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Review Articles

A Unified Library of Nonlinear Solution Schemes

[+] Author and Article Information
Sofie E. Leon

Department of Civil and Environmental Engineering,  University of Illinois at Urbana-Champaign, 205 North Matthews Avenue, Urbana, IL 61801

Glaucio H. Paulino1

Department of Civil and Environmental Engineering,  University of Illinois at Urbana-Champaign, 205 North Matthews Avenue, Urbana, IL 61801paulino@illinois.edu

Anderson Pereira, Ivan F. M. Menezes

Group of Technology in Computer Graphics,Tecgraf, Pontifical Catholic  University of Rio de Janeiro, Rio de Janeiro, Rio de Janeiro, 22451, Brazil

Eduardo N. Lages

Center for Technology,  Federal University of Alagoas, Maceio, Alagoas, 57072, Brazil

1

Corresponding author.

Appl. Mech. Rev 64(4), 040803 (Aug 20, 2012) (26 pages) doi:10.1115/1.4006992 History: Received November 26, 2011; Accepted April 02, 2012; Published August 17, 2012; Online August 20, 2012

Nonlinear problems are prevalent in structural and continuum mechanics, and there is high demand for computational tools to solve these problems. Despite efforts to develop efficient and effective algorithms, one single algorithm may not be capable of solving any and all nonlinear problems. A brief review of recent nonlinear solution techniques is first presented. Emphasis, however, is placed on the review of load, displacement, arc length, work, generalized displacement, and orthogonal residual control algorithms, which are unified into a single framework. Each of these solution schemes differs in the use of a constraint equation for the incremental-iterative procedure. The governing finite element equations and constraint equation for each solution scheme are combined into a single matrix equation, which characterizes the unified approach. This conceptual model leads naturally to an effective object-oriented implementation. Within the unified framework, the strengths and weaknesses of the various solution schemes are examined through numerical examples.

Copyright © 2011 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Critical points in nonlinear equilibrium paths

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Figure 2

Snap through behavior

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Figure 3

Purely incremental procedure

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Figure 4

Incremental-iterative procedure

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Figure 5

(a) Standard and (b) modified updates to the tangent matrix

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Figure 6

Load control method

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Figure 7

Displacement control method

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Figure 8

Arc length control method

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Figure 9

(a) Spherical, (b) cylindrical and (c) elliptical version of the arc length control method

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Figure 10

(a) Schematic of the linearized arc length control method, (b) updated and fixed normal plane versions

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Figure 11

Generalized stiffness parameter used in the generalized displacement control method [4]

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Figure 12

(a) Orthogonality constraint for the orthogonal residual procedure, (b) error induced without the orthogonality constraint

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Figure 13

Comparison of original ORP load factor and unified approach load factor

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Figure 14

Incremental-iterative procedure of the unified scheme

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Figure 15

NLS++ class organization

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Figure 16

Unidimensional function results

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Figure 17

Solution to the two-dimensional function example; labels indicate where the solution schemes failed. LCM fails at the first load limit point, the DCM fails at the displacement limit point in the degree of freedom corresponding to the control, the WCM and ORP fail near the first displacement limit point, and the variable DCM, ALCM, and GDCM capture the full equilibrium path.

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Solution to the two-dimensional function example using the LCM. The method snaps through the first load limit point and fails at the second load limit points.

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Figure 19

Solution to the two-dimensional function example using the variable DCM. The labels indicate the locations where the control degree of freedom changes automatically.

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Figure 20

von Mises truss schematic

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Figure 21

Equilibrium paths for the von Mises truss with varying spring stiffness, C

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Figure 22

Solution to the von Mises truss without snap-back; labels indicate where the solution schemes failed. LCM failed at the first load limit point, WCM failed at the first displacement limit point, and the variable DCM, ALCM, GDCM, and ORP captured the full equilibrium path.

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Figure 23

Solution to the von Mises truss with snap-back; labels indicate where the solution schemes failed. LCM failed at the first load limit point, WCM failed at the first displacement limit point, and the variable DCM, ALCM, GDCM, and ORP captured the full equilibrium path.

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Figure 24

Solution to the von Mises truss example with snap-back using the DCM with u1 as the control displacement. The method snaps through at the displacement limit point and does not capture the full behavior.

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Figure 25

Deformed shape of the von Mises truss with material and geometric nonlinearity

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Figure 26

Analytical solution and solution obtained with the GDCM for the elasto-plastic case of the von Mises truss: (a) C = 0.04 and σY=0.05; (b) C = 0.04 and σY=0.1; (c) C = 0.02 and σY=0.05; (d) C = 0.02 and σY=0.1

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Figure 27

Solution to the elasto-plastic case of the von Mises truss; labels indicate with the solution schemes failed. σY=0.05, C=0.02.

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Figure 28

Twelve-bar truss schematic (a) 3D view, (b) x-z view, (c) y-z view

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Figure 29

Twelve bar truss results: Markers indicate where the solution schemes failed. LCM failed at the first load limit point, WCM failed at the first displacement limit point, and the variable DCM, ALCM, GDCM, and ORP captured the full equilibrium path. (a) DOF u1, (b) DOF u2, (c) DOF u3.

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Figure 30

Displacement-displacement curves for the twelve bar truss (a) u2 versus u1, (b) u1 versus u3, (c) u2 versus u3

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Figure 31

Lee frame schematic

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Figure 32

Deformed shape of the Lee frame corresponding to (b) label 1, (c) label 2, (d) label 3, (e) label 4, (f) label 5 in (a) load factor versus displacement curve for DOFs u1 and u2 (see Fig. 3)

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Figure 33

Lee frame results: Markers indicate where the solution schemes failed. LCM failed at the first load limit point, WCM failed at the first displacement limit point, and the variable DCM, ALCM, GDCM, and ORP captured the full equilibrium path.

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Figure 34

Lee frame results using the ORP with different values for the control factor, δλ, and scale factor, β, which are indicated by the marker labels

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Figure 35

Screen shot (shot of www.ghpaulino.com/NLS_tutorial.html)

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