International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Topology optimization of linear structural system under stationary stochastic excitation Zhu, Mu; Yang, Yang; Shields, Michael D.; Guest, James K. Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  1 Topology Optimization of Linear Structural System Under Stationary Stochastic Excitation Mu Zhu Graduate Student, Dept. of Civil Engineering, Johns Hopkins University, Baltimore, USA Yang Yang Graduate Student, Dept. of Civil Engineering, Johns Hopkins University, Baltimore, USA Michael D. Shields Assistant Professor, Dept. of Civil Engineering, Johns Hopkins University, Baltimore, USA James K. Guest Associate Professor, Dept. of Civil Engineering, Johns Hopkins University, Baltimore, USA  ABSTRACT: Topology optimization as a form-finding methodology for design has developed rapidly in recent years. Its applications can vary from small-scale material microstructures to large-scale building systems. While deterministic optimization of structural systems under static loading is well understood, optimization under dynamic loading is less studied, and the more challenging problem of finding optimal structural systems under stochastic dynamic loading has not yet been addressed. In this work, topology optimization is used to design the lateral load system of structures such that the expected system responses to stationary stochastic excitations are optimized. More specifically, we design the size and location of bracings for multi-story building to minimize the variance of relative deformations under stationary base excitations. To evaluate the variance, we first solve this problem in frequency domain by using the autospectral density function of the relative displacement for the covariant stationary situation. This is compared to analysis in time domain through use of the impulse response function. The resulting stochastic dynamic topology optimization problem is solved using the gradient-based optimizer Method of Moving Asymptotes (MMA), with sensitivities provided via the adjoint method. The popular Solid Isotropic Material with Penalization (SIMP) is used to prevent existence of low-area bracing members and provide clear indications of bracing patterns. Numerical results are presented and comparisons of time and frequency domain methods are given.  Topology optimization as a form-finding methodology for structural design has received rapidly growing interest in recent years. The goal is to achieve specific structural performance objectives by optimally distributing structural members through the building system. While problems of systems under static loading are generally well understood, the more challenging problems of dynamic response under stochastic excitation is far less investigated. Some progress has been made in optimization for free vibration problems (Du & Olhoff, 2007) and forced vibrations by deterministic dynamic transient loading through time history analysis (Kang et al, 2006). In this paper, topology optimization is used to design structures such that the expected system responses to stationary stochastic excitations are optimized. For example, we seek to find the optimal lateral load resisting system by means of optimizing location of bracings for a multi-story building that minimizes the variance of its deformation under stationary base excitation described by the well-known Kanai–Tajimi model. Response variance is first computed by frequency domain method, which requires 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  2 integration of the autospectral density function of the relative displacement. The time domain method is also discussed through use of the impulse response function since it may be generalized, thus offering an avenue towards optimizing structures for non-stationary excitations. However, it will show double integrations have to be solved for this method. The design goal of the topology optimization is to determine where bracing members are to be located.  Although such existence problems are discrete, we use continuous values for design variables for the optimization procedure and drive them to binary solutions using the popular Solid Isotropic Material with Penalization (SIMP) by Bendsøe (1989).  Although typically used in continuum domains, including to design conceptual lateral bracing in buildings (e.g., Mijar and Swan 1998), the approach is applied herein to frame members. Sensitivities are computed using the adjoint method, and the optimization problem is solved using the gradient-based algorithm Method of Moving Asymptotes (MMA) developed by Svanberg (1987).  