Computational & Technology Resources
an online resource for computational,
engineering & technology publications
engineering & technology publications
Civil-Comp Proceedings
ISSN 1759-3433
ISSN 1759-3433
CCP: 73
PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON CIVIL AND STRUCTURAL ENGINEERING COMPUTING
PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON CIVIL AND STRUCTURAL ENGINEERING COMPUTING
Edited by: B.H.V. Topping
Paper 88
Sensitivity of Inverse Boundary Element Techniques to Errors in 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, "Sensitivity of Inverse Boundary Element Techniques to Errors in Photoelastic Measurements", in B.H.V. Topping, (Editor), "Proceedings of the Eighth International Conference on Civil and Structural Engineering Computing", Civil-Comp Press, Stirlingshire, UK, Paper 88, 2001. doi:10.4203/ccp.73.88
Keywords: boundary element method, photoelasticity, stress separation, inverse problem.
Summary
The application of computational mechanics techniques to inverse problems has
been receiving increasing attention in various branches of science and engineering.
In solid mechanics it may 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.
Photoelastic readings provide information regarding the differences of the
principal stresses and their orientations at interior points. However, it is often
difficult to separate the individual Cartesian stress components from the photoelastic
data The 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.
where
and 
are matrices that contain the numerical integrations involving the
surface kernels. After applying the boundary conditions of the problem, the relevant
elements of the matrices and are swapped such that all the unknowns end up
on the left hand side of the equation and the known variables are on the right hand
side, as follows:
where
is the solution matrix, 
contains all unknown variables (displacements
and tractions), and 
contains all prescribed values of displacements and
tractions. After solving the equations, all tractions and displacements on the
boundary are known. The interior integral equation is as follows:
where
and 
are matrices that contain the numerical integrations involving the
interior kernels. Therefore, in the forward BE approach, the boundary displacements
and tractions are computed first, and then subsequently used to obtain the interior
variables.
The inverse method, as the name implies, uses the interior stress values to
compute the boundary values. By applying the prescribed boundary conditions and
swapping the relevant elements of the matrices and , Equation 88.3 becomes
Previous research has shown that inverse BE methods relying on the interior kernels only cannot be used to obtain the solution, and that it is necessary to use both interior integral equation 88.4 and surface integral equation 88.2. In the unknown region
, both the displacements 
and tractions 
are
unknown. If one set of them is obtained first the remaining unknowns can be solved
as the forward BE procedure. The one which is to be found first will appear on the
right-hand side in equations 88.2 and 88.4 respectively, as follows
Subscript
denotes the related partitions belonging to the unknown region
of that
matrix or vector. Combining equations 88.5 and 88.6 and eliminating from them,
an equation with 
in region
as the only unknowns, can be derived as follows:
where
and
The Singular Value Decomposition (SVD) method is used to solve equation 88.7. It is clear that once is computed from equation 88.7, the remaining unknowns can
be solved straightforwardly using the forward BE equation 88.5.
The present method to deal with photoelastic stress analysis by employing inverse
BEM described above is to create the interior equations at a number of chosen points
in equation 88.6 by using the difference of Cartesian direct stresses and the magnitude
of the Cartesian shear stress
. Accordingly, the coefficients in the matrices
and are replaced by
and
respectively.
This paper also investigates the sensitivity of the inverse BE technique to errors of
photoelastic data at the interior points, discusses how to choose the number and
locations of the interior points and how to improve the numerical algorithm. The
study indicates that the number of measurement points and the location of these
points are important in constructing an accurate system matrix and dealing with the
measurement errors. A criterion for locating the interior measurement points is
given. Examples show good agreement with other solutions even if up to 5% 'noise'
is added to the photoelastic measurement data.
This paper presents an inverse boundary element (BE) technique to reconstruct the boundary conditions in the unknown regions using photoelastic measurements taken from the photoelastic isochromatic fringe order and the isoclinic angle at interior points. The separate individual Cartesian stress components are then obtained using the forward BE method The BE approach is based on two integral equations: a surface (boundary) integral equation relating displacements and tractions on the surface, and an interior integral equation relating the stresses at the interior points to the surface displacements and surface tractions. The surface (boundary) integral equation is as follows:
 |
(88.1) |
where
and are matrices that contain the numerical integrations involving the
surface kernels. After applying the boundary conditions of the problem, the relevant
elements of the matrices and are swapped such that all the unknowns end up
on the left hand side of the equation and the known variables are on the right hand
side, as follows:
where
is the solution matrix, contains all unknown variables (displacements
and tractions), and contains all prescribed values of displacements and
tractions. After solving the equations, all tractions and displacements on the
boundary are known. The interior integral equation is as follows:
where
and are matrices that contain the numerical integrations involving the
interior kernels. Therefore, in the forward BE approach, the boundary displacements
and tractions are computed first, and then subsequently used to obtain the interior
variables.
The inverse method, as the name implies, uses the interior stress values to
compute the boundary values. By applying the prescribed boundary conditions and
swapping the relevant elements of the matrices and , Equation 88.3 becomes
Previous research has shown that inverse BE methods relying on the interior kernels only cannot be used to obtain the solution, and that it is necessary to use both interior integral equation 88.4 and surface integral equation 88.2. In the unknown region
, both the displacements and tractions are
unknown. If one set of them is obtained first the remaining unknowns can be solved
as the forward BE procedure. The one which is to be found first will appear on the
right-hand side in equations 88.2 and 88.4 respectively, as follows
Subscript
denotes the related partitions belonging to the unknown region
of that
matrix or vector. Combining equations 88.5 and 88.6 and eliminating from them,
an equation with in region
as the only unknowns, can be derived as follows:
where
 |
(88.8) |
and
 |
(88.9) |
The Singular Value Decomposition (SVD) method is used to solve equation 88.7. It is clear that once
is computed from equation 88.7, the remaining unknowns can
be solved straightforwardly using the forward BE equation 88.5.
The present method to deal with photoelastic stress analysis by employing inverse
BEM described above is to create the interior equations at a number of chosen points
in equation 88.6 by using the difference of Cartesian direct stresses and the magnitude
of the Cartesian shear stress
. Accordingly, the coefficients in the matrices
and are replaced by
and
respectively.
This paper also investigates the sensitivity of the inverse BE technique to errors of
photoelastic data at the interior points, discusses how to choose the number and
locations of the interior points and how to improve the numerical algorithm. The
study indicates that the number of measurement points and the location of these
points are important in constructing an accurate system matrix and dealing with the
measurement errors. A criterion for locating the interior measurement points is
given. Examples show good agreement with other solutions even if up to 5% 'noise'
is added to the photoelastic measurement data.
purchase the full-text of this paper (price £20)
go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £122 +P&P)