referenced have RTYPE with FR prefix. Arguments are still positional. The extra arguments in the DEQATN must appear at the end of the argument list. The discrepancy is resolved internally with the forcing frequency(ies) associated with DRESP1s. An follows:

S10-93NAS105, Section 10, May 2005 DRESP2 (Cont.) In the above example, the DEQATN has two more additional terms than have been defined on the DRESP2. The first additional term is the forcing frequency (in hertz) of the first DRESP1 ID on the DRESP2. The second additional term is the forcing frequency of second DRESP1 ID in the list. When all DRESP1s involved have the same frequency, the user is not required to name all the additional terms in the argument list of DEQATN.

S10-94NAS105, Section 10, May 2005 DRESP2 BULK DATA ENTRY n DRESP2 is used to define a synthetic response that is not directly available from the MSC.NASTRAN analysis capabilities n Examples: u Generation of a new stress or strain failure criterion that is not available in MSC.NASTRAN u Imposing local buckling criteria based on element sizes as well as stress components u Programming proprietary design-sizing equations u Generation of nonlinear displacement responses such as displacement magnitudes: u More complex relations among displacements could be formed by combining with MPC and RBE3 capabilities

S10-95NAS105, Section 10, May 2005 DOPTPRM Overrides default values of parameters used in design optimization. Format: Example: FieldContents PARAMiName of the design optimization parameter. Allowable names are given in Table 5. (Character.) VALiValue of the parameter. (Real or Integer, see Table 5). Remarks: 1. Only one DOPTPRM entry is allowed in the Bulk Data Section. I

S10-96NAS105, Section 10, May 2005 DOPTPRM (Cont.) NameDescription, Type, and Default Value APRCODApproximation method to be used. 1 = Direct Linearization; 2 = Mixed Method based on response type; 3 = Convex Linearization. APRCOD = 1 is recommended for shape optimization problems. (Integer 1, 2, or 3; Default = 2) CONV1Relative criterion to detect convergence. If the relative change in objective between two optimization cycles is less than CONV1, then optimization is terminated. (Real > 0.0; Default = 0.001) CONV2Absolute criterion to detect convergence. If the absolute change in objective between two optimization cycles is less than CONV2, then optimization is terminated. (Real > 0.0; Default = 1.0E20) CONVDVRelative convergence criterion on design variables. (Real > 0.0; Default = 0.001) CONVPRRelative convergence criterion on properties. (Real > 0.0; Default = 0.001) CTConstraint tolerance. Constraint is considered active if current value is greater than CT. (Real < 0.0; Default = –0.03) CTMINConstraint is considered violated if current value is greater than CTMIN. (Real > 0.0; Default = 0.003) DABOBJMaximum absolute change in objective between ITRMOP consecutive iterations (see ITRMOP) to indicate convergence at optimizer level. F0 is the initial objective function value. (Real > 0.0; Default = MAX[0.001 * ABS(F0), ]) DELBRelative finite difference move parameter. (Real > 0.0; Default = ) Table 5. PARAMi Names and Description.

S10-97NAS105, Section 10, May 2005 DOPTPRM (Cont.) Table 5. PARAMi Names and Description. (Cont.) NameDescription, Type, and Default Value DELOBJMaximum relative change in objective between ITRMOP consecutive iterations to indicate convergence at optimizer level. (Real > 0.0; Default = 0.001) DELPFractional change allowed in each property during any optimization design cycle. This provides constraints on property moves. (Real > 0.0; Default = 0.2) DELXFractional change allowed in each design variable during any optimization cycle. (Real > 0.0; Default = 1.0) DESMAXMaximum number of design cycles (not including FSD cycle) to be performed. (Integer > 0; Default = 5) DISCODDiscrete Processing Method: (Integer 1, 2, 3 or 4; Default = 1) 1: Design of Experiments 2: Conservative Discrete Design 3: Rounding up to the nearest design variable 4: Rounded off to the nearest design variable DISBEGDesign cycle ID for discrete variable processing initiation. Discrete variable processing analysis is carried out for every design cycle after DISBEG. (Integer >=0, default = 0 = the last design cycle) DOBJ1Relative change in objective attempted on the first optimization iteration. Used to estimate initial move in the one-dimensional search. Updated as the optimization progresses. (Real > 0.0; Default = 0.1) DOBJ2Absolute change in objective attempted on the first optimization iteration. (Real > 0.0; Default = 0.2 * (F0)) DPMINMinimum move limit imposed. (Real > 0.0; Default = 0.01) DX1Maximum relative change in a design variable attempted on the first optimization iteration. Used to estimate the initial move in the one dimensional search. Updated as the optimization progresses. (Real > 0.0; Default = 0.01)

