Chapter 34: Topology Optimization MBB Beam and Torsion

34 Topology Optimization MBB Beam and Torsion

Summary - Beam 527

Introduction 528

Solution Requirements 528

Modeling Tips 531

Summary - Torsion 533

Introduction 534

Solution Requirements 534

Modeling Tips 539

Input File(s) 540

527CHAPTER 34

Topology Optimization MBB Beam and Torsion

Summary - BeamTitle Chapter 34: Topology Optimization MBB Beam and Torsion

Topology optimization features

• Compliance minimization• Mass target• Checkerboard free• Minimum member size control• Mirror symmetry constraints

Geometry

Material properties Young’s Modulus = 2x105 MPa, Poisson’s ratio = 0.3

Analysis type Static analysis

Boundary conditions Supported on rollers at one point and fixed support at another point.

Applied loads A concentrated force = 100.0 N (half model)

Element type 4-node liner QUAD elements

Topology result Material distribution)

(Symmetry) (Mesh 4800 Elements)

Units: m12 x 2 x 0.01 Plate

P = 200.0 N

P = 200.0 N

MD Demonstration Problems

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IntroductionAn MBB beam example (a half model shown in Figure 34-1) is used to demonstrate (a) basic MD Nastran topology optimization capabilities without manufacturing constraints, (b) minimum member size control, and (c) mirror symmetry constraints. The structural compliance (i.e., total strain energy) is minimized with a mass target 0.5 (i.e., 50% material savings). The loading and boundary conditions are shown in Figure 34-1. The structure is modeled with 4800 CQUAD4 elements.

Figure 34-1 MBB Beam

Solution RequirementsThis MBB beam is well accepted by academic and industry for topology optimization validation.

Design Model Description

These solutions demonstrate:

• A distinct design can be obtained by MD Nastran topology optimization with checkerboard free algorithm (as default)

• The minimum member size is mainly used to control the size of members in topology optimal designs. Preventing thin members enhances the simplicity of the design and, hence, its manufacturability. Minimum member size is more like quality control than quantity control.

• By using symmetry constraints in topology optimization, a symmetric design can be obtained regardless of the boundary conditions or loads.

Objective: Minimize compliance

Topology design region: PSHELL

Constraints: Mass target = 0.5 (i.e., mass savings 50%)

(a) Minimum member size control and/or(b) Mirror symmetry constraints

P = 100.0 N

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Optimization Solution

Basic compliance minimization

The input data for this example related to topology optimization model is given in Listing 1. A TOPVAR =1 Bulk Data entry is used to define a topological design region. XINIT=0.5 on the TOPVAR entry matches the mass target constraint so that the initial design is feasible. The rest values on the TOPVAR entry are default values that are recommended for general topology optimization applications. Type one design responses DRESP1 = 1 and 2 identify compliance and fractional mass, respectively. DCONSTR= 1 specifies the mass target. DESOBJ=1 in Case Control Command selects DRESP1=1 entry to be used as a design objective (minimization as default) and DESGLB selects the design constraint DCONSTR= 1 to be applied in this topology optimization task.

Listing 1 Input File for MBB Beam

DESOBJ = 1DESGLB = 1SUBCASE 1$ Subcase name : Default SUBTITLE=Default SPC = 2 LOAD = 2 ANALYSIS = STATICSBEGIN BULKDCONSTR 1 2 .5TOPVAR, 1 , Tshel, Pshell, , , , , 1DRESP1 1 COMPL COMP DRESP1 2 FRMASS FRMASS

Figure 34-2 shows the topology optimized result that is smoothed and remeshed by using Patran. This optimal design is very clear without any checkerboard effect. It is noticed that there are some small members.

Figure 34-2 MBB Beam Topology Design

Minimum Member Size Control

The MBB beam (shown in Figure 34-1) is used here to demonstrate the minimum member size control capability.

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The input data for this example related to topology optimization with “minimum member size” is given in Listing 2. The minimum member size value is defined by the TDMIN = 0.5 parameter on the DOPTPRM entry and corresponds to the length of 10 elements.

