PROGRESS IN TOPOLOGY OPTIMIZATION WITH MANUFACTURING CONSTRAINTS

8/22/2019 PROGRESS IN TOPOLOGY OPTIMIZATION WITH MANUFACTURING CONSTRAINTS

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PROGRESS IN TOPOLOGY OPTIMIZATION WITH MANUFACTURING

CONSTRAINTS

M. Zhou, R. Fleury, Y.K. Shyy, H. Thomas, J.M. Brennan

Altair Engineering, Inc.2445 MacCabe Way, Suite 100, Irvine CA92614

[email protected]

Abstract

Topology optimization has been shown to be an

extremely powerful tool in generating efficient design

concepts in the early stage of a design process.

Unfortunately, very often designs suggested by

topology optimization turn out to be infeasible for

certain manufacturing process. At such occasions, it is

often very difficult, if not impossible, to transform a

design proposal to a manufacturable design. In this

paper, design requirements for casting and extrusion

production are addressed for topology optimization.

Introduction

Topology optimization has seen rapid development

during the last decade, both in research and industrial

applications [1-20]. The power of this technology lies

in its early impact in a design process. It has been

shown that topology optimization can help to create

highly efficient design concepts. This very often leads

to much more significant design improvement

compared to sizing and shape optimization that can beonly applied to a structure with given layout. The fast

development of commercial software contributed to a

rapid penetration of this technology in the industry.

Most commercial FEA software have added certain

capabilities of topology optimization. In this

development, Altair OptiStruct [23], a commercial

product that specializes in design optimization, has

emerged to be a front runner in this direction. Several

capabilities are unique to this product. It uses a general

setting of a multiple constrained optimization problem,

in which topology, shape and sizing optimization can

be handled simultaneously [18]. Minimum member size

control provides a means to control the degree ofsimplicity of the solution of topology optimization,

which could be interpreted as a general purpose

manufacturing constraint [17].

Copyright 2002 by M. Zhou, R. Fleury, Y.K. Shyy,

H. Thomas, J.M. Brennan

Published by the American Institute of Aeronautics and

Astronautics, Inc., with permission.

At the conceptual design stage, manufacturing

feasibility is one of the key requirements. For example,

casting is a very popular manufacturing means for mass

production of machine parts. To be feasible for casting,

cavities in the structure has to be open and lined up

with the sliding direction of the dies. Unfortunately,

topology optimization very often creates cavities that

would prohibit drawing the dies. In many cases, it is

impossible to transfer such a design concept into one

that is feasible for casting production without losing its

constructive merits. In a recent paper by Zhou, Shyy

and Thomas [19], a general mathematical approach has

been proposed for casting manufacturing constraints. In

this paper, practical applications of the proposed

method are shown. This capability has been

implemented in the commercial software Altair

OptiStruct. It is applicable to any 3D finite element

model with arbitrary mesh. This capability is recently

released with OptiStruct 5.1 [23].

Furthermore, requirements for extrusion, which is alsoa popular manufacturing procedure for mass

production, are addressed in this paper. For extruded

parts, the cross-section has to be constant along the path

of extrusion.

Topology optimization problem

The general optimization problem can be stated

mathematically as follows

Nii

MjUjgjg

f

,...,1,10

,...,1,0)(Subject to

)(Minimize

=

=

(1)

Where )(f represents the objective function, )(jg

andUjg represent the j-th constraint response and its

upper bound, respectively. M is the total number of

constraints; i is the normalized material density of the

i-th element. Note that the problem in (1) is a relaxation

formulation of the topology problem, where the density


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should only take the value 0 or 1. To enforce the design

to be close to a 0/1 solution, a penalty is introduced to

reduce the efficiency of elements with intermediate

densities. For the SIMP approach [6][7] the

penalization is achieved by the following power law

formulation:

)( i

p

iiiKK = (2)

where Ki and Ki represent the penalized and the real

stiffness matrix at full density of the i-th element

respectively, and p is the penalization factor that is

bigger than 1. Typicallyp takes value between 2 and 4.