S10-98NAS105, Section 10, May 2005 DOPTPRM (Cont.) Table 5. PARAMi Names and Description. (Cont.) Specifies the number of Fully Stressed Design Cycles that are to be performed (Integer, Default = 0) DX2 Absolute change in a design variable attempted on the first optimization iteration. (Real > 0.0; Default = 0.2 * MAX[X(I)]) DXMIN Minimum design variable move limit (Real > 0.0; Default = 0.05) NameDescription, Type, and Default Value FSDALPRelaxation parameter applied in Fully Stressed Design (Real, 0.0 < FSDMAX 0.0; Default = 0.005) GSCALConstraint normalization factor. See Remarks under the DSCREEN and DCONSTR entries. (Real > 0.0; Default = 0.001) IGMAXIf IGMAX = 0, only gradients of active and violated constraints are calculated. If IGMAX > 0, up to NCOLA gradients are calculated including active, violated, and near active constraints. (Integer > 0; Default = 0) IPRINT Print control during approximate optimization phase. Increasing values represent increasing levels of optimizer information. (0

S10-99NAS105, Section 10, May 2005 DOPTPRM (Cont.) Table 5. PARAMi Names and Description. (Cont.) ISCALDesign variables are rescaled every ISCAL iterations. Set ISCAL= –1 to turn off scaling. (Integer; Default = NDV (number of design variables)) IPRINT1If IPRNT1 = 1, print scaling factors for design variable vector. (Integer 0 or 1; Default = 0) IPRINT2If IPRNT2 = 1, print miscellaneous search information. If IPRNT2 = 2, turn on print during one- dimensional search process. (Warning: This may lead to excessive output.) (Integer 0, 1, or 2; Default = 0) ITMAXMaximum number of iterations allowed at optimizer level during each design cycle. (Integer; Default = 40) ITRMOPNumber of consecutive iterations for which convergence criteria must be satisfied to indicate convergence at the optimizer level. (Integer; Default = 2) ITRMSTNumber of consecutive iterations for which convergence criteria must be met at the optimizer level to indicate convergence in the Sequential Linear Programming Method. (Integer > 0; Default = 2) IWRITEFORTRAN unit for print during approximate optimization phase. Default value for IWRITE is set to the FORTRAN unit for standard output. (Integer > 0, Default = 6 or value of SYSTME (2).) JTMAX Maximum number of iterations allowed at the optimizer level for the Sequential Linear Programming Method. This is the number of linearized sub-problems solved. (Integer 0; Default = 20) JPRINTSequential Linear Programming sub-problem print. If JPRINT > 0, IPRINT is turned on during the approximate linear sub-problem. (Default = 0) JWRITEIf JWRITE > 0, file number on which iteration history will be written. (Integer > 0; Default = 0)

S10-100NAS105, Section 10, May 2005 DOPTPRM (Cont.) Table 5. PARAMi Names and Description. (Cont.) METHOD Optimization Method: (Integer 1, 2, or 3; Default = 1) 1:Modified Method of Feasible Directions. (Default) 2:Sequential Linear Programming 3:Sequential Quadratic Programming P1Print control items specified for P2. (Integer >= 0; Default = 0) Initial results are always printed prior to the first approximate optimization. If an optimization task is performed, final results are always printed for the final analysis unless PARAM,SOFTEXIT,YES is specified. These two sets of print are not controllable. n: Print at every n-th design cycle. P2 PLVIOL Flag for handling of property limit violation. By default, the job will terminate with a user fatal message if the property derived from design model (DVPRELi, DVMRELi, DVCRELi) exceeds the property limits. Setting PLVIOL to a non-zero number will cause the program to issue a user warning message by ignoring the property limits violation and proceed with the analysis. (Integer; Default = 0) Items to be printed according to P1: (Integer; Default = 1) 0: No print. 1: Print objective and design variables. (Default) 2: Print properties. 4: Print constraints. 8: Print responses. 10: Print weight as a function of a material ID (note that there is not a design quantity so that only inputs to the approximate design are available) n: Sum of desired items. For example, P2 = 10 means print properties and responses. PTOLMaximum tolerance on differences allowed between the property values on property entries and the property values calculated from the design variable values on the DESVAR entry (through DVPRELi relations). PTOL is provided to trap ill-posed design models. (The minimum tolerance may be specified on user parameter DPEPS. (Real > 0.0; Default = 1.0E+35) STPSCLScaling factor for shape finite difference step sizes, to be applied to all shape design variables. (Real > 0.0; Default = 1.0)

