A topology optimization method based on element ... - Springer Link

J. Cent. South Univ. (2014) 21: 558−566 DOI: 10.1007/s11771-014-1974-8

A topology optimization method based on element independent nodal density YI Ji-jun(易继军)1, 2, ZENG Tao(曾韬)1, RONG Jian-hua(荣见华)2, LI Yan-mei(李艳梅)3 1. School of Mechanical and Electrical Engineering, Central South University, Changsha 410083, China; 2. School of Automobile and Mechanical Engineering, Changsha University of Science and Technology, Changsha 410004, China; 3. Hunan Technical College of Water Resources and Hydro Power, Changsha 410131, China © Central South University Press and Springer-Verlag Berlin Heidelberg 2014 Abstract: A methodology for topology optimization based on element independent nodal density (EIND) is developed. Nodal densities are implemented as the design variables and interpolated onto element space to determine the density of any point with Shepard interpolation function. The influence of the diameter of interpolation is discussed which shows good robustness. The new approach is demonstrated on the minimum volume problem subjected to a displacement constraint. The rational approximation for material properties (RAMP) method and a dual programming optimization algorithm are used to penalize the intermediate density point to achieve nearly 0−1 solutions. Solutions are shown to meet stability, mesh dependence or non-checkerboard patterns of topology optimization without additional constraints. Finally, the computational efficiency is greatly improved by multithread parallel computing with OpenMP. Key words: topology optimization; element independent nodal density; Shepard interpolation; parallel computation

1 Introduction Structural optimization seeks to achieve the best performance for a structure while satisfying various constraints such as a given displacement. The types of structural optimization may be classified into three categories, i.e., size, shape and topology optimizations. Compared with other types of structural optimization, topology optimization of continuum structures is technically by far the most challenging and at the same time is economically the most rewarding. Topology optimization may greatly enhance the performance of structures for many engineering applications. Starting with the landmark paper of BENDSØE and KIKUCHI [1], numerical methods for topology optimization of continuum structures have been investigated extensively. Most of these methods are based on finite element analysis (FEA) where the design domain is discretized into a fine mesh of elements. Popular methods for topology optimization include the homogenization design method (HDM) [2−4], the solid isotropic material with penalty (SIMP) method [5−7],

which is a modification of the original HDM, the bi-directional evolutionary structural optimization (BESO) method [8−11], and the level set method [12−15]. In these methods, the elements within the design domain are used for discretizing both the material distribution and the displacement fields. Elemental densities which are assumed uniform within each element are taken as design variables. The element-wise topology optimization performed by these methods exhibits various numerical problems, such as grey-scale, checkerboard pattern and mesh dependency. SIGMUND and PETERSSON [16] surveyed many of the numerical instability problems encountered in continuum topology optimization. The usage of higher order finite elements, the imposition of perimeter constraints and spatial filtering techniques were proved to be effective in suppressing the numerical instabilities. Filtering techniques may be most widely used. Since there can be an optimal design associated with each filter strength, the design obtained will strongly depend on the filter characteristics. With the purpose to overcome the numerical instability and to generate more distinct and

Foundation item: Projects(11372055, 11302033) supported by the National Natural Science Foundation of China; Project supported by the Huxiang Scholar Foundation from Changsha University of Science and Technology, China; Project(2012KFJJ02) supported by the Key Labortory of Lightweight and Reliability Technology for Engineering Velicle, Education Department of Hunan Province, China Received date: 2012−08−09; Accepted date: 2012−10−10 Corresponding author: YI Ji-jun, PhD; Tel: +86−15874980592; E-mail: [email protected]

