Structural Shape Optimization Based on Parametric Dimension-driving and Cad Software Integration_x+p

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6th

World Congress on Structural and Multidisciplinary Optimization

Rio de Janeiro, 30 May - 03 June 2005, Brazil

Structural Shape Optimization based on Parametric Dimension-Driving and

CAD Software Integration*

Lei DAI*1, Yuanxian GU1, Guozhong ZHAO1, Yingqiao Guo2

1State Key Laboratory of Structural Analysis for Industrial Equipment,

Dept. of Engineering Mechanics, Dalian University of Technology, Dalian 116024, China

Tel & Fax: 86-411-470-8769, Email: [email protected]. GMMS. University of Reims, Faculty of Sciences, BP 1039-51687, Reims, France

1. AbstractThe structural shape optimization is one of the powerful tools to decrease stress concentration and increase structural safety and lifetime.However, among all categories of structural optimization, the shape optimization is a more complicated one due to the shape

description and control, and the associated adaptive mesh regeneration. It’s particularly true when dealing with 3D objects andremarkable shape modifications. Parametric design and dimension-driving technologies have been extensively developed in CAD

software since 1990s. The dimension-driven and parametric design of most of prevailing CAD systems, provide a practical technologyfor the realization of parametric structural shape optimization. The integration between structural shape optimization and such kind of

CAD software is a direct way to make use of these technologies in shape description and control, and then enhance the shapeoptimization in complicated engineering applications. In this paper, parametric structural shape optimization based on the dimensiondriving is developed by integrating with the commercial 3D parametric solid modeling CAD software - Mechanical Desktop® R6.0,

produced by Autodesk company (simplified as MDT6 hereafter). In MDT6, driving dimensions of solids as well as surfaces are directlyused as design variables to control the geometric and computational models. Through ObjectARX - the user Application Programming

Interface (API) of MDT6, the finite element mesh generator, the finite element analysis tool and the optimizer are integrated togetherwith the solid modeling tool in an object-oriented C++ application program interface, and are executed automatically during iteration ofdesign optimization. With geometric constraint and dimension-driving facilities of MDT6, an effective parametric shape optimization

system is accomplished.Key words: parametric design, dimension-driving, shape optimization

2. IntroductionStructural shape optimization is a powerful tool to improve structural performances through changing the shape of structure. Different

from sizing design, structural shape optimization is more complicated, which control the geometry of the structure and typically require

refinement of finite element model during the course of the optimization. The difficulties in structural shape optimization are mostlyencountered in two broad areas [1]. Firstly, because of the continuously changing finite element model, it is difficult to ensure the

accuracy of the finite element analysis remains adequate throughout the design process. Second, it is more expensive to obtain goodsensitivity derivatives with respect to shape design variables than with respect to sizing variables.Most of the prevailing commercial software of parametric 3D solid modeling, being extensively developed in CAD software since

1990s, which can provide users with convenient facilities of interactive solid modeling operation, entity drawing, data management,and also user extended programming tools. With the extended programming tools, users can freely embed other tools on the CADsoftware to realize any specific functionality.

In the present paper, the structural parametric shape optimization system is developed and embedded in a prevailing three-dimensionalsolid modeling CAD software. Through the API of the CAD software, the finite element mesh generator, the finite element analysis

tool and the optimizer are integrated together into the system. During the optimization iteration, structural parametric dimensions,which are defined as design variables of structural shape optimization, will drive the geometric model to automatically renew. Then thefinite element mesh generator, the finite element analysis tool and the optimizer are invoked respectively to renew the finite element

mesh model, compute the structure response and design sensitivity, optimize the design without the users interaction up to theconvergence of the optimization. In the following, the main difficulties of structural shape optimization and available solutions are

described more detailed in the Section 3. Section 4 and 5 of the paper introduce the integrated parametric solid modeling software andthe parametric structural shape optimization system. Numerical examples are illustrated in Section 6. Some conclusions are given inSection 7.

3. Structural Shape Optimization 3.1 The description of design variables in structural shape optimizationThe first difficulty of the structural shape optimization is mainly caused by the un-proper methods of structural shape description andmesh generator. It’s also the main barrier for application of shape optimization design into the industry of engineering and

manufacturing. The design variables in structural shape optimization are selected to descript and control the shape of structure. In along period, the coordinates of the boundary nodes of the finite element model are used as design variables in structural shape

optimization, which is tested to be a un-general method because of hard shape control, un available optimized result and huge amountof design variable etc. Although substantial researches are executed and bring forwards lots of improvements, such as polynomial