S10-101NAS105, Section 10, May 2005 DYNAMIC RESPONSE OPTIMIZATION n Available dynamic analysis disciplines in Solution 200 u Direct Frequency u Modal Frequency u Modal Transient u Acoustic (Fluid-Structure Interaction) n Available response types: (see also the DRESP1 entry) u Displacement u Velocity u Acceleration u SPC Force u Stress u Element force u Equations (DRESP2 + DEQATN) u Weight u Volume

S10-102NAS105, Section 10, May 2005 DYNAMIC RESPONSE OPTIMIZATION (Cont.) Limitations in Dynamic Response Sensitivity u F is assumed zero when calculating sensitivities in direct and modal frequency and modal transient This assumption is usually good, except for those situations in which the following may be significant: l Gravity (or other mass-related) loads l Follower forces (shape sensitivity) l Thermal loading

S10-103NAS105, Section 10, May 2005 SAMPLE PROBLEM XXX – MINIMIZATION OF DRIVER RESPONSE TO A ROTATING IMBALANCE FOR THE CAR MODEL n This uses the model from Sample 17a. The automobile has the front left wheel out of balance. The amount of mass is.3 units and the radius to the mass is 10 units. Apply this rotating loading to the car and determine the response. The frequency range of interest is from hz. n We want to minimize the driver response to the input excitation.

S10-104NAS105, Section 10, May 2005 INPUT FOR OPTIMIZATION n Sample xxx – dynamic optimization of the car model n Executive and Case Control $ samplexxx.dat – linear bushings in the car model $ SOL 200 TIME 200 CEND TITLE = Samplexxx – dynamic analysis model SUBTITLE = Rotating force due to tire out of balance LABEL = perform optimization to minimize driver response set 999 = 358,471 DISP(phase) = 999 SUBCASE 1 ANALYSIS = MFREQ DESSUB = 100 $ constraints DESOBJ(min) = 300 $ design objective – minimize driver response DLOAD = 1 METHOD = 10 FREQ = 14 BEGIN BULK

S10-105NAS105, Section 10, May 2005 INPUT FOR OPTIMIZATION (Cont.) n Model and loading $ $ following param will exit after calculating sensitivities $ $param,optexit,4 $ get 40 modes eigrl,10,,,40,0 include car.dat include linspring.dat $ DLOAD RLOAD RLOAD DPHASE DAREA DAREA TABLED ENDT FREQ $

S10-106NAS105, Section 10, May 2005 INPUT FOR OPTIMIZATION (Cont.) n Design variablesspring stiffness and shock absorber damping $ $ data for design sensitivity $ $ define design variables $ $ allow more than 5 cycles in the optimization $ doptprm,desmax,25 $ desvar,1,frntdamp,10.,1.,100. desvar,2,reardamp,5.,1.,100. desvar,3,frntstif,10.,4.,20. desvar,4,rearstif,8.,4.,20. desvar,5,frntstfx,10.,4.,20. desvar,6,rearstfx,10.,4.,20. $ $ relation between properties and variables $ dvprel1,101,pvisc,2001,3,1.,,,,+dv101 +dv101,1,1. dvprel1,102,pvisc,2002,3,1.,,,,+dv102 +dv102,2,1. dvprel1,103,prod,1001,4,400.,,,,+dv103 +dv103,3,100. dvprel1,104,prod,1002,4,400.,,,,+dv104 +dv104,4,100. dvprel1,105,pelas,2001,3,4000.,,,,+dv105 +dv105,5,1000. dvprel1,106,pelas,2002,3,4000.,,,,+dv106 +dv106,6,1000. $

