Information Content, Finite Element Analysis, and
Interval Methods

Acknowledgements

Why do engineers calculate?

Why do engineers calculate?

What information is provided by a finite element analysis?

Information provided by FEM

Historical Considerations

On Computations

Significant Figures

Sources of uncertainty in engineering analysis

Impact of uncertainty in
engineering analysis

Interval arithmetic – Background

Interval arithmetic – Background

Outline

Weak Form of PDE

FEA Example
One Dimensional Truss

Example of FEA –  1 D  Element

Example of FEA

Outline

Element Stiffness

Assembling System of Equations

Statically Determinate

Statically Determinant

Outline

Sharp Results – Overestimation

Sharp Results – Overestimation

Sharp Results – Overestimation

Sharp Results – Overestimation

Outline

Interval Finite Elements

Interval Finite Elements

Interval Finite Elements

Interval Finite Elements

Interval Finite Elements

Outline

Finite Element – Sharp Results

Finite Element – Sharp Results

Finite Element – Sharp Results

Finite Element – Sharp Results

Finite Element – Sharp Results

Finite Element – Sharp Results

Finite Element – Present Formulation

Finite Element – Present Formulation

Finite Element – Present Formulation

Finite Element – Present Formulation

Finite Element – Present Formulation

Finite Element – Present Formulation

Finite Element – Present Formulation

Finite Element – Present Formulation

Finite Element – Present Formulation

Finite Element-Present Formulation

Finite Element-Present Formulation

Finite Element-Present Formulation

Finite Element – Present Formulation

Outline

Examples

Examples – Load Uncertainty

Examples – Load Uncertainty

Examples – Load Uncertainty

Examples – Load Uncertainty

Examples – Continuum Load Uncertainty

Examples – Continuum Load Uncertainty

Examples – Stiffness Uncertainty

Examples – Stiffness Uncertainty

Examples – Stiffness Uncertainty

Example -  Rao problem

“Rao’s Problem"

Interval Loads for Rao’s Problem

Comparison of Results

Conclusions

Conclusions

Conclusions

"Thank you"

Examples – Load Uncertainty

Conclusions

Interpretation of Intervals

Extension to Unknown PDF

Histogram - interval

Histogram

Examples – Load Uncertainty

Examples – Load Uncertainty

Historical Considerations