An effective algorithm for reliability-Based optimization of stiffened mindlin plate

Volume 35 Number 4 4 Vietnam Journal of Mechanics, VAST, Vol. 35, No. 4 (2013), pp. 335 – 346 AN EFFECTIVE ALGORITHM FOR RELIABILITY-BASED OPTIMIZATION OF STIFFENED MINDLIN PLATE Nguyen Thoi Trung1,2,∗, Bui Xuan Thang1, Ho Huu Vinh2, Lam Phat Thuan3, Ngo Thanh Phong1 1University of Science, VNU-HCM, Ho Chi Minh City, Vietnam 2Ton Duc Thang University, Ho Chi Minh City, Vietnam 3University of Technical Education, Ho Chi Minh City, Vietnam ∗E-mail: ngttrung@hcmus.edu.vn Abstract. Nowadays, stiffened plates have been widely used in many branches of struc- tural engineering such as aircraft, ships, bridges, buildings etc... In comparision with common bending plate structures, stiffened plates not only have larger bending stiffness but also use less amount of material. Hence, it usually has higher economic efficiency. However, to obtain high effectiveness in solving the design problems of the stiffened plate, the reliability-based optimization problems need to be established together with the ordinary numerical computing methods. Therefore, the paper presents an approach to establish and solve the reliability-based optimization problem for the stiffened Mindlin plate. To analyze the behavior of Mindlin plate, we use the recently proposed CS-DSG3 element. The random variables are chosen to be elastic modulus, density of mass and external force. The design variables are the thickness, the width and the height of the stiff- ened plate. The objective function can be the strain energy or the mass of the structure and subjected to the constraints of displacement or vibration frequency. The reliability- based optimization algorithm used in this paper is a three-step closed loop: 1) Estimating the random variables by the Reliability Index (RI) method; 2) Solving the optimization problem using Sequential Quadratic Programming (SQP) method; 3) Checking and es- timating the reliability by the first-order reliability method (FORM) in which the limit state function is the limit of displacement or vibration frequency of the structure. Keywords: Stiffened plate, reliability-based optimization, sequential quadratic program- ming - SQP, cell-based smoothed discrete shear gap method (CS-DSG3), reliability index method, first order reliability method. 1. INTRODUCTION Together with many effective applications in many branches of structural engineer- ing such as aircraft, ships, bridges, buildings etc..., the researches on numerical simulation of stiffened plate has been developed quickly, in which the analyses of static respond, free vibration and compressive stabilization play a major role. Some researches can be mentioned as: Bhimaraddi et al. [1] presented a method using FEM to analyze the static 336 Nguyen Thoi Trung, Bui Xuan Thang, Ho Huu Vinh, Lam Phat Thuan, Ngo Thanh Phong respond and free vibration of the ring-sectional plate stiffened with orthogonal beams; Mukhopadhyay [2] applied the Finite Difference Method to analyze the vibration and the stabilization of stiffened plate; Manoranjan Barik [3] used FEM for the static, dynamic and stable analysis of randomly stiffened plate; Bui Xuan Thang et al. [4] proposed a method using CS-DSG3 elements to solve the problem of static, dynamic and stable analysis of stiffened plate, etc... Recently, together with the successive development of computer science and many effective optimization methods, the optimization problems of stiffened plates has been carried out and become one of the interesting research tendencies. Some noticeable works can be listed as: Ravi Bellur Ramaswamy [5] solved the design optimization problem of stiffened plate in which the objective function is the minimum mass of the structure under the respond constraints of frequency and ultimate load. Karoly Jarmai [6] presented a method for optimizing stiffened plates by Massonnet and Gience technique. In this work, the objective function is the minimum cost of the construction work, etc... However so far, the reliability has not been considered in optimization