AN ADAPTIVE RESPONSE SURFACE METHOD UTILIZING ERROR ESTIMATES

Transcription

1 8 th ASCE Specialty Conference on Probabilistic Mechanics and Structural Reliability PMC Abstract AN ADAPTIVE RESPONSE SURFACE METHOD UTILIZING ERROR ESTIMATES Michael Macke, Dirk Roos and Jörg Riedel Institut für Strukturmechanik, Bauhaus-Universität Weimar, Germany A response surface method is proposed which allows adaptation by reducing the local error between the approximated and the actual limit state surface. The response surface is constructed as a set of n-simplices which interpolate linearly the conditional failure probabilities between determined points of the limit state surface. The local errors are estimated via the difference between the interpolated and the actual conditional failure probabilities in the directions of the centroids of the n-simplices. Adaptation is then performed by successive subdivisions of the response surface such that the computational effort is concentrated at response surface parts contributing most to the absolute error. The proposed method is illustrated for a frame structure discretized by finite elements and a limit state function with convex failure domain. Introduction Nonlinear analysis of realistic engineering structures requires in general the utilization of numerical methods like finite elements to determine failure or limit states. Thereby the limit state function is available only in implicit form, i.e. it is the outcome of extensive, iterative numerical calculations. Imbedding these methods in reliability assessment procedures is still computational demanding. Therefore, in the past the response surface method has been proposed, which allows to approximate the limit state surface for a small number of determined limit state points. The response surface is constructed either by fitting globally a second or higher order polynomial (Grigoriu, 1982; Rackwitz, 1982), or by utilizing local information at the limit state points to determine tangent hyper-planes, secant hyper-planes or segment hyperspheres (Katsuki and Frangopol, 1994; Guan and Melchers, 1997; Roos et al., 2000). However, so far no techniques for improving these response surfaces based on error estimates of the different approximations have been provided. Reliability Analysis Given is a vector X =[X 1,...,X n ]ofnmutually independent, standardized, normal random variables X i. The n-dimensional probability space is separated by the limit state function g(x) into a failure domain F = {x : g(x) 0} and a safe domain S = {x : g(x) > 0}. The failure probability P (F ) is defined as P (F )= ϕ n (x) dx (1) g(x) 0 with ϕ n ( ) asthen-dimensional normal probability density. For evaluating the integral of eq. (1) standard techniques like the first or second order reliability method, or Monte Carlo simulation can be utilized. Macke, Roos and Riedel 1

2 However, when we define failure as the loss of structural stability and apply a numerical technique like the finite element method, the limit state function g(x) can be evaluated in the safe domain only. Moreover, utilizing e.g. the smallest eigenvalue of the tangent stiffness matrix as a criterion for the loss of structural stability will in general result in highly nonlinear limit state functions. Approximating these limit state functions via limit state points evaluated around the origin can be entirely misleading with respect to the limit state surface g(x) = 0. Therefore, it is more advisable to employ only points on the limit state surface for constructing response surfaces. For this purpose the random vector X is replaced by a random directional unit vector A =[A 1,...,A n ] and a random radius R, i.e. X = AR. Therewith eq. (1) can be written as [ P (F )= 1 χ 2 n (r 2 (a)) ] f A (a)ds(a), with g(ar (a)) = 0 (2) r(a)=1 whereby f A ( ) is the probability density of the random directional unit vector A, χ 2 n ( ) the χ 2 distribution function with n degrees of freedom, and ds(a) denotes integration over the unit hyper-sphere r(a) = 1. For evaluating eq. (2) only the distances r (a) of the limit state surface from the origin in the direction of a have to be known. In approximating now the limit state surface g(x) = 0 by a response surface g(x)=0 two questions arise. What type of response surface to choose? And how to select the limit state points for constructing the response surface? In the classical response surface approach the actual limit state surface is approximated by a second order polynomial of the form g(x) =a 0 + a i x i + b ij x i x j =0 (3) j=1 which requires at least n+n(n+1)/2 limit state points to determine the coefficients a i and b ij. Additional limit state points can be included by performing linear regression. However, there is no rational criterion available for choosing directions for additional limit state points. Moreover, the whole approach is entirely motivated by geometrical considerations. In other words, determining the response surface g(x) in different random variable spaces will result in different approximating response surfaces and therewith different failure probabilities. Moreover, the inclusion of additional limit state points does not necessarily result in a better approximation since the shape of the response surface does not take into account the probability contents of the surrounding spaces of the limit state points. From the above considerations follows that any approximation strategy should yield correct probability results if applied repeatedly and should be independent of the random variable space chosen. Therefore, we propose in the following a local adaptive response surface based on n-simplices, which requires in principle only (n+1) starting points. Utilizing n-simplices allows a high flexibility to local variations of the limit state surface. Interpolating the conditional probabilities instead of the coordinates of Macke, Roos and Riedel 2

