Keywords: finite element methods, dislocation energy minimization, strain fitting, displacement fitting, strain-assumed elements, straingages, individual element test.

We present a procedure for matching a displacement field to a given strain field, or vice-versa, over an arbitrary domain, which can be a finite element. The fitting criterion used is minimization of a dislocation energy functional. The strain field, whether given or fitted, need not be compatible. The method has four immediate applications: (i) finite element stiffness formulation based on fitting assumed-natural-strain (ANS) fields to node displacements; (ii) pointwise recovery of an internal displacement field in ANS elements as required for consistent mass, body load or geometric stiffness computations; (iii) recovery of smoothed strains from node displacements for stress post-processing, and (iv) system identification and damage detection from experimental data. The article focuses on application (i) for the strain fitting (SF) problem and (ii) for the displacement fitting (DF) problem. The separation of mean and deviatoric strains is emphasized whenever it is found convenient to simplify calculations.

We are given the strain field e(x) in the volume of a body or finite element, which contains free or specified parameters. This strain field is not necessarily compatible (derivable from a continuous displacement field). The source of e(x) could be experimental, from interpolation of strain gage readings. Or it may be one of the primary fields in strain-assumed finite element formulations. Two related problems are studied:

Strain fitting, or SF problem. Given a continuous displacement field u(x) and a strain field form e(x) that contains free parameters, find the parameters that best fit . The SF problem is trivial if e(x) is left completely free since if so e(x)Du(x) is obviously the solution.

Displacement fitting, or DF problem. Given e(x), find an associated displacement field u(x) in so that the displacement-derived strain field e(x)Du(x) matches over in the sense discussed below. Here D is the appropriate strain-displacement operator. The fitted displacement field is specified only within a rigid body motion. The symbol e follows the field-dependence notation developed for Parametrized Variational Principles [1-4].

Both problems DF and SF occur in finite element technology. Problem SF is important in the development of stiffness equations of assumed-strain elements, as well as in the recovery of smooth strain fields from node displacement information for postprocessing. Problem DF occurs when fitting a displacement field to an assumed strain element for constructing consistent masses, body-loads node forces and geometric stiffnesses. Problem DF also occurs in system identification and damage detection [5,6], but this application is not addressed in the paper.

The paper correlates variants of this method with the Free Formulation of Bergan and Nygård [7], application of the free-free flexibility to handle rigid body motions [8,9] use of Barlow points as optimal straingage locations [10] the Individual Element Test of Bergan and Hanssen [11], energy orthogonality [12], and the Assumed Natural Deviatoric Strain (ANDES) formulation [13,14]. It concludes with four examples of the DF problem in bars, plane beams [15,16], and the 4-node bilinear quadrilateral.

1

C. A. Felippa and C. Militello, " Developments in Variational Methods for High Performance Plate and Shell Elements," in Analytical and Computational Models for Shells, CED Vol. 3, ed. by A. K. Noor, T. Belytschko and J. C. Simo, ASME, New York, 191-216, 1989.

2

C. A. Felippa, " A Survey of Parametrized Variational Principles and Applications to Computational Mechanics," Comp. Meths. Appl. Mech. Engrg., 113, 109-139, 1994. doi:10.1016/0045-7825(94)90214-3

3

C. A. Felippa, " Parametrized Unification of Matrix Structural Analysis: Classical Formulation and d-Connected Mixed Elements," Fin. Elem. Anal. Des., 21, 45-74, 1995. doi:10.1016/0168-874X(95)00027-8

4

C. A. Felippa, " Recent Developments in Parametrized Variational Principles for Mechanics," Comput. Mech., 18, 159-174, 1996. doi:10.1007/BF00369934

5

K. C. Park, G. W. Reich and K. F. Alvin, " Damage Detection Using Localized Flexibilities," in Structural Health Monitoring, Current Status and Perspectives, ed. by F.-K. Chang, Technomic Pub., 125-139, 1997.

6

K. C. Park and G. W. Reich, " A Theory for Strain-based Structural System Identification, Proc. 9th International Conference on Adaptive Structures and Technologies, 14-16 October 1998, Cambridge, MA.

7

P. G. Bergan and M. K. Nygård, "Finite Elements with Increased Freedom in Choosing Shape Functions," Int. J. Numer. Meth. Engrg., 20, 643-664, 1984. doi:10.1002/nme.1620200405

8

C. A. Felippa and K. C. Park, " The Construction of Free-Free Flexibility Matrices as Generalized Stiffness Inverses," Computers & Structures, 68, 411-418, 1998. doi:10.1016/S0045-7949(98)00068-6

9

C. A. Felippa, K. C. Park and M. R. Justino F., " The Construction of Free-Free Flexibility Matrices for Multilevel Structural Analysis," Comp. Meths. Appl. Mech. Engrg., 191, 2139-2168, 2002. doi:10.1016/S0045-7825(01)00379-6

10

B. M. Irons and S. Ahmad, " Techniques of Finite Elements," Ellis Horwood Ltd, 1980.

11

P. G. Bergan and L. Hanssen, " A New Approach for Deriving `Good' Finite Elements," in The Mathematics of Finite Elements and Applications - Volume II, ed. by J. R. Whiteman, Academic Press, London, 483-497, 1975.

12

P. G. Bergan, " Finite Elements Based on Energy Orthogonal Functions," Int. J. Numer. Meth. Engrg., 15, 1141-1555, 1980. doi:10.1002/nme.1620151009

13

C. Militello and C. A. Felippa, " The First ANDES Elements: 9-DOF Plate Bending Triangles," Comp. Meths. Appl. Mech. Engrg., 93, 217-246, 1991. doi:10.1016/0045-7825(91)90152-V

14

C. A. Felippa and C. Militello, " Membrane Triangles with Corner Drilling Freedoms: II. The ANDES Element," Fin. Elem. Anal. Des., 12, 189-201, 1992. doi:10.1016/0168-874X(92)90034-A

15

C. A. Felippa, " Customizing the Mass and Geometric Stiffness of Plane Thin Beam Elements by Fourier Methods," Engrg. Comput., 18, 286-303, 2001. doi:10.1108/02644400110365914

16

C. A. Felippa, " Customizing High Performance Elements by Fourier Methods," Trends in Computational Mechanics, ed. by W. A. Wall, K.-U. Bleitzinger and K. Schweizerhof, CIMNE, Barcelona, Spain, 283-296, 2001.

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