Shape optimization of thin-walled beam-like structures

Thin-Walled Structures 39 (2001) 611–630 www.elsevier.com/locate/tws

Shape optimization of thin-walled beam-like structures P. Vinot *, S. Cogan, J. Piranda R. Chale´at Applied Mechanics Laboratory, 24, rue de l’Epitaphe, 25000 Besanc¸on, France Received 14 July 2000; received in revised form 13 April 2001; accepted 1 May 2001

Abstract This article presents a methodology for optimizing the shape of thin-walled structures having a beam-like dynamic behavior. The equivalent nominal beam characteristics (quadratic moments of inertia, torsional rigidity, etc.) of a refined finite element model are determined from a direct calculation based on explicit equations which are functions of the nodal coordinates defining the cross-sectional geometry. A subset of these coordinates is then taken as the design variables for a nonlinear optimization problem where new target physical properties are sought. A complementary model reduction procedure is introduced to improve the precision of the proposed method for beams with variable cross-section or topological accidents.  2001 Elsevier Science Ltd. All rights reserved. Keywords: Shape optimization; Parametric identification; Model reduction; Beams

1. Introduction A knowledge of the dynamic behavior of mechanical structures is essential for their design and optimization. In the context of the present article, we are interested in optimizing the response behavior of complex automobile assemblies with respect to a set of physical design variables. While the application of finite element methods to obtain and optimize discrete elastodynamic models of mechanical systems is now commonplace, the cost of analyzing the resulting large order models is often prohibitive. It is thus necessary to develop a strategy to reduce the size of the model in question while preserving its fidelity. The literature is replete with different method* Corresponding author. Fax: +33-3-81-66-67-00. E-mail address: [email protected] (P. Vinot). 0263-8231/01/$ - see front matter  2001 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 3 1 ( 0 1 ) 0 0 0 2 4 - 6

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Nomenclature B, BT matrix B, transpose of B (x, y, z) Cartesian coordinates in the local reference frame →

xb y0, z0 Y,Z q

ai G Gi C IY, IZ ky, kz A J

neutral axis of the beam global reference frame in the plane of a cross-section principal bending directions angle between the global reference frame and the principal bending directions angle between the ith segment and the principal bending direction Y mass center of the section of the beam mass center of the ith segment shear center of the section of the beam bending moments of inertia respectively about Y and Z at point G shear coefficients cross-sectional area torsion rigidity constant

ologies for reducing the order of these discrete models via a generalized transformation matrix as exemplified by the component mode synthesis approaches (e.g. [1– 3]). However, these reduction methods, which have the disadvantage of not preserving the topological structure of the initial model, are severely limiting due to the inaccessibility of the physical design parameters (local stiffness and mass modifications). An alternative reduction strategy consists of simplifying the refined model by replacing certain zones by dynamically equivalent but less finely meshed topologies. For example, a methodology has recently been developed for the automobile industry [4], which is particularly well adapted to structures having a beamlike dynamic behavior. A direct consequence of this strategy is the possibility of performing a parametric optimization on the resulting simplified model of the complete body-in-white. However, once the optimal parameters have been determined based on the simplified model, it is necessary to transform back to the real geometrical shapes that satisfy a number of relatively complex topological constraints. This article addresses the shape optimization problem which determines the actual refined geometry of the thin-walled beam cross-section satisfying the complete set of optimized design constraints, both mechanical and topological. The presentation is organized as follows. Firstly, a brief review of the basic equivalent beam formulation will be given based on [4]. The principle consists of replacing the three-dimensional mesh by an equivalent model composed only of beam finite elements. The method identifies the set of physical beam parameters to be introduced in the equivalent model based on a straight beam of constant cross-section. The equivalent beam results are taken as reference values and are used to establish correction coefficients for the direct approach that follows.

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Secondly, the beam properties are calculated using an explicit formula based on the position of the nodes defining the beam cross-section. The potential design parameters include the coordinates of the nodes in the plane, the wall thickness, and the orientation of the principal inertial directions. Thirdly, the shape optimization is performed by minimizing a cost function computed from target and direct values. The design parameters in this final optimization consist of the nodal coordinates defining the cross-sectional geometry of the thinwalled beam. The proposed method establishes a simple relation between the two design spaces [5], the first containing the properties computed with the equivalent beam method and the second containing properties determined explicitly based on purely geometric arguments.

