Transactions on the Built Environment vol 37 © 1999 WIT Press, www.witpress.com, ISSN 1743-3509
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, IL
67&07, U&4
Email: [email protected], [email protected]
Abstract
A single-termed monomial power function is presented as an alternative to the
standard Taylor series linearization for purposes of approximating nonlinear
structural behavior. This monomial approximation becomes linear after a
logarithmic transformation; consequently, computational tools based on linear
algebra remain useful and effective. The monomial approximation has been
shown to have several useful properties, not shared by the standard Taylor
linearization, that are responsible for improved performance when applied to
structural optimization problems.
Introduction
We usually think of nonlinear structural behavior as relating to structures that
possess geometric or material nonlinearities. But most problems in structural
optimization are inherently nonlinear, even though the response of the structure
to a load can be adequately analyzed by linear methods. The behavior of a
structure or structural member in response to a change in shape or cross-sectional
properties is highly nonlinear and must be treated by methods that take into
account this nonlinear behavior.
One common way to treat nonlinearities, whether for structural optimization
or for structural analysis purposes, is to solve a sequence of simpler problems
that (hopefully) converge to the solution of the nonlinear problem. In many
cases, these simpler problems make use of linearization resulting from the linear
part of the Taylor series expansion. This paper demonstrates that there is an
alternative type of linearization that appears to have significant advantages when
applied to structural optimization.
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Computer Aided Optimum Design of Structures
Briefly stated, this alternative linearization can be thought of as a
"monomialization." This monomial (single-termed power function) approximation more faithfully models the relationship between the design variables and the
structural response. Conveniently, it becomes a linear form when transformed
into log space. Thus, computational tools based on linear algebra remain useful
and effective. Since the monomial approximation generally provides a higher
quality approximation to nonlinear phenomena exhibited, each step of the iterative method becomes more effective.
In the following sections, the monomial approximation is presented and some
of its unusual properties are discussed. The it is applied to the member sizing of
a 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 can
be found in the literature as early as 1967, with the advent of geometric programming theory.' Duffnf introduced a process called "condensation" which approximates a generalized polynomial with positive coefficients and positive variables, £^,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 must
be 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 positive
functions, P-Q. Condensation can be applied to each group separately to yield a
difference of two monomials. In the context of solving systems of equations, P0=0, the monomial approximation of Q can be transferred to the right-hand side
of the equation and then be divided through to yield a ratio of monomials equal to
unity. The ratio of two monomials is itself a monomial. After a transformation
of variables, .%/ = exp(z^), the logarithm of the monomial approximation becomes 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 positive
variables, P(X)-Q(X)=Q. All algebraic systems of equations can be cast in this
form/ as can many non-algebraic systems. It has been shown* that the linear
system 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 current operating point), \Dp 1 and \D^ are diagonal matrices of P and Q values
Transactions on the Built Environment vol 37 © 1999 WIT Press, www.witpress.com, ISSN 1743-3509
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 possible because of their positive nature.
Note the similarity of this linear system to that produced by Newton's
method
Despite the similarities, the "monomial method" of solving a system on nonlinear equations has been shown to have significant performance-enhancing properties, not shared by Newton's method. These include (1) a built-in, automatic scaling and equilibrating property, (2) asymptotic characteristics that often draw distant starting points very close to a solution on the first iteration, and (3) a rejection of spurious, meaningless solutions. Further discussion of these properties is
available in the literature.^ The monomial method has proven to be especially
effective when applied to structural design problems. Other applications in which
the monomial method has been shown to be highly effective include chemical
equilibrium analysis/ biochemical systems analysis,*^ and cash flow rate of return computations.'°
Member sizing of frame structures
An application of the monomial that has proven to be very effective is the member sizing of frame structures. Specifically, we are seeking a fully-stressed design
(FSD)—a common goal for a structural engineer. It is possible for multiple
FSDs to exist and the common methods for finding them, such as the stress ratio
method, are not able to converge to some of them." Here we will locate them by
forming 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 distributed load acting downward along the top member. For the sake of simplicity, the
cross 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 by
two 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 of
member sizes that produces maximum stresses in each member equal to the
maximum 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 the
following 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
Transactions on the Built Environment vol 37 © 1999 WIT Press, www.witpress.com, ISSN 1743-3509
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Computer Aided Optimum Design of Structures
,,
,, 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 and
reasonable FSDs. The fourth one is a meaningless, spurious solution.
To assess the performance of Newton's method and the monomial method
from a global perspective, consider a mapping of all points in a portion of the
design space onto their positions after one iteration of each method. Figure 2(a)
shows a "checkerboard" pattern superimposed over a portion of the design space
spanning 0"<;t,<900" and 0"<%2^600". The small rectangle in the lower-left
corner of Figure 2(a) encloses all four of the solutions to the system of equations. Figure 2(b) shows the action of one iteration of Newton's method. There
is an obvious contraction toward the solution. Figure 2(c) shows the action of
one iteration of the monomial method. All points in the checkerboard are mapped
into 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 converging
to the meaningless solution, or diverging to a floating point overflow condition.
In contrast with the monomial method, none of the starting points converge to
the meaningless solution, and only a tiny fraction encounter numerical problems.
A more thorough investigation of the performance of these methods can be found
in the literature.^
Multiple fully-stressed designs are surprisingly prevalent, even when multiple
loading conditions are acting. Figure 3 shows three FSDs for a two-story, twobay frame with two loading conditions. When we consider allowing the member
size to go to zero in this problem, an additional 14 FSDs are introduced.
Transactions on the Built Environment vol 37 © 1999 WIT Press, www.witpress.com, ISSN 1743-3509
Computer Aided Optimum Design of Structures 81
(a)
Figure 2. Mapping of one iteration of (b) Newton's method and (c)
the monomial method for a large region of starting points.
