Nonlinear Interval Finite Elements for Structural Mechanics Problems Rafi Muhanna School of Civil and Environmental Engineering Georgia Institute of Technology, Atlanta, GA , USA Robert L. Mullen Department of Civil and Environmental Engineering University of South Carolina, Columbia, SC USA M. V. Rama Rao Vasavi College of Engineering, Hyderabad INDIA REC 2012, June 13-15, 2012, Brno, Czech Republic

Outline  Interval FEM development  Barriers to non-linear FEM Improved interval sharpness for secondary quantities Loss of sharpness in iterative updates  Results  Conclusions

Finite Element Analysis  Approximate methods for solving PDE.  Reduces continuum problem into a discrete system of equations.  Interval extension to Finite Element methods have been developed by Muhanna, Zhang, Rao, Modares, Berke, Qiu, Elishakoff, Pownuk, Neumaier, Dessombz, Moens, Mullen and others.

 We will use a two dimensional truss as an exemplar for the development of non-linear interval finite element methods

Required Improvements to Linear Interval Finite Element Methods  Sharp solutions to systems of equations  Improve sharpness of Secondary quantities (stress/strain).  Prevent accumulation of errors in iterative correctors

Outline  Interval FEM development  Barriers to non-linear FEM Improved interval sharpness for secondary quantities Loss of sharpness in iterative updates  Results  Conclusions

Error in secondary quantities Conventional Finite Element Secondary quantities such as stress/strain calculated from displacement have shown significant overestimation of interval bounds

Use constraints to augment original variational  Indirect terms  Direct terms

Does it work?

Outline  Interval FEM development  Barriers to non-linear FEM Improved interval sharpness for secondary quantities Loss of sharpness in iterative updates  Formulation  Results  Conclusions

Nonlinear equation solving Interval Style  Extend Newton methods to Interval systems  Alternative methods with sharper results

Interval Modified Newton- Raphson Method

The ‘out of balance’ force vector can now be introduced as

Containment as a stopping criterion

 count = 1: countmax  Kc(U) U = P  U = K -1 (U) P : Obtain solution based on interval methods discussed.  for e = 1: number of elements  max (σ) = a×sup(ε)+b×(sup(ε))3  max(Es) = max (σ ) / sup (ε (e))  min (σ) = a×inf(ε)+b×(inf(ε))3  min(Es) = min (σ ) / inf (ε (e))  Es (e) = infsup (min (Es), max (Es))  end : of loop on elements

Kc:update K with the new values of Es end : of loop on count For the stopping criterion the sum of the L1 norms of the following relative change of the secant lower and upper bounds is required to be less than a specified small value

Outline  Interval FEM development  Barriers to non-linear FEM Improved interval sharpness for secondary quantities Loss of sharpness in iterative updates  Results  Conclusions

Example of nonlinear interval Monte Carlo  Non-linear IFEM can be used to construct Probability bounds using interval Monte Carlo

Example problem 1

Nonlinear Parameters  Loading from Pbox  Lognormal bounds  Cov=.1 +/- 12.5% Secant formulation

Horizontal displacement

Summary  Nonlinear analysis of structures with interval parameters for loading and stiffness parameters can be calculated with reasonably sharp bounds on interval response quantities.  Computational effort required is similar to non-interval problems. Method can be used for Monte Carlo simulations.

Rene Magritte, Clairvoyance, 1936