In general, the optimization problem in Eq.(1) involves

a very large number of design variables. However, the

number of active constraints is usually small, if local

constraints such as stress constraints are excluded.

Because of this characteristic, the problem can be

solved very efficiently by the dual method of nonlinear

programming [21], [22]).

Casting Manufacturing Constraints

The mathematical formulation for casting constraints

has been presented in [19], which is summarized below.

Considering the finite element model shown in Fig.1,

where the mesh is perfectly lined up in the sliding

direction, the following constraints can sufficiently

prevent the creation of cavities and boundaries that

prevent the slide of the die:

...nji

(3)

where nji ,...,, represent densities of elements that

are along the same line in the sliding direction, as

shown in Fig.1.

Figure.1 Illustrative finite element model

Thus the optimization problem shown in (1) should be

modified in the following manner:

Kkknji

MmU

mg

mg

f

,...,1,)1...0(

,...,1,0)(S.t.

)(Min

=

=

(4)

where Krepresents the number of sets of elements that

are lined up in the sliding direction of the die. It can be

shown that the additional single variable linear

constraints can be handled efficiently in the dual

optimizer similar to side constraints [21].

A more typical scenario in casting is to have two dies

pulling apart along the sliding direction. For this case,

there are two sequences of variables along each line,

separated at the die splitting point. Therefore, the above

formulation does cover this situation when the die

splitting surface is known. In OptiStruct, a method has

been developed to optimize the die splitting surface

under this two-die scenario [23]. Also handling of

irregular mesh, e.g. tetra mesh, has been implemented.

The details for these techniques are not presented in this

paper. Different draw directions can be applied to sub-

domains of the packaging space for topology

optimization. This can be useful in achieving so-called

casting patches that are sometimes applied to

complicated parts.

Extrusion Manufacturing Constraints

For the model shown in Fig.1, the requirement forextrusion in the sliding direction is

...nji

=== (5)

Which represents a simple variable linking. Thus the

optimization problem (1) is modified as follows:

Kkknji

MmU

mg

mg

f

,...,1,)1...0(

,...,1,0)(S.t.

)(Min

=

===

=

(4)

where Krepresents the number of sets of elements that

are lined up along the path of the extrusion. This

problem actually represents a simplification of the

original problem (1) in that the number of independent

design variables is significantly reduced. This method is

implemented in OptiStruct for the forthcoming release.


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Numerical Examples

Four numerical examples are presented to illustrate the

impact of manufacturing constraints discussed in this

paper.

Example 1 Beam under Torsion with Casting

manufacturing constraints: A coarse mesh of the beam

has been used in reference [19] for study during the

research phase. Here a finer finite element mesh, shown

in Fig.2, is used and the structure is optimized using the

released code OptiStruct 5.1 [23]. The beam has a

dimension of 4x4x16, and 2048 hexagonal elements are

used. One end of the beam is fixed and a pair of

twisting forces is applied on the free end. The

compliance is minimized subject to a volume fraction

constraint of 0.3. Without casting manufacturing

constraints we obtained a tube like structure shown in

Fig.3, which is reached after 12 iterations. This design

is indeed the optimal topology under torsion. However,

if the beam has to be produced with a single die

drawing upwards (in the Z direction), the tube solutiondoes not give any idea for a casting feasible design. The

solution under the given casting constraints is shown in

Fig.4, which is achieved after 73 iterations. Periodical

Fig.2 FE model of the beam under torsion

Fig.3 Optimized topology without casting constraints

Fig.4 Optimized topology with casting constraints

X cells are formed to carry the twisting forces to

the supported end.

