shape optimisation for structural design by means of

Shape Optimisation For Structural Design By Means Of Finite Elements Method

Marco Evangelos Biancolini, Carlo Brutti, Eugenio Pezzuti

Università degli Studi di Roma “Tor Vergata” Dipartimento di Ingegneria Meccanica Via di Tor Vergata 110, 00133 Roma E-mail: [email protected]

Abstract

In this paper a shape optimisation method is exposed. The optimum shape of a three-dimensional structure was searched to obtain the minimum weight with prescribed constraint on stiffness and maximum stress. The calculations were performed directly on the FEM model of the structure without links to the original geometric model: shape evolution was generated as a linear combination of vectors belonging to the FEM model nodal positions space. A suitable vector basis is obtained by a set of fictitious FEM analyses in which the new shapes that respect design symmetry and fixed dimensions result imposing proper constraints, materials and load conditions. Of course vector basis plays a crucial role on the results both for structural performance and for aesthetic. A typical application of the method was exposed concerning the optimisation of a motorbike frame.

1. INTRODUCTION

Structure design is a continuos challenge toward the best for strength, minimum weight and cost. Actual power of calculation available even on inexpensive personal computer has introduced powerful simulation tools as CAD, FEA, MB in every design fields. New tools are able to reproduce reality in a more and more reliable way so that the engineer is able to know physical quantities in a detail degree that in many cases is not accessible even experimentally. Simulation tools are the keys to reduce time to market, getting the best performance reducing the number of hard prototypes; the term “Virtual Prototyping” explain very well the actual design trend. Obviously the optimisation of a “virtual prototype” could be conducted with lower efforts saving time and money against the “hard prototype”. Shape optimisation is a very interesting tool in many fields of mechanical engineering for structural aspects (FEM), fluidodynamic (CFD), acoustic, and many other. Shape optimisation, together with emerging new techniques of topological optimisation can lead to new design solutions that at a first glance could not even supposed by the engineer. Simulation of physic, combined with geometric and topologic optimisation, can lead to new shapes and new ideas in design. Functional optimisation has to be performed under expert designer judgement, taking into account aesthetic aspects and construction feasibility.

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XII ADM International Conference - Grand Hotel - Rimini – Italy - Sept. 5th-7th, 2001

Shape evolution need a parameterisation of a starting model. This parameterisation could be achieved at various level. Geometric parameterisation is a standard tool available in advanced solid modeller and is very useful for integrated design CAD/CAE/CAM. Many FEA packages are directly connected with solid modeller [1]. A different approach consists in the local parameterisation of boundary surfaces, subdividing the boundary in regular patches connected at moveable key points; prescribed continuity order could be imposed in the local displacements fields [2]. A simpler approach consists in the parameterisation of the discrete representation of the structure. Within a FEA model of a structure, a lot of starting geometry information are preserved, and overall model belongs to a finite dimension space (order 3*N in the space, if N is the number of nodes). Parameterisation could be easily achieved by means of a base for a subspace of the model (order<<3*N). This approach preserve starting topology of discrete model and is free of troublesome coupling between geometry and FEM model; on the counterpart topology preserving could lead to distorted meshes for strong shape variations. In this paper shape optimisation based on discrete model will be exposed; a commercial package devoted to this task (MSC/Nastran 70.5) will be exploited to perform numerical optimisation. A brief description of implemented procedures will be conducted, introducing minimisation procedures gradient based, and sensitivity coefficients. Furthermore, a typical application of shape optimisation in structural analysis will be presented, about the optimisation of frame for bikes or motorbikes [3,4]. Great attention was recently devoted on this matter [6,7]. The application consists in the definition of preliminary geometry of the frame within a solid modeller, conducted imposing the proper geometric constraints. Finite elements model is obtained by means of a devoted preprocessor (Femap 7.0). Shape optimisation is conducted in order to reduce the weight of the structure with respect of constraints on torsional stiffness, maximum stress, byke parameter and layout compatibility with engine and other components.