3 the limit state points results in an approximation scheme independent of the random variable space chosen. Moreover, by utilizing the difference between the interpolated and the actual conditional probability a rational criterion for the response surface adaptation is provided. Adaptive Response Surface Given is a n-simplex s j with vertices v j1, v j2,...,v jn representing the limit state points. The centroid of the n-simplex is determined by c j = 1 n v ji (4) However, we want to divide the n-simplex such that we obtain equal hyper-sphere segments. Therefore we choose as the direction for a possible new limit state point the direction of the centroid c j for the n-simplex s j projected on a unit hyper-sphere, i.e. c j = 1 n vji = 1 n v ji v ji with denoting the Euclidian norm. In a next step we linearly interpolate the conditional probabilities (5) p ji =1 χ 2 n ( v ji 2 ) (6) over the n-simplex and calculate the conditional probability p j in the direction of the centroid c j, i.e. we define another n-simplex s j with vertices p j1 vj1,p j2 vj2,...,p jn vjn. Therewith the weight p j is part of the solution of the equation system [ pj2 vj2 p j1vj1,...,p jnvjn p j1vj1, ][ ] c j α1,...,α n 1,p T j = p j1 vj1 (7) whereby the point p j c j is a point of the n-simplex s j if the conditions n 1 α i 1 and α i 0 (i =1,...,n 1) (8) are fulfilled (Roos et al., 2000). Having determined an actual limit state point r j in the direction of the centroid c j the local error for the response surface adaptation is calculated as [( e j = γ j 1 χ 2 n ( r j 2 ) ) ] p j (9) whereby γ j denotes weighting with respect to the size of the hyper-sphere segment defined by the n-simplex s j and the origin. The limit state point r j with the largest absolute error e j is chosen as the next point to be inserted in the response surface. Macke, Roos and Riedel 3

4 Numerical Examples Frame Structure The first example is a frame structure subjected to a horizontal load H and vertical load V (see Figure 1). The frame is discretized by 32 finite elements of beam type with rectangular cross section (width w = 0.04 l, height h = 0.08 l). The limit state is determined by plastic collapse of the structure. A typical failure mode is depicted in Figure 2. The constitutive model of the beams is rate-independent perfect plasticity with Huber-von Mises yield condition (Young s modulus E = MP a, flow stress σ Y = 240 MP a). The loads are modeled as independent and normally distributed random variables H N(1.8 M pl /l, 0.5 M pl /l) andv N(1.4 M pl /l, 0.4 M pl /l), whereby the plastic moment M pl is given as M pl =3σ Y wh 2 /12. H V l H V l l Figure 1. Simple frame structure discretized by finite elements. Figure 2. Failure mode of frame structure for H =3.23 M pl /l and V =3.19 M pl /l. vertical load V [Mpl/l] mean horizontal load H [M pl /l] failure probability ˆP (F ) [ 10 7 ] exact number of limit state points Figure 3. Adaptive response surface (solid line), second order polynomial (dashed line), exact limit state (dotted line) and limit state points ( : original, : adapted). Figure 4. Convergence behavior of estimated failure probabilities of frame structure for adaptive response surface () and second order polynomial (). As starting points for constructing the response surface the 4 limit state points in the direction of the main axes of the basic random variables H and V are chosen. Macke, Roos and Riedel 4

