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Smooth spline-like finite-element differentiation of full-field experimental data over arbitrary geometry

A new, more-flexible numerical technique for representing and differentiating experimental data is described and demonstrated

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Abstract

A new smooth two-dimensional numerical method is presented for representing and differentiating discrete fullfield experimental information. The method involves minimizing a positive definite functional by use of finite-elements to ‘best-fit’ the data. Experimental data points may occur in any configuration, boundary derivatives need not be specified, and arbitrarily shaped regions can be handled readily. The method is demonstrated by photomechanical strain analysis.

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Abbreviations

{A}:

quintic polynomial coefficients

[B], [B 1], [B 2], [C]:

geometric matrices

D :

number of data points inR

E :

dimension of regionR

F i :

measured experimental value at ith data point

\(\bar F_j\) :

approximating function over the jth element

\(f(\bar F)\) :

least-squares defining functional

J :

number of data points in jth element

[K]:

stiffness matrix

M :

number of elements inR

N :

number of nodal parameters inR [nodal degrees of freedom (NDOF)]

[N] j :

shape function for the jth element

{P}:

load vector

R :

total region of interest

R j :

region of the jth element

RES :

residual value

{U}:

nodal parameters for entire regionR

W :

transverse plate deflection in fringes

[X]:

spatial variable matrix

{u} j :

nodal parameters for the jth element

w i :

weighting factor of the ith data point

\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} _i\) :

location of ith data point

x′, y′ :

global coordinates

x, y :

local coordinates

δ:

cubic-spline smoothing parameter

∈:

smoothing parameter

η:

coordinate normal to an element edge

θ:

angle defining orientation of an element edge with respect to thex′ axis

ξ:

coordinate along an element edge

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Author information

Authors and Affiliations

  1. Engineering Mechanics Dept., General Motors Research Lab., 48090, Warren, MI

    D. J. Segalman (Associate Senior Research Engineer)

  2. Systems Technology Dept., Electric Boat Div., General Dynamics Corp., 06340, Groton, CT

    D. B. Woyak (Engineer)

  3. Department of Engineering Mechanics, Univ. of Wisconsin, 53706, Madison, WI

    R. E. Rowlands (Professor)

Authors
  1. D. J. Segalman
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  2. D. B. Woyak
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  3. R. E. Rowlands
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Additional information

D.J. Segalman and D.B. Woyak were associated with the Dept. of Engineering Mechanics, Univ. of Wisconsin, Madison at the time of this work.

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Segalman, D.J., Woyak, D.B. & Rowlands, R.E. Smooth spline-like finite-element differentiation of full-field experimental data over arbitrary geometry. Experimental Mechanics 19, 429–437 (1979). https://doi.org/10.1007/BF02326046

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  • DOI: https://doi.org/10.1007/BF02326046

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