Topology optimization of continuum structures with

6th World Congresses of Structural and Multidisciplinary Optimization Rio de Janeiro, 30 May - 03 June 2005, Brazil

Topology optimization of continuum structures with respect to simple and multiple eigenfrequencies Jianbin Du, Niels Olhoff Institute of Mechanical Engineering, Aalborg University, Aalborg, Denmark, [email protected], [email protected] 1. Abstract A frequent goal of the design of vibrating structures is to avoid resonance of the structure in a given interval for external excitation frequencies. This can be achieved by, e.g., maximizing the fundamental eigenfrequency or the gap between two consecutive eigenfrequencies of given order. This problem is often complicated by the fact that the eigenfrequencies in question may be multiple, and this is particularly the case in topology optimization. In the present paper, different approaches are considered and discussed for topology optimization involving simple and multiple eigenfrequencies of linearly elastic structures without damping. The mathematical formulations of these topology optimization problems and several illustrative results are presented. 2. Keywords: eigenfrequency design, multiple eigenvalues, topology optimization, bound formulation 3. Introduction Problems of passive design against vibrations and noise were already undertaken some decades ago in the papers [1,2] in the form of shape optimization of transversely vibrating beams with respect to fundamental and higher order eigenfrequencies. By optimizing with respect to the fundamental eigenfrequency, minimum cost designs against vibration resonance were obtained subject to all external excitation frequencies within the large range from zero and up to the particular optimum fundamental eigenfrequency. Optimization with respect to a higher order eigenfrequency was found to produce a considerable gap between the subject eigenfrequency and the adjacent lower eigenfrequency, and this approach offered even more competitive designs for avoidance of resonance in problems where external exitation frequencies are confined within a large interval with finite lower and upper limits. In the subsequent paper [3], the design objective was directly formulated as maximization of the separation (gap) between two consecutive eigenfrequencies of the beam. Topology optimization with respect to eigenfrequencies of structural vibration was first considered by Dias and Kikuchi [4], who dealt with single frequency design of plane disks. Subsequently, Ma et al. [5,6], Dias et al. [7], and Kosaka and Swan [8] presented different formulations for simultaneous maximization of several frequencies of free vibration of disk and plate structures, defining the objective function as a scalar weighted function of the eigenfrequencies. In contrast to this, the papers by Krog and Olhoff [9] and Jensen and Pedersen [10] apply a variable bound formulation which facilitates proper treatment of multiple eigenfrequencies that very often result from the optimization. Ref. [9] treats optimization of fundamental and higher order eigenfrequencies of disk and plate structures, and [10] deals with maximization of the separation of adjacent eigenfrequencies for bi-material plates. A recent paper by Pedersen [11] deals with maximum fundamental eigenfrequency design of plates, and includes a technique to avoid spurious localized modes. Methods for optimization of simple eigenvalues/eigenfrequencies in shape and sizing design problems are well established and can be implemented directly in topology optimization. However, particularly in topology optimization it is often found that, although an eigenfrequency is simple during the initial stage of the iterative design procedure, it may become multiple due to the coincidence of this eigenfrequency with one (or more) of its adjacent eigenfrequencies. In order to capture this behaviour, it is necessary to apply a more general solution procedure that allows for the multiplicity of the eigenfrequency because such an eigenfrequency does not possess usual differentiability properties. In the present paper, different approaches are presented for topology optimization problems involving simple and multiple eigenfrequencies. The first is based on the method presented by Seyranian, Lund and Olhoff [12], and presents a new OC-based algorithm including an extension for handling the additional constraints on the design variables in topology optimization. In the second approach, the eigenfrequency optimization problem is formulated by a bound formulation (Bendsøe, Olhoff and Taylor [13], Taylor and Bendsøe [14], Olhoff [15]) which can be solved efficiently by mathematical programming or the MMA method (Svanberg [16]). Moreover, the procedure of treating the multiple eigenvalues can be greatly simplified by using the increments of the design variables as unknowns (Krog and Olhoff [9]). The material of our paper is organized as follows. Section 4 gives a brief account of the SIMP (Solid Isotropic Microstructure with Penalty) material models used in this paper. An extended form of the SIMP model for handling of bi-material topology design is also included. In Section 5, structural topology optimization subject to prescribed volume of material are first considered for problems of maximizing the fundamental or a higher order eigenfrequency, and then problems of maximizing the distance (gap) between two consecutive eigenfrequencies are considered. Section 5 also presents topological design sensitivity results for simple and multiple eigenvalues. Several numerical solutions to the types of problems considered in Section 5 are then presented in Sections 6, and Section 7 concludes the paper. 4. Material interpolation for topology optimization of vibrating structures Topology optimization is basically a problem of discrete optimization, but this difficulty is avoided by introducing relationships between stiffness components and the volumetric density of material ρ, which is a continuous variable defined between limits 0 (corresponding to void) and 1 (corresponding to solid elastic material) over the admissible design domain. The aim of the optimization process is to determine the optimum zero(void)-one(solid) distribution of a prescribed amount of the given material over the admissible design domain. To achieve this goal, many different material models have been developed (see e.g. [17,18]), among which the SIMP (Solid Isotropic Microstructure with Penalty) model (see, e.g., Bendsøe [19]; Rozvany, et al. [20]; Bendsøe and