Listing 2 Input File for MBB Beam with Minimum Member Size

DESOBJ = 1DESGLB = 1SUBCASE 1$ Subcase name : Default SUBTITLE=Default SPC = 2 LOAD = 2 ANALYSIS = STATICSBEGIN BULKDOPTPRM, TDMIN, 0.5DCONSTR 1 2 .5TOPVAR, 1 , Tshel, Pshell, , , , , 1DRESP1 1 COMPL COMP DRESP1 2 FRMASS FRMASS

The Figure 34-3shows the topology optimized result with “minimum member size” TDMIN=0.5. Compared the design shown in Figure 34-2, this design with “minimum member size” is obviously much simpler and there are no tiny members at all.

Figure 34-3 MBB Beam Topology Design with “Minimum Member Size”

Mirror Symmetric Constraints

Since the loads applied on the MBB beam are not symmetric, the topology optimized designs Figures 34-2 and 34-3 are not symmetric. The MBB beam is employed again to demonstrate the mirror symmetric constraint capability that enforces the design to be symmetric about a given plane.

To apply symmetric constraints on designed properties, users need to create a reference coordinate system using a rectangular coordinate system CORD1R or CORD2R. In this example, grid 10001 (location x=3, y=1, and z=0) is defined as the origin. Grid 10002 (x=3, y=1, and z=1) lies on the z-axis, and grid 1003 (x=4, y=1, and z=0) lies in the x-z plane. CORD1R CID=1 defines a reference coordinate system. A continuation line “SYM” enforces the property PSHELL=1 to be symmetric about the planes YZ and ZX in the reference coordinate system CID=1. In addition, a minimum member size TDMIN=0.15 is applied. The input data for this example is given in Listing 3.

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Listing 3 Input File for MBB Beam with Mirror Symmetry Constraints

DESOBJ = 1DESGLB = 1SUBCASE 1$ Subcase name : Default SUBTITLE=Default SPC = 2 LOAD = 2 ANALYSIS = STATICSBEGIN BULKCORD1R 1 10001 10002 10003GRID 10001 3. 1. 0.0GRID 10002 3. 1. 1.0GRID 10003 4. 1. 0.0TOPVAR, 1 , Tshel, Pshell, , , , , 1 , SYM , 1 , YZ , ZX , TDMIN, 0.15 DRESP1 1 COMPL COMP DRESP1 2 FRMASS FRMASS DCONSTR 1 2 .5

Figure 34-3 shows the topology optimal result with symmetric constraints and minimum member size.

Figure 34-4 MBB Beam with Symmetric Constraints and Minimum Member Size

Modeling TipsThe quality of the results of a topology optimization task is a strong function of how the problem is posed in MD Nastran. This section contains a number of tips:

• A DRESP1=COMP is introduced to define the compliance of structures for topology optimizations. The response is usually used as an objective to maximize structural stiffness in static analysis problems.

• A DRESP1=FRMASS is introduced to define the mass fraction of topology designed elements. The DRESP1=WEIGHT is the total weight of all structural and nonstructural mass. For topology optimization tasks, a DRESP1=FRMASS response is recommended to define a mass reduction target in a design constraint.

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• The POWER field on the TOPVAR entry has a large influence on the solution of topology optimization problems. A lower POWER often produces a solution that contains large “grey” areas (area with intermediate densities 0.3 – 0.7). A higher value produces more distinct black and white (solid and void) designs. However, near singularities often occur when a high POWER is selected.

• A TCHECK parameter on DOPTPRM is used to turn on/off the checkerboard free algorithm. This default normally results in a better design for general finite element mesh. However, if high order elements and/or a coarser mesh is used, turning off the filtering algorithm may produce a better result.

• The TDMIN parameter is mainly used to control the degree of simplicity in terms of manufacturing considerations. It is common to see some members with smaller size than TDMIN at the final design since the small members have contributions to the objective. Minimum member size is more like quality control than quantity control. It is in general recommended that TDMIN should not be less than the length of 3 elements.

