A Multi-point constraints based integrated layout and ...

Struct Multidisc Optim DOI 10.1007/s00158-014-1134-7

RESEARCH PAPER

A Multi-point constraints based integrated layout and topology optimization design of multi-component systems Ji-Hong Zhu · Huan-Huan Gao · Wei-Hong Zhang · Ying Zhou

Received: 16 April 2013 / Revised: 17 March 2014 / Accepted: 9 July 2014 © Springer-Verlag Berlin Heidelberg 2014

Abstract The integrated layout and topology optimization is to find proper layout of movable components and topology patterns of their supporting structures, where two kinds of design variables, i.e. the structural pseudo-densities and the components’ locations are optimized simultaneously. The purpose of this paper is to demonstrate a new multipoint constraints (MPC) based method where the rivets or bolts connections between the components and their supporting structures are introduced. The displacement consistence involved in the MPC is strictly maintained which makes the components to carry the loads together with the supporting structures. Moreover, more benefits like avoidance of finite element remeshing and precise geometry of the components can be obtained. In particular, sensitivities with respect to the components’ locations can be analytically and efficiently achieved by deriving the MPC equations. Finally, several numerical examples are tested and discussed to demonstrate the validity of the proposed method. Keywords Multi-component systems · Multi-point constraints · Layout optimization · Topology optimization

J.-H. Zhu · H.-H. Gao · W.-H. Zhang · Y. Zhou Engineering Simulation and Aerospace Computing (ESAC), Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China e-mail: [email protected] H.-H. Gao e-mail: Gao [email protected]

1 Introduction Topology optimization stepped into its rapid developing age since the homogenization-based method for continuum structures was proposed (Bendsøe and Kikuchi 1988). More excellent works were put forward later and obtained great success in both theoretical studies (See Bendsoe and Sigmund 2003; Bruyneel and Duysinx 2004; Allaire et al. 2004; Yan et al. 2008a; Guo and Cheng 2010; Deaton and Grandhi 2014) and practical applications (e.g. Maute and Allen 2004; Remouchamps et al. 2011). Recently, techniques of topology optimization have been developed to solve more complicated problems with multidisciplinary objectives (e.g. Maute and Allen 2004; Yan et al. 2008b; Yang and Li 2012) or complicated structural systems (e.g. Niu et al. 2009; Remouchamps et al. 2011). Among others, simultaneous design of structural configuration, elastic supports or fasteners, components etc. has become one of the most challenging works and gained a lot of attention. For instance, Chickermane and Gea (1997), Li et al. (2001), Qian and Ananthasuresh (2004) proposed a density method to design the topologies of several independent components and locations of interconnections modeled by spring elements. A simultaneous optimization procedure was further developed to deal with the number, position, stiffness of the supports, as well as the proper configuration and cross-sections of beam structures (Bojczuk and Mroz 1998). Thereafter, simultaneous layout design of continuum structures and involved elastic supports was studied to find efficient compliant mechanisms or to maximize the global stiffness (Buhl 2001). Ma et al. (2006) simultaneously optimized several separated design domains working as different components with different material properties. Another concept of integrated layout design of multicomponent systems was initially proposed and implemented

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for the components’ positions and supporting structures. The structural topology and components’ locations are designated as design variables and optimized simultaneously. The idea can be illustratively expressed as shown in Fig. 1. To solve the problem, one of the key difficulties lies in the description of the interfaces between the components and supporting structures. Two different strategies have been presented in the past ten years. For example, Qian and Ananthasuresh (2004) carried out the layout design of multicomponent systems with exponential functions describing the components geometry. The material distribution of the components is thus interpolated on the components’ boundaries. Shan (2008) and Zhang et al. (2013) took advantage of level-set functions to describe the shapes of components, together with the material distributions of both topology design domains and embedded components. Besides, a non-overlapping method was also proposed to avoid the geometry collision based on level-set functions in their work Later, Kang and Wang (2013) further developed the level-set based method during the simultaneous design of structural topology and movable holes in the design domain. Xia et al. (2012a, 2013) used R-functions and compress implicit functions to describe components’ geometry and material distributions. Zhang et al. (2012) adopted XFEM to handle material distribution of the components, especially on the boundary of the components where the elements were of multiple material phases. Above all, with the updating of structural topology and components’ locations being regarded as the changes of material properties, these methods hold the advantages of remesh-free and sensitivity analyticity, but have some difficulties to describe exactly the real material distribution around the interfaces between components and supporting structures Alternatively, Zhu et al. (2006, 2008, 2009 and 2010) and Xia et al. (2012b) proposed more techniques to use precise material distributions of the components. The layout design of the components was considered as a kind of packing optimization which was solved with a new Finite-circle Method (FCM) (Zhang and Zhu 2006; Zhu et al. 2012). New modeling techniques including density points and embedded