2 Sigmund [21]) is a simple and effective one which is widely used in optimum topology design. 4.1 SIMP model for topology optimization of single-material structures The SIMP model is normally applied together with a filter technique (see, e.g., Sigmund [22]), as this prevents checkerboard formation and dependency of optimum topology solutions on finite element mesh-refinement. Thus, the finite element elasticity matrix Ee is expressed in terms of the element volumetric material density ρe, 0 ≤ ρe ≤ 1, and the penalization power p, p ≥ 1, as (1) E e ( ρ e ) = ρ ep E *e , where E*e is the elasticity matrix of a corresponding element with the fully solid elastic material the structure is to be made of. By analogy with (1), for a vibrating structure the finite element mass matrix may be expressed as (2) M e ( ρ e ) = ρ eq M *e , where M *e represents the element mass matrix corresponding to fully solid material, and the power q ≥ 1. Apart from exceptions briefly discussed in the following section, normally q = 1 is chosen. The global stiffness matrix K and mass matrix M for the finite element based structural response analyses behind the optimization can now be calculated by NE

K = ∑ ρ ep K *e , e =1

NE

M = ∑ ρ eq M *e ,

(3)

e =1

where K *e is the stiffness matrix of a finite element fully solid material, and NE denotes the total number of finite elements in the admissible design domain. 4.2. Localized eigenmodes With values assigned to p and q as stated above, application of the SIMP model for problems of topology optimization with respect to eigenfrequencies may lead to the occurrence of spurious, localized eigenmodes associated with very low values of corresponding eigenfrequencies. The localized eigenmodes may occur in sub-regions of the design domain with low values of the material density (e.g. ρe ≤ 0.1), where the ratio between the stiffness (with, say, p = 3 in the interpolation formula) and the mass (with q = 1) is very small. To eliminate these spurious eigenmodes, we have set the mass very low via a high value of the penalization power in subregions with low material density (Pedersen [11]; Tcherniak [23]). Thus, the interpolation formula (2) for the finite element mass matrix was modified as

⎧ ρ M * , ( ρ e > 0.1) . M e ( ρ e ) = ⎨ er e* ⎩ρ e M e , ( ρ e ≤ 0.1)

(4)

where the penalization power r is chosen to be about r = 6, i.e., much larger than the penalization power p for the stiffness, which is kept unchanged at a value about p = 3. 4.3 SIMP model for topology optimization of bi-material structures The SIMP model for topology optimization of structures made of two different solid elastic materials can be easily obtained by an extension of the SIMP model for single-material design. Following Bendsøe and Sigmund [24], the finite element elasticity matrix for the bi-material problem can be expressed as (5) E e ( ρ e ) = ρ ep E*e1 + (1 − ρ ep )E*e2 , where E *e1 and E *e2 denote the element elasticity matrices corresponding to the two given solid, elastic materials *1 and *2. Here, material *1 is assumed to be the stiffer one. The penalization power p in (5) was assigned the value 3 in this paper. The element mass matrix of the bi-material model may be stated as the simple linear interpolation (6) M e ( ρ e ) = ρ e M *e1 + (1 − ρ e )M *e2 , where M *e1 and M *e2 are the element mass matrices corresponding to the two different, given solid elastic materials *1 and *2. When bi-material design is treated via the problem formulations in Sections 5 and 6, then V* denotes the total volume

