nonlinear structural behavior Scott A. Burns, Keith M ... · Member sizing of frame structures An...

A monomial-based method for approximating

nonlinear structural behavior

Scott A. Burns, Keith M. Mueller

University of Illinois at Urbana-Champaign, 104 S. Mathews, Urbana, IL67&07, U&4Email: [email protected], [email protected]

Abstract

A single-termed monomial power function is presented as an alternative to thestandard Taylor series linearization for purposes of approximating nonlinearstructural behavior. This monomial approximation becomes linear after alogarithmic transformation; consequently, computational tools based on linearalgebra remain useful and effective. The monomial approximation has beenshown to have several useful properties, not shared by the standard Taylorlinearization, that are responsible for improved performance when applied tostructural optimization problems.

Introduction

We usually think of nonlinear structural behavior as relating to structures thatpossess geometric or material nonlinearities. But most problems in structuraloptimization are inherently nonlinear, even though the response of the structureto a load can be adequately analyzed by linear methods. The behavior of astructure or structural member in response to a change in shape or cross-sectionalproperties is highly nonlinear and must be treated by methods that take intoaccount this nonlinear behavior.

One common way to treat nonlinearities, whether for structural optimizationor for structural analysis purposes, is to solve a sequence of simpler problemsthat (hopefully) converge to the solution of the nonlinear problem. In manycases, these simpler problems make use of linearization resulting from the linearpart of the Taylor series expansion. This paper demonstrates that there is analternative type of linearization that appears to have significant advantages whenapplied to structural optimization.

Transactions on the Built Environment vol 37 © 1999 WIT Press, www.witpress.com, ISSN 1743-3509

78 Computer Aided Optimum Design of Structures

Briefly stated, this alternative linearization can be thought of as a"monomialization." This monomial (single-termed power function) approxima-tion more faithfully models the relationship between the design variables and thestructural response. Conveniently, it becomes a linear form when transformedinto log space. Thus, computational tools based on linear algebra remain usefuland effective. Since the monomial approximation generally provides a higherquality approximation to nonlinear phenomena exhibited, each step of the itera-tive method becomes more effective.

In the following sections, the monomial approximation is presented and someof its unusual properties are discussed. The it is applied to the member sizing ofa frame structure and to the shape optimization of a continuum structure.

The monomial approximation

The idea of using a monomial function to approximate a nonlinear function canbe found in the literature as early as 1967, with the advent of geometric pro-gramming theory.' Duffnf introduced a process called "condensation" which ap-proximates a generalized polynomial with positive coefficients and positive vari-

ables, £ ,c/IIyLi*/ , with a monomial, <?JJ .xfj , by applying the weighted

arithmetic-geometric mean inequality, X/ i"/ -II/lil";/ /) » after a suitable

choice of weights, 5, is made based on the current operating point (which mustbe positive). If the original function contains both positive and negative terms,the terms can be grouped according to sign to produce a difference of two positivefunctions, P-Q. Condensation can be applied to each group separately to yield adifference of two monomials. In the context of solving systems of equations, P-0=0, the monomial approximation of Q can be transferred to the right-hand sideof the equation and then be divided through to yield a ratio of monomials equal tounity. The ratio of two monomials is itself a monomial. After a transformationof variables, .%/ = exp(z ), the logarithm of the monomial approximation be-

comes linear in z.This idea can be generalized to any systems of equations that can be expressed

in a form containing the difference of two strictly positive functions of positivevariables, P(X)-Q(X)=Q. All algebraic systems of equations can be cast in thisform/ as can many non-algebraic systems. It has been shown* that the linearsystem produced by the process of condensation described above is equivalent to

= -(in(P)-ln(G)) (1)

where [jp] and [JQ] are Jacobian matrices of P and Q, evaluated at X (the cur-

rent operating point), \Dp 1 and \D^ are diagonal matrices of P and Q values


Computer Aided Optimum Design of Structures 79

evaluated at the operating point, and fD^ 1 is a diagonal matrix of X operating

point values. Inversion of the diagonal matrix of function values is always pos-sible because of their positive nature.

