Interval Finite Element Methods for Uncertainty Treatment in Structural Engineering Mechanics Rafi L. Muhanna Georgia Institute of Technology USA

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1 Interval Finite Element Methods for Uncertainty Treatment in Structural Engineering Mechanics Rafi L. Muhanna Georgia Institute of Technology USA Second Scandinavian Workshop on INTERVAL METHODS AND THEIR APPLICATIONS August 25-27, 27, 2003,Technical University of Denmark, Copenhagen, Denmark

2 Outline Introduction Interval Finite Elements Element-By By-Element Examples Conclusions

3 Acknowledgement Professor Bob Mullen Dr. Hao Zhang

4 Center for Reliable Engineering Computing (REC) We handle computations with care

5 Outline Introduction Interval Finite Elements Element-by by-element Examples Conclusions

6 Introduction- Uncertainty Uncertainty is unavoidable in engineering system structural mechanics entails uncertainties in material, geometry and load parameters Probabilistic approach is the traditional approach requires sufficient information to validate the probabilistic model criticism of the credibility of probabilistic approach when data is insufficient (Elishakoff, 1995; Ferson and Ginzburg, 1996)

7 Introduction- Interval Approach Nonprobabilistic approach for uncertainty modeling when only range information (tolerance) is available t = t0 ± δ Represents an uncertain quantity by giving a range of possible values t = [ t δ, t + δ ] 0 0 How to define bounds on the possible ranges of uncertainty? experimental data, measurements, statistical analysis, expert knowledge

8 Introduction- Why Interval? Simple and elegant Conforms to practical tolerance concept Describes the uncertainty that can not be appropriately modeled by probabilistic approach Computational basis for other uncertainty approaches (e.g., fuzzy set, random set) Provides guaranteed enclosures

9 Introduction- Finite Element Method Finite Element Method (FEM) is a numerical method that provides approximate solutions to partial differential equations

10 Introduction- Uncertainty & Errors Mathematical model (validation) Discretization of the mathematical model into a computational framework Parameter uncertainty (loading, material properties) Rounding errors

11 Outline Introduction Interval Finite Elements Element-by by-element Examples Conclusions

12 Interval Finite Elements Uncertain Data Geometry Materials Loads Interval Stiffness Matrix T K B CB dv Interval Load Vector = F = N t da i i Element Level K U = F

13 Interval Finite Elements K U = F K = B T C B dv = Interval element stiffness matrix B = Interval strain-displacement matrix C = Interval elasticity matrix F = [F 1,... F i,... F n ] = Interval element load vector (traction) F = N t da i i N i = Shape function corresponding to the i-th DOF t = Surface traction

14 Interval Finite Elements (IFEM) Follows conventional FEM Loads, geometry and material property are expressed as interval quantities System response is a function of the interval variables and therefore varies in an interval Computing the exact response range is proven NP-hard The problem is to estimate the bounds on the unknown exact response range based on the bounds of the parameters

15 IFEM- Inner-Bound Methods Combinatorial method (Muhanna and Mullen 1995, Rao and Berke 1997) Sensitivity analysis method (Pownuk 2004) Perturbation (Mc William 2000) Monte Carlo sampling method Need for alternative methods that achieve Rigorousness guaranteed enclosure Accuracy sharp enclosure Scalability large scale problem Efficiency

16 IFEM- Enclosure Linear static finite element Muhanna, Mullen, 1995, 1999, 2001,and Zhang 2004 Popova 2003, and Kramer 2004 Neumaier and Pownuk 2004 Corliss, Foley, and Kearfott 2004 Dynamic Dessombz, 2000 Free vibration-buckling Modares, Mullen 2004, and Billini and Muhanna 2005

17 Interval arithmetic Interval number: Interval vector and interval matrix, e.g., T x = ( x, x ) = ([0,1], [ 2,1]) 1 2 midpoint, width, absolute value: defined componentwise Notations x = [ x, x] ( midpoint: x = ( x+ x) / 2, width: wid( x) = x x, absolute value: x = max{ x, x }. intervals: boldface, e.g., x, b, A real: non-boldface, x x, A A T