Because of the existence of semi-dense elements, a

comparison of the performance of the final designs

should be done with new FE models recovered from the

final results. This could be done by post processing a

result using OSSmooth embedded in Altair HyperMesh

[24] to generate iso-surfaces for a given density

threshold. During this process, OSSmooth also

generates a tetra mesh for reanalysis. However, it is

very difficult to find appropriate density thresholds for

two designs that would result in the same material

volume for comparison. As an alternative, analyzing thefinal result without penalty for intermediate density can

be a reasonable approximation. This implies from

Eq.(2) that the stiffness of an semi-dense element is

considered to be linearly proportional to its material

density. Eliminating penalty the compliance for the

designs with and without casting constraints are

1.37688 and 2.44428, respectively. This means that the

casting feasible design has a torsional stiffness of 56%

of the design without casting constraints. Despite of the

significant loss in stiffness compared to the tube design,

the casting feasible design does appear to be an optimal

concept under the given manufacturing constraints. The

concept of using X cells to transform the twisting forceseliminates bending at the system level.

Example 2 Engine Bracket with Casting

Manufacturing Constraints: An aluminum engine

bracket of a car has been optimized using earlier release

of Altair OptiStruct and a reduction of weight from 950

g to 739 g has been achieved in the finalized product

[20][17]. Since this is an aluminum cast part, it

represents a good example for investigating the


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influence of the casting constraints during topology

optimization. The finite element model of the design

domain is shown in Fig. 5, in which 9046 elements are

used and the design domain is shown in blue color. Six

load cases were considered, which reflect the following

driving and service status: 1) start; 2) backwards; 3)

into a pothole; 4) out of a pothole; 5) loads from an

attaching part and 6) loads during engine transport. The

sum of compliance of all load cases is minimized for a

given volume fraction of 0.3. The result without casting

manufacturing constraints is shown in Fig. 6, which is

reached after 25 iterations. An minimum member size

constraint ofdmin=15 mm is used, which is close to the

default minimum member size of 16.50 mm determined

by the average mesh size when casting constraints are

applied. With a casting draw direction upwards (in the

Fig.5 Finite element model of the engine bracket

Fig.6 Topology of the engine bracket without casting

constraints

Z direction), the result of a single die is shown in Fig. 7

and the result of two dies splitting is shown in Fig. 8,

which needed 26 and 25 iterations, respectively.

Reanalyzing the final designs without penalty for

intermediate density provides a reasonable baseline for

performance comparison. The compliance of the design

without casting constraints is 52.7582. The compliance

of the designs with casting constraints are 55.0611 for

the single die option and 51.4573 for the two die

splitting option. Comparing the reciprocals of the

combined compliance, the values for the casting

designs with a single die option and two die splitting

option are 95.8% and 102.5%, respectively, of the value

of the design without casting constraints. Note that the

Fig.7 Topology of the engine bracket with casting

constraints single die


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relatively small differences in compliance may fall into

the range of errors due to allowance of semi-dense

elements in the model. Nevertheless, the results indicate

that for this design problem, the additional casting

manufacturing constraints did not reduce the

performance of the optimized design. Clearly, the

design proposals with casting manufacturing constraints

are much easier to interpret.

Fig.8 Topology of the engine bracket with casting

constraints two dies splitting

Example 3 Rail with Extrusion Requirements: A

curved beam is considered to be a rail over which a

moving load is applied (Fig. 9). 59151 elements, mostly

hexagons, are used in the FE model. Both ends of the

beam are simply supported. The moving load is

simulated as a point load applied over the length of the

rail as five independent load cases. The rail should be

manufactured through extrusion. The objective is to

minimize the sum of the compliance under all load

cases. The material volume fraction is constrained

Fig.9 FE model of a rail under moving loads

Fig.10 Topology of the rail without extrusion

constraints


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at 0.3. The optimized topologies without and with

extrusion constraints are shown in Fig.10 and Fig.11,

respectively. 27 and 35 iterations are needed,

respectively.

Reanalyzing the final designs without penalty for

intermediate density, the compliance for the designs

without and with extrusion constraints are 29.9396 and

37.4377, respectively. This implies a 20% loss in

performance due to extrusion constraints. Note that the

extrusion design represents a clean proposal that

requires little refinement. On the other hand, the design

obtained without manufacturing constraints may require

significant modification that could cause efficiency loss

in performance.