problems of stiffened plate. Therefore, in this paper, we attempt to contribute an approach to establish and solve the reliability-based optimization problem for stiffened Mindlin plate. To analyze the respond of Mindlin plate, we used the CS-DSG3 element recently proposed by Nguyen Thoi Trung et al. [7]. The reliability-based optimization algorithm used in the paper is a three-step closed loop, in which the random variables chosen are elastic modulus, density of mass and external force. The design variables are the thickness, the width and the height of the stiffened plate. The objective function can be the strain energy or the mass of the structure and subjected to the constraints of displacement or vibration frequency. 2. THEORETICAL FUNDAMENTAL The reliability-based optimization problem can be defined by the following mathe- matical model: min x f(x) s.t.   hi(x) = 0 i = 1, ..., l. gj(x) ≤ 0 j = 1, ..., m. βt ≥ β (1) where x is the vector of design variables such as material type, geometrical form, or size of the section, etc... ; hi(x) = 0 and gj(x) ≤ 0 are inequality and equality constraints; l, m are the number of inequality and equality constraints, respectively; β ≥ βt is the constraint on reliability; f(x) is the objective function which can be the function of mass, cost or other characteristics of the structure. The aim of this problem is to find the values of design variables in design space such that the objective function is minimum. In this paper, the Sequential Quadratic Programming method will be applied to solve the optimization problem to find out the suitable value of design variables. 2.1. Brief on the sequential quadratic programming method Sequential Quadratic Programming Method proposed by Wilson [8] is one of many effective algorithms applied for solving nonlinear optimization problems. This algorithm gives out the solution with high convergence, reliable results and has been built in many An effective algorithm for reliability-based optimization of stiffened Mindlin plate 337 optimization software such as: NPSOL, NLPQL, OPSYC, OPTIMA, MATLAB, etc... The main idea of SQP is presented as follows: Consider the structural optimization problem with the objective function and con- straints as follows min x f(x) s.t. { hi(x) = 0 i = 1, ..., l. gj(x) ≤ 0 j = 1, ..., m. (2) where x ∈ Rn, f : Rn→ R, h : Rn→ Rl, and g : Rn→ Rm are second-order differentiable functions. At k-th iterate, the SQP algorithm will generate a search direction dx by solving the Quadratic Programming subproblem as follows min dx ∇f(xk)dx + 1 2 dTxBkdx s.t { hi(xk) +∇hi(xk)dx = 0 i = 1, ..., l. gj(xk) +∇gj(xk)dx ≤ 0 j = 1, ..., m. (3) where Bk is the positive definite Hessian matrix of the Lagrangian L(x, λ, µ) = f(x) + l∑ i=1 λTi hi(x) + m∑ j=1 µTj gj(x) (4) Hessian matrixBk will be updated throughout the optimization process by Broyden- Fletcher-Goldfarb-Shanno algorithm. Bk+1 = Bk + yˆkyˆ T k yˆTk dk − Bkdk(Bkdk) T dTkBkdk (5) where yˆk = tyk + (1− t)Bkdk with yk = ∇L(xk+1, λk+1, µk+1)−∇L(xk, λk+1, µk+1), t is reduced factor, t =   1, if dTk yk > 0.2d T kBkdk 0.8dTkBkdk dTkBkdk − dTk yk , otherwise so that Bk is always positive definite. Bk, dk and yk must satisfy Newton equation: Bkd T k = yk (6) Optimization process will be executed at the next iterate with xk+1 = xk + αkdk, in which αk is chosen such that merit function always decreases ϕ(xk + αkdk) < ϕ(xk), where the merit function is expressed by ϕ(x) = f(x) + r   l∑ i=1 |hi(x)|+ m∑ j=1 max(0, gj(x))   , r > max(λi,µj) (7) 2.2. Reliability index method The solution of the optimization problem without reliability always lies on the limit state surface between the safe and failure domains. As a consequence, the stability of the structure may not be ensured if the input data such as load or material parameters, etc... oscillates with certain distribution rules. To ensure the random variables oscillating in the safe domain, the Reliability Index (RI) method [9] was proposed with the aim 338 Nguyen Thoi Trung, Bui Xuan Thang, Ho Huu Vinh, Lam Phat Thuan, Ngo Thanh Phong of determining