18 Linear interval equation Linear interval equation Ax = b ( A A, b b) Solution set Σ(A, b) = {x R A A b b: Ax = b} Hull of the solution set Σ(A, b) A H b := Σ(A, b)

19 Linear interval equation Example 2 [ 1, 0] x1 1.2 = [ 1, 0] 2 x Hull x Solution set 1.0 Enclosure x 2

20 Outline Introduction Interval Finite Elements Element-by by-element Examples Conclusions

21 Naïve interval FEA E1, A1, L1 2 p 1 E2, A2, L2 3 p 2 EA/ L = k = [0.95, 1.05], EA p / L = k = [1.9, 2.1], = 0.5, p = k1+ k2 k2 u1 p1 [2.85, 3.15] [ 2.1, 1.9] u1 0.5 = = k k u p [ 2.1, 1.9] [1.9, 2.1] u exact solution: u 1 = [1.429, 1.579], u 2 = [1.905, 2.105] naïve solution: u 1 = [-0.052, 3.052], u 2 = [0.098, 3.902] interval arithmetic assumes that all coefficients are independent uncertainty in the response is severely overestimated

22 Element-By-Element Element-By-Element (EBE) technique elements are detached no element coupling structure stiffness matrix is block-diagonal (k 1,, k Ne ) the size of the system is increased u = (u 1,, u Ne ) T need to impose necessary constraints for compatibility and equilibrium u 1 u 3 u 4 u 2 E 1, A 1, L 1 E 2, A 2, L 2 Element-By-Element model

23 Element-By-Element Suppose the modulus of elasticity is interval: ( E = E(1 + δ) δ: zero-midpoint interval The element stiffness matrix can be split into two parts, ( ( ( k = k( I + d) = k + kd ( ( k : deterministic part, element stiffness matrix evalued using E, ( kd : interval part d: interval diagonal matrix, diag( δ,..., δ).

24 Element-By-Element : : ( Element stiffness matrix: k = k( I + d) Structure stiffness matrix: ( ( ( K = K( I + D) = K + KD or K ( k 1 k 1 d 1 = O = O I + O ( N k k e N N e d e

25 Constraints Impose necessary constraints for compatibility and equilibrium Penalty method Lagrange multiplier method u 1 u 2 u 3 u 4 E 1, A 1, L 1 E 2, A 2, L 2 Element-By-Element model

26 Constraints penalty method Constraint conditions: cu = 0 Using the penalty method: ( K + Q) u = p T Q: penalty matrix, Q= c ηc η: diagonal matrix of penalty number η Requires a careful choice of the penatly number i E 1, A 1, L 1 E 2, A 2, L 2 A spring of large stiffness is added to force node 2 and node 3 to have the same displacement.

27 Constraints Lagrange multiplier Constraint conditions: cu = 0 Using the Lagrange multiplier method: T K c u p = c 0 λ 0 λ : Lagrange multiplier vector, introdued as new unknowns

28 Load in EBE Nodal load applied by elements p b T where p = N φ( x) dx i p = ( p,..., p ) Suppose the surface traction φ( x) is described by j an interval function: φ( x) = a x. p b can be rewritten as b p 1 b W: deterministic matrix F: interval vector containing the interval coefficients of the surface tractiton m j= 0 N j = WF e T

29 Fixed point iteration For the interval equation Ax = b, preconditioning: RAx = Rb, R is the preconditioning matrix transform it into g (x * ) = x * : R b RA x 0 + (I RA) x * = x *, x = x * +x 0 Theorem (Rump, 1990): for some interval vector x *, if g (x * ) int (x * ) then A H b x * +x 0 Iteration algorithm: *( l+ 1) *( l) iterate: x = z + G( ε x ) ( ( ( 1 where z = Rb RAx0, G = I RA, R= A, Ax0 = b No dependency handling

30 Fixed point iteration, Interval FEA calls for a modified method which exploits the special form of the structure equations ( ( ( K + Q) u= p with K = K + KD ( 1, Choose R= ( K + Q), construct iterations: *( l+ 1) *( l) u = Rp R( K + Q) u0 + ( I R( K + Q) )( ε u ) ( *( l) = Rp u0 RKD ( u0+ε u ) ( () l = Rp u0 RKM ( + + if u int( u ), then u= u + u = Rp RKM *( l 1) *( l) *( l 1) ( l) 0 : interval vector, =( δ,..., δ ) 1 1 N e The interval variables δ,..., δ appear only once in each iteration. Most sources of dependence are eliminated. N e T

