Reliability-Based Design Sensitivity Analysis and Optimization for the Hyper-Elastic Structure Using the Meshfree Method

In this paper, efficient design sensitivity analysis (DSA) and optimization methods are presented to support reliability-based design for a hyper-elastic structure with frictional contact using a meshfree method. For the structural reliability analysis, the firstorder reliability method (FORM) is utilized. The continuum-based DSA method is employed to search the most probable point (MPP) in the standard normal random variable space for structural reliability analysis. To develop the continuum-based DSA for the hyper-elastic constitutive relation and penalized contact formulation, the material derivative of continuum mechanics is utilized. The sensitivity equation is solved at each converged load step using the same tangent stiffness of response analysis due to the path dependency of the frictional contact problem. For the reliability-based structural DSA and optimization method, analytical reliability-based DSA is used to calculate the design sensitivity of reliability indices with respect to probabilistic design variables. A numerical result is presented to validate the proposed method. NOMEMNCLATURE X Random system parameter; X = [Xi] T (i = 1, 2,...,n) x Outcomes of the random system parameter; x = [xi] T (i = 1, 2,...,n) x , x Lower and upper tolerance limits of the system parameter; x x x L U ≤ ≤ Φ( ) • Standard normal cumulative distribution function (CDF) G(x) System performance function; system fails if G(x) < 0 FG g ( ) CDF of the system performance function G(x); FG(g) = P(G(x) < g) Pf Failure probability; Pf = FG(0) = P(G(x) < 0) Pf Prescribed failure probability limit βG General probability index; βG(g) = −Φ (FG(g)) βs Reliability index; βs = −Φ (FG(0)) g Target probabilistic performance measure; P(G(x) < g) = Pf u MPP corresponding to G(u) = 0 in the u-space; βs = ||ug=0 * || gn Normal gap function for contact gt Tangential slip function for contact aΩ( , ) z z Structural variational form ′ aV ( , ) z z Structural fictitious load form aΩ ∆ * ( ; , ) z z z Linearized structural variational form Ω( ) z External load form ′ V ( ) z External fictitious load form bΓ ( , ) z z Contact variational form bN ( , ) z z Normal contact variational form bT ( , ) z z Tangential slip variational form bΓ ∆ * ( ; , ) z z z Linearized contact variational form ′ bV ( , ) z z Contact fictitious load form INTRODUCTION In engineering design, the traditional deterministic design optimization model (Arora, 1989; Haftka and Gurdal, 1991) has been successfully applied to systematically reduce the cost and improve quality. However, the existence of uncertainties in either engineering simulations or manufacturing processes requires a reliability-based design optimization (RBDO) model for robust and cost-effective designs. In the RBDO model for robust system parameter design, the mean values of random system parameters are chosen as design variables, and the cost function is minimized subject to prescribed probabilistic constraints. The probabilistic constraint can be directly prescribed by the reliability index evaluated in the traditional first-order reliability analysis (Enevoldsen, 1994; Enevoldsen and Sorensen, 1994; Chandu and Grandi, 1995; Frangopol and Corotis, 1996; ; Wu and Wang, 1996; Yu et al., 1997; Grandhi and Wang, 1998). The probabilistic constraint can also be evaluated using the performance measure approach (Tu et al., 1999). The first-order reliability method (FORM) which introduces the first-order approximation at the most probable point (MPP) to obtain structural reliability is practically very efficient. However, for FORM, the first-order sensitivities of the structural performance with respect to random variables are required. Therefore, an accurate and efficient approach for sensitivity analysis of failure functions is highly desirable for the general application of FORM for structural problems. In this paper, the frictional contact problem for hyper-elastic structural systems using the meshfree method is considered for RBDO. For nonlinear analysis of the hyper-elastic material, an effective numerical method, which can handle material incompressibility under large deformation, is highly desirable for the analysis of rubber components. The meshfree method is an ideal choice since, unlike the conventional FEA method, it is not affected by the mesh distortion