1.1 Numerical procedure

Shape optimisation was conducted by means of the software MSC/Nastran V70.5 [8,9]. This Finite Elements Package is a very powerful tool for structural analysis and includes an internal optimisation kernel gradient based. Internal optimisation solution (SOL200) takes the advantage to exploit all the calculation feature available for the FEM model; goal function and constraint functions could be evaluated by means of every analysis implemented in the Nastran solver . For instance, in the same optimisation run we can minimise the weight of the structure imposing to have a prescribed stress limit for different load and constraint condition, to have a buckling multiplier less than a prescribed value for a particular load, to have the first natural mode upper then a prescribed value and so on. Each FEM parameters can be elected as optimisation variable (Beam Area, Plate Thickness) or can be expressed in term of optimisation variables.

1.2 Minimisation strategy

Minimisation algorithm implemented in MSC/Nastran is based on gradient search direction. For this reason partial derivatives of the response quantities with respect to the design variables are required in order to evaluate the gradients of objective and constraint functions. A crude incremental calculation for such quantities should be very expensive in

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terms of computational effort: for this reason an approximated approach is adopted based on sensitivity matrix obtained exploiting the results of the previous calculation step [10]. For the static analysis we have: [ ]{ } { }PYK = (1) where [K] is the stiffness matrix, Y the displacements vector and P the load vector. Differentiating eq. (1) we obtain:

[ ]{ } [ ] { } { }

iii

PYKY

K

λλλ ∂∂=

∂∂+

∂∂

(2)

where λi is the perturbed design parameter. Sensitivity terms of displacements with respect of design parameters are obtained as follows:

{ } [ ] { } [ ]{ }

∂∂−

∂∂=

∂∂ − Y

KPK

Y

iii λλλ1 (3)

Being the matrix [K]-1 already available from previous calculation step we need only to

evaluate { } [ ]

ii

KP

λλ ∂∂

∂∂

, terms. This evaluation could be performed numerically as follows:

{ } [ ] { } { } [ ] [ ]

i

new

i

new

ii

KKPPKP

λλλλ ∆−

∆−

=∂∂

∂∂

,, (4)

furthermore, being the stresses related to the displacements by the recovery matrix: { } [ ]{ }YR=σ (5) sensitivity terms of stresses could be easily evaluated as:

{ } [ ] [ ]

ii

YR

λλσ

∂∂=

∂∂

. (6)

Analogous considerations could be extended for modal analysis, dynamic analysis, buckling analysis and so on. Optimum search in Nastran is performed using The Modified Feasible Direction Algorithm applied to the approximate problem while convergence is verified according to the Kuhn Tucker criterion. Let { }( )λf to be the objective function , subject to

active constraints { }( ) 0≤λjg . Feasible direction search{ }S is determined imposing the

minimisation of

{ } { }( )λfS T ∇ subjected to { } { }( ) 0≤∇ λjT gS . (7)

Optimisation problem, once determined the feasible direction is reduced to a mono dimensional problem as follows: { } ( ) { } { }Soldnew ξλξλ += (8) Implementation is performed numerically according to the following steps: • Evaluate { }( )λf { }( )λjg for starting { }λ .

• Identify the set of critical and near critical constraints J.• Calculate { }( )λf∇ and { }( )λg∇ for active constraints.

• Determine a usable-feasible search direction { }S

• Perform a one-dimensional search to find ξ *• Set { } ( ) { } { }Soldnew

*ξλξλ +=• Check for convergence to the optimum

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1.3 Optimisation parameter in shape optimisation

In order to achieve shape optimisation the nodal positions of Finite Element Model have to be connected to optimisation parameters; one way to have this dependence is to express the new model positions (e.g. new shape) as the original model perturbed by a linear combination of a vector basis, in which the weights of the combination become the optimisation variables. Assuming that starting geometry is expressed by nodal positions (x,y,z) and that basis vectors are obtained displacing each node in a convenient direction (δx, δy, δz), overall geometry results in the following linear combination:

∑

+

=

ii

startz

y

x

z

y

x

z

y

x

δδδ

λ for each node. (9)

In this way structural model is linked to design variables λi .