S10-107NAS105, Section 10, May 2005 INPUT FOR OPTIMIZATION (Cont.) n Responses – drivers seat (GRID 471) and wheel (GRID 358) Note that only one response is defined for the wheel displacement, while separate responses are defined for each loading frequency for the drivers seat. The wheel response will internally generate a response quantity for each excitation frequency, since no frequency is provided on the DRESP1 entry. Since a DRESP2 is going to be used to define the SRSS value of the drivers seat displacements, it is necessary to have a separate response specified for each excitation frequency. Note that only the first 51 are specified here– it may be necessary to also define responses for the higher frequencies. $ select displacement Y at driver seat and mount point as $ response quantities $ $ mount point – only one response – this will be constrained $ dresp1,200,disp,frdisp,,,2,,358 $ $ define drivers seat disp as a response – $ select the first 51 excitation frequencies $ this will be used as both a constraint and the objective $ dresp1,201,driver,frdisp,,,2,.5,471 =,*(1),=,=,=,=,=,*(.5),== =(49) $

S10-108NAS105, Section 10, May 2005 INPUT FOR OPTIMIZATION (Cont.) n Constraints u Wheeldisplacement must be less than 0.5 u Drivermeasured displacements must be less than 0.25 $ add constraints $ $ require that maximum tire displacement be.5 inches $ dconstr,101,200,–.5,.5 $ $ require that maximum driver displacement be.25 inches $ dconstr,102,201,–.25,.25 =,*(1),*(1),== =(49) $ $ combine constraints into set 100 $ dconadd,100,101,102,103,104,105,106,107,+dc100a +dc100a,108,109,110,111,112,113,114,115,+dc100b *(1),*(8),*(8),*(8),*(8),*(8),*(8),*(8),*(8),*(8) =(3) +dc100f,148,149,150,151,152 $

S10-109NAS105, Section 10, May 2005 INPUT FOR OPTIMIZATION (Cont.) n Objectiveminimize driver response – over the range 0–25hz $ $ define objective = minimize srss of response $ dresp2,300,srss, 301,,,,,,+dr300a +dr300a,dresp1,201,202,203,204,205,206,207,+dr300b +dr300b,,208,209,210,211,212,213,214,+dr300c *(1),=,*(7),*(7),*(7),*(7),*(7),*(7),*(7),*(7) =(4) +dr300h,,250,251 $ deqatn 301 resp(a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p, q,r,s,t,u,v,w,x,y,z,aa,bb,cc,dd,ee,ff,gg,hh,ii, jj,kk,ll,mm,nn,oo,pp,qq,rr,ss,tt,uu,vv,ww,xx,yy)= sqrt(a**2+b**2+c**2+d**2+e**2+f**2+g**2+h**2+ j**2+i**2+l**2+m**2+n**2+o**2+p**2+q**2+r**2+ s**2+t**2+u**2+v**2+w**2+x**2+y**2+z**2+ aa**2+bb**2+cc**2+dd**2+ee**2+ff**2+gg**2+ hh**2+ii**2+jj**2+kk**2+ll**2+mm**2+nn**2+oo**2+ pp**2+qq**2+rr**2+ss**2+tt**2+uu**2+vv**2+ww**2+ xx**2+yy**2) $ $ end of optimization input

S10-110NAS105, Section 10, May 2005 OPTIMIZATION RESULTS (Cont.) n Resultsgraphical

S10-111NAS105, Section 10, May 2005 OPTIMIZATION RESULTS (Cont.) n Resultsgraphical

S10-112NAS105, Section 10, May 2005 SHAPE OPTIMIZATION n MSC.NASTRAN allows shape changes in your model to be used as design variables. n For instance, consider the finite element model shown below with the loads shown. Find a shape of the hole that will minimize weight but keep stresses below the yield point of the material. n In order to minimize weight, increase the size of the hole n Tell the optimizer how to change the shape of the hole by defining shape design variables.

S10-113NAS105, Section 10, May 2005 SHAPE OPTIMIZATION (Cont.) n The shape design variables shown below tell the optimizer that elliptical shape changes to the hole in both the x and the y directions are allowable.

S10-114NAS105, Section 10, May 2005 SHAPE OPTIMIZATION (Cont.) n The before and after optimization shape of the model are shown below: n The weight was decreased while the stresses were kept below the yield point. BeforeAfter

S10-115NAS105, Section 10, May 2005 HOW DO I CREATE SHAPE DESIGN VARIABLES? n Four different methods for generating shape basis vectors have been implemented: u Manual grid variation u Direct input of shapes u Geometric boundary shapes u Analytic boundary shapes n In these notes, only the manual grid variation is shown.