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smoother structural topology, a topology optimization method based on nodal design variables was developed. An essential issue for a material distribution based topology optimization approach is how to construct a material density field using the continuous values of the density design variables. In some studies, conventional displacement shape functions have been used to interpolate the nodal density values to the element-wise material density distribution [17−20]. However, these approaches are restricted to the case of bi-linear elements, since higher-order finite element (e.g., 8-node quadrilateral element) shape functions possess no rangerestricted property. POULSEN [21] used wavelet basis functions to interpolate the design field, and achieve designs that are nearly free of checkerboarding and one-node connected hinges. However, in these interpolation methods, the nodal design variables are converted to element-wise constant densities when evaluating elemental stiffness matrices. The processes also obtain the result of optimization, but the extracted contour of optimization is not smooth. Topology optimization methods based on continuous distribution of material densities can overcome checkerboard patterns by virtue of the continuous approximation and obtain smooth topology. The basic idea is construction of the continuous density field from the design variable points within the design domain rather than within each element by the interpolation scheme. The parameters interpolation was introduced by JOG and HABER [22] and they found both the Q4/U and Q4/Q4 implementations to be unstable, even with the Q9/U and Q8/U implementations (Q4/U is that displacement field is discretized by four points and design variable field is discretized by one point when a quadrilateral finite element is used in two dimensions). MATSUI and TERADA [18] presented a so-called CAMD (continuous approximation of material distribution) model into the homogenization topology optimization method. RAHMATALLA and SWAN [19] proposed a Q4/Q4 implementation of the SIMP method, in which the checkerboard patterns can be effectively avoided. Meanwhile, a new problem, namely the so-called “islanding” phenomenon, was observed. KANG and WANG [23] presented a non-local density interpolation method, in which a non-local Shepard interpolation scheme and higher-order elements were applied to eliminate the numerical instabilities, e.g., the checkerboards. WANG et al [24] proposed a multilevel nodal density-based approximation scheme for topology optimization of structures based on the concept of SIMP method, which was efficient in finite element implementation and effective in the elimination of numerical instabilities. An adaptive density point refinement approach for continuum topology

559

optimization on the basis of an analysis-mesh separated material density field description based on nodal design variables was presented by WANG et al [25], which could effectively reduce the number of design variables, and consequently reduce the computational time involved in the optimization. A topology optimization method based on the element independent nodal design variables was developed in this work. Regular B8 (8-node hexahedral) elements were applied to discretized the displacement field, and the nodal densities of each B8 element were considered as design variables. The developed element independent nodal density method was applied to the topology optimization of continuum solids with the objective of minimizing the volume subject to a displacement constraint.

2 Nodal design variables and interpolation scheme Element independent nodal densities are the design variable in this method. Thus, the relative density at any point is interpolated by an interpolation scheme. They determine the topology, stiffness and volume of material. 2.1 Identifying nodes As proposed by GUEST et al [23, 26] and KANG and WANG [23], we set the scale parameter rmin to identify the nodes that influence the density of point x. Nodes are included in the influence if they are located within a distance rmin of the point x. This can be visualized by drawing a sphere of radius rmin centered at the point x, thus generating the spherical subdomain Ωx. Nodes located inside Ωx contribute to the computation of density ρ(x) of point x. As the mesh is refined, rmin and Ωx consequently do not change. The only difference between the two meshes is the number of nodes located inside Ωx, which is included in the interpolation function. This is essential to generating mesh-independent solutions. 2.2 Shepard interpolation scheme Interpolation provide continuum of density field and mesh-independent, which might alleviate numerical instability and checkerboard effects [27]. In implementing continuum structural topology optimization formulations, many functions are available to interpolate nodal density onto the points inside the element space, for example, the standard C0 shape functions used in the finite element method. However, each node’s shape function influences only the elements connected to that node. Mesh-independent can not be obtained, when interpolation functions are influenced by mesh size. They should be based on a physical length

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scale that does not change with mesh refinement. In this work, Shepard interpolation method [23, 28−29] is used to achieve mesh independency. Let ρi (i= 1, 2, …, n) denote a set of density of node inside Ωx at the associated point x=(X, Y, Z), where (X, Y, Z) define the point x location in a Cartesian coordinate system. Thus, the relative density at point x is interpolated by the nodal densities inside the influence domain Ωx with Shepard interpolation method.    N i ( x) i (x  xi )  ( x)  iS x   i (x  xi )

(1)

Ri ( x) n

 Ri ( x)

(i  1, 2,, n)

(1)

i 1

where Ri ( x) 

1 , and ri ( x)  x  xi ri ( x)

2

is the education

distance from the points x to xi. n is the number of node inside the influence domain Ωx. In the method of element independent nodal variable density, the density in the element space is not constant, and the global density field of the structure has C0 continuity. It is also easy to know from the bounded property of Shepard interpolation that 0≤ρx≤1 holds if 0  i  1 (i  S x ). Moreover, the property Ni(x)≥0 also guarantees that the derivative of the density with respect to the design variable will be always non-negative. This property is essential for guaranteeing a correct searching direction in seeking the optimal material distribution by a gradient-based algorithm.

