Optimization of mechanical structures under ... - Wiley Online Library

GAMM-Mitt. 30, No. 2, 300 – 325 (2007)

Optimization of mechanical structures under special consideration of materials Thomas Vietor∗1 and Simon van den Akker1 1

Ford-Werke GmbH D-50725 K¨oln, Germany

Received 09 February 2007 Structural and Mechanical Optimization Problems are well distributed and described in literature. A high number of applications from different industry fields are publicated. Due to increasing demands for performance improvement together with reduced weight and costs the specific consideration of material laws and uncertainties into optimization problems in mechanical engineering is required. This paper presents an overview about Stochastic Structural Optimization as far as required to consider specific material laws. The application of structural stochastic optimization with specific material laws is illustrated at different test and industrial examples: • Ten-Bar System as standard test application with elastic material law and scattering material parameters • Ceramic Mirror with brittle ceramic materials • Powertrain Mounting System with nonlinear rubber material • Large scale Vehicle Body Structure with plastic material. Here the application will be presented and one of the commercial software systems. The details of this example will be publicated in the future. The numerical evaluation of this example is very expensive and time consuming. This is the one reason why stochastic optimization is not well distributed already. The presented examples are results of different projects the authors has worked at in the last 12 years at university and industry. About a decade ago stochastic optimization routines were still programmed in small working groups at universities. To increase the application to different industry branches, commercial software has to be used. There are a number of software packages on the market claiming the capability of stochastic optimization as well. In the next time these packages will be tested at different applications. This will be part of the planned publication in the future. The Vehicle Body Structure example in this paper will outline the task without going into the details. c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


1

Introduction

Extensive research has been conducted in the fields of structural analysis and structural optimization, as well as in material sciences with respect to advanced materials. The application of optimization methods to the layout of components made of advanced materials, however, has comparatively recently gained increased importance, as the latter field is strongly dependent on the availability of high-performance computers. Yet the recent substantial progress ∗

Corresponding author: e-mail: [email protected], Phone: +49 (0)221 903 7006, Fax: +49 (0)221 903 7673 c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


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in the development of hard- and software certainly gives impetus to this particular area of research, where efficient optimization procedures have to be linked to corresponding structural analyses. Owing to the character of the material laws, material parameters, and design variables, the hitherto applied scalar, deterministic continuous optimization models have proved to be insufficient, although past research projects have already attempted to extend the models to problems with several conflicting objectives and/or discrete design variables. In many cases, however, these activities are still unable to cover advanced materials. Their treatment requires the introduction of a stochastic optimization model as well as an augmentation of the respective solution procedures. The introduction of stochastical quantities to the optimization problem is covered firstly in detail between 1960 and 1980. In the first papers the minimization of a deterministic objective with constrained failure probability is investigated [22], [11]. The demonstrated methods are used also for the optimization of mechanical structures. In [11] the difference between single and system failure is included. The use of stochastic optimization for structures with numerical structural analysis is introduced later [12], [13]. The reason is the high numerical effort for the calculation of failure probabilities and sensitivities. In mathematics the formulation of stochastical optimization problems are well defined. The methods are used for applications from operations research [7] The use of the developed methods for structural mechanics is often not possible. In [21] the application of optimality criterion is shown at an example from vehicle engineering. Especially convergence criteria are difficult to evaluate because of unsufficient informations about the mathematical character of constraints and objectives. In [9] [10] mathematical methods are applied to mechanical structures. The three existing methods to solve a stochastic optimization problem are described in more detail in [23]. The introduction of a FOSM method to the optimization loop [8] allows the solution of the stochastic optimization problem without greater extensions of existing programming codes. The optimization problem defined in equation (1) can be solved by means of different procedures. By integrating a stochastic optimization procedure into the optimization procedure SAPOP [3] a complete and extensive optimization environment is available that additionally allows to use further optimization strategies in combination with stochastic optimization. In the last time a number of relevant publications describe the calculation of reliabilities and the extension of the deterministic to the stochastic optimization problem [17] [18] [19] [20]. The introduction of reliability in the design process is described in literature as well [24] and introduced in several industrial areas.