2. Background A methodology for simplifying thin-walled members having a beam-like behavior has been developed for the automobile industry [4]. The following sections provide a brief review of the equivalent beam approach. 2.1. Equivalent beam formulation The proposed equivalent beam method can only be applied to beams with closed or solid end cross-sections. However, the beam can have small openings along its length. This restriction allows warping effects to be neglected and hence the method can be based on Saint-Venant’s principle. Let us consider a straight beam defined by the length L and the constant cross-sectional area A. In each cross-section of the beam, two geometric characteristics can be defined, namely the mass center G (denoted (yG, zG)) and the shear center C (denoted (yC, zC)). Using a Timoshenko formulation of a beam for an elastic isotropic material, we obtain the element matrices Kb and Mb. However, since beam structures may have cross-sections of arbitrary shape, the coordinates of G and C are a priori unknown. Hence, we define two points O1 and O2 such that O1 (resp. O2) lies in the plane defined by the end sections of the beam (S1 and S2) with O1O2 parallel to the axis → x b of the beam. Let K b0 be the stiffness matrix of the beam element expressed in the reference frame defined by O1O2 and the principal beam axes. This matrix is deduced from Kb on the basis of several rigid body transformations. In order to obtain the most general formulation of the equivalent beam, the previous element matrices must be expressed in arbitrary directions with respect to the global reference frame. We note the stiffness and mass matrices expressed in the reference frame defined by O1O2 and the principal beam axes, KO and MO. 2.2. Parametric identification ˜ be the assembled stiffness and mass matrices of the initial model. Let K˜ and M In a first step, the Guyan condensation Tg is applied to this model [3] to obtain the

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condensed stiffness and mass matrices on the degrees of freedom at the nodes of the beam extremities. According to the hypothesis of non-deformability of the crosssections, a rigid body transformation TS can be established between O1 (resp. O2) ˜ O and K˜ O correand each node of S1 (resp. S2). Finally, the numerical matrices M sponding to the 12-dof condensed model expressed in the reference frame defined by O1O2 and the principal beam axes, can be expressed as: ˜ O⫽T TST TgM ˜ TgTS; K˜ O⫽T TST TgK˜ TgTS M

(1)

The identification procedure consists of computing the terms of the analytical matrix KO based on the numerical values of the condensed matrix. The principal directions and the six geometric parameters A, J, IY, IZ, ky and kz can be identified. The equivalent mass density is determined separately so as to preserve the total mass of the model.

3. Formulation of the direct method The methodology consists of expressing the different geometric properties A, J, IY and IZ as functions of the nodal coordinates of the end sections of the beam using an explicit formulation. We then optimize the shape of these two sections with respect to the target beam properties. This section is devoted to a description of the analytical functions relating these properties to the nodal coordinates of the beam end sections. Two cross-sectional topologies will be considered independently, namely with and without an interior partition resulting in single or two-cell box beams. The partition represents a technologically acceptable modification of the beam cross-section and allows added liberty in attaining design targets without changing the external shape. Practically speaking, this partition can be inserted at the time of assembly of the box beam and is spot welded together with the outer surfaces. 3.1. Direct method without partition Let a cross-section (Figs. 1 and 2) be defined by n segments, where Gi is the mass center of the ith segment, L the circumference of the section, (Y, Z) the principal bending directions, q the angle between the global reference frame and the principal bending directions and e the constant thickness. Let (y1i, z1i) and (y2i, z2i) be the coordinates of the end points of the ith segment. The length of a segment is: lgi⫽冑(y2i−y1i)2+(z2i−z1i)2 and the characteristics of the section are as follows.

(2)

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Fig. 1.

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Definition of the axes for the ith segment.

Fig. 2.

Definition of the section.