Note the similarity of this linear system to that produced by Newton'smethod

Despite the similarities, the "monomial method" of solving a system on nonlin-ear equations has been shown to have significant performance-enhancing proper-ties, not shared by Newton's method. These include (1) a built-in, automatic scal-ing and equilibrating property, (2) asymptotic characteristics that often draw dis-tant starting points very close to a solution on the first iteration, and (3) a rejec-tion of spurious, meaningless solutions. Further discussion of these properties isavailable in the literature. The monomial method has proven to be especiallyeffective when applied to structural design problems. Other applications in whichthe monomial method has been shown to be highly effective include chemicalequilibrium analysis/ biochemical systems analysis,*^ and cash flow rate of re-turn computations.'°

Member sizing of frame structures

An application of the monomial that has proven to be very effective is the mem-ber sizing of frame structures. Specifically, we are seeking a fully-stressed design(FSD)—a common goal for a structural engineer. It is possible for multipleFSDs to exist and the common methods for finding them, such as the stress ratiomethod, are not able to converge to some of them." Here we will locate them byforming a system of nonlinear equations that describe the fully-stressed condition,and compare Newton's method and the monomial method as solution techniques.

Figure 1 presents a one-story, one-bay portal frame structure with a distrib-uted load acting downward along the top member. For the sake of simplicity, thecross sections are all assumed to be square, and the two columns to be identical.Thus, the cross-sectional properties of all members can be described uniquely bytwo positive variables, x^ and x^ as described in Figure 1. A common goal in

structural design is to produce a fully-stressed design, that is, a selection ofmember sizes that produces maximum stresses in each member equal to themaximum allowable stress. For the particular case /=18 ft (5.49 m), ct=0.6, w=\k/ft (175 kN/m), and maximum allowable stress=1.8 ksi (12.4 MPa), then thefollowing set of equations will have solutions that are fully-stressed designs:

11664*1*2 +54xfx2~* +21.6*f* -10.8*,%-* " .32 -0

*i4*2 +18*,<*2~* +8398*2~* -12.96V*2~* -5.184 = 0



,, ,, 175kN/m(1000lb/ft) ,,

All

cross

sections

3.29m

(10.8ft)

-o-

5.49m (18 ft)

Figure 1. One-story, one-bay frame structure.

There are four solutions to these equations: (jc,, ) = (2.92", 11.72"), (6.45",11.15"), (9.12", 9.71"), and (-1.98", 11.74"). The first three are meaningful andreasonable FSDs. The fourth one is a meaningless, spurious solution.

To assess the performance of Newton's method and the monomial methodfrom a global perspective, consider a mapping of all points in a portion of thedesign space onto their positions after one iteration of each method. Figure 2(a)shows a "checkerboard" pattern superimposed over a portion of the design spacespanning 0"<;t,<900" and 0"<%2 600". The small rectangle in the lower-leftcorner of Figure 2(a) encloses all four of the solutions to the system of equa-tions. Figure 2(b) shows the action of one iteration of Newton's method. Thereis an obvious contraction toward the solution. Figure 2(c) shows the action ofone iteration of the monomial method. All points in the checkerboard are mappedinto the tiny specks appearing in the vicinity of the three meaningful solutions.With Newton's method, well over half of the starting points end up convergingto the meaningless solution, or diverging to a floating point overflow condition.In contrast with the monomial method, none of the starting points converge tothe meaningless solution, and only a tiny fraction encounter numerical problems.A more thorough investigation of the performance of these methods can be foundin the literature.

Multiple fully-stressed designs are surprisingly prevalent, even when multipleloading conditions are acting. Figure 3 shows three FSDs for a two-story, two-bay frame with two loading conditions. When we consider allowing the membersize to go to zero in this problem, an additional 14 FSDs are introduced.