1. STOCHASTIC EXCITATION 1.1. Modeling of an N-DOF system An n-story frame building subject to earthquake is considered and modeled as a linear n-degree of freedom (n-DOF) structural system under stationary base ground motion. The mass distributed throughout the building is idealized as concentrated at the floor levels, forming the diagonal mass matrix M. The DOF’s of the structure to which zero mass is assigned is eliminated from dynamic analysis using static condensation method, leaving only lateral DOF’s associated with each floor level and form the dynamic stiffness matrix K.  Assuming modal damping ratios ξ? are given, we first identify the natural frequencies and mode shapes of the structure by solving:                 KΦ = Ω2MΦ      (1) where Ω is the natural frequencies and Φ is natural modes. When subject to stationary stochastic excitation P(t), the modal equations of motion can be written as:         !!qn + 2ω nξn !qn +ω n2qn = Pn (t)     (2)  1.2. Generating stochastic excitation Stochastic excitation simulating general earthquake ground motion is assumed to follow the stationary Kanai–Tajimi model with Power Spectral Density (PSD) given by:     (3) where       (4)  and  and  are the characteristic frequency and damping of the ground and is the variance of the excitation. With values of and  selected as 8 and 0.2 respectively, the PSD of the ground motion is plotted as follows:  Figure 1: Kanai-Tajimi power spectral density function for ground motion.  SP(ω ) = S01+ 4ξg2ωωg⎛⎝⎜⎞⎠⎟21−ωωg⎛⎝⎜⎞⎠⎟2⎡⎣⎢⎢⎤⎦⎥⎥2+ 4ξg2ωωg⎛⎝⎜⎞⎠⎟2S0=σ 2πωg2ξg+12ξg⎛⎝⎜⎞⎠⎟2ωgξg σ 2 ωgξg12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  3 1.3. Dynamic and sensitivity analysis 1.3.1. Frequency domain method The system response quantity z(t) of interest is the x-direction displacement at each floor level, which can be written as follows in modal coordinates:  Z(t) = Bnqn(t)n∑                       (5) where the coefficients  are determined as the product of the location indication vector 𝐿 and the modal shape vector nφ . For example, the coefficient at the roof for a 4-story building is expressed by:               = =<1,0,0,0>n n nB Lφ φ                    (6)  The performance of the structures is evaluated by the variance of response quantity z t , which is equal to the integral of the response PSD Sz(w):                      σ Z 2 = SZ (ω )−∞∞∫ dω                       (7)  The response PSD is computed from the modal response cross-spectral density terms as:                    SZ (ω ) = SZmZnn∑ (m∑ ω )      (8) where    SZmZn (ω ) = BmBnHm (−iω )Hn (iω )SPmPn (ω )       (9)  with modal frequency response functions:  Hm(−iω ) = 1Km1− 2iξm(ωωm)− (ωωm)2⎡⎣⎢⎤⎦⎥   (10)  Hn(iω ) = 1Kn1+ 2iξn(ωωn)− (ωωn)2⎡⎣⎢⎤⎦⎥       (11) generalized modal stiffness terms:                             Kn = φnTKφn                  (12) and cross spectral density matrix according to the modal (and cross-modal) contributions:                      SPmPn = φmTSp (ω )φn                     (13)  Minimizing the objective function with gradient-based optimizers requires the sensitivity (derivative) of the response quantity (in this case variance 𝜎??) to the design variables (𝜌?) must be computed. Using the frequency domain method, the sensitivity is computed as follows  21 1( ) ( ) ( )m nnddf nddfm n m n P Pm nie eB B H i H i Sω ω ωσρ ρ= =⎛ ⎞⎡ ⎤∂ −⎜ ⎟⎣ ⎦∂ ⎝ ⎠=∂ ∂∑∑       (14) where                              m me eBLφρ ρ∂ ∂=∂ ∂                       (15)           ( )22 22 2( )2m mm mm e ee m m miH iiω ωω ξ ωω ρ ρρ ω ξ ωω ω∂ ∂− +∂ − ∂ ∂=∂ − −    (16)          ( )22 22 2( )2n nn nn e ee n n niH iiω ωω ξ ωω ρ ρρ ω ξ ωω ω∂ ∂− +∂ − ∂ ∂=∂ − −      (17)  The sensitivity of the eigenvalues and eigenvectors can be computed as follows: { } { }22KTmm m me e eMωφ ω φρ ρ ρ⎛ ⎞⎡ ⎤ ⎡ ⎤∂ ∂ ∂= −⎜ ⎟⎢ ⎥ ⎢ ⎥⎜ ⎟∂ ∂ ∂⎣ ⎦ ⎣ ⎦⎝ ⎠          (18) Bn12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  4  ∂ φm{ }∂ρe= −12φm{ }T∂M∂ρeφm{ }⎛⎝⎜⎞⎠⎟φm{ }+φs{ }T∂K∂ρe⎡⎣⎢⎤⎦⎥−ωm2∂M∂ρe⎡⎣⎢⎤⎦⎥⎛⎝⎜⎞⎠⎟φm{ }ωm2−ωs2( )⎛⎝⎜⎜⎜⎜⎜⎞⎠⎟⎟⎟⎟⎟φs{ }s=1s≠mndf∑    (19) where subscripts m and s represent the m-th and s-th mode respectively. The sensitivity of the stiffness matrix and mass matrix are denoted by eKρ∂∂and eMρ∂∂respectively. 1.3.2. Time domain method An alternative method to calculate the variance of response is the time domain method. For stationary stochastic excitation, the response is also stationary. In that case, the variance of response is not a function of time and calculation of variance begins with computation of the autocorrelation function of the excitation from its power spectral density using the inverse Fourier transform.                