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, 2003,Technical University of Denmark, Copenhagen, Denmark

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

Acknowledgement Professor Bob Mullen Professor Bob Mullen Dr. Hao Zhang Dr. Hao Zhang

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

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

 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) Introduction- Uncertainty

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

 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) Introduction- Why Interval?  Provides guaranteed enclosures

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

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

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

Uncertain Data Geometry Materials Loads Interval Stiffness Matrix Interval Load Vector Element Level K U = F Interval Finite Elements

= 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) N i = Shape function corresponding to the i-th DOF t = Surface traction K U = F Interval Finite Elements

 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 Interval Finite Elements (IFEM)

 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 IFEM- Inner-Bound Methods

 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 IFEM- Enclosure

Interval arithmetic  Interval number:  Interval vector and interval matrix, e.g.,  Notations

Linear interval equation  Linear interval equation Ax = b ( A  A, b  b)  Solution set  (A, b) = {x  R |  A  A  b  b: Ax = b}  Hull of the solution set  (A, b) A H b := ◊  (A, b)

Linear interval equation  Example x2x2  1.0 x1x1  Enclosure Solution set Hull

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

Naïve interval FEA 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

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 Element-By-Element model

Element-By-Element

:: ::

Constraints Impose necessary constraints for compatibility and equilibrium  Penalty method  Lagrange multiplier method Element-By-Element model

Constraints – penalty method

Constraints – Lagrange multiplier

Load in EBE Nodal load applied by elements p b

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: No dependency handling

Fixed point iteration ,, ,,

Convergence of fixed point The algorithm converges if and only if To minimize ρ(|G|): : 1

Stress calculation Conventional method: Present method:

Element nodal force calculation Conventional method: Present method:

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

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

Truss structure

Truss structure - results Methodu 5 (LB)u 5 (UB)N 7 (LB)N 7 (UB) Combinatorial Naïve IFEA– –  %146.18%362%327% Present IFEA  0.19%0.11%0.19%0.15% unit: u 5 (m), N 7 (kN). LB: lower bound; UB: upper bound.

Truss structure – results

Frame structure MemberShape C1C1 W12×19 C2C2 W14×132 C3C3 W14×109 C4C4 W10×12 C5C5 W14×109 C6C6 B1B1 W27×84 B2B2 W36×135 B3B3 W18×40 B4B4 W27×94

Frame structure – case 1 CombinatorialPresent IFEA Nodal forceLBUBLBUB Axial (kN) Shear (kN) Moment (kN·m)

Frame structure – case 2 Monte Carlo sampling * Present IFEA Nodal forceLBUBLBUB Axial (kN) Shear (kN) Moment (kN.m) * 10 6 samples are made.

Truss with a large number of interval variables story×bayNeNe NvNv 3× × × × × × × × × ×

Scalability study Sensitivity Analysis Present IFEA Story×bayLB * UB * LBUB  LB  UB wid/d 0 3× %0.10%2.64% 4× %0.16%2.84% 5× %0.22%3.02% 6× %0.29%3.19% 7× %0.37%3.37% 8× %0.45%3.56%  LB = |LB － LB * |/ LB *,  LB = |UB － UB * |/ UB *,  LB = (LB － LB * )/ LB *

Efficiency study Story×bayNvNv Iteratio n titi trtr tti/tti/ttr/ttr/t 3× %78.4% 4× %80.5% 5× %89.1% 6× %89.7% 7× %90.1% 8× %90.0% t i : iteration time, t r : CPU time for matrix inversion, t : total comp. CPU time

Efficiency study NvNv Sens.Present

Plate with quarter-circle cutout

Plate – case 1 Monte Carlo sampling * Present IFEA ResponseLBUBLBUB u A (10 － 5 m) v E (10 － 5 m) － － － － σ xx (MPa) σ yy (MPa) * 10 6 samples are made.

Plate – case 2

CombinatorialPresent IFEA ResponseLBUBLBUB u A (10 － 5 m) v E (10 － 5 m) － － － － σ xx (MPa) σ yy (MPa)

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

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

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%)

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