*

Project supported by the NSFC (10421002, 10228206) and the Doctor Science Research Start-up Foundation of Liaoning Province(20021063)



representation of boundary, Spline representation of boundary and the design element concept etc., there is still not satisfied and popular resolution with this kind of design variables. For two-dimensional problems, Bennet and Botkin[2] presented a parameterizeddesign description method based on boundary element method. Then Botkin [3] extended the idea to three-dimensional and developed

a library of design features, which included the shape parameters similar to the current CAD tools. In these researches, parametricdesign is taken into account. But without integrated into the parametric modeling technology based commercial CAD tools, theimplementations are strongly limited. Edwin[4] proposed a design parameterization method based on commercial CAD tools. Guan[5,

6] have also realized a structural parametric shape optimization system with another commercial CAD tools. These works overcome the

limitations of Botkin’s method. Those commercial CAD tools are based upon the feature-based design and parametric modelingtechnology. The parametric modeling technology can provide the engineer with the most accurate and convenient method for

representation of geometry and solid design and also allow for repeated design refinement through pre-defined design variables tillsatisfied, which are used to be time consuming and costly. Applying the parametric solid modeling technology into structural shape

optimization, structural shape dimensions can be directly used as design variables and the dimension driving can also used to controlthe modification of the geometry of the structure during the optimization iterations. In the present work, automatic finite element meshgenerating method is adopted. Therefore, during the optimization iterating, the dimension and density of the finite element can be

pre-limited to ensure the accuracy of the result of finite element analysis.

3.2 The finite element modeling in structural shape optimization

The difficulty of the accuracy of the finite element analysis can also be influenced by the mesh generation method applied in shapeoptimization. There are two basic solutions to the problem of adapting the mesh to the changing boundary. The first is use kinds

modification rules for deforming the initial mesh according the changed boundary. The second is based on the use of sophisticatedautomatically mesh generation techniques, which generate a mesh and adaptively improve it based on the calculated response. It has be

discovered that, with the first method of mesh generation, as the design variables changed the boundary of the model, unacceptableelement distortions would occur after even small design changes, maybe just because the un-proportional changes between the designvariables. For the second mesh generation method, adaptive mesh refinement is taken either the form of adding additional elements in

the area to be refined or of increasing the order of the finite element. Bonnett and Botkin [7,8,9] used the strain energy gradients toidentify regions, which require mesh refinement and extend the concept to three-dimensional structures. However, those improvementsin the efficiency of the mesh generator reduced its cost to a few percent to the whole system. And changing of the topology of the finite

element may also lead to the un-continuous of sensitivity analysis by differential of the discredited finite element system or thecontinuum structural constructive equation. A suitable method of mesh generator for sensitivity analysis is to provide automatic mesh

generation while keeping the finite element distortion to a minimum.

3.3 Sensitivity analysis in structural shape optimization

Another difficulty in structural shape optimization is the calculation of the sensitivity derivative. The design sensitivity analysis is thecalculation of quantitative information on how the response of a structure. There are mainly two kinds of method for sensitivity

analysis.

db

dU

U

g

b

g

db

dg

b

F U b

K

b

U K

T

⎟ ⎠

⎞⎜⎝

⎛

∂

∂+

∂

∂=

∂

∂

=∂

∂

+∂

∂

(1)

The first is based on differentiation of the discretized finite element system, and the second based on differentiating the continuum

equations. For the first method, the governing equation for displacement is used to get the bU ∂∂ , and then the derivative of response

function ( )bug , to design variables { }b is obtained with direction method or adjoint method Eq.(1). Actually it’s difficult to obtain the

explicit relationship between the element stiffness matrix and shape design variables. In most of application of this method, finitedifference has been utilized to compute the design derivatives. Thus even a small change of the boundary would lead to expensive

computation.

uvb

uu ∇⋅+

∂

∂=& (2)

The second method of the sensitivity analysis is based on the concept of material derivative from continuum mechanics. The design

velocity field v , as showed in Eq.(2), is taken into account in the derivative of displacement to design variables and also in theresponse function. The design velocity field computation is very crucial to the accurate of this method [10], which are closed linkedwith the regeneration of finite element mesh. In practically, design velocity field are either used to directing regeneration of finite

element mesh or computed from regenerated finite element mesh. Inappropriate design velocity field may lead to erroneous designsensitivity analysis or element distortion. Besides, the design velocity field must depend linearly on the variation of shape parameters.In generally, both of the sensitivity analysis methods desire to keep the topology of the finite element mesh, in other words means

keeping the number of nodes and elements, during a consistent iteration, which may increase the difficulty of the optimization forcomplicated three-dimensional structure.In the present work, based on the parametric solid modeling a parametric finite element modeling method with automatically mesh

regeneration is used to refine the finite element in each of iteration. Topology of finite element mesh is not limited to obtain moreaccurate finite element analysis result while permitting a more greatly change of design variables, which control the structure shape.Accordingly, finite difference algorithm Eq.(3) is used in sensitivity analysis. In sensitivity analysis, only the values of structural

response functions are taken into account. The numerical examples will show the effectiveness and robustness of such a simple method,which is mainly fit for free change of shape design variables of complicated 3D structure. These applications, mentioned here above,

are integrated into Mechanical Desktop6, through the user Application Programming Interface.