NE

∑ρ V e =1

e

e

*

of the stiffer material *1 available for the structure, while the total volume of material *2 is given by V0 - V , where V0 is the volume of the admissible design domain. In figures in Section 6 presenting optimum topologies of bi-material structures, material *1 is shown in black and material *2 in grey. 5. Formulations of eigenfrequency optimization problems 5.1 Maximization of the fundamental eigenfrequency Problems of topology design for maximization of fundamental eigenfrequencies of vibrating elastic structures have, e.g., been considered in [4-9,11]. Assuming that damping can be neglected, such a design problem can be formulated as a max-min problem as follows,

3 max {min{ω 2j }} ρe

(7 a )

j

Subject to : Kφ j = ω 2j Mφ j ,

( j = 1, ..., J ),

( 7b )

φ Mφ k = δ jk ,

( j ≥ k , k , j =1, ..., J ),

(7 c )

T j

Ne

∑ρ V e =1

e

e

(V * = αV0 ),

(7 d )

(e = 1,L , N E ).

( 7 e)

−V * ≤ 0 ,

0 < ρ ≤ ρe ≤ 1 ,

Here ωj is the jth eigenfrequency and ϕj the corresponding eigenvector, and K and M are the symmetric and positive definite stiffness and mass matrices of the finite element based, generalized structural eigenvalue problem in the constraint (7b). The J candidate eigenfrequencies considered will all be real and can be numbered such that (8) 0 < ω1 ≤ ω 2 ≤ L ≤ ω J ,

and it will be assumed that the corresponding eigenvectors are M-orthonormalized, cf. (7c) where δjk is Kronecker’s delta. In problem (7a-e), NE denotes the total number of finite elements in the admissible design domain. The design variables ρe, e = 1,…,NE, represent the volumetric material densities of the finite elements, and (7e) specify lower and upper limits ρ and 1 for ρe. To avoid singularity of the stiffness matrix, ρ is not zero, but taken to be a small positive value like ρ = 10-3. In (7d), the symbol α

defines the volume fraction V * /V0 , where V0 is the volume of the admissible design domain, and V * the given available volume of solid material and of solid material *1, respectively, for a single-material and a bi-material design problem, cf. Sections 4.1 and 4.3. 5.2 Sensitivity analysis of a unimodal eigenfrequency If the jth eigenfrequency ωj is unimodal (also called simple or distinct), i.e.,

ω j −1 < ω j < ω j +1 , then the corresponding eigenvector

φ j will be unique and differentiable with respect to the design variables ρe, e = 1,…,NE. The sensitivity (λ j )′ρ of the eigenvalue e

λ j = ω 2j with respect to a particular design variable ρe, is then given by (see Williams [25] and Lancaster [26]) (λ j )′ρ e = φ Tj (K ′ρ e − λ j M ′ρe ) φ j , (e = 1,L, N E ) .

(9)

Here ( )′ρ = ∂( ) ∂ρ e . The derivatives of the matrices K and M can be calculated explicitly from the material models in Section 4. e The optimality condition for the maximization of a unimodal eigenvalue λ j = ω 2j of given order j (j = 1, 2, …) now follows from (9) and usage of the Lagrange multiplier method, and takes the form

(λ j )′ρe − γ 0Ve = 0 , (e = 1,L, N E ) ,

(10)

where γ0 (≥ 0) is the Lagrange multiplier corresponding to the material volume constraint, and the side constraints for ρe have been ignored. With this sensitivity result and optimality condition, the design problem (7a-e) may be solved for a unimodal optimum eigenfrequency by using an OC (Optimality Criterion) based method, e.g., the fixed point method (see [27]), or a mathematical programming method, e.g., MMA [16]. 5.3 Bound formulations for maximization of the nth eigenfrequency or the distance between two consecutive eigenfrequencies In this section we first consider the more general problem of maximizing the nth eigenfrequency of a vibrating structure, i.e., the fundamental eigenfrequency (n = 1) or a higher order eigenfrequency (n > 1). Employing a bound formulation [13-15] involving a scalar variable β which plays both the role of an objective function to be maximized and a variable lower bound for the nth and higher order eigenfrequencies (counted with possible multiplicity), the above problem can be formulated as max {β } β , ρe

(11)

Subject to :