2

Calculation of the failure probability of mechanical structures

In the following, the essential terms of calculating the reliability of mechanical structures are defined, and the respective calculation procedures are introduced. We treat the definitions and procedures in a very general way so that transfer to other technical disciplines is possible. 2.1

Definitions

Definition 2.1 Vector of the stochastic variables: This vector combines all stochastic quantities of a mechanical structure, which entails both the stochastic design variables that can be varied during optimization, and the stochastic parameters like material and geometry c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


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ZT = (XT , PT )

(1)

where X denotes the stochastic design variables and P the stochastic parameters. A distinction must be made between the stochastic variable vector Z and the realization of the variables z. Definition 2.2 Boundary state function: The equation of a boundary state function reads g = g(Z)

(2)

The function g = g(Z) of the stochastic variables Z described the structural behaviour with respect to failure. We define Failure of the structure| g(z) < 0

(3)

Survival of the structure| g(z) ≥ 0

(4)

By this function, the space of the stochastic variables is devided into the failure range Df = {z | g(z) < 0}

(5)

which contains all realizations of the stochastic variable vector leading to failure, and into the survival or save range Ds = {z | g(z) ≥ 0}

(6)

The definition g(z) = 0 is called the boundary state surface. If several boundary state functions exists, the statements are correspondingly valid for each of them. Definition 2.3 Failure probability: On the basis of definition (2.2) we obtain for the failure probability with the boundary state function g

Pf = P [g(Z) < 0]
Pf = d(z)dz

(7)

Df

with g(Z) boundary state function (Z) stochastic variables d(z) cumulative density function of the stochastic variables By this, the calculation of the failure probability is retransformed to the calculation of a multidimensional integral using a density function. 2.2

Introduction of different procedures

The difficulty of solving a multiple integral with a nonlinear description of the integration bounds is a familiar occurence in many applications. In the following, possible solution procedures shall be categorized, where only those approaches are introduced that are based on the second moment method. c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


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• Analytical solution • Numerical integration • Simulation • Special procedures that are adapted to the given task, employing the character of the integrand and/or the range of integration – Single-Check-point procedures – Multiple-Check-Point procedures • Response Surface method 2.3

Second Moment Method

When calculating the failure probability according to equation (7) one has to know the density function di (zi ) or the joint density function d(z). In spite of recent substantial progress in this field, these functions cannot be given in all cases, owing to a lack of data. The literature provides extensive tables listing information on the functions as well as the corresponding parameters (in particular material parameters). Due to its flexibility and versatile applicability, the Weibull distribution has gained high importance. Insufficient information on the density functions has led to the development of the Second Moment Method as the first two moments of a stochastic variable are generally available. General Advanced First Order Second Moment (AFOSM)-procedure With independent, normally distributed variables Z, linear boundary state functions, and variables Y in standard normal distribution given, one defines ˜ ) = a0 + aT Y g(Z) = g(Z(Y )) = G(Y

(8)

˜ )) = a0 µG := E(g(Z)) = E(G(Y

(9)

with

2 ˜ )) = |a|2 := V (g(Z)) = V (G(Y σG

(10)

˜ ) is normally distributed with the parameters µG , σ 2 ; we then obtain for the failure Hence G(Y G probability   0 − µG ˜ (11) Pf = P [g(Z) < 0] = P [G(Y ) < 0] = Φ σG Pf = Φ(−β)

(12)

with β :=

µg a0 = σG |a|

(13) c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


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Equation (12) is exact in terms of the given assumptions of a linear boundary state function and normally distributed variables. For non-normally distributed variables Y it is assumed that a transformation to a variable Y in standard normal distribution can be carried out: Z = Z(Y)

(14)

At the point of minimum distance to the origin the boundary state function is linearized in the variable space. The latter point is termed the design point with a distance |β| to the origin. Thus Y ∗ is to be determined as the perpendicular of the origin 0 to the hyperplane g˜(y) = 0. Assuming that g˜ is linear in y, one can define the following optimization problem for y Min |y|2 with respect to a0 + aT y = 0

(15)

The LAGRANGE-function L(y, λ) := |y|2 + λ(a0 + aT y)

(16)

a0 λ λ ∇y L = 2y + λa = 0 −→ y∗ = − a, = 2 2 |a|2

(17)

yields

and we thus obtain for the design point and the reliability index y∗ =

a0 |a0 | = |β| a ⇒ |y|∗ = |a|2 |a|

(18)

and  |β| =

β|a0 ≥ 0 −β|a0 < 0

(19)

Fig. 2.1 illustrates the linear boundary state surface g(y) = 0 with the vector y∗ mapping the origin onto the boundary state function. 2.4