1.4 Vector basis generation

Discrete representation of a structure by finite elements method consists in the subdivision of overall domain in a set of elements connected at shared nodal positions. If the structure is discretised by N nodes, in the space we have a 3N dimensional vector (2N in the plane). A straightforward method to generate a vector in this space consists in the use of a fictitious FEM model, topologically identical to the original one, but loaded and constrained in order to obtain a suitable displacements field. In order to achieve a good parameterisation we have to select properly the vector basis. Design constraints as fixed curves, points or surfaces are implemented by means of rigid constraints on auxiliary model. Furthermore desired symmetry of the model has to be conserved imposing proper constraint and loads. Excessive mesh distortion during optimisation could be reduced optimising each vector assuming a non-uniform Young modulus distribution in auxiliary model [4].

2. SHAPE OPTIMISATION FOR A MOTORBIKE FRAME

Numerical procedure described was exploited to optimise the weight of a motorbike frame, subject to constraints on torsional stiffness and structural integrity.

2.1 Design parameters

Prescribed functional data for the frame that we are going to analyse are summarised in the following table [3,11].

Span (mm) 1400

Steering bar angle (°) 24.5

Front run (mm) 97

Wheels diameter (mm) 508 (front), 458 (rear)

Saddle height (mm) 820

Max width (mm) 420

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Fork length (mm) 660

Pitch / Fork length ratio 1:3

Table 1 – Functional data for the frame

Starting from design parameters we are able to define a path in the space for the backbone of the frame represented in fig.1. Now we have to define a proper cross section for lateral beams. We need a profile with good torsional stiffness, high strength and minimum, weight so, according to actual trends in frame design, a octagonal closed section was chosen and id represented in fig 2.

Figure 1: Design parameters Figure 2: Cross section

Of course starting geometry is only a guess for our shape optimisation: we need it to properly define structure topology. According to the previous sketch a full 3D model was obtained lofting the cross section and is shown in fig. 3

Figure 3: Preliminary frame

A typical material for such applications was chosen. In particular we decided to use an aluminium alloy ANTICORODAL 100 (P-Al Si Mg 7 UNI EN 568-2) with the following properties: • Young Modulus E = 68670 MPa • Poisson Modulus ν = 0.33• Ultimate stress Ru = 310 MPa • Yielding stress Ry = 260 MPa • Density ρ = 2690 kg / m3

Furthermore a prescription on torsional stiffness is required because high stiffness guaranties a neutral behaviour of the frame preserving assigned geometric parameters (assuming a

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perfect rigid frame). Torsional stiffness is determined experimentally constraining the steering bar and loading the fork joints by means of a rigid bar [7]. Reference value was chosen to be 650 kgm/° (APRILIA RSV1000).

2.2 Starting structure FEM model

In order to reproduce the behaviour of a boxed structure, mean surface of solid model was meshed with shell elements NASTRAN CQUAD4 including both in plane and out of plane behaviour (NASTRAN PSHELL). Three-dimensional model was created by means of the preprocessor Femap 7.0 [12]. FEM model is represented in fig. 4, and was realised with 3821 nodes and 3887 elements. Preliminary analyses show that the stiffness requirement is more severe than stress requirement, so a frame that meets stiffness requirement results under stressed if stress concentrator are controlled. Torsional stiffness was obtained numerically imposing fixed boundary in the positions of the bearings of the steering bar and loading fork joints with two concentrated loads in order to have a prescribed torque. Displacements of loaded nodes are then used to obtain the rotation and so the torsional stiffness. Deformed shape is reproduced in fig 5.

Figure 4 - FEM Model Figure 5 - Deformed shape under torsion

2.3 Vector basis

As previously described, each vector in the basis was obtained by a fictitious FEM analysis. Overall basis consists in four vectors that are represented, together with fictitious models, in fig. 6. The first vector controls cross section height. The second vector controls the orientation of cross section. The third vector was chosen in order to minimise the path between steering bar and rear suspension connections. Fourth vector impose a variation in the distance between lateral beams.

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Figure 6: Basis vectors (fictitious loads on the left, deformed shape on the right)

2.4 Optimisation results

A first result was obtained bounding shape evolution to a low extent in order to achieve an optimised shape easy to manufacture. Weight reduction was about 10% and hard convergence was reached after 8 calculation cycles.