suitable values of the random variables before solving the optimization problem. Specifically, the values of these random variables are determined by solving the problem of finding the minimizer of distance function as follows β = min √ uTu subject to β ≥ βt (8) where βt is the objective reliability index and relates to the failure probability of the structure by the following formula Pf ≈ ϕ(−βt) ⇒ βt ≈ ϕ−1Pf , (9) where ϕ(.) is the standard Guassian cumulated function. Depending on the requirement of the problem and the importance of the structure, the value of βt will be determined specifically; u is the vector of normalized variables transformed from the vector of physical variables y as follows ui = (yi − µi)/σi; (10) where µi, σi are respectively the mean value and standard-deviation of random variables yi. 2.3. First-order reliability method – FORM Reliability analysis is a tool for computing the failure probability of the structure, in which all random variables related to design process are taken into account. In normalized space, the failure probability of the structure is calculated by Pf = P{g(U) < 0} = ∫ g(U)<0 φU(u)du (11) The direct evaluation of the integral (11) is not easy because the number of the random variables is large and the limit state function g(U) and the density function of the random variables are usually high order nonlinear functions. Hence, many methods were proposed to the compute the reliability index instead of evaluating the integral (11). Among these methods, the First-Order Reliability Method (FORM) [10] is the most pop- ular. The failure probability Pf then can be calculated easily from the RI by the relation (9). In the FORM, the limit state function g(U) is approximated by a linear function based on first order Taylor series expansion as follows g(U) = g(u∗) +∇g(u∗)(U− u∗)T ; (12) where u∗ = (u∗1, u ∗ 2, ..., u ∗ n) is the vector of the present values of the normalized variables ui; ∇g(u∗) is the gradient of g(U) at u∗, and determined by∇g(u∗) = [ ∂g ∂U1 , ∂g ∂U2 , ..., ∂g ∂Ui ]∣∣∣∣ u∗ . In the FORM, one needs to search for the Most Probable Point (MPP) which has the minimum distance β from the origin of coordinate O to the limit state function g(U) in normalized space. This problem can be illustrated in Fig. 1 and presented in the form of β = mind(u) = √ u21 + u 2 2 + ...+ u 2 n subjected to g(u) = 0; (13) An effective algorithm for reliability-based optimization of stiffened Mindlin plate 339 Fig. 1. Most Probable Point u∗ in normalized space U where β is the reliability index used to compute failure probability of the structure by Eq. (9). The algorithms used to solve (13) can be found in the work of Hasofer and Lind [10]. After obtaining the failure probability, the reliability of the structure will be determined by the following formula R = 1− Pf = 1−Φ(−β) = Φ(β) (14) 2.4. A simple and effective reliability-based optimization algorithm Combining the RI method with the SQP method and the FORM, we propose a three-step algorithm for the reliability-based design optimization problem as illustrated in Fig. 2. The content of each step is presented as follows: + Step 1: Give a reliability index and the values of the random variables for the optimization problem are determined by the RI method without using the limit state function. Fig. 2. A reliability-based optimization algorithm 340 Nguyen Thoi Trung, Bui Xuan Thang, Ho Huu Vinh, Lam Phat Thuan, Ngo Thanh Phong + Step 2: Using the set of the random variables obtained in step 1, we use the SQP method to solve the optimization problem to find the optimum solution of the design variables. + Step 3: From the optimum value of the design variables in step 2, we carry out checking and estimating the reliability level of the limit state functions by the FORM. These limit state functions are the constraints in step 2. If the reliability value satisfies the condition β ≥ βt, the loop stops. Otherwise, go back to step 1 and increase the given reliability value. 