Transformations

The transformation according to equation (14) can be carried out in different ways. In this paper these methods are listed with references and not described. Instead of a detailed description, we will briefly state some reasons for the necessity of the transformations, where a distinction shall be made in two cases: 1. Determination of the reliability index β in a reduced space Here, transformation to variables with standard normal distribution is required, using • ROSENBLATT-transformation or • orthogonal transformation c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


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Fig. 1 Linear boundary state function and reliability index β

2. Determination of the reliability index β in the original space In this case no variable transformation is carried out. However, one reqiures the parameters µz , σz of an equivalent normal distribution in order to define β. The parameters can be determined by means of various methods that differ with respect to effort and accuracy. All methods are derived from the Normal-Tail-Approximation: • Determination of a 2-parameter equivalent normal distribution: a) Normal-Tail-Approximation [2] b) Multiplication-Factor-Method c) ”Inaccurate Procedure” • Determination of a 3-paramter equivalent normal distribution d) CHEN-LIND method [1] e) WU-WIRSHING method [15],[16] Using the parameters µz , σz of the equivalent normal distribution, the design point and its corresponding reliability can be determined by means of the general FOSM-procedure. 2.5

Evaluation of the Second Moment Method [14]

Generally, one can distinguish between two different applications of the reliabiliy calculations: • Application in structural mechanics for single or multiple calculation of a given structure. In this case, the efficiency in terms of accuracy of applied procedures is of predominant interest, while the numerical effort is of secondary importance. • Application in structural optimization for the calculation of objective functions and constraints. Depending on the employed optimization procedure, objective functions and constraints have to be computated at least once in each iteration, and thus the performance of the optimization calculation depends on the numerical effort when determining these values. Hence, the suitability of a procedure is judged on the grounds of accuracy c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


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Fig. 2 Procedure of Normal-Tail-Approximation with iterative determination of the design point

and numerical effort. Finally, several optimization procedures require the calculation of sensitivity information so that the methods must also be evaluated in terms of their ability to efficiently process these quantities. In the following, the most important features of the methods of second moment are summarized in order to compare their efficiency and to allow for a choice of a procedure most suitable adapted to the given application. 2.5.1

Comparison of first and second order procedures

The accuracy of the FOSM-method largely depends on the dimension n of the vector of the stochastic variables Z and on the reliability index β. In addition the value of the curvature of g(Z) at the point of maximum likelihood strongly determines the difference between linear and quadratic approximation. With increasing curvature and with an increasing dimension n of the vector of the stochastic quantities z the deviation increases too, whereas the deviation decreases if the reliability index β gets larger. In terms of practical application this means that the curvature must not be ”too large” at the point of calculation. 2.5.2

Selection of procedures in structural mechanics

The following criteria are essential for the use of the procedures in structural mechanics: • If possible, the number of stochastic variables n should not exceed 10. For many applications, this demand imposes no limitations. • Application to optimization: Owing to their moderate numerical effort, FOSM-methods with normal-tail-approximation (NTA) prove to be suitable. • Use in accurate post-calculation: FOSM-methods with WU-WIRSHING-transformations • Prior to an application in optimization test calculations are required in order to determine the range for which the simpler NTA provides sufficient accuracy. c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


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307

Definition and solution of a stochastic optimization problem

3.1

Definition of a stochastic optimization problem

Definition 3.1 Continuous, stochastic optimization problem

”Min” fA (Z) x∈D

(20)

    fAi (Z) = k1 E fi (Z) + k2 V fi (Z) , ZT = (X T , P T )

(21)

with

D = {x ∈ Rn |hi = 0 ∀i = 1, . . . , mst ; gj ≥ 0∀j = 1, . . . , ng1 ;   Pf k = P gk Z < 0 < Pkmax ∀k = ng1 + 1, . . . , ng1 , +ng2 , xkl ≤ xk ≤ xku ∀k = 1, . . . , n} (22) partitioning of the vector of inequality constraints gT = (g1 , g2 , . . . , gng1 , gng1 +1 , . . . , gng1 +ng2 )