31 Convergence of fixed point The algorithm converges if and only if To minimize ρ( G ): ( 1 choose R = A so that G = I RA : 1 ρ ( G ) < 1 smaller ρ( G ) less iterations required, and less overestimation in results has a small spectral radius reduce the overestimation in G ( ( ( ( 1 G= I RA=I ( K + Q) ( K + Q+ KD) = RKD

32 Stress calculation Conventional method: σ = CBu Present method: e, (severe overestimation) C: elasticity matrix, B: strain-displacement matrix ( ( E = (1 + δ) E, C = (1 + δ) C σ = CBLu ( () l =CBL( Rp RKM ) ( ( ( = + δ p M L: Boolean matrix, L u= ue () l (1 )( CBLR CBLRK )

33 Element nodal force calculation Conventional method: f = ( ku p ), (severe overestimation) T e e e Present method: ( Te) 1( k1( ue) 1 ( p ) 1) e in the EBE model, T ( Ku pb ) = M ( e) N ( ( ) ( ) ) T k e N u e e N p e e Ne from ( K + Q) u= p + p T( Ku p ) = T( p Qu) Calculate T( p for all elements. c c b b c Qu) to obtain the element nodal forces

34 Outline Introduction Interval Finite Elements Element-by by-element Examples Conclusions

35 Numerical example Examine the rigorousness, accuracy, scalability, and efficiency of the present method Comparison with the alternative methods the combinatorial method, sensitivity analysis method, and Monte Carlo sampling method these alternative methods give inner estimation x i x x i xo x: exact solution, x: inner bound, x : outer bound o

36 Truss structure P m P 2 P 1 P 3 4.5m 4.5m 4.5m 4.5m 2 1, 2, 3 13, 14, 15: [9.95, 10.05] cm (1% uncertainty) A A A,A A A cross-sectional area 2 of all other elements: [5.97, 6.03] cm (1% uncertainty) modulus of elasticity of all elements: 200,000 MPa p p = [190, 210] kn, p = [95,105] kn 1 2 = [95,105] kn, p = [85.5,94.5] kn (10% uncertainty) 3 4

37 Truss structure - results Table: results of selected responses Method u 5 (LB) u 5 (UB) N 7 (LB) N 7 (UB) Combinatorial Naïve IFEA δ Present IFEA δ % % % % % % % % unit: u 5 (m), N 7 (kn). LB: lower bound; UB: upper bound.

38 Truss structure results u 5 (m) Comb. LB Comb. UB IFEA LB IFEA UB % 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% Uncertainty in cross-sectional area for moderate uncertainty ( 5%), very sharp bounds are obtained for relatively large uncertainty, reasonable bounds are obtained in the case of 10% uncertainty: Comb.: u5 = [ , ], I u5 FEM: = [ , ] (relative difference: 2.59%, 1.80% for LB, UB, respectively)

39 Frame structure w 3 w B 8 3 B 4 C 4 C 5 C 6 w 1 w B 1 5 B 2 C 1 C 2 C m 14.63m 4.57m 6.1m Member Shape C 1 W12 19 C 2 C 3 W C 4 W C 5 W10 12 C 6 W B 1 W W27 84 B 2 W B 3 W18 40 W27 94 results listed: nodal forces at the left node of member B 2 B 4

40 Frame structure case 1 Case 1: load uncertainty w w = [105.8,113.1] kn/m, w = [105.8,113.1] kn/m, 1 2 = [49.255,52.905] kn/m, w = [49.255,52.905] kn/m, 3 4 Table: Nodal forces at the left node of member B 2 Nodal force Axial (kn) Shear (kn) Moment (kn m) Combinatorial LB UB Present IFEA LB UB exact solution is obtained in the case of load uncertainty

41 Frame structure case 2 Case 2: stiffness uncertainty and load uncertainty 1% uncertainty introduced to AI,, and Eof each element. Number of interval variables: 34. Table: Nodal forces at the left node of member B 2 Nodal force Axial (kn) Shear (kn) Moment (kn.m) Monte Carlo sampling * LB UB Present IFEA LB UB * 10 6 samples are made.