• Maximum design cycle DESMAX=30 (as default) is often required to produce a reasonable result. More design cycles may be required to achieve a clear 0/1 material distribution, particularly when manufacturability constraints are used.

• There are many solutions to a topology optimization: one global and many local minimization. It is not unusual to see different solutions to the same problem with the same discretization by using different optimization solvers or the same optimization solver with different starting values of design variables.

• In a multiple subcase problem, a DRSPAN Case Control Command can be used to construct a weighting function via a DRESP2 or DRESP3. For example, a static and normal mode combined problem, the objective can be defined as

where and are two weighting factors. is the calculated compliance and 1 is the calculated eigenvalue via DRESP1 definition. and 0 are the initial value of these responses.

• To obtain a rib pattern by topology optimization, a core non-designable shell element thickness must be defined together with two designable above and below the core thicknesses. That is, add two designable elements for each regular element.

• If some elements are disconnected on the final topology design proposal, the mass target may be too small to fill the design space.

obj weight1c1

c0----- weight2

0

1----- +=

weight1 weight2 c1

c0

533CHAPTER 34

Topology Optimization MBB Beam and Torsion

Summary - TorsionTitle Chapter 34: A Torsion Beam

Topology optimization features

• Compliance minimization• Mass target• Checkerboard free• Minimum member size control• Mirror symmetry constraints

Geometry

Material properties Young’s Modulus = 2.1x105MPa, Poisson’s ratio = 0.3, density = 1.0

Analysis type Static analysis

Boundary conditions Cantilever

Applied loads A pair of twisting forces = 1000.0 N at the free end

Element type 8-node HEXA elements

Topology result Material distribution)

Units: mLength = 16 and width = 4 and height = 4

+

+

P = 1000P = 1000

Z

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IntroductionA torsion beam is used here to demonstrate the extrusion and casting constraints. Figure 34-5 shows the FEM model of the torsion beam. A pair of twisting forces is applied on one end while the other end is fixed. 2048 CHEXA elements are used for this model. The objective is to minimize the structural compliance with mass target of 0.3 (i.e., 70% material savings).

Figure 34-5 Torsion Beam

Solution RequirementsThis torsion beam is utilized to show MD Nastran topology optimization extrusion and casting constraint capabilities.

Design Model Description

Three solutions demonstrate:

• By using extrusion constraints in topology optimization, a constant cross-section design along the given extrusion direction can be obtained regardless of the boundary conditions or loads.

• The use of casting constraints can prevent hollow profiles in topology optimization so that the die can slide in a given direction. One or two die options are available for selection.

• Some combined manufacturing constraints are allowed in topology optimization to achieve design goal.

Objective: Minimize compliance

Topology design region: PSOLID

Constraints: Mass target = 0.3 (i.e., mass savings 70%)

(a) Extrusion constraints(b) Casting constraints with one or two dies

Units: mLength = 16 and width = 4 and height = 4

+

+

P = 1000P = 1000

Z

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Optimization Solution

Extrusion Constraints With One Die

If is often to see some topology optimized designs can contain cavities that are not achievable or require a high cost manufacturing process. For example, the result from the torsion beam without manufacturing constraints is shown in Figure 34-6. Clearly, this topology design proposal is not achievable by casting.

Figure 34-6 Torsion Beam without Manufacturing Constraints

The extrusion constraints enforce a constant cross-section design along the given extrusion direction. The input data related to imposing an extrusion constraint along the z-axis in the basic coordinate system (as the default option) is given in Listing 4.

Listing 4 Input File for Torsion Beam with Extrusion

DESOBJ = 1DESGLB = 1SUBCASE 1$ Subcase name : Default SUBTITLE=Default ANALYSIS = STATICS SPC = 2 LOAD = 2$ Direct Text Input for this SubcaseBEGIN BULKDRESP1 2 Frmass FRMASS

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DRESP1 1 COMPL COMPDCONSTR 1 2 .3TOPVAR, 1 , TSOLID, PSOLID, .3, , , , 1 , EXT , , ZPSOLID 1 1 0

Figure 34-7 shows the topology optimized result with extrusion constraints. It is obvious that the design has a constant cross-section along the z-axis.