(a)

Figure 2. Mapping of one iteration of (b) Newton's method and (c)the monomial method for a large region of starting points.



1.05 MN/m (6 kip/ft)t

1I

f t t

.05 MN/mt t t*3

3.05m _(10 ft) __

t

(6kt

f t

up/ft)t t

3.05m(10ft)

f

x

t

X

4

1

12.74m(9ft)

t2.74m(9ft)

J

Loading condition 1allowable stress = 12.4 MPa (1.8 ksi)

0.53 MN/m (3 kip/ft)44.5 kN — -(10 kips)

178kN ^(40 kips)

> t t t t

0.53 MN/mMM

*3

t t t t '%6

(3 kip/ft)MM

*3

Loading condition 2allowable stress = 16.5 MPa (2.4 ksi)

FSD

#i#2#3

%1cm (in)

13.55 (5.33)30.48 (12.00)13.66 (5.38)

*2cm (in)

54.46 (21.44)45.32 (17.84)55.96 (22.03)

%3cm (in)

38.57 (15.19)37.61 (14.81)39.30 (15.47)

4cm (in)24.26 (9.55)9.97 (3.93)8.98 (3.54)

*5cm (in)

15.99 (6.30)29.53 (11.63)30.18 (11.88)

*6cm (in)

35.54 (13.99)37.48 (14.75)39.72 (15.64)

Figure 3. Three fully-stressed designs.

Shape Optimization of Continuum Structures

Another useful application of the monomial method is to find the boundary shapeof a continuum structure that minimizes volume while satisfying constraints onstress and displacement. Space limitation prevent complete presentation of theformulation, but it is available in the literature. Briefly, the problem is modeledusing finite elements and a mathematical program is cast in integrated form, us-ing nodal coordinates, nodal displacements, inter-element forces, and elementstresses as variables. The structural behavior is modeled as a set of nonlinearequality constraints (stiffness equations) relating the four sets of variables. Ine-quality constraints are imposed to limit stress and displacement, or any otherbehavior that can be modeled with the four sets of variables. Bounds on dis-placement variables represent support conditions. The monomial method fitsseamlessly within the optimization method of generalized geometric program-ming.^ Since GGP operates in the space of the logarithm of the variables, themonomial method is implemented by appending a set of linear equations to theGGP system at each iteration.

Consider a fixed-ended beam with variable depth, having a 60 inch spanlength, a selfweight of 150 pounds per cubic foot, and a distributed load of 1kip/ft applied along the top surface. The beam has a uniform thickness of 12inches. The finite element mesh consists of 48 elements, but symmetry is usedto reduce it to 24. The x-coordinates of all corner nodes are fixed. The y-coordinates of the interior corner nodes are constrained to lie at the centroid of the



corner nodes above and below them. The y-coordinates of the corner nodes on thetop and bottom surface of the beam are unrestricted. Principal stresses are con-strained between -4 ksi and 4 ksi everywhere throughout the beam; no constraintsare placed on maximum displacement. The modulus of elasticity and Poisson'sratio of this hypothetical material are specified as 4000 ksi and 0.15, respec-tively.

Figure 4 illustrates the iterative solution sequence using the modified GGPalgorithm to solve the integrated formulation. The solution is obtained in sixiterations. Experience has demonstrated that the number of iterations to a solu-tion is independent of the size of the problem. A second solution of this problemwas performed, this time with the allowable tensile stress reduced from 4 ksi to 3ksi. The allowable compressive stress was kept at -4 ksi. The sequence of itera-tions are presented in Figure 5. This time, the structure evolves into an archwhich experiences uniform axial compression everywhere. The solution processhas completely avoided the more restrictive allowable tensile stress constraints,which are not active in this solution at all.