( )[ ] ( )[ ]

b

bbU gbbbbU g

db

dg

∆

−∆+∆+=

,, (3)

The numerical examples will reveal the effectiveness and robustness of the parametric structural shape optimization system.

4. Parametric Dimension-Driven Technology of Solid Modeling Software and APIMDT6 is a feature-based and parametric modeling CAD software. It provides parametric solid modeling method and dimension-driven

for solid refinement within user’s design intent. User can apply the pre-defined design variables to dimensions in the whole design process from drawing a two-dimensional sketch on a plane, to defining the operation of extruding, revolving, lofting or sweeping thesketch to create a solid object and refining the solid object with additional features including: hole, shell, draft, fillet, chamfer, rib.

Therefore the design intent can be captured through the relations between geometric dimensions and design variables. Those pre-defined design variables can be documented in family table and system database. User can modify those design variables byvisiting the family table or system database and update the solid object to incorporate design changes. Besides dimension constraints,

geometric constraints can be imposed on the two dimensional profiles to maintain the design intent.MDT6 also provide API toolkit - ObjectARX® to extend capabilities of the software and embed developer’s algorithms and

applications. ObjectARX® programming environment provides an object-oriented C++ application programming interface fordevelopers to use, customize, and extend AutoCAD® and AutoCAD-based products like Autodesk Map™ 3D, Autodesk® LandDesktop, Autodesk® Architectural Desktop, and AutoCAD® Mechanical Desktop. ObjectARX libraries comprise a versatile set of

tools for application developers to take advantage of the open architecture of AutoCAD, providing direct access to AutoCAD databasestructures, the graphics system, and native command definition. With the developing toolkit, developer’s application can be integrated

into the software system and loaded onto the user graphic interface as menus, toolbars and executive commands. User can call those plug-ins as freely as other native commands in the CAD software.

5. Parametric Structural Shape Optimization System based on MDT6The system architecture and flow chart of the parametric structural shape optimization system developed in the present work is showed

in Figure1. The system mainly includes two parts. I. User Graphic Interface of the three-dimensional CAD software, where user cancreate geometric model of structure and define design variables to concretize the design intent. The parametric structural shapeoptimization system can be running within the user graphic interface of the CAD software, including geometric modeling, finite

element model definition, finite element analysis, structural shape optimization model definition. II. User Developing Interface with

programming tools – ObjectARX, through which the finite element meshing tool – AutoFEM, the finite analysis tool of the

commercial structural analysis and design optimization system JIFEX and optimization tools - DOT are introduced into the CADsoftware. Combined with essential numerical algorithms these tools can be utilized as Plug-ins with embedded Commands/Menus and buildthe parametric structural shape optimization system within MDT6. During the optimization iteration, design variables are modified

after each optimization and drive the geometric model to regenerating, then the mesh generator will be evoked automatically to update previous finite element model. The system will read the present data of objection function and constrain functions from the results afterfinite element analysis computation and start a new optimization loop until the optimizer runs into a convergence and end the iteration.

Figure1. System architecture and flow chart

The proposed system has two major characteristics. The first is that the driving dimensions are directly used as design variables in

structural shape optimization without other additional operations. Most of common 3D mechanical parts such as cube, cylinder, cone,sphere, tube, pipe, and the assembly of these primitive geometry elements can be used as optimization object. In the process of partmodeling, the user pre-defined design variables, which are imposed as solid object dimension parameters in CAD software, are used to

driving the refinement of the structure. All user familiar geometric dimensions including length, breadth, thickness, height,diameter/radius, degree, relative position and so on can be defined as design variables. Once the design variables are modified within

the structural shape optimization, MDT will regenerate the model automatically. The second characteristic of this system is the

topologically un-consistent of the finite element mesh. In the shape optimization the topology of geometric model is fixed, while thenumber of nodes and elements are updated in each regenerated finite element model. The finite element model including finite element



mesh and its property information such as loads, boundary conditions, material and geometric properties, is automatically updated with

the dimension-driving parameters. With this scheme, element distortions that are encountered in remarkable shape modification inconventional structural shape optimization are avoided. Therefore, large modifications of shape design variables are permitted in the proposed structural shape optimization system.

