Sharp Interval Estimates for Finite Element Solutions with Fuzzy Material Properties

Transcription

1 Sharp Interval Estimates for Finite Element Solutions with Fuzzy Material Properties R. L. Mullen, F.ASCE Department of Civil Engineering, Case Western Reserve University, Cleveland, Ohio R. L. Muhanna, M.ASCE Department of Civil Engineering, University of Maryland, College Park, Maryland ABSTRACT A new algorithm for calculating sharp hulls on interval solutions to finite element problems is presented. The algorithm treats both load and stiffness uncertainty. Examples using the algorithm are presented along with comparisons to other interval solution methods. INTRODUCTION One of the most difficult problems facing design engineers is the selection of the correct boundary conditions and constitutive models for an analysis. The routine application of sophisticated engineering analysis tools has been limited due to the cost of generating the required input data. For example, generation of a constitutive model for one material that includes temperature, loading rate, time, and multi-axial effects can require several years of data collection and analysis. Inclusion of uncertainty also requires extensive effort to define problem parameters. In the limiting case, each problem parameter (real number) in a deterministic analysis must be replaced with a probability density function (noncountable set of real numbers). In this paper, we address the use of interval methods for representation of uncertain parameters. Interval solutions do not provide the detail of a probability density description; however each uncertain parameter can be defined using only two real numbers: the upper and lower limit of the parameter. Interval arithmetic was introduced by Moore (1966) for the treatment of truncation errors in computations. Koyluoglu, Cakmak and Nielsen (1995) extended the original application to the treatment of pattern loading and structural uncertainties. The solutions for the system of linear interval equations were obtained utilized the triangle inequalities and linear programming. Rao, S.S. and Sawyer, J.P. (1995), Rao, S.S. and Berke L (1997), and Rao, S.S. and Li Chen (1998) have developed different versions of interval based finite element methods; these works were mainly restricted to narrow intervals and approximate numerical results. Abdel-Tawab and Noor (1999), recently used parameters interval representation for analysis of dynamic thermo-elasto-viscoplastic damage response. The parameters number was limited; they used the combinatorial solution for the interval system to construct fuzzy membership of the response values. The authors (Muhanna and Mullen 1995; Mullen and Muhanna 1996; Muhanna and Mullen 1999) have developed a finite element analysis procedure utilizing the concept of fuzzy sets through interval calculations. We also have computed the response of different structural

2 systems due to geometric and loading uncertainties. Results were exact in the case of load uncertainty and sharp for geometric uncertainty. In the present work, we introduce a new algorithm for interval-based calculations that is numerically efficient and results in a very sharp solution enclosure. INTERVAL ALGEBRAIC PROPERTIES An interval number is a closed set in R that includes the possible range of an unknown real number where R denotes the set of real numbers. Therefore, a real interval is a set of the form x [x l, x u ] : = { x ~ R x l x ~ x u } (1) where x l and x u are the lower and upper bounds of the interval number x respectively, and the bounds are elements of R with x l x u. Based on the above-mentioned definition, interval arithmetic is defined on sets of intervals, rather than on sets of real numbers, and interval mathematics can be considered as a generalization of real numbers mathematics. Definition of real intervals and operations with intervals could be found in a number of references (Hansen 1965; Moore 1966; Alefeld and Herzberger 1983; Neumaier 1990). Overestimation is a major drawback in interval computations. One reason of such overestimation is that only some of the algebraic laws valid for real numbers remain valid for intervals; other laws only hold in a weaker form (Neumaier 1990, pp 19-21). For example, distributive and cancellation laws do not hold in interval arithmetic. The other difficultly in interval calculations is dependency. The dependency problem arises when one or several variables occur more than once in an interval expression. Dependency may lead to catastrophic overestimation in interval computations. For example, if we subtract the interval x = [a, b] = [1, 2] from itself, as if we are evaluating the function f = x x, we obtain [a b, b a] = [ 1, 1] as a result. The result is not the interval [0, 0] as we might expect. FORMULATION In the presented work, an interval finite element approach is used for uncertainty treatment in linear solid and structural mechanics. A formulation based on an elementby-element (EBE) technique is introduced. Stiffness and load parameters are considered to have values known only to be within an interval. Using the EBE technique, where elements are kept disassembled we delay the coupling that usually occurs in the conventional FEM formulation. Lagrange multiplier method is used to impose the necessary constraints to ensure compatibility and equilibrium (Neumaier 1999). In steady-state analysis, the variational formulation for non-interval case of a discrete structural model is given in the following form * 1 T T T Π = U KU U P + λ ( CU V ) (2) 2 with the constraint (CU = V) and the stationary conditions Π = 0 for all i (3) U i