Coordinates of the mass center

冘

冘

n

n

(y2i+(y1i) (z2i+(z1i) lgi lgi 2 2 i⫽1 i⫽1 yG⫽ ; zG⫽ n n

冘

i⫽1

lgi

冘

i⫽1

lgi

(3)

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Constant cross-sectional area

冘 n

A⫽e

lgi

(4)

i⫽1

Bending moments of inertia. In the reference axes GiYZ, the bending moments of inertia for the ith segment are given by: lgie(lg2i sin2 ai+e2 cos2 ai) lgie(lg2i cos2 ai+e2 sin2 ai) IG,Y⫽ ; IG,Z⫽ 12 12

(5)

where ai is the angle between the ith segment and the principal inertia direction Y. It is determined from the relation (6): ai⫽q⫺bi

(6)

where bi is the angle between the ith segment and the reference frame, and cos bi, sin bi are functions of the coordinates at the extremities. This leads to the following expressions:

再 冋 册

册

再 冋 册

册

冋

elgi e2(z2i−z1i)2 sin2 q (y2i⫺y1i)2⫹ ⫹… cos2 q (z2i⫺z1i)2 IG,Y⫽ 12 lg2i

冉

(7)

冊冎

e2(y2i−y1i)2 e2 ⫹ ⫹2 sin q cos q(y ⫺y )(z ⫺z ) ⫺1⫹ 2i 1i 2i 1i lg2i lg2i

冋

elgi e2(y2i−y1i)2 IG,Y⫽ sin2 q (z2i⫺z1i)2⫹ ⫹… cos2 q (y2i⫺y1i)2 12 lg2i

冉

e2 e2(z2i−z1i)2 ⫹2 sin q cos q(y2i⫺y1i)(z2i⫺z1i) ⫺1⫹ 2 ⫹ 2 lgi lgi

(8)

冊冎

The bending moments of inertia of the section in the reference GYZ are given by:

冘 n

IY⫽

冘 n

(IGiY⫹lgied 2Yi); Iz⫽

i⫽1

(IGiZ⫹lgied 2zi)

(9)

i⫽1

where dYi is the distance between the principal axis Y and the mass center of the ith segment (GiH in Fig. 1) and dZi is the distance between the principal axis Z and the mass center of the ith segment (GiK in Fig. 1). d 2Yi⫽

冋冉

冊 冉 冊 冉

冊册

1 z2i−z1i y2i−y1i ⫺zG ⫺AY ⫺yG 2 1⫹AY 2 2

d 2Zi⫽

冋冉

2

(10)

冊册

1 z2i−z1i y2i−y1i ⫺zG ⫺AZ ⫺yG 2 1⫹AZ 2 2

2

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with AY=sin q/cos q and AZ=⫺cos q/sin q. Torsional rigidity constant. The torsional rigidity constant J is determined for a thin-walled beam based on Bredt’s formula (22) [6]: 4A2inte J⫽ L

(11)

where Aint is the area enclosed by the section. The segments extending beyond the spot welds do not have a significant influence on the torsional rigidity and are not included in the calculation. 3.2. Direct method with partition The partition has in most cases a thickness different from that of the main section. Hence, Eqs. (2)–(11) must be redefined for a non-constant thickness section. The initial partition is shown in Fig. 6. Let e1 be the thickness of the outline, eC the thickness of the partition and ei the thickness of segment i (=e1 or eC). Moreover, let S1 and S2 be the two areas resulting from the partition (see Fig. 6), L1 and L2 the length of the exterior outlines of S1 and S2 and LC the length of the partition. The new beam properties are as follows. Coordinates of the mass center

冘

冘

n

n

(y2i+(y1i) (z2i+(z1i) lgiei lgiei 2 2 i⫽1 i⫽1 ; zG⫽ yG⫽ n n

冘

lgiei

i⫽1

冘

(12)

lgiei

i⫽1

Cross-sectional area

冘冑 n

A⫽

ei (y2i−y1i)2+(z2i−z1i)2

(13)

i⫽1

Torsional rigidity constant 4 J⫽ [e21LC(S1⫹S2)2⫹eCe1(L2S 21⫹L1S 22)] with D⫽eCL1L2⫹LCe1(L1⫹L2) D

(14)

Bending moments of inertia. In the reference GiYZ, the bending moment of inertia of the ith segment are: IGiY⫽

再 冋

册

冋

eilgi e2i (z2i−z1i)2 sin2 q (y2i⫺y1i)2⫹ ⫹… cos2 q (z2i⫺z1i)2 12 lg2i

(15)