S10-116NAS105, Section 10, May 2005 MANUAL GRID VARIATION n Since the model is a coarse model, the manual input of shapes is used n A DVGRID Bulk Data entry defines the direction and magnitude of a grid variation for a given change in a design variable. Its format is: DVGRID, DVID, GID, CID, COEFF, N1, N2, N3 COEFF times the vector {N1,N2,N3} determines the direction and magnitude of a grid variation G as shown in the figure below: n The figure below shows that with manual grid variation, if you want the horizontal movement of the left boundary of your model to be a shape variable, then only the specified grids will be allowed to be variable Effect of manual grid variation shape variables {G} I { G} i

S10-117NAS105, Section 10, May 2005 MANUAL GRID VARIATION (Cont.) n All possible shape design variable movements must be explicitly defined. A single DVGRID Bulk Data entry is required for every grid movement for each shape design variable. n An example DVGRID Bulk Data entry is: dvgrid, 41, 116, 2, 1., 1., 0.0, 0.0 n This data entry says that grid 116 is controlled by design variable 41. It can move in the direction of the {1.0, 0.0, 0.0} vector in coordinate system 2. n The DESVAR Bulk Data entry defines the controlling design variable. An example is: n This data entry says that design variable 41, labeled OHUBRAD, has an initial value of 8.68, a lower limit of 8.0, an upper limit of 9.6 and it can move in increments of.08, where.08 is the fraction of the current value. In this case the maximum movement at the next design cycle is.08 times 8.68 or.07. At.07 increments it will take three design cycles to exceed the upper limit of 9.6. If you want more cycles, you can decrease this value. If you put 0.0 as an initial value, then a small default movement will be used.

S10-118NAS105, Section 10, May 2005 MANUAL GRID VARIATION (Cont.) n DVGRID entries can be used in combination with the direct input of shapes approach but not in combination with geometric boundary shapes and analytic boundary shapes. n Advantages: u Since this can be considered the lowest-level approach, its strength lies in its generality. Using DVGRIDs alone, the designer has direct control over every designed grid point in the model (that is, every grid point with a location that is to change during shape optimization). n Disadvantages: u In all but the simplest of problems, the data input can be formidable without a preprocessor. The resultant basis vectors are treated as constant and not updated with each design cycle. Therefore, the risks of mesh distortion are higher than with the geometric boundary shape and analytic methods.

S10-119NAS105, Section 10, May 2005 INPUT FILE STRUCTURE: MANUAL GRID VARIATION sol 200 (Executive) Analysis = static dessub = 20 (Case Control) desobj = dresp1, 10, wtmin, weight (Bulk Data) dconstr, 20, 31, , dresp1, 31, Vmhexa, stress, psolid,,13,,2 desvar, 41, ohubrad, 8.68, 8.0, 9.6,.08 dvgrid, 41, 116, 2, 1., 1., 0.0, 0.0 dvgrid, 41, 1141, 2, 1., 1., 0.0, n Executive Section u Only the SOL 200 statement is required n Executive Section Only the SOL 200 statement is required

S10-120NAS105, Section 10, May 2005 INPUT FILE STRUCTURE: MANUAL GRID VARIATION (Cont.) n Case Control Section u The ANALYSIS request specifies which type of analysis you are going to perform (in this case, static analysis). u The DESOBJ request specifies that your objective is to minimize weight. You can specify minimum or maximum, with minimum the default. The DESOBJ references the DRESP1 Bulk Data entry with rtype=weight. u The DESSUB references the DCONSTR Bulk Data entry, which in turn references a DRESP1 Bulk Data entry. These three Bulk Data entries define a constraint. n Bulk Data Section u The DRESP1 Bulk Data entry, which invokes a weight response. u The DCONSTR Bulk Data entry references another DRESP1 Bulk Data entry, which together define a constraint such that the von Mises stress is less than psi for all the elements. u The shape variables are defined with the combination of a DESVAR Bulk Data entry and a set of DVGRID Bulk Data entries.

