Structural Optimization Considering Fatigue Requirements

Structural Optimization Considering Fatigue Requirements

J Ming Zhou, Cheng-Au Xiong, Kang Zhao Raphael Fleury, Yaw-Kang Shyy Altair Engineering Engineering Irvine, California, USA [email protected]

First Published at: th 8 World Congress on Structural and Multidisciplinary Optimization June 1- 5,2009, Lisbon, Portugal

www.altairproductdesign.com copyright Altair Engineering, Inc. 2013

www.altairproductdesign.com

1.0 Abstract Durability requirements are important for all structures subject to cyclical loading. An efficient optimization framework is introduced to address fatigue life constraints. This includes utilization of high quality approximation of fatigue life through stresses as intermediate responses, and integrated implementation of FE and fatigue analysis.

2.0 Keywords Structural Optimization, fatigue, sizing optimization, shape optimization, durability design.

3.0 Introduction In the past decade structural optimization has become increasingly a standard CAE tool for structural design across virtually every engineering field. As many structures are designed to function cyclical loading. Durability requirements are often among the key driven design criteria. For fatigue analysis, S-N (stress-life) method is typically applied for high cycle fatigue whilst E-N (strain-life) is suitable for low cycle fatigue where plastic strain results in significant life reduction, though plastic strains are needed for E-N fatigue analysis, they are usually not based on accurate nonlinear FE analysis. Instead, approximate plastic strains are obtained by Neuber correction and cyclical stress-strain relationship. Therefore, from theoretical perspective considering fatigue constraints is straight forward as sensitivity of fatigue life can be simply derived by chain rules from stress sensitivity. However, for broad engineering application s efficient CAE software is essential. In this paper a comprehensive process is developed for efficient handling of fatigue requirements for size and shape optimization of large structures. In order to enable efficient treatment of a general optimization problem Involving fatigue and other constraints, S-N and E-N fatigue analysis is implemented as all integrated part of the FEA and optimization code OptiStruct [1] [26-34]. To enhance accuracy, fatigue life or damage is approximated through approximation of stresses.

4.0 Optimization Problem The general optimization problem was first introduced by Schmit in 1960 (2], which can be stated mathematically as follows:

Copyright Altair Engineering, Inc., 2013

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Where f(X) represents the objective function, g j(X) and represent the j-th constraint response and its upper bound, respectively. M is the total number of constraints; X I is the i-th design variable, represent its lower and upper bounds, respectively. respectively . The total number of design variables is N. In the problem considered in this paper, the design variables include: (1) sizing variables that define the cross-sectional dimensions of I-D elements (rods and beams) and 2-D elements (plates and shells); (2) shape variables that define the shape variation of existing boundaries. Sizing and shape variables combined in a design formulation, the objective function and design constraints can be any of the following responses: volumes or weights of structural parts, compliance. Eigen frequencies, displacements, stresses, fatigue life or damage etc. An equation utility has also been developed that allows users to formulate any custom response using the supported responses and design variables [1]. Shape variations in this work arc defined as a linear combination of predefined vectors of shape perturbation:

where Z is the vector of nodal coordinates, Z o o is the vector of nodal coordinates at the initial design, PV i i is the i-th grid perturbation vector. K is the total number of shape design variables. Note that the vector Z must also include internal nodes of the finite element mesh in order to avoid mesh distortion. This approach is easy to implement since it neither needs remeshing capability nor mesh smoothing algorithm during the iterative process. However, it may encounter mesh distortion for large shape changes. Literature on shape optimization Is very extensive and reviews can be found in survey articles and some research papers (see, e.g., Haftka and Grandhi [4], Ding [5J, Kikuchi [6], Chang and Choi [7], Yang e/ al. [8], Schramm and Pilkey [9], Schleupen et al. [10]). An overview of sizing optimization can be found in textbooks and review articles (see, e.g., Schmit [11], Haftka and Glirdal [12], Kirsch [13]).

5.0 Fatigue Analysis Fatigue analysis based on S-N (stress-life) and E-N approaches are used in fatigue life prediction in this paper. An overview of fatigue analysis theory C'1I1 be found in many text books (e.g., Lee et al. [3]). Wohler's pioneering work in the 1800s established the fundamental methodology of fatigue prediction. Based on observations Wohler was the first to establish a relationship between stress and number of cycles to failure, known as Wohler or S-N curve (shown in Fig. 1). S-N method is well suited for high cycle fatigue where stresses are within elastic range.


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When S-N testing data arc presented in log-log plot of alternating nominal stress versus cycles to failure N, the relationship between S and N can be typically described by one segment or two segment straight lines. A two segment S-N relationship is given in Eq. (3), which is illustrated ill Fig. 2.