5.1 Finite element modeling in present system

The finite element mesh generator used in the system is AutoFEM, which is the pre-process application of commercial structural

analysis and design optimization system JIFEX. AutoFEM can create finite element model base on B-rep (boundary representation)data of geometric model. With the user extended programming tool provided by MDT6, AutoFEM is embedded in and permitted to

visit the Brep data of the geometric model. When the geometric model is update with changed design variables, the finite element meshcan be regenerated automatically according to the Brep data. Besides finite element mesh, finite element model also include its

properties such as loads, boundary condition, structural material and element properties. In the earlier structural optimization, these properties are used to be assigned to the finite element nodes and elements directly. In the present work, these properties are assigned tothe geometric model including surface, curve and point, and then transfer to finite element mesh before finite element analysis through

an additional data transformation algorithm. When changes are made to geometric model, they can be immediately automaticallyre-assigned without user interaction during the iterations. It will be helpful for design-related load and boundary condition problem tocertain extent. In the present system, the finite element model of solid, plate, shell and beam structures is constructed with tetrahedron

element.

5.2 Definition of structural shape optimization in present system

In addition to static structural responses, such as nodal stresses and nodal displacements, structural dynamic responses, including

structural natural frequency, can also be considered in this system. The modified feasible direction algorithm and sequentiallinear/quadratic programming algorithms can be used as optimization algorithms in this system. In terms of the above-mentioned

un-limited topology of finite element model, the global finite difference method is used to obtain the sensitivity analysis information.During the optimization, multiple numbers of a same constraint are imposed on specific structure response field to prevent the jumping

of the minimal/maximal value from nodes or elements.

6. ExamplesIn this part, two numerical examples illustrate the effectiveness and robustness of the present parametric structural shape optimizationsystem.

6.1 Shape optimization for minimizing structure weight of horizontal lathe-bed

A numerical example of parametric shape optimization of horizontal lathe-bed is given to testify the practicability of the system. Thestructure is the middle part of a horizontal lathe, which is also the trajectory as showed in Figure 1. To simulate the actual constraintrelation with the supporting body of the lathe at each side, one side of the structure is clasped at x, y and z direction while the other side

just be clasped at x and y direction. The structure subjected to three uniform surface pressures, which is to simulate the load condition

of slipping work-platform on the trajectory. There are totally 154 dimensions used to descript the shape of the structure, which arelabeled in the two profile of structure in Figure 3. Among them, 6 dimensions are set to be design variables of shape optimizationamong others, namely Dim01 to Dim06 respectively. These design variables and optimized result are showed within Table 1. Theobject function history in optimization is showed in Figure 4. The system spent half and two hours on this numerical example. Structure

weight is to be minimized. The constraint function is the quantity of maximal displacement within 3.0e-3mm.

Figure 1. Geometric model of initial design



Figure 2. Structural displacement cloud of optimum design

Profile of left view

Profile of front viewFigure 3. Descriptions of structural design variables



Table1. Optimization result of numerical example 1

Design Items Initial design Optimized design

Thick01 19.000 11.650

Thick02 20.000 11.000

Thick03 13.000 11.000

Deepth01 20.000 20.000

Height01 80.000 40.000

Design variables

(mm)

Height02 198.000 170.000

Objective (kg) Structure weight 159.5 119.6 ↓25.01%

1st Displacement -2.516e-2 -2.97-e-2 ↑18.04%Constrains (mm)

2nd Displacement -2.504e-2 -2.96e-2 ↑18.13%

Node number 6264 5746Finite elementmesh model Element number 19404 18803

obj ect f uncti on hi story

0.000

20.000

40.000

60.000

80.000

100.000

120.000

140.000

160.000

180.000

1 4 7 1 0

1 3

1 6

1 9

2 2

2 5

2 8

3 1

3 4

3 7

4 0

4 3

4 6

i t erat i on

s t r u c t u r e w e i g h t

desi gn var i abl es hi st ory

0. 000

50. 000

100. 000

150. 000

200. 000

250. 000

1 4 7 1 0

1 3

1 6

1 9

2 2

2 5

2 8

3 1

3 4

3 7

4 0

4 3

4 6

i t erati on

d e s i g n v a r i a b l e s

DeVar1

DeVar2

DeVar3

DeVar4

DeVar5

DeVar6

Figure 4. Object function history in optimization

6.2 Shape optimization for maximizing structure minimal natural frequency of flange

The initial design and optimized design are showed in Figure 5. In the optimization, four dimensions of the sweeping profile, showed in

Figure 6., are defined as design variables, optimization object function is maximize the structural minimal natural frequency andstructure weight and maximal nodal displacement are selected as constraint functions to be within the initial design during the iteration.