3 Formulation of topology optimization and numerical implementation Similar to the SIMP (solid isotropic microstructures with penalization) method [2−3, 30], the discrete topological variables ρi that only take value 0 or 1 are replaced by continuous topological variables between 0 and 1. Consequently, the difficulty of the discrete optimization is avoided by penalization. Adopting the rational approximation for material properties (RAMP), which was proposed by STOLPE and SVANBERG [31], we assume f (  ( x)) 

 ( x) 1  q[1   ( x )]

E ( x)  f (  x ) E 0

(3)

(4)

where E0 is elastic modulus of the fully solid material. The function f(ρx) has the following properties: f (  ( x))  0

where Sx is the sub-domain of design variable located within the influence domain Ωx of point x, and ρi is the density value of the i-th node. xi is the position of the point associated with the i-th node. The corresponding interpolation function Ni(x) is defined as

Ni ( x) 

where q is the penalization factor (q is a parameter which in some sense corresponds to p in the SIMP approach). In the SIMP method with a power law, p>3 is suitable for a wide range of applications. The function figure of p=3 is similar to the q=5 in RAMP. So, q=5 is set to make the intermediate density approach either 0 (void) or 1(solid). The relation between the elastic modulus and the material density at point x is expressed by

df (  ( x)) / d  ( x) 

as  ( x)  0 1  0 as  ( x)  0 1 q

The volume of an element is given by Vi  

Vi0

 ( x)dV

(5)

where Vi is the volume of the i-th element, Vi0 is the original volume of the i-th element. 3.1 Optimal model In continuum structures, topology optimization aims to optimize the material densities which are considered design variables in a design domain. Minimum volume with a reference domain Ω in R2 or R3 is considered while satisfying displacement constraints. The problem of structural optimization in terms of a minimum volume approach with displacement constraints can be stated as N el  min V   Vi i 1   s.t. u j  U j ( j  1, , J )   i  i  1 (iq  1, 2, , Q)   

(6)

where V is the structural volume being optimized, Vi is the volume of the i-th element, Nel is the number of all elements, uj is the displacement of the j-th degree of freedom of the structure, Uj is its constraint limit, J is the number of the displacement constraints, and ρi is the density of i-th node,  i is its lower limit. Here, the small positive lower bound  i =0.0001 is set so that the structure optimized is always kept to be non-singular in optimization process. 3.2 Sensitivity analysis The solution of the gradient-based optimization problem requires the computation of sensitivities of the objective function and the constraints. In a finite element analysis, the static behavior of a

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structure for any load case can be expressed by the stiffness equation as Ku=F

(7)

where K is the global stiffness matrix of a structure being optimized and, u and F are the global nodal displacement and nodal load vectors, respectively. It is assumed that the nodal density ρi has no effect on the load vector. Assuming that only ti=1/ρi is changed and is treated as a variable, we obtain the partial derivative of the displacement vector with respect to ti from Eq. (7) as u K   K 1 u  ti ti

(8)

The sensitivity of the j-th displacement is calculated using the adjoint method [32]. To find the partial derivative of uj with respect to ti, a unit virtual load vector Fj is introduced, in which only the j-th component is equal to unity and all the others are equal to zero. Multiplying Eq. (8) by F jT , following equations can be obtained  uj  ti

N con

   (uej )T e 1

K e e u ti

K e 1 K e  2 ti ti i

(9) (10)

where uj is the displacement due to the unit virtual load Fj, and ue and uej are the element displacement vectors containing the entries of u and uj, respectively, which are related to the e-th element. Ncon is the number of element influenced by ρi. K 0e is the element stiffness matrix of the e-th element of the solid material. The sensitivity of the displacement requires the computation of the sensitivity of the stiffness matrix with respect to the design variable. The derivative of the elemental stiffness matrix with respect to the design variable is expressed by

Ke   K

e

i

e



B T D(  ) B de f (  ( x))

f (  ( x)) i

i

e



B T D0 B de

(11)

f (  ( x))  ( x) (1  q )  N i (13)  ( x) i [(1   ( x))q  1]2

E ( x)(1   )  (1   )(1  2  )





1 

1 

0

0

0

0

1

0

0

0

0

1  2 2 1   

0

0

0

0

1  2 2 1   

0

0

0

0

1

 1 

 1 

    0    0    0    0   1  2   2 1     (14) 0

Since the stiffness matrix integrand is evaluated at the Gauss points, the densities at these Gauss points are directly computed from the design variables using interpolation function. Numerical quadrature, such as Gaussian quadrature, is commonly reduced to the evaluation and summation of the stiffness integrand at specific Gauss points. The sensitivity analysis of the objective function in Eq. (6) is calculated similarly to Eq. (12). The derivative of the total material volume with respect to the design variables can be computed by Gauss quadrature method over the influence-domain.