Fig.11 Topology of the rail with extrusion constraints

Example 4 Layout of Stiffening Ribs of a Folding

Chair: The design problem is to find the optimal layoutof stiffening ribs underneath the sitting surface of a

folding chair. For this purpose, the shell structure is

filled with solids in the FE model shown in Fig.12. A

single load case simulating the sitting pressure load is

considered. The dimension of the model is about

10x8.5x1.2 in. The compliance is minimized subject to

a volume fraction constraint of 0.1 for the solid domain.

For the run without draw direction constraints, a

Fig.12 FE model of sitting area of a folding chair

minimum member size of 0.2 in is used. For the

run with a draw direction perpendicular to the sitting

surface, a default minimum member size of 0.46 in is

active, which is calculated based on the average meshsize by OptiStruct [23]. The optimized topology

without draw direction constraints is achieved after 38

iterations, which is shown in Fig.13. It can be seen that

the structure forms a sandwich box underneath a

portion of the area under pressure load. Some rather

arbitrary rods are formed to connect the top and the

bottom shells. It is clear that this design proposal does

not give many clues about rib layout.

The optimized topology with draw direction constraints

is reached after 52 iterations, which is shown in Fig.14.

The result provides us with a clear design of the layout

of stiffening panels. For a comparison of their structuralperformance, both final designs are analyzed without

penalization of intermediate density. The compliance

for the structures without and with draw direction

constraints are 1.00449 and 0.94585, respectively. This

implies, to our surprise, a 6% higher performance for

the design with draw direction constraints.

In general, it is true that additional constraints should

reduce the feasible design region and hence result in a


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higher objective function at the optimum. However, the

topology optimization problem in (1) is a highly none

convex problem due to the penalty applied. The

existence of local optima could prevent the iterative

process to achieve a better design, although the

formulation without draw direction constraints does not

exclude the design shown in Fig.14. The same

phenomenon has also been found with the additional

constraint on minimum member size [17].

Fig.13 Topology of the folding chair without draw

direction constraints

Concluding Remarks

It has been shown that manufacturing constraints are

important aspects that require attention during the

conceptual design stage. Since topology optimizationserves as a tool for generating design concepts,

incorporation of manufacturing constraints has

significant impact in shortening the distance between

concept and reality.

In this paper, extension and applications of casting

manufacturing constraints are presented. This capability

has been implemented in the commercial software

Fig.14 Topology of the folding chair with draw

direction constraints

Altair OptiStruct 5.1 that has been released in June

2002 [23]. In addition, the formulation and solution of

extrusion constraints are introduced. This capability has

also been implemented in OptiStruct and will be

available in the forthcoming release. Four applications

demonstrate that solutions under manufacturing

constraints are usually non-intuitive and could not be

easily derived from topology designs obtained without

taking manufacturing constraints into consideration.

It is worth mentioning that both casting and extrusion

manufacturing constraints could be used for conceptual

design study of structures that do not need to be

manufactured using the corresponding procedures.

Those requirements could be regarded as specific

geometric constraints and can be used for any design

that desires such characteristics. Example 4 showed that

the casting constraints could be used to locate the

layout of rib patterns of a shell structure. If it is desired

to have ribs going through the entire depth of a solid

domain, extrusion constraints can be used to determine

the layout of the stiffening panels.

It is strongly recommended that when manufacturing

constraints are desired, the user should run a baseline

comparison with a formulation without those additional

constraints. This could help to assess the relative

efficiency loss due to the manufacturing constraints.

This provides a trade-off study between cost and

structural performance in the decision process for the


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right manufacturing procedure for the structure under

consideration.

There are more manufacturing constraints that should

be addressed during topology optimization. One that

has been frequently requested by users of OptiStruct is

a maximum member size control. For example, this

requirement is important for casting or extrusion parts,

where the thermal dissipation requirements impose

restrictions on member thickness. It appears that the

more manufacturing constraints are considered for

topology optimization, the closer to the final product

reality the solution can be. A virtual prototyping

directly out of topology optimization represents the

ultimate challenge.

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