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册

冉

冊冎

e2i (y2i−y1i)2 e2i ⫹ ⫹2 sin q cos q(y ⫺y )(z ⫺z ) ⫺1⫹ 2i 1i 2i 1i lg2i lg2i

IGiZ⫽

再 冋 册

册

冋

eilgi e2i (y2i−y1i)2 sin2 q (z2i⫺z1i)2⫹ ⫹… cos2 q (y2i⫺y1i)2 12 lg2i

冉

e2i (z2i−z1i)2 e2i ⫹ ⫹2 sin q cos q(y2i⫺y1i)(z2i⫺z1i) ⫺1⫹ 2 2 lgi lgi

(16)

冊冎

In the reference GYZ, the bending moment of inertia of the section are:

冘 n

IY⫽

冘 n

IGiY⫹lgieid 2Yi; IZ⫽

i⫽1

IGiZ⫹lgieid 2Zi

(17)

i⫽1

4. Shape optimization Two cases of shape optimization will be considered, namely for constant and variable section beams. The following paragraphs provide the definitions of the cost function which is minimized for the shape optimization. For a non-constant section beam, the use of correction coefficients allows the values obtained by the direct method to be calibrated with respect to the reference results provided by the equivalent beam method. For variable section beams, the presence of sharp discontinuities in section along the length of the beam should be avoided by dividing the beam into continuous segments and applying the proposed procedure to each segment independently. However, discontinuities in the walls of the beam, such as holes or flanges, do not require any special treatment. 4.1. Definition of the cost function 4.1.1. Constant section beams First, the initial properties are determined based on the equivalent beam method. The same characteristics are then computed with the direct method based on the nodal coordinates in the plane, the thickness, and the principal inertia direction which were defined by the first computation. The optimization of the nodal coordinates is carried out by minimizing a cost function F based on relative differences Pi between the target values (subscript c) and those obtained by the direct method, that is to say: F⫽

冘

wiP2i

i

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冊 冉 冊 冉 冊 冉 冊 冉 冊

F⫽w1 ⫹w5

冉

YG−YGc YGc

Iy−Iyc

2

Iyc

2

⫹w2

⫹…w6

ZG−ZGc ZGc

Iz−Izc

2

⫹w3

619

冉 冊

A−Ac 2 J−Jc ⫹…w4 Ac Jc

2

(18)

2

Iz c

where wi are weighting coefficients. In practice, the center of gravity of the box beam is rarely an issue and w1 and w2 are usually set to zero. In order to avoid obtaining a technologically impracticable section (for example, avoiding the acute angle between two consecutive segments or crossing of segments), geometric constraints are imposed on the length and the slope of segments. They are expressed as functions of the nodal coordinates and included in the cost function to be minimized. The validity of the final properties are verified by the equivalent beam method. In practice, the principal inertial directions vary only slightly and are assumed to be constant in the course of the iterations. 4.1.2. Non-constant beam section In the case of non-constant beam sections, the physical beam properties computed by the direct method based on a single section do not generally represent the characteristics due to section variation or discontinuities (hole, …). However, the direct method can be calibrated with respect to the equivalent beam method considered as a reference. First, based on the initial shape of the beam, the coefficients a0C are introduced to relate the characteristics computed by the direct method of the two sections extremities (A01, A02, J01, J02, I0y1, I0y2, I0z1, I0z2), to those obtained by the equivalent beam method (Ap, Jp, Iyp, Izp) which are taken as reference values. Notice that only the characteristics of those two end sections need to be computed. Hence, we define a linear domain between the two sets of characteristics [5]:  A01+A02 A0p=a0A 2

  



J01+J22 J0p=a0J 2 I0yp=a0Iy I0zp=a0Iz

(19)

I0Y1+I0y2 2

I0z1+I0z2 2

The optimization parameters are now the coordinates of the nodes in the extremities of the two sections of the beam. The cost function becomes:

冢

a0A

F⫽w3

冣 冢

冣 冢

冣

A1+A2 J1+J2 Iy1+Iy2 2 2 a0J a0Iy −Ac −Jc −Iyc 2 2 2 ⫹w4 ⫹…w5 Ac Jc Iy c

2

(20)