β − ω 2j ≤ 0 ,

( j = n, n + 1, ..., J ),

Constraints: 7(b-e). Here, J is the highest order of an eigenfrequency to be considered a candidate to exchange its order with the nth eigenfrequency or to coalesce with this eigenfrequency during the design process. Note that the bound formulation (11) allows for multiplicity, as none of the conditions impose unimodality on the eigenfrequencies. Also note that introducing the additional parameter β ensures that we have a standard differentiable problem even if there are multiple eigenvalues. For example, the max-min objective function (7a) of the fundamental eigenfrequency optimization problem in Section 5.1 clearly is non-differentiable, while the bound formulation of the same problem obtained by setting n = 1 in (11) is differentiable. The problem of maximizing the distance (gap) between two consecutive eigenfrequencies of orders n and n - 1 may now be written in the following extended bound formulation, where two bound parameters are used (see also [3,10]):

4 max {β 2 − β1 }

β 1, β 2, ρe

(12)

Subject to :

β 2 − ω 2j ≤ 0 ,

( j = n, n + 1, ..., J ),

ω − β1 ≤ 0 ,

( j = 1, ..., n − 1),

2 j

Constraints: 7(b-e). Note that if we remove the bound variable β1 and the corresponding set of constraints from the eigenfrequency gap maximization problem (12), then this problem reduces to the nth eigenfrequency maximization problem (11), and in particular, for n = 1, to the problem of maximizing the fundamental eigenfrequency. 5.4 Sensitivity analysis of multiple eigenfrequencies Multiple eigenfrequencies may manifest themselves in different ways in structural optimization problems. One possibility is that an eigenfrequency subject to optimization is multiple from the beginning of the design process, e.g., due to structural symmetries, but an originally unimodal eigenfrequency may also become multiple during the optimization process due to coalescence with one (or more) of its adjacent eigenfrequencies. In these cases, sensitivities of the multiple eigenfrequency cannot be calculated straightforwardly from (9) due to lack of usual differentiability properties of the eigenspace spanned by the eigenvectors associated with the multiple eigenfrequency. Investigations of sensitivity analysis of multiple eigenvalues (or eigenfrequencies) are available in many papers (see, e.g., [9,12]). Following [12], let us assume that an eigenvalue λn has multiplicity N, i.e., λ n = λ n +1 = L = λ n + N −1 = ω n2 , and that, as is always possible, we have determined N eigenvectors

φ n which correspond to the eigenvalues of magnitude λn and satisfy the

orthonormality conditions in (7), (11) and (12) for k,j = 1,…,n+N-1. Based on perturbation analysis, it was found in [12] that the sensitivities of the N fold eigenvalue λn with respect to the design variable ρe may be obtained by solving the following subeigenvalue problems (in matrix form), (13) f ρsk T ≡ Φ T (K ′ρ − ω n2 M ′ρ )Φ T = TΛ ′ρ , (s, k = n, …, n+N-1),

[ ] e

e

e

e

where Λ ′ρ = diag{(λ n )′ρ , (λn +1 )′ρ , L, (λ n + N −1 )′ρ } is a diagonal matrix consisting of the sensitivities of the members of the N fold e e e e

eigenvalue with respect to the design variable ρe, and where the matrix Φ = [φ n , φ n +1 , L , φ n + N −1 ] consists of the eigenvectors

associated with the members of the N fold eigenvalue. The symbols K ′ρ and M ′ρ denote the partial derivatives of the stiffness and e e

mass matrices with respect to the design variable ρe, and we have defined f ρsk = φ Ts (K ′ρ − ω n2 M ′ρ )φ k (s, k = n, …, n+N-1) as the e e e

generalized gradients. The matrix T consists of the N orthonormalized eigenvectors of the sub-eigenvalue problem (13), i.e. TTT = I, where I is the N×N identity matrix. In order to obtain the increments of the multiple eigenvalue λn with respect to changes of all design variables, we need to consider the directional derivatives of the eigenvalue. For this purpose, the vectors fsk (s, k = n, …, n+N-1) of generalized gradients are defined as follows

{ = {φ

f sk = f ρsk1 , L, T s

f ρskN e

}

T

}

(14)