(23)

and fAi gj gk hi Z X P  E f (Z) V f (Z) k1 , k2 D Pf k Pkmax n g1 n g2 xkl , xku 3.2

augmented objective function, inequality constraints as a function of deterministic variables only, inequality constraints as a function of stochastic variables, equality constraints hi = hi (µ) calculated by means of the expected values, vector of the stochastic variables, vector of the stochastic design variables, vector of the stochastic parameters, expected value of the objective function, variance of the objective function, weighting factors, feasible design space, failure probability of the k-th inequality constraint, feasible value of the failure probability, number of deterministic ineqality constraints, number of stochastic inequality constraints, lower and upper bounds of the design variables, respectively. Stochastic optimization procedures

The optimization problem defined in equation (20) can be solved by means of different procedures some of which are briefly introduced in the following, and are then evaluated in c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


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terms of their applicability in structural mechanics. Special emphasis is put on the question whether these methods can be integrated into an optimization procedure, using mathematical programming methods. By integrating a stochastic optimization procedure into the optimization procedure SAPOP [3], a complete and extensive optimization environment is available that additionally allows to use further optimization strategies in combination with stochastic optimization. Instead of using gradient methods other authors are utilizing a similar optimization loop but using genetic algorithms for the solution of the optimization problem [5] [26] [27]. The application of genetic algorithms is practical for industrial relevant examples with the limitation of the number of design variables. With increased computing power this number has been extended to a number of 10. The integration of stochastic optimization into an optimization procedure brings about a certain inevitable ambiguity with respect to the notation of variables. E.g., both the transformation variables and the stochastic variables are denoted by z and the analysis variables as well as the transformed variables in standard normal distribution are denoted by y, and finally β describes both the LAGRANGE-multipliers and the reliability index. However, in the following we maintain this standard of notation as it is generally used in the literature, and since the respective meaning largely follows from the context. The following, most important stochastic optimization procedures are referenced in the literature only the Taylor-expansion will be explained here: 1. Procedure of stochastic search [9] 2. Stochastic approximation [10] 3. Taylor-expansion [12] The expansion of the objective function in a Taylor-series with respect to the stochastic variables     (24) ω ) − z0 + . . . F (¯ ω ) = f Z(¯ ω ) = f (z0 ) + ∇z f (z0 ) Z(¯ For applications in structural mechanics we suggest the following first approximation: • Set k2 = 0 in equation (21) • The series in equation (24) be terminated after the 0-th term, • Define z0 = E(Z) as expansion point in equation (24) Thus, the stochastic optimization problem in (20)) is transformed into a quasi-deterministic optimization problem with reliability constraints. Here, fulfillment of constraints is determined by stating probabilities, which is considered during optimization. On the other hand, the deviation of the objective function owing to the stochastic distribution of the design variables and constraints is neglected. This can be counterbalanced imposing an additional constraint, in the form of defining a reliability of remaining under a demand level. As demand level, one may use the corresponding expected value, calculated by means of equation (24) The additional constraint then reads for the i-th component of the objective function vector f   (25) 1 − P fi (Z) < E(fi ) < Pfimax c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


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Sensitivity analysis [6]

3.3

Sensitivity analysis is an essential feature of structural optimization, and the efficiency of a practically applied optimization procedure strongly depends on the effective performance of the analysis. It can be shown that the sensitivities of the reliability indices can be obtained semi-analytically from the determination of the indices. Augmented optimization procedure and sequential realization [3]

3.4

Fig. (3) shows a flow-chart of the optimization loop that has been augmented by an Advanced First Order Second Moment Method (AFOSM) - procedure for calculating the reliability indices. The figure shows that by integrating the procedure, one obtains two interlocked optimization loops. The same linkage would occur with a nonlinear structural analysis that is also performed iteratively. The outer loop comprises the quasi-stochastic optimization described above, while in the inner loop the reliability indices and the sensitivities are calculated. In the present case, the inner optimization problem is solved by a modified generalized reduced gradient (GRG) - procedure that appears to be suitable here since the determination of the reliability index βi requires the solution of the minimization problem in (15), considering an equality constraint. The optimization procedure SAPOP [3] is extended with this AFOSM method. The modular character of the procedure allows the extension by the required modules with a moderate effort. With the use of the so-called three columns of optimization the separation of the optimization problem in the structural model, the stochastic optimization model and optimization algorithms is possible. In the following sections this optimization procedure is applied to structural optimization examples. For one of the examples a different approach is taken and a commercial software system is used.

