EVOLUTIONARY STRENGTH OPTIMIZATION OF ON- SERT AND ...

EVOLUTIONARY METHODS FOR DESIGN, OPTIMIZATION AND CONTROL P. Neittaanm¨ aki, J. P´ eriaux and T. Tuovinen (Eds.) c CIMNE, Barcelona, Spain 2007

EVOLUTIONARY STRENGTH OPTIMIZATION OF ONSERT AND INSERT SHAPES FOR LOCAL LOAD INTRODUCTION David Keller

Gerald Kress

Paolo Ermanni

Centre of Structure Technologies, ETHZ ETH Zentrum, LEO C4, Leonhardstr. 27 8092 Zurich, Switzerland Email: [email protected] Web page: http://www.structures.ethz.ch

Centre of Structure Technologies, ETHZ ETH Zentrum, LEO C2, Leonhardstr. 27 8092 Zurich, Switzerland Email: [email protected]

Centre of Structure Technologies, ETHZ ETH Zentrum, LEO C3.2, Leonhardstr. 27 8092 Zurich, Switzerland Email: [email protected]

Abstract. Local load introduction into fiber reinforced panels is achieved by metallic insert or onsert elements. These are either inserted into the substrate or simply bonded onto its surface. The shape of these elements has much influence on the load capcity of the system. Strength optimal shapes depend on the loading and the mechanical properties of the substrate. This study introduces a method for the automatic shape optimization of load introduction elements. The geometry is represented by a non-uniform rational B-spline (NURBS). The coordinates of the control points of this spline are adapted by an Evolutionary Algorithm (EA). Candidate solutions are simulated with the Finite Element Method (FEM). Numerical examples illustrate the capabilities of the method. Key words: Shape Optimization, Structural Optimization, Evolutionary Algorithms, Constraint Handling Technics, Finite Element Analysis, NURBS

1

INTRODUCTION

Along with the requirement for ever increasing product performance, the need for new joining technologies in multi-material systems may arise. This leads to a demand in bonding technologies that can connect components from different materials. Nowadays, local load introduction into flat panels is often achieved by the use of metallic inserts which are inserted into the core of a sandwich structure or directly integrated into the laminate. Thus, they lead to local imperfections in the substrate. Especially when it comes to applications with fiber reinforced materials a requirement for minimal damages caused by mountage may arise. Hence, a so 1

D. KELLER et al./Evolutionary Strength Optimization of Onsert and Insert Shapes

F

F

(a) Onsert system

(b) Insert system

Figure 1: Onsert and insert system with loading and support

called onsert has been proposed by Kress et al. 1 . This new joining element is simply bonded onto the surface of the otherwise unharmed structure. Prior investigations by Kress et al. 2, 3 point out that the shape of onsert systems has much influence on the load capacity of the load introduction system. 1.1

Onsert and insert systems

Two different situations are studied: An onsert and an insert system. The onsert is simply bonded onto the surface of the substrate (Fig. 1(a)). The tension load is perpendicular to the substrates surface. It is introduced through a screw. The insert is integrated into the core of a sandwich structure (Fig. 1(b)). It is bonded on the inner surface of one face sheet. A hole is required in one sheet in order to introduce the load with a screw. 2 2.1

SHAPE OPTIMIZATION METHOD Geometry Model and Shape Parameterization

A spline-based parameterization-concept is developed. Splines are a kind of piecewise polynomials which found wide applications in computational shape and topology optimization (e.g. Braibant and Fleury 4 , Eschenauer et al. 5 , Cervera and Trevelyan 6 , 7 ). NURBS – non-uniform rational B-splines – are homogenous images of higher-dimensional B-splines and able to exactly represent conic sections. These are required to form the cylindrical hole of the onsert or insert. The shape is encoded by a web of control points defining one NURBS body in three dimensions. The coordinates of the control points of this body are subject to change in the optimization. The geometry model is completed with a circular disc forming the relevant part of the substrate and a body in between the onsert system and the substrate forming the bond layer. 2.2

Analysis Model

The Finite Element Method (FEM, cf. Zienkiewicz and Taylor 8 ) is employed to simulate the structural response of the system to loading. The finite element mesh – i.e. the position of the nodes and the coincidence table – is directly constructed from the NURBS representation of the shape. 2.3

Objective and Constraint

The numerical optimization is carried out in order to maximize the load capacity of the bonded system. Thus, the objective can be represented by an ultimate load Fult of a load introduction element subject to a constraint on its maximal mass m, i.e. maximize Fult subject to m ≤ mlimit . Fult denotes the load at which the

EVOLUTIONARY METHODS FOR DESIGN, OPTIMIZATION AND CONTROL (EUROGEN 2007)