T

(K ′ρ1 − ω n2 M ′ρ1 )φ k , L, φ Ts (K ′ρ N − ω n2 M ′ρ N )φ k . E

E

For a small arbitrary increment vector ∆ρ = {∆ρ1 , L , ∆ρ N }T of the design variables, the vector of increments ∆(ω n2 ) of the members E of the multiple eigenfrequency ω n2 is determined approximately by the following equation, see [12],

det [f skT ∆ρ − δ sk ∆(ω n2 )] = 0 , (s, k = n, …, n+N-1),

(15)

where δsk is Kronecker’s delta. As is to be expected, for N = 1, i.e. s = k = n, (15) reduces to the case of a simple eigenfrequency NE

f nnT ∆ρ − ∆(ω n2 ) ≡ ∑ f ρnne ∆ρ e − ∆(ω n2 ) = 0 ,

(16)

e =1

where f ρnn = φ Tn (K ′ρ − ω n2 M ′ρ )φ n = (λ n ) ′ρ , cf. (9) and (14). e e e e According to the results of Seyranian et al. [12], the necessary optimality conditions for the multiple eigenfrequency

λn = λ n +1 = L = λn + N −1 = ω n2 (N-fold) to be maximized may be stated in the following form n + N −1

∑ γ~ s =n

ss

~ ss f ρe − γ 0Ve = 0 ,

(e =1,L, N E ) , (γ~ss ≥ 0, γ 0 ≥ 0) ,

(17)

~ where ~ f ρsse (s = n, …, n+N-1) are the generalized gradients associated with the basis matrix ( Φ = ΦT , cf. (13)) of eigenvectors

[~ ]

~T ~ corresponding to the N-fold eigenfrequency, i.e. f ρss = Φ (K ′ρ e − ω n2 M ′ρ e )Φ = Λ ′ρ e (cf. (13)). Similarly, for N = 1, i.e. s = n, (17) e

reduces to the optimality condition (10) for the case of a simple eigenfrequency. 5.5 Optimization algorithm for multiple eigenfrequencies According to the results of the sensitivity analysis in Section 5.4, more efficient formulations may be obtained if we rewrite the bound formulations (11) and (12) as follows in terms of increments, with a view to facilitate solution of topology optimization

5 problems with multiple eigenfrequencies,

max

max {β }

β 1, β 2, ∆ρe

β , ∆ρe

Subject to :

Subject to :

β − [ω 2j + ∆(ω 2j )] ≤ 0 ,

( j = n, ..., J ),

det [f skT ∆ρ − δ sk ∆(ω 2j )]= 0, ( j = n, ..., J ), NE

∑ (ρ e =1

+ ∆ρ e )Ve − V ≤ 0, (V = αV0 ), *

e

{β 2 − β1 }

0 < ρ ≤ ρ e + ∆ρ e ≤ 1,

*

(e = 1, L , N E ),

β 2 − [ω 2j + ∆(ω 2j )] ≤ 0 , ( j = n, ..., J ),

[ω

2 j −1

+ ∆(ω

2 j −1

)]− β1 ≤ 0 , ( j = 2, ..., n),

(18)

det [f ∆ρ − δ sk ∆(ω 2j )]= 0, ( j = 1, ..., J ), T sk

NE

∑ (ρ e =1

e

+ ∆ρ e )Ve − V * ≤ 0, (V * = αV0 ),

0 < ρ ≤ ρ e + ∆ρ e ≤ 1,

(e = 1, L , N E ).

For brevity, we have omitted in (18) to write the structural eigenvalue problem (7b) and the orthonormality conditions (7c) among the constraints in (18). The unknowns (design variables) of both problems in (18) are the increments ∆ρ e of the material volume densities ρe, e = 1,…,NE, which are determined by a loop of inner iterations subject to fixed values of the eigenfrequencies ω j and material volume densities ρe. The new values of ω j and ρe, e = 1,…,NE, for the next global iteration are obtained from the last iterative step of the inner loop. Specifically, if we introduce the additional constraints f skT ∆ρ = 0 (for s ≠ k ), (15) will be linearized, and as a result, the optimization problems (18) reduce to linear programming problems, see [9]. Problems (18) or their linearized formulations can be solved using the MMA method [16] or a linear programming algorithm. 6. Numerical examples 6.1 Single material topology design of beam-like 2D structure 6.1.1 Maximization of the fundamental eigenfrequency

Admissible design domain

Admissible design domain

Admissible design domain

(a) (b) (c) Figure 1(a-c) – Admissible design domains of beam-like 2D structures with three different sets of boundary conditions. (a) Simply supported ends. (b) One end clamped, the other simply supported. (c) Clamped ends. The fundamental eigenfrequencies of the 3 initial designs (uniform distribution of material with density ρ = 0.5) are all unimodal with values ω10a = 68.7 , ω10b = 104.1 and