Fig. 1 Illustration of integrated layout design of multicomponent systems. a Typical definition b Optimized Design

meshing were adopted to implement components movements and pseudo-densities updating during optimization iterations. Elements on the components’ boundaries were properly meshed to ensure direct nodal connections with supporting structures. However, updating of components’ positions led to unsatisfied finite element remeshing and more difficulties during sensitivity analysis. Till now, all above efforts have been dedicated to the integrated layout optimization with components embedded in the design domain. As illustrated in Fig. 2a, local element remeshing or material interpolation is needed to ensure the direct nodal connections. But in many other practical industrial cases, as shown in Fig. 2b, components and supporting structures are mostly assembled together by rivets or bolts with the connecting positions designated in advance. The components are actually floating on the surface of the topological design domain and direct nodal connections are not applicable any more. In this paper, we propose to use multi-point constraints (MPC) to define rivets or bolts connections. The displacement consistence is strictly maintained by satisfying the MPC equations. When the components move, only the MPC connections need to be rebuilt at new positions. In this way, the advantages of remesh-free, analyticity as well as the precise material description are simultaneously maintained.

2 Basic definition of integrated layout design with MPC 2.1 Formulation of MPC To use MPC as the connections, the topological design domain and the components are discretized respectively into finite elements as illustrated in Fig. 3. Assume M1 is one of the connecting nodes on the component, which is projected to the point M∗1 inside the structural element e1 . Then we enable the following MPC equation.   uM1 = u∗M1= Ne1 M∗1 · ue1 uM1 − Ne1 M∗1 · ue1 = 0

(1)

A Multi-point constraints based integrated layout and topology optimization design Fig. 2 Connections between components and supporting structures. a Directly embedding b Bonding with MPC

where uM1 and u∗M1 denote the displacement   vectors of node M1 and point M∗1 . ue1 and Ne1 M∗1 are the displacement vectors of the element e1 and the shape function coefficient matrix at the point M∗1 . Note that the MPC equation is actually a linear combination of the nodal displacements. When we have multiple connections, these MPC equations as well as the boundary conditions can be organized as the following linear equations.

where K and F are respectively the global stiffness matrix and the global nodal load vector. λ is the Lagrange multiplier vector. The stiffness matrix can be expressed as

Hu = 0

Ks is the stiffness matrix of the supporting structure. K1 , K2 and Kn respectively stand for the stiffness matrices of the first, the second and the nth components. Similar definitions are used for the global load vector and displacement vector. Then we apply the stationary conditions to (3) and obtain

Ku + HT λ = F (5) Hu = 0

(2)

where H is the coefficient matrix determined by the shape functions of the structural elements, the MPC connection positions and the boundary conditions. u is the global displacement vector. Considering the above displacement constraints, the revised form of the overall potential energy of the global system can be express as (u, λ) =

1 T u Ku − FT u + λT Hu 2

(3)

⎡

Ks ⎢0 ⎢ ⎢ K = ⎢0 ⎢ ⎣0 0

0 0 0 K1 0 0 0 K2 0 .. . 0 0 0

0

0

0 0 0

⎤

⎥ ⎥ ⎥ ⎥ ⎥ 0 ⎦ Kn

(4)

By solving the above equations, the displacement vector u and Lagrange multiplier vector λ can be finally obtained. Similar explanations of MPC equations can be found in some existing works, e.g. Ainsworth (2001) and Yoon et al. (2004). Typically, to move the components during the optimization iteration, we only have to relocate the finite element models of the components at the new positions and rebuild the MPC equations. Since there are no direct nodal connections between the components and the supporting structures, the element remeshing is avoided in this procedure. 2.2 Optimization model

Fig. 3 Definition of MPC connections

In the integrated optimization, we choose to minimize the global strain energy to have better stiffness with a prescribed material volume constraint. The design variables are