Do

Dc 1

1

0.5

0.1 O Oestim

Oinit

(a) Objective mapping function

0.01

C Climit

Climit + Ctolerance

(b) Constraint mapping function

Figure 2: The objective mapping function introduced by König 9 is parameterized by an upper and a lower bound Oinit and Oestim . A smoothed step function is used for the mapping of the constraint value. It is defined by a limit value Climit and some tolerance Ctolerance .

first component of the system suffers irreversible deformations. A scalar fitness value F is accumulated from two weighted demands: an objective term Do rating the load capacity and a penalty term Dc for the mass constraint. This allows to substitute the original problem with an unconstrained formulation: minimize F = wo Do + wc Dc . Two alternative objectives are examined. The first one founds on the original problem formulation whereas the second one corresponds to the definition used by Kress et al. 3 . Objective A: maximal load capacity Depending on the characteristics of the system, i.e. the thickness and assembly of the substrate and the material and shape of the load introduction element, the first failure to be expected may be in either the onsert/insert, the bond layer or the substrate. Different failure scenarios are tested, one for each component. The deformation and stress fields are computed from the Finite Element model for a predefined load Finit . From the stress field a material utilization coefficient ui can be calculated for each component. A maximal stress criterion in this direction is applied to determine the load capacity of the substrate. The maximal utilization umax = max ui over all components is used to estimate the 1 ultimate load Fult = λ · Finit = umax · Finit . The ultimate load Fult is directly mapped to a fitness portion Do by the objective mapping function developed by König 9 (see fig. 2(a)). Objective B: minimal stress deviation Kress et al. 3 used a stress deviation measure in the bond layer to rate different candidate solutions. However, this deviation measure does not take into account possible failures in the other components. The objective term finally takes into account a linear load factor λ = umax −1 as well as the characteristics of the stress distribution in the bond layer. The portion of the objective to the fitness is defined as: sR 2n 1 2n A (σeqv,h − σ eqv ) dA , (1) O= · λ A where A denotes the bottom area of the bond layer, σeqv,h is the equivalent von Mises stress averaged over the height of the bond layer, and σ eqv is the mean value


of the equivalent von Mises stress in the bond layer. A predefined power n (n ∈ N) is introduced in order to adjust the penalty of local peaks in the stress distribution. The objective term O is mapped to a fitness portion Do according to the definition of König 9 . Constraint The constrained mass value is directly mapped to a penalty portion Dc in the range (0, 1) by the upper-limit function introduced by König 9 (see fig. 2(b)). Constraint Adaption The load capacity of the system depends on the bonding area and therewith the mass of the load introduction element: With increasing area (and mass) the load capacity increases. Therefore, one could expect optimal solutions to be found on the constraint with a mass of mlimit . In order to reduce the risk to achieve infeasible solutions a self-adaptive strategy developed by Giger 10 is engaged. So the weight wc as well as the slope of the penalty function for the constraint term Dc are adapted on the run according to the values of the best candidate solution in the current population. 2.4

Optimization Algorithm

The consideration of different failure scenarios in the objective formulation leads to local discontinuities in the fitness landscape in the regions where the first failing component changes. Furthermore, the consideration of different onsert materials introduces another discrete parameter and thus renders the parameterization heterogeneous. Thus, an Evolutionary Algorithm is engaged. The gene definitions and genetic variation operators are taken from the universal genotype concept introduced by König 9 . 3

NUMERICAL EXAMPLES

Three different cases are examined. The material models employed can be seen from tab. 1. Case A: Rotationally symmetric onsert on isotropic substrate The substrate is formed by a plate of Al 2 (thickness 2mm). The onsert is to be manufactured from Steel. Two optimization runs are presented whereas for the first one objective A is minimized and objective B for the second one, respectively. The best shapes found after 240 generations with 60 individuals each are illustrated in fig. 3(a) and fig. 3(b). The highest ultimate load found for objective A is 656.216 N at a weight of 15.7284 gr while for objective B the values are 639.117 N at 14.666 gr. Case B: Quarter-symmetric insert in orthotropic sandwich The sandwich consists of two face sheets of unidirectional CFRP 2 (thickness 2mm each) and 26mm Core material in between. The insert is to be manufactured from steel. An optimization for objective A is carried out and the best solution after 330 generations with 60 individuals each is illustrated in fig. 3(c). An ultimate load of 832.352 N with a weight of 14.7024 grams is achieved. The first failure has to be expected in the bond layer.