2.5. Brief on the behavior equation of Mindlin plate stiffened with Timoshenko beam Stiffened plate can be seen as the combination between Mindlin elements and the stiffening Timoshenko beam elements, as illustrated in Fig. 3. The stiffening beam is set parallelly with the axes in the surface of plate and the centroid of beam has a distance e from the middle plane of plate. The plate-beam system is discretized by a set of node. The degree of freedom (DOF) of each node of the plate is d = [u, v, w, βx, βy] T , in which u, v, w are the displacements at the middle of the plate and βx, βy are the rotations around the y-axis and x-axis. The DOF of each node of the beam is ds = [usc, vsc, wsc, βsx, βsy] T , in which usc, vsc, wsc are respectively the centroidal displacements of beam and are expressed by the middle surface displacements of plate as usc = u+ eβx, vsc = v + eβy, wsc = w, (15) and βsx, βsy are the rotations of beam around y-axis and x-axis. Fig. 3. A plate stiffened by an x-direction stiffener To connect these DOFs of plate and beams, we used the compatible conditions on displacement proposed by Peng et al. [11] as follows ds = Td (16) where T = 5Nsn × 5Npn is the transforming matrix of beam nodes and plate nodes; Npn, Nsn are the total nodes of plate and beam, respectively. An effective algorithm for reliability-based optimization of stiffened Mindlin plate 341 2.5.1. Energy equation of plates, beams and stiffened plates The strain energy of Mindlin plate is given as UP = 1 2 ∫∫ A ε T 0D m ε0dA+ 1 2 ∫∫ A κ T bD b κbdA+ 1 2 ∫∫ A γ TDsγdA, (17) where ε0, κb ,γ are respectively the membrane, bending and shear strains of plate, and are expressed as follows ε0 = [u,x, v,y, u,y + v,x] T ;κb = [βx,x, βy,y, βx,y + βy,x] T ;γ = [w,x + βx, w,y + βy] T . (18) Dm,Db and Ds are the material matrices involving with the strains of membrane, bending, and shearing components of the plate. The kinetic energy of Mindlin plate is given by the following formula TP = 1 2 ∫∫ A u˙TmP u˙ dA, (19) where u˙ is the derivative with respect to time of the vector of displacement field and the matrix mp has the form of mp = ρ diag(t, t, t, t 3/12, t3/12), (20) where ρ is the density of plate. The strain and kinetic energy of beam are expressed by the centroidal displacements as Us = 1 2 ∫ l εTstD stεTstdx; Ts = 1 2 ∫ l u˙Tmstu˙dx, (21) where εst = [ u,x + eβx,x βx,x w,x + βx βy,x ]T ;Dst = diag(EsAs, EsIsy, kGsAs, GsJs) and mst = ρsAs   1 0 0 e 0 0 1 0 0 e 0 0 1 0 0 e 0 0 e2 + Isy/As 0 0 e 0 0 e2 + Isx/As   ; (22) where As is the sectional area of beam; Js is the torsion moment; Isx, Isy are the second moments of the cross-sectional area of the stiffness about the axis parallel to the x-axis and y-axis. Using the superposition principle, the total strain and kinetic energy of the stiffened plate are obtained by U = UP + Nstx∑ i=1 U isx + Nsty∑ i=1 U isy; T = TP + Nstx∑ i=1 T isx + Nsty∑ i=1 T isy (23) where Nstx, Nsty are the number of stiffness in x and y directions, respectively. 342 Nguyen Thoi Trung, Bui Xuan Thang, Ho Huu Vinh, Lam Phat Thuan, Ngo Thanh Phong 2.5.2. Finite element method for stiffened plate problem FEM is applied independently for each element of plate and beam in stiffened plate. First, the displacement of plate is approximated by a three-node triangular element as d = NPn∑ i=1 Nidi = NPn∑ i=1 Ni(x)I5di, (24) where di = [ui, vi,wi, βxi, βyi] T is the vector of displacement field of plate at ith node. Ni(x) is the linear shape function of triangular element. The displacement field of beam is approximated by a two-node bar element as d = Nsn∑ i=1 ϕidsi = Nsn∑ i=1 ϕi(x)I5dsi, (25) where dsi = [usi, vsi, wsi, βxsi, βysi] T is the vector of displacement field of beam at node i-th; ϕi(x) is the linear shape function of bar element. Substituting Eqs. (25) and (24) into Eq. (23), we obtain the total strain and kinetic energy of the stiffened plate as U = 1 2 dTKd; T = 1 2 d˙TMd˙, (26) where K, M are respectively the stiffness matrix and mass matrix of the stiffened plate and computed by K = KP +T T xKsxT + xT T yKsyTy ; M =MP +T T xMsxT + xT T yMsyTy, (27) where Tx,Ty are transforming matrices and Kp,Ksx,Ksy,Mp,Msx,Msy are respectively the