3 where Π, K, U, and P are total potential energy, stiffness matrix, displacement vector, and load vector respectively. Invoking the stationarity of Π *, that is δπ * = 0, we obtain T K C U P = (4) C 0 λ V The proposed interval formulation, as noted earlier, is based on EBE technique: each element has its own set of nodes, but the set of elements is disassembled, so that a node belongs to a single element. A set of additional constraints is introduced to force unknowns associated with coincident nodes to have identical values. Thus, the constraint equation CU = V takes the form C ~ U = 0 (5) If we express K ( n n) in the form D S ~ and substitute in equation (4) ~ ~ T DSU = P C λ (6) where D (n n) is interval diagonal matrix, its diagonal entries are the positive interval multipliers associated with stiffness uncertainty in each element, and n = degrees of freedom per element number of elements in the structure. S ~ (n n) is a non-interval T singular matrix (fixed point matrix). If we multiply equation (5) by D C ~ and add the result to equation (6), we get ~ ~ ~ ~ D( SU + C T CU) = ( P C T λ) (7) or ~ ~ ~ D( SU + QU) = ( P C T λ) ~ ~ ~ D( S + Q) U = ( P C T λ) ~ ~ DRU = ( P C T λ) (8) where R ~ is a non-interval positive definite matrix Note that the vector λ, in the present formulation, represents the vector of internal forces, and in statically determinate structures the internal forces are independent of the structural stiffness and the use of non-interval λ results in the exact hull of the solution. In the case of statically indeterminate structures, values of λ are intervals. To start our calculations, we use the internal forces from the centered values to calculate a predicted value of displacement. Starting with results of Eq. (8) using the centered values of λ as a predictor and using the inclusion theory developed in the works of Gay 1982, Neumaier 1987, 1989 and Rump 1990, we calculate bounds on the hull of the solutions. EXAMPLES The effectiveness of the algorithm is illustrated by numerical solutions for a sequence of truss problems with increasing number of bays (Figs. 1 and 2).

4 kn 2 5 m 10 m 10 m 20 m Figure (1). Two-Bay Truss kn 2 20 kn 3 20 kn m 10 m 10 m 10 m 10 m 40 m Figure (2). Four-Bay Truss. For each problem, three solutions are presented: combinatorial (exact), present formulation, and an interval solution without overestimation handling, which we will be calling the interval solution. The truss elements have the following data: A = 0.01 m 2, E = 200 GPa, and 1% uncertainty in the modulus of elasticity, i.e. E = [199, 201] GPa. The results for displacements of selected nodes are given in Tables (1), (2). As it is clear from the tables, the results obtained using the present formulation show stability in the solution sharpness with the increase of the problem size. In the two-bay truss problem (11 elements), the upper and lower bounds on displacements were obtained within a range of 0.065% to 0.7% of the exact displacements diameters. However, the interval solution overestimates the exact diameter by 300% to In the four-bay truss problem (21 elements), the upper and lower bounds on displacements were obtained within a range of 0.013% to 0.78% of the exact displacements diameters. However, the interval solution overestimates the exact diameter by 6800% to 56500%. Evidently, in spite the differences in the geometry and loading of the three problems, the present approach keeps the same level of the solution sharpness. However, the interval solution without overestimation handling results in meaningless values and reversed signs. Table 1. Two bay truss (11 elements) with 1% uncertainty in Modulus of Elasticity, E = [199, 201] Gpa Node 2 4 Displacements U (m) V (m) U (m) U (LB) U (UB) V (LB) V (UB) U (LB) U (UB) Present Outer Comb.(exact) Present Inner Present B-diam * B-diam / D E Interval solution ** D I / D E * B-diam. = outer bound inner bound for each of lower and upper bounds. ** D I, D E are the diameter of the interval solution and the diameter exact solution respectively.