S10-121NAS105, Section 10, May 2005 INPUT FILE STRUCTURE: MANUAL GRID VARIATION (Cont.) n Design variable 41 defines the change in thickness of the outer hub. As shown in the figure below, all the grids in the inner part of the hub are included in the set of DVGRIDS. In addition, the grids that represent the fillet from the stiffener to the hub are also included, and thus move with the hub. This keeps fillet radius constant as the hub and stiffener design variables move relative to one another. n Only the first and last DVGRID are shown for the design variable. In reality, all the grids (approximately 150) defining the part of the model that are moving for a particular design variable are present in the model file for this design variable alone.

S10-122NAS105, Section 10, May 2005 INPUT FOR SHAPE OPTIMIZATION n Use the size of the front beams as design variables. n Two new design variables are to be created, one for the upper beams and the other for the lower beams. n Each variable will represent a change in the depth of the member.

S10-123NAS105, Section 10, May 2005 UPPER BEAMS

S10-124NAS105, Section 10, May 2005 UPPER BEAMS (Cont.)

S10-125NAS105, Section 10, May 2005 LOWER BEAMS

S10-126NAS105, Section 10, May 2005 LOWER BEAMS (Cont.)

S10-127NAS105, Section 10, May 2005 OPTIMIZATION INPUT $ $ define shape variables $ desvar,7,topchnl,1.,–10.,10. desvar,8,botbox,1.,–10.,10. $ $ define shapes $ $ top beams $ dvgrid,7,542,,.1,0.,1.,0. dvgrid,7,551,,.1,0.,1.,0. dvgrid,7,894,,.1,0.,1.,0. dvgrid,7,896,,.1,0.,1.,0. dvgrid,7,893,,.1,0.,1.,0. dvgrid,7,895,,.1,0.,1.,0. dvgrid,7,795,,.1,0.,1.,0. dvgrid,7,794,,.1,0.,1.,0. dvgrid,7,541,,.1,0.,1.,0. dvgrid,7,573,,.1,0.,1.,0. $ dvgrid,7,48,,.1,0.,1.,0. dvgrid,7,39,,.1,0.,1.,0. dvgrid,7,429,,.1,0.,1.,0. dvgrid,7,427,,.1,0.,1.,0. dvgrid,7,428,,.1,0.,1.,0. dvgrid,7,426,,.1,0.,1.,0. dvgrid,7,323,,.1,0.,1.,0. dvgrid,7,324,,.1,0.,1.,0. dvgrid,7,70,,.1,0.,1.,0. dvgrid,7,38,,.1,0.,1.,0.

S10-128NAS105, Section 10, May 2005 OPTIMIZATION INPUT (Cont.) $ $ bottom beams $ dvgrid,8,361,,.1,0.,1.,0. dvgrid,8,360,,.1,0.,1.,0. dvgrid,8,366,,.1,0.,1.,0. dvgrid,8,367,,.1,0.,1.,0. dvgrid,8,362,,.1,0.,1.,0. dvgrid,8,363,,.1,0.,1.,0. dvgrid,8,370,,.1,0.,1.,0. dvgrid,8,371,,.1,0.,1.,0. dvgrid,8,343,,.1,0.,1.,0. dvgrid,8,363,,.1,0.,1.,0. dvgrid,8,342,,.1,0.,1.,0. dvgrid,8,365,,.1,0.,1.,0. dvgrid,8,827,,.1,0.,1.,0. dvgrid,8,828,,.1,0.,1.,0. dvgrid,8,834,,.1,0.,1.,0. dvgrid,8,833,,.1,0.,1.,0. dvgrid,8,830,,.1,0.,1.,0. dvgrid,8,829,,.1,0.,1.,0. dvgrid,8,838,,.1,0.,1.,0. dvgrid,8,837,,.1,0.,1.,0. dvgrid,8,744,,.1,0.,1.,0. dvgrid,8,810,,.1,0.,1.,0. dvgrid,8,746,,.1,0.,1.,0. dvgrid,8,809,,.1,0.,1.,0. $

S10-129NAS105, Section 10, May 2005 SHAPE-OPTIMIZATION RESULTS

S10-130NAS105, Section 10, May 2005 OPTMIZATION SUMMARY n Selection of design variables should be based on an understanding of the problem. n Proper selection of design variables can result in noticeable improvements in the design and reductions in weight. n For the example shown, the shape variables did not have a significant effect on the final response, but the weight of the model was reduced by changing the shapes of the beams. n MSC.NASTRAN optimization is a powerful tool which can result in significant improvement in the design without a noticeable amount of effort by the user.