Rp(t) = Sp(ω )exp−∞∞∫(iωτ )dω            (20) For the Kanai-Tajimi excitation described in the previous section, the autocorrelation plot of the excitation is given in Figure 2.   Figure 2: Autocorrelation of the Kanai-Tajimi ground motion excitation.   The autocorrelation function of the structural response 𝑧(𝑡) is defined by:                        Rz(τ ) = E[z(t)z(t +τ )]               (21)  For a linear system, using modal expansion representation, the autocorrelation can be expanded as:   Rz (τ ) = E[ BmBnqm (t)qn (t +τ )n∑m∑ ]  (22)  which is equal to the following:  Rz(τ ) = E[BmBnqm(t)qn(t +τ )n∑m∑]= Rzmzn(τ )n∑m∑       (23) where Rzmzn(τ )  is the autocorrelation from the modal response.        For an arbitrary excitation, Duhamel’s integral is implemented to find the response. One can take advantage of the linear system so that a modal response associated with mode n can be calculated as               qn (t) = Pn (τ )hn−∞t∫ (t −τ )dτ                  (24)  where the impulse response function          hn(t) =1ωDnMnexp(−ςnωnt)sin(ωDnt)       (25)  and the damped natural frequency for under-damped system is given by:                      ωDn =ω n 1−ς n2                       (26)  One can thus obtain the closed form of the autocorrelation by substituting Eq. (24) into Eq. (23) as:  12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  5 Rzmzn(τ ) = BmBn−∞t+τ∫−∞t∫Pm(θ1)Pn(θ2)hm(t −θ1)hn(t +τ −θ2)dθ1dθ2      (27) Through change of variables, the above equation becomes  Rzmzn(τ ) = BmBn0∞∫0∞∫RPmPn(τ − t2+ t1)hm(t1)hn(t2)dt1dt2      (28) Note that, although expressed for stationary excitation in its current form, Eq. 28 may be generalized to compute the autocorrelation function for non-stationary response.        For stationary response discussed in this paper, the variance is not a function of 𝜏. In another words, when 𝜏 = 0, the autocorrelation of the response is equal to the variance used in the objective function, which can be calculated as:   σ 2z= BmBn0∞∫0∞∫RPmPn(τ − t2+ t1)hm(t1)hn(t2)dt1dt2n∑m∑       (29)        In time domain analysis, the sensitivity of this variance to the design variables necessary for gradient-based optimization algorithms is given by:   ∂σi2∂ρe=∂ BmBn0∞∫0∞∫RPmPn(τ − t2+ t1)hm(t1)hn(t2)dt1dt2n∑m∑⎛⎝⎜⎞⎠⎟∂ρe (30)       Using the chain rule, the derivative of the coefficients Bm and Bn can be calculated in the same way as in Equation 15 by evaluating the sensitivity of the eigenvectors.        The derivative of the remainder of the integrand involves computing the sensitivity of the impulse response function as follows:     ∂0∞∫0∞∫Raa(τ − t2+ t1)hm(t1)hn(t2)dt1dt2⎛⎝⎜⎞⎠⎟∂ρe=0∞∫0∞∫Raa(τ − t2+ t1)(∂hm(t1)∂ρ hn(t2 )+ hm(t1) ∂hn(t1)∂ρ )dt1dt2  (31) where the sensitivity   ∂hn(t1)∂ρe=∂∂ρe1ωDnMnexp(−ςnωnt)sin(ωDnt)⎡⎣⎢⎢⎤⎦⎥⎥          (32)  which simplifies to evaluating the sensitivity of eigenvalues calculated previously.  2. TOPOLOGY OPTIMIZATION The design problem is formulated to minimize the variance of the structural response under stochastic excitations subject to a constraint on the structural mass (volume) of the lateral bracing system. As is typical in topology optimization, the design variables are defined to indicate the existence of each bracing member, with ρe=1 indicating the member e should exist in the final topology and ρe=0 indicating it should not.  The design optimization problem is stated formally as:       minρ    σ Z 2s.t.     (K −ωi2M )φi= 0,  for i = 1,...,n         ρ e ⋅ve ≤Vallowede∑          0 ≤ ρ e ≤1   (30) where K is the global stiffness matrix, and M is the global mass matrix, ve is the element length, and Vallowed is the total allowable volume of bracing material. Here we consider the design objective of minimizing the variance of the 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  6 displacement σZ2  at roof level because lateral motion at roof is often governing. Although topology optimization seeks to determine the existence of each design variable, problems are typically large scale making it computationally prohibitive to use binary programming.  We instead use continuous values between 0 and 1 to enable calculation of sensitivities of objective function and constraints, thereby allowing use of the effective gradient-based optimization algorithm Method of Moving Asymptotes (MMA) (Svanberg 1987). The Solid Isotropic Material with Penalization (SIMP), which is typically used for continuum domains (Bendsøe 1989), is implemented to penalize the intermediate values of design variables to push the final solution to be a 0-1 representation.   