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Computer Aided Optimum Design of Structures
1.05 MN/m (6 kip/ft)
t f t t t f t f
1
x4 2.74m
(9ft)
1 .05 MN/m (6kup/ft)
I t t t t t t t
*3
t
2.74m
X 1 (9ft)
3.05m _ 3.05m J
(10 ft) __ (10ft)
Loading condition 1
allowable stress = 12.4 MPa (1.8 ksi)
FSD
#i
#2
#3
%1
cm (in)
13.55 (5.33)
30.48 (12.00)
13.66 (5.38)
*2
cm (in)
54.46 (21.44)
45.32 (17.84)
55.96 (22.03)
0.53 MN/m (3 kip/ft)
44.5 kN — - > t t t t t t t t '
%6
(10 kips)
0.53 MN/m (3 kip/ft)
178kN ^ M M M M
*3
*3
(40 kips)
Loading condition 2
allowable stress = 16.5 MPa (2.4 ksi)
4
%3
cm (in) cm (in)
38.57 (15.19) 24.26 (9.55)
37.61 (14.81) 9.97 (3.93)
39.30 (15.47) 8.98 (3.54)
*5
*6
cm (in) cm (in)
15.99 (6.30) 35.54 (13.99)
29.53 (11.63) 37.48 (14.75)
30.18 (11.88) 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 shape
of a continuum structure that minimizes volume while satisfying constraints on
stress and displacement. Space limitation prevent complete presentation of the
formulation, but it is available in the literature.^ Briefly, the problem is modeled
using finite elements and a mathematical program is cast in integrated form, using nodal coordinates, nodal displacements, inter-element forces, and element
stresses as variables. The structural behavior is modeled as a set of nonlinear
equality constraints (stiffness equations) relating the four sets of variables. Inequality constraints are imposed to limit stress and displacement, or any other
behavior that can be modeled with the four sets of variables. Bounds on displacement variables represent support conditions. The monomial method fits
seamlessly within the optimization method of generalized geometric programming.^ Since GGP operates in the space of the logarithm of the variables, the
monomial method is implemented by appending a set of linear equations to the
GGP system at each iteration. ^
Consider a fixed-ended beam with variable depth, having a 60 inch span
length, a selfweight of 150 pounds per cubic foot, and a distributed load of 1
kip/ft applied along the top surface. The beam has a uniform thickness of 12
inches. The finite element mesh consists of 48 elements, but symmetry is used
to reduce it to 24. The x-coordinates of all corner nodes are fixed. The ycoordinates of the interior corner nodes are constrained to lie at the centroid of the
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corner nodes above and below them. The y-coordinates of the corner nodes on the
top and bottom surface of the beam are unrestricted. Principal stresses are constrained between -4 ksi and 4 ksi everywhere throughout the beam; no constraints
are placed on maximum displacement. The modulus of elasticity and Poisson's
ratio of this hypothetical material are specified as 4000 ksi and 0.15, respectively.
Figure 4 illustrates the iterative solution sequence using the modified GGP
algorithm to solve the integrated formulation. The solution is obtained in six
iterations. Experience has demonstrated that the number of iterations to a solution is independent of the size of the problem. A second solution of this problem
was performed, this time with the allowable tensile stress reduced from 4 ksi to 3
ksi. The allowable compressive stress was kept at -4 ksi. The sequence of iterations are presented in Figure 5. This time, the structure evolves into an arch
which experiences uniform axial compression everywhere. The solution process
has 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 the
first solution. Since the second problem is more highly constrained than the first,
this indicates that the first solution was a not the global minimum. Tightening
the allowable tensile stress constraints forced the solution out of the symmetric
solution and into the arch-shaped global minimum. Because the allowable tensile
stress constraints are not binding in the arch solution, this indicates that the arch
solution is an alternative solution to the first problem with equal tensile and
compressive allowable stresses. Had the first problem been initiated with a starting configuration more closely resembling the arch-shaped solution, the solution
process could have converged to the arch solution instead. A third optimal solution to the first problem is a downward sweeping arch in uniform axial tension.
This problem suggests that local minima are certainly possible, and probably
quite prevalent, in shape optimization.
Conclusions
A monomial-based approximation has been presented as an alternative to standard
Taylor series linearization. In the context of solving systems of nonlinear equations, the monomial approximation leads to the monomial method; Taylor
linearization leads to Newton's method. The monomial method has been shown
to generally outperform Newton's method, sometimes quite dramatically, when
applied to structural design applications. The method is very general in its application. The main requirement is that the system of equations being solved be
separable into strictly positive and negative parts, and that the variables be positive. Current research is being conducted into generalizing the method to include
variables that are not restricted in sign. This will further enhance the applicability
of the method. In the case of structural applications, current research is being
conducted into applying the monomial method to the analysis of geometrically
Transactions on the Built Environment vol 37 © 1999 WIT Press, www.witpress.com, ISSN 1743-3509
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Computer Aided Optimum Design of Structures
Figure 4. First six iterations of shape optimization with equal
allowable tensile and compressive stress constraints.
nonlinear structures. Incorporating the monomial approximation into an incremental method may allow larger load increments to be taken, improving the performance of such methods.
Acknowledgments
This material is based on work supported by the National Science Foundation
under award No. CMS 97-14069.
References
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Wiley & Sons, New York, 1967.
Transactions on the Built Environment vol 37 © 1999 WIT Press, www.witpress.com, ISSN 1743-3509
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Figure 5. Same shape optimization problem with tightened
allowable tensile stress constraints.
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Computer Aided Optimum Design of Structures
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