problem. To accelerate computations for FORM and RBDO, a continuum-based DSA method was employed to perform an accurate and efficient sensitivity analysis of the failure function. A continuum-based shape design sensitivity formulation for a hyper-elastic structure with frictional contact has been developed using the material derivative of continuum mechanics and penalized contact formulation (Kim et al., 2000) and is utilized in this paper. For numerical analysis of the frictional contact problem, the reproducing kernel particle method (RKPM) (Liu et al., 1995; Chen et al., 1998) is utilized, and, thus, sensitivity calculation. To handle material incompressibility under large deformation, a pressure projection method (Chen et al., 1996) which is a generalization of the B-bar method (Hughes, 1987) for linear problems to avoid volumetric locking for nearly incompressible materials is used. GENERAL DEFINITION OF THE RBDO MODEL For the random system parameter X = [Xi] T (i = 1, 2,..., n), the system performance criteria are described by the system performance functions G(x) such that the system fails if G(x) < 0. The statistic description of G(x) is characterized by its cumulative distribution function (CDF) FG(g) as FG(g) = P(G(x) < g) = ... ... f dx X ( ) G( ) x 1 dxn g < x , x x x L U ≤ ≤ (1) where fX( ) x is the joint probability density function (JPDF) of all random system parameters and g is named the probabilistic performance measure. The probability analysis of the system performance function is to evaluate the non-decreasing FG(g)~g relationship (Tu et al., 1999), which is performed in the probability integration domain bounded by the system parameter tolerance limits given in Eq. (1). A generalized probability index βG, which is a non-increasing function of g, is introduced (Madsen et al., 1986) as FG(g) = Φ(−βG) (2) which can be expressed in two ways using the following inverse transformations (Rubinstein, 1981; Tu et al., 1999), respectively, as βG(g) = −Φ (FG(g)) (3a) g(βG) = FG −1 (Φ(−βG)) (3b) Thus, the non-increasing βG~g relationship represents a one-to-one mapping of FG(g)~g and also completely describes the probability distribution of the performance function. In the robust system parameter design, the RBDO model (Enevoldsen and Sorensen, 1994; Chandu and Grandi, 1995; Wu and Wang, 1996; Yu et al., 1997; Grandhi and Wang, 1998) can generally be defined as minimize Cost(d) (4a) subject to Pf,j=P(Gj(x) < 0) ≤ Pf, j , j = 1, 2,..., np (4b) d ≤ d ≤ d (4c) where the cost can be any function of the design variable d= [di ] T ≡ [μ i ] T (i = 1, 2, ..., n), and each prescribed failure probability limit Pf is often represented by the reliability target index as βt = −Φ(Pf ). Hence, any probabilistic constraint in Eq. (4b) can be rewritten using Eq. (1) as FG(0) ≤ Φ(−βt) (5) which can also be expressed in two ways through inverse transformations as βs = −Φ (FG(0)) ≥ βt (6a) g = FG −1 (Φ(−βt)) ≥ 0 (6b) where βs is traditionally called the reliability index and g is named the target probabilistic performance measure. To date, most researchers have used the reliability index approach (RIA) of Eq. (6a). In this paper, RIA is used to directly prescribe the probabilistic constraint as βs(d) ≥ βt (7a) At a given design d = [di k ] ≡ [μ i k ], the evaluation of reliability index βs(dk ) for RIA is performed using the well-developed reliability analysis (Madsen et al., 1986) as βs(d k ) = − − < Φ ( ... f dx ...dxn X ( ) ) G( ) 0 x 1 x , x x x L U ≤ ≤ (7b) FORM FOR APPROXIMATE PROBABILITY INTEGRATION The evaluation of Eq. (7a) requires reliability analysis where the multiple integration is involved as shown in Eq. (7b), and the exact probability integration is in general extremely complicated to compute. The Monte Carlo simulation (MCS) (Rubinstein, 1981) provides a convenient approximation for reliability because it directly approximates the βG~g relationship. However, MCS becomes prohibitively expensive for many engineering applications. Some approximate probability integration methods have been developed to provide efficient solutions (Breitung, 1984; Madsen et al., 1986; Tvedt, 1990), such as FORM or the asymptotic SORM. The FORM often provides adequate accuracy and is widely accepted for RBDO applications. The RIA can be used effectively with FORM or SORM in the probabilistic constraint evaluation. This paper focuses on RBDO using FORM for approximate probability integration. General