Figure 34-7 Torsion Beam with Extrusion Constraints in Z-Axis

Casting Constraints with One Die

A torsion beam (shown in Figure 34-5 is used here to demonstrate the combination of one die casting manufacturability constraints and mirror symmetric constraints.

The casting constraints with one die option enforce that the material can only be added to the region by “filling up” in the given draw direction from the bottom (or, stated another way, that voids extend from the top surface and do not reappear in the die direction). To apply casting constraints and symmetric constraints on designed properties, a reference coordinate system CID=1 is defined by using a rectangular coordinate system CORD1R. A CAST continuation line defines casting constraints in the Y direction and one die is a default option. Another SYM continuation line defines symmetric constraints about the YZ plane. The input data related to the topology optimization model is given in Listing 5.

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Listing 5 Input File for Torsion with One Die

DESOBJ = 1DESGLB = 1SUBCASE 1$ Subcase name : Default SUBTITLE=Default ANALYSIS = STATICS SPC = 2 LOAD = 2$ Direct Text Input for this SubcaseBEGIN BULKDRESP1 2 Frmass FRMASSDRESP1 1 COMPL COMPDCONSTR 1 2 .3CORD1R 1 5 167 7PSOLID 1 1 0TOPVAR, 1 , TSOLID, PSOLID, .3, , , , 1 , CAST, 1 , Y, , YES , SYM, 1 , YZ

Figure 34-8 shows the topology optimized result with one die casting constraints. It is observed that the design material is added by “filling up” in the Y direction from the bottom. In addition, the design is symmetric about the YZ plane in the reference coordinate system CID=1.

Figure 34-8 Torsion Beam with One Die Casting Constraints in Y Direction

MD Demonstration Problems

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Casting Constraints with Two Dies

A torsion beam (shown in Figure 34-5 is also used here to demonstrate two die casting manufacturability constraints.

The input for two die casting constraints is similar to the one die option in Example 5. Here, the difference is that 2 are selected for the DIE field on the TOPVAR entry. The input data related to imposing two die casting constraints is given in Listing 6.

Listing 6 Input File for Torsion with Two Dies

DESOBJ = 1DESGLB = 1SUBCASE 1$ Subcase name : Default SUBTITLE=Default ANALYSIS = STATICS SPC = 2 LOAD = 2$ Direct Text Input for this SubcaseBEGIN BULKDRESP1 2 Frmass FRMASSDRESP1 1 COMPL COMPDCONSTR 1 2 .3CORD1R 1 5 167 7PSOLID 1 1 0

TOPVAR, 1 , TSOLID, PSOLID, , , , , 1 , CAST, 1 , Y, 2, YES , SYM , 1 , YZPSOLID 1 1 0

Figure 34-9 shows the topology optimized result with two die casting constraints. It is observed that the design material grows from the splitting plane in opposite directions along the y-axis specified in the reference coordinate system CID=1. The splitting plane is determined by optimization and in this case corresponds to the

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Figure 34-9 Torsion Beam with Two Die Casting Constraints in Y-Axis

Modeling Tips• It is recommended that a base line topology optimization job (without any manufacturability constraints) be

carried out before a topology optimization solution with manufacturability constraints. Benefits are:

a. a topology optimization without restriction may result in a better design

b. the design proposal from the no restriction run may give some hints for imposing manufacturability constraints.

• Topology optimization with manufacturability constraints often needs more material to fill the design space. Therefore, the design with manufacturability constraints usually requires a relatively bigger mass target (less material savings) than the one without manufacturability constraints.

• The casting constraints may have difficulty dealing with a design model that has one or more non-smoothed boundary surfaces to be designed. It is recommended to use smooth surfaces for topology designed boundary surfaces.

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Input File(s)

File Description

nug_34a.dat Basic compliance minimization

nug_34b.dat Minimum member size

nug_34c.dat Mirror symmetry constraints

nug_34d.dat Extrusion constraints

nug_34e.dat One die casting constraints

nug_34f.dat Casting constraints with two dies