5 Additional limit state points are inserted one after the other according to the adaptive response surface procedure described above. As can be seen from Figure 3, the additional points are inserted in the response surface in such a way that the exact limit state surface as provided by finite element analysis is approximated more and more accurately by the adaptive response surface in the two regions contributing most to the estimated failure probability ˆP (F ). Thereby the adaptive response surface shows an excellent agreement due to its local adaptability, whereas the shape of the second order polynomial is dominated strongly by two of the four original limit state points lying in unimportant regions of the random variable space. In Figure 4 the convergence behavior of the adaptive response surface and the second order polynomial are compared. Whereas for the second order polynomial the inclusion of additional limit state points does not lead to an improvement of the estimated failure probability ˆP (F ) due to its general in-flexibility to local variations, the adaptive response surface converges already with 9 limit state points to the exact result and remains stable from there on. It should be noted, that although only m points are utilized for estimating the failure probability P (F ), there have been performed (m 1) additional limit state evaluations in order to be able to decide where the response surface should be apdated. Convex Failure Domain Given is a limit state function with a convex failure domain (Katsuki and Frangopol, 1994) g(x) =0.1(x 1 x 2 ) (x 1 + x 2 )+β (10) whereby X 1 and X 2 are independent, standardized, normal random variables and β is the reliability index. In the following we choose β =3.0, which corresponds to a random variable X failure domain random variable X 1 failure probability ˆP (F ) [ 10 4 ] exact number of limit state points Figure 5. Adaptive response surface (solid line), second order polynomial (dashed line), exact limit state (dotted line) and limit state points ( : original, : adapted). Figure 6. Convergence behavior of estimated failure probabilities of convex problem for adaptive response surface () and second order polynomial (). Macke, Roos and Riedel 5

6 failure probability of P (F )= (as determined by adaptive sampling). As starting points for the response surface construction again the 4 limit state points in the direction of the main axes are chosen. Since in these directions no failure occurs we introduce artificial failure states at a distance of 8 standard deviations from the origin. In Figure 5 the adaptive response surface and the limit state surface g(x) = 0 are compared for m = 10 utilized limit state points showing again an excellent agreement in the region of interest, i.e. nearest to the origin. This is also the number of limit state points where the second order polynomial fitted by linear regression diverges as can be seen in Figure 6. The adaptive response surface, however, reaches already with 8 utilized limit state points the exact value and remains again stable from there on. Conclusions An adaptive response surface has been proposed which shows a high flexibility to local variations of the limit state surface as required in reliability analysis. In addition, a procedure for improving the approximation by utilizing local error estimates has been provided. The examples given herein indicate a sufficient convergence behavior of the proposed approach with a reasonable, i.e. small number of limit state evaluations. In case parts of the limit state function are known this information can be taken into account by excluding the known region from adaptation in subsequent steps. Future research will focus on the question whether the relative error should be included in the adaptation, on a random start procedure and adaptation scheme to avoid systematic neglect of certain sub-domains, and on a criterion for deciding when a sufficient accuracy of the estimated failure probability has been obtained. Acknowledgement The work reported in this paper has been supported by the Deutsche Forschungsgemeinschaft (DFG) through Sonderforschungsbereich 524 (first and third author) and under Contract No. Be-2160/3-1 (second author) which is gratefully acknowledged by the authors. References Grigoriu, M. (1982), Methods for Approximate Reliability Analysis, Structural Safety, 1, Guan, X. L. and R. E. Melchers (1997), Multitangent-Plane Surface Method for Reliability Calculation, J. Engrg. Mech., ASCE, 123, Katsuki, S. and D. M. Frangopol (1994), Hyperspace Division Method for Structural Reliability, J. Engrg. Mech., ASCE, 120, Rackwitz, R. (1982), Response Surfaces in Structural Reliability, Volume67ofBerichte zur Zuverlässigkeitstheorie der Bauwerke, TUMünchen, München. Roos, D., C. Bucher and V. Bayer (2000), Polyhedral Response Surfaces for Structural Reliability Assessment, in: R. E. Melchers and M. G. Stewart (eds.), Proc. 8th Int. Conf. on Applications of Statistics and Probability, Sydney, Australia, December 12 15, Balkema, Rotterdam, Macke, Roos and Riedel 6