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New shape is represented in fig. 7.

Figure 7: First optimisation results (original model on the left, new model on the right)

A second result was obtained imposing a strong shape evolution. Quadrilateral elements suffer distortion during evolution because connected nodes can considerably go out from original plane. For this reason each quadrilateral element was splitted in two triangular elements (CTRIA3). Preserving assigned stiffness requirement we obtain a weight reduction of 34% . New shape is represented in fig 8.

Figure 8: Second optimisation results (original model on the left, new model on the right)

4. CONCLUSIONS

In this paper a shape optimisation method for structural design was studied numerically. First a brief summary of theoretical basis of the optimisation method adopted was exposed. The minimisation strategy is addressed to structure modelled by means of finite

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elements method defining the objective and constraint functions.. Optimum search was performed according to gradient method starting from a guess solution in parameters space; feasible direction is obtained by an incremental analysis that takes advantage of sensitivity coefficients, easily available from the stiffness matrix computed based on the guess solution. In order to connect model shape to design variables a direct method was chosen that consists on a vector basic generation suitable to span overall geometry within an allowable domain. The boundaries of this domain are defined taking into account geometrical constraints that so are implicitly satisfied, without increasing the complexity of the optimisation process. Fictitious static analysis are exploited to generate a vector for the basis. Proper loads and constraints are able to produce a deformed shape that fulfils prescribed conditions on geometry and symmetry. This optimisation process was tested on an engineering problem regarding a motorbike frame using FEM models for Nastran code. Starting geometry was defined imposing bike dimensions chosen for a good handling behaviour of the vehicle. The constraint of optimisation regarded the stiffness and the strength while objective was to reduce the weight. Geometric dimensions where modified in the optimisation process according to the selected basis. Each vector of the basis was tailored to modify a particular geometric detail of the structure highlighted from the study of starting solution. Optimisation was then carried out in two ways with different boundaries in shape modification extent. The first one produces an easy to manufacture frame, reducing the weight of about 10%. The second solution proposed brings to a new shape that beside a weight reduction of about 34%, shows to be interesting from aesthetic point of view although related manufacturing process is quite difficult.

REFERENCES

[1] “CAD/CAM All’Opera” Enzo Guaglione, Il Progettista Industriale, Giugno 2000. [2] “Sviluppo di una nuova metodologia per la gestione delle variabili di forma in problemi di

ottimizzazione multidisciplinare” G.Chiandussi, G. Belingardi,G. Bugeda, E. Onate , XXVIII Convegno Nazionale AIAS, Vicenza 1999

[3] “Simulazione Dinamica e analisi tensionale d un telaio per motociclo” Giuseppe Polucci ,Tesi di Laurea in Ingegneria Meccanica

[4] “Ottimizzazione strutturale di forma nel campo delle costruzioni di macchine”, Roberto Cimini, Tesi di Laurea in Ingegneria Meccanica

[5] “Comportamento a fatica di telai motociclistici in condizioni di esercizio” R. Tovo, N. Facchin, XXVII Convegno Nazionale AIAS, Perugia 1998.

[6] “Ottimizzazione del progetto di un forcellone monobraccio per motocilco” G. Bartolozzi, D. Croccolo, M. Pesaresi, XXVII Convegno Nazionale AIAS, Perugia 1998.

[7] “Analisi strutturale di un telaio motociclistico” G. Bartolozzi, D.Croccolo, A.Loiero, XXVIII Convegno Nazionale AIAS, Vicenza 1999

[8] MSC/NASTRAN Version 68 Design Sensitivity and Optimization User`s Guide, Gregory J. Moore, Ph.D. by The MacNeal-Schwendler Corporation, 1995

[9] MSC/NASTRAN V70.5 QUICK REFERENCE GUIDE BY THE MACNEAL-SCHWENDLER CORPORATION, 1998

[10] Engineering Optimization Theory and Practice Third Edition, Singiresu S. Rao, Wiley Interscience Publication, 1996.

[11] “Cinematica e dinamica della motocicletta” V. Crossalter, Edizioni Progetto, 1997 [12] FEMAP Version 7.0 User Guide.

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shape optimisation for structural design by means of

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