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⫹w6

冢

a0Iz

冣

Iz1+Iz2 −Izc 2 Iz c

2

At the end of the first optimization, we compare the values of the characteristics obtained by the equivalent-beam method with those obtained using the a coefficients. If the two sets of values are different, we define a new relation between these two sets: C1+C2 ⫹a2C Cp⫽a1C 2

(21)

where C represents the characteristics A, J, IY, IZ. The coefficients a11,2C are computed from the values of the characteristics of the first two iterations (superscripts 0 and 1): C0p⫽a11C

C01C02 1 ⫹a2C 2

(22)

C11C12 1 C1p⫽a11C ⫹a2C 2 If the two sets of values are still different at the ith iteration, then the reference values can be expressed as a polynomial function of the characteristics of each end section:

冉 冊

Cp⫽ai1C

冉 冊

C1C2 C1C2 ⫹…⫹aii−1C ⫹aiiC 2 2

(23)

In practice, for all the cases we treated, the linear domain defined in Eq. (19) is sufficient to calibrate the direct method. Insofar as the modifications of the characteristics of the beam do not exceed a certain threshold, in practice about 30%, the coefficients a0C are nearly constant. 4.2. Optimization algorithm An optimization algorithm based on a sequential quadratic programming (SQP) method is used to minimize the cost function with respect to the beam characteristics which themselves depend explicitly on the coordinates of the nodes defining the end sections of the box beam. A quadratic programming (QP) sub-problem is solved at each iteration and an estimate of the Hessian of the Lagrangian is updated at each iteration using the BFGS formula. Practically speaking, the proposed methodology is implemented in MATLAB and the routine constr is used to find an optimal solution. The flow chart in Fig. 16 represents an overview of the method.

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Table 1 Initial, target and attained characteristics (percent variation) Characteristics J (mm4) IY (mm4) IZ (mm4)

Initial 42,631 108,580 47,926

Desired

108,580 (0) 57,990 (21.3)

Attained 57,127 (34.1) 110,700 (1.95) 58,132 (21.3)

Table 2 Initial, target and attained characteristics (percent variation) Characteristics J (mm4) IY (mm4) IZ (mm4)

Initial 45,083 148,190 51,722

Desired 45,083 (0) 148,190 (0) 62,066 (20)

Attained 44,554 (⫺1.17) 155,640 (5.03) 64,250 (24.2)

5. Numerical applications The shape optimization method is applied to a straight 660 mm long steel beam whose section is defined in Fig. 2. The optimization is performed using the proposed procedure. The initial and attained beam properties, calculated with the equivalent beam method, are reported in Tables 1–3 (values in bold represent the subset of beam characteristics whose target values are different from the nominal values). In the first example, geometric constraints are imposed so as to preserve the shape of several external surfaces. The objective is to obtain a 10.5% increase in the first eigenfrequency while maintaining the second eigenfrequency constant. These constraints can be satisfied by a shape change which results in a 21% increase in the moment of inertia IZ while leaving IY unchanged. The constant of torsional rigidity is left unconstrained. The evolution of the cost function is shown in Fig. 3. The final directions of inertia and section are shown in Figs. 4 and 5. Initial, target and attained characteristics are given in Table 1. Although there is no reason to believe that the solution obtained is unique, it is clearly feasible from a physical point of view in the sense that section nodes moved away from the principal axes in order to increase the moment of inertia. Table 3 Initial, target and attained characteristics (percent variation) Characteristics J (mm4) IY (mm4) IZ (mm4)

Initial 3856 13,755 5711

Desired 3856 (0) 17,194 (25) 5711 (0)

Attained 4386 (13.7) 17,374 (26.3) 5683 (⫺0.5)

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P. Vinot et al. / Thin-Walled Structures 39 (2001) 611–630

Fig. 3.

Fig. 4.

Cost function.

Initial and modified sections at iteration 3.

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Fig. 5.

Fig. 6.

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Initial and modified sections.

Definition of the initial partitioned section.

An increase in the constant of torsional rigidity of about 34% could not be avoided. Indeed, the geometric constraints on a significant part of the section severely limited the space of feasible modifications. In order to avoid this phenomenon, an alternative and allowable modification is introduced via a partition fixed between the two welded half sections of the beam. In the second example, we apply the methodology to a box beam with an internal

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Fig. 7.

Fig. 8.