4

Application examples

This chapter introduces a number of examples in order to illustrate an engineering-related application of the theories, computation methods, and program systems that were introduced in the previous chapters. On the one hand, the reliability computation of mechanical structures is presented and, on the other hand, a stochastic and a deterministic optimization are illustrated and their results are compared. 4.1

Ten-bar-system with scattering elastic parameters

In [20] a truss system is investigated for a space station. Beside some practical examples a truss system can be used to demonstrate the effect of stochastic parameters with a structrual analysis which requires a small numerical effort. 4.1.1

Structural model and material behaviour

The ten-bar-system depicted in (4) the material and geometry data given in Table (1). The bar forces of the statically indeterminate bar system can be solved analytically, and from the forces we then obtain the stresses. With the stochastic cross-sectional areas X1 , . . . , X10 the c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


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Fig. 3 Flow-chart of the optimization loop for stochastic optimization

boundary state functions then follow as gi (Z) = 1 −

σi (Z) Fi (Z) =1− ∀i = 1, . . . , 10 σf eas Xi σf eas

(26)

gi (Z) = 1 +

σi (Z) Fi (Z) =1+ ∀i = 11, . . . , 20 σf eas Xi σf eas

(27)

These equations describe failure by exceeding the tension strength gi , i = 1, . . . , 10, exceeding the compression strength gi , i = 11, . . . , 20,

4.1.2

Optimization model

The mass m of the ten-bar-system is to be minimized where stress constraints shall be fulfilled simultaneously. The optimization model then reads: Minf (Z) x∈D

(28) c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


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Fig. 4 Ten-bar-system

length Young’s modulus density

material and geometry information 360 mm 210000 N/mm**2 7.85 * 10**(-6) kg/mm**3 stochastical data

cross sectional areas design variables standard deviations load F expected value standard deviation feasible stress expected value standard deviation

expected values 0.1 * expected value normal, lognormal, gumbel distributed 10000 N 1000 N = 10 % expected value lognormal distributed 250 N/mm**2 25 N/mm**2 = 10 % expected value

Table 1 Material and geometry data of the ten-bar-system

with f (Z) = m(Z) and ZT = (XT , PT ),   D = {x ∈ Rn |Pf i = P gi (Z) < 0 < Pimax ∀i = 1, . . . , 20; PfS f eas }, D = {x ∈ Rn |βi > βimin ∀i = 1, . . . , 20; Pf s < PfS f eas }, ZT = (XT , PT ) = (A1 , . . . , A10 ; F, ωf eas )

(29)

The defined optimization model is substantially simplified for the optimization calculations. In order to generally illustrate the influence the stochastic character of the design variables and c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


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parameters, is chosen as the only reliability constraint to be considered, whereas the remaining ones are treated as deterministic constraints.

4.1.3

Optimization results

Fig. (5) shows the influence of different distribution functions of the load F on the failure probability, where the assumed type of distribution of F F gains a substantial influence on the index β1 . In addition, no statement can be made as to an over- or underevaluation of β if one assumes a normal distribution instead of the real distribution. It shall be mentioned at this point that no equivalent normal distribution is calculated. In the present example, the reliability index for a load with weibull-distribution is higher for small expected values of F than with a normally distributed load, and smaller for large expected values. A vice versa statement holds for a load with gumbel distribution. This clearly proves that actual type of distribution of the variables and parameters is of vital importance for applications in structural mechanics.

Fig. 5 Failure probability Pf 1 of bar 1 in dependency of the stochastic load F

Fig. (6) presents the solutions for the optimization problem, where the cross-sections X1 , . . . , X10 are assumed to be normally distributed, the force F gumbel-distributed, and the feasible stress σf eas log-normally distributed; the reliablity constraint of bar 1 only is considered. It is shown in how far the optimum mass depends on the maximum admissible failure probability for various factors Vσ from which the standard deviation of the design variables are calculated with the relation σ = Vσ µ where the latter factor gains a strong influence on the optimal mass. The difference in the masses increases with growing reliability index. One should also note the large range of the failure probabilities extending over nearly 12 powers of ten. c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


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Fig. 6 Optimal results for the ten-bar-system

4.1.4

Ceramic Mirror plate with brittle materials

Optical astronomy and laser technology call for highly focussing mirrors. It is the task of structural optimization to find an optimal layout for the mirror structure that fulfills demands for highest accuracy by achieving minimum weight at the same time. The deformation of the mirror surface is covered by an rms-value that describes the deviation of the deformed from the ideal mirror surface. For optical applications, the rms-value must only amount to a fraction of the wavelength of light, a fact that calls for a stiff structure with a corresponding use of material. On the other hand, one has to consider deformations due to the deadweight at different mirror positions as well as economical demands for a small total mass. In this section, an optimal layout for an optical mirror shall be determined with respect to the mass or the rms-value of the deflections. 4.1.5