Ve  

e

 ( x)de

(15)

So, the total volume is N el

V   e 1

e

 Ni ( x) i de

(16)

iS x

The derivatives of V with respect to the design variables are

V Nel  Ni de i e 1  e V 1  ti (ti ) 2

(17)

 (  Ni de )

eS x

(18)

e

(12)

where B is the usual displacement-strain matrix and D0 corresponds to the constitutive matrix of the solid material. For example, the formulation of the constitutive matrix for 3D isotropic solid structures is D( x) 

  1    1      1      0    0    0 

3.3

Topological optimization with varying displacement limits In order to make the approximation functions of displacement constraints in Eq. (6) hold true at each iteration process and make the optimum topology obtained be of good 0−1 distribution topology variable property at the same time, an equivalent optimization model (Eq. (19)) with varying displacement constraint limits is built. And these varying displacement constraint limits are of the function of design variable trust-regions. This approach can approximately make structural

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design variables move in some small domains at every loop iteration step, and make some topological variables approach 0, and a lot of topological variables of the structure optimized be 1.0. N el

 min V    e  de  e 1   l ( j  1, , J ; l  1, 2, )  s.t. u j  U j  i  i  1 (i  1, 2, , N nod )

(19)

where U lj is expressed as U lj





 u k  min  u k , ( U  u k ) , u k  U j j j j j  j   u kj  min  u kj , U j  u kj , u kj  U j 





N nod

 (u j )t

i

i 1

tik

(20)

(ti  tik )

 uj

u j  u kj 

 ti

tik

N nod

N nod

i 1

i 1

 c(tik )tik 

 c(tik )t

where Nnod is the number of the nodes in the design domain. Thus, the j-th displacement in the next iteration, u kj 1 , can be estimated by the j-th displacement in the current iteration. Let

Ci j  c(tik )sign(u kj ) (i  1,, N nod ) Ai j  c(tik )tik sign(u kj ) (i  1, , N nod ) then Eq. (19) can be transferred into Eq. (21) N el  min : V    1 t de e  e 1  N nod N nod  l k k s.t.  Ci j ti  U j  u j sign(u j )   Ai j i 1 i 1   ( j  1,  , J ; l  1, 2, )   1  ti  ti (i  1, 2, , N nod )

ai 

3

(22)

(21)

bi 

N el

 (   Ni de ) (t ( k ) ) 2 e 1

i

Let c(tik ) 

 t  R N nod  N nod  min :  (bi (ti )2  ai ti )   i 1  N nod N nod l  k k    s.t. C t U u sign( u )  ij i j j  Ai j j  1  i i 1   1  ti  ti 

where

where j=1, …, J; l=1, 2, …. In the above, β is a displacement limit changing factor. Typical values of β between 0.01 and 0.20 have been used for displacement constraints in the example problems in this work. u kj is the displacement of the j-th degree of freedom of the structure at the k-th loop iteration step. U lj ( j  1, 2, , J ) are varied by Eq. (20) at every iteration step. An first-order series expansion for the displacement function uj at ti (i  1, 2,, N nod ) can expressed as u j  u kj 

If the constant items in the objective function are omitted, solving Eq. (21) can be transferred to solving Eq. (22)

1

e

N el

 (  Ni de )

(ti( k ) )3 e 1

e

The programming solving problem of Eq. (22) can be transferred into solving dual programming problem by using the dual theory [33−34].

4 Numerical examples This section illustrates the proposed approach with 3D applications. These examples including the cantilever beam and the MMB benchmark examples are investigated. For simplicity, all the quantities are dimensionless. In addition, elastic modulus is chosen as 2.1×1011 and Poison ratio as 0.3 for all examples. 4.1 Cantilever beam Figure 1(a) shows a 3D cantilever beam with length of 10, height of 6, and unit width. The beam is fixed at the left end area and a line load P=4000 is applied downward at the midline of the right end. The initial displacement at the constraint point, the center of the right end area, is 4.758×10−7. A displacement constraint is taken as 1.4×10−6. The cantilever domain is discretized into mesh size of 40×24×4 B8 elements which results in a total of 38400 elements as shown in Fig. 1(b). There are 41×25×5 design variable points distributed within the initial design domain. The results obtained from element independent nodal density and element-based approaches are shown in Figs. 1(c) and (d), respectively. These figures show that for the same displacement mesh size, the topology obtained from element independent nodal density has a much better resolution and smoother than that of the element-based approach. Let d denote the length of cuboid element diagonal line. To investigate the influence of length scale, we vary the length scales from 0.5d, 1.0d to 1.5d for above-mentioned approaches while keeping the same displacement mesh size, as shown in Fig. 2. When the

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P

(a)

(b)