Interpretation of FORM In FORM, the transformation (Hohenbichler and Rackwitz, 1981; Madsen et al., 1986) from the nonnormal random system parameter X (x-space) to the independent and standard normal variable U (u-space) is required. If all system parameters are mutually independent, the transformations can be simplified as ui = Φ 1(FXi (xi)), i = 1, 2, ..., n (8a) xi = FXi −1 (Φ(ui)), i = 1, 2, ..., n (8b) The performance function G(x) can then be represented as GU(u) in the u-space. The point on the hypersurface GU(u) =0 with the maximum joint probability density is the point with the minimum distance from the origin and is named the most probable point (MPP) ug=0 * . The minimum distance, named the first-order reliability index βs,FORM, is an approximation of the generalized probability index corresponding to ga as βs,FORM ≈ βs = βG(0) (9) Thus, the first-order reliability analysis is to find the MPP on the hypersurface GU(u) = 0 in the u-space, and MPP ug=0 * is found by performing first-order reliability analysis in RIA. First-Order Reliability Analysis In traditional first-order reliability analysis (Madsen et al., 1986), the first-order reliability index βs,FORM is the solution of a nonlinear optimization problem minimize ||u|| (10a) subject to GU(u) =0 (10b) where the optimum is the MPP ug=0 * and thus βs,FORM = ||ug=0 * ||. Many MPP search algorithms (such as HL-RF, Modified HL-RF, AMVFO) and general optimization algorithms (such as SLP, SQP, MFD, augmented Lagrangian method, etc.) can be used to find the MPP (Wu and Wirsching, 1987; Wu et al., 1990; Wang and Grandhi, 1994). DESIGN SENSITIVITY ANALYSIS OF MULTIBODY FRICTIONAL CONTACT PROBLEM Response Analysis of Contact Problem For the multibody contact problem, the contact point depends on the motion of a slave body and a master body together since the second body also moves as it deforms. The normal contact condition prevents penetration of one body into another and the tangential slip represents frictional behavior of the contact surface. A regularized Coulomb friction law proposed by Wriggers et al. (1990) is utilized in this paper. Contact Condition Figure 1 shows a general contact condition between two bodies in R. Body 1 is referred to as the slave body and body 2 as the master body. The surface coordinate of the master body xc c ∈Γ 2 can be represented by a natural coordinate ξ along the master surface. As the point x ∈Γc 1 on the slave surface is in contact with the point xc c ∈Γ 2 on the master surface, xc can be represented using the natural coordinate ξc at the contact point as xc c ( ) ξ . The contact point moves as the slave body is deformed by the change of ξc in addition to the deformation of the master body. The tangential vector at xc(ξc) along the master surface can be obtained by t x ( ) , ξ ξ c c = where comma represents the partial derivative. The normal contact condition can be imposed on the structure by measuring the distance between parts of the boundaries Γ Γ c c and 1 2 . The impenetration condition can be defined, using the normal gap function gn which measures the normal distance, as gn c c T n c ≡ − ≥ ( ( )) ( ) x x e ξ ξ 0 , x x ∈ ∈ Γ Γ c c c 1 2 , (11) where e e e n c t ( ) ξ = × 3 is the unit outward normal vector of the master surface at the contact point, e t t t = / is the unit tangential vector, and e3 is the unit vector out of plane direction fixed in R . The contact point xc c ∈Γ 2 corresponding to the slave surface point x ∈Γc 1 is determined by solving the following contact consistency condition φ ξ ξ ξ ( ) ( ( )) ( ) c c c T t c = − = x x e 0 (12) Note that, in Eq. (12), xc c ( ) ξ is the closest projection point of x ∈Γc 1 onto the master surface. As the contact point moves along the master surface, a frictional force that resists the tangential relative movement exists along the tangential direction of the surface of the master body. The tangential slip function gt is the measure of the relative movement of the contact point along the master surface as gt c c ≡ − t 0 0 ( ) ξ ξ (13) where t 0 and c ξ are the tangential vector and natural coordinate of the previous converged time step, respectively. If there exists a region Γc which violates the impenetration conditions of Eq. (11), it is penalized by the penalty function. Similarly, the tangential movement of Eq. (13) can also be penalized. Define the contact penalty function for the violated region by P = + 12 12 2 2 ω ω n n t t g d g d