3. NUMERICAL EXAMPLE Consider a four story braced frame structure as depicted in Figure 3. The frame is made up of steel beam and column structural members with moderate cross sectional properties (A=1000, I=100000), while the bracing elements are the design variables with maximum cross-sectional area A=1000. Mass is concentrated at each floor, assuming concrete slab with densitiy of 150 lb/ft3 is used. The optimization takes a ground structure approach, where bracing members are distributed throughout the design domain in a uniform, low magnitude manner.   Figure 3: Initial ground structure with low volume density.  The ground structure is shown in Figure 3, with uniformly distributed low density volume bracing members connecting every node pair. The ground structures serves as the initial design for the optimization procedure. The material usage constraint is chosen as 10% of the total amount of material of all potential bracings. A uniform initial design is used where every member has a volume fraction ρe of 0.1. This topology is then optimized using frequency domain analysis and time domain analysis, and the resulting bracing topologies are shown in Figure 4a and Figure 4b, respectively. The optimized topologies redistribute the structural material in a way of forming chevron shaped pattern bracing at each floor as well as crossing story bracings connecting the from the mid height to the foot of the frame to provide maximum lateral stiffness. Bracings of red color indicate the cross-sectional areas reach the maximum value, while the lighter color indicate slender member size. In addition, the evolutions of objective functions of using two different methods are displayed in Figure 5a and Figure 5b, respectively. These plots proved the optimality of the final results found iteratively from the initial ground structure.       These results show strong agreement between the two approaches as expected, but note that the computational cost of time domain method is larger than frequency domain method due to calculation of double integration.                           (a)                   (b) Figure 4. Optimized results using frequency domain and time domain method, respectively. 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  7                                 (a)                               (b) Figure 5: Evolution of objective function using frequency domain method (a) and time domain method (b), respectively.  4. CONCLUSIONS Topology optimization is used to design the lateral bracing systems of linear structural systems under stochastic excitation. Specifically, the excitation simulates earthquake ground motion generated by Kanai-Tajimi model and the design objective is to minimize the variance of building roof displacement. Both time and frequency domain methods give nearly identical results.  Although the time domain method requires larger computational effort compared to the frequency domain approach due to the double integration, it potentially offers an avenue to design problems governed by non-stationary processes. The proposed approach shows great promise as a design tool to guide the optimal design of bracing systems, and is straightforward to combine with recent advances in improving constructability (Zhu et al. 2014, Asadpoure et al. 2015) of building frames to create more useful, cost effective design solutions.  5. REFERENCES Asadpoure, A. et al. 2014 Incorporating fabrication cost into topology optimization of discrete structures and lattices. Struct. Multidisc Optim. 1-12 Bendsøe M.P. 1989. Optimal shape design as a material distribution problem, Structural Optimization 1: 193–202. Du J.B. & Olhoff N. 2007. Topological design of freely vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps. Structural and Multidisciplinary Optimization 34(2): 91-110. Kang B.S. et al. 2006. A review of  optimization of structures subjected to transient loads. Struct. Multidisc Optim 31:81-95 Mijar, A. R., and Swan, C. C. (1998). “Continuum topology optimization for concept design of frame bracing system.” J. Struct. Eng., 124:541-550 Svanberg K. 1987. The method of moving asymptotes—a new method for structural optimization. International Journal for Numerical Methods in Engineering 24(2): 359–373. Zhu M. et al. 2014. Considering constructability in structural topology optimization. Proc. Structures Congress, ASCE, Reston, VA:2754–2764.  

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