42 Truss with a large number of interval variables p p p p C D p p n@l p B A A i E i m@l = [0.995,1.005] A, = [0.995,1.005] E for i = 1,..., N 0 0 e story bay N e N v

43 Scalability study vertical displacement at right upper corner (node D): v D Table: displacement at node D = a PL EA 0 0 Sensitivity Analysis Present IFEA Story ba y LB * UB * LB UB δ LB δ UB wid/d % 0.10% 2.64% % 0.16% 2.84% % 0.22% 3.02% % 0.29% 3.19% % 0.37% 3.37% % 0.45% 3.56% δ LB = LB- LB * / LB *, δ LB = UB- UB * / UB *, δ LB = (LB- LB * )/ LB *

44 Efficiency study Table: CPU time for the analyses with the present method (unit: seconds) Story bay N v Iteratio n t i t t i /t t r /t t r % 12.4% 10.2% 9.7% 9.1% 9.2% 78.4% 80.5% 89.1% 89.7% 90.1% 90.0% t i : iteration time, t r : CPU time for matrix inversion, t : total comp. CPU time majority of time is spent on matrix inversion

45 Efficiency study CPU time (sec) Computational time: a comparison of the senstitivity analysis method and the present method Sensitivity Analysis method Present interval FEA Number of interval variables Computational time (seconds) N v Sens Present

46 Plate with quarter-circle cutout thickness: 0.005m Possion ratio: 0.3 load: 100kN/m modulus of elasticity: E = [199, 201]GPa number of element: 352 element type: six-node isoparametric quadratic triangle results presented: ua, ve, σxx and σyy at node F

47 Plate case 1 Case 1: the modulus of elasticity for each element varies independently in the interval [199, 201] GPa. Table: results of selected responses Monte Carlo sampling * Response LB UB u A (10-5 m) v E (10-5 m) σ xx (MPa) σ yy (MPa) * 10 6 samples are made. Present IFEA LB UB

48 Plate case 2 Case 2: each subdomain has an independent modulus of elasticity. E i = [199, 201] GPa, for i = 1,..., m 0.025m 0.025m 0.025m E 1 E2 E 5 E m E 3 E 4 E 7 E m

49 Plate case 2 Table: results of selected responses Response u A (10-5 m) v E (10-5 m) σ xx (MPa) σ yy (MPa) Combinatorial LB UB Present IFEA LB UB

50 Outline Introduction Interval Finite Elements Element-by by-element Examples Conclusions

51 Conclusions Development and implementation of IFEM uncertain material, geometry and load parameters are described by interval variables interval arithmetic is used to guarantee an enclosure of response Enhanced dependence problem control use Element-By-Element technique use the penalty method or Lagrange multiplier method to impose constraints modify and enhance fixed point iteration to take into account the dependence problem develop special algorithms to calculate stress and element nodal force

52 Conclusions The method is generally applicable to linear static FEM, regardless of element type Evaluation of the present method Rigorousness: in all the examples, the results obtained by the present method enclose those from the alternative methods Accuracy: sharp results are obtained for moderate parameter uncertainty (no more than 5%); reasonable results are obtained for relatively large parameter uncertainty (5%~10%)

53 Conclusions Scalability: the accuracy of the method remains at the same level with increase of the problem scale Efficiency: the present method is significantly superior to the conventional methods such as the combinatorial, Monte Carlo sampling, and sensitivity analysis method The present IFEM represents an efficient method to handle uncertainty in engineering applications

Uncertainty is unavoidable in engineering system. Probabilistic approach is the traditional approach

Uncertainty is unavoidable in engineering system. Probabilistic approach is the traditional approach Interval Finite Elements as a Basis for Generalized Models of Uncertainty in Engineering Analysis Rafi L. Muhanna Georgia Institute of Technology USA Collaborative Research Center 528 German National Science