global stiffness and mass matrices of plate and beam with respect to x and y directions and computed by KP = 1 2 ∫∫ A (BTmD mBm +B T b D bBb +B T sD sBs) dA;MP = 1 2 ∫∫ A NTmPN dA (28) Ksti = 1 2 ∫ li BTstD stBst dA; Msti = 1 2 ∫ li ϕTmstiϕdl, i = x, y (29) For the problem of static analysis, we solve the following discretized equations Kd = f (30) where f is the load vector, d is the vector of displacement. For the problem of free vibration analysis, we solve the following system of equations (M− ω2K)d = 0, (31) where ω is the eigen-frequencies that need to be determined. In this paper, we use CS-DSG3 element recently proposed by Nguyen Thoi Trung et al. [7] to analyze the behavior of Mindlin plate. According to this method, each three-node triangular elements will be divided into 3 sub-triangular elements and these sub-elements are connected to each other through the centroid of the triangle as illustrated in Fig. 4. In each sub-element, the Discrete Shear Gap (DSG) method is applied to compute bending, shearing and membrane components. Then, the Cell-based smoothed technique An effective algorithm for reliability-based optimization of stiffened Mindlin plate 343 Fig. 4. Three sub-triangles ∆1,∆2,∆3 created from triangle 123 in the CS-DSG3 [12] will be used to smooth the bending, shearing and membrane strain fields of sub- triangles. The smoothed strain components on the element are then used to compute the bending, shearing and membrane stiffness matrices, respectively. And the remaining computing process is carried out as in the standard FEM. 3. NUMERICAL RESULTS 3.1. Static problem Consider a Mindlin clamped plate stiffened with Timoshenko beam in two directions x, y and subjected to a uniformly distributed load of q = 0.4 (N/mm2) as shown in Fig. 5a. The parameters for the problem are given as follows: The sizes of plate L = 700 (mm), H = 1500 (mm); Young’s modulus E = 2.06845× 105 (N/mm2); Possion’s ratio ν = 0.3. Fig. 5. a) Plate stiffened in two directions x, y; b) Plate stiffened in x-direction The optimization problem is then established as min ti U(ti) s.t.   u(x) ≤ 6 (mm); β(u) ≥ βt; βt = 2.3263 ti ∈ X = {ti ∈ Rn, 9 ≤ t1 ≤ 15; 15≤ t2 ≤ 25 (mm)} , i = 1÷ 4 t3 = 3t2; t4 = 2t2 (32) 344 Nguyen Thoi Trung, Bui Xuan Thang, Ho Huu Vinh, Lam Phat Thuan, Ngo Thanh Phong where U is the strain energy function; t1, t2 are respectively the design variables of the thickness of plate and the width of beam. t3, t4 are the height of beam along the directions of x and y; u(x) is the displacement function of the structure; βt is the objective reliability index corresponding to the given reliability index of 99%. 3.2. Dynamic problem Consider a Reissner-Mindlin clamped plate stiffened with Timoshenko beam along x-direction, as shown in Fig. 5b. The parameters for the free vibration problem are given as: the sizes of plate L = 0.41, H= 0.61 (m); Young’s modulus E = 211 × 109 (kg/m2); Possion’s ratio ν = 0.3, the density of plate ρ = 7830 (kg/m3). The optimization problem is presented as min ti M(ti) s.t.   ω(x) ≤ 220; β(u) ≥ βt; βt = 2.3263 ti ∈ X = {ti ∈ Rn, 5 ≤ t1 ≤ 10; 8 ≤ t2 ≤ 20 (mm)} , i = 1÷ 3 t3 = 2t2 (33) where M is the mass function of the structure; t1, t2, t3 are design variables of thickness, width and height of beam along x-direction; ω(x) is the eigen-frequency function of the structure; βt is the objective reliability index corresponding to the given reliability index of 99%. The stiffened plates used in two above problems are simply supported plates, and are discretized by a mesh of 8× 8× 2 three-node triangular elements. 