Cost function.

Initial and modified sections at iteration 5.

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Fig. 9.

Initial and modified sections at iteration 16.

Fig. 10. Initial and modified sections.

625

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Fig. 11.

Definition of the beam.

Fig. 12.

Cost function section 1.

partition. The process is the same as before with the characteristic values determined by the equivalent beam method. Note that the initial characteristics of the box beam are different from those in the first example. The method is applied to the section defined in Fig. 6 in which the partition includes five nodes. The number of nodes nC is defined before the shape optimization (nC=4 in Fig. 6). In the present example, a minimum of five nodes is defined in order to obtain a sufficiently rich space of feasible modifications. The thickness of the partition is initially set to be identical with that of the main section, that is to say 0.7 mm. However, it could be modified

P. Vinot et al. / Thin-Walled Structures 39 (2001) 611–630

Fig. 13.

Fig. 14.

Cost function section 2.

Initial and modified section 1.

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Fig. 15.

Initial and modified section 2.

in order to obtain target properties and reduce the mass. The shape optimization is carried out by minimizing the same cost function and the properties are computed by the direct method. The modified parameters are now the nodal coordinates of the partition. Geometric constraints are applied on the length and the slope of segments. In this example, a 10% increase of the second eigenfrequency corresponds to an augmentation of 20% in the bending moment of inertia IZ. Initial, target and attained characteristics are given in Table 2. The cost function is shown in Fig. 7. The evolutions at iterations 5 and 16 are shown in Figs. 8 and 9. Initial and modified sections are shown in Fig. 10. Note that the dashed line corresponds to the new geometry of the internal partition. The external surfaces have not been modified in this example. In the third example, the method is applied to a 800 mm long steel beam of variable section with a opening in one face (see Fig. 11). The wall thickness of the beam is 0.7 mm. The objective here is to obtain a 25% increase in the first bending moment of inertia while restricting all geometric modifications to the vertical and lower box beam surfaces. The target values of the two extremity sections are calculated with Eqs. (19). The shape optimization of the two section extremities are computed independently. The cost functions are shown in Figs. 12 and 13. Initial and modified sections are shown in Figs. 14 and 15. Initial, target and attained characteristics are given in Table 3. An increase in the torsional rigidity could not be avoided. Indeed, the geometric constraints on a significant part of the section severely limited the space of feasible modifications. The efficiency of the coefficients ai is shown by this example. We note that the hole

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Fig. 16.

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Flowchart of the method.

severely reduces the torsional constant J. The relative difference between the value of J obtained by the direct method and the one obtained by the reference (equivalent beam) method without the coefficient aJ is about 224%. By using the coefficient aJ, this error is only about 5%.

6. Conclusion A method for optimizing the shape of thin-walled beam-like structures has been presented (Fig. 16). The strategy can be applied to structures having both constant and variable cross-sections and containing local topological accidents. A direct calculation procedure is formulated and correction coefficients used to calibrate the beam property estimations on the basis of reference properties obtained by a finite element

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method based on equivalent beam procedure. The proposed procedure can take into account both target equivalent beam properties and constraints on the final shape of the thin-walled section. The approach is illustrated by three examples on both constant and variable section profiles, with or without partition. The characteristics of the modified section are very close to te target properties, except when a major part of the section is fixed by severe geometrical constraints. In this case, the incorporation of a partition proves to be a particularly effective solution.

References [1] Guyan RG. Reduction of stiffness and mass matrices. Am Inst Aeronaut J 1965;3:380. [2] Craig RR, Bampton MCC. Coupling of substructures for dynamic analyses. Am Inst Aeronaut Astronaut J 1968;6:1313–9. [3] Craig RR, Chang CJ. On the use attachment modes in substructure coupling for dynamics analysis. In: AIAA/ASME 18th Structures, Structural Dynamics and Materials Conference, vol. B, 1977. p. 89–99. [4] Corn S, Bouhaddi N, Piranda J. Transverse vibrations of short beams: finite element models obtained by a condensation method. J Sound Vib 1997;201:353–63. [5] Biegler LT et al. Large scale optimization with applications. Berlin: Springer, 1997. [6] Vlassov BZ. Pie`ces longues en voiles minces. Paris: Eyrolles, 1992.