Structural Optimization Strategy Using Reduced Equivalent

results in the fast computation of proper orthogonal modeΦ.

3.3 Structural Optimization Strategy Using Reduced

For example, we require the maximum displacement be less than a allowed limit such that

c₁= max(|u(t;µ)|)−u≤0. (3.21) Also, the maximum stress constraint is expressed as

c₂ = max(|σ(t;µ)|)−σ≤0. (3.22) For the characteristic of dynamics, the first natural frequency condition is assigned:

c₃ =ω₁−ω₁(µ)≤0. (3.23) Also, other natural frequencies and buckling constraints could be imposed.

The dynamic equation is solved by the time-discretized methods, explic- itly or implicitly. In this study, Newmark-Beta scheme was chosen to solve the equation under certain initial conditions. The initial condition and the ex- ternal dynamic force are assumed to be independent to the design variables.

The external force is discretized as follows:

fi =f(ti), i= 1,2,· · · , NT. (3.24) Thus, the displacement field at every time steps can be obtained by solving Eq. (3.16) and (3.17). After the displacement solution is obtained, and the column matrices of the external force and the displacement field are written as

F= [ f₁ f₂ · · · f_NT ] (3.25)

U= [ u1 u2 · · · u_N ], (3.26)

Finally, the displacement solution obtained and the design variable are substituted to the constraints shown in Eq. (3.20).

3.3.2 Optimization Strategy Using Equivalent Static Load

For the structural optimization under dynamic loadings, the FOM presented in Eq. (3.16) have to be solved at every iterations and sensitivity compu- tations. The problem is solving Eq. (3.16) requires a lot of computational resources. To reduce the computational burden, the Equivalent Static Loads (ESL) algorithm was developed by Choi and his co-workers (Ref. [53, 54, 55]).

The ESL is a static load set that makes the same displacement field as that under a dynamic load. Thus, the optimization algorithm using ESL executes multiple static optimizations instead of solving dynamic equation at every function evaluations.

From the Eq. (3.16), the inertia and damping parts are moved to the RHS. Then the sum of RHS is a equivalent load set such that

K(µ)u(t;µ) =f(t)−M(µ)¨u(t;µ)−C(µ) ˙u(t;µ)

=f_eq(t;µ). (3.27)

Once the displacement field U is computed from the time integration, the ESL is calculated by multiplying the stiffness matrix to the displacement field obtained. Thus, Eq. (3.27) is changed as follows:

K(µ)U(µ) =Feq(µ). (3.28) Note that the ESL can be calculated after obtaining the displacement field from the time integration.

The main characteristic of the optimization using ESL algorithm is that the update of the design variable of stiffness matrix is discriminated to that of the design variable of ESL. Although this process can be questionable, the discrimination of the design variable is nothing but an engineering as- sumption. Actually, Stolpe [56] indicated that the optima of the ESL-based method and that of full transient analysis could not be the same in general.

However, the optimal solution obtained by the ESL-based method satisfies all the constraints under a dynamic loading condition. Therefore, eventhough the procedure of optimization using ESL was not fully proved by the mathemat- ical tool, we can use the ESL-based optimization considering the efficiency of the algorithm.

The design variables in Eq. (3.28) are divided to µ and µ. During theˆ static optimization,µˆis fixed while onlyµis varied. Thus, the static solution using each indices depend on the design variables can be represented as

U(µ^m,k) =K⁻¹(µ^m,k)Feq(µˆ^m), (3.29) where the superscriptkdenotes the iteration of design variable for the static optimization. The superscript m represents the update of ESL. So, during the static optimization process,mis fixed. Initially, all variables are the same as follows:

µ^1,1 =µˆ¹ =µ_ini. (3.30) After the first static optimization process, theµ^1,k is converged to the optima µ^1,∗ which is the initial design variable of the second static optimization process such that

In Ref. [53, 54, 55], the convergence criteria was set as the convergence of ESL, which is the physically same criteria of the convergence of the design variables under the assumption of a linear elastic material.

3.3.3 Mode of Equivalent Static Load

The inner loop of the ESL optimization algorithm is exactly same to the static optimization under a multiple loading condition. At the front of this chapter, we proposed to calculate the mode of external loads to employ the reduced basis method under the multiple loading condition. Thus, the global-POD method combined with the reduced basis method can be employed to the static optimization of the ESL algorithm. This combination of two methods can reduce computational costs of the ESL optimization algorithm. Before that, we need to clarify the meaning of the mode of external loads.

The meaning of the mode of ESL is clear. As shown in Eq. (3.27), the ESL is composed of the external load, inertia and damping parts. The mode of external load is straightforward. Since the magnitude does not affect to the mode of external load, the direction vector of external load is the mode of the external load. The inertia and damping parts are also clear. The inertia is a multiplication of the mass matrix and the acceleration. The mass matrix is not a function of time, so the mode of inertia is the same as that of acceleration. Of course the mode of damping part is the same as that of velocity. Therefore, by superposing the three modes, the mode of ESL is easily derived. The good point of this method is that the acceleration and velocity are necessarily computed during the time integration for updating ESL. So the mode of ESL is easily obtained without additional, heavy computations.

Another way to obtain the mode of ESL is applying POD to the ESL directly. In other words, each column of the ESL is regarded as a snapshot. So, from the Eq. (3.11), the POM of the load is calculated. During the transient analysis, the ∆tis constant, which results in all the same weighting factors.

δ1=δ2=· · ·= ∆t. (3.32)

Thus, the snapshots matrix is expressed as follows:

Feq= [ feq,1 feq,2 · · · feq,N_T ] (3.33)

=Γ_eqΛ⁻

1

eq2A^T_eq, (3.34)

where

Γeq=

γ₁ γ₂ · · · γ_Nγ

, Nγ≪NT. (3.35) The rest of the procedure is exactly same to Eqs. (3.13)∼(3.15) and Eqs.

(3.5)∼(3.10). To sum up, the static optimization part of the ESL algorithm is reduced by applying the POD to the ESL and the reduced basis method.