It is interesting to note that the second solution has far less volume than thefirst solution. Since the second problem is more highly constrained than the first,this indicates that the first solution was a not the global minimum. Tighteningthe allowable tensile stress constraints forced the solution out of the symmetricsolution and into the arch-shaped global minimum. Because the allowable tensilestress constraints are not binding in the arch solution, this indicates that the archsolution is an alternative solution to the first problem with equal tensile andcompressive allowable stresses. Had the first problem been initiated with a start-ing configuration more closely resembling the arch-shaped solution, the solutionprocess could have converged to the arch solution instead. A third optimal solu-tion to the first problem is a downward sweeping arch in uniform axial tension.This problem suggests that local minima are certainly possible, and probablyquite prevalent, in shape optimization.

Conclusions

A monomial-based approximation has been presented as an alternative to standardTaylor series linearization. In the context of solving systems of nonlinear equa-tions, the monomial approximation leads to the monomial method; Taylorlinearization leads to Newton's method. The monomial method has been shownto generally outperform Newton's method, sometimes quite dramatically, whenapplied to structural design applications. The method is very general in its appli-cation. The main requirement is that the system of equations being solved beseparable into strictly positive and negative parts, and that the variables be posi-tive. Current research is being conducted into generalizing the method to includevariables that are not restricted in sign. This will further enhance the applicabilityof the method. In the case of structural applications, current research is beingconducted into applying the monomial method to the analysis of geometrically



Figure 4. First six iterations of shape optimization with equalallowable tensile and compressive stress constraints.

nonlinear structures. Incorporating the monomial approximation into an incre-mental method may allow larger load increments to be taken, improving the per-formance of such methods.

Acknowledgments

This material is based on work supported by the National Science Foundationunder award No. CMS 97-14069.

References

1. Duffin, R. J., Peterson, E., & Zener, C., Geometric Programming, JohnWiley & Sons, New York, 1967.



Figure 5. Same shape optimization problem with tightenedallowable tensile stress constraints.

2. Duffin, R. J. Linearizing geometric programs, SI AM Review, 12, 211-227, 1970.

3. Duffin, R. J. & Peterson, E. L. Geometric programming with sig-nomials, /. Opt. Theory and AppL, 11, 3-35, 1973.



4. Burns, S. A. & Locascio, A. A monomial-based method for solving sys-tems of nonlinear algebraic equations, Int. J. Num. Meth. Engr., 31,1295-1318, 1991.

5. Burns, S. A. The monomial method: extensions, variations, and perform-ance issues, Int. J. Num. Meth. Engr., 37, 2093-2107, 1994.

6. Burns, S. A. The monomial method and asymptotic properties of alge-braic systems, Int. J. Num. Meth. Engr., 37, 3939-3948, 1994.

7. Wall,T. W., Greening, D. & Woolsey, R. E. D. Solving complex chemi-cal equilibria using a geometric-programming based technique, OperationsResearch, 34, 345-355, 1986.

8. Savageau, M. A. Finding multiple roots of nonlinear algebraic equationsusing S-system methodology, Appl. Math, and Computation, 55, 187-199, 1993.

9. Mueller, K. M., Burns, S. A. & Savageau, M. A. A comparison of themonomial method and the S-system method for solving systems of alge-braic equations, Appl. Math, and Computation, 90, 167-180, 1998.

10. Baker, D. W., Application of the geometric inequality to the solution ofsystems of nonlinear equations, Ph.D. thesis, Colorado School of Mines,Golden, CO, 1980.

11. Burns, S. A., Orlet, M. W., & Raschke, M. A. Multiple fully stressedstructural designs and the stress ratio method, Microcomputers in CivilEngrg., 10, 63-76, 1995.

12. Burns, S. A. Application of the monomial method to shape optimization,ASMEJ. of Mechanical Design, 116, 1013-1018, 1994.

13. Avriel, M., Dembo, R. & Passy, U. Solution of generalized geometricprograms, Int. J. Num. Meth. Engrg., 9, 149-168, 1975.

14. Burns, S. A. Generalized geometric programming with many equalityconstraints, Int. J. Num. Meth. Engrg., 24, 725-741, 1987.


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