Initial design

Optimum designFigure 5. The initial and the optimized design of flange



Figure 6. Sweeping profile of structure

Table2. Optimization result of numerical example 2

Design Items Initial design Optimized design

Dim01 36.5 10.0

Dim02 20.0 25.0

Dim03 25.0 10.0

Design variables

(mm)

Dim04 40.0 15.2

Objective Minimal nature

frequency

6.298 12.839 ↑103.86%

Structure weight 4.91e3 1.91e3 ↓61.10%Constrains(g, mm) 1st Displacement 3.824e-3 3.783e-3 ↓1.07%

Node number 1508 1500Finite elementmesh model Element number 5055 4328

desi gn var i abl es hi st or y

0. 000

5. 000

10. 000

15. 000

20. 000

25. 000

30. 000

35. 000

40. 000

45. 000

1 3 5 7 9 11 13 15 17 19 21 23

i t er at i on

d e s

i g n

v a r

i a

b l e s

Figure 7. The design variables history in optimization

After the optimization, the structural minimal natural frequency is increased 103.86%, while structure weigh is reduced 61.10% at the

same time and structure maximal nodal displacement is a little reduced also. The system spent half hour on this numerical example.

7. ConclusionIn the present paper, integration with prevailing commercial 3D parametric solid modeling CAD software is to facilitate the applicationof structural shape optimization in the industry of engineering and manufacturing. The fully automatically mesh regeneration withun-limited topology is utilized in finite element modeling and the finite difference is utilized in sensitivity analysis, all adoption are for

the purpose of simplify operations and avoiding the above-mentioned difficulties in mechanical structure shape optimization,especially complicated 3D structure with remarkable shape change during design process. Certainly, these adoptions have to achieved



at the expense of efficiency. But such a loss is endurable in practical implementation of those numerical examples. In such way, this

parametric shape optimization system can also be integrated with other prevailing commercial 3D parametric solid modeling CADsoftware, being familiar with those design engineers, and therefore, as an effective and suitable tool this system can be made availablefor them in their early prototype design phase. The numerical examples are also given to illustrate the effectiveness and robustness of

the proposed system. Besides the parametric structural shape optimization of solid object, similar work on surface object can also be putinto practice.

8. AcknowledgeThe supports for the projects of the National Natural Science Foundation of China (Grand No.s 10421002, 10421002, 10228206) and

the Doctor Science Research Start-up Foundation of Liaoning Province (No.20021063) gratefully acknowledged.

9. References1. R.T. Haftka and R.V. Grandhi. Structural shape optimization–A survey. Computer Methods in Applied Mechanics and

Engineering. 1986, 57: 91-106

2. J.A. Bennett and M.E. Botkin. Shape optimization of 2-D structure with geometric problem description and adaptive meshrefinement. AIAA J., 1985, 23(3): 258-264

3. M.E. Botkin. Shape design modeling using fully automatic three-dimension mesh generation. Finite Element in Analysis and

Design. 1991, 10: 165-1814. Edwin Hardee, K.H. Chang, etc. A CAD-based design parameterization for shape optimization of elastic solids. Advanced in

Engineering Software. 1999, 30: 185-199

5. Z.Q. Guan, X.F. Sui, Y.X. Gu, etc. A CAD-based parameterization method of finite element modeling for structural shape

optimization, Proc. WCSMO-4, 2001, China, 418-4196. Z.Q. Guan, X.Y. Du, Y.X. Gu, etc. Parameterization technique for elastic-plastic analysis and shape optimization design of

mechanical structures. Journal of Mechanical Strength. 2003, 25(4): 420-4257. J.A. Bennett and M.E. Botkin. Shape optimization of two-dimensional structures with geometric problem description and adaptive

mesh refinement. AIAA/ASME/ASCE/AHS 24th Structures, Structural Dynamics and Materials Conference. AIAA-83-0941,Lake Tahoe, NV, 1983

8. J.A. Bennett and M.E. Botkin. Structural shape optimization with geometric description and adaptive refinement. AIAA J. 1984,

23: 458-4649. M.E. Botkin, J.A. Bennett. Shape optimization of three-dimensional folded plate structures. AIAA J.1985, 23(11): 1804-1810

10. K.K. Choi, K.H. Chang. A study of design velocity field computation for shape optimal design. Finite Elements in Analysis andDesign. 1994, 15: 317-341

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Structural Shape Optimization Based on Parametric Dimension-driving and Cad Software Integration_x+p