Where is tile Nominal stress range, N f f total fatigue cycles to failure, b 1, b 2 2 slopes of the two segment life lines, and SRI1, SRI2 the stress range intercepts of the two segment life lines. When plastic behavior occurs during cyclical loading, typically the fatigue life is highly reduced, and this case is characterized as low cycle fatigue. For this case, the fatigue life was found 10 correlate with total strain consistently. The E-N (Strain-Life) method was introduced in the 1950s [3]. A typical E-N curve is shown in Fig. 2. Though plastic strains are needed for E-N fatigue analysis, they are usually not based on accurate nonlinear FE analysis. Instead, approximate plastic strains are obtained by Neuber correction and cyclical stress-strain relationship. Please refer to [1, 3] for details on S-N and E-N methods, complex load cycle counting and other correction formulations for influencing factors such as mean stress, certainty of survival etc. In practice no structure is subject to constant loading cycles. The most popular damage accumulation model is the Palmgren-Miner linear damage summation rule [3] given in Eq. (4).

where, D is the total damage caused by one block of all applied loading cycles, N i,f i,f the material fatigue life (number of cycles to failure) from its S-N or E-N curve lit a combination of stress amplitude and means stress level i , n j is number of applied stress cycles at load leveI i , D i i the cycle ratio (cumulative damage). Fatigue life limit is reached when overall accumulative damage equals to 1.0. Therefore, the fatigue life N as the number of repeats of the given loading cycle block is simply the inverse of total damage D caused by passing tile given loading cycle block, i.e., N=1/D . The linear damage summation rule does not take into account


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the effect of load sequence on the accumulation of damage due to cyclic fatigue loading. However it has been proved to work well for many applications.

6.0 Approximation Formulations The general approach for the optimization problem In eq.(1) is the approximation concept approach pioneered by Schmit and Farshi [14]. In this approach, the optimization problem is solved by solving a serious of explicit approximate problems. The overall efficiency of this approach is determined by the accuracy of the approximation. Typical approximation formulations used in structural optimization are linear approximation shown in eq.(5), reciprocal approximation in eq.(6) (Schmit and Farshi [14]) and convex approximation in eq.(7) (Haftka and Starnes [15]):


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The formulation in terms of mixed variables in eq.(7) is also termed conservative approximation since it has been shown by Haflka and Starnes [15] that this formulation gives a more conservative approximation of the constraint compared with both Linear and reciprocal approximations. Because this approximation is convex and separable, it is used to create an efficient dual method by Fluery and Brabaint [16]. Advanced Advanced approximation approximation techniques techniques developed developed in the late 1980s utilizes the use of intermediate variables and intermediate responses to enhance the quality of approximation (Vanderplaats and Salajeh [17]. Zhou and Xia [18], Zhou [191. Canfield [20], Vanderplaats and Thomas [211. Zhou and Thomas [22]). For sizing problems, the intermediate variables are the following cross-sectional properties:

where A is the cross-sectional area, I1, I2, J are moments of inertia of the i-th beam properties. NSM stands for non-structural mass. t, D and t s s, are the thickness, bending stiffness and shear thickness of the i-th shell properties, respectively. The above intermediate variables Y can be explicitly expressed as functions of sizing variables X , i.e.,Y = Y(X) . The quality of the approximation of displacements has been shown to be highly enhanced when reciprocal approximation is formulated in terms of intermediate variables Y : :

It can be shown that the above approximation is exact for statically determinate structures, For stresses constraints, the relevant element forces, termed intermediate responses herein, are approximated as follows:

Then the approximate stresses are recovered using exact stress recovery relationships:


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Through S-N or E-N relationship discussed in section 5, total fatigue damage D or life (1/D) can be expressed as explicit function of stresses under load cases involved. Hence approximate fatigue damage can be calculated from approximate stresses using exact S-N or E-N procedure, expressed in Eq. (11). This highly enhances the approximation accuracy as the nonlinearity in stress-life relationship is captured exactly.

The advanced approximation techniques summarized herein are implemented in the Altair OptiStruct code for the integrated problem covering sizing and shape optimization. Note that many approximation approaches that are well suited for other types of optimization problems are not mentioned in this paper. For a review, see the paper by Barthelemy and Haflka [23].