(c) (d) Fig. 1 3D cantilever beam: (a) Design domain and boundary condition; (b) Finite element mesh; (c) Topology optimization results obtained with element independent nodal density; (d) Topology optimization results obtained with element-wise method

(a)

(b)

(c) Fig. 2 Topology optimization results obtained with various radii of interpolations: (a) rmin=0.5d; (b) rmin=1.0d; (c) rmin=1.5d

radius is larger than d, the optimized topologies are almost the same. When a line load P=4000 is applied downward at the line of the lower-right edge, the initial displacement at the constraint point, the midpoint of the lower-right edge, becomes 5.810×10−7. A displacement constraint is taken as 1.4×10−6. The results obtained from element

independent nodal density approach are shown in Fig. 3. A 3D cantilever beam with length of 15, height of 6, and unit width is fixed at the left end area and a line load P=4000 is applied downward at the midline of the right end. The initial displacement of constraint point, the center of the right end area, is 1.355×10−6. A displacement constraint is taken as 3.2×10−6. The

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the top area, is 6.023×10−6. A displacement constraint is taken as 2.0×10−5. The MMB domain is discretized into mesh size of 120×20×4 B8 elements. The results obtained from NDEI approach are shown in Fig. 6. As shown in Fig. 6, when the rmin is larger, the result of topology optimization is more stable. When the same mesh is used, computational cost of element independent nodal density is much more than the element-based approach, which is mainly due to the number of density nodes in the influence domain, meanwhile, the resolution of result topology is higher. The efficiency of topology optimization is more important when 3D large-scale problems, in which the finite element analysis cost is advanced by parallel programming technique, are considered. Now that, all commodity processors are becoming multicore. OpenMP provides one of the few programming models that allow computational scientists to easily take advantage of the parallelism offered by these processors. By using multiple slower cores, it is possible to achieve higher clock-speeds more efficiently than by designing super-fast individual processors [35]. Nowadays, personal computers using multi-core processing technology become widely available to consumers. In order to use a multicore processor at full capacity the applications run on the system must be multithreaded. The current application is redesigned to run optimally on a multicore system. These improvements will provide faster programs and a better computing experience.

cantilever domain is discretized into mesh size of 60× 24×4=5760 B8 elements. There are 61×25×5=7625 design variable points distributed within the initial design domain. The results obtained from element independent nodal density approach are shown in Fig. 4.

Fig. 3 Topology optimization result (rmin=1.0d)

Fig. 4 Topology optimization result (rmin=1.0d)

4.2 MMB beam Figure 5 shows a 3D MMB beam with length of 60, height of 10, and width of 2. An external force P=4000 is applied to the top-middle line, as shown in Fig. 5. The initial displacement at the constraint point, the center of

P

Fig. 5 Design domain and boundary condition

(a)

(b)

(c) Fig. 6 Topology optimization results obtained from NDEI approach with various radii of interpolations: (a) rmin=0.5d; (b) rmin=1.0d; (c) rmin=1.5d

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4.3 Cubic block A cubic domain shown in Fig. 7(a) is discretized into mesh size of 40×40×40 B8 elements which results in a total of 64000 elements. An external force P=4000 is applied to the center of top area, the four corners of the ground area is fixed, as shown in Fig. 7(a). The initial displacement at the constraint point, the center of the top area, is 1.52×10−7. A displacement constraint is taken as 3.048×10−7. The results obtained from NDEI approach are shown in Fig. 7(b). When the four-core processor (2.66 GHz) is used, the computational efficiency of parallel computation is approximately three times of serial computation, which is listed in Table 1.

P

(a)

within a finite element is used. In contrast to the element-based procedure, point density values are interpolated by Shepard functions. It avoids checkerboard patterns and mesh-dependency for low order finite elements. The method shows good robustness. These density values are used in order to determine a smooth iso-line to describe the boundary of the optimization layout. As a result, a smooth topology can be obtained. 2) The topology optimization problem is treated as the material distribution problem. A node with 0 relative density represents a void and a node with a relative density of 1 represents a solid node. To solve such a topology optimization problem, sequential quadratic programming algorithm and dual programming is employed with an iterative scheme for updating the design variables. Numerical examples demonstrate the effectiveness of the proposed method with respect to optimal solutions and convergence compared with the typical solutions of element-based topology optimization and nodal-based topology optimization. The numerical instability problems related to a finite element mesh do not exist in the proposed method. 3) A multithread parallel computing with OpenMP has been implemented which has significantly improved the computational efficiency, especially for the 3D optimization problems.

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(Edited by DENG Lü-xiang)