Structural model and material behaviour

Fig. (7) shows a mirror construction applied in optical applications. The mirror consists of the actual mirror surface (that is assumed to be plane for the initial investigations), a honeycomb core, a bottom layer and a boundary stiffener. The quadratic core cells are prescribed for manufacturing reasons. The mirror is supported at small points that are equally arranged on a support circle; owing to their small surface, the support points are idealized as point supports. The design variables to be defined lateron are also given in Fig. (7). Mirrors of this type are generally made of quartz glass and glass ceramics. With the deformations the rms-value is calculated as

rms= A1 u2 dA (30) A

with the vector u denoting the deviation of the deformed surface from a reference surface, and the plate surface A. Since the stresses in the calculated examples are small they will be neglected. A finite-element-based structural analysis will be employed. c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


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4.1.6

Optimization model

In this example, the mass of the mirror and the rms-value of the deformations are defined as objective functions. Hence, the optimization problem reads: ”Min” f (x) x∈D

(31)

with the vector objective function f T = (f1 , f2 ) = (m, rms)

(32)

Fig. 7 Mirror plate with design variables

Application of constraint-oriented transformation (Trade-off method) with the primary objective f1 = m and with the secondary objective function f2 = rms transformed into a constraint yields the scalar optimization problem ”Min” f1 (Z) x∈D

(33)

with ZT = (XT , PT ) = (X1 , . . . , X6 ; E, P ) and the design variables shown in Fig. (7), the stochastic Youngs modulus E, and the stochastic pressure load P . Continuous, stochastic optimization model     n D = x ∈ R |Pf 1 = P g1 (Z) < 0 < P1max (34)  D=

 x ∈ Rn |β1 > β1min

(35) c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


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ZT = (XT , PT ) = (X1 , . . . , X6 ; E, P )

(36)

The normally distributed variables X1 , . . . , X6 with the expected value µxi as design variables are defined as stochastic model. In the following we will state whenever the distribution function of the stochastic parameters differs from the normal distribution. The expected value of the mass E(m) is calculated by means of a simple Taylor expansion at the point of the expected value of the stochastic varaibles. 4.1.7

Calculation of failure probabilities

Table (2) lists the results of the reliability calculations for different distribution functions of the compression load, based upon the following expected values of the design variables and parameters: µx1 = 5mm, µp1 = µE = 72500N/nm2 µx2 = 5mm, µp2 = µP = 6.5 · 10−5 N/nm2 µx3 = 30mm, µx4 = 2mm µx5 = 15mm, µx6 = 100mm

(37)

Here, again, the AFOSM-procedure, the normal-tail approximation, and the WW-procedure are compared with each other. The results show that it is of vital to have an exact stochastic description of the load. In case of the AFOSM-procedure the reliability indices are too large, which results in a non-conservative evaluation of the reliability. Fig. (8) illustrates the influence of the expected value on the failure probability of the mirror plate. 4.1.8

Optimization results

Fig. (9) compares the results for the deterministic and the stochastic optimization problem, where treatment is confined to six supporting points. The calculation is based on the condition that βmin = 3.0. The optimal masses determined for the stochastic optimization model are substantially higher than for the deterministic problem, and the increase of the optimal mass is stronger for smaller rms-values. For values over 20 nm, the results are also low-efficient. normal

β1 Pf

3.72 9.87 · 10−5

β1 Pf

3.54 1.99 · 10−4

distribution lognormal gumbel FOSM-method 3.73 3.75 9.67 · 10−5 8.93 · 10−5 WW-method 3.54 3.51 2.04 · 10−4 2.23 · 10−4

weibull

2.69 3.56 · 10−3 2.54 55.4 · 10−3

Table 2 Reliability indices and failure probabilities of a mirror plate for different distribution functions of the compression load P

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Fig. 8 Reliability index βi of the mirror plate

Fig. 9 Optimization results for the mirror plate

4.2 4.2.1

Powertrain Mounting System with nonlinear rubber material Structural model and material behaviour