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the pseudo-densities describing the structural distribution and the geometry variables for the components’ locations and orientations. More geometrical constraints have to be imposed to avoid the components’ overlapping and keep all the components inside the design domain. The optimization model can be mathematically expressed as Find: ηi , i = 1, 2, · · · · · · , Nd ; (ξj x , ξjy , ξj θ ), j = 1, 2, · · · · · · Nc min: C = 12 uT Ku subject to: Eq.(5) V ≤ VU j 1 ∩ j 2 = ∅, j 1 = 1, 2, · · · · · · Nc , j 2 = 1, 2, · · · · · · Nc , j 1 = j 2 j ⊆ d , j = 1, 2, · · · · · · Nc

(6) Fig. 4 An example for the proposed FCM

3 Sensitivity analysis where Nd and Nc are the numbers of the pseudo-density variables and the components, respectively. ξj x , ξjy and ξj θ are the geometry design variables i.e. the location and orientation related to the j th component. C is the strain energy of the structure system and K is the global stiffness matrix. V is the material volume fraction of the design domain with an upper limit of VU . j , j 1 and j 2 are the areas of the j th, j 1th and j 2th components, respectively. d denotes the area of the global design domain. To introduce design dependent loads such as the gravity to the topology design, the popularly used SIMP scheme will unfortunately lead to localized deformation in the low density areas (see Bruyneel and Duysinx 2004). To avoid the problem, it is necessary to improve the material interpolation model. Here we use a previously proposed polynomial material interpolation (Zhu et al. 2009). 

p

E(i) = (1 − α) ηi + αηi



E0 (i)

(7)

where ηi denotes the ith pseudo-density. E(i) and E0 (i) represents the corresponding Young’s modulus and the value of a solid element. p is the penalty factor and α is polynomial coefficient factor. In this study, p and α are set to be 4 and 1/16. To avoid the components’ overlapping and keep all the components inside the design domain, the previously proposed FCM (See Zhu et al. 2008) is applied here. As illustrated in Fig. 4, a series of circles with different radii are introduced to approximate the contours of the components and the design domain. The overlapping constraints between the components or those between the components and the boundary of the design domain can be analytically evaluated by the distances of the circles according to their center coordinates and radii.

Considering the static equation in (5), the differentiation with respect to the pseudo-density variable ηi can be written as ∂K ∂u ∂(F − HT λ) u+K = ∂ηi ∂ηi ∂ηi

(8)

Assuming F = f + G, where f and G stand for design independent external loads and design dependent inertial loads, respectively, we also have ∂(F − HT λ) ∂G ∂(HT λ) = − ∂ηi ∂ηi ∂ηi

(9)

Then we have the derivative of the overall strain energy ∂C ∂u 1 ∂K = uT K + uT u ∂ηi ∂ηi 2 ∂ηi

(10)

Substituting (8) and (9) into (10) and recalling ∂HT /∂ηi = 0, we yield ∂C ∂G ∂λ 1 ∂K = uT − uT HT − uT u ∂ηi ∂ηi ∂ηi 2 ∂ηi

(11)

As uT HT = 0, (11) can be simplified as ∂C ∂G 1 T ∂K = uT − u u ∂ηi ∂ηi 2 ∂ηi

(12)

where the derivatives of the inertial load vector G and the stiffness matrix K can be easily obtained according to

A Multi-point constraints based integrated layout and topology optimization design

the material interpolation model for the element mass and stiffness. The derivative of the strain energy with respect to the geometry design variable ξj is similarly written as

K∗j is the initial stiffness matrix of the j th component and Aj is the corresponding rotational transformation matrix. Suppose uj is the displacement vector of the j th component, then we can obtain

∂HT ∂C 1 ∂K = −uT λ − uT u ∂ξj ∂ξj 2 ∂ξj

uT

(13)

Suppose ξj is a translational variable of the j th component, both the stiffness matrix of the supporting structures and the components will remain unchanged after a translational moving. So we yield ∂C ∂HT = −uT λ ∂ξj ∂ξj

(14)

If ξj is a rotational variable, the stiffness matrix of the j th component after the rotation can be expressed as Kj = ATj K∗j Aj

(15)

⎧ ⎨

∂C = ⎩ −uT ∂ξj

∂HT ∂ξj λ 1 T T ∗ dAj 2 uj (Aj Kj dξj

−uT

∂HT ∂ξj

λ−

dATj ∗ ∂Kj dAj ∂K u = uTj uj = uTj (ATj K∗j + K Aj )uj ∂ξj ∂ξj dξj dξj j (16)