ω10c = 146.1 . As a first example, we consider the topology optimization of a single-material beam-like structure modeled by 2D plane stress elements. The admissible design domain is specified, and three different cases (a), (b) and (c) of boundary conditions as shown in Fig. 1 and defined in the caption, are considered. The design objective is to maximize the fundamental eigenfrequency for a prescribed material volume fraction α = 50%, and in the initial design the available material is uniformly distributed over the admissible design domain. The material is isotropic with Young’s modulus E = 107, Poisson’s ratio υ = 0.3 and mass density ρm = 1 (SI units are used throughout). The fundamental eigenfrequencies of the initial designs with the three cases (a), (b) and (c) of boundary conditions are given in the caption of Fig. 1. The optimized topologies are shown in Figs. 2(a-c), and the corresponding optimum fundamental eigenfrequencies are all bimodal with values given in the caption of Fig. 2. Iteration histories of the first 3 eigenfrequencies for each of the three cases of boundary conditions are given in Fig. 3.

(a) (b) (c) Figure 2(a-c) – Optimized single-material topologies (50% volume fraction) for the three different sets of boundary conditions , (b) defined in Figs.1(a-c). The optimum fundamental eigenfrequencies are found to be all bimodal with values (a) ω1opt a = 174.7 , and (c) ω1opt , implying that they are increased by (a) 154%, (b) 177% and (c) 212% relative to the initial ω1opt b = 288.7 c = 456.4 designs.

6

Eigenfrequencies

Eigenfrequencies

500 400

ω3

300

ω2

200

(Maximized) ω1

20

40

Iteration number

60

ω3

500

400

ω2

300

ω1 (Maximized)

200

100 0 0

700

600

80

Eigenfrequencies

600

100 0

600

ω3

500

ω2 ω1 (Maximized)

400 300 200

20

40

60

80

100

Iteration number

100 0

20

40

60

80

100

Iteration number

(a) (b) (c) Figure 3(a-c) – Iteration histories of the first 3 eigenfrequencies associated with the design process leading to the results in Figs.2(ac). It is seen that for each of the three cases, the fundamental eigenfrequency is simple for the initial design, but soon coalesces with the second eigenfrequency. Thus, the maximized fundamental eigenfrequencies for all three cases are bimodal. 6.1.2 Maximization of the second eigenfrequency We now present an example of topology optimization of single material beam-like structures for maximum value of the second eigenfrequency. The initial data and the three sets of boundary conditions in this example are the same as for the first example in Section 6.1.1. The resulting optimum topologies are shown in Figs. 4(a-c).

(a) (b) (c) Figure 4(a-c) – Optimized single-material topologies (50% volume fraction) for the three different sets of boundary conditions in Figs.1(a-c). The optimum second eigenfrequencies are found to be (a) ω 2opt , (b) ω 2opt , and (c) ω 2opt . a = 598.3 b = 732.8 c = 849.0 6.2 Topology design of plate-like 3D structures 6.2.1 Maximization of the fundamental eigenfrequency of single-material structures

(a) (b) (c) Figure 5 – Plate-like 3D structure (a=20, b= 20 and t=1) with three different cases of boundary conditions and attachment of a concentrated nonstructural mass. (a) Simple supports at four corners and concentrated mass mc at the center of the structure ( mc = m0 / 3 , m0 the total structural mass of the plate). (b) Four edges clamped and concentrated mass mc at the center ( mc = m0 / 10 ). (c) One edge clamped, other edges free, and concentrated mass mc attached at the mid-point of the edge opposite to the clamped one.( mc = m0 / 10 ). The fundamental eigenfrequencies for the 3 initial designs (uniform distribution of material with density ρ = 0.5) are ω10a = 8.13 , ω10b = 31.1 and ω10c = 3.46 . In this section, we consider the topology optimization of a single-material plate-like structure modeled by 8-node 3D brick elements with Wilson incompatible displacement models to improve precision. The admissible design domain is specified, and three different cases (a), (b) and (c) of boundary conditions and attached concentrated, nonstructural masses as shown in Fig. 5 and defined in the caption, are considered. The design objective is to maximize the fundamental eigenfrequency for a prescribed material volume fraction α = 50%, and in the initial design the available material is uniformly distributed over the admissible design domain. The material is isotropic with Young’s modulus E = 1011, Poisson’s ratio υ = 0.3 and mass density ρm = 7800 (SI units are used throughout). The fundamental eigenfrequencies of the initial designs with the three cases (a), (b) and (c) of boundary/mass conditions are given in the caption of Fig. 5. The optimized plate topologies are shown in Figs. 6(a-c), and the corresponding optimum fundamental eigenfrequencies are all unimodal with values given in the caption of Fig. 6.