EVOLUTIONARY METHODS FOR DESIGN, OPTIMIZATION AND CONTROL (EUROGEN 2007)

Onsert or Insert Young’s modulus/GP a Poisson’s ratio Failure criteria Stress at failure/MP a Density/kgm−3

PA-CF40 25 0.35 vM 60 1185

Mg 42 0.35 vM 90 1740

Al 1 69 0.3 vM 130 2700

Ti 102 0.35 vM 220 4500

Steel 200 0.3 vM 1000 7850

CFRP 1 CFRP 2 Core 90.5 / 6.6 135 / 10 0.36 0.3 / 0.38 0.27 / 0.3 0.28 mZ mZ vM 37 55 3.5

Bond 1.6 0.4 vM 16.6

Substrate and Bond Young’s modulus/GP a Poisson’s ratio Failure criteria Stress at failure/MP a

Al 2 72 0.33 vM 290

Table 1: Material models employed in the numerical examples (vM: von Mises, mZ: maximal stress in z-direction).

(a) Strength optimal (b) Strength optimal (c) Strength optimal (d) Strength optimal onsert (case A, objec- onsert (case A, objec- insert (case B) onsert (case C) tive A) tive B) Figure 3: Strength optimal insert (case B) and free-matieral onsert (case C). The colors in the substrate illustrate the 0-direction of the laminate.

Case C: Quarter-symmetric onsert with variable material on orthotropic laminate The substrate consists of a symmetric laminate of CFRP 1 with an overall thickness of 2mm. The ply orientation is defined as follows: [0, 0, 90, 0, 0, 90]s. The onsert material is to be chosen from PA-CF40, Mg, Al 1, Ti, or steel. Objective A is employed. The optimization is carried out with a population size of 150 individuals. The best result achieves an ultimate load of 362.678 N with a weight of 14.9132 grams. It is illustrated in fig. 3(d) and consists of PA-CF40. The bond layer is the first failing compontent. 4

CONCLUSIONS

The numerical examples illustrate the capabilities of the method. Feasible solutions are generated for each case study. The constraint adaption mechanism tolerates infeasible solutions at the beginning of the optimization process but increases the penalty on the run. Optimal solutions could be expected on the constraint with a mass of exactly 15 grams. However, these are not likely to be found for three reasons: Once a near-optimum solution is found the convergence of the method is slow what makes the last tuning steps computationally expensive. Furthermore, the fitness adaption mechanism with fixed threshold values and amplification/reduction factors may be too coarse to find the exact trade off between objective and penalty. And finally, the difference in the mass and strength values of a near-optimum solu-


tion and a solution exactly on the constraint may be lower than the numerical noise in the simulation. ACKNOWLEDGEMENTS The present study results from a cooperation with the Hochschule f¨ ur Technik Rapperswil (Switzerland) and the Hochschule Ravensburg-Weingarten (Germany). It has been suggested by M. Henne. The pleasant collaboration with M. Henne, M. Niedermeier, and M. Di Domenico is gratefully acknowledged. This study is part of project no. GRS-060/05 supported by the Gebert-R¨ uf Foundation. REFERENCES [1] G. Kress, P. Naeff, M. Niedermeier, and P. Ermanni. The onsert: A new joining technology for sandwich structures. Composite Structures, 73:196–207, 2006. [2] G. Kress, P. Naeff, M. Niedermeier, and P. Ermanni. Onsert strength design. International Journal of Adhesion and Adhesives, 24:201–209, 2004. [3] G. Kress, D. Fritsche, and P. Ermanni. Failure criteria and onsert shape optimization. International Journal of Adhesion and Adhesives, 25:109–120, 2005. [4] V. Braibant and C. Fleury. Shape optimal design using B-splines. Computer Methods in Applied Mechanics and Engineering, 44:247–267, 1984. [5] H. A. Eschenauer, V. Kobelev, and A. Schumacher. Bubble method of topology and shape optimization of structures. Struct. Optim., 8(1):42–51, 1994. [6] E. Cervera and J. Trevelyan. Evolutionary structural optimisation based on boundary representation of NURBS. part I: 2d algorithms. Computers & Structures, 83:1902–1916, 2005. [7] E. Cervera and J. Trevelyan. Evolutionary structural optimisation based on boundary representation of NURBS. part II: 3D algorithms. Computers & Structures, 83(23-24):1917–1929, September 2005. [8] O.C. Zienkiewicz and R.L. Taylor. The finite element method, volume 2. Oxford, 5 edition, 2000. ISBN 0-7506-5055-9. [9] O. König. Evolutionary Design Optimization: Tools and Applications. PhD thesis, Swiss Federal Institute of Technology ETH, Z¨ urich, 2004. Diss. ETH NO. 15486. [10] M. Giger. Representation Concepts in Evolutionary Algorithm-Based Structural Optimization. PhD thesis, Swiss Federal Institute of Technology ETH, Z¨ urich, 2007. Diss. ETH NO. 17017.