7.0 Sensitivity Analysis Discrete sensitivity analysis that is directly formulated on the basis of the discrete finite element formulation is used. An overview of sensitivity analysis can be found in textbooks and review papers (see, e.g., Adelman and Haftka [24], Haftka and Giirdal [12], Kirsch [13]), For static analysis, responses such as displacements, stresses and forces can be expressed as a function of the displacement vector U as follows:

The derivatives of the response can be expressed as:

From the stiffness equation

where K is the stiffness matrix and P is the load vector, the following equation can be derived for the calculation of the displacement sensitivities,


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In eq.(15), the vector

can be interpreted as the displacement vector corresponding to a

load vector where is termed the pseudo load vector, For N y y intermediate variables, N y y pseudo load vectors need to be solved for each loading case in order to calculate the derivatives of any number of responses. Substitution of eq.(15) into eq.(13) yields the following expression:

With

The method using equation (16) for sensitivity analysis is called the adjoint method, The vectors and Q j arc called the adjoint displacement vector and the adjoint load vector, respectively. The solution of one adjoint vector is needed for calculation the derivatives of each response, For N R R responses involved in the approximate problem, the total number of adjoint load vectors is N R R, which is independent of the number of design variables. It is easy to see that for a specific load case, the adjoint method is more efficient than the direct method if N R R related to this load case is smaller than N y y , and vice versa. Both methods arc implemented in OptjStruct and the favourable one is automatically selected according to this rule. For sensitivity with respect to shape design variables, the so-called semi-analytical method is used. In this approach, the derivatives of the stiffness matrix is calculated using central [mite differences as follows:

It has been shown that very large errors can occur when this method Is used. This phenomenon has stimulated intensive research effort in revealing the reasons for the errors and developing methods to eliminate them (see, e.g., Barthelemy and Haftka [24], 011hoff et al.[25]). For fatigue damage or life response, its sensitivity can be simply obtained from cq. (J 1) through simple chain rules once sensitivity of forces and stresses is obtained.

7.0 Example An automotive automotive connector connector component component is considered. considered. Due to symmetry only half of the part is modeled, which is shown in Fig 4. Only a single load case consists of loading from the bearing is considered. The blue area is defined as shape variation zone. The unique Free-Shape feature of OptiStruct is use, with which OptiStruct creates shape parameterization


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automatically with a proprietary technique [1]. The optimization problem is to minimize the fatigue damage on the blue area surface where stress concentration (see Fig. 5) affects fatigue life negatively. The load time history shown in Fig. 6 is applied, which consists of 2544 load time points. Feature line grids are kept within their respective planes. The optimization run converged after 11 iterations. The Initial and the final design shape and damage are shown in Fig. 7 and Fig. 8, respectively. The initial fatigue damage of 8.177x10 4 caused by one loading pass ofthe load lime history shown In Fig. 5 is reduced to 2.657x10 4. This means that the component's fatigue life is increased 308% from enduring 1223 to 3764 loading blocks. The final volume Increased 1.5% compared to the initial design.


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8.0 Concluding Remarks The design of structures under cyclical loading is often durability driven. Therefore, it is important that fatigue requirements are directly addressed in optimization for such design applications. An advanced technique is proposed ill this paper. During each iteration approximate fatigue damage or life is calculated based on approximated stresses. This high fldelity approximation technique allows the optimization process to converge quickly and stably. Moreover, in order to carry out this optimization framework efficiently, finite element analysis and fatigue analysis have to be implementation in an integrated software. This development is carried out in the commercial software OptiStruct Release 10.0 [1]. Industrial examples have shown that this capability is highly effective for enhancing durability of structures, It is expected that its availability within a widely distributed commercial software could generate a significant impact to the durability and safety of automotives and a wide range of consumer products.


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9.0 References [1] Altair Engineering, Inc., OptiStruct Release 10 - Users manual. Altair Engineering, lnc., Troy, MI, 2009. [2] L.A. Schmit, Structural design by systematic synthesis. Proc. 200 COIlf. on Electornic Computatlon., ASCE, New York, 1960. [3] Y.L. Lee, J. Pan, R.B. Hathaway and M. Markey, Fatigue Testing and Analysis - Theory and Practice. Elsevier, New York, 2005. [4] R.T. Haftka and R. Grandhi, Structural shape optimization -a survey. Compo Meth. App/. Mech. Engrg., 57, 91·106,1986. [5] Y. Ding, Shape optimization of structures – a literature survey. Compo Struct., 24, 9851004, 1986. [6] N. Kikuchi, K.Y. Chung, T. Torigaki and E. Taylor, adaptive finite element methods for shape optimization. Camp. Mel". Appl. Mech. Engrg., 57, 67-89,1986. [7] KH. Chang and KK Choi, A geometry-based parameterization method for shape design of elastic solids. Meeh. Struct. & Mach., 20, 215-252,1992. [8] R.J. Yang, A. Lee and D.T. McGeen, Application of basis function concept to practical shape optimization problems. Struct. Optim., 5, 55-63, 1992. [9] U. Schramm and W.D. Pilkey, The coupling of geometric descriptions and finite elements using NURBs – a study ill shape optimization. Finite Elements in Analysis and Design, 15,11 34,1993. [10] A.Schleupen, K Maute and E. Ramm, Adaptive FE-procedures in shape optimization. Struct. Opt/m.,19, 282-302, 1995. [I 1]L.A. Schmit, Structural synthesis- its genesis and development. AIAA J., 19, 1249-1263, 1981. [12] R.T. Haftka and Z. Giirda I, Elements of structural optimization. Klu wee Acad, Publ., Dordrecht, 1992. [13] U. Kirsch, Structural optimization: fundamentals and applications. Springer-Verlag, Berlin, 1993. [14] L.A. Schmit and B. Farshi, Some approximation concepts for structural synthesis. AIAA J., 12, 692-699, 1974.