The derivative of the rotational transformation matrix can be easily calculated according to the coordinate transformation matrix. Consequently, (13) can be written as T dATj ∗ ∂C 1 T T ∗ dAj T ∂H = −u λ − uj (Aj Kj + K Aj )uj ∂ξj ∂ξj 2 dξj dξj j (17)

Now the final geometry sensitivities can be concluded as

when ξj = ξj x , ξjy +

dATj dξj

K∗j Aj )uj

Note that H consists of the shape function coefficient matrix of the corresponding elements, which depend upon the connection positions and further the geometrical design variables describing the components’ locations and orientations.

4 Numerical examples Some numerical examples are tested to verify the effects of the proposed optimization model and corresponding methods. 4.1 A rectangle structure with four components As shown in Fig. 5a, four components with different shapes are introduced into a 1.8m × 0.6m rectangle design domain and certain connections (the bolding component boundaries) are defined by MPC. The sizes of each component are shown in Fig. 5b. At first, only external loads are considered. Different material properties are set for the supporting structure and components as follows. For the supporting structure: the elastic modulus Es = 7 × 1010 pa, the Poisson’s ratio μs = 0.3, the density ρs = 2700kg/m3 .

when ξj = ξj θ

(18)

For the four components: the elastic modulus Ec = 2 × 1011 pa, the Poisson’s ratio μc = 0.3, the density ρc = 7800kg/m3 . With volume fraction upper bound of 0.5 and FCM nonoverlap constraints, the gradient-based algorithm GCMMA (Svanberg 2007) within the multidisciplinary optimization platform Boss-Quattro (Radovcic and Remouchamps 2002) is used to solve the problem. The structural layout is automatically optimized and all the components find proper positions to replace or reinforce the structural material locally as shown in Fig. 6. The optimization problem converges after 45 iterations. Strain energy value decreases from initial 0.32258J to final 0.03550J. In the following tests, different values of the elastic modulus are assigned for the components and everything else keeps unchanged. Figure 7 shows the different optimized design and their objective function values. The components, when they are stiff enough as shown in Fig. 7a and b, can become integrated parts in key load carrying path and replace some structural material. As the components become weaker, they generally change their locations from load carrying path to less important positions. Some of them are even just mounted on the structure, although they are still active in carrying loads.

J.-H. Zhu et al. Fig. 5 Optimization problem with four different components under only external loads. a The load case and initial states b Detailed sizes of four components of components

Fig. 6 Design iterations with four components under only external loads. a iteration 10 b iteration 20 c iteration 30 d iteration 40 e iteration 45

A Multi-point constraints based integrated layout and topology optimization design Fig. 7 Different optimization results due to different components’ stiffness. a Ec = 2 × 1011 pa, C = 0.03550J b Ec = 7.0 × 1010 pa, C = 0.03823J c Ec = 3.5 × 1010 pa, C = 0.03931J d Ec = 1.4 × 1010 pa, C = 0.03990J e Ec = 7.0 × 109 pa, C = 0.04073J

Fig. 8 Different optimized design due to different initial component layouts. a C = 0.03536J (b) C = 0.03550J (c) C = 0.03722J

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Fig. 9 Optimization problem with four components under both external and inertial loads

To have in-depth understandings of the integrated layout and topology optimization problems, three different initial locations of the four components and the corresponding optimized design are presented in the following test as shown in Fig. 8. Although the final objective function values are close, the optimized component layouts and structural topologies are different, which indicates that the different solutions we have achieved are local optima. As we can easily prove that the integrated optimization is non-convex, the gradient-based optimization algorithm (GCMMA) we used here can only guarantee local solutions. The optimized designs are extremely dependent upon the initial designs. Nevertheless, these local optima provide more possibility and selectivity to improve the optimized designs. The next test will present the effect of the inertial loads. As in Fig. 9, an acceleration of a = 50m/s 2 is applied to the whole structure system. With all other configurations of the problem unchanged, the optimization converges after 64