5 Table 2. Four bay truss(21 elements) with 1% uncertainty in Modulus of Elasticity, E = [199, 201] GPa Node 3 6 Displacements V(m) U (m) U (m) U (LB) U (UB) V (LB) V (UB) U (LB) U (UB) Present Outer Comb.(exact) Present Inner Present B-diam * B-diam / D E Interval solution ** D I / D E *B-diam. = outer bound inner bound for each of lower and upper bounds. ** D I, D E are the diameter of the interval solution and the diameter exact solution respectively. CONCLUSIONS An interval-based finite element formulation for uncertainty in solid and structural mechanics has been developed. The method allows the treatment of uncertainty in a form of intervals (tolerances) for both load and stiffness terms. The present formulation avoids most sources of overestimation and computes very sharp hull for the solution set of interval linear equations in the field of solid and structural mechanics, even with wide interval quantities. The method allows the engineering practice to account for uncertainty in load and stiffness and to calculate very sharp bounds on the system response for all possible scenarios in a single analysis. ACKNOWLEDGEMENTS The authors acknowledge support from the National Science Foundation's Grant DMI and Dr. George Hazelrigg for his continuous support and encouragement. REFERENCES Abdel-Tawab, K. Noor, A. K., (1999). " A Fuzzy-set Analysis for a Dynamic Thermoelasto-viscoplastic Damage Response, Computers and Structures Vol. 70, pp Gay, D. M. (1982). "Solving Interval Linear Equations, " SIAM Journal on Numerical Analysis, Vol. 19, 4, pp Hansen, E.(1965). "Interval arithmetic in matrix computation, "J. S. I. A. M., series B, Numerical Analysis, part I, 2, Jansson, C.(1991). "Interval Linear System with Symmetric Matricies, Skew-Symmetric Matricies, and Dependencies in the Right Hand Side", Computing, Vol. 46, pp Koyluoglu, U. and Elishakoff, I. (1998). "A Comparison of Stochastic and Interval Finite Elements Applied to Shear Frames with Uncertain Stiffness Properties, " Computers and Structures, Vol. 67, No. 1-3, pp Koyluoglu, U., Cakmak, S., Ahmet, N., and Soren R. K. (1995). "Interval Algebra to Deal with Pattern Loading and Structural Uncertainty, " Journal of Engineering Mechanics, November, Moor, R.,E.(1966). Interval Analysis, Prentice-Hall, Inc., Englewood Cliffs, N. J.

6 Muhanna, R. L. and Mullen, R. L.(1995). "Development of Interval Based Methods for Fuzziness in Continuum Mechanics, " Proc., ISUMA-NAFIPS 95, September 17-20, Muhanna, R. L.and Mullen, R. L. (1999), "Formulation of Fuzzy Finite Element Methods for Mechanics Problems, " Computer-Aided Civil and Infrastructure Engineering (previously Microcomputers in Civil Engineering), Vol.14, pp Mullen, R. L. and Muhanna, R. L.(1996). "Structural Analysis with Fuzzy-Based Load Uncertainty, " Proc, 7th ASCE EMD/STD Joint Specialty Conference on Probabilistic Mechanics and Structural Reliability, WPI, MA, August 7-9, Neumaier, A. (1987). "Overestimation in Linear Interval Equations, " SIAM Journal on Numerical Analysis, Vol. 24, 1, pp Neumaier, A. (1989). "Rigorous Sensitivity Analysis for Parameter-Dependent Systems of Equations, "Journal of Mathematical Analysis and Applications, Vol. 144, pp Neumaier, A.(1990). Interval methods for systems of equations, Cambridge University Press. Rao, S. S., Sawyer, P. (1995). " Fuzzy Finite Element Approach for Analysis of Imprecisely Defined Systems, " AIAA Journal, Vol. 33, No. 12, pp Rao, S. S., Berke, L. (1997). " Analysis of Uncertain Structural Systems Using Interval Analysis, " AIAA Journal, Vol. 35, No. 4, pp Rao, S. S., LI Chen, (1998). " Numerical Solution of Fuzzy Linear Equations In Engineering Analysis, " Int. J. Numer. Meth. Engng. Vol. 43, pp Rump, S. M. (1990). " Rigorous Sensitivity Analysis for Systems of Linear and Nonlinear Equations, " Mathematics of Computations, Vol. 54, 190, pp