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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 76
PROCEEDINGS OF THE THIRD INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY
Edited by: B.H.V. Topping and Z. Bittnar
Paper 16

Evaluation of Inverse Boundary Element Techniques using Experimental Photoelastic Measurements

P. Wang, A.A. Becker, I.A. Jones and T.H. Hyde

School of Mechanical, Materials, Manufacturing Engineering and Management, University of Nottingham, United Kingdom

Full Bibliographic Reference for this paper
P. Wang, A.A. Becker, I.A. Jones, T.H. Hyde, "Evaluation of Inverse Boundary Element Techniques using Experimental Photoelastic Measurements", in B.H.V. Topping, Z. Bittnar, (Editors), "Proceedings of the Third International Conference on Engineering Computational Technology", Civil-Comp Press, Stirlingshire, UK, Paper 16, 2002. doi:10.4203/ccp.76.16
Keywords: boundary element technique, inverse problem, photoelasticity, stress separation, image processing.

Summary
Measurements from photoelastic specimens provide information regarding the differences of the principal stresses and their orientations at interior points. However, in many practical applications, it is often difficult to separate the individual Cartesian stress components from the photoelastic data. The inverse boundary element (BE) technique is an ideal companion to photoelastic analysis since, unlike the finite element (FE) method, the interior solutions can be represented by unconnected points rather than discretised elements. Inverse BE methods provide non-destructive evaluation tools which can be used to identify unknown variables at the surface, such as the stress distribution in a contact region between contacting bodies.

The inverse BE technique can be used to reconstruct the boundary conditions using photoelastic measurements taken from the photoelastic isochromatic fringe orders and the isoclinic angles at interior points. The separate individual Cartesian stress components can then be computed using the conventional `forward' BE method.

In the conventional BE approach, either traction or displacement is known at each element on the boundary. After applying the boundary conditions and swapping the relevant coefficients of the system matrices, equations are obtained with all the unknowns on the left-hand side and the known variables on the right- hand side, as follows:

(16.1)

After solving the equations, all tractions and displacements on the boundary are known. The interior stresses can then be calculated using:

(16.2)

For the inverse problem, both the displacements and tractions may be unknown on part of the boundary, . If the unknowns are arranged in the same way as equation (16.1), i.e. displacements are placed on the left-hand side, the remaining unknowns appear on the right-hand side, as follows:

(16.3)

Equation (16.2) becomes an equation containing unknowns in both sides:

(16.4)

Note that subscript 0 denotes the coefficients and variables belonging to the unknown region of the boundary. Combining equations (16.3) and (16.4) and eliminating from them, an equation with in the unknown region as the only unknown, can be derived as follows:

(16.5)

where, , and .

To deal with photoelastic stress analysis by the inverse BE technique, the interior equation (16.4) at a number of chosen points is created by using the difference of Cartesian direct stresses and shear stress . Accordingly, the coefficients in the matrices and are replaced by and respectively. The Singular Value Decomposition (SVD) method is used to solve equation (16.5). Once is obtained from equation (16.5), the unknown vector can be computed using the conventional forward BE.

Previous research [1] has shown that results from inverse BE methods are in good agreement with the corresponding exact solutions. Provided that the locations of the interior points are carefully chosen to be close to the unknown region, the inverse BE technique has been shown to be reliable for random errors in the interior points of up to 10%. Further numerical improvements such as smoothing displacements or adding some constrains at intermediate stages of the inverse BE method are considered and are shown to result in a significant improvement in the solutions.

This paper presents experimental evaluations of the inverse BE technique using actual photoelastic images from two examples; a compressed circular disc and a more complex case study of the contact of spur gear teeth. The photoelastic images of the isoclinic and isochromatic data inside the solution domain are automatically processed to provide the input for the inverse BE program. The results demonstrate the reliability of the inverse BE method in practical photoelasticity problems where measurement errors can exist.

References
1
Chen, D., Becker, A.A., Jones, I.A,, Hyde, T.H. and Wang, P., "Development of New Inverse Boundary Element Techniques in Photoelasticity"; J Strain Anal, 36, 253-264, 2001. doi:10.1243/0309324011514449

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