Fig. 10 Design iterations with four components under both external and inertial loads. a iteration 10 b iteration 20 c iteration 30 d iteration 40 e iteration 64

iterations. The evolution of the structural layout is shown in Fig. 10. Affected by the inertial loads, more material is distributed in the right part of the design domain to get close to the boundary conditions and the final configuration is locally different from former design in Fig. 6e. The four components are relocated to adapt to the new structure pattern. In Fig. 10a and b, the components are found moving rather quickly at the first several iterations when there are hardly any clear structures in the design domain. As the structure becomes clear, more limitations are actually brought to the components’ movement. Any improper movements will break the structural loading path and lead to the decrease of structural stiffness. Both optimization procedures thus evolved to reasonable local optima. The convergence histories of the two tests are rather stable and have been plotted in Fig. 11. In both optimized designs shown in Figs. 6e and 10e, we can find not all the MPC are interconnected with solid structure. It shows that in some cases the designated connection positions are probably not adaptive to the optimized structure topology. As a result, the integrated layout optimization just chooses to use a part of the connections which are more effective in composing the load carrying path. 4.2 A 3D solid beam with four components under external and inertial loads. Here a 3D model is presented considering both external and inertial loads. A solid beam with its size shown in Fig. 12a is discretized into 150 × 60 × 5 solid hexahedron elements

A Multi-point constraints based integrated layout and topology optimization design

Fig. 11 Iteration histories of the objective functions

and acts as the design domain. Four 0.01m thick rectangle components with stiffeners as shown in Fig. 12b are bonded onto the front and back surfaces of the beam with numbers of MPC, respectively. The location and orientation of the

Fig. 12 An optimization problem considering both external and inertial loads. a The load case of solid beam with four components b Sizes of components and connection positions

components 1 and 2 are decided by one group of geometry design variables, while those of the components 3 and 4 are decided by another group. In Fig. 12a, some downward external forces with the same value of 200N are applied to the nodes of the two 0.05m × 0.05m areas at the right end of the beam. An upward acceleration of 100m/s 2 is applied to generate the inertial loads. For the supporting structure: the elastic modulus Es = 7 × 1010 pa, the Poisson’s ratio μs = 0.3, the density ρs = 2700kg/m3 . For the four components: the elastic modulus Ec = 1.1 × 1011 pa, the Poisson’s ratio μc = 0.3, the density ρc = 4500kg/m3 . During the optimization, the upper bound of volume fraction constraint is set to be 0.3 and non-overlap constraints are also assigned with FCM. After 49 iterations, the optimization problem finally converges. Figure 13 gives the evolution history of the structure patterns. During the optimization iterations, we can find the two groups of

J.-H. Zhu et al. Fig. 13 Design iterations of four components and topology configuration. a iteration 9 b iteration 16 c iteration 23 d iteration 30 e iteration 49

components are now taking two critical positions in the load carrying path and acting as some indispensable parts of the structure system. The value of objective function decreases from initial 99.20J to final 6.63J as plotted in the curve in Fig. 14 while the volume fraction of the structure reaches the predefined upper bound.

Fig. 14 Iteration histories of global strain energy and material volume fraction

5 Conclusions In this paper, a new MPC based integrated layout and topology optimization method is proposed to have the layout of the movable components and the topology of the supporting structures designed simultaneously. By introducing MPC as the designated connections between the components and the supporting structure, the components are now floating and bonded on the surface of the topology design domain, which is different from the previous works of the integrated layout and topology optimization. Nodal coincidence and mesh regeneration are not needed any more which simplifies the finite element model itself as well as its modeling procedure. Moreover, sensitivities with respect to both topology and geometry design variables can be analytically obtained with the benefits of the fixed finite element mesh and MPC equations. To verify the validity of the proposed method, several numerical examples with external and inertial loads are taken into account and implemented. During the design procedure, the optimization algorithm pushes the components to proper positions and some clear and reasonable structures

A Multi-point constraints based integrated layout and topology optimization design

are generated to give firm supports. The obtained optimized results have evidently shown the components and the structures are now designed as integrities to carry the loads. As a conclusion, by introducing MPC to define the connections between the components and the structures, the proposed method and optimization procedure can be considered as an effective extension of the previous works of integrated layout and topology optimization.

Acknowledgments This work is supported by National Natural Science Foundation of China (90916027, 51275424, 11172236), 973 Program (2011CB610304), the 111 Project(B07050), Science and Technology Research and development projects in Shaanxi Province(2014KJXX-37), the Fundamental Research Funds for the Central Universities (3102014JC02020505).

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