An illustration of a typical powertrain mounting system as used in vehicle engineering is shown in Fig. (12). The mounting system is the elastic mounting with 3 elastic mounts of the c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


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powertrain against the vehicle car body. In the considered frequency range, the powertrain behaves rigid. The lowest powertrain elastic eigenfreqency is substantially higher (> 250 Hz) than the 6 eigenfrequencies of the mounted mass (< 30 Hz). Important is the elastic behaviour of the vehicle car body which can have eigenfequencies in the elatic mounting range (< 30 Hz). It is necessary for customer satisfaction to design and manufacture a robust vehicle. A great scattering of different customer relevant requirements of the vehicle is not acceptable. Identified main parameters responsible for the scattering of the interior noise and vibrations at driver contact points in the vehicle related to the powertrain mounting system are: • Engine mount idle stiffnesses. Here a tolerance level of ±5% is realistic for prototype parts and ±10% for production parts (0 < f < 25 Hz, no preload on the mounts). • Engine mount dynamic stiffnesses in the relevant frequency range. Here ±20% seems to be a realistic value (25Hz < f < 250 Hz, mounts preloaded). • Maximum preload values due to weight of the powertrain and/or engine torque. A deviation of ±(5 . . . 10)% seems to be a realistic tolerance band. Because of the preload dependency of the dynamical stiffness this is a critical quantity. • Load variation in operating condition with air condition, power steering and electrical consumers on/off. • Mounting positions. Because of production tolerances the mounting positions are stochastic quantities as well but assumed as deterministic in this paper. This assumption limits the number of stochastic quantities to the material parameters which are the main focus in this publication. 4.2.2

Optimization Model

As objective function the mass of the engine mount system is defined Min f (Z) with f (Z) = m(Z) and ZT = (XT , PT ) x∈D ZT = (XT , PT ) = (X1 , . . . , X17 ),

(38)

where X1 , . . . , X8 engine mount stiffness of 3 mounts in global coordinates normal, weibull, gumbel distributed, one stiffness is assumed to be 0 because of a kinematic relation. X9 , . . . , X17 coordinates of the engine mount locations in 3 directions (deterministic) c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


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As deterministic contraints: maximum deflections, maximum stresses, location of eigenfrequencies. As stochastic constraints 

   x ∈ Rn |Pi = P gi (Z) < 0 < Pimax ∀i = 1, . . . , 3; Pf s < Pf sf eas ,   D = x ∈ Rn |βi > βimin ∀i = 1, . . . , 3; Pf s < Pf sf eas D=

with gi < N V Hindexi ∀i = 1, . . . , 3

4.2.3

(39)

Failure Probabilities and Optimization Results

Calculated reliability indices and failure probabilities are shown in Fig. (11) for one NVHIndex as measure for the response at one contact point and the variation of the expected value of x3 of one mount. The results are calculated for different distributions of the stochastic variables. With the increase of the stiffness the reliability index is decreasing respectively the failure probability increasing. The change of the distribution of the stiffnesses has a small influence in the range around the nominal value of 250N/mm2 and can be ignored. The optimal results are shown as functional efficient boundaries in Fig. (12). Normal distributed variables are assumed. As parameter the standard variance of the normally distributed variables is used. From this chart an optimal value for the adjusted NVH-Index can be found together with the corresponding reliability index. With higher required reliability the NVH-Index is getting higher respectively the response at the contact point.

Fig. 10 Adjusted Response Curves for 8 variables

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Impact of the results to the mass production of a passenger vehicle

The detailed results of this application were applied to the development of a vehicle which is currently in production. The calculated sensitivities and optimal results were used to investigate required actions from the mount supplier to reduce tolerances in the production process. With the results the most sensitive parameters could be identified. This reduced the costs of the tolerance reduction substantially. Originally it was planned to measure and control all elastic mounting parameters. In addition the scattering of the vibrations in the vehicle at idle could be reduced and the reliability of achieving the demanded vibrations could be calculated. This was the first example in vehicle engineering where stochastic optimization methods were applied to influence mass production of a vehicle.