7

(a) (b) (c) Figure 6(a-c) – Optimized single-material topologies (50% volume fraction) for the three different cases of boundary conditions and mass attachment in Figs.5(a-c). The optimum fundamental eigenfrequencies are found to be (a) ω1opt , (b) ω1opt , and (c) a = 16.4 b = 65.4 , implying that they are increased by (a) 102%, (b) 110% and (c) 180% relative to the initial designs. ω1opt c = 9.7 6.2.2 Maximization of higher order eigenfrequencies of bi-material structures We now consider topology optimization of bi-material plate-like structures with respect to higher order eigenfrequencies. Both of the two materials are isotropic. The stiffer material *1 with elasticity and mass matrices E *e1 , M *e1 , see Section 4.3, is indicated by black in Fig. 7, and is the same as that used for optimization with a single-material in the preceding examples. The weaker material *2 is indicated by grey in Fig. 7, and has the properties E*e2 = 0.1E*e1 and M *e2 = 0.1M *e1 . We take the volume fraction of material *1 to be 50%, and present results of optimizing the topologies of a bi-material quadratic plate with the same boundary conditions and attachment of a concentrated mass as shown in Fig. 5(b). Figs. 7(a-c) show the optimized plate topologies associated with maximum values of the 4th, 5th and 6th eigenfrequencies.

(a) (b) (c) Figure 7 – Optimized topologies of bi-material plate with all edges clamped and a concentrated mass attached to the center, cf. Fig. 5(b). The topologies correspond to maximum values of the (a) 4th, (b) 5th and (c) 6th eigenfrequency. The stiffer and the weaker material are indicated by black and grey, respectively, and the volume fraction of the stiffer material *1 is 50%. 6.2.3 Maximization of the distance (gap) between two consecutive eigenfrequencies of bi-material structures This example also concerns topology optimization of bi-material plate-like structures, and we use the same materials and volume fractions as in the previous example. The design objective considered is to maximize the distance (gap) between the 2nd and 3rd eigenfrequencies of the structure. We use the same admissible design domain as in Fig. 5 for the plate structure, and choose the cases (a) and (c) of boundary conditions and concentrated mass attachment as shown in Fig. 5. The results are given in Fig. 8 and 9. 80

ω5 ω4 ω3

Eigenfrequencies

70 60 50

(Maximized) Gap: ω3 − ω2

40

ω2

30 20 10 0

ω1 10

20

30

Iteration number

40

50

(a) (b) Figure 8 – (a) Optimized topology of the plate-like structure with simple supports at four corners and a concentrated mass at the center, cf. Fig. 5(a). The gap between the 2nd and the 3rd eigenfrequencies is maximized. (b) Iteration histories for the first five eigenfrequencies associated with the process leading to the design (a). It shows that the 2nd and the 3rd eigenfrequencies form a double eigenfrequency for the initial design, but that they split as the design process proceeds, and the 3rd and the 4th eigenfrequencies end up being a double eigenfrequency of the final design.

8

(c) ω3 = 50.1 (b) ω 2 = 18.4 Figure 9 – (a) Optimized topology of the plate-like structure with the upper horizontal edge clamped, other edges free, and a concentrated mass attached at the mid-point of the lower horizontal edge, cf. Fig. 5(c). (b), (c) The 2nd and 3rd eigenmodes of the optimized bi-material plate topology. The gap between the 2nd and the 3rd eigenfrequencies is maximized, and ∆ωopt = ω3-ω2 = 31.7. (a)

7. Conclusions Problems of topology optimization with respect to eigenfrequencies of free vibrations of structures are presented in this paper. The design objectives are maximization of specific eigenfrequencies and distances between two consecutive eigenfrequencies of the structures. Different approaches have been developed and applied to handle topology optimization problems associated with multiple eigenfrequencies. Several numerical examples of topology optimization of simple single- and bi-material 2D and plate-like 3D structures are carried out with these design objectives and presented in the paper. The results witness that the techniques enable us, in a cost-efficient manner, to move structural resonance frequencies far away from external excitation frequencies and thereby avoid high vibration levels. 8. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

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