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[15] R.T. Haftka and J.H. Starnes, Application of a quadratic extended interior penalty function for structural optimization. AIAA J., 14,718-724,1976. [16] C. Fleury and V. Braibant, Structural optimization: a new dual method using mixed variables. 111/. J. Num. Metll. Engrg., 23, 409-428, 1986. [17] G.N. Vanderplaats and E. Salajegheh, A ncw approximation method for stress constraints in structural synthesis. AIM J., 27, 252-358, 1989: [18] M. Zhou and R.W. Xia, Two-level approximation concept ill structural synthesis. Int. J. NUIIl. Meth. Engrg., 29,1681-1699,1990. [19] M. Zhou, Geometrical optimization of trusses by 8 two-level approximation concept. Struct. Optim., 1, 235-240,1990. [20] R,A. Cllufield, High quality approximation of eigenvalues in structural optimization, AIAA J., 28, 1143-1149, 1990. [21] G,N, Vanderplaats and H,L. Thomas, An improved approximation for stress constraints in plate structures, Struct. Optlm. 6,1-6,1993, [22] M. Zhou and H,L. Thomas, Alternative approximation for stresses in plate structures. AIAA J., 31, 31, 2169-2174, 2169-2174, 1993. 1993. [23] J.M, Barthelemy and R.T. Haftka, Recent advances in approximation concepts for optimum struc lura1design. Stntct. Optim. 12,129-144, 1993. [24] H.M. Adelman and R,T. Haftka, Sensitivity analysis of discrete structural systems, AIAA J. 24(5): 823-832, 1986. [25] N, Ollhoff, J. Rasmussen and E. L1111d, Method of "exact" numerical differentiation for error elimination in finite element based semi-analytical sensitivity analysis. Mech. Struct. Macn., 21, 1-66,1993, [26] M, Zhou, Y,K, Shyy and H.L. Thomas, Checkerboard and minimum member size control in topology optimization. Struct. OpUII/" 21, 152-158,2001. [27] H.L, Thomas, M. Zh011 and U. Schramm, Issues of commercial optimization software development. Struct. Multidisc. Optim., 23, 97-110, 2002. [28] M, Zhou, N. Pagaldipti, H.L. Thomas and Y,K. Shyy, An Integrated Approach for Topology, Sizing, and Shape Optimization. Struct. Multidisc, Optim., 26, 308-317,2004. [291M. Zhou, Y.K. Sh~ and H.L. Thomas, Topology Optimization with Manufacturing Constraints, Proceedings of the 4' World Congress oj Structural and Multidisciplinary Optimization, Dalian, China, 2000. [30] M. Zhou, R. Fleury, Y.K. Shyy, H,L. Thomas and J. Brennan, Progress in Topology Optimization with Manufacturing Constraints, AIAA-2002-4901, Proceedings of the 911JAIM MDO Conference, Atlanta, GA, 2002. [31] M. Zhou, Topology optimization of shell structures with linear buckling responses. Proc, 61hWorld congress of computational Mechanics, September 5-10, Beijing, China, 2004. [32] U. Schramm, M. Zhou, P.S. Tang and C.G. Harte, Topology Layout of Structural Designs and Buckling, AIAA-2004-4636, Proc. 101h AIAAIISSMO Symposium on Multidisciplinary Analysis and and Optimization, Optimization, Albany, Albany, NY, 2004, 2004, [33] P. Cervellera, M, Zh01l lind U, Schramm, Optimization Driven Design of shell Structures under Stiffness; Strength and Stability Requirements, Proceedings of the 6'h WCSMO, Rio de Janeiro, BRA, 2005, [34] M, Zhou, R, Fleury and W. Dias, Composite Design Optimization - from Concept to PlyBook Details, Proceedings of the 8'" WCSMO, Lisbon, Portugal, 2008.


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Structural Optimization Considering Fatigue Requirements

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