Fig. 11 Reliability indices and failure probability for interior noise as function of engine mount dynamical stiffness (Quantitative results of a demonstration model)

4.3

Large Scale Vehicle Body Structure with Plastic Material

For this application example it is not intended to apply the described extended optimization loop according Fig. (3). Because of the high effort for the numerical structural analysis and the highly nonlinearity in material behaviour and structural behaviour FOSM methods are not the first choice. For this example commercial software should be applied and tested. 4.3.1

Structural Model and material behaviour

Fig. (13) shows a total vehicle crash model as Finite-Element model with a very high number of degrees of freedom. Beside the size of the model, the nonlinear material behaviour in c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


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Fig. 12 Engine Mount System and Optimal Results for the Engine Mount System as Functional Efficient Boundary (Quantitative results of demonstration model)

typical vehicle crash situtations have to be taken into account. To reduce the numerical model a sub-model of the toal vehicle has been created as illustrated in Fig. (13). A validation of the sub-model results was done with the initial full vehicle model. This is shown in Fig. 4.11 graphically for the stresses. Stresses and calculated section forces are well reflected by the sub-model. As load case the frontal offset crash according EURO-NCAP is used. 4.3.2

Optimization Model and Optimization Procedure

Fig. (15) shows the defintion of the design variables which are assumed to be stochastic variables. In Fig. (16) the optimization model is described and Fig. (17) shows the optimization procedure applied for the vehicle body structure optimization problem. This example is a real world industry example where the structural model as finite element model has more than 500.000 dof. The size and applied modelling of the structural model requires the use of available standard tools or the combination of these tools. Typical for industry relevant safety models is the huge required calculation time for one structural analysis. Because of the costs of one analysis the number of iterations for the optimization must be limited. This is one of the main acceptance criteria of stochastic optimization produres in industry. 4.3.3

Optimization Results - Next Steps

For the presented model deterministic optimizations were performed. From the time required for one structural analysis of 2-3 hours on a workstation it is obvious that the deterministic optimization required 60-80 hours. The extension to the stochastic optimization is the next step. Here different commercial tools will be tested. Main criteria will be • possibility use different distribution functions • ease of use and the preperation of the required input deck • numerical stability and required total CPU time c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


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Fig. 13 Total Vehicle Model and definition of sub-model

Fig. 14 Definition of a sub model and comparision with the results of the total vehicle model

Beside the software evaluation the sensitivity of crash results to scattering variables is of high interest. This sensitivity is not described in publications by now. There are different c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


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Fig. 15 Definition of design variables

Fig. 16 Formulation of the Optimization Model

groups in industry working on this but up to now the calculated examples were test examples only with limited numerical effort. c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


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Fig. 17 Optimization Procedure used with Isight as commercial tool

5

Conclusions

In this paper an overview is given of possible solution procedures for problems in structural mechanics, where special emphasis is put on the calculation of failure probabilities and/or the application of specific material laws. By applying the procedures to examples from structural mechanics it is shown that the inclusion of stochastic quantities plays a decisive role for the optimal layout. However, it can be proved that stochastic optimization can efficiently be applied for these examples. For industrial applications the use of the presented software tool SAPOP is limited because this tool is a non-commercial tool and no support is available. In addition SAPOP is limited to stochastic optimization based on AFOSM with limited applicability to highly non-linear problems. To extend the application range some authors apply Genetic Algorithms [26] [27] [28] as tool for the stochastic optimization problem. Recently different companies claim for their software the capability to deal with stochastic optimization problems as well. In an ongoing project the author will assess several software systems. The selected application example was presented in this paper. The project is planned for finalization 2nd quarter of 2007.

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[25] Vietor, T. 1997: Stochastic Optimization for Mechanical Structures. In: Marti, K. (ed.): Issue on Struct. Rel. and Stochastic Structural Optimization. Mathematical Methods of Operations Research, Vol 46, Issue 3, Physica Verlag, Heidelberg, pp. 377-408. [26] Rajan, S.D.; Nguyen, D.T. 2004: Design optimization of discrete structural systems using MPIenabled genetic algorithm. Struct. Multidisciplinary Optim. 28 (5), 340-348. [27] Salajegheh, E.; Heidari, A. 2004: Optimum design of structures against earthquake by adaptive genetic algorithm using wavelet networks. Struct. Multidisciplinary Optim. 28 (4), 277-285. [28] Vietor, T.; Hilmann, J., et.al.: A new Method of Automatic Concept Model Generation for Optimization and Robust Design. 7th International Conference on ”Computational Structures Technology”, Session on ”Computational Stochastic Structural Analyis and Optimization”, Lisbon, Portugal, September 7-9, 2004. Accepted for publication. Civil-Comp Press, 2004.

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