3.3. Numerical results The results of two problems obtained by the SQP and compared with Genetic Al- gorithm (GA - a global optimization method based on the laws of natural evolution) are presented in Tab. 1 for two cases: a) without considering the reliability index; and b) with reliability index Table 1. Results for the two problems by the SQP and GA Result Problem Method optimization Strain Energy Mass of structure Thickness of plate Width of beam Computational cost (seconds) W (N.m) M(kg) t1(mm) t2(mm) Without reliability Static SQP 457748 —– 9 15 7 GA 457750 —– 9 15 8250 Dynamic SQP —– 10.43 0.5 1 9 GA —– 10.47 0.51 1 6432 With the reliability of 98% Static SQP 579568 —– 10.4 15 45 GA 583230 —– 10.37 15.1 15655 Dynamic SQP —– 12.93 0.5 1 30 GA —– 12.93 0.5 1 7545 According to the results in Tab. 1, the solutions by the SQP agree very well with those by the GA. In addition, in comparison about the objective function value and com- putational cost, it is seen that the objective function values of problems by the SQP are smaller than that by GA, and the computational cost by the SQP is much less than that An effective algorithm for reliability-based optimization of stiffened Mindlin plate 345 by the GA. These results hence illustrate the efficiency of the SQP compared with those of the GA. Note that, for both problems, the values of the objective function of the problem with reliability are larger than those of the problem without the reliability. This reflects correctly the logic of the problems because the problem with reliability tends toward the safety of the structure. These results hence illustrate the efficiency of the proposed three- step algorithm for the reliability-based design optimization. Also note that the SQP is a gradient-based optimization method and hence the process of finding the global minimum solution will depend strongly on the initial trial value of the solution, especially for the complicated optimization problems with many local extreme points and many design variables. For such complicated optimization problems, it is hence advised to combine the SQP with a global optimization method for finding the suitable initial trial value. This hence will increase significantly the computational cost of the SQP. 4. CONCLUSIONS AND REMARKS The paper presents an approach to establish and solve two reliability-based de- sign optimization problems for Mindlin plate stiffened by Timoshenko beams. In the first problem, static analysis is considered. The objective function is the minimum of the strain energy with constraints on displacement. In the second problem, free vibration analysis is considered. The objective function is the minimum of the mass of the structure under the constraints of eigen-frequencies. The design variables in two problems are the thickness of plate and the width, the height of stiffened beam. The random variables are chosen to be elastic modulus, density and external load. A three-step reliability-based optimization algorithm was proposed including: 1) Estimating the random variables by the Reliability Index (RI) method; 2) Solving the op- timization problem using Sequential Quadratic Programming (SQP) method; 3) Checking and estimating the reliability by the first-order reliability method (FORM). Numerical results of two reliability-based design optimization problems for Mindlin plate stiffened by Timoshenko beams illustrated the efficiency of the SQP compared with those of the GA and the efficiency of the proposed three-step algorithm for the reliability- based design optimization. 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Vu Lam Dong, Pham Duc Chinh, Construction of bounds on the effective shear modulus of isotropic multicomponent materials. 275 3. Dao Van Dung, Nguyen Thi Nga, Nonlinear buckling and post-buckling of eccentrically stiffened functionally graded cylindrical shells surrounded by an elastic medium based on the first order shear deformation theory. 285 4. N. T. Khiem, L. K. Toan, N. T. L. Khue, Change in mode shape nodes of multiple cracked bar: II. The numerical analysis. 299 5. Tran Van Lien, Trinh Anh Hao, Determination of mode shapes of a multiple cracked beam element and its application for free vibration analysis of a multi- span continuous beam. 313 6. Phan Anh Tuan, Pham Thi Thanh Huong, Vu Duy Quang, A method of skin frictional resistant reduction by creating small bubbles at bottom of ships. 325 7. Nguyen Thoi Trung, Bui Xuan Thang, Ho Huu Vinh, Lam Phat Thuan, Ngo Thanh Phong, An effective